LMFDB - Embedded newform 471.4.b.a.313.8

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			313.8
Character	$\chi$	$=$	471.313
Dual form			471.4.b.a.313.33

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.78788i q^{2} -3.00000 q^{3} -6.34802 q^{4} -9.26420i q^{5} +11.3636i q^{6} -4.30898i q^{7} -6.25748i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-3.78788i q^{2} -3.00000 q^{3} -6.34802 q^{4} -9.26420i q^{5} +11.3636i q^{6} -4.30898i q^{7} -6.25748i q^{8} +9.00000 q^{9} -35.0916 q^{10} +9.78893 q^{11} +19.0441 q^{12} +35.0642 q^{13} -16.3219 q^{14} +27.7926i q^{15} -74.4868 q^{16} -102.476 q^{17} -34.0909i q^{18} -92.8250 q^{19} +58.8093i q^{20} +12.9270i q^{21} -37.0793i q^{22} -177.997i q^{23} +18.7724i q^{24} +39.1747 q^{25} -132.819i q^{26} -27.0000 q^{27} +27.3535i q^{28} -88.6337i q^{29} +105.275 q^{30} -25.9566 q^{31} +232.087i q^{32} -29.3668 q^{33} +388.166i q^{34} -39.9193 q^{35} -57.1322 q^{36} -315.562 q^{37} +351.610i q^{38} -105.193 q^{39} -57.9705 q^{40} +81.5457i q^{41} +48.9657 q^{42} +301.218i q^{43} -62.1404 q^{44} -83.3778i q^{45} -674.231 q^{46} +342.173 q^{47} +223.460 q^{48} +324.433 q^{49} -148.389i q^{50} +307.427 q^{51} -222.589 q^{52} +38.5884i q^{53} +102.273i q^{54} -90.6866i q^{55} -26.9634 q^{56} +278.475 q^{57} -335.734 q^{58} +590.710i q^{59} -176.428i q^{60} +739.881i q^{61} +98.3204i q^{62} -38.7809i q^{63} +283.223 q^{64} -324.842i q^{65} +111.238i q^{66} -490.213 q^{67} +650.519 q^{68} +533.991i q^{69} +151.209i q^{70} -234.314 q^{71} -56.3173i q^{72} +39.4206i q^{73} +1195.31i q^{74} -117.524 q^{75} +589.255 q^{76} -42.1804i q^{77} +398.457i q^{78} +21.1853i q^{79} +690.060i q^{80} +81.0000 q^{81} +308.885 q^{82} -775.721i q^{83} -82.0606i q^{84} +949.356i q^{85} +1140.98 q^{86} +265.901i q^{87} -61.2541i q^{88} +198.127 q^{89} -315.825 q^{90} -151.091i q^{91} +1129.93i q^{92} +77.8698 q^{93} -1296.11i q^{94} +859.949i q^{95} -696.261i q^{96} +482.840i q^{97} -1228.91i q^{98} +88.1004 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	3.78788i		−	1.33922i	−0.742714	−	0.669609i	$-0.766462\pi$
$2$		−	3.78788i		−	1.33922i	0.742714	−	0.669609i	$-0.233538\pi$
$3$	−3.00000			−0.577350
$3$	−3.00000			−0.577350
$4$	−6.34802			−0.793503
$5$		−	9.26420i		−	0.828615i	−0.910137	−	0.414307i	$-0.864024\pi$
$5$		−	9.26420i		−	0.828615i	0.910137	−	0.414307i	$-0.135976\pi$
$6$			11.3636i			0.773197i
$7$		−	4.30898i		−	0.232663i	−0.993210	−	0.116332i	$-0.962886\pi$
$7$		−	4.30898i		−	0.232663i	0.993210	−	0.116332i	$-0.0371135\pi$
$8$		−	6.25748i		−	0.276544i
$9$	9.00000			0.333333
$10$	−35.0916			−1.10970
$11$	9.78893			0.268316			0.134158	−	0.990960i	$-0.457167\pi$
$11$	9.78893			0.268316			0.134158	+	0.990960i	$0.457167\pi$
$12$	19.0441			0.458129
$13$	35.0642			0.748082			0.374041	−	0.927412i	$-0.377972\pi$
$13$	35.0642			0.748082			0.374041	+	0.927412i	$0.377972\pi$
$14$	−16.3219			−0.311587
$15$			27.7926i			0.478401i
$16$	−74.4868			−1.16386
$17$	−102.476			−1.46200			−0.731001	−	0.682377i	$-0.760946\pi$
$17$	−102.476			−1.46200			−0.731001	+	0.682377i	$0.760946\pi$
$18$		−	34.0909i		−	0.446406i
$19$	−92.8250			−1.12082			−0.560408	−	0.828216i	$-0.689356\pi$
$19$	−92.8250			−1.12082			−0.560408	+	0.828216i	$0.689356\pi$
$20$			58.8093i			0.657508i
$21$			12.9270i			0.134328i
$22$		−	37.0793i		−	0.359333i
$23$		−	177.997i		−	1.61369i	−0.590760	−	0.806847i	$-0.701172\pi$
$23$		−	177.997i		−	1.61369i	0.590760	−	0.806847i	$-0.298828\pi$
$24$			18.7724i			0.159663i
$25$	39.1747			0.313397
$26$		−	132.819i		−	1.00184i
$27$	−27.0000			−0.192450
$28$			27.3535i			0.184619i
$29$		−	88.6337i		−	0.567547i	−0.958891	−	0.283774i	$-0.908414\pi$
$29$		−	88.6337i		−	0.567547i	0.958891	−	0.283774i	$-0.0915864\pi$
$30$	105.275			0.640683
$31$	−25.9566			−0.150385			−0.0751926	−	0.997169i	$-0.523957\pi$
$31$	−25.9566			−0.150385			−0.0751926	+	0.997169i	$0.523957\pi$
$32$			232.087i			1.28211i
$33$	−29.3668			−0.154912
$34$			388.166i			1.95794i
$35$	−39.9193			−0.192788
$36$	−57.1322			−0.264501
$37$	−315.562			−1.40211			−0.701055	−	0.713107i	$-0.747287\pi$
$37$	−315.562			−1.40211			−0.701055	+	0.713107i	$0.747287\pi$
$38$			351.610i			1.50102i
$39$	−105.193			−0.431905
$40$	−57.9705			−0.229149
$41$			81.5457i			0.310617i	0.987866	+	0.155308i	$0.0496372\pi$
$41$			81.5457i			0.310617i	−0.987866	+	0.155308i	$0.950363\pi$
$42$	48.9657			0.179895
$43$			301.218i			1.06826i	0.845402	+	0.534131i	$0.179361\pi$
$43$			301.218i			1.06826i	−0.845402	+	0.534131i	$0.820639\pi$
$44$	−62.1404			−0.212909
$45$		−	83.3778i		−	0.276205i
$46$	−674.231			−2.16109
$47$	342.173			1.06194			0.530968	−	0.847392i	$-0.321828\pi$
$47$	342.173			1.06194			0.530968	+	0.847392i	$0.321828\pi$
$48$	223.460			0.671953
$49$	324.433			0.945868
$50$		−	148.389i		−	0.419707i
$51$	307.427			0.844087
$52$	−222.589			−0.593605
$53$			38.5884i			0.100010i	0.998749	+	0.0500049i	$0.0159237\pi$
$53$			38.5884i			0.100010i	−0.998749	+	0.0500049i	$0.984076\pi$
$54$			102.273i			0.257732i
$55$		−	90.6866i		−	0.222331i
$56$	−26.9634			−0.0643417
$57$	278.475			0.647104
$58$	−335.734			−0.760069
$59$			590.710i			1.30346i	0.758453	+	0.651728i	$0.225955\pi$
$59$			590.710i			1.30346i	−0.758453	+	0.651728i	$0.774045\pi$
$60$		−	176.428i		−	0.379613i
$61$			739.881i			1.55298i	0.630127	+	0.776492i	$0.283003\pi$
$61$			739.881i			1.55298i	−0.630127	+	0.776492i	$0.716997\pi$
$62$			98.3204i			0.201399i
$63$		−	38.7809i		−	0.0775544i
$64$	283.223			0.553170
$65$		−	324.842i		−	0.619872i
$66$			111.238i			0.207461i
$67$	−490.213			−0.893866			−0.446933	−	0.894567i	$-0.647484\pi$
$67$	−490.213			−0.893866			−0.446933	+	0.894567i	$0.647484\pi$
$68$	650.519			1.16010
$69$			533.991i			0.931667i
$70$			151.209i			0.258185i
$71$	−234.314			−0.391662			−0.195831	−	0.980638i	$-0.562740\pi$
$71$	−234.314			−0.391662			−0.195831	+	0.980638i	$0.562740\pi$
$72$		−	56.3173i		−	0.0921814i
$73$			39.4206i			0.0632032i	0.999501	+	0.0316016i	$0.0100608\pi$
$73$			39.4206i			0.0632032i	−0.999501	+	0.0316016i	$0.989939\pi$
$74$			1195.31i			1.87773i
$75$	−117.524			−0.180940
$76$	589.255			0.889371
$77$		−	42.1804i		−	0.0624273i
$78$			398.457i			0.578415i
$79$			21.1853i			0.0301713i	0.999886	+	0.0150857i	$0.00480210\pi$
$79$			21.1853i			0.0301713i	−0.999886	+	0.0150857i	$0.995198\pi$
$80$			690.060i			0.964388i
$81$	81.0000			0.111111
$82$	308.885			0.415984
$83$		−	775.721i		−	1.02586i	−0.858430	−	0.512930i	$-0.828560\pi$
$83$		−	775.721i		−	1.02586i	0.858430	−	0.512930i	$-0.171440\pi$
$84$		−	82.0606i		−	0.106590i
$85$			949.356i			1.21144i
$86$	1140.98			1.43064
$87$			265.901i			0.327674i
$88$		−	61.2541i		−	0.0742012i
$89$	198.127			0.235972			0.117986	−	0.993015i	$-0.462356\pi$
$89$	198.127			0.235972			0.117986	+	0.993015i	$0.462356\pi$
$90$	−315.825			−0.369898
$91$		−	151.091i		−	0.174051i
$92$			1129.93i			1.28047i
$93$	77.8698			0.0868250
$94$		−	1296.11i		−	1.42216i
$95$			859.949i			0.928725i
$96$		−	696.261i		−	0.740228i
$97$			482.840i			0.505412i	0.967543	+	0.252706i	$0.0813205\pi$
$97$			482.840i			0.505412i	−0.967543	+	0.252706i	$0.918680\pi$
$98$		−	1228.91i		−	1.26672i
$99$	88.1004			0.0894386
$100$	−248.682			−0.248682
$101$	1464.20			1.44251			0.721256	−	0.692669i	$-0.243565\pi$
$101$	1464.20			1.44251			0.721256	+	0.692669i	$0.243565\pi$
$102$		−	1164.50i		−	1.13042i
$103$		−	1431.03i		−	1.36897i	−0.729027	−	0.684485i	$-0.760027\pi$
$103$		−	1431.03i		−	1.36897i	0.729027	−	0.684485i	$-0.239973\pi$
$104$		−	219.414i		−	0.206878i
$105$	119.758			0.111306
$106$	146.168			0.133935
$107$			66.2092i			0.0598194i	0.999553	+	0.0299097i	$0.00952198\pi$
$107$			66.2092i			0.0598194i	−0.999553	+	0.0299097i	$0.990478\pi$
$108$	171.397			0.152710
$109$	−2106.11			−1.85072			−0.925361	−	0.379088i	$-0.876238\pi$
$109$	−2106.11			−1.85072			−0.925361	+	0.379088i	$0.876238\pi$
$110$	−343.510			−0.297749
$111$	946.686			0.809509
$112$			320.962i			0.270787i
$113$	−898.368			−0.747888			−0.373944	−	0.927451i	$-0.621995\pi$
$113$	−898.368			−0.747888			−0.373944	+	0.927451i	$0.621995\pi$
$114$		−	1054.83i		−	0.866613i
$115$	−1649.00			−1.33713
$116$			562.649i			0.450350i
$117$	315.578			0.249361
$118$	2237.54			1.74561
$119$			441.566i			0.340154i
$120$	173.912			0.132299
$121$	−1235.18			−0.928007
$122$	2802.58			2.07978
$123$		−	244.637i		−	0.179335i
$124$	164.773			0.119331
$125$		−	1520.95i		−	1.08830i
$126$	−146.897			−0.103862
$127$	2218.49			1.55007			0.775036	−	0.631917i	$-0.217732\pi$
$127$	2218.49			1.55007			0.775036	+	0.631917i	$0.217732\pi$
$128$			783.881i			0.541296i
$129$		−	903.653i		−	0.616762i
$130$	−1230.46			−0.830143
$131$		−	304.479i		−	0.203072i	−0.994832	−	0.101536i	$-0.967624\pi$
$131$		−	304.479i		−	0.203072i	0.994832	−	0.101536i	$-0.0323758\pi$
$132$	186.421			0.122923
$133$			399.981i			0.260773i
$134$			1856.87i			1.19708i
$135$			250.133i			0.159467i
$136$			641.240i			0.404308i
$137$			659.721i			0.411414i	0.978614	+	0.205707i	$0.0659494\pi$
$137$			659.721i			0.411414i	−0.978614	+	0.205707i	$0.934051\pi$
$138$	2022.69			1.24770
$139$			702.314i			0.428558i	0.976773	+	0.214279i	$0.0687401\pi$
$139$			702.314i			0.428558i	−0.976773	+	0.214279i	$0.931260\pi$
$140$	253.409			0.152978
$141$	−1026.52			−0.613110
$142$			887.554i			0.524520i
$143$	343.241			0.200722
$144$	−670.381			−0.387952
$145$	−821.120			−0.470278
$146$	149.320			0.0846428
$147$	−973.298			−0.546097
$148$	2003.20			1.11258
$149$		−	244.879i		−	0.134640i	−0.997731	−	0.0673198i	$-0.978555\pi$
$149$		−	244.879i		−	0.134640i	0.997731	−	0.0673198i	$-0.0214448\pi$
$150$			445.167i			0.242318i
$151$		−	3259.52i		−	1.75666i	−0.478054	−	0.878331i	$-0.658658\pi$
$151$		−	3259.52i		−	1.75666i	0.478054	−	0.878331i	$-0.341342\pi$
$152$			580.851i			0.309955i
$153$	−922.282			−0.487334
$154$	−159.774			−0.0836037
$155$			240.467i			0.124611i
$156$	667.766			0.342718
$157$	353.559	−	1935.17i	0.179727	−	0.983717i
$157$	353.559	−	1935.17i	0.179727	−	0.983717i
$158$	80.2474			0.0404060
$159$		−	115.765i		−	0.0577407i
$160$	2150.10			1.06238
$161$	−766.987			−0.375447
$162$		−	306.818i		−	0.148802i
$163$		−	1314.52i		−	0.631664i	−0.948815	−	0.315832i	$-0.897716\pi$
$163$		−	1314.52i		−	0.631664i	0.948815	−	0.315832i	$-0.102284\pi$
$164$		−	517.654i		−	0.246476i
$165$			272.060i			0.128363i
$166$	−2938.34			−1.37385
$167$	−1257.81			−0.582828			−0.291414	−	0.956597i	$-0.594126\pi$
$167$	−1257.81			−0.582828			−0.291414	+	0.956597i	$0.594126\pi$
$168$	80.8902			0.0371477
$169$	−967.500			−0.440373
$170$	3596.04			1.62238
$171$	−835.425			−0.373606
$172$		−	1912.14i		−	0.847670i
$173$	−3727.21			−1.63800			−0.819001	−	0.573792i	$-0.805472\pi$
$173$	−3727.21			−1.63800			−0.819001	+	0.573792i	$0.805472\pi$
$174$	1007.20			0.438826
$175$		−	168.803i		−	0.0729161i
$176$	−729.146			−0.312281
$177$		−	1772.13i		−	0.752550i
$178$		−	750.483i		−	0.316017i
$179$		−	244.841i		−	0.102236i	−0.998693	−	0.0511181i	$-0.983722\pi$
$179$		−	244.841i		−	0.102236i	0.998693	−	0.0511181i	$-0.0162785\pi$
$180$			529.284i			0.219169i
$181$			100.875i			0.0414253i	0.999785	+	0.0207127i	$0.00659352\pi$
$181$			100.875i			0.0414253i	−0.999785	+	0.0207127i	$0.993406\pi$
$182$	−572.315			−0.233092
$183$		−	2219.64i		−	0.896615i
$184$	−1113.81			−0.446258
$185$			2923.43i			1.16181i
$186$		−	294.961i		−	0.116277i
$187$	−1003.13			−0.392278
$188$	−2172.12			−0.842650
$189$			116.343i			0.0447761i
$190$	3257.38			1.24376
$191$			2595.88i			0.983410i	0.870762	+	0.491705i	$0.163626\pi$
$191$			2595.88i			0.983410i	−0.870762	+	0.491705i	$0.836374\pi$
$192$	−849.670			−0.319373
$193$	−4100.81			−1.52944			−0.764722	−	0.644360i	$-0.777124\pi$
$193$	−4100.81			−1.52944			−0.764722	+	0.644360i	$0.777124\pi$
$194$	1828.94			0.676856
$195$			974.526i			0.357883i
$196$	−2059.51			−0.750549
$197$	5076.44			1.83595			0.917974	−	0.396641i	$-0.129824\pi$
$197$	5076.44			1.83595			0.917974	+	0.396641i	$0.129824\pi$
$198$		−	333.714i		−	0.119778i
$199$	−3916.13			−1.39501			−0.697506	−	0.716579i	$-0.745707\pi$
$199$	−3916.13			−1.39501			−0.697506	+	0.716579i	$0.745707\pi$
$200$		−	245.135i		−	0.0866683i
$201$	1470.64			0.516074
$202$		−	5546.23i		−	1.93184i
$203$	−381.921			−0.132047
$204$	−1951.56			−0.669786
$205$	755.455			0.257382
$206$	−5420.58			−1.83335
$207$		−	1601.97i		−	0.537898i
$208$	−2611.82			−0.870660
$209$	−908.658			−0.300733
$210$		−	453.628i		−	0.149063i
$211$			1571.41i			0.512705i	0.966583	+	0.256352i	$0.0825207\pi$
$211$			1571.41i			0.512705i	−0.966583	+	0.256352i	$0.917479\pi$
$212$		−	244.960i		−	0.0793581i
$213$	702.943			0.226126
$214$	250.792			0.0801112
$215$	2790.54			0.885178
$216$			168.952i			0.0532210i
$217$			111.847i			0.0349891i
$218$			7977.68i			2.47852i
$219$		−	118.262i		−	0.0364904i
$220$			575.681i			0.176420i
$221$	−3593.23			−1.09370
$222$		−	3585.93i		−	1.08411i
$223$		−	5711.70i		−	1.71517i	−0.514339	−	0.857587i	$-0.671963\pi$
$223$		−	5711.70i		−	1.71517i	0.514339	−	0.857587i	$-0.328037\pi$
$224$	1000.06			0.298300
$225$	352.572			0.104466
$226$			3402.91i			1.00158i
$227$			1444.26i			0.422286i	0.977455	+	0.211143i	$0.0677187\pi$
$227$			1444.26i			0.422286i	−0.977455	+	0.211143i	$0.932281\pi$
$228$	−1767.77			−0.513479
$229$		−	207.743i		−	0.0599477i	−0.999551	−	0.0299739i	$-0.990458\pi$
$229$		−	207.743i		−	0.0599477i	0.999551	−	0.0299739i	$-0.00954240\pi$
$230$			6246.21i			1.79071i
$231$			126.541i			0.0360424i
$232$	−554.624			−0.156952
$233$	−5359.63			−1.50696			−0.753478	−	0.657473i	$-0.771625\pi$
$233$	−5359.63			−1.50696			−0.753478	+	0.657473i	$0.771625\pi$
$234$		−	1195.37i		−	0.333948i
$235$		−	3169.96i		−	0.879937i
$236$		−	3749.84i		−	1.03430i
$237$		−	63.5560i		−	0.0174194i
$238$	1672.60			0.455540
$239$	−881.970			−0.238702			−0.119351	−	0.992852i	$-0.538081\pi$
$239$	−881.970			−0.238702			−0.119351	+	0.992852i	$0.538081\pi$
$240$		−	2070.18i		−	0.556790i
$241$			6421.81i			1.71645i	0.513272	+	0.858226i	$0.328433\pi$
$241$			6421.81i			1.71645i	−0.513272	+	0.858226i	$0.671567\pi$
$242$			4678.70i			1.24280i
$243$	−243.000			−0.0641500
$244$		−	4696.78i		−	1.23230i
$245$		−	3005.61i		−	0.783760i
$246$	−926.655			−0.240168
$247$	−3254.84			−0.838463
$248$			162.423i			0.0415882i
$249$			2327.16i			0.592281i
$250$	−5761.16			−1.45747
$251$		−	3179.41i		−	0.799533i	−0.916617	−	0.399767i	$-0.869091\pi$
$251$		−	3179.41i		−	0.799533i	0.916617	−	0.399767i	$-0.130909\pi$
$252$			246.182i			0.0615397i
$253$		−	1742.40i		−	0.432980i
$254$		−	8403.37i		−	2.07588i
$255$		−	2848.07i		−	0.699423i
$256$	5235.03			1.27808
$257$	1358.30			0.329682			0.164841	−	0.986320i	$-0.447289\pi$
$257$	1358.30			0.329682			0.164841	+	0.986320i	$0.447289\pi$
$258$	−3422.93			−0.825978
$259$			1359.75i			0.326220i
$260$			2062.10i			0.491870i
$261$		−	797.703i		−	0.189182i
$262$	−1153.33			−0.271958
$263$	−3670.54			−0.860589			−0.430295	−	0.902689i	$-0.641590\pi$
$263$	−3670.54			−0.860589			−0.430295	+	0.902689i	$0.641590\pi$
$264$			183.762i			0.0428401i
$265$	357.490			0.0828697
$266$	1515.08			0.349232
$267$	−594.382			−0.136238
$268$	3111.88			0.709286
$269$		−	5975.95i		−	1.35450i	−0.735754	−	0.677249i	$-0.763172\pi$
$269$		−	5975.95i		−	1.35450i	0.735754	−	0.677249i	$-0.236828\pi$
$270$	947.475			0.213561
$271$		−	4194.55i		−	0.940224i	−0.882607	−	0.470112i	$-0.844214\pi$
$271$		−	4194.55i		−	0.940224i	0.882607	−	0.470112i	$-0.155786\pi$
$272$	7633.09			1.70156
$273$			453.274i			0.100489i
$274$	2498.94			0.550973
$275$	383.478			0.0840895
$276$		−	3389.79i		−	0.739280i
$277$	5254.41			1.13974			0.569868	−	0.821737i	$-0.306995\pi$
$277$	5254.41			1.13974			0.569868	+	0.821737i	$0.306995\pi$
$278$	2660.28			0.573932
$279$	−233.609			−0.0501284
$280$			249.794i			0.0533145i
$281$	−176.537			−0.0374781			−0.0187390	−	0.999824i	$-0.505965\pi$
$281$	−176.537			−0.0374781			−0.0187390	+	0.999824i	$0.505965\pi$
$282$			3888.33i			0.821087i
$283$	4715.28			0.990439			0.495219	−	0.868768i	$-0.335088\pi$
$283$	4715.28			0.990439			0.495219	+	0.868768i	$0.335088\pi$
$284$	1487.43			0.310785
$285$		−	2579.85i		−	0.536200i
$286$		−	1300.16i		−	0.268811i
$287$	351.379			0.0722692
$288$			2088.78i			0.427371i
$289$	5588.28			1.13745
$290$			3110.30i			0.629804i
$291$		−	1448.52i		−	0.291800i
$292$		−	250.243i		−	0.0501519i
$293$		−	2651.34i		−	0.528646i	−0.964434	−	0.264323i	$-0.914852\pi$
$293$		−	2651.34i		−	0.528646i	0.964434	−	0.264323i	$-0.0851484\pi$
$294$			3686.73i			0.731343i
$295$	5472.45			1.08006
$296$			1974.62i			0.387746i
$297$	−264.301			−0.0516374
$298$	−927.573			−0.180312
$299$		−	6241.33i		−	1.20718i
$300$	746.045			0.143577
$301$	1297.94			0.248545
$302$	−12346.7			−2.35255
$303$	−4392.61			−0.832835
$304$	6914.24			1.30447
$305$	6854.40			1.28683
$306$			3493.49i			0.652646i
$307$		−	3967.78i		−	0.737633i	−0.929502	−	0.368817i	$-0.879763\pi$
$307$		−	3967.78i		−	0.737633i	0.929502	−	0.368817i	$-0.120237\pi$
$308$			267.762i			0.0495362i
$309$			4293.10i			0.790375i
$310$	910.860			0.166882
$311$	2624.74			0.478570			0.239285	−	0.970949i	$-0.423087\pi$
$311$	2624.74			0.478570			0.239285	+	0.970949i	$0.423087\pi$
$312$			658.241i			0.119441i
$313$	1943.23			0.350920			0.175460	−	0.984487i	$-0.443859\pi$
$313$	1943.23			0.350920			0.175460	+	0.984487i	$0.443859\pi$
$314$	−7330.20	−	1339.24i	−1.31741	−	0.240693i
$315$	−359.273			−0.0642627
$316$		−	134.485i		−	0.0239410i
$317$	−2024.78			−0.358747			−0.179374	−	0.983781i	$-0.557407\pi$
$317$	−2024.78			−0.358747			−0.179374	+	0.983781i	$0.557407\pi$
$318$	−438.504			−0.0773274
$319$		−	867.630i		−	0.152282i
$320$		−	2623.84i		−	0.458365i
$321$		−	198.627i		−	0.0345368i
$322$			2905.25i			0.502806i
$323$	9512.31			1.63864
$324$	−514.190			−0.0881670
$325$	1373.63			0.234447
$326$	−4979.25			−0.845936
$327$	6318.32			1.06851
$328$	510.271			0.0858993
$329$		−	1474.42i		−	0.247074i
$330$	1030.53			0.171905
$331$	−8402.00			−1.39521			−0.697607	−	0.716481i	$-0.745752\pi$
$331$	−8402.00			−1.39521			−0.697607	+	0.716481i	$0.745752\pi$
$332$			4924.29i			0.814023i
$333$	−2840.06			−0.467370
$334$			4764.43i			0.780533i
$335$			4541.43i			0.740671i
$336$		−	962.887i		−	0.156339i
$337$			6838.80i			1.10544i	0.833367	+	0.552720i	$0.186410\pi$
$337$			6838.80i			1.10544i	−0.833367	+	0.552720i	$0.813590\pi$
$338$			3664.77i			0.589755i
$339$	2695.10			0.431793
$340$		−	6026.53i		−	0.961278i
$341$	−254.087			−0.0403507
$342$			3164.49i			0.500339i
$343$		−	2875.96i		−	0.452732i
$344$	1884.87			0.295422
$345$	4947.00			0.771993
$346$			14118.2i			2.19364i
$347$	7738.48			1.19719			0.598593	−	0.801054i	$-0.295727\pi$
$347$	7738.48			1.19719			0.598593	+	0.801054i	$0.295727\pi$
$348$		−	1687.95i		−	0.260010i
$349$	3530.63			0.541519			0.270760	−	0.962647i	$-0.412725\pi$
$349$	3530.63			0.541519			0.270760	+	0.962647i	$0.412725\pi$
$350$	−639.406			−0.0976505
$351$	−946.734			−0.143968
$352$			2271.88i			0.344011i
$353$	3139.66			0.473392			0.236696	−	0.971584i	$-0.423935\pi$
$353$	3139.66			0.473392			0.236696	+	0.971584i	$0.423935\pi$
$354$	−6712.61			−1.00783
$355$			2170.73i			0.324537i
$356$	−1257.72			−0.187244
$357$		−	1324.70i		−	0.196388i
$358$	−927.429			−0.136917
$359$		−	5569.51i		−	0.818795i	−0.912356	−	0.409398i	$-0.865739\pi$
$359$		−	5569.51i		−	0.818795i	0.912356	−	0.409398i	$-0.134261\pi$
$360$	−521.735			−0.0763829
$361$	1757.48			0.256230
$362$	382.103			0.0554775
$363$	3705.53			0.535785
$364$			959.131i			0.138110i
$365$	365.200			0.0523711
$366$	−8407.73			−1.20076
$367$		−	8896.10i		−	1.26532i	−0.774430	−	0.632660i	$-0.781963\pi$
$367$		−	8896.10i		−	1.26532i	0.774430	−	0.632660i	$-0.218037\pi$
$368$			13258.4i			1.87811i
$369$			733.911i			0.103539i
$370$	11073.6			1.55592
$371$	166.277			0.0232686
$372$	−494.319			−0.0688959
$373$		−	10206.4i		−	1.41680i	−0.705813	−	0.708398i	$-0.749418\pi$
$373$		−	10206.4i		−	1.41680i	0.705813	−	0.708398i	$-0.250582\pi$
$374$			3799.73i			0.525346i
$375$			4562.84i			0.628331i
$376$		−	2141.14i		−	0.293673i
$377$		−	3107.87i		−	0.424572i
$378$	440.692			0.0599649
$379$		−	1246.32i		−	0.168915i	−0.996427	−	0.0844577i	$-0.973084\pi$
$379$		−	1246.32i		−	0.168915i	0.996427	−	0.0844577i	$-0.0269158\pi$
$380$		−	5458.98i		−	0.736946i
$381$	−6655.47			−0.894934
$382$	9832.88			1.31700
$383$		−	1669.22i		−	0.222697i	−0.993781	−	0.111349i	$-0.964483\pi$
$383$		−	1669.22i		−	0.222697i	0.993781	−	0.111349i	$-0.0355170\pi$
$384$		−	2351.64i		−	0.312518i
$385$	−390.767			−0.0517281
$386$			15533.4i			2.04826i
$387$			2710.96i			0.356087i
$388$		−	3065.08i		−	0.401046i
$389$	−9483.42			−1.23606			−0.618031	−	0.786153i	$-0.712070\pi$
$389$	−9483.42			−1.23606			−0.618031	+	0.786153i	$0.712070\pi$
$390$	3691.38			0.479283
$391$			18240.4i			2.35922i
$392$		−	2030.13i		−	0.261574i
$393$			913.438i			0.117244i
$394$		−	19229.0i		−	2.45873i
$395$	196.265			0.0250004
$396$	−559.264			−0.0709698
$397$			12749.5i			1.61178i	0.592062	+	0.805892i	$0.298314\pi$
$397$			12749.5i			1.61178i	−0.592062	+	0.805892i	$0.701686\pi$
$398$			14833.8i			1.86822i
$399$		−	1199.94i		−	0.150557i
$400$	−2918.00			−0.364749
$401$		−	7624.82i		−	0.949539i	−0.880110	−	0.474769i	$-0.842531\pi$
$401$		−	7624.82i		−	0.949539i	0.880110	−	0.474769i	$-0.157469\pi$
$402$		−	5570.60i		−	0.691135i
$403$	−910.148			−0.112501
$404$	−9294.80			−1.14464
$405$		−	750.400i		−	0.0920683i
$406$			1446.67i			0.176840i
$407$	−3089.02			−0.376208
$408$		−	1923.72i		−	0.233427i
$409$			15937.7i			1.92682i	0.268026	+	0.963412i	$0.413629\pi$
$409$			15937.7i			1.92682i	−0.268026	+	0.963412i	$0.586371\pi$
$410$		−	2861.57i		−	0.344690i
$411$		−	1979.16i		−	0.237530i
$412$			9084.23i			1.08628i
$413$	2545.36			0.303266
$414$	−6068.08			−0.720362
$415$	−7186.43			−0.850043
$416$			8137.95i			0.959125i
$417$		−	2106.94i		−	0.247428i
$418$			3441.89i			0.402747i
$419$	−10612.1			−1.23731			−0.618656	−	0.785662i	$-0.712323\pi$
$419$	−10612.1			−1.23731			−0.618656	+	0.785662i	$0.712323\pi$
$420$	−760.226			−0.0883219
$421$		−	16152.3i		−	1.86987i	−0.354817	−	0.934936i	$-0.615457\pi$
$421$		−	16152.3i		−	1.86987i	0.354817	−	0.934936i	$-0.384543\pi$
$422$	5952.33			0.686623
$423$	3079.56			0.353979
$424$	241.466			0.0276572
$425$	−4014.46			−0.458188
$426$		−	2662.66i		−	0.302832i
$427$	3188.13			0.361322
$428$		−	420.297i		−	0.0474669i
$429$	−1029.72			−0.115887
$430$		−	10570.2i		−	1.18545i
$431$	14923.0			1.66779			0.833894	−	0.551925i	$-0.186107\pi$
$431$	14923.0			1.66779			0.833894	+	0.551925i	$0.186107\pi$
$432$	2011.14			0.223984
$433$		−	2212.49i		−	0.245555i	−0.992434	−	0.122778i	$-0.960820\pi$
$433$		−	2212.49i		−	0.245555i	0.992434	−	0.122778i	$-0.0391802\pi$
$434$	423.661			0.0468580
$435$	2463.36			0.271515
$436$	13369.6			1.46855
$437$			16522.6i			1.80866i
$438$	−447.961			−0.0488686
$439$		−	14414.0i		−	1.56706i	−0.621351	−	0.783532i	$-0.713416\pi$
$439$		−	14414.0i		−	1.56706i	0.621351	−	0.783532i	$-0.286584\pi$
$440$	−567.470			−0.0614842
$441$	2919.89			0.315289
$442$			13610.7i			1.46470i
$443$		−	9896.63i		−	1.06141i	−0.847558	−	0.530703i	$-0.821928\pi$
$443$		−	9896.63i		−	1.06141i	0.847558	−	0.530703i	$-0.178072\pi$
$444$	−6009.59			−0.642348
$445$		−	1835.49i		−	0.195530i
$446$	−21635.2			−2.29699
$447$			734.638i			0.0777342i
$448$		−	1220.40i		−	0.128702i
$449$			4171.88i			0.438493i	0.975669	+	0.219247i	$0.0703599\pi$
$449$			4171.88i			0.438493i	−0.975669	+	0.219247i	$0.929640\pi$
$450$		−	1335.50i		−	0.139902i
$451$			798.245i			0.0833435i
$452$	5702.86			0.593451
$453$			9778.56i			1.01421i
$454$	5470.69			0.565533
$455$	−1399.74			−0.144221
$456$		−	1742.55i		−	0.178953i
$457$	−2357.23			−0.241284			−0.120642	−	0.992696i	$-0.538495\pi$
$457$	−2357.23			−0.241284			−0.120642	+	0.992696i	$0.538495\pi$
$458$	−786.904			−0.0802830
$459$	2766.85			0.281362
$460$	10467.9			1.06102
$461$	4888.50			0.493883			0.246941	−	0.969030i	$-0.420574\pi$
$461$	4888.50			0.493883			0.246941	+	0.969030i	$0.420574\pi$
$462$	479.322			0.0482686
$463$			1965.72i			0.197311i	0.995122	+	0.0986553i	$0.0314541\pi$
$463$			1965.72i			0.197311i	−0.995122	+	0.0986553i	$0.968546\pi$
$464$			6602.04i			0.660543i
$465$		−	721.401i		−	0.0719444i
$466$			20301.6i			2.01814i
$467$	2586.82			0.256325			0.128162	−	0.991753i	$-0.459092\pi$
$467$	2586.82			0.256325			0.128162	+	0.991753i	$0.459092\pi$
$468$	−2003.30			−0.197868
$469$			2112.32i			0.207970i
$470$	−12007.4			−1.17843
$471$	−1060.68	+	5805.51i	−0.103765	+	0.567949i
$472$	3696.36			0.360463
$473$			2948.60i			0.286632i
$474$	−240.742			−0.0233284
$475$	−3636.39			−0.351261
$476$		−	2803.07i		−	0.269913i
$477$			347.296i			0.0333366i
$478$			3340.80i			0.319675i
$479$		−	10714.9i		−	1.02208i	−0.859556	−	0.511041i	$-0.829260\pi$
$479$		−	10714.9i		−	1.02208i	0.859556	−	0.511041i	$-0.170740\pi$
$480$	−6450.30			−0.613364
$481$	−11064.9			−1.04889
$482$	24325.0			2.29870
$483$	2300.96			0.216765
$484$	7840.93			0.736376
$485$	4473.12			0.418792
$486$			920.455i			0.0859108i
$487$	−5787.42			−0.538507			−0.269254	−	0.963069i	$-0.586777\pi$
$487$	−5787.42			−0.538507			−0.269254	+	0.963069i	$0.586777\pi$
$488$	4629.79			0.429469
$489$			3943.57i			0.364692i
$490$	−11384.9			−1.04963
$491$		−	6688.38i		−	0.614750i	−0.951588	−	0.307375i	$-0.900549\pi$
$491$		−	6688.38i		−	0.614750i	0.951588	−	0.307375i	$-0.0994506\pi$
$492$			1552.96i			0.142303i
$493$			9082.81i			0.829755i
$494$			12328.9i			1.12288i
$495$		−	816.179i		−	0.0741102i
$496$	1933.42			0.175027
$497$			1009.66i			0.0911253i
$498$	8815.01			0.793193
$499$			16806.6i			1.50775i	0.657017	+	0.753876i	$0.271818\pi$
$499$			16806.6i			1.50775i	−0.657017	+	0.753876i	$0.728182\pi$
$500$			9655.00i			0.863570i
$501$	3773.43			0.336496
$502$	−12043.2			−1.07075
$503$		−	15662.6i		−	1.38839i	−0.719787	−	0.694195i	$-0.755761\pi$
$503$		−	15662.6i		−	1.38839i	0.719787	−	0.694195i	$-0.244239\pi$
$504$	−242.671			−0.0214472
$505$		−	13564.7i		−	1.19529i
$506$	−6600.01			−0.579854
$507$	2902.50			0.254250
$508$	−14083.0			−1.22999
$509$		−	15819.0i		−	1.37754i	−0.724981	−	0.688768i	$-0.758152\pi$
$509$		−	15819.0i		−	1.37754i	0.724981	−	0.688768i	$-0.241848\pi$
$510$	−10788.1			−0.936679
$511$	169.863			0.0147051
$512$		−	13558.6i		−	1.17034i
$513$	2506.27			0.215701
$514$		−	5145.06i		−	0.441516i
$515$	−13257.4			−1.13435
$516$			5736.41i			0.489402i
$517$	3349.51			0.284935
$518$	5150.58			0.436879
$519$	11181.6			0.945701
$520$	−2032.69			−0.171422
$521$			915.345i			0.0769712i	0.999259	+	0.0384856i	$0.0122534\pi$
$521$			915.345i			0.0769712i	−0.999259	+	0.0384856i	$0.987747\pi$
$522$	−3021.60			−0.253356
$523$	−11224.6			−0.938462			−0.469231	−	0.883075i	$-0.655469\pi$
$523$	−11224.6			−0.938462			−0.469231	+	0.883075i	$0.655469\pi$
$524$			1932.84i			0.161139i
$525$			506.409i			0.0420981i
$526$			13903.5i			1.15252i
$527$	2659.92			0.219863
$528$	2187.44			0.180296
$529$	−19516.0			−1.60401
$530$		−	1354.13i		−	0.110981i
$531$			5316.39i			0.434485i
$532$		−	2539.09i		−	0.206924i
$533$			2859.34i			0.232367i
$534$			2251.45i			0.182453i
$535$	613.375			0.0495673
$536$			3067.50i			0.247194i
$537$			734.523i			0.0590261i
$538$	−22636.2			−1.81397
$539$	3175.85			0.253791
$540$		−	1587.85i		−	0.126538i
$541$			6768.33i			0.537880i	0.963157	+	0.268940i	$0.0866734\pi$
$541$			6768.33i			0.537880i	−0.963157	+	0.268940i	$0.913327\pi$
$542$	−15888.4			−1.25916
$543$		−	302.625i		−	0.0239169i
$544$		−	23783.3i		−	1.87445i
$545$			19511.4i			1.53353i
$546$	1716.95			0.134576
$547$	6092.05			0.476192			0.238096	−	0.971242i	$-0.423477\pi$
$547$	6092.05			0.476192			0.238096	+	0.971242i	$0.423477\pi$
$548$		−	4187.92i		−	0.326458i
$549$			6658.92i			0.517661i
$550$		−	1452.57i		−	0.112614i
$551$			8227.42i			0.636116i
$552$	3341.44			0.257647
$553$	91.2872			0.00701976
$554$		−	19903.0i		−	1.52635i
$555$		−	8770.29i		−	0.670771i
$556$		−	4458.31i		−	0.340062i
$557$	−14459.9			−1.09997			−0.549987	−	0.835173i	$-0.685367\pi$
$557$	−14459.9			−1.09997			−0.549987	+	0.835173i	$0.685367\pi$
$558$			884.884i			0.0671328i
$559$			10562.0i			0.799148i
$560$	2973.46			0.224378
$561$	3009.39			0.226482
$562$			668.702i			0.0501913i
$563$		−	2155.98i		−	0.161392i	−0.996739	−	0.0806962i	$-0.974286\pi$
$563$		−	2155.98i		−	0.161392i	0.996739	−	0.0806962i	$-0.0257143\pi$
$564$	6516.36			0.486504
$565$			8322.65i			0.619711i
$566$		−	17860.9i		−	1.32641i
$567$		−	349.028i		−	0.0258515i
$568$			1466.22i			0.108312i
$569$			7574.39i			0.558058i	0.960283	+	0.279029i	$0.0900126\pi$
$569$			7574.39i			0.558058i	−0.960283	+	0.279029i	$0.909987\pi$
$570$	−9772.15			−0.718088
$571$	−23970.5			−1.75680			−0.878402	−	0.477923i	$-0.841390\pi$
$571$	−23970.5			−1.75680			−0.878402	+	0.477923i	$0.841390\pi$
$572$	−2178.90			−0.159274
$573$		−	7787.64i		−	0.567772i
$574$		−	1330.98i		−	0.0967841i
$575$		−	6972.98i		−	0.505728i
$576$	2549.01			0.184390
$577$	11135.1			0.803394			0.401697	−	0.915773i	$-0.368420\pi$
$577$	11135.1			0.803394			0.401697	+	0.915773i	$0.368420\pi$
$578$		−	21167.7i		−	1.52329i
$579$	12302.4			0.883025
$580$	5212.49			0.373167
$581$	−3342.57			−0.238680
$582$	−5486.81			−0.390783
$583$			377.739i			0.0268342i
$584$	246.674			0.0174785
$585$		−	2923.58i		−	0.206624i
$586$	−10043.0			−0.707972
$587$		−	1167.44i		−	0.0820872i	−0.999157	−	0.0410436i	$-0.986932\pi$
$587$		−	1167.44i		−	0.0820872i	0.999157	−	0.0410436i	$-0.0130683\pi$
$588$	6178.52			0.433330
$589$	2409.42			0.168554
$590$		−	20729.0i		−	1.44644i
$591$	−15229.3			−1.05998
$592$	23505.2			1.63185
$593$	7844.29			0.543214			0.271607	−	0.962408i	$-0.412445\pi$
$593$	7844.29			0.543214			0.271607	+	0.962408i	$0.412445\pi$
$594$			1001.14i			0.0691537i
$595$	4090.76			0.281857
$596$			1554.50i			0.106837i
$597$	11748.4			0.805410
$598$	−23641.4			−1.61667
$599$		−	6836.02i		−	0.466298i	−0.972441	−	0.233149i	$-0.925097\pi$
$599$		−	6836.02i		−	0.466298i	0.972441	−	0.233149i	$-0.0749030\pi$
$600$			735.405i			0.0500379i
$601$	−11192.3			−0.759642			−0.379821	−	0.925060i	$-0.624014\pi$
$601$	−11192.3			−0.759642			−0.379821	+	0.925060i	$0.624014\pi$
$602$		−	4916.45i		−	0.332856i
$603$	−4411.92			−0.297955
$604$			20691.5i			1.39392i
$605$			11442.9i			0.768960i
$606$			16638.7i			1.11535i
$607$			877.288i			0.0586623i	0.999570	+	0.0293312i	$0.00933774\pi$
$607$			877.288i			0.0586623i	−0.999570	+	0.0293312i	$0.990662\pi$
$608$		−	21543.5i		−	1.43701i
$609$	1145.76			0.0762376
$610$		−	25963.6i		−	1.72334i
$611$	11998.0			0.794416
$612$	5854.67			0.386701
$613$			24787.6i			1.63322i	0.577193	+	0.816608i	$0.304148\pi$
$613$			24787.6i			1.63322i	−0.577193	+	0.816608i	$0.695852\pi$
$614$	−15029.5			−0.987851
$615$	−2266.37			−0.148599
$616$	−263.943			−0.0172639
$617$	18986.6			1.23885			0.619426	−	0.785055i	$-0.287365\pi$
$617$	18986.6			1.23885			0.619426	+	0.785055i	$0.287365\pi$
$618$	16261.7			1.05848
$619$	−1383.90			−0.0898607			−0.0449304	−	0.998990i	$-0.514307\pi$
$619$	−1383.90			−0.0898607			−0.0449304	+	0.998990i	$0.514307\pi$
$620$		−	1526.49i		−	0.0988796i
$621$			4805.92i			0.310556i
$622$		−	9942.18i		−	0.640909i
$623$		−	853.728i		−	0.0549019i
$624$	7835.46			0.502676
$625$	−9193.51			−0.588385
$626$		−	7360.72i		−	0.469958i
$627$	2725.97			0.173628
$628$	−2244.40	+	12284.5i	−0.142614	+	0.780582i
$629$	32337.5			2.04989
$630$			1360.88i			0.0860618i
$631$	−23124.2			−1.45889			−0.729446	−	0.684038i	$-0.760222\pi$
$631$	−23124.2			−1.45889			−0.729446	+	0.684038i	$0.760222\pi$
$632$	132.567			0.00834371
$633$		−	4714.24i		−	0.296010i
$634$			7669.61i			0.480440i
$635$		−	20552.5i		−	1.28441i
$636$			734.880i			0.0458174i
$637$	11376.0			0.707587
$638$	−3286.48			−0.203939
$639$	−2108.83			−0.130554
$640$	7262.03			0.448526
$641$	−23601.6			−1.45430			−0.727151	−	0.686478i	$-0.759156\pi$
$641$	−23601.6			−1.45430			−0.727151	+	0.686478i	$0.759156\pi$
$642$	−752.377			−0.0462522
$643$		−	821.290i		−	0.0503709i	−0.999683	−	0.0251855i	$-0.991982\pi$
$643$		−	821.290i		−	0.0503709i	0.999683	−	0.0251855i	$-0.00801763\pi$
$644$	4868.85			0.297919
$645$	−8371.62			−0.511058
$646$		−	36031.5i		−	2.19449i
$647$	−12206.0			−0.741681			−0.370840	−	0.928697i	$-0.620930\pi$
$647$	−12206.0			−0.741681			−0.370840	+	0.928697i	$0.620930\pi$
$648$		−	506.856i		−	0.0307271i
$649$			5782.42i			0.349738i
$650$		−	5203.14i		−	0.313976i
$651$		−	335.540i		−	0.0202010i
$652$			8344.62i			0.501228i
$653$	7970.83			0.477676			0.238838	−	0.971059i	$-0.423233\pi$
$653$	7970.83			0.477676			0.238838	+	0.971059i	$0.423233\pi$
$654$		−	23933.0i		−	1.43097i
$655$	−2820.76			−0.168269
$656$		−	6074.07i		−	0.361513i
$657$			354.786i			0.0210677i
$658$	−5584.91			−0.330885
$659$	18563.9			1.09734			0.548669	−	0.836039i	$-0.315135\pi$
$659$	18563.9			1.09734			0.548669	+	0.836039i	$0.315135\pi$
$660$		−	1727.04i		−	0.101856i
$661$	7893.42			0.464476			0.232238	−	0.972659i	$-0.425395\pi$
$661$	7893.42			0.464476			0.232238	+	0.972659i	$0.425395\pi$
$662$			31825.7i			1.86849i
$663$	10779.7			0.631446
$664$	−4854.06			−0.283696
$665$	3705.51			0.216080
$666$			10757.8i			0.625910i
$667$	−15776.5			−0.915848
$668$	7984.61			0.462476
$669$			17135.1i			0.990256i
$670$	17202.4			0.991919
$671$			7242.64i			0.416690i
$672$	−3000.18			−0.172224
$673$		−	19843.7i		−	1.13658i	−0.822829	−	0.568289i	$-0.807606\pi$
$673$		−	19843.7i		−	1.13658i	0.822829	−	0.568289i	$-0.192394\pi$
$674$	25904.6			1.48042
$675$	−1057.72			−0.0603134
$676$	6141.71			0.349437
$677$	30043.7			1.70557			0.852785	−	0.522262i	$-0.174912\pi$
$677$	30043.7			1.70557			0.852785	+	0.522262i	$0.174912\pi$
$678$		−	10208.7i		−	0.578265i
$679$	2080.55			0.117591
$680$	5940.58			0.335016
$681$		−	4332.79i		−	0.243807i
$682$			962.452i			0.0540384i
$683$		−	23993.2i		−	1.34418i	−0.740471	−	0.672088i	$-0.765398\pi$
$683$		−	23993.2i		−	1.34418i	0.740471	−	0.672088i	$-0.234602\pi$
$684$	5303.30			0.296457
$685$	6111.78			0.340904
$686$	−10893.8			−0.606307
$687$			623.228i			0.0346108i
$688$		−	22436.7i		−	1.24330i
$689$			1353.07i			0.0748156i
$690$		−	18738.6i		−	1.03387i
$691$			20328.3i			1.11914i	0.828783	+	0.559570i	$0.189034\pi$
$691$			20328.3i			1.11914i	−0.828783	+	0.559570i	$0.810966\pi$
$692$	23660.4			1.29976
$693$		−	379.623i		−	0.0208091i
$694$		−	29312.4i		−	1.60329i
$695$	6506.38			0.355109
$696$	1663.87			0.0906162
$697$		−	8356.46i		−	0.454122i
$698$		−	13373.6i		−	0.725212i
$699$	16078.9			0.870042
$700$			1071.57i			0.0578591i
$701$		−	7884.71i		−	0.424824i	−0.977180	−	0.212412i	$-0.931868\pi$
$701$		−	7884.71i		−	0.424824i	0.977180	−	0.212412i	$-0.0681318\pi$
$702$			3586.11i			0.192805i
$703$	29292.0			1.57151
$704$	2772.45			0.148424
$705$			9509.87i			0.508032i
$706$		−	11892.7i		−	0.633975i
$707$		−	6309.23i		−	0.335620i
$708$			11249.5i			0.597151i
$709$	9632.31			0.510224			0.255112	−	0.966911i	$-0.417888\pi$
$709$	9632.31			0.510224			0.255112	+	0.966911i	$0.417888\pi$
$710$	8222.47			0.434625
$711$			190.668i			0.0100571i
$712$		−	1239.78i		−	0.0652566i
$713$			4620.20i			0.242676i
$714$	−5017.80			−0.263006
$715$		−	3179.86i		−	0.166321i
$716$			1554.26i			0.0811248i
$717$	2645.91			0.137815
$718$	−21096.6			−1.09654
$719$		−	32323.2i		−	1.67657i	−0.545236	−	0.838283i	$-0.683560\pi$
$719$		−	32323.2i		−	1.67657i	0.545236	−	0.838283i	$-0.316440\pi$
$720$			6210.54i			0.321463i
$721$	−6166.30			−0.318509
$722$		−	6657.12i		−	0.343147i
$723$		−	19265.4i		−	0.990994i
$724$		−	640.358i		−	0.0328711i
$725$		−	3472.20i		−	0.177868i
$726$		−	14036.1i		−	0.717532i
$727$	−24711.4			−1.26066			−0.630328	−	0.776329i	$-0.717080\pi$
$727$	−24711.4			−1.26066			−0.630328	+	0.776329i	$0.717080\pi$
$728$	−945.451			−0.0481329
$729$	729.000			0.0370370
$730$		−	1383.33i		−	0.0701363i
$731$		−	30867.5i		−	1.56180i
$732$			14090.3i			0.711467i
$733$	11363.3			0.572596			0.286298	−	0.958141i	$-0.407575\pi$
$733$	11363.3			0.572596			0.286298	+	0.958141i	$0.407575\pi$
$734$	−33697.3			−1.69454
$735$			9016.82i			0.452504i
$736$	41310.8			2.06894
$737$	−4798.66			−0.239838
$738$	2779.97			0.138661
$739$	−15734.2			−0.783211			−0.391605	−	0.920133i	$-0.628080\pi$
$739$	−15734.2			−0.783211			−0.391605	+	0.920133i	$0.628080\pi$
$740$		−	18558.0i		−	0.921899i
$741$	9764.51			0.484087
$742$		−	629.836i		−	0.0311617i
$743$	−1684.14			−0.0831560			−0.0415780	−	0.999135i	$-0.513239\pi$
$743$	−1684.14			−0.0831560			−0.0415780	+	0.999135i	$0.513239\pi$
$744$		−	487.269i		−	0.0240109i
$745$	−2268.61			−0.111564
$746$	−38660.4			−1.89740
$747$		−	6981.49i		−	0.341954i
$748$	6367.88			0.311274
$749$	285.294			0.0139178
$750$	17283.5			0.841471
$751$			1788.03i			0.0868790i	0.999056	+	0.0434395i	$0.0138316\pi$
$751$			1788.03i			0.0868790i	−0.999056	+	0.0434395i	$0.986168\pi$
$752$	−25487.4			−1.23594
$753$			9538.24i			0.461611i
$754$	−11772.2			−0.568594
$755$	−30196.8			−1.45560
$756$		−	738.546i		−	0.0355299i
$757$			24951.1i			1.19797i	0.800760	+	0.598985i	$0.204429\pi$
$757$			24951.1i			1.19797i	−0.800760	+	0.598985i	$0.795571\pi$
$758$	−4720.89			−0.226214
$759$			5227.21i			0.249981i
$760$	5381.12			0.256834
$761$		−	11200.4i		−	0.533525i	−0.963762	−	0.266762i	$-0.914046\pi$
$761$		−	11200.4i		−	0.533525i	0.963762	−	0.266762i	$-0.0859539\pi$
$762$			25210.1i			1.19851i
$763$			9075.19i			0.430595i
$764$		−	16478.7i		−	0.780339i
$765$			8544.20i			0.403812i
$766$	−6322.80			−0.298240
$767$			20712.8i			0.975092i
$768$	−15705.1			−0.737902
$769$	20323.8			0.953049			0.476524	−	0.879161i	$-0.341896\pi$
$769$	20323.8			0.953049			0.476524	+	0.879161i	$0.341896\pi$
$770$			1480.18i			0.0692752i
$771$	−4074.89			−0.190342
$772$	26032.0			1.21362
$773$	11534.0			0.536674			0.268337	−	0.963325i	$-0.413526\pi$
$773$	11534.0			0.536674			0.268337	+	0.963325i	$0.413526\pi$
$774$	10268.8			0.476879
$775$	−1016.84			−0.0471303
$776$	3021.36			0.139769
$777$		−	4079.26i		−	0.188343i
$778$			35922.0i			1.65536i
$779$		−	7569.48i		−	0.348145i
$780$		−	6186.31i		−	0.283981i
$781$	−2293.69			−0.105089
$782$	69092.4			3.15951
$783$			2393.11i			0.109225i
$784$	−24165.9			−1.10085
$785$	−17927.8	−	3275.44i	−0.815122	−	0.148924i
$786$	3459.99			0.157015
$787$		−	6299.34i		−	0.285321i	−0.989772	−	0.142660i	$-0.954434\pi$
$787$		−	6299.34i		−	0.285321i	0.989772	−	0.142660i	$-0.0455657\pi$
$788$	−32225.4			−1.45683
$789$	11011.6			0.496861
$790$		−	743.428i		−	0.0334810i
$791$			3871.05i			0.174006i
$792$		−	551.287i		−	0.0247337i
$793$			25943.3i			1.16176i
$794$	48293.5			2.15853
$795$	−1072.47			−0.0478448
$796$	24859.7			1.10695
$797$	−25799.5			−1.14663			−0.573316	−	0.819334i	$-0.694343\pi$
$797$	−25799.5			−1.14663			−0.573316	+	0.819334i	$0.694343\pi$
$798$	−4545.24			−0.201629
$799$	−35064.4			−1.55255
$800$			9091.93i			0.401811i
$801$	1783.15			0.0786572
$802$	−28881.9			−1.27164
$803$			385.886i			0.0169584i
$804$	−9335.65			−0.409506
$805$			7105.52i			0.311101i
$806$			3447.53i			0.150663i
$807$			17927.8i			0.782020i
$808$		−	9162.23i		−	0.398918i
$809$		−	29808.4i		−	1.29543i	−0.761881	−	0.647717i	$-0.775724\pi$
$809$		−	29808.4i		−	1.29543i	0.761881	−	0.647717i	$-0.224276\pi$
$810$	−2842.42			−0.123299
$811$			35846.5i			1.55209i	0.630680	+	0.776043i	$0.282776\pi$
$811$			35846.5i			1.55209i	−0.630680	+	0.776043i	$0.717224\pi$
$812$	2424.45			0.104780
$813$			12583.6i			0.542838i
$814$			11700.8i			0.503825i
$815$	−12178.0			−0.523407
$816$	−22899.3			−0.982396
$817$		−	27960.5i		−	1.19733i
$818$	60370.2			2.58043
$819$		−	1359.82i		−	0.0580171i
$820$	−4795.65			−0.204233
$821$	−17169.3			−0.729856			−0.364928	−	0.931036i	$-0.618906\pi$
$821$	−17169.3			−0.729856			−0.364928	+	0.931036i	$0.618906\pi$
$822$	−7496.82			−0.318104
$823$		−	30889.8i		−	1.30832i	−0.756354	−	0.654162i	$-0.773021\pi$
$823$		−	30889.8i		−	1.30832i	0.756354	−	0.654162i	$-0.226979\pi$
$824$	−8954.66			−0.378581
$825$	−1150.43			−0.0485491
$826$		−	9641.51i		−	0.406139i
$827$	−5605.75			−0.235709			−0.117854	−	0.993031i	$-0.537602\pi$
$827$	−5605.75			−0.235709			−0.117854	+	0.993031i	$0.537602\pi$
$828$			10169.4i			0.426824i
$829$	3902.78			0.163509			0.0817546	−	0.996652i	$-0.473948\pi$
$829$	3902.78			0.163509			0.0817546	+	0.996652i	$0.473948\pi$
$830$			27221.3i			1.13839i
$831$	−15763.2			−0.658026
$832$	9931.00			0.413817
$833$	−33246.5			−1.38286
$834$	−7980.85			−0.331360
$835$			11652.6i			0.482940i
$836$	5768.18			0.238632
$837$	700.828			0.0289417
$838$			40197.3i			1.65703i
$839$		−	12279.5i		−	0.505287i	−0.967559	−	0.252643i	$-0.918700\pi$
$839$		−	12279.5i		−	0.505287i	0.967559	−	0.252643i	$-0.0812999\pi$
$840$		−	749.382i		−	0.0307811i
$841$	16533.1			0.677890
$842$	−61183.0			−2.50416
$843$	529.612			0.0216380
$844$		−	9975.38i		−	0.406833i
$845$			8963.11i			0.364900i
$846$		−	11665.0i		−	0.474055i
$847$			5322.36i			0.215913i
$848$		−	2874.33i		−	0.116397i
$849$	−14145.8			−0.571830
$850$			15206.3i			0.613613i
$851$			56169.1i			2.26258i
$852$	−4462.30			−0.179432
$853$	26037.9			1.04516			0.522579	−	0.852591i	$-0.324970\pi$
$853$	26037.9			1.04516			0.522579	+	0.852591i	$0.324970\pi$
$854$		−	12076.3i		−	0.483889i
$855$			7739.54i			0.309575i
$856$	414.303			0.0165427
$857$		−	41335.3i		−	1.64759i	−0.566887	−	0.823796i	$-0.691852\pi$
$857$		−	41335.3i		−	1.64759i	0.566887	−	0.823796i	$-0.308148\pi$
$858$			3900.47i			0.155198i
$859$			10402.0i			0.413169i	0.978429	+	0.206585i	$0.0662349\pi$
$859$			10402.0i			0.413169i	−0.978429	+	0.206585i	$0.933765\pi$
$860$	−17714.4			−0.702392
$861$	−1054.14			−0.0417246
$862$		−	56526.6i		−	2.23353i
$863$			6145.10i			0.242389i	0.992629	+	0.121195i	$0.0386725\pi$
$863$			6145.10i			0.242389i	−0.992629	+	0.121195i	$0.961328\pi$
$864$		−	6266.35i		−	0.246743i
$865$			34529.6i			1.35727i
$866$	−8380.64			−0.328852
$867$	−16764.9			−0.656706
$868$		−	710.005i		−	0.0277640i
$869$			207.382i			0.00809545i
$870$		−	9330.91i		−	0.363618i
$871$	−17188.9			−0.668685
$872$			13178.9i			0.511806i
$873$			4345.56i			0.168471i
$874$	62585.5			2.42218
$875$	−6553.73			−0.253208
$876$			750.729i			0.0289552i
$877$			41796.7i			1.60932i	0.593735	+	0.804660i	$0.297653\pi$
$877$			41796.7i			1.60932i	−0.593735	+	0.804660i	$0.702347\pi$
$878$	−54598.4			−2.09864
$879$			7954.03i			0.305214i
$880$			6754.95i			0.258761i
$881$			47705.9i			1.82435i	0.409799	+	0.912176i	$0.365599\pi$
$881$			47705.9i			1.82435i	−0.409799	+	0.912176i	$0.634401\pi$
$882$		−	11060.2i		−	0.422241i
$883$		−	14762.0i		−	0.562605i	−0.959619	−	0.281303i	$-0.909234\pi$
$883$		−	14762.0i		−	0.562605i	0.959619	−	0.281303i	$-0.0907665\pi$
$884$	22809.9			0.867852
$885$	−16417.4			−0.623574
$886$	−37487.2			−1.42145
$887$		−	9294.30i		−	0.351829i	−0.984405	−	0.175914i	$-0.943712\pi$
$887$		−	9294.30i		−	0.351829i	0.984405	−	0.175914i	$-0.0562882\pi$
$888$		−	5923.87i		−	0.223865i
$889$		−	9559.43i		−	0.360645i
$890$	−6952.62			−0.261857
$891$	792.904			0.0298129
$892$			36258.0i			1.36100i
$893$	−31762.2			−1.19024
$894$	2782.72			0.104103
$895$	−2268.26			−0.0847145
$896$	3377.73			0.125940
$897$			18724.0i			0.696963i
$898$	15802.6			0.587237
$899$			2300.63i			0.0853507i
$900$	−2238.14			−0.0828939
$901$		−	3954.38i		−	0.146215i
$902$	3023.66			0.111615
$903$	−3893.83			−0.143498
$904$			5621.52i			0.206824i
$905$	934.527			0.0343257
$906$	37040.0			1.35825
$907$	52961.8			1.93888			0.969441	−	0.245323i	$-0.0788941\pi$
$907$	52961.8			1.93888			0.969441	+	0.245323i	$0.0788941\pi$
$908$		−	9168.21i		−	0.335086i
$909$	13177.8			0.480837
$910$			5302.04i			0.193144i
$911$	5029.58			0.182917			0.0914585	−	0.995809i	$-0.470847\pi$
$911$	5029.58			0.182917			0.0914585	+	0.995809i	$0.470847\pi$
$912$	−20742.7			−0.753136
$913$		−	7593.48i		−	0.275255i
$914$			8928.91i			0.323132i
$915$	−20563.2			−0.742949
$916$			1318.76i			0.0475687i
$917$	−1312.00			−0.0472475
$918$		−	10480.5i		−	0.376805i
$919$		−	25259.7i		−	0.906683i	−0.891337	−	0.453342i	$-0.850232\pi$
$919$		−	25259.7i		−	0.906683i	0.891337	−	0.453342i	$-0.149768\pi$
$920$			10318.6i			0.369776i
$921$			11903.4i			0.425873i
$922$		−	18517.0i		−	0.661417i
$923$	−8216.05			−0.292995
$924$		−	803.286i		−	0.0285997i
$925$	−12362.0			−0.439418
$926$	7445.92			0.264242
$927$		−	12879.3i		−	0.456323i
$928$	20570.7			0.727659
$929$	−14634.6			−0.516843			−0.258421	−	0.966032i	$-0.583202\pi$
$929$	−14634.6			−0.516843			−0.258421	+	0.966032i	$0.583202\pi$
$930$	−2732.58			−0.0963492
$931$	−30115.5			−1.06014
$932$	34023.0			1.19577
$933$	−7874.21			−0.276302
$934$		−	9798.55i		−	0.343274i
$935$			9293.18i			0.325048i
$936$		−	1974.72i		−	0.0689593i
$937$			15348.2i			0.535117i	0.963542	+	0.267558i	$0.0862168\pi$
$937$			15348.2i			0.535117i	−0.963542	+	0.267558i	$0.913783\pi$
$938$	8001.21			0.278517
$939$	−5829.69			−0.202604
$940$			20123.0i			0.698232i
$941$	−45264.0			−1.56808			−0.784040	−	0.620710i	$-0.786844\pi$
$941$	−45264.0			−1.56808			−0.784040	+	0.620710i	$0.786844\pi$
$942$	21990.6	+	4017.71i	0.760607	+	0.138964i
$943$	14514.9			0.501241
$944$		−	44000.1i		−	1.51703i
$945$	1077.82			0.0371021
$946$	11168.9			0.383862
$947$		−	1577.70i		−	0.0541377i	−0.999634	−	0.0270688i	$-0.991383\pi$
$947$		−	1577.70i		−	0.0541377i	0.999634	−	0.0270688i	$-0.00861733\pi$
$948$			403.455i			0.0138224i
$949$			1382.25i			0.0472812i
$950$			13774.2i			0.470415i
$951$	6074.33			0.207123
$952$	2763.09			0.0940677
$953$	39399.2			1.33921			0.669604	−	0.742719i	$-0.266464\pi$
$953$	39399.2			1.33921			0.669604	+	0.742719i	$0.266464\pi$
$954$	1315.51			0.0446450
$955$	24048.7			0.814868
$956$	5598.77			0.189411
$957$			2602.89i			0.0879200i
$958$	−40586.8			−1.36879
$959$	2842.73			0.0957210
$960$			7871.51i			0.264637i
$961$	−29117.3			−0.977384
$962$			41912.7i			1.40470i
$963$			595.882i			0.0199398i
$964$		−	40765.8i		−	1.36201i
$965$			37990.7i			1.26732i
$966$		−	8715.76i		−	0.290295i
$967$	24850.8			0.826421			0.413210	−	0.910636i	$-0.364407\pi$
$967$	24850.8			0.826421			0.413210	+	0.910636i	$0.364407\pi$
$968$			7729.10i			0.256635i
$969$	−28536.9			−0.946067
$970$		−	16943.6i		−	0.560853i
$971$			32717.4i			1.08131i	0.841245	+	0.540655i	$0.181823\pi$
$971$			32717.4i			1.08131i	−0.841245	+	0.540655i	$0.818177\pi$
$972$	1542.57			0.0509032
$973$	3026.26			0.0997097
$974$			21922.0i			0.721178i
$975$	−4120.89			−0.135358
$976$		−	55111.3i		−	1.80745i
$977$	−16974.0			−0.555832			−0.277916	−	0.960605i	$-0.589644\pi$
$977$	−16974.0			−0.555832			−0.277916	+	0.960605i	$0.589644\pi$
$978$	14937.7			0.488401
$979$	1939.46			0.0633149
$980$			19079.7i			0.621916i
$981$	−18955.0			−0.616907
$982$	−25334.8			−0.823284
$983$			5204.52i			0.168869i	0.996429	+	0.0844345i	$0.0269084\pi$
$983$			5204.52i			0.168869i	−0.996429	+	0.0844345i	$0.973092\pi$
$984$	−1530.81			−0.0495940
$985$		−	47029.2i		−	1.52129i
$986$	34404.6			1.11122
$987$			4423.25i			0.142648i
$988$	20661.8			0.665323
$989$	53615.9			1.72385
$990$	−3091.59			−0.0992496
$991$	53491.9			1.71466			0.857329	−	0.514769i	$-0.172122\pi$
$991$	53491.9			1.71466			0.857329	+	0.514769i	$0.172122\pi$
$992$		−	6024.19i		−	0.192811i
$993$	25206.0			0.805527
$994$	3824.45			0.122037
$995$			36279.8i			1.15593i
$996$		−	14772.9i		−	0.469977i
$997$			56206.6i			1.78544i	0.450614	+	0.892719i	$0.351205\pi$
$997$			56206.6i			1.78544i	−0.450614	+	0.892719i	$0.648795\pi$
$998$	63661.5			2.01921
$999$	8520.18			0.269836

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.4.b.a.313.8	✓	40
157.156	even	2	inner	471.4.b.a.313.33	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.4.b.a.313.8	✓	40	1.1	even	1	trivial
471.4.b.a.313.33	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)

\(f(q)\)	\(=\)	\(q-3.78788i q^{2} -3.00000 q^{3} -6.34802 q^{4} -9.26420i q^{5} +11.3636i q^{6} -4.30898i q^{7} -6.25748i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-3.78788i q^{2} -3.00000 q^{3} -6.34802 q^{4} -9.26420i q^{5} +11.3636i q^{6} -4.30898i q^{7} -6.25748i q^{8} +9.00000 q^{9} -35.0916 q^{10} +9.78893 q^{11} +19.0441 q^{12} +35.0642 q^{13} -16.3219 q^{14} +27.7926i q^{15} -74.4868 q^{16} -102.476 q^{17} -34.0909i q^{18} -92.8250 q^{19} +58.8093i q^{20} +12.9270i q^{21} -37.0793i q^{22} -177.997i q^{23} +18.7724i q^{24} +39.1747 q^{25} -132.819i q^{26} -27.0000 q^{27} +27.3535i q^{28} -88.6337i q^{29} +105.275 q^{30} -25.9566 q^{31} +232.087i q^{32} -29.3668 q^{33} +388.166i q^{34} -39.9193 q^{35} -57.1322 q^{36} -315.562 q^{37} +351.610i q^{38} -105.193 q^{39} -57.9705 q^{40} +81.5457i q^{41} +48.9657 q^{42} +301.218i q^{43} -62.1404 q^{44} -83.3778i q^{45} -674.231 q^{46} +342.173 q^{47} +223.460 q^{48} +324.433 q^{49} -148.389i q^{50} +307.427 q^{51} -222.589 q^{52} +38.5884i q^{53} +102.273i q^{54} -90.6866i q^{55} -26.9634 q^{56} +278.475 q^{57} -335.734 q^{58} +590.710i q^{59} -176.428i q^{60} +739.881i q^{61} +98.3204i q^{62} -38.7809i q^{63} +283.223 q^{64} -324.842i q^{65} +111.238i q^{66} -490.213 q^{67} +650.519 q^{68} +533.991i q^{69} +151.209i q^{70} -234.314 q^{71} -56.3173i q^{72} +39.4206i q^{73} +1195.31i q^{74} -117.524 q^{75} +589.255 q^{76} -42.1804i q^{77} +398.457i q^{78} +21.1853i q^{79} +690.060i q^{80} +81.0000 q^{81} +308.885 q^{82} -775.721i q^{83} -82.0606i q^{84} +949.356i q^{85} +1140.98 q^{86} +265.901i q^{87} -61.2541i q^{88} +198.127 q^{89} -315.825 q^{90} -151.091i q^{91} +1129.93i q^{92} +77.8698 q^{93} -1296.11i q^{94} +859.949i q^{95} -696.261i q^{96} +482.840i q^{97} -1228.91i q^{98} +88.1004 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 120 q^{3} - 164 q^{4} + 360 q^{9}+O(q^{10}) \) 40 * q - 120 * q^3 - 164 * q^4 + 360 * q^9	\( 40 q - 120 q^{3} - 164 q^{4} + 360 q^{9} - 174 q^{10} + 110 q^{11} + 492 q^{12} - 194 q^{13} - 78 q^{14} + 796 q^{16} - 150 q^{17} + 172 q^{19} - 668 q^{25} - 1080 q^{27} + 522 q^{30} + 66 q^{31} - 330 q^{33} - 400 q^{35} - 1476 q^{36} - 142 q^{37} + 582 q^{39} + 1160 q^{40} + 234 q^{42} - 1182 q^{44} + 132 q^{46} - 244 q^{47} - 2388 q^{48} - 3786 q^{49} + 450 q^{51} + 1596 q^{52} - 256 q^{56} - 516 q^{57} - 1780 q^{58} - 1790 q^{64} - 320 q^{67} + 1646 q^{68} + 712 q^{71} + 2004 q^{75} - 3004 q^{76} + 3240 q^{81} + 4112 q^{82} - 4198 q^{86} + 366 q^{89} - 1566 q^{90} - 198 q^{93} + 990 q^{99}+O(q^{100}) \) 40 * q - 120 * q^3 - 164 * q^4 + 360 * q^9 - 174 * q^10 + 110 * q^11 + 492 * q^12 - 194 * q^13 - 78 * q^14 + 796 * q^16 - 150 * q^17 + 172 * q^19 - 668 * q^25 - 1080 * q^27 + 522 * q^30 + 66 * q^31 - 330 * q^33 - 400 * q^35 - 1476 * q^36 - 142 * q^37 + 582 * q^39 + 1160 * q^40 + 234 * q^42 - 1182 * q^44 + 132 * q^46 - 244 * q^47 - 2388 * q^48 - 3786 * q^49 + 450 * q^51 + 1596 * q^52 - 256 * q^56 - 516 * q^57 - 1780 * q^58 - 1790 * q^64 - 320 * q^67 + 1646 * q^68 + 712 * q^71 + 2004 * q^75 - 3004 * q^76 + 3240 * q^81 + 4112 * q^82 - 4198 * q^86 + 366 * q^89 - 1566 * q^90 - 198 * q^93 + 990 * q^99

Introduction

L-functions

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