LMFDB - Embedded newform 471.4.b.a.313.20

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			313.20
Character	$\chi$	$=$	471.313
Dual form			471.4.b.a.313.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.275605i q^{2} -3.00000 q^{3} +7.92404 q^{4} -19.4532i q^{5} +0.826816i q^{6} -15.2471i q^{7} -4.38875i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-0.275605i q^{2} -3.00000 q^{3} +7.92404 q^{4} -19.4532i q^{5} +0.826816i q^{6} -15.2471i q^{7} -4.38875i q^{8} +9.00000 q^{9} -5.36141 q^{10} +43.0465 q^{11} -23.7721 q^{12} +61.4201 q^{13} -4.20219 q^{14} +58.3597i q^{15} +62.1828 q^{16} -27.8030 q^{17} -2.48045i q^{18} +113.323 q^{19} -154.148i q^{20} +45.7414i q^{21} -11.8638i q^{22} -42.1549i q^{23} +13.1663i q^{24} -253.428 q^{25} -16.9277i q^{26} -27.0000 q^{27} -120.819i q^{28} +88.4551i q^{29} +16.0842 q^{30} -16.5221 q^{31} -52.2479i q^{32} -129.140 q^{33} +7.66265i q^{34} -296.606 q^{35} +71.3164 q^{36} +146.063 q^{37} -31.2324i q^{38} -184.260 q^{39} -85.3753 q^{40} +86.5465i q^{41} +12.6066 q^{42} +67.8646i q^{43} +341.102 q^{44} -175.079i q^{45} -11.6181 q^{46} -366.968 q^{47} -186.548 q^{48} +110.525 q^{49} +69.8460i q^{50} +83.4090 q^{51} +486.696 q^{52} -192.075i q^{53} +7.44134i q^{54} -837.393i q^{55} -66.9159 q^{56} -339.969 q^{57} +24.3787 q^{58} -297.310i q^{59} +462.444i q^{60} +761.717i q^{61} +4.55359i q^{62} -137.224i q^{63} +483.062 q^{64} -1194.82i q^{65} +35.5915i q^{66} -343.054 q^{67} -220.312 q^{68} +126.465i q^{69} +81.7461i q^{70} -781.308 q^{71} -39.4988i q^{72} +274.214i q^{73} -40.2557i q^{74} +760.283 q^{75} +897.975 q^{76} -656.336i q^{77} +50.7831i q^{78} -331.333i q^{79} -1209.66i q^{80} +81.0000 q^{81} +23.8527 q^{82} -145.699i q^{83} +362.457i q^{84} +540.858i q^{85} +18.7038 q^{86} -265.365i q^{87} -188.920i q^{88} -208.027 q^{89} -48.2527 q^{90} -936.481i q^{91} -334.037i q^{92} +49.5664 q^{93} +101.138i q^{94} -2204.49i q^{95} +156.744i q^{96} -1512.84i q^{97} -30.4613i q^{98} +387.419 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	0.275605i		−	0.0974412i	−0.998812	−	0.0487206i	$-0.984486\pi$
$2$		−	0.275605i		−	0.0974412i	0.998812	−	0.0487206i	$-0.0155144\pi$
$3$	−3.00000			−0.577350
$3$	−3.00000			−0.577350
$4$	7.92404			0.990505
$5$		−	19.4532i		−	1.73995i	−0.493097	−	0.869974i	$-0.664135\pi$
$5$		−	19.4532i		−	1.73995i	0.493097	−	0.869974i	$-0.335865\pi$
$6$			0.826816i			0.0562577i
$7$		−	15.2471i		−	0.823268i	−0.911349	−	0.411634i	$-0.864958\pi$
$7$		−	15.2471i		−	0.823268i	0.911349	−	0.411634i	$-0.135042\pi$
$8$		−	4.38875i		−	0.193957i
$9$	9.00000			0.333333
$10$	−5.36141			−0.169543
$11$	43.0465			1.17991			0.589955	−	0.807436i	$-0.299145\pi$
$11$	43.0465			1.17991			0.589955	+	0.807436i	$0.299145\pi$
$12$	−23.7721			−0.571868
$13$	61.4201			1.31038			0.655188	−	0.755466i	$-0.272590\pi$
$13$	61.4201			1.31038			0.655188	+	0.755466i	$0.272590\pi$
$14$	−4.20219			−0.0802202
$15$			58.3597i			1.00456i
$16$	62.1828			0.971606
$17$	−27.8030			−0.396660			−0.198330	−	0.980135i	$-0.563552\pi$
$17$	−27.8030			−0.396660			−0.198330	+	0.980135i	$0.563552\pi$
$18$		−	2.48045i		−	0.0324804i
$19$	113.323			1.36832			0.684159	−	0.729333i	$-0.260169\pi$
$19$	113.323			1.36832			0.684159	+	0.729333i	$0.260169\pi$
$20$		−	154.148i		−	1.72343i
$21$			45.7414i			0.475314i
$22$		−	11.8638i		−	0.114972i
$23$		−	42.1549i		−	0.382170i	−0.981574	−	0.191085i	$-0.938799\pi$
$23$		−	42.1549i		−	0.382170i	0.981574	−	0.191085i	$-0.0612006\pi$
$24$			13.1663i			0.111981i
$25$	−253.428			−2.02742
$26$		−	16.9277i		−	0.127685i
$27$	−27.0000			−0.192450
$28$		−	120.819i		−	0.815451i
$29$			88.4551i			0.566404i	0.959060	+	0.283202i	$0.0913966\pi$
$29$			88.4551i			0.566404i	−0.959060	+	0.283202i	$0.908603\pi$
$30$	16.0842			0.0978855
$31$	−16.5221			−0.0957246			−0.0478623	−	0.998854i	$-0.515241\pi$
$31$	−16.5221			−0.0957246			−0.0478623	+	0.998854i	$0.515241\pi$
$32$		−	52.2479i		−	0.288632i
$33$	−129.140			−0.681221
$34$			7.66265i			0.0386510i
$35$	−296.606			−1.43244
$36$	71.3164			0.330168
$37$	146.063			0.648990			0.324495	−	0.945887i	$-0.394806\pi$
$37$	146.063			0.648990			0.324495	+	0.945887i	$0.394806\pi$
$38$		−	31.2324i		−	0.133331i
$39$	−184.260			−0.756545
$40$	−85.3753			−0.337476
$41$			86.5465i			0.329666i	0.986322	+	0.164833i	$0.0527085\pi$
$41$			86.5465i			0.329666i	−0.986322	+	0.164833i	$0.947292\pi$
$42$	12.6066			0.0463151
$43$			67.8646i			0.240680i	0.992733	+	0.120340i	$0.0383985\pi$
$43$			67.8646i			0.240680i	−0.992733	+	0.120340i	$0.961601\pi$
$44$	341.102			1.16871
$45$		−	175.079i		−	0.579983i
$46$	−11.6181			−0.0372391
$47$	−366.968			−1.13889			−0.569444	−	0.822030i	$-0.692841\pi$
$47$	−366.968			−1.13889			−0.569444	+	0.822030i	$0.692841\pi$
$48$	−186.548			−0.560957
$49$	110.525			0.322230
$50$			69.8460i			0.197554i
$51$	83.4090			0.229012
$52$	486.696			1.29793
$53$		−	192.075i		−	0.497802i	−0.968529	−	0.248901i	$-0.919931\pi$
$53$		−	192.075i		−	0.497802i	0.968529	−	0.248901i	$-0.0800694\pi$
$54$			7.44134i			0.0187526i
$55$		−	837.393i		−	2.05298i
$56$	−66.9159			−0.159679
$57$	−339.969			−0.789999
$58$	24.3787			0.0551911
$59$		−	297.310i		−	0.656043i	−0.944670	−	0.328021i	$-0.893618\pi$
$59$		−	297.310i		−	0.656043i	0.944670	−	0.328021i	$-0.106382\pi$
$60$			462.444i			0.995022i
$61$			761.717i			1.59882i	0.600788	+	0.799409i	$0.294854\pi$
$61$			761.717i			1.59882i	−0.600788	+	0.799409i	$0.705146\pi$
$62$			4.55359i			0.00932752i
$63$		−	137.224i		−	0.274423i
$64$	483.062			0.943481
$65$		−	1194.82i		−	2.27999i
$66$			35.5915i			0.0663790i
$67$	−343.054			−0.625534			−0.312767	−	0.949830i	$-0.601256\pi$
$67$	−343.054			−0.625534			−0.312767	+	0.949830i	$0.601256\pi$
$68$	−220.312			−0.392893
$69$			126.465i			0.220646i
$70$			81.7461i			0.139579i
$71$	−781.308			−1.30598			−0.652988	−	0.757369i	$-0.726485\pi$
$71$	−781.308			−1.30598			−0.652988	+	0.757369i	$0.726485\pi$
$72$		−	39.4988i		−	0.0646524i
$73$			274.214i			0.439648i	0.975540	+	0.219824i	$0.0705483\pi$
$73$			274.214i			0.439648i	−0.975540	+	0.219824i	$0.929452\pi$
$74$		−	40.2557i		−	0.0632383i
$75$	760.283			1.17053
$76$	897.975			1.35533
$77$		−	656.336i		−	0.971382i
$78$			50.7831i			0.0737187i
$79$		−	331.333i		−	0.471871i	−0.971769	−	0.235936i	$-0.924185\pi$
$79$		−	331.333i		−	0.471871i	0.971769	−	0.235936i	$-0.0758155\pi$
$80$		−	1209.66i		−	1.69054i
$81$	81.0000			0.111111
$82$	23.8527			0.0321230
$83$		−	145.699i		−	0.192682i	−0.995348	−	0.0963409i	$-0.969286\pi$
$83$		−	145.699i		−	0.192682i	0.995348	−	0.0963409i	$-0.0307139\pi$
$84$			362.457i			0.470801i
$85$			540.858i			0.690168i
$86$	18.7038			0.0234522
$87$		−	265.365i		−	0.327013i
$88$		−	188.920i		−	0.228852i
$89$	−208.027			−0.247762			−0.123881	−	0.992297i	$-0.539534\pi$
$89$	−208.027			−0.247762			−0.123881	+	0.992297i	$0.539534\pi$
$90$	−48.2527			−0.0565142
$91$		−	936.481i		−	1.07879i
$92$		−	334.037i		−	0.378541i
$93$	49.5664			0.0552666
$94$			101.138i			0.110975i
$95$		−	2204.49i		−	2.38080i
$96$			156.744i			0.166642i
$97$		−	1512.84i		−	1.58356i	−0.610807	−	0.791779i	$-0.709155\pi$
$97$		−	1512.84i		−	1.58356i	0.610807	−	0.791779i	$-0.290845\pi$
$98$		−	30.4613i		−	0.0313985i
$99$	387.419			0.393303
$100$	−2008.17			−2.00817
$101$	1679.34			1.65446			0.827229	−	0.561864i	$-0.189916\pi$
$101$	1679.34			1.65446			0.827229	+	0.561864i	$0.189916\pi$
$102$		−	22.9880i		−	0.0223152i
$103$			1488.84i			1.42427i	0.702044	+	0.712134i	$0.252271\pi$
$103$			1488.84i			1.42427i	−0.702044	+	0.712134i	$0.747729\pi$
$104$		−	269.558i		−	0.254157i
$105$	889.817			0.827022
$106$	−52.9369			−0.0485065
$107$			1349.97i			1.21969i	0.792521	+	0.609845i	$0.208768\pi$
$107$			1349.97i			1.21969i	−0.792521	+	0.609845i	$0.791232\pi$
$108$	−213.949			−0.190623
$109$	−1342.62			−1.17982			−0.589908	−	0.807470i	$-0.700836\pi$
$109$	−1342.62			−1.17982			−0.589908	+	0.807470i	$0.700836\pi$
$110$	−230.790			−0.200045
$111$	−438.189			−0.374694
$112$		−	948.109i		−	0.799892i
$113$	−1169.79			−0.973848			−0.486924	−	0.873444i	$-0.661881\pi$
$113$	−1169.79			−0.973848			−0.486924	+	0.873444i	$0.661881\pi$
$114$			93.6971i			0.0769784i
$115$	−820.048			−0.664956
$116$			700.922i			0.561026i
$117$	552.781			0.436792
$118$	−81.9403			−0.0639256
$119$			423.916i			0.326557i
$120$	256.126			0.194842
$121$	522.002			0.392188
$122$	209.933			0.155791
$123$		−	259.639i		−	0.190333i
$124$	−130.922			−0.0948157
$125$			2498.33i			1.78766i
$126$	−37.8197			−0.0267401
$127$	−1152.77			−0.805446			−0.402723	−	0.915322i	$-0.631936\pi$
$127$	−1152.77			−0.805446			−0.402723	+	0.915322i	$0.631936\pi$
$128$		−	551.118i		−	0.380566i
$129$		−	203.594i		−	0.138957i
$130$	−329.298			−0.222165
$131$			2527.35i			1.68562i	0.538214	+	0.842808i	$0.319099\pi$
$131$			2527.35i			1.68562i	−0.538214	+	0.842808i	$0.680901\pi$
$132$	−1023.31			−0.674753
$133$		−	1727.85i		−	1.12649i
$134$			94.5476i			0.0609528i
$135$			525.237i			0.334853i
$136$			122.020i			0.0769350i
$137$		−	2563.65i		−	1.59874i	−0.600839	−	0.799370i	$-0.705167\pi$
$137$		−	2563.65i		−	1.59874i	0.600839	−	0.799370i	$-0.294833\pi$
$138$	34.8543			0.0215000
$139$			245.163i			0.149600i	0.997199	+	0.0748000i	$0.0238319\pi$
$139$			245.163i			0.149600i	−0.997199	+	0.0748000i	$0.976168\pi$
$140$	−2350.32			−1.41884
$141$	1100.90			0.657537
$142$			215.333i			0.127256i
$143$	2643.92			1.54612
$144$	559.645			0.323869
$145$	1720.74			0.985514
$146$	75.5747			0.0428398
$147$	−331.575			−0.186040
$148$	1157.41			0.642828
$149$		−	1334.40i		−	0.733677i	−0.930285	−	0.366839i	$-0.880440\pi$
$149$		−	1334.40i		−	0.733677i	0.930285	−	0.366839i	$-0.119560\pi$
$150$		−	209.538i		−	0.114058i
$151$			2865.33i			1.54422i	0.635488	+	0.772111i	$0.280799\pi$
$151$			2865.33i			1.54422i	−0.635488	+	0.772111i	$0.719201\pi$
$152$		−	497.346i		−	0.265395i
$153$	−250.227			−0.132220
$154$	−180.890			−0.0946526
$155$			321.409i			0.166556i
$156$	−1460.09			−0.749362
$157$	−429.026	+	1919.85i	−0.218089	+	0.975929i
$157$	−429.026	+	1919.85i	−0.218089	+	0.975929i
$158$	−91.3170			−0.0459797
$159$			576.225i			0.287406i
$160$	−1016.39			−0.502204
$161$	−642.741			−0.314628
$162$		−	22.3240i		−	0.0108268i
$163$			3247.05i			1.56030i	0.625593	+	0.780150i	$0.284857\pi$
$163$			3247.05i			1.56030i	−0.625593	+	0.780150i	$0.715143\pi$
$164$			685.798i			0.326536i
$165$			2512.18i			1.18529i
$166$	−40.1555			−0.0187751
$167$	1357.41			0.628980			0.314490	−	0.949261i	$-0.398167\pi$
$167$	1357.41			0.628980			0.314490	+	0.949261i	$0.398167\pi$
$168$	200.748			0.0921905
$169$	1575.43			0.717083
$170$	149.063			0.0672508
$171$	1019.91			0.456106
$172$			537.762i			0.238395i
$173$	554.774			0.243807			0.121904	−	0.992542i	$-0.461100\pi$
$173$	554.774			0.243807			0.121904	+	0.992542i	$0.461100\pi$
$174$	−73.1361			−0.0318646
$175$			3864.05i			1.66911i
$176$	2676.75			1.14641
$177$			891.931i			0.378766i
$178$			57.3333i			0.0241422i
$179$		−	1302.78i		−	0.543990i	−0.962299	−	0.271995i	$-0.912317\pi$
$179$		−	1302.78i		−	0.543990i	0.962299	−	0.271995i	$-0.0876835\pi$
$180$		−	1387.33i		−	0.574476i
$181$			433.713i			0.178108i	0.996027	+	0.0890542i	$0.0283844\pi$
$181$			433.713i			0.178108i	−0.996027	+	0.0890542i	$0.971616\pi$
$182$	−258.099			−0.105119
$183$		−	2285.15i		−	0.923078i
$184$	−185.007			−0.0741246
$185$		−	2841.40i		−	1.12921i
$186$		−	13.6608i		−	0.00538524i
$187$	−1196.82			−0.468023
$188$	−2907.87			−1.12807
$189$			411.673i			0.158438i
$190$	−607.570			−0.231988
$191$			2771.48i			1.04993i	0.851122	+	0.524967i	$0.175922\pi$
$191$			2771.48i			1.04993i	−0.851122	+	0.524967i	$0.824078\pi$
$192$	−1449.19			−0.544719
$193$	−658.586			−0.245627			−0.122814	−	0.992430i	$-0.539192\pi$
$193$	−658.586			−0.245627			−0.122814	+	0.992430i	$0.539192\pi$
$194$	−416.945			−0.154304
$195$			3584.46i			1.31635i
$196$	875.804			0.319171
$197$	−330.043			−0.119363			−0.0596817	−	0.998217i	$-0.519009\pi$
$197$	−330.043			−0.119363			−0.0596817	+	0.998217i	$0.519009\pi$
$198$		−	106.775i		−	0.0383239i
$199$	197.017			0.0701819			0.0350909	−	0.999384i	$-0.488828\pi$
$199$	197.017			0.0701819			0.0350909	+	0.999384i	$0.488828\pi$
$200$			1112.23i			0.393233i
$201$	1029.16			0.361152
$202$		−	462.834i		−	0.161212i
$203$	1348.69			0.466302
$204$	660.936			0.226837
$205$	1683.61			0.573601
$206$	410.332			0.138782
$207$		−	379.394i		−	0.127390i
$208$	3819.27			1.27317
$209$	4878.15			1.61449
$210$		−	245.238i		−	0.0805860i
$211$			1089.79i			0.355566i	0.984070	+	0.177783i	$0.0568925\pi$
$211$			1089.79i			0.355566i	−0.984070	+	0.177783i	$0.943108\pi$
$212$		−	1522.01i		−	0.493076i
$213$	2343.93			0.754005
$214$	372.060			0.118848
$215$	1320.19			0.418772
$216$			118.496i			0.0373271i
$217$			251.915i			0.0788069i
$218$			370.034i			0.114963i
$219$		−	822.641i		−	0.253831i
$220$		−	6635.54i		−	2.03349i
$221$	−1707.66			−0.519773
$222$			120.767i			0.0365107i
$223$		−	2598.72i		−	0.780372i	−0.920736	−	0.390186i	$-0.872411\pi$
$223$		−	2598.72i		−	0.780372i	0.920736	−	0.390186i	$-0.127589\pi$
$224$	−796.631			−0.237621
$225$	−2280.85			−0.675807
$226$			322.401i			0.0948929i
$227$		−	1286.02i		−	0.376019i	−0.982167	−	0.188010i	$-0.939796\pi$
$227$		−	1286.02i		−	0.376019i	0.982167	−	0.188010i	$-0.0602036\pi$
$228$	−2693.92			−0.782498
$229$			5016.05i			1.44747i	0.690079	+	0.723734i	$0.257576\pi$
$229$			5016.05i			1.44747i	−0.690079	+	0.723734i	$0.742424\pi$
$230$			226.010i			0.0647941i
$231$			1969.01i			0.560828i
$232$	388.208			0.109858
$233$	1167.76			0.328336			0.164168	−	0.986432i	$-0.447506\pi$
$233$	1167.76			0.328336			0.164168	+	0.986432i	$0.447506\pi$
$234$		−	152.349i		−	0.0425615i
$235$			7138.70i			1.98161i
$236$		−	2355.90i		−	0.649814i
$237$			993.998i			0.272435i
$238$	116.833			0.0318201
$239$	−5957.48			−1.61237			−0.806187	−	0.591660i	$-0.798473\pi$
$239$	−5957.48			−1.61237			−0.806187	+	0.591660i	$0.798473\pi$
$240$			3628.97i			0.976036i
$241$		−	4376.72i		−	1.16983i	−0.811094	−	0.584916i	$-0.801128\pi$
$241$		−	4376.72i		−	1.16983i	0.811094	−	0.584916i	$-0.198872\pi$
$242$		−	143.866i		−	0.0382152i
$243$	−243.000			−0.0641500
$244$			6035.88i			1.58364i
$245$		−	2150.07i		−	0.560664i
$246$	−71.5580			−0.0185462
$247$	6960.30			1.79301
$248$			72.5115i			0.0185665i
$249$			437.098i			0.111245i
$250$	688.554			0.174192
$251$		−	1866.24i		−	0.469307i	−0.972079	−	0.234654i	$-0.924604\pi$
$251$		−	1866.24i		−	0.469307i	0.972079	−	0.234654i	$-0.0753956\pi$
$252$		−	1087.37i		−	0.271817i
$253$		−	1814.62i		−	0.450926i
$254$			317.709i			0.0784836i
$255$		−	1622.57i		−	0.398468i
$256$	3712.61			0.906398
$257$	−5018.09			−1.21798			−0.608988	−	0.793180i	$-0.708424\pi$
$257$	−5018.09			−1.21798			−0.608988	+	0.793180i	$0.708424\pi$
$258$	−56.1115			−0.0135401
$259$		−	2227.04i		−	0.534292i
$260$		−	9467.80i		−	2.25834i
$261$			796.096i			0.188801i
$262$	696.551			0.164248
$263$	−1755.26			−0.411536			−0.205768	−	0.978601i	$-0.565969\pi$
$263$	−1755.26			−0.411536			−0.205768	+	0.978601i	$0.565969\pi$
$264$			566.761i			0.132128i
$265$	−3736.48			−0.866151
$266$	−476.204			−0.109767
$267$	624.081			0.143045
$268$	−2718.38			−0.619595
$269$			6512.13i			1.47603i	0.674785	+	0.738014i	$0.264236\pi$
$269$			6512.13i			1.47603i	−0.674785	+	0.738014i	$0.735764\pi$
$270$	144.758			0.0326285
$271$		−	4650.68i		−	1.04247i	−0.853414	−	0.521234i	$-0.825472\pi$
$271$		−	4650.68i		−	1.04247i	0.853414	−	0.521234i	$-0.174528\pi$
$272$	−1728.87			−0.385397
$273$			2809.44i			0.622839i
$274$	−706.556			−0.155783
$275$	−10909.2			−2.39218
$276$			1002.11i			0.218551i
$277$	2694.04			0.584366			0.292183	−	0.956362i	$-0.405618\pi$
$277$	2694.04			0.584366			0.292183	+	0.956362i	$0.405618\pi$
$278$	67.5681			0.0145772
$279$	−148.699			−0.0319082
$280$			1301.73i			0.277833i
$281$	2394.94			0.508434			0.254217	−	0.967147i	$-0.418182\pi$
$281$	2394.94			0.508434			0.254217	+	0.967147i	$0.418182\pi$
$282$		−	303.415i		−	0.0640712i
$283$	7921.84			1.66397			0.831987	−	0.554795i	$-0.187203\pi$
$283$	7921.84			1.66397			0.831987	+	0.554795i	$0.187203\pi$
$284$	−6191.12			−1.29358
$285$			6613.48i			1.37456i
$286$		−	728.679i		−	0.150656i
$287$	1319.59			0.271403
$288$		−	470.231i		−	0.0962105i
$289$	−4139.99			−0.842661
$290$		−	474.244i		−	0.0960296i
$291$			4538.51i			0.914268i
$292$			2172.88i			0.435473i
$293$			5455.54i			1.08777i	0.839160	+	0.543884i	$0.183047\pi$
$293$			5455.54i			1.08777i	−0.839160	+	0.543884i	$0.816953\pi$
$294$			91.3838i			0.0181279i
$295$	−5783.64			−1.14148
$296$		−	641.034i		−	0.125876i
$297$	−1162.26			−0.227074
$298$	−367.766			−0.0714904
$299$		−	2589.16i		−	0.500786i
$300$	6024.52			1.15942
$301$	1034.74			0.198144
$302$	789.701			0.150471
$303$	−5038.01			−0.955202
$304$	7046.73			1.32947
$305$	14817.8			2.78186
$306$			68.9639i			0.0128837i
$307$			7761.13i			1.44284i	0.692499	+	0.721419i	$0.256510\pi$
$307$			7761.13i			1.44284i	−0.692499	+	0.721419i	$0.743490\pi$
$308$		−	5200.83i		−	0.962159i
$309$		−	4466.51i		−	0.822301i
$310$	88.5819			0.0162294
$311$	10.6111			0.00193473			0.000967363	−	1.00000i	$-0.499692\pi$
$311$	10.6111			0.00193473			0.000967363		1.00000i	$0.499692\pi$
$312$			808.673i			0.146737i
$313$	−10179.5			−1.83826			−0.919132	−	0.393950i	$-0.871108\pi$
$313$	−10179.5			−1.83826			−0.919132	+	0.393950i	$0.871108\pi$
$314$	529.121	+	118.242i	0.0950957	+	0.0212509i
$315$	−2669.45			−0.477481
$316$		−	2625.49i		−	0.467391i
$317$	−4803.68			−0.851110			−0.425555	−	0.904933i	$-0.639921\pi$
$317$	−4803.68			−0.851110			−0.425555	+	0.904933i	$0.639921\pi$
$318$	158.811			0.0280052
$319$			3807.68i			0.668305i
$320$		−	9397.12i		−	1.64161i
$321$		−	4049.92i		−	0.704188i
$322$			177.143i			0.0306577i
$323$	−3150.71			−0.542757
$324$	641.847			0.110056
$325$	−15565.6			−2.65668
$326$	894.905			0.152037
$327$	4027.87			0.681167
$328$	379.831			0.0639410
$329$			5595.21i			0.937610i
$330$	692.370			0.115496
$331$	8575.19			1.42397			0.711987	−	0.702193i	$-0.247796\pi$
$331$	8575.19			1.42397			0.711987	+	0.702193i	$0.247796\pi$
$332$		−	1154.53i		−	0.190852i
$333$	1314.57			0.216330
$334$		−	374.110i		−	0.0612886i
$335$			6673.51i			1.08840i
$336$			2844.33i			0.461818i
$337$			4641.92i			0.750330i	0.926958	+	0.375165i	$0.122414\pi$
$337$			4641.92i			0.750330i	−0.926958	+	0.375165i	$0.877586\pi$
$338$		−	434.197i		−	0.0698734i
$339$	3509.38			0.562252
$340$			4285.78i			0.683615i
$341$	−711.220			−0.112946
$342$		−	281.091i		−	0.0444435i
$343$		−	6914.96i		−	1.08855i
$344$	297.841			0.0466817
$345$	2460.15			0.383912
$346$		−	152.899i		−	0.0237569i
$347$	−5148.28			−0.796467			−0.398233	−	0.917284i	$-0.630377\pi$
$347$	−5148.28			−0.796467			−0.398233	+	0.917284i	$0.630377\pi$
$348$		−	2102.77i		−	0.323908i
$349$	10911.5			1.67358			0.836789	−	0.547526i	$-0.184430\pi$
$349$	10911.5			1.67358			0.836789	+	0.547526i	$0.184430\pi$
$350$	1064.95			0.162640
$351$	−1658.34			−0.252182
$352$		−	2249.09i		−	0.340559i
$353$	11839.9			1.78520			0.892600	−	0.450850i	$-0.148879\pi$
$353$	11839.9			1.78520			0.892600	+	0.450850i	$0.148879\pi$
$354$	245.821			0.0369074
$355$			15199.0i			2.27233i
$356$	−1648.41			−0.245410
$357$		−	1271.75i		−	0.188538i
$358$	−359.053			−0.0530071
$359$		−	5039.27i		−	0.740843i	−0.928864	−	0.370422i	$-0.879213\pi$
$359$		−	5039.27i		−	0.740843i	0.928864	−	0.370422i	$-0.120787\pi$
$360$	−768.378			−0.112492
$361$	5983.07			0.872294
$362$	119.534			0.0173551
$363$	−1566.00			−0.226430
$364$		−	7420.71i		−	1.06855i
$365$	5334.34			0.764965
$366$	−629.800			−0.0899458
$367$			3311.95i			0.471069i	0.971866	+	0.235535i	$0.0756841\pi$
$367$			3311.95i			0.471069i	−0.971866	+	0.235535i	$0.924316\pi$
$368$		−	2621.31i		−	0.371318i
$369$			778.918i			0.109889i
$370$	−783.104			−0.110031
$371$	−2928.59			−0.409825
$372$	392.766			0.0547419
$373$		−	3272.28i		−	0.454242i	−0.973867	−	0.227121i	$-0.927069\pi$
$373$		−	3272.28i		−	0.454242i	0.973867	−	0.227121i	$-0.0729313\pi$
$374$			329.850i			0.0456047i
$375$		−	7495.00i		−	1.03211i
$376$			1610.53i			0.220896i
$377$			5432.93i			0.742201i
$378$	113.459			0.0154384
$379$			5502.26i			0.745731i	0.927885	+	0.372866i	$0.121625\pi$
$379$			5502.26i			0.745731i	−0.927885	+	0.372866i	$0.878375\pi$
$380$		−	17468.5i		−	2.35820i
$381$	3458.30			0.465024
$382$	763.836			0.102307
$383$			2736.77i			0.365124i	0.983194	+	0.182562i	$0.0584390\pi$
$383$			2736.77i			0.365124i	−0.983194	+	0.182562i	$0.941561\pi$
$384$			1653.35i			0.219720i
$385$	−12767.8			−1.69015
$386$			181.510i			0.0239342i
$387$			610.782i			0.0802268i
$388$		−	11987.8i		−	1.56852i
$389$	8363.41			1.09008			0.545041	−	0.838409i	$-0.316514\pi$
$389$	8363.41			1.09008			0.545041	+	0.838409i	$0.316514\pi$
$390$	987.895			0.128267
$391$			1172.03i			0.151591i
$392$		−	485.066i		−	0.0624989i
$393$		−	7582.05i		−	0.973191i
$394$			90.9616i			0.0116309i
$395$	−6445.49			−0.821032
$396$	3069.92			0.389569
$397$		−	14547.3i		−	1.83907i	−0.393012	−	0.919533i	$-0.628567\pi$
$397$		−	14547.3i		−	1.83907i	0.393012	−	0.919533i	$-0.371433\pi$
$398$		−	54.2990i		−	0.00683860i
$399$			5183.55i			0.650381i
$400$	−15758.8			−1.96986
$401$		−	12934.2i		−	1.61074i	−0.592776	−	0.805368i	$-0.701968\pi$
$401$		−	12934.2i		−	1.61074i	0.592776	−	0.805368i	$-0.298032\pi$
$402$		−	283.643i		−	0.0351911i
$403$	−1014.79			−0.125435
$404$	13307.1			1.63875
$405$		−	1575.71i		−	0.193328i
$406$		−	371.705i		−	0.0454370i
$407$	6287.50			0.765749
$408$		−	366.061i		−	0.0444184i
$409$			5609.29i			0.678146i	0.940760	+	0.339073i	$0.110113\pi$
$409$			5609.29i			0.678146i	−0.940760	+	0.339073i	$0.889887\pi$
$410$		−	464.011i		−	0.0558924i
$411$			7690.95i			0.923033i
$412$			11797.6i			1.41074i
$413$	−4533.13			−0.540099
$414$	−104.563			−0.0124130
$415$	−2834.32			−0.335256
$416$		−	3209.07i		−	0.378216i
$417$		−	735.488i		−	0.0863717i
$418$		−	1344.44i		−	0.157318i
$419$	−3298.35			−0.384571			−0.192285	−	0.981339i	$-0.561590\pi$
$419$	−3298.35			−0.384571			−0.192285	+	0.981339i	$0.561590\pi$
$420$	7050.95			0.819169
$421$		−	14222.5i		−	1.64647i	−0.567703	−	0.823233i	$-0.692168\pi$
$421$		−	14222.5i		−	1.64647i	0.567703	−	0.823233i	$-0.307832\pi$
$422$	300.353			0.0346468
$423$	−3302.71			−0.379629
$424$	−842.969			−0.0965523
$425$	7046.05			0.804197
$426$		−	645.998i		−	0.0734712i
$427$	11614.0			1.31625
$428$			10697.2i			1.20811i
$429$	−7931.76			−0.892655
$430$		−	363.850i		−	0.0408056i
$431$	−4957.88			−0.554090			−0.277045	−	0.960857i	$-0.589355\pi$
$431$	−4957.88			−0.554090			−0.277045	+	0.960857i	$0.589355\pi$
$432$	−1678.93			−0.186986
$433$		−	14680.7i		−	1.62935i	−0.579919	−	0.814674i	$-0.696916\pi$
$433$		−	14680.7i		−	1.62935i	0.579919	−	0.814674i	$-0.303084\pi$
$434$	69.4291			0.00767904
$435$	−5162.21			−0.568987
$436$	−10639.0			−1.16861
$437$		−	4777.11i		−	0.522930i
$438$	−226.724			−0.0247336
$439$		−	577.435i		−	0.0627778i	−0.999507	−	0.0313889i	$-0.990007\pi$
$439$		−	577.435i		−	0.0627778i	0.999507	−	0.0313889i	$-0.00999304\pi$
$440$	−3675.11			−0.398191
$441$	994.725			0.107410
$442$			470.641i			0.0506473i
$443$		−	14704.2i		−	1.57702i	−0.615025	−	0.788508i	$-0.710854\pi$
$443$		−	14704.2i		−	1.57702i	0.615025	−	0.788508i	$-0.289146\pi$
$444$	−3472.23			−0.371137
$445$			4046.79i			0.431093i
$446$	−716.220			−0.0760404
$447$			4003.19i			0.423589i
$448$		−	7365.32i		−	0.776738i
$449$		−	16365.2i		−	1.72009i	−0.510218	−	0.860045i	$-0.670435\pi$
$449$		−	16365.2i		−	1.72009i	0.510218	−	0.860045i	$-0.329565\pi$
$450$			628.614i			0.0658515i
$451$			3725.52i			0.388976i
$452$	−9269.49			−0.964602
$453$		−	8596.00i		−	0.891557i
$454$	−354.435			−0.0366397
$455$	−18217.6			−1.87704
$456$			1492.04i			0.153226i
$457$	−3493.68			−0.357609			−0.178804	−	0.983885i	$-0.557223\pi$
$457$	−3493.68			−0.357609			−0.178804	+	0.983885i	$0.557223\pi$
$458$	1382.45			0.141043
$459$	750.681			0.0763372
$460$	−6498.10			−0.658642
$461$	11732.1			1.18529			0.592644	−	0.805465i	$-0.298084\pi$
$461$	11732.1			1.18529			0.592644	+	0.805465i	$0.298084\pi$
$462$	542.669			0.0546477
$463$		−	11020.0i		−	1.10614i	−0.833135	−	0.553069i	$-0.813457\pi$
$463$		−	11020.0i		−	1.10614i	0.833135	−	0.553069i	$-0.186543\pi$
$464$			5500.39i			0.550321i
$465$		−	964.226i		−	0.0961611i
$466$		−	321.840i		−	0.0319935i
$467$	17126.6			1.69705			0.848527	−	0.529152i	$-0.177490\pi$
$467$	17126.6			1.69705			0.848527	+	0.529152i	$0.177490\pi$
$468$	4380.26			0.432644
$469$			5230.60i			0.514982i
$470$	1967.46			0.193090
$471$	1287.08	−	5759.55i	0.125914	−	0.563453i
$472$	−1304.82			−0.127244
$473$			2921.33i			0.283981i
$474$	273.951			0.0265464
$475$	−28719.2			−2.77416
$476$			3359.13i			0.323457i
$477$		−	1728.67i		−	0.165934i
$478$			1641.91i			0.157112i
$479$			13463.5i			1.28427i	0.766592	+	0.642134i	$0.221951\pi$
$479$			13463.5i			1.28427i	−0.766592	+	0.642134i	$0.778049\pi$
$480$	3049.17			0.289948
$481$	8971.21			0.850420
$482$	−1206.25			−0.113990
$483$	1928.22			0.181651
$484$	4136.36			0.388464
$485$	−29429.5			−2.75531
$486$			66.9721i			0.00625086i
$487$	−2935.82			−0.273172			−0.136586	−	0.990628i	$-0.543613\pi$
$487$	−2935.82			−0.273172			−0.136586	+	0.990628i	$0.543613\pi$
$488$	3342.99			0.310102
$489$		−	9741.16i		−	0.900839i
$490$	−592.570			−0.0546318
$491$			11265.0i			1.03540i	0.855562	+	0.517701i	$0.173212\pi$
$491$			11265.0i			1.03540i	−0.855562	+	0.517701i	$0.826788\pi$
$492$		−	2057.39i		−	0.188525i
$493$		−	2459.32i		−	0.224670i
$494$		−	1918.30i		−	0.174713i
$495$		−	7536.54i		−	0.684328i
$496$	−1027.39			−0.0930065
$497$			11912.7i			1.07517i
$498$	120.467			0.0108398
$499$		−	10523.4i		−	0.944071i	−0.881580	−	0.472036i	$-0.843519\pi$
$499$		−	10523.4i		−	0.944071i	0.881580	−	0.472036i	$-0.156481\pi$
$500$			19796.9i			1.77069i
$501$	−4072.23			−0.363142
$502$	−514.346			−0.0457299
$503$			1711.58i			0.151720i	0.997118	+	0.0758602i	$0.0241703\pi$
$503$			1711.58i			0.151720i	−0.997118	+	0.0758602i	$0.975830\pi$
$504$	−602.243			−0.0532262
$505$		−	32668.5i		−	2.87867i
$506$	−500.119			−0.0439387
$507$	−4726.29			−0.414008
$508$	−9134.58			−0.797798
$509$		−	18997.6i		−	1.65433i	−0.561962	−	0.827163i	$-0.689953\pi$
$509$		−	18997.6i		−	1.65433i	0.561962	−	0.827163i	$-0.310047\pi$
$510$	−447.190			−0.0388272
$511$	4180.97			0.361948
$512$		−	5432.16i		−	0.468886i
$513$	−3059.72			−0.263333
$514$			1383.01i			0.118681i
$515$	28962.7			2.47815
$516$		−	1613.29i		−	0.137638i
$517$	−15796.7			−1.34379
$518$	−613.785			−0.0520621
$519$	−1664.32			−0.140762
$520$	−5243.76			−0.442220
$521$			19834.8i			1.66790i	0.551839	+	0.833951i	$0.313926\pi$
$521$			19834.8i			1.66790i	−0.551839	+	0.833951i	$0.686074\pi$
$522$	219.408			0.0183970
$523$	14751.8			1.23337			0.616684	−	0.787211i	$-0.288475\pi$
$523$	14751.8			1.23337			0.616684	+	0.787211i	$0.288475\pi$
$524$			20026.8i			1.66961i
$525$		−	11592.1i		−	0.963662i
$526$			483.759i			0.0401006i
$527$	459.364			0.0379701
$528$	−8030.25			−0.661879
$529$	10390.0			0.853946
$530$			1029.79i			0.0843988i
$531$		−	2675.79i		−	0.218681i
$532$		−	13691.5i		−	1.11580i
$533$			5315.70i			0.431986i
$534$		−	172.000i		−	0.0139385i
$535$	26261.3			2.12220
$536$			1505.58i			0.121327i
$537$			3908.34i			0.314073i
$538$	1794.78			0.143826
$539$	4757.71			0.380203
$540$			4162.00i			0.331674i
$541$		−	1141.64i		−	0.0907263i	−0.998971	−	0.0453631i	$-0.985556\pi$
$541$		−	1141.64i		−	0.0907263i	0.998971	−	0.0453631i	$-0.0144445\pi$
$542$	−1281.75			−0.101579
$543$		−	1301.14i		−	0.102831i
$544$			1452.65i			0.114489i
$545$			26118.4i			2.05282i
$546$	774.297			0.0606902
$547$	8264.00			0.645965			0.322983	−	0.946405i	$-0.395314\pi$
$547$	8264.00			0.645965			0.322983	+	0.946405i	$0.395314\pi$
$548$		−	20314.5i		−	1.58356i
$549$			6855.45i			0.532939i
$550$			3006.63i			0.233096i
$551$			10024.0i			0.775020i
$552$	555.022			0.0427958
$553$	−5051.87			−0.388476
$554$		−	742.493i		−	0.0569413i
$555$			8524.19i			0.651949i
$556$			1942.68i			0.148180i
$557$	−5156.39			−0.392250			−0.196125	−	0.980579i	$-0.562836\pi$
$557$	−5156.39			−0.392250			−0.196125	+	0.980579i	$0.562836\pi$
$558$			40.9823i			0.00310917i
$559$			4168.25i			0.315382i
$560$	−18443.8			−1.39177
$561$	3590.46			0.270213
$562$		−	660.057i		−	0.0495424i
$563$			4428.93i			0.331541i	0.986164	+	0.165770i	$0.0530110\pi$
$563$			4428.93i			0.331541i	−0.986164	+	0.165770i	$0.946989\pi$
$564$	8723.60			0.651294
$565$			22756.2i			1.69445i
$566$		−	2183.30i		−	0.162140i
$567$		−	1235.02i		−	0.0914742i
$568$			3428.97i			0.253303i
$569$			25834.8i			1.90343i	0.306984	+	0.951715i	$0.400680\pi$
$569$			25834.8i			1.90343i	−0.306984	+	0.951715i	$0.599320\pi$
$570$	1822.71			0.133939
$571$	19122.1			1.40146			0.700731	−	0.713426i	$-0.252857\pi$
$571$	19122.1			1.40146			0.700731	+	0.713426i	$0.252857\pi$
$572$	20950.5			1.53144
$573$		−	8314.45i		−	0.606180i
$574$		−	363.685i		−	0.0264458i
$575$			10683.2i			0.774819i
$576$	4347.56			0.314494
$577$	18508.8			1.33541			0.667704	−	0.744427i	$-0.267277\pi$
$577$	18508.8			1.33541			0.667704	+	0.744427i	$0.267277\pi$
$578$			1141.00i			0.0821099i
$579$	1975.76			0.141813
$580$	13635.2			0.976156
$581$	−2221.50			−0.158629
$582$	1250.84			0.0890873
$583$		−	8268.16i		−	0.587362i
$584$	1203.46			0.0852728
$585$		−	10753.4i		−	0.759995i
$586$	1503.58			0.105993
$587$			1580.72i			0.111147i	0.998455	+	0.0555735i	$0.0176987\pi$
$587$			1580.72i			0.111147i	−0.998455	+	0.0555735i	$0.982301\pi$
$588$	−2627.41			−0.184273
$589$	−1872.33			−0.130982
$590$			1594.00i			0.111227i
$591$	990.129			0.0689145
$592$	9082.60			0.630562
$593$	21217.2			1.46928			0.734641	−	0.678456i	$-0.237350\pi$
$593$	21217.2			1.46928			0.734641	+	0.678456i	$0.237350\pi$
$594$			320.324i			0.0221263i
$595$	8246.53			0.568193
$596$		−	10573.8i		−	0.726711i
$597$	−591.052			−0.0405195
$598$	−713.586			−0.0487972
$599$			14304.4i			0.975731i	0.872919	+	0.487865i	$0.162224\pi$
$599$			14304.4i			0.975731i	−0.872919	+	0.487865i	$0.837776\pi$
$600$		−	3336.69i		−	0.227033i
$601$	−94.2674			−0.00639808			−0.00319904	−	0.999995i	$-0.501018\pi$
$601$	−94.2674			−0.00639808			−0.00319904	+	0.999995i	$0.501018\pi$
$602$		−	285.180i		−	0.0193074i
$603$	−3087.49			−0.208511
$604$			22705.0i			1.52956i
$605$		−	10154.6i		−	0.682386i
$606$			1388.50i			0.0930761i
$607$		−	4752.84i		−	0.317812i	−0.987294	−	0.158906i	$-0.949203\pi$
$607$		−	4752.84i		−	0.317812i	0.987294	−	0.158906i	$-0.0507967\pi$
$608$		−	5920.88i		−	0.394940i
$609$	−4046.06			−0.269220
$610$		−	4083.88i		−	0.271068i
$611$	−22539.2			−1.49237
$612$	−1982.81			−0.130964
$613$		−	807.332i		−	0.0531939i	−0.999646	−	0.0265969i	$-0.991533\pi$
$613$		−	807.332i		−	0.0531939i	0.999646	−	0.0265969i	$-0.00846706\pi$
$614$	2139.01			0.140592
$615$	−5050.82			−0.331169
$616$	−2880.49			−0.188406
$617$	4350.88			0.283889			0.141945	−	0.989875i	$-0.454664\pi$
$617$	4350.88			0.283889			0.141945	+	0.989875i	$0.454664\pi$
$618$	−1230.99			−0.0801260
$619$	18048.5			1.17194			0.585968	−	0.810334i	$-0.300714\pi$
$619$	18048.5			1.17194			0.585968	+	0.810334i	$0.300714\pi$
$620$			2546.85i			0.164974i
$621$			1138.18i			0.0735486i
$622$		−	2.92447i		−	0.000188522i
$623$			3171.81i			0.203974i
$624$	−11457.8			−0.735064
$625$	16922.2			1.08302
$626$			2805.51i			0.179123i
$627$	−14634.5			−0.932127
$628$	−3399.62	+	15213.0i	−0.216018	+	0.966663i
$629$	−4060.99			−0.257428
$630$			735.715i			0.0465263i
$631$	−23377.8			−1.47489			−0.737446	−	0.675406i	$-0.763968\pi$
$631$	−23377.8			−1.47489			−0.737446	+	0.675406i	$0.763968\pi$
$632$	−1454.14			−0.0915228
$633$		−	3269.38i		−	0.205286i
$634$			1323.92i			0.0829331i
$635$			22425.0i			1.40143i
$636$			4566.03i			0.284677i
$637$	6788.46			0.422242
$638$	1049.42			0.0651205
$639$	−7031.78			−0.435325
$640$	−10721.0			−0.662165
$641$	13046.2			0.803890			0.401945	−	0.915664i	$-0.368334\pi$
$641$	13046.2			0.803890			0.401945	+	0.915664i	$0.368334\pi$
$642$	−1116.18			−0.0686170
$643$			16465.8i			1.00987i	0.863157	+	0.504936i	$0.168484\pi$
$643$			16465.8i			1.00987i	−0.863157	+	0.504936i	$0.831516\pi$
$644$	−5093.11			−0.311641
$645$	−3960.56			−0.241778
$646$			868.353i			0.0528868i
$647$	−19053.5			−1.15776			−0.578881	−	0.815412i	$-0.696510\pi$
$647$	−19053.5			−1.15776			−0.578881	+	0.815412i	$0.696510\pi$
$648$		−	355.489i		−	0.0215508i
$649$		−	12798.2i		−	0.774071i
$650$			4289.95i			0.258870i
$651$		−	755.745i		−	0.0454992i
$652$			25729.8i			1.54548i
$653$	14464.5			0.866830			0.433415	−	0.901195i	$-0.357309\pi$
$653$	14464.5			0.866830			0.433415	+	0.901195i	$0.357309\pi$
$654$		−	1110.10i		−	0.0663738i
$655$	49165.1			2.93289
$656$			5381.70i			0.320305i
$657$			2467.92i			0.146549i
$658$	1542.07			0.0913618
$659$	−27642.6			−1.63399			−0.816997	−	0.576641i	$-0.804363\pi$
$659$	−27642.6			−1.63399			−0.816997	+	0.576641i	$0.804363\pi$
$660$			19906.6i			1.17404i
$661$	−24233.5			−1.42598			−0.712990	−	0.701174i	$-0.752660\pi$
$661$	−24233.5			−1.42598			−0.712990	+	0.701174i	$0.752660\pi$
$662$		−	2363.37i		−	0.138754i
$663$	5122.99			0.300091
$664$	−639.438			−0.0373720
$665$	−33612.2			−1.96004
$666$		−	362.302i		−	0.0210794i
$667$	3728.82			0.216462
$668$	10756.2			0.623008
$669$			7796.15i			0.450548i
$670$	1839.26			0.106055
$671$			32789.3i			1.88646i
$672$	2389.89			0.137191
$673$		−	36.4202i		−	0.00208603i	−0.999999	−	0.00104301i	$-0.999668\pi$
$673$		−	36.4202i		−	0.00208603i	0.999999	−	0.00104301i	$-0.000332002\pi$
$674$	1279.34			0.0731131
$675$	6842.55			0.390178
$676$	12483.8			0.710274
$677$	−25464.1			−1.44559			−0.722796	−	0.691061i	$-0.757144\pi$
$677$	−25464.1			−1.44559			−0.722796	+	0.691061i	$0.757144\pi$
$678$		−	967.203i		−	0.0547865i
$679$	−23066.4			−1.30369
$680$	2373.69			0.133863
$681$			3858.07i			0.217095i
$682$			196.016i			0.0110056i
$683$		−	3485.35i		−	0.195261i	−0.995223	−	0.0976305i	$-0.968874\pi$
$683$		−	3485.35i		−	0.195261i	0.995223	−	0.0976305i	$-0.0311263\pi$
$684$	8081.77			0.451775
$685$	−49871.2			−2.78173
$686$	−1905.80			−0.106070
$687$		−	15048.2i		−	0.835696i
$688$			4220.01i			0.233846i
$689$		−	11797.3i		−	0.652308i
$690$		−	678.029i		−	0.0374089i
$691$		−	29608.8i		−	1.63006i	−0.579418	−	0.815031i	$-0.696720\pi$
$691$		−	29608.8i		−	1.63006i	0.579418	−	0.815031i	$-0.303280\pi$
$692$	4396.05			0.241492
$693$		−	5907.02i		−	0.323794i
$694$			1418.89i			0.0776086i
$695$	4769.20			0.260297
$696$	−1164.62			−0.0634266
$697$		−	2406.25i		−	0.130765i
$698$		−	3007.26i		−	0.163075i
$699$	−3503.27			−0.189565
$700$			30618.9i			1.65326i
$701$			25752.6i			1.38754i	0.720199	+	0.693768i	$0.244050\pi$
$701$			25752.6i			1.38754i	−0.720199	+	0.693768i	$0.755950\pi$
$702$			457.048i			0.0245729i
$703$	16552.3			0.888024
$704$	20794.1			1.11322
$705$		−	21416.1i		−	1.14408i
$706$		−	3263.15i		−	0.173952i
$707$		−	25605.1i		−	1.36206i
$708$			7067.70i			0.375170i
$709$	−19293.9			−1.02200			−0.511000	−	0.859581i	$-0.670725\pi$
$709$	−19293.9			−1.02200			−0.511000	+	0.859581i	$0.670725\pi$
$710$	4188.92			0.221419
$711$		−	2981.99i		−	0.157290i
$712$			912.978i			0.0480552i
$713$			696.488i			0.0365830i
$714$	−350.500			−0.0183714
$715$		−	51432.8i		−	2.69018i
$716$		−	10323.3i		−	0.538825i
$717$	17872.5			0.930905
$718$	−1388.85			−0.0721887
$719$			26083.8i			1.35294i	0.736471	+	0.676469i	$0.236491\pi$
$719$			26083.8i			1.35294i	−0.736471	+	0.676469i	$0.763509\pi$
$720$		−	10886.9i		−	0.563515i
$721$	22700.5			1.17255
$722$		−	1648.96i		−	0.0849974i
$723$			13130.2i			0.675402i
$724$			3436.76i			0.176417i
$725$		−	22417.0i		−	1.14834i
$726$			431.599i			0.0220636i
$727$	−2195.79			−0.112019			−0.0560093	−	0.998430i	$-0.517838\pi$
$727$	−2195.79			−0.112019			−0.0560093	+	0.998430i	$0.517838\pi$
$728$	−4109.98			−0.209239
$729$	729.000			0.0370370
$730$		−	1470.17i		−	0.0745391i
$731$		−	1886.84i		−	0.0954682i
$732$		−	18107.6i		−	0.914313i
$733$	−37906.2			−1.91009			−0.955045	−	0.296460i	$-0.904194\pi$
$733$	−37906.2			−1.91009			−0.955045	+	0.296460i	$0.904194\pi$
$734$	912.792			0.0459016
$735$			6450.20i			0.323700i
$736$	−2202.50			−0.110306
$737$	−14767.3			−0.738074
$738$	214.674			0.0107077
$739$	13982.2			0.695998			0.347999	−	0.937495i	$-0.386861\pi$
$739$	13982.2			0.695998			0.347999	+	0.937495i	$0.386861\pi$
$740$		−	22515.3i		−	1.11849i
$741$	−20880.9			−1.03519
$742$			807.136i			0.0399338i
$743$	4238.72			0.209291			0.104646	−	0.994510i	$-0.466629\pi$
$743$	4238.72			0.209291			0.104646	+	0.994510i	$0.466629\pi$
$744$		−	217.534i		−	0.0107194i
$745$	−25958.3			−1.27656
$746$	−901.858			−0.0442619
$747$		−	1311.29i		−	0.0642273i
$748$	−9483.66			−0.463579
$749$	20583.2			1.00413
$750$	−2065.66			−0.100570
$751$			12516.8i			0.608181i	0.952643	+	0.304091i	$0.0983525\pi$
$751$			12516.8i			0.608181i	−0.952643	+	0.304091i	$0.901647\pi$
$752$	−22819.1			−1.10655
$753$			5598.73i			0.270955i
$754$	1497.34			0.0723210
$755$	55739.9			2.68687
$756$			3262.11i			0.156934i
$757$		−	16898.9i		−	0.811364i	−0.914014	−	0.405682i	$-0.867034\pi$
$757$		−	16898.9i		−	0.811364i	0.914014	−	0.405682i	$-0.132966\pi$
$758$	1516.45			0.0726649
$759$			5443.86i			0.260342i
$760$	−9674.98			−0.461774
$761$			219.744i			0.0104674i	0.999986	+	0.00523371i	$0.00166595\pi$
$761$			219.744i			0.0104674i	−0.999986	+	0.00523371i	$0.998334\pi$
$762$		−	953.127i		−	0.0453125i
$763$			20471.2i			0.971305i
$764$			21961.4i			1.03997i
$765$			4867.72i			0.230056i
$766$	754.268			0.0355781
$767$		−	18260.8i		−	0.859662i
$768$	−11137.8			−0.523309
$769$	−10469.5			−0.490950			−0.245475	−	0.969403i	$-0.578944\pi$
$769$	−10469.5			−0.490950			−0.245475	+	0.969403i	$0.578944\pi$
$770$			3518.89i			0.164691i
$771$	15054.3			0.703198
$772$	−5218.66			−0.243295
$773$	6643.75			0.309132			0.154566	−	0.987982i	$-0.450602\pi$
$773$	6643.75			0.309132			0.154566	+	0.987982i	$0.450602\pi$
$774$	168.335			0.00781740
$775$	4187.17			0.194074
$776$	−6639.46			−0.307143
$777$			6681.13i			0.308474i
$778$		−	2305.00i		−	0.106219i
$779$			9807.69i			0.451087i
$780$			28403.4i			1.30385i
$781$	−33632.6			−1.54093
$782$	323.018			0.0147712
$783$		−	2388.29i		−	0.109004i
$784$	6872.75			0.313081
$785$	37347.3	+	8345.94i	1.69807	+	0.379464i
$786$	−2089.65			−0.0948289
$787$		−	1799.34i		−	0.0814989i	−0.999169	−	0.0407495i	$-0.987025\pi$
$787$		−	1799.34i		−	0.0814989i	0.999169	−	0.0407495i	$-0.0129745\pi$
$788$	−2615.27			−0.118230
$789$	5265.78			0.237601
$790$			1776.41i			0.0800023i
$791$			17836.0i			0.801738i
$792$		−	1700.28i		−	0.0762840i
$793$			46784.7i			2.09505i
$794$	−4009.32			−0.179201
$795$	11209.4			0.500072
$796$	1561.17			0.0695155
$797$	3384.78			0.150433			0.0752165	−	0.997167i	$-0.476035\pi$
$797$	3384.78			0.150433			0.0752165	+	0.997167i	$0.476035\pi$
$798$	1428.61			0.0633739
$799$	10202.8			0.451751
$800$			13241.1i			0.585178i
$801$	−1872.24			−0.0825873
$802$	−3564.74			−0.156952
$803$			11803.9i			0.518745i
$804$	8155.13			0.357723
$805$			12503.4i			0.547437i
$806$			279.682i			0.0122225i
$807$		−	19536.4i		−	0.852186i
$808$		−	7370.19i		−	0.320894i
$809$		−	21704.6i		−	0.943257i	−0.881797	−	0.471628i	$-0.843666\pi$
$809$		−	21704.6i		−	0.943257i	0.881797	−	0.471628i	$-0.156334\pi$
$810$	−434.274			−0.0188381
$811$		−	34844.7i		−	1.50871i	−0.656468	−	0.754354i	$-0.727950\pi$
$811$		−	34844.7i		−	1.50871i	0.656468	−	0.754354i	$-0.272050\pi$
$812$	10687.1			0.461874
$813$			13952.0i			0.601869i
$814$		−	1732.87i		−	0.0746155i
$815$	63165.6			2.71484
$816$	5186.60			0.222509
$817$			7690.61i			0.329327i
$818$	1545.95			0.0660793
$819$		−	8428.33i		−	0.359597i
$820$	13341.0			0.568155
$821$	−32860.7			−1.39689			−0.698444	−	0.715665i	$-0.746124\pi$
$821$	−32860.7			−1.39689			−0.698444	+	0.715665i	$0.746124\pi$
$822$	2119.67			0.0899414
$823$		−	3592.50i		−	0.152159i	−0.997102	−	0.0760794i	$-0.975760\pi$
$823$		−	3592.50i		−	0.152159i	0.997102	−	0.0760794i	$-0.0242402\pi$
$824$	6534.14			0.276247
$825$	32727.5			1.38112
$826$			1249.35i			0.0526279i
$827$	−26784.7			−1.12623			−0.563117	−	0.826377i	$-0.690398\pi$
$827$	−26784.7			−1.12623			−0.563117	+	0.826377i	$0.690398\pi$
$828$		−	3006.33i		−	0.126180i
$829$	−1648.00			−0.0690439			−0.0345220	−	0.999404i	$-0.510991\pi$
$829$	−1648.00			−0.0690439			−0.0345220	+	0.999404i	$0.510991\pi$
$830$			781.154i			0.0326678i
$831$	−8082.13			−0.337384
$832$	29669.7			1.23631
$833$	−3072.92			−0.127816
$834$	−202.704			−0.00841616
$835$		−	26406.0i		−	1.09439i
$836$	38654.7			1.59916
$837$	446.097			0.0184222
$838$			909.043i			0.0374730i
$839$			3890.69i			0.160097i	0.996791	+	0.0800485i	$0.0255075\pi$
$839$			3890.69i			0.160097i	−0.996791	+	0.0800485i	$0.974492\pi$
$840$		−	3905.19i		−	0.160407i
$841$	16564.7			0.679187
$842$	−3919.80			−0.160434
$843$	−7184.81			−0.293545
$844$			8635.56i			0.352190i
$845$		−	30647.2i		−	1.24769i
$846$			910.244i			0.0369915i
$847$		−	7959.03i		−	0.322875i
$848$		−	11943.8i		−	0.483668i
$849$	−23765.5			−0.960696
$850$		−	1941.93i		−	0.0783619i
$851$		−	6157.27i		−	0.248024i
$852$	18573.4			0.746846
$853$	13058.3			0.524159			0.262079	−	0.965046i	$-0.415592\pi$
$853$	13058.3			0.524159			0.262079	+	0.965046i	$0.415592\pi$
$854$		−	3200.88i		−	0.128257i
$855$		−	19840.4i		−	0.793601i
$856$	5924.70			0.236568
$857$		−	10083.3i		−	0.401911i	−0.979600	−	0.200955i	$-0.935595\pi$
$857$		−	10083.3i		−	0.401911i	0.979600	−	0.200955i	$-0.0644046\pi$
$858$			2186.04i			0.0869814i
$859$			18270.1i			0.725691i	0.931849	+	0.362845i	$0.118195\pi$
$859$			18270.1i			0.725691i	−0.931849	+	0.362845i	$0.881805\pi$
$860$	10461.2			0.414795
$861$	−3958.76			−0.156695
$862$			1366.42i			0.0539911i
$863$			49114.1i			1.93727i	0.248487	+	0.968635i	$0.420067\pi$
$863$			49114.1i			1.93727i	−0.248487	+	0.968635i	$0.579933\pi$
$864$			1410.69i			0.0555472i
$865$		−	10792.1i		−	0.424212i
$866$	−4046.07			−0.158766
$867$	12420.0			0.486511
$868$			1996.19i			0.0780587i
$869$		−	14262.7i		−	0.556766i
$870$			1422.73i			0.0554427i
$871$	−21070.4			−0.819684
$872$			5892.44i			0.228834i
$873$		−	13615.5i		−	0.527853i
$874$	−1316.60			−0.0509549
$875$	38092.4			1.47172
$876$		−	6518.64i		−	0.251421i
$877$		−	49002.7i		−	1.88678i	−0.331691	−	0.943388i	$-0.607619\pi$
$877$		−	49002.7i		−	1.88678i	0.331691	−	0.943388i	$-0.392381\pi$
$878$	−159.144			−0.00611715
$879$		−	16366.6i		−	0.628023i
$880$		−	52071.4i		−	1.99469i
$881$		−	11265.5i		−	0.430809i	−0.976525	−	0.215405i	$-0.930893\pi$
$881$		−	11265.5i		−	0.430809i	0.976525	−	0.215405i	$-0.0691070\pi$
$882$		−	274.151i		−	0.0104662i
$883$		−	2131.79i		−	0.0812463i	−0.999175	−	0.0406232i	$-0.987066\pi$
$883$		−	2131.79i		−	0.0812463i	0.999175	−	0.0406232i	$-0.0129343\pi$
$884$	−13531.6			−0.514838
$885$	17350.9			0.659034
$886$	−4052.56			−0.153666
$887$			4087.04i			0.154712i	0.997004	+	0.0773559i	$0.0246478\pi$
$887$			4087.04i			0.154712i	−0.997004	+	0.0773559i	$0.975352\pi$
$888$			1923.10i			0.0726747i
$889$			17576.4i			0.663097i
$890$	1115.32			0.0420062
$891$	3486.77			0.131101
$892$		−	20592.3i		−	0.772962i
$893$	−41585.8			−1.55836
$894$	1103.30			0.0412750
$895$	−25343.2			−0.946515
$896$	−8402.97			−0.313307
$897$			7767.48i			0.289129i
$898$	−4510.33			−0.167608
$899$		−	1461.47i		−	0.0542187i
$900$	−18073.6			−0.669391
$901$			5340.26i			0.197458i
$902$	1026.77			0.0379023
$903$	−3104.22			−0.114399
$904$			5133.93i			0.188885i
$905$	8437.11			0.309899
$906$	−2369.10			−0.0868743
$907$	−30449.7			−1.11474			−0.557368	−	0.830266i	$-0.688189\pi$
$907$	−30449.7			−1.11474			−0.557368	+	0.830266i	$0.688189\pi$
$908$		−	10190.5i		−	0.372449i
$909$	15114.0			0.551486
$910$			5020.86i			0.182901i
$911$	−12326.8			−0.448305			−0.224152	−	0.974554i	$-0.571961\pi$
$911$	−12326.8			−0.448305			−0.224152	+	0.974554i	$0.571961\pi$
$912$	−21140.2			−0.767567
$913$		−	6271.85i		−	0.227347i
$914$			962.876i			0.0348458i
$915$	−44453.5			−1.60611
$916$			39747.4i			1.43372i
$917$	38534.9			1.38771
$918$		−	206.892i		−	0.00743839i
$919$			50482.8i			1.81205i	0.423223	+	0.906025i	$0.360899\pi$
$919$			50482.8i			1.81205i	−0.423223	+	0.906025i	$0.639101\pi$
$920$			3598.99i			0.128973i
$921$		−	23283.4i		−	0.833023i
$922$		−	3233.42i		−	0.115496i
$923$	−47988.1			−1.71132
$924$			15602.5i			0.555503i
$925$	−37016.4			−1.31578
$926$	−3037.17			−0.107783
$927$			13399.5i			0.474756i
$928$	4621.60			0.163482
$929$	−46595.6			−1.64559			−0.822795	−	0.568338i	$-0.807586\pi$
$929$	−46595.6			−1.64559			−0.822795	+	0.568338i	$0.807586\pi$
$930$	−265.746			−0.00937005
$931$	12525.0			0.440913
$932$	9253.36			0.325219
$933$	−31.8333			−0.00111701
$934$		−	4720.18i		−	0.165363i
$935$			23282.0i			0.814336i
$936$		−	2426.02i		−	0.0847189i
$937$		−	20055.8i		−	0.699246i	−0.936891	−	0.349623i	$-0.886310\pi$
$937$		−	20055.8i		−	0.699246i	0.936891	−	0.349623i	$-0.113690\pi$
$938$	1441.58			0.0501805
$939$	30538.4			1.06132
$940$			56567.4i			1.96279i
$941$	2521.30			0.0873456			0.0436728	−	0.999046i	$-0.486094\pi$
$941$	2521.30			0.0873456			0.0436728	+	0.999046i	$0.486094\pi$
$942$	−1587.36	−	354.726i	−0.0549035	−	0.0122692i
$943$	3648.36			0.125988
$944$		−	18487.6i		−	0.637415i
$945$	8008.36			0.275674
$946$	805.135			0.0276715
$947$		−	11798.4i		−	0.404855i	−0.979297	−	0.202428i	$-0.935117\pi$
$947$		−	11798.4i		−	0.404855i	0.979297	−	0.202428i	$-0.0648831\pi$
$948$			7876.48i			0.269848i
$949$			16842.2i			0.576103i
$950$			7915.15i			0.270317i
$951$	14411.1			0.491388
$952$	1860.46			0.0633381
$953$	37252.4			1.26624			0.633118	−	0.774055i	$-0.281775\pi$
$953$	37252.4			1.26624			0.633118	+	0.774055i	$0.281775\pi$
$954$	−476.432			−0.0161688
$955$	53914.3			1.82683
$956$	−47207.4			−1.59707
$957$		−	11423.1i		−	0.385846i
$958$	3710.62			0.125141
$959$	−39088.3			−1.31619
$960$			28191.4i			0.947783i
$961$	−29518.0			−0.990837
$962$		−	2472.51i		−	0.0828659i
$963$			12149.8i			0.406563i
$964$		−	34681.3i		−	1.15872i
$965$			12811.6i			0.427379i
$966$		−	531.429i		−	0.0177002i
$967$	−51903.0			−1.72605			−0.863024	−	0.505163i	$-0.831432\pi$
$967$	−51903.0			−1.72605			−0.863024	+	0.505163i	$0.831432\pi$
$968$		−	2290.93i		−	0.0760676i
$969$	9452.14			0.313361
$970$			8110.93i			0.268481i
$971$		−	38768.9i		−	1.28131i	−0.767828	−	0.640656i	$-0.778663\pi$
$971$		−	38768.9i		−	1.28131i	0.767828	−	0.640656i	$-0.221337\pi$
$972$	−1925.54			−0.0635409
$973$	3738.03			0.123161
$974$			809.129i			0.0266182i
$975$	46696.7			1.53384
$976$			47365.7i			1.55342i
$977$	−10445.7			−0.342055			−0.171028	−	0.985266i	$-0.554709\pi$
$977$	−10445.7			−0.342055			−0.171028	+	0.985266i	$0.554709\pi$
$978$	−2684.71			−0.0877788
$979$	−8954.83			−0.292337
$980$		−	17037.2i		−	0.555341i
$981$	−12083.6			−0.393272
$982$	3104.69			0.100891
$983$			47335.7i			1.53589i	0.640519	+	0.767943i	$0.278719\pi$
$983$			47335.7i			1.53589i	−0.640519	+	0.767943i	$0.721281\pi$
$984$	−1139.49			−0.0369164
$985$			6420.40i			0.207686i
$986$	−677.801			−0.0218921
$987$		−	16785.6i		−	0.541329i
$988$	55153.7			1.77599
$989$	2860.83			0.0919808
$990$	−2077.11			−0.0666817
$991$	−24953.6			−0.799876			−0.399938	−	0.916542i	$-0.630968\pi$
$991$	−24953.6			−0.799876			−0.399938	+	0.916542i	$0.630968\pi$
$992$			863.246i			0.0276291i
$993$	−25725.6			−0.822132
$994$	3283.21			0.104766
$995$		−	3832.62i		−	0.122113i
$996$			3463.58i			0.110189i
$997$		−	5755.42i		−	0.182824i	−0.995813	−	0.0914122i	$-0.970862\pi$
$997$		−	5755.42i		−	0.182824i	0.995813	−	0.0914122i	$-0.0291381\pi$
$998$	−2900.30			−0.0919914
$999$	−3943.70			−0.124898

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.4.b.a.313.20	✓	40
157.156	even	2	inner	471.4.b.a.313.21	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.4.b.a.313.20	✓	40	1.1	even	1	trivial
471.4.b.a.313.21	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)

\(f(q)\)	\(=\)	\(q-0.275605i q^{2} -3.00000 q^{3} +7.92404 q^{4} -19.4532i q^{5} +0.826816i q^{6} -15.2471i q^{7} -4.38875i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-0.275605i q^{2} -3.00000 q^{3} +7.92404 q^{4} -19.4532i q^{5} +0.826816i q^{6} -15.2471i q^{7} -4.38875i q^{8} +9.00000 q^{9} -5.36141 q^{10} +43.0465 q^{11} -23.7721 q^{12} +61.4201 q^{13} -4.20219 q^{14} +58.3597i q^{15} +62.1828 q^{16} -27.8030 q^{17} -2.48045i q^{18} +113.323 q^{19} -154.148i q^{20} +45.7414i q^{21} -11.8638i q^{22} -42.1549i q^{23} +13.1663i q^{24} -253.428 q^{25} -16.9277i q^{26} -27.0000 q^{27} -120.819i q^{28} +88.4551i q^{29} +16.0842 q^{30} -16.5221 q^{31} -52.2479i q^{32} -129.140 q^{33} +7.66265i q^{34} -296.606 q^{35} +71.3164 q^{36} +146.063 q^{37} -31.2324i q^{38} -184.260 q^{39} -85.3753 q^{40} +86.5465i q^{41} +12.6066 q^{42} +67.8646i q^{43} +341.102 q^{44} -175.079i q^{45} -11.6181 q^{46} -366.968 q^{47} -186.548 q^{48} +110.525 q^{49} +69.8460i q^{50} +83.4090 q^{51} +486.696 q^{52} -192.075i q^{53} +7.44134i q^{54} -837.393i q^{55} -66.9159 q^{56} -339.969 q^{57} +24.3787 q^{58} -297.310i q^{59} +462.444i q^{60} +761.717i q^{61} +4.55359i q^{62} -137.224i q^{63} +483.062 q^{64} -1194.82i q^{65} +35.5915i q^{66} -343.054 q^{67} -220.312 q^{68} +126.465i q^{69} +81.7461i q^{70} -781.308 q^{71} -39.4988i q^{72} +274.214i q^{73} -40.2557i q^{74} +760.283 q^{75} +897.975 q^{76} -656.336i q^{77} +50.7831i q^{78} -331.333i q^{79} -1209.66i q^{80} +81.0000 q^{81} +23.8527 q^{82} -145.699i q^{83} +362.457i q^{84} +540.858i q^{85} +18.7038 q^{86} -265.365i q^{87} -188.920i q^{88} -208.027 q^{89} -48.2527 q^{90} -936.481i q^{91} -334.037i q^{92} +49.5664 q^{93} +101.138i q^{94} -2204.49i q^{95} +156.744i q^{96} -1512.84i q^{97} -30.4613i q^{98} +387.419 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 120 q^{3} - 164 q^{4} + 360 q^{9}+O(q^{10}) \) 40 * q - 120 * q^3 - 164 * q^4 + 360 * q^9	\( 40 q - 120 q^{3} - 164 q^{4} + 360 q^{9} - 174 q^{10} + 110 q^{11} + 492 q^{12} - 194 q^{13} - 78 q^{14} + 796 q^{16} - 150 q^{17} + 172 q^{19} - 668 q^{25} - 1080 q^{27} + 522 q^{30} + 66 q^{31} - 330 q^{33} - 400 q^{35} - 1476 q^{36} - 142 q^{37} + 582 q^{39} + 1160 q^{40} + 234 q^{42} - 1182 q^{44} + 132 q^{46} - 244 q^{47} - 2388 q^{48} - 3786 q^{49} + 450 q^{51} + 1596 q^{52} - 256 q^{56} - 516 q^{57} - 1780 q^{58} - 1790 q^{64} - 320 q^{67} + 1646 q^{68} + 712 q^{71} + 2004 q^{75} - 3004 q^{76} + 3240 q^{81} + 4112 q^{82} - 4198 q^{86} + 366 q^{89} - 1566 q^{90} - 198 q^{93} + 990 q^{99}+O(q^{100}) \) 40 * q - 120 * q^3 - 164 * q^4 + 360 * q^9 - 174 * q^10 + 110 * q^11 + 492 * q^12 - 194 * q^13 - 78 * q^14 + 796 * q^16 - 150 * q^17 + 172 * q^19 - 668 * q^25 - 1080 * q^27 + 522 * q^30 + 66 * q^31 - 330 * q^33 - 400 * q^35 - 1476 * q^36 - 142 * q^37 + 582 * q^39 + 1160 * q^40 + 234 * q^42 - 1182 * q^44 + 132 * q^46 - 244 * q^47 - 2388 * q^48 - 3786 * q^49 + 450 * q^51 + 1596 * q^52 - 256 * q^56 - 516 * q^57 - 1780 * q^58 - 1790 * q^64 - 320 * q^67 + 1646 * q^68 + 712 * q^71 + 2004 * q^75 - 3004 * q^76 + 3240 * q^81 + 4112 * q^82 - 4198 * q^86 + 366 * q^89 - 1566 * q^90 - 198 * q^93 + 990 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more