LMFDB - Embedded newform 354.5.d.a.235.31

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [354,5,Mod(235,354)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(354, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("354.235");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 354 = 2 \cdot 3 \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	354.d (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:	$36.5929669317$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			235.31
Character	$\chi$	$=$	354.235
Dual form			354.5.d.a.235.32

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} +5.19615 q^{3} -8.00000 q^{4} +13.0356 q^{5} -14.6969i q^{6} +71.9157 q^{7} +22.6274i q^{8} +27.0000 q^{9} +O(q^{10})$	\(q-2.82843i q^{2} +5.19615 q^{3} -8.00000 q^{4} +13.0356 q^{5} -14.6969i q^{6} +71.9157 q^{7} +22.6274i q^{8} +27.0000 q^{9} -36.8702i q^{10} +76.6106i q^{11} -41.5692 q^{12} +118.606i q^{13} -203.408i q^{14} +67.7349 q^{15} +64.0000 q^{16} +31.5143 q^{17} -76.3675i q^{18} +461.021 q^{19} -104.285 q^{20} +373.685 q^{21} +216.687 q^{22} +901.986i q^{23} +117.576i q^{24} -455.073 q^{25} +335.469 q^{26} +140.296 q^{27} -575.325 q^{28} -1161.22 q^{29} -191.583i q^{30} +457.352i q^{31} -181.019i q^{32} +398.080i q^{33} -89.1358i q^{34} +937.463 q^{35} -216.000 q^{36} -440.417i q^{37} -1303.96i q^{38} +616.295i q^{39} +294.962i q^{40} +2196.09 q^{41} -1056.94i q^{42} +1247.28i q^{43} -612.885i q^{44} +351.961 q^{45} +2551.20 q^{46} +1319.13i q^{47} +332.554 q^{48} +2770.86 q^{49} +1287.14i q^{50} +163.753 q^{51} -948.848i q^{52} +53.4409 q^{53} -396.817i q^{54} +998.664i q^{55} +1627.27i q^{56} +2395.54 q^{57} +3284.41i q^{58} +(729.257 - 3403.75i) q^{59} -541.879 q^{60} -235.014i q^{61} +1293.59 q^{62} +1941.72 q^{63} -512.000 q^{64} +1546.10i q^{65} +1125.94 q^{66} -5947.06i q^{67} -252.114 q^{68} +4686.85i q^{69} -2651.55i q^{70} +7826.66 q^{71} +610.940i q^{72} -8267.19i q^{73} -1245.69 q^{74} -2364.63 q^{75} -3688.17 q^{76} +5509.50i q^{77} +1743.15 q^{78} +519.406 q^{79} +834.278 q^{80} +729.000 q^{81} -6211.47i q^{82} +6918.96i q^{83} -2989.48 q^{84} +410.807 q^{85} +3527.85 q^{86} -6033.85 q^{87} -1733.50 q^{88} -1209.41i q^{89} -995.496i q^{90} +8529.63i q^{91} -7215.88i q^{92} +2376.47i q^{93} +3731.07 q^{94} +6009.69 q^{95} -940.604i q^{96} -10730.3i q^{97} -7837.18i q^{98} +2068.49i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 320 q^{4} - 80 q^{7} + 1080 q^{9}+O(q^{10}) $ 40 * q - 320 * q^4 - 80 * q^7 + 1080 * q^9	$ 40 q - 320 q^{4} - 80 q^{7} + 1080 q^{9} - 144 q^{15} + 2560 q^{16} + 480 q^{17} - 792 q^{19} - 1024 q^{22} + 3400 q^{25} + 768 q^{26} + 640 q^{28} + 1608 q^{29} - 5760 q^{35} - 8640 q^{36} + 6264 q^{41} + 7040 q^{46} + 17912 q^{49} + 1296 q^{51} - 1104 q^{53} + 5040 q^{57} + 13584 q^{59} + 1152 q^{60} - 12288 q^{62} - 2160 q^{63} - 20480 q^{64} + 1152 q^{66} - 3840 q^{68} + 35352 q^{71} + 4608 q^{74} + 3168 q^{75} + 6336 q^{76} - 12672 q^{78} - 15720 q^{79} + 29160 q^{81} - 26872 q^{85} + 18432 q^{86} + 7776 q^{87} + 8192 q^{88} - 18432 q^{94} - 19128 q^{95}+O(q^{100}) $ 40 * q - 320 * q^4 - 80 * q^7 + 1080 * q^9 - 144 * q^15 + 2560 * q^16 + 480 * q^17 - 792 * q^19 - 1024 * q^22 + 3400 * q^25 + 768 * q^26 + 640 * q^28 + 1608 * q^29 - 5760 * q^35 - 8640 * q^36 + 6264 * q^41 + 7040 * q^46 + 17912 * q^49 + 1296 * q^51 - 1104 * q^53 + 5040 * q^57 + 13584 * q^59 + 1152 * q^60 - 12288 * q^62 - 2160 * q^63 - 20480 * q^64 + 1152 * q^66 - 3840 * q^68 + 35352 * q^71 + 4608 * q^74 + 3168 * q^75 + 6336 * q^76 - 12672 * q^78 - 15720 * q^79 + 29160 * q^81 - 26872 * q^85 + 18432 * q^86 + 7776 * q^87 + 8192 * q^88 - 18432 * q^94 - 19128 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/354\mathbb{Z}\right)^\times$.

$n$	$61$	$119$
$\chi(n)$	$-1$	$1$

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 $

$2$		−	2.82843i		−	0.707107i
$2$		−	2.82843i		−	0.707107i
$3$	5.19615			0.577350
$3$	5.19615			0.577350
$4$	−8.00000			−0.500000
$5$	13.0356			0.521424			0.260712	−	0.965417i	$-0.416043\pi$
$5$	13.0356			0.521424			0.260712	+	0.965417i	$0.416043\pi$
$6$		−	14.6969i		−	0.408248i
$7$	71.9157			1.46767			0.733833	−	0.679330i	$-0.237729\pi$
$7$	71.9157			1.46767			0.733833	+	0.679330i	$0.237729\pi$
$8$			22.6274i			0.353553i
$9$	27.0000			0.333333
$10$		−	36.8702i		−	0.368702i
$11$			76.6106i			0.633145i	0.948568	+	0.316573i	$0.102532\pi$
$11$			76.6106i			0.633145i	−0.948568	+	0.316573i	$0.897468\pi$
$12$	−41.5692			−0.288675
$13$			118.606i			0.701811i	0.936411	+	0.350905i	$0.114126\pi$
$13$			118.606i			0.701811i	−0.936411	+	0.350905i	$0.885874\pi$
$14$		−	203.408i		−	1.03780i
$15$	67.7349			0.301044
$16$	64.0000			0.250000
$17$	31.5143			0.109046			0.0545230	−	0.998513i	$-0.482636\pi$
$17$	31.5143			0.109046			0.0545230	+	0.998513i	$0.482636\pi$
$18$		−	76.3675i		−	0.235702i
$19$	461.021			1.27707			0.638534	−	0.769594i	$-0.279541\pi$
$19$	461.021			1.27707			0.638534	+	0.769594i	$0.279541\pi$
$20$	−104.285			−0.260712
$21$	373.685			0.847358
$22$	216.687			0.447701
$23$			901.986i			1.70508i	0.522665	+	0.852538i	$0.324938\pi$
$23$			901.986i			1.70508i	−0.522665	+	0.852538i	$0.675062\pi$
$24$			117.576i			0.204124i
$25$	−455.073			−0.728117
$26$	335.469			0.496255
$27$	140.296			0.192450
$28$	−575.325			−0.733833
$29$	−1161.22			−1.38076			−0.690378	−	0.723449i	$-0.742556\pi$
$29$	−1161.22			−1.38076			−0.690378	+	0.723449i	$0.742556\pi$
$30$		−	191.583i		−	0.212870i
$31$			457.352i			0.475912i	0.971276	+	0.237956i	$0.0764774\pi$
$31$			457.352i			0.475912i	−0.971276	+	0.237956i	$0.923523\pi$
$32$		−	181.019i		−	0.176777i
$33$			398.080i			0.365547i
$34$		−	89.1358i		−	0.0771071i
$35$	937.463			0.765276
$36$	−216.000			−0.166667
$37$		−	440.417i		−	0.321707i	−0.986978	−	0.160853i	$-0.948575\pi$
$37$		−	440.417i		−	0.321707i	0.986978	−	0.160853i	$-0.0514247\pi$
$38$		−	1303.96i		−	0.903023i
$39$			616.295i			0.405191i
$40$			294.962i			0.184351i
$41$	2196.09			1.30642			0.653208	−	0.757179i	$-0.273423\pi$
$41$	2196.09			1.30642			0.653208	+	0.757179i	$0.273423\pi$
$42$		−	1056.94i		−	0.599172i
$43$			1247.28i			0.674572i	0.941402	+	0.337286i	$0.109509\pi$
$43$			1247.28i			0.674572i	−0.941402	+	0.337286i	$0.890491\pi$
$44$		−	612.885i		−	0.316573i
$45$	351.961			0.173808
$46$	2551.20			1.20567
$47$			1319.13i			0.597162i	0.954384	+	0.298581i	$0.0965133\pi$
$47$			1319.13i			0.597162i	−0.954384	+	0.298581i	$0.903487\pi$
$48$	332.554			0.144338
$49$	2770.86			1.15405
$50$			1287.14i			0.514857i
$51$	163.753			0.0629577
$52$		−	948.848i		−	0.350905i
$53$	53.4409			0.0190249			0.00951244	−	0.999955i	$-0.496972\pi$
$53$	53.4409			0.0190249			0.00951244	+	0.999955i	$0.496972\pi$
$54$		−	396.817i		−	0.136083i
$55$			998.664i			0.330137i
$56$			1627.27i			0.518899i
$57$	2395.54			0.737315
$58$			3284.41i			0.976341i
$59$	729.257	−	3403.75i	0.209496	−	0.977809i
$59$	729.257	−	3403.75i	0.209496	−	0.977809i
$60$	−541.879			−0.150522
$61$		−	235.014i		−	0.0631589i	−0.999501	−	0.0315794i	$-0.989946\pi$
$61$		−	235.014i		−	0.0631589i	0.999501	−	0.0315794i	$-0.0100537\pi$
$62$	1293.59			0.336521
$63$	1941.72			0.489222
$64$	−512.000			−0.125000
$65$			1546.10i			0.365941i
$66$	1125.94			0.258480
$67$		−	5947.06i		−	1.32481i	−0.749147	−	0.662404i	$-0.769536\pi$
$67$		−	5947.06i		−	1.32481i	0.749147	−	0.662404i	$-0.230464\pi$
$68$	−252.114			−0.0545230
$69$			4686.85i			0.984426i
$70$		−	2651.55i		−	0.541132i
$71$	7826.66			1.55260			0.776300	−	0.630363i	$-0.217094\pi$
$71$	7826.66			1.55260			0.776300	+	0.630363i	$0.217094\pi$
$72$			610.940i			0.117851i
$73$		−	8267.19i		−	1.55136i	−0.631127	−	0.775680i	$-0.717407\pi$
$73$		−	8267.19i		−	1.55136i	0.631127	−	0.775680i	$-0.282593\pi$
$74$	−1245.69			−0.227481
$75$	−2364.63			−0.420379
$76$	−3688.17			−0.638534
$77$			5509.50i			0.929246i
$78$	1743.15			0.286513
$79$	519.406			0.0832248			0.0416124	−	0.999134i	$-0.486751\pi$
$79$	519.406			0.0832248			0.0416124	+	0.999134i	$0.486751\pi$
$80$	834.278			0.130356
$81$	729.000			0.111111
$82$		−	6211.47i		−	0.923776i
$83$			6918.96i			1.00435i	0.864766	+	0.502174i	$0.167466\pi$
$83$			6918.96i			1.00435i	−0.864766	+	0.502174i	$0.832534\pi$
$84$	−2989.48			−0.423679
$85$	410.807			0.0568591
$86$	3527.85			0.476994
$87$	−6033.85			−0.797179
$88$	−1733.50			−0.223851
$89$		−	1209.41i		−	0.152684i	−0.997082	−	0.0763422i	$-0.975676\pi$
$89$		−	1209.41i		−	0.152684i	0.997082	−	0.0763422i	$-0.0243241\pi$
$90$		−	995.496i		−	0.122901i
$91$			8529.63i			1.03002i
$92$		−	7215.88i		−	0.852538i
$93$			2376.47i			0.274768i
$94$	3731.07			0.422257
$95$	6009.69			0.665893
$96$		−	940.604i		−	0.102062i
$97$		−	10730.3i		−	1.14043i	−0.821497	−	0.570213i	$-0.806861\pi$
$97$		−	10730.3i		−	1.14043i	0.821497	−	0.570213i	$-0.193139\pi$
$98$		−	7837.18i		−	0.816033i
$99$			2068.49i			0.211048i
$100$	3640.59			0.364059
$101$		−	3057.38i		−	0.299714i	−0.988708	−	0.149857i	$-0.952119\pi$
$101$		−	3057.38i		−	0.299714i	0.988708	−	0.149857i	$-0.0478813\pi$
$102$		−	463.163i		−	0.0445178i
$103$		−	413.512i		−	0.0389775i	−0.999810	−	0.0194887i	$-0.993796\pi$
$103$		−	413.512i		−	0.0389775i	0.999810	−	0.0194887i	$-0.00620385\pi$
$104$	−2683.75			−0.248128
$105$	4871.20			0.441832
$106$		−	151.154i		−	0.0134526i
$107$	−488.624			−0.0426783			−0.0213391	−	0.999772i	$-0.506793\pi$
$107$	−488.624			−0.0426783			−0.0213391	+	0.999772i	$0.506793\pi$
$108$	−1122.37			−0.0962250
$109$		−	12377.5i		−	1.04179i	−0.853620	−	0.520896i	$-0.825598\pi$
$109$		−	12377.5i		−	1.04179i	0.853620	−	0.520896i	$-0.174402\pi$
$110$	2824.65			0.233442
$111$		−	2288.47i		−	0.185738i
$112$	4602.60			0.366917
$113$		−	58.5659i		−	0.00458657i	−0.999997	−	0.00229328i	$-0.999270\pi$
$113$		−	58.5659i		−	0.00458657i	0.999997	−	0.00229328i	$-0.000729976\pi$
$114$		−	6775.60i		−	0.521360i
$115$			11757.9i			0.889067i
$116$	9289.72			0.690378
$117$			3202.36i			0.233937i
$118$	−9627.27	−	2062.65i	−0.691416	−	0.148136i
$119$	2266.37			0.160043
$120$			1532.67i			0.106435i
$121$	8771.82			0.599127
$122$	−664.720			−0.0446601
$123$	11411.2			0.754260
$124$		−	3658.81i		−	0.237956i
$125$	−14079.4			−0.901081
$126$		−	5492.02i		−	0.345932i
$127$	6961.62			0.431621			0.215811	−	0.976435i	$-0.430761\pi$
$127$	6961.62			0.431621			0.215811	+	0.976435i	$0.430761\pi$
$128$			1448.15i			0.0883883i
$129$			6481.07i			0.389464i
$130$	4373.03			0.258759
$131$		−	13594.7i		−	0.792185i	−0.918211	−	0.396092i	$-0.870366\pi$
$131$		−	13594.7i		−	0.792185i	0.918211	−	0.396092i	$-0.129634\pi$
$132$		−	3184.64i		−	0.182773i
$133$	33154.6			1.87431
$134$	−16820.8			−0.936780
$135$	1828.84			0.100348
$136$			713.087i			0.0385536i
$137$	−21151.7			−1.12695			−0.563475	−	0.826133i	$-0.690536\pi$
$137$	−21151.7			−1.12695			−0.563475	+	0.826133i	$0.690536\pi$
$138$	13256.4			0.696095
$139$	20763.3			1.07465			0.537324	−	0.843376i	$-0.319435\pi$
$139$	20763.3			1.07465			0.537324	+	0.843376i	$0.319435\pi$
$140$	−7499.71			−0.382638
$141$			6854.41i			0.344772i
$142$		−	22137.1i		−	1.09785i
$143$	−9086.48			−0.444348
$144$	1728.00			0.0833333
$145$	−15137.1			−0.719959
$146$	−23383.2			−1.09698
$147$	14397.8			0.666288
$148$			3523.33i			0.160853i
$149$			6501.17i			0.292832i	0.989223	+	0.146416i	$0.0467738\pi$
$149$			6501.17i			0.292832i	−0.989223	+	0.146416i	$0.953226\pi$
$150$			6688.18i			0.297253i
$151$			4605.38i			0.201982i	0.994887	+	0.100991i	$0.0322013\pi$
$151$			4605.38i			0.201982i	−0.994887	+	0.100991i	$0.967799\pi$
$152$			10431.7i			0.451511i
$153$	850.886			0.0363487
$154$	15583.2			0.657076
$155$			5961.85i			0.248152i
$156$		−	4930.36i		−	0.202595i
$157$		−	8564.91i		−	0.347475i	−0.984792	−	0.173737i	$-0.944416\pi$
$157$		−	8564.91i		−	0.347475i	0.984792	−	0.173737i	$-0.0555844\pi$
$158$		−	1469.10i		−	0.0588488i
$159$	277.687			0.0109840
$160$		−	2359.69i		−	0.0921756i
$161$			64866.9i			2.50248i
$162$		−	2061.92i		−	0.0785674i
$163$	−1698.22			−0.0639173			−0.0319587	−	0.999489i	$-0.510174\pi$
$163$	−1698.22			−0.0639173			−0.0319587	+	0.999489i	$0.510174\pi$
$164$	−17568.7			−0.653208
$165$			5189.21i			0.190605i
$166$	19569.8			0.710182
$167$	−40686.7			−1.45888			−0.729439	−	0.684046i	$-0.760219\pi$
$167$	−40686.7			−1.45888			−0.729439	+	0.684046i	$0.760219\pi$
$168$			8455.52i			0.299586i
$169$	14493.6			0.507462
$170$		−	1161.94i		−	0.0402055i
$171$	12447.6			0.425689
$172$		−	9978.26i		−	0.337286i
$173$		−	373.836i		−	0.0124908i	−0.999980	−	0.00624538i	$-0.998012\pi$
$173$		−	373.836i		−	0.0124908i	0.999980	−	0.00624538i	$-0.00198798\pi$
$174$			17066.3i			0.563691i
$175$	−32726.9			−1.06863
$176$			4903.08i			0.158286i
$177$	3789.33	−	17686.4i	0.120953	−	0.564539i
$178$	−3420.74			−0.107964
$179$			48418.3i			1.51114i	0.655071	+	0.755568i	$0.272639\pi$
$179$			48418.3i			1.51114i	−0.655071	+	0.755568i	$0.727361\pi$
$180$	−2815.69			−0.0869040
$181$	−46967.3			−1.43363			−0.716817	−	0.697262i	$-0.754402\pi$
$181$	−46967.3			−1.43363			−0.716817	+	0.697262i	$0.754402\pi$
$182$	24125.4			0.728337
$183$		−	1221.17i		−	0.0364648i
$184$	−20409.6			−0.602836
$185$		−	5741.09i		−	0.167746i
$186$	6721.67			0.194290
$187$			2414.33i			0.0690419i
$188$		−	10553.0i		−	0.298581i
$189$	10089.5			0.282453
$190$		−	16998.0i		−	0.470858i
$191$		−	35697.0i		−	0.978509i	−0.872141	−	0.489255i	$-0.837269\pi$
$191$		−	35697.0i		−	0.978509i	0.872141	−	0.489255i	$-0.162731\pi$
$192$	−2660.43			−0.0721688
$193$	25170.3			0.675730			0.337865	−	0.941195i	$-0.390295\pi$
$193$	25170.3			0.675730			0.337865	+	0.941195i	$0.390295\pi$
$194$	−30349.8			−0.806403
$195$			8033.77i			0.211276i
$196$	−22166.9			−0.577023
$197$	−68867.3			−1.77452			−0.887260	−	0.461270i	$-0.847394\pi$
$197$	−68867.3			−1.77452			−0.887260	+	0.461270i	$0.847394\pi$
$198$	5850.56			0.149234
$199$	−37635.2			−0.950361			−0.475180	−	0.879888i	$-0.657617\pi$
$199$	−37635.2			−0.950361			−0.475180	+	0.879888i	$0.657617\pi$
$200$		−	10297.1i		−	0.257428i
$201$		−	30901.8i		−	0.764878i
$202$	−8647.57			−0.211929
$203$	−83509.6			−2.02649
$204$	−1310.02			−0.0314789
$205$	28627.3			0.681196
$206$	−1169.59			−0.0275612
$207$			24353.6i			0.568359i
$208$			7590.79i			0.175453i
$209$			35319.1i			0.808569i
$210$		−	13777.8i		−	0.312423i
$211$			46880.1i			1.05299i	0.850179	+	0.526494i	$0.176494\pi$
$211$			46880.1i			1.05299i	−0.850179	+	0.526494i	$0.823506\pi$
$212$	−427.527			−0.00951244
$213$	40668.5			0.896394
$214$			1382.04i			0.0301781i
$215$			16259.1i			0.351738i
$216$			3174.54i			0.0680414i
$217$			32890.7i			0.698480i
$218$	−35008.9			−0.736658
$219$		−	42957.6i		−	0.895678i
$220$		−	7989.31i		−	0.165068i
$221$			3737.78i			0.0765296i
$222$	−6472.78			−0.131336
$223$	53767.4			1.08121			0.540604	−	0.841277i	$-0.318196\pi$
$223$	53767.4			1.08121			0.540604	+	0.841277i	$0.318196\pi$
$224$		−	13018.1i		−	0.259449i
$225$	−12287.0			−0.242706
$226$	−165.649			−0.00324319
$227$		−	13393.3i		−	0.259918i	−0.991519	−	0.129959i	$-0.958515\pi$
$227$		−	13393.3i		−	0.259918i	0.991519	−	0.129959i	$-0.0414846\pi$
$228$	−19164.3			−0.368658
$229$		−	5622.87i		−	0.107223i	−0.998562	−	0.0536114i	$-0.982927\pi$
$229$		−	5622.87i		−	0.107223i	0.998562	−	0.0536114i	$-0.0170732\pi$
$230$	33256.4			0.628666
$231$			28628.2i			0.536501i
$232$		−	26275.3i		−	0.488171i
$233$			104467.i			1.92428i	0.272549	+	0.962142i	$0.412133\pi$
$233$			104467.i			1.92428i	−0.272549	+	0.962142i	$0.587867\pi$
$234$	9057.65			0.165418
$235$			17195.7i			0.311374i
$236$	−5834.05	+	27230.0i	−0.104748	+	0.488905i
$237$	2698.91			0.0480499
$238$		−	6410.26i		−	0.113168i
$239$	−84162.7			−1.47341			−0.736706	−	0.676213i	$-0.763620\pi$
$239$	−84162.7			−1.47341			−0.736706	+	0.676213i	$0.763620\pi$
$240$	4335.04			0.0752610
$241$	−53479.0			−0.920766			−0.460383	−	0.887720i	$-0.652288\pi$
$241$	−53479.0			−0.920766			−0.460383	+	0.887720i	$0.652288\pi$
$242$		−	24810.5i		−	0.423647i
$243$	3788.00			0.0641500
$244$			1880.11i			0.0315794i
$245$	36119.8			0.601747
$246$		−	32275.7i		−	0.533342i
$247$			54679.9i			0.896260i
$248$	−10348.7			−0.168260
$249$			35952.0i			0.579861i
$250$			39822.5i			0.637161i
$251$	8610.91			0.136679			0.0683395	−	0.997662i	$-0.478230\pi$
$251$	8610.91			0.136679			0.0683395	+	0.997662i	$0.478230\pi$
$252$	−15533.8			−0.244611
$253$	−69101.6			−1.07956
$254$		−	19690.4i		−	0.305202i
$255$	2134.62			0.0328276
$256$	4096.00			0.0625000
$257$	−24174.3			−0.366005			−0.183003	−	0.983112i	$-0.558582\pi$
$257$	−24174.3			−0.366005			−0.183003	+	0.983112i	$0.558582\pi$
$258$	18331.2			0.275393
$259$		−	31672.9i		−	0.472159i
$260$		−	12368.8i		−	0.182970i
$261$	−31352.8			−0.460252
$262$	−38451.6			−0.560159
$263$	15619.3			0.225814			0.112907	−	0.993606i	$-0.463984\pi$
$263$	15619.3			0.225814			0.112907	+	0.993606i	$0.463984\pi$
$264$	−9007.53			−0.129240
$265$	696.634			0.00992002
$266$		−	93775.5i		−	1.32534i
$267$		−	6284.29i		−	0.0881524i
$268$			47576.5i			0.662404i
$269$		−	91681.8i		−	1.26701i	−0.773741	−	0.633503i	$-0.781617\pi$
$269$		−	91681.8i		−	1.26701i	0.773741	−	0.633503i	$-0.218383\pi$
$270$		−	5172.75i		−	0.0709568i
$271$	−23909.2			−0.325557			−0.162779	−	0.986663i	$-0.552046\pi$
$271$	−23909.2			−0.325557			−0.162779	+	0.986663i	$0.552046\pi$
$272$	2016.91			0.0272615
$273$			44321.3i			0.594685i
$274$			59826.1i			0.796874i
$275$		−	34863.4i		−	0.461004i
$276$		−	37494.8i		−	0.492213i
$277$	145442.			1.89553			0.947767	−	0.318964i	$-0.103335\pi$
$277$	145442.			1.89553			0.947767	+	0.318964i	$0.103335\pi$
$278$		−	58727.4i		−	0.759891i
$279$			12348.5i			0.158637i
$280$			21212.4i			0.270566i
$281$	−50468.8			−0.639161			−0.319581	−	0.947559i	$-0.603542\pi$
$281$	−50468.8			−0.639161			−0.319581	+	0.947559i	$0.603542\pi$
$282$	19387.2			0.243790
$283$		−	148907.i		−	1.85927i	−0.368488	−	0.929633i	$-0.620124\pi$
$283$		−	148907.i		−	1.85927i	0.368488	−	0.929633i	$-0.379876\pi$
$284$	−62613.3			−0.776300
$285$	31227.2			0.384454
$286$			25700.4i			0.314202i
$287$	157933.			1.91738
$288$		−	4887.52i		−	0.0589256i
$289$	−82527.9			−0.988109
$290$			42814.3i			0.509088i
$291$		−	55756.1i		−	0.658425i
$292$			66137.5i			0.775680i
$293$	63017.4			0.734049			0.367024	−	0.930211i	$-0.380377\pi$
$293$	63017.4			0.734049			0.367024	+	0.930211i	$0.380377\pi$
$294$		−	40723.2i		−	0.471137i
$295$	9506.29	−	44370.0i	0.109236	−	0.509853i
$296$	9965.49			0.113741
$297$			10748.2i			0.121849i
$298$	18388.1			0.207064
$299$	−106981.			−1.19664
$300$	18917.0			0.210189
$301$			89699.2i			0.990046i
$302$	13026.0			0.142823
$303$		−	15886.6i		−	0.173040i
$304$	29505.4			0.319267
$305$		−	3063.55i		−	0.0329325i
$306$		−	2406.67i		−	0.0257024i
$307$	9766.49			0.103624			0.0518122	−	0.998657i	$-0.483500\pi$
$307$	9766.49			0.103624			0.0518122	+	0.998657i	$0.483500\pi$
$308$		−	44076.0i		−	0.464623i
$309$		−	2148.67i		−	0.0225037i
$310$	16862.7			0.175470
$311$	57858.3			0.598198			0.299099	−	0.954222i	$-0.403314\pi$
$311$	57858.3			0.598198			0.299099	+	0.954222i	$0.403314\pi$
$312$	−13945.2			−0.143257
$313$		−	167799.i		−	1.71278i	−0.516331	−	0.856389i	$-0.672703\pi$
$313$		−	167799.i		−	1.71278i	0.516331	−	0.856389i	$-0.327297\pi$
$314$	−24225.2			−0.245702
$315$	25311.5			0.255092
$316$	−4155.25			−0.0416124
$317$	45806.8			0.455839			0.227919	−	0.973680i	$-0.426808\pi$
$317$	45806.8			0.455839			0.227919	+	0.973680i	$0.426808\pi$
$318$		−	785.417i		−	0.00776687i
$319$		−	88961.4i		−	0.874219i
$320$	−6674.22			−0.0651780
$321$	−2538.96			−0.0246403
$322$	183471.			1.76952
$323$	14528.8			0.139259
$324$	−5832.00			−0.0555556
$325$		−	53974.4i		−	0.511001i
$326$			4803.29i			0.0451964i
$327$		−	64315.5i		−	0.601479i
$328$			49691.7i			0.461888i
$329$			94866.2i			0.876435i
$330$	14677.3			0.134778
$331$	161647.			1.47540			0.737702	−	0.675127i	$-0.235911\pi$
$331$	161647.			1.47540			0.737702	+	0.675127i	$0.235911\pi$
$332$		−	55351.7i		−	0.502174i
$333$		−	11891.3i		−	0.107236i
$334$			115079.i			1.03158i
$335$		−	77523.5i		−	0.690786i
$336$	23915.8			0.211839
$337$			62124.9i			0.547024i	0.961869	+	0.273512i	$0.0881853\pi$
$337$			62124.9i			0.547024i	−0.961869	+	0.273512i	$0.911815\pi$
$338$		−	40994.1i		−	0.358829i
$339$		−	304.317i		−	0.00264806i
$340$	−3286.46			−0.0284296
$341$	−35038.0			−0.301321
$342$		−	35207.1i		−	0.301008i
$343$	26598.9			0.226087
$344$	−28222.8			−0.238497
$345$			61095.9i			0.513303i
$346$	−1057.37			−0.00883230
$347$		−	14358.0i		−	0.119243i	−0.998221	−	0.0596217i	$-0.981011\pi$
$347$		−	14358.0i		−	0.119243i	0.998221	−	0.0596217i	$-0.0189894\pi$
$348$	48270.8			0.398590
$349$		−	169000.i		−	1.38751i	−0.720212	−	0.693754i	$-0.755955\pi$
$349$		−	169000.i		−	1.38751i	0.720212	−	0.693754i	$-0.244045\pi$
$350$			92565.7i			0.755638i
$351$			16640.0i			0.135064i
$352$	13868.0			0.111925
$353$			187823.i			1.50730i	0.657278	+	0.753648i	$0.271708\pi$
$353$			187823.i			1.50730i	−0.657278	+	0.753648i	$0.728292\pi$
$354$	−50024.8	−	10717.8i	−0.399189	−	0.0855265i
$355$	102025.			0.809563
$356$			9675.30i			0.0763422i
$357$	11776.4			0.0924009
$358$	136948.			1.06853
$359$	−203966.			−1.58259			−0.791297	−	0.611432i	$-0.790594\pi$
$359$	−203966.			−1.58259			−0.791297	+	0.611432i	$0.790594\pi$
$360$			7963.97i			0.0614504i
$361$	82219.6			0.630900
$362$			132843.i			1.01373i
$363$	45579.7			0.345906
$364$		−	68237.1i		−	0.515012i
$365$		−	107768.i		−	0.808915i
$366$	−3453.99			−0.0257845
$367$			21743.1i			0.161432i	0.996737	+	0.0807159i	$0.0257207\pi$
$367$			21743.1i			0.161432i	−0.996737	+	0.0807159i	$0.974279\pi$
$368$			57727.1i			0.426269i
$369$	59294.3			0.435472
$370$	−16238.3			−0.118614
$371$	3843.24			0.0279222
$372$		−	19011.7i		−	0.137384i
$373$	47541.9			0.341711			0.170855	−	0.985296i	$-0.445347\pi$
$373$	47541.9			0.341711			0.170855	+	0.985296i	$0.445347\pi$
$374$	6828.75			0.0488200
$375$	−73158.7			−0.520240
$376$	−29848.5			−0.211129
$377$		−	137727.i		−	0.969029i
$378$		−	28537.4i		−	0.199724i
$379$	−87143.6			−0.606676			−0.303338	−	0.952883i	$-0.598101\pi$
$379$	−87143.6			−0.606676			−0.303338	+	0.952883i	$0.598101\pi$
$380$	−48077.5			−0.332947
$381$	36173.6			0.249197
$382$	−100966.			−0.691910
$383$	−20889.8			−0.142408			−0.0712042	−	0.997462i	$-0.522684\pi$
$383$	−20889.8			−0.142408			−0.0712042	+	0.997462i	$0.522684\pi$
$384$			7524.83i			0.0510310i
$385$			71819.6i			0.484531i
$386$		−	71192.2i		−	0.477813i
$387$			33676.6i			0.224857i
$388$			85842.2i			0.570213i
$389$	−226227.			−1.49501			−0.747506	−	0.664255i	$-0.768749\pi$
$389$	−226227.			−1.49501			−0.747506	+	0.664255i	$0.768749\pi$
$390$	22722.9			0.149395
$391$			28425.4i			0.185932i
$392$			62697.5i			0.408017i
$393$		−	70640.1i		−	0.457368i
$394$			194786.i			1.25477i
$395$	6770.77			0.0433954
$396$		−	16547.9i		−	0.105524i
$397$		−	258128.i		−	1.63777i	−0.573956	−	0.818886i	$-0.694592\pi$
$397$		−	258128.i		−	1.63777i	0.573956	−	0.818886i	$-0.305408\pi$
$398$			106449.i			0.672006i
$399$	172277.			1.08213
$400$	−29124.7			−0.182029
$401$			121353.i			0.754678i	0.926075	+	0.377339i	$0.123161\pi$
$401$			121353.i			0.754678i	−0.926075	+	0.377339i	$0.876839\pi$
$402$	−87403.6			−0.540850
$403$	−54244.6			−0.334000
$404$			24459.0i			0.149857i
$405$	9502.95			0.0579360
$406$			236201.i			1.43294i
$407$	33740.6			0.203687
$408$			3705.31i			0.0222589i
$409$		−	127039.i		−	0.759434i	−0.925103	−	0.379717i	$-0.876021\pi$
$409$		−	127039.i		−	0.759434i	0.925103	−	0.379717i	$-0.123979\pi$
$410$		−	80970.2i		−	0.481678i
$411$	−109908.			−0.650645
$412$			3308.10i			0.0194887i
$413$	52445.0	−	244783.i	0.307471	−	1.43510i
$414$	68882.4			0.401890
$415$			90192.7i			0.523691i
$416$	21470.0			0.124064
$417$	107889.			0.620448
$418$	99897.5			0.571745
$419$		−	66337.2i		−	0.377858i	−0.981991	−	0.188929i	$-0.939498\pi$
$419$		−	66337.2i		−	0.377858i	0.981991	−	0.188929i	$-0.0605017\pi$
$420$	−38969.6			−0.220916
$421$			316689.i			1.78677i	0.449290	+	0.893386i	$0.351677\pi$
$421$			316689.i			1.78677i	−0.449290	+	0.893386i	$0.648323\pi$
$422$	132597.			0.744575
$423$			35616.5i			0.199054i
$424$			1209.23i			0.00672631i
$425$	−14341.3			−0.0793982
$426$		−	115028.i		−	0.633847i
$427$		−	16901.2i		−	0.0926962i
$428$	3908.99			0.0213391
$429$	−47214.7			−0.256545
$430$	45987.6			0.248716
$431$		−	19159.7i		−	0.103142i	−0.998669	−	0.0515709i	$-0.983577\pi$
$431$		−	19159.7i		−	0.103142i	0.998669	−	0.0515709i	$-0.0164228\pi$
$432$	8978.95			0.0481125
$433$	59077.3			0.315097			0.157549	−	0.987511i	$-0.449641\pi$
$433$	59077.3			0.315097			0.157549	+	0.987511i	$0.449641\pi$
$434$	93029.1			0.493900
$435$	−78654.8			−0.415668
$436$			99020.2i			0.520896i
$437$			415834.i			2.17750i
$438$	−121502.			−0.633340
$439$	16622.3			0.0862507			0.0431253	−	0.999070i	$-0.486269\pi$
$439$	16622.3			0.0862507			0.0431253	+	0.999070i	$0.486269\pi$
$440$	−22597.2			−0.116721
$441$	74813.3			0.384682
$442$	10572.0			0.0541146
$443$		−	166081.i		−	0.846278i	−0.906065	−	0.423139i	$-0.860928\pi$
$443$		−	166081.i		−	0.846278i	0.906065	−	0.423139i	$-0.139072\pi$
$444$			18307.8i			0.0928688i
$445$		−	15765.4i		−	0.0796132i
$446$		−	152077.i		−	0.764530i
$447$			33781.0i			0.169067i
$448$	−36820.8			−0.183458
$449$	231721.			1.14940			0.574702	−	0.818363i	$-0.305118\pi$
$449$	231721.			1.14940			0.574702	+	0.818363i	$0.305118\pi$
$450$			34752.8i			0.171619i
$451$			168243.i			0.827151i
$452$			468.527i			0.00229328i
$453$			23930.3i			0.116614i
$454$	−37882.1			−0.183790
$455$			111189.i			0.537079i
$456$			54204.8i			0.260680i
$457$			58668.2i			0.280912i	0.990087	+	0.140456i	$0.0448569\pi$
$457$			58668.2i			0.280912i	−0.990087	+	0.140456i	$0.955143\pi$
$458$	−15903.9			−0.0758179
$459$	4421.33			0.0209859
$460$		−	94063.3i		−	0.444534i
$461$	92384.8			0.434709			0.217355	−	0.976093i	$-0.430257\pi$
$461$	92384.8			0.434709			0.217355	+	0.976093i	$0.430257\pi$
$462$	80972.8			0.379363
$463$		−	348577.i		−	1.62606i	−0.582220	−	0.813031i	$-0.697816\pi$
$463$		−	348577.i		−	1.62606i	0.582220	−	0.813031i	$-0.302184\pi$
$464$	−74317.8			−0.345189
$465$			30978.7i			0.143271i
$466$	295479.			1.36067
$467$		−	270751.i		−	1.24147i	−0.784021	−	0.620734i	$-0.786835\pi$
$467$		−	270751.i		−	1.24147i	0.784021	−	0.620734i	$-0.213165\pi$
$468$		−	25618.9i		−	0.116968i
$469$		−	427687.i		−	1.94438i
$470$	48636.7			0.220175
$471$		−	44504.6i		−	0.200615i
$472$	77018.2	+	16501.2i	0.345708	+	0.0740681i
$473$	−95555.1			−0.427102
$474$		−	7633.68i		−	0.0339764i
$475$	−209798.			−0.929855
$476$	−18131.0			−0.0800216
$477$	1442.90			0.00634163
$478$			238048.i			1.04186i
$479$	265996.			1.15932			0.579660	−	0.814858i	$-0.303185\pi$
$479$	265996.			1.15932			0.579660	+	0.814858i	$0.303185\pi$
$480$		−	12261.3i		−	0.0532176i
$481$	52236.1			0.225777
$482$			151262.i			0.651080i
$483$			337058.i			1.44481i
$484$	−70174.6			−0.299564
$485$		−	139875.i		−	0.594645i
$486$		−	10714.1i		−	0.0453609i
$487$	179950.			0.758743			0.379371	−	0.925245i	$-0.376140\pi$
$487$	179950.			0.758743			0.379371	+	0.925245i	$0.376140\pi$
$488$	5317.76			0.0223300
$489$	−8824.21			−0.0369027
$490$		−	102162.i		−	0.425499i
$491$	180154.			0.747277			0.373639	−	0.927574i	$-0.378110\pi$
$491$	180154.			0.747277			0.373639	+	0.927574i	$0.378110\pi$
$492$	−91289.5			−0.377130
$493$	−36594.9			−0.150566
$494$	154658.			0.633751
$495$			26963.9i			0.110046i
$496$			29270.5i			0.118978i
$497$	562859.			2.27870
$498$	101687.			0.410024
$499$	348732.			1.40052			0.700261	−	0.713887i	$-0.253067\pi$
$499$	348732.			1.40052			0.700261	+	0.713887i	$0.253067\pi$
$500$	112635.			0.450541
$501$	−211414.			−0.842284
$502$		−	24355.3i		−	0.0966466i
$503$		−	337691.i		−	1.33470i	−0.744744	−	0.667350i	$-0.767428\pi$
$503$		−	337691.i		−	1.33470i	0.744744	−	0.667350i	$-0.232572\pi$
$504$			43936.2i			0.172966i
$505$		−	39854.7i		−	0.156278i
$506$			195449.i			0.763365i
$507$	75311.0			0.292983
$508$	−55692.9			−0.215811
$509$			423913.i			1.63622i	0.575062	+	0.818110i	$0.304978\pi$
$509$			423913.i			1.63622i	−0.575062	+	0.818110i	$0.695022\pi$
$510$		−	6037.61i		−	0.0232126i
$511$		−	594541.i		−	2.27688i
$512$		−	11585.2i		−	0.0441942i
$513$	64679.5			0.245772
$514$			68375.2i			0.258805i
$515$		−	5390.38i		−	0.0203238i
$516$		−	51848.6i		−	0.194732i
$517$	−101059.			−0.378090
$518$	−89584.4			−0.333867
$519$		−	1942.51i		−	0.00721155i
$520$	−34984.2			−0.129380
$521$	124425.			0.458387			0.229194	−	0.973381i	$-0.426391\pi$
$521$	124425.			0.458387			0.229194	+	0.973381i	$0.426391\pi$
$522$			88679.1i			0.325447i
$523$	−515652.			−1.88518			−0.942591	−	0.333950i	$-0.891618\pi$
$523$	−515652.			−1.88518			−0.942591	+	0.333950i	$0.891618\pi$
$524$			108757.i			0.396092i
$525$	−170054.			−0.616976
$526$		−	44178.1i		−	0.159675i
$527$			14413.1i			0.0518963i
$528$			25477.1i			0.0913867i
$529$	−533737.			−1.90729
$530$		−	1970.38i		−	0.00701452i
$531$	19689.9	−	91901.4i	0.0698321	−	0.325936i
$532$	−265237.			−0.937154
$533$			260469.i			0.916857i
$534$	−17774.7			−0.0623331
$535$	−6369.50			−0.0222535
$536$	134567.			0.468390
$537$			251589.i			0.872454i
$538$	−259315.			−0.895908
$539$			212277.i			0.730678i
$540$	−14630.7			−0.0501740
$541$			273373.i			0.934032i	0.884249	+	0.467016i	$0.154671\pi$
$541$			273373.i			0.934032i	−0.884249	+	0.467016i	$0.845329\pi$
$542$			67625.5i			0.230204i
$543$	−244049.			−0.827709
$544$		−	5704.69i		−	0.0192768i
$545$		−	161348.i		−	0.543215i
$546$	125359.			0.420506
$547$	−382095.			−1.27702			−0.638508	−	0.769615i	$-0.720448\pi$
$547$	−382095.			−1.27702			−0.638508	+	0.769615i	$0.720448\pi$
$548$	169214.			0.563475
$549$		−	6345.38i		−	0.0210530i
$550$	−98608.7			−0.325979
$551$	−535345.			−1.76332
$552$	−106051.			−0.348047
$553$	37353.4			0.122146
$554$		−	411373.i		−	1.34034i
$555$		−	29831.6i		−	0.0968480i
$556$	−166106.			−0.537324
$557$	67859.1			0.218724			0.109362	−	0.994002i	$-0.465119\pi$
$557$	67859.1			0.218724			0.109362	+	0.994002i	$0.465119\pi$
$558$	34926.8			0.112174
$559$	−147935.			−0.473422
$560$	59997.7			0.191319
$561$			12545.2i			0.0398614i
$562$			142747.i			0.451955i
$563$		−	181249.i		−	0.571818i	−0.958257	−	0.285909i	$-0.907704\pi$
$563$		−	181249.i		−	0.571818i	0.958257	−	0.285909i	$-0.0922956\pi$
$564$		−	54835.2i		−	0.172386i
$565$		−	763.441i		−	0.00239155i
$566$	−421172.			−1.31470
$567$	52426.5			0.163074
$568$			177097.i			0.548927i
$569$		−	162103.i		−	0.500687i	−0.968157	−	0.250344i	$-0.919456\pi$
$569$		−	162103.i		−	0.500687i	0.968157	−	0.250344i	$-0.0805436\pi$
$570$		−	88324.0i		−	0.271850i
$571$		−	646936.i		−	1.98422i	−0.125386	−	0.992108i	$-0.540017\pi$
$571$		−	646936.i		−	1.98422i	0.125386	−	0.992108i	$-0.459983\pi$
$572$	72691.8			0.222174
$573$		−	185487.i		−	0.564942i
$574$		−	446702.i		−	1.35579i
$575$		−	410470.i		−	1.24150i
$576$	−13824.0			−0.0416667
$577$	−249691.			−0.749984			−0.374992	−	0.927028i	$-0.622355\pi$
$577$	−249691.			−0.749984			−0.374992	+	0.927028i	$0.622355\pi$
$578$			233424.i			0.698699i
$579$	130788.			0.390133
$580$	121097.			0.359979
$581$			497581.i			1.47405i
$582$	−157702.			−0.465577
$583$			4094.14i			0.0120455i
$584$	187065.			0.548488
$585$			41744.7i			0.121980i
$586$		−	178240.i		−	0.519051i
$587$			124990.i			0.362744i	0.983415	+	0.181372i	$0.0580539\pi$
$587$			124990.i			0.362744i	−0.983415	+	0.181372i	$0.941946\pi$
$588$	−115183.			−0.333144
$589$			210849.i			0.607772i
$590$	−125497.	−	26887.9i	−0.360521	−	0.0772417i
$591$	−357845.			−1.02452
$592$		−	28186.7i		−	0.0804267i
$593$	−562164.			−1.59865			−0.799326	−	0.600898i	$-0.794810\pi$
$593$	−562164.			−1.59865			−0.799326	+	0.600898i	$0.794810\pi$
$594$	30400.4			0.0861602
$595$	29543.5			0.0834503
$596$		−	52009.3i		−	0.146416i
$597$	−195558.			−0.548691
$598$			302588.i			0.846153i
$599$	−128620.			−0.358471			−0.179235	−	0.983806i	$-0.557362\pi$
$599$	−128620.			−0.358471			−0.179235	+	0.983806i	$0.557362\pi$
$600$		−	53505.5i		−	0.148626i
$601$			443026.i			1.22654i	0.789874	+	0.613269i	$0.210146\pi$
$601$			443026.i			1.22654i	−0.789874	+	0.613269i	$0.789854\pi$
$602$	253708.			0.700068
$603$		−	160571.i		−	0.441602i
$604$		−	36843.1i		−	0.100991i
$605$	114346.			0.312399
$606$	−44934.1			−0.122358
$607$	174622.			0.473938			0.236969	−	0.971517i	$-0.423846\pi$
$607$	174622.			0.473938			0.236969	+	0.971517i	$0.423846\pi$
$608$		−	83453.8i		−	0.225756i
$609$	−433928.			−1.16999
$610$	−8665.02			−0.0232868
$611$	−156457.			−0.419095
$612$	−6807.08			−0.0181743
$613$			528180.i			1.40560i	0.711389	+	0.702799i	$0.248067\pi$
$613$			528180.i			1.40560i	−0.711389	+	0.702799i	$0.751933\pi$
$614$		−	27623.8i		−	0.0732735i
$615$	148752.			0.393289
$616$	−124666.			−0.328538
$617$	274885.			0.722072			0.361036	−	0.932552i	$-0.382423\pi$
$617$	274885.			0.722072			0.361036	+	0.932552i	$0.382423\pi$
$618$	−6077.36			−0.0159125
$619$	−372699.			−0.972695			−0.486347	−	0.873766i	$-0.661671\pi$
$619$	−372699.			−0.972695			−0.486347	+	0.873766i	$0.661671\pi$
$620$		−	47694.8i		−	0.124076i
$621$			126545.i			0.328142i
$622$		−	163648.i		−	0.422990i
$623$		−	86975.7i		−	0.224090i
$624$			39442.9i			0.101298i
$625$	100888.			0.258272
$626$	−474608.			−1.21112
$627$			183523.i			0.466828i
$628$			68519.3i			0.173737i
$629$		−	13879.4i		−	0.0350808i
$630$		−	71591.8i		−	0.180377i
$631$	−654254.			−1.64319			−0.821594	−	0.570072i	$-0.806915\pi$
$631$	−654254.			−1.64319			−0.821594	+	0.570072i	$0.806915\pi$
$632$			11752.8i			0.0294244i
$633$			243596.i			0.607943i
$634$		−	129561.i		−	0.322327i
$635$	90748.8			0.225057
$636$	−2221.50			−0.00549201
$637$			328641.i			0.809922i
$638$	−251621.			−0.618166
$639$	211320.			0.517534
$640$			18877.6i			0.0460878i
$641$	140139.			0.341069			0.170535	−	0.985352i	$-0.445451\pi$
$641$	140139.			0.341069			0.170535	+	0.985352i	$0.445451\pi$
$642$			7181.27i			0.0174233i
$643$	290047.			0.701531			0.350765	−	0.936463i	$-0.385921\pi$
$643$	290047.			0.701531			0.350765	+	0.936463i	$0.385921\pi$
$644$		−	518935.i		−	1.25124i
$645$			84484.6i			0.203076i
$646$		−	41093.5i		−	0.0984710i
$647$	379096.			0.905608			0.452804	−	0.891610i	$-0.350424\pi$
$647$	379096.			0.905608			0.452804	+	0.891610i	$0.350424\pi$
$648$			16495.4i			0.0392837i
$649$	260764.	+	55868.8i	0.619095	+	0.132642i
$650$	−152663.			−0.361332
$651$			170905.i			0.403268i
$652$	13585.8			0.0319587
$653$	3248.56			0.00761842			0.00380921	−	0.999993i	$-0.498787\pi$
$653$	3248.56			0.00761842			0.00380921	+	0.999993i	$0.498787\pi$
$654$	−181912.			−0.425310
$655$		−	177215.i		−	0.413064i
$656$	140549.			0.326604
$657$		−	223214.i		−	0.517120i
$658$	268322.			0.619733
$659$			530315.i			1.22113i	0.791965	+	0.610567i	$0.209058\pi$
$659$			530315.i			1.22113i	−0.791965	+	0.610567i	$0.790942\pi$
$660$		−	41513.7i		−	0.0953023i
$661$	−241571.			−0.552894			−0.276447	−	0.961029i	$-0.589157\pi$
$661$	−241571.			−0.552894			−0.276447	+	0.961029i	$0.589157\pi$
$662$		−	457206.i		−	1.04327i
$663$			19422.1i			0.0441844i
$664$	−156558.			−0.355091
$665$	432191.			0.977309
$666$	−33633.5			−0.0758271
$667$		−	1.04740e6i		−	2.35429i
$668$	325493.			0.729439
$669$	279384.			0.624236
$670$	−219269.			−0.488460
$671$	18004.6			0.0399887
$672$		−	67644.2i		−	0.149793i
$673$			720933.i			1.59171i	0.605485	+	0.795857i	$0.292979\pi$
$673$			720933.i			1.59171i	−0.605485	+	0.795857i	$0.707021\pi$
$674$	175716.			0.386804
$675$	−63845.0			−0.140126
$676$	−115949.			−0.253731
$677$	53229.2			0.116138			0.0580688	−	0.998313i	$-0.481506\pi$
$677$	53229.2			0.116138			0.0580688	+	0.998313i	$0.481506\pi$
$678$	−860.739			−0.00187246
$679$		−	771675.i		−	1.67377i
$680$			9295.51i			0.0201027i
$681$		−	69593.8i		−	0.150064i
$682$			99102.3i			0.213066i
$683$			116305.i			0.249320i	0.992200	+	0.124660i	$0.0397840\pi$
$683$			116305.i			0.249320i	−0.992200	+	0.124660i	$0.960216\pi$
$684$	−99580.6			−0.212845
$685$	−275725.			−0.587619
$686$		−	75233.2i		−	0.159868i
$687$		−	29217.3i		−	0.0619051i
$688$			79826.1i			0.168643i
$689$			6338.41i			0.0133519i
$690$	172805.			0.362960
$691$		−	363295.i		−	0.760858i	−0.924810	−	0.380429i	$-0.875776\pi$
$691$		−	363295.i		−	0.760858i	0.924810	−	0.380429i	$-0.124224\pi$
$692$			2990.69i			0.00624538i
$693$			148757.i			0.309749i
$694$	−40610.5			−0.0843178
$695$	270661.			0.560347
$696$		−	136530.i		−	0.281846i
$697$	69208.0			0.142459
$698$	−478004.			−0.981117
$699$			542829.i			1.11099i
$700$	261815.			0.534317
$701$			795382.i			1.61860i	0.587395	+	0.809300i	$0.300153\pi$
$701$			795382.i			1.61860i	−0.587395	+	0.809300i	$0.699847\pi$
$702$	47064.9			0.0955044
$703$		−	203042.i		−	0.410841i
$704$		−	39224.6i		−	0.0791432i
$705$			89351.2i			0.179772i
$706$	531243.			1.06582
$707$		−	219873.i		−	0.439880i
$708$	−30314.6	+	141491.i	−0.0604764	+	0.282269i
$709$	557121.			1.10830			0.554150	−	0.832417i	$-0.313044\pi$
$709$	557121.			1.10830			0.554150	+	0.832417i	$0.313044\pi$
$710$		−	288571.i		−	0.572447i
$711$	14024.0			0.0277416
$712$	27365.9			0.0539821
$713$	−412524.			−0.811467
$714$		−	33308.7i		−	0.0653373i
$715$	−118448.			−0.231694
$716$		−	387346.i		−	0.755568i
$717$	−437322.			−0.850675
$718$			576904.i			1.11906i
$719$			755892.i			1.46218i	0.682279	+	0.731092i	$0.260989\pi$
$719$			755892.i			1.46218i	−0.682279	+	0.731092i	$0.739011\pi$
$720$	22525.5			0.0434520
$721$		−	29738.0i		−	0.0572060i
$722$		−	232552.i		−	0.446114i
$723$	−277885.			−0.531605
$724$	375738.			0.716817
$725$	528438.			1.00535
$726$		−	128919.i		−	0.244593i
$727$	−496616.			−0.939620			−0.469810	−	0.882768i	$-0.655678\pi$
$727$	−496616.			−0.939620			−0.469810	+	0.882768i	$0.655678\pi$
$728$	−193004.			−0.364169
$729$	19683.0			0.0370370
$730$	−304813.			−0.571990
$731$			39307.2i			0.0735593i
$732$			9769.35i			0.0182324i
$733$	−456465.			−0.849570			−0.424785	−	0.905294i	$-0.639650\pi$
$733$	−456465.			−0.849570			−0.424785	+	0.905294i	$0.639650\pi$
$734$	61498.7			0.114150
$735$	187684.			0.347419
$736$	163277.			0.301418
$737$	455608.			0.838796
$738$		−	167710.i		−	0.307925i
$739$			264440.i			0.484214i	0.970250	+	0.242107i	$0.0778386\pi$
$739$			264440.i			0.484214i	−0.970250	+	0.242107i	$0.922161\pi$
$740$			45928.8i			0.0838728i
$741$			284125.i			0.517456i
$742$		−	10870.3i		−	0.0197440i
$743$	−100102.			−0.181329			−0.0906645	−	0.995881i	$-0.528899\pi$
$743$	−100102.			−0.181329			−0.0906645	+	0.995881i	$0.528899\pi$
$744$	−53773.3			−0.0971451
$745$			84746.5i			0.152690i
$746$		−	134469.i		−	0.241626i
$747$			186812.i			0.334783i
$748$		−	19314.6i		−	0.0345210i
$749$	−35139.7			−0.0626375
$750$			206924.i			0.367865i
$751$		−	112757.i		−	0.199923i	−0.994991	−	0.0999616i	$-0.968128\pi$
$751$		−	112757.i		−	0.199923i	0.994991	−	0.0999616i	$-0.0318720\pi$
$752$			84424.4i			0.149291i
$753$	44743.6			0.0789116
$754$	−389551.			−0.685207
$755$			60033.9i			0.105318i
$756$	−80715.9			−0.141226
$757$	594305.			1.03709			0.518547	−	0.855049i	$-0.326473\pi$
$757$	594305.			1.03709			0.518547	+	0.855049i	$0.326473\pi$
$758$			246479.i			0.428985i
$759$	−359063.			−0.623285
$760$			135984.i			0.235429i
$761$	−493882.			−0.852813			−0.426407	−	0.904532i	$-0.640221\pi$
$761$	−493882.			−0.852813			−0.426407	+	0.904532i	$0.640221\pi$
$762$		−	102314.i		−	0.176209i
$763$		−	890138.i		−	1.52900i
$764$			285576.i			0.489255i
$765$	11091.8			0.0189530
$766$			59085.1i			0.100698i
$767$	403706.	+	86494.2i	0.686237	+	0.147027i
$768$	21283.4			0.0360844
$769$			526058.i			0.889572i	0.895637	+	0.444786i	$0.146720\pi$
$769$			526058.i			0.889572i	−0.895637	+	0.444786i	$0.853280\pi$
$770$	203137.			0.342615
$771$	−125613.			−0.211313
$772$	−201362.			−0.337865
$773$			820920.i			1.37386i	0.726725	+	0.686929i	$0.241042\pi$
$773$			820920.i			1.37386i	−0.726725	+	0.686929i	$0.758958\pi$
$774$	95251.9			0.158998
$775$		−	208128.i		−	0.346520i
$776$	242798.			0.403202
$777$		−	164577.i		−	0.272601i
$778$			639866.i			1.05713i
$779$	1.01244e6			1.66838
$780$		−	64270.2i		−	0.105638i
$781$			599605.i			0.983022i
$782$	80399.2			0.131474
$783$	−162914.			−0.265726
$784$	177335.			0.288511
$785$		−	111649.i		−	0.181182i
$786$	−199800.			−0.323408
$787$	38812.3			0.0626643			0.0313321	−	0.999509i	$-0.490025\pi$
$787$	38812.3			0.0626643			0.0313321	+	0.999509i	$0.490025\pi$
$788$	550939.			0.887260
$789$	81160.4			0.130374
$790$		−	19150.6i		−	0.0306852i
$791$		−	4211.80i		−	0.00673155i
$792$	−46804.5			−0.0746169
$793$	27874.1			0.0443256
$794$	−730095.			−1.15808
$795$	3619.81			0.00572733
$796$	301082.			0.475180
$797$		−	209085.i		−	0.329160i	−0.986364	−	0.164580i	$-0.947373\pi$
$797$		−	209085.i		−	0.329160i	0.986364	−	0.164580i	$-0.0526269\pi$
$798$		−	487272.i		−	0.765183i
$799$			41571.5i			0.0651181i
$800$			82377.1i			0.128714i
$801$		−	32654.1i		−	0.0508948i
$802$	343238.			0.533638
$803$	633354.			0.982236
$804$			247215.i			0.382439i
$805$			845578.i			1.30485i
$806$			153427.i			0.236174i
$807$		−	476392.i		−	0.731506i
$808$	69180.6			0.105965
$809$		−	1.28940e6i		−	1.97011i	−0.172233	−	0.985056i	$-0.555098\pi$
$809$		−	1.28940e6i		−	1.97011i	0.172233	−	0.985056i	$-0.444902\pi$
$810$		−	26878.4i		−	0.0409669i
$811$		−	448438.i		−	0.681806i	−0.940099	−	0.340903i	$-0.889267\pi$
$811$		−	448438.i		−	0.681806i	0.940099	−	0.340903i	$-0.110733\pi$
$812$	668077.			1.01324
$813$	−124236.			−0.187960
$814$		−	95432.8i		−	0.144029i
$815$	−22137.3			−0.0333280
$816$	10480.2			0.0157394
$817$			575024.i			0.861473i
$818$	−359320.			−0.537001
$819$			230300.i			0.343341i
$820$	−229018.			−0.340598
$821$			259012.i			0.384268i	0.981369	+	0.192134i	$0.0615408\pi$
$821$			259012.i			0.384268i	−0.981369	+	0.192134i	$0.938459\pi$
$822$			310866.i			0.460076i
$823$			837577.i			1.23659i	0.785947	+	0.618294i	$0.212176\pi$
$823$			837577.i			1.23659i	−0.785947	+	0.618294i	$0.787824\pi$
$824$	9356.71			0.0137806
$825$		−	181156.i		−	0.266161i
$826$	−692352.	−	148337.i	−1.01477	−	0.217415i
$827$	739452.			1.08118			0.540592	−	0.841285i	$-0.318200\pi$
$827$	739452.			1.08118			0.540592	+	0.841285i	$0.318200\pi$
$828$		−	194829.i		−	0.284179i
$829$	1.21521e6			1.76824			0.884122	−	0.467257i	$-0.154758\pi$
$829$	1.21521e6			1.76824			0.884122	+	0.467257i	$0.154758\pi$
$830$	255104.			0.370306
$831$	755741.			1.09439
$832$		−	60726.3i		−	0.0877264i
$833$	87321.8			0.125844
$834$		−	305156.i		−	0.438723i
$835$	−530375.			−0.760694
$836$		−	282553.i		−	0.404285i
$837$			64164.6i			0.0915893i
$838$	−187630.			−0.267186
$839$		−	1.25452e6i		−	1.78219i	−0.453818	−	0.891095i	$-0.649938\pi$
$839$		−	1.25452e6i		−	1.78219i	0.453818	−	0.891095i	$-0.350062\pi$
$840$			110223.i			0.156211i
$841$	641140.			0.906485
$842$	895732.			1.26344
$843$	−262244.			−0.369020
$844$		−	375040.i		−	0.526494i
$845$	188933.			0.264602
$846$	100739.			0.140752
$847$	630831.			0.879319
$848$	3420.22			0.00475622
$849$		−	773742.i		−	1.07345i
$850$			40563.3i			0.0561430i
$851$	397250.			0.548535
$852$	−325348.			−0.448197
$853$	−860547.			−1.18270			−0.591352	−	0.806413i	$-0.701406\pi$
$853$	−860547.			−1.18270			−0.591352	+	0.806413i	$0.701406\pi$
$854$	−47803.8			−0.0655461
$855$	162262.			0.221964
$856$		−	11056.3i		−	0.0150891i
$857$			250066.i			0.340481i	0.985403	+	0.170240i	$0.0544544\pi$
$857$			250066.i			0.340481i	−0.985403	+	0.170240i	$0.945546\pi$
$858$			133543.i			0.181404i
$859$		−	1.05336e6i		−	1.42755i	−0.700374	−	0.713776i	$-0.746983\pi$
$859$		−	1.05336e6i		−	1.42755i	0.700374	−	0.713776i	$-0.253017\pi$
$860$		−	130073.i		−	0.175869i
$861$	820644.			1.10700
$862$	−54191.9			−0.0729323
$863$			284135.i			0.381507i	0.981638	+	0.190754i	$0.0610931\pi$
$863$			284135.i			0.381507i	−0.981638	+	0.190754i	$0.938907\pi$
$864$		−	25396.3i		−	0.0340207i
$865$		−	4873.18i		−	0.00651298i
$866$		−	167096.i		−	0.222808i
$867$	−428827.			−0.570485
$868$		−	263126.i		−	0.349240i
$869$			39792.0i			0.0526934i
$870$			222469.i			0.293922i
$871$	705357.			0.929764
$872$	280071.			0.368329
$873$		−	289717.i		−	0.380142i
$874$	1.17616e6			1.53972
$875$	−1.01253e6			−1.32249
$876$			343661.i			0.447839i
$877$	650765.			0.846106			0.423053	−	0.906105i	$-0.360958\pi$
$877$	650765.			0.846106			0.423053	+	0.906105i	$0.360958\pi$
$878$		−	47015.0i		−	0.0609884i
$879$	327448.			0.423803
$880$			63914.5i			0.0825342i
$881$		−	804733.i		−	1.03681i	−0.855135	−	0.518406i	$-0.826526\pi$
$881$		−	804733.i		−	1.03681i	0.855135	−	0.518406i	$-0.173474\pi$
$882$		−	211604.i		−	0.272011i
$883$	−1.12964e6			−1.44883			−0.724415	−	0.689364i	$-0.757890\pi$
$883$	−1.12964e6			−1.44883			−0.724415	+	0.689364i	$0.757890\pi$
$884$		−	29902.3i		−	0.0382648i
$885$	49396.1	−	230553.i	0.0630676	−	0.294364i
$886$	−469748.			−0.598409
$887$		−	163030.i		−	0.207214i	−0.994618	−	0.103607i	$-0.966962\pi$
$887$		−	163030.i		−	0.207214i	0.994618	−	0.103607i	$-0.0330385\pi$
$888$	51782.2			0.0656682
$889$	500649.			0.633476
$890$	−44591.3			−0.0562951
$891$			55849.1i			0.0703495i
$892$	−430139.			−0.540604
$893$			608147.i			0.762616i
$894$	95547.2			0.119548
$895$			631161.i			0.787942i
$896$			104145.i			0.129725i
$897$	−555889.			−0.690881
$898$		−	655406.i		−	0.812751i
$899$		−	531084.i		−	0.657118i
$900$	98295.8			0.121353
$901$	1684.15			0.00207459
$902$	475864.			0.584884
$903$			466091.i			0.571603i
$904$	1325.19			0.00162160
$905$	−612246.			−0.747530
$906$	67685.1			0.0824587
$907$	32431.9			0.0394237			0.0197119	−	0.999806i	$-0.493725\pi$
$907$	32431.9			0.0394237			0.0197119	+	0.999806i	$0.493725\pi$
$908$			107147.i			0.129959i
$909$		−	82549.2i		−	0.0999045i
$910$	314489.			0.379772
$911$	−542118.			−0.653216			−0.326608	−	0.945160i	$-0.605906\pi$
$911$	−542118.			−0.653216			−0.326608	+	0.945160i	$0.605906\pi$
$912$	153314.			0.184329
$913$	−530065.			−0.635899
$914$	165939.			0.198635
$915$		−	15918.7i		−	0.0190136i
$916$			44983.0i			0.0536114i
$917$		−	977671.i		−	1.16266i
$918$		−	12505.4i		−	0.0148393i
$919$			631129.i			0.747286i	0.927573	+	0.373643i	$0.121891\pi$
$919$			631129.i			0.747286i	−0.927573	+	0.373643i	$0.878109\pi$
$920$	−266051.			−0.314333
$921$	50748.2			0.0598276
$922$		−	261304.i		−	0.307386i
$923$			928289.i			1.08963i
$924$		−	229026.i		−	0.268250i
$925$			200422.i			0.234240i
$926$	−985926.			−1.14980
$927$		−	11164.8i		−	0.0129925i
$928$			210202.i			0.244085i
$929$		−	43280.6i		−	0.0501490i	−0.999686	−	0.0250745i	$-0.992018\pi$
$929$		−	43280.6i		−	0.0501490i	0.999686	−	0.0250745i	$-0.00798230\pi$
$930$	87620.9			0.101308
$931$	1.27743e6			1.47379
$932$		−	835739.i		−	0.962142i
$933$	300641.			0.345370
$934$	−765799.			−0.877851
$935$			31472.2i			0.0360001i
$936$	−72461.2			−0.0827092
$937$			399631.i			0.455176i	0.973757	+	0.227588i	$0.0730840\pi$
$937$			399631.i			0.455176i	−0.973757	+	0.227588i	$0.926916\pi$
$938$	−1.20968e6			−1.37488
$939$		−	871910.i		−	0.988873i
$940$		−	137565.i		−	0.155687i
$941$			927158.i			1.04707i	0.852005	+	0.523533i	$0.175386\pi$
$941$			927158.i			1.04707i	−0.852005	+	0.523533i	$0.824614\pi$
$942$	−125878.			−0.141856
$943$			1.98084e6i			2.22754i
$944$	46672.4	−	217840.i	0.0523741	−	0.244452i
$945$	131522.			0.147277
$946$			270271.i			0.302007i
$947$	528847.			0.589698			0.294849	−	0.955544i	$-0.404731\pi$
$947$	528847.			0.589698			0.294849	+	0.955544i	$0.404731\pi$
$948$	−21591.3			−0.0240249
$949$	980539.			1.08876
$950$			593400.i			0.657507i
$951$	238019.			0.263179
$952$			51282.1i			0.0565838i
$953$	−594717.			−0.654824			−0.327412	−	0.944882i	$-0.606176\pi$
$953$	−594717.			−0.654824			−0.327412	+	0.944882i	$0.606176\pi$
$954$		−	4081.15i		−	0.00448421i
$955$		−	465331.i		−	0.510218i
$956$	673302.			0.736706
$957$		−	462257.i		−	0.504730i
$958$		−	752349.i		−	0.819763i
$959$	−1.52114e6			−1.65399
$960$	−34680.3			−0.0376305
$961$	714351.			0.773508
$962$		−	147746.i		−	0.159649i
$963$	−13192.8			−0.0142261
$964$	427832.			0.460383
$965$	328109.			0.352341
$966$	953345.			1.02163
$967$			1.36543e6i			1.46021i	0.683336	+	0.730105i	$0.260529\pi$
$967$			1.36543e6i			1.46021i	−0.683336	+	0.730105i	$0.739471\pi$
$968$			198484.i			0.211823i
$969$	75493.6			0.0804012
$970$	−395628.			−0.420478
$971$	185537.			0.196785			0.0983924	−	0.995148i	$-0.468630\pi$
$971$	185537.			0.196785			0.0983924	+	0.995148i	$0.468630\pi$
$972$	−30304.0			−0.0320750
$973$	1.49320e6			1.57722
$974$		−	508976.i		−	0.536512i
$975$		−	280459.i		−	0.295026i
$976$		−	15040.9i		−	0.0157897i
$977$		−	1.42229e6i		−	1.49004i	−0.667040	−	0.745022i	$-0.732439\pi$
$977$		−	1.42229e6i		−	1.49004i	0.667040	−	0.745022i	$-0.267561\pi$
$978$			24958.6i			0.0260941i
$979$	92653.8			0.0966714
$980$	−288959.			−0.300873
$981$		−	334193.i		−	0.347264i
$982$		−	509554.i		−	0.528405i
$983$			688290.i			0.712303i	0.934428	+	0.356151i	$0.115911\pi$
$983$			688290.i			0.712303i	−0.934428	+	0.356151i	$0.884089\pi$
$984$			258206.i			0.266671i
$985$	−897727.			−0.925277
$986$			103506.i			0.106466i
$987$			492939.i			0.506010i
$988$		−	437439.i		−	0.448130i
$989$	−1.12503e6			−1.15020
$990$	76265.5			0.0778140
$991$			817163.i			0.832073i	0.909348	+	0.416036i	$0.136581\pi$
$991$			817163.i			0.832073i	−0.909348	+	0.416036i	$0.863419\pi$
$992$	82789.5			0.0841302
$993$	839941.			0.851824
$994$		−	1.59201e6i		−	1.61128i
$995$	−490598.			−0.495541
$996$		−	287616.i		−	0.289930i
$997$	−1.58760e6			−1.59716			−0.798582	−	0.601886i	$-0.794416\pi$
$997$	−1.58760e6			−1.59716			−0.798582	+	0.601886i	$0.794416\pi$
$998$		−	986362.i		−	0.990319i
$999$		−	61788.8i		−	0.0619125i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.5.d.a.235.31	✓	40
3.2	odd	2		1062.5.d.b.235.20		40
59.58	odd	2	inner	354.5.d.a.235.32	yes	40
177.176	even	2		1062.5.d.b.235.19		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.5.d.a.235.31	✓	40	1.1	even	1	trivial
354.5.d.a.235.32	yes	40	59.58	odd	2	inner
1062.5.d.b.235.19		40	177.176	even	2
1062.5.d.b.235.20		40	3.2	odd	2

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:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	354.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} +5.19615 q^{3} -8.00000 q^{4} +13.0356 q^{5} -14.6969i q^{6} +71.9157 q^{7} +22.6274i q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q-2.82843i q^{2} +5.19615 q^{3} -8.00000 q^{4} +13.0356 q^{5} -14.6969i q^{6} +71.9157 q^{7} +22.6274i q^{8} +27.0000 q^{9} -36.8702i q^{10} +76.6106i q^{11} -41.5692 q^{12} +118.606i q^{13} -203.408i q^{14} +67.7349 q^{15} +64.0000 q^{16} +31.5143 q^{17} -76.3675i q^{18} +461.021 q^{19} -104.285 q^{20} +373.685 q^{21} +216.687 q^{22} +901.986i q^{23} +117.576i q^{24} -455.073 q^{25} +335.469 q^{26} +140.296 q^{27} -575.325 q^{28} -1161.22 q^{29} -191.583i q^{30} +457.352i q^{31} -181.019i q^{32} +398.080i q^{33} -89.1358i q^{34} +937.463 q^{35} -216.000 q^{36} -440.417i q^{37} -1303.96i q^{38} +616.295i q^{39} +294.962i q^{40} +2196.09 q^{41} -1056.94i q^{42} +1247.28i q^{43} -612.885i q^{44} +351.961 q^{45} +2551.20 q^{46} +1319.13i q^{47} +332.554 q^{48} +2770.86 q^{49} +1287.14i q^{50} +163.753 q^{51} -948.848i q^{52} +53.4409 q^{53} -396.817i q^{54} +998.664i q^{55} +1627.27i q^{56} +2395.54 q^{57} +3284.41i q^{58} +(729.257 - 3403.75i) q^{59} -541.879 q^{60} -235.014i q^{61} +1293.59 q^{62} +1941.72 q^{63} -512.000 q^{64} +1546.10i q^{65} +1125.94 q^{66} -5947.06i q^{67} -252.114 q^{68} +4686.85i q^{69} -2651.55i q^{70} +7826.66 q^{71} +610.940i q^{72} -8267.19i q^{73} -1245.69 q^{74} -2364.63 q^{75} -3688.17 q^{76} +5509.50i q^{77} +1743.15 q^{78} +519.406 q^{79} +834.278 q^{80} +729.000 q^{81} -6211.47i q^{82} +6918.96i q^{83} -2989.48 q^{84} +410.807 q^{85} +3527.85 q^{86} -6033.85 q^{87} -1733.50 q^{88} -1209.41i q^{89} -995.496i q^{90} +8529.63i q^{91} -7215.88i q^{92} +2376.47i q^{93} +3731.07 q^{94} +6009.69 q^{95} -940.604i q^{96} -10730.3i q^{97} -7837.18i q^{98} +2068.49i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} - 80 q^{7} + 1080 q^{9}+O(q^{10}) \) 40 * q - 320 * q^4 - 80 * q^7 + 1080 * q^9	\( 40 q - 320 q^{4} - 80 q^{7} + 1080 q^{9} - 144 q^{15} + 2560 q^{16} + 480 q^{17} - 792 q^{19} - 1024 q^{22} + 3400 q^{25} + 768 q^{26} + 640 q^{28} + 1608 q^{29} - 5760 q^{35} - 8640 q^{36} + 6264 q^{41} + 7040 q^{46} + 17912 q^{49} + 1296 q^{51} - 1104 q^{53} + 5040 q^{57} + 13584 q^{59} + 1152 q^{60} - 12288 q^{62} - 2160 q^{63} - 20480 q^{64} + 1152 q^{66} - 3840 q^{68} + 35352 q^{71} + 4608 q^{74} + 3168 q^{75} + 6336 q^{76} - 12672 q^{78} - 15720 q^{79} + 29160 q^{81} - 26872 q^{85} + 18432 q^{86} + 7776 q^{87} + 8192 q^{88} - 18432 q^{94} - 19128 q^{95}+O(q^{100}) \) 40 * q - 320 * q^4 - 80 * q^7 + 1080 * q^9 - 144 * q^15 + 2560 * q^16 + 480 * q^17 - 792 * q^19 - 1024 * q^22 + 3400 * q^25 + 768 * q^26 + 640 * q^28 + 1608 * q^29 - 5760 * q^35 - 8640 * q^36 + 6264 * q^41 + 7040 * q^46 + 17912 * q^49 + 1296 * q^51 - 1104 * q^53 + 5040 * q^57 + 13584 * q^59 + 1152 * q^60 - 12288 * q^62 - 2160 * q^63 - 20480 * q^64 + 1152 * q^66 - 3840 * q^68 + 35352 * q^71 + 4608 * q^74 + 3168 * q^75 + 6336 * q^76 - 12672 * q^78 - 15720 * q^79 + 29160 * q^81 - 26872 * q^85 + 18432 * q^86 + 7776 * q^87 + 8192 * q^88 - 18432 * q^94 - 19128 * q^95

Introduction

L-functions

Modular forms

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