LMFDB - Embedded newform 471.4.b.a.313.9

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			313.9
Character	$\chi$	$=$	471.313
Dual form			471.4.b.a.313.32

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.68048i q^{2} -3.00000 q^{3} -5.54594 q^{4} +17.9346i q^{5} +11.0414i q^{6} -17.3085i q^{7} -9.03212i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-3.68048i q^{2} -3.00000 q^{3} -5.54594 q^{4} +17.9346i q^{5} +11.0414i q^{6} -17.3085i q^{7} -9.03212i q^{8} +9.00000 q^{9} +66.0080 q^{10} +20.7561 q^{11} +16.6378 q^{12} -29.5708 q^{13} -63.7035 q^{14} -53.8038i q^{15} -77.6101 q^{16} -9.01400 q^{17} -33.1243i q^{18} -21.6815 q^{19} -99.4643i q^{20} +51.9254i q^{21} -76.3925i q^{22} +180.126i q^{23} +27.0963i q^{24} -196.650 q^{25} +108.835i q^{26} -27.0000 q^{27} +95.9917i q^{28} -76.1422i q^{29} -198.024 q^{30} +105.751 q^{31} +213.385i q^{32} -62.2684 q^{33} +33.1759i q^{34} +310.420 q^{35} -49.9135 q^{36} +128.425 q^{37} +79.7982i q^{38} +88.7123 q^{39} +161.987 q^{40} +291.460i q^{41} +191.110 q^{42} +460.761i q^{43} -115.112 q^{44} +161.411i q^{45} +662.949 q^{46} +70.6371 q^{47} +232.830 q^{48} +43.4173 q^{49} +723.767i q^{50} +27.0420 q^{51} +163.998 q^{52} -77.8678i q^{53} +99.3730i q^{54} +372.253i q^{55} -156.332 q^{56} +65.0444 q^{57} -280.240 q^{58} +290.914i q^{59} +298.393i q^{60} +190.808i q^{61} -389.215i q^{62} -155.776i q^{63} +164.481 q^{64} -530.340i q^{65} +229.178i q^{66} -450.368 q^{67} +49.9911 q^{68} -540.377i q^{69} -1142.50i q^{70} -75.6856 q^{71} -81.2890i q^{72} +442.127i q^{73} -472.666i q^{74} +589.951 q^{75} +120.244 q^{76} -359.256i q^{77} -326.504i q^{78} +836.048i q^{79} -1391.91i q^{80} +81.0000 q^{81} +1072.71 q^{82} +353.930i q^{83} -287.975i q^{84} -161.663i q^{85} +1695.82 q^{86} +228.427i q^{87} -187.472i q^{88} +1444.09 q^{89} +594.072 q^{90} +511.824i q^{91} -998.966i q^{92} -317.253 q^{93} -259.978i q^{94} -388.848i q^{95} -640.156i q^{96} +628.064i q^{97} -159.797i q^{98} +186.805 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	3.68048i		−	1.30125i	−0.759401	−	0.650623i	$-0.774508\pi$
$2$		−	3.68048i		−	1.30125i	0.759401	−	0.650623i	$-0.225492\pi$
$3$	−3.00000			−0.577350
$3$	−3.00000			−0.577350
$4$	−5.54594			−0.693243
$5$			17.9346i			1.60412i	0.597243	+	0.802060i	$0.296263\pi$
$5$			17.9346i			1.60412i	−0.597243	+	0.802060i	$0.703737\pi$
$6$			11.0414i			0.751275i
$7$		−	17.3085i		−	0.934569i	−0.884107	−	0.467284i	$-0.845232\pi$
$7$		−	17.3085i		−	0.934569i	0.884107	−	0.467284i	$-0.154768\pi$
$8$		−	9.03212i		−	0.399167i
$9$	9.00000			0.333333
$10$	66.0080			2.08736
$11$	20.7561			0.568928			0.284464	−	0.958687i	$-0.408184\pi$
$11$	20.7561			0.568928			0.284464	+	0.958687i	$0.408184\pi$
$12$	16.6378			0.400244
$13$	−29.5708			−0.630881			−0.315440	−	0.948945i	$-0.602152\pi$
$13$	−29.5708			−0.630881			−0.315440	+	0.948945i	$0.602152\pi$
$14$	−63.7035			−1.21610
$15$		−	53.8038i		−	0.926139i
$16$	−77.6101			−1.21266
$17$	−9.01400			−0.128601			−0.0643005	−	0.997931i	$-0.520482\pi$
$17$	−9.01400			−0.128601			−0.0643005	+	0.997931i	$0.520482\pi$
$18$		−	33.1243i		−	0.433749i
$19$	−21.6815			−0.261793			−0.130896	−	0.991396i	$-0.541786\pi$
$19$	−21.6815			−0.261793			−0.130896	+	0.991396i	$0.541786\pi$
$20$		−	99.4643i		−	1.11204i
$21$			51.9254i			0.539574i
$22$		−	76.3925i		−	0.740315i
$23$			180.126i			1.63299i	0.577352	+	0.816495i	$0.304086\pi$
$23$			180.126i			1.63299i	−0.577352	+	0.816495i	$0.695914\pi$
$24$			27.0963i			0.230459i
$25$	−196.650			−1.57320
$26$			108.835i			0.820932i
$27$	−27.0000			−0.192450
$28$			95.9917i			0.647883i
$29$		−	76.1422i		−	0.487561i	−0.969830	−	0.243780i	$-0.921612\pi$
$29$		−	76.1422i		−	0.487561i	0.969830	−	0.243780i	$-0.0783876\pi$
$30$	−198.024			−1.20514
$31$	105.751			0.612692			0.306346	−	0.951920i	$-0.400894\pi$
$31$	105.751			0.612692			0.306346	+	0.951920i	$0.400894\pi$
$32$			213.385i			1.17880i
$33$	−62.2684			−0.328471
$34$			33.1759i			0.167342i
$35$	310.420			1.49916
$36$	−49.9135			−0.231081
$37$	128.425			0.570620			0.285310	−	0.958435i	$-0.407903\pi$
$37$	128.425			0.570620			0.285310	+	0.958435i	$0.407903\pi$
$38$			79.7982i			0.340657i
$39$	88.7123			0.364239
$40$	161.987			0.640312
$41$			291.460i			1.11021i	0.831782	+	0.555103i	$0.187321\pi$
$41$			291.460i			1.11021i	−0.831782	+	0.555103i	$0.812679\pi$
$42$	191.110			0.702118
$43$			460.761i			1.63408i	0.576581	+	0.817040i	$0.304387\pi$
$43$			460.761i			1.63408i	−0.576581	+	0.817040i	$0.695613\pi$
$44$	−115.112			−0.394405
$45$			161.411i			0.534707i
$46$	662.949			2.12492
$47$	70.6371			0.219223			0.109611	−	0.993975i	$-0.465039\pi$
$47$	70.6371			0.219223			0.109611	+	0.993975i	$0.465039\pi$
$48$	232.830			0.700128
$49$	43.4173			0.126581
$50$			723.767i			2.04712i
$51$	27.0420			0.0742478
$52$	163.998			0.437354
$53$		−	77.8678i		−	0.201811i	−0.994896	−	0.100905i	$-0.967826\pi$
$53$		−	77.8678i		−	0.201811i	0.994896	−	0.100905i	$-0.0321739\pi$
$54$			99.3730i			0.250425i
$55$			372.253i			0.912629i
$56$	−156.332			−0.373049
$57$	65.0444			0.151146
$58$	−280.240			−0.634437
$59$			290.914i			0.641929i	0.947091	+	0.320965i	$0.104007\pi$
$59$			290.914i			0.641929i	−0.947091	+	0.320965i	$0.895993\pi$
$60$			298.393i			0.642039i
$61$			190.808i			0.400500i	0.979745	+	0.200250i	$0.0641755\pi$
$61$			190.808i			0.400500i	−0.979745	+	0.200250i	$0.935825\pi$
$62$		−	389.215i		−	0.797263i
$63$		−	155.776i		−	0.311523i
$64$	164.481			0.321251
$65$		−	530.340i		−	1.01201i
$66$			229.178i			0.427421i
$67$	−450.368			−0.821212			−0.410606	−	0.911813i	$-0.634683\pi$
$67$	−450.368			−0.821212			−0.410606	+	0.911813i	$0.634683\pi$
$68$	49.9911			0.0891517
$69$		−	540.377i		−	0.942807i
$70$		−	1142.50i		−	1.95078i
$71$	−75.6856			−0.126510			−0.0632551	−	0.997997i	$-0.520148\pi$
$71$	−75.6856			−0.126510			−0.0632551	+	0.997997i	$0.520148\pi$
$72$		−	81.2890i		−	0.133056i
$73$			442.127i			0.708863i	0.935082	+	0.354432i	$0.115326\pi$
$73$			442.127i			0.708863i	−0.935082	+	0.354432i	$0.884674\pi$
$74$		−	472.666i		−	0.742518i
$75$	589.951			0.908288
$76$	120.244			0.181486
$77$		−	359.256i		−	0.531702i
$78$		−	326.504i		−	0.473965i
$79$			836.048i			1.19067i	0.803478	+	0.595334i	$0.202980\pi$
$79$			836.048i			1.19067i	−0.803478	+	0.595334i	$0.797020\pi$
$80$		−	1391.91i		−	1.94525i
$81$	81.0000			0.111111
$82$	1072.71			1.44465
$83$			353.930i			0.468058i	0.972230	+	0.234029i	$0.0751911\pi$
$83$			353.930i			0.468058i	−0.972230	+	0.234029i	$0.924809\pi$
$84$		−	287.975i		−	0.374055i
$85$		−	161.663i		−	0.206291i
$86$	1695.82			2.12634
$87$			228.427i			0.281493i
$88$		−	187.472i		−	0.227097i
$89$	1444.09			1.71992			0.859960	−	0.510361i	$-0.170488\pi$
$89$	1444.09			1.71992			0.859960	+	0.510361i	$0.170488\pi$
$90$	594.072			0.695785
$91$			511.824i			0.589602i
$92$		−	998.966i		−	1.13206i
$93$	−317.253			−0.353738
$94$		−	259.978i		−	0.285263i
$95$		−	388.848i		−	0.419947i
$96$		−	640.156i		−	0.680580i
$97$			628.064i			0.657425i	0.944430	+	0.328713i	$0.106615\pi$
$97$			628.064i			0.657425i	−0.944430	+	0.328713i	$0.893385\pi$
$98$		−	159.797i		−	0.164713i
$99$	186.805			0.189643
$100$	1090.61			1.09061
$101$	156.183			0.153869			0.0769345	−	0.997036i	$-0.475487\pi$
$101$	156.183			0.153869			0.0769345	+	0.997036i	$0.475487\pi$
$102$		−	99.5276i		−	0.0966147i
$103$		−	115.035i		−	0.110046i	−0.998485	−	0.0550228i	$-0.982477\pi$
$103$		−	115.035i		−	0.110046i	0.998485	−	0.0550228i	$-0.0175232\pi$
$104$			267.086i			0.251827i
$105$	−931.261			−0.865541
$106$	−286.591			−0.262605
$107$		−	1074.51i		−	0.970810i	−0.874290	−	0.485405i	$-0.838672\pi$
$107$		−	1074.51i		−	0.970810i	0.874290	−	0.485405i	$-0.161328\pi$
$108$	149.740			0.133415
$109$	−465.901			−0.409406			−0.204703	−	0.978824i	$-0.565623\pi$
$109$	−465.901			−0.409406			−0.204703	+	0.978824i	$0.565623\pi$
$110$	1370.07			1.18755
$111$	−385.275			−0.329448
$112$			1343.31i			1.13331i
$113$	1240.38			1.03261			0.516306	−	0.856404i	$-0.327307\pi$
$113$	1240.38			1.03261			0.516306	+	0.856404i	$0.327307\pi$
$114$		−	239.395i		−	0.196679i
$115$	−3230.48			−2.61951
$116$			422.280i			0.337998i
$117$	−266.137			−0.210294
$118$	1070.70			0.835308
$119$			156.019i			0.120186i
$120$	−485.962			−0.369684
$121$	−900.183			−0.676321
$122$	702.267			0.521150
$123$		−	874.381i		−	0.640978i
$124$	−586.489			−0.424744
$125$		−	1285.02i		−	0.919484i
$126$	−573.331			−0.405368
$127$	−1256.01			−0.877581			−0.438791	−	0.898589i	$-0.644593\pi$
$127$	−1256.01			−0.877581			−0.438791	+	0.898589i	$0.644593\pi$
$128$			1101.72i			0.760772i
$129$		−	1382.28i		−	0.943437i
$130$	−1951.91			−1.31687
$131$			1238.57i			0.826061i	0.910717	+	0.413030i	$0.135530\pi$
$131$			1238.57i			0.826061i	−0.910717	+	0.413030i	$0.864470\pi$
$132$	345.337			0.227710
$133$			375.273i			0.244664i
$134$			1657.57i			1.06860i
$135$		−	484.234i		−	0.308713i
$136$			81.4155i			0.0513333i
$137$		−	2063.51i		−	1.28685i	−0.765511	−	0.643423i	$-0.777514\pi$
$137$		−	2063.51i		−	1.28685i	0.765511	−	0.643423i	$-0.222486\pi$
$138$	−1988.85			−1.22682
$139$			3038.64i			1.85420i	0.374812	+	0.927101i	$0.377707\pi$
$139$			3038.64i			1.85420i	−0.374812	+	0.927101i	$0.622293\pi$
$140$	−1721.57			−1.03928
$141$	−211.911			−0.126568
$142$			278.559i			0.164621i
$143$	−613.774			−0.358926
$144$	−698.491			−0.404219
$145$	1365.58			0.782106
$146$	1627.24			0.922406
$147$	−130.252			−0.0730816
$148$	−712.238			−0.395578
$149$			103.630i			0.0569777i	0.999594	+	0.0284889i	$0.00906952\pi$
$149$			103.630i			0.0569777i	−0.999594	+	0.0284889i	$0.990930\pi$
$150$		−	2171.30i		−	1.18191i
$151$			3146.61i			1.69581i	0.530148	+	0.847905i	$0.322136\pi$
$151$			3146.61i			1.69581i	−0.530148	+	0.847905i	$0.677864\pi$
$152$			195.829i			0.104499i
$153$	−81.1260			−0.0428670
$154$	−1322.24			−0.691876
$155$			1896.60i			0.982831i
$156$	−491.993			−0.252506
$157$	1883.45	−	567.905i	0.957424	−	0.288686i
$157$	1883.45	−	567.905i	0.957424	−	0.288686i
$158$	3077.06			1.54935
$159$			233.603i			0.116515i
$160$	−3826.98			−1.89094
$161$	3117.70			1.52614
$162$		−	298.119i		−	0.144583i
$163$		−	363.543i		−	0.174692i	−0.996178	−	0.0873462i	$-0.972161\pi$
$163$		−	363.543i		−	0.174692i	0.996178	−	0.0873462i	$-0.0278386\pi$
$164$		−	1616.42i		−	0.769642i
$165$		−	1116.76i		−	0.526906i
$166$	1302.63			0.609059
$167$	1821.21			0.843888			0.421944	−	0.906622i	$-0.361348\pi$
$167$	1821.21			0.843888			0.421944	+	0.906622i	$0.361348\pi$
$168$	468.996			0.215380
$169$	−1322.57			−0.601989
$170$	−594.996			−0.268436
$171$	−195.133			−0.0872643
$172$		−	2555.36i		−	1.13281i
$173$	−1253.34			−0.550808			−0.275404	−	0.961329i	$-0.588812\pi$
$173$	−1253.34			−0.550808			−0.275404	+	0.961329i	$0.588812\pi$
$174$	840.720			0.366292
$175$			3403.71i			1.47027i
$176$	−1610.88			−0.689914
$177$		−	872.743i		−	0.370618i
$178$		−	5314.94i		−	2.23804i
$179$		−	553.203i		−	0.230996i	−0.993308	−	0.115498i	$-0.963154\pi$
$179$		−	553.203i		−	0.230996i	0.993308	−	0.115498i	$-0.0368464\pi$
$180$		−	895.179i		−	0.370682i
$181$			1450.94i			0.595842i	0.954591	+	0.297921i	$0.0962932\pi$
$181$			1450.94i			0.595842i	−0.954591	+	0.297921i	$0.903707\pi$
$182$	1883.76			0.767217
$183$		−	572.425i		−	0.231229i
$184$	1626.92			0.651836
$185$			2303.25i			0.915344i
$186$			1167.64i			0.460300i
$187$	−187.096			−0.0731647
$188$	−391.749			−0.151975
$189$			467.328i			0.179858i
$190$	−1431.15			−0.546455
$191$		−	2231.68i		−	0.845437i	−0.906261	−	0.422718i	$-0.861076\pi$
$191$		−	2231.68i		−	0.845437i	0.906261	−	0.422718i	$-0.138924\pi$
$192$	−493.442			−0.185474
$193$	763.964			0.284929			0.142465	−	0.989800i	$-0.454497\pi$
$193$	763.964			0.284929			0.142465	+	0.989800i	$0.454497\pi$
$194$	2311.58			0.855472
$195$			1591.02i			0.584284i
$196$	−240.790			−0.0877514
$197$	−1363.44			−0.493101			−0.246550	−	0.969130i	$-0.579297\pi$
$197$	−1363.44			−0.493101			−0.246550	+	0.969130i	$0.579297\pi$
$198$		−	687.533i		−	0.246772i
$199$	−510.400			−0.181816			−0.0909079	−	0.995859i	$-0.528977\pi$
$199$	−510.400			−0.181816			−0.0909079	+	0.995859i	$0.528977\pi$
$200$			1776.17i			0.627970i
$201$	1351.10			0.474127
$202$		−	574.828i		−	0.200222i
$203$	−1317.90			−0.455659
$204$	−149.973			−0.0514718
$205$	−5227.23			−1.78090
$206$	−423.383			−0.143197
$207$			1621.13i			0.544330i
$208$	2294.99			0.765042
$209$	−450.023			−0.148941
$210$			3427.49i			1.12628i
$211$			2407.77i			0.785583i	0.919628	+	0.392792i	$0.128491\pi$
$211$			2407.77i			0.785583i	−0.919628	+	0.392792i	$0.871509\pi$
$212$			431.850i			0.139904i
$213$	227.057			0.0730407
$214$	−3954.71			−1.26326
$215$	−8263.58			−2.62126
$216$			243.867i			0.0768197i
$217$		−	1830.39i		−	0.572602i
$218$			1714.74i			0.532738i
$219$		−	1326.38i		−	0.409262i
$220$		−	2064.49i		−	0.632673i
$221$	266.551			0.0811319
$222$			1418.00i			0.428693i
$223$		−	5084.14i		−	1.52672i	−0.645972	−	0.763361i	$-0.723548\pi$
$223$		−	5084.14i		−	1.52672i	0.645972	−	0.763361i	$-0.276452\pi$
$224$	3693.37			1.10167
$225$	−1769.85			−0.524401
$226$		−	4565.19i		−	1.34368i
$227$		−	6075.09i		−	1.77629i	−0.459563	−	0.888145i	$-0.651994\pi$
$227$		−	6075.09i		−	1.77629i	0.459563	−	0.888145i	$-0.348006\pi$
$228$	−360.732			−0.104781
$229$		−	461.623i		−	0.133209i	−0.997779	−	0.0666047i	$-0.978783\pi$
$229$		−	461.623i		−	0.133209i	0.997779	−	0.0666047i	$-0.0212166\pi$
$230$			11889.7i			3.40863i
$231$			1077.77i			0.306978i
$232$	−687.725			−0.194618
$233$	−4291.87			−1.20674			−0.603368	−	0.797463i	$-0.706175\pi$
$233$	−4291.87			−1.20674			−0.603368	+	0.797463i	$0.706175\pi$
$234$			979.512i			0.273644i
$235$			1266.85i			0.351660i
$236$		−	1613.39i		−	0.445013i
$237$		−	2508.14i		−	0.687432i
$238$	574.223			0.156392
$239$	−5041.24			−1.36440			−0.682199	−	0.731167i	$-0.738976\pi$
$239$	−5041.24			−1.36440			−0.682199	+	0.731167i	$0.738976\pi$
$240$			4175.72i			1.12309i
$241$		−	4095.42i		−	1.09464i	−0.836923	−	0.547321i	$-0.815648\pi$
$241$		−	4095.42i		−	1.09464i	0.836923	−	0.547321i	$-0.184352\pi$
$242$			3313.11i			0.880061i
$243$	−243.000			−0.0641500
$244$		−	1058.21i		−	0.277644i
$245$			778.672i			0.203051i
$246$	−3218.14			−0.834070
$247$	641.137			0.165160
$248$		−	955.155i		−	0.244566i
$249$		−	1061.79i		−	0.270234i
$250$	−4729.49			−1.19648
$251$			5802.63i			1.45920i	0.683876	+	0.729599i	$0.260293\pi$
$251$			5802.63i			1.45920i	−0.683876	+	0.729599i	$0.739707\pi$
$252$			863.925i			0.215961i
$253$			3738.71i			0.929054i
$254$			4622.72i			1.14195i
$255$			484.988i			0.119102i
$256$	5370.69			1.31120
$257$	3349.64			0.813015			0.406507	−	0.913647i	$-0.366747\pi$
$257$	3349.64			0.813015			0.406507	+	0.913647i	$0.366747\pi$
$258$	−5087.47			−1.22764
$259$		−	2222.84i		−	0.533284i
$260$			2941.23i			0.701568i
$261$		−	685.280i		−	0.162520i
$262$	4558.52			1.07491
$263$	2988.46			0.700671			0.350335	−	0.936624i	$-0.386068\pi$
$263$	2988.46			0.700671			0.350335	+	0.936624i	$0.386068\pi$
$264$			562.415i			0.131115i
$265$	1396.53			0.323728
$266$	1381.18			0.318368
$267$	−4332.26			−0.992997
$268$	2497.72			0.569300
$269$		−	788.431i		−	0.178704i	−0.996000	−	0.0893522i	$-0.971520\pi$
$269$		−	788.431i		−	0.178704i	0.996000	−	0.0893522i	$-0.0284797\pi$
$270$	−1782.22			−0.401712
$271$			2117.35i			0.474612i	0.971435	+	0.237306i	$0.0762644\pi$
$271$			2117.35i			0.474612i	−0.971435	+	0.237306i	$0.923736\pi$
$272$	699.577			0.155949
$273$		−	1535.47i		−	0.340407i
$274$	−7594.72			−1.67450
$275$	−4081.70			−0.895038
$276$			2996.90i			0.653594i
$277$	−59.7999			−0.0129712			−0.00648561	−	0.999979i	$-0.502064\pi$
$277$	−59.7999			−0.0129712			−0.00648561	+	0.999979i	$0.502064\pi$
$278$	11183.7			2.41277
$279$	951.759			0.204231
$280$		−	2803.75i		−	0.598415i
$281$	1102.37			0.234028			0.117014	−	0.993130i	$-0.462668\pi$
$281$	1102.37			0.234028			0.117014	+	0.993130i	$0.462668\pi$
$282$			779.935i			0.164697i
$283$	−7390.35			−1.55233			−0.776167	−	0.630527i	$-0.782839\pi$
$283$	−7390.35			−1.55233			−0.776167	+	0.630527i	$0.782839\pi$
$284$	419.748			0.0877023
$285$			1166.55i			0.242457i
$286$			2258.98i			0.467051i
$287$	5044.73			1.03756
$288$			1920.47i			0.392933i
$289$	−4831.75			−0.983462
$290$		−	5026.00i		−	1.01771i
$291$		−	1884.19i		−	0.379565i
$292$		−	2452.01i		−	0.491414i
$293$		−	2310.82i		−	0.460749i	−0.973102	−	0.230374i	$-0.926005\pi$
$293$		−	2310.82i		−	0.460749i	0.973102	−	0.230374i	$-0.0739951\pi$
$294$			479.390i			0.0950972i
$295$	−5217.44			−1.02973
$296$		−	1159.95i		−	0.227773i
$297$	−560.415			−0.109490
$298$	381.407			0.0741421
$299$		−	5326.45i		−	1.03022i
$300$	−3271.83			−0.629664
$301$	7975.07			1.52716
$302$	11581.0			2.20667
$303$	−468.548			−0.0888363
$304$	1682.70			0.317465
$305$	−3422.07			−0.642451
$306$			298.583i			0.0557805i
$307$		−	403.480i		−	0.0750091i	−0.999296	−	0.0375045i	$-0.988059\pi$
$307$		−	403.480i		−	0.0750091i	0.999296	−	0.0375045i	$-0.0119409\pi$
$308$			1992.42i			0.368599i
$309$			345.104i			0.0635349i
$310$	6980.41			1.27891
$311$	−8689.49			−1.58436			−0.792179	−	0.610288i	$-0.791054\pi$
$311$	−8689.49			−1.58436			−0.792179	+	0.610288i	$0.791054\pi$
$312$		−	801.259i		−	0.145392i
$313$	−194.408			−0.0351073			−0.0175537	−	0.999846i	$-0.505588\pi$
$313$	−194.408			−0.0351073			−0.0175537	+	0.999846i	$0.505588\pi$
$314$	−2090.16	−	6932.00i	−0.375652	−	1.24584i
$315$	2793.78			0.499720
$316$		−	4636.67i		−	0.825422i
$317$	−2908.45			−0.515315			−0.257657	−	0.966236i	$-0.582951\pi$
$317$	−2908.45			−0.515315			−0.257657	+	0.966236i	$0.582951\pi$
$318$	859.772			0.151615
$319$		−	1580.42i		−	0.277387i
$320$			2949.90i			0.515326i
$321$			3223.52i			0.560497i
$322$		−	11474.6i		−	1.98589i
$323$	195.437			0.0336668
$324$	−449.221			−0.0770270
$325$	5815.10			0.992503
$326$	−1338.01			−0.227318
$327$	1397.70			0.236370
$328$	2632.50			0.443158
$329$		−	1222.62i		−	0.204879i
$330$	−4110.21			−0.685635
$331$	1237.34			0.205469			0.102734	−	0.994709i	$-0.467241\pi$
$331$	1237.34			0.205469			0.102734	+	0.994709i	$0.467241\pi$
$332$		−	1962.87i		−	0.324478i
$333$	1155.83			0.190207
$334$		−	6702.92i		−	1.09811i
$335$		−	8077.18i		−	1.31732i
$336$		−	4029.93i		−	0.654318i
$337$		−	10041.3i		−	1.62310i	−0.584284	−	0.811549i	$-0.698625\pi$
$337$		−	10041.3i		−	1.62310i	0.584284	−	0.811549i	$-0.301375\pi$
$338$			4867.70i			0.783336i
$339$	−3721.14			−0.596178
$340$			896.572i			0.143010i
$341$	2194.98			0.348577
$342$			718.184i			0.113552i
$343$		−	6688.29i		−	1.05287i
$344$	4161.65			0.652271
$345$	9691.45			1.51238
$346$			4612.90i			0.716736i
$347$	939.091			0.145282			0.0726412	−	0.997358i	$-0.476857\pi$
$347$	939.091			0.145282			0.0726412	+	0.997358i	$0.476857\pi$
$348$		−	1266.84i		−	0.195143i
$349$	5941.09			0.911230			0.455615	−	0.890177i	$-0.349419\pi$
$349$	5941.09			0.911230			0.455615	+	0.890177i	$0.349419\pi$
$350$	12527.3			1.91318
$351$	798.410			0.121413
$352$			4429.05i			0.670652i
$353$	−2498.71			−0.376751			−0.188376	−	0.982097i	$-0.560322\pi$
$353$	−2498.71			−0.376751			−0.188376	+	0.982097i	$0.560322\pi$
$354$	−3212.11			−0.482265
$355$		−	1357.39i		−	0.202938i
$356$	−8008.82			−1.19232
$357$		−	468.056i		−	0.0693897i
$358$	−2036.05			−0.300583
$359$			6121.59i			0.899959i	0.893039	+	0.449980i	$0.148569\pi$
$359$			6121.59i			0.899959i	−0.893039	+	0.449980i	$0.851431\pi$
$360$	1457.89			0.213437
$361$	−6388.91			−0.931464
$362$	5340.15			0.775337
$363$	2700.55			0.390474
$364$		−	2838.55i		−	0.408737i
$365$	−7929.37			−1.13710
$366$	−2106.80			−0.300886
$367$		−	1728.54i		−	0.245856i	−0.992416	−	0.122928i	$-0.960772\pi$
$367$		−	1728.54i		−	0.245856i	0.992416	−	0.122928i	$-0.0392285\pi$
$368$		−	13979.6i		−	1.98026i
$369$			2623.14i			0.370069i
$370$	8477.08			1.19109
$371$	−1347.77			−0.188606
$372$	1759.47			0.245226
$373$			13511.8i			1.87564i	0.347120	+	0.937821i	$0.387160\pi$
$373$			13511.8i			1.87564i	−0.347120	+	0.937821i	$0.612840\pi$
$374$			688.602i			0.0952053i
$375$			3855.06i			0.530865i
$376$		−	638.002i		−	0.0875065i
$377$			2251.58i			0.307593i
$378$	1719.99			0.234039
$379$			1044.19i			0.141521i	0.997493	+	0.0707605i	$0.0225426\pi$
$379$			1044.19i			0.141521i	−0.997493	+	0.0707605i	$0.977457\pi$
$380$			2156.53i			0.291125i
$381$	3768.03			0.506672
$382$	−8213.64			−1.10012
$383$			9664.99i			1.28945i	0.764416	+	0.644723i	$0.223027\pi$
$383$			9664.99i			1.28945i	−0.764416	+	0.644723i	$0.776973\pi$
$384$		−	3305.15i		−	0.439232i
$385$	6443.12			0.852914
$386$		−	2811.76i		−	0.370763i
$387$			4146.85i			0.544693i
$388$		−	3483.21i		−	0.455755i
$389$	9961.10			1.29832			0.649162	−	0.760651i	$-0.275120\pi$
$389$	9961.10			1.29832			0.649162	+	0.760651i	$0.275120\pi$
$390$	5855.72			0.760297
$391$		−	1623.65i		−	0.210004i
$392$		−	392.150i		−	0.0505270i
$393$		−	3715.70i		−	0.476926i
$394$			5018.10i			0.641646i
$395$	−14994.2			−1.90997
$396$	−1036.01			−0.131468
$397$		−	12882.6i		−	1.62861i	−0.580435	−	0.814306i	$-0.697118\pi$
$397$		−	12882.6i		−	1.62861i	0.580435	−	0.814306i	$-0.302882\pi$
$398$			1878.52i			0.236587i
$399$		−	1125.82i		−	0.141257i
$400$	15262.0			1.90775
$401$		−	2519.55i		−	0.313766i	−0.987617	−	0.156883i	$-0.949855\pi$
$401$		−	2519.55i		−	0.313766i	0.987617	−	0.156883i	$-0.0501446\pi$
$402$		−	4972.71i		−	0.616956i
$403$	−3127.14			−0.386535
$404$	−866.181			−0.106669
$405$			1452.70i			0.178236i
$406$			4850.52i			0.592925i
$407$	2665.61			0.324642
$408$		−	244.247i		−	0.0296373i
$409$			10886.1i			1.31609i	0.752977	+	0.658047i	$0.228617\pi$
$409$			10886.1i			1.31609i	−0.752977	+	0.658047i	$0.771383\pi$
$410$			19238.7i			2.31740i
$411$			6190.54i			0.742961i
$412$			637.976i			0.0762884i
$413$	5035.28			0.599927
$414$	5966.54			0.708308
$415$	−6347.59			−0.750822
$416$		−	6309.97i		−	0.743682i
$417$		−	9115.92i		−	1.07052i
$418$			1656.30i			0.193809i
$419$	12621.9			1.47164			0.735821	−	0.677176i	$-0.236797\pi$
$419$	12621.9			1.47164			0.735821	+	0.677176i	$0.236797\pi$
$420$	5164.72			0.600030
$421$			7853.67i			0.909179i	0.890701	+	0.454589i	$0.150214\pi$
$421$			7853.67i			0.909179i	−0.890701	+	0.454589i	$0.849786\pi$
$422$	8861.77			1.02224
$423$	635.734			0.0730743
$424$	−703.311			−0.0805561
$425$	1772.61			0.202315
$426$		−	835.678i		−	0.0950440i
$427$	3302.60			0.374295
$428$			5959.16i			0.673007i
$429$	1841.32			0.207226
$430$			30413.9i			3.41091i
$431$	8305.24			0.928189			0.464094	−	0.885786i	$-0.346380\pi$
$431$	8305.24			0.928189			0.464094	+	0.885786i	$0.346380\pi$
$432$	2095.47			0.233376
$433$		−	13157.9i		−	1.46034i	−0.683266	−	0.730169i	$-0.739441\pi$
$433$		−	13157.9i		−	1.46034i	0.683266	−	0.730169i	$-0.260559\pi$
$434$	−6736.70			−0.745097
$435$	−4096.74			−0.451549
$436$	2583.86			0.283817
$437$		−	3905.38i		−	0.427505i
$438$	−4881.72			−0.532551
$439$		−	18121.2i		−	1.97011i	−0.172253	−	0.985053i	$-0.555105\pi$
$439$		−	18121.2i		−	1.97011i	0.172253	−	0.985053i	$-0.444895\pi$
$440$	3362.23			0.364291
$441$	390.756			0.0421937
$442$		−	981.036i		−	0.105573i
$443$			3478.78i			0.373097i	0.982446	+	0.186548i	$0.0597301\pi$
$443$			3478.78i			0.373097i	−0.982446	+	0.186548i	$0.940270\pi$
$444$	2136.71			0.228387
$445$			25899.1i			2.75896i
$446$	−18712.1			−1.98664
$447$		−	310.889i		−	0.0328961i
$448$		−	2846.91i		−	0.300231i
$449$		−	16449.4i		−	1.72895i	−0.502678	−	0.864474i	$-0.667652\pi$
$449$		−	16449.4i		−	1.72895i	0.502678	−	0.864474i	$-0.332348\pi$
$450$			6513.91i			0.682374i
$451$			6049.58i			0.631627i
$452$	−6879.07			−0.715850
$453$		−	9439.82i		−	0.979076i
$454$	−22359.3			−2.31139
$455$	−9179.37			−0.945792
$456$		−	587.488i		−	0.0603326i
$457$	4265.49			0.436611			0.218306	−	0.975880i	$-0.429947\pi$
$457$	4265.49			0.436611			0.218306	+	0.975880i	$0.429947\pi$
$458$	−1699.00			−0.173338
$459$	243.378			0.0247493
$460$	17916.1			1.81596
$461$	5358.88			0.541405			0.270703	−	0.962663i	$-0.412744\pi$
$461$	5358.88			0.541405			0.270703	+	0.962663i	$0.412744\pi$
$462$	3966.71			0.399455
$463$		−	10030.8i		−	1.00685i	−0.864038	−	0.503426i	$-0.832073\pi$
$463$		−	10030.8i		−	1.00685i	0.864038	−	0.503426i	$-0.167927\pi$
$464$			5909.40i			0.591244i
$465$		−	5689.81i		−	0.567438i
$466$			15796.1i			1.57026i
$467$	6401.52			0.634320			0.317160	−	0.948372i	$-0.397271\pi$
$467$	6401.52			0.634320			0.317160	+	0.948372i	$0.397271\pi$
$468$	1475.98			0.145785
$469$			7795.18i			0.767480i
$470$	4662.61			0.457596
$471$	−5650.34	+	1703.71i	−0.552769	+	0.166673i
$472$	2627.57			0.256237
$473$			9563.62i			0.929674i
$474$	−9231.18			−0.894519
$475$	4263.66			0.411853
$476$		−	865.270i		−	0.0833184i
$477$		−	700.810i		−	0.0672702i
$478$			18554.2i			1.77542i
$479$		−	943.761i		−	0.0900241i	−0.998986	−	0.0450120i	$-0.985667\pi$
$479$		−	943.761i		−	0.0900241i	0.998986	−	0.0450120i	$-0.0143326\pi$
$480$	11481.0			1.09173
$481$	−3797.63			−0.359993
$482$	−15073.1			−1.42440
$483$	−9353.09			−0.881118
$484$	4992.36			0.468855
$485$	−11264.1			−1.05459
$486$			894.357i			0.0834750i
$487$	−206.853			−0.0192472			−0.00962361	−	0.999954i	$-0.503063\pi$
$487$	−206.853			−0.0192472			−0.00962361	+	0.999954i	$0.503063\pi$
$488$	1723.40			0.159866
$489$			1090.63i			0.100859i
$490$	2865.89			0.264220
$491$			20663.2i			1.89922i	0.313427	+	0.949612i	$0.398523\pi$
$491$			20663.2i			1.89922i	−0.313427	+	0.949612i	$0.601477\pi$
$492$			4849.27i			0.444353i
$493$			686.346i			0.0627008i
$494$		−	2359.69i		−	0.214914i
$495$			3350.28i			0.304210i
$496$	−8207.34			−0.742985
$497$			1310.00i			0.118233i
$498$	−3907.90			−0.351641
$499$		−	15927.5i		−	1.42888i	−0.699696	−	0.714441i	$-0.746681\pi$
$499$		−	15927.5i		−	1.42888i	0.699696	−	0.714441i	$-0.253319\pi$
$500$			7126.64i			0.637426i
$501$	−5463.62			−0.487219
$502$	21356.5			1.89878
$503$		−	12484.3i		−	1.10666i	−0.832963	−	0.553329i	$-0.813357\pi$
$503$		−	12484.3i		−	1.10666i	0.832963	−	0.553329i	$-0.186643\pi$
$504$	−1406.99			−0.124350
$505$			2801.08i			0.246824i
$506$	13760.2			1.20893
$507$	3967.71			0.347559
$508$	6965.76			0.608377
$509$			15849.2i			1.38016i	0.723732	+	0.690081i	$0.242425\pi$
$509$			15849.2i			1.38016i	−0.723732	+	0.690081i	$0.757575\pi$
$510$	1784.99			0.154982
$511$	7652.53			0.662481
$512$		−	10953.0i		−	0.945427i
$513$	585.399			0.0503821
$514$		−	12328.3i		−	1.05793i
$515$	2063.10			0.176527
$516$			7666.07i			0.654031i
$517$	1466.15			0.124722
$518$	−8181.12			−0.693934
$519$	3760.02			0.318009
$520$	−4790.09			−0.403960
$521$		−	2320.87i		−	0.195162i	−0.995228	−	0.0975808i	$-0.968890\pi$
$521$		−	2320.87i		−	0.195162i	0.995228	−	0.0975808i	$-0.0311104\pi$
$522$	−2522.16			−0.211479
$523$	−13804.8			−1.15419			−0.577097	−	0.816675i	$-0.695815\pi$
$523$	−13804.8			−1.15419			−0.577097	+	0.816675i	$0.695815\pi$
$524$		−	6869.01i		−	0.572660i
$525$		−	10211.1i		−	0.848858i
$526$		−	10999.0i		−	0.911746i
$527$	−953.240			−0.0787927
$528$	4832.65			0.398322
$529$	−20278.2			−1.66666
$530$		−	5139.89i		−	0.421250i
$531$			2618.23i			0.213976i
$532$		−	2081.24i		−	0.169611i
$533$		−	8618.70i		−	0.700408i
$534$			15944.8i			1.29213i
$535$	19270.9			1.55730
$536$			4067.78i			0.327801i
$537$			1659.61i			0.133366i
$538$	−2901.81			−0.232539
$539$	901.174			0.0720155
$540$			2685.54i			0.214013i
$541$			18967.2i			1.50732i	0.657262	+	0.753662i	$0.271715\pi$
$541$			18967.2i			1.50732i	−0.657262	+	0.753662i	$0.728285\pi$
$542$	7792.87			0.617587
$543$		−	4352.82i		−	0.344010i
$544$		−	1923.46i		−	0.151595i
$545$		−	8355.75i		−	0.656736i
$546$	−5651.28			−0.442953
$547$	−20521.1			−1.60406			−0.802028	−	0.597286i	$-0.796246\pi$
$547$	−20521.1			−1.60406			−0.802028	+	0.597286i	$0.796246\pi$
$548$			11444.1i			0.892096i
$549$			1717.28i			0.133500i
$550$			15022.6i			1.16467i
$551$			1650.87i			0.127640i
$552$	−4880.75			−0.376338
$553$	14470.7			1.11276
$554$			220.092i			0.0168788i
$555$		−	6909.76i		−	0.528474i
$556$		−	16852.1i		−	1.28541i
$557$	21723.9			1.65255			0.826277	−	0.563264i	$-0.190455\pi$
$557$	21723.9			1.65255			0.826277	+	0.563264i	$0.190455\pi$
$558$		−	3502.93i		−	0.265754i
$559$		−	13625.1i		−	1.03091i
$560$	−24091.7			−1.81797
$561$	561.287			0.0422417
$562$		−	4057.25i		−	0.304528i
$563$		−	5760.34i		−	0.431207i	−0.976481	−	0.215603i	$-0.930828\pi$
$563$		−	5760.34i		−	0.431207i	0.976481	−	0.215603i	$-0.0691719\pi$
$564$	1175.25			0.0877426
$565$			22245.7i			1.65643i
$566$			27200.0i			2.01997i
$567$		−	1401.99i		−	0.103841i
$568$			683.601i			0.0504987i
$569$			4555.49i			0.335635i	0.985818	+	0.167817i	$0.0536719\pi$
$569$			4555.49i			0.335635i	−0.985818	+	0.167817i	$0.946328\pi$
$570$	4293.45			0.315496
$571$	4294.23			0.314725			0.157363	−	0.987541i	$-0.449701\pi$
$571$	4294.23			0.314725			0.157363	+	0.987541i	$0.449701\pi$
$572$	3403.96			0.248823
$573$			6695.03i			0.488113i
$574$		−	18567.0i		−	1.35013i
$575$		−	35421.7i		−	2.56902i
$576$	1480.33			0.107084
$577$	−1970.57			−0.142177			−0.0710884	−	0.997470i	$-0.522647\pi$
$577$	−1970.57			−0.142177			−0.0710884	+	0.997470i	$0.522647\pi$
$578$			17783.2i			1.27973i
$579$	−2291.89			−0.164504
$580$	−7573.43			−0.542189
$581$	6125.98			0.437433
$582$	−6934.73			−0.493907
$583$		−	1616.23i		−	0.114816i
$584$	3993.34			0.282955
$585$		−	4773.06i		−	0.337336i
$586$	−8504.92			−0.599548
$587$			2903.91i			0.204186i	0.994775	+	0.102093i	$0.0325540\pi$
$587$			2903.91i			0.204186i	−0.994775	+	0.102093i	$0.967446\pi$
$588$	722.369			0.0506633
$589$	−2292.83			−0.160398
$590$			19202.7i			1.33993i
$591$	4090.31			0.284692
$592$	−9967.08			−0.691967
$593$	−9481.89			−0.656618			−0.328309	−	0.944570i	$-0.606479\pi$
$593$	−9481.89			−0.656618			−0.328309	+	0.944570i	$0.606479\pi$
$594$			2062.60i			0.142474i
$595$	−2798.13			−0.192794
$596$		−	574.725i		−	0.0394994i
$597$	1531.20			0.104971
$598$	−19603.9			−1.34057
$599$			5976.99i			0.407701i	0.979002	+	0.203851i	$0.0653457\pi$
$599$			5976.99i			0.407701i	−0.979002	+	0.203851i	$0.934654\pi$
$600$		−	5328.50i		−	0.362559i
$601$	8985.48			0.609859			0.304930	−	0.952375i	$-0.401367\pi$
$601$	8985.48			0.609859			0.304930	+	0.952375i	$0.401367\pi$
$602$		−	29352.1i		−	1.98721i
$603$	−4053.31			−0.273737
$604$		−	17450.9i		−	1.17561i
$605$		−	16144.4i		−	1.08490i
$606$			1724.48i			0.115598i
$607$		−	7452.31i		−	0.498320i	−0.968462	−	0.249160i	$-0.919846\pi$
$607$		−	7452.31i		−	0.498320i	0.968462	−	0.249160i	$-0.0801545\pi$
$608$		−	4626.51i		−	0.308601i
$609$	3953.71			0.263075
$610$			12594.9i			0.835987i
$611$	−2088.79			−0.138304
$612$	449.920			0.0297172
$613$			4838.55i			0.318804i	0.987214	+	0.159402i	$0.0509567\pi$
$613$			4838.55i			0.318804i	−0.987214	+	0.159402i	$0.949043\pi$
$614$	−1485.00			−0.0976053
$615$	15681.7			1.02821
$616$	−3244.85			−0.212238
$617$	−18821.7			−1.22810			−0.614048	−	0.789269i	$-0.710460\pi$
$617$	−18821.7			−1.22810			−0.614048	+	0.789269i	$0.710460\pi$
$618$	1270.15			0.0826746
$619$	27567.7			1.79004			0.895022	−	0.446021i	$-0.147159\pi$
$619$	27567.7			1.79004			0.895022	+	0.446021i	$0.147159\pi$
$620$		−	10518.4i		−	0.681340i
$621$		−	4863.39i		−	0.314269i
$622$			31981.5i			2.06164i
$623$		−	24994.9i		−	1.60738i
$624$	−6884.96			−0.441697
$625$	−1534.97			−0.0982381
$626$			715.515i			0.0456833i
$627$	1350.07			0.0859913
$628$	−10445.5	+	3149.57i	−0.663727	+	0.200130i
$629$	−1157.62			−0.0733823
$630$		−	10282.5i		−	0.650259i
$631$	1556.70			0.0982115			0.0491057	−	0.998794i	$-0.484363\pi$
$631$	1556.70			0.0982115			0.0491057	+	0.998794i	$0.484363\pi$
$632$	7551.28			0.475275
$633$		−	7223.32i		−	0.453557i
$634$			10704.5i			0.670551i
$635$		−	22526.0i		−	1.40775i
$636$		−	1295.55i		−	0.0807734i
$637$	−1283.88			−0.0798576
$638$	−5816.70			−0.360949
$639$	−681.170			−0.0421701
$640$	−19758.8			−1.22037
$641$	18312.1			1.12837			0.564185	−	0.825648i	$-0.309190\pi$
$641$	18312.1			1.12837			0.564185	+	0.825648i	$0.309190\pi$
$642$	11864.1			0.729345
$643$			25006.7i			1.53370i	0.641828	+	0.766849i	$0.278176\pi$
$643$			25006.7i			1.53370i	−0.641828	+	0.766849i	$0.721824\pi$
$644$	−17290.6			−1.05799
$645$	24790.7			1.51339
$646$		−	719.301i		−	0.0438089i
$647$	−11011.7			−0.669108			−0.334554	−	0.942377i	$-0.608586\pi$
$647$	−11011.7			−0.669108			−0.334554	+	0.942377i	$0.608586\pi$
$648$		−	731.601i		−	0.0443519i
$649$			6038.25i			0.365211i
$650$		−	21402.3i		−	1.29149i
$651$			5491.16i			0.330592i
$652$			2016.19i			0.121104i
$653$	−26366.3			−1.58008			−0.790039	−	0.613056i	$-0.789940\pi$
$653$	−26366.3			−1.58008			−0.790039	+	0.613056i	$0.789940\pi$
$654$		−	5144.22i		−	0.307576i
$655$	−22213.2			−1.32510
$656$		−	22620.3i		−	1.34630i
$657$			3979.14i			0.236288i
$658$	−4499.82			−0.266598
$659$	3109.00			0.183778			0.0918889	−	0.995769i	$-0.470710\pi$
$659$	3109.00			0.183778			0.0918889	+	0.995769i	$0.470710\pi$
$660$			6193.48i			0.365274i
$661$	6646.58			0.391107			0.195554	−	0.980693i	$-0.437350\pi$
$661$	6646.58			0.391107			0.195554	+	0.980693i	$0.437350\pi$
$662$		−	4553.99i		−	0.267365i
$663$	−799.653			−0.0468415
$664$	3196.73			0.186833
$665$	−6730.37			−0.392470
$666$		−	4253.99i		−	0.247506i
$667$	13715.2			0.796182
$668$	−10100.3			−0.585019
$669$			15252.4i			0.881453i
$670$	−29727.9			−1.71416
$671$			3960.44i			0.227856i
$672$	−11080.1			−0.636049
$673$		−	45.5835i		−	0.00261087i	−0.999999	−	0.00130543i	$-0.999584\pi$
$673$		−	45.5835i		−	0.00261087i	0.999999	−	0.00130543i	$-0.000415533\pi$
$674$	−36956.8			−2.11205
$675$	5309.56			0.302763
$676$	7334.90			0.417325
$677$	−25701.5			−1.45907			−0.729534	−	0.683945i	$-0.760263\pi$
$677$	−25701.5			−1.45907			−0.729534	+	0.683945i	$0.760263\pi$
$678$			13695.6i			0.775775i
$679$	10870.8			0.614409
$680$	−1460.16			−0.0823447
$681$			18225.3i			1.02554i
$682$		−	8078.58i		−	0.453585i
$683$		−	7013.46i		−	0.392918i	−0.980512	−	0.196459i	$-0.937056\pi$
$683$		−	7013.46i		−	0.392918i	0.980512	−	0.196459i	$-0.0629442\pi$
$684$	1082.20			0.0604954
$685$	37008.3			2.06425
$686$	−24616.1			−1.37004
$687$			1384.87i			0.0769084i
$688$		−	35759.7i		−	1.98158i
$689$			2302.61i			0.127318i
$690$		−	35669.2i		−	1.96797i
$691$		−	23302.8i		−	1.28290i	−0.767166	−	0.641448i	$-0.778334\pi$
$691$		−	23302.8i		−	1.28290i	0.767166	−	0.641448i	$-0.221666\pi$
$692$	6950.95			0.381843
$693$		−	3233.31i		−	0.177234i
$694$		−	3456.31i		−	0.189048i
$695$	−54496.8			−2.97436
$696$	2063.18			0.112363
$697$		−	2627.22i		−	0.142774i
$698$		−	21866.1i		−	1.18573i
$699$	12875.6			0.696709
$700$		−	18876.8i		−	1.01925i
$701$			5593.52i			0.301376i	0.988581	+	0.150688i	$0.0481488\pi$
$701$			5593.52i			0.301376i	−0.988581	+	0.150688i	$0.951851\pi$
$702$		−	2938.53i		−	0.157988i
$703$	−2784.44			−0.149384
$704$	3413.98			0.182769
$705$		−	3800.54i		−	0.203031i
$706$			9196.47i			0.490246i
$707$		−	2703.28i		−	0.143801i
$708$			4840.18i			0.256928i
$709$	−7428.60			−0.393493			−0.196747	−	0.980454i	$-0.563038\pi$
$709$	−7428.60			−0.393493			−0.196747	+	0.980454i	$0.563038\pi$
$710$	−4995.85			−0.264072
$711$			7524.43i			0.396889i
$712$		−	13043.2i		−	0.686535i
$713$			19048.5i			1.00052i
$714$	−1722.67			−0.0902931
$715$		−	11007.8i		−	0.575760i
$716$			3068.03i			0.160136i
$717$	15123.7			0.787735
$718$	22530.4			1.17107
$719$		−	23600.4i		−	1.22413i	−0.790809	−	0.612063i	$-0.790340\pi$
$719$		−	23600.4i		−	1.22413i	0.790809	−	0.612063i	$-0.209660\pi$
$720$		−	12527.2i		−	0.648416i
$721$	−1991.07			−0.102845
$722$			23514.3i			1.21206i
$723$			12286.2i			0.631992i
$724$		−	8046.82i		−	0.413063i
$725$			14973.4i			0.767031i
$726$		−	9939.32i		−	0.508103i
$727$	34110.6			1.74016			0.870078	−	0.492915i	$-0.164068\pi$
$727$	34110.6			1.74016			0.870078	+	0.492915i	$0.164068\pi$
$728$	4622.86			0.235349
$729$	729.000			0.0370370
$730$			29183.9i			1.47965i
$731$		−	4153.30i		−	0.210144i
$732$			3174.64i			0.160298i
$733$	33471.6			1.68663			0.843317	−	0.537416i	$-0.180600\pi$
$733$	33471.6			1.68663			0.843317	+	0.537416i	$0.180600\pi$
$734$	−6361.87			−0.319920
$735$		−	2336.02i		−	0.117232i
$736$	−38436.2			−1.92497
$737$	−9347.90			−0.467211
$738$	9654.43			0.481551
$739$	−35121.5			−1.74826			−0.874131	−	0.485689i	$-0.838569\pi$
$739$	−35121.5			−1.74826			−0.874131	+	0.485689i	$0.838569\pi$
$740$		−	12773.7i		−	0.634555i
$741$	−1923.41			−0.0953553
$742$			4960.45i			0.245423i
$743$	34672.3			1.71198			0.855992	−	0.516989i	$-0.172947\pi$
$743$	34672.3			1.71198			0.855992	+	0.516989i	$0.172947\pi$
$744$			2865.47i			0.141200i
$745$	−1858.56			−0.0913991
$746$	49729.9			2.44067
$747$			3185.37i			0.156019i
$748$	1037.62			0.0507209
$749$	−18598.1			−0.907288
$750$	14188.5			0.690786
$751$			20576.3i			0.999787i	0.866087	+	0.499894i	$0.166628\pi$
$751$			20576.3i			0.999787i	−0.866087	+	0.499894i	$0.833372\pi$
$752$	−5482.15			−0.265842
$753$		−	17407.9i		−	0.842468i
$754$	8286.91			0.400254
$755$	−56433.2			−2.72028
$756$		−	2591.78i		−	0.124685i
$757$			34007.6i			1.63280i	0.577489	+	0.816398i	$0.304033\pi$
$757$			34007.6i			1.63280i	−0.577489	+	0.816398i	$0.695967\pi$
$758$	3843.12			0.184154
$759$		−	11216.1i		−	0.536389i
$760$	−3512.12			−0.167629
$761$			9841.15i			0.468780i	0.972143	+	0.234390i	$0.0753092\pi$
$761$			9841.15i			0.468780i	−0.972143	+	0.234390i	$0.924691\pi$
$762$		−	13868.2i		−	0.659305i
$763$			8064.03i			0.382618i
$764$			12376.7i			0.586093i
$765$		−	1454.96i		−	0.0687638i
$766$	35571.8			1.67789
$767$		−	8602.56i		−	0.404981i
$768$	−16112.1			−0.757024
$769$	24096.1			1.12995			0.564973	−	0.825110i	$-0.308887\pi$
$769$	24096.1			1.12995			0.564973	+	0.825110i	$0.308887\pi$
$770$		−	23713.8i		−	1.10985i
$771$	−10048.9			−0.469394
$772$	−4236.90			−0.197525
$773$	12484.5			0.580901			0.290451	−	0.956890i	$-0.406195\pi$
$773$	12484.5			0.580901			0.290451	+	0.956890i	$0.406195\pi$
$774$	15262.4			0.708780
$775$	−20796.0			−0.963887
$776$	5672.75			0.262422
$777$			6668.52i			0.307892i
$778$		−	36661.6i		−	1.68944i
$779$		−	6319.28i		−	0.290644i
$780$		−	8823.70i		−	0.405050i
$781$	−1570.94			−0.0719752
$782$	−5975.82			−0.273267
$783$			2055.84i			0.0938311i
$784$	−3369.62			−0.153499
$785$	10185.1	+	33778.9i	0.463087	+	1.53582i
$786$	−13675.5			−0.620599
$787$			27898.7i			1.26364i	0.775116	+	0.631819i	$0.217691\pi$
$787$			27898.7i			1.26364i	−0.775116	+	0.631819i	$0.782309\pi$
$788$	7561.54			0.341838
$789$	−8965.38			−0.404533
$790$			55185.9i			2.48535i
$791$		−	21469.1i		−	0.965046i
$792$		−	1687.25i		−	0.0756990i
$793$		−	5642.35i		−	0.252668i
$794$	−47414.2			−2.11923
$795$	−4189.58			−0.186905
$796$	2830.65			0.126042
$797$	6474.60			0.287757			0.143878	−	0.989595i	$-0.454043\pi$
$797$	6474.60			0.287757			0.143878	+	0.989595i	$0.454043\pi$
$798$	−4143.55			−0.183810
$799$	−636.723			−0.0281923
$800$		−	41962.3i		−	1.85449i
$801$	12996.8			0.573307
$802$	−9273.15			−0.408287
$803$			9176.83i			0.403292i
$804$	−7493.15			−0.328685
$805$			55914.7i			2.44812i
$806$			11509.4i			0.502978i
$807$			2365.29i			0.103175i
$808$		−	1410.66i		−	0.0614194i
$809$			7436.74i			0.323191i	0.986857	+	0.161596i	$0.0516641\pi$
$809$			7436.74i			0.323191i	−0.986857	+	0.161596i	$0.948336\pi$
$810$	5346.65			0.231928
$811$		−	3133.04i		−	0.135655i	−0.997697	−	0.0678274i	$-0.978393\pi$
$811$		−	3133.04i		−	0.135655i	0.997697	−	0.0678274i	$-0.0216067\pi$
$812$	7309.02			0.315882
$813$		−	6352.05i		−	0.274017i
$814$		−	9810.71i		−	0.422439i
$815$	6519.99			0.280228
$816$	−2098.73			−0.0900372
$817$		−	9989.98i		−	0.427791i
$818$	40066.0			1.71256
$819$			4606.42i			0.196534i
$820$	28989.9			1.23460
$821$	9896.69			0.420703			0.210351	−	0.977626i	$-0.432539\pi$
$821$	9896.69			0.420703			0.210351	+	0.977626i	$0.432539\pi$
$822$	22784.2			0.966775
$823$			14890.0i			0.630658i	0.948982	+	0.315329i	$0.102115\pi$
$823$			14890.0i			0.630658i	−0.948982	+	0.315329i	$0.897885\pi$
$824$	−1039.01			−0.0439266
$825$	12245.1			0.516751
$826$		−	18532.2i		−	0.780653i
$827$	−15532.0			−0.653082			−0.326541	−	0.945183i	$-0.605883\pi$
$827$	−15532.0			−0.653082			−0.326541	+	0.945183i	$0.605883\pi$
$828$		−	8990.69i		−	0.377353i
$829$	−14245.2			−0.596810			−0.298405	−	0.954439i	$-0.596455\pi$
$829$	−14245.2			−0.596810			−0.298405	+	0.954439i	$0.596455\pi$
$830$			23362.2i			0.977004i
$831$	179.400			0.00748894
$832$	−4863.82			−0.202671
$833$	−391.364			−0.0162784
$834$	−33551.0			−1.39302
$835$			32662.6i			1.35370i
$836$	2495.80			0.103252
$837$	−2855.28			−0.117913
$838$		−	46454.5i		−	1.91497i
$839$			4778.43i			0.196626i	0.995156	+	0.0983132i	$0.0313447\pi$
$839$			4778.43i			0.196626i	−0.995156	+	0.0983132i	$0.968655\pi$
$840$			8411.26i			0.345495i
$841$	18591.4			0.762285
$842$	28905.3			1.18307
$843$	−3307.11			−0.135116
$844$		−	13353.4i		−	0.544600i
$845$		−	23719.8i		−	0.965663i
$846$		−	2339.81i		−	0.0950877i
$847$			15580.8i			0.632069i
$848$			6043.32i			0.244727i
$849$	22171.0			0.896241
$850$		−	6524.04i		−	0.263262i
$851$			23132.6i			0.931817i
$852$	−1259.24			−0.0506349
$853$	−38298.7			−1.53731			−0.768653	−	0.639666i	$-0.779073\pi$
$853$	−38298.7			−1.53731			−0.768653	+	0.639666i	$0.779073\pi$
$854$		−	12155.2i		−	0.487050i
$855$		−	3499.64i		−	0.139982i
$856$	−9705.08			−0.387515
$857$		−	30021.7i		−	1.19664i	−0.801257	−	0.598321i	$-0.795835\pi$
$857$		−	30021.7i		−	1.19664i	0.801257	−	0.598321i	$-0.204165\pi$
$858$		−	6776.95i		−	0.269652i
$859$		−	35732.2i		−	1.41929i	−0.704561	−	0.709643i	$-0.748856\pi$
$859$		−	35732.2i		−	1.41929i	0.704561	−	0.709643i	$-0.251144\pi$
$860$	45829.3			1.81717
$861$	−15134.2			−0.599038
$862$		−	30567.3i		−	1.20780i
$863$			22804.5i			0.899508i	0.893152	+	0.449754i	$0.148488\pi$
$863$			22804.5i			0.899508i	−0.893152	+	0.449754i	$0.851512\pi$
$864$		−	5761.41i		−	0.226860i
$865$		−	22478.2i		−	0.883561i
$866$	−48427.3			−1.90026
$867$	14495.2			0.567802
$868$			10151.2i			0.396952i
$869$			17353.1i			0.677404i
$870$			15078.0i			0.587577i
$871$	13317.7			0.518087
$872$			4208.07i			0.163421i
$873$			5652.58i			0.219142i
$874$	−14373.7			−0.556290
$875$	−22241.7			−0.859322
$876$			7356.03i			0.283718i
$877$			30159.2i			1.16124i	0.814176	+	0.580618i	$0.197189\pi$
$877$			30159.2i			1.16124i	−0.814176	+	0.580618i	$0.802811\pi$
$878$	−66694.6			−2.56359
$879$			6932.45i			0.266013i
$880$		−	28890.6i		−	1.10671i
$881$			15693.7i			0.600151i	0.953915	+	0.300076i	$0.0970119\pi$
$881$			15693.7i			0.600151i	−0.953915	+	0.300076i	$0.902988\pi$
$882$		−	1438.17i		−	0.0549044i
$883$		−	4003.32i		−	0.152573i	−0.997086	−	0.0762867i	$-0.975694\pi$
$883$		−	4003.32i		−	0.152573i	0.997086	−	0.0762867i	$-0.0243064\pi$
$884$	−1478.28			−0.0562441
$885$	15652.3			0.594516
$886$	12803.6			0.485491
$887$			29451.5i			1.11486i	0.830223	+	0.557432i	$0.188213\pi$
$887$			29451.5i			1.11486i	−0.830223	+	0.557432i	$0.811787\pi$
$888$			3479.85i			0.131505i
$889$			21739.6i			0.820160i
$890$	95321.3			3.59009
$891$	1681.25			0.0632142
$892$			28196.3i			1.05839i
$893$	−1531.51			−0.0573910
$894$	−1144.22			−0.0428060
$895$	9921.47			0.370546
$896$	19069.0			0.710994
$897$			15979.3i			0.594799i
$898$	−60541.9			−2.24979
$899$		−	8052.12i		−	0.298724i
$900$	9815.50			0.363537
$901$			701.900i			0.0259530i
$902$	22265.4			0.821903
$903$	−23925.2			−0.881707
$904$		−	11203.2i		−	0.412184i
$905$	−26022.0			−0.955802
$906$	−34743.1			−1.27402
$907$	27873.3			1.02042			0.510208	−	0.860051i	$-0.329568\pi$
$907$	27873.3			1.02042			0.510208	+	0.860051i	$0.329568\pi$
$908$			33692.1i			1.23140i
$909$	1405.65			0.0512897
$910$			33784.5i			1.23071i
$911$	19571.6			0.711785			0.355893	−	0.934527i	$-0.384177\pi$
$911$	19571.6			0.711785			0.355893	+	0.934527i	$0.384177\pi$
$912$	−5048.10			−0.183289
$913$			7346.21i			0.266291i
$914$		−	15699.1i		−	0.568139i
$915$	10266.2			0.370919
$916$			2560.14i			0.0923464i
$917$	21437.7			0.772010
$918$		−	895.749i		−	0.0322049i
$919$			21914.5i			0.786608i	0.919408	+	0.393304i	$0.128668\pi$
$919$			21914.5i			0.786608i	−0.919408	+	0.393304i	$0.871332\pi$
$920$			29178.1i			1.04562i
$921$			1210.44i			0.0433065i
$922$		−	19723.3i		−	0.704502i
$923$	2238.08			0.0798129
$924$		−	5977.25i		−	0.212811i
$925$	−25254.8			−0.897701
$926$	−36918.3			−1.31016
$927$		−	1035.31i		−	0.0366819i
$928$	16247.6			0.574736
$929$	36554.0			1.29096			0.645479	−	0.763778i	$-0.276658\pi$
$929$	36554.0			1.29096			0.645479	+	0.763778i	$0.276658\pi$
$930$	−20941.2			−0.738376
$931$	−941.350			−0.0331380
$932$	23802.4			0.836561
$933$	26068.5			0.914730
$934$		−	23560.7i		−	0.825406i
$935$		−	3355.49i		−	0.117365i
$936$			2403.78i			0.0839423i
$937$		−	7071.68i		−	0.246555i	−0.992372	−	0.123277i	$-0.960660\pi$
$937$		−	7071.68i		−	0.246555i	0.992372	−	0.123277i	$-0.0393405\pi$
$938$	28690.0			0.998680
$939$	583.224			0.0202692
$940$		−	7025.87i		−	0.243786i
$941$	35782.7			1.23962			0.619810	−	0.784752i	$-0.287210\pi$
$941$	35782.7			1.23962			0.619810	+	0.784752i	$0.287210\pi$
$942$	6270.49	+	20796.0i	0.216883	+	0.719289i
$943$	−52499.5			−1.81296
$944$		−	22577.9i		−	0.778440i
$945$	−8381.35			−0.288514
$946$	35198.7			1.20973
$947$			29101.5i			0.998596i	0.866430	+	0.499298i	$0.166409\pi$
$947$			29101.5i			0.998596i	−0.866430	+	0.499298i	$0.833591\pi$
$948$			13910.0i			0.476558i
$949$		−	13074.0i		−	0.447208i
$950$		−	15692.3i		−	0.535922i
$951$	8725.35			0.297517
$952$	1409.18			0.0479745
$953$	42846.9			1.45640			0.728198	−	0.685367i	$-0.240358\pi$
$953$	42846.9			1.45640			0.728198	+	0.685367i	$0.240358\pi$
$954$	−2579.32			−0.0875351
$955$	40024.2			1.35618
$956$	27958.4			0.945858
$957$			4741.25i			0.160149i
$958$	−3473.49			−0.117144
$959$	−35716.2			−1.20265
$960$		−	8849.69i		−	0.297523i
$961$	−18607.7			−0.624609
$962$			13977.1i			0.468440i
$963$		−	9670.57i		−	0.323603i
$964$			22712.9i			0.758853i
$965$			13701.4i			0.457061i
$966$			34423.9i			1.14655i
$967$	−38386.4			−1.27655			−0.638275	−	0.769808i	$-0.720352\pi$
$967$	−38386.4			−1.27655			−0.638275	+	0.769808i	$0.720352\pi$
$968$			8130.56i			0.269965i
$969$	−586.310			−0.0194376
$970$			41457.2i			1.37228i
$971$			45003.2i			1.48736i	0.668538	+	0.743678i	$0.266920\pi$
$971$			45003.2i			1.48736i	−0.668538	+	0.743678i	$0.733080\pi$
$972$	1347.66			0.0444715
$973$	52594.2			1.73288
$974$			761.318i			0.0250454i
$975$	−17445.3			−0.573022
$976$		−	14808.7i		−	0.485670i
$977$	32421.2			1.06167			0.530833	−	0.847477i	$-0.321879\pi$
$977$	32421.2			1.06167			0.530833	+	0.847477i	$0.321879\pi$
$978$	4014.03			0.131242
$979$	29973.6			0.978511
$980$		−	4318.47i		−	0.140764i
$981$	−4193.11			−0.136469
$982$	76050.7			2.47136
$983$			55056.5i			1.78640i	0.449662	+	0.893199i	$0.351545\pi$
$983$			55056.5i			1.78640i	−0.449662	+	0.893199i	$0.648455\pi$
$984$	−7897.51			−0.255857
$985$		−	24452.7i		−	0.790993i
$986$	2526.09			0.0815892
$987$			3667.86i			0.118287i
$988$	−3555.71			−0.114496
$989$	−82994.9			−2.66844
$990$	12330.6			0.395852
$991$	−36863.3			−1.18164			−0.590818	−	0.806805i	$-0.701195\pi$
$991$	−36863.3			−1.18164			−0.590818	+	0.806805i	$0.701195\pi$
$992$			22565.7i			0.722240i
$993$	−3712.01			−0.118627
$994$	4821.43			0.153850
$995$		−	9153.83i		−	0.291654i
$996$			5888.62i			0.187337i
$997$		−	40871.1i		−	1.29830i	−0.760662	−	0.649148i	$-0.775126\pi$
$997$		−	40871.1i		−	1.29830i	0.760662	−	0.649148i	$-0.224874\pi$
$998$	−58620.8			−1.85933
$999$	−3467.48			−0.109816

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.4.b.a.313.9	✓	40
157.156	even	2	inner	471.4.b.a.313.32	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.4.b.a.313.9	✓	40	1.1	even	1	trivial
471.4.b.a.313.32	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)

\(f(q)\)	\(=\)	\(q-3.68048i q^{2} -3.00000 q^{3} -5.54594 q^{4} +17.9346i q^{5} +11.0414i q^{6} -17.3085i q^{7} -9.03212i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-3.68048i q^{2} -3.00000 q^{3} -5.54594 q^{4} +17.9346i q^{5} +11.0414i q^{6} -17.3085i q^{7} -9.03212i q^{8} +9.00000 q^{9} +66.0080 q^{10} +20.7561 q^{11} +16.6378 q^{12} -29.5708 q^{13} -63.7035 q^{14} -53.8038i q^{15} -77.6101 q^{16} -9.01400 q^{17} -33.1243i q^{18} -21.6815 q^{19} -99.4643i q^{20} +51.9254i q^{21} -76.3925i q^{22} +180.126i q^{23} +27.0963i q^{24} -196.650 q^{25} +108.835i q^{26} -27.0000 q^{27} +95.9917i q^{28} -76.1422i q^{29} -198.024 q^{30} +105.751 q^{31} +213.385i q^{32} -62.2684 q^{33} +33.1759i q^{34} +310.420 q^{35} -49.9135 q^{36} +128.425 q^{37} +79.7982i q^{38} +88.7123 q^{39} +161.987 q^{40} +291.460i q^{41} +191.110 q^{42} +460.761i q^{43} -115.112 q^{44} +161.411i q^{45} +662.949 q^{46} +70.6371 q^{47} +232.830 q^{48} +43.4173 q^{49} +723.767i q^{50} +27.0420 q^{51} +163.998 q^{52} -77.8678i q^{53} +99.3730i q^{54} +372.253i q^{55} -156.332 q^{56} +65.0444 q^{57} -280.240 q^{58} +290.914i q^{59} +298.393i q^{60} +190.808i q^{61} -389.215i q^{62} -155.776i q^{63} +164.481 q^{64} -530.340i q^{65} +229.178i q^{66} -450.368 q^{67} +49.9911 q^{68} -540.377i q^{69} -1142.50i q^{70} -75.6856 q^{71} -81.2890i q^{72} +442.127i q^{73} -472.666i q^{74} +589.951 q^{75} +120.244 q^{76} -359.256i q^{77} -326.504i q^{78} +836.048i q^{79} -1391.91i q^{80} +81.0000 q^{81} +1072.71 q^{82} +353.930i q^{83} -287.975i q^{84} -161.663i q^{85} +1695.82 q^{86} +228.427i q^{87} -187.472i q^{88} +1444.09 q^{89} +594.072 q^{90} +511.824i q^{91} -998.966i q^{92} -317.253 q^{93} -259.978i q^{94} -388.848i q^{95} -640.156i q^{96} +628.064i q^{97} -159.797i q^{98} +186.805 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 120 q^{3} - 164 q^{4} + 360 q^{9}+O(q^{10}) \) 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

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