LMFDB - Embedded newform 471.4.b.a.313.10

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			313.10
Character	$\chi$	$=$	471.313
Dual form			471.4.b.a.313.31

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.30421i q^{2} -3.00000 q^{3} -2.91779 q^{4} +13.7388i q^{5} +9.91262i q^{6} +33.0358i q^{7} -16.7927i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-3.30421i q^{2} -3.00000 q^{3} -2.91779 q^{4} +13.7388i q^{5} +9.91262i q^{6} +33.0358i q^{7} -16.7927i q^{8} +9.00000 q^{9} +45.3958 q^{10} +17.8062 q^{11} +8.75336 q^{12} +14.1529 q^{13} +109.157 q^{14} -41.2164i q^{15} -78.8288 q^{16} -19.3632 q^{17} -29.7379i q^{18} -93.0887 q^{19} -40.0869i q^{20} -99.1075i q^{21} -58.8354i q^{22} -92.0967i q^{23} +50.3781i q^{24} -63.7545 q^{25} -46.7642i q^{26} -27.0000 q^{27} -96.3915i q^{28} +264.372i q^{29} -136.187 q^{30} -67.0293 q^{31} +126.125i q^{32} -53.4186 q^{33} +63.9800i q^{34} -453.873 q^{35} -26.2601 q^{36} -196.837 q^{37} +307.584i q^{38} -42.4588 q^{39} +230.711 q^{40} +97.6935i q^{41} -327.472 q^{42} -36.0530i q^{43} -51.9547 q^{44} +123.649i q^{45} -304.306 q^{46} -273.402 q^{47} +236.486 q^{48} -748.366 q^{49} +210.658i q^{50} +58.0896 q^{51} -41.2952 q^{52} -260.964i q^{53} +89.2136i q^{54} +244.636i q^{55} +554.760 q^{56} +279.266 q^{57} +873.541 q^{58} -669.127i q^{59} +120.261i q^{60} -266.218i q^{61} +221.479i q^{62} +297.323i q^{63} -213.886 q^{64} +194.444i q^{65} +176.506i q^{66} -97.0936 q^{67} +56.4976 q^{68} +276.290i q^{69} +1499.69i q^{70} +338.801 q^{71} -151.134i q^{72} +518.608i q^{73} +650.390i q^{74} +191.263 q^{75} +271.613 q^{76} +588.243i q^{77} +140.293i q^{78} -32.8801i q^{79} -1083.01i q^{80} +81.0000 q^{81} +322.800 q^{82} +1326.67i q^{83} +289.175i q^{84} -266.027i q^{85} -119.127 q^{86} -793.117i q^{87} -299.014i q^{88} -66.9332 q^{89} +408.562 q^{90} +467.554i q^{91} +268.718i q^{92} +201.088 q^{93} +903.377i q^{94} -1278.93i q^{95} -378.376i q^{96} -605.071i q^{97} +2472.76i q^{98} +160.256 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	3.30421i		−	1.16821i	−0.811677	−	0.584107i	$-0.801445\pi$
$2$		−	3.30421i		−	1.16821i	0.811677	−	0.584107i	$-0.198555\pi$
$3$	−3.00000			−0.577350
$3$	−3.00000			−0.577350
$4$	−2.91779			−0.364723
$5$			13.7388i			1.22884i	0.788981	+	0.614418i	$0.210609\pi$
$5$			13.7388i			1.22884i	−0.788981	+	0.614418i	$0.789391\pi$
$6$			9.91262i			0.674469i
$7$			33.0358i			1.78377i	0.452265	+	0.891884i	$0.350616\pi$
$7$			33.0358i			1.78377i	−0.452265	+	0.891884i	$0.649384\pi$
$8$		−	16.7927i		−	0.742139i
$9$	9.00000			0.333333
$10$	45.3958			1.43554
$11$	17.8062			0.488070			0.244035	−	0.969766i	$-0.421529\pi$
$11$	17.8062			0.488070			0.244035	+	0.969766i	$0.421529\pi$
$12$	8.75336			0.210573
$13$	14.1529			0.301947			0.150974	−	0.988538i	$-0.451759\pi$
$13$	14.1529			0.301947			0.150974	+	0.988538i	$0.451759\pi$
$14$	109.157			2.08382
$15$		−	41.2164i		−	0.709468i
$16$	−78.8288			−1.23170
$17$	−19.3632			−0.276251			−0.138125	−	0.990415i	$-0.544108\pi$
$17$	−19.3632			−0.276251			−0.138125	+	0.990415i	$0.544108\pi$
$18$		−	29.7379i		−	0.389405i
$19$	−93.0887			−1.12400			−0.562001	−	0.827137i	$-0.689968\pi$
$19$	−93.0887			−1.12400			−0.562001	+	0.827137i	$0.689968\pi$
$20$		−	40.0869i		−	0.448185i
$21$		−	99.1075i		−	1.02986i
$22$		−	58.8354i		−	0.570170i
$23$		−	92.0967i		−	0.834934i	−0.908692	−	0.417467i	$-0.862918\pi$
$23$		−	92.0967i		−	0.834934i	0.908692	−	0.417467i	$-0.137082\pi$
$24$			50.3781i			0.428474i
$25$	−63.7545			−0.510036
$26$		−	46.7642i		−	0.352739i
$27$	−27.0000			−0.192450
$28$		−	96.3915i		−	0.650582i
$29$			264.372i			1.69285i	0.532507	+	0.846426i	$0.321250\pi$
$29$			264.372i			1.69285i	−0.532507	+	0.846426i	$0.678750\pi$
$30$	−136.187			−0.828811
$31$	−67.0293			−0.388349			−0.194174	−	0.980967i	$-0.562203\pi$
$31$	−67.0293			−0.388349			−0.194174	+	0.980967i	$0.562203\pi$
$32$			126.125i			0.696750i
$33$	−53.4186			−0.281787
$34$			63.9800i			0.322720i
$35$	−453.873			−2.19196
$36$	−26.2601			−0.121574
$37$	−196.837			−0.874589			−0.437294	−	0.899318i	$-0.644063\pi$
$37$	−196.837			−0.874589			−0.437294	+	0.899318i	$0.644063\pi$
$38$			307.584i			1.31307i
$39$	−42.4588			−0.174329
$40$	230.711			0.911966
$41$			97.6935i			0.372126i	0.982538	+	0.186063i	$0.0595728\pi$
$41$			97.6935i			0.372126i	−0.982538	+	0.186063i	$0.940427\pi$
$42$	−327.472			−1.20309
$43$		−	36.0530i		−	0.127861i	−0.997954	−	0.0639306i	$-0.979636\pi$
$43$		−	36.0530i		−	0.127861i	0.997954	−	0.0639306i	$-0.0203636\pi$
$44$	−51.9547			−0.178011
$45$			123.649i			0.409612i
$46$	−304.306			−0.975381
$47$	−273.402			−0.848506			−0.424253	−	0.905544i	$-0.639463\pi$
$47$	−273.402			−0.848506			−0.424253	+	0.905544i	$0.639463\pi$
$48$	236.486			0.711122
$49$	−748.366			−2.18183
$50$			210.658i			0.595831i
$51$	58.0896			0.159493
$52$	−41.2952			−0.110127
$53$		−	260.964i		−	0.676342i	−0.941085	−	0.338171i	$-0.890192\pi$
$53$		−	260.964i		−	0.676342i	0.941085	−	0.338171i	$-0.109808\pi$
$54$			89.2136i			0.224823i
$55$			244.636i			0.599758i
$56$	554.760			1.32380
$57$	279.266			0.648942
$58$	873.541			1.97761
$59$		−	669.127i		−	1.47649i	−0.674533	−	0.738245i	$-0.735655\pi$
$59$		−	669.127i		−	1.47649i	0.674533	−	0.738245i	$-0.264345\pi$
$60$			120.261i			0.258760i
$61$		−	266.218i		−	0.558783i	−0.960177	−	0.279391i	$-0.909867\pi$
$61$		−	266.218i		−	0.558783i	0.960177	−	0.279391i	$-0.0901326\pi$
$62$			221.479i			0.453674i
$63$			297.323i			0.594589i
$64$	−213.886			−0.417747
$65$			194.444i			0.371044i
$66$			176.506i			0.329188i
$67$	−97.0936			−0.177043			−0.0885214	−	0.996074i	$-0.528214\pi$
$67$	−97.0936			−0.177043			−0.0885214	+	0.996074i	$0.528214\pi$
$68$	56.4976			0.100755
$69$			276.290i			0.482049i
$70$			1499.69i			2.56067i
$71$	338.801			0.566314			0.283157	−	0.959074i	$-0.408618\pi$
$71$	338.801			0.566314			0.283157	+	0.959074i	$0.408618\pi$
$72$		−	151.134i		−	0.247380i
$73$			518.608i			0.831486i	0.909482	+	0.415743i	$0.136478\pi$
$73$			518.608i			0.831486i	−0.909482	+	0.415743i	$0.863522\pi$
$74$			650.390i			1.02171i
$75$	191.263			0.294469
$76$	271.613			0.409949
$77$			588.243i			0.870603i
$78$			140.293i			0.203654i
$79$		−	32.8801i		−	0.0468267i	−0.999726	−	0.0234133i	$-0.992547\pi$
$79$		−	32.8801i		−	0.0468267i	0.999726	−	0.0234133i	$-0.00745337\pi$
$80$		−	1083.01i		−	1.51356i
$81$	81.0000			0.111111
$82$	322.800			0.434722
$83$			1326.67i			1.75447i	0.480057	+	0.877237i	$0.340616\pi$
$83$			1326.67i			1.75447i	−0.480057	+	0.877237i	$0.659384\pi$
$84$			289.175i			0.375613i
$85$		−	266.027i		−	0.339467i
$86$	−119.127			−0.149369
$87$		−	793.117i		−	0.977368i
$88$		−	299.014i		−	0.362216i
$89$	−66.9332			−0.0797180			−0.0398590	−	0.999205i	$-0.512691\pi$
$89$	−66.9332			−0.0797180			−0.0398590	+	0.999205i	$0.512691\pi$
$90$	408.562			0.478514
$91$			467.554i			0.538604i
$92$			268.718i			0.304520i
$93$	201.088			0.224213
$94$			903.377i			0.991236i
$95$		−	1278.93i		−	1.38121i
$96$		−	378.376i		−	0.402269i
$97$		−	605.071i		−	0.633357i	−0.948533	−	0.316679i	$-0.897432\pi$
$97$		−	605.071i		−	0.633357i	0.948533	−	0.316679i	$-0.102568\pi$
$98$			2472.76i			2.54884i
$99$	160.256			0.162690
$100$	186.022			0.186022
$101$	−1260.14			−1.24147			−0.620735	−	0.784020i	$-0.713166\pi$
$101$	−1260.14			−1.24147			−0.620735	+	0.784020i	$0.713166\pi$
$102$		−	191.940i		−	0.186322i
$103$			1068.83i			1.02248i	0.859439	+	0.511238i	$0.170813\pi$
$103$			1068.83i			1.02248i	−0.859439	+	0.511238i	$0.829187\pi$
$104$		−	237.666i		−	0.224087i
$105$	1361.62			1.26553
$106$	−862.279			−0.790112
$107$			537.956i			0.486039i	0.970021	+	0.243019i	$0.0781379\pi$
$107$			537.956i			0.486039i	−0.970021	+	0.243019i	$0.921862\pi$
$108$	78.7802			0.0701910
$109$	−499.540			−0.438965			−0.219483	−	0.975616i	$-0.570437\pi$
$109$	−499.540			−0.438965			−0.219483	+	0.975616i	$0.570437\pi$
$110$	808.327			0.700645
$111$	590.510			0.504944
$112$		−	2604.18i		−	2.19707i
$113$	748.230			0.622899			0.311449	−	0.950263i	$-0.399186\pi$
$113$	748.230			0.622899			0.311449	+	0.950263i	$0.399186\pi$
$114$		−	922.753i		−	0.758103i
$115$	1265.30			1.02600
$116$		−	771.382i		−	0.617423i
$117$	127.376			0.100649
$118$	−2210.93			−1.72485
$119$		−	639.679i		−	0.492767i
$120$	−692.134			−0.526524
$121$	−1013.94			−0.761788
$122$	−879.640			−0.652777
$123$		−	293.080i		−	0.214847i
$124$	195.577			0.141640
$125$			841.440i			0.602085i
$126$	982.415			0.694607
$127$	−1445.32			−1.00986			−0.504928	−	0.863161i	$-0.668481\pi$
$127$	−1445.32			−1.00986			−0.504928	+	0.863161i	$0.668481\pi$
$128$			1715.73i			1.18477i
$129$			108.159i			0.0738206i
$130$	642.484			0.433458
$131$			2713.54i			1.80979i	0.425632	+	0.904897i	$0.360052\pi$
$131$			2713.54i			1.80979i	−0.425632	+	0.904897i	$0.639948\pi$
$132$	155.864			0.102774
$133$		−	3075.26i		−	2.00496i
$134$			320.817i			0.206824i
$135$		−	370.947i		−	0.236489i
$136$			325.160i			0.205016i
$137$			2042.83i			1.27395i	0.770885	+	0.636974i	$0.219814\pi$
$137$			2042.83i			1.27395i	−0.770885	+	0.636974i	$0.780186\pi$
$138$	912.919			0.563137
$139$		−	1072.90i		−	0.654690i	−0.944905	−	0.327345i	$-0.893846\pi$
$139$		−	1072.90i		−	0.654690i	0.944905	−	0.327345i	$-0.106154\pi$
$140$	1324.30			0.799458
$141$	820.206			0.489885
$142$		−	1119.47i		−	0.661576i
$143$	252.010			0.147371
$144$	−709.459			−0.410567
$145$	−3632.16			−2.08024
$146$	1713.59			0.971353
$147$	2245.10			1.25968
$148$	574.328			0.318983
$149$		−	828.063i		−	0.455285i	−0.973745	−	0.227643i	$-0.926898\pi$
$149$		−	828.063i		−	0.455285i	0.973745	−	0.227643i	$-0.0731018\pi$
$150$		−	631.974i		−	0.344003i
$151$		−	2542.13i		−	1.37004i	−0.728526	−	0.685018i	$-0.759794\pi$
$151$		−	2542.13i		−	1.37004i	0.728526	−	0.685018i	$-0.240206\pi$
$152$			1563.21i			0.834165i
$153$	−174.269			−0.0920836
$154$	1943.68			1.01705
$155$		−	920.902i		−	0.477217i
$156$	123.886			0.0635820
$157$	−497.110	−	1903.36i	−0.252699	−	0.967545i
$157$	−497.110	−	1903.36i	−0.252699	−	0.967545i
$158$	−108.643			−0.0547035
$159$			782.891i			0.390486i
$160$	−1732.81			−0.856191
$161$	3042.49			1.48933
$162$		−	267.641i		−	0.129802i
$163$		−	868.318i		−	0.417251i	−0.977996	−	0.208626i	$-0.933101\pi$
$163$		−	868.318i		−	0.417251i	0.977996	−	0.208626i	$-0.0668990\pi$
$164$		−	285.049i		−	0.135723i
$165$		−	733.907i		−	0.346270i
$166$	4383.61			2.04960
$167$	1008.53			0.467322			0.233661	−	0.972318i	$-0.424929\pi$
$167$	1008.53			0.467322			0.233661	+	0.972318i	$0.424929\pi$
$168$	−1664.28			−0.764298
$169$	−1996.69			−0.908828
$170$	−879.008			−0.396570
$171$	−837.799			−0.374667
$172$			105.195i			0.0466339i
$173$	2570.81			1.12980			0.564899	−	0.825160i	$-0.308915\pi$
$173$	2570.81			1.12980			0.564899	+	0.825160i	$0.308915\pi$
$174$	−2620.62			−1.14178
$175$		−	2106.18i		−	0.909785i
$176$	−1403.64			−0.601156
$177$			2007.38i			0.852451i
$178$			221.161i			0.0931277i
$179$		−	498.993i		−	0.208360i	−0.994558	−	0.104180i	$-0.966778\pi$
$179$		−	498.993i		−	0.208360i	0.994558	−	0.104180i	$-0.0332218\pi$
$180$		−	360.782i		−	0.149395i
$181$			3136.96i			1.28822i	0.764931	+	0.644112i	$0.222773\pi$
$181$			3136.96i			1.28822i	−0.764931	+	0.644112i	$0.777227\pi$
$182$	1544.89			0.629205
$183$			798.654i			0.322613i
$184$	−1546.55			−0.619637
$185$		−	2704.30i		−	1.07473i
$186$		−	664.436i		−	0.261929i
$187$	−344.785			−0.134830
$188$	797.729			0.309470
$189$		−	891.968i		−	0.343286i
$190$	−4225.84			−1.61355
$191$			609.457i			0.230884i	0.993314	+	0.115442i	$0.0368284\pi$
$191$			609.457i			0.230884i	−0.993314	+	0.115442i	$0.963172\pi$
$192$	641.659			0.241186
$193$	534.931			0.199509			0.0997543	−	0.995012i	$-0.468194\pi$
$193$	534.931			0.199509			0.0997543	+	0.995012i	$0.468194\pi$
$194$	−1999.28			−0.739897
$195$		−	583.333i		−	0.214222i
$196$	2183.57			0.795763
$197$	−2341.23			−0.846731			−0.423366	−	0.905959i	$-0.639151\pi$
$197$	−2341.23			−0.846731			−0.423366	+	0.905959i	$0.639151\pi$
$198$		−	529.518i		−	0.190057i
$199$	1220.90			0.434910			0.217455	−	0.976070i	$-0.430224\pi$
$199$	1220.90			0.434910			0.217455	+	0.976070i	$0.430224\pi$
$200$			1070.61i			0.378517i
$201$	291.281			0.102216
$202$			4163.76i			1.45030i
$203$	−8733.76			−3.01965
$204$	−169.493			−0.0581710
$205$	−1342.19			−0.457281
$206$	3531.64			1.19447
$207$		−	828.870i		−	0.278311i
$208$	−1115.66			−0.371909
$209$	−1657.56			−0.548591
$210$		−	4499.07i		−	1.47841i
$211$		−	1070.22i		−	0.349179i	−0.984641	−	0.174590i	$-0.944140\pi$
$211$		−	1070.22i		−	0.349179i	0.984641	−	0.174590i	$-0.0558599\pi$
$212$			761.437i			0.246678i
$213$	−1016.40			−0.326962
$214$	1777.52			0.567797
$215$	495.324			0.157120
$216$			453.403i			0.142825i
$217$		−	2214.37i		−	0.692724i
$218$			1650.58i			0.512805i
$219$		−	1555.82i		−	0.480058i
$220$		−	713.795i		−	0.218746i
$221$	−274.046			−0.0834132
$222$		−	1951.17i		−	0.589882i
$223$		−	4607.65i		−	1.38364i	−0.722071	−	0.691819i	$-0.756810\pi$
$223$		−	4607.65i		−	1.38364i	0.722071	−	0.691819i	$-0.243190\pi$
$224$	−4166.65			−1.24284
$225$	−573.790			−0.170012
$226$		−	2472.31i		−	0.727679i
$227$			1387.43i			0.405671i	0.979213	+	0.202835i	$0.0650156\pi$
$227$			1387.43i			0.405671i	−0.979213	+	0.202835i	$0.934984\pi$
$228$	−814.839			−0.236684
$229$			5274.33i			1.52200i	0.648754	+	0.760998i	$0.275291\pi$
$229$			5274.33i			1.52200i	−0.648754	+	0.760998i	$0.724709\pi$
$230$		−	4180.80i		−	1.19858i
$231$		−	1764.73i		−	0.502643i
$232$	4439.52			1.25633
$233$	3914.93			1.10075			0.550377	−	0.834916i	$-0.314484\pi$
$233$	3914.93			1.10075			0.550377	+	0.834916i	$0.314484\pi$
$234$		−	420.878i		−	0.117580i
$235$		−	3756.21i		−	1.04267i
$236$			1952.37i			0.538510i
$237$			98.6404i			0.0270354i
$238$	−2113.63			−0.575657
$239$	2586.28			0.699970			0.349985	−	0.936755i	$-0.386187\pi$
$239$	2586.28			0.699970			0.349985	+	0.936755i	$0.386187\pi$
$240$			3249.04i			0.873852i
$241$			1667.63i			0.445732i	0.974849	+	0.222866i	$0.0715412\pi$
$241$			1667.63i			0.445732i	−0.974849	+	0.222866i	$0.928459\pi$
$242$			3350.27i			0.889931i
$243$	−243.000			−0.0641500
$244$			776.768i			0.203801i
$245$		−	10281.7i		−	2.68110i
$246$	−968.399			−0.250987
$247$	−1317.48			−0.339389
$248$			1125.60i			0.288209i
$249$		−	3980.02i		−	1.01295i
$250$	2780.29			0.703364
$251$		−	3437.81i		−	0.864514i	−0.901751	−	0.432257i	$-0.857717\pi$
$251$		−	3437.81i		−	0.864514i	0.901751	−	0.432257i	$-0.142283\pi$
$252$		−	867.524i		−	0.216861i
$253$		−	1639.89i		−	0.407506i
$254$			4775.65i			1.17973i
$255$			798.081i			0.195991i
$256$	3958.03			0.966315
$257$	−3635.81			−0.882472			−0.441236	−	0.897391i	$-0.645460\pi$
$257$	−3635.81			−0.882472			−0.441236	+	0.897391i	$0.645460\pi$
$258$	357.380			0.0862383
$259$		−	6502.67i		−	1.56006i
$260$		−	567.347i		−	0.135328i
$261$			2379.35i			0.564284i
$262$	8966.09			2.11423
$263$	−4986.17			−1.16905			−0.584526	−	0.811375i	$-0.698720\pi$
$263$	−4986.17			−1.16905			−0.584526	+	0.811375i	$0.698720\pi$
$264$			897.042i			0.209125i
$265$	3585.33			0.831113
$266$	−10161.3			−2.34222
$267$	200.800			0.0460252
$268$	283.298			0.0645717
$269$		−	737.473i		−	0.167154i	−0.996501	−	0.0835772i	$-0.973365\pi$
$269$		−	737.473i		−	0.167154i	0.996501	−	0.0835772i	$-0.0266345\pi$
$270$	−1225.69			−0.276270
$271$		−	2806.49i		−	0.629086i	−0.949243	−	0.314543i	$-0.898149\pi$
$271$		−	2806.49i		−	0.629086i	0.949243	−	0.314543i	$-0.101851\pi$
$272$	1526.38			0.340258
$273$		−	1402.66i		−	0.310963i
$274$	6749.94			1.48824
$275$	−1135.22			−0.248933
$276$		−	806.155i		−	0.175815i
$277$	−4219.86			−0.915331			−0.457665	−	0.889125i	$-0.651314\pi$
$277$	−4219.86			−0.915331			−0.457665	+	0.889125i	$0.651314\pi$
$278$	−3545.07			−0.764818
$279$	−603.264			−0.129450
$280$			7621.74i			1.62674i
$281$	3827.96			0.812658			0.406329	−	0.913727i	$-0.366809\pi$
$281$	3827.96			0.812658			0.406329	+	0.913727i	$0.366809\pi$
$282$		−	2710.13i		−	0.572291i
$283$	4553.62			0.956483			0.478242	−	0.878228i	$-0.341274\pi$
$283$	4553.62			0.956483			0.478242	+	0.878228i	$0.341274\pi$
$284$	−988.550			−0.206548
$285$			3836.78i			0.797443i
$286$		−	832.693i		−	0.172161i
$287$	−3227.39			−0.663786
$288$			1135.13i			0.232250i
$289$	−4538.07			−0.923686
$290$			12001.4i			2.43016i
$291$			1815.21i			0.365669i
$292$		−	1513.19i		−	0.303262i
$293$			7137.32i			1.42309i	0.702638	+	0.711547i	$0.252005\pi$
$293$			7137.32i			1.42309i	−0.702638	+	0.711547i	$0.747995\pi$
$294$		−	7418.27i		−	1.47157i
$295$	9192.99			1.81436
$296$			3305.42i			0.649066i
$297$	−480.767			−0.0939291
$298$	−2736.09			−0.531871
$299$		−	1303.44i		−	0.252106i
$300$	−558.066			−0.107400
$301$	1191.04			0.228074
$302$	−8399.72			−1.60049
$303$	3780.42			0.716763
$304$	7338.07			1.38443
$305$	3657.52			0.686652
$306$			575.820i			0.107573i
$307$			10053.2i			1.86895i	0.356025	+	0.934476i	$0.384132\pi$
$307$			10053.2i			1.86895i	−0.356025	+	0.934476i	$0.615868\pi$
$308$		−	1716.37i		−	0.317529i
$309$		−	3206.49i		−	0.590326i
$310$	−3042.85			−0.557491
$311$	6845.83			1.24820			0.624102	−	0.781343i	$-0.285465\pi$
$311$	6845.83			1.24820			0.624102	+	0.781343i	$0.285465\pi$
$312$			712.997i			0.129377i
$313$	9879.91			1.78417			0.892085	−	0.451867i	$-0.149242\pi$
$313$	9879.91			1.78417			0.892085	+	0.451867i	$0.149242\pi$
$314$	−6289.09	+	1642.55i	−1.13030	+	0.295206i
$315$	−4084.85			−0.730652
$316$			95.9373i			0.0170788i
$317$	6318.56			1.11951			0.559756	−	0.828657i	$-0.310895\pi$
$317$	6318.56			1.11951			0.559756	+	0.828657i	$0.310895\pi$
$318$	2586.84			0.456172
$319$			4707.46i			0.826230i
$320$		−	2938.54i		−	0.513342i
$321$		−	1613.87i		−	0.280615i
$322$		−	10053.0i		−	1.73985i
$323$	1802.49			0.310506
$324$	−236.341			−0.0405248
$325$	−902.313			−0.154004
$326$	−2869.10			−0.487439
$327$	1498.62			0.253437
$328$	1640.54			0.276169
$329$		−	9032.06i		−	1.51354i
$330$	−2424.98			−0.404518
$331$	−1358.48			−0.225585			−0.112792	−	0.993619i	$-0.535980\pi$
$331$	−1358.48			−0.225585			−0.112792	+	0.993619i	$0.535980\pi$
$332$		−	3870.95i		−	0.639898i
$333$	−1771.53			−0.291530
$334$		−	3332.41i		−	0.545932i
$335$		−	1333.95i		−	0.217556i
$336$			7812.53i			1.26848i
$337$			11235.2i			1.81608i	0.418883	+	0.908040i	$0.362422\pi$
$337$			11235.2i			1.81608i	−0.418883	+	0.908040i	$0.637578\pi$
$338$			6597.49i			1.06171i
$339$	−2244.69			−0.359631
$340$			776.210i			0.123811i
$341$	−1193.54			−0.189541
$342$			2768.26i			0.437691i
$343$		−	13391.6i		−	2.10810i
$344$	−605.426			−0.0948907
$345$	−3795.89			−0.592359
$346$		−	8494.49i		−	1.31985i
$347$	−61.1506			−0.00946034			−0.00473017	−	0.999989i	$-0.501506\pi$
$347$	−61.1506			−0.00946034			−0.00473017	+	0.999989i	$0.501506\pi$
$348$			2314.15i			0.356469i
$349$	−2336.50			−0.358367			−0.179184	−	0.983816i	$-0.557346\pi$
$349$	−2336.50			−0.358367			−0.179184	+	0.983816i	$0.557346\pi$
$350$	−6959.26			−1.06282
$351$	−382.129			−0.0581098
$352$			2245.81i			0.340063i
$353$	10172.1			1.53373			0.766866	−	0.641807i	$-0.221815\pi$
$353$	10172.1			1.53373			0.766866	+	0.641807i	$0.221815\pi$
$354$	6632.80			0.995845
$355$			4654.72i			0.695907i
$356$	195.297			0.0290750
$357$			1919.04i			0.284499i
$358$	−1648.78			−0.243409
$359$		−	11278.7i		−	1.65813i	−0.559154	−	0.829064i	$-0.688874\pi$
$359$		−	11278.7i		−	1.65813i	0.559154	−	0.829064i	$-0.311126\pi$
$360$	2076.40			0.303989
$361$	1806.51			0.263378
$362$	10365.2			1.50492
$363$	3041.82			0.439818
$364$		−	1364.22i		−	0.196441i
$365$	−7125.05			−1.02176
$366$	2638.92			0.376881
$367$			5876.98i			0.835901i	0.908470	+	0.417951i	$0.137251\pi$
$367$			5876.98i			0.835901i	−0.908470	+	0.417951i	$0.862749\pi$
$368$			7259.87i			1.02839i
$369$			879.241i			0.124042i
$370$	−8935.57			−1.25551
$371$	8621.16			1.20644
$372$	−586.731			−0.0817758
$373$			7494.98i			1.04042i	0.854039	+	0.520208i	$0.174146\pi$
$373$			7494.98i			1.04042i	−0.854039	+	0.520208i	$0.825854\pi$
$374$			1139.24i			0.157510i
$375$		−	2524.32i		−	0.347614i
$376$			4591.15i			0.629709i
$377$			3741.64i			0.511152i
$378$	−2947.25			−0.401032
$379$		−	9164.43i		−	1.24207i	−0.783782	−	0.621036i	$-0.786712\pi$
$379$		−	9164.43i		−	1.24207i	0.783782	−	0.621036i	$-0.213288\pi$
$380$			3731.64i			0.503760i
$381$	4335.97			0.583041
$382$	2013.77			0.269721
$383$		−	856.861i		−	0.114317i	−0.998365	−	0.0571587i	$-0.981796\pi$
$383$		−	856.861i		−	0.114317i	0.998365	−	0.0571587i	$-0.0182041\pi$
$384$		−	5147.18i		−	0.684026i
$385$	−8081.74			−1.06983
$386$		−	1767.52i		−	0.233069i
$387$		−	324.477i		−	0.0426204i
$388$			1765.47i			0.231000i
$389$	10947.2			1.42685			0.713425	−	0.700731i	$-0.247143\pi$
$389$	10947.2			1.42685			0.713425	+	0.700731i	$0.247143\pi$
$390$	−1927.45			−0.250257
$391$			1783.28i			0.230651i
$392$			12567.1i			1.61922i
$393$		−	8140.61i		−	1.04488i
$394$			7735.93i			0.989163i
$395$	451.734			0.0575422
$396$	−467.592			−0.0593368
$397$		−	12103.0i		−	1.53006i	−0.643997	−	0.765028i	$-0.722725\pi$
$397$		−	12103.0i		−	1.53006i	0.643997	−	0.765028i	$-0.277275\pi$
$398$		−	4034.10i		−	0.508068i
$399$			9225.79i			1.15756i
$400$	5025.69			0.628211
$401$			11574.6i			1.44142i	0.693236	+	0.720710i	$0.256184\pi$
$401$			11574.6i			1.44142i	−0.693236	+	0.720710i	$0.743816\pi$
$402$		−	962.452i		−	0.119410i
$403$	−948.661			−0.117261
$404$	3676.82			0.452793
$405$			1112.84i			0.136537i
$406$			28858.1i			3.52760i
$407$	−3504.92			−0.426860
$408$		−	975.480i		−	0.118366i
$409$			12077.2i			1.46010i	0.683395	+	0.730049i	$0.260503\pi$
$409$			12077.2i			1.46010i	−0.683395	+	0.730049i	$0.739497\pi$
$410$			4434.88i			0.534202i
$411$		−	6128.49i		−	0.735514i
$412$		−	3118.62i		−	0.372921i
$413$	22105.2			2.63371
$414$	−2738.76			−0.325127
$415$	−18226.9			−2.15596
$416$			1785.04i			0.210382i
$417$			3218.69i			0.377986i
$418$			5476.91i			0.640872i
$419$	1962.64			0.228833			0.114416	−	0.993433i	$-0.463500\pi$
$419$	1962.64			0.228833			0.114416	+	0.993433i	$0.463500\pi$
$420$	−3972.91			−0.461567
$421$			4070.67i			0.471241i	0.971845	+	0.235621i	$0.0757122\pi$
$421$			4070.67i			0.471241i	−0.971845	+	0.235621i	$0.924288\pi$
$422$	−3536.22			−0.407916
$423$	−2460.62			−0.282835
$424$	−4382.28			−0.501940
$425$	1234.49			0.140898
$426$			3358.41i			0.381961i
$427$	8794.74			0.996738
$428$		−	1569.64i		−	0.177270i
$429$	−756.029			−0.0850850
$430$		−	1636.65i		−	0.183550i
$431$	−17177.2			−1.91972			−0.959859	−	0.280484i	$-0.909505\pi$
$431$	−17177.2			−1.91972			−0.959859	+	0.280484i	$0.909505\pi$
$432$	2128.38			0.237041
$433$		−	327.521i		−	0.0363502i	−0.999835	−	0.0181751i	$-0.994214\pi$
$433$		−	327.521i		−	0.0363502i	0.999835	−	0.0181751i	$-0.00578563\pi$
$434$	−7316.73			−0.809250
$435$	10896.5			1.20102
$436$	1457.55			0.160101
$437$			8573.16i			0.938467i
$438$	−5140.76			−0.560811
$439$			10466.9i			1.13794i	0.822357	+	0.568972i	$0.192659\pi$
$439$			10466.9i			1.13794i	−0.822357	+	0.568972i	$0.807341\pi$
$440$	4108.09			0.445103
$441$	−6735.30			−0.727275
$442$			905.504i			0.0974444i
$443$		−	15677.8i		−	1.68143i	−0.541475	−	0.840717i	$-0.682134\pi$
$443$		−	15677.8i		−	1.68143i	0.541475	−	0.840717i	$-0.317866\pi$
$444$	−1722.98			−0.184165
$445$		−	919.581i		−	0.0979603i
$446$	−15224.6			−1.61638
$447$			2484.19i			0.262859i
$448$		−	7065.92i		−	0.745164i
$449$			7538.97i			0.792396i	0.918165	+	0.396198i	$0.129671\pi$
$449$			7538.97i			0.792396i	−0.918165	+	0.396198i	$0.870329\pi$
$450$			1895.92i			0.198610i
$451$			1739.55i			0.181623i
$452$	−2183.18			−0.227186
$453$			7626.38i			0.790991i
$454$	4584.37			0.473910
$455$	−6423.63			−0.661855
$456$		−	4689.63i		−	0.481605i
$457$	−7654.48			−0.783505			−0.391752	−	0.920071i	$-0.628131\pi$
$457$	−7654.48			−0.783505			−0.391752	+	0.920071i	$0.628131\pi$
$458$	17427.5			1.77802
$459$	522.806			0.0531645
$460$	−3691.87			−0.374205
$461$	−5656.38			−0.571461			−0.285731	−	0.958310i	$-0.592236\pi$
$461$	−5656.38			−0.571461			−0.285731	+	0.958310i	$0.592236\pi$
$462$	−5831.03			−0.587195
$463$		−	12715.4i		−	1.27632i	−0.769905	−	0.638158i	$-0.779697\pi$
$463$		−	12715.4i		−	1.27632i	0.769905	−	0.638158i	$-0.220303\pi$
$464$		−	20840.2i		−	2.08509i
$465$			2762.70i			0.275521i
$466$		−	12935.7i		−	1.28592i
$467$	16712.6			1.65603			0.828016	−	0.560704i	$-0.189470\pi$
$467$	16712.6			1.65603			0.828016	+	0.560704i	$0.189470\pi$
$468$	−371.657			−0.0367091
$469$		−	3207.57i		−	0.315803i
$470$	−12411.3			−1.21807
$471$	1491.33	+	5710.08i	0.145896	+	0.558612i
$472$	−11236.4			−1.09576
$473$		−	641.966i		−	0.0624052i
$474$	325.928			0.0315831
$475$	5934.82			0.573281
$476$			1866.45i			0.179724i
$477$		−	2348.67i		−	0.225447i
$478$		−	8545.62i		−	0.817715i
$479$		−	17363.8i		−	1.65631i	−0.560497	−	0.828157i	$-0.689390\pi$
$479$		−	17363.8i		−	1.65631i	0.560497	−	0.828157i	$-0.310610\pi$
$480$	5198.43			0.494322
$481$	−2785.82			−0.264080
$482$	5510.19			0.520710
$483$	−9127.47			−0.859864
$484$	2958.46			0.277842
$485$	8312.95			0.778292
$486$			802.922i			0.0749409i
$487$	−2294.53			−0.213501			−0.106751	−	0.994286i	$-0.534045\pi$
$487$	−2294.53			−0.213501			−0.106751	+	0.994286i	$0.534045\pi$
$488$	−4470.52			−0.414694
$489$			2604.96i			0.240900i
$490$	−33972.7			−3.13210
$491$		−	4212.56i		−	0.387190i	−0.981082	−	0.193595i	$-0.937985\pi$
$491$		−	4212.56i		−	0.387190i	0.981082	−	0.193595i	$-0.0620148\pi$
$492$			855.146i			0.0783597i
$493$		−	5119.09i		−	0.467652i
$494$			4353.22i			0.396479i
$495$			2201.72i			0.199919i
$496$	5283.84			0.478329
$497$			11192.6i			1.01017i
$498$	−13150.8			−1.18334
$499$		−	1905.47i		−	0.170943i	−0.996341	−	0.0854717i	$-0.972760\pi$
$499$		−	1905.47i		−	0.170943i	0.996341	−	0.0854717i	$-0.0272397\pi$
$500$		−	2455.14i		−	0.219595i
$501$	−3025.60			−0.269808
$502$	−11359.3			−1.00994
$503$			17213.3i			1.52585i	0.646485	+	0.762926i	$0.276238\pi$
$503$			17213.3i			1.52585i	−0.646485	+	0.762926i	$0.723762\pi$
$504$	4992.84			0.441268
$505$		−	17312.8i		−	1.52556i
$506$	−5418.54			−0.476054
$507$	5990.08			0.524712
$508$	4217.15			0.368318
$509$			11592.8i			1.00952i	0.863261	+	0.504758i	$0.168418\pi$
$509$			11592.8i			1.00952i	−0.863261	+	0.504758i	$0.831582\pi$
$510$	2637.02			0.228960
$511$	−17132.6			−1.48318
$512$			647.676i			0.0559053i
$513$	2513.40			0.216314
$514$			12013.5i			1.03092i
$515$	−14684.4			−1.25645
$516$		−	315.585i		−	0.0269241i
$517$	−4868.25			−0.414130
$518$	−21486.2			−1.82249
$519$	−7712.43			−0.652289
$520$	3265.24			0.275366
$521$			15586.2i			1.31064i	0.755351	+	0.655320i	$0.227466\pi$
$521$			15586.2i			1.31064i	−0.755351	+	0.655320i	$0.772534\pi$
$522$	7861.87			0.659204
$523$	13888.9			1.16122			0.580612	−	0.814181i	$-0.302813\pi$
$523$	13888.9			1.16122			0.580612	+	0.814181i	$0.302813\pi$
$524$		−	7917.52i		−	0.660074i
$525$			6318.55i			0.525265i
$526$			16475.4i			1.36570i
$527$	1297.90			0.107282
$528$	4210.92			0.347078
$529$	3685.21			0.302885
$530$		−	11846.7i		−	0.970918i
$531$		−	6022.14i		−	0.492163i
$532$			8972.96i			0.731254i
$533$			1382.65i			0.112362i
$534$		−	663.483i		−	0.0537673i
$535$	−7390.86			−0.597261
$536$			1630.46i			0.131390i
$537$			1496.98i			0.120297i
$538$	−2436.77			−0.195272
$539$	−13325.6			−1.06488
$540$			1082.35i			0.0862532i
$541$			8463.40i			0.672588i	0.941757	+	0.336294i	$0.109174\pi$
$541$			8463.40i			0.672588i	−0.941757	+	0.336294i	$0.890826\pi$
$542$	−9273.24			−0.734907
$543$		−	9410.89i		−	0.743757i
$544$		−	2442.19i		−	0.192478i
$545$		−	6863.07i		−	0.539416i
$546$	−4634.68			−0.363271
$547$	13779.8			1.07712			0.538559	−	0.842588i	$-0.318969\pi$
$547$	13779.8			1.07712			0.538559	+	0.842588i	$0.318969\pi$
$548$		−	5960.55i		−	0.464638i
$549$		−	2395.96i		−	0.186261i
$550$			3751.02i			0.290807i
$551$		−	24610.1i		−	1.90277i
$552$	4639.65			0.357748
$553$	1086.22			0.0835279
$554$			13943.3i			1.06930i
$555$			8112.90i			0.620493i
$556$			3130.48i			0.238781i
$557$	4693.69			0.357052			0.178526	−	0.983935i	$-0.442867\pi$
$557$	4693.69			0.357052			0.178526	+	0.983935i	$0.442867\pi$
$558$			1993.31i			0.151225i
$559$		−	510.255i		−	0.0386073i
$560$	35778.2			2.69983
$561$	1034.35			0.0778440
$562$		−	12648.4i		−	0.949358i
$563$			1434.49i			0.107383i	0.998558	+	0.0536913i	$0.0170987\pi$
$563$			1434.49i			0.107383i	−0.998558	+	0.0536913i	$0.982901\pi$
$564$	−2393.19			−0.178673
$565$			10279.8i			0.765440i
$566$		−	15046.1i		−	1.11738i
$567$			2675.90i			0.198196i
$568$		−	5689.38i		−	0.420284i
$569$		−	22341.6i		−	1.64606i	−0.568000	−	0.823029i	$-0.692283\pi$
$569$		−	22341.6i		−	1.64606i	0.568000	−	0.823029i	$-0.307717\pi$
$570$	12677.5			0.931584
$571$	3738.57			0.274000			0.137000	−	0.990571i	$-0.456254\pi$
$571$	3738.57			0.274000			0.137000	+	0.990571i	$0.456254\pi$
$572$	−735.311			−0.0537498
$573$		−	1828.37i		−	0.133301i
$574$			10664.0i			0.775444i
$575$			5871.57i			0.425846i
$576$	−1924.98			−0.139249
$577$	24776.6			1.78763			0.893817	−	0.448431i	$-0.148017\pi$
$577$	24776.6			1.78763			0.893817	+	0.448431i	$0.148017\pi$
$578$			14994.7i			1.07906i
$579$	−1604.79			−0.115186
$580$	10597.9			0.758711
$581$	−43827.8			−3.12957
$582$	5997.84			0.427180
$583$		−	4646.77i		−	0.330102i
$584$	8708.82			0.617078
$585$			1750.00i			0.123681i
$586$	23583.2			1.66248
$587$			19619.3i			1.37951i	0.724042	+	0.689756i	$0.242282\pi$
$587$			19619.3i			1.37951i	−0.724042	+	0.689756i	$0.757718\pi$
$588$	−6550.72			−0.459434
$589$	6239.67			0.436505
$590$		−	30375.6i		−	2.11956i
$591$	7023.70			0.488861
$592$	15516.4			1.07723
$593$	16875.1			1.16860			0.584298	−	0.811539i	$-0.301370\pi$
$593$	16875.1			1.16860			0.584298	+	0.811539i	$0.301370\pi$
$594$			1588.55i			0.109729i
$595$	8788.42			0.605529
$596$			2416.11i			0.166053i
$597$	−3662.69			−0.251095
$598$	−4306.83			−0.294514
$599$		−	1491.14i		−	0.101714i	−0.998706	−	0.0508569i	$-0.983805\pi$
$599$		−	1491.14i		−	0.101714i	0.998706	−	0.0508569i	$-0.0161952\pi$
$600$		−	3211.83i		−	0.218537i
$601$	−18501.4			−1.25572			−0.627859	−	0.778327i	$-0.716069\pi$
$601$	−18501.4			−1.25572			−0.627859	+	0.778327i	$0.716069\pi$
$602$		−	3935.44i		−	0.266440i
$603$	−873.842			−0.0590143
$604$			7417.39i			0.499684i
$605$		−	13930.3i		−	0.936111i
$606$		−	12491.3i		−	0.837333i
$607$			21601.7i			1.44446i	0.691655	+	0.722228i	$0.256882\pi$
$607$			21601.7i			1.44446i	−0.691655	+	0.722228i	$0.743118\pi$
$608$		−	11740.8i		−	0.783148i
$609$	26201.3			1.74340
$610$		−	12085.2i		−	0.802156i
$611$	−3869.44			−0.256204
$612$	508.479			0.0335850
$613$		−	4793.51i		−	0.315837i	−0.987452	−	0.157919i	$-0.949522\pi$
$613$		−	4793.51i		−	0.315837i	0.987452	−	0.157919i	$-0.0504784\pi$
$614$	33218.0			2.18334
$615$	4026.57			0.264011
$616$	9878.17			0.646109
$617$	26411.2			1.72330			0.861650	−	0.507503i	$-0.169431\pi$
$617$	26411.2			1.72330			0.861650	+	0.507503i	$0.169431\pi$
$618$	−10594.9			−0.689627
$619$	4013.08			0.260580			0.130290	−	0.991476i	$-0.458409\pi$
$619$	4013.08			0.260580			0.130290	+	0.991476i	$0.458409\pi$
$620$			2686.99i			0.174052i
$621$			2486.61i			0.160683i
$622$		−	22620.0i		−	1.45817i
$623$		−	2211.19i		−	0.142198i
$624$	3346.98			0.214722
$625$	−19529.7			−1.24990
$626$		−	32645.3i		−	2.08429i
$627$	4972.67			0.316729
$628$	1450.46	+	5553.60i	0.0921651	+	0.352886i
$629$	3811.39			0.241606
$630$			13497.2i			0.853558i
$631$	684.974			0.0432146			0.0216073	−	0.999767i	$-0.493122\pi$
$631$	684.974			0.0432146			0.0216073	+	0.999767i	$0.493122\pi$
$632$	−552.146			−0.0347519
$633$			3210.65i			0.201599i
$634$		−	20877.8i		−	1.30783i
$635$		−	19857.0i		−	1.24095i
$636$		−	2284.31i		−	0.142419i
$637$	−10591.6			−0.658797
$638$	15554.4			0.965213
$639$	3049.21			0.188771
$640$	−23572.0			−1.45588
$641$	−13797.7			−0.850194			−0.425097	−	0.905148i	$-0.639760\pi$
$641$	−13797.7			−0.850194			−0.425097	+	0.905148i	$0.639760\pi$
$642$	−5332.55			−0.327818
$643$			15594.0i			0.956403i	0.878250	+	0.478202i	$0.158711\pi$
$643$			15594.0i			0.956403i	−0.878250	+	0.478202i	$0.841289\pi$
$644$	−8877.34			−0.543193
$645$	−1485.97			−0.0907134
$646$		−	5955.82i		−	0.362738i
$647$	−27092.9			−1.64626			−0.823130	−	0.567853i	$-0.807774\pi$
$647$	−27092.9			−1.64626			−0.823130	+	0.567853i	$0.807774\pi$
$648$		−	1360.21i		−	0.0824599i
$649$		−	11914.6i		−	0.720630i
$650$			2981.43i			0.179910i
$651$			6643.10i			0.399944i
$652$			2533.57i			0.152181i
$653$	−16553.7			−0.992028			−0.496014	−	0.868314i	$-0.665204\pi$
$653$	−16553.7			−0.992028			−0.496014	+	0.868314i	$0.665204\pi$
$654$		−	4951.75i		−	0.296068i
$655$	−37280.7			−2.22394
$656$		−	7701.06i		−	0.458347i
$657$			4667.47i			0.277162i
$658$	−29843.8			−1.76813
$659$	13852.0			0.818813			0.409406	−	0.912352i	$-0.365736\pi$
$659$	13852.0			0.818813			0.409406	+	0.912352i	$0.365736\pi$
$660$			2141.38i			0.126293i
$661$	8917.41			0.524731			0.262365	−	0.964969i	$-0.415497\pi$
$661$	8917.41			0.524731			0.262365	+	0.964969i	$0.415497\pi$
$662$			4488.69i			0.263531i
$663$	822.137			0.0481586
$664$	22278.4			1.30206
$665$	42250.4			2.46376
$666$			5853.51i			0.340569i
$667$	24347.8			1.41342
$668$	−2942.69			−0.170443
$669$			13823.0i			0.798844i
$670$	−4407.64			−0.254152
$671$		−	4740.33i		−	0.272725i
$672$	12500.0			0.717554
$673$			9288.34i			0.532005i	0.963972	+	0.266003i	$0.0857029\pi$
$673$			9288.34i			0.532005i	−0.963972	+	0.266003i	$0.914297\pi$
$674$	37123.4			2.12157
$675$	1721.37			0.0981564
$676$	5825.93			0.331471
$677$	20320.4			1.15359			0.576793	−	0.816890i	$-0.304304\pi$
$677$	20320.4			1.15359			0.576793	+	0.816890i	$0.304304\pi$
$678$			7416.92i			0.420126i
$679$	19989.0			1.12976
$680$	−4467.31			−0.251931
$681$		−	4162.30i		−	0.234214i
$682$			3943.69i			0.221425i
$683$		−	1168.11i		−	0.0654415i	−0.999465	−	0.0327208i	$-0.989583\pi$
$683$		−	1168.11i		−	0.0654415i	0.999465	−	0.0327208i	$-0.0104172\pi$
$684$	2444.52			0.136650
$685$	−28066.0			−1.56547
$686$	−44248.7			−2.46271
$687$		−	15823.0i		−	0.878725i
$688$			2842.01i			0.157487i
$689$		−	3693.40i		−	0.204220i
$690$			12542.4i			0.692002i
$691$			3559.77i			0.195977i	0.995188	+	0.0979885i	$0.0312408\pi$
$691$			3559.77i			0.195977i	−0.995188	+	0.0979885i	$0.968759\pi$
$692$	−7501.07			−0.412064
$693$			5294.18i			0.290201i
$694$			202.054i			0.0110517i
$695$	14740.3			0.804506
$696$	−13318.6			−0.725343
$697$		−	1891.66i		−	0.102800i
$698$			7720.29i			0.418650i
$699$	−11744.8			−0.635520
$700$			6145.39i			0.331820i
$701$		−	3769.81i		−	0.203115i	−0.994830	−	0.101558i	$-0.967617\pi$
$701$		−	3769.81i		−	0.203115i	0.994830	−	0.101558i	$-0.0323826\pi$
$702$			1262.63i			0.0678847i
$703$	18323.3			0.983038
$704$	−3808.50			−0.203890
$705$			11268.6i			0.601988i
$706$		−	33610.8i		−	1.79173i
$707$		−	41629.7i		−	2.21449i
$708$		−	5857.11i		−	0.310909i
$709$	−21123.1			−1.11889			−0.559445	−	0.828867i	$-0.688986\pi$
$709$	−21123.1			−1.11889			−0.559445	+	0.828867i	$0.688986\pi$
$710$	15380.2			0.812968
$711$		−	295.921i		−	0.0156089i
$712$			1123.99i			0.0591618i
$713$			6173.17i			0.324246i
$714$	6340.90			0.332356
$715$			3462.31i			0.181095i
$716$			1455.96i			0.0759938i
$717$	−7758.85			−0.404128
$718$	−37267.2			−1.93705
$719$		−	21879.3i		−	1.13486i	−0.823423	−	0.567428i	$-0.807939\pi$
$719$		−	21879.3i		−	1.13486i	0.823423	−	0.567428i	$-0.192061\pi$
$720$		−	9747.12i		−	0.504519i
$721$	−35309.7			−1.82386
$722$		−	5969.09i		−	0.307682i
$723$		−	5002.89i		−	0.257343i
$724$		−	9152.99i		−	0.469845i
$725$		−	16854.9i		−	0.863415i
$726$		−	10050.8i		−	0.513802i
$727$	21602.4			1.10205			0.551023	−	0.834490i	$-0.314238\pi$
$727$	21602.4			1.10205			0.551023	+	0.834490i	$0.314238\pi$
$728$	7851.48			0.399719
$729$	729.000			0.0370370
$730$			23542.6i			1.19363i
$731$			698.100i			0.0353217i
$732$		−	2330.30i		−	0.117665i
$733$	−18680.3			−0.941298			−0.470649	−	0.882321i	$-0.655980\pi$
$733$	−18680.3			−0.941298			−0.470649	+	0.882321i	$0.655980\pi$
$734$	19418.8			0.976511
$735$			30845.0i			1.54794i
$736$	11615.7			0.581740
$737$	−1728.87			−0.0864093
$738$	2905.20			0.144907
$739$	3522.52			0.175342			0.0876711	−	0.996149i	$-0.472058\pi$
$739$	3522.52			0.175342			0.0876711	+	0.996149i	$0.472058\pi$
$740$			7890.57i			0.391977i
$741$	3952.43			0.195946
$742$		−	28486.1i		−	1.40938i
$743$	12794.8			0.631756			0.315878	−	0.948800i	$-0.397701\pi$
$743$	12794.8			0.631756			0.315878	+	0.948800i	$0.397701\pi$
$744$		−	3376.81i		−	0.166397i
$745$	11376.6			0.559471
$746$	24765.0			1.21543
$747$			11940.1i			0.584825i
$748$	1006.01			0.0491755
$749$	−17771.8			−0.866980
$750$	−8340.87			−0.406087
$751$		−	26999.6i		−	1.31189i	−0.754809	−	0.655944i	$-0.772271\pi$
$751$		−	26999.6i		−	1.31189i	0.754809	−	0.655944i	$-0.227729\pi$
$752$	21552.0			1.04510
$753$			10313.4i			0.499127i
$754$	12363.2			0.597135
$755$	34925.8			1.68355
$756$			2602.57i			0.125204i
$757$			38470.2i			1.84706i	0.383525	+	0.923530i	$0.374710\pi$
$757$			38470.2i			1.84706i	−0.383525	+	0.923530i	$0.625290\pi$
$758$	−30281.2			−1.45101
$759$			4919.67i			0.235274i
$760$	−21476.6			−1.02505
$761$		−	20853.0i		−	0.993327i	−0.867943	−	0.496663i	$-0.834558\pi$
$761$		−	20853.0i		−	0.993327i	0.867943	−	0.496663i	$-0.165442\pi$
$762$		−	14326.9i		−	0.681116i
$763$		−	16502.7i		−	0.783012i
$764$		−	1778.27i		−	0.0842087i
$765$		−	2394.24i		−	0.113156i
$766$	−2831.25			−0.133547
$767$		−	9470.10i		−	0.445822i
$768$	−11874.1			−0.557902
$769$	−38632.7			−1.81161			−0.905807	−	0.423691i	$-0.860734\pi$
$769$	−38632.7			−1.81161			−0.905807	+	0.423691i	$0.860734\pi$
$770$			26703.8i			1.24979i
$771$	10907.4			0.509496
$772$	−1560.81			−0.0727655
$773$	3943.97			0.183512			0.0917559	−	0.995782i	$-0.470752\pi$
$773$	3943.97			0.183512			0.0917559	+	0.995782i	$0.470752\pi$
$774$	−1072.14			−0.0497897
$775$	4273.42			0.198072
$776$	−10160.8			−0.470039
$777$			19508.0i			0.900702i
$778$		−	36171.8i		−	1.66687i
$779$		−	9094.16i		−	0.418270i
$780$			1702.04i			0.0781318i
$781$	6032.76			0.276401
$782$	5892.34			0.269450
$783$		−	7138.05i		−	0.325789i
$784$	58992.8			2.68736
$785$	26149.9	−	6829.69i	1.18895	−	0.310525i
$786$	−26898.3			−1.22065
$787$			15991.4i			0.724308i	0.932118	+	0.362154i	$0.117959\pi$
$787$			15991.4i			0.724308i	−0.932118	+	0.362154i	$0.882041\pi$
$788$	6831.22			0.308823
$789$	14958.5			0.674953
$790$		−	1492.62i		−	0.0672216i
$791$			24718.4i			1.11111i
$792$		−	2691.12i		−	0.120739i
$793$		−	3767.77i		−	0.168723i
$794$	−39990.8			−1.78743
$795$	−10756.0			−0.479843
$796$	−3562.32			−0.158622
$797$	−28245.5			−1.25534			−0.627670	−	0.778480i	$-0.715991\pi$
$797$	−28245.5			−1.25534			−0.627670	+	0.778480i	$0.715991\pi$
$798$	30483.9			1.35228
$799$	5293.93			0.234400
$800$		−	8041.05i		−	0.355368i
$801$	−602.399			−0.0265727
$802$	38245.0			1.68389
$803$			9234.43i			0.405823i
$804$	−849.895			−0.0372805
$805$			41800.1i			1.83014i
$806$			3134.57i			0.136986i
$807$			2212.42i			0.0965067i
$808$			21161.1i			0.921344i
$809$			28572.5i			1.24173i	0.783919	+	0.620863i	$0.213218\pi$
$809$			28572.5i			1.24173i	−0.783919	+	0.620863i	$0.786782\pi$
$810$	3677.06			0.159505
$811$			37204.6i			1.61089i	0.592673	+	0.805444i	$0.298073\pi$
$811$			37204.6i			1.61089i	−0.592673	+	0.805444i	$0.701927\pi$
$812$	25483.2			1.10134
$813$			8419.48i			0.363203i
$814$			11581.0i			0.498664i
$815$	11929.6			0.512733
$816$	−4579.13			−0.196448
$817$			3356.13i			0.143716i
$818$	39905.6			1.70571
$819$			4207.98i			0.179535i
$820$	3916.23			0.166781
$821$	−39480.8			−1.67831			−0.839154	−	0.543894i	$-0.816949\pi$
$821$	−39480.8			−1.67831			−0.839154	+	0.543894i	$0.816949\pi$
$822$	−20249.8			−0.859238
$823$			26657.1i			1.12905i	0.825416	+	0.564526i	$0.190941\pi$
$823$			26657.1i			1.12905i	−0.825416	+	0.564526i	$0.809059\pi$
$824$	17948.5			0.758819
$825$	3405.67			0.143722
$826$		−	73040.0i		−	3.07674i
$827$	−5186.18			−0.218067			−0.109033	−	0.994038i	$-0.534776\pi$
$827$	−5186.18			−0.218067			−0.109033	+	0.994038i	$0.534776\pi$
$828$			2418.47i			0.101507i
$829$	−35253.9			−1.47698			−0.738491	−	0.674263i	$-0.764461\pi$
$829$	−35253.9			−1.47698			−0.738491	+	0.674263i	$0.764461\pi$
$830$			60225.5i			2.51862i
$831$	12659.6			0.528466
$832$	−3027.12			−0.126138
$833$	14490.8			0.602731
$834$	10635.2			0.441568
$835$			13856.0i			0.574261i
$836$	4836.40			0.200084
$837$	1809.79			0.0747378
$838$		−	6484.95i		−	0.267326i
$839$		−	8705.89i		−	0.358237i	−0.983828	−	0.179118i	$-0.942675\pi$
$839$		−	8705.89i		−	0.358237i	0.983828	−	0.179118i	$-0.0573245\pi$
$840$		−	22865.2i		−	0.939196i
$841$	−45503.7			−1.86575
$842$	13450.4			0.550510
$843$	−11483.9			−0.469188
$844$			3122.67i			0.127354i
$845$		−	27432.2i		−	1.11680i
$846$			8130.39i			0.330412i
$847$		−	33496.3i		−	1.35885i
$848$			20571.5i			0.833051i
$849$	−13660.9			−0.552226
$850$		−	4079.01i		−	0.164599i
$851$			18128.0i			0.730224i
$852$	2965.65			0.119251
$853$	−32628.2			−1.30969			−0.654847	−	0.755761i	$-0.727267\pi$
$853$	−32628.2			−1.30969			−0.654847	+	0.755761i	$0.727267\pi$
$854$		−	29059.6i		−	1.16440i
$855$		−	11510.3i		−	0.460404i
$856$	9033.72			0.360708
$857$			20769.2i			0.827845i	0.910312	+	0.413922i	$0.135842\pi$
$857$			20769.2i			0.827845i	−0.910312	+	0.413922i	$0.864158\pi$
$858$			2498.08i			0.0993974i
$859$		−	43032.5i		−	1.70925i	−0.519242	−	0.854627i	$-0.673786\pi$
$859$		−	43032.5i		−	1.70925i	0.519242	−	0.854627i	$-0.326214\pi$
$860$	−1445.25			−0.0573054
$861$	9682.16			0.383237
$862$			56757.1i			2.24264i
$863$		−	20371.8i		−	0.803549i	−0.915739	−	0.401775i	$-0.868393\pi$
$863$		−	20371.8i		−	0.803549i	0.915739	−	0.401775i	$-0.131607\pi$
$864$		−	3405.38i		−	0.134090i
$865$			35319.8i			1.38834i
$866$	−1082.20			−0.0424648
$867$	13614.2			0.533290
$868$			6461.05i			0.252653i
$869$		−	585.470i		−	0.0228547i
$870$		−	36004.2i		−	1.40305i
$871$	−1374.16			−0.0534576
$872$			8388.61i			0.325773i
$873$		−	5445.64i		−	0.211119i
$874$	28327.5			1.09633
$875$	−27797.7			−1.07398
$876$			4539.56i			0.175089i
$877$			924.471i			0.0355954i	0.999842	+	0.0177977i	$0.00566548\pi$
$877$			924.471i			0.0355954i	−0.999842	+	0.0177977i	$0.994335\pi$
$878$	34584.8			1.32936
$879$		−	21412.0i		−	0.821624i
$880$		−	19284.3i		−	0.738722i
$881$		−	2139.40i		−	0.0818141i	−0.999163	−	0.0409071i	$-0.986975\pi$
$881$		−	2139.40i		−	0.0818141i	0.999163	−	0.0409071i	$-0.0130248\pi$
$882$			22254.8i			0.849613i
$883$			40787.5i			1.55448i	0.629201	+	0.777242i	$0.283382\pi$
$883$			40787.5i			1.55448i	−0.629201	+	0.777242i	$0.716618\pi$
$884$	799.607			0.0304227
$885$	−27579.0			−1.04752
$886$	−51802.7			−1.96427
$887$		−	29135.8i		−	1.10291i	−0.834203	−	0.551457i	$-0.814072\pi$
$887$		−	29135.8i		−	1.10291i	0.834203	−	0.551457i	$-0.185928\pi$
$888$		−	9916.26i		−	0.374739i
$889$		−	47747.5i		−	1.80135i
$890$	−3038.49			−0.114439
$891$	1442.30			0.0542300
$892$			13444.1i			0.504645i
$893$	25450.6			0.953722
$894$	8208.27			0.307076
$895$	6855.56			0.256040
$896$	−56680.5			−2.11335
$897$			3910.31i			0.145554i
$898$	24910.3			0.925688
$899$		−	17720.7i		−	0.657417i
$900$	1674.20			0.0620073
$901$			5053.09i			0.186840i
$902$	5747.83			0.212175
$903$	−3573.12			−0.131679
$904$		−	12564.8i		−	0.462278i
$905$	−43098.1			−1.58302
$906$	25199.2			0.924046
$907$	19937.7			0.729902			0.364951	−	0.931027i	$-0.381086\pi$
$907$	19937.7			0.729902			0.364951	+	0.931027i	$0.381086\pi$
$908$		−	4048.24i		−	0.147958i
$909$	−11341.3			−0.413824
$910$			21225.0i			0.773189i
$911$	−44226.6			−1.60845			−0.804223	−	0.594328i	$-0.797418\pi$
$911$	−44226.6			−1.60845			−0.804223	+	0.594328i	$0.797418\pi$
$912$	−22014.2			−0.799302
$913$			23623.0i			0.856307i
$914$			25292.0i			0.915301i
$915$	−10972.5			−0.396438
$916$		−	15389.4i		−	0.555108i
$917$	−89644.0			−3.22825
$918$		−	1727.46i		−	0.0621075i
$919$			19415.6i			0.696911i	0.937325	+	0.348456i	$0.113294\pi$
$919$			19415.6i			0.696911i	−0.937325	+	0.348456i	$0.886706\pi$
$920$		−	21247.7i		−	0.761432i
$921$		−	30159.7i		−	1.07904i
$922$			18689.8i			0.667589i
$923$	4795.03			0.170997
$924$			5149.10i			0.183326i
$925$	12549.2			0.446072
$926$	−42014.3			−1.49101
$927$			9619.47i			0.340825i
$928$	−33344.0			−1.17949
$929$	53659.5			1.89506			0.947531	−	0.319664i	$-0.103570\pi$
$929$	53659.5			1.89506			0.947531	+	0.319664i	$0.103570\pi$
$930$	9128.55			0.321868
$931$	69664.5			2.45237
$932$	−11422.9			−0.401470
$933$	−20537.5			−0.720650
$934$		−	55221.9i		−	1.93460i
$935$		−	4736.93i		−	0.165683i
$936$		−	2138.99i		−	0.0746956i
$937$			39683.0i			1.38355i	0.722112	+	0.691776i	$0.243171\pi$
$937$			39683.0i			1.38355i	−0.722112	+	0.691776i	$0.756829\pi$
$938$	−10598.5			−0.368926
$939$	−29639.7			−1.03009
$940$			10959.8i			0.380288i
$941$	−15317.7			−0.530653			−0.265326	−	0.964159i	$-0.585480\pi$
$941$	−15317.7			−0.530653			−0.265326	+	0.964159i	$0.585480\pi$
$942$	18867.3	−	4927.66i	0.652579	−	0.170437i
$943$	8997.24			0.310700
$944$			52746.5i			1.81859i
$945$	12254.6			0.421842
$946$	−2121.19			−0.0729026
$947$		−	17243.8i		−	0.591710i	−0.955233	−	0.295855i	$-0.904395\pi$
$947$		−	17243.8i		−	0.591710i	0.955233	−	0.295855i	$-0.0956045\pi$
$948$		−	287.812i		−	0.00986043i
$949$			7339.82i			0.251065i
$950$		−	19609.9i		−	0.669715i
$951$	−18955.7			−0.646351
$952$	−10741.9			−0.365702
$953$	−286.183			−0.00972757			−0.00486379	−	0.999988i	$-0.501548\pi$
$953$	−286.183			−0.00972757			−0.00486379	+	0.999988i	$0.501548\pi$
$954$	−7760.51			−0.263371
$955$	−8373.21			−0.283718
$956$	−7546.23			−0.255295
$957$		−	14122.4i		−	0.477024i
$958$	−57373.7			−1.93493
$959$	−67486.6			−2.27243
$960$			8815.63i			0.296378i
$961$	−25298.1			−0.849185
$962$			9204.92i			0.308502i
$963$			4841.60i			0.162013i
$964$		−	4865.78i		−	0.162569i
$965$			7349.31i			0.245163i
$966$			30159.1i			1.00450i
$967$	−10160.0			−0.337872			−0.168936	−	0.985627i	$-0.554033\pi$
$967$	−10160.0			−0.337872			−0.168936	+	0.985627i	$0.554033\pi$
$968$			17026.8i			0.565352i
$969$	−5407.48			−0.179271
$970$		−	27467.7i		−	0.909211i
$971$			1887.47i			0.0623808i	0.999513	+	0.0311904i	$0.00992983\pi$
$971$			1887.47i			0.0623808i	−0.999513	+	0.0311904i	$0.990070\pi$
$972$	709.022			0.0233970
$973$	35444.0			1.16781
$974$			7581.61i			0.249415i
$975$	2706.94			0.0889143
$976$			20985.7i			0.688253i
$977$	−7438.78			−0.243590			−0.121795	−	0.992555i	$-0.538865\pi$
$977$	−7438.78			−0.243590			−0.121795	+	0.992555i	$0.538865\pi$
$978$	8607.31			0.281423
$979$	−1191.83			−0.0389080
$980$			29999.7i			0.977861i
$981$	−4495.86			−0.146322
$982$	−13919.2			−0.452321
$983$		−	15234.1i		−	0.494294i	−0.968978	−	0.247147i	$-0.920507\pi$
$983$		−	15234.1i		−	0.494294i	0.968978	−	0.247147i	$-0.0794931\pi$
$984$	−4921.61			−0.159446
$985$		−	32165.7i		−	1.04049i
$986$	−16914.5			−0.546317
$987$			27096.2i			0.873841i
$988$	3844.12			0.123783
$989$	−3320.36			−0.106756
$990$	7274.94			0.233548
$991$	−49576.2			−1.58914			−0.794572	−	0.607170i	$-0.792305\pi$
$991$	−49576.2			−1.58914			−0.794572	+	0.607170i	$0.792305\pi$
$992$		−	8454.09i		−	0.270582i
$993$	4075.43			0.130242
$994$	36982.6			1.18010
$995$			16773.6i			0.534433i
$996$			11612.9i			0.369445i
$997$			10207.7i			0.324253i	0.986770	+	0.162126i	$0.0518353\pi$
$997$			10207.7i			0.324253i	−0.986770	+	0.162126i	$0.948165\pi$
$998$	−6296.08			−0.199698
$999$	5314.59			0.168315

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.4.b.a.313.10	✓	40
157.156	even	2	inner	471.4.b.a.313.31	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.4.b.a.313.10	✓	40	1.1	even	1	trivial
471.4.b.a.313.31	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)

\(f(q)\)	\(=\)	\(q-3.30421i q^{2} -3.00000 q^{3} -2.91779 q^{4} +13.7388i q^{5} +9.91262i q^{6} +33.0358i q^{7} -16.7927i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-3.30421i q^{2} -3.00000 q^{3} -2.91779 q^{4} +13.7388i q^{5} +9.91262i q^{6} +33.0358i q^{7} -16.7927i q^{8} +9.00000 q^{9} +45.3958 q^{10} +17.8062 q^{11} +8.75336 q^{12} +14.1529 q^{13} +109.157 q^{14} -41.2164i q^{15} -78.8288 q^{16} -19.3632 q^{17} -29.7379i q^{18} -93.0887 q^{19} -40.0869i q^{20} -99.1075i q^{21} -58.8354i q^{22} -92.0967i q^{23} +50.3781i q^{24} -63.7545 q^{25} -46.7642i q^{26} -27.0000 q^{27} -96.3915i q^{28} +264.372i q^{29} -136.187 q^{30} -67.0293 q^{31} +126.125i q^{32} -53.4186 q^{33} +63.9800i q^{34} -453.873 q^{35} -26.2601 q^{36} -196.837 q^{37} +307.584i q^{38} -42.4588 q^{39} +230.711 q^{40} +97.6935i q^{41} -327.472 q^{42} -36.0530i q^{43} -51.9547 q^{44} +123.649i q^{45} -304.306 q^{46} -273.402 q^{47} +236.486 q^{48} -748.366 q^{49} +210.658i q^{50} +58.0896 q^{51} -41.2952 q^{52} -260.964i q^{53} +89.2136i q^{54} +244.636i q^{55} +554.760 q^{56} +279.266 q^{57} +873.541 q^{58} -669.127i q^{59} +120.261i q^{60} -266.218i q^{61} +221.479i q^{62} +297.323i q^{63} -213.886 q^{64} +194.444i q^{65} +176.506i q^{66} -97.0936 q^{67} +56.4976 q^{68} +276.290i q^{69} +1499.69i q^{70} +338.801 q^{71} -151.134i q^{72} +518.608i q^{73} +650.390i q^{74} +191.263 q^{75} +271.613 q^{76} +588.243i q^{77} +140.293i q^{78} -32.8801i q^{79} -1083.01i q^{80} +81.0000 q^{81} +322.800 q^{82} +1326.67i q^{83} +289.175i q^{84} -266.027i q^{85} -119.127 q^{86} -793.117i q^{87} -299.014i q^{88} -66.9332 q^{89} +408.562 q^{90} +467.554i q^{91} +268.718i q^{92} +201.088 q^{93} +903.377i q^{94} -1278.93i q^{95} -378.376i q^{96} -605.071i q^{97} +2472.76i q^{98} +160.256 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 120 q^{3} - 164 q^{4} + 360 q^{9}+O(q^{10}) \) 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

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