LMFDB - Embedded newform 76.8.i.a.73.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,8,Mod(5,76)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 16]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("76.5");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	76.i (of order $9$, degree $6$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$23.7412619368$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$12$ over $\Q(\zeta_{9})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			73.7
Character	$\chi$	$=$	76.73
Dual form			76.8.i.a.25.7

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(13.7721 - 5.01263i) q^{3} +(38.0867 - 216.001i) q^{5} +(-872.699 - 1511.56i) q^{7} +(-1510.80 + 1267.71i) q^{9} +O(q^{10})$	\(q+(13.7721 - 5.01263i) q^{3} +(38.0867 - 216.001i) q^{5} +(-872.699 - 1511.56i) q^{7} +(-1510.80 + 1267.71i) q^{9} +(-3051.04 + 5284.56i) q^{11} +(10897.0 + 3966.19i) q^{13} +(-558.197 - 3165.69i) q^{15} +(-2776.50 - 2329.76i) q^{17} +(-29034.4 + 7132.73i) q^{19} +(-19595.8 - 16442.8i) q^{21} +(4938.68 + 28008.6i) q^{23} +(28207.9 + 10266.8i) q^{25} +(-30478.5 + 52790.4i) q^{27} +(-94547.2 + 79334.5i) q^{29} +(11743.5 + 20340.4i) q^{31} +(-15529.7 + 88073.2i) q^{33} +(-359736. + 130933. i) q^{35} +16945.1 q^{37} +169956. q^{39} +(-525042. + 191100. i) q^{41} +(40971.2 - 232359. i) q^{43} +(216284. + 374615. i) q^{45} +(-292934. + 245801. i) q^{47} +(-1.11144e6 + 1.92506e6i) q^{49} +(-49916.4 - 18168.1i) q^{51} +(-208955. - 1.18504e6i) q^{53} +(1.02526e6 + 860298. i) q^{55} +(-364111. + 243771. i) q^{57} +(-1.88498e6 - 1.58169e6i) q^{59} +(-355162. - 2.01423e6i) q^{61} +(3.23468e6 + 1.17733e6i) q^{63} +(1.27173e6 - 2.20270e6i) q^{65} +(-1.92395e6 + 1.61438e6i) q^{67} +(208413. + 360982. i) q^{69} +(-59237.5 + 335952. i) q^{71} +(1.68185e6 - 612144. i) q^{73} +439945. q^{75} +1.06506e7 q^{77} +(-2.19403e6 + 798560. i) q^{79} +(593846. - 3.36787e6i) q^{81} +(-1.47049e6 - 2.54697e6i) q^{83} +(-608977. + 510992. i) q^{85} +(-904438. + 1.56653e6i) q^{87} +(-3.40627e6 - 1.23978e6i) q^{89} +(-3.51469e6 - 1.99328e7i) q^{91} +(263692. + 221264. i) q^{93} +(434848. + 6.54311e6i) q^{95} +(4.94372e6 + 4.14827e6i) q^{97} +(-2.08978e6 - 1.18517e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q - 39 q^{3} + 591 q^{7} - 1677 q^{9}+O(q^{10}) $ 72 * q - 39 * q^3 + 591 * q^7 - 1677 * q^9	$ 72 q - 39 q^{3} + 591 q^{7} - 1677 q^{9} - 5793 q^{11} + 22083 q^{13} - 70707 q^{15} + 29409 q^{17} + 12756 q^{19} - 40419 q^{21} - 116796 q^{23} + 26934 q^{25} + 439347 q^{27} - 148005 q^{29} + 141366 q^{31} + 1206654 q^{33} - 531651 q^{35} - 1508004 q^{37} - 2140644 q^{39} + 494157 q^{41} + 923511 q^{43} + 1790028 q^{45} - 662955 q^{47} - 4191369 q^{49} + 3753363 q^{51} - 246513 q^{53} + 2650185 q^{55} + 5831286 q^{57} - 2244432 q^{59} - 10302306 q^{61} - 12102117 q^{63} + 3985635 q^{65} + 11688288 q^{67} + 12632535 q^{69} + 10541736 q^{71} + 560730 q^{73} - 17718750 q^{75} - 22266192 q^{77} + 9424767 q^{79} + 16585437 q^{81} + 2359839 q^{83} - 20859762 q^{85} + 2348574 q^{87} - 4397130 q^{89} - 12653112 q^{91} + 52259766 q^{93} + 46248669 q^{95} - 34112766 q^{97} - 93757926 q^{99}+O(q^{100}) $ 72 * q - 39 * q^3 + 591 * q^7 - 1677 * q^9 - 5793 * q^11 + 22083 * q^13 - 70707 * q^15 + 29409 * q^17 + 12756 * q^19 - 40419 * q^21 - 116796 * q^23 + 26934 * q^25 + 439347 * q^27 - 148005 * q^29 + 141366 * q^31 + 1206654 * q^33 - 531651 * q^35 - 1508004 * q^37 - 2140644 * q^39 + 494157 * q^41 + 923511 * q^43 + 1790028 * q^45 - 662955 * q^47 - 4191369 * q^49 + 3753363 * q^51 - 246513 * q^53 + 2650185 * q^55 + 5831286 * q^57 - 2244432 * q^59 - 10302306 * q^61 - 12102117 * q^63 + 3985635 * q^65 + 11688288 * q^67 + 12632535 * q^69 + 10541736 * q^71 + 560730 * q^73 - 17718750 * q^75 - 22266192 * q^77 + 9424767 * q^79 + 16585437 * q^81 + 2359839 * q^83 - 20859762 * q^85 + 2348574 * q^87 - 4397130 * q^89 - 12653112 * q^91 + 52259766 * q^93 + 46248669 * q^95 - 34112766 * q^97 - 93757926 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$e\left(\frac{2}{9}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	13.7721	−	5.01263i	0.294493	−	0.107187i	−0.190548	−	0.981678i	$-0.561027\pi$
$3$	13.7721	−	5.01263i	0.294493	−	0.107187i	0.485042	+	0.874491i	$0.338804\pi$
$4$		0			0
$5$	38.0867	−	216.001i	0.136263	−	0.772787i	−0.837709	−	0.546118i	$-0.816105\pi$
$5$	38.0867	−	216.001i	0.136263	−	0.772787i	0.973972	−	0.226669i	$-0.0727837\pi$
$6$		0			0
$7$	−872.699	−	1511.56i	−0.961660	−	1.66564i	−0.718334	−	0.695699i	$-0.755095\pi$
$7$	−872.699	−	1511.56i	−0.961660	−	1.66564i	−0.243326	−	0.969945i	$-0.578238\pi$
$8$		0			0
$9$	−1510.80	+	1267.71i	−0.690807	+	0.579656i
$10$		0			0
$11$	−3051.04	+	5284.56i	−0.691152	+	1.19711i	0.280308	+	0.959910i	$0.409563\pi$
$11$	−3051.04	+	5284.56i	−0.691152	+	1.19711i	−0.971461	+	0.237201i	$0.923770\pi$
$12$		0			0
$13$	10897.0	+	3966.19i	1.37564	+	0.500693i	0.920855	−	0.389906i	$-0.127492\pi$
$13$	10897.0	+	3966.19i	1.37564	+	0.500693i	0.454789	+	0.890599i	$0.349715\pi$
$14$		0			0
$15$	−558.197	−	3165.69i	−0.0427040	−	0.242186i
$16$		0			0
$17$	−2776.50	−	2329.76i	−0.137065	−	0.115011i	0.571678	−	0.820478i	$-0.306293\pi$
$17$	−2776.50	−	2329.76i	−0.137065	−	0.115011i	−0.708743	+	0.705467i	$0.750737\pi$
$18$		0			0
$19$	−29034.4	+	7132.73i	−0.971125	+	0.238571i
$19$	−29034.4	+	7132.73i	−0.971125	+	0.238571i
$20$		0			0
$21$	−19595.8	−	16442.8i	−0.461737	−	0.387444i
$22$		0			0
$23$	4938.68	+	28008.6i	0.0846376	+	0.480004i	0.997434	+	0.0715891i	$0.0228070\pi$
$23$	4938.68	+	28008.6i	0.0846376	+	0.480004i	−0.912797	+	0.408414i	$0.866082\pi$
$24$		0			0
$25$	28207.9	+	10266.8i	0.361061	+	0.131415i
$26$		0			0
$27$	−30478.5	+	52790.4i	−0.298003	+	0.516157i
$28$		0			0
$29$	−94547.2	+	79334.5i	−0.719873	+	0.604045i	−0.927350	−	0.374195i	$-0.877919\pi$
$29$	−94547.2	+	79334.5i	−0.719873	+	0.604045i	0.207478	+	0.978240i	$0.433475\pi$
$30$		0			0
$31$	11743.5	+	20340.4i	0.0708000	+	0.122629i	0.899252	−	0.437431i	$-0.144111\pi$
$31$	11743.5	+	20340.4i	0.0708000	+	0.122629i	−0.828452	+	0.560060i	$0.810778\pi$
$32$		0			0
$33$	−15529.7	+	88073.2i	−0.0752253	+	0.426624i
$34$		0			0
$35$	−359736.	+	130933.i	−1.41823	+	0.516192i
$36$		0			0
$37$	16945.1			0.0549969			0.0274984	−	0.999622i	$-0.491246\pi$
$37$	16945.1			0.0549969			0.0274984	+	0.999622i	$0.491246\pi$
$38$		0			0
$39$	169956.			0.458786
$40$		0			0
$41$	−525042.	+	191100.i	−1.18973	+	0.433028i	−0.859630	−	0.510916i	$-0.829306\pi$
$41$	−525042.	+	191100.i	−1.18973	+	0.433028i	−0.330104	+	0.943944i	$0.607084\pi$
$42$		0			0
$43$	40971.2	−	232359.i	0.0785848	−	0.445677i	−0.919973	−	0.391983i	$-0.871789\pi$
$43$	40971.2	−	232359.i	0.0785848	−	0.445677i	0.998557	−	0.0536941i	$-0.0170996\pi$
$44$		0			0
$45$	216284.	+	374615.i	0.353819	+	0.612833i
$46$		0			0
$47$	−292934.	+	245801.i	−0.411554	+	0.345335i	−0.824939	−	0.565221i	$-0.808791\pi$
$47$	−292934.	+	245801.i	−0.411554	+	0.345335i	0.413385	+	0.910556i	$0.364346\pi$
$48$		0			0
$49$	−1.11144e6	+	1.92506e6i	−1.34958	+	2.33754i
$50$		0			0
$51$	−49916.4	−	18168.1i	−0.0526924	−	0.0191785i
$52$		0			0
$53$	−208955.	−	1.18504e6i	−0.192791	−	1.09337i	−0.915529	−	0.402252i	$-0.868228\pi$
$53$	−208955.	−	1.18504e6i	−0.192791	−	1.09337i	0.722738	−	0.691122i	$-0.242883\pi$
$54$		0			0
$55$	1.02526e6	+	860298.i	0.830933	+	0.697236i
$56$		0			0
$57$	−364111.	+	243771.i	−0.260418	+	0.174349i
$58$		0			0
$59$	−1.88498e6	−	1.58169e6i	−1.19488	−	1.00263i	−0.999761	−	0.0218465i	$-0.993045\pi$
$59$	−1.88498e6	−	1.58169e6i	−1.19488	−	1.00263i	−0.195121	−	0.980779i	$-0.562510\pi$
$60$		0			0
$61$	−355162.	−	2.01423e6i	−0.200342	−	1.13620i	−0.904603	−	0.426255i	$-0.859832\pi$
$61$	−355162.	−	2.01423e6i	−0.200342	−	1.13620i	0.704261	−	0.709941i	$-0.251279\pi$
$62$		0			0
$63$	3.23468e6	+	1.17733e6i	1.62982	+	0.593206i
$64$		0			0
$65$	1.27173e6	−	2.20270e6i	0.574379	−	0.994853i
$66$		0			0
$67$	−1.92395e6	+	1.61438e6i	−0.781503	+	0.655759i	−0.943627	−	0.331011i	$-0.892610\pi$
$67$	−1.92395e6	+	1.61438e6i	−0.781503	+	0.655759i	0.162123	+	0.986770i	$0.448166\pi$
$68$		0			0
$69$	208413.	+	360982.i	0.0763752	+	0.132286i
$70$		0			0
$71$	−59237.5	+	335952.i	−0.0196423	+	0.111397i	−0.993053	−	0.117672i	$-0.962457\pi$
$71$	−59237.5	+	335952.i	−0.0196423	+	0.111397i	0.973410	+	0.229069i	$0.0735681\pi$
$72$		0			0
$73$	1.68185e6	−	612144.i	0.506008	−	0.184172i	−0.0763862	−	0.997078i	$-0.524338\pi$
$73$	1.68185e6	−	612144.i	0.506008	−	0.184172i	0.582394	+	0.812906i	$0.302116\pi$
$74$		0			0
$75$	439945.			0.120416
$76$		0			0
$77$	1.06506e7			2.65861
$78$		0			0
$79$	−2.19403e6	+	798560.i	−0.500665	+	0.182227i	−0.579993	−	0.814621i	$-0.696945\pi$
$79$	−2.19403e6	+	798560.i	−0.500665	+	0.182227i	0.0793282	+	0.996849i	$0.474722\pi$
$80$		0			0
$81$	593846.	−	3.36787e6i	0.124158	−	0.704138i
$82$		0			0
$83$	−1.47049e6	−	2.54697e6i	−0.282286	−	0.488934i	0.689661	−	0.724132i	$-0.257759\pi$
$83$	−1.47049e6	−	2.54697e6i	−0.282286	−	0.488934i	−0.971947	+	0.235198i	$0.924426\pi$
$84$		0			0
$85$	−608977.	+	510992.i	−0.107556	+	0.0902502i
$86$		0			0
$87$	−904438.	+	1.56653e6i	−0.147252	+	0.255048i
$88$		0			0
$89$	−3.40627e6	−	1.23978e6i	−0.512170	−	0.186415i	0.0729898	−	0.997333i	$-0.476746\pi$
$89$	−3.40627e6	−	1.23978e6i	−0.512170	−	0.186415i	−0.585160	+	0.810918i	$0.698968\pi$
$90$		0			0
$91$	−3.51469e6	−	1.99328e7i	−0.488924	−	2.77283i
$92$		0			0
$93$	263692.	+	221264.i	0.0339943	+	0.0285246i
$94$		0			0
$95$	434848.	+	6.54311e6i	0.0520361	+	0.782981i
$96$		0			0
$97$	4.94372e6	+	4.14827e6i	0.549987	+	0.461494i	0.874937	−	0.484237i	$-0.160903\pi$
$97$	4.94372e6	+	4.14827e6i	0.549987	+	0.461494i	−0.324950	+	0.945731i	$0.605347\pi$
$98$		0			0
$99$	−2.08978e6	−	1.18517e7i	−0.216460	−	1.22760i
$100$		0			0
$101$	1.38651e7	+	5.04648e6i	1.33905	+	0.487376i	0.909514	−	0.415673i	$-0.136454\pi$
$101$	1.38651e7	+	5.04648e6i	1.33905	+	0.487376i	0.429539	+	0.903048i	$0.358676\pi$
$102$		0			0
$103$	8.17691e6	−	1.41628e7i	0.737325	−	1.27708i	−0.216371	−	0.976311i	$-0.569422\pi$
$103$	8.17691e6	−	1.41628e7i	0.737325	−	1.27708i	0.953696	−	0.300773i	$-0.0972446\pi$
$104$		0			0
$105$	−4.29799e6	+	3.60645e6i	−0.362329	+	0.304030i
$106$		0			0
$107$	1.00681e7	+	1.74385e7i	0.794523	+	1.37615i	0.923142	+	0.384460i	$0.125612\pi$
$107$	1.00681e7	+	1.74385e7i	0.794523	+	1.37615i	−0.128619	+	0.991694i	$0.541054\pi$
$108$		0			0
$109$	−3.45914e6	+	1.96178e7i	−0.255844	+	1.45097i	0.538052	+	0.842912i	$0.319160\pi$
$109$	−3.45914e6	+	1.96178e7i	−0.255844	+	1.45097i	−0.793896	+	0.608053i	$0.791951\pi$
$110$		0			0
$111$	233369.	−	84939.5i	0.0161962	−	0.00589494i
$112$		0			0
$113$	−3.58605e6			−0.233798			−0.116899	−	0.993144i	$-0.537295\pi$
$113$	−3.58605e6			−0.233798			−0.116899	+	0.993144i	$0.537295\pi$
$114$		0			0
$115$	6.23798e6			0.382473
$116$		0			0
$117$	−2.14911e7	+	7.82213e6i	−1.24053	+	0.451518i
$118$		0			0
$119$	−1.09852e6	+	6.23002e6i	−0.0597577	+	0.338903i
$120$		0			0
$121$	−8.87413e6	−	1.53704e7i	−0.455383	−	0.788747i
$122$		0			0
$123$	−6.27301e6	+	5.26368e6i	−0.303954	+	0.255048i
$124$		0			0
$125$	1.18597e7	−	2.05415e7i	0.543110	−	0.940693i
$126$		0			0
$127$	−2.90325e7	−	1.05670e7i	−1.25768	−	0.457759i	−0.374692	−	0.927149i	$-0.622252\pi$
$127$	−2.90325e7	−	1.05670e7i	−1.25768	−	0.457759i	−0.882991	+	0.469390i	$0.844474\pi$
$128$		0			0
$129$	−600472.	−	3.40544e6i	−0.0246280	−	0.139672i
$130$		0			0
$131$	−3.81718e7	−	3.20299e7i	−1.48352	−	1.24482i	−0.902327	−	0.431053i	$-0.858142\pi$
$131$	−3.81718e7	−	3.20299e7i	−1.48352	−	1.24482i	−0.581192	−	0.813767i	$-0.697413\pi$
$132$		0			0
$133$	3.61198e7	+	3.76625e7i	1.33127	+	1.38812i
$134$		0			0
$135$	1.02419e7	+	8.59399e6i	0.358272	+	0.300626i
$136$		0			0
$137$	6.76068e6	+	3.83417e7i	0.224630	+	1.27394i	0.863391	+	0.504536i	$0.168337\pi$
$137$	6.76068e6	+	3.83417e7i	0.224630	+	1.27394i	−0.638760	+	0.769406i	$0.720552\pi$
$138$		0			0
$139$	−1.31092e7	−	4.77137e6i	−0.414024	−	0.150692i	0.126606	−	0.991953i	$-0.459592\pi$
$139$	−1.31092e7	−	4.77137e6i	−0.414024	−	0.150692i	−0.540630	+	0.841261i	$0.681814\pi$
$140$		0			0
$141$	−2.80220e6	+	4.85356e6i	−0.0841846	+	0.145812i
$142$		0			0
$143$	−5.42068e7	+	4.54849e7i	−1.55017	+	1.30074i
$144$		0			0
$145$	1.35353e7	+	2.34438e7i	0.368706	+	0.638617i
$146$		0			0
$147$	−5.65716e6	+	3.20834e7i	−0.146889	+	0.833046i
$148$		0			0
$149$	−3.26197e6	+	1.18726e6i	−0.0807845	+	0.0294031i	−0.382096	−	0.924123i	$-0.624798\pi$
$149$	−3.26197e6	+	1.18726e6i	−0.0807845	+	0.0294031i	0.301312	+	0.953526i	$0.402576\pi$
$150$		0			0
$151$	−4.24341e7			−1.00299			−0.501494	−	0.865161i	$-0.667216\pi$
$151$	−4.24341e7			−1.00299			−0.501494	+	0.865161i	$0.667216\pi$
$152$		0			0
$153$	7.14818e6			0.161352
$154$		0			0
$155$	4.84081e6	−	1.76191e6i	0.104414	−	0.0380035i
$156$		0			0
$157$	−1.45374e6	+	8.24455e6i	−0.0299804	+	0.170027i	−0.996122	−	0.0879864i	$-0.971957\pi$
$157$	−1.45374e6	+	8.24455e6i	−0.0299804	+	0.170027i	0.966141	+	0.258013i	$0.0830679\pi$
$158$		0			0
$159$	−8.81793e6	−	1.52731e7i	−0.173971	−	0.301327i
$160$		0			0
$161$	3.80267e7	−	3.19082e7i	0.718122	−	0.602576i
$162$		0			0
$163$	1.41933e7	−	2.45834e7i	0.256700	−	0.444617i	−0.708656	−	0.705554i	$-0.750698\pi$
$163$	1.41933e7	−	2.45834e7i	0.256700	−	0.444617i	0.965356	+	0.260937i	$0.0840315\pi$
$164$		0			0
$165$	1.84324e7	+	6.70884e6i	0.319439	+	0.116266i
$166$		0			0
$167$	−5.39334e6	−	3.05872e7i	−0.0896088	−	0.508196i	−0.996267	−	0.0863298i	$-0.972486\pi$
$167$	−5.39334e6	−	3.05872e7i	−0.0896088	−	0.508196i	0.906658	−	0.421867i	$-0.138625\pi$
$168$		0			0
$169$	5.49462e7	+	4.61053e7i	0.875657	+	0.734764i
$170$		0			0
$171$	3.48228e7	−	4.75832e7i	0.532571	−	0.727725i
$172$		0			0
$173$	−5.29328e7	−	4.44159e7i	−0.777255	−	0.652194i	0.165301	−	0.986243i	$-0.447141\pi$
$173$	−5.29328e7	−	4.44159e7i	−0.777255	−	0.652194i	−0.942556	+	0.334049i	$0.891585\pi$
$174$		0			0
$175$	−9.09806e6	−	5.15977e7i	−0.128326	−	0.727775i
$176$		0			0
$177$	−3.38886e7	−	1.23344e7i	−0.459353	−	0.167191i
$178$		0			0
$179$	−5.95338e7	+	1.03116e8i	−0.775850	+	1.34381i	0.158465	+	0.987365i	$0.449345\pi$
$179$	−5.95338e7	+	1.03116e8i	−0.775850	+	1.34381i	−0.934315	+	0.356447i	$0.883988\pi$
$180$		0			0
$181$	6.55296e7	−	5.49858e7i	0.821415	−	0.689249i	−0.131888	−	0.991265i	$-0.542104\pi$
$181$	6.55296e7	−	5.49858e7i	0.821415	−	0.689249i	0.953303	+	0.302016i	$0.0976596\pi$
$182$		0			0
$183$	−1.49879e7	−	2.59598e7i	−0.180785	−	0.313128i
$184$		0			0
$185$	645383.	−	3.66015e6i	0.00749405	−	0.0425009i
$186$		0			0
$187$	2.07830e7	−	7.56438e6i	0.232414	−	0.0845917i
$188$		0			0
$189$	1.06394e8			1.14631
$190$		0			0
$191$	−2.33628e7			−0.242610			−0.121305	−	0.992615i	$-0.538708\pi$
$191$	−2.33628e7			−0.242610			−0.121305	+	0.992615i	$0.538708\pi$
$192$		0			0
$193$	−1.40724e6	+	512195.i	−0.0140902	+	0.00512843i	−0.349056	−	0.937102i	$-0.613498\pi$
$193$	−1.40724e6	+	512195.i	−0.0140902	+	0.00512843i	0.334965	+	0.942230i	$0.391275\pi$
$194$		0			0
$195$	6.47306e6	−	3.67105e7i	0.0625156	−	0.354544i
$196$		0			0
$197$	1.35415e7	+	2.34546e7i	0.126193	+	0.218573i	0.922199	−	0.386716i	$-0.126391\pi$
$197$	1.35415e7	+	2.34546e7i	0.126193	+	0.218573i	−0.796005	+	0.605289i	$0.793057\pi$
$198$		0			0
$199$	1.39854e8	−	1.17351e8i	1.25802	−	1.05561i	0.262134	−	0.965031i	$-0.415574\pi$
$199$	1.39854e8	−	1.17351e8i	1.25802	−	1.05561i	0.995890	−	0.0905756i	$-0.0288707\pi$
$200$		0			0
$201$	−1.84044e7	+	3.18774e7i	−0.159859	+	0.276884i
$202$		0			0
$203$	2.02430e8	+	7.36785e7i	1.69840	+	0.618165i
$204$		0			0
$205$	2.12805e7	+	1.20688e8i	0.172521	+	0.978417i
$206$		0			0
$207$	−4.29681e7	−	3.60545e7i	−0.336705	−	0.282529i
$208$		0			0
$209$	5.08918e7	−	1.75196e8i	0.385599	−	1.32743i
$210$		0			0
$211$	−8.94145e7	−	7.50277e7i	−0.655269	−	0.549836i	0.253396	−	0.967363i	$-0.418453\pi$
$211$	−8.94145e7	−	7.50277e7i	−0.655269	−	0.549836i	−0.908665	+	0.417527i	$0.862897\pi$
$212$		0			0
$213$	868181.	+	4.92370e6i	0.00615576	+	0.0349111i
$214$		0			0
$215$	−4.86292e7	−	1.76996e7i	−0.333705	−	0.121459i
$216$		0			0
$217$	2.04971e7	−	3.55021e7i	0.136171	−	0.235855i
$218$		0			0
$219$	2.00942e7	−	1.68610e7i	0.129275	−	0.108475i
$220$		0			0
$221$	−2.10153e7	−	3.63996e7i	−0.130967	−	0.226842i
$222$		0			0
$223$	3.71336e7	−	2.10595e8i	0.224233	−	1.27169i	−0.639912	−	0.768448i	$-0.721029\pi$
$223$	3.71336e7	−	2.10595e8i	0.224233	−	1.27169i	0.864146	−	0.503242i	$-0.167860\pi$
$224$		0			0
$225$	−5.56316e7	+	2.02483e7i	−0.325599	+	0.118508i
$226$		0			0
$227$	−1.22800e8			−0.696798			−0.348399	−	0.937346i	$-0.613275\pi$
$227$	−1.22800e8			−0.696798			−0.348399	+	0.937346i	$0.613275\pi$
$228$		0			0
$229$	−2.20371e8			−1.21263			−0.606317	−	0.795223i	$-0.707354\pi$
$229$	−2.20371e8			−1.21263			−0.606317	+	0.795223i	$0.707354\pi$
$230$		0			0
$231$	1.46681e8	−	5.33874e7i	0.782944	−	0.284968i
$232$		0			0
$233$	−4.57266e7	+	2.59329e8i	−0.236823	+	1.34309i	0.601918	+	0.798558i	$0.294403\pi$
$233$	−4.57266e7	+	2.59329e8i	−0.236823	+	1.34309i	−0.838741	+	0.544531i	$0.816708\pi$
$234$		0			0
$235$	4.19362e7	+	7.26356e7i	0.210791	+	0.365100i
$236$		0			0
$237$	−2.62134e7	+	2.19957e7i	−0.127910	+	0.107329i
$238$		0			0
$239$	−1.62642e8	+	2.81705e8i	−0.770622	+	1.33476i	0.166601	+	0.986024i	$0.446721\pi$
$239$	−1.62642e8	+	2.81705e8i	−0.770622	+	1.33476i	−0.937223	+	0.348731i	$0.886613\pi$
$240$		0			0
$241$	2.12793e8	+	7.74504e7i	0.979260	+	0.356422i	0.781552	−	0.623840i	$-0.214428\pi$
$241$	2.12793e8	+	7.74504e7i	0.979260	+	0.356422i	0.197708	+	0.980261i	$0.436650\pi$
$242$		0			0
$243$	−3.18530e7	−	1.80647e8i	−0.142406	−	0.807624i
$244$		0			0
$245$	3.73484e8	+	3.13390e8i	1.62252	+	1.36146i
$246$		0			0
$247$	−3.44678e8	−	3.74305e7i	−1.45537	−	0.158047i
$248$		0			0
$249$	−3.30187e7	−	2.77060e7i	−0.135539	−	0.113730i
$250$		0			0
$251$	3.68450e7	+	2.08959e8i	0.147069	+	0.834069i	0.965683	+	0.259725i	$0.0836319\pi$
$251$	3.68450e7	+	2.08959e8i	0.147069	+	0.834069i	−0.818614	+	0.574345i	$0.805257\pi$
$252$		0			0
$253$	−1.63081e8	−	5.93568e7i	−0.633115	−	0.230435i
$254$		0			0
$255$	−5.82547e6	+	1.00900e7i	−0.0220009	+	0.0381067i
$256$		0			0
$257$	3.73174e8	−	3.13130e8i	1.37134	−	1.15069i	0.399046	−	0.916931i	$-0.369341\pi$
$257$	3.73174e8	−	3.13130e8i	1.37134	−	1.15069i	0.972294	−	0.233760i	$-0.0751030\pi$
$258$		0			0
$259$	−1.47880e7	−	2.56135e7i	−0.0528883	−	0.0916052i
$260$		0			0
$261$	4.22685e7	−	2.39716e8i	0.147155	−	0.834557i
$262$		0			0
$263$	−6.68187e7	+	2.43200e7i	−0.226492	+	0.0824364i	−0.452773	−	0.891626i	$-0.649565\pi$
$263$	−6.68187e7	+	2.43200e7i	−0.226492	+	0.0824364i	0.226281	+	0.974062i	$0.427343\pi$
$264$		0			0
$265$	−2.63928e8			−0.871215
$266$		0			0
$267$	−5.31260e7			−0.170812
$268$		0			0
$269$	−4.98927e8	+	1.81594e8i	−1.56280	+	0.568813i	−0.971376	−	0.237546i	$-0.923657\pi$
$269$	−4.98927e8	+	1.81594e8i	−1.56280	+	0.568813i	−0.591425	+	0.806360i	$0.701435\pi$
$270$		0			0
$271$	−8.70557e7	+	4.93717e8i	−0.265708	+	1.50691i	0.501303	+	0.865272i	$0.332854\pi$
$271$	−8.70557e7	+	4.93717e8i	−0.265708	+	1.50691i	−0.767011	+	0.641634i	$0.778257\pi$
$272$		0			0
$273$	−1.48320e8	−	2.56898e8i	−0.441196	−	0.764173i
$274$		0			0
$275$	−1.40319e8	+	1.17742e8i	−0.406867	+	0.341402i
$276$		0			0
$277$	5.23094e7	−	9.06025e7i	0.147877	−	0.256130i	−0.782566	−	0.622568i	$-0.786089\pi$
$277$	5.23094e7	−	9.06025e7i	0.147877	−	0.256130i	0.930443	+	0.366438i	$0.119423\pi$
$278$		0			0
$279$	−4.35278e7	−	1.58428e7i	−0.119992	−	0.0436735i
$280$		0			0
$281$	1.19512e8	+	6.77783e8i	0.321320	+	1.82230i	0.534366	+	0.845253i	$0.320551\pi$
$281$	1.19512e8	+	6.77783e8i	0.321320	+	1.82230i	−0.213046	+	0.977042i	$0.568338\pi$
$282$		0			0
$283$	−2.27296e8	−	1.90724e8i	−0.596129	−	0.500211i	0.294070	−	0.955784i	$-0.404990\pi$
$283$	−2.27296e8	−	1.90724e8i	−0.596129	−	0.500211i	−0.890199	+	0.455572i	$0.849435\pi$
$284$		0			0
$285$	3.87869e7	+	8.79325e7i	0.0992495	+	0.225005i
$286$		0			0
$287$	7.47061e8	+	6.26859e8i	1.86539	+	1.56525i
$288$		0			0
$289$	−6.89734e7	−	3.91168e8i	−0.168089	−	0.953280i
$290$		0			0
$291$	8.88791e7	+	3.23493e7i	0.211434	+	0.0769555i
$292$		0			0
$293$	9.26208e6	−	1.60424e7i	0.0215116	−	0.0372591i	−0.855069	−	0.518514i	$-0.826485\pi$
$293$	9.26208e6	−	1.60424e7i	0.0215116	−	0.0372591i	0.876581	+	0.481255i	$0.159819\pi$
$294$		0			0
$295$	−4.13438e8	+	3.46916e8i	−0.937635	+	0.786769i
$296$		0			0
$297$	−1.85983e8	−	3.22131e8i	−0.411931	−	0.713486i
$298$		0			0
$299$	−5.72707e7	+	3.24798e8i	−0.123903	+	0.702691i
$300$		0			0
$301$	−3.86980e8	+	1.40849e8i	−0.817910	+	0.297695i
$302$		0			0
$303$	2.16247e8			0.446583
$304$		0			0
$305$	−4.48601e8			−0.905337
$306$		0			0
$307$	8.49133e8	−	3.09059e8i	1.67491	−	0.609617i	0.682311	−	0.731062i	$-0.260975\pi$
$307$	8.49133e8	−	3.09059e8i	1.67491	−	0.609617i	0.992598	+	0.121445i	$0.0387527\pi$
$308$		0			0
$309$	4.16201e7	−	2.36039e8i	0.0802507	−	0.455124i
$310$		0			0
$311$	−1.02913e8	−	1.78250e8i	−0.194003	−	0.336023i	0.752570	−	0.658512i	$-0.228814\pi$
$311$	−1.02913e8	−	1.78250e8i	−0.194003	−	0.336023i	−0.946573	+	0.322489i	$0.895480\pi$
$312$		0			0
$313$	−1.91962e8	+	1.61076e8i	−0.353843	+	0.296910i	−0.802331	−	0.596879i	$-0.796407\pi$
$313$	−1.91962e8	+	1.61076e8i	−0.353843	+	0.296910i	0.448488	+	0.893789i	$0.351963\pi$
$314$		0			0
$315$	3.77502e8	−	6.53853e8i	0.680507	−	1.17867i
$316$		0			0
$317$	−1.03216e9	−	3.75675e8i	−1.81986	−	0.662376i	−0.995327	−	0.0965665i	$-0.969214\pi$
$317$	−1.03216e9	−	3.75675e8i	−1.81986	−	0.662376i	−0.824536	−	0.565809i	$-0.808564\pi$
$318$		0			0
$319$	−1.30781e8	−	7.41693e8i	−0.225567	−	1.27925i
$320$		0			0
$321$	2.26072e8	+	1.89697e8i	0.381487	+	0.320106i
$322$		0			0
$323$	9.72315e7	+	4.78391e7i	0.160546	+	0.0789905i
$324$		0			0
$325$	2.66661e8	+	2.23755e8i	0.430892	+	0.361561i
$326$		0			0
$327$	5.06970e7	+	2.87517e8i	0.0801799	+	0.454723i
$328$		0			0
$329$	6.27185e8	+	2.28277e8i	0.970980	+	0.353408i
$330$		0			0
$331$	2.76222e8	−	4.78431e8i	0.418659	−	0.725139i	−0.577146	−	0.816641i	$-0.695833\pi$
$331$	2.76222e8	−	4.78431e8i	0.418659	−	0.725139i	0.995805	+	0.0915021i	$0.0291668\pi$
$332$		0			0
$333$	−2.56006e7	+	2.14814e7i	−0.0379922	+	0.0318793i
$334$		0			0
$335$	2.75431e8	+	4.77060e8i	0.400272	+	0.693291i
$336$		0			0
$337$	1.44333e8	−	8.18554e8i	0.205429	−	1.16505i	−0.691334	−	0.722535i	$-0.742977\pi$
$337$	1.44333e8	−	8.18554e8i	0.205429	−	1.16505i	0.896763	−	0.442510i	$-0.145912\pi$
$338$		0			0
$339$	−4.93874e7	+	1.79755e7i	−0.0688520	+	0.0250601i
$340$		0			0
$341$	−1.43320e8			−0.195734
$342$		0			0
$343$	2.44239e9			3.26802
$344$		0			0
$345$	8.59100e7	−	3.12687e7i	0.112636	−	0.0409961i
$346$		0			0
$347$	5.85083e7	−	3.31817e8i	0.0751734	−	0.426330i	−0.923874	−	0.382696i	$-0.874996\pi$
$347$	5.85083e7	−	3.31817e8i	0.0751734	−	0.426330i	0.999048	−	0.0436333i	$-0.0138933\pi$
$348$		0			0
$349$	2.59627e7	+	4.49688e7i	0.0326935	+	0.0566268i	0.881909	−	0.471419i	$-0.156258\pi$
$349$	2.59627e7	+	4.49688e7i	0.0326935	+	0.0566268i	−0.849216	+	0.528046i	$0.822925\pi$
$350$		0			0
$351$	−5.41502e8	+	4.54374e8i	−0.668383	+	0.560840i
$352$		0			0
$353$	1.09729e8	−	1.90057e8i	0.132773	−	0.229970i	−0.791971	−	0.610558i	$-0.790945\pi$
$353$	1.09729e8	−	1.90057e8i	0.132773	−	0.229970i	0.924745	+	0.380588i	$0.124278\pi$
$354$		0			0
$355$	7.03097e7	+	2.55906e7i	0.0834096	+	0.0303586i
$356$		0			0
$357$	1.60999e7	+	9.13069e7i	0.0187277	+	0.106210i
$358$		0			0
$359$	−5.90841e8	−	4.95775e8i	−0.673970	−	0.565528i	0.240268	−	0.970707i	$-0.422765\pi$
$359$	−5.90841e8	−	4.95775e8i	−0.673970	−	0.565528i	−0.914237	+	0.405179i	$0.867209\pi$
$360$		0			0
$361$	7.92120e8	−	4.14189e8i	0.886168	−	0.463365i
$362$		0			0
$363$	−1.99262e8	−	1.67200e8i	−0.218651	−	0.183470i
$364$		0			0
$365$	−6.81672e7	−	3.86595e8i	−0.0733754	−	0.416132i
$366$		0			0
$367$	5.98999e8	+	2.18018e8i	0.632550	+	0.230229i	0.638341	−	0.769754i	$-0.279621\pi$
$367$	5.98999e8	+	2.18018e8i	0.632550	+	0.230229i	−0.00579087	+	0.999983i	$0.501843\pi$
$368$		0			0
$369$	5.50972e8	−	9.54311e8i	0.570870	−	0.988776i
$370$		0			0
$371$	−1.60891e9	+	1.35003e9i	−1.63577	+	1.37257i
$372$		0			0
$373$	−4.15553e7	−	7.19759e7i	−0.0414615	−	0.0718135i	0.844550	−	0.535477i	$-0.179868\pi$
$373$	−4.15553e7	−	7.19759e7i	−0.0414615	−	0.0718135i	−0.886011	+	0.463663i	$0.846535\pi$
$374$		0			0
$375$	6.03652e7	−	3.42348e8i	0.0591122	−	0.335242i
$376$		0			0
$377$	−1.34494e9	+	4.89517e8i	−1.29273	+	0.470515i
$378$		0			0
$379$	−3.10283e8			−0.292766			−0.146383	−	0.989228i	$-0.546763\pi$
$379$	−3.10283e8			−0.292766			−0.146383	+	0.989228i	$0.546763\pi$
$380$		0			0
$381$	−4.52806e8			−0.419445
$382$		0			0
$383$	5.17209e8	−	1.88249e8i	0.470403	−	0.171213i	−0.0959318	−	0.995388i	$-0.530583\pi$
$383$	5.17209e8	−	1.88249e8i	0.470403	−	0.171213i	0.566335	+	0.824175i	$0.308361\pi$
$384$		0			0
$385$	4.05645e8	−	2.30053e9i	0.362271	−	2.05454i
$386$		0			0
$387$	2.32664e8	+	4.02986e8i	0.204052	+	0.353429i
$388$		0			0
$389$	8.35270e8	−	7.00875e8i	0.719454	−	0.603694i	−0.207780	−	0.978176i	$-0.566624\pi$
$389$	8.35270e8	−	7.00875e8i	0.719454	−	0.603694i	0.927234	+	0.374482i	$0.122179\pi$
$390$		0			0
$391$	5.15411e7	−	8.92719e7i	0.0436049	−	0.0755259i
$392$		0			0
$393$	−6.86259e8	−	2.49778e8i	−0.570315	−	0.207578i
$394$		0			0
$395$	8.89262e7	+	5.04325e8i	0.0726005	+	0.411738i
$396$		0			0
$397$	1.77844e8	+	1.49229e8i	0.142651	+	0.119698i	0.711321	−	0.702868i	$-0.248097\pi$
$397$	1.77844e8	+	1.49229e8i	0.142651	+	0.119698i	−0.568670	+	0.822566i	$0.692542\pi$
$398$		0			0
$399$	6.86234e8	+	3.37636e8i	0.540838	+	0.266099i
$400$		0			0
$401$	2.97974e8	+	2.50030e8i	0.230767	+	0.193636i	0.750838	−	0.660487i	$-0.229650\pi$
$401$	2.97974e8	+	2.50030e8i	0.230767	+	0.193636i	−0.520071	+	0.854123i	$0.674094\pi$
$402$		0			0
$403$	4.72956e7	+	2.68227e8i	0.0359959	+	0.204143i
$404$		0			0
$405$	−7.04844e8	−	2.56542e8i	−0.527230	−	0.191896i
$406$		0			0
$407$	−5.17002e7	+	8.95474e7i	−0.0380112	+	0.0658374i
$408$		0			0
$409$	−1.02821e9	+	8.62772e8i	−0.743106	+	0.623540i	−0.933670	−	0.358135i	$-0.883413\pi$
$409$	−1.02821e9	+	8.62772e8i	−0.743106	+	0.623540i	0.190564	+	0.981675i	$0.438968\pi$
$410$		0			0
$411$	2.85302e8	+	4.94157e8i	0.202702	+	0.351090i
$412$		0			0
$413$	−7.45793e8	+	4.22960e9i	−0.520946	+	2.95443i
$414$		0			0
$415$	−6.06152e8	+	2.20621e8i	−0.416307	+	0.151523i
$416$		0			0
$417$	−2.04459e8			−0.138080
$418$		0			0
$419$	1.66439e9			1.10537			0.552683	−	0.833392i	$-0.313604\pi$
$419$	1.66439e9			1.10537			0.552683	+	0.833392i	$0.313604\pi$
$420$		0			0
$421$	−3.24298e8	+	1.18035e8i	−0.211815	+	0.0770944i	−0.445749	−	0.895158i	$-0.647062\pi$
$421$	−3.24298e8	+	1.18035e8i	−0.211815	+	0.0770944i	0.233933	+	0.972253i	$0.424840\pi$
$422$		0			0
$423$	1.30960e8	−	7.42709e8i	0.0841291	−	0.477120i
$424$		0			0
$425$	−5.43999e7	−	9.42234e7i	−0.0343745	−	0.0595384i
$426$		0			0
$427$	−2.73467e9	+	2.29466e9i	−1.69984	+	1.42633i
$428$		0			0
$429$	−5.18542e8	+	8.98142e8i	−0.317091	+	0.549217i
$430$		0			0
$431$	2.17665e9	+	7.92237e8i	1.30954	+	0.476634i	0.900092	−	0.435699i	$-0.143499\pi$
$431$	2.17665e9	+	7.92237e8i	1.30954	+	0.476634i	0.409449	+	0.912333i	$0.365721\pi$
$432$		0			0
$433$	8.67010e7	+	4.91706e8i	0.0513235	+	0.291070i	0.999657	−	0.0262065i	$-0.00834274\pi$
$433$	8.67010e7	+	4.91706e8i	0.0513235	+	0.291070i	−0.948333	+	0.317277i	$0.897232\pi$
$434$		0			0
$435$	3.03925e8	+	2.55023e8i	0.177033	+	0.148548i
$436$		0			0
$437$	−3.43169e8	−	7.77987e8i	−0.196709	−	0.445951i
$438$		0			0
$439$	−2.26745e9	−	1.90262e9i	−1.27912	−	1.07331i	−0.993365	−	0.115007i	$-0.963311\pi$
$439$	−2.26745e9	−	1.90262e9i	−1.27912	−	1.07331i	−0.285756	−	0.958302i	$-0.592245\pi$
$440$		0			0
$441$	−7.61266e8	−	4.31735e9i	−0.422670	−	2.39708i
$442$		0			0
$443$	2.10947e9	+	7.67785e8i	1.15282	+	0.419592i	0.846527	−	0.532346i	$-0.178689\pi$
$443$	2.10947e9	+	7.67785e8i	1.15282	+	0.419592i	0.306292	+	0.951938i	$0.400912\pi$
$444$		0			0
$445$	−3.97527e8	+	6.88537e8i	−0.213849	+	0.370397i
$446$		0			0
$447$	−3.89728e7	+	3.27021e7i	−0.0206389	+	0.0173181i
$448$		0			0
$449$	1.22304e9	+	2.11838e9i	0.637647	+	1.10444i	0.985948	+	0.167054i	$0.0534254\pi$
$449$	1.22304e9	+	2.11838e9i	0.637647	+	1.10444i	−0.348301	+	0.937383i	$0.613241\pi$
$450$		0			0
$451$	5.92047e8	−	3.35767e9i	0.303905	−	1.72353i
$452$		0			0
$453$	−5.84406e8	+	2.12706e8i	−0.295373	+	0.107507i
$454$		0			0
$455$	−4.43935e9			−2.20943
$456$		0			0
$457$	−2.96384e9			−1.45260			−0.726302	−	0.687376i	$-0.758763\pi$
$457$	−2.96384e9			−1.45260			−0.726302	+	0.687376i	$0.758763\pi$
$458$		0			0
$459$	2.07613e8	−	7.55648e7i	0.100210	−	0.0364733i
$460$		0			0
$461$	−3.61921e8	+	2.05256e9i	−0.172052	+	0.975758i	0.769439	+	0.638720i	$0.220536\pi$
$461$	−3.61921e8	+	2.05256e9i	−0.172052	+	0.975758i	−0.941491	+	0.337037i	$0.890575\pi$
$462$		0			0
$463$	−1.73350e9	−	3.00250e9i	−0.811688	−	1.40589i	−0.911682	−	0.410897i	$-0.865216\pi$
$463$	−1.73350e9	−	3.00250e9i	−0.811688	−	1.40589i	0.0999936	−	0.994988i	$-0.468118\pi$
$464$		0			0
$465$	5.78363e7	−	4.85304e7i	0.0266756	−	0.0223835i
$466$		0			0
$467$	−1.08197e9	+	1.87402e9i	−0.491592	+	0.851463i	−0.999953	−	0.00968132i	$-0.996918\pi$
$467$	−1.08197e9	+	1.87402e9i	−0.491592	+	0.851463i	0.508361	+	0.861144i	$0.330252\pi$
$468$		0			0
$469$	4.11926e9	+	1.49929e9i	1.84380	+	0.671089i
$470$		0			0
$471$	2.13059e7	+	1.20832e8i	0.00939564	+	0.0532853i
$472$		0			0
$473$	1.10291e9	+	9.25452e8i	0.479211	+	0.402105i
$474$		0			0
$475$	−8.92228e8	−	9.68920e7i	−0.381987	−	0.0414820i
$476$		0			0
$477$	1.81798e9	+	1.52546e9i	0.766962	+	0.643558i
$478$		0			0
$479$	4.41524e8	+	2.50401e9i	0.183561	+	1.04103i	0.927791	+	0.373101i	$0.121706\pi$
$479$	4.41524e8	+	2.50401e9i	0.183561	+	1.04103i	−0.744230	+	0.667924i	$0.767183\pi$
$480$		0			0
$481$	1.84651e8	+	6.72075e7i	0.0756561	+	0.0275366i
$482$		0			0
$483$	3.63763e8	−	6.30057e8i	0.146894	−	0.254428i
$484$		0			0
$485$	1.08432e9	−	9.09851e8i	0.431579	−	0.362138i
$486$		0			0
$487$	2.54957e8	+	4.41598e8i	0.100026	+	0.173251i	0.911695	−	0.410867i	$-0.134774\pi$
$487$	2.54957e8	+	4.41598e8i	0.100026	+	0.173251i	−0.811669	+	0.584118i	$0.801441\pi$
$488$		0			0
$489$	7.22431e7	−	4.09711e8i	0.0279393	−	0.158452i
$490$		0			0
$491$	2.11184e9	−	7.68645e8i	0.805146	−	0.293049i	0.0935290	−	0.995617i	$-0.470185\pi$
$491$	2.11184e9	−	7.68645e8i	0.805146	−	0.293049i	0.711617	+	0.702567i	$0.247963\pi$
$492$		0			0
$493$	4.47341e8			0.168141
$494$		0			0
$495$	−2.63957e9			−0.978171
$496$		0			0
$497$	5.59508e8	−	2.03644e8i	0.204437	−	0.0744089i
$498$		0			0
$499$	−9.26286e8	+	5.25323e9i	−0.333728	+	1.89267i	0.105712	+	0.994397i	$0.466288\pi$
$499$	−9.26286e8	+	5.25323e9i	−0.333728	+	1.89267i	−0.439441	+	0.898272i	$0.644823\pi$
$500$		0			0
$501$	−2.27600e8	−	3.94214e8i	−0.0808611	−	0.140056i
$502$		0			0
$503$	2.25206e9	−	1.88970e9i	0.789028	−	0.662073i	−0.156477	−	0.987682i	$-0.550014\pi$
$503$	2.25206e9	−	1.88970e9i	0.789028	−	0.662073i	0.945505	+	0.325609i	$0.105569\pi$
$504$		0			0
$505$	1.61812e9	−	2.80266e9i	0.559101	−	0.968392i
$506$		0			0
$507$	9.87833e8	+	3.59542e8i	0.336632	+	0.122524i
$508$		0			0
$509$	5.23918e8	+	2.97128e9i	0.176097	+	0.998693i	0.936871	+	0.349676i	$0.113708\pi$
$509$	5.23918e8	+	2.97128e9i	0.176097	+	0.998693i	−0.760774	+	0.649017i	$0.775180\pi$
$510$		0			0
$511$	−2.39304e9	−	2.00800e9i	−0.793372	−	0.665718i
$512$		0			0
$513$	5.08387e8	−	1.75013e9i	0.166258	−	0.572348i
$514$		0			0
$515$	−2.74775e9	−	2.30563e9i	−0.886444	−	0.743815i
$516$		0			0
$517$	−4.05195e8	−	2.29797e9i	−0.128958	−	0.731355i
$518$		0			0
$519$	−9.51636e8	−	3.46367e8i	−0.298803	−	0.108755i
$520$		0			0
$521$	1.35536e9	−	2.34754e9i	0.419876	−	0.727247i	−0.576051	−	0.817414i	$-0.695407\pi$
$521$	1.35536e9	−	2.34754e9i	0.419876	−	0.727247i	0.995927	+	0.0901674i	$0.0287402\pi$
$522$		0			0
$523$	5.63190e8	−	4.72572e8i	0.172147	−	0.144448i	−0.552644	−	0.833418i	$-0.686381\pi$
$523$	5.63190e8	−	4.72572e8i	0.172147	−	0.144448i	0.724791	+	0.688969i	$0.241937\pi$
$524$		0			0
$525$	−3.83940e8	−	6.65003e8i	−0.115799	−	0.200570i
$526$		0			0
$527$	1.47823e7	−	8.38347e7i	0.00439952	−	0.0249509i
$528$		0			0
$529$	2.43940e9	−	8.87868e8i	0.716453	−	0.260767i
$530$		0			0
$531$	4.85294e9			1.40661
$532$		0			0
$533$	−6.47932e9			−1.85347
$534$		0			0
$535$	4.15020e9	−	1.51055e9i	1.17174	−	0.426478i
$536$		0			0
$537$	−3.03025e8	+	1.71854e9i	−0.0844438	+	0.478905i
$538$		0			0
$539$	−6.78208e9	−	1.17469e10i	−1.86553	−	3.23119i
$540$		0			0
$541$	1.41044e9	−	1.18350e9i	0.382970	−	0.321350i	−0.430897	−	0.902401i	$-0.641803\pi$
$541$	1.41044e9	−	1.18350e9i	0.382970	−	0.321350i	0.813867	+	0.581051i	$0.197358\pi$
$542$		0			0
$543$	6.26856e8	−	1.08575e9i	0.168023	−	0.291024i
$544$		0			0
$545$	4.10570e9	+	1.49435e9i	1.08642	+	0.395426i
$546$		0			0
$547$	1.02112e9	+	5.79104e9i	0.266760	+	1.51287i	0.763978	+	0.645243i	$0.223244\pi$
$547$	1.02112e9	+	5.79104e9i	0.266760	+	1.51287i	−0.497218	+	0.867626i	$0.665645\pi$
$548$		0			0
$549$	3.09003e9	+	2.59284e9i	0.797001	+	0.668763i
$550$		0			0
$551$	2.17925e9	−	2.97781e9i	0.554979	−	0.758344i
$552$		0			0
$553$	3.12180e9	+	2.61950e9i	0.784994	+	0.658689i
$554$		0			0
$555$	−9.45870e6	−	5.36430e7i	−0.00234858	−	0.0133195i
$556$		0			0
$557$	4.21016e9	+	1.53237e9i	1.03230	+	0.375726i	0.801956	−	0.597383i	$-0.203793\pi$
$557$	4.21016e9	+	1.53237e9i	1.03230	+	0.375726i	0.230344	+	0.973109i	$0.426015\pi$
$558$		0			0
$559$	1.36804e9	−	2.36952e9i	0.331252	−	0.573746i
$560$		0			0
$561$	2.48308e8	−	2.08355e8i	0.0593772	−	0.0498234i
$562$		0			0
$563$	−6.48704e8	−	1.12359e9i	−0.153203	−	0.265356i	0.779200	−	0.626775i	$-0.215626\pi$
$563$	−6.48704e8	−	1.12359e9i	−0.153203	−	0.265356i	−0.932403	+	0.361420i	$0.882292\pi$
$564$		0			0
$565$	−1.36581e8	+	7.74588e8i	−0.0318581	+	0.180676i
$566$		0			0
$567$	−5.60898e9	+	2.04150e9i	−1.29224	+	0.470337i
$568$		0			0
$569$	9.54822e8			0.217285			0.108642	−	0.994081i	$-0.465350\pi$
$569$	9.54822e8			0.217285			0.108642	+	0.994081i	$0.465350\pi$
$570$		0			0
$571$	−1.66483e9			−0.374235			−0.187117	−	0.982338i	$-0.559914\pi$
$571$	−1.66483e9			−0.374235			−0.187117	+	0.982338i	$0.559914\pi$
$572$		0			0
$573$	−3.21755e8	+	1.17109e8i	−0.0714470	+	0.0260046i
$574$		0			0
$575$	−1.48250e8	+	8.40768e8i	−0.0325205	+	0.184433i
$576$		0			0
$577$	−6.58097e8	−	1.13986e9i	−0.142618	−	0.247022i	0.785864	−	0.618400i	$-0.212219\pi$
$577$	−6.58097e8	−	1.13986e9i	−0.142618	−	0.247022i	−0.928482	+	0.371378i	$0.878885\pi$
$578$		0			0
$579$	−1.68132e7	+	1.41080e7i	−0.00359978	+	0.00302058i
$580$		0			0
$581$	−2.56659e9	+	4.44547e9i	−0.542926	+	0.940375i
$582$		0			0
$583$	6.89996e9	+	2.51138e9i	1.44214	+	0.524895i
$584$		0			0
$585$	8.71058e8	+	4.94002e9i	0.179888	+	1.02019i
$586$		0			0
$587$	−1.93320e9	−	1.62215e9i	−0.394497	−	0.331022i	0.423865	−	0.905725i	$-0.360673\pi$
$587$	−1.93320e9	−	1.62215e9i	−0.394497	−	0.331022i	−0.818362	+	0.574703i	$0.805118\pi$
$588$		0			0
$589$	−4.86049e8	−	5.06808e8i	−0.0980114	−	0.102197i
$590$		0			0
$591$	3.04065e8	+	2.55141e8i	0.0605913	+	0.0508421i
$592$		0			0
$593$	−1.21163e9	−	6.87147e9i	−0.238604	−	1.35319i	−0.834890	−	0.550417i	$-0.814469\pi$
$593$	−1.21163e9	−	6.87147e9i	−0.238604	−	1.35319i	0.596286	−	0.802772i	$-0.296642\pi$
$594$		0			0
$595$	1.30385e9	+	4.74562e8i	0.253757	+	0.0923600i
$596$		0			0
$597$	1.33784e9	−	2.31721e9i	0.257332	−	0.445713i
$598$		0			0
$599$	−3.30400e9	+	2.77239e9i	−0.628126	+	0.527060i	−0.900346	−	0.435175i	$-0.856687\pi$
$599$	−3.30400e9	+	2.77239e9i	−0.628126	+	0.527060i	0.272220	+	0.962235i	$0.412242\pi$
$600$		0			0
$601$	−3.50830e9	−	6.07655e9i	−0.659228	−	1.14182i	−0.980816	−	0.194937i	$-0.937550\pi$
$601$	−3.50830e9	−	6.07655e9i	−0.659228	−	1.14182i	0.321587	−	0.946880i	$-0.395784\pi$
$602$		0			0
$603$	8.60123e8	−	4.87800e9i	0.159753	−	0.906006i
$604$		0			0
$605$	−3.65801e9	+	1.33141e9i	−0.671585	+	0.244437i
$606$		0			0
$607$	−2.63330e9			−0.477904			−0.238952	−	0.971031i	$-0.576804\pi$
$607$	−2.63330e9			−0.477904			−0.238952	+	0.971031i	$0.576804\pi$
$608$		0			0
$609$	3.15721e9			0.566425
$610$		0			0
$611$	−4.16700e9	+	1.51666e9i	−0.739059	+	0.268995i
$612$		0			0
$613$	−1.13148e9	+	6.41697e9i	−0.198398	+	1.12517i	0.709098	+	0.705110i	$0.249102\pi$
$613$	−1.13148e9	+	6.41697e9i	−0.198398	+	1.12517i	−0.907496	+	0.420060i	$0.862009\pi$
$614$		0			0
$615$	8.98039e8	+	1.55545e9i	0.155680	+	0.269645i
$616$		0			0
$617$	6.52066e9	−	5.47148e9i	1.11762	−	0.937793i	0.119136	−	0.992878i	$-0.461987\pi$
$617$	6.52066e9	−	5.47148e9i	1.11762	−	0.937793i	0.998482	+	0.0550850i	$0.0175430\pi$
$618$		0			0
$619$	2.47342e9	−	4.28408e9i	0.419160	−	0.726007i	−0.576695	−	0.816959i	$-0.695658\pi$
$619$	2.47342e9	−	4.28408e9i	0.419160	−	0.726007i	0.995855	+	0.0909528i	$0.0289912\pi$
$620$		0			0
$621$	−1.62911e9	−	5.92948e8i	−0.272979	−	0.0993564i
$622$		0			0
$623$	1.09865e9	+	6.23073e9i	0.182033	+	1.03236i
$624$		0			0
$625$	−2.18878e9	−	1.83661e9i	−0.358610	−	0.300910i
$626$		0			0
$627$	−1.77307e8	−	2.66792e9i	−0.0287270	−	0.432252i
$628$		0			0
$629$	−4.70480e7	−	3.94780e7i	−0.00753814	−	0.00632525i
$630$		0			0
$631$	1.16992e9	+	6.63497e9i	0.185377	+	1.05132i	0.925471	+	0.378820i	$0.123670\pi$
$631$	1.16992e9	+	6.63497e9i	0.185377	+	1.05132i	−0.740094	+	0.672503i	$0.765219\pi$
$632$		0			0
$633$	−1.60751e9	−	5.85086e8i	−0.251907	−	0.0916868i
$634$		0			0
$635$	−3.38822e9	+	5.86857e9i	−0.525126	+	0.909545i
$636$		0			0
$637$	−1.97465e10	+	1.65693e10i	−3.02693	+	2.53989i
$638$		0			0
$639$	−3.36394e8	−	5.82651e8i	−0.0510029	−	0.0883396i
$640$		0			0
$641$	−1.74396e9	+	9.89049e9i	−0.261537	+	1.48325i	0.517180	+	0.855876i	$0.326982\pi$
$641$	−1.74396e9	+	9.89049e9i	−0.261537	+	1.48325i	−0.778718	+	0.627375i	$0.784129\pi$
$642$		0			0
$643$	−9.81952e9	+	3.57401e9i	−1.45664	+	0.530173i	−0.944437	−	0.328692i	$-0.893392\pi$
$643$	−9.81952e9	+	3.57401e9i	−1.45664	+	0.530173i	−0.512202	+	0.858865i	$0.671170\pi$
$644$		0			0
$645$	−7.58448e8			−0.111293
$646$		0			0
$647$	2.74731e8			0.0398788			0.0199394	−	0.999801i	$-0.493653\pi$
$647$	2.74731e8			0.0398788			0.0199394	+	0.999801i	$0.493653\pi$
$648$		0			0
$649$	1.41097e10	−	5.13551e9i	2.02610	−	0.737440i
$650$		0			0
$651$	1.04330e8	−	5.91683e8i	0.0148209	−	0.0840534i
$652$		0			0
$653$	−5.53018e9	−	9.57855e9i	−0.777218	−	1.34618i	−0.933539	−	0.358475i	$-0.883297\pi$
$653$	−5.53018e9	−	9.57855e9i	−0.777218	−	1.34618i	0.156321	−	0.987706i	$-0.450037\pi$
$654$		0			0
$655$	−8.37232e9	+	7.02521e9i	−1.16413	+	0.976821i
$656$		0			0
$657$	−1.76491e9	+	3.05692e9i	−0.242798	+	0.420538i
$658$		0			0
$659$	−1.85913e9	−	6.76668e8i	−0.253052	−	0.0921036i	0.212379	−	0.977187i	$-0.431879\pi$
$659$	−1.85913e9	−	6.76668e8i	−0.253052	−	0.0921036i	−0.465432	+	0.885084i	$0.654101\pi$
$660$		0			0
$661$	1.05673e9	+	5.99301e9i	0.142318	+	0.807123i	0.969482	+	0.245163i	$0.0788415\pi$
$661$	1.05673e9	+	5.99301e9i	0.142318	+	0.807123i	−0.827164	+	0.561960i	$0.810047\pi$
$662$		0			0
$663$	−4.71882e8	−	3.95956e8i	−0.0628834	−	0.0527655i
$664$		0			0
$665$	9.51080e9	−	6.36746e9i	1.25413	−	0.839635i
$666$		0			0
$667$	−2.68899e9	−	2.25633e9i	−0.350872	−	0.294416i
$668$		0			0
$669$	−5.44228e8	−	3.08647e9i	−0.0702732	−	0.398539i
$670$		0			0
$671$	1.17279e10	+	4.26861e9i	1.49862	+	0.545453i
$672$		0			0
$673$	4.92275e9	−	8.52645e9i	0.622522	−	1.07824i	−0.366492	−	0.930421i	$-0.619441\pi$
$673$	4.92275e9	−	8.52645e9i	0.622522	−	1.07824i	0.989014	−	0.147819i	$-0.0472253\pi$
$674$		0			0
$675$	−1.40172e9	+	1.17619e9i	−0.175428	+	0.147202i
$676$		0			0
$677$	6.36394e9	+	1.10227e10i	0.788253	+	1.36529i	0.927036	+	0.374972i	$0.122348\pi$
$677$	6.36394e9	+	1.10227e10i	0.788253	+	1.36529i	−0.138783	+	0.990323i	$0.544319\pi$
$678$		0			0
$679$	1.95598e9	−	1.10929e10i	0.239784	−	1.35988i
$680$		0			0
$681$	−1.69121e9	+	6.15550e8i	−0.205202	+	0.0746875i
$682$		0			0
$683$	−2.11529e9			−0.254037			−0.127019	−	0.991900i	$-0.540541\pi$
$683$	−2.11529e9			−0.254037			−0.127019	+	0.991900i	$0.540541\pi$
$684$		0			0
$685$	8.53933e9			1.01509
$686$		0			0
$687$	−3.03497e9	+	1.10464e9i	−0.357113	+	0.129978i
$688$		0			0
$689$	2.42312e9	−	1.37422e10i	0.282233	−	1.60062i
$690$		0			0
$691$	4.90549e8	+	8.49656e8i	0.0565600	+	0.0979648i	0.892919	−	0.450217i	$-0.148653\pi$
$691$	4.90549e8	+	8.49656e8i	0.0565600	+	0.0979648i	−0.836359	+	0.548182i	$0.815320\pi$
$692$		0			0
$693$	−1.60908e10	+	1.35018e10i	−1.83659	+	1.54108i
$694$		0			0
$695$	−1.52991e9	+	2.64988e9i	−0.172869	+	0.299419i
$696$		0			0
$697$	1.90299e9	+	6.92633e8i	0.212874	+	0.0774798i
$698$		0			0
$699$	6.70168e8	+	3.80071e9i	0.0742187	+	0.420915i
$700$		0			0
$701$	1.27949e9	+	1.07362e9i	0.140289	+	0.117717i	0.710232	−	0.703968i	$-0.248590\pi$
$701$	1.27949e9	+	1.07362e9i	0.140289	+	0.117717i	−0.569942	+	0.821685i	$0.693035\pi$
$702$		0			0
$703$	−4.91990e8	+	1.20865e8i	−0.0534088	+	0.0131207i
$704$		0			0
$705$	9.41644e8	+	7.90133e8i	0.101210	+	0.0849256i
$706$		0			0
$707$	−4.47200e9	−	2.53620e10i	−0.475920	−	2.69907i
$708$		0			0
$709$	8.90093e9	+	3.23967e9i	0.937937	+	0.341381i	0.765351	−	0.643613i	$-0.222566\pi$
$709$	8.90093e9	+	3.23967e9i	0.937937	+	0.341381i	0.172586	+	0.984994i	$0.444788\pi$
$710$		0			0
$711$	2.30238e9	−	3.98785e9i	0.240234	−	0.416097i
$712$		0			0
$713$	−5.11709e8	+	4.29375e8i	−0.0528701	+	0.0443633i
$714$		0			0
$715$	7.76021e9	+	1.34411e10i	0.793967	+	1.37519i
$716$		0			0
$717$	−8.27843e8	+	4.69493e9i	−0.0838747	+	0.475677i
$718$		0			0
$719$	−4.06940e9	+	1.48114e9i	−0.408300	+	0.148609i	−0.538001	−	0.842944i	$-0.680820\pi$
$719$	−4.06940e9	+	1.48114e9i	−0.408300	+	0.148609i	0.129701	+	0.991553i	$0.458598\pi$
$720$		0			0
$721$	−2.85439e10			−2.83622
$722$		0			0
$723$	3.31884e9			0.326589
$724$		0			0
$725$	−3.48149e9	+	1.26716e9i	−0.339298	+	0.123494i
$726$		0			0
$727$	2.18732e9	−	1.24049e10i	0.211126	−	1.19736i	−0.676378	−	0.736555i	$-0.736451\pi$
$727$	2.18732e9	−	1.24049e10i	0.211126	−	1.19736i	0.887504	−	0.460800i	$-0.152438\pi$
$728$		0			0
$729$	2.39538e9	+	4.14892e9i	0.228996	+	0.396633i
$730$		0			0
$731$	−6.55097e8	+	5.49692e8i	−0.0620290	+	0.0520485i
$732$		0			0
$733$	4.70043e9	−	8.14138e9i	0.440833	−	0.763544i	−0.556919	−	0.830567i	$-0.688017\pi$
$733$	4.70043e9	−	8.14138e9i	0.440833	−	0.763544i	0.997751	+	0.0670225i	$0.0213499\pi$
$734$		0			0
$735$	6.71456e9	+	2.44390e9i	0.623752	+	0.227027i
$736$		0			0
$737$	−2.66126e9	−	1.50928e10i	−0.244879	−	1.38878i
$738$		0			0
$739$	1.27109e10	+	1.06657e10i	1.15857	+	0.972155i	0.999885	−	0.0151746i	$-0.00483041\pi$
$739$	1.27109e10	+	1.06657e10i	1.15857	+	0.972155i	0.158684	+	0.987329i	$0.449275\pi$
$740$		0			0
$741$	−4.93456e9	+	1.21225e9i	−0.445538	+	0.109453i
$742$		0			0
$743$	1.03802e10	+	8.71004e9i	0.928422	+	0.779039i	0.975534	−	0.219851i	$-0.0705571\pi$
$743$	1.03802e10	+	8.71004e9i	0.928422	+	0.779039i	−0.0471112	+	0.998890i	$0.515002\pi$
$744$		0			0
$745$	1.32211e8	+	7.49806e8i	0.0117144	+	0.0664358i
$746$		0			0
$747$	5.45042e9	+	1.98379e9i	0.478418	+	0.174130i
$748$		0			0
$749$	1.75729e10	−	3.04372e10i	1.52812	−	2.64678i
$750$		0			0
$751$	−9.86961e8	+	8.28158e8i	−0.0850276	+	0.0713466i	−0.684311	−	0.729191i	$-0.739897\pi$
$751$	−9.86961e8	+	8.28158e8i	−0.0850276	+	0.0713466i	0.599283	+	0.800537i	$0.295452\pi$
$752$		0			0
$753$	1.55487e9	+	2.69311e9i	0.132712	+	0.229864i
$754$		0			0
$755$	−1.61618e9	+	9.16579e9i	−0.136670	+	0.775096i
$756$		0			0
$757$	−2.09504e10	+	7.62533e9i	−1.75532	+	0.638885i	−0.999867	−	0.0163147i	$-0.994807\pi$
$757$	−2.09504e10	+	7.62533e9i	−1.75532	+	0.638885i	−0.755456	+	0.655200i	$0.772584\pi$
$758$		0			0
$759$	−2.54351e9			−0.211148
$760$		0			0
$761$	5.34881e9			0.439957			0.219979	−	0.975505i	$-0.429401\pi$
$761$	5.34881e9			0.439957			0.219979	+	0.975505i	$0.429401\pi$
$762$		0			0
$763$	3.26722e10	−	1.18917e10i	2.66283	−	0.969189i
$764$		0			0
$765$	2.72251e8	−	1.54401e9i	0.0219864	−	0.124691i
$766$		0			0
$767$	−1.42674e10	−	2.47119e10i	−1.14172	−	1.97753i
$768$		0			0
$769$	−1.50562e10	+	1.26337e10i	−1.19392	+	1.00181i	−0.194133	+	0.980975i	$0.562189\pi$
$769$	−1.50562e10	+	1.26337e10i	−1.19392	+	1.00181i	−0.999783	+	0.0208392i	$0.993366\pi$
$770$		0			0
$771$	3.56978e9	−	6.18304e9i	0.280512	−	0.485860i
$772$		0			0
$773$	8.79592e9	+	3.20145e9i	0.684941	+	0.249298i	0.660968	−	0.750414i	$-0.270146\pi$
$773$	8.79592e9	+	3.20145e9i	0.684941	+	0.249298i	0.0239735	+	0.999713i	$0.492368\pi$
$774$		0			0
$775$	1.22429e8	+	6.94328e8i	0.00944773	+	0.0535807i
$776$		0			0
$777$	−3.32052e8	−	2.78625e8i	−0.0253941	−	0.0213082i
$778$		0			0
$779$	1.38812e10	−	9.29344e9i	1.05207	−	0.704361i
$780$		0			0
$781$	−1.59462e9	−	1.33805e9i	−0.119779	−	0.100506i
$782$		0			0
$783$	−1.30644e9	−	7.40918e9i	−0.0972574	−	0.551574i
$784$		0			0
$785$	1.72546e9	+	6.28016e8i	0.127309	+	0.0463369i
$786$		0			0
$787$	−3.14760e9	+	5.45181e9i	−0.230180	+	0.398684i	−0.957861	−	0.287232i	$-0.907265\pi$
$787$	−3.14760e9	+	5.45181e9i	−0.230180	+	0.398684i	0.727681	+	0.685916i	$0.240598\pi$
$788$		0			0
$789$	−7.98326e8	+	6.69875e8i	−0.0578643	+	0.0485539i
$790$		0			0
$791$	3.12954e9	+	5.42052e9i	0.224834	+	0.389425i
$792$		0			0
$793$	4.11859e9	−	2.33577e10i	0.293287	−	1.66331i
$794$		0			0
$795$	−3.63485e9	+	1.32298e9i	−0.256567	+	0.0933828i
$796$		0			0
$797$	−1.64437e10			−1.15053			−0.575263	−	0.817969i	$-0.695100\pi$
$797$	−1.64437e10			−1.15053			−0.575263	+	0.817969i	$0.695100\pi$
$798$		0			0
$799$	1.38599e9			0.0961270
$800$		0			0
$801$	6.71785e9	−	2.44510e9i	0.461867	−	0.168106i
$802$		0			0
$803$	−1.89649e9	+	1.07555e10i	−0.129255	+	0.733039i
$804$		0			0
$805$	−5.44388e9	−	9.42907e9i	−0.367809	−	0.637064i
$806$		0			0
$807$	−5.96100e9	+	5.00187e9i	−0.399265	+	0.335023i
$808$		0			0
$809$	9.37250e9	−	1.62336e10i	0.622351	−	1.07794i	−0.366695	−	0.930341i	$-0.619511\pi$
$809$	9.37250e9	−	1.62336e10i	0.622351	−	1.07794i	0.989047	−	0.147603i	$-0.0471558\pi$
$810$		0			0
$811$	1.10238e10	+	4.01232e9i	0.725699	+	0.264133i	0.678343	−	0.734745i	$-0.262698\pi$
$811$	1.10238e10	+	4.01232e9i	0.725699	+	0.264133i	0.0473560	+	0.998878i	$0.484920\pi$
$812$		0			0
$813$	1.27588e9	+	7.23590e9i	0.0832711	+	0.472254i
$814$		0			0
$815$	−4.76946e9	−	4.00205e9i	−0.308616	−	0.258959i
$816$		0			0
$817$	4.67781e8	+	7.03864e9i	0.0300099	+	0.451556i
$818$		0			0
$819$	3.05789e10	+	2.56588e10i	1.94504	+	1.63208i
$820$		0			0
$821$	6.13388e8	+	3.47870e9i	0.0386843	+	0.219389i	0.998022	−	0.0628728i	$-0.0200263\pi$
$821$	6.13388e8	+	3.47870e9i	0.0386843	+	0.219389i	−0.959337	+	0.282262i	$0.908915\pi$
$822$		0			0
$823$	3.52827e9	+	1.28418e9i	0.220629	+	0.0803023i	0.449970	−	0.893044i	$-0.351435\pi$
$823$	3.52827e9	+	1.28418e9i	0.220629	+	0.0803023i	−0.229341	+	0.973346i	$0.573657\pi$
$824$		0			0
$825$	−1.34229e9	+	2.32492e9i	−0.0832258	+	0.144151i
$826$		0			0
$827$	−1.29048e10	+	1.08285e10i	−0.793384	+	0.665728i	−0.946581	−	0.322467i	$-0.895488\pi$
$827$	−1.29048e10	+	1.08285e10i	−0.793384	+	0.665728i	0.153196	+	0.988196i	$0.451043\pi$
$828$		0			0
$829$	2.22068e9	+	3.84634e9i	0.135377	+	0.234480i	0.925742	−	0.378157i	$-0.123442\pi$
$829$	2.22068e9	+	3.84634e9i	0.135377	+	0.234480i	−0.790364	+	0.612637i	$0.790109\pi$
$830$		0			0
$831$	2.66253e8	−	1.50999e9i	0.0160950	−	0.0912791i
$832$		0			0
$833$	7.57084e9	−	2.75556e9i	0.453823	−	0.165178i
$834$		0			0
$835$	−6.81226e9			−0.404938
$836$		0			0
$837$	−1.43170e9			−0.0843945
$838$		0			0
$839$	−6.76825e9	+	2.46344e9i	−0.395649	+	0.144004i	−0.532180	−	0.846631i	$-0.678627\pi$
$839$	−6.76825e9	+	2.46344e9i	−0.395649	+	0.144004i	0.136531	+	0.990636i	$0.456405\pi$
$840$		0			0
$841$	−3.50204e8	+	1.98610e9i	−0.0203018	+	0.115137i
$842$		0			0
$843$	5.04340e9	+	8.73543e9i	0.289953	+	0.502213i
$844$		0			0
$845$	1.20515e10	−	1.01124e10i	0.687136	−	0.576575i
$846$		0			0
$847$	−1.54889e10	+	2.68276e10i	−0.875848	+	1.51701i
$848$		0			0
$849$	−4.08638e9	−	1.48732e9i	−0.229172	−	0.0834118i
$850$		0			0
$851$	8.36863e7	+	4.74609e8i	0.00465480	+	0.0263987i
$852$		0			0
$853$	−1.41947e9	−	1.19108e9i	−0.0783076	−	0.0657079i	0.602794	−	0.797897i	$-0.294054\pi$
$853$	−1.41947e9	−	1.19108e9i	−0.0783076	−	0.0657079i	−0.681101	+	0.732189i	$0.738499\pi$
$854$		0			0
$855$	−8.95171e9	−	9.33403e9i	−0.489807	−	0.510726i
$856$		0			0
$857$	−2.57579e10	−	2.16135e10i	−1.39791	−	1.17298i	−0.962019	−	0.272981i	$-0.911990\pi$
$857$	−2.57579e10	−	2.16135e10i	−1.39791	−	1.17298i	−0.435887	−	0.900001i	$-0.643565\pi$
$858$		0			0
$859$	−2.56585e9	−	1.45516e10i	−0.138119	−	0.783314i	−0.972637	−	0.232332i	$-0.925364\pi$
$859$	−2.56585e9	−	1.45516e10i	−0.138119	−	0.783314i	0.834517	−	0.550982i	$-0.185747\pi$
$860$		0			0
$861$	1.34308e10	+	4.88842e9i	0.717119	+	0.261010i
$862$		0			0
$863$	6.97383e9	−	1.20790e10i	0.369346	−	0.639727i	−0.620117	−	0.784509i	$-0.712915\pi$
$863$	6.97383e9	−	1.20790e10i	0.369346	−	0.639727i	0.989463	+	0.144783i	$0.0462483\pi$
$864$		0			0
$865$	−1.16099e10	+	9.74186e9i	−0.609919	+	0.511782i
$866$		0			0
$867$	−2.91069e9	−	5.04146e9i	−0.151680	−	0.262718i
$868$		0			0
$869$	2.47403e9	−	1.40309e10i	0.127890	−	0.725298i
$870$		0			0
$871$	−2.73682e10	+	9.96121e9i	−1.40340	+	0.510797i
$872$		0			0
$873$	−1.27277e10			−0.647443
$874$		0			0
$875$	−4.13997e10			−2.08915
$876$		0			0
$877$	1.52043e9	−	5.53393e8i	0.0761148	−	0.0277035i	−0.303682	−	0.952773i	$-0.598216\pi$
$877$	1.52043e9	−	5.53393e8i	0.0761148	−	0.0277035i	0.379797	+	0.925070i	$0.375994\pi$
$878$		0			0
$879$	4.71436e7	−	2.67365e8i	0.00234132	−	0.0132783i
$880$		0			0
$881$	−1.95269e10	−	3.38216e10i	−0.962095	−	1.66640i	−0.717225	−	0.696841i	$-0.754588\pi$
$881$	−1.95269e10	−	3.38216e10i	−0.962095	−	1.66640i	−0.244870	−	0.969556i	$-0.578745\pi$
$882$		0			0
$883$	−9.34435e9	+	7.84084e9i	−0.456759	+	0.383266i	−0.841937	−	0.539576i	$-0.818584\pi$
$883$	−9.34435e9	+	7.84084e9i	−0.456759	+	0.383266i	0.385178	+	0.922842i	$0.374140\pi$
$884$		0			0
$885$	−3.95495e9	+	6.85017e9i	−0.191796	+	0.332200i
$886$		0			0
$887$	2.62221e10	+	9.54407e9i	1.26164	+	0.459199i	0.884319	−	0.466884i	$-0.154623\pi$
$887$	2.62221e10	+	9.54407e9i	1.26164	+	0.459199i	0.377320	+	0.926083i	$0.376846\pi$
$888$		0			0
$889$	9.36404e9	+	5.31061e10i	0.446999	+	2.53506i
$890$		0			0
$891$	1.59859e10	+	1.34137e10i	0.757119	+	0.635298i
$892$		0			0
$893$	6.75193e9	−	9.22609e9i	0.317284	−	0.433548i
$894$		0			0
$895$	2.00056e10	+	1.67867e10i	0.932761	+	0.782679i
$896$		0			0
$897$	8.39357e8	+	4.76023e9i	0.0388305	+	0.220219i
$898$		0			0
$899$	−2.72401e9	−	9.91460e8i	−0.125040	−	0.0455110i
$900$		0			0
$901$	−2.18070e9	+	3.77709e9i	−0.0993252	+	0.172036i
$902$		0			0
$903$	−4.62350e9	+	3.87958e9i	−0.208960	+	0.175338i
$904$		0			0
$905$	−9.38117e9	−	1.62487e10i	−0.420714	−	0.728698i
$906$		0			0
$907$	−3.38407e9	+	1.91920e10i	−0.150596	+	0.854073i	0.812106	+	0.583510i	$0.198321\pi$
$907$	−3.38407e9	+	1.91920e10i	−0.150596	+	0.854073i	−0.962702	+	0.270563i	$0.912790\pi$
$908$		0			0
$909$	−2.73448e10	+	9.95268e9i	−1.20754	+	0.439508i
$910$		0			0
$911$	3.12147e10			1.36787			0.683936	−	0.729542i	$-0.260267\pi$
$911$	3.12147e10			1.36787			0.683936	+	0.729542i	$0.260267\pi$
$912$		0			0
$913$	1.79461e10			0.780410
$914$		0			0
$915$	−6.17817e9	+	2.24867e9i	−0.266616	+	0.0970402i
$916$		0			0
$917$	−1.51027e10	+	8.56514e10i	−0.646786	+	3.66811i
$918$		0			0
$919$	−3.08644e9	−	5.34587e9i	−0.131176	−	0.227203i	0.792954	−	0.609281i	$-0.208542\pi$
$919$	−3.08644e9	−	5.34587e9i	−0.131176	−	0.227203i	−0.924130	+	0.382078i	$0.875209\pi$
$920$		0			0
$921$	1.01451e10	−	8.51278e9i	0.427907	−	0.359056i
$922$		0			0
$923$	−1.97796e9	+	3.42593e9i	−0.0827965	+	0.143408i
$924$		0			0
$925$	4.77985e8	+	1.73972e8i	0.0198572	+	0.00722743i
$926$		0			0
$927$	5.60068e9	+	3.17631e10i	0.230920	+	1.30961i
$928$		0			0
$929$	1.87031e9	+	1.56938e9i	0.0765347	+	0.0642203i	0.680252	−	0.732978i	$-0.261870\pi$
$929$	1.87031e9	+	1.56938e9i	0.0765347	+	0.0642203i	−0.603718	+	0.797198i	$0.706315\pi$
$930$		0			0
$931$	1.85389e10	−	6.38206e10i	0.752940	−	2.59201i
$932$		0			0
$933$	−2.31083e9	−	1.93901e9i	−0.0931498	−	0.0781619i
$934$		0			0
$935$	−8.42356e8	−	4.77724e9i	−0.0337019	−	0.191133i
$936$		0			0
$937$	−2.06596e10	−	7.51949e9i	−0.820416	−	0.298607i	−0.102496	−	0.994733i	$-0.532683\pi$
$937$	−2.06596e10	−	7.51949e9i	−0.820416	−	0.298607i	−0.717919	+	0.696127i	$0.754905\pi$
$938$		0			0
$939$	−1.83631e9	+	3.18059e9i	−0.0723797	+	0.125365i
$940$		0			0
$941$	−2.76617e10	+	2.32110e10i	−1.08222	+	0.908091i	−0.996103	−	0.0881942i	$-0.971890\pi$
$941$	−2.76617e10	+	2.32110e10i	−1.08222	+	0.908091i	−0.0861172	+	0.996285i	$0.527446\pi$
$942$		0			0
$943$	−7.94545e9	−	1.37619e10i	−0.308551	−	0.534426i
$944$		0			0
$945$	4.05221e9	−	2.29812e10i	0.156200	−	0.885854i
$946$		0			0
$947$	−9.72183e9	+	3.53846e9i	−0.371983	+	0.135391i	−0.521245	−	0.853407i	$-0.674532\pi$
$947$	−9.72183e9	+	3.53846e9i	−0.371983	+	0.135391i	0.149262	+	0.988798i	$0.452310\pi$
$948$		0			0
$949$	2.07550e10			0.788301
$950$		0			0
$951$	−1.60981e10			−0.606935
$952$		0			0
$953$	1.75637e9	−	6.39266e8i	0.0657341	−	0.0239253i	−0.308944	−	0.951080i	$-0.599976\pi$
$953$	1.75637e9	−	6.39266e8i	0.0657341	−	0.0239253i	0.374678	+	0.927155i	$0.377753\pi$
$954$		0			0
$955$	−8.89813e8	+	5.04638e9i	−0.0330588	+	0.187486i
$956$		0			0
$957$	−5.51896e9	−	9.55911e9i	−0.203547	−	0.352554i
$958$		0			0
$959$	5.20557e10	−	4.36800e10i	1.90592	−	1.59925i
$960$		0			0
$961$	1.34805e10	−	2.33489e10i	0.489975	−	0.848661i
$962$		0			0
$963$	−3.73179e10	−	1.35826e10i	−1.34656	−	0.490107i
$964$		0			0
$965$	5.70370e7	+	3.23473e8i	0.00204320	+	0.0115876i
$966$		0			0
$967$	−6.72885e9	−	5.64618e9i	−0.239303	−	0.200799i	0.515247	−	0.857042i	$-0.327700\pi$
$967$	−6.72885e9	−	5.64618e9i	−0.239303	−	0.200799i	−0.754550	+	0.656243i	$0.772145\pi$
$968$		0			0
$969$	1.57888e9	+	1.71459e8i	0.0557463	+	0.00605380i
$970$		0			0
$971$	−2.55792e10	−	2.14635e10i	−0.896643	−	0.752373i	0.0728882	−	0.997340i	$-0.476778\pi$
$971$	−2.55792e10	−	2.14635e10i	−0.896643	−	0.752373i	−0.969531	+	0.244967i	$0.921223\pi$
$972$		0			0
$973$	4.22821e9	+	2.39794e10i	0.147150	+	0.834531i
$974$		0			0
$975$	4.79409e9	+	1.74491e9i	0.165649	+	0.0602915i
$976$		0			0
$977$	−2.16460e10	+	3.74920e10i	−0.742586	+	1.28620i	0.208728	+	0.977974i	$0.433068\pi$
$977$	−2.16460e10	+	3.74920e10i	−0.742586	+	1.28620i	−0.951314	+	0.308223i	$0.900266\pi$
$978$		0			0
$979$	1.69444e10	−	1.42180e10i	0.577146	−	0.484283i
$980$		0			0
$981$	−1.96435e10	−	3.40236e10i	−0.664322	−	1.15064i
$982$		0			0
$983$	−1.72992e8	+	9.81088e8i	−0.00580884	+	0.0329436i	−0.987574	−	0.157152i	$-0.949769\pi$
$983$	−1.72992e8	+	9.81088e8i	−0.00580884	+	0.0329436i	0.981766	+	0.190096i	$0.0608798\pi$
$984$		0			0
$985$	5.58197e9	−	2.03167e9i	0.186106	−	0.0677371i
$986$		0			0
$987$	9.78192e9			0.323828
$988$		0			0
$989$	6.71040e9			0.220578
$990$		0			0
$991$	5.42698e9	−	1.97526e9i	0.177133	−	0.0644713i	−0.251931	−	0.967745i	$-0.581066\pi$
$991$	5.42698e9	−	1.97526e9i	0.177133	−	0.0644713i	0.429064	+	0.903274i	$0.358843\pi$
$992$		0			0
$993$	1.40596e9	−	7.97360e9i	0.0455670	−	0.258423i
$994$		0			0
$995$	−2.00214e10	−	3.46780e10i	−0.644337	−	1.11602i
$996$		0			0
$997$	1.76790e10	−	1.48345e10i	0.564969	−	0.474066i	−0.315003	−	0.949091i	$-0.602005\pi$
$997$	1.76790e10	−	1.48345e10i	0.564969	−	0.474066i	0.879972	+	0.475025i	$0.157561\pi$
$998$		0			0
$999$	−5.16462e8	+	8.94538e8i	−0.0163892	+	0.0283870i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.8.i.a.73.7	yes	72
19.6	even	9	inner	76.8.i.a.25.7	✓	72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.8.i.a.25.7	✓	72	19.6	even	9	inner
76.8.i.a.73.7	yes	72	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	76.i (of order \(9\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(13.7721 - 5.01263i) q^{3} +(38.0867 - 216.001i) q^{5} +(-872.699 - 1511.56i) q^{7} +(-1510.80 + 1267.71i) q^{9} +O(q^{10})\)	\(q+(13.7721 - 5.01263i) q^{3} +(38.0867 - 216.001i) q^{5} +(-872.699 - 1511.56i) q^{7} +(-1510.80 + 1267.71i) q^{9} +(-3051.04 + 5284.56i) q^{11} +(10897.0 + 3966.19i) q^{13} +(-558.197 - 3165.69i) q^{15} +(-2776.50 - 2329.76i) q^{17} +(-29034.4 + 7132.73i) q^{19} +(-19595.8 - 16442.8i) q^{21} +(4938.68 + 28008.6i) q^{23} +(28207.9 + 10266.8i) q^{25} +(-30478.5 + 52790.4i) q^{27} +(-94547.2 + 79334.5i) q^{29} +(11743.5 + 20340.4i) q^{31} +(-15529.7 + 88073.2i) q^{33} +(-359736. + 130933. i) q^{35} +16945.1 q^{37} +169956. q^{39} +(-525042. + 191100. i) q^{41} +(40971.2 - 232359. i) q^{43} +(216284. + 374615. i) q^{45} +(-292934. + 245801. i) q^{47} +(-1.11144e6 + 1.92506e6i) q^{49} +(-49916.4 - 18168.1i) q^{51} +(-208955. - 1.18504e6i) q^{53} +(1.02526e6 + 860298. i) q^{55} +(-364111. + 243771. i) q^{57} +(-1.88498e6 - 1.58169e6i) q^{59} +(-355162. - 2.01423e6i) q^{61} +(3.23468e6 + 1.17733e6i) q^{63} +(1.27173e6 - 2.20270e6i) q^{65} +(-1.92395e6 + 1.61438e6i) q^{67} +(208413. + 360982. i) q^{69} +(-59237.5 + 335952. i) q^{71} +(1.68185e6 - 612144. i) q^{73} +439945. q^{75} +1.06506e7 q^{77} +(-2.19403e6 + 798560. i) q^{79} +(593846. - 3.36787e6i) q^{81} +(-1.47049e6 - 2.54697e6i) q^{83} +(-608977. + 510992. i) q^{85} +(-904438. + 1.56653e6i) q^{87} +(-3.40627e6 - 1.23978e6i) q^{89} +(-3.51469e6 - 1.99328e7i) q^{91} +(263692. + 221264. i) q^{93} +(434848. + 6.54311e6i) q^{95} +(4.94372e6 + 4.14827e6i) q^{97} +(-2.08978e6 - 1.18517e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 39 q^{3} + 591 q^{7} - 1677 q^{9}+O(q^{10}) \) 72 * q - 39 * q^3 + 591 * q^7 - 1677 * q^9	\( 72 q - 39 q^{3} + 591 q^{7} - 1677 q^{9} - 5793 q^{11} + 22083 q^{13} - 70707 q^{15} + 29409 q^{17} + 12756 q^{19} - 40419 q^{21} - 116796 q^{23} + 26934 q^{25} + 439347 q^{27} - 148005 q^{29} + 141366 q^{31} + 1206654 q^{33} - 531651 q^{35} - 1508004 q^{37} - 2140644 q^{39} + 494157 q^{41} + 923511 q^{43} + 1790028 q^{45} - 662955 q^{47} - 4191369 q^{49} + 3753363 q^{51} - 246513 q^{53} + 2650185 q^{55} + 5831286 q^{57} - 2244432 q^{59} - 10302306 q^{61} - 12102117 q^{63} + 3985635 q^{65} + 11688288 q^{67} + 12632535 q^{69} + 10541736 q^{71} + 560730 q^{73} - 17718750 q^{75} - 22266192 q^{77} + 9424767 q^{79} + 16585437 q^{81} + 2359839 q^{83} - 20859762 q^{85} + 2348574 q^{87} - 4397130 q^{89} - 12653112 q^{91} + 52259766 q^{93} + 46248669 q^{95} - 34112766 q^{97} - 93757926 q^{99}+O(q^{100}) \) 72 * q - 39 * q^3 + 591 * q^7 - 1677 * q^9 - 5793 * q^11 + 22083 * q^13 - 70707 * q^15 + 29409 * q^17 + 12756 * q^19 - 40419 * q^21 - 116796 * q^23 + 26934 * q^25 + 439347 * q^27 - 148005 * q^29 + 141366 * q^31 + 1206654 * q^33 - 531651 * q^35 - 1508004 * q^37 - 2140644 * q^39 + 494157 * q^41 + 923511 * q^43 + 1790028 * q^45 - 662955 * q^47 - 4191369 * q^49 + 3753363 * q^51 - 246513 * q^53 + 2650185 * q^55 + 5831286 * q^57 - 2244432 * q^59 - 10302306 * q^61 - 12102117 * q^63 + 3985635 * q^65 + 11688288 * q^67 + 12632535 * q^69 + 10541736 * q^71 + 560730 * q^73 - 17718750 * q^75 - 22266192 * q^77 + 9424767 * q^79 + 16585437 * q^81 + 2359839 * q^83 - 20859762 * q^85 + 2348574 * q^87 - 4397130 * q^89 - 12653112 * q^91 + 52259766 * q^93 + 46248669 * q^95 - 34112766 * q^97 - 93757926 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more