LMFDB - Embedded newform 312.4.m.a.181.64

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [312,4,Mod(181,312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(312, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("312.181");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 312 = 2^{3} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	312.m (of order $2$, degree $1$, 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:	$18.4085959218$
Analytic rank:	$0$
Dimension:	$84$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.64
Character	$\chi$	$=$	312.181
Dual form			312.4.m.a.181.63

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.09150 + 1.90411i) q^{2} +3.00000i q^{3} +(0.748758 + 7.96488i) q^{4} -5.98228 q^{5} +(-5.71232 + 6.27450i) q^{6} -13.5739i q^{7} +(-13.6000 + 18.0843i) q^{8} -9.00000 q^{9} +(-12.5120 - 11.3909i) q^{10} -38.8640 q^{11} +(-23.8946 + 2.24627i) q^{12} +(-17.8282 - 43.3492i) q^{13} +(25.8461 - 28.3898i) q^{14} -17.9469i q^{15} +(-62.8787 + 11.9275i) q^{16} +10.7249 q^{17} +(-18.8235 - 17.1370i) q^{18} -50.0052 q^{19} +(-4.47928 - 47.6482i) q^{20} +40.7217 q^{21} +(-81.2842 - 74.0013i) q^{22} +58.0103 q^{23} +(-54.2528 - 40.7999i) q^{24} -89.2123 q^{25} +(45.2539 - 124.612i) q^{26} -27.0000i q^{27} +(108.115 - 10.1636i) q^{28} +140.989i q^{29} +(34.1727 - 37.5359i) q^{30} -49.2610i q^{31} +(-154.222 - 94.7813i) q^{32} -116.592i q^{33} +(22.4312 + 20.4214i) q^{34} +81.2029i q^{35} +(-6.73882 - 71.6839i) q^{36} -109.159 q^{37} +(-104.586 - 95.2152i) q^{38} +(130.048 - 53.4845i) q^{39} +(81.3588 - 108.185i) q^{40} -200.982i q^{41} +(85.1695 + 77.5384i) q^{42} +53.5585i q^{43} +(-29.0998 - 309.548i) q^{44} +53.8406 q^{45} +(121.329 + 110.458i) q^{46} -95.3945i q^{47} +(-35.7826 - 188.636i) q^{48} +158.749 q^{49} +(-186.588 - 169.870i) q^{50} +32.1748i q^{51} +(331.923 - 174.457i) q^{52} +385.163i q^{53} +(51.4109 - 56.4705i) q^{54} +232.496 q^{55} +(245.474 + 184.604i) q^{56} -150.016i q^{57} +(-268.459 + 294.879i) q^{58} -305.930 q^{59} +(142.945 - 13.4378i) q^{60} +164.474i q^{61} +(93.7981 - 103.029i) q^{62} +122.165i q^{63} +(-142.082 - 491.891i) q^{64} +(106.653 + 259.327i) q^{65} +(222.004 - 243.853i) q^{66} -962.386 q^{67} +(8.03037 + 85.4228i) q^{68} +174.031i q^{69} +(-154.619 + 169.836i) q^{70} +195.268i q^{71} +(122.400 - 162.759i) q^{72} +317.283i q^{73} +(-228.307 - 207.851i) q^{74} -267.637i q^{75} +(-37.4418 - 398.285i) q^{76} +527.537i q^{77} +(373.835 + 135.762i) q^{78} -221.141 q^{79} +(376.158 - 71.3539i) q^{80} +81.0000 q^{81} +(382.691 - 420.354i) q^{82} -445.937 q^{83} +(30.4907 + 324.344i) q^{84} -64.1596 q^{85} +(-101.981 + 112.018i) q^{86} -422.968 q^{87} +(528.549 - 702.828i) q^{88} -523.463i q^{89} +(112.608 + 102.518i) q^{90} +(-588.418 + 241.998i) q^{91} +(43.4357 + 462.045i) q^{92} +147.783 q^{93} +(181.641 - 199.518i) q^{94} +299.145 q^{95} +(284.344 - 462.667i) q^{96} -147.749i q^{97} +(332.024 + 302.275i) q^{98} +349.776 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 756 q^{9} - 36 q^{10} + 12 q^{12} - 208 q^{14} - 148 q^{16} - 104 q^{17} + 620 q^{22} + 2188 q^{25} + 444 q^{26} - 204 q^{30} - 40 q^{38} - 1924 q^{40} + 192 q^{42} + 624 q^{48} - 3396 q^{49} - 1292 q^{52}+ \cdots + 2480 q^{95}+O(q^{100}) $ 84 * q - 756 * q^9 - 36 * q^10 + 12 * q^12 - 208 * q^14 - 148 * q^16 - 104 * q^17 + 620 * q^22 + 2188 * q^25 + 444 * q^26 - 204 * q^30 - 40 * q^38 - 1924 * q^40 + 192 * q^42 + 624 * q^48 - 3396 * q^49 - 1292 * q^52 - 1616 * q^55 + 608 * q^56 - 2120 * q^62 - 2856 * q^64 + 696 * q^65 - 396 * q^66 - 2536 * q^68 - 3936 * q^74 - 156 * q^78 + 3160 * q^79 + 6804 * q^81 + 4276 * q^82 - 2088 * q^87 + 1780 * q^88 + 324 * q^90 + 4792 * q^92 - 860 * q^94 + 2480 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/312\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$157$	$209$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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 $

$2$	2.09150	+	1.90411i	0.739457	+	0.673203i
$2$	2.09150	+	1.90411i	0.739457	+	0.673203i
$3$			3.00000i			0.577350i
$3$			3.00000i			0.577350i
$4$	0.748758	+	7.96488i	0.0935947	+	0.995610i
$5$	−5.98228			−0.535072			−0.267536	−	0.963548i	$-0.586209\pi$
$5$	−5.98228			−0.535072			−0.267536	+	0.963548i	$0.586209\pi$
$6$	−5.71232	+	6.27450i	−0.388674	+	0.426926i
$7$		−	13.5739i		−	0.732922i	−0.930433	−	0.366461i	$-0.880569\pi$
$7$		−	13.5739i		−	0.732922i	0.930433	−	0.366461i	$-0.119431\pi$
$8$	−13.6000	+	18.0843i	−0.601039	+	0.799220i
$9$	−9.00000			−0.333333
$10$	−12.5120	−	11.3909i	−0.395663	−	0.360212i
$11$	−38.8640			−1.06527			−0.532634	−	0.846346i	$-0.678798\pi$
$11$	−38.8640			−1.06527			−0.532634	+	0.846346i	$0.678798\pi$
$12$	−23.8946	+	2.24627i	−0.574816	+	0.0540369i
$13$	−17.8282	−	43.3492i	−0.380357	−	0.924840i
$13$	−17.8282	−	43.3492i	−0.380357	−	0.924840i
$14$	25.8461	−	28.3898i	0.493405	−	0.541964i
$15$		−	17.9469i		−	0.308924i
$16$	−62.8787	+	11.9275i	−0.982480	+	0.186368i
$17$	10.7249			0.153010			0.0765052	−	0.997069i	$-0.475624\pi$
$17$	10.7249			0.153010			0.0765052	+	0.997069i	$0.475624\pi$
$18$	−18.8235	−	17.1370i	−0.246486	−	0.224401i
$19$	−50.0052			−0.603788			−0.301894	−	0.953341i	$-0.597619\pi$
$19$	−50.0052			−0.603788			−0.301894	+	0.953341i	$0.597619\pi$
$20$	−4.47928	−	47.6482i	−0.0500799	−	0.532723i
$21$	40.7217			0.423153
$22$	−81.2842	−	74.0013i	−0.787720	−	0.717142i
$23$	58.0103			0.525913			0.262956	−	0.964808i	$-0.415302\pi$
$23$	58.0103			0.525913			0.262956	+	0.964808i	$0.415302\pi$
$24$	−54.2528	−	40.7999i	−0.461430	−	0.347010i
$25$	−89.2123			−0.713698
$26$	45.2539	−	124.612i	0.341347	−	0.939937i
$27$		−	27.0000i		−	0.192450i
$28$	108.115	−	10.1636i	0.729704	−	0.0685976i
$29$			140.989i			0.902795i	0.892323	+	0.451398i	$0.149074\pi$
$29$			140.989i			0.902795i	−0.892323	+	0.451398i	$0.850926\pi$
$30$	34.1727	−	37.5359i	0.207969	−	0.228436i
$31$		−	49.2610i		−	0.285404i	−0.989766	−	0.142702i	$-0.954421\pi$
$31$		−	49.2610i		−	0.285404i	0.989766	−	0.142702i	$-0.0455791\pi$
$32$	−154.222	−	94.7813i	−0.851966	−	0.523598i
$33$		−	116.592i		−	0.615033i
$34$	22.4312	+	20.4214i	0.113145	+	0.103007i
$35$			81.2029i			0.392166i
$36$	−6.73882	−	71.6839i	−0.0311982	−	0.331870i
$37$	−109.159			−0.485018			−0.242509	−	0.970149i	$-0.577970\pi$
$37$	−109.159			−0.485018			−0.242509	+	0.970149i	$0.577970\pi$
$38$	−104.586	−	95.2152i	−0.446476	−	0.406472i
$39$	130.048	−	53.4845i	0.533956	−	0.219599i
$40$	81.3588	−	108.185i	0.321599	−	0.427640i
$41$		−	200.982i		−	0.765564i	−0.923839	−	0.382782i	$-0.874966\pi$
$41$		−	200.982i		−	0.765564i	0.923839	−	0.382782i	$-0.125034\pi$
$42$	85.1695	+	77.5384i	0.312903	+	0.284868i
$43$			53.5585i			0.189944i	0.995480	+	0.0949721i	$0.0302762\pi$
$43$			53.5585i			0.189944i	−0.995480	+	0.0949721i	$0.969724\pi$
$44$	−29.0998	−	309.548i	−0.0997035	−	1.06059i
$45$	53.8406			0.178357
$46$	121.329	+	110.458i	0.388890	+	0.354046i
$47$		−	95.3945i		−	0.296058i	−0.988983	−	0.148029i	$-0.952707\pi$
$47$		−	95.3945i		−	0.296058i	0.988983	−	0.148029i	$-0.0472929\pi$
$48$	−35.7826	−	188.636i	−0.107599	−	0.567235i
$49$	158.749			0.462826
$50$	−186.588	−	169.870i	−0.527749	−	0.480464i
$51$			32.1748i			0.0883406i
$52$	331.923	−	174.457i	0.885180	−	0.465248i
$53$			385.163i			0.998231i	0.866535	+	0.499116i	$0.166342\pi$
$53$			385.163i			0.998231i	−0.866535	+	0.499116i	$0.833658\pi$
$54$	51.4109	−	56.4705i	0.129558	−	0.142309i
$55$	232.496			0.569995
$56$	245.474	+	184.604i	0.585766	+	0.440514i
$57$		−	150.016i		−	0.348597i
$58$	−268.459	+	294.879i	−0.607765	+	0.667579i
$59$	−305.930			−0.675063			−0.337531	−	0.941314i	$-0.609592\pi$
$59$	−305.930			−0.675063			−0.337531	+	0.941314i	$0.609592\pi$
$60$	142.945	−	13.4378i	0.307568	−	0.0289136i
$61$			164.474i			0.345224i	0.984990	+	0.172612i	$0.0552207\pi$
$61$			164.474i			0.345224i	−0.984990	+	0.172612i	$0.944779\pi$
$62$	93.7981	−	103.029i	0.192135	−	0.211044i
$63$			122.165i			0.244307i
$64$	−142.082	−	491.891i	−0.277505	−	0.960724i
$65$	106.653	+	259.327i	0.203519	+	0.494856i
$66$	222.004	−	243.853i	0.414042	−	0.454791i
$67$	−962.386			−1.75484			−0.877419	−	0.479724i	$-0.840737\pi$
$67$	−962.386			−1.75484			−0.877419	+	0.479724i	$0.840737\pi$
$68$	8.03037	+	85.4228i	0.0143210	+	0.152339i
$69$			174.031i			0.303636i
$70$	−154.619	+	169.836i	−0.264007	+	0.289990i
$71$			195.268i			0.326395i	0.986593	+	0.163198i	$0.0521808\pi$
$71$			195.268i			0.326395i	−0.986593	+	0.163198i	$0.947819\pi$
$72$	122.400	−	162.759i	0.200346	−	0.266407i
$73$			317.283i			0.508701i	0.967112	+	0.254351i	$0.0818617\pi$
$73$			317.283i			0.508701i	−0.967112	+	0.254351i	$0.918138\pi$
$74$	−228.307	−	207.851i	−0.358650	−	0.326516i
$75$		−	267.637i		−	0.412054i
$76$	−37.4418	−	398.285i	−0.0565114	−	0.601138i
$77$			527.537i			0.780758i
$78$	373.835	+	135.762i	0.542673	+	0.197077i
$79$	−221.141			−0.314940			−0.157470	−	0.987524i	$-0.550334\pi$
$79$	−221.141			−0.314940			−0.157470	+	0.987524i	$0.550334\pi$
$80$	376.158	−	71.3539i	0.525697	−	0.0997201i
$81$	81.0000			0.111111
$82$	382.691	−	420.354i	0.515380	−	0.566102i
$83$	−445.937			−0.589734			−0.294867	−	0.955538i	$-0.595275\pi$
$83$	−445.937			−0.589734			−0.294867	+	0.955538i	$0.595275\pi$
$84$	30.4907	+	324.344i	0.0396048	+	0.421295i
$85$	−64.1596			−0.0818716
$86$	−101.981	+	112.018i	−0.127871	+	0.140456i
$87$	−422.968			−0.521229
$88$	528.549	−	702.828i	0.640267	−	0.851383i
$89$		−	523.463i		−	0.623449i	−0.950173	−	0.311724i	$-0.899093\pi$
$89$		−	523.463i		−	0.623449i	0.950173	−	0.311724i	$-0.100907\pi$
$90$	112.608	+	102.518i	0.131888	+	0.120071i
$91$	−588.418	+	241.998i	−0.677835	+	0.278772i
$92$	43.4357	+	462.045i	0.0492226	+	0.523604i
$93$	147.783			0.164778
$94$	181.641	−	199.518i	0.199307	−	0.218922i
$95$	299.145			0.323070
$96$	284.344	−	462.667i	0.302299	−	0.491883i
$97$		−	147.749i		−	0.154656i	−0.997006	−	0.0773278i	$-0.975361\pi$
$97$		−	147.749i		−	0.154656i	0.997006	−	0.0773278i	$-0.0246388\pi$
$98$	332.024	+	302.275i	0.342240	+	0.311576i
$99$	349.776			0.355089
$100$	−66.7984	−	710.565i	−0.0667984	−	0.710565i
$101$			720.923i			0.710243i	0.934820	+	0.355122i	$0.115561\pi$
$101$			720.923i			0.710243i	−0.934820	+	0.355122i	$0.884439\pi$
$102$	−61.2642	+	67.2936i	−0.0594712	+	0.0653241i
$103$	−1277.90			−1.22248			−0.611240	−	0.791445i	$-0.709329\pi$
$103$	−1277.90			−1.22248			−0.611240	+	0.791445i	$0.709329\pi$
$104$	1026.40	+	267.138i	0.967760	+	0.251875i
$105$	−243.609			−0.226417
$106$	−733.392	+	805.570i	−0.672013	+	0.738150i
$107$			1574.80i			1.42282i	0.702778	+	0.711409i	$0.251943\pi$
$107$			1574.80i			1.42282i	−0.702778	+	0.711409i	$0.748057\pi$
$108$	215.052	−	20.2165i	0.191605	−	0.0180123i
$109$	713.898			0.627330			0.313665	−	0.949534i	$-0.398443\pi$
$109$	713.898			0.627330			0.313665	+	0.949534i	$0.398443\pi$
$110$	486.265	+	442.697i	0.421487	+	0.383722i
$111$		−	327.478i		−	0.280025i
$112$	161.903	+	853.510i	0.136593	+	0.720081i
$113$	504.714			0.420173			0.210086	−	0.977683i	$-0.432625\pi$
$113$	504.714			0.420173			0.210086	+	0.977683i	$0.432625\pi$
$114$	285.646	−	313.758i	0.234677	−	0.257773i
$115$	−347.034			−0.281401
$116$	−1122.96	+	105.567i	−0.898832	+	0.0844969i
$117$	160.454	+	390.143i	0.126786	+	0.308280i
$118$	−639.853	−	582.523i	−0.499180	−	0.454454i
$119$		−	145.579i		−	0.112145i
$120$	324.556	+	244.076i	0.246898	+	0.185675i
$121$	179.413			0.134796
$122$	−313.175	+	343.997i	−0.232406	+	0.255279i
$123$	602.946			0.441999
$124$	392.358	−	36.8845i	0.284151	−	0.0267123i
$125$	1281.48			0.916952
$126$	−232.615	+	255.509i	−0.164468	+	0.180655i
$127$	−1704.14			−1.19069			−0.595347	−	0.803469i	$-0.702985\pi$
$127$	−1704.14			−1.19069			−0.595347	+	0.803469i	$0.702985\pi$
$128$	639.447	−	1299.33i	0.441560	−	0.897232i
$129$	−160.676			−0.109664
$130$	−270.722	+	745.463i	−0.182645	+	0.502934i
$131$			1672.35i			1.11537i	0.830052	+	0.557686i	$0.188311\pi$
$131$			1672.35i			1.11537i	−0.830052	+	0.557686i	$0.811689\pi$
$132$	928.643	−	87.2993i	0.612333	−	0.0575638i
$133$			678.765i			0.442529i
$134$	−2012.83	−	1832.49i	−1.29763	−	1.18136i
$135$			161.522i			0.102975i
$136$	−145.859	+	193.953i	−0.0919652	+	0.122289i
$137$		−	1338.74i		−	0.834866i	−0.908708	−	0.417433i	$-0.862930\pi$
$137$		−	1338.74i		−	0.834866i	0.908708	−	0.417433i	$-0.137070\pi$
$138$	−331.374	+	363.986i	−0.204409	+	0.224526i
$139$		−	1263.33i		−	0.770896i	−0.922729	−	0.385448i	$-0.874047\pi$
$139$		−	1263.33i		−	0.770896i	0.922729	−	0.385448i	$-0.125953\pi$
$140$	−646.772	+	60.8013i	−0.390444	+	0.0367046i
$141$	286.184			0.170929
$142$	−371.811	+	408.403i	−0.219730	+	0.241355i
$143$	692.875	+	1684.73i	0.405183	+	0.985202i
$144$	565.909	−	107.348i	0.327493	−	0.0621226i
$145$		−	843.438i		−	0.483060i
$146$	−604.141	+	663.598i	−0.342459	+	0.376163i
$147$			476.248i			0.267213i
$148$	−81.7338	−	869.441i	−0.0453951	−	0.482889i
$149$	1083.84			0.595917			0.297959	−	0.954579i	$-0.403694\pi$
$149$	1083.84			0.595917			0.297959	+	0.954579i	$0.403694\pi$
$150$	509.609	−	559.763i	0.277396	−	0.304696i
$151$			2821.74i			1.52073i	0.649498	+	0.760363i	$0.274979\pi$
$151$			2821.74i			1.52073i	−0.649498	+	0.760363i	$0.725021\pi$
$152$	680.068	−	904.308i	0.362900	−	0.482559i
$153$	−96.5243			−0.0510035
$154$	−1004.49	+	1103.34i	−0.525609	+	0.577337i
$155$			294.693i			0.152712i
$156$	523.372	+	995.768i	0.268611	+	0.511059i
$157$			142.839i			0.0726102i	0.999341	+	0.0363051i	$0.0115588\pi$
$157$			142.839i			0.0726102i	−0.999341	+	0.0363051i	$0.988441\pi$
$158$	−462.516	−	421.075i	−0.232885	−	0.212019i
$159$	−1155.49			−0.576329
$160$	922.601	+	567.009i	0.455863	+	0.280162i
$161$		−	787.426i		−	0.385453i
$162$	169.412	+	154.233i	0.0821619	+	0.0748004i
$163$	−2758.34			−1.32546			−0.662729	−	0.748860i	$-0.730602\pi$
$163$	−2758.34			−1.32546			−0.662729	+	0.748860i	$0.730602\pi$
$164$	1600.80	−	150.487i	0.762204	−	0.0716528i
$165$			697.487i			0.329087i
$166$	−932.677	−	849.111i	−0.436083	−	0.397011i
$167$			3245.29i			1.50376i	0.659300	+	0.751880i	$0.270853\pi$
$167$			3245.29i			1.50376i	−0.659300	+	0.751880i	$0.729147\pi$
$168$	−553.813	+	736.423i	−0.254331	+	0.338192i
$169$	−1561.31	+	1545.68i	−0.710656	+	0.703539i
$170$	−134.190	−	122.167i	−0.0605405	−	0.0551162i
$171$	450.047			0.201263
$172$	−426.588	+	40.1024i	−0.189110	+	0.0177778i
$173$		−	2974.07i		−	1.30702i	−0.756918	−	0.653510i	$-0.773296\pi$
$173$		−	2974.07i		−	1.30702i	0.756918	−	0.653510i	$-0.226704\pi$
$174$	−884.638	−	805.376i	−0.385427	−	0.350893i
$175$			1210.96i			0.523085i
$176$	2443.72	−	463.552i	1.04660	−	0.198532i
$177$		−	917.790i		−	0.389748i
$178$	996.728	−	1094.82i	0.419708	−	0.461014i
$179$		−	3536.59i		−	1.47674i	−0.674395	−	0.738371i	$-0.735595\pi$
$179$		−	3536.59i		−	1.47674i	0.674395	−	0.738371i	$-0.264405\pi$
$180$	40.3135	+	428.834i	0.0166933	+	0.177574i
$181$			2464.94i			1.01225i	0.862460	+	0.506125i	$0.168923\pi$
$181$			2464.94i			1.01225i	−0.862460	+	0.506125i	$0.831077\pi$
$182$	−1691.47	−	614.272i	−0.688901	−	0.250181i
$183$	−493.421			−0.199315
$184$	−788.938	+	1049.08i	−0.316094	+	0.420320i
$185$	653.022			0.259519
$186$	309.088	+	281.394i	0.121846	+	0.110929i
$187$	−416.814			−0.162997
$188$	759.806	−	71.4274i	0.294758	−	0.0277095i
$189$	−366.495			−0.141051
$190$	625.663	+	569.604i	0.238897	+	0.217492i
$191$	4406.00			1.66914			0.834572	−	0.550898i	$-0.185715\pi$
$191$	4406.00			1.66914			0.834572	+	0.550898i	$0.185715\pi$
$192$	1475.67	−	426.247i	0.554674	−	0.160217i
$193$		−	204.930i		−	0.0764311i	−0.999270	−	0.0382156i	$-0.987833\pi$
$193$		−	204.930i		−	0.0764311i	0.999270	−	0.0382156i	$-0.0121674\pi$
$194$	281.329	−	309.016i	0.104115	−	0.114361i
$195$	−777.982	+	319.960i	−0.285705	+	0.117501i
$196$	118.865	+	1264.42i	0.0433181	+	0.460794i
$197$	1230.05			0.444861			0.222430	−	0.974949i	$-0.428601\pi$
$197$	1230.05			0.444861			0.222430	+	0.974949i	$0.428601\pi$
$198$	731.558	+	666.011i	0.262573	+	0.239047i
$199$	188.319			0.0670832			0.0335416	−	0.999437i	$-0.489321\pi$
$199$	188.319			0.0670832			0.0335416	+	0.999437i	$0.489321\pi$
$200$	1213.28	−	1613.34i	0.428960	−	0.570402i
$201$		−	2887.16i		−	1.01316i
$202$	−1372.71	+	1507.81i	−0.478138	+	0.525195i
$203$	1913.77			0.661678
$204$	−256.268	+	24.0911i	−0.0879528	+	0.00826821i
$205$			1202.33i			0.409632i
$206$	−2672.74	−	2433.26i	−0.903973	−	0.822978i
$207$	−522.093			−0.175304
$208$	1638.06	+	2513.10i	0.546054	+	0.837750i
$209$	1943.40			0.643196
$210$	−509.508	−	463.857i	−0.167426	−	0.152425i
$211$		−	2513.40i		−	0.820044i	−0.912076	−	0.410022i	$-0.865521\pi$
$211$		−	2513.40i		−	0.820044i	0.912076	−	0.410022i	$-0.134479\pi$
$212$	−3067.78	+	288.394i	−0.993850	+	0.0934292i
$213$	−585.804			−0.188444
$214$	−2998.58	+	3293.69i	−0.957846	+	1.05211i
$215$		−	320.402i		−	0.101634i
$216$	488.276	+	367.199i	0.153810	+	0.115670i
$217$	−668.663			−0.209179
$218$	1493.12	+	1359.34i	0.463884	+	0.422321i
$219$	−951.850			−0.293699
$220$	174.083	+	1851.80i	0.0533485	+	0.567493i
$221$	−191.206	−	464.917i	−0.0581986	−	0.141510i
$222$	623.552	−	684.920i	0.188514	−	0.207067i
$223$		−	1884.34i		−	0.565852i	−0.959142	−	0.282926i	$-0.908695\pi$
$223$		−	1884.34i		−	0.565852i	0.959142	−	0.282926i	$-0.0913051\pi$
$224$	−1286.55	+	2093.40i	−0.383756	+	0.624424i
$225$	802.910			0.237899
$226$	1055.61	+	961.030i	0.310700	+	0.282862i
$227$	−4267.64			−1.24781			−0.623906	−	0.781499i	$-0.714455\pi$
$227$	−4267.64			−1.24781			−0.623906	+	0.781499i	$0.714455\pi$
$228$	1194.86	−	112.325i	0.347067	−	0.0326269i
$229$	6270.47			1.80945			0.904725	−	0.425996i	$-0.140076\pi$
$229$	6270.47			1.80945			0.904725	+	0.425996i	$0.140076\pi$
$230$	−725.823	−	660.790i	−0.208084	−	0.189440i
$231$	−1582.61			−0.450771
$232$	−2549.69	−	1917.45i	−0.721532	−	0.542615i
$233$	5767.97			1.62177			0.810885	−	0.585206i	$-0.198986\pi$
$233$	5767.97			1.62177			0.810885	+	0.585206i	$0.198986\pi$
$234$	−407.285	+	1121.51i	−0.113782	+	0.313312i
$235$			570.677i			0.158412i
$236$	−229.067	−	2436.70i	−0.0631823	−	0.672099i
$237$		−	663.422i		−	0.181831i
$238$	277.198	−	304.479i	0.0754961	−	0.0829262i
$239$			3607.47i			0.976351i	0.872745	+	0.488176i	$0.162337\pi$
$239$			3607.47i			0.976351i	−0.872745	+	0.488176i	$0.837663\pi$
$240$	214.062	+	1128.48i	0.0575734	+	0.303512i
$241$		−	1114.14i		−	0.297792i	−0.988853	−	0.148896i	$-0.952428\pi$
$241$		−	1114.14i		−	0.297792i	0.988853	−	0.148896i	$-0.0475720\pi$
$242$	375.243	+	341.622i	0.0996758	+	0.0907450i
$243$			243.000i			0.0641500i
$244$	−1310.01	+	123.151i	−0.343709	+	0.0323112i
$245$	−949.683			−0.247645
$246$	1261.06	+	1148.07i	0.326839	+	0.297555i
$247$	891.501	+	2167.69i	0.229655	+	0.558407i
$248$	890.849	+	669.947i	0.228101	+	0.171539i
$249$		−	1337.81i		−	0.340483i
$250$	2680.21	+	2440.07i	0.678047	+	0.617295i
$251$		−	5483.73i		−	1.37900i	−0.724284	−	0.689502i	$-0.757829\pi$
$251$		−	5483.73i		−	1.37900i	0.724284	−	0.689502i	$-0.242171\pi$
$252$	−973.031	+	91.4721i	−0.243235	+	0.0228659i
$253$	−2254.52			−0.560238
$254$	−3564.21	−	3244.87i	−0.880467	−	0.801579i
$255$		−	192.479i		−	0.0472686i
$256$	3811.47	−	1499.98i	0.930534	−	0.366205i
$257$	5673.89			1.37715			0.688576	−	0.725165i	$-0.258236\pi$
$257$	5673.89			1.37715			0.688576	+	0.725165i	$0.258236\pi$
$258$	−336.053	−	305.943i	−0.0810921	−	0.0738264i
$259$			1481.72i			0.355480i
$260$	−1985.66	+	1043.65i	−0.473635	+	0.248941i
$261$		−	1268.90i		−	0.300932i
$262$	−3184.33	+	3497.72i	−0.750873	+	0.824771i
$263$	−2408.95			−0.564799			−0.282400	−	0.959297i	$-0.591130\pi$
$263$	−2408.95			−0.564799			−0.282400	+	0.959297i	$0.591130\pi$
$264$	2108.48	+	1585.65i	0.491546	+	0.369659i
$265$		−	2304.16i		−	0.534126i
$266$	−1292.44	+	1419.64i	−0.297912	+	0.327232i
$267$	1570.39			0.359948
$268$	−720.594	−	7665.29i	−0.164244	−	1.74714i
$269$			1125.02i			0.254995i	0.991839	+	0.127497i	$0.0406944\pi$
$269$			1125.02i			0.254995i	−0.991839	+	0.127497i	$0.959306\pi$
$270$	−307.554	+	337.823i	−0.0693228	+	0.0761453i
$271$		−	5686.90i		−	1.27474i	−0.770557	−	0.637371i	$-0.780022\pi$
$271$		−	5686.90i		−	1.27474i	0.770557	−	0.637371i	$-0.219978\pi$
$272$	−674.370	+	127.922i	−0.150330	+	0.0285162i
$273$	−725.994	−	1765.25i	−0.160949	−	0.391348i
$274$	2549.11	−	2799.98i	0.562034	−	0.617348i
$275$	3467.15			0.760280
$276$	−1386.14	+	130.307i	−0.302303	+	0.0284187i
$277$		−	9173.89i		−	1.98991i	−0.100307	−	0.994957i	$-0.531983\pi$
$277$		−	9173.89i		−	1.98991i	0.100307	−	0.994957i	$-0.468017\pi$
$278$	2405.52	−	2642.27i	0.518970	−	0.570045i
$279$			443.349i			0.0951347i
$280$	−1468.50	−	1104.36i	−0.313427	−	0.235707i
$281$		−	3169.15i		−	0.672796i	−0.941720	−	0.336398i	$-0.890791\pi$
$281$		−	3169.15i		−	0.672796i	0.941720	−	0.336398i	$-0.109209\pi$
$282$	598.553	+	544.924i	0.126395	+	0.115070i
$283$		−	8235.73i		−	1.72991i	−0.501853	−	0.864953i	$-0.667348\pi$
$283$		−	8235.73i		−	1.72991i	0.501853	−	0.864953i	$-0.332652\pi$
$284$	−1555.29	+	146.208i	−0.324962	+	0.0305489i
$285$			897.436i			0.186525i
$286$	−1758.75	+	4842.92i	−0.363626	+	1.00129i
$287$	−2728.11			−0.561099
$288$	1388.00	+	853.032i	0.283989	+	0.174533i
$289$	−4797.98			−0.976588
$290$	1606.00	−	1764.05i	0.325198	−	0.357202i
$291$	443.246			0.0892905
$292$	−2527.12	+	237.568i	−0.506468	+	0.0476118i
$293$	6334.32			1.26299			0.631493	−	0.775382i	$-0.282443\pi$
$293$	6334.32			1.26299			0.631493	+	0.775382i	$0.282443\pi$
$294$	−906.826	+	996.073i	−0.179888	+	0.197592i
$295$	1830.16			0.361207
$296$	1484.56	−	1974.07i	0.291515	−	0.387636i
$297$			1049.33i			0.205011i
$298$	2266.85	+	2063.75i	0.440656	+	0.401174i
$299$	−1034.22	−	2514.70i	−0.200035	−	0.486385i
$300$	2131.70	−	200.395i	0.410245	−	0.0385661i
$301$	726.998			0.139214
$302$	−5372.89	+	5901.67i	−1.02376	+	1.12451i
$303$	−2162.77			−0.410059
$304$	3144.26	−	596.439i	0.593210	−	0.112527i
$305$		−	983.928i		−	0.184720i
$306$	−201.881	−	183.793i	−0.0377149	−	0.0343357i
$307$	−1452.38			−0.270005			−0.135003	−	0.990845i	$-0.543104\pi$
$307$	−1452.38			−0.270005			−0.135003	+	0.990845i	$0.543104\pi$
$308$	−4201.77	+	394.997i	−0.777331	+	0.0730748i
$309$		−	3833.71i		−	0.705800i
$310$	−561.127	+	616.351i	−0.102806	+	0.112924i
$311$	−1397.02			−0.254719			−0.127359	−	0.991857i	$-0.540650\pi$
$311$	−1397.02			−0.254719			−0.127359	+	0.991857i	$0.540650\pi$
$312$	−801.414	+	3079.21i	−0.145420	+	0.558736i
$313$	−7577.17			−1.36833			−0.684164	−	0.729328i	$-0.739833\pi$
$313$	−7577.17			−1.36833			−0.684164	+	0.729328i	$0.739833\pi$
$314$	−271.981	+	298.748i	−0.0488814	+	0.0536921i
$315$		−	730.826i		−	0.130722i
$316$	−165.581	−	1761.36i	−0.0294767	−	0.313557i
$317$	−9498.82			−1.68299			−0.841493	−	0.540268i	$-0.818323\pi$
$317$	−9498.82			−1.68299			−0.841493	+	0.540268i	$0.818323\pi$
$318$	−2416.71	−	2200.18i	−0.426171	−	0.387987i
$319$		−	5479.41i		−	0.961719i
$320$	849.977	+	2942.63i	0.148485	+	0.514056i
$321$	−4724.40			−0.821464
$322$	1499.34	−	1646.90i	0.259488	−	0.285026i
$323$	−536.302			−0.0923859
$324$	60.6494	+	645.156i	0.0103994	+	0.110623i
$325$	1590.49	+	3867.28i	0.271460	+	0.660056i
$326$	−5769.06	−	5252.17i	−0.980119	−	0.892302i
$327$			2141.69i			0.362189i
$328$	3634.62	+	2733.35i	0.611854	+	0.460134i
$329$	−1294.88			−0.216987
$330$	−1328.09	+	1458.80i	−0.221542	+	0.243346i
$331$	−6822.95			−1.13300			−0.566500	−	0.824062i	$-0.691703\pi$
$331$	−6822.95			−1.13300			−0.566500	+	0.824062i	$0.691703\pi$
$332$	−333.899	−	3551.83i	−0.0551960	−	0.587145i
$333$	982.433			0.161673
$334$	−6179.37	+	6787.52i	−1.01234	+	1.11197i
$335$	5757.27			0.938965
$336$	−2560.53	+	485.710i	−0.415739	+	0.0788620i
$337$	6372.16			1.03001			0.515005	−	0.857187i	$-0.327790\pi$
$337$	6372.16			1.03001			0.515005	+	0.857187i	$0.327790\pi$
$338$	−6208.62	+	259.878i	−0.999125	+	0.0418210i
$339$			1514.14i			0.242587i
$340$	−48.0400	−	511.023i	−0.00766275	−	0.0815122i
$341$			1914.48i			0.304032i
$342$	941.273	+	856.937i	0.148825	+	0.135491i
$343$		−	6810.69i		−	1.07214i
$344$	−968.568	−	728.394i	−0.151807	−	0.114164i
$345$		−	1041.10i		−	0.162467i
$346$	5662.95	−	6220.27i	0.879890	−	0.966485i
$347$			1212.23i			0.187539i	0.995594	+	0.0937697i	$0.0298917\pi$
$347$			1212.23i			0.187539i	−0.995594	+	0.0937697i	$0.970108\pi$
$348$	−316.701	−	3368.89i	−0.0487843	−	0.518941i
$349$	5662.28			0.868467			0.434234	−	0.900800i	$-0.357019\pi$
$349$	5662.28			0.868467			0.434234	+	0.900800i	$0.357019\pi$
$350$	−2305.79	+	2532.72i	−0.352142	+	0.386799i
$351$	−1170.43	+	481.361i	−0.177985	+	0.0731998i
$352$	5993.70	+	3683.58i	0.907572	+	0.557772i
$353$			8108.34i			1.22256i	0.791415	+	0.611279i	$0.209345\pi$
$353$			8108.34i			1.22256i	−0.791415	+	0.611279i	$0.790655\pi$
$354$	1747.57	−	1919.56i	0.262379	−	0.288202i
$355$		−	1168.15i		−	0.174645i
$356$	4169.32	−	391.947i	0.620712	−	0.0583515i
$357$	436.737			0.0647467
$358$	6734.03	−	7396.77i	0.994148	−	1.09199i
$359$		−	668.684i		−	0.0983058i	−0.998791	−	0.0491529i	$-0.984348\pi$
$359$		−	668.684i		−	0.0983058i	0.998791	−	0.0491529i	$-0.0156522\pi$
$360$	−732.229	+	973.668i	−0.107200	+	0.142547i
$361$	−4358.48			−0.635440
$362$	−4693.50	+	5155.42i	−0.681451	+	0.748517i
$363$			538.240i			0.0778244i
$364$	−2368.07	−	4505.48i	−0.340990	−	0.648768i
$365$		−	1898.08i		−	0.272192i
$366$	−1031.99	−	939.526i	−0.147385	−	0.134180i
$367$	0.283026			4.02556e−5			2.01278e−5	−	1.00000i	$-0.499994\pi$
$367$	0.283026			4.02556e−5			2.01278e−5		1.00000i	$0.499994\pi$
$368$	−3647.62	+	691.920i	−0.516699	+	0.0980132i
$369$			1808.84i			0.255188i
$370$	1365.80	+	1243.42i	0.191904	+	0.174709i
$371$	5228.17			0.731626
$372$	110.654	+	1177.07i	0.0154224	+	0.164055i
$373$		−	2513.80i		−	0.348954i	−0.984661	−	0.174477i	$-0.944177\pi$
$373$		−	2513.80i		−	0.348954i	0.984661	−	0.174477i	$-0.0558234\pi$
$374$	−871.767	−	793.658i	−0.120529	−	0.109730i
$375$			3844.44i			0.529402i
$376$	1725.14	+	1297.36i	0.236615	+	0.177942i
$377$	6111.78	−	2513.58i	0.834941	−	0.343385i
$378$	−766.526	−	697.846i	−0.104301	−	0.0949559i
$379$	−2611.98			−0.354006			−0.177003	−	0.984210i	$-0.556640\pi$
$379$	−2611.98			−0.354006			−0.177003	+	0.984210i	$0.556640\pi$
$380$	223.987	+	2382.66i	0.0302376	+	0.321652i
$381$		−	5112.42i		−	0.687447i
$382$	9215.15	+	8389.48i	1.23426	+	1.12367i
$383$			3771.33i			0.503149i	0.967838	+	0.251575i	$0.0809484\pi$
$383$			3771.33i			0.503149i	−0.967838	+	0.251575i	$0.919052\pi$
$384$	3897.99	+	1918.34i	0.518017	+	0.254935i
$385$		−	3155.87i		−	0.417762i
$386$	390.209	−	428.612i	0.0514537	−	0.0565176i
$387$		−	482.027i		−	0.0633147i
$388$	1176.80	−	110.628i	0.153977	−	0.0144750i
$389$		−	3406.59i		−	0.444013i	−0.975045	−	0.222006i	$-0.928739\pi$
$389$		−	3406.59i		−	0.444013i	0.975045	−	0.222006i	$-0.0712606\pi$
$390$	−2236.39	−	812.165i	−0.290369	−	0.105450i
$391$	622.157			0.0804701
$392$	−2158.98	+	2870.87i	−0.278176	+	0.369900i
$393$	−5017.05			−0.643961
$394$	2572.65	+	2342.15i	0.328955	+	0.299482i
$395$	1322.93			0.168515
$396$	261.898	+	2785.93i	0.0332345	+	0.353531i
$397$	7698.79			0.973277			0.486639	−	0.873603i	$-0.338223\pi$
$397$	7698.79			0.973277			0.486639	+	0.873603i	$0.338223\pi$
$398$	393.869	+	358.579i	0.0496052	+	0.0451606i
$399$	−2036.30			−0.255494
$400$	5609.55	−	1064.08i	0.701194	−	0.133010i
$401$			6769.22i			0.842990i	0.906831	+	0.421495i	$0.138494\pi$
$401$			6769.22i			0.842990i	−0.906831	+	0.421495i	$0.861506\pi$
$402$	5497.46	−	6038.50i	0.682060	−	0.749186i
$403$	−2135.43	+	878.233i	−0.263953	+	0.108556i
$404$	−5742.07	+	539.797i	−0.707126	+	0.0664750i
$405$	−484.565			−0.0594524
$406$	4002.66	+	3644.03i	0.489283	+	0.445444i
$407$	4242.37			0.516674
$408$	−581.858	−	437.576i	−0.0706036	−	0.0530961i
$409$			15510.5i			1.87517i	0.347750	+	0.937587i	$0.386946\pi$
$409$			15510.5i			1.87517i	−0.347750	+	0.937587i	$0.613054\pi$
$410$	−2289.37	+	2514.68i	−0.275765	+	0.302905i
$411$	4016.23			0.482010
$412$	−956.840	−	10178.3i	−0.114418	−	1.21711i
$413$			4152.66i			0.494768i
$414$	−1091.96	−	994.121i	−0.129630	−	0.118015i
$415$	2667.72			0.315550
$416$	−1359.20	+	8375.19i	−0.160192	+	0.987086i
$417$	3790.00			0.445077
$418$	4064.63	+	3700.45i	0.475616	+	0.433002i
$419$			8562.24i			0.998312i	0.866512	+	0.499156i	$0.166357\pi$
$419$			8562.24i			0.998312i	−0.866512	+	0.499156i	$0.833643\pi$
$420$	−182.404	−	1940.32i	−0.0211914	−	0.225423i
$421$	9584.46			1.10954			0.554772	−	0.832003i	$-0.312806\pi$
$421$	9584.46			1.10954			0.554772	+	0.832003i	$0.312806\pi$
$422$	4785.78	−	5256.77i	0.552057	−	0.606388i
$423$			858.551i			0.0986860i
$424$	−6965.40	−	5238.21i	−0.797806	−	0.599976i
$425$	−956.795			−0.109203
$426$	−1225.21	−	1115.43i	−0.139347	−	0.126861i
$427$	2232.55			0.253022
$428$	−12543.1	+	1179.14i	−1.41657	+	0.133168i
$429$	−5054.18	+	2078.62i	−0.568807	+	0.233932i
$430$	610.080	−	670.122i	0.0684202	−	0.0751539i
$431$			1393.38i			0.155724i	0.996964	+	0.0778618i	$0.0248093\pi$
$431$			1393.38i			0.155724i	−0.996964	+	0.0778618i	$0.975191\pi$
$432$	322.043	+	1697.73i	0.0358665	+	0.189078i
$433$	6186.20			0.686582			0.343291	−	0.939229i	$-0.388458\pi$
$433$	6186.20			0.686582			0.343291	+	0.939229i	$0.388458\pi$
$434$	−1398.51	−	1273.21i	−0.154679	−	0.140820i
$435$	2530.31			0.278895
$436$	534.537	+	5686.11i	0.0587148	+	0.624577i
$437$	−2900.82			−0.317540
$438$	−1990.80	−	1812.42i	−0.217178	−	0.197719i
$439$	−5154.74			−0.560416			−0.280208	−	0.959939i	$-0.590403\pi$
$439$	−5154.74			−0.560416			−0.280208	+	0.959939i	$0.590403\pi$
$440$	−3161.93	+	4204.52i	−0.342589	+	0.455551i
$441$	−1428.74			−0.154275
$442$	485.345	−	1336.45i	0.0522296	−	0.143820i
$443$			245.694i			0.0263505i	0.999913	+	0.0131752i	$0.00419393\pi$
$443$			245.694i			0.0263505i	−0.999913	+	0.0131752i	$0.995806\pi$
$444$	2608.32	−	245.202i	0.278796	−	0.0262089i
$445$			3131.50i			0.333590i
$446$	3587.99	−	3941.11i	0.380934	−	0.418424i
$447$			3251.52i			0.344053i
$448$	−6676.88	+	1928.61i	−0.704136	+	0.203389i
$449$			1111.53i			0.116830i	0.998292	+	0.0584148i	$0.0186046\pi$
$449$			1111.53i			0.116830i	−0.998292	+	0.0584148i	$0.981395\pi$
$450$	1679.29	+	1528.83i	0.175916	+	0.160155i
$451$			7810.98i			0.815531i
$452$	377.909	+	4019.99i	0.0393260	+	0.418328i
$453$	−8465.21			−0.877992
$454$	−8925.78	−	8126.04i	−0.922704	−	0.840031i
$455$	3520.08	−	1447.70i	0.362690	−	0.149163i
$456$	2712.92	+	2040.20i	0.278606	+	0.209520i
$457$			14132.1i			1.44655i	0.690560	+	0.723275i	$0.257364\pi$
$457$			14132.1i			1.44655i	−0.690560	+	0.723275i	$0.742636\pi$
$458$	13114.7	+	11939.6i	1.33801	+	1.21813i
$459$		−	289.573i		−	0.0294469i
$460$	−259.845	−	2764.09i	−0.0263376	−	0.280166i
$461$	−11456.8			−1.15748			−0.578739	−	0.815513i	$-0.696455\pi$
$461$	−11456.8			−1.15748			−0.578739	+	0.815513i	$0.696455\pi$
$462$	−3310.03	−	3013.46i	−0.333326	−	0.303460i
$463$		−	15209.1i		−	1.52663i	−0.646029	−	0.763313i	$-0.723571\pi$
$463$		−	15209.1i		−	1.52663i	0.646029	−	0.763313i	$-0.276429\pi$
$464$	−1681.66	−	8865.23i	−0.168252	−	0.886978i
$465$	−884.079			−0.0881682
$466$	12063.7	+	10982.8i	1.19923	+	1.09178i
$467$			13191.1i			1.30709i	0.756887	+	0.653546i	$0.226719\pi$
$467$			13191.1i			1.30709i	−0.756887	+	0.653546i	$0.773281\pi$
$468$	−2987.30	+	1570.12i	−0.295060	+	0.155083i
$469$			13063.3i			1.28616i
$470$	−1086.63	+	1193.57i	−0.106644	+	0.117139i
$471$	−428.517			−0.0419215
$472$	4160.63	−	5532.52i	0.405739	−	0.539523i
$473$		−	2081.50i		−	0.202341i
$474$	1263.23	−	1387.55i	0.122409	−	0.134456i
$475$	4461.08			0.430922
$476$	1159.52	−	109.003i	0.111652	−	0.0104961i
$477$		−	3466.47i		−	0.332744i
$478$	−6869.01	+	7545.03i	−0.657283	+	0.721970i
$479$			10982.4i			1.04760i	0.851843	+	0.523798i	$0.175485\pi$
$479$			10982.4i			1.04760i	−0.851843	+	0.523798i	$0.824515\pi$
$480$	−1701.03	+	2767.80i	−0.161752	+	0.263192i
$481$	1946.11	+	4731.97i	0.184480	+	0.448564i
$482$	2121.44	−	2330.22i	0.200475	−	0.220205i
$483$	2362.28			0.222541
$484$	134.337	+	1429.01i	0.0126162	+	0.134204i
$485$			883.874i			0.0827519i
$486$	−462.698	+	508.235i	−0.0431860	+	0.0474362i
$487$		−	933.005i		−	0.0868141i	−0.999057	−	0.0434071i	$-0.986179\pi$
$487$		−	933.005i		−	0.0868141i	0.999057	−	0.0434071i	$-0.0138212\pi$
$488$	−2974.39	−	2236.83i	−0.275910	−	0.207493i
$489$		−	8275.01i		−	0.765253i
$490$	−1986.26	−	1808.30i	−0.183123	−	0.166715i
$491$			2730.02i			0.250925i	0.992098	+	0.125462i	$0.0400414\pi$
$491$			2730.02i			0.250925i	−0.992098	+	0.125462i	$0.959959\pi$
$492$	451.461	+	4802.40i	0.0413687	+	0.440059i
$493$			1512.10i			0.138137i
$494$	−2262.93	+	6231.23i	−0.206101	+	0.567523i
$495$	−2092.46			−0.189998
$496$	587.562	+	3097.47i	0.0531901	+	0.280404i
$497$	2650.55			0.239222
$498$	2547.33	−	2798.03i	0.229214	−	0.251773i
$499$	17407.9			1.56169			0.780847	−	0.624723i	$-0.214788\pi$
$499$	17407.9			1.56169			0.780847	+	0.624723i	$0.214788\pi$
$500$	959.517	+	10206.8i	0.0858218	+	0.912926i
$501$	−9735.86			−0.868196
$502$	10441.6	−	11469.2i	0.928350	−	1.01971i
$503$	9323.78			0.826495			0.413247	−	0.910619i	$-0.364394\pi$
$503$	9323.78			0.826495			0.413247	+	0.910619i	$0.364394\pi$
$504$	−2209.27	−	1661.44i	−0.195255	−	0.146838i
$505$		−	4312.77i		−	0.380031i
$506$	−4715.32	−	4292.84i	−0.414272	−	0.377154i
$507$	−4637.03	−	4683.94i	−0.406189	−	0.410298i
$508$	−1275.99	−	13573.3i	−0.111443	−	1.18547i
$509$	−2250.01			−0.195933			−0.0979667	−	0.995190i	$-0.531234\pi$
$509$	−2250.01			−0.195933			−0.0979667	+	0.995190i	$0.531234\pi$
$510$	366.500	−	402.569i	0.0318214	−	0.0349531i
$511$	4306.77			0.372838
$512$	10827.8	+	4120.24i	0.934621	+	0.355645i
$513$			1350.14i			0.116199i
$514$	11867.0	+	10803.7i	1.01834	+	0.927103i
$515$	7644.78			0.654115
$516$	−120.307	−	1279.76i	−0.0102640	−	0.109183i
$517$			3707.42i			0.315381i
$518$	−2821.35	+	3099.01i	−0.239310	+	0.262863i
$519$	8922.21			0.754608
$520$	−6140.23	−	1598.10i	−0.517821	−	0.134771i
$521$	−6504.14			−0.546932			−0.273466	−	0.961882i	$-0.588170\pi$
$521$	−6504.14			−0.546932			−0.273466	+	0.961882i	$0.588170\pi$
$522$	2416.13	−	2653.91i	0.202588	−	0.222526i
$523$		−	863.189i		−	0.0721695i	−0.999349	−	0.0360847i	$-0.988511\pi$
$523$		−	863.189i		−	0.0721695i	0.999349	−	0.0360847i	$-0.0114886\pi$
$524$	−13320.1	+	1252.18i	−1.11048	+	0.104393i
$525$	−3632.88			−0.302003
$526$	−5038.32	−	4586.90i	−0.417645	−	0.380225i
$527$		−	528.320i		−	0.0436698i
$528$	1390.66	+	7331.16i	0.114622	+	0.604257i
$529$	−8801.80			−0.723416
$530$	4387.36	−	4819.15i	0.359575	−	0.394963i
$531$	2753.37			0.225021
$532$	−5406.29	+	508.231i	−0.440587	+	0.0414184i
$533$	−8712.42	+	3583.14i	−0.708024	+	0.291188i
$534$	3284.47	+	2990.19i	0.266166	+	0.242318i
$535$		−	9420.89i		−	0.761310i
$536$	13088.4	−	17404.1i	1.05473	−	1.40250i
$537$	10609.8			0.852597
$538$	−2142.15	+	2352.98i	−0.171663	+	0.188558i
$539$	−6169.64			−0.493033
$540$	−1286.50	+	120.941i	−0.102523	+	0.00963788i
$541$	6178.96			0.491043			0.245522	−	0.969391i	$-0.421041\pi$
$541$	6178.96			0.491043			0.245522	+	0.969391i	$0.421041\pi$
$542$	10828.5	−	11894.2i	0.858160	−	0.942617i
$543$	−7394.81			−0.584423
$544$	−1654.02	−	1016.52i	−0.130360	−	0.0801159i
$545$	−4270.74			−0.335667
$546$	1842.82	−	5074.40i	0.144442	−	0.397737i
$547$			10494.5i			0.820319i	0.912014	+	0.410160i	$0.134527\pi$
$547$			10494.5i			0.820319i	−0.912014	+	0.410160i	$0.865473\pi$
$548$	10662.9	−	1002.39i	0.831201	−	0.0781390i
$549$		−	1480.26i		−	0.115075i
$550$	7251.55	+	6601.82i	0.562195	+	0.511823i
$551$		−	7050.20i		−	0.545097i
$552$	−3147.23	−	2366.81i	−0.242672	−	0.182497i
$553$			3001.74i			0.230826i
$554$	17468.1	−	19187.2i	1.33962	−	1.47146i
$555$			1959.06i			0.149834i
$556$	10062.3	−	945.931i	0.767512	−	0.0721518i
$557$	−21135.1			−1.60776			−0.803882	−	0.594789i	$-0.797235\pi$
$557$	−21135.1			−1.60776			−0.803882	+	0.594789i	$0.797235\pi$
$558$	−844.183	+	927.264i	−0.0640450	+	0.0703481i
$559$	2321.72	−	954.851i	0.175668	−	0.0722467i
$560$	−968.551	−	5105.94i	−0.0730871	−	0.385295i
$561$		−	1250.44i		−	0.0941064i
$562$	6034.40	−	6628.28i	0.452928	−	0.497504i
$563$		−	2509.76i		−	0.187876i	−0.995578	−	0.0939378i	$-0.970055\pi$
$563$		−	2509.76i		−	0.187876i	0.995578	−	0.0939378i	$-0.0299455\pi$
$564$	214.282	+	2279.42i	0.0159981	+	0.170179i
$565$	−3019.34			−0.224823
$566$	15681.7	−	17225.0i	1.16458	−	1.27919i
$567$		−	1099.49i		−	0.0814357i
$568$	−3531.28	−	2655.64i	−0.260861	−	0.196176i
$569$	−16497.7			−1.21550			−0.607751	−	0.794127i	$-0.707928\pi$
$569$	−16497.7			−1.21550			−0.607751	+	0.794127i	$0.707928\pi$
$570$	−1708.81	+	1876.99i	−0.125569	+	0.137927i
$571$			1739.60i			0.127496i	0.997966	+	0.0637478i	$0.0203053\pi$
$571$			1739.60i			0.127496i	−0.997966	+	0.0637478i	$0.979695\pi$
$572$	−12899.9	+	6780.12i	−0.942954	+	0.495614i
$573$			13218.0i			0.963681i
$574$	−5705.85	−	5194.61i	−0.414909	−	0.377733i
$575$	−5175.23			−0.375343
$576$	1278.74	+	4427.02i	0.0925015	+	0.320241i
$577$			13777.0i			0.994011i	0.867747	+	0.497005i	$0.165567\pi$
$577$			13777.0i			0.994011i	−0.867747	+	0.497005i	$0.834433\pi$
$578$	−10035.0	−	9135.86i	−0.722145	−	0.657442i
$579$	614.791			0.0441275
$580$	6717.89	−	631.531i	0.480940	−	0.0452119i
$581$			6053.10i			0.432229i
$582$	927.049	+	843.987i	0.0660265	+	0.0601106i
$583$		−	14969.0i		−	1.06338i
$584$	−5737.84	−	4315.04i	−0.406564	−	0.305749i
$585$	−959.879	−	2333.95i	−0.0678395	−	0.164952i
$586$	13248.2	+	12061.2i	0.933924	+	0.850246i
$587$	10322.4			0.725814			0.362907	−	0.931825i	$-0.381784\pi$
$587$	10322.4			0.725814			0.362907	+	0.931825i	$0.381784\pi$
$588$	−3793.26	+	356.594i	−0.266040	+	0.0250097i
$589$			2463.30i			0.172324i
$590$	3827.78	+	3484.82i	0.267097	+	0.243166i
$591$			3690.15i			0.256840i
$592$	6863.79	−	1302.00i	0.476521	−	0.0903917i
$593$		−	13699.1i		−	0.948656i	−0.880348	−	0.474328i	$-0.842691\pi$
$593$		−	13699.1i		−	0.948656i	0.880348	−	0.474328i	$-0.157309\pi$
$594$	−1998.03	+	2194.67i	−0.138014	+	0.151597i
$595$			870.896i			0.0600054i
$596$	811.534	+	8632.67i	0.0557747	+	0.593302i
$597$			564.956i			0.0387305i
$598$	2625.19	−	7228.77i	0.179519	−	0.494325i
$599$	−27000.4			−1.84174			−0.920872	−	0.389865i	$-0.872522\pi$
$599$	−27000.4			−1.84174			−0.920872	+	0.389865i	$0.872522\pi$
$600$	4840.02	+	3639.85i	0.329322	+	0.247660i
$601$	−28619.0			−1.94242			−0.971209	−	0.238228i	$-0.923434\pi$
$601$	−28619.0			−1.94242			−0.971209	+	0.238228i	$0.923434\pi$
$602$	1520.52	+	1384.28i	0.102943	+	0.0937195i
$603$	8661.48			0.584946
$604$	−22474.8	+	2112.80i	−1.51405	+	0.142332i
$605$	−1073.30			−0.0721255
$606$	−4523.44	−	4118.14i	−0.303221	−	0.276053i
$607$	14875.0			0.994658			0.497329	−	0.867562i	$-0.334314\pi$
$607$	14875.0			0.994658			0.497329	+	0.867562i	$0.334314\pi$
$608$	7711.91	+	4739.56i	0.514407	+	0.316142i
$609$			5741.32i			0.382020i
$610$	1873.50	−	2057.89i	0.124354	−	0.136592i
$611$	−4135.28	+	1700.71i	−0.273806	+	0.112608i
$612$	−72.2734	−	768.805i	−0.00477366	−	0.0507796i
$613$	−14835.8			−0.977510			−0.488755	−	0.872421i	$-0.662549\pi$
$613$	−14835.8			−0.977510			−0.488755	+	0.872421i	$0.662549\pi$
$614$	−3037.65	−	2765.49i	−0.199658	−	0.181769i
$615$	−3607.00			−0.236501
$616$	−9540.12	−	7174.47i	−0.623997	−	0.469266i
$617$			4086.98i			0.266670i	0.991071	+	0.133335i	$0.0425687\pi$
$617$			4086.98i			0.266670i	−0.991071	+	0.133335i	$0.957431\pi$
$618$	7299.79	−	8018.21i	0.475147	−	0.521909i
$619$	−1121.89			−0.0728475			−0.0364238	−	0.999336i	$-0.511597\pi$
$619$	−1121.89			−0.0728475			−0.0364238	+	0.999336i	$0.511597\pi$
$620$	−2347.20	+	220.654i	−0.152041	+	0.0142930i
$621$		−	1566.28i		−	0.101212i
$622$	−2921.86	−	2660.07i	−0.188354	−	0.171478i
$623$	−7105.43			−0.456939
$624$	−7539.29	+	4914.19i	−0.483675	+	0.315264i
$625$	3485.36			0.223063
$626$	−15847.7	−	14427.7i	−1.01182	−	0.921163i
$627$			5830.21i			0.371349i
$628$	−1137.70	+	106.952i	−0.0722914	+	0.00679593i
$629$	−1170.72			−0.0742128
$630$	1391.57	−	1528.52i	0.0880024	−	0.0966633i
$631$		−	9262.65i		−	0.584375i	−0.956361	−	0.292187i	$-0.905617\pi$
$631$		−	9262.65i		−	0.584375i	0.956361	−	0.292187i	$-0.0943831\pi$
$632$	3007.50	−	3999.17i	0.189291	−	0.251706i
$633$	7540.19			0.473453
$634$	−19866.8	−	18086.8i	−1.24450	−	1.13299i
$635$	10194.7			0.637106
$636$	−865.182	−	9203.35i	−0.0539414	−	0.573799i
$637$	−2830.21	−	6881.66i	−0.176039	−	0.428040i
$638$	10433.4	−	11460.2i	0.647432	−	0.711150i
$639$		−	1757.41i		−	0.108798i
$640$	−3825.35	+	7772.96i	−0.236266	+	0.480083i
$641$	−16241.4			−1.00078			−0.500388	−	0.865801i	$-0.666809\pi$
$641$	−16241.4			−1.00078			−0.500388	+	0.865801i	$0.666809\pi$
$642$	−9881.08	−	8995.75i	−0.607438	−	0.553013i
$643$	−23823.8			−1.46115			−0.730574	−	0.682834i	$-0.760747\pi$
$643$	−23823.8			−1.46115			−0.730574	+	0.682834i	$0.760747\pi$
$644$	6271.76	−	589.592i	0.383761	−	0.0360763i
$645$	961.207			0.0586783
$646$	−1121.68	−	1021.18i	−0.0683154	−	0.0621945i
$647$	11713.0			0.711723			0.355861	−	0.934539i	$-0.384188\pi$
$647$	11713.0			0.711723			0.355861	+	0.934539i	$0.384188\pi$
$648$	−1101.60	+	1464.83i	−0.0667821	+	0.0888022i
$649$	11889.7			0.719122
$650$	−4037.20	+	11116.9i	−0.243619	+	0.670832i
$651$		−	2005.99i		−	0.120770i
$652$	−2065.33	−	21969.8i	−0.124056	−	1.31964i
$653$		−	29517.9i		−	1.76895i	−0.466587	−	0.884475i	$-0.654516\pi$
$653$		−	29517.9i		−	1.76895i	0.466587	−	0.884475i	$-0.345484\pi$
$654$	−4078.01	+	4479.36i	−0.243827	+	0.267824i
$655$		−	10004.5i		−	0.596805i
$656$	2397.22	+	12637.5i	0.142676	+	0.752152i
$657$		−	2855.55i		−	0.169567i
$658$	−2708.23	−	2465.58i	−0.160453	−	0.146077i
$659$			12155.7i			0.718541i	0.933234	+	0.359270i	$0.116974\pi$
$659$			12155.7i			0.718541i	−0.933234	+	0.359270i	$0.883026\pi$
$660$	−5555.40	+	522.249i	−0.327642	+	0.0308008i
$661$	−24245.7			−1.42670			−0.713349	−	0.700809i	$-0.752822\pi$
$661$	−24245.7			−1.42670			−0.713349	+	0.700809i	$0.752822\pi$
$662$	−14270.2	−	12991.6i	−0.837806	−	0.762739i
$663$	1394.75	−	573.618i	0.0817009	−	0.0336010i
$664$	6064.72	−	8064.45i	0.354453	−	0.471327i
$665$		−	4060.57i		−	0.236785i
$666$	2054.76	+	1870.66i	0.119550	+	0.108839i
$667$			8178.84i			0.474791i
$668$	−25848.3	+	2429.93i	−1.49716	+	0.140744i
$669$	5653.03			0.326695
$670$	12041.3	+	10962.5i	0.694325	+	0.632114i
$671$		−	6392.11i		−	0.367756i
$672$	−6280.19	−	3859.66i	−0.360511	−	0.221562i
$673$	8527.42			0.488422			0.244211	−	0.969722i	$-0.421471\pi$
$673$	8527.42			0.488422			0.244211	+	0.969722i	$0.421471\pi$
$674$	13327.4	+	12133.3i	0.761649	+	0.693406i
$675$			2408.73i			0.137351i
$676$	−13480.2	−	11278.3i	−0.766965	−	0.641689i
$677$		−	6576.03i		−	0.373320i	−0.982425	−	0.186660i	$-0.940234\pi$
$677$		−	6576.03i		−	0.373320i	0.982425	−	0.186660i	$-0.0597662\pi$
$678$	−2883.09	+	3166.83i	−0.163310	+	0.179383i
$679$	−2005.52			−0.113350
$680$	872.567	−	1160.28i	0.0492080	−	0.0654334i
$681$		−	12802.9i		−	0.720425i
$682$	−3645.37	+	4004.14i	−0.204675	+	0.224819i
$683$	20606.9			1.15447			0.577234	−	0.816579i	$-0.304132\pi$
$683$	20606.9			1.15447			0.577234	+	0.816579i	$0.304132\pi$
$684$	336.976	+	3584.57i	0.0188371	+	0.200379i
$685$			8008.74i			0.446713i
$686$	12968.3	−	14244.6i	0.721766	−	0.792800i
$687$			18811.4i			1.04469i
$688$	−638.822	−	3367.69i	−0.0353995	−	0.186616i
$689$	16696.5	−	6866.76i	0.923204	−	0.379685i
$690$	1982.37	−	2177.47i	0.109373	−	0.120137i
$691$	−21339.5			−1.17481			−0.587403	−	0.809294i	$-0.699850\pi$
$691$	−21339.5			−1.17481			−0.587403	+	0.809294i	$0.699850\pi$
$692$	23688.1	−	2226.86i	1.30128	−	0.122330i
$693$		−	4747.83i		−	0.260253i
$694$	−2308.22	+	2535.39i	−0.126252	+	0.138677i
$695$			7557.62i			0.412485i
$696$	5752.35	−	7649.07i	0.313279	−	0.416577i
$697$		−	2155.52i		−	0.117139i
$698$	11842.7	+	10781.6i	0.642194	+	0.584655i
$699$			17303.9i			0.936329i
$700$	−9645.14	+	906.715i	−0.520789	+	0.0489580i
$701$		−	11788.5i		−	0.635155i	−0.948232	−	0.317578i	$-0.897131\pi$
$701$		−	11788.5i		−	0.635155i	0.948232	−	0.317578i	$-0.102869\pi$
$702$	−3364.52	−	1221.86i	−0.180891	−	0.0656922i
$703$	5458.53			0.292848
$704$	5521.89	+	19116.9i	0.295617	+	1.02343i
$705$	−1712.03			−0.0914593
$706$	−15439.1	+	16958.6i	−0.823031	+	0.904030i
$707$	9785.74			0.520553
$708$	7310.09	−	687.202i	0.388037	−	0.0364783i
$709$	10555.8			0.559140			0.279570	−	0.960125i	$-0.409808\pi$
$709$	10555.8			0.559140			0.279570	+	0.960125i	$0.409808\pi$
$710$	2224.28	−	2443.19i	0.117571	−	0.129142i
$711$	1990.27			0.104980
$712$	9466.44	+	7119.07i	0.498272	+	0.374717i
$713$		−	2857.64i		−	0.150098i
$714$	913.437	+	831.594i	0.0478775	+	0.0435877i
$715$	−4144.97	−	10078.5i	−0.216802	−	0.527154i
$716$	28168.5	−	2648.05i	1.47026	−	0.138215i
$717$	−10822.4			−0.563697
$718$	1273.25	−	1398.55i	0.0661798	−	0.0726930i
$719$	−13395.8			−0.694826			−0.347413	−	0.937712i	$-0.612940\pi$
$719$	−13395.8			−0.694826			−0.347413	+	0.937712i	$0.612940\pi$
$720$	−3385.43	+	642.185i	−0.175232	+	0.0332400i
$721$			17346.1i			0.895983i
$722$	−9115.77	−	8299.01i	−0.469881	−	0.427780i
$723$	3342.41			0.171930
$724$	−19632.9	+	1845.64i	−1.00781	+	0.0947414i
$725$		−	12578.0i		−	0.644323i
$726$	−1024.87	+	1125.73i	−0.0523917	+	0.0575479i
$727$	29111.3			1.48512			0.742558	−	0.669782i	$-0.233613\pi$
$727$	29111.3			1.48512			0.742558	+	0.669782i	$0.233613\pi$
$728$	3626.10	−	13932.3i	0.184605	−	0.709292i
$729$	−729.000			−0.0370370
$730$	3614.14	−	3969.83i	0.183240	−	0.201274i
$731$			574.411i			0.0290634i
$732$	−369.453	−	3930.04i	−0.0186549	−	0.198440i
$733$	890.638			0.0448792			0.0224396	−	0.999748i	$-0.492857\pi$
$733$	890.638			0.0448792			0.0224396	+	0.999748i	$0.492857\pi$
$734$	0.591949	+	0.538911i	2.97673e−5	+	2.71002e-5i
$735$		−	2849.05i		−	0.142978i
$736$	−8946.48	−	5498.30i	−0.448059	−	0.275367i
$737$	37402.2			1.86937
$738$	−3444.22	+	3783.19i	−0.171793	+	0.188701i
$739$	−13313.5			−0.662715			−0.331358	−	0.943505i	$-0.607507\pi$
$739$	−13313.5			−0.662715			−0.331358	+	0.943505i	$0.607507\pi$
$740$	488.955	+	5201.24i	0.0242897	+	0.258380i
$741$	−6503.06	+	2674.50i	−0.322397	+	0.132592i
$742$	10934.7	+	9954.99i	0.541006	+	0.492533i
$743$		−	30168.6i		−	1.48961i	−0.667282	−	0.744805i	$-0.732543\pi$
$743$		−	30168.6i		−	1.48961i	0.667282	−	0.744805i	$-0.267457\pi$
$744$	−2009.84	+	2672.55i	−0.0990381	+	0.131694i
$745$	−6483.84			−0.318859
$746$	4786.55	−	5257.62i	0.234917	−	0.258037i
$747$	4013.43			0.196578
$748$	−312.093	−	3319.87i	−0.0152557	−	0.162282i
$749$	21376.2			1.04281
$750$	−7320.22	+	8040.64i	−0.356395	+	0.391470i
$751$	−12030.7			−0.584562			−0.292281	−	0.956333i	$-0.594414\pi$
$751$	−12030.7			−0.584562			−0.292281	+	0.956333i	$0.594414\pi$
$752$	1137.82	+	5998.29i	0.0551757	+	0.290871i
$753$	16451.2			0.796168
$754$	17568.9	+	6380.32i	0.848571	+	0.308166i
$755$		−	16880.4i		−	0.813698i
$756$	−274.416	−	2919.09i	−0.0132016	−	0.140432i
$757$		−	4839.04i		−	0.232335i	−0.993230	−	0.116168i	$-0.962939\pi$
$757$		−	4839.04i		−	0.232335i	0.993230	−	0.116168i	$-0.0370610\pi$
$758$	−5462.96	−	4973.49i	−0.261773	−	0.238318i
$759$		−	6763.55i		−	0.323453i
$760$	−4068.36	+	5409.83i	−0.194178	+	0.258204i
$761$		−	10293.1i		−	0.490309i	−0.969484	−	0.245155i	$-0.921161\pi$
$761$		−	10293.1i		−	0.490309i	0.969484	−	0.245155i	$-0.0788388\pi$
$762$	9734.60	−	10692.6i	0.462792	−	0.508338i
$763$		−	9690.38i		−	0.459784i
$764$	3299.02	+	35093.2i	0.156223	+	1.66182i
$765$	577.436			0.0272905
$766$	−7181.02	+	7887.75i	−0.338722	+	0.372057i
$767$	5454.17	+	13261.8i	0.256765	+	0.624325i
$768$	4499.93	+	11434.4i	0.211429	+	0.537244i
$769$			29234.3i			1.37089i	0.728124	+	0.685445i	$0.240392\pi$
$769$			29234.3i			1.37089i	−0.728124	+	0.685445i	$0.759608\pi$
$770$	6009.12	−	6600.51i	0.281238	−	0.308917i
$771$			17021.7i			0.795099i
$772$	1632.25	−	153.443i	0.0760956	−	0.00715355i
$773$	9451.29			0.439766			0.219883	−	0.975526i	$-0.429432\pi$
$773$	9451.29			0.439766			0.219883	+	0.975526i	$0.429432\pi$
$774$	917.830	−	1008.16i

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:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	312.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.09150 + 1.90411i) q^{2} +3.00000i q^{3} +(0.748758 + 7.96488i) q^{4} -5.98228 q^{5} +(-5.71232 + 6.27450i) q^{6} -13.5739i q^{7} +(-13.6000 + 18.0843i) q^{8} -9.00000 q^{9} +(-12.5120 - 11.3909i) q^{10} -38.8640 q^{11} +(-23.8946 + 2.24627i) q^{12} +(-17.8282 - 43.3492i) q^{13} +(25.8461 - 28.3898i) q^{14} -17.9469i q^{15} +(-62.8787 + 11.9275i) q^{16} +10.7249 q^{17} +(-18.8235 - 17.1370i) q^{18} -50.0052 q^{19} +(-4.47928 - 47.6482i) q^{20} +40.7217 q^{21} +(-81.2842 - 74.0013i) q^{22} +58.0103 q^{23} +(-54.2528 - 40.7999i) q^{24} -89.2123 q^{25} +(45.2539 - 124.612i) q^{26} -27.0000i q^{27} +(108.115 - 10.1636i) q^{28} +140.989i q^{29} +(34.1727 - 37.5359i) q^{30} -49.2610i q^{31} +(-154.222 - 94.7813i) q^{32} -116.592i q^{33} +(22.4312 + 20.4214i) q^{34} +81.2029i q^{35} +(-6.73882 - 71.6839i) q^{36} -109.159 q^{37} +(-104.586 - 95.2152i) q^{38} +(130.048 - 53.4845i) q^{39} +(81.3588 - 108.185i) q^{40} -200.982i q^{41} +(85.1695 + 77.5384i) q^{42} +53.5585i q^{43} +(-29.0998 - 309.548i) q^{44} +53.8406 q^{45} +(121.329 + 110.458i) q^{46} -95.3945i q^{47} +(-35.7826 - 188.636i) q^{48} +158.749 q^{49} +(-186.588 - 169.870i) q^{50} +32.1748i q^{51} +(331.923 - 174.457i) q^{52} +385.163i q^{53} +(51.4109 - 56.4705i) q^{54} +232.496 q^{55} +(245.474 + 184.604i) q^{56} -150.016i q^{57} +(-268.459 + 294.879i) q^{58} -305.930 q^{59} +(142.945 - 13.4378i) q^{60} +164.474i q^{61} +(93.7981 - 103.029i) q^{62} +122.165i q^{63} +(-142.082 - 491.891i) q^{64} +(106.653 + 259.327i) q^{65} +(222.004 - 243.853i) q^{66} -962.386 q^{67} +(8.03037 + 85.4228i) q^{68} +174.031i q^{69} +(-154.619 + 169.836i) q^{70} +195.268i q^{71} +(122.400 - 162.759i) q^{72} +317.283i q^{73} +(-228.307 - 207.851i) q^{74} -267.637i q^{75} +(-37.4418 - 398.285i) q^{76} +527.537i q^{77} +(373.835 + 135.762i) q^{78} -221.141 q^{79} +(376.158 - 71.3539i) q^{80} +81.0000 q^{81} +(382.691 - 420.354i) q^{82} -445.937 q^{83} +(30.4907 + 324.344i) q^{84} -64.1596 q^{85} +(-101.981 + 112.018i) q^{86} -422.968 q^{87} +(528.549 - 702.828i) q^{88} -523.463i q^{89} +(112.608 + 102.518i) q^{90} +(-588.418 + 241.998i) q^{91} +(43.4357 + 462.045i) q^{92} +147.783 q^{93} +(181.641 - 199.518i) q^{94} +299.145 q^{95} +(284.344 - 462.667i) q^{96} -147.749i q^{97} +(332.024 + 302.275i) q^{98} +349.776 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 756 q^{9} - 36 q^{10} + 12 q^{12} - 208 q^{14} - 148 q^{16} - 104 q^{17} + 620 q^{22} + 2188 q^{25} + 444 q^{26} - 204 q^{30} - 40 q^{38} - 1924 q^{40} + 192 q^{42} + 624 q^{48} - 3396 q^{49} - 1292 q^{52}+ \cdots + 2480 q^{95}+O(q^{100}) \) 84 * q - 756 * q^9 - 36 * q^10 + 12 * q^12 - 208 * q^14 - 148 * q^16 - 104 * q^17 + 620 * q^22 + 2188 * q^25 + 444 * q^26 - 204 * q^30 - 40 * q^38 - 1924 * q^40 + 192 * q^42 + 624 * q^48 - 3396 * q^49 - 1292 * q^52 - 1616 * q^55 + 608 * q^56 - 2120 * q^62 - 2856 * q^64 + 696 * q^65 - 396 * q^66 - 2536 * q^68 - 3936 * q^74 - 156 * q^78 + 3160 * q^79 + 6804 * q^81 + 4276 * q^82 - 2088 * q^87 + 1780 * q^88 + 324 * q^90 + 4792 * q^92 - 860 * q^94 + 2480 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more