LMFDB - Embedded newform 471.4.b.a.313.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [471,4,Mod(313,471)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(471, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("471.313");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 471 = 3 \cdot 157 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	471.b (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:	$27.7898996127$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			313.16
Character	$\chi$	$=$	471.313
Dual form			471.4.b.a.313.25

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.38921i q^{2} -3.00000 q^{3} +6.07011 q^{4} +4.30021i q^{5} +4.16762i q^{6} -36.7534i q^{7} -19.5463i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-1.38921i q^{2} -3.00000 q^{3} +6.07011 q^{4} +4.30021i q^{5} +4.16762i q^{6} -36.7534i q^{7} -19.5463i q^{8} +9.00000 q^{9} +5.97388 q^{10} -64.0690 q^{11} -18.2103 q^{12} +37.9498 q^{13} -51.0580 q^{14} -12.9006i q^{15} +21.4070 q^{16} -114.416 q^{17} -12.5029i q^{18} +66.8100 q^{19} +26.1027i q^{20} +110.260i q^{21} +89.0050i q^{22} +90.8994i q^{23} +58.6388i q^{24} +106.508 q^{25} -52.7201i q^{26} -27.0000 q^{27} -223.097i q^{28} -7.45942i q^{29} -17.9216 q^{30} -236.798 q^{31} -186.109i q^{32} +192.207 q^{33} +158.947i q^{34} +158.047 q^{35} +54.6309 q^{36} -184.469 q^{37} -92.8129i q^{38} -113.849 q^{39} +84.0531 q^{40} -11.4862i q^{41} +153.174 q^{42} +290.703i q^{43} -388.905 q^{44} +38.7019i q^{45} +126.278 q^{46} +7.84607 q^{47} -64.2211 q^{48} -1007.81 q^{49} -147.962i q^{50} +343.248 q^{51} +230.359 q^{52} -54.2257i q^{53} +37.5086i q^{54} -275.510i q^{55} -718.392 q^{56} -200.430 q^{57} -10.3627 q^{58} -457.026i q^{59} -78.3082i q^{60} -410.690i q^{61} +328.962i q^{62} -330.780i q^{63} -87.2877 q^{64} +163.192i q^{65} -267.015i q^{66} -161.081 q^{67} -694.516 q^{68} -272.698i q^{69} -219.560i q^{70} -285.924 q^{71} -175.917i q^{72} +1079.22i q^{73} +256.266i q^{74} -319.525 q^{75} +405.544 q^{76} +2354.75i q^{77} +158.160i q^{78} -212.352i q^{79} +92.0547i q^{80} +81.0000 q^{81} -15.9567 q^{82} +463.346i q^{83} +669.291i q^{84} -492.012i q^{85} +403.847 q^{86} +22.3783i q^{87} +1252.31i q^{88} -753.141 q^{89} +53.7649 q^{90} -1394.78i q^{91} +551.769i q^{92} +710.395 q^{93} -10.8998i q^{94} +287.297i q^{95} +558.327i q^{96} +1153.23i q^{97} +1400.06i q^{98} -576.621 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 120 q^{3} - 164 q^{4} + 360 q^{9}+O(q^{10}) $ 40 * q - 120 * q^3 - 164 * q^4 + 360 * q^9	$ 40 q - 120 q^{3} - 164 q^{4} + 360 q^{9} - 174 q^{10} + 110 q^{11} + 492 q^{12} - 194 q^{13} - 78 q^{14} + 796 q^{16} - 150 q^{17} + 172 q^{19} - 668 q^{25} - 1080 q^{27} + 522 q^{30} + 66 q^{31} - 330 q^{33} - 400 q^{35} - 1476 q^{36} - 142 q^{37} + 582 q^{39} + 1160 q^{40} + 234 q^{42} - 1182 q^{44} + 132 q^{46} - 244 q^{47} - 2388 q^{48} - 3786 q^{49} + 450 q^{51} + 1596 q^{52} - 256 q^{56} - 516 q^{57} - 1780 q^{58} - 1790 q^{64} - 320 q^{67} + 1646 q^{68} + 712 q^{71} + 2004 q^{75} - 3004 q^{76} + 3240 q^{81} + 4112 q^{82} - 4198 q^{86} + 366 q^{89} - 1566 q^{90} - 198 q^{93} + 990 q^{99}+O(q^{100}) $ 40 * q - 120 * q^3 - 164 * q^4 + 360 * q^9 - 174 * q^10 + 110 * q^11 + 492 * q^12 - 194 * q^13 - 78 * q^14 + 796 * q^16 - 150 * q^17 + 172 * q^19 - 668 * q^25 - 1080 * q^27 + 522 * q^30 + 66 * q^31 - 330 * q^33 - 400 * q^35 - 1476 * q^36 - 142 * q^37 + 582 * q^39 + 1160 * q^40 + 234 * q^42 - 1182 * q^44 + 132 * q^46 - 244 * q^47 - 2388 * q^48 - 3786 * q^49 + 450 * q^51 + 1596 * q^52 - 256 * q^56 - 516 * q^57 - 1780 * q^58 - 1790 * q^64 - 320 * q^67 + 1646 * q^68 + 712 * q^71 + 2004 * q^75 - 3004 * q^76 + 3240 * q^81 + 4112 * q^82 - 4198 * q^86 + 366 * q^89 - 1566 * q^90 - 198 * q^93 + 990 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/471\mathbb{Z}\right)^\times$.

$n$	$158$	$319$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	1.38921i		−	0.491159i	−0.969376	−	0.245579i	$-0.921022\pi$
$2$		−	1.38921i		−	0.491159i	0.969376	−	0.245579i	$-0.0789782\pi$
$3$	−3.00000			−0.577350
$3$	−3.00000			−0.577350
$4$	6.07011			0.758763
$5$			4.30021i			0.384622i	0.981334	+	0.192311i	$0.0615983\pi$
$5$			4.30021i			0.384622i	−0.981334	+	0.192311i	$0.938402\pi$
$6$			4.16762i			0.283571i
$7$		−	36.7534i		−	1.98450i	−0.124274	−	0.992248i	$-0.539660\pi$
$7$		−	36.7534i		−	1.98450i	0.124274	−	0.992248i	$-0.460340\pi$
$8$		−	19.5463i		−	0.863832i
$9$	9.00000			0.333333
$10$	5.97388			0.188911
$11$	−64.0690			−1.75614			−0.878069	−	0.478533i	$-0.841169\pi$
$11$	−64.0690			−1.75614			−0.878069	+	0.478533i	$0.841169\pi$
$12$	−18.2103			−0.438072
$13$	37.9498			0.809644			0.404822	−	0.914396i	$-0.367333\pi$
$13$	37.9498			0.809644			0.404822	+	0.914396i	$0.367333\pi$
$14$	−51.0580			−0.974702
$15$		−	12.9006i		−	0.222062i
$16$	21.4070			0.334485
$17$	−114.416			−1.63235			−0.816174	−	0.577806i	$-0.803909\pi$
$17$	−114.416			−1.63235			−0.816174	+	0.577806i	$0.803909\pi$
$18$		−	12.5029i		−	0.163720i
$19$	66.8100			0.806699			0.403349	−	0.915046i	$-0.367846\pi$
$19$	66.8100			0.806699			0.403349	+	0.915046i	$0.367846\pi$
$20$			26.1027i			0.291837i
$21$			110.260i			1.14575i
$22$			89.0050i			0.862543i
$23$			90.8994i			0.824080i	0.911166	+	0.412040i	$0.135184\pi$
$23$			90.8994i			0.824080i	−0.911166	+	0.412040i	$0.864816\pi$
$24$			58.6388i			0.498733i
$25$	106.508			0.852066
$26$		−	52.7201i		−	0.397664i
$27$	−27.0000			−0.192450
$28$		−	223.097i		−	1.50576i
$29$		−	7.45942i		−	0.0477648i	−0.999715	−	0.0238824i	$-0.992397\pi$
$29$		−	7.45942i		−	0.0477648i	0.999715	−	0.0238824i	$-0.00760273\pi$
$30$	−17.9216			−0.109068
$31$	−236.798			−1.37194			−0.685971	−	0.727628i	$-0.740623\pi$
$31$	−236.798			−1.37194			−0.685971	+	0.727628i	$0.740623\pi$
$32$		−	186.109i		−	1.02812i
$33$	192.207			1.01391
$34$			158.947i			0.801742i
$35$	158.047			0.763282
$36$	54.6309			0.252921
$37$	−184.469			−0.819637			−0.409819	−	0.912167i	$-0.634408\pi$
$37$	−184.469			−0.819637			−0.409819	+	0.912167i	$0.634408\pi$
$38$		−	92.8129i		−	0.396217i
$39$	−113.849			−0.467448
$40$	84.0531			0.332249
$41$		−	11.4862i		−	0.0437522i	−0.999761	−	0.0218761i	$-0.993036\pi$
$41$		−	11.4862i		−	0.0437522i	0.999761	−	0.0218761i	$-0.00696394\pi$
$42$	153.174			0.562745
$43$			290.703i			1.03097i	0.856898	+	0.515486i	$0.172389\pi$
$43$			290.703i			1.03097i	−0.856898	+	0.515486i	$0.827611\pi$
$44$	−388.905			−1.33249
$45$			38.7019i			0.128207i
$46$	126.278			0.404754
$47$	7.84607			0.0243504			0.0121752	−	0.999926i	$-0.496124\pi$
$47$	7.84607			0.0243504			0.0121752	+	0.999926i	$0.496124\pi$
$48$	−64.2211			−0.193115
$49$	−1007.81			−2.93822
$50$		−	147.962i		−	0.418499i
$51$	343.248			0.942437
$52$	230.359			0.614328
$53$		−	54.2257i		−	0.140537i	−0.997528	−	0.0702686i	$-0.977614\pi$
$53$		−	54.2257i		−	0.140537i	0.997528	−	0.0702686i	$-0.0223856\pi$
$54$			37.5086i			0.0945235i
$55$		−	275.510i		−	0.675450i
$56$	−718.392			−1.71427
$57$	−200.430			−0.465748
$58$	−10.3627			−0.0234601
$59$		−	457.026i		−	1.00847i	−0.863566	−	0.504235i	$-0.831774\pi$
$59$		−	457.026i		−	1.00847i	0.863566	−	0.504235i	$-0.168226\pi$
$60$		−	78.3082i		−	0.168492i
$61$		−	410.690i		−	0.862025i	−0.902346	−	0.431012i	$-0.858157\pi$
$61$		−	410.690i		−	0.862025i	0.902346	−	0.431012i	$-0.141843\pi$
$62$			328.962i			0.673842i
$63$		−	330.780i		−	0.661499i
$64$	−87.2877			−0.170484
$65$			163.192i			0.311407i
$66$		−	267.015i		−	0.497989i
$67$	−161.081			−0.293719			−0.146859	−	0.989157i	$-0.546917\pi$
$67$	−161.081			−0.293719			−0.146859	+	0.989157i	$0.546917\pi$
$68$	−694.516			−1.23857
$69$		−	272.698i		−	0.475783i
$70$		−	219.560i		−	0.374892i
$71$	−285.924			−0.477929			−0.238964	−	0.971028i	$-0.576808\pi$
$71$	−285.924			−0.477929			−0.238964	+	0.971028i	$0.576808\pi$
$72$		−	175.917i		−	0.287944i
$73$			1079.22i			1.73032i	0.501494	+	0.865161i	$0.332784\pi$
$73$			1079.22i			1.73032i	−0.501494	+	0.865161i	$0.667216\pi$
$74$			256.266i			0.402572i
$75$	−319.525			−0.491940
$76$	405.544			0.612093
$77$			2354.75i			3.48505i
$78$			158.160i			0.229591i
$79$		−	212.352i		−	0.302423i	−0.988501	−	0.151212i	$-0.951682\pi$
$79$		−	212.352i		−	0.302423i	0.988501	−	0.151212i	$-0.0483175\pi$
$80$			92.0547i			0.128650i
$81$	81.0000			0.111111
$82$	−15.9567			−0.0214893
$83$			463.346i			0.612757i	0.951910	+	0.306378i	$0.0991172\pi$
$83$			463.346i			0.612757i	−0.951910	+	0.306378i	$0.900883\pi$
$84$			669.291i			0.869352i
$85$		−	492.012i		−	0.627838i
$86$	403.847			0.506371
$87$			22.3783i			0.0275770i
$88$			1252.31i			1.51701i
$89$	−753.141			−0.896997			−0.448499	−	0.893783i	$-0.648041\pi$
$89$	−753.141			−0.896997			−0.448499	+	0.893783i	$0.648041\pi$
$90$	53.7649			0.0629702
$91$		−	1394.78i		−	1.60674i
$92$			551.769i			0.625281i
$93$	710.395			0.792092
$94$		−	10.8998i		−	0.0119599i
$95$			287.297i			0.310274i
$96$			558.327i			0.593584i
$97$			1153.23i			1.20714i	0.797311	+	0.603569i	$0.206255\pi$
$97$			1153.23i			1.20714i	−0.797311	+	0.603569i	$0.793745\pi$
$98$			1400.06i			1.44313i
$99$	−576.621			−0.585380
$100$	646.516			0.646516
$101$	−82.0104			−0.0807954			−0.0403977	−	0.999184i	$-0.512862\pi$
$101$	−82.0104			−0.0807954			−0.0403977	+	0.999184i	$0.512862\pi$
$102$		−	476.842i		−	0.462886i
$103$		−	900.523i		−	0.861467i	−0.902479	−	0.430734i	$-0.858255\pi$
$103$		−	900.523i		−	0.861467i	0.902479	−	0.430734i	$-0.141745\pi$
$104$		−	741.777i		−	0.699396i
$105$	−474.142			−0.440681
$106$	−75.3307			−0.0690261
$107$		−	1777.39i		−	1.60586i	−0.596073	−	0.802930i	$-0.703273\pi$
$107$		−	1777.39i		−	1.60586i	0.596073	−	0.802930i	$-0.296727\pi$
$108$	−163.893			−0.146024
$109$	−1050.00			−0.922676			−0.461338	−	0.887224i	$-0.652630\pi$
$109$	−1050.00			−0.922676			−0.461338	+	0.887224i	$0.652630\pi$
$110$	−382.740			−0.331753
$111$	553.408			0.473218
$112$		−	786.781i		−	0.663784i
$113$	−1194.03			−0.994029			−0.497015	−	0.867742i	$-0.665570\pi$
$113$	−1194.03			−0.994029			−0.497015	+	0.867742i	$0.665570\pi$
$114$			278.439i			0.228756i
$115$	−390.887			−0.316960
$116$		−	45.2795i		−	0.0362422i
$117$	341.548			0.269881
$118$	−634.904			−0.495319
$119$			4205.17i			3.23939i
$120$	−252.159			−0.191824
$121$	2773.83			2.08402
$122$	−570.534			−0.423391
$123$			34.4586i			0.0252604i
$124$	−1437.39			−1.04098
$125$			995.534i			0.712346i
$126$	−459.522			−0.324901
$127$	−351.099			−0.245315			−0.122658	−	0.992449i	$-0.539142\pi$
$127$	−351.099			−0.245315			−0.122658	+	0.992449i	$0.539142\pi$
$128$		−	1367.61i		−	0.944382i
$129$		−	872.110i		−	0.595232i
$130$	226.707			0.152950
$131$		−	2039.99i		−	1.36057i	−0.732948	−	0.680285i	$-0.761856\pi$
$131$		−	2039.99i		−	1.36057i	0.732948	−	0.680285i	$-0.238144\pi$
$132$	1166.72			0.769315
$133$		−	2455.49i		−	1.60089i
$134$			223.775i			0.144263i
$135$		−	116.106i		−	0.0740206i
$136$			2236.41i			1.41007i
$137$			720.667i			0.449422i	0.974425	+	0.224711i	$0.0721438\pi$
$137$			720.667i			0.449422i	−0.974425	+	0.224711i	$0.927856\pi$
$138$	−378.834			−0.233685
$139$		−	2122.49i		−	1.29516i	−0.761996	−	0.647581i	$-0.775781\pi$
$139$		−	2122.49i		−	1.29516i	0.761996	−	0.647581i	$-0.224219\pi$
$140$	959.363			0.579150
$141$	−23.5382			−0.0140587
$142$			397.208i			0.234739i
$143$	−2431.40			−1.42185
$144$	192.663			0.111495
$145$	32.0771			0.0183714
$146$	1499.26			0.849863
$147$	3023.43			1.69638
$148$	−1119.75			−0.621910
$149$		−	1952.92i		−	1.07376i	−0.843660	−	0.536878i	$-0.819604\pi$
$149$		−	1952.92i		−	1.07376i	0.843660	−	0.536878i	$-0.180396\pi$
$150$			443.886i			0.241621i
$151$		−	3187.80i		−	1.71801i	−0.511966	−	0.859005i	$-0.671083\pi$
$151$		−	3187.80i		−	1.71801i	0.511966	−	0.859005i	$-0.328917\pi$
$152$		−	1305.89i		−	0.696852i
$153$	−1029.74			−0.544116
$154$	3271.24			1.71171
$155$		−	1018.28i		−	0.527680i
$156$	−691.077			−0.354682
$157$	−1806.04	−	779.821i	−0.918073	−	0.396411i
$157$	−1806.04	−	779.821i	−0.918073	−	0.396411i
$158$	−295.001			−0.148538
$159$			162.677i			0.0811392i
$160$	800.308			0.395437
$161$	3340.86			1.63538
$162$		−	112.526i		−	0.0545732i
$163$			1321.36i			0.634948i	0.948267	+	0.317474i	$0.102835\pi$
$163$			1321.36i			0.634948i	−0.948267	+	0.317474i	$0.897165\pi$
$164$		−	69.7224i		−	0.0331976i
$165$			826.530i			0.389971i
$166$	643.683			0.300961
$167$	2218.39			1.02793			0.513964	−	0.857812i	$-0.328176\pi$
$167$	2218.39			1.02793			0.513964	+	0.857812i	$0.328176\pi$
$168$	2155.18			0.989735
$169$	−756.815			−0.344477
$170$	−683.507			−0.308368
$171$	601.290			0.268900
$172$			1764.60i			0.782264i
$173$	3633.36			1.59676			0.798379	−	0.602155i	$-0.205691\pi$
$173$	3633.36			1.59676			0.798379	+	0.602155i	$0.205691\pi$
$174$	31.0880			0.0135447
$175$		−	3914.54i		−	1.69092i
$176$	−1371.53			−0.587402
$177$			1371.08i			0.582241i
$178$			1046.27i			0.440568i
$179$			460.950i			0.192475i	0.995358	+	0.0962375i	$0.0306808\pi$
$179$			460.950i			0.192475i	−0.995358	+	0.0962375i	$0.969319\pi$
$180$			234.925i			0.0972791i
$181$			3973.49i			1.63175i	0.578227	+	0.815876i	$0.303745\pi$
$181$			3973.49i			1.63175i	−0.578227	+	0.815876i	$0.696255\pi$
$182$	−1937.64			−0.789162
$183$			1232.07i			0.497690i
$184$	1776.75			0.711866
$185$		−	793.257i		−	0.315251i
$186$		−	986.885i		−	0.389043i
$187$	7330.51			2.86663
$188$	47.6265			0.0184762
$189$			992.341i			0.381916i
$190$	399.115			0.152394
$191$		−	187.445i		−	0.0710108i	−0.999369	−	0.0355054i	$-0.988696\pi$
$191$		−	187.445i		−	0.0710108i	0.999369	−	0.0355054i	$-0.0113041\pi$
$192$	261.863			0.0984288
$193$	754.373			0.281352			0.140676	−	0.990056i	$-0.455072\pi$
$193$	754.373			0.281352			0.140676	+	0.990056i	$0.455072\pi$
$194$	1602.07			0.592896
$195$		−	489.576i		−	0.179791i
$196$	−6117.52			−2.22942
$197$	−2621.77			−0.948188			−0.474094	−	0.880474i	$-0.657224\pi$
$197$	−2621.77			−0.948188			−0.474094	+	0.880474i	$0.657224\pi$
$198$			801.045i			0.287514i
$199$	3217.16			1.14602			0.573011	−	0.819548i	$-0.305775\pi$
$199$	3217.16			1.14602			0.573011	+	0.819548i	$0.305775\pi$
$200$		−	2081.84i		−	0.736041i
$201$	483.243			0.169579
$202$			113.929i			0.0396834i
$203$	−274.159			−0.0947891
$204$	2083.55			0.715086
$205$	49.3930			0.0168281
$206$	−1251.01			−0.423117
$207$			818.095i			0.274693i
$208$	812.392			0.270814
$209$	−4280.45			−1.41667
$210$			658.681i			0.216444i
$211$		−	3021.62i		−	0.985861i	−0.870069	−	0.492930i	$-0.835926\pi$
$211$		−	3021.62i		−	0.985861i	0.870069	−	0.492930i	$-0.164074\pi$
$212$		−	329.156i		−	0.106634i
$213$	857.772			0.275932
$214$	−2469.16			−0.788732
$215$	−1250.08			−0.396535
$216$			527.750i			0.166244i
$217$			8703.14i			2.72262i
$218$			1458.67i			0.453180i
$219$		−	3237.67i		−	0.999002i
$220$		−	1672.38i		−	0.512507i
$221$	−4342.06			−1.32162
$222$		−	768.798i		−	0.232425i
$223$		−	3781.66i		−	1.13560i	−0.823167	−	0.567800i	$-0.807795\pi$
$223$		−	3781.66i		−	1.13560i	0.823167	−	0.567800i	$-0.192205\pi$
$224$	−6840.14			−2.04029
$225$	958.574			0.284022
$226$			1658.76i			0.488226i
$227$			3264.32i			0.954453i	0.878780	+	0.477226i	$0.158358\pi$
$227$			3264.32i			0.954453i	−0.878780	+	0.477226i	$0.841642\pi$
$228$	−1216.63			−0.353392
$229$			1235.69i			0.356581i	0.983978	+	0.178290i	$0.0570566\pi$
$229$			1235.69i			0.356581i	−0.983978	+	0.178290i	$0.942943\pi$
$230$			543.022i			0.155677i
$231$		−	7064.25i		−	2.01209i
$232$	−145.804			−0.0412608
$233$	954.461			0.268364			0.134182	−	0.990957i	$-0.457159\pi$
$233$	954.461			0.268364			0.134182	+	0.990957i	$0.457159\pi$
$234$		−	474.481i		−	0.132555i
$235$			33.7398i			0.00936570i
$236$		−	2774.20i		−	0.765190i
$237$			637.056i			0.174604i
$238$	5841.85			1.59105
$239$	6447.29			1.74494			0.872470	−	0.488667i	$-0.162517\pi$
$239$	6447.29			1.74494			0.872470	+	0.488667i	$0.162517\pi$
$240$		−	276.164i		−	0.0742763i
$241$		−	5024.64i		−	1.34301i	−0.741000	−	0.671505i	$-0.765648\pi$
$241$		−	5024.64i		−	1.34301i	0.741000	−	0.671505i	$-0.234352\pi$
$242$		−	3853.43i		−	1.02359i
$243$	−243.000			−0.0641500
$244$		−	2492.93i		−	0.654073i
$245$		−	4333.80i		−	1.13011i
$246$	47.8701			0.0124068
$247$	2535.43			0.653139
$248$			4628.53i			1.18513i
$249$		−	1390.04i		−	0.353775i
$250$	1383.00			0.349875
$251$		−	1647.46i		−	0.414290i	−0.978310	−	0.207145i	$-0.933583\pi$
$251$		−	1647.46i		−	0.414290i	0.978310	−	0.207145i	$-0.0664172\pi$
$252$		−	2007.87i		−	0.501921i
$253$		−	5823.83i		−	1.44720i
$254$			487.750i			0.120489i
$255$			1476.04i			0.362482i
$256$	−2598.20			−0.634325
$257$	6568.70			1.59434			0.797168	−	0.603758i	$-0.206331\pi$
$257$	6568.70			1.59434			0.797168	+	0.603758i	$0.206331\pi$
$258$	−1211.54			−0.292353
$259$			6779.87i			1.62657i
$260$			990.592i			0.236284i
$261$		−	67.1348i		−	0.0159216i
$262$	−2833.97			−0.668256
$263$	7108.08			1.66655			0.833275	−	0.552858i	$-0.186463\pi$
$263$	7108.08			1.66655			0.833275	+	0.552858i	$0.186463\pi$
$264$		−	3756.93i		−	0.875845i
$265$	233.182			0.0540538
$266$	−3411.19			−0.786291
$267$	2259.42			0.517882
$268$	−977.778			−0.222863
$269$			8465.81i			1.91885i	0.281972	+	0.959423i	$0.409012\pi$
$269$			8465.81i			1.91885i	−0.281972	+	0.959423i	$0.590988\pi$
$270$	−161.295			−0.0363559
$271$		−	3605.73i		−	0.808238i	−0.914707	−	0.404119i	$-0.867578\pi$
$271$		−	3605.73i		−	0.808238i	0.914707	−	0.404119i	$-0.132422\pi$
$272$	−2449.30			−0.545996
$273$			4184.35i			0.927649i
$274$	1001.16			0.220737
$275$	−6823.87			−1.49635
$276$		−	1655.31i		−	0.361006i
$277$	5850.81			1.26910			0.634551	−	0.772881i	$-0.281185\pi$
$277$	5850.81			1.26910			0.634551	+	0.772881i	$0.281185\pi$
$278$	−2948.58			−0.636130
$279$	−2131.18			−0.457314
$280$		−	3089.24i		−	0.659347i
$281$	−4765.46			−1.01168			−0.505842	−	0.862626i	$-0.668818\pi$
$281$	−4765.46			−1.01168			−0.505842	+	0.862626i	$0.668818\pi$
$282$			32.6994i			0.00690505i
$283$	−4688.88			−0.984893			−0.492446	−	0.870343i	$-0.663897\pi$
$283$	−4688.88			−0.984893			−0.492446	+	0.870343i	$0.663897\pi$
$284$	−1735.59			−0.362635
$285$		−	861.892i		−	0.179137i
$286$			3377.72i			0.698353i
$287$	−422.156			−0.0868261
$288$		−	1674.98i		−	0.342706i
$289$	8177.99			1.66456
$290$		−	44.5617i		−	0.00902328i
$291$		−	3459.68i		−	0.696941i
$292$			6551.00i			1.31290i
$293$		−	8207.67i		−	1.63651i	−0.574856	−	0.818254i	$-0.694942\pi$
$293$		−	8207.67i		−	1.63651i	0.574856	−	0.818254i	$-0.305058\pi$
$294$		−	4200.17i		−	0.833194i
$295$	1965.31			0.387880
$296$			3605.69i			0.708029i
$297$	1729.86			0.337969
$298$	−2713.01			−0.527385
$299$			3449.61i			0.667211i
$300$	−1939.55			−0.373266
$301$	10684.3			2.04596
$302$	−4428.51			−0.843816
$303$	246.031			0.0466473
$304$	1430.20			0.269828
$305$	1766.05			0.331554
$306$			1430.53i			0.267247i
$307$			1018.23i			0.189295i	0.995511	+	0.0946474i	$0.0301724\pi$
$307$			1018.23i			0.189295i	−0.995511	+	0.0946474i	$0.969828\pi$
$308$			14293.6i			2.64433i
$309$			2701.57i			0.497368i
$310$	−1414.60			−0.259175
$311$	775.181			0.141339			0.0706696	−	0.997500i	$-0.477486\pi$
$311$	775.181			0.141339			0.0706696	+	0.997500i	$0.477486\pi$
$312$			2225.33i			0.403797i
$313$	−697.041			−0.125876			−0.0629378	−	0.998017i	$-0.520047\pi$
$313$	−697.041			−0.125876			−0.0629378	+	0.998017i	$0.520047\pi$
$314$	−1083.33	+	2508.96i	−0.194701	+	0.450920i
$315$	1422.43			0.254427
$316$		−	1289.00i		−	0.229468i
$317$	−1266.98			−0.224482			−0.112241	−	0.993681i	$-0.535803\pi$
$317$	−1266.98			−0.224482			−0.112241	+	0.993681i	$0.535803\pi$
$318$	225.992			0.0398522
$319$			477.918i			0.0838816i
$320$		−	375.355i		−	0.0655719i
$321$			5332.18i			0.927144i
$322$		−	4641.14i		−	0.803232i
$323$	−7644.13			−1.31681
$324$	491.679			0.0843070
$325$	4041.96			0.689870
$326$	1835.64			0.311860
$327$	3150.00			0.532707
$328$	−224.512			−0.0377946
$329$		−	288.370i		−	0.0483232i
$330$	1148.22			0.191538
$331$	−7226.33			−1.19998			−0.599992	−	0.800006i	$-0.704830\pi$
$331$	−7226.33			−1.19998			−0.599992	+	0.800006i	$0.704830\pi$
$332$			2812.56i			0.464937i
$333$	−1660.22			−0.273212
$334$		−	3081.80i		−	0.504876i
$335$		−	692.682i		−	0.112971i
$336$			2360.34i			0.383236i
$337$		−	3292.98i		−	0.532285i	−0.963934	−	0.266143i	$-0.914251\pi$
$337$		−	3292.98i		−	0.532285i	0.963934	−	0.266143i	$-0.0857492\pi$
$338$			1051.37i			0.169193i
$339$	3582.10			0.573903
$340$		−	2986.57i		−	0.476380i
$341$	15171.4			2.40932
$342$		−	835.317i		−	0.132072i
$343$			24434.0i			3.84640i
$344$	5682.17			0.890587
$345$	1172.66			0.182997
$346$		−	5047.49i		−	0.784262i
$347$	5550.69			0.858722			0.429361	−	0.903133i	$-0.358739\pi$
$347$	5550.69			0.858722			0.429361	+	0.903133i	$0.358739\pi$
$348$			135.838i			0.0209244i
$349$	466.355			0.0715285			0.0357642	−	0.999360i	$-0.488613\pi$
$349$	466.355			0.0715285			0.0357642	+	0.999360i	$0.488613\pi$
$350$	−5438.10			−0.830510
$351$	−1024.64			−0.155816
$352$			11923.8i			1.80552i
$353$	−3487.44			−0.525829			−0.262915	−	0.964819i	$-0.584684\pi$
$353$	−3487.44			−0.525829			−0.262915	+	0.964819i	$0.584684\pi$
$354$	1904.71			0.285973
$355$		−	1229.53i		−	0.183822i
$356$	−4571.64			−0.680609
$357$		−	12615.5i		−	1.87026i
$358$	640.355			0.0945357
$359$		−	5321.31i		−	0.782306i	−0.920326	−	0.391153i	$-0.872076\pi$
$359$		−	5321.31i		−	0.782306i	0.920326	−	0.391153i	$-0.127924\pi$
$360$	756.478			0.110750
$361$	−2395.42			−0.349237
$362$	5520.00			0.801449
$363$	−8321.50			−1.20321
$364$		−	8466.48i		−	1.21913i
$365$	−4640.89			−0.665521
$366$	1711.60			0.244445
$367$			4482.13i			0.637508i	0.947837	+	0.318754i	$0.103264\pi$
$367$			4482.13i			0.637508i	−0.947837	+	0.318754i	$0.896736\pi$
$368$			1945.89i			0.275642i
$369$		−	103.376i		−	0.0145841i
$370$	−1102.00			−0.154838
$371$	−1992.98			−0.278896
$372$	4312.17			0.601010
$373$		−	2514.01i		−	0.348983i	−0.984659	−	0.174491i	$-0.944172\pi$
$373$		−	2514.01i		−	0.348983i	0.984659	−	0.174491i	$-0.0558281\pi$
$374$		−	10183.6i		−	1.40797i
$375$		−	2986.60i		−	0.411273i
$376$		−	153.362i		−	0.0210346i
$377$		−	283.083i		−	0.0386725i
$378$	1378.57			0.187582
$379$		−	9474.17i		−	1.28405i	−0.766683	−	0.642026i	$-0.778094\pi$
$379$		−	9474.17i		−	1.28405i	0.766683	−	0.642026i	$-0.221906\pi$
$380$			1743.92i			0.235425i
$381$	1053.30			0.141633
$382$	−260.400			−0.0348776
$383$			3410.98i			0.455073i	0.973770	+	0.227536i	$0.0730671\pi$
$383$			3410.98i			0.455073i	−0.973770	+	0.227536i	$0.926933\pi$
$384$			4102.83i			0.545239i
$385$	−10125.9			−1.34043
$386$		−	1047.98i		−	0.138189i
$387$			2616.33i			0.343658i
$388$			7000.21i			0.915932i
$389$	−1935.25			−0.252240			−0.126120	−	0.992015i	$-0.540252\pi$
$389$	−1935.25			−0.252240			−0.126120	+	0.992015i	$0.540252\pi$
$390$	−680.122			−0.0883059
$391$		−	10400.3i		−	1.34519i
$392$			19699.0i			2.53813i
$393$			6119.97i			0.785525i
$394$			3642.17i			0.465711i
$395$	913.158			0.116319
$396$	−3500.15			−0.444164
$397$		−	12046.3i		−	1.52289i	−0.648231	−	0.761444i	$-0.724491\pi$
$397$		−	12046.3i		−	1.52289i	0.648231	−	0.761444i	$-0.275509\pi$
$398$		−	4469.30i		−	0.562878i
$399$			7366.48i			0.924274i
$400$	2280.02			0.285003
$401$		−	12395.9i		−	1.54369i	−0.635808	−	0.771847i	$-0.719333\pi$
$401$		−	12395.9i		−	1.54369i	0.635808	−	0.771847i	$-0.280667\pi$
$402$		−	671.324i		−	0.0832901i
$403$	−8986.44			−1.11079
$404$	−497.812			−0.0613046
$405$			348.317i			0.0427358i
$406$			380.863i			0.0465565i
$407$	11818.8			1.43940
$408$		−	6709.22i		−	0.814107i
$409$			4046.36i			0.489192i	0.969625	+	0.244596i	$0.0786553\pi$
$409$			4046.36i			0.489192i	−0.969625	+	0.244596i	$0.921345\pi$
$410$		−	68.6171i		−	0.00826526i
$411$		−	2162.00i		−	0.259474i
$412$		−	5466.27i		−	0.653650i
$413$	−16797.3			−2.00131
$414$	1136.50			0.134918
$415$	−1992.48			−0.235680
$416$		−	7062.79i		−	0.832409i
$417$			6367.48i			0.747762i
$418$			5946.43i			0.695812i
$419$	−15558.6			−1.81405			−0.907023	−	0.421082i	$-0.861650\pi$
$419$	−15558.6			−1.81405			−0.907023	+	0.421082i	$0.861650\pi$
$420$	−2878.09			−0.334372
$421$			8362.10i			0.968037i	0.875058	+	0.484019i	$0.160823\pi$
$421$			8362.10i			0.968037i	−0.875058	+	0.484019i	$0.839177\pi$
$422$	−4197.65			−0.484214
$423$	70.6146			0.00811679
$424$	−1059.91			−0.121401
$425$	−12186.2			−1.39087
$426$		−	1191.62i		−	0.135527i
$427$	−15094.3			−1.71068
$428$		−	10789.0i		−	1.21847i
$429$	7294.21			0.820904
$430$			1736.63i			0.194762i
$431$	5913.80			0.660922			0.330461	−	0.943820i	$-0.392796\pi$
$431$	5913.80			0.660922			0.330461	+	0.943820i	$0.392796\pi$
$432$	−577.990			−0.0643716
$433$			13063.1i			1.44982i	0.688843	+	0.724910i	$0.258119\pi$
$433$			13063.1i			1.44982i	−0.688843	+	0.724910i	$0.741881\pi$
$434$	12090.5			1.33724
$435$	−96.2312			−0.0106067
$436$	−6373.61			−0.700093
$437$			6072.99i			0.664784i
$438$	−4497.79			−0.490669
$439$			7163.45i			0.778800i	0.921069	+	0.389400i	$0.127318\pi$
$439$			7163.45i			0.778800i	−0.921069	+	0.389400i	$0.872682\pi$
$440$	−5385.20			−0.583475
$441$	−9070.30			−0.979408
$442$			6032.01i			0.649126i
$443$		−	9852.55i		−	1.05668i	−0.849033	−	0.528339i	$-0.822815\pi$
$443$		−	9852.55i		−	1.05668i	0.849033	−	0.528339i	$-0.177185\pi$
$444$	3359.25			0.359060
$445$		−	3238.66i		−	0.345005i
$446$	−5253.51			−0.557760
$447$			5858.77i			0.619933i
$448$			3208.12i			0.338324i
$449$		−	15230.1i		−	1.60078i	−0.599479	−	0.800390i	$-0.704626\pi$
$449$		−	15230.1i		−	1.60078i	0.599479	−	0.800390i	$-0.295374\pi$
$450$		−	1331.66i		−	0.139500i
$451$			735.909i			0.0768350i
$452$	−7247.91			−0.754233
$453$			9563.41i			0.991894i
$454$	4534.82			0.468788
$455$	5997.86			0.617986
$456$			3917.66i			0.402328i
$457$	−13226.8			−1.35388			−0.676942	−	0.736037i	$-0.736695\pi$
$457$	−13226.8			−1.35388			−0.676942	+	0.736037i	$0.736695\pi$
$458$	1716.63			0.175138
$459$	3089.23			0.314146
$460$	−2372.72			−0.240497
$461$	−2335.79			−0.235984			−0.117992	−	0.993015i	$-0.537646\pi$
$461$	−2335.79			−0.235984			−0.117992	+	0.993015i	$0.537646\pi$
$462$	−9813.71			−0.988258
$463$			5432.79i			0.545320i	0.962110	+	0.272660i	$0.0879034\pi$
$463$			5432.79i			0.545320i	−0.962110	+	0.272660i	$0.912097\pi$
$464$		−	159.684i		−	0.0159766i
$465$			3054.85i			0.304656i
$466$		−	1325.94i		−	0.131809i
$467$	10001.8			0.991067			0.495533	−	0.868589i	$-0.334973\pi$
$467$	10001.8			0.991067			0.495533	+	0.868589i	$0.334973\pi$
$468$	2073.23			0.204776
$469$			5920.27i			0.582884i
$470$	46.8715			0.00460004
$471$	5418.11	+	2339.46i	0.530050	+	0.228868i
$472$	−8933.16			−0.871149
$473$		−	18625.1i		−	1.81053i
$474$	885.002			0.0857584
$475$	7115.82			0.687360
$476$			25525.8i			2.45793i
$477$		−	488.031i		−	0.0468458i
$478$		−	8956.62i		−	0.857042i
$479$			12321.4i			1.17533i	0.809106	+	0.587663i	$0.199952\pi$
$479$			12321.4i			1.17533i	−0.809106	+	0.587663i	$0.800048\pi$
$480$	−2400.92			−0.228306
$481$	−7000.57			−0.663614
$482$	−6980.26			−0.659631
$483$	−10022.6			−0.944189
$484$	16837.5			1.58128
$485$	−4959.12			−0.464292
$486$			337.577i			0.0315078i
$487$	−4078.77			−0.379521			−0.189761	−	0.981830i	$-0.560771\pi$
$487$	−4078.77			−0.379521			−0.189761	+	0.981830i	$0.560771\pi$
$488$	−8027.47			−0.744644
$489$		−	3964.07i		−	0.366587i
$490$	−6020.54			−0.555062
$491$			16586.1i			1.52449i	0.647291	+	0.762243i	$0.275902\pi$
$491$			16586.1i			1.52449i	−0.647291	+	0.762243i	$0.724098\pi$
$492$			209.167i			0.0191666i
$493$			853.476i			0.0779688i
$494$		−	3522.23i		−	0.320795i
$495$		−	2479.59i		−	0.225150i
$496$	−5069.15			−0.458894
$497$			10508.7i			0.948448i
$498$	−1931.05			−0.173760
$499$		−	5507.40i		−	0.494078i	−0.969005	−	0.247039i	$-0.920542\pi$
$499$		−	5507.40i		−	0.494078i	0.969005	−	0.247039i	$-0.0794576\pi$
$500$			6043.00i			0.540502i
$501$	−6655.16			−0.593475
$502$	−2288.66			−0.203482
$503$		−	17446.5i		−	1.54652i	−0.634088	−	0.773261i	$-0.718624\pi$
$503$		−	17446.5i		−	1.54652i	0.634088	−	0.773261i	$-0.281376\pi$
$504$	−6465.53			−0.571424
$505$		−	352.662i		−	0.0310757i
$506$	−8090.50			−0.710804
$507$	2270.45			0.198884
$508$	−2131.21			−0.186136
$509$		−	3996.62i		−	0.348030i	−0.984743	−	0.174015i	$-0.944326\pi$
$509$		−	3996.62i		−	0.348030i	0.984743	−	0.174015i	$-0.0556741\pi$
$510$	2050.52			0.178036
$511$	39665.1			3.43382
$512$		−	7331.46i		−	0.632828i
$513$	−1803.87			−0.155249
$514$		−	9125.28i		−	0.783072i
$515$	3872.44			0.331340
$516$		−	5293.80i		−	0.451640i
$517$	−502.690			−0.0427626
$518$	9418.64			0.798902
$519$	−10900.1			−0.921889
$520$	3189.80			0.269003
$521$			7869.59i			0.661753i	0.943674	+	0.330876i	$0.107344\pi$
$521$			7869.59i			0.661753i	−0.943674	+	0.330876i	$0.892656\pi$
$522$	−93.2641			−0.00782003
$523$	−1774.06			−0.148325			−0.0741626	−	0.997246i	$-0.523628\pi$
$523$	−1774.06			−0.148325			−0.0741626	+	0.997246i	$0.523628\pi$
$524$		−	12382.9i		−	1.03235i
$525$			11743.6i			0.976253i
$526$		−	9874.59i		−	0.818541i
$527$	27093.5			2.23949
$528$	4114.58			0.339136
$529$	3904.30			0.320893
$530$		−	323.938i		−	0.0265490i
$531$		−	4113.24i		−	0.336157i
$532$		−	14905.1i		−	1.21470i
$533$		−	435.898i		−	0.0354237i
$534$		−	3138.80i		−	0.254362i
$535$	7643.16			0.617650
$536$			3148.53i			0.253724i
$537$		−	1382.85i		−	0.111125i
$538$	11760.8			0.942458
$539$	64569.4			5.15993
$540$		−	704.774i		−	0.0561641i
$541$		−	12693.2i		−	1.00873i	−0.863490	−	0.504367i	$-0.831726\pi$
$541$		−	12693.2i		−	1.00873i	0.863490	−	0.504367i	$-0.168274\pi$
$542$	−5009.10			−0.396973
$543$		−	11920.5i		−	0.942093i
$544$			21293.8i			1.67825i
$545$		−	4515.22i		−	0.354882i
$546$	5812.92			0.455623
$547$	−20496.1			−1.60210			−0.801049	−	0.598599i	$-0.795725\pi$
$547$	−20496.1			−1.60210			−0.801049	+	0.598599i	$0.795725\pi$
$548$			4374.53i			0.341005i
$549$		−	3696.21i		−	0.287342i
$550$			9479.77i			0.734943i
$551$		−	498.364i		−	0.0385318i
$552$	−5330.24			−0.410996
$553$	−7804.65			−0.600158
$554$		−	8127.99i		−	0.623331i
$555$			2379.77i			0.182010i
$556$		−	12883.8i		−	0.982722i
$557$	−1363.90			−0.103753			−0.0518765	−	0.998654i	$-0.516520\pi$
$557$	−1363.90			−0.103753			−0.0518765	+	0.998654i	$0.516520\pi$
$558$			2960.66i			0.224614i
$559$			11032.1i			0.834721i
$560$	3383.32			0.255306
$561$	−21991.5			−1.65505
$562$			6620.20i			0.496897i
$563$		−	13860.8i		−	1.03759i	−0.854899	−	0.518795i	$-0.826381\pi$
$563$		−	13860.8i		−	1.03759i	0.854899	−	0.518795i	$-0.173619\pi$
$564$	−142.879			−0.0106672
$565$		−	5134.60i		−	0.382326i
$566$			6513.82i			0.483739i
$567$		−	2977.02i		−	0.220500i
$568$			5588.75i			0.412850i
$569$		−	1569.14i		−	0.115610i	−0.998328	−	0.0578048i	$-0.981590\pi$
$569$		−	1569.14i		−	0.115610i	0.998328	−	0.0578048i	$-0.0184101\pi$
$570$	−1197.35			−0.0879847
$571$	−6402.00			−0.469204			−0.234602	−	0.972092i	$-0.575379\pi$
$571$	−6402.00			−0.469204			−0.234602	+	0.972092i	$0.575379\pi$
$572$	−14758.9			−1.07885
$573$			562.336i			0.0409981i
$574$			586.462i			0.0426454i
$575$			9681.53i			0.702170i
$576$	−785.589			−0.0568279
$577$	−16572.1			−1.19568			−0.597839	−	0.801616i	$-0.703974\pi$
$577$	−16572.1			−1.19568			−0.597839	+	0.801616i	$0.703974\pi$
$578$		−	11360.9i		−	0.817564i
$579$	−2263.12			−0.162439
$580$	194.711			0.0139396
$581$	17029.5			1.21601
$582$	−4806.21			−0.342309
$583$			3474.19i			0.246803i
$584$	21094.8			1.49471
$585$			1468.73i			0.103802i
$586$	−11402.1			−0.803785
$587$			27902.7i			1.96196i	0.194113	+	0.980979i	$0.437817\pi$
$587$			27902.7i			1.96196i	−0.194113	+	0.980979i	$0.562183\pi$
$588$	18352.6			1.28715
$589$	−15820.5			−1.10674
$590$		−	2730.22i		−	0.190511i
$591$	7865.30			0.547437
$592$	−3948.94			−0.274156
$593$	22619.4			1.56638			0.783192	−	0.621779i	$-0.213590\pi$
$593$	22619.4			1.56638			0.783192	+	0.621779i	$0.213590\pi$
$594$		−	2403.14i		−	0.165996i
$595$	−18083.1			−1.24594
$596$		−	11854.4i		−	0.814727i
$597$	−9651.47			−0.661656
$598$	4792.22			0.327707
$599$		−	13436.3i		−	0.916516i	−0.888819	−	0.458258i	$-0.848474\pi$
$599$		−	13436.3i		−	0.916516i	0.888819	−	0.458258i	$-0.151526\pi$
$600$			6245.52i			0.424954i
$601$	−14988.0			−1.01726			−0.508629	−	0.860986i	$-0.669848\pi$
$601$	−14988.0			−1.01726			−0.508629	+	0.860986i	$0.669848\pi$
$602$		−	14842.7i		−	1.00489i
$603$	−1449.73			−0.0979063
$604$		−	19350.3i		−	1.30356i
$605$			11928.1i			0.801562i
$606$		−	341.788i		−	0.0229112i
$607$		−	6049.84i		−	0.404539i	−0.979330	−	0.202270i	$-0.935168\pi$
$607$		−	6049.84i		−	0.404539i	0.979330	−	0.202270i	$-0.0648318\pi$
$608$		−	12434.0i		−	0.829381i
$609$	822.477			0.0547265
$610$		−	2453.41i		−	0.162846i
$611$	297.757			0.0197151
$612$	−6250.65			−0.412855
$613$		−	10645.6i		−	0.701425i	−0.936483	−	0.350713i	$-0.885939\pi$
$613$		−	10645.6i		−	0.701425i	0.936483	−	0.350713i	$-0.114061\pi$
$614$	1414.53			0.0929738
$615$	−148.179			−0.00971570
$616$	46026.6			3.01050
$617$	−11026.0			−0.719431			−0.359715	−	0.933062i	$-0.617126\pi$
$617$	−11026.0			−0.719431			−0.359715	+	0.933062i	$0.617126\pi$
$618$	3753.04			0.244287
$619$	8843.78			0.574252			0.287126	−	0.957893i	$-0.407300\pi$
$619$	8843.78			0.574252			0.287126	+	0.957893i	$0.407300\pi$
$620$		−	6181.08i		−	0.400384i
$621$		−	2454.28i		−	0.158594i
$622$		−	1076.89i		−	0.0694199i
$623$			27680.5i			1.78009i
$624$	−2437.17			−0.156354
$625$	9032.52			0.578081
$626$			968.334i			0.0618249i
$627$	12841.4			0.817918
$628$	−10962.8	−	4733.60i	−0.696600	−	0.300782i
$629$	21106.2			1.33793
$630$		−	1976.04i		−	0.124964i
$631$	23063.9			1.45509			0.727544	−	0.686061i	$-0.240661\pi$
$631$	23063.9			1.45509			0.727544	+	0.686061i	$0.240661\pi$
$632$	−4150.69			−0.261243
$633$			9064.85i			0.569187i
$634$			1760.10i			0.110256i
$635$		−	1509.80i		−	0.0943538i
$636$			987.467i			0.0615655i
$637$	−38246.2			−2.37892
$638$	663.926			0.0411992
$639$	−2573.32			−0.159310
$640$	5881.02			0.363231
$641$	−145.366			−0.00895727			−0.00447864	−	0.999990i	$-0.501426\pi$
$641$	−145.366			−0.00895727			−0.00447864	+	0.999990i	$0.501426\pi$
$642$	7407.49			0.455375
$643$		−	24841.0i		−	1.52353i	−0.647851	−	0.761767i	$-0.724332\pi$
$643$		−	24841.0i		−	1.52353i	0.647851	−	0.761767i	$-0.275668\pi$
$644$	20279.4			1.24087
$645$	3750.25			0.228940
$646$			10619.3i			0.646764i
$647$	9442.31			0.573749			0.286874	−	0.957968i	$-0.407384\pi$
$647$	9442.31			0.573749			0.286874	+	0.957968i	$0.407384\pi$
$648$		−	1583.25i		−	0.0959813i
$649$			29281.2i			1.77101i
$650$		−	5615.12i		−	0.338835i
$651$		−	26109.4i		−	1.57190i
$652$			8020.77i			0.481775i
$653$	−26649.3			−1.59704			−0.798521	−	0.601966i	$-0.794384\pi$
$653$	−26649.3			−1.59704			−0.798521	+	0.601966i	$0.794384\pi$
$654$		−	4376.00i		−	0.261644i
$655$	8772.38			0.523306
$656$		−	245.885i		−	0.0146345i
$657$			9713.01i			0.576774i
$658$	−400.605			−0.0237344
$659$	−6584.95			−0.389246			−0.194623	−	0.980878i	$-0.562348\pi$
$659$	−6584.95			−0.389246			−0.194623	+	0.980878i	$0.562348\pi$
$660$			5017.13i			0.295896i
$661$	14058.3			0.827236			0.413618	−	0.910451i	$-0.364265\pi$
$661$	14058.3			0.827236			0.413618	+	0.910451i	$0.364265\pi$
$662$			10038.9i			0.589383i
$663$	13026.2			0.763038
$664$	9056.68			0.529319
$665$	10559.1			0.615738
$666$			2306.39i			0.134191i
$667$	678.057			0.0393620
$668$	13465.8			0.779954
$669$			11345.0i			0.655639i
$670$	−962.278			−0.0554866
$671$			26312.5i			1.51383i
$672$	20520.4			1.17796
$673$			4710.61i			0.269808i	0.990859	+	0.134904i	$0.0430726\pi$
$673$			4710.61i			0.269808i	−0.990859	+	0.134904i	$0.956927\pi$
$674$	−4574.63			−0.261437
$675$	−2875.72			−0.163980
$676$	−4593.95			−0.261376
$677$	11904.2			0.675800			0.337900	−	0.941182i	$-0.390283\pi$
$677$	11904.2			0.675800			0.337900	+	0.941182i	$0.390283\pi$
$678$		−	4976.28i		−	0.281877i
$679$	42385.0			2.39556
$680$	−9617.01			−0.542346
$681$		−	9792.97i		−	0.551053i
$682$		−	21076.2i		−	1.18336i
$683$		−	5034.94i		−	0.282074i	−0.990004	−	0.141037i	$-0.954956\pi$
$683$		−	5034.94i		−	0.282074i	0.990004	−	0.141037i	$-0.0450436\pi$
$684$	3649.90			0.204031
$685$	−3099.02			−0.172858
$686$	33943.9			1.88919
$687$		−	3707.08i		−	0.205872i
$688$			6223.09i			0.344845i
$689$		−	2057.85i		−	0.113785i
$690$		−	1629.07i		−	0.0898804i
$691$		−	19680.5i		−	1.08348i	−0.840547	−	0.541739i	$-0.817766\pi$
$691$		−	19680.5i		−	1.08348i	0.840547	−	0.541739i	$-0.182234\pi$
$692$	22054.9			1.21156
$693$			21192.8i			1.16168i
$694$		−	7711.05i		−	0.421769i
$695$	9127.17			0.498149
$696$	437.412			0.0238219
$697$			1314.20i			0.0714189i
$698$		−	647.864i		−	0.0351318i
$699$	−2863.38			−0.154940
$700$		−	23761.6i		−	1.28301i
$701$			16453.5i			0.886506i	0.896396	+	0.443253i	$0.146176\pi$
$701$			16453.5i			0.886506i	−0.896396	+	0.443253i	$0.853824\pi$
$702$			1423.44i			0.0765304i
$703$	−12324.4			−0.661200
$704$	5592.43			0.299393
$705$		−	101.219i		−	0.00540729i
$706$			4844.78i			0.258266i
$707$			3014.16i			0.160338i
$708$			8322.59i			0.441783i
$709$	−25484.5			−1.34992			−0.674959	−	0.737856i	$-0.735839\pi$
$709$	−25484.5			−1.34992			−0.674959	+	0.737856i	$0.735839\pi$
$710$	−1708.08			−0.0902858
$711$		−	1911.17i		−	0.100808i
$712$			14721.1i			0.774855i
$713$		−	21524.8i		−	1.13059i
$714$	−17525.5			−0.918596
$715$		−	10455.5i		−	0.546874i
$716$			2798.01i			0.146043i
$717$	−19341.9			−1.00744
$718$	−7392.39			−0.384236
$719$			11888.3i			0.616631i	0.951284	+	0.308316i	$0.0997653\pi$
$719$			11888.3i			0.616631i	−0.951284	+	0.308316i	$0.900235\pi$
$720$			828.492i			0.0428835i
$721$	−33097.2			−1.70958
$722$			3327.73i			0.171531i
$723$			15073.9i			0.775388i
$724$			24119.5i			1.23811i
$725$		−	794.490i		−	0.0406988i
$726$			11560.3i			0.590968i
$727$	9404.68			0.479780			0.239890	−	0.970800i	$-0.422889\pi$
$727$	9404.68			0.479780			0.239890	+	0.970800i	$0.422889\pi$
$728$	−27262.8			−1.38795
$729$	729.000			0.0370370
$730$			6447.15i			0.326876i
$731$		−	33261.1i		−	1.68291i
$732$			7478.80i			0.377629i
$733$	−20421.7			−1.02905			−0.514525	−	0.857475i	$-0.672032\pi$
$733$	−20421.7			−1.02905			−0.514525	+	0.857475i	$0.672032\pi$
$734$	6226.61			0.313118
$735$			13001.4i			0.652468i
$736$	16917.2			0.847250
$737$	10320.3			0.515811
$738$	−143.610			−0.00716310
$739$	−13354.1			−0.664733			−0.332366	−	0.943150i	$-0.607847\pi$
$739$	−13354.1			−0.664733			−0.332366	+	0.943150i	$0.607847\pi$
$740$		−	4815.15i		−	0.239201i
$741$	−7606.28			−0.377090
$742$			2768.66i			0.136982i
$743$	−24790.6			−1.22407			−0.612033	−	0.790833i	$-0.709648\pi$
$743$	−24790.6			−1.22407			−0.612033	+	0.790833i	$0.709648\pi$
$744$		−	13885.6i		−	0.684234i
$745$	8397.98			0.412991
$746$	−3492.48			−0.171406
$747$			4170.11i			0.204252i
$748$	44497.0			2.17509
$749$	−65325.2			−3.18682
$750$	−4149.01			−0.202000
$751$			31612.6i			1.53603i	0.640431	+	0.768016i	$0.278756\pi$
$751$			31612.6i			1.53603i	−0.640431	+	0.768016i	$0.721244\pi$
$752$	167.961			0.00814483
$753$			4942.39i			0.239191i
$754$	−393.261			−0.0189943
$755$	13708.2			0.660786
$756$			6023.62i			0.289784i
$757$		−	32398.4i		−	1.55553i	−0.628553	−	0.777767i	$-0.716353\pi$
$757$		−	32398.4i		−	1.55553i	0.628553	−	0.777767i	$-0.283647\pi$
$758$	−13161.6			−0.630673
$759$			17471.5i			0.835540i
$760$	5615.59			0.268025
$761$		−	28081.0i		−	1.33763i	−0.743428	−	0.668815i	$-0.766802\pi$
$761$		−	28081.0i		−	1.33763i	0.743428	−	0.668815i	$-0.233198\pi$
$762$		−	1463.25i		−	0.0695642i
$763$			38591.0i			1.83105i
$764$		−	1137.81i		−	0.0538804i
$765$		−	4428.11i		−	0.209279i
$766$	4738.55			0.223513
$767$		−	17344.0i		−	0.816502i
$768$	7794.59			0.366228
$769$	14421.9			0.676288			0.338144	−	0.941094i	$-0.390201\pi$
$769$	14421.9			0.676288			0.338144	+	0.941094i	$0.390201\pi$
$770$			14067.0i			0.658363i
$771$	−19706.1			−0.920490
$772$	4579.13			0.213480
$773$	14491.8			0.674298			0.337149	−	0.941451i	$-0.390537\pi$
$773$	14491.8			0.674298			0.337149	+	0.941451i	$0.390537\pi$
$774$	3634.62			0.168790
$775$	−25221.0			−1.16899
$776$	22541.3			1.04276
$777$		−	20339.6i		−	0.939099i
$778$			2688.47i			0.123890i
$779$		−	767.393i		−	0.0352949i
$780$		−	2971.78i		−	0.136419i
$781$	18318.9			0.839309
$782$	−14448.2			−0.660700
$783$			201.404i			0.00919234i
$784$	−21574.2			−0.982791
$785$	3353.39	−	7766.34i	0.152469	−	0.353112i
$786$	8501.90			0.385818
$787$			4217.24i			0.191015i	0.995429	+	0.0955073i	$0.0304473\pi$
$787$			4217.24i			0.191015i	−0.995429	+	0.0955073i	$0.969553\pi$
$788$	−15914.4			−0.719451
$789$	−21324.2			−0.962184
$790$		−	1268.56i		−	0.0571310i
$791$			43884.8i			1.97265i
$792$			11270.8i			0.505669i
$793$		−	15585.6i		−	0.697933i
$794$	−16734.8			−0.747980
$795$	−699.546			−0.0312080
$796$	19528.5			0.869559
$797$	13680.3			0.608007			0.304003	−	0.952671i	$-0.401677\pi$
$797$	13680.3			0.608007			0.304003	+	0.952671i	$0.401677\pi$
$798$	10233.6			0.453965
$799$	−897.715			−0.0397483
$800$		−	19822.1i		−	0.876023i
$801$	−6778.27			−0.298999
$802$	−17220.5			−0.758199
$803$		−	69144.7i		−	3.03869i
$804$	2933.34			0.128670
$805$			14366.4i			0.629005i
$806$			12484.0i			0.545572i
$807$		−	25397.4i		−	1.10785i
$808$			1603.00i			0.0697936i
$809$			10137.2i			0.440552i	0.975438	+	0.220276i	$0.0706958\pi$
$809$			10137.2i			0.440552i	−0.975438	+	0.220276i	$0.929304\pi$
$810$	483.884			0.0209901
$811$		−	25429.9i		−	1.10106i	−0.834814	−	0.550532i	$-0.814425\pi$
$811$		−	25429.9i		−	1.10106i	0.834814	−	0.550532i	$-0.185575\pi$
$812$	−1664.17			−0.0719225
$813$			10817.2i			0.466636i
$814$		−	16418.7i		−	0.706972i
$815$	−5682.11			−0.244215
$816$	7347.91			0.315231
$817$			19421.9i			0.831684i
$818$	5621.23			0.240271
$819$		−	12553.0i		−	0.535578i
$820$	299.821			0.0127685
$821$	−9364.39			−0.398075			−0.199038	−	0.979992i	$-0.563782\pi$
$821$	−9364.39			−0.398075			−0.199038	+	0.979992i	$0.563782\pi$
$822$	−3003.47			−0.127443
$823$		−	26057.3i		−	1.10364i	−0.833962	−	0.551822i	$-0.813933\pi$
$823$		−	26057.3i		−	1.10364i	0.833962	−	0.551822i	$-0.186067\pi$
$824$	−17601.9			−0.744163
$825$	20471.6			0.863915
$826$			23334.9i			0.982958i
$827$	−22827.6			−0.959845			−0.479923	−	0.877311i	$-0.659335\pi$
$827$	−22827.6			−0.959845			−0.479923	+	0.877311i	$0.659335\pi$
$828$			4965.92i			0.208427i
$829$	20222.4			0.847231			0.423616	−	0.905842i	$-0.360761\pi$
$829$	20222.4			0.847231			0.423616	+	0.905842i	$0.360761\pi$
$830$			2767.97i			0.115756i
$831$	−17552.4			−0.732717
$832$	−3312.55			−0.138031
$833$	115310.			4.79621
$834$	8845.75			0.367270
$835$			9539.53i			0.395364i
$836$	−25982.8			−1.07492
$837$	6393.55			0.264031
$838$			21614.0i			0.890984i
$839$		−	26890.1i		−	1.10649i	−0.833018	−	0.553247i	$-0.813389\pi$
$839$		−	26890.1i		−	1.10649i	0.833018	−	0.553247i	$-0.186611\pi$
$840$			9267.71i			0.380674i
$841$	24333.4			0.997719
$842$	11616.7			0.475460
$843$	14296.4			0.584096
$844$		−	18341.5i		−	0.748035i
$845$		−	3254.46i		−	0.132493i
$846$		−	98.0983i		−	0.00398663i
$847$		−	101948.i		−	4.13574i
$848$		−	1160.81i		−	0.0470076i
$849$	14066.6			0.568628
$850$			16929.2i			0.683137i
$851$		−	16768.2i		−	0.675446i
$852$	5206.77			0.209367
$853$	16872.1			0.677243			0.338622	−	0.940923i	$-0.390039\pi$
$853$	16872.1			0.677243			0.338622	+	0.940923i	$0.390039\pi$
$854$			20969.0i			0.840217i
$855$			2585.68i			0.103425i
$856$	−34741.4			−1.38719
$857$		−	19018.0i		−	0.758041i	−0.925388	−	0.379021i	$-0.876261\pi$
$857$		−	19018.0i		−	0.758041i	0.925388	−	0.379021i	$-0.123739\pi$
$858$		−	10133.2i		−	0.403194i
$859$		−	32027.4i		−	1.27213i	−0.771636	−	0.636065i	$-0.780561\pi$
$859$		−	32027.4i		−	1.27213i	0.771636	−	0.636065i	$-0.219439\pi$
$860$	−7588.15			−0.300876
$861$	1266.47			0.0501291
$862$		−	8215.49i		−	0.324618i
$863$		−	12476.9i		−	0.492142i	−0.969252	−	0.246071i	$-0.920860\pi$
$863$		−	12476.9i		−	0.492142i	0.969252	−	0.246071i	$-0.0791396\pi$
$864$			5024.94i			0.197861i
$865$			15624.2i			0.614149i
$866$	18147.3			0.712092
$867$	−24534.0			−0.961035
$868$			52829.0i			2.06582i
$869$			13605.2i			0.531097i
$870$			133.685i			0.00520960i
$871$	−6112.99			−0.237808
$872$			20523.6i			0.797037i
$873$			10379.0i			0.402379i
$874$	8436.64			0.326514
$875$	36589.2			1.41365
$876$		−	19653.0i		−	0.758006i
$877$			42862.1i			1.65034i	0.564884	+	0.825170i	$0.308921\pi$
$877$			42862.1i			1.65034i	−0.564884	+	0.825170i	$0.691079\pi$
$878$	9951.52			0.382514
$879$			24623.0i			0.944839i
$880$		−	5897.85i		−	0.225928i
$881$		−	20066.1i		−	0.767361i	−0.923466	−	0.383680i	$-0.874656\pi$
$881$		−	20066.1i		−	0.767361i	0.923466	−	0.383680i	$-0.125344\pi$
$882$			12600.5i			0.481045i
$883$			38887.0i			1.48205i	0.671476	+	0.741026i	$0.265661\pi$
$883$			38887.0i			1.48205i	−0.671476	+	0.741026i	$0.734339\pi$
$884$	−26356.7			−1.00280
$885$	−5895.93			−0.223943
$886$	−13687.2			−0.518997
$887$			29977.3i			1.13477i	0.823453	+	0.567384i	$0.192045\pi$
$887$			29977.3i			1.13477i	−0.823453	+	0.567384i	$0.807955\pi$
$888$		−	10817.1i		−	0.408780i
$889$			12904.1i			0.486827i
$890$	−4499.17			−0.169452
$891$	−5189.59			−0.195127
$892$		−	22955.1i		−	0.861652i
$893$	524.196			0.0196434
$894$	8139.04			0.304486
$895$	−1982.18			−0.0740302
$896$	−50264.3			−1.87412
$897$		−	10348.8i		−	0.385215i
$898$	−21157.7			−0.786237
$899$			1766.38i			0.0655306i
$900$	5818.64			0.215505
$901$			6204.28i			0.229406i
$902$	1022.33			0.0377382
$903$	−32053.0			−1.18124
$904$			23338.9i			0.858674i
$905$	−17086.8			−0.627609
$906$	13285.5			0.487177
$907$	−2345.13			−0.0858531			−0.0429265	−	0.999078i	$-0.513668\pi$
$907$	−2345.13			−0.0858531			−0.0429265	+	0.999078i	$0.513668\pi$
$908$			19814.8i			0.724203i
$909$	−738.093			−0.0269318
$910$		−	8332.26i		−	0.303529i
$911$	−27223.2			−0.990060			−0.495030	−	0.868876i	$-0.664843\pi$
$911$	−27223.2			−0.990060			−0.495030	+	0.868876i	$0.664843\pi$
$912$	−4290.61			−0.155786
$913$		−	29686.1i		−	1.07609i
$914$			18374.8i			0.664971i
$915$	−5298.16			−0.191423
$916$			7500.80i			0.270560i
$917$	−74976.5			−2.70005
$918$		−	4291.58i		−	0.154295i
$919$			6042.68i			0.216898i	0.994102	+	0.108449i	$0.0345885\pi$
$919$			6042.68i			0.216898i	−0.994102	+	0.108449i	$0.965412\pi$
$920$			7640.38i			0.273800i
$921$		−	3054.69i		−	0.109289i
$922$			3244.89i			0.115905i
$923$	−10850.8			−0.386952
$924$		−	42880.8i		−	1.52670i
$925$	−19647.5			−0.698385
$926$	7547.26			0.267839
$927$		−	8104.70i		−	0.287156i
$928$	−1388.27			−0.0491078
$929$	26352.6			0.930680			0.465340	−	0.885132i	$-0.345932\pi$
$929$	26352.6			0.930680			0.465340	+	0.885132i	$0.345932\pi$
$930$	4243.81			0.149635
$931$	−67331.9			−2.37026
$932$	5793.68			0.203625
$933$	−2325.54			−0.0816022
$934$		−	13894.6i		−	0.486771i
$935$			31522.7i			1.10257i
$936$		−	6675.99i		−	0.233132i
$937$			6979.24i			0.243332i	0.992571	+	0.121666i	$0.0388236\pi$
$937$			6979.24i			0.243332i	−0.992571	+	0.121666i	$0.961176\pi$
$938$	8224.48			0.286289
$939$	2091.12			0.0726744
$940$			204.804i			0.00710635i
$941$	11543.9			0.399916			0.199958	−	0.979804i	$-0.435919\pi$
$941$	11543.9			0.399916			0.199958	+	0.979804i	$0.435919\pi$
$942$	3250.00	−	7526.88i	0.112410	−	0.260339i
$943$	1044.09			0.0360553
$944$		−	9783.57i		−	0.337318i
$945$	−4267.28			−0.146894
$946$	−25874.0			−0.889258
$947$			7246.87i			0.248671i	0.992240	+	0.124336i	$0.0396800\pi$
$947$			7246.87i			0.248671i	−0.992240	+	0.124336i	$0.960320\pi$
$948$			3866.99i			0.132483i
$949$			40956.3i			1.40095i
$950$		−	9885.34i		−	0.337603i
$951$	3800.94			0.129605
$952$	82195.4			2.79829
$953$	−6699.83			−0.227732			−0.113866	−	0.993496i	$-0.536324\pi$
$953$	−6699.83			−0.227732			−0.113866	+	0.993496i	$0.536324\pi$
$954$	−677.976			−0.0230087
$955$	806.054			0.0273124
$956$	39135.7			1.32400
$957$		−	1433.75i		−	0.0484291i
$958$	17117.0			0.577272
$959$	26487.0			0.891876
$960$			1126.07i			0.0378579i
$961$	26282.4			0.882227
$962$			9725.24i			0.325940i
$963$		−	15996.5i		−	0.535287i
$964$		−	30500.1i		−	1.01903i
$965$			3243.96i			0.108214i
$966$			13923.4i			0.463746i
$967$	−15796.2			−0.525307			−0.262654	−	0.964890i	$-0.584598\pi$
$967$	−15796.2			−0.525307			−0.262654	+	0.964890i	$0.584598\pi$
$968$		−	54218.2i		−	1.80025i
$969$	22932.4			0.760263
$970$			6889.23i			0.228041i
$971$			37937.9i			1.25385i	0.779081	+	0.626923i	$0.215686\pi$
$971$			37937.9i			1.25385i	−0.779081	+	0.626923i	$0.784314\pi$
$972$	−1475.04			−0.0486747
$973$	−78008.8			−2.57024
$974$			5666.26i			0.186405i
$975$	−12125.9			−0.398296
$976$		−	8791.66i		−	0.288334i
$977$	1020.23			0.0334084			0.0167042	−	0.999860i	$-0.494683\pi$
$977$	1020.23			0.0334084			0.0167042	+	0.999860i	$0.494683\pi$
$978$	−5506.91			−0.180053
$979$	48253.0			1.57525
$980$		−	26306.6i		−	0.857484i
$981$	−9449.99			−0.307559
$982$	23041.6			0.748764
$983$		−	43007.5i		−	1.39545i	−0.716366	−	0.697725i	$-0.754196\pi$
$983$		−	43007.5i		−	1.39545i	0.716366	−	0.697725i	$-0.245804\pi$
$984$	673.537			0.0218207
$985$		−	11274.1i		−	0.364695i
$986$	1185.65			0.0382951
$987$			865.109i			0.0278994i
$988$	15390.3			0.495578
$989$	−26424.7			−0.849604
$990$	−3444.66			−0.110584
$991$	−47874.1			−1.53458			−0.767291	−	0.641299i	$-0.778396\pi$
$991$	−47874.1			−1.53458			−0.767291	+	0.641299i	$0.778396\pi$
$992$			44070.3i			1.41052i
$993$	21679.0			0.692811
$994$	14598.7			0.465838
$995$			13834.5i			0.440786i
$996$		−	8437.67i		−	0.268432i
$997$			23291.6i			0.739872i	0.929057	+	0.369936i	$0.120620\pi$
$997$			23291.6i			0.739872i	−0.929057	+	0.369936i	$0.879380\pi$
$998$	−7650.92			−0.242671
$999$	4980.67			0.157739

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.4.b.a.313.16	✓	40
157.156	even	2	inner	471.4.b.a.313.25	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.4.b.a.313.16	✓	40	1.1	even	1	trivial
471.4.b.a.313.25	yes	40	157.156	even	2	inner

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:	\( N \)	\(=\)	\( 471 = 3 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	471.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.38921i q^{2} -3.00000 q^{3} +6.07011 q^{4} +4.30021i q^{5} +4.16762i q^{6} -36.7534i q^{7} -19.5463i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-1.38921i q^{2} -3.00000 q^{3} +6.07011 q^{4} +4.30021i q^{5} +4.16762i q^{6} -36.7534i q^{7} -19.5463i q^{8} +9.00000 q^{9} +5.97388 q^{10} -64.0690 q^{11} -18.2103 q^{12} +37.9498 q^{13} -51.0580 q^{14} -12.9006i q^{15} +21.4070 q^{16} -114.416 q^{17} -12.5029i q^{18} +66.8100 q^{19} +26.1027i q^{20} +110.260i q^{21} +89.0050i q^{22} +90.8994i q^{23} +58.6388i q^{24} +106.508 q^{25} -52.7201i q^{26} -27.0000 q^{27} -223.097i q^{28} -7.45942i q^{29} -17.9216 q^{30} -236.798 q^{31} -186.109i q^{32} +192.207 q^{33} +158.947i q^{34} +158.047 q^{35} +54.6309 q^{36} -184.469 q^{37} -92.8129i q^{38} -113.849 q^{39} +84.0531 q^{40} -11.4862i q^{41} +153.174 q^{42} +290.703i q^{43} -388.905 q^{44} +38.7019i q^{45} +126.278 q^{46} +7.84607 q^{47} -64.2211 q^{48} -1007.81 q^{49} -147.962i q^{50} +343.248 q^{51} +230.359 q^{52} -54.2257i q^{53} +37.5086i q^{54} -275.510i q^{55} -718.392 q^{56} -200.430 q^{57} -10.3627 q^{58} -457.026i q^{59} -78.3082i q^{60} -410.690i q^{61} +328.962i q^{62} -330.780i q^{63} -87.2877 q^{64} +163.192i q^{65} -267.015i q^{66} -161.081 q^{67} -694.516 q^{68} -272.698i q^{69} -219.560i q^{70} -285.924 q^{71} -175.917i q^{72} +1079.22i q^{73} +256.266i q^{74} -319.525 q^{75} +405.544 q^{76} +2354.75i q^{77} +158.160i q^{78} -212.352i q^{79} +92.0547i q^{80} +81.0000 q^{81} -15.9567 q^{82} +463.346i q^{83} +669.291i q^{84} -492.012i q^{85} +403.847 q^{86} +22.3783i q^{87} +1252.31i q^{88} -753.141 q^{89} +53.7649 q^{90} -1394.78i q^{91} +551.769i q^{92} +710.395 q^{93} -10.8998i q^{94} +287.297i q^{95} +558.327i q^{96} +1153.23i q^{97} +1400.06i q^{98} -576.621 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 120 q^{3} - 164 q^{4} + 360 q^{9}+O(q^{10}) \) 40 * q - 120 * q^3 - 164 * q^4 + 360 * q^9	\( 40 q - 120 q^{3} - 164 q^{4} + 360 q^{9} - 174 q^{10} + 110 q^{11} + 492 q^{12} - 194 q^{13} - 78 q^{14} + 796 q^{16} - 150 q^{17} + 172 q^{19} - 668 q^{25} - 1080 q^{27} + 522 q^{30} + 66 q^{31} - 330 q^{33} - 400 q^{35} - 1476 q^{36} - 142 q^{37} + 582 q^{39} + 1160 q^{40} + 234 q^{42} - 1182 q^{44} + 132 q^{46} - 244 q^{47} - 2388 q^{48} - 3786 q^{49} + 450 q^{51} + 1596 q^{52} - 256 q^{56} - 516 q^{57} - 1780 q^{58} - 1790 q^{64} - 320 q^{67} + 1646 q^{68} + 712 q^{71} + 2004 q^{75} - 3004 q^{76} + 3240 q^{81} + 4112 q^{82} - 4198 q^{86} + 366 q^{89} - 1566 q^{90} - 198 q^{93} + 990 q^{99}+O(q^{100}) \) 40 * q - 120 * q^3 - 164 * q^4 + 360 * q^9 - 174 * q^10 + 110 * q^11 + 492 * q^12 - 194 * q^13 - 78 * q^14 + 796 * q^16 - 150 * q^17 + 172 * q^19 - 668 * q^25 - 1080 * q^27 + 522 * q^30 + 66 * q^31 - 330 * q^33 - 400 * q^35 - 1476 * q^36 - 142 * q^37 + 582 * q^39 + 1160 * q^40 + 234 * q^42 - 1182 * q^44 + 132 * q^46 - 244 * q^47 - 2388 * q^48 - 3786 * q^49 + 450 * q^51 + 1596 * q^52 - 256 * q^56 - 516 * q^57 - 1780 * q^58 - 1790 * q^64 - 320 * q^67 + 1646 * q^68 + 712 * q^71 + 2004 * q^75 - 3004 * q^76 + 3240 * q^81 + 4112 * q^82 - 4198 * q^86 + 366 * q^89 - 1566 * q^90 - 198 * q^93 + 990 * q^99

Introduction

L-functions

Modular forms

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