LMFDB - Embedded newform 76.7.h.a.69.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,7,Mod(65,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("76.65");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	76.h (of order $6$, degree $2$, 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:	$17.4841103551$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 9460 x^{18} + 36670708 x^{16} + 75655761912 x^{14} + 90488614064544 x^{12} + \cdots + 10\!\cdots\!00 $ x^20 + 9460x^18 + 36670708x^16 + 75655761912x^14 + 90488614064544x^12 + 64290781142478432x^10 + 26741969359067432592x^8 + 6092313787108272225600x^6 + 632563499034568748983104x^4 + 15152233769669432690883072*x^2 + 103372598905826219735270400
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{8}\cdot 19^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			69.2
Root			$35.1361i$ of defining polynomial
Character	$\chi$	$=$	76.69
Dual form			76.7.h.a.65.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-31.9288 + 18.4341i) q^{3} +(71.3406 + 123.566i) q^{5} +421.828 q^{7} +(315.131 - 545.823i) q^{9} +O(q^{10})$	\(q+(-31.9288 + 18.4341i) q^{3} +(71.3406 + 123.566i) q^{5} +421.828 q^{7} +(315.131 - 545.823i) q^{9} +1066.00 q^{11} +(99.2772 + 57.3177i) q^{13} +(-4555.64 - 2630.20i) q^{15} +(1346.17 + 2331.63i) q^{17} +(1614.83 + 6666.20i) q^{19} +(-13468.5 + 7776.01i) q^{21} +(-8939.31 + 15483.3i) q^{23} +(-2366.47 + 4098.85i) q^{25} -3640.28i q^{27} +(-25331.6 - 14625.2i) q^{29} +40694.9i q^{31} +(-34036.0 + 19650.7i) q^{33} +(30093.5 + 52123.4i) q^{35} -73561.3i q^{37} -4226.40 q^{39} +(22575.6 - 13034.0i) q^{41} +(-1774.87 - 3074.17i) q^{43} +89926.6 q^{45} +(-74936.6 + 129794. i) q^{47} +60289.9 q^{49} +(-85963.1 - 49630.8i) q^{51} +(7948.56 + 4589.10i) q^{53} +(76049.1 + 131721. i) q^{55} +(-174445. - 183076. i) q^{57} +(-211634. + 122187. i) q^{59} +(57062.6 - 98835.3i) q^{61} +(132931. - 230243. i) q^{63} +16356.3i q^{65} +(-26072.9 - 15053.2i) q^{67} -659152. i q^{69} +(470303. - 271530. i) q^{71} +(196387. + 340152. i) q^{73} -174495. i q^{75} +449668. q^{77} +(559140. - 322819. i) q^{79} +(296836. + 514135. i) q^{81} -465849. q^{83} +(-192073. + 332681. i) q^{85} +1.07841e6 q^{87} +(-501206. - 289372. i) q^{89} +(41877.9 + 24178.2i) q^{91} +(-750173. - 1.29934e6i) q^{93} +(-708510. + 675108. i) q^{95} +(-310070. + 179019. i) q^{97} +(335930. - 581847. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 30 q^{3} - 56 q^{5} + 464 q^{7} + 2200 q^{9}+O(q^{10}) $ 20 * q - 30 * q^3 - 56 * q^5 + 464 * q^7 + 2200 * q^9	$ 20 q - 30 q^{3} - 56 q^{5} + 464 q^{7} + 2200 q^{9} - 3644 q^{11} - 7140 q^{13} + 9168 q^{15} + 1132 q^{17} + 2110 q^{19} - 8748 q^{21} + 832 q^{23} - 27698 q^{25} - 10920 q^{29} - 30306 q^{33} + 4172 q^{35} + 81144 q^{39} + 109206 q^{41} + 110740 q^{43} - 785440 q^{45} + 107080 q^{47} + 136092 q^{49} + 199872 q^{51} + 254796 q^{53} + 354840 q^{55} + 212268 q^{57} - 610638 q^{59} + 47864 q^{61} - 254476 q^{63} - 839562 q^{67} + 366660 q^{71} + 854482 q^{73} + 763088 q^{77} + 1718592 q^{79} - 1054142 q^{81} + 439612 q^{83} - 400236 q^{85} - 1604736 q^{87} + 478032 q^{89} + 599856 q^{91} + 829380 q^{93} - 1055660 q^{95} - 191286 q^{97} - 2336728 q^{99}+O(q^{100}) $ 20 * q - 30 * q^3 - 56 * q^5 + 464 * q^7 + 2200 * q^9 - 3644 * q^11 - 7140 * q^13 + 9168 * q^15 + 1132 * q^17 + 2110 * q^19 - 8748 * q^21 + 832 * q^23 - 27698 * q^25 - 10920 * q^29 - 30306 * q^33 + 4172 * q^35 + 81144 * q^39 + 109206 * q^41 + 110740 * q^43 - 785440 * q^45 + 107080 * q^47 + 136092 * q^49 + 199872 * q^51 + 254796 * q^53 + 354840 * q^55 + 212268 * q^57 - 610638 * q^59 + 47864 * q^61 - 254476 * q^63 - 839562 * q^67 + 366660 * q^71 + 854482 * q^73 + 763088 * q^77 + 1718592 * q^79 - 1054142 * q^81 + 439612 * q^83 - 400236 * q^85 - 1604736 * q^87 + 478032 * q^89 + 599856 * q^91 + 829380 * q^93 - 1055660 * q^95 - 191286 * q^97 - 2336728 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$21$	$39$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$

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
$3$	−31.9288	+	18.4341i	−1.18255	+	0.682744i	−0.956602	−	0.291396i	$-0.905880\pi$
$3$	−31.9288	+	18.4341i	−1.18255	+	0.682744i	−0.225945	+	0.974140i	$0.572547\pi$
$4$		0			0
$5$	71.3406	+	123.566i	0.570725	+	0.988525i	0.996492	+	0.0836918i	$0.0266711\pi$
$5$	71.3406	+	123.566i	0.570725	+	0.988525i	−0.425767	+	0.904833i	$0.639996\pi$
$6$		0			0
$7$	421.828			1.22982			0.614910	−	0.788598i	$-0.289192\pi$
$7$	421.828			1.22982			0.614910	+	0.788598i	$0.289192\pi$
$8$		0			0
$9$	315.131	−	545.823i	0.432279	−	0.748729i
$10$		0			0
$11$	1066.00			0.800901			0.400450	−	0.916318i	$-0.368854\pi$
$11$	1066.00			0.800901			0.400450	+	0.916318i	$0.368854\pi$
$12$		0			0
$13$	99.2772	+	57.3177i	0.0451876	+	0.0260891i	0.522424	−	0.852686i	$-0.325028\pi$
$13$	99.2772	+	57.3177i	0.0451876	+	0.0260891i	−0.477236	+	0.878775i	$0.658361\pi$
$14$		0			0
$15$	−4555.64	−	2630.20i	−1.34982	−	0.779318i
$16$		0			0
$17$	1346.17	+	2331.63i	0.274002	+	0.474585i	0.969883	−	0.243572i	$-0.0783193\pi$
$17$	1346.17	+	2331.63i	0.274002	+	0.474585i	−0.695881	+	0.718157i	$0.744986\pi$
$18$		0			0
$19$	1614.83	+	6666.20i	0.235432	+	0.971891i
$19$	1614.83	+	6666.20i	0.235432	+	0.971891i
$20$		0			0
$21$	−13468.5	+	7776.01i	−1.45432	+	0.839652i
$22$		0			0
$23$	−8939.31	+	15483.3i	−0.734718	+	1.27257i	0.220129	+	0.975471i	$0.429352\pi$
$23$	−8939.31	+	15483.3i	−0.734718	+	1.27257i	−0.954847	+	0.297098i	$0.903981\pi$
$24$		0			0
$25$	−2366.47	+	4098.85i	−0.151454	+	0.262326i
$26$		0			0
$27$		−	3640.28i		−	0.184945i
$28$		0			0
$29$	−25331.6	−	14625.2i	−1.03865	−	0.599664i	−0.119199	−	0.992870i	$-0.538033\pi$
$29$	−25331.6	−	14625.2i	−1.03865	−	0.599664i	−0.919450	+	0.393206i	$0.871366\pi$
$30$		0			0
$31$			40694.9i			1.36601i	0.730413	+	0.683006i	$0.239328\pi$
$31$			40694.9i			1.36601i	−0.730413	+	0.683006i	$0.760672\pi$
$32$		0			0
$33$	−34036.0	+	19650.7i	−0.947103	+	0.546810i
$34$		0			0
$35$	30093.5	+	52123.4i	0.701889	+	1.21571i
$36$		0			0
$37$		−	73561.3i		−	1.45226i	−0.687558	−	0.726129i	$-0.741317\pi$
$37$		−	73561.3i		−	1.45226i	0.687558	−	0.726129i	$-0.258683\pi$
$38$		0			0
$39$	−4226.40			−0.0712487
$40$		0			0
$41$	22575.6	−	13034.0i	0.327557	−	0.189115i	−0.327199	−	0.944956i	$-0.606105\pi$
$41$	22575.6	−	13034.0i	0.327557	−	0.189115i	0.654756	+	0.755840i	$0.272771\pi$
$42$		0			0
$43$	−1774.87	−	3074.17i	−0.0223235	−	0.0386654i	0.854648	−	0.519208i	$-0.173773\pi$
$43$	−1774.87	−	3074.17i	−0.0223235	−	0.0386654i	−0.876971	+	0.480543i	$0.840440\pi$
$44$		0			0
$45$	89926.6			0.986849
$46$		0			0
$47$	−74936.6	+	129794.i	−0.721773	+	1.25015i	0.238516	+	0.971139i	$0.423339\pi$
$47$	−74936.6	+	129794.i	−0.721773	+	1.25015i	−0.960289	+	0.279009i	$0.909994\pi$
$48$		0			0
$49$	60289.9			0.512455
$50$		0			0
$51$	−85963.1	−	49630.8i	−0.648040	−	0.374146i
$52$		0			0
$53$	7948.56	+	4589.10i	0.0533901	+	0.0308248i	0.526457	−	0.850201i	$-0.323520\pi$
$53$	7948.56	+	4589.10i	0.0533901	+	0.0308248i	−0.473067	+	0.881026i	$0.656853\pi$
$54$		0			0
$55$	76049.1	+	131721.i	0.457094	+	0.791710i
$56$		0			0
$57$	−174445.	−	183076.i	−0.941962	−	0.988567i
$58$		0			0
$59$	−211634.	+	122187.i	−1.03045	+	0.594933i	−0.917115	−	0.398624i	$-0.869488\pi$
$59$	−211634.	+	122187.i	−1.03045	+	0.594933i	−0.113339	+	0.993556i	$0.536155\pi$
$60$		0			0
$61$	57062.6	−	98835.3i	0.251398	−	0.435434i	−0.712513	−	0.701659i	$-0.752443\pi$
$61$	57062.6	−	98835.3i	0.251398	−	0.435434i	0.963911	+	0.266225i	$0.0857763\pi$
$62$		0			0
$63$	132931.	−	230243.i	0.531625	−	0.920801i
$64$		0			0
$65$			16356.3i			0.0595588i
$66$		0			0
$67$	−26072.9	−	15053.2i	−0.0866893	−	0.0500501i	0.456029	−	0.889965i	$-0.349271\pi$
$67$	−26072.9	−	15053.2i	−0.0866893	−	0.0500501i	−0.542718	+	0.839915i	$0.682605\pi$
$68$		0			0
$69$		−	659152.i		−	2.00650i
$70$		0			0
$71$	470303.	−	271530.i	1.31402	−	0.758652i	0.331263	−	0.943538i	$-0.392525\pi$
$71$	470303.	−	271530.i	1.31402	−	0.758652i	0.982760	+	0.184887i	$0.0591918\pi$
$72$		0			0
$73$	196387.	+	340152.i	0.504828	+	0.874388i	0.999984	+	0.00558433i	$0.00177756\pi$
$73$	196387.	+	340152.i	0.504828	+	0.874388i	−0.495156	+	0.868804i	$0.664889\pi$
$74$		0			0
$75$		−	174495.i		−	0.413618i
$76$		0			0
$77$	449668.			0.984963
$78$		0			0
$79$	559140.	−	322819.i	1.13407	−	0.654754i	0.189113	−	0.981955i	$-0.439439\pi$
$79$	559140.	−	322819.i	1.13407	−	0.654754i	0.944955	+	0.327201i	$0.106105\pi$
$80$		0			0
$81$	296836.	+	514135.i	0.558549	+	0.967435i
$82$		0			0
$83$	−465849.			−0.814725			−0.407363	−	0.913267i	$-0.633551\pi$
$83$	−465849.			−0.814725			−0.407363	+	0.913267i	$0.633551\pi$
$84$		0			0
$85$	−192073.	+	332681.i	−0.312759	+	0.541715i
$86$		0			0
$87$	1.07841e6			1.63767
$88$		0			0
$89$	−501206.	−	289372.i	−0.710962	−	0.410474i	0.100455	−	0.994942i	$-0.467970\pi$
$89$	−501206.	−	289372.i	−0.710962	−	0.410474i	−0.811417	+	0.584467i	$0.801304\pi$
$90$		0			0
$91$	41877.9	+	24178.2i	0.0555726	+	0.0320849i
$92$		0			0
$93$	−750173.	−	1.29934e6i	−0.932637	−	1.61537i
$94$		0			0
$95$	−708510.	+	675108.i	−0.826371	+	0.787413i
$96$		0			0
$97$	−310070.	+	179019.i	−0.339738	+	0.196148i	−0.660156	−	0.751128i	$-0.729510\pi$
$97$	−310070.	+	179019.i	−0.339738	+	0.196148i	0.320418	+	0.947276i	$0.396177\pi$
$98$		0			0
$99$	335930.	−	581847.i	0.346212	−	0.599657i
$100$		0			0
$101$	−131410.	+	227608.i	−0.127545	+	0.220914i	−0.922725	−	0.385459i	$-0.874043\pi$
$101$	−131410.	+	227608.i	−0.127545	+	0.220914i	0.795180	+	0.606374i	$0.207376\pi$
$102$		0			0
$103$			1.73394e6i			1.58680i	0.608702	+	0.793399i	$0.291690\pi$
$103$			1.73394e6i			1.58680i	−0.608702	+	0.793399i	$0.708310\pi$
$104$		0			0
$105$	−1.92170e6	−	1.10949e6i	−1.66003	−	0.958420i
$106$		0			0
$107$		−	131037.i		−	0.106966i	−0.998569	−	0.0534828i	$-0.982968\pi$
$107$		−	131037.i		−	0.106966i	0.998569	−	0.0534828i	$-0.0170322\pi$
$108$		0			0
$109$	−579151.	+	334373.i	−0.447211	+	0.258197i	−0.706652	−	0.707562i	$-0.749795\pi$
$109$	−579151.	+	334373.i	−0.447211	+	0.258197i	0.259441	+	0.965759i	$0.416462\pi$
$110$		0			0
$111$	1.35603e6	+	2.34872e6i	0.991521	+	1.71736i
$112$		0			0
$113$			1.25098e6i			0.866991i	0.901156	+	0.433496i	$0.142720\pi$
$113$			1.25098e6i			0.866991i	−0.901156	+	0.433496i	$0.857280\pi$
$114$		0			0
$115$	−2.55095e6			−1.67729
$116$		0			0
$117$	62570.7	−	36125.2i	0.0390673	−	0.0225555i
$118$		0			0
$119$	567852.	+	983549.i	0.336972	+	0.583653i
$120$		0			0
$121$	−635207.			−0.358558
$122$		0			0
$123$	−480540.	+	832320.i	−0.258235	+	0.447275i
$124$		0			0
$125$	1.55409e6			0.795695
$126$		0			0
$127$	−2.08424e6	−	1.20334e6i	−1.01751	−	0.587457i	−0.104125	−	0.994564i	$-0.533204\pi$
$127$	−2.08424e6	−	1.20334e6i	−1.01751	−	0.587457i	−0.913380	+	0.407107i	$0.866537\pi$
$128$		0			0
$129$	113339.	+	65436.3i	0.0527971	+	0.0304824i
$130$		0			0
$131$	1.66921e6	+	2.89116e6i	0.742503	+	1.28605i	0.951352	+	0.308105i	$0.0996947\pi$
$131$	1.66921e6	+	2.89116e6i	0.742503	+	1.28605i	−0.208850	+	0.977948i	$0.566972\pi$
$132$		0			0
$133$	681180.	+	2.81199e6i	0.289539	+	1.19525i
$134$		0			0
$135$	449813.	−	259700.i	0.182823	−	0.105553i
$136$		0			0
$137$	1.06961e6	−	1.85261e6i	0.415970	−	0.720481i	−0.579560	−	0.814930i	$-0.696775\pi$
$137$	1.06961e6	−	1.85261e6i	0.415970	−	0.720481i	0.995530	+	0.0944487i	$0.0301088\pi$
$138$		0			0
$139$	2.56807e6	−	4.44803e6i	0.956230	−	1.65624i	0.224703	−	0.974427i	$-0.427859\pi$
$139$	2.56807e6	−	4.44803e6i	0.956230	−	1.65624i	0.731527	−	0.681812i	$-0.238808\pi$
$140$		0			0
$141$		−	5.52555e6i		−	1.97114i
$142$		0			0
$143$	105829.	+	61100.6i	0.0361908	+	0.0208948i
$144$		0			0
$145$		−	4.17349e6i		−	1.36897i
$146$		0			0
$147$	−1.92498e6	+	1.11139e6i	−0.606003	+	0.349876i
$148$		0			0
$149$	1.52848e6	+	2.64741e6i	0.462064	+	0.800319i	0.999064	−	0.0432639i	$-0.0137756\pi$
$149$	1.52848e6	+	2.64741e6i	0.462064	+	0.800319i	−0.536999	+	0.843583i	$0.680442\pi$
$150$		0			0
$151$			3.72742e6i			1.08262i	0.840822	+	0.541311i	$0.182072\pi$
$151$			3.72742e6i			1.08262i	−0.840822	+	0.541311i	$0.817928\pi$
$152$		0			0
$153$	1.69688e6			0.473780
$154$		0			0
$155$	−5.02849e6	+	2.90320e6i	−1.35034	+	0.779617i
$156$		0			0
$157$	−2.06213e6	−	3.57172e6i	−0.532865	−	0.922949i	−0.999263	−	0.0383746i	$-0.987782\pi$
$157$	−2.06213e6	−	3.57172e6i	−0.532865	−	0.922949i	0.466398	−	0.884575i	$-0.345551\pi$
$158$		0			0
$159$	−338384.			−0.0841818
$160$		0			0
$161$	−3.77085e6	+	6.53131e6i	−0.903570	+	1.56503i
$162$		0			0
$163$	3.05686e6			0.705852			0.352926	−	0.935651i	$-0.385187\pi$
$163$	3.05686e6			0.705852			0.352926	+	0.935651i	$0.385187\pi$
$164$		0			0
$165$	−4.85631e6	−	2.80379e6i	−1.08107	−	0.624157i
$166$		0			0
$167$	4.50995e6	+	2.60382e6i	0.968328	+	0.559065i	0.898726	−	0.438510i	$-0.144494\pi$
$167$	4.50995e6	+	2.60382e6i	0.968328	+	0.559065i	0.0696021	+	0.997575i	$0.477827\pi$
$168$		0			0
$169$	−2.40683e6	−	4.16876e6i	−0.498639	−	0.863668i
$170$		0			0
$171$	4.14745e6	+	1.21932e6i	0.829455	+	0.243853i
$172$		0			0
$173$	7.97278e6	−	4.60309e6i	1.53983	−	0.889019i	0.540977	−	0.841037i	$-0.318054\pi$
$173$	7.97278e6	−	4.60309e6i	1.53983	−	0.889019i	0.998848	−	0.0479816i	$-0.0152789\pi$
$174$		0			0
$175$	−998244.	+	1.72901e6i	−0.186261	+	0.322614i
$176$		0			0
$177$	4.50480e6	−	7.80254e6i	0.812373	−	1.40707i
$178$		0			0
$179$		−	3.25528e6i		−	0.567582i	−0.958886	−	0.283791i	$-0.908408\pi$
$179$		−	3.25528e6i		−	0.567582i	0.958886	−	0.283791i	$-0.0915922\pi$
$180$		0			0
$181$	−5.46323e6	−	3.15420e6i	−0.921327	−	0.531928i	−0.0372688	−	0.999305i	$-0.511866\pi$
$181$	−5.46323e6	−	3.15420e6i	−0.921327	−	0.531928i	−0.884058	+	0.467377i	$0.845199\pi$
$182$		0			0
$183$			4.20759e6i			0.686562i
$184$		0			0
$185$	9.08964e6	−	5.24791e6i	1.43559	−	0.828841i
$186$		0			0
$187$	1.43502e6	+	2.48552e6i	0.219448	+	0.380095i
$188$		0			0
$189$		−	1.53557e6i		−	0.227449i
$190$		0			0
$191$	1.91518e6			0.274859			0.137430	−	0.990512i	$-0.456116\pi$
$191$	1.91518e6			0.274859			0.137430	+	0.990512i	$0.456116\pi$
$192$		0			0
$193$	−134631.	+	77729.2i	−0.0187272	+	0.0108122i	−0.509334	−	0.860569i	$-0.670108\pi$
$193$	−134631.	+	77729.2i	−0.0187272	+	0.0108122i	0.490607	+	0.871381i	$0.336775\pi$
$194$		0			0
$195$	−301514.	−	522237.i	−0.0406634	−	0.0704311i
$196$		0			0
$197$	−8.92512e6			−1.16739			−0.583694	−	0.811974i	$-0.698393\pi$
$197$	−8.92512e6			−1.16739			−0.583694	+	0.811974i	$0.698393\pi$
$198$		0			0
$199$	7.77164e6	−	1.34609e7i	0.986174	−	1.70810i	0.349575	−	0.936908i	$-0.386326\pi$
$199$	7.77164e6	−	1.34609e7i	0.986174	−	1.70810i	0.636599	−	0.771195i	$-0.280341\pi$
$200$		0			0
$201$	1.10997e6			0.136686
$202$		0			0
$203$	−1.06856e7	−	6.16933e6i	−1.27735	−	0.737479i
$204$		0			0
$205$	3.22111e6	+	1.85971e6i	0.373890	+	0.215866i
$206$		0			0
$207$	5.63411e6	+	9.75857e6i	0.635206	+	1.10021i
$208$		0			0
$209$	1.72141e6	+	7.10616e6i	0.188558	+	0.778388i
$210$		0			0
$211$	1.52497e7	−	8.80440e6i	1.62335	−	0.937243i	0.637339	−	0.770584i	$-0.280035\pi$
$211$	1.52497e7	−	8.80440e6i	1.62335	−	0.937243i	0.986014	−	0.166660i	$-0.0532981\pi$
$212$		0			0
$213$	−1.00108e7	+	1.73392e7i	−1.03593	+	1.79428i
$214$		0			0
$215$	253241.	−	438627.i	0.0254811	−	0.0441346i
$216$		0			0
$217$			1.71662e7i			1.67995i
$218$		0			0
$219$	−1.25408e7	−	7.24042e6i	−1.19397	−	0.689337i
$220$		0			0
$221$			308638.i			0.0285938i
$222$		0			0
$223$	1.01495e7	−	5.85980e6i	0.915227	−	0.528407i	0.0331178	−	0.999451i	$-0.489456\pi$
$223$	1.01495e7	−	5.85980e6i	0.915227	−	0.528407i	0.882109	+	0.471045i	$0.156123\pi$
$224$		0			0
$225$	1.49150e6	+	2.58335e6i	0.130941	+	0.226796i
$226$		0			0
$227$			387204.i			0.0331026i	0.999863	+	0.0165513i	$0.00526869\pi$
$227$			387204.i			0.0331026i	−0.999863	+	0.0165513i	$0.994731\pi$
$228$		0			0
$229$	796769.			0.0663477			0.0331739	−	0.999450i	$-0.489438\pi$
$229$	796769.			0.0663477			0.0331739	+	0.999450i	$0.489438\pi$
$230$		0			0
$231$	−1.43574e7	+	8.28922e6i	−1.16477	+	0.672478i
$232$		0			0
$233$	3.30974e6	+	5.73264e6i	0.261653	+	0.453197i	0.966681	−	0.255983i	$-0.0823990\pi$
$233$	3.30974e6	+	5.73264e6i	0.261653	+	0.453197i	−0.705028	+	0.709179i	$0.749066\pi$
$234$		0			0
$235$	−2.13841e7			−1.64774
$236$		0			0
$237$	−1.19018e7	+	2.06145e7i	−0.894059	+	1.54856i
$238$		0			0
$239$	−1.15759e7			−0.847928			−0.423964	−	0.905679i	$-0.639362\pi$
$239$	−1.15759e7			−0.847928			−0.423964	+	0.905679i	$0.639362\pi$
$240$		0			0
$241$	−5.64835e6	−	3.26108e6i	−0.403525	−	0.232975i	0.284479	−	0.958682i	$-0.408179\pi$
$241$	−5.64835e6	−	3.26108e6i	−0.403525	−	0.232975i	−0.688004	+	0.725707i	$0.741513\pi$
$242$		0			0
$243$	−1.66570e7	−	9.61691e6i	−1.16085	−	0.670219i
$244$		0			0
$245$	4.30112e6	+	7.44975e6i	0.292471	+	0.506575i
$246$		0			0
$247$	−221776.	+	754360.i	−0.0147171	+	0.0500596i
$248$		0			0
$249$	1.48740e7	−	8.58750e6i	0.963451	−	0.556249i
$250$		0			0
$251$	−1.09405e7	+	1.89496e7i	−0.691858	+	1.19833i	0.279370	+	0.960184i	$0.409874\pi$
$251$	−1.09405e7	+	1.89496e7i	−0.691858	+	1.19833i	−0.971228	+	0.238150i	$0.923459\pi$
$252$		0			0
$253$	−9.52930e6	+	1.65052e7i	−0.588436	+	1.01920i
$254$		0			0
$255$		−	1.41628e7i		−	0.854138i
$256$		0			0
$257$	1.05616e7	+	6.09775e6i	0.622201	+	0.359228i	0.777725	−	0.628604i	$-0.216373\pi$
$257$	1.05616e7	+	6.09775e6i	0.622201	+	0.359228i	−0.155525	+	0.987832i	$0.549707\pi$
$258$		0			0
$259$		−	3.10302e7i		−	1.78602i
$260$		0			0
$261$	−1.59656e7	+	9.21772e6i	−0.897972	+	0.518444i
$262$		0			0
$263$	1.35492e7	+	2.34679e7i	0.744810	+	1.29005i	0.950283	+	0.311386i	$0.100793\pi$
$263$	1.35492e7	+	2.34679e7i	0.744810	+	1.29005i	−0.205473	+	0.978663i	$0.565873\pi$
$264$		0			0
$265$			1.30956e6i			0.0703699i
$266$		0			0
$267$	2.13372e7			1.12100
$268$		0			0
$269$	1.91038e7	−	1.10296e7i	0.981436	−	0.566632i	0.0787324	−	0.996896i	$-0.474913\pi$
$269$	1.91038e7	−	1.10296e7i	0.981436	−	0.566632i	0.902703	+	0.430264i	$0.141579\pi$
$270$		0			0
$271$	−1.87150e7	−	3.24153e7i	−0.940331	−	1.62870i	−0.764839	−	0.644221i	$-0.777182\pi$
$271$	−1.87150e7	−	3.24153e7i	−0.940331	−	1.62870i	−0.175492	−	0.984481i	$-0.556152\pi$
$272$		0			0
$273$	−1.78281e6			−0.0876230
$274$		0			0
$275$	−2.52266e6	+	4.36937e6i	−0.121300	+	0.210097i
$276$		0			0
$277$	1.24196e7			0.584344			0.292172	−	0.956366i	$-0.405622\pi$
$277$	1.24196e7			0.584344			0.292172	+	0.956366i	$0.405622\pi$
$278$		0			0
$279$	2.22122e7	+	1.28242e7i	1.02277	+	0.590498i
$280$		0			0
$281$	9.93810e6	+	5.73776e6i	0.447903	+	0.258597i	0.706944	−	0.707269i	$-0.250073\pi$
$281$	9.93810e6	+	5.73776e6i	0.447903	+	0.258597i	−0.259041	+	0.965866i	$0.583406\pi$
$282$		0			0
$283$	1.26650e7	+	2.19364e7i	0.558786	+	0.967845i	0.997598	+	0.0692665i	$0.0220659\pi$
$283$	1.26650e7	+	2.19364e7i	0.558786	+	0.967845i	−0.438813	+	0.898579i	$0.644601\pi$
$284$		0			0
$285$	1.01769e7	−	3.46161e7i	0.439622	−	1.49535i
$286$		0			0
$287$	9.52301e6	−	5.49811e6i	0.402836	−	0.232578i
$288$		0			0
$289$	8.44444e6	−	1.46262e7i	0.349846	−	0.605951i
$290$		0			0
$291$	6.60010e6	−	1.14317e7i	0.267838	−	0.463909i
$292$		0			0
$293$			3.69103e7i			1.46739i	0.679481	+	0.733693i	$0.262205\pi$
$293$			3.69103e7i			1.46739i	−0.679481	+	0.733693i	$0.737795\pi$
$294$		0			0
$295$	−3.01961e7	−	1.74338e7i	−1.17621	−	0.679086i
$296$		0			0
$297$		−	3.88054e6i		−	0.148123i
$298$		0			0
$299$	−1.77494e6	+	1.02476e6i	−0.0664003	+	0.0383362i
$300$		0			0
$301$	−748691.	−	1.29677e6i	−0.0274538	−	0.0475515i
$302$		0			0
$303$		−	9.68967e6i		−	0.348322i
$304$		0			0
$305$	1.62835e7			0.573917
$306$		0			0
$307$	−3.11621e7	+	1.79915e7i	−1.07699	+	0.621801i	−0.930083	−	0.367349i	$-0.880266\pi$
$307$	−3.11621e7	+	1.79915e7i	−1.07699	+	0.621801i	−0.146908	+	0.989150i	$0.546932\pi$
$308$		0			0
$309$	−3.19635e7	−	5.53625e7i	−1.08338	−	1.87646i
$310$		0			0
$311$	4.31039e7			1.43296			0.716482	−	0.697606i	$-0.245751\pi$
$311$	4.31039e7			1.43296			0.716482	+	0.697606i	$0.245751\pi$
$312$		0			0
$313$	1.23205e6	−	2.13398e6i	0.0401787	−	0.0695916i	−0.845237	−	0.534392i	$-0.820541\pi$
$313$	1.23205e6	−	2.13398e6i	0.0401787	−	0.0695916i	0.885415	+	0.464800i	$0.153874\pi$
$314$		0			0
$315$	3.79336e7			1.21365
$316$		0			0
$317$	−4.39334e6	−	2.53650e6i	−0.137917	−	0.0796263i	0.429454	−	0.903089i	$-0.358706\pi$
$317$	−4.39334e6	−	2.53650e6i	−0.137917	−	0.0796263i	−0.567371	+	0.823462i	$0.692039\pi$
$318$		0			0
$319$	−2.70035e7	−	1.55905e7i	−0.831855	−	0.480272i
$320$		0			0
$321$	2.41556e6	+	4.18386e6i	0.0730301	+	0.126492i
$322$		0			0
$323$	−1.33693e7	+	1.27390e7i	−0.396736	+	0.378032i
$324$		0			0
$325$	−469873.	+	271281.i	−0.0136877	+	0.00790260i
$326$		0			0
$327$	1.23277e7	−	2.13522e7i	0.352565	−	0.610661i
$328$		0			0
$329$	−3.16104e7	+	5.47508e7i	−0.887650	+	1.53745i
$330$		0			0
$331$		−	4.71417e7i		−	1.29993i	−0.759963	−	0.649967i	$-0.774783\pi$
$331$		−	4.71417e7i		−	1.29993i	0.759963	−	0.649967i	$-0.225217\pi$
$332$		0			0
$333$	−4.01514e7	−	2.31814e7i	−1.08735	−	0.627780i
$334$		0			0
$335$		−	4.29562e6i		−	0.114259i
$336$		0			0
$337$	1.19346e7	−	6.89043e6i	0.311829	−	0.180035i	−0.335915	−	0.941892i	$-0.609046\pi$
$337$	1.19346e7	−	6.89043e6i	0.311829	−	0.180035i	0.647745	+	0.761857i	$0.275712\pi$
$338$		0			0
$339$	−2.30607e7	−	3.99422e7i	−0.591933	−	1.02526i
$340$		0			0
$341$			4.33807e7i			1.09404i
$342$		0			0
$343$	−2.41957e7			−0.599592
$344$		0			0
$345$	8.14486e7	−	4.70244e7i	1.98347	−	1.14516i
$346$		0			0
$347$	2.80464e7	+	4.85778e7i	0.671257	+	1.16265i	0.977548	+	0.210713i	$0.0675784\pi$
$347$	2.80464e7	+	4.85778e7i	0.671257	+	1.16265i	−0.306292	+	0.951938i	$0.599088\pi$
$348$		0			0
$349$	4.06910e7			0.957242			0.478621	−	0.878021i	$-0.341137\pi$
$349$	4.06910e7			0.957242			0.478621	+	0.878021i	$0.341137\pi$
$350$		0			0
$351$	208653.	−	361397.i	0.00482506	−	0.00835724i
$352$		0			0
$353$	5.19218e7			1.18039			0.590196	−	0.807260i	$-0.299051\pi$
$353$	5.19218e7			1.18039			0.590196	+	0.807260i	$0.299051\pi$
$354$		0			0
$355$	6.71035e7	+	3.87422e7i	1.49989	+	0.865963i
$356$		0			0
$357$	−3.62616e7	−	2.09357e7i	−0.796972	−	0.460132i
$358$		0			0
$359$	−1.46164e7	−	2.53163e7i	−0.315905	−	0.547163i	0.663725	−	0.747977i	$-0.268975\pi$
$359$	−1.46164e7	−	2.53163e7i	−0.315905	−	0.547163i	−0.979629	+	0.200814i	$0.935641\pi$
$360$		0			0
$361$	−4.18305e7	+	2.15295e7i	−0.889144	+	0.457628i
$362$		0			0
$363$	2.02814e7	−	1.17095e7i	0.424011	−	0.244803i
$364$		0			0
$365$	−2.80207e7	+	4.85333e7i	−0.576236	+	0.998071i
$366$		0			0
$367$	−3.48495e7	+	6.03612e7i	−0.705016	+	1.22112i	0.261670	+	0.965157i	$0.415727\pi$
$367$	−3.48495e7	+	6.03612e7i	−0.705016	+	1.22112i	−0.966686	+	0.255966i	$0.917607\pi$
$368$		0			0
$369$		−	1.64297e7i		−	0.327002i
$370$		0			0
$371$	3.35293e6	+	1.93581e6i	0.0656602	+	0.0379089i
$372$		0			0
$373$		−	1.64339e7i		−	0.316675i	−0.987385	−	0.158338i	$-0.949387\pi$
$373$		−	1.64339e7i		−	0.316675i	0.987385	−	0.158338i	$-0.0506135\pi$
$374$		0			0
$375$	−4.96203e7	+	2.86483e7i	−0.940947	+	0.543256i
$376$		0			0
$377$	−1.67657e6	−	2.90390e6i	−0.0312894	−	0.0541948i
$378$		0			0
$379$			1.75646e7i			0.322643i	0.986902	+	0.161321i	$0.0515756\pi$
$379$			1.75646e7i			0.322643i	−0.986902	+	0.161321i	$0.948424\pi$
$380$		0			0
$381$	8.87296e7			1.60433
$382$		0			0
$383$	−2.35091e7	+	1.35730e7i	−0.418446	+	0.241590i	−0.694412	−	0.719578i	$-0.744335\pi$
$383$	−2.35091e7	+	1.35730e7i	−0.418446	+	0.241590i	0.275966	+	0.961167i	$0.411002\pi$
$384$		0			0
$385$	3.20796e7	+	5.55635e7i	0.562143	+	0.973661i
$386$		0			0
$387$	−2.23727e6			−0.0385999
$388$		0			0
$389$	3.89835e7	−	6.75214e7i	0.662266	−	1.14708i	−0.317753	−	0.948173i	$-0.602928\pi$
$389$	3.89835e7	−	6.75214e7i	0.662266	−	1.14708i	0.980019	−	0.198904i	$-0.0637382\pi$
$390$		0			0
$391$	−4.81353e7			−0.805256
$392$		0			0
$393$	−1.06592e8	−	6.15409e7i	−1.75609	−	1.01388i
$394$		0			0
$395$	7.97787e7	+	4.60603e7i	1.29448	+	0.747369i
$396$		0			0
$397$	−3.79322e7	−	6.57005e7i	−0.606229	−	1.05002i	−0.991856	−	0.127364i	$-0.959348\pi$
$397$	−3.79322e7	−	6.57005e7i	−0.606229	−	1.05002i	0.385627	−	0.922655i	$-0.373985\pi$
$398$		0			0
$399$	−7.35857e7	−	7.72264e7i	−1.15844	−	1.21576i
$400$		0			0
$401$	−4.97608e7	+	2.87294e7i	−0.771709	+	0.445547i	−0.833484	−	0.552543i	$-0.813657\pi$
$401$	−4.97608e7	+	2.87294e7i	−0.771709	+	0.445547i	0.0617746	+	0.998090i	$0.480324\pi$
$402$		0			0
$403$	−2.33254e6	+	4.04007e6i	−0.0356380	+	0.0617268i
$404$		0			0
$405$	−4.23529e7	+	7.33574e7i	−0.637556	+	1.10428i
$406$		0			0
$407$		−	7.84163e7i		−	1.16312i
$408$		0			0
$409$	6.44982e7	+	3.72381e7i	0.942710	+	0.544274i	0.890809	−	0.454379i	$-0.150139\pi$
$409$	6.44982e7	+	3.72381e7i	0.942710	+	0.544274i	0.0519010	+	0.998652i	$0.483472\pi$
$410$		0			0
$411$			7.88688e7i			1.13600i
$412$		0			0
$413$	−8.92730e7	+	5.15418e7i	−1.26727	+	0.731660i
$414$		0			0
$415$	−3.32340e7	−	5.75629e7i	−0.464984	−	0.805376i
$416$		0			0
$417$			1.89360e8i			2.61144i
$418$		0			0
$419$	−3.77011e7			−0.512521			−0.256260	−	0.966608i	$-0.582490\pi$
$419$	−3.77011e7			−0.512521			−0.256260	+	0.966608i	$0.582490\pi$
$420$		0			0
$421$	−4.91676e7	+	2.83869e7i	−0.658920	+	0.380428i	−0.791865	−	0.610696i	$-0.790890\pi$
$421$	−4.91676e7	+	2.83869e7i	−0.658920	+	0.380428i	0.132945	+	0.991123i	$0.457557\pi$
$422$		0			0
$423$	4.72297e7	+	8.18043e7i	0.624014	+	1.08082i
$424$		0			0
$425$	−1.27427e7			−0.165995
$426$		0			0
$427$	2.40706e7	−	4.16915e7i	0.309174	−	0.535505i
$428$		0			0
$429$	−4.50534e6			−0.0570631
$430$		0			0
$431$	5.17615e6	+	2.98845e6i	0.0646510	+	0.0373262i	0.531977	−	0.846759i	$-0.321449\pi$
$431$	5.17615e6	+	2.98845e6i	0.0646510	+	0.0373262i	−0.467326	+	0.884085i	$0.654783\pi$
$432$		0			0
$433$	7.16448e7	+	4.13641e7i	0.882512	+	0.509519i	0.871486	−	0.490421i	$-0.163157\pi$
$433$	7.16448e7	+	4.13641e7i	0.882512	+	0.509519i	0.0110263	+	0.999939i	$0.496490\pi$
$434$		0			0
$435$	7.69344e7	+	1.33254e8i	0.934659	+	1.61888i
$436$		0			0
$437$	−1.17651e8	−	3.45883e7i	−1.40977	−	0.414462i
$438$		0			0
$439$	−5.32222e7	+	3.07279e7i	−0.629071	+	0.363194i	−0.780392	−	0.625290i	$-0.784981\pi$
$439$	−5.32222e7	+	3.07279e7i	−0.629071	+	0.363194i	0.151321	+	0.988485i	$0.451647\pi$
$440$		0			0
$441$	1.89992e7	−	3.29076e7i	0.221523	−	0.383690i
$442$		0			0
$443$	−1.25683e7	+	2.17689e7i	−0.144566	+	0.250395i	−0.929211	−	0.369550i	$-0.879512\pi$
$443$	−1.25683e7	+	2.17689e7i	−0.144566	+	0.250395i	0.784645	+	0.619945i	$0.212845\pi$
$444$		0			0
$445$		−	8.25758e7i		−	0.937072i
$446$		0			0
$447$	−9.76053e7	−	5.63524e7i	−1.09283	−	0.630943i
$448$		0			0
$449$			237975.i			0.00262901i	0.999999	+	0.00131450i	$0.000418420\pi$
$449$			237975.i			0.00262901i	−0.999999	+	0.00131450i	$0.999582\pi$
$450$		0			0
$451$	2.40655e7	−	1.38942e7i	0.262341	−	0.151463i
$452$		0			0
$453$	−6.87115e7	−	1.19012e8i	−0.739154	−	1.28025i
$454$		0			0
$455$			6.89956e6i			0.0732465i
$456$		0			0
$457$	1.75422e8			1.83795			0.918977	−	0.394312i	$-0.129017\pi$
$457$	1.75422e8			1.83795			0.918977	+	0.394312i	$0.129017\pi$
$458$		0			0
$459$	8.48780e6	−	4.90044e6i	0.0877723	−	0.0506753i
$460$		0			0
$461$	−4.46179e7	−	7.72806e7i	−0.455414	−	0.788801i	0.543297	−	0.839540i	$-0.317176\pi$
$461$	−4.46179e7	−	7.72806e7i	−0.455414	−	0.788801i	−0.998712	+	0.0507393i	$0.983842\pi$
$462$		0			0
$463$	1.61029e8			1.62242			0.811208	−	0.584758i	$-0.198811\pi$
$463$	1.61029e8			1.62242			0.811208	+	0.584758i	$0.198811\pi$
$464$		0			0
$465$	1.07036e8	−	1.85391e8i	1.06456	−	1.84387i
$466$		0			0
$467$	9.21922e7			0.905198			0.452599	−	0.891714i	$-0.350497\pi$
$467$	9.21922e7			0.905198			0.452599	+	0.891714i	$0.350497\pi$
$468$		0			0
$469$	−1.09983e7	−	6.34986e6i	−0.106612	−	0.0615525i
$470$		0			0
$471$	1.31683e8	+	7.60270e7i	1.26028	+	0.727621i
$472$		0			0
$473$	−1.89201e6	−	3.27706e6i	−0.0178789	−	0.0309672i
$474$		0			0
$475$	−3.11452e7	−	9.15643e6i	−0.290610	−	0.0854369i
$476$		0			0
$477$	5.00968e6	−	2.89234e6i	0.0461588	−	0.0266498i
$478$		0			0
$479$	6.32801e7	−	1.09604e8i	0.575785	−	0.997290i	−0.420170	−	0.907445i	$-0.638030\pi$
$479$	6.32801e7	−	1.09604e8i	0.575785	−	0.997290i	0.995956	−	0.0898444i	$-0.0286370\pi$
$480$		0			0
$481$	4.21636e6	−	7.30296e6i	0.0378881	−	0.0656241i
$482$		0			0
$483$		−	2.78049e8i		−	2.46763i
$484$		0			0
$485$	−4.42412e7	−	2.55427e7i	−0.387794	−	0.223893i
$486$		0			0
$487$			1.24619e8i			1.07894i	0.842006	+	0.539468i	$0.181375\pi$
$487$			1.24619e8i			1.07894i	−0.842006	+	0.539468i	$0.818625\pi$
$488$		0			0
$489$	−9.76019e7	+	5.63505e7i	−0.834703	+	0.481916i
$490$		0			0
$491$	2.89737e7	+	5.01839e7i	0.244771	+	0.423955i	0.962067	−	0.272813i	$-0.0879540\pi$
$491$	2.89737e7	+	5.01839e7i	0.244771	+	0.423955i	−0.717296	+	0.696768i	$0.754621\pi$
$492$		0			0
$493$		−	7.87521e7i		−	0.657236i
$494$		0			0
$495$	9.58617e7			0.790368
$496$		0			0
$497$	1.98387e8	−	1.14539e8i	1.61601	−	0.933004i
$498$		0			0
$499$	−4.56868e7	−	7.91319e7i	−0.367696	−	0.636868i	0.621509	−	0.783407i	$-0.286520\pi$
$499$	−4.56868e7	−	7.91319e7i	−0.367696	−	0.636868i	−0.989205	+	0.146539i	$0.953187\pi$
$500$		0			0
$501$	−1.91996e8			−1.52679
$502$		0			0
$503$	5.13318e7	−	8.89092e7i	0.403350	−	0.698623i	−0.590778	−	0.806834i	$-0.701179\pi$
$503$	5.13318e7	−	8.89092e7i	0.403350	−	0.698623i	0.994128	+	0.108211i	$0.0345123\pi$
$504$		0			0
$505$	−3.74994e7			−0.291172
$506$		0			0
$507$	1.53695e8	+	8.87356e7i	1.17933	+	0.680885i
$508$		0			0
$509$	1.93760e8	+	1.11868e8i	1.46930	+	0.848304i	0.999408	−	0.0344176i	$-0.0109576\pi$
$509$	1.93760e8	+	1.11868e8i	1.46930	+	0.848304i	0.469897	+	0.882721i	$0.344291\pi$
$510$		0			0
$511$	8.28415e7	+	1.43486e8i	0.620848	+	1.07534i
$512$		0			0
$513$	2.42668e7	−	5.87843e6i	0.179747	−	0.0435421i
$514$		0			0
$515$	−2.14255e8	+	1.23700e8i	−1.56859	+	0.905625i
$516$		0			0
$517$	−7.98824e7	+	1.38360e8i	−0.578069	+	1.00124i
$518$		0			0
$519$	−1.69707e8	+	2.93942e8i	−1.21394	+	2.10261i
$520$		0			0
$521$		−	2.68923e8i		−	1.90158i	−0.309830	−	0.950792i	$-0.600272\pi$
$521$		−	2.68923e8i		−	1.90158i	0.309830	−	0.950792i	$-0.399728\pi$
$522$		0			0
$523$	−2.00521e8	−	1.15771e8i	−1.40170	−	0.809272i	−0.407132	−	0.913369i	$-0.633471\pi$
$523$	−2.00521e8	−	1.15771e8i	−1.40170	−	0.809272i	−0.994567	+	0.104098i	$0.966805\pi$
$524$		0			0
$525$		−	7.36068e7i		−	0.508675i
$526$		0			0
$527$	−9.48856e7	+	5.47822e7i	−0.648289	+	0.374290i
$528$		0			0
$529$	−8.58048e7	−	1.48618e8i	−0.579621	−	1.00393i
$530$		0			0
$531$			1.54019e8i			1.02871i
$532$		0			0
$533$	2.98832e6			0.0197354
$534$		0			0
$535$	1.61917e7	−	9.34829e6i	0.105738	−	0.0610479i
$536$		0			0
$537$	6.00080e7	+	1.03937e8i	0.387513	+	0.671193i
$538$		0			0
$539$	6.42689e7			0.410426
$540$		0			0
$541$	8.24377e7	−	1.42786e8i	0.520636	−	0.901767i	−0.479076	−	0.877773i	$-0.659028\pi$
$541$	8.24377e7	−	1.42786e8i	0.520636	−	0.901767i	0.999712	−	0.0239943i	$-0.00763834\pi$
$542$		0			0
$543$	2.32579e8			1.45268
$544$		0			0
$545$	−8.26340e7	−	4.77088e7i	−0.510469	−	0.294719i
$546$		0			0
$547$	−1.38115e8	−	7.97409e7i	−0.843879	−	0.487214i	0.0147021	−	0.999892i	$-0.495320\pi$
$547$	−1.38115e8	−	7.97409e7i	−0.843879	−	0.487214i	−0.858581	+	0.512678i	$0.828653\pi$
$548$		0			0
$549$	−3.59644e7	−	6.22922e7i	−0.217348	−	0.376458i
$550$		0			0
$551$	5.65884e7	−	1.92483e8i	0.338277	−	1.15063i
$552$		0			0
$553$	2.35861e8	−	1.36174e8i	1.39470	−	0.805229i
$554$		0			0
$555$	−1.93481e8	+	3.35119e8i	−1.13177	+	1.96029i
$556$		0			0
$557$	−3.47132e7	+	6.01251e7i	−0.200877	+	0.347929i	−0.948811	−	0.315844i	$-0.897712\pi$
$557$	−3.47132e7	+	6.01251e7i	−0.200877	+	0.347929i	0.747934	+	0.663773i	$0.231046\pi$
$558$		0			0
$559$		−	406927.i		−	0.00232960i
$560$		0			0
$561$	−9.16366e7	−	5.29064e7i	−0.519016	−	0.299654i
$562$		0			0
$563$		−	1.90241e8i		−	1.06605i	−0.846098	−	0.533027i	$-0.821054\pi$
$563$		−	1.90241e8i		−	1.06605i	0.846098	−	0.533027i	$-0.178946\pi$
$564$		0			0
$565$	−1.54578e8	+	8.92456e7i	−0.857042	+	0.494814i
$566$		0			0
$567$	1.25214e8	+	2.16876e8i	0.686914	+	1.18977i
$568$		0			0
$569$			2.09862e8i			1.13919i	0.821924	+	0.569597i	$0.192901\pi$
$569$			2.09862e8i			1.13919i	−0.821924	+	0.569597i	$0.807099\pi$
$570$		0			0
$571$	−8.50161e7			−0.456660			−0.228330	−	0.973584i	$-0.573326\pi$
$571$	−8.50161e7			−0.456660			−0.228330	+	0.973584i	$0.573326\pi$
$572$		0			0
$573$	−6.11495e7	+	3.53047e7i	−0.325034	+	0.187659i
$574$		0			0
$575$	−4.23093e7	−	7.32818e7i	−0.222552	−	0.385472i
$576$		0			0
$577$	2.21647e8			1.15381			0.576905	−	0.816811i	$-0.304260\pi$
$577$	2.21647e8			1.15381			0.576905	+	0.816811i	$0.304260\pi$
$578$		0			0
$579$	2.86573e6	−	4.96360e6i	0.0147639	−	0.0255718i
$580$		0			0
$581$	−1.96508e8			−1.00196
$582$		0			0
$583$	8.47316e6	+	4.89198e6i	0.0427602	+	0.0246876i
$584$		0			0
$585$	8.92766e6	+	5.15439e6i	0.0445934	+	0.0257460i
$586$		0			0
$587$	6.67888e7	+	1.15682e8i	0.330209	+	0.571940i	0.982553	−	0.185984i	$-0.0595473\pi$
$587$	6.67888e7	+	1.15682e8i	0.330209	+	0.571940i	−0.652343	+	0.757924i	$0.726214\pi$
$588$		0			0
$589$	−2.71280e8	+	6.57152e7i	−1.32761	+	0.321603i
$590$		0			0
$591$	2.84968e8	−	1.64526e8i	1.38049	−	0.797027i
$592$		0			0
$593$	1.54384e8	−	2.67401e8i	0.740351	−	1.28233i	−0.211984	−	0.977273i	$-0.567993\pi$
$593$	1.54384e8	−	2.67401e8i	0.740351	−	1.28233i	0.952335	−	0.305053i	$-0.0986741\pi$
$594$		0			0
$595$	−8.10219e7	+	1.40334e8i	−0.384637	+	0.666211i
$596$		0			0
$597$			5.73052e8i			2.69322i
$598$		0			0
$599$	−2.36171e8	−	1.36353e8i	−1.09887	−	0.634432i	−0.162945	−	0.986635i	$-0.552099\pi$
$599$	−2.36171e8	−	1.36353e8i	−1.09887	−	0.634432i	−0.935924	+	0.352203i	$0.885433\pi$
$600$		0			0
$601$		−	5.00777e7i		−	0.230686i	−0.993326	−	0.115343i	$-0.963203\pi$
$601$		−	5.00777e7i		−	0.230686i	0.993326	−	0.115343i	$-0.0367967\pi$
$602$		0			0
$603$	−1.64328e7	+	9.48747e6i	−0.0749478	+	0.0432711i
$604$		0			0
$605$	−4.53161e7	−	7.84897e7i	−0.204638	−	0.354443i
$606$		0			0
$607$			3.69542e8i			1.65234i	0.563425	+	0.826168i	$0.309484\pi$
$607$			3.69542e8i			1.65234i	−0.563425	+	0.826168i	$0.690516\pi$
$608$		0			0
$609$	4.54904e8			2.01404
$610$		0			0
$611$	−1.48790e7	+	8.59039e6i	−0.0652304	+	0.0376608i
$612$		0			0
$613$	−540896.	−	936860.i	−0.00234819	−	0.00406718i	0.864849	−	0.502032i	$-0.167414\pi$
$613$	−540896.	−	936860.i	−0.00234819	−	0.00406718i	−0.867197	+	0.497965i	$0.834081\pi$
$614$		0			0
$615$	−1.37128e8			−0.589524
$616$		0			0
$617$	−1.31382e8	+	2.27560e8i	−0.559345	+	0.968815i	0.438206	+	0.898875i	$0.355614\pi$
$617$	−1.31382e8	+	2.27560e8i	−0.559345	+	0.968815i	−0.997551	+	0.0699400i	$0.977719\pi$
$618$		0			0
$619$	2.19171e8			0.924083			0.462042	−	0.886858i	$-0.347117\pi$
$619$	2.19171e8			0.924083			0.462042	+	0.886858i	$0.347117\pi$
$620$		0			0
$621$	5.63637e7	+	3.25416e7i	0.235356	+	0.135883i
$622$		0			0
$623$	−2.11423e8	−	1.22065e8i	−0.874355	−	0.504809i
$624$		0			0
$625$	1.47846e8	+	2.56077e8i	0.605577	+	1.04889i
$626$		0			0
$627$	−1.85958e8	−	1.95159e8i	−0.754418	−	0.791744i
$628$		0			0
$629$	1.71518e8	−	9.90260e7i	0.689220	−	0.397921i
$630$		0			0
$631$	−7.94087e7	+	1.37540e8i	−0.316068	+	0.547445i	−0.979664	−	0.200646i	$-0.935696\pi$
$631$	−7.94087e7	+	1.37540e8i	−0.316068	+	0.547445i	0.663596	+	0.748091i	$0.269029\pi$
$632$		0			0
$633$	−3.24602e8	+	5.62227e8i	−1.27979	+	2.21667i
$634$		0			0
$635$		−	3.43387e8i		−	1.34111i
$636$		0			0
$637$	5.98541e6	+	3.45568e6i	0.0231566	+	0.0133695i
$638$		0			0
$639$		−	3.42270e8i		−	1.31180i
$640$		0			0
$641$	3.26755e8	−	1.88652e8i	1.24065	−	0.716289i	0.271422	−	0.962460i	$-0.412506\pi$
$641$	3.26755e8	−	1.88652e8i	1.24065	−	0.716289i	0.969226	+	0.246172i	$0.0791728\pi$
$642$		0			0
$643$	−2.80619e7	−	4.86046e7i	−0.105556	−	0.182829i	0.808409	−	0.588621i	$-0.200329\pi$
$643$	−2.80619e7	−	4.86046e7i	−0.105556	−	0.182829i	−0.913965	+	0.405792i	$0.866996\pi$
$644$		0			0
$645$			1.86731e7i			0.0695884i
$646$		0			0
$647$	2.36422e8			0.872920			0.436460	−	0.899724i	$-0.356232\pi$
$647$	2.36422e8			0.872920			0.436460	+	0.899724i	$0.356232\pi$
$648$		0			0
$649$	−2.25601e8	+	1.30251e8i	−0.825291	+	0.476482i
$650$		0			0
$651$	−3.16444e8	−	5.48097e8i	−1.14697	−	1.98662i
$652$		0			0
$653$	−4.57874e8			−1.64439			−0.822197	−	0.569203i	$-0.807252\pi$
$653$	−4.57874e8			−1.64439			−0.822197	+	0.569203i	$0.807252\pi$
$654$		0			0
$655$	−2.38165e8	+	4.12515e8i	−0.847530	+	1.46796i
$656$		0			0
$657$	2.47550e8			0.872906
$658$		0			0
$659$	−3.25023e8	−	1.87652e8i	−1.13568	−	0.655687i	−0.190325	−	0.981721i	$-0.560954\pi$
$659$	−3.25023e8	−	1.87652e8i	−1.13568	−	0.655687i	−0.945358	+	0.326034i	$0.894288\pi$
$660$		0			0
$661$	−1.73210e8	−	1.00003e8i	−0.599749	−	0.346265i	0.169194	−	0.985583i	$-0.445884\pi$
$661$	−1.73210e8	−	1.00003e8i	−0.599749	−	0.346265i	−0.768943	+	0.639318i	$0.779217\pi$
$662$		0			0
$663$	−5.68945e6	−	9.85442e6i	−0.0195222	−	0.0338135i
$664$		0			0
$665$	−2.98869e8	+	2.84779e8i	−1.01629	+	0.968375i
$666$		0			0
$667$	4.52895e8	−	2.61479e8i	1.52623	−	0.881169i
$668$		0			0
$669$	−2.16040e8	+	3.74192e8i	−0.721533	+	1.24973i
$670$		0			0
$671$	6.08287e7	−	1.05358e8i	0.201345	−	0.348740i
$672$		0			0
$673$			2.08520e8i			0.684074i	0.939686	+	0.342037i	$0.111117\pi$
$673$			2.08520e8i			0.684074i	−0.939686	+	0.342037i	$0.888883\pi$
$674$		0			0
$675$	1.49210e7	+	8.61462e6i	0.0485160	+	0.0280107i
$676$		0			0
$677$			2.71485e7i			0.0874943i	0.999043	+	0.0437471i	$0.0139296\pi$
$677$			2.71485e7i			0.0874943i	−0.999043	+	0.0437471i	$0.986070\pi$
$678$		0			0
$679$	−1.30796e8	+	7.55152e7i	−0.417817	+	0.241227i
$680$		0			0
$681$	−7.13776e6	−	1.23630e7i	−0.0226006	−	0.0391454i
$682$		0			0
$683$		−	3.34847e8i		−	1.05096i	−0.850807	−	0.525478i	$-0.823886\pi$
$683$		−	3.34847e8i		−	1.05096i	0.850807	−	0.525478i	$-0.176114\pi$
$684$		0			0
$685$	3.05225e8			0.949618
$686$		0			0
$687$	−2.54399e7	+	1.46877e7i	−0.0784593	+	0.0452985i
$688$		0			0
$689$	526074.	+	911187.i	0.00160838	+	0.00278580i
$690$		0			0
$691$	6.02055e8			1.82474			0.912372	−	0.409362i	$-0.134249\pi$
$691$	6.02055e8			1.82474			0.912372	+	0.409362i	$0.134249\pi$
$692$		0			0
$693$	1.41704e8	−	2.45439e8i	0.425779	−	0.737470i
$694$		0			0
$695$	7.32831e8			2.18298
$696$		0			0
$697$	6.07811e7	+	3.50920e7i	0.179502	+	0.103636i
$698$		0			0
$699$	−2.11352e8	−	1.22024e8i	−0.618834	−	0.357284i
$700$		0			0
$701$	−1.18742e8	−	2.05667e8i	−0.344707	−	0.597050i	0.640594	−	0.767880i	$-0.278688\pi$
$701$	−1.18742e8	−	2.05667e8i	−0.344707	−	0.597050i	−0.985300	+	0.170830i	$0.945355\pi$
$702$		0			0
$703$	4.90374e8	−	1.18789e8i	1.41144	−	0.341908i
$704$		0			0
$705$	6.82768e8	−	3.94196e8i	1.94852	−	1.12498i
$706$		0			0
$707$	−5.54323e7	+	9.60116e7i	−0.156857	+	0.271685i
$708$		0			0
$709$	1.49887e8	−	2.59611e8i	0.420556	−	0.728425i	−0.575438	−	0.817846i	$-0.695168\pi$
$709$	1.49887e8	−	2.59611e8i	0.420556	−	0.728425i	0.995994	+	0.0894210i	$0.0285016\pi$
$710$		0			0
$711$		−	4.06922e8i		−	1.13215i
$712$		0			0
$713$	−6.30093e8	−	3.63784e8i	−1.73835	−	1.00363i
$714$		0			0
$715$			1.74358e7i			0.0477007i
$716$		0			0
$717$	3.69603e8	−	2.13390e8i	1.00272	−	0.578918i
$718$		0			0
$719$	3.06247e8	+	5.30435e8i	0.823919	+	1.42707i	0.902743	+	0.430181i	$0.141550\pi$
$719$	3.06247e8	+	5.30435e8i	0.823919	+	1.42707i	−0.0788237	+	0.996889i	$0.525116\pi$
$720$		0			0
$721$			7.31423e8i			1.95147i
$722$		0			0
$723$	2.40460e8			0.636250
$724$		0			0
$725$	1.19893e8	−	6.92203e7i	0.314615	−	0.181643i
$726$		0			0
$727$	−3.16196e8	−	5.47667e8i	−0.822911	−	1.42532i	−0.903505	−	0.428577i	$-0.859015\pi$
$727$	−3.16196e8	−	5.47667e8i	−0.822911	−	1.42532i	0.0805941	−	0.996747i	$-0.474318\pi$
$728$		0			0
$729$	2.76329e8			0.713254
$730$		0			0
$731$	4.77856e6	−	8.27671e6i	0.0122333	−	0.0211888i
$732$		0			0
$733$	4.36325e8			1.10789			0.553946	−	0.832552i	$-0.313121\pi$
$733$	4.36325e8			1.10789			0.553946	+	0.832552i	$0.313121\pi$
$734$		0			0
$735$	−2.74659e8	−	1.58574e8i	−0.691722	−	0.399366i
$736$		0			0
$737$	−2.77937e7	−	1.60467e7i	−0.0694295	−	0.0400851i
$738$		0			0
$739$	8.65823e7	+	1.49965e8i	0.214534	+	0.371584i	0.953128	−	0.302566i	$-0.0978434\pi$
$739$	8.65823e7	+	1.49965e8i	0.214534	+	0.371584i	−0.738594	+	0.674150i	$0.764510\pi$
$740$		0			0
$741$	−6.82491e6	−	2.81740e7i	−0.0167742	−	0.0692459i
$742$		0			0
$743$	−6.64859e7	+	3.83856e7i	−0.162093	+	0.0935842i	−0.578852	−	0.815433i	$-0.696499\pi$
$743$	−6.64859e7	+	3.83856e7i	−0.162093	+	0.0935842i	0.416759	+	0.909017i	$0.363166\pi$
$744$		0			0
$745$	−2.18086e8	+	3.77736e8i	−0.527423	+	0.913524i
$746$		0			0
$747$	−1.46804e8	+	2.54271e8i	−0.352188	+	0.610008i
$748$		0			0
$749$		−	5.52753e7i		−	0.131548i
$750$		0			0
$751$	−3.71691e8	−	2.14596e8i	−0.877531	−	0.506643i	−0.00768760	−	0.999970i	$-0.502447\pi$
$751$	−3.71691e8	−	2.14596e8i	−0.877531	−	0.506643i	−0.869844	+	0.493328i	$0.835780\pi$
$752$		0			0
$753$		−	8.06715e8i		−	1.88945i
$754$		0			0
$755$	−4.60581e8	+	2.65916e8i	−1.07020	+	0.617880i
$756$		0			0
$757$	2.17121e8	+	3.76064e8i	0.500511	+	0.866910i	1.00000		0.000589841i	$0.000187752\pi$
$757$	2.17121e8	+	3.76064e8i	0.500511	+	0.866910i	−0.499489	+	0.866320i	$0.666479\pi$
$758$		0			0
$759$		−	7.02656e8i		−	1.60701i
$760$		0			0
$761$	−9.00184e7			−0.204257			−0.102129	−	0.994771i	$-0.532565\pi$
$761$	−9.00184e7			−0.204257			−0.102129	+	0.994771i	$0.532565\pi$
$762$		0			0
$763$	−2.44302e8	+	1.41048e8i	−0.549988	+	0.317536i
$764$		0			0
$765$	1.21057e8	+	2.09676e8i	0.270398	+	0.468343i
$766$		0			0
$767$	−2.80138e7			−0.0620850
$768$		0			0
$769$	5.21029e6	−	9.02448e6i	0.0114573	−	0.0198446i	−0.860240	−	0.509889i	$-0.829686\pi$
$769$	5.21029e6	−	9.02448e6i	0.0114573	−	0.0198446i	0.871697	+	0.490045i	$0.163020\pi$
$770$		0			0
$771$	−4.49625e8			−0.981042
$772$		0			0
$773$	−3.35683e8	−	1.93807e8i	−0.726761	−	0.419596i	0.0904752	−	0.995899i	$-0.471161\pi$
$773$	−3.35683e8	−	1.93807e8i	−0.726761	−	0.419596i	−0.817236	+	0.576303i	$0.804495\pi$
$774$		0			0
$775$	−1.66802e8	−	9.63032e7i	−0.358341	−	0.206888i
$776$		0			0
$777$	5.72013e8	+	9.90756e8i	1.21939	+	2.11205i
$778$		0			0
$779$	1.23343e8	+	1.29446e8i	0.260917	+	0.273826i
$780$		0			0
$781$	5.01343e8	−	2.89450e8i	1.05240	−	0.607605i
$782$		0			0
$783$	−5.32399e7	+	9.22142e7i	−0.110905	+	0.192093i
$784$		0			0
$785$	2.94227e8	−	5.09617e8i	0.608239	−	1.05350i
$786$		0			0
$787$		−	2.28533e8i		−	0.468840i	−0.972135	−	0.234420i	$-0.924681\pi$
$787$		−	2.28533e8i		−	0.468840i	0.972135	−	0.234420i	$-0.0753191\pi$
$788$		0			0
$789$	−8.65217e8	−	4.99533e8i	−1.76155	−	1.01703i
$790$		0			0
$791$			5.27698e8i			1.06624i
$792$		0			0
$793$	1.13300e7	−	6.54139e6i	0.0227202	−	0.0131175i
$794$		0			0
$795$	−2.41405e7	−	4.18126e7i	−0.0480447	−	0.0832158i
$796$		0			0
$797$			5.29025e8i			1.04496i	0.852651	+	0.522482i	$0.174994\pi$
$797$			5.29025e8i			1.04496i	−0.852651	+	0.522482i	$0.825006\pi$
$798$		0			0
$799$	−4.03510e8			−0.791068
$800$		0			0
$801$	−3.15891e8	+	1.82380e8i	−0.614668	+	0.354879i
$802$		0			0
$803$	2.09348e8	+	3.62602e8i	0.404318	+	0.700299i
$804$		0			0
$805$	−1.07606e9			−2.06276
$806$		0			0
$807$	−4.06640e8	+	7.04320e8i	−0.773729	+	1.34014i
$808$		0			0
$809$	−4.72773e8			−0.892909			−0.446455	−	0.894806i	$-0.647314\pi$
$809$	−4.72773e8			−0.892909			−0.446455	+	0.894806i	$0.647314\pi$
$810$		0			0
$811$	9.87486e7	+	5.70125e7i	0.185126	+	0.106883i	0.589699	−	0.807623i	$-0.299246\pi$
$811$	9.87486e7	+	5.70125e7i	0.185126	+	0.106883i	−0.404573	+	0.914506i	$0.632580\pi$
$812$		0			0
$813$	1.19509e9	+	6.89986e8i	2.22397	+	1.28401i
$814$		0			0
$815$	2.18079e8	+	3.77723e8i	0.402847	+	0.697752i
$816$		0			0
$817$	1.76269e7	−	1.67959e7i	0.0323229	−	0.0307991i
$818$		0			0
$819$	2.63941e7	−	1.52386e7i	0.0480457	−	0.0277392i
$820$		0			0
$821$	−2.13097e8	+	3.69094e8i	−0.385076	+	0.666972i	−0.991780	−	0.127956i	$-0.959158\pi$
$821$	−2.13097e8	+	3.69094e8i	−0.385076	+	0.666972i	0.606703	+	0.794928i	$0.292492\pi$
$822$		0			0
$823$	−7.61813e7	+	1.31950e8i	−0.136662	+	0.236706i	−0.926231	−	0.376956i	$-0.876971\pi$
$823$	−7.61813e7	+	1.31950e8i	−0.136662	+	0.236706i	0.789569	+	0.613662i	$0.210304\pi$
$824$		0			0
$825$		−	1.86011e8i		−	0.331267i
$826$		0			0
$827$	−3.63598e8	−	2.09923e8i	−0.642843	−	0.371146i	0.142866	−	0.989742i	$-0.454368\pi$
$827$	−3.63598e8	−	2.09923e8i	−0.642843	−	0.371146i	−0.785709	+	0.618596i	$0.787702\pi$
$828$		0			0
$829$		−	4.91296e8i		−	0.862343i	−0.902270	−	0.431171i	$-0.858100\pi$
$829$		−	4.91296e8i		−	0.862343i	0.902270	−	0.431171i	$-0.141900\pi$
$830$		0			0
$831$	−3.96543e8	+	2.28944e8i	−0.691014	+	0.398957i
$832$		0			0
$833$	8.11604e7	+	1.40574e8i	0.140414	+	0.243203i
$834$		0			0
$835$			7.43033e8i			1.27629i
$836$		0			0
$837$	1.48141e8			0.252638
$838$		0			0
$839$	6.88100e8	−	3.97275e8i	1.16511	−	0.672674i	0.212584	−	0.977143i	$-0.431812\pi$
$839$	6.88100e8	−	3.97275e8i	1.16511	−	0.672674i	0.952522	+	0.304468i	$0.0984788\pi$
$840$		0			0
$841$	1.30382e8	+	2.25829e8i	0.219195	+	0.379657i
$842$		0			0
$843$	−4.23082e8			−0.706223
$844$		0			0
$845$	3.43410e8	−	5.94804e8i	0.569171	−	0.985833i
$846$		0			0
$847$	−2.67948e8			−0.440961
$848$		0			0
$849$	−8.08755e8	−	4.66935e8i	−1.32158	−	0.763015i
$850$		0			0
$851$	1.13897e9	+	6.57587e8i	1.84810	+	1.06700i
$852$		0			0
$853$	−4.21584e8	−	7.30206e8i	−0.679262	−	1.17652i	−0.975203	−	0.221311i	$-0.928967\pi$
$853$	−4.21584e8	−	7.30206e8i	−0.679262	−	1.17652i	0.295941	−	0.955206i	$-0.404367\pi$
$854$		0			0
$855$	1.45216e8	+	5.99469e8i	0.232336	+	0.959109i
$856$		0			0
$857$	−5.75052e8	+	3.32006e8i	−0.913618	+	0.527478i	−0.881594	−	0.472009i	$-0.843529\pi$
$857$	−5.75052e8	+	3.32006e8i	−0.913618	+	0.527478i	−0.0320246	+	0.999487i	$0.510195\pi$
$858$		0			0
$859$	3.69259e8	−	6.39576e8i	0.582575	−	1.00905i	−0.412598	−	0.910913i	$-0.635379\pi$
$859$	3.69259e8	−	6.39576e8i	0.582575	−	1.00905i	0.995173	−	0.0981364i	$-0.0312881\pi$
$860$		0			0
$861$	−2.02705e8	+	3.51096e8i	−0.317582	+	0.550068i
$862$		0			0
$863$		−	3.52500e8i		−	0.548437i	−0.961667	−	0.274219i	$-0.911581\pi$
$863$		−	3.52500e8i		−	0.548437i	0.961667	−	0.274219i	$-0.0884192\pi$
$864$		0			0
$865$	1.13757e9	+	6.56774e8i	1.75763	+	1.01477i
$866$		0			0
$867$			6.22662e8i			0.955422i
$868$		0			0
$869$	5.96042e8	−	3.44125e8i	0.908276	−	0.524393i
$870$		0			0
$871$	−1.72563e6	−	2.98888e6i	−0.00261152	−	0.00452329i
$872$		0			0
$873$			2.25658e8i			0.339162i
$874$		0			0
$875$	6.55560e8			0.978561
$876$		0			0
$877$	−3.74974e8	+	2.16491e8i	−0.555907	+	0.320953i	−0.751501	−	0.659732i	$-0.770670\pi$
$877$	−3.74974e8	+	2.16491e8i	−0.555907	+	0.320953i	0.195594	+	0.980685i	$0.437336\pi$
$878$		0			0
$879$	−6.80407e8	−	1.17850e9i	−1.00185	−	1.73525i
$880$		0			0
$881$	−1.16456e9			−1.70307			−0.851535	−	0.524297i	$-0.824328\pi$
$881$	−1.16456e9			−1.70307			−0.851535	+	0.524297i	$0.824328\pi$
$882$		0			0
$883$	4.15350e7	−	7.19407e7i	0.0603298	−	0.104494i	−0.834283	−	0.551336i	$-0.814118\pi$
$883$	4.15350e7	−	7.19407e7i	0.0603298	−	0.104494i	0.894613	+	0.446842i	$0.147451\pi$
$884$		0			0
$885$	1.28550e9			1.85457
$886$		0			0
$887$	−7.84912e7	−	4.53169e7i	−0.112473	−	0.0649366i	0.442708	−	0.896666i	$-0.354018\pi$
$887$	−7.84912e7	−	4.53169e7i	−0.112473	−	0.0649366i	−0.555181	+	0.831729i	$0.687351\pi$
$888$		0			0
$889$	−8.79191e8	−	5.07601e8i	−1.25135	−	0.722466i
$890$		0			0
$891$	3.16427e8	+	5.48067e8i	0.447342	+	0.774820i
$892$		0			0
$893$	−9.86243e8	−	2.89947e8i	−1.38493	−	0.407160i
$894$		0			0
$895$	4.02240e8	−	2.32233e8i	0.561069	−	0.323933i
$896$		0			0
$897$	3.77811e7	−	6.54388e7i	0.0523477	−	0.0906688i
$898$		0			0
$899$	5.95171e8	−	1.03087e9i	0.819149	−	1.41881i
$900$		0			0
$901$			2.47109e7i			0.0337842i
$902$		0			0
$903$	4.78096e7	+	2.76029e7i	0.0649309	+	0.0374879i
$904$		0			0
$905$		−	9.00090e8i		−	1.21434i
$906$		0			0
$907$	7.15302e7	−	4.12980e7i	0.0958666	−	0.0553486i	−0.451300	−	0.892372i	$-0.649040\pi$
$907$	7.15302e7	−	4.12980e7i	0.0958666	−	0.0553486i	0.547167	+	0.837024i	$0.315706\pi$
$908$		0			0
$909$	8.28226e7	+	1.43453e8i	0.110270	+	0.190993i
$910$		0			0
$911$			5.00459e8i			0.661932i	0.943643	+	0.330966i	$0.107374\pi$
$911$			5.00459e8i			0.661932i	−0.943643	+	0.330966i	$0.892626\pi$
$912$		0			0
$913$	−4.96595e8			−0.652514
$914$		0			0
$915$	−5.19913e8	+	3.00172e8i	−0.678684	+	0.391838i
$916$		0			0
$917$	7.04121e8	+	1.21957e9i	0.913144	+	1.58161i
$918$		0			0
$919$	−5.70505e8			−0.735044			−0.367522	−	0.930015i	$-0.619794\pi$
$919$	−5.70505e8			−0.735044			−0.367522	+	0.930015i	$0.619794\pi$
$920$		0			0
$921$	6.63313e8	−	1.14889e9i	0.849062	−	1.47062i
$922$		0			0
$923$	6.22539e7			0.0791701
$924$		0			0
$925$	3.01516e8	+	1.74081e8i	0.380966	+	0.219951i
$926$		0			0
$927$	9.46423e8	+	5.46417e8i	1.18808	+	0.685939i
$928$		0			0
$929$	−5.00837e8	−	8.67475e8i	−0.624668	−	1.08196i	−0.988605	−	0.150533i	$-0.951901\pi$
$929$	−5.00837e8	−	8.67475e8i	−0.624668	−	1.08196i	0.363937	−	0.931424i	$-0.381432\pi$
$930$		0			0
$931$	9.73578e7	+	4.01904e8i	0.120648	+	0.498051i
$932$		0			0
$933$	−1.37625e9	+	7.94581e8i	−1.69455	+	0.978347i
$934$		0			0
$935$	−2.04750e8	+	3.54637e8i	−0.250489	+	0.433860i
$936$		0			0
$937$	−1.32969e8	+	2.30309e8i	−0.161634	+	0.279958i	−0.935455	−	0.353446i	$-0.885010\pi$
$937$	−1.32969e8	+	2.30309e8i	−0.161634	+	0.279958i	0.773821	+	0.633404i	$0.218343\pi$
$938$		0			0
$939$			9.08470e7i			0.109727i
$940$		0			0
$941$	−3.41304e8	−	1.97052e8i	−0.409611	−	0.236489i	0.281011	−	0.959704i	$-0.409330\pi$
$941$	−3.41304e8	−	1.97052e8i	−0.409611	−	0.236489i	−0.690623	+	0.723215i	$0.742663\pi$
$942$		0			0
$943$			4.66061e8i			0.555786i
$944$		0			0
$945$	1.89744e8	−	1.09549e8i	0.224839	−	0.129811i
$946$		0			0
$947$	3.37430e8	+	5.84445e8i	0.397314	+	0.688167i	0.993393	−	0.114758i	$-0.0366092\pi$
$947$	3.37430e8	+	5.84445e8i	0.397314	+	0.688167i	−0.596080	+	0.802925i	$0.703276\pi$
$948$		0			0
$949$			4.50258e7i			0.0526820i
$950$		0			0
$951$	1.87032e8			0.217458
$952$		0			0
$953$	1.83731e7	−	1.06077e7i	0.0212278	−	0.0122559i	−0.489349	−	0.872088i	$-0.662765\pi$
$953$	1.83731e7	−	1.06077e7i	0.0212278	−	0.0122559i	0.510576	+	0.859832i	$0.329432\pi$
$954$		0			0
$955$	1.36630e8	+	2.36651e8i	0.156869	+	0.271705i
$956$		0			0
$957$	1.14958e9			1.31161
$958$		0			0
$959$	4.51190e8	−	7.81483e8i	0.511568	−	0.886061i
$960$		0			0
$961$	−7.68569e8			−0.865990
$962$		0			0
$963$	−7.15233e7	−	4.12940e7i	−0.0800882	−	0.0462389i
$964$		0			0
$965$	−1.92093e7	−	1.10905e7i	−0.0213762	−	0.0123415i
$966$		0			0
$967$	−5.55692e8	−	9.62486e8i	−0.614546	−	1.06443i	−0.990464	−	0.137772i	$-0.956006\pi$
$967$	−5.55692e8	−	9.62486e8i	−0.614546	−	1.06443i	0.375918	−	0.926653i	$-0.377328\pi$
$968$		0			0
$969$	1.92033e8	−	6.53193e8i	0.211060	−	0.717910i
$970$		0			0
$971$	−2.32260e8	+	1.34095e8i	−0.253697	+	0.146472i	−0.621456	−	0.783449i	$-0.713459\pi$
$971$	−2.32260e8	+	1.34095e8i	−0.253697	+	0.146472i	0.367759	+	0.929921i	$0.380125\pi$
$972$		0			0
$973$	1.08328e9	−	1.87630e9i	1.17599	−	2.03688i
$974$		0			0
$975$	1.00017e7	−	1.73234e7i	0.0107909	−	0.0186904i
$976$		0			0
$977$			1.32741e8i			0.142339i	0.997464	+	0.0711693i	$0.0226731\pi$
$977$			1.32741e8i			0.142339i	−0.997464	+	0.0711693i	$0.977327\pi$
$978$		0			0
$979$	−5.34286e8	−	3.08470e8i	−0.569410	−	0.328749i
$980$		0			0
$981$			4.21485e8i			0.446453i
$982$		0			0
$983$	−7.64205e8	+	4.41214e8i	−0.804543	+	0.464503i	−0.845057	−	0.534676i	$-0.820434\pi$
$983$	−7.64205e8	+	4.41214e8i	−0.804543	+	0.464503i	0.0405144	+	0.999179i	$0.487100\pi$
$984$		0			0
$985$	−6.36724e8	−	1.10284e9i	−0.666258	−	1.15399i
$986$		0			0
$987$		−	2.33083e9i		−	2.42415i
$988$		0			0
$989$	6.34646e7			0.0656059
$990$		0			0
$991$	1.34320e9	−	7.75496e8i	1.38013	−	0.796817i	0.387954	−	0.921679i	$-0.373182\pi$
$991$	1.34320e9	−	7.75496e8i	1.38013	−	0.796817i	0.992174	+	0.124861i	$0.0398486\pi$
$992$		0			0
$993$	8.69014e8	+	1.50518e9i	0.887522	+	1.53723i
$994$		0			0
$995$	2.21774e9			2.25134
$996$		0			0
$997$	−1.09290e8	+	1.89296e8i	−0.110279	+	0.191009i	−0.915883	−	0.401446i	$-0.868508\pi$
$997$	−1.09290e8	+	1.89296e8i	−0.110279	+	0.191009i	0.805604	+	0.592455i	$0.201841\pi$
$998$		0			0
$999$	−2.67784e8			−0.268589

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.7.h.a.69.2	yes	20
3.2	odd	2		684.7.y.c.145.3		20
19.8	odd	6	inner	76.7.h.a.65.2	✓	20
57.8	even	6		684.7.y.c.217.3		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.7.h.a.65.2	✓	20	19.8	odd	6	inner
76.7.h.a.69.2	yes	20	1.1	even	1	trivial
684.7.y.c.145.3		20	3.2	odd	2
684.7.y.c.217.3		20	57.8	even	6

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:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	76.h (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-31.9288 + 18.4341i) q^{3} +(71.3406 + 123.566i) q^{5} +421.828 q^{7} +(315.131 - 545.823i) q^{9} +O(q^{10})\)	\(q+(-31.9288 + 18.4341i) q^{3} +(71.3406 + 123.566i) q^{5} +421.828 q^{7} +(315.131 - 545.823i) q^{9} +1066.00 q^{11} +(99.2772 + 57.3177i) q^{13} +(-4555.64 - 2630.20i) q^{15} +(1346.17 + 2331.63i) q^{17} +(1614.83 + 6666.20i) q^{19} +(-13468.5 + 7776.01i) q^{21} +(-8939.31 + 15483.3i) q^{23} +(-2366.47 + 4098.85i) q^{25} -3640.28i q^{27} +(-25331.6 - 14625.2i) q^{29} +40694.9i q^{31} +(-34036.0 + 19650.7i) q^{33} +(30093.5 + 52123.4i) q^{35} -73561.3i q^{37} -4226.40 q^{39} +(22575.6 - 13034.0i) q^{41} +(-1774.87 - 3074.17i) q^{43} +89926.6 q^{45} +(-74936.6 + 129794. i) q^{47} +60289.9 q^{49} +(-85963.1 - 49630.8i) q^{51} +(7948.56 + 4589.10i) q^{53} +(76049.1 + 131721. i) q^{55} +(-174445. - 183076. i) q^{57} +(-211634. + 122187. i) q^{59} +(57062.6 - 98835.3i) q^{61} +(132931. - 230243. i) q^{63} +16356.3i q^{65} +(-26072.9 - 15053.2i) q^{67} -659152. i q^{69} +(470303. - 271530. i) q^{71} +(196387. + 340152. i) q^{73} -174495. i q^{75} +449668. q^{77} +(559140. - 322819. i) q^{79} +(296836. + 514135. i) q^{81} -465849. q^{83} +(-192073. + 332681. i) q^{85} +1.07841e6 q^{87} +(-501206. - 289372. i) q^{89} +(41877.9 + 24178.2i) q^{91} +(-750173. - 1.29934e6i) q^{93} +(-708510. + 675108. i) q^{95} +(-310070. + 179019. i) q^{97} +(335930. - 581847. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 30 q^{3} - 56 q^{5} + 464 q^{7} + 2200 q^{9}+O(q^{10}) \) 20 * q - 30 * q^3 - 56 * q^5 + 464 * q^7 + 2200 * q^9	\( 20 q - 30 q^{3} - 56 q^{5} + 464 q^{7} + 2200 q^{9} - 3644 q^{11} - 7140 q^{13} + 9168 q^{15} + 1132 q^{17} + 2110 q^{19} - 8748 q^{21} + 832 q^{23} - 27698 q^{25} - 10920 q^{29} - 30306 q^{33} + 4172 q^{35} + 81144 q^{39} + 109206 q^{41} + 110740 q^{43} - 785440 q^{45} + 107080 q^{47} + 136092 q^{49} + 199872 q^{51} + 254796 q^{53} + 354840 q^{55} + 212268 q^{57} - 610638 q^{59} + 47864 q^{61} - 254476 q^{63} - 839562 q^{67} + 366660 q^{71} + 854482 q^{73} + 763088 q^{77} + 1718592 q^{79} - 1054142 q^{81} + 439612 q^{83} - 400236 q^{85} - 1604736 q^{87} + 478032 q^{89} + 599856 q^{91} + 829380 q^{93} - 1055660 q^{95} - 191286 q^{97} - 2336728 q^{99}+O(q^{100}) \) 20 * q - 30 * q^3 - 56 * q^5 + 464 * q^7 + 2200 * q^9 - 3644 * q^11 - 7140 * q^13 + 9168 * q^15 + 1132 * q^17 + 2110 * q^19 - 8748 * q^21 + 832 * q^23 - 27698 * q^25 - 10920 * q^29 - 30306 * q^33 + 4172 * q^35 + 81144 * q^39 + 109206 * q^41 + 110740 * q^43 - 785440 * q^45 + 107080 * q^47 + 136092 * q^49 + 199872 * q^51 + 254796 * q^53 + 354840 * q^55 + 212268 * q^57 - 610638 * q^59 + 47864 * q^61 - 254476 * q^63 - 839562 * q^67 + 366660 * q^71 + 854482 * q^73 + 763088 * q^77 + 1718592 * q^79 - 1054142 * q^81 + 439612 * q^83 - 400236 * q^85 - 1604736 * q^87 + 478032 * q^89 + 599856 * q^91 + 829380 * q^93 - 1055660 * q^95 - 191286 * q^97 - 2336728 * q^99

Introduction

L-functions

Modular forms

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Database

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