LMFDB - Embedded newform 76.6.e.a.45.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,6,Mod(45,76)]

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, 2]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("76.45");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	76.e (of order $3$, 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:	$12.1891703058$
Analytic rank:	$0$
Dimension:	$18$
Relative dimension:	$9$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 2 x^{17} + 1540 x^{16} - 768 x^{15} + 1608492 x^{14} - 1027368 x^{13} + 897054160 x^{12} + \cdots + 80\!\cdots\!36 $ x^18 - 2x^17 + 1540x^16 - 768x^15 + 1608492x^14 - 1027368x^13 + 897054160x^12 - 1275481376x^11 + 361098181456x^10 - 863969476320x^9 + 79755165392064x^8 - 375077568148992x^7 + 12736924096193536x^6 - 57314532742553600x^5 + 977121800205220864x^4 - 4977732006498379776x^3 + 53672321824823513088x^2 - 185653809995679793152*x + 804303742853852430336
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{3}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			45.5
Root			$2.80322 - 4.85531i$ of defining polynomial
Character	$\chi$	$=$	76.45
Dual form			76.6.e.a.49.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.30322 - 5.72134i) q^{3} +(12.6095 + 21.8402i) q^{5} -40.7176 q^{7} +(99.6775 - 172.647i) q^{9} +O(q^{10})$	\(q+(-3.30322 - 5.72134i) q^{3} +(12.6095 + 21.8402i) q^{5} -40.7176 q^{7} +(99.6775 - 172.647i) q^{9} +324.832 q^{11} +(-48.8600 + 84.6280i) q^{13} +(83.3036 - 144.286i) q^{15} +(-1057.36 - 1831.41i) q^{17} +(806.259 - 1351.31i) q^{19} +(134.499 + 232.959i) q^{21} +(1506.33 - 2609.05i) q^{23} +(1244.50 - 2155.54i) q^{25} -2922.39 q^{27} +(-527.733 + 914.060i) q^{29} +7602.00 q^{31} +(-1072.99 - 1858.48i) q^{33} +(-513.427 - 889.282i) q^{35} -2057.16 q^{37} +645.581 q^{39} +(230.415 + 399.091i) q^{41} +(-188.582 - 326.633i) q^{43} +5027.52 q^{45} +(-8354.76 + 14470.9i) q^{47} -15149.1 q^{49} +(-6985.41 + 12099.1i) q^{51} +(5234.06 - 9065.67i) q^{53} +(4095.96 + 7094.41i) q^{55} +(-10394.6 - 149.205i) q^{57} +(23733.3 + 41107.2i) q^{59} +(-13715.7 + 23756.3i) q^{61} +(-4058.63 + 7029.76i) q^{63} -2464.39 q^{65} +(17228.2 - 29840.0i) q^{67} -19903.0 q^{69} +(1879.84 + 3255.97i) q^{71} +(6145.42 + 10644.2i) q^{73} -16443.5 q^{75} -13226.4 q^{77} +(26717.1 + 46275.4i) q^{79} +(-14568.3 - 25233.1i) q^{81} -1317.09 q^{83} +(26665.6 - 46186.2i) q^{85} +6972.87 q^{87} +(51736.4 - 89610.0i) q^{89} +(1989.46 - 3445.85i) q^{91} +(-25111.1 - 43493.7i) q^{93} +(39679.5 + 569.563i) q^{95} +(-52995.4 - 91790.7i) q^{97} +(32378.5 - 56081.1i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 11 q^{3} + 11 q^{5} + 336 q^{7} - 902 q^{9}+O(q^{10}) $ 18 * q - 11 * q^3 + 11 * q^5 + 336 * q^7 - 902 * q^9	$ 18 q - 11 q^{3} + 11 q^{5} + 336 q^{7} - 902 q^{9} - 320 q^{11} + 227 q^{13} - 101 q^{15} + 179 q^{17} - 868 q^{19} - 5700 q^{21} - 3425 q^{23} - 7054 q^{25} + 14722 q^{27} - 7349 q^{29} - 9960 q^{31} - 2998 q^{33} + 15888 q^{35} + 26444 q^{37} - 30246 q^{39} - 7311 q^{41} - 8283 q^{43} - 62164 q^{45} + 37603 q^{47} + 124738 q^{49} + 47227 q^{51} - 20337 q^{53} + 716 q^{55} - 57555 q^{57} - 74455 q^{59} - 7569 q^{61} - 52544 q^{63} + 188998 q^{65} - 26177 q^{67} + 116282 q^{69} - 53463 q^{71} - 14103 q^{73} + 120912 q^{75} - 31960 q^{77} + 31825 q^{79} - 21137 q^{81} + 82600 q^{83} - 50787 q^{85} - 339766 q^{87} - 155197 q^{89} - 2800 q^{91} - 46460 q^{93} + 49315 q^{95} + 111241 q^{97} - 193544 q^{99}+O(q^{100}) $ 18 * q - 11 * q^3 + 11 * q^5 + 336 * q^7 - 902 * q^9 - 320 * q^11 + 227 * q^13 - 101 * q^15 + 179 * q^17 - 868 * q^19 - 5700 * q^21 - 3425 * q^23 - 7054 * q^25 + 14722 * q^27 - 7349 * q^29 - 9960 * q^31 - 2998 * q^33 + 15888 * q^35 + 26444 * q^37 - 30246 * q^39 - 7311 * q^41 - 8283 * q^43 - 62164 * q^45 + 37603 * q^47 + 124738 * q^49 + 47227 * q^51 - 20337 * q^53 + 716 * q^55 - 57555 * q^57 - 74455 * q^59 - 7569 * q^61 - 52544 * q^63 + 188998 * q^65 - 26177 * q^67 + 116282 * q^69 - 53463 * q^71 - 14103 * q^73 + 120912 * q^75 - 31960 * q^77 + 31825 * q^79 - 21137 * q^81 + 82600 * q^83 - 50787 * q^85 - 339766 * q^87 - 155197 * q^89 - 2800 * q^91 - 46460 * q^93 + 49315 * q^95 + 111241 * q^97 - 193544 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$21$	$39$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

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
$3$	−3.30322	−	5.72134i	−0.211901	−	0.367024i	0.740408	−	0.672158i	$-0.234632\pi$
$3$	−3.30322	−	5.72134i	−0.211901	−	0.367024i	−0.952310	+	0.305133i	$0.901299\pi$
$4$		0			0
$5$	12.6095	+	21.8402i	0.225565	+	0.390690i	0.956489	−	0.291769i	$-0.0942439\pi$
$5$	12.6095	+	21.8402i	0.225565	+	0.390690i	−0.730924	+	0.682459i	$0.760911\pi$
$6$		0			0
$7$	−40.7176			−0.314078			−0.157039	−	0.987592i	$-0.550195\pi$
$7$	−40.7176			−0.314078			−0.157039	+	0.987592i	$0.550195\pi$
$8$		0			0
$9$	99.6775	−	172.647i	0.410196	−	0.710479i
$10$		0			0
$11$	324.832			0.809426			0.404713	−	0.914444i	$-0.367371\pi$
$11$	324.832			0.809426			0.404713	+	0.914444i	$0.367371\pi$
$12$		0			0
$13$	−48.8600	+	84.6280i	−0.0801854	+	0.138885i	−0.903329	−	0.428947i	$-0.858885\pi$
$13$	−48.8600	+	84.6280i	−0.0801854	+	0.138885i	0.823144	+	0.567833i	$0.192218\pi$
$14$		0			0
$15$	83.3036	−	144.286i	0.0955951	−	0.165575i
$16$		0			0
$17$	−1057.36	−	1831.41i	−0.887365	−	1.53696i	−0.842978	−	0.537947i	$-0.819200\pi$
$17$	−1057.36	−	1831.41i	−0.887365	−	1.53696i	−0.0443869	−	0.999014i	$-0.514133\pi$
$18$		0			0
$19$	806.259	−	1351.31i	0.512378	−	0.858760i
$19$	806.259	−	1351.31i	0.512378	−	0.858760i
$20$		0			0
$21$	134.499	+	232.959i	0.0665536	+	0.115274i
$22$		0			0
$23$	1506.33	−	2609.05i	0.593748	−	1.02840i	−0.399975	−	0.916526i	$-0.630981\pi$
$23$	1506.33	−	2609.05i	0.593748	−	1.02840i	0.993722	−	0.111875i	$-0.0356856\pi$
$24$		0			0
$25$	1244.50	−	2155.54i	0.398241	−	0.689774i
$26$		0			0
$27$	−2922.39			−0.771487
$28$		0			0
$29$	−527.733	+	914.060i	−0.116525	+	0.201827i	−0.918388	−	0.395680i	$-0.870509\pi$
$29$	−527.733	+	914.060i	−0.116525	+	0.201827i	0.801863	+	0.597507i	$0.203842\pi$
$30$		0			0
$31$	7602.00			1.42077			0.710385	−	0.703813i	$-0.248521\pi$
$31$	7602.00			1.42077			0.710385	+	0.703813i	$0.248521\pi$
$32$		0			0
$33$	−1072.99	−	1858.48i	−0.171519	−	0.297079i
$34$		0			0
$35$	−513.427	−	889.282i	−0.0708449	−	0.122707i
$36$		0			0
$37$	−2057.16			−0.247037			−0.123519	−	0.992342i	$-0.539418\pi$
$37$	−2057.16			−0.247037			−0.123519	+	0.992342i	$0.539418\pi$
$38$		0			0
$39$	645.581			0.0679656
$40$		0			0
$41$	230.415	+	399.091i	0.0214068	+	0.0370777i	0.876530	−	0.481347i	$-0.159852\pi$
$41$	230.415	+	399.091i	0.0214068	+	0.0370777i	−0.855124	+	0.518424i	$0.826519\pi$
$42$		0			0
$43$	−188.582	−	326.633i	−0.0155535	−	0.0269395i	0.858144	−	0.513409i	$-0.171618\pi$
$43$	−188.582	−	326.633i	−0.0155535	−	0.0269395i	−0.873697	+	0.486470i	$0.838284\pi$
$44$		0			0
$45$	5027.52			0.370103
$46$		0			0
$47$	−8354.76	+	14470.9i	−0.551682	+	0.955542i	0.446471	+	0.894798i	$0.352681\pi$
$47$	−8354.76	+	14470.9i	−0.551682	+	0.955542i	−0.998153	+	0.0607440i	$0.980653\pi$
$48$		0			0
$49$	−15149.1			−0.901355
$50$		0			0
$51$	−6985.41	+	12099.1i	−0.376068	+	0.651369i
$52$		0			0
$53$	5234.06	−	9065.67i	0.255947	−	0.443313i	−0.709206	−	0.705002i	$-0.750946\pi$
$53$	5234.06	−	9065.67i	0.255947	−	0.443313i	0.965152	+	0.261689i	$0.0842795\pi$
$54$		0			0
$55$	4095.96	+	7094.41i	0.182578	+	0.316235i
$56$		0			0
$57$	−10394.6	−	149.205i	−0.423759	−	0.00608268i
$58$		0			0
$59$	23733.3	+	41107.2i	0.887621	+	1.53740i	0.842680	+	0.538415i	$0.180977\pi$
$59$	23733.3	+	41107.2i	0.887621	+	1.53740i	0.0449408	+	0.998990i	$0.485690\pi$
$60$		0			0
$61$	−13715.7	+	23756.3i	−0.471948	+	0.817438i	−0.999485	−	0.0320943i	$-0.989782\pi$
$61$	−13715.7	+	23756.3i	−0.471948	+	0.817438i	0.527537	+	0.849532i	$0.323116\pi$
$62$		0			0
$63$	−4058.63	+	7029.76i	−0.128833	+	0.223146i
$64$		0			0
$65$	−2464.39			−0.0723480
$66$		0			0
$67$	17228.2	−	29840.0i	0.468869	−	0.812105i	−0.530498	−	0.847686i	$-0.677995\pi$
$67$	17228.2	−	29840.0i	0.468869	−	0.812105i	0.999367	+	0.0355811i	$0.0113282\pi$
$68$		0			0
$69$	−19903.0			−0.503264
$70$		0			0
$71$	1879.84	+	3255.97i	0.0442562	+	0.0766541i	0.887305	−	0.461183i	$-0.152575\pi$
$71$	1879.84	+	3255.97i	0.0442562	+	0.0766541i	−0.843049	+	0.537837i	$0.819242\pi$
$72$		0			0
$73$	6145.42	+	10644.2i	0.134972	+	0.233779i	0.925587	−	0.378535i	$-0.123572\pi$
$73$	6145.42	+	10644.2i	0.134972	+	0.233779i	−0.790615	+	0.612314i	$0.790239\pi$
$74$		0			0
$75$	−16443.5			−0.337551
$76$		0			0
$77$	−13226.4			−0.254223
$78$		0			0
$79$	26717.1	+	46275.4i	0.481639	+	0.834223i	0.999778	−	0.0210731i	$-0.00670827\pi$
$79$	26717.1	+	46275.4i	0.481639	+	0.834223i	−0.518139	+	0.855297i	$0.673375\pi$
$80$		0			0
$81$	−14568.3	−	25233.1i	−0.246716	−	0.427325i
$82$		0			0
$83$	−1317.09			−0.0209856			−0.0104928	−	0.999945i	$-0.503340\pi$
$83$	−1317.09			−0.0209856			−0.0104928	+	0.999945i	$0.503340\pi$
$84$		0			0
$85$	26665.6	−	46186.2i	0.400317	−	0.693369i
$86$		0			0
$87$	6972.87			0.0987673
$88$		0			0
$89$	51736.4	−	89610.0i	0.692342	−	1.19917i	−0.278726	−	0.960371i	$-0.589912\pi$
$89$	51736.4	−	89610.0i	0.692342	−	1.19917i	0.971068	−	0.238801i	$-0.0767545\pi$
$90$		0			0
$91$	1989.46	−	3445.85i	0.0251845	−	0.0436208i
$92$		0			0
$93$	−25111.1	−	43493.7i	−0.301063	−	0.521457i
$94$		0			0
$95$	39679.5	+	569.563i	0.451083	+	0.00647489i
$96$		0			0
$97$	−52995.4	−	91790.7i	−0.571885	−	0.990533i	−0.996372	−	0.0850998i	$-0.972879\pi$
$97$	−52995.4	−	91790.7i	−0.571885	−	0.990533i	0.424488	−	0.905434i	$-0.360454\pi$
$98$		0			0
$99$	32378.5	−	56081.1i	0.332023	−	0.575081i
$100$		0			0
$101$	−34282.4	+	59378.8i	−0.334401	+	0.579200i	−0.983370	−	0.181616i	$-0.941867\pi$
$101$	−34282.4	+	59378.8i	−0.334401	+	0.579200i	0.648969	+	0.760815i	$0.275201\pi$
$102$		0			0
$103$	−35253.0			−0.327419			−0.163709	−	0.986509i	$-0.552346\pi$
$103$	−35253.0			−0.327419			−0.163709	+	0.986509i	$0.552346\pi$
$104$		0			0
$105$	−3391.92	+	5874.98i	−0.0300243	+	0.0520036i
$106$		0			0
$107$	126916.			1.07166			0.535832	−	0.844325i	$-0.319998\pi$
$107$	126916.			1.07166			0.535832	+	0.844325i	$0.319998\pi$
$108$		0			0
$109$	−107794.	−	186705.i	−0.869019	−	1.50519i	−0.863000	−	0.505204i	$-0.831417\pi$
$109$	−107794.	−	186705.i	−0.869019	−	1.50519i	−0.00601921	−	0.999982i	$-0.501916\pi$
$110$		0			0
$111$	6795.23	+	11769.7i	0.0523476	+	0.0906687i
$112$		0			0
$113$	25767.6			0.189835			0.0949177	−	0.995485i	$-0.469741\pi$
$113$	25767.6			0.189835			0.0949177	+	0.995485i	$0.469741\pi$
$114$		0			0
$115$	75976.3			0.535714
$116$		0			0
$117$	9740.49	+	16871.0i	0.0657834	+	0.113940i
$118$		0			0
$119$	43053.4	+	74570.6i	0.278702	+	0.482726i
$120$		0			0
$121$	−55535.1			−0.344829
$122$		0			0
$123$	1522.22	−	2636.57i	0.00907226	−	0.0157136i
$124$		0			0
$125$	141579.			0.810446
$126$		0			0
$127$	−59314.9	+	102736.i	−0.326328	+	0.565216i	−0.981780	−	0.190020i	$-0.939145\pi$
$127$	−59314.9	+	102736.i	−0.326328	+	0.565216i	0.655452	+	0.755237i	$0.272478\pi$
$128$		0			0
$129$	−1245.85	+	2157.88i	−0.00659163	+	0.0114170i
$130$		0			0
$131$	−27739.5	−	48046.2i	−0.141228	−	0.244614i	0.786731	−	0.617295i	$-0.211772\pi$
$131$	−27739.5	−	48046.2i	−0.141228	−	0.244614i	−0.927959	+	0.372682i	$0.878438\pi$
$132$		0			0
$133$	−32829.0	+	55022.2i	−0.160927	+	0.269717i
$134$		0			0
$135$	−36849.7	−	63825.6i	−0.174020	−	0.301412i
$136$		0			0
$137$	−159895.	+	276946.i	−0.727835	+	1.26065i	0.229962	+	0.973200i	$0.426140\pi$
$137$	−159895.	+	276946.i	−0.727835	+	1.26065i	−0.957797	+	0.287447i	$0.907193\pi$
$138$		0			0
$139$	−105811.	+	183269.i	−0.464506	+	0.804549i	−0.999179	−	0.0405104i	$-0.987102\pi$
$139$	−105811.	+	183269.i	−0.464506	+	0.804549i	0.534673	+	0.845059i	$0.320435\pi$
$140$		0			0
$141$	110390.			0.467609
$142$		0			0
$143$	−15871.3	+	27489.9i	−0.0649041	+	0.112417i
$144$		0			0
$145$	−26617.7			−0.105136
$146$		0			0
$147$	50040.7	+	86673.0i	0.190998	+	0.330819i
$148$		0			0
$149$	36052.9	+	62445.4i	0.133037	+	0.230428i	0.924846	−	0.380342i	$-0.124194\pi$
$149$	36052.9	+	62445.4i	0.133037	+	0.230428i	−0.791809	+	0.610769i	$0.790860\pi$
$150$		0			0
$151$	265545.			0.947753			0.473877	−	0.880591i	$-0.342854\pi$
$151$	265545.			0.947753			0.473877	+	0.880591i	$0.342854\pi$
$152$		0			0
$153$	−421582.			−1.45597
$154$		0			0
$155$	95857.2	+	166029.i	0.320476	+	0.555080i
$156$		0			0
$157$	−159838.	−	276848.i	−0.517526	−	0.896381i	−0.999793	−	0.0203568i	$-0.993520\pi$
$157$	−159838.	−	276848.i	−0.517526	−	0.896381i	0.482267	−	0.876024i	$-0.339814\pi$
$158$		0			0
$159$	−69157.0			−0.216942
$160$		0			0
$161$	−61334.4	+	106234.i	−0.186483	+	0.322998i
$162$		0			0
$163$	−202591.			−0.597244			−0.298622	−	0.954371i	$-0.596527\pi$
$163$	−202591.			−0.597244			−0.298622	+	0.954371i	$0.596527\pi$
$164$		0			0
$165$	27059.7	−	46868.7i	0.0773771	−	0.134021i
$166$		0			0
$167$	−151820.	+	262959.i	−0.421247	+	0.729621i	−0.996062	−	0.0886623i	$-0.971741\pi$
$167$	−151820.	+	262959.i	−0.421247	+	0.729621i	0.574815	+	0.818284i	$0.305074\pi$
$168$		0			0
$169$	180872.	+	313279.i	0.487141	+	0.843752i
$170$		0			0
$171$	−152933.	−	273893.i	−0.399956	−	0.716294i
$172$		0			0
$173$	328662.	+	569260.i	0.834900	+	1.44609i	0.894112	+	0.447844i	$0.147808\pi$
$173$	328662.	+	569260.i	0.834900	+	1.44609i	−0.0592118	+	0.998245i	$0.518859\pi$
$174$		0			0
$175$	−50673.2	+	87768.6i	−0.125079	+	0.216643i
$176$		0			0
$177$	156792.	−	271572.i	0.376176	−	0.651557i
$178$		0			0
$179$	265873.			0.620215			0.310108	−	0.950701i	$-0.399635\pi$
$179$	265873.			0.620215			0.310108	+	0.950701i	$0.399635\pi$
$180$		0			0
$181$	75575.2	−	130900.i	0.171468	−	0.296991i	−0.767465	−	0.641090i	$-0.778482\pi$
$181$	75575.2	−	130900.i	0.171468	−	0.296991i	0.938933	+	0.344099i	$0.111816\pi$
$182$		0			0
$183$	181224.			0.400026
$184$		0			0
$185$	−25939.6	−	44928.7i	−0.0557230	−	0.0965150i
$186$		0			0
$187$	−343466.	−	594901.i	−0.718257	−	1.24406i
$188$		0			0
$189$	118993.			0.242307
$190$		0			0
$191$	−120491.			−0.238986			−0.119493	−	0.992835i	$-0.538127\pi$
$191$	−120491.			−0.238986			−0.119493	+	0.992835i	$0.538127\pi$
$192$		0			0
$193$	38385.0	+	66484.8i	0.0741769	+	0.128478i	0.900728	−	0.434384i	$-0.143034\pi$
$193$	38385.0	+	66484.8i	0.0741769	+	0.128478i	−0.826551	+	0.562862i	$0.809700\pi$
$194$		0			0
$195$	8140.43	+	14099.6i	0.0153307	+	0.0265535i
$196$		0			0
$197$	581905.			1.06828			0.534141	−	0.845395i	$-0.320635\pi$
$197$	581905.			1.06828			0.534141	+	0.845395i	$0.320635\pi$
$198$		0			0
$199$	83536.9	−	144690.i	0.149536	−	0.259004i	−0.781520	−	0.623880i	$-0.785555\pi$
$199$	83536.9	−	144690.i	0.149536	−	0.259004i	0.931056	+	0.364876i	$0.118889\pi$
$200$		0			0
$201$	−227633.			−0.397416
$202$		0			0
$203$	21488.0	−	37218.4i	0.0365979	−	0.0633895i
$204$		0			0
$205$	−5810.82	+	10064.6i	−0.00965724	+	0.0167268i
$206$		0			0
$207$	−300295.	−	520127.i	−0.487105	−	0.843691i
$208$		0			0
$209$	261899.	−	438950.i	0.414732	−	0.695103i
$210$		0			0
$211$	−360222.	−	623923.i	−0.557011	−	0.964772i	−0.997744	−	0.0671336i	$-0.978615\pi$
$211$	−360222.	−	623923.i	−0.557011	−	0.964772i	0.440733	−	0.897638i	$-0.354719\pi$
$212$		0			0
$213$	12419.0	−	21510.4i	0.0187559	−	0.0324862i
$214$		0			0
$215$	4755.83	−	8237.33i	0.00701665	−	0.0121532i
$216$		0			0
$217$	−309536.			−0.446233
$218$		0			0
$219$	40599.3	−	70320.1i	0.0572016	−	0.0990761i
$220$		0			0
$221$	206651.			0.284615
$222$		0			0
$223$	−676428.	−	1.17161e6i	−0.910876	−	1.57768i	−0.812830	−	0.582501i	$-0.802074\pi$
$223$	−676428.	−	1.17161e6i	−0.910876	−	1.57768i	−0.0980458	−	0.995182i	$-0.531259\pi$
$224$		0			0
$225$	−248098.	−	429718.i	−0.326713	−	0.565884i
$226$		0			0
$227$	944144.			1.21611			0.608056	−	0.793894i	$-0.291950\pi$
$227$	944144.			1.21611			0.608056	+	0.793894i	$0.291950\pi$
$228$		0			0
$229$	1.23700e6			1.55877			0.779383	−	0.626547i	$-0.215532\pi$
$229$	1.23700e6			1.55877			0.779383	+	0.626547i	$0.215532\pi$
$230$		0			0
$231$	43689.6	+	75672.7i	0.0538702	+	0.0933059i
$232$		0			0
$233$	495767.	+	858694.i	0.598258	+	1.03621i	0.993078	+	0.117454i	$0.0374733\pi$
$233$	495767.	+	858694.i	0.598258	+	1.03621i	−0.394821	+	0.918758i	$0.629193\pi$
$234$		0			0
$235$	−421396.			−0.497761
$236$		0			0
$237$	176505.	−	305715.i	0.204120	−	0.353546i
$238$		0			0
$239$	−256516.			−0.290482			−0.145241	−	0.989396i	$-0.546396\pi$
$239$	−256516.			−0.290482			−0.145241	+	0.989396i	$0.546396\pi$
$240$		0			0
$241$	−398215.	+	689728.i	−0.441646	+	0.764954i	−0.997812	−	0.0661177i	$-0.978939\pi$
$241$	−398215.	+	689728.i	−0.441646	+	0.764954i	0.556166	+	0.831072i	$0.312272\pi$
$242$		0			0
$243$	−451315.	+	781701.i	−0.490303	+	0.849229i
$244$		0			0
$245$	−191022.	−	330859.i	−0.203314	−	0.352150i
$246$		0			0
$247$	74965.1	+	134257.i	0.0781838	+	0.140022i
$248$		0			0
$249$	4350.64	+	7535.53i	0.00444687	+	0.00770221i
$250$		0			0
$251$	−292660.	+	506902.i	−0.293210	+	0.507855i	−0.974567	−	0.224097i	$-0.928057\pi$
$251$	−292660.	+	506902.i	−0.293210	+	0.507855i	0.681357	+	0.731951i	$0.261390\pi$
$252$		0			0
$253$	489306.	−	847503.i	0.480595	−	0.832415i
$254$		0			0
$255$	−352329.			−0.339311
$256$		0			0
$257$	−436841.	+	756630.i	−0.412563	+	0.714580i	−0.995169	−	0.0981747i	$-0.968700\pi$
$257$	−436841.	+	756630.i	−0.412563	+	0.714580i	0.582606	+	0.812754i	$0.302033\pi$
$258$		0			0
$259$	83762.5			0.0775890
$260$		0			0
$261$	105206.	+	182222.i	0.0955961	+	0.165577i
$262$		0			0
$263$	146951.	+	254527.i	0.131004	+	0.226905i	0.924064	−	0.382238i	$-0.124847\pi$
$263$	146951.	+	254527.i	0.131004	+	0.226905i	−0.793060	+	0.609144i	$0.791513\pi$
$264$		0			0
$265$	263995.			0.230930
$266$		0			0
$267$	−683586.			−0.586833
$268$		0			0
$269$	1.16327e6	+	2.01484e6i	0.980163	+	1.69769i	0.661722	+	0.749749i	$0.269826\pi$
$269$	1.16327e6	+	2.01484e6i	0.980163	+	1.69769i	0.318440	+	0.947943i	$0.396841\pi$
$270$		0			0
$271$	130705.	+	226387.i	0.108111	+	0.187253i	0.915005	−	0.403443i	$-0.132187\pi$
$271$	130705.	+	226387.i	0.108111	+	0.187253i	−0.806894	+	0.590696i	$0.798853\pi$
$272$		0			0
$273$	−26286.5			−0.0213465
$274$		0			0
$275$	404255.	−	700189.i	0.322347	−	0.558321i
$276$		0			0
$277$	−1.76335e6			−1.38083			−0.690415	−	0.723414i	$-0.742572\pi$
$277$	−1.76335e6			−1.38083			−0.690415	+	0.723414i	$0.742572\pi$
$278$		0			0
$279$	757749.	−	1.31246e6i	0.582794	−	1.00943i
$280$		0			0
$281$	−477850.	+	827661.i	−0.361016	+	0.625297i	−0.988128	−	0.153631i	$-0.950903\pi$
$281$	−477850.	+	827661.i	−0.361016	+	0.625297i	0.627113	+	0.778929i	$0.284237\pi$
$282$		0			0
$283$	172840.	+	299368.i	0.128286	+	0.222197i	0.923013	−	0.384770i	$-0.125719\pi$
$283$	172840.	+	299368.i	0.128286	+	0.222197i	−0.794727	+	0.606967i	$0.792386\pi$
$284$		0			0
$285$	−127811.	−	228901.i	−0.0932088	−	0.166930i
$286$		0			0
$287$	−9381.96	−	16250.0i	−0.00672340	−	0.0116453i
$288$		0			0
$289$	−1.52611e6	+	2.64330e6i	−1.07483	+	1.86167i
$290$		0			0
$291$	−350110.	+	606409.i	−0.242366	+	0.419791i
$292$		0			0
$293$	1.17269e6			0.798023			0.399011	−	0.916946i	$-0.369353\pi$
$293$	1.17269e6			0.798023			0.399011	+	0.916946i	$0.369353\pi$
$294$		0			0
$295$	−598527.	+	1.03668e6i	−0.400432	+	0.693569i
$296$		0			0
$297$	−949286.			−0.624462
$298$		0			0
$299$	147199.	+	254956.i	0.0952198	+	0.164925i
$300$		0			0
$301$	7678.60	+	13299.7i	0.00488501	+	0.00846109i
$302$		0			0
$303$	452969.			0.283440
$304$		0			0
$305$	−691791.			−0.425819
$306$		0			0
$307$	−160062.	−	277235.i	−0.0969264	−	0.167881i	0.813485	−	0.581586i	$-0.197568\pi$
$307$	−160062.	−	277235.i	−0.0969264	−	0.167881i	−0.910411	+	0.413705i	$0.864234\pi$
$308$		0			0
$309$	116448.	+	201695.i	0.0693805	+	0.120171i
$310$		0			0
$311$	1.45223e6			0.851400			0.425700	−	0.904864i	$-0.360028\pi$
$311$	1.45223e6			0.851400			0.425700	+	0.904864i	$0.360028\pi$
$312$		0			0
$313$	965219.	−	1.67181e6i	0.556884	−	0.964552i	−0.440870	−	0.897571i	$-0.645330\pi$
$313$	965219.	−	1.67181e6i	0.556884	−	0.964552i	0.997754	−	0.0669807i	$-0.0213366\pi$
$314$		0			0
$315$	−204709.			−0.116241
$316$		0			0
$317$	757768.	−	1.31249e6i	0.423534	−	0.733582i	−0.572749	−	0.819731i	$-0.694123\pi$
$317$	757768.	−	1.31249e6i	0.423534	−	0.733582i	0.996282	+	0.0861494i	$0.0274562\pi$
$318$		0			0
$319$	−171425.	+	296916.i	−0.0943184	+	0.163364i
$320$		0			0
$321$	−419233.	−	726132.i	−0.227087	−	0.393326i
$322$		0			0
$323$	−3.32732e6	−	47760.6i	−1.77455	−	0.0254720i
$324$		0			0
$325$	121613.	+	210640.i	0.0638662	+	0.110620i
$326$		0			0
$327$	−712136.	+	1.23346e6i	−0.368293	+	0.637902i
$328$		0			0
$329$	340186.	−	589219.i	0.173271	−	0.300115i
$330$		0			0
$331$	3.41462e6			1.71306			0.856531	−	0.516096i	$-0.172615\pi$
$331$	3.41462e6			1.71306			0.856531	+	0.516096i	$0.172615\pi$
$332$		0			0
$333$	−205052.	+	355161.i	−0.101334	+	0.175515i
$334$		0			0
$335$	868951.			0.423042
$336$		0			0
$337$	−1.59676e6	−	2.76567e6i	−0.765888	−	1.32656i	−0.939776	−	0.341791i	$-0.888967\pi$
$337$	−1.59676e6	−	2.76567e6i	−0.765888	−	1.32656i	0.173889	−	0.984765i	$-0.444367\pi$
$338$		0			0
$339$	−85115.9	−	147425.i	−0.0402264	−	0.0696742i
$340$		0			0
$341$	2.46938e6			1.15001
$342$		0			0
$343$	1.30118e6			0.597174
$344$		0			0
$345$	−250966.	−	434686.i	−0.113519	−	0.196620i
$346$		0			0
$347$	405681.	+	702659.i	0.180868	+	0.313272i	0.942176	−	0.335118i	$-0.108776\pi$
$347$	405681.	+	702659.i	0.180868	+	0.313272i	−0.761309	+	0.648390i	$0.775443\pi$
$348$		0			0
$349$	−491593.			−0.216044			−0.108022	−	0.994148i	$-0.534452\pi$
$349$	−491593.			−0.216044			−0.108022	+	0.994148i	$0.534452\pi$
$350$		0			0
$351$	142788.	−	247316.i	0.0618620	−	0.107148i
$352$		0			0
$353$	−1.82830e6			−0.780929			−0.390465	−	0.920618i	$-0.627686\pi$
$353$	−1.82830e6			−0.780929			−0.390465	+	0.920618i	$0.627686\pi$
$354$		0			0
$355$	−47407.5	+	82112.2i	−0.0199653	+	0.0345809i
$356$		0			0
$357$	284429.	−	492646.i	0.118115	−	0.204581i
$358$		0			0
$359$	−1.35760e6	−	2.35143e6i	−0.555950	−	0.962934i	−0.997829	−	0.0658593i	$-0.979021\pi$
$359$	−1.35760e6	−	2.35143e6i	−0.555950	−	0.962934i	0.441879	−	0.897075i	$-0.354312\pi$
$360$		0			0
$361$	−1.17599e6	−	2.17902e6i	−0.474937	−	0.880020i
$362$		0			0
$363$	183444.	+	317735.i	0.0730698	+	0.126561i
$364$		0			0
$365$	−154981.	+	268435.i	−0.0608900	+	0.105465i
$366$		0			0
$367$	1.32754e6	−	2.29937e6i	0.514498	−	0.891136i	−0.485361	−	0.874314i	$-0.661312\pi$
$367$	1.32754e6	−	2.29937e6i	0.514498	−	0.891136i	0.999858	−	0.0168222i	$-0.00535492\pi$
$368$		0			0
$369$	91868.9			0.0351239
$370$		0			0
$371$	−213119.	+	369132.i	−0.0803872	+	0.139235i
$372$		0			0
$373$	4.33938e6			1.61494			0.807468	−	0.589911i	$-0.200837\pi$
$373$	4.33938e6			1.61494			0.807468	+	0.589911i	$0.200837\pi$
$374$		0			0
$375$	−467667.	−	810023.i	−0.171735	−	0.297453i
$376$		0			0
$377$	−51570.1	−	89322.0i	−0.0186872	−	0.0323672i
$378$		0			0
$379$	3.27824e6			1.17231			0.586155	−	0.810199i	$-0.300641\pi$
$379$	3.27824e6			1.17231			0.586155	+	0.810199i	$0.300641\pi$
$380$		0			0
$381$	783719.			0.276597
$382$		0			0
$383$	1.41742e6	+	2.45505e6i	0.493745	+	0.855191i	0.999974	−	0.00720770i	$-0.00229430\pi$
$383$	1.41742e6	+	2.45505e6i	0.493745	+	0.855191i	−0.506229	+	0.862399i	$0.668961\pi$
$384$		0			0
$385$	−166778.	−	288867.i	−0.0573437	−	0.0993223i
$386$		0			0
$387$	−75189.4			−0.0255199
$388$		0			0
$389$	1.39366e6	−	2.41389e6i	0.466963	−	0.808804i	−0.532324	−	0.846540i	$-0.678681\pi$
$389$	1.39366e6	−	2.41389e6i	0.466963	−	0.808804i	0.999288	+	0.0377363i	$0.0120147\pi$
$390$		0			0
$391$	−6.37098e6			−2.10748
$392$		0			0
$393$	−183259.	+	317414.i	−0.0598528	+	0.103668i
$394$		0			0
$395$	−673777.	+	1.16702e6i	−0.217282	+	0.376343i
$396$		0			0
$397$	−1.84860e6	−	3.20187e6i	−0.588663	−	1.01959i	−0.994408	−	0.105607i	$-0.966321\pi$
$397$	−1.84860e6	−	3.20187e6i	−0.588663	−	1.01959i	0.405745	−	0.913986i	$-0.367012\pi$
$398$		0			0
$399$	423242.	+	6075.26i	0.133093	+	0.00191044i
$400$		0			0
$401$	−2.13618e6	−	3.69998e6i	−0.663404	−	1.14905i	−0.979715	−	0.200394i	$-0.935778\pi$
$401$	−2.13618e6	−	3.69998e6i	−0.663404	−	1.14905i	0.316312	−	0.948655i	$-0.397555\pi$
$402$		0			0
$403$	−371434.	+	643343.i	−0.113925	+	0.197324i
$404$		0			0
$405$	367398.	−	636352.i	0.111301	−	0.192779i
$406$		0			0
$407$	−668230.			−0.199959
$408$		0			0
$409$	−2.89320e6	+	5.01117e6i	−0.855205	+	1.48126i	0.0212492	+	0.999774i	$0.493236\pi$
$409$	−2.89320e6	+	5.01117e6i	−0.855205	+	1.48126i	−0.876454	+	0.481485i	$0.840098\pi$
$410$		0			0
$411$	2.11267e6			0.616917
$412$		0			0
$413$	−966362.	−	1.67379e6i	−0.278782	−	0.482865i
$414$		0			0
$415$	−16607.8	−	28765.6i	−0.00473361	−	0.00819885i
$416$		0			0
$417$	1.39806e6			0.393718
$418$		0			0
$419$	3.36575e6			0.936585			0.468293	−	0.883573i	$-0.344869\pi$
$419$	3.36575e6			0.936585			0.468293	+	0.883573i	$0.344869\pi$
$420$		0			0
$421$	2.36298e6	+	4.09279e6i	0.649761	+	1.12542i	0.983180	+	0.182641i	$0.0584646\pi$
$421$	2.36298e6	+	4.09279e6i	0.649761	+	1.12542i	−0.333418	+	0.942779i	$0.608202\pi$
$422$		0			0
$423$	1.66556e6	+	2.88484e6i	0.452595	+	0.783918i
$424$		0			0
$425$	−5.26357e6			−1.41354
$426$		0			0
$427$	558472.	−	967301.i	0.148228	−	0.256739i
$428$		0			0
$429$	209705.			0.0550131
$430$		0			0
$431$	212460.	−	367991.i	0.0550914	−	0.0954211i	−0.837164	−	0.546951i	$-0.815788\pi$
$431$	212460.	−	367991.i	0.0550914	−	0.0954211i	0.892256	+	0.451530i	$0.149122\pi$
$432$		0			0
$433$	−3.10444e6	+	5.37705e6i	−0.795726	+	1.37824i	0.126651	+	0.991947i	$0.459577\pi$
$433$	−3.10444e6	+	5.37705e6i	−0.795726	+	1.37824i	−0.922377	+	0.386291i	$0.873756\pi$
$434$		0			0
$435$	87924.1	+	152289.i	0.0222784	+	0.0385874i
$436$		0			0
$437$	−2.31114e6	−	4.13910e6i	−0.578926	−	1.03682i
$438$		0			0
$439$	3.48593e6	+	6.03781e6i	0.863291	+	1.49526i	0.868734	+	0.495279i	$0.164934\pi$
$439$	3.48593e6	+	6.03781e6i	0.863291	+	1.49526i	−0.00544290	+	0.999985i	$0.501733\pi$
$440$		0			0
$441$	−1.51002e6	+	2.61543e6i	−0.369732	+	0.640394i
$442$		0			0
$443$	−1.32950e6	+	2.30277e6i	−0.321870	+	0.557495i	−0.980874	−	0.194645i	$-0.937645\pi$
$443$	−1.32950e6	+	2.30277e6i	−0.321870	+	0.557495i	0.659004	+	0.752139i	$0.270978\pi$
$444$		0			0
$445$	2.60947e6			0.624672
$446$		0			0
$447$	238181.	−	412541.i	0.0563817	−	0.0976559i
$448$		0			0
$449$	−636809.			−0.149071			−0.0745355	−	0.997218i	$-0.523747\pi$
$449$	−636809.			−0.149071			−0.0745355	+	0.997218i	$0.523747\pi$
$450$		0			0
$451$	74846.3	+	129638.i	0.0173272	+	0.0300116i
$452$		0			0
$453$	−877152.	−	1.51927e6i	−0.200830	−	0.347848i
$454$		0			0
$455$	100344.			0.0227229
$456$		0			0
$457$	−1.62218e6			−0.363337			−0.181668	−	0.983360i	$-0.558150\pi$
$457$	−1.62218e6			−0.363337			−0.181668	+	0.983360i	$0.558150\pi$
$458$		0			0
$459$	3.09003e6	+	5.35209e6i	0.684591	+	1.18575i
$460$		0			0
$461$	1.88236e6	+	3.26034e6i	0.412525	+	0.714514i	0.995165	−	0.0982159i	$-0.0313136\pi$
$461$	1.88236e6	+	3.26034e6i	0.412525	+	0.714514i	−0.582640	+	0.812730i	$0.697980\pi$
$462$		0			0
$463$	8.56420e6			1.85667			0.928334	−	0.371748i	$-0.121241\pi$
$463$	8.56420e6			1.85667			0.928334	+	0.371748i	$0.121241\pi$
$464$		0			0
$465$	633274.	−	1.09686e6i	0.135819	−	0.235245i
$466$		0			0
$467$	−8.68072e6			−1.84189			−0.920944	−	0.389694i	$-0.872581\pi$
$467$	−8.68072e6			−1.84189			−0.920944	+	0.389694i	$0.872581\pi$
$468$		0			0
$469$	−701489.	+	1.21502e6i	−0.147261	+	0.255064i
$470$		0			0
$471$	−1.05596e6	+	1.82898e6i	−0.219329	+	0.379889i
$472$		0			0
$473$	−61257.4	−	106101.i	−0.0125894	−	0.0218055i
$474$		0			0
$475$	−1.90942e6	−	3.41964e6i	−0.388300	−	0.695418i
$476$		0			0
$477$	−1.04344e6	−	1.80729e6i	−0.209976	−	0.363690i
$478$		0			0
$479$	−4.82656e6	+	8.35985e6i	−0.961167	+	1.66479i	−0.241589	+	0.970379i	$0.577669\pi$
$479$	−4.82656e6	+	8.35985e6i	−0.961167	+	1.66479i	−0.719578	+	0.694412i	$0.755665\pi$
$480$		0			0
$481$	100513.	−	174093.i	0.0198088	−	0.0343098i
$482$		0			0
$483$	810403.			0.158064
$484$		0			0
$485$	1.33649e6	−	2.31486e6i	0.257994	−	0.446859i
$486$		0			0
$487$	−3.07551e6			−0.587617			−0.293809	−	0.955864i	$-0.594923\pi$
$487$	−3.07551e6			−0.587617			−0.293809	+	0.955864i	$0.594923\pi$
$488$		0			0
$489$	669203.	+	1.15909e6i	0.126557	+	0.219203i
$490$		0			0
$491$	−3.46681e6	−	6.00470e6i	−0.648973	−	1.12405i	−0.983368	−	0.181622i	$-0.941865\pi$
$491$	−3.46681e6	−	6.00470e6i	−0.648973	−	1.12405i	0.334395	−	0.942433i	$-0.391468\pi$
$492$		0			0
$493$	2.23202e6			0.413601
$494$		0			0
$495$	1.63310e6			0.299571
$496$		0			0
$497$	−76542.5	−	132576.i	−0.0138999	−	0.0240753i
$498$		0			0
$499$	272382.	+	471779.i	0.0489696	+	0.0848178i	0.889471	−	0.456991i	$-0.151073\pi$
$499$	272382.	+	471779.i	0.0489696	+	0.0848178i	−0.840502	+	0.541809i	$0.817740\pi$
$500$		0			0
$501$	2.00597e6			0.357052
$502$		0			0
$503$	3.01810e6	−	5.22751e6i	0.531880	−	0.921244i	−0.467427	−	0.884032i	$-0.654819\pi$
$503$	3.01810e6	−	5.22751e6i	0.531880	−	0.921244i	0.999307	−	0.0372121i	$-0.0118477\pi$
$504$		0			0
$505$	−1.72913e6			−0.301716
$506$		0			0
$507$	1.19492e6	−	2.06966e6i	0.206452	−	0.357585i
$508$		0			0
$509$	292272.	−	506231.i	0.0500027	−	0.0866072i	−0.839941	−	0.542678i	$-0.817410\pi$
$509$	292272.	−	506231.i	0.0500027	−	0.0866072i	0.889943	+	0.456071i	$0.150744\pi$
$510$		0			0
$511$	−250227.	−	433406.i	−0.0423918	−	0.0734247i
$512$		0			0
$513$	−2.35620e6	+	3.94906e6i	−0.395293	+	0.662522i
$514$		0			0
$515$	−444522.	−	769934.i	−0.0738542	−	0.127919i
$516$		0			0
$517$	−2.71389e6	+	4.70060e6i	−0.446546	+	0.773441i
$518$		0			0
$519$	2.17129e6	−	3.76078e6i	0.353833	−	0.612857i
$520$		0			0
$521$	5.06982e6			0.818272			0.409136	−	0.912473i	$-0.365830\pi$
$521$	5.06982e6			0.818272			0.409136	+	0.912473i	$0.365830\pi$
$522$		0			0
$523$	2.09028e6	−	3.62047e6i	0.334157	−	0.578777i	−0.649166	−	0.760647i	$-0.724882\pi$
$523$	2.09028e6	−	3.62047e6i	0.334157	−	0.578777i	0.983323	+	0.181870i	$0.0582151\pi$
$524$		0			0
$525$	669539.			0.106017
$526$		0			0
$527$	−8.03809e6	−	1.39224e7i	−1.26074	−	2.18367i
$528$		0			0
$529$	−1.31992e6	−	2.28617e6i	−0.205073	−	0.355196i
$530$		0			0
$531$	9.46269e6			1.45639
$532$		0			0
$533$	−45032.4			−0.00686605
$534$		0			0
$535$	1.60035e6	+	2.77188e6i	0.241730	+	0.418688i
$536$		0			0
$537$	−878238.	−	1.52115e6i	−0.131425	−	0.227634i
$538$		0			0
$539$	−4.92091e6			−0.729580
$540$		0			0
$541$	2.36120e6	−	4.08971e6i	0.346848	−	0.600758i	−0.638840	−	0.769340i	$-0.720585\pi$
$541$	2.36120e6	−	4.08971e6i	0.346848	−	0.600758i	0.985688	+	0.168582i	$0.0539187\pi$
$542$		0			0
$543$	−998565.			−0.145337
$544$		0			0
$545$	2.71845e6	−	4.70850e6i	0.392040	−	0.679034i
$546$		0			0
$547$	−2.59488e6	+	4.49447e6i	−0.370808	+	0.642259i	−0.989690	−	0.143225i	$-0.954253\pi$
$547$	−2.59488e6	+	4.49447e6i	−0.370808	+	0.642259i	0.618882	+	0.785484i	$0.287586\pi$
$548$		0			0
$549$	2.73430e6	+	4.73594e6i	0.387182	+	0.670619i
$550$		0			0
$551$	809691.	+	1.45010e6i	0.113616	+	0.203479i
$552$		0			0
$553$	−1.08786e6	−	1.88422e6i	−0.151272	−	0.262011i
$554$		0			0
$555$	−171368.	+	296819.i	−0.0236156	+	0.0409033i
$556$		0			0
$557$	3.55878e6	−	6.16399e6i	0.486030	−	0.841829i	−0.513841	−	0.857886i	$-0.671778\pi$
$557$	3.55878e6	−	6.16399e6i	0.486030	−	0.841829i	0.999871	+	0.0160563i	$0.00511109\pi$
$558$		0			0
$559$	36856.4			0.00498866
$560$		0			0
$561$	−2.26909e6	+	3.93017e6i	−0.304399	+	0.527235i
$562$		0			0
$563$	6.55995e6			0.872227			0.436114	−	0.899892i	$-0.356355\pi$
$563$	6.55995e6			0.872227			0.436114	+	0.899892i	$0.356355\pi$
$564$		0			0
$565$	324915.	+	562769.i	0.0428202	+	0.0741668i
$566$		0			0
$567$	593188.	+	1.02743e6i	0.0774881	+	0.134213i
$568$		0			0
$569$	1.10777e7			1.43440			0.717200	−	0.696868i	$-0.245424\pi$
$569$	1.10777e7			1.43440			0.717200	+	0.696868i	$0.245424\pi$
$570$		0			0
$571$	−5.64684e6			−0.724795			−0.362398	−	0.932024i	$-0.618042\pi$
$571$	−5.64684e6			−0.724795			−0.362398	+	0.932024i	$0.618042\pi$
$572$		0			0
$573$	398009.	+	689372.i	0.0506415	+	0.0877136i
$574$		0			0
$575$	−3.74928e6	−	6.49394e6i	−0.472909	−	0.819103i
$576$		0			0
$577$	2.99850e6			0.374942			0.187471	−	0.982270i	$-0.439971\pi$
$577$	2.99850e6			0.374942			0.187471	+	0.982270i	$0.439971\pi$
$578$		0			0
$579$	253588.	−	439227.i	0.0314364	−	0.0544494i
$580$		0			0
$581$	53628.9			0.00659110
$582$		0			0
$583$	1.70019e6	−	2.94482e6i	0.207170	−	0.358829i
$584$		0			0
$585$	−245645.	+	425469.i	−0.0296768	+	0.0514018i
$586$		0			0
$587$	−7.21676e6	−	1.24998e7i	−0.864465	−	1.49730i	−0.867578	−	0.497301i	$-0.834324\pi$
$587$	−7.21676e6	−	1.24998e7i	−0.864465	−	1.49730i	0.00311341	−	0.999995i	$-0.499009\pi$
$588$		0			0
$589$	6.12919e6	−	1.02727e7i	0.727972	−	1.22010i
$590$		0			0
$591$	−1.92216e6	−	3.32927e6i	−0.226371	−	0.392086i
$592$		0			0
$593$	−3.75784e6	+	6.50878e6i	−0.438836	+	0.760086i	−0.997600	−	0.0692407i	$-0.977942\pi$
$593$	−3.75784e6	+	6.50878e6i	−0.438836	+	0.760086i	0.558764	+	0.829327i	$0.311276\pi$
$594$		0			0
$595$	−1.08576e6	+	1.88059e6i	−0.125731	+	0.217772i
$596$		0			0
$597$	−1.10376e6			−0.126748
$598$		0			0
$599$	−8.09527e6	+	1.40214e7i	−0.921858	+	1.59671i	−0.125321	+	0.992116i	$0.539996\pi$
$599$	−8.09527e6	+	1.40214e7i	−0.921858	+	1.59671i	−0.796538	+	0.604589i	$0.793337\pi$
$600$		0			0
$601$	1.99094e6			0.224839			0.112419	−	0.993661i	$-0.464140\pi$
$601$	1.99094e6			0.224839			0.112419	+	0.993661i	$0.464140\pi$
$602$		0			0
$603$	−3.43452e6	−	5.94876e6i	−0.384656	−	0.666244i
$604$		0			0
$605$	−700267.	−	1.21290e6i	−0.0777814	−	0.134721i
$606$		0			0
$607$	1.47599e6			0.162596			0.0812982	−	0.996690i	$-0.474093\pi$
$607$	1.47599e6			0.162596			0.0812982	+	0.996690i	$0.474093\pi$
$608$		0			0
$609$	−283919.			−0.0310206
$610$		0			0
$611$	−816427.	−	1.41409e6i	−0.0884737	−	0.153241i
$612$		0			0
$613$	6.12920e6	+	1.06161e7i	0.658798	+	1.14107i	0.980927	+	0.194376i	$0.0622683\pi$
$613$	6.12920e6	+	1.06161e7i	0.658798	+	1.14107i	−0.322129	+	0.946696i	$0.604398\pi$
$614$		0			0
$615$	76777.6			0.00818553
$616$		0			0
$617$	−3.89936e6	+	6.75389e6i	−0.412364	+	0.714235i	−0.995148	−	0.0983924i	$-0.968630\pi$
$617$	−3.89936e6	+	6.75389e6i	−0.412364	+	0.714235i	0.582784	+	0.812627i	$0.301963\pi$
$618$		0			0
$619$	8.39100e6			0.880212			0.440106	−	0.897946i	$-0.354941\pi$
$619$	8.39100e6			0.880212			0.440106	+	0.897946i	$0.354941\pi$
$620$		0			0
$621$	−4.40210e6	+	7.62466e6i	−0.458069	+	0.793398i
$622$		0			0
$623$	−2.10658e6	+	3.64871e6i	−0.217449	+	0.376633i
$624$		0			0
$625$	−2.10384e6	−	3.64395e6i	−0.215433	−	0.373141i
$626$		0			0
$627$	−3.37649e6	−	48466.5i	−0.343002	−	0.00492348i
$628$		0			0
$629$	2.17516e6	+	3.76749e6i	0.219212	+	0.379687i
$630$		0			0
$631$	725688.	−	1.25693e6i	0.0725565	−	0.125672i	−0.827465	−	0.561518i	$-0.810218\pi$
$631$	725688.	−	1.25693e6i	0.0725565	−	0.125672i	0.900021	+	0.435846i	$0.143551\pi$
$632$		0			0
$633$	−2.37978e6	+	4.12191e6i	−0.236063	+	0.408873i
$634$		0			0
$635$	−2.99171e6			−0.294432
$636$		0			0
$637$	740184.	−	1.28204e6i	0.0722755	−	0.125185i
$638$		0			0
$639$	749510.			0.0726148
$640$		0			0
$641$	−4.20230e6	−	7.27859e6i	−0.403963	−	0.699684i	0.590237	−	0.807230i	$-0.299034\pi$
$641$	−4.20230e6	−	7.27859e6i	−0.403963	−	0.699684i	−0.994200	+	0.107545i	$0.965701\pi$
$642$		0			0
$643$	2.07760e6	+	3.59850e6i	0.198168	+	0.343237i	0.947934	−	0.318465i	$-0.103167\pi$
$643$	2.07760e6	+	3.59850e6i	0.198168	+	0.343237i	−0.749766	+	0.661703i	$0.769834\pi$
$644$		0			0
$645$	−62838.1			−0.00594736
$646$		0			0
$647$	−5.48842e6			−0.515450			−0.257725	−	0.966218i	$-0.582973\pi$
$647$	−5.48842e6			−0.515450			−0.257725	+	0.966218i	$0.582973\pi$
$648$		0			0
$649$	7.70933e6	+	1.33529e7i	0.718464	+	1.24442i
$650$		0			0
$651$	1.02246e6	+	1.77096e6i	0.0945573	+	0.163778i
$652$		0			0
$653$	6.39658e6			0.587036			0.293518	−	0.955954i	$-0.405174\pi$
$653$	6.39658e6			0.587036			0.293518	+	0.955954i	$0.405174\pi$
$654$		0			0
$655$	699560.	−	1.21167e6i	0.0637120	−	0.110353i
$656$		0			0
$657$	2.45024e6			0.221460
$658$		0			0
$659$	891088.	−	1.54341e6i	0.0799295	−	0.138442i	−0.823290	−	0.567621i	$-0.807864\pi$
$659$	891088.	−	1.54341e6i	0.0799295	−	0.138442i	0.903219	+	0.429179i	$0.141197\pi$
$660$		0			0
$661$	−8.15028e6	+	1.41167e7i	−0.725553	+	1.25669i	0.233193	+	0.972430i	$0.425083\pi$
$661$	−8.15028e6	+	1.41167e7i	−0.725553	+	1.25669i	−0.958746	+	0.284264i	$0.908251\pi$
$662$		0			0
$663$	−682614.	−	1.18232e6i	−0.0603103	−	0.104461i
$664$		0			0
$665$	−1.61565e6	−	23191.2i	−0.141675	−	0.00203362i
$666$		0			0
$667$	1.58988e6	+	2.75376e6i	0.138373	+	0.239669i
$668$		0			0
$669$	−4.46877e6	+	7.74014e6i	−0.386032	+	0.668627i
$670$		0			0
$671$	−4.45531e6	+	7.71682e6i	−0.382007	+	0.661656i
$672$		0			0
$673$	3.81795e6			0.324932			0.162466	−	0.986714i	$-0.448055\pi$
$673$	3.81795e6			0.324932			0.162466	+	0.986714i	$0.448055\pi$
$674$		0			0
$675$	−3.63692e6	+	6.29934e6i	−0.307238	+	0.532152i
$676$		0			0
$677$	1.93449e6			0.162216			0.0811081	−	0.996705i	$-0.474154\pi$
$677$	1.93449e6			0.162216			0.0811081	+	0.996705i	$0.474154\pi$
$678$		0			0
$679$	2.15785e6	+	3.73750e6i	0.179616	+	0.311105i
$680$		0			0
$681$	−3.11871e6	−	5.40177e6i	−0.257696	−	0.446342i
$682$		0			0
$683$	732465.			0.0600808			0.0300404	−	0.999549i	$-0.490436\pi$
$683$	732465.			0.0600808			0.0300404	+	0.999549i	$0.490436\pi$
$684$		0			0
$685$	−8.06474e6			−0.656696
$686$		0			0
$687$	−4.08608e6	−	7.07730e6i	−0.330305	−	0.572105i
$688$		0			0
$689$	511473.	+	885897.i	0.0410464	+	0.0710944i
$690$		0			0
$691$	−5.91673e6			−0.471397			−0.235698	−	0.971826i	$-0.575738\pi$
$691$	−5.91673e6			−0.471397			−0.235698	+	0.971826i	$0.575738\pi$
$692$		0			0
$693$	−1.31837e6	+	2.28349e6i	−0.104281	+	0.180620i
$694$		0			0
$695$	−5.33685e6			−0.419105
$696$		0			0
$697$	487266.	−	843969.i	0.0379913	−	0.0658028i
$698$		0			0
$699$	3.27525e6	−	5.67291e6i	0.253543	−	0.439150i
$700$		0			0
$701$	8.07558e6	+	1.39873e7i	0.620696	+	1.07508i	0.989356	+	0.145513i	$0.0464831\pi$
$701$	8.07558e6	+	1.39873e7i	0.620696	+	1.07508i	−0.368661	+	0.929564i	$0.620184\pi$
$702$		0			0
$703$	−1.65860e6	+	2.77986e6i	−0.126577	+	0.212146i
$704$		0			0
$705$	1.39196e6	+	2.41095e6i	0.105476	+	0.182690i
$706$		0			0
$707$	1.39590e6	−	2.41777e6i	0.105028	−	0.181914i
$708$		0			0
$709$	3.50009e6	−	6.06234e6i	0.261495	−	0.452923i	−0.705144	−	0.709064i	$-0.749118\pi$
$709$	3.50009e6	−	6.06234e6i	0.261495	−	0.452923i	0.966640	+	0.256141i	$0.0824510\pi$
$710$		0			0
$711$	1.06524e7			0.790265
$712$		0			0
$713$	1.14512e7	−	1.98340e7i	0.843579	−	1.46112i
$714$		0			0
$715$	−800514.			−0.0585604
$716$		0			0
$717$	847328.	+	1.46762e6i	0.0615536	+	0.106614i
$718$		0			0
$719$	6.55155e6	+	1.13476e7i	0.472630	+	0.818620i	0.999509	−	0.0313203i	$-0.00997121\pi$
$719$	6.55155e6	+	1.13476e7i	0.472630	+	0.818620i	−0.526879	+	0.849940i	$0.676638\pi$
$720$		0			0
$721$	1.43542e6			0.102835
$722$		0			0
$723$	5.26156e6			0.374342
$724$		0			0
$725$	1.31353e6	+	2.27510e6i	0.0928101	+	0.160752i
$726$		0			0
$727$	−7.15352e6	−	1.23903e7i	−0.501977	−	0.869449i	−0.999997	−	0.00228403i	$-0.999273\pi$
$727$	−7.15352e6	−	1.23903e7i	−0.501977	−	0.869449i	0.498021	−	0.867165i	$-0.334060\pi$
$728$		0			0
$729$	−1.11705e6			−0.0778490
$730$		0			0
$731$	−398799.	+	690741.i	−0.0276033	+	0.0478103i
$732$		0			0
$733$	−2.46017e7			−1.69124			−0.845620	−	0.533785i	$-0.820769\pi$
$733$	−2.46017e7			−1.69124			−0.845620	+	0.533785i	$0.820769\pi$
$734$		0			0
$735$	−1.26197e6	+	2.18580e6i	−0.0861651	+	0.149242i
$736$		0			0
$737$	5.59626e6	−	9.69300e6i	0.379515	−	0.657339i
$738$		0			0
$739$	3.19096e6	+	5.52691e6i	0.214937	+	0.372281i	0.953253	−	0.302173i	$-0.0977122\pi$
$739$	3.19096e6	+	5.52691e6i	0.214937	+	0.372281i	−0.738316	+	0.674455i	$0.764379\pi$
$740$		0			0
$741$	520506.	−	872382.i	0.0348241	−	0.0583661i
$742$		0			0
$743$	−401987.	−	696262.i	−0.0267141	−	0.0462701i	0.852359	−	0.522957i	$-0.175171\pi$
$743$	−401987.	−	696262.i	−0.0267141	−	0.0462701i	−0.879073	+	0.476686i	$0.841838\pi$
$744$		0			0
$745$	−909214.	+	1.57480e6i	−0.0600172	+	0.103953i
$746$		0			0
$747$	−131284.	+	227391.i	−0.00860819	+	0.0149098i
$748$		0			0
$749$	−5.16774e6			−0.336586
$750$		0			0
$751$	1.22115e7	−	2.11510e7i	0.790078	−	1.36846i	−0.135839	−	0.990731i	$-0.543373\pi$
$751$	1.22115e7	−	2.11510e7i	0.790078	−	1.36846i	0.925918	−	0.377725i	$-0.123294\pi$
$752$		0			0
$753$	3.86688e6			0.248527
$754$		0			0
$755$	3.34837e6	+	5.79956e6i	0.213780	+	0.370277i
$756$		0			0
$757$	−1.52156e7	−	2.63542e7i	−0.965050	−	1.67152i	−0.709480	−	0.704726i	$-0.751070\pi$
$757$	−1.52156e7	−	2.63542e7i	−0.965050	−	1.67152i	−0.255570	−	0.966790i	$-0.582263\pi$
$758$		0			0
$759$	−6.46514e6			−0.407355
$760$		0			0
$761$	6.72001e6			0.420638			0.210319	−	0.977633i	$-0.432550\pi$
$761$	6.72001e6			0.420638			0.210319	+	0.977633i	$0.432550\pi$
$762$		0			0
$763$	4.38913e6	+	7.60219e6i	0.272940	+	0.472745i
$764$		0			0
$765$	−5.31592e6	−	9.20744e6i	−0.328416	−	0.568834i
$766$		0			0
$767$	−4.63843e6			−0.284697
$768$		0			0
$769$	4.61439e6	−	7.99236e6i	0.281384	−	0.487371i	−0.690342	−	0.723483i	$-0.742540\pi$
$769$	4.61439e6	−	7.99236e6i	0.281384	−	0.487371i	0.971726	+	0.236112i	$0.0758734\pi$
$770$		0			0
$771$	5.77192e6			0.349691
$772$		0			0
$773$	−2207.17	+	3822.92i	−0.000132858	+	0.000230116i	−0.866092	−	0.499885i	$-0.833376\pi$
$773$	−2207.17	+	3822.92i	−0.000132858	+	0.000230116i	0.865959	+	0.500115i	$0.166709\pi$
$774$		0			0
$775$	9.46072e6	−	1.63864e7i	0.565809	−	0.980010i
$776$		0			0
$777$	−276686.	−	479234.i	−0.0164412	−	0.0284770i
$778$		0			0
$779$	725071.	+	10407.7i	0.0428092	+	0.000614487i
$780$		0			0
$781$	610632.	+	1.05765e6i	0.0358222	+	0.0620458i
$782$		0			0
$783$	1.54224e6	−	2.67124e6i	0.0898975	−	0.155707i
$784$		0			0
$785$	4.03095e6	−	6.98181e6i	0.233471	−	0.404384i
$786$		0			0
$787$	−1.44974e7			−0.834360			−0.417180	−	0.908824i	$-0.636982\pi$
$787$	−1.44974e7			−0.834360			−0.417180	+	0.908824i	$0.636982\pi$
$788$		0			0
$789$	970823.	−	1.68152e6i	0.0555198	−	0.0961631i
$790$		0			0
$791$	−1.04919e6			−0.0596231
$792$		0			0
$793$	−1.34030e6	−	2.32147e6i	−0.0756867	−	0.131093i
$794$		0			0
$795$	−872032.	−	1.51040e6i	−0.0489345	−	0.0847570i
$796$		0			0
$797$	−1.24082e7			−0.691929			−0.345964	−	0.938248i	$-0.612448\pi$
$797$	−1.24082e7			−0.691929			−0.345964	+	0.938248i	$0.612448\pi$
$798$		0			0
$799$	3.53361e7			1.95818
$800$		0			0
$801$	−1.03139e7	−	1.78642e7i	−0.567991	−	0.983790i
$802$		0			0
$803$	1.99623e6	+	3.45757e6i	0.109250	+	0.189227i
$804$		0			0
$805$	−3.09357e6			−0.168256
$806$		0			0
$807$	7.68504e6	−	1.33109e7i	0.415396	−	0.719487i
$808$		0			0
$809$	2.16700e7			1.16409			0.582047	−	0.813155i	$-0.302252\pi$
$809$	2.16700e7			1.16409			0.582047	+	0.813155i	$0.302252\pi$
$810$		0			0
$811$	2.06018e6	−	3.56833e6i	0.109990	−	0.190508i	−0.805776	−	0.592220i	$-0.798252\pi$
$811$	2.06018e6	−	3.56833e6i	0.109990	−	0.190508i	0.915766	+	0.401712i	$0.131585\pi$
$812$		0			0
$813$	863493.	−	1.49561e6i	0.0458176	−	0.0793584i
$814$		0			0
$815$	−2.55457e6	−	4.42464e6i	−0.134717	−	0.233337i
$816$		0			0
$817$	−593429.	−	8518.14i	−0.0311038	−	0.000446467i
$818$		0			0
$819$	−396610.	−	686948.i	−0.0206611	−	0.0357861i
$820$		0			0
$821$	1.20488e7	−	2.08691e7i	0.623858	−	1.08055i	−0.364902	−	0.931046i	$-0.618898\pi$
$821$	1.20488e7	−	2.08691e7i	0.623858	−	1.08055i	0.988760	−	0.149509i	$-0.0477692\pi$
$822$		0			0
$823$	−1.56576e7	+	2.71197e7i	−0.805796	+	1.39568i	0.109957	+	0.993936i	$0.464929\pi$
$823$	−1.56576e7	+	2.71197e7i	−0.805796	+	1.39568i	−0.915753	+	0.401743i	$0.868405\pi$
$824$		0			0
$825$	−5.34136e6			−0.273223
$826$		0			0
$827$	−6.78298e6	+	1.17485e7i	−0.344871	+	0.597334i	−0.985330	−	0.170658i	$-0.945411\pi$
$827$	−6.78298e6	+	1.17485e7i	−0.344871	+	0.597334i	0.640459	+	0.767992i	$0.278744\pi$
$828$		0			0
$829$	−2.34177e7			−1.18347			−0.591737	−	0.806131i	$-0.701557\pi$
$829$	−2.34177e7			−1.18347			−0.591737	+	0.806131i	$0.701557\pi$
$830$		0			0
$831$	5.82474e6	+	1.00887e7i	0.292600	+	0.506798i
$832$		0			0
$833$	1.60181e7	+	2.77442e7i	0.799831	+	1.38535i
$834$		0			0
$835$	−7.65746e6			−0.380074
$836$		0			0
$837$	−2.22160e7			−1.09611
$838$		0			0
$839$	1.43756e7	+	2.48992e7i	0.705051	+	1.22118i	0.966673	+	0.256014i	$0.0824093\pi$
$839$	1.43756e7	+	2.48992e7i	0.705051	+	1.22118i	−0.261622	+	0.965170i	$0.584257\pi$
$840$		0			0
$841$	9.69857e6	+	1.67984e7i	0.472844	+	0.818990i
$842$		0			0
$843$	6.31377e6			0.305999
$844$		0			0
$845$	−4.56139e6	+	7.90056e6i	−0.219764	+	0.380642i
$846$		0			0
$847$	2.26126e6			0.108303
$848$		0			0
$849$	1.14186e6	−	1.97775e6i	0.0543679	−	0.0941679i
$850$		0			0
$851$	−3.09876e6	+	5.36722e6i	−0.146678	+	0.254054i
$852$		0			0
$853$	−8.39592e6	−	1.45422e7i	−0.395090	−	0.684315i	0.598023	−	0.801479i	$-0.295953\pi$
$853$	−8.39592e6	−	1.45422e7i	−0.395090	−	0.684315i	−0.993113	+	0.117164i	$0.962620\pi$
$854$		0			0
$855$	4.05348e6	−	6.79375e6i	0.189633	−	0.317829i
$856$		0			0
$857$	−5.57351e6	−	9.65361e6i	−0.259225	−	0.448991i	0.706810	−	0.707404i	$-0.250134\pi$
$857$	−5.57351e6	−	9.65361e6i	−0.259225	−	0.448991i	−0.966035	+	0.258413i	$0.916800\pi$
$858$		0			0
$859$	1.21896e7	−	2.11131e7i	0.563648	−	0.976266i	−0.433526	−	0.901141i	$-0.642731\pi$
$859$	1.21896e7	−	2.11131e7i	0.563648	−	0.976266i	0.997174	−	0.0751256i	$-0.0239358\pi$
$860$		0			0
$861$	−61981.3	+	107355.i	−0.00284940	+	0.00493530i
$862$		0			0
$863$	1.13867e7			0.520440			0.260220	−	0.965549i	$-0.416205\pi$
$863$	1.13867e7			0.520440			0.260220	+	0.965549i	$0.416205\pi$
$864$		0			0
$865$	−8.28850e6	+	1.43561e7i	−0.376648	+	0.652374i
$866$		0			0
$867$	2.01643e7			0.911036
$868$		0			0
$869$	8.67858e6	+	1.50317e7i	0.389851	+	0.675242i
$870$		0			0
$871$	1.68354e6	+	2.91597e6i	0.0751929	+	0.130238i
$872$		0			0
$873$	−2.11298e7			−0.938338
$874$		0			0
$875$	−5.76477e6			−0.254543
$876$		0			0
$877$	−1.95522e7	−	3.38654e7i	−0.858414	−	1.48682i	−0.873441	−	0.486930i	$-0.838117\pi$
$877$	−1.95522e7	−	3.38654e7i	−0.858414	−	1.48682i	0.0150273	−	0.999887i	$-0.495216\pi$
$878$		0			0
$879$	−3.87366e6	−	6.70938e6i	−0.169102	−	0.292894i
$880$		0			0
$881$	1.09362e7			0.474708			0.237354	−	0.971423i	$-0.423720\pi$
$881$	1.09362e7			0.474708			0.237354	+	0.971423i	$0.423720\pi$
$882$		0			0
$883$	−2.14771e7	+	3.71994e7i	−0.926986	+	1.60559i	−0.138651	+	0.990341i	$0.544277\pi$
$883$	−2.14771e7	+	3.71994e7i	−0.926986	+	1.60559i	−0.788335	+	0.615246i	$0.789057\pi$
$884$		0			0
$885$	7.90826e6			0.339409
$886$		0			0
$887$	−1.08256e7	+	1.87505e7i	−0.462001	+	0.800209i	−0.999061	−	0.0433353i	$-0.986202\pi$
$887$	−1.08256e7	+	1.87505e7i	−0.462001	+	0.800209i	0.537060	+	0.843544i	$0.319535\pi$
$888$		0			0
$889$	2.41516e6	−	4.18318e6i	0.102492	−	0.177522i
$890$		0			0
$891$	−4.73227e6	−	8.19653e6i	−0.199699	−	0.345888i
$892$		0			0
$893$	1.28186e7	+	2.29572e7i	0.537911	+	0.963362i
$894$		0			0
$895$	3.35252e6	+	5.80673e6i	0.139899	+	0.242312i
$896$		0			0
$897$	972461.	−	1.68435e6i	0.0403544	−	0.0698959i
$898$		0			0
$899$	−4.01183e6	+	6.94869e6i	−0.165555	+	0.286750i
$900$		0			0
$901$	−2.21373e7			−0.908473
$902$		0			0
$903$	50728.2	−	87863.8i	0.00207028	−	0.00358584i
$904$		0			0
$905$	3.81185e6			0.154708
$906$		0			0
$907$	7.14753e6	+	1.23799e7i	0.288495	+	0.499687i	0.973451	−	0.228897i	$-0.0735119\pi$
$907$	7.14753e6	+	1.23799e7i	0.288495	+	0.499687i	−0.684956	+	0.728584i	$0.740179\pi$
$908$		0			0
$909$	6.83437e6	+	1.18375e7i	0.274340	+	0.475170i
$910$		0			0
$911$	−1.67784e7			−0.669816			−0.334908	−	0.942251i	$-0.608705\pi$
$911$	−1.67784e7			−0.669816			−0.334908	+	0.942251i	$0.608705\pi$
$912$		0			0
$913$	−427834.			−0.0169863
$914$		0			0
$915$	2.28514e6	+	3.95797e6i	0.0902318	+	0.156286i
$916$		0			0
$917$	1.12949e6	+	1.95633e6i	0.0443565	+	0.0768278i
$918$		0			0
$919$	−3.40604e7			−1.33034			−0.665168	−	0.746694i	$-0.731640\pi$
$919$	−3.40604e7			−1.33034			−0.665168	+	0.746694i	$0.731640\pi$
$920$		0			0
$921$	−1.05744e6	+	1.83154e6i	−0.0410777	+	0.0711486i
$922$		0			0
$923$	−367396.			−0.0141948
$924$		0			0
$925$	−2.56014e6	+	4.43429e6i	−0.0983804	+	0.170400i
$926$		0			0
$927$	−3.51394e6	+	6.08632e6i	−0.134306	+	0.232624i
$928$		0			0
$929$	1.16519e7	+	2.01817e7i	0.442953	+	0.767216i	0.997907	−	0.0646644i	$-0.0205977\pi$
$929$	1.16519e7	+	2.01817e7i	0.442953	+	0.767216i	−0.554955	+	0.831881i	$0.687264\pi$
$930$		0			0
$931$	−1.22141e7	+	2.04711e7i	−0.461835	+	0.774048i
$932$		0			0
$933$	−4.79703e6	−	8.30869e6i	−0.180413	−	0.312485i
$934$		0			0
$935$	8.66184e6	−	1.50027e7i	0.324027	−	0.561231i
$936$		0			0
$937$	1.82963e7	−	3.16902e7i	0.680794	−	1.17917i	−0.293946	−	0.955822i	$-0.594968\pi$
$937$	1.82963e7	−	3.16902e7i	0.680794	−	1.17917i	0.974739	−	0.223347i	$-0.0716982\pi$
$938$		0			0
$939$	−1.27533e7			−0.472018
$940$		0			0
$941$	−1.27910e7	+	2.21547e7i	−0.470902	+	0.815626i	−0.999446	−	0.0332800i	$-0.989405\pi$
$941$	−1.27910e7	+	2.21547e7i	−0.470902	+	0.815626i	0.528544	+	0.848906i	$0.322738\pi$
$942$		0			0
$943$	1.38833e6			0.0508409
$944$		0			0
$945$	1.50043e6	+	2.59883e6i	0.0546560	+	0.0946669i
$946$		0			0
$947$	1.62172e7	+	2.80890e7i	0.587627	+	1.01780i	0.994542	+	0.104333i	$0.0332709\pi$
$947$	1.62172e7	+	2.80890e7i	0.587627	+	1.01780i	−0.406916	+	0.913466i	$0.633396\pi$
$948$		0			0
$949$	−1.20106e6			−0.0432912
$950$		0			0
$951$	−1.00123e7			−0.358990
$952$		0			0
$953$	−2.43185e7	−	4.21208e7i	−0.867369	−	1.50233i	−0.864675	−	0.502331i	$-0.832476\pi$
$953$	−2.43185e7	−	4.21208e7i	−0.867369	−	1.50233i	−0.00269410	−	0.999996i	$-0.500858\pi$
$954$		0			0
$955$	−1.51933e6	−	2.63156e6i	−0.0539068	−	0.0933694i
$956$		0			0
$957$	2.26501e6			0.0799448
$958$		0			0
$959$	6.51053e6	−	1.12766e7i	0.228597	−	0.395941i
$960$		0			0
$961$	2.91613e7			1.01859
$962$		0			0
$963$	1.26507e7	−	2.19117e7i	0.439592	−	0.761395i
$964$		0			0
$965$	−968028.	+	1.67667e6i	−0.0334634	+	0.0579603i
$966$		0			0
$967$	2.29008e7	+	3.96653e7i	0.787561	+	1.36410i	0.927457	+	0.373930i	$0.121990\pi$
$967$	2.29008e7	+	3.96653e7i	0.787561	+	1.36410i	−0.139895	+	0.990166i	$0.544677\pi$
$968$		0			0
$969$	1.07176e7	+	1.91945e7i	0.366680	+	0.656699i
$970$		0			0
$971$	1.65694e7	+	2.86990e7i	0.563973	+	0.976830i	0.997144	+	0.0755181i	$0.0240611\pi$
$971$	1.65694e7	+	2.86990e7i	0.563973	+	0.976830i	−0.433172	+	0.901311i	$0.642606\pi$
$972$		0			0
$973$	4.30835e6	−	7.46229e6i	0.145891	−	0.252691i
$974$		0			0
$975$	803428.	−	1.39158e6i	0.0270667	−	0.0468809i
$976$		0			0
$977$	2.87294e7			0.962920			0.481460	−	0.876468i	$-0.340107\pi$
$977$	2.87294e7			0.962920			0.481460	+	0.876468i	$0.340107\pi$
$978$		0			0
$979$	1.68056e7	−	2.91082e7i	0.560400	−	0.970641i
$980$		0			0
$981$	−4.29787e7			−1.42587
$982$		0			0
$983$	1.41284e7	+	2.44712e7i	0.466348	+	0.807739i	0.999261	−	0.0384309i	$-0.0122359\pi$
$983$	1.41284e7	+	2.44712e7i	0.466348	+	0.807739i	−0.532913	+	0.846170i	$0.678903\pi$
$984$		0			0
$985$	7.33750e6	+	1.27089e7i	0.240967	+	0.417367i
$986$		0			0
$987$	−4.49483e6			−0.146866
$988$		0			0
$989$	−1.13627e6			−0.0369395
$990$		0			0
$991$	−2.23566e7	−	3.87228e7i	−0.723140	−	1.25251i	−0.959735	−	0.280907i	$-0.909365\pi$
$991$	−2.23566e7	−	3.87228e7i	−0.723140	−	1.25251i	0.236595	−	0.971608i	$-0.423968\pi$
$992$		0			0
$993$	−1.12792e7	−	1.95362e7i	−0.363000	−	0.628735i
$994$		0			0
$995$	4.21342e6			0.134920
$996$		0			0
$997$	−2.35342e7	+	4.07625e7i	−0.749828	+	1.29874i	0.198077	+	0.980187i	$0.436531\pi$
$997$	−2.35342e7	+	4.07625e7i	−0.749828	+	1.29874i	−0.947905	+	0.318554i	$0.896803\pi$
$998$		0			0
$999$	6.01181e6			0.190586

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.6.e.a.45.5	✓	18
3.2	odd	2		684.6.k.f.577.4		18
4.3	odd	2		304.6.i.d.273.5		18
19.11	even	3	inner	76.6.e.a.49.5	yes	18
57.11	odd	6		684.6.k.f.505.4		18
76.11	odd	6		304.6.i.d.49.5		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.e.a.45.5	✓	18	1.1	even	1	trivial
76.6.e.a.49.5	yes	18	19.11	even	3	inner
304.6.i.d.49.5		18	76.11	odd	6
304.6.i.d.273.5		18	4.3	odd	2
684.6.k.f.505.4		18	57.11	odd	6
684.6.k.f.577.4		18	3.2	odd	2

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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	76.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.30322 - 5.72134i) q^{3} +(12.6095 + 21.8402i) q^{5} -40.7176 q^{7} +(99.6775 - 172.647i) q^{9} +O(q^{10})\)	\(q+(-3.30322 - 5.72134i) q^{3} +(12.6095 + 21.8402i) q^{5} -40.7176 q^{7} +(99.6775 - 172.647i) q^{9} +324.832 q^{11} +(-48.8600 + 84.6280i) q^{13} +(83.3036 - 144.286i) q^{15} +(-1057.36 - 1831.41i) q^{17} +(806.259 - 1351.31i) q^{19} +(134.499 + 232.959i) q^{21} +(1506.33 - 2609.05i) q^{23} +(1244.50 - 2155.54i) q^{25} -2922.39 q^{27} +(-527.733 + 914.060i) q^{29} +7602.00 q^{31} +(-1072.99 - 1858.48i) q^{33} +(-513.427 - 889.282i) q^{35} -2057.16 q^{37} +645.581 q^{39} +(230.415 + 399.091i) q^{41} +(-188.582 - 326.633i) q^{43} +5027.52 q^{45} +(-8354.76 + 14470.9i) q^{47} -15149.1 q^{49} +(-6985.41 + 12099.1i) q^{51} +(5234.06 - 9065.67i) q^{53} +(4095.96 + 7094.41i) q^{55} +(-10394.6 - 149.205i) q^{57} +(23733.3 + 41107.2i) q^{59} +(-13715.7 + 23756.3i) q^{61} +(-4058.63 + 7029.76i) q^{63} -2464.39 q^{65} +(17228.2 - 29840.0i) q^{67} -19903.0 q^{69} +(1879.84 + 3255.97i) q^{71} +(6145.42 + 10644.2i) q^{73} -16443.5 q^{75} -13226.4 q^{77} +(26717.1 + 46275.4i) q^{79} +(-14568.3 - 25233.1i) q^{81} -1317.09 q^{83} +(26665.6 - 46186.2i) q^{85} +6972.87 q^{87} +(51736.4 - 89610.0i) q^{89} +(1989.46 - 3445.85i) q^{91} +(-25111.1 - 43493.7i) q^{93} +(39679.5 + 569.563i) q^{95} +(-52995.4 - 91790.7i) q^{97} +(32378.5 - 56081.1i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 11 q^{3} + 11 q^{5} + 336 q^{7} - 902 q^{9}+O(q^{10}) \) 18 * q - 11 * q^3 + 11 * q^5 + 336 * q^7 - 902 * q^9	\( 18 q - 11 q^{3} + 11 q^{5} + 336 q^{7} - 902 q^{9} - 320 q^{11} + 227 q^{13} - 101 q^{15} + 179 q^{17} - 868 q^{19} - 5700 q^{21} - 3425 q^{23} - 7054 q^{25} + 14722 q^{27} - 7349 q^{29} - 9960 q^{31} - 2998 q^{33} + 15888 q^{35} + 26444 q^{37} - 30246 q^{39} - 7311 q^{41} - 8283 q^{43} - 62164 q^{45} + 37603 q^{47} + 124738 q^{49} + 47227 q^{51} - 20337 q^{53} + 716 q^{55} - 57555 q^{57} - 74455 q^{59} - 7569 q^{61} - 52544 q^{63} + 188998 q^{65} - 26177 q^{67} + 116282 q^{69} - 53463 q^{71} - 14103 q^{73} + 120912 q^{75} - 31960 q^{77} + 31825 q^{79} - 21137 q^{81} + 82600 q^{83} - 50787 q^{85} - 339766 q^{87} - 155197 q^{89} - 2800 q^{91} - 46460 q^{93} + 49315 q^{95} + 111241 q^{97} - 193544 q^{99}+O(q^{100}) \) 18 * q - 11 * q^3 + 11 * q^5 + 336 * q^7 - 902 * q^9 - 320 * q^11 + 227 * q^13 - 101 * q^15 + 179 * q^17 - 868 * q^19 - 5700 * q^21 - 3425 * q^23 - 7054 * q^25 + 14722 * q^27 - 7349 * q^29 - 9960 * q^31 - 2998 * q^33 + 15888 * q^35 + 26444 * q^37 - 30246 * q^39 - 7311 * q^41 - 8283 * q^43 - 62164 * q^45 + 37603 * q^47 + 124738 * q^49 + 47227 * q^51 - 20337 * q^53 + 716 * q^55 - 57555 * q^57 - 74455 * q^59 - 7569 * q^61 - 52544 * q^63 + 188998 * q^65 - 26177 * q^67 + 116282 * q^69 - 53463 * q^71 - 14103 * q^73 + 120912 * q^75 - 31960 * q^77 + 31825 * q^79 - 21137 * q^81 + 82600 * q^83 - 50787 * q^85 - 339766 * q^87 - 155197 * q^89 - 2800 * q^91 - 46460 * q^93 + 49315 * q^95 + 111241 * q^97 - 193544 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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