LMFDB - Embedded newform 350.6.c.g.99.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,6,Mod(99,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("350.99");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	350.c (of order $2$, degree $1$, not 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:	$56.1343369345$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	350.99
Dual form			350.6.c.g.99.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} +49.0000i q^{7} -64.0000i q^{8} +162.000 q^{9} +O(q^{10})$	\(q+4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} +49.0000i q^{7} -64.0000i q^{8} +162.000 q^{9} -187.000 q^{11} +144.000i q^{12} +627.000i q^{13} -196.000 q^{14} +256.000 q^{16} -1813.00i q^{17} +648.000i q^{18} -258.000 q^{19} +441.000 q^{21} -748.000i q^{22} +2970.00i q^{23} -576.000 q^{24} -2508.00 q^{26} -3645.00i q^{27} -784.000i q^{28} -1299.00 q^{29} +1916.00 q^{31} +1024.00i q^{32} +1683.00i q^{33} +7252.00 q^{34} -2592.00 q^{36} -6578.00i q^{37} -1032.00i q^{38} +5643.00 q^{39} +6676.00 q^{41} +1764.00i q^{42} +3178.00i q^{43} +2992.00 q^{44} -11880.0 q^{46} +22001.0i q^{47} -2304.00i q^{48} -2401.00 q^{49} -16317.0 q^{51} -10032.0i q^{52} +26168.0i q^{53} +14580.0 q^{54} +3136.00 q^{56} +2322.00i q^{57} -5196.00i q^{58} -3932.00 q^{59} -48740.0 q^{61} +7664.00i q^{62} +7938.00i q^{63} -4096.00 q^{64} -6732.00 q^{66} +44832.0i q^{67} +29008.0i q^{68} +26730.0 q^{69} +63736.0 q^{71} -10368.0i q^{72} +60470.0i q^{73} +26312.0 q^{74} +4128.00 q^{76} -9163.00i q^{77} +22572.0i q^{78} +43721.0 q^{79} +6561.00 q^{81} +26704.0i q^{82} +97276.0i q^{83} -7056.00 q^{84} -12712.0 q^{86} +11691.0i q^{87} +11968.0i q^{88} -45560.0 q^{89} -30723.0 q^{91} -47520.0i q^{92} -17244.0i q^{93} -88004.0 q^{94} +9216.00 q^{96} +57295.0i q^{97} -9604.00i q^{98} -30294.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} + 72 q^{6} + 324 q^{9}+O(q^{10}) $ 2 * q - 32 * q^4 + 72 * q^6 + 324 * q^9	$ 2 q - 32 q^{4} + 72 q^{6} + 324 q^{9} - 374 q^{11} - 392 q^{14} + 512 q^{16} - 516 q^{19} + 882 q^{21} - 1152 q^{24} - 5016 q^{26} - 2598 q^{29} + 3832 q^{31} + 14504 q^{34} - 5184 q^{36} + 11286 q^{39} + 13352 q^{41} + 5984 q^{44} - 23760 q^{46} - 4802 q^{49} - 32634 q^{51} + 29160 q^{54} + 6272 q^{56} - 7864 q^{59} - 97480 q^{61} - 8192 q^{64} - 13464 q^{66} + 53460 q^{69} + 127472 q^{71} + 52624 q^{74} + 8256 q^{76} + 87442 q^{79} + 13122 q^{81} - 14112 q^{84} - 25424 q^{86} - 91120 q^{89} - 61446 q^{91} - 176008 q^{94} + 18432 q^{96} - 60588 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 + 72 * q^6 + 324 * q^9 - 374 * q^11 - 392 * q^14 + 512 * q^16 - 516 * q^19 + 882 * q^21 - 1152 * q^24 - 5016 * q^26 - 2598 * q^29 + 3832 * q^31 + 14504 * q^34 - 5184 * q^36 + 11286 * q^39 + 13352 * q^41 + 5984 * q^44 - 23760 * q^46 - 4802 * q^49 - 32634 * q^51 + 29160 * q^54 + 6272 * q^56 - 7864 * q^59 - 97480 * q^61 - 8192 * q^64 - 13464 * q^66 + 53460 * q^69 + 127472 * q^71 + 52624 * q^74 + 8256 * q^76 + 87442 * q^79 + 13122 * q^81 - 14112 * q^84 - 25424 * q^86 - 91120 * q^89 - 61446 * q^91 - 176008 * q^94 + 18432 * q^96 - 60588 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	4.00000i	0.707107i
$2$	4.00000i	0.707107i
$3$	− 9.00000i	− 0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 9.00000i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	36.0000	0.408248
$7$	49.0000i	0.377964i
$7$	49.0000i	0.377964i
$8$	− 64.0000i	− 0.353553i
$9$	162.000	0.666667
$10$	0	0
$11$	−187.000	−0.465972	−0.232986	−	0.972480i	$-0.574850\pi$
$11$	−187.000	−0.465972	−0.232986	+	0.972480i	$0.574850\pi$
$12$	144.000i	0.288675i
$13$	627.000i	1.02899i	0.857495	+	0.514493i	$0.172020\pi$
$13$	627.000i	1.02899i	−0.857495	+	0.514493i	$0.827980\pi$
$14$	−196.000	−0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	− 1813.00i	− 1.52151i	−0.649038	−	0.760756i	$-0.724828\pi$
$17$	− 1813.00i	− 1.52151i	0.649038	−	0.760756i	$-0.275172\pi$
$18$	648.000i	0.471405i
$19$	−258.000	−0.163959	−0.0819796	−	0.996634i	$-0.526124\pi$
$19$	−258.000	−0.163959	−0.0819796	+	0.996634i	$0.526124\pi$
$20$	0	0
$21$	441.000	0.218218
$22$	− 748.000i	− 0.329492i
$23$	2970.00i	1.17068i	0.810789	+	0.585338i	$0.199038\pi$
$23$	2970.00i	1.17068i	−0.810789	+	0.585338i	$0.800962\pi$
$24$	−576.000	−0.204124
$25$	0	0
$26$	−2508.00	−0.727602
$27$	− 3645.00i	− 0.962250i
$28$	− 784.000i	− 0.188982i
$29$	−1299.00	−0.286823	−0.143412	−	0.989663i	$-0.545807\pi$
$29$	−1299.00	−0.286823	−0.143412	+	0.989663i	$0.545807\pi$
$30$	0	0
$31$	1916.00	0.358089	0.179045	−	0.983841i	$-0.442699\pi$
$31$	1916.00	0.358089	0.179045	+	0.983841i	$0.442699\pi$
$32$	1024.00i	0.176777i
$33$	1683.00i	0.269029i
$34$	7252.00	1.07587
$35$	0	0
$36$	−2592.00	−0.333333
$37$	− 6578.00i	− 0.789932i	−0.918696	−	0.394966i	$-0.870756\pi$
$37$	− 6578.00i	− 0.789932i	0.918696	−	0.394966i	$-0.129244\pi$
$38$	− 1032.00i	− 0.115937i
$39$	5643.00	0.594085
$40$	0	0
$41$	6676.00	0.620236	0.310118	−	0.950698i	$-0.399632\pi$
$41$	6676.00	0.620236	0.310118	+	0.950698i	$0.399632\pi$
$42$	1764.00i	0.154303i
$43$	3178.00i	0.262109i	0.991375	+	0.131055i	$0.0418364\pi$
$43$	3178.00i	0.262109i	−0.991375	+	0.131055i	$0.958164\pi$
$44$	2992.00	0.232986
$45$	0	0
$46$	−11880.0	−0.827793
$47$	22001.0i	1.45277i	0.687286	+	0.726387i	$0.258802\pi$
$47$	22001.0i	1.45277i	−0.687286	+	0.726387i	$0.741198\pi$
$48$	− 2304.00i	− 0.144338i
$49$	−2401.00	−0.142857
$50$	0	0
$51$	−16317.0	−0.878446
$52$	− 10032.0i	− 0.514493i
$53$	26168.0i	1.27962i	0.768533	+	0.639810i	$0.220987\pi$
$53$	26168.0i	1.27962i	−0.768533	+	0.639810i	$0.779013\pi$
$54$	14580.0	0.680414
$55$	0	0
$56$	3136.00	0.133631
$57$	2322.00i	0.0946619i
$58$	− 5196.00i	− 0.202815i
$59$	−3932.00	−0.147056	−0.0735281	−	0.997293i	$-0.523426\pi$
$59$	−3932.00	−0.147056	−0.0735281	+	0.997293i	$0.523426\pi$
$60$	0	0
$61$	−48740.0	−1.67711	−0.838554	−	0.544819i	$-0.816598\pi$
$61$	−48740.0	−1.67711	−0.838554	+	0.544819i	$0.816598\pi$
$62$	7664.00i	0.253207i
$63$	7938.00i	0.251976i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	−6732.00	−0.190232
$67$	44832.0i	1.22012i	0.792357	+	0.610058i	$0.208854\pi$
$67$	44832.0i	1.22012i	−0.792357	+	0.610058i	$0.791146\pi$
$68$	29008.0i	0.760756i
$69$	26730.0	0.675890
$70$	0	0
$71$	63736.0	1.50051	0.750255	−	0.661148i	$-0.229931\pi$
$71$	63736.0	1.50051	0.750255	+	0.661148i	$0.229931\pi$
$72$	− 10368.0i	− 0.235702i
$73$	60470.0i	1.32811i	0.747685	+	0.664053i	$0.231165\pi$
$73$	60470.0i	1.32811i	−0.747685	+	0.664053i	$0.768835\pi$
$74$	26312.0	0.558566
$75$	0	0
$76$	4128.00	0.0819796
$77$	− 9163.00i	− 0.176121i
$78$	22572.0i	0.420081i
$79$	43721.0	0.788174	0.394087	−	0.919073i	$-0.371061\pi$
$79$	43721.0	0.788174	0.394087	+	0.919073i	$0.371061\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	26704.0i	0.438573i
$83$	97276.0i	1.54992i	0.632008	+	0.774962i	$0.282231\pi$
$83$	97276.0i	1.54992i	−0.632008	+	0.774962i	$0.717769\pi$
$84$	−7056.00	−0.109109
$85$	0	0
$86$	−12712.0	−0.185339
$87$	11691.0i	0.165597i
$88$	11968.0i	0.164746i
$89$	−45560.0	−0.609689	−0.304845	−	0.952402i	$-0.598605\pi$
$89$	−45560.0	−0.609689	−0.304845	+	0.952402i	$0.598605\pi$
$90$	0	0
$91$	−30723.0	−0.388920
$92$	− 47520.0i	− 0.585338i
$93$	− 17244.0i	− 0.206743i
$94$	−88004.0	−1.02727
$95$	0	0
$96$	9216.00	0.102062
$97$	57295.0i	0.618283i	0.951016	+	0.309142i	$0.100042\pi$
$97$	57295.0i	0.618283i	−0.951016	+	0.309142i	$0.899958\pi$
$98$	− 9604.00i	− 0.101015i
$99$	−30294.0	−0.310648
$100$	0	0
$101$	−44970.0	−0.438651	−0.219326	−	0.975652i	$-0.570386\pi$
$101$	−44970.0	−0.438651	−0.219326	+	0.975652i	$0.570386\pi$
$102$	− 65268.0i	− 0.621155i
$103$	− 101405.i	− 0.941817i	−0.882182	−	0.470908i	$-0.843926\pi$
$103$	− 101405.i	− 0.941817i	0.882182	−	0.470908i	$-0.156074\pi$
$104$	40128.0	0.363801
$105$	0	0
$106$	−104672.	−0.904828
$107$	166002.i	1.40170i	0.713311	+	0.700848i	$0.247195\pi$
$107$	166002.i	1.40170i	−0.713311	+	0.700848i	$0.752805\pi$
$108$	58320.0i	0.481125i
$109$	−8289.00	−0.0668245	−0.0334123	−	0.999442i	$-0.510637\pi$
$109$	−8289.00	−0.0668245	−0.0334123	+	0.999442i	$0.510637\pi$
$110$	0	0
$111$	−59202.0	−0.456067
$112$	12544.0i	0.0944911i
$113$	263206.i	1.93910i	0.244898	+	0.969549i	$0.421245\pi$
$113$	263206.i	1.93910i	−0.244898	+	0.969549i	$0.578755\pi$
$114$	−9288.00	−0.0669360
$115$	0	0
$116$	20784.0	0.143412
$117$	101574.i	0.685990i
$118$	− 15728.0i	− 0.103984i
$119$	88837.0	0.575078
$120$	0	0
$121$	−126082.	−0.782870
$122$	− 194960.i	− 1.18589i
$123$	− 60084.0i	− 0.358093i
$124$	−30656.0	−0.179045
$125$	0	0
$126$	−31752.0	−0.178174
$127$	− 30052.0i	− 0.165335i	−0.996577	−	0.0826674i	$-0.973656\pi$
$127$	− 30052.0i	− 0.165335i	0.996577	−	0.0826674i	$-0.0263439\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	28602.0	0.151329
$130$	0	0
$131$	−120050.	−0.611201	−0.305600	−	0.952160i	$-0.598857\pi$
$131$	−120050.	−0.611201	−0.305600	+	0.952160i	$0.598857\pi$
$132$	− 26928.0i	− 0.134515i
$133$	− 12642.0i	− 0.0619707i
$134$	−179328.	−0.862752
$135$	0	0
$136$	−116032.	−0.537936
$137$	− 31776.0i	− 0.144643i	−0.997381	−	0.0723216i	$-0.976959\pi$
$137$	− 31776.0i	− 0.144643i	0.997381	−	0.0723216i	$-0.0230408\pi$
$138$	106920.i	0.477927i
$139$	200162.	0.878708	0.439354	−	0.898314i	$-0.355207\pi$
$139$	200162.	0.878708	0.439354	+	0.898314i	$0.355207\pi$
$140$	0	0
$141$	198009.	0.838759
$142$	254944.i	1.06102i
$143$	− 117249.i	− 0.479478i
$144$	41472.0	0.166667
$145$	0	0
$146$	−241880.	−0.939113
$147$	21609.0i	0.0824786i
$148$	105248.i	0.394966i
$149$	−309642.	−1.14260	−0.571300	−	0.820741i	$-0.693561\pi$
$149$	−309642.	−1.14260	−0.571300	+	0.820741i	$0.693561\pi$
$150$	0	0
$151$	−208657.	−0.744716	−0.372358	−	0.928089i	$-0.621451\pi$
$151$	−208657.	−0.744716	−0.372358	+	0.928089i	$0.621451\pi$
$152$	16512.0i	0.0579683i
$153$	− 293706.i	− 1.01434i
$154$	36652.0	0.124536
$155$	0	0
$156$	−90288.0	−0.297042
$157$	− 36010.0i	− 0.116593i	−0.998299	−	0.0582967i	$-0.981433\pi$
$157$	− 36010.0i	− 0.116593i	0.998299	−	0.0582967i	$-0.0185669\pi$
$158$	174884.i	0.557324i
$159$	235512.	0.738789
$160$	0	0
$161$	−145530.	−0.442474
$162$	26244.0i	0.0785674i
$163$	175670.i	0.517879i	0.965893	+	0.258940i	$0.0833731\pi$
$163$	175670.i	0.517879i	−0.965893	+	0.258940i	$0.916627\pi$
$164$	−106816.	−0.310118
$165$	0	0
$166$	−389104.	−1.09596
$167$	157413.i	0.436767i	0.975863	+	0.218383i	$0.0700783\pi$
$167$	157413.i	0.436767i	−0.975863	+	0.218383i	$0.929922\pi$
$168$	− 28224.0i	− 0.0771517i
$169$	−21836.0	−0.0588107
$170$	0	0
$171$	−41796.0	−0.109306
$172$	− 50848.0i	− 0.131055i
$173$	− 23471.0i	− 0.0596233i	−0.999556	−	0.0298117i	$-0.990509\pi$
$173$	− 23471.0i	− 0.0596233i	0.999556	−	0.0298117i	$-0.00949076\pi$
$174$	−46764.0	−0.117095
$175$	0	0
$176$	−47872.0	−0.116493
$177$	35388.0i	0.0849030i
$178$	− 182240.i	− 0.431116i
$179$	−612228.	−1.42817	−0.714086	−	0.700058i	$-0.753158\pi$
$179$	−612228.	−1.42817	−0.714086	+	0.700058i	$0.753158\pi$
$180$	0	0
$181$	528832.	1.19983	0.599917	−	0.800062i	$-0.295200\pi$
$181$	528832.	1.19983	0.599917	+	0.800062i	$0.295200\pi$
$182$	− 122892.i	− 0.275008i
$183$	438660.i	0.968279i
$184$	190080.	0.413897
$185$	0	0
$186$	68976.0	0.146189
$187$	339031.i	0.708982i
$188$	− 352016.i	− 0.726387i
$189$	178605.	0.363696
$190$	0	0
$191$	540369.	1.07178	0.535892	−	0.844287i	$-0.319976\pi$
$191$	540369.	1.07178	0.535892	+	0.844287i	$0.319976\pi$
$192$	36864.0i	0.0721688i
$193$	960320.i	1.85576i	0.372874	+	0.927882i	$0.378372\pi$
$193$	960320.i	1.85576i	−0.372874	+	0.927882i	$0.621628\pi$
$194$	−229180.	−0.437192
$195$	0	0
$196$	38416.0	0.0714286
$197$	− 761944.i	− 1.39881i	−0.714728	−	0.699403i	$-0.753449\pi$
$197$	− 761944.i	− 1.39881i	0.714728	−	0.699403i	$-0.246551\pi$
$198$	− 121176.i	− 0.219661i
$199$	−125084.	−0.223908	−0.111954	−	0.993713i	$-0.535711\pi$
$199$	−125084.	−0.223908	−0.111954	+	0.993713i	$0.535711\pi$
$200$	0	0
$201$	403488.	0.704434
$202$	− 179880.i	− 0.310173i
$203$	− 63651.0i	− 0.108409i
$204$	261072.	0.439223
$205$	0	0
$206$	405620.	0.665965
$207$	481140.i	0.780451i
$208$	160512.i	0.257246i
$209$	48246.0	0.0764004
$210$	0	0
$211$	627547.	0.970376	0.485188	−	0.874410i	$-0.338751\pi$
$211$	627547.	0.970376	0.485188	+	0.874410i	$0.338751\pi$
$212$	− 418688.i	− 0.639810i
$213$	− 573624.i	− 0.866320i
$214$	−664008.	−0.991149
$215$	0	0
$216$	−233280.	−0.340207
$217$	93884.0i	0.135345i
$218$	− 33156.0i	− 0.0472521i
$219$	544230.	0.766782
$220$	0	0
$221$	1.13675e6	1.56561
$222$	− 236808.i	− 0.322488i
$223$	− 1.22110e6i	− 1.64433i	−0.569248	−	0.822166i	$-0.692766\pi$
$223$	− 1.22110e6i	− 1.64433i	0.569248	−	0.822166i	$-0.307234\pi$
$224$	−50176.0	−0.0668153
$225$	0	0
$226$	−1.05282e6	−1.37115
$227$	− 390547.i	− 0.503047i	−0.967851	−	0.251524i	$-0.919068\pi$
$227$	− 390547.i	− 0.503047i	0.967851	−	0.251524i	$-0.0809316\pi$
$228$	− 37152.0i	− 0.0473309i
$229$	712124.	0.897360	0.448680	−	0.893692i	$-0.351894\pi$
$229$	712124.	0.897360	0.448680	+	0.893692i	$0.351894\pi$
$230$	0	0
$231$	−82467.0	−0.101683
$232$	83136.0i	0.101407i
$233$	561576.i	0.677671i	0.940846	+	0.338835i	$0.110033\pi$
$233$	561576.i	0.677671i	−0.940846	+	0.338835i	$0.889967\pi$
$234$	−406296.	−0.485068
$235$	0	0
$236$	62912.0	0.0735281
$237$	− 393489.i	− 0.455053i
$238$	355348.i	0.406641i
$239$	1.36084e6	1.54103	0.770515	−	0.637421i	$-0.219999\pi$
$239$	1.36084e6	1.54103	0.770515	+	0.637421i	$0.219999\pi$
$240$	0	0
$241$	530050.	0.587860	0.293930	−	0.955827i	$-0.405037\pi$
$241$	530050.	0.587860	0.293930	+	0.955827i	$0.405037\pi$
$242$	− 504328.i	− 0.553573i
$243$	− 944784.i	− 1.02640i
$244$	779840.	0.838554
$245$	0	0
$246$	240336.	0.253210
$247$	− 161766.i	− 0.168712i
$248$	− 122624.i	− 0.126604i
$249$	875484.	0.894849
$250$	0	0
$251$	990330.	0.992192	0.496096	−	0.868268i	$-0.334766\pi$
$251$	990330.	0.992192	0.496096	+	0.868268i	$0.334766\pi$
$252$	− 127008.i	− 0.125988i
$253$	− 555390.i	− 0.545503i
$254$	120208.	0.116909
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.81643e6i	1.71548i	0.514083	+	0.857740i	$0.328132\pi$
$257$	1.81643e6i	1.71548i	−0.514083	+	0.857740i	$0.671868\pi$
$258$	114408.i	0.107006i
$259$	322322.	0.298566
$260$	0	0
$261$	−210438.	−0.191215
$262$	− 480200.i	− 0.432184i
$263$	− 1.95847e6i	− 1.74594i	−0.487777	−	0.872968i	$-0.662192\pi$
$263$	− 1.95847e6i	− 1.74594i	0.487777	−	0.872968i	$-0.337808\pi$
$264$	107712.	0.0951162
$265$	0	0
$266$	50568.0	0.0438199
$267$	410040.i	0.352004i
$268$	− 717312.i	− 0.610058i
$269$	−218034.	−0.183715	−0.0918573	−	0.995772i	$-0.529280\pi$
$269$	−218034.	−0.183715	−0.0918573	+	0.995772i	$0.529280\pi$
$270$	0	0
$271$	1.26265e6	1.04438	0.522192	−	0.852828i	$-0.325114\pi$
$271$	1.26265e6	1.04438	0.522192	+	0.852828i	$0.325114\pi$
$272$	− 464128.i	− 0.380378i
$273$	276507.i	0.224543i
$274$	127104.	0.102278
$275$	0	0
$276$	−427680.	−0.337945
$277$	1.10264e6i	0.863443i	0.902007	+	0.431721i	$0.142094\pi$
$277$	1.10264e6i	0.863443i	−0.902007	+	0.431721i	$0.857906\pi$
$278$	800648.i	0.621340i
$279$	310392.	0.238726
$280$	0	0
$281$	−998213.	−0.754149	−0.377075	−	0.926183i	$-0.623070\pi$
$281$	−998213.	−0.754149	−0.377075	+	0.926183i	$0.623070\pi$
$282$	792036.i	0.593092i
$283$	− 386371.i	− 0.286773i	−0.989667	−	0.143387i	$-0.954201\pi$
$283$	− 386371.i	− 0.286773i	0.989667	−	0.143387i	$-0.0457992\pi$
$284$	−1.01978e6	−0.750255
$285$	0	0
$286$	468996.	0.339042
$287$	327124.i	0.234427i
$288$	165888.i	0.117851i
$289$	−1.86711e6	−1.31500
$290$	0	0
$291$	515655.	0.356966
$292$	− 967520.i	− 0.664053i
$293$	− 783571.i	− 0.533224i	−0.963804	−	0.266612i	$-0.914096\pi$
$293$	− 783571.i	− 0.533224i	0.963804	−	0.266612i	$-0.0859041\pi$
$294$	−86436.0	−0.0583212
$295$	0	0
$296$	−420992.	−0.279283
$297$	681615.i	0.448382i
$298$	− 1.23857e6i	− 0.807940i
$299$	−1.86219e6	−1.20461
$300$	0	0
$301$	−155722.	−0.0990681
$302$	− 834628.i	− 0.526594i
$303$	404730.i	0.253255i
$304$	−66048.0	−0.0409898
$305$	0	0
$306$	1.17482e6	0.717248
$307$	− 2.81773e6i	− 1.70629i	−0.521670	−	0.853147i	$-0.674691\pi$
$307$	− 2.81773e6i	− 1.70629i	0.521670	−	0.853147i	$-0.325309\pi$
$308$	146608.i	0.0880604i
$309$	−912645.	−0.543758
$310$	0	0
$311$	−847398.	−0.496806	−0.248403	−	0.968657i	$-0.579906\pi$
$311$	−847398.	−0.496806	−0.248403	+	0.968657i	$0.579906\pi$
$312$	− 361152.i	− 0.210041i
$313$	364955.i	0.210561i	0.994443	+	0.105281i	$0.0335741\pi$
$313$	364955.i	0.210561i	−0.994443	+	0.105281i	$0.966426\pi$
$314$	144040.	0.0824440
$315$	0	0
$316$	−699536.	−0.394087
$317$	− 1.93744e6i	− 1.08288i	−0.840740	−	0.541439i	$-0.817880\pi$
$317$	− 1.93744e6i	− 1.08288i	0.840740	−	0.541439i	$-0.182120\pi$
$318$	942048.i	0.522402i
$319$	242913.	0.133652
$320$	0	0
$321$	1.49402e6	0.809270
$322$	− 582120.i	− 0.312876i
$323$	467754.i	0.249466i
$324$	−104976.	−0.0555556
$325$	0	0
$326$	−702680.	−0.366196
$327$	74601.0i	0.0385812i
$328$	− 427264.i	− 0.219286i
$329$	−1.07805e6	−0.549097
$330$	0	0
$331$	−61460.0	−0.0308335	−0.0154167	−	0.999881i	$-0.504907\pi$
$331$	−61460.0	−0.0308335	−0.0154167	+	0.999881i	$0.504907\pi$
$332$	− 1.55642e6i	− 0.774962i
$333$	− 1.06564e6i	− 0.526621i
$334$	−629652.	−0.308841
$335$	0	0
$336$	112896.	0.0545545
$337$	3.74116e6i	1.79445i	0.441574	+	0.897225i	$0.354420\pi$
$337$	3.74116e6i	1.79445i	−0.441574	+	0.897225i	$0.645580\pi$
$338$	− 87344.0i	− 0.0415854i
$339$	2.36885e6	1.11954
$340$	0	0
$341$	−358292.	−0.166860
$342$	− 167184.i	− 0.0772911i
$343$	− 117649.i	− 0.0539949i
$344$	203392.	0.0926697
$345$	0	0
$346$	93884.0	0.0421601
$347$	− 211334.i	− 0.0942206i	−0.998890	−	0.0471103i	$-0.984999\pi$
$347$	− 211334.i	− 0.0942206i	0.998890	−	0.0471103i	$-0.0150012\pi$
$348$	− 187056.i	− 0.0827987i
$349$	−3.39558e6	−1.49228	−0.746140	−	0.665789i	$-0.768095\pi$
$349$	−3.39558e6	−1.49228	−0.746140	+	0.665789i	$0.768095\pi$
$350$	0	0
$351$	2.28542e6	0.990142
$352$	− 191488.i	− 0.0823730i
$353$	− 3.88094e6i	− 1.65768i	−0.559489	−	0.828838i	$-0.689003\pi$
$353$	− 3.88094e6i	− 1.65768i	0.559489	−	0.828838i	$-0.310997\pi$
$354$	−141552.	−0.0600355
$355$	0	0
$356$	728960.	0.304845
$357$	− 799533.i	− 0.332021i
$358$	− 2.44891e6i	− 1.00987i
$359$	−3.24210e6	−1.32767	−0.663836	−	0.747878i	$-0.731073\pi$
$359$	−3.24210e6	−1.32767	−0.663836	+	0.747878i	$0.731073\pi$
$360$	0	0
$361$	−2.40954e6	−0.973117
$362$	2.11533e6i	0.848411i
$363$	1.13474e6i	0.451990i
$364$	491568.	0.194460
$365$	0	0
$366$	−1.75464e6	−0.684676
$367$	− 1.44430e6i	− 0.559749i	−0.960037	−	0.279874i	$-0.909707\pi$
$367$	− 1.44430e6i	− 0.559749i	0.960037	−	0.279874i	$-0.0902928\pi$
$368$	760320.i	0.292669i
$369$	1.08151e6	0.413490
$370$	0	0
$371$	−1.28223e6	−0.483651
$372$	275904.i	0.103371i
$373$	3.43542e6i	1.27852i	0.768991	+	0.639260i	$0.220759\pi$
$373$	3.43542e6i	1.27852i	−0.768991	+	0.639260i	$0.779241\pi$
$374$	−1.35612e6	−0.501326
$375$	0	0
$376$	1.40806e6	0.513633
$377$	− 814473.i	− 0.295137i
$378$	714420.i	0.257172i
$379$	1.68635e6	0.603044	0.301522	−	0.953459i	$-0.402505\pi$
$379$	1.68635e6	0.603044	0.301522	+	0.953459i	$0.402505\pi$
$380$	0	0
$381$	−270468.	−0.0954560
$382$	2.16148e6i	0.757865i
$383$	2.64354e6i	0.920850i	0.887699	+	0.460425i	$0.152303\pi$
$383$	2.64354e6i	0.920850i	−0.887699	+	0.460425i	$0.847697\pi$
$384$	−147456.	−0.0510310
$385$	0	0
$386$	−3.84128e6	−1.31222
$387$	514836.i	0.174740i
$388$	− 916720.i	− 0.309142i
$389$	452099.	0.151481	0.0757407	−	0.997128i	$-0.475868\pi$
$389$	452099.	0.151481	0.0757407	+	0.997128i	$0.475868\pi$
$390$	0	0
$391$	5.38461e6	1.78120
$392$	153664.i	0.0505076i
$393$	1.08045e6i	0.352877i
$394$	3.04778e6	0.989105
$395$	0	0
$396$	484704.	0.155324
$397$	1.95530e6i	0.622641i	0.950305	+	0.311321i	$0.100771\pi$
$397$	1.95530e6i	0.622641i	−0.950305	+	0.311321i	$0.899229\pi$
$398$	− 500336.i	− 0.158327i
$399$	−113778.	−0.0357788
$400$	0	0
$401$	−4.76737e6	−1.48053	−0.740266	−	0.672314i	$-0.765300\pi$
$401$	−4.76737e6	−1.48053	−0.740266	+	0.672314i	$0.765300\pi$
$402$	1.61395e6i	0.498110i
$403$	1.20133e6i	0.368469i
$404$	719520.	0.219326
$405$	0	0
$406$	254604.	0.0766567
$407$	1.23009e6i	0.368086i
$408$	1.04429e6i	0.310577i
$409$	4.13199e6	1.22138	0.610690	−	0.791870i	$-0.290892\pi$
$409$	4.13199e6	1.22138	0.610690	+	0.791870i	$0.290892\pi$
$410$	0	0
$411$	−285984.	−0.0835097
$412$	1.62248e6i	0.470908i
$413$	− 192668.i	− 0.0555820i
$414$	−1.92456e6	−0.551862
$415$	0	0
$416$	−642048.	−0.181901
$417$	− 1.80146e6i	− 0.507322i
$418$	192984.i	0.0540232i
$419$	−190512.	−0.0530136	−0.0265068	−	0.999649i	$-0.508438\pi$
$419$	−190512.	−0.0530136	−0.0265068	+	0.999649i	$0.508438\pi$
$420$	0	0
$421$	−5.19186e6	−1.42764	−0.713818	−	0.700332i	$-0.753035\pi$
$421$	−5.19186e6	−1.42764	−0.713818	+	0.700332i	$0.753035\pi$
$422$	2.51019e6i	0.686160i
$423$	3.56416e6i	0.968515i
$424$	1.67475e6	0.452414
$425$	0	0
$426$	2.29450e6	0.612581
$427$	− 2.38826e6i	− 0.633887i
$428$	− 2.65603e6i	− 0.700848i
$429$	−1.05524e6	−0.276827
$430$	0	0
$431$	−4.21781e6	−1.09369	−0.546845	−	0.837234i	$-0.684171\pi$
$431$	−4.21781e6	−1.09369	−0.546845	+	0.837234i	$0.684171\pi$
$432$	− 933120.i	− 0.240563i
$433$	− 4.86027e6i	− 1.24578i	−0.782310	−	0.622890i	$-0.785959\pi$
$433$	− 4.86027e6i	− 1.24578i	0.782310	−	0.622890i	$-0.214041\pi$
$434$	−375536.	−0.0957034
$435$	0	0
$436$	132624.	0.0334123
$437$	− 766260.i	− 0.191943i
$438$	2.17692e6i	0.542197i
$439$	2.03113e6	0.503011	0.251505	−	0.967856i	$-0.419074\pi$
$439$	2.03113e6	0.503011	0.251505	+	0.967856i	$0.419074\pi$
$440$	0	0
$441$	−388962.	−0.0952381
$442$	4.54700e6i	1.10706i
$443$	2.84199e6i	0.688038i	0.938963	+	0.344019i	$0.111789\pi$
$443$	2.84199e6i	0.688038i	−0.938963	+	0.344019i	$0.888211\pi$
$444$	947232.	0.228034
$445$	0	0
$446$	4.88440e6	1.16272
$447$	2.78678e6i	0.659680i
$448$	− 200704.i	− 0.0472456i
$449$	4.59682e6	1.07607	0.538037	−	0.842921i	$-0.319166\pi$
$449$	4.59682e6	1.07607	0.538037	+	0.842921i	$0.319166\pi$
$450$	0	0
$451$	−1.24841e6	−0.289012
$452$	− 4.21130e6i	− 0.969549i
$453$	1.87791e6i	0.429962i
$454$	1.56219e6	0.355708
$455$	0	0
$456$	148608.	0.0334680
$457$	− 4.93367e6i	− 1.10504i	−0.833498	−	0.552522i	$-0.813665\pi$
$457$	− 4.93367e6i	− 1.10504i	0.833498	−	0.552522i	$-0.186335\pi$
$458$	2.84850e6i	0.634530i
$459$	−6.60838e6	−1.46408
$460$	0	0
$461$	−4.75667e6	−1.04244	−0.521220	−	0.853422i	$-0.674523\pi$
$461$	−4.75667e6	−1.04244	−0.521220	+	0.853422i	$0.674523\pi$
$462$	− 329868.i	− 0.0719011i
$463$	4.08619e6i	0.885862i	0.896556	+	0.442931i	$0.146061\pi$
$463$	4.08619e6i	0.885862i	−0.896556	+	0.442931i	$0.853939\pi$
$464$	−332544.	−0.0717058
$465$	0	0
$466$	−2.24630e6	−0.479186
$467$	4.15932e6i	0.882531i	0.897377	+	0.441266i	$0.145470\pi$
$467$	4.15932e6i	0.882531i	−0.897377	+	0.441266i	$0.854530\pi$
$468$	− 1.62518e6i	− 0.342995i
$469$	−2.19677e6	−0.461160
$470$	0	0
$471$	−324090.	−0.0673152
$472$	251648.i	0.0519922i
$473$	− 594286.i	− 0.122136i
$474$	1.57396e6	0.321771
$475$	0	0
$476$	−1.42139e6	−0.287539
$477$	4.23922e6i	0.853080i
$478$	5.44335e6i	1.08967i
$479$	3.36040e6	0.669195	0.334597	−	0.942361i	$-0.391400\pi$
$479$	3.36040e6	0.669195	0.334597	+	0.942361i	$0.391400\pi$
$480$	0	0
$481$	4.12441e6	0.812828
$482$	2.12020e6i	0.415680i
$483$	1.30977e6i	0.255463i
$484$	2.01731e6	0.391435
$485$	0	0
$486$	3.77914e6	0.725775
$487$	− 7.05243e6i	− 1.34746i	−0.738977	−	0.673730i	$-0.764691\pi$
$487$	− 7.05243e6i	− 1.34746i	0.738977	−	0.673730i	$-0.235309\pi$
$488$	3.11936e6i	0.592947i
$489$	1.58103e6	0.298998
$490$	0	0
$491$	−83937.0	−0.0157127	−0.00785633	−	0.999969i	$-0.502501\pi$
$491$	−83937.0	−0.0157127	−0.00785633	+	0.999969i	$0.502501\pi$
$492$	961344.i	0.179047i
$493$	2.35509e6i	0.436405i
$494$	647064.	0.119297
$495$	0	0
$496$	490496.	0.0895223
$497$	3.12306e6i	0.567140i
$498$	3.50194e6i	0.632754i
$499$	7.10526e6	1.27741	0.638703	−	0.769454i	$-0.279471\pi$
$499$	7.10526e6	1.27741	0.638703	+	0.769454i	$0.279471\pi$
$500$	0	0
$501$	1.41672e6	0.252167
$502$	3.96132e6i	0.701586i
$503$	2.89147e6i	0.509564i	0.966999	+	0.254782i	$0.0820037\pi$
$503$	2.89147e6i	0.509564i	−0.966999	+	0.254782i	$0.917996\pi$
$504$	508032.	0.0890871
$505$	0	0
$506$	2.22156e6	0.385729
$507$	196524.i	0.0339544i
$508$	480832.i	0.0826674i
$509$	−1.03548e6	−0.177153	−0.0885764	−	0.996069i	$-0.528232\pi$
$509$	−1.03548e6	−0.177153	−0.0885764	+	0.996069i	$0.528232\pi$
$510$	0	0
$511$	−2.96303e6	−0.501977
$512$	262144.i	0.0441942i
$513$	940410.i	0.157770i
$514$	−7.26572e6	−1.21303
$515$	0	0
$516$	−457632.	−0.0756645
$517$	− 4.11419e6i	− 0.676952i
$518$	1.28929e6i	0.211118i
$519$	−211239.	−0.0344236
$520$	0	0
$521$	−7.49715e6	−1.21005	−0.605023	−	0.796208i	$-0.706836\pi$
$521$	−7.49715e6	−1.21005	−0.605023	+	0.796208i	$0.706836\pi$
$522$	− 841752.i	− 0.135210i
$523$	3.53223e6i	0.564670i	0.959316	+	0.282335i	$0.0911089\pi$
$523$	3.53223e6i	0.564670i	−0.959316	+	0.282335i	$0.908891\pi$
$524$	1.92080e6	0.305600
$525$	0	0
$526$	7.83390e6	1.23456
$527$	− 3.47371e6i	− 0.544837i
$528$	430848.i	0.0672573i
$529$	−2.38456e6	−0.370483
$530$	0	0
$531$	−636984.	−0.0980375
$532$	202272.i	0.0309854i
$533$	4.18585e6i	0.638213i
$534$	−1.64016e6	−0.248905
$535$	0	0
$536$	2.86925e6	0.431376
$537$	5.51005e6i	0.824556i
$538$	− 872136.i	− 0.129906i
$539$	448987.	0.0665674
$540$	0	0
$541$	−4.99188e6	−0.733281	−0.366641	−	0.930363i	$-0.619492\pi$
$541$	−4.99188e6	−0.733281	−0.366641	+	0.930363i	$0.619492\pi$
$542$	5.05061e6i	0.738491i
$543$	− 4.75949e6i	− 0.692725i
$544$	1.85651e6	0.268968
$545$	0	0
$546$	−1.10603e6	−0.158776
$547$	− 5.12634e6i	− 0.732553i	−0.930506	−	0.366277i	$-0.880632\pi$
$547$	− 5.12634e6i	− 0.732553i	0.930506	−	0.366277i	$-0.119368\pi$
$548$	508416.i	0.0723216i
$549$	−7.89588e6	−1.11807
$550$	0	0
$551$	335142.	0.0470273
$552$	− 1.71072e6i	− 0.238963i
$553$	2.14233e6i	0.297902i
$554$	−4.41055e6	−0.610546
$555$	0	0
$556$	−3.20259e6	−0.439354
$557$	− 8.86866e6i	− 1.21121i	−0.795765	−	0.605606i	$-0.792931\pi$
$557$	− 8.86866e6i	− 1.21121i	0.795765	−	0.605606i	$-0.207069\pi$
$558$	1.24157e6i	0.168805i
$559$	−1.99261e6	−0.269707
$560$	0	0
$561$	3.05128e6	0.409331
$562$	− 3.99285e6i	− 0.533264i
$563$	9.07277e6i	1.20634i	0.797613	+	0.603169i	$0.206096\pi$
$563$	9.07277e6i	1.20634i	−0.797613	+	0.603169i	$0.793904\pi$
$564$	−3.16814e6	−0.419379
$565$	0	0
$566$	1.54548e6	0.202779
$567$	321489.i	0.0419961i
$568$	− 4.07910e6i	− 0.530510i
$569$	−2.08310e6	−0.269730	−0.134865	−	0.990864i	$-0.543060\pi$
$569$	−2.08310e6	−0.269730	−0.134865	+	0.990864i	$0.543060\pi$
$570$	0	0
$571$	−5.46368e6	−0.701286	−0.350643	−	0.936509i	$-0.614037\pi$
$571$	−5.46368e6	−0.701286	−0.350643	+	0.936509i	$0.614037\pi$
$572$	1.87598e6i	0.239739i
$573$	− 4.86332e6i	− 0.618794i
$574$	−1.30850e6	−0.165765
$575$	0	0
$576$	−663552.	−0.0833333
$577$	− 7.66246e6i	− 0.958140i	−0.877777	−	0.479070i	$-0.840974\pi$
$577$	− 7.66246e6i	− 0.958140i	0.877777	−	0.479070i	$-0.159026\pi$
$578$	− 7.46845e6i	− 0.929845i
$579$	8.64288e6	1.07143
$580$	0	0
$581$	−4.76652e6	−0.585816
$582$	2.06262e6i	0.252413i
$583$	− 4.89342e6i	− 0.596267i
$584$	3.87008e6	0.469556
$585$	0	0
$586$	3.13428e6	0.377046
$587$	1.57465e7i	1.88620i	0.332510	+	0.943100i	$0.392104\pi$
$587$	1.57465e7i	1.88620i	−0.332510	+	0.943100i	$0.607896\pi$
$588$	− 345744.i	− 0.0412393i
$589$	−494328.	−0.0587120
$590$	0	0
$591$	−6.85750e6	−0.807601
$592$	− 1.68397e6i	− 0.197483i
$593$	− 1.62409e7i	− 1.89658i	−0.317398	−	0.948292i	$-0.602809\pi$
$593$	− 1.62409e7i	− 1.89658i	0.317398	−	0.948292i	$-0.397191\pi$
$594$	−2.72646e6	−0.317054
$595$	0	0
$596$	4.95427e6	0.571300
$597$	1.12576e6i	0.129273i
$598$	− 7.44876e6i	− 0.851787i
$599$	1.90793e6	0.217268	0.108634	−	0.994082i	$-0.465352\pi$
$599$	1.90793e6	0.217268	0.108634	+	0.994082i	$0.465352\pi$
$600$	0	0
$601$	3.52970e6	0.398613	0.199306	−	0.979937i	$-0.436131\pi$
$601$	3.52970e6	0.398613	0.199306	+	0.979937i	$0.436131\pi$
$602$	− 622888.i	− 0.0700517i
$603$	7.26278e6i	0.813411i
$604$	3.33851e6	0.372358
$605$	0	0
$606$	−1.61892e6	−0.179079
$607$	− 3.37799e6i	− 0.372123i	−0.982538	−	0.186061i	$-0.940428\pi$
$607$	− 3.37799e6i	− 0.372123i	0.982538	−	0.186061i	$-0.0595724\pi$
$608$	− 264192.i	− 0.0289842i
$609$	−572859.	−0.0625899
$610$	0	0
$611$	−1.37946e7	−1.49488
$612$	4.69930e6i	0.507171i
$613$	1.20412e6i	0.129425i	0.997904	+	0.0647127i	$0.0206131\pi$
$613$	1.20412e6i	0.129425i	−0.997904	+	0.0647127i	$0.979387\pi$
$614$	1.12709e7	1.20653
$615$	0	0
$616$	−586432.	−0.0622681
$617$	5.47330e6i	0.578810i	0.957207	+	0.289405i	$0.0934575\pi$
$617$	5.47330e6i	0.578810i	−0.957207	+	0.289405i	$0.906543\pi$
$618$	− 3.65058e6i	− 0.384495i
$619$	−3.22662e6	−0.338471	−0.169236	−	0.985576i	$-0.554130\pi$
$619$	−3.22662e6	−0.338471	−0.169236	+	0.985576i	$0.554130\pi$
$620$	0	0
$621$	1.08256e7	1.12648
$622$	− 3.38959e6i	− 0.351295i
$623$	− 2.23244e6i	− 0.230441i
$624$	1.44461e6	0.148521
$625$	0	0
$626$	−1.45982e6	−0.148889
$627$	− 434214.i	− 0.0441098i
$628$	576160.i	0.0582967i
$629$	−1.19259e7	−1.20189
$630$	0	0
$631$	1.36282e7	1.36259	0.681297	−	0.732007i	$-0.261416\pi$
$631$	1.36282e7	1.36259	0.681297	+	0.732007i	$0.261416\pi$
$632$	− 2.79814e6i	− 0.278662i
$633$	− 5.64792e6i	− 0.560247i
$634$	7.74975e6	0.765711
$635$	0	0
$636$	−3.76819e6	−0.369394
$637$	− 1.50543e6i	− 0.146998i
$638$	971652.i	0.0945059i
$639$	1.03252e7	1.00034
$640$	0	0
$641$	−1.92472e7	−1.85021	−0.925106	−	0.379710i	$-0.876024\pi$
$641$	−1.92472e7	−1.85021	−0.925106	+	0.379710i	$0.876024\pi$
$642$	5.97607e6i	0.572240i
$643$	− 1.28399e7i	− 1.22472i	−0.790580	−	0.612358i	$-0.790221\pi$
$643$	− 1.28399e7i	− 1.22472i	0.790580	−	0.612358i	$-0.209779\pi$
$644$	2.32848e6	0.221237
$645$	0	0
$646$	−1.87102e6	−0.176399
$647$	− 2.00233e7i	− 1.88050i	−0.340481	−	0.940251i	$-0.610590\pi$
$647$	− 2.00233e7i	− 1.88050i	0.340481	−	0.940251i	$-0.389410\pi$
$648$	− 419904.i	− 0.0392837i
$649$	735284.	0.0685241
$650$	0	0
$651$	844956.	0.0781415
$652$	− 2.81072e6i	− 0.258940i
$653$	− 7.23655e6i	− 0.664124i	−0.943258	−	0.332062i	$-0.892256\pi$
$653$	− 7.23655e6i	− 0.664124i	0.943258	−	0.332062i	$-0.107744\pi$
$654$	−298404.	−0.0272810
$655$	0	0
$656$	1.70906e6	0.155059
$657$	9.79614e6i	0.885404i
$658$	− 4.31220e6i	− 0.388270i
$659$	−1.42474e7	−1.27798	−0.638989	−	0.769216i	$-0.720647\pi$
$659$	−1.42474e7	−1.27798	−0.638989	+	0.769216i	$0.720647\pi$
$660$	0	0
$661$	1.49265e7	1.32878	0.664391	−	0.747385i	$-0.268691\pi$
$661$	1.49265e7	1.32878	0.664391	+	0.747385i	$0.268691\pi$
$662$	− 245840.i	− 0.0218026i
$663$	− 1.02308e7i	− 0.903908i
$664$	6.22566e6	0.547981
$665$	0	0
$666$	4.26254e6	0.372377
$667$	− 3.85803e6i	− 0.335777i
$668$	− 2.51861e6i	− 0.218383i
$669$	−1.09899e7	−0.949355
$670$	0	0
$671$	9.11438e6	0.781485
$672$	451584.i	0.0385758i
$673$	− 1.55062e7i	− 1.31967i	−0.751409	−	0.659837i	$-0.770625\pi$
$673$	− 1.55062e7i	− 1.31967i	0.751409	−	0.659837i	$-0.229375\pi$
$674$	−1.49646e7	−1.26887
$675$	0	0
$676$	349376.	0.0294053
$677$	7.80065e6i	0.654122i	0.945003	+	0.327061i	$0.106058\pi$
$677$	7.80065e6i	0.654122i	−0.945003	+	0.327061i	$0.893942\pi$
$678$	9.47542e6i	0.791633i
$679$	−2.80745e6	−0.233689
$680$	0	0
$681$	−3.51492e6	−0.290434
$682$	− 1.43317e6i	− 0.117988i
$683$	1.58547e7i	1.30049i	0.759727	+	0.650243i	$0.225333\pi$
$683$	1.58547e7i	1.30049i	−0.759727	+	0.650243i	$0.774667\pi$
$684$	668736.	0.0546531
$685$	0	0
$686$	470596.	0.0381802
$687$	− 6.40912e6i	− 0.518091i
$688$	813568.i	0.0655274i
$689$	−1.64073e7	−1.31671
$690$	0	0
$691$	2.03656e7	1.62257	0.811284	−	0.584652i	$-0.198769\pi$
$691$	2.03656e7	1.62257	0.811284	+	0.584652i	$0.198769\pi$
$692$	375536.i	0.0298117i
$693$	− 1.48441e6i	− 0.117414i
$694$	845336.	0.0666240
$695$	0	0
$696$	748224.	0.0585475
$697$	− 1.21036e7i	− 0.943696i
$698$	− 1.35823e7i	− 1.05520i
$699$	5.05418e6	0.391253
$700$	0	0
$701$	2.48036e7	1.90643	0.953213	−	0.302300i	$-0.0977543\pi$
$701$	2.48036e7	1.90643	0.953213	+	0.302300i	$0.0977543\pi$
$702$	9.14166e6i	0.700136i
$703$	1.69712e6i	0.129517i
$704$	765952.	0.0582465
$705$	0	0
$706$	1.55237e7	1.17215
$707$	− 2.20353e6i	− 0.165795i
$708$	− 566208.i	− 0.0424515i
$709$	−1.81917e7	−1.35912	−0.679560	−	0.733620i	$-0.737829\pi$
$709$	−1.81917e7	−1.35912	−0.679560	+	0.733620i	$0.737829\pi$
$710$	0	0
$711$	7.08280e6	0.525450
$712$	2.91584e6i	0.215558i
$713$	5.69052e6i	0.419207i
$714$	3.19813e6	0.234774
$715$	0	0
$716$	9.79565e6	0.714086
$717$	− 1.22475e7i	− 0.889715i
$718$	− 1.29684e7i	− 0.938806i
$719$	1.66202e7	1.19899	0.599493	−	0.800380i	$-0.295369\pi$
$719$	1.66202e7	1.19899	0.599493	+	0.800380i	$0.295369\pi$
$720$	0	0
$721$	4.96884e6	0.355973
$722$	− 9.63814e6i	− 0.688098i
$723$	− 4.77045e6i	− 0.339401i
$724$	−8.46131e6	−0.599917
$725$	0	0
$726$	−4.53895e6	−0.319605
$727$	1.57591e7i	1.10585i	0.833231	+	0.552925i	$0.186489\pi$
$727$	1.57591e7i	1.10585i	−0.833231	+	0.552925i	$0.813511\pi$
$728$	1.96627e6i	0.137504i
$729$	−6.90873e6	−0.481481
$730$	0	0
$731$	5.76171e6	0.398803
$732$	− 7.01856e6i	− 0.484139i
$733$	− 2.15238e6i	− 0.147965i	−0.997260	−	0.0739827i	$-0.976429\pi$
$733$	− 2.15238e6i	− 0.147965i	0.997260	−	0.0739827i	$-0.0235710\pi$
$734$	5.77721e6	0.395802
$735$	0	0
$736$	−3.04128e6	−0.206948
$737$	− 8.38358e6i	− 0.568540i
$738$	4.32605e6i	0.292382i
$739$	2.27267e7	1.53083	0.765413	−	0.643540i	$-0.222535\pi$
$739$	2.27267e7	1.53083	0.765413	+	0.643540i	$0.222535\pi$
$740$	0	0
$741$	−1.45589e6	−0.0974057
$742$	− 5.12893e6i	− 0.341993i
$743$	1.21153e7i	0.805123i	0.915393	+	0.402561i	$0.131880\pi$
$743$	1.21153e7i	0.805123i	−0.915393	+	0.402561i	$0.868120\pi$
$744$	−1.10362e6	−0.0730947
$745$	0	0
$746$	−1.37417e7	−0.904050
$747$	1.57587e7i	1.03328i
$748$	− 5.42450e6i	− 0.354491i
$749$	−8.13410e6	−0.529791
$750$	0	0
$751$	−2.07590e7	−1.34310	−0.671549	−	0.740961i	$-0.734371\pi$
$751$	−2.07590e7	−1.34310	−0.671549	+	0.740961i	$0.734371\pi$
$752$	5.63226e6i	0.363193i
$753$	− 8.91297e6i	− 0.572842i
$754$	3.25789e6	0.208693
$755$	0	0
$756$	−2.85768e6	−0.181848
$757$	1.86222e7i	1.18111i	0.806997	+	0.590556i	$0.201091\pi$
$757$	1.86222e7i	1.18111i	−0.806997	+	0.590556i	$0.798909\pi$
$758$	6.74539e6i	0.426417i
$759$	−4.99851e6	−0.314946
$760$	0	0
$761$	2.65336e7	1.66087	0.830434	−	0.557117i	$-0.188093\pi$
$761$	2.65336e7	1.66087	0.830434	+	0.557117i	$0.188093\pi$
$762$	− 1.08187e6i	− 0.0674976i
$763$	− 406161.i	− 0.0252573i
$764$	−8.64590e6	−0.535892
$765$	0	0
$766$	−1.05742e7	−0.651139
$767$	− 2.46536e6i	− 0.151319i
$768$	− 589824.i	− 0.0360844i
$769$	2.01595e7	1.22931	0.614657	−	0.788794i	$-0.289294\pi$
$769$	2.01595e7	1.22931	0.614657	+	0.788794i	$0.289294\pi$
$770$	0	0
$771$	1.63479e7	0.990433
$772$	− 1.53651e7i	− 0.927882i
$773$	− 5.86488e6i	− 0.353029i	−0.984298	−	0.176514i	$-0.943518\pi$
$773$	− 5.86488e6i	− 0.353029i	0.984298	−	0.176514i	$-0.0564822\pi$
$774$	−2.05934e6	−0.123560
$775$	0	0
$776$	3.66688e6	0.218596
$777$	− 2.90090e6i	− 0.172377i
$778$	1.80840e6i	0.107114i
$779$	−1.72241e6	−0.101693
$780$	0	0
$781$	−1.19186e7	−0.699196
$782$	2.15384e7i	1.25950i
$783$	4.73486e6i	0.275996i
$784$	−614656.	−0.0357143
$785$	0	0
$786$	−4.32180e6	−0.249522
$787$	1.63347e6i	0.0940100i	0.998895	+	0.0470050i	$0.0149677\pi$
$787$	1.63347e6i	0.0940100i	−0.998895	+	0.0470050i	$0.985032\pi$
$788$	1.21911e7i	0.699403i
$789$	−1.76263e7	−1.00802
$790$	0	0
$791$	−1.28971e7	−0.732910
$792$	1.93882e6i	0.109831i
$793$	− 3.05600e7i	− 1.72572i
$794$	−7.82121e6	−0.440274
$795$	0	0
$796$	2.00134e6	0.111954
$797$	− 2.07673e7i	− 1.15807i	−0.815303	−	0.579034i	$-0.803430\pi$
$797$	− 2.07673e7i	− 1.15807i	0.815303	−	0.579034i	$-0.196570\pi$
$798$	− 455112.i	− 0.0252994i
$799$	3.98878e7	2.21041
$800$	0	0
$801$	−7.38072e6	−0.406460
$802$	− 1.90695e7i	− 1.04689i
$803$	− 1.13079e7i	− 0.618860i
$804$	−6.45581e6	−0.352217
$805$	0	0
$806$	−4.80533e6	−0.260547
$807$	1.96231e6i	0.106068i
$808$	2.87808e6i	0.155087i
$809$	−3.53936e6	−0.190131	−0.0950656	−	0.995471i	$-0.530306\pi$
$809$	−3.53936e6	−0.190131	−0.0950656	+	0.995471i	$0.530306\pi$
$810$	0	0
$811$	−2.11480e7	−1.12906	−0.564530	−	0.825412i	$-0.690943\pi$
$811$	−2.11480e7	−1.12906	−0.564530	+	0.825412i	$0.690943\pi$
$812$	1.01842e6i	0.0542045i
$813$	− 1.13639e7i	− 0.602976i
$814$	−4.92034e6	−0.260276
$815$	0	0
$816$	−4.17715e6	−0.219611
$817$	− 819924.i	− 0.0429753i
$818$	1.65280e7i	0.863646i
$819$	−4.97713e6	−0.259280
$820$	0	0
$821$	265389.	0.0137412	0.00687061	−	0.999976i	$-0.497813\pi$
$821$	265389.	0.0137412	0.00687061	+	0.999976i	$0.497813\pi$
$822$	− 1.14394e6i	− 0.0590503i
$823$	3.09261e7i	1.59157i	0.605579	+	0.795785i	$0.292942\pi$
$823$	3.09261e7i	1.59157i	−0.605579	+	0.795785i	$0.707058\pi$
$824$	−6.48992e6	−0.332982
$825$	0	0
$826$	770672.	0.0393024
$827$	2.84152e7i	1.44473i	0.691510	+	0.722367i	$0.256946\pi$
$827$	2.84152e7i	1.44473i	−0.691510	+	0.722367i	$0.743054\pi$
$828$	− 7.69824e6i	− 0.390225i
$829$	3.33547e7	1.68566	0.842832	−	0.538177i	$-0.180887\pi$
$829$	3.33547e7	1.68566	0.842832	+	0.538177i	$0.180887\pi$
$830$	0	0
$831$	9.92374e6	0.498509
$832$	− 2.56819e6i	− 0.128623i
$833$	4.35301e6i	0.217359i
$834$	7.20583e6	0.358731
$835$	0	0
$836$	−771936.	−0.0382002
$837$	− 6.98382e6i	− 0.344572i
$838$	− 762048.i	− 0.0374863i
$839$	5.66205e6	0.277695	0.138848	−	0.990314i	$-0.455660\pi$
$839$	5.66205e6	0.277695	0.138848	+	0.990314i	$0.455660\pi$
$840$	0	0
$841$	−1.88237e7	−0.917732
$842$	− 2.07674e7i	− 1.00949i
$843$	8.98392e6i	0.435408i
$844$	−1.00408e7	−0.485188
$845$	0	0
$846$	−1.42566e7	−0.684844
$847$	− 6.17802e6i	− 0.295897i
$848$	6.69901e6i	0.319905i
$849$	−3.47734e6	−0.165569
$850$	0	0
$851$	1.95367e7	0.924754
$852$	9.17798e6i	0.433160i
$853$	2.19983e7i	1.03518i	0.855628	+	0.517592i	$0.173171\pi$
$853$	2.19983e7i	1.03518i	−0.855628	+	0.517592i	$0.826829\pi$
$854$	9.55304e6	0.448226
$855$	0	0
$856$	1.06241e7	0.495574
$857$	2.17568e7i	1.01191i	0.862559	+	0.505956i	$0.168860\pi$
$857$	2.17568e7i	1.01191i	−0.862559	+	0.505956i	$0.831140\pi$
$858$	− 4.22096e6i	− 0.195746i
$859$	4.09384e7	1.89299	0.946494	−	0.322721i	$-0.104598\pi$
$859$	4.09384e7	1.89299	0.946494	+	0.322721i	$0.104598\pi$
$860$	0	0
$861$	2.94412e6	0.135347
$862$	− 1.68712e7i	− 0.773355i
$863$	5.65597e6i	0.258512i	0.991611	+	0.129256i	$0.0412588\pi$
$863$	5.65597e6i	0.258512i	−0.991611	+	0.129256i	$0.958741\pi$
$864$	3.73248e6	0.170103
$865$	0	0
$866$	1.94411e7	0.880899
$867$	1.68040e7i	0.759216i
$868$	− 1.50214e6i	− 0.0676725i
$869$	−8.17583e6	−0.367267
$870$	0	0
$871$	−2.81097e7	−1.25548
$872$	530496.i	0.0236260i
$873$	9.28179e6i	0.412189i
$874$	3.06504e6	0.135724
$875$	0	0
$876$	−8.70768e6	−0.383391
$877$	− 2.61067e7i	− 1.14618i	−0.819493	−	0.573089i	$-0.805745\pi$
$877$	− 2.61067e7i	− 1.14618i	0.819493	−	0.573089i	$-0.194255\pi$
$878$	8.12454e6i	0.355682i
$879$	−7.05214e6	−0.307857
$880$	0	0
$881$	1.44294e6	0.0626339	0.0313170	−	0.999510i	$-0.490030\pi$
$881$	1.44294e6	0.0626339	0.0313170	+	0.999510i	$0.490030\pi$
$882$	− 1.55585e6i	− 0.0673435i
$883$	− 1.52432e7i	− 0.657921i	−0.944344	−	0.328960i	$-0.893302\pi$
$883$	− 1.52432e7i	− 0.657921i	0.944344	−	0.328960i	$-0.106698\pi$
$884$	−1.81880e7	−0.782807
$885$	0	0
$886$	−1.13679e7	−0.486517
$887$	3.31500e7i	1.41473i	0.706847	+	0.707366i	$0.250117\pi$
$887$	3.31500e7i	1.41473i	−0.706847	+	0.707366i	$0.749883\pi$
$888$	3.78893e6i	0.161244i
$889$	1.47255e6	0.0624907
$890$	0	0
$891$	−1.22691e6	−0.0517747
$892$	1.95376e7i	0.822166i
$893$	− 5.67626e6i	− 0.238195i
$894$	−1.11471e7	−0.466464
$895$	0	0
$896$	802816.	0.0334077
$897$	1.67597e7i	0.695481i
$898$	1.83873e7i	0.760899i
$899$	−2.48888e6	−0.102708
$900$	0	0
$901$	4.74426e7	1.94696
$902$	− 4.99365e6i	− 0.204363i
$903$	1.40150e6i	0.0571970i
$904$	1.68452e7	0.685575
$905$	0	0
$906$	−7.51165e6	−0.304029
$907$	− 1.16963e7i	− 0.472096i	−0.971741	−	0.236048i	$-0.924148\pi$
$907$	− 1.16963e7i	− 0.472096i	0.971741	−	0.236048i	$-0.0758523\pi$
$908$	6.24875e6i	0.251524i
$909$	−7.28514e6	−0.292434
$910$	0	0
$911$	2.89321e7	1.15501	0.577503	−	0.816389i	$-0.304027\pi$
$911$	2.89321e7	1.15501	0.577503	+	0.816389i	$0.304027\pi$
$912$	594432.i	0.0236655i
$913$	− 1.81906e7i	− 0.722221i
$914$	1.97347e7	0.781384
$915$	0	0
$916$	−1.13940e7	−0.448680
$917$	− 5.88245e6i	− 0.231012i
$918$	− 2.64335e7i	− 1.03526i
$919$	4.57838e7	1.78823	0.894115	−	0.447838i	$-0.147806\pi$
$919$	4.57838e7	1.78823	0.894115	+	0.447838i	$0.147806\pi$
$920$	0	0
$921$	−2.53596e7	−0.985129
$922$	− 1.90267e7i	− 0.737116i
$923$	3.99625e7i	1.54400i
$924$	1.31947e6	0.0508417
$925$	0	0
$926$	−1.63448e7	−0.626399
$927$	− 1.64276e7i	− 0.627878i
$928$	− 1.33018e6i	− 0.0507036i
$929$	−2.46947e7	−0.938782	−0.469391	−	0.882990i	$-0.655526\pi$
$929$	−2.46947e7	−0.938782	−0.469391	+	0.882990i	$0.655526\pi$
$930$	0	0
$931$	619458.	0.0234227
$932$	− 8.98522e6i	− 0.338835i
$933$	7.62658e6i	0.286831i
$934$	−1.66373e7	−0.624044
$935$	0	0
$936$	6.50074e6	0.242534
$937$	1.98926e7i	0.740187i	0.928994	+	0.370094i	$0.120674\pi$
$937$	1.98926e7i	0.740187i	−0.928994	+	0.370094i	$0.879326\pi$
$938$	− 8.78707e6i	− 0.326090i
$939$	3.28460e6	0.121568
$940$	0	0
$941$	−3.73454e7	−1.37488	−0.687438	−	0.726243i	$-0.741265\pi$
$941$	−3.73454e7	−1.37488	−0.687438	+	0.726243i	$0.741265\pi$
$942$	− 1.29636e6i	− 0.0475991i
$943$	1.98277e7i	0.726095i
$944$	−1.00659e6	−0.0367641
$945$	0	0
$946$	2.37714e6	0.0863630
$947$	− 5.10396e7i	− 1.84941i	−0.380689	−	0.924703i	$-0.624313\pi$
$947$	− 5.10396e7i	− 1.84941i	0.380689	−	0.924703i	$-0.375687\pi$
$948$	6.29582e6i	0.227526i
$949$	−3.79147e7	−1.36660
$950$	0	0
$951$	−1.74369e7	−0.625200
$952$	− 5.68557e6i	− 0.203321i
$953$	254832.i	0.00908912i	0.999990	+	0.00454456i	$0.00144658\pi$
$953$	254832.i	0.00908912i	−0.999990	+	0.00454456i	$0.998553\pi$
$954$	−1.69569e7	−0.603218
$955$	0	0
$956$	−2.17734e7	−0.770515
$957$	− 2.18622e6i	− 0.0771638i
$958$	1.34416e7i	0.473192i
$959$	1.55702e6	0.0546700
$960$	0	0
$961$	−2.49581e7	−0.871772
$962$	1.64976e7i	0.574756i
$963$	2.68923e7i	0.934464i
$964$	−8.48080e6	−0.293930
$965$	0	0
$966$	−5.23908e6	−0.180639
$967$	− 1.17012e7i	− 0.402405i	−0.979550	−	0.201203i	$-0.935515\pi$
$967$	− 1.17012e7i	− 0.402405i	0.979550	−	0.201203i	$-0.0644850\pi$
$968$	8.06925e6i	0.276786i
$969$	4.20979e6	0.144029
$970$	0	0
$971$	3.59080e7	1.22220	0.611101	−	0.791553i	$-0.290727\pi$
$971$	3.59080e7	1.22220	0.611101	+	0.791553i	$0.290727\pi$
$972$	1.51165e7i	0.513200i
$973$	9.80794e6i	0.332120i
$974$	2.82097e7	0.952799
$975$	0	0
$976$	−1.24774e7	−0.419277
$977$	5.50592e7i	1.84541i	0.385504	+	0.922706i	$0.374028\pi$
$977$	5.50592e7i	1.84541i	−0.385504	+	0.922706i	$0.625972\pi$
$978$	6.32412e6i	0.211423i
$979$	8.51972e6	0.284098
$980$	0	0
$981$	−1.34282e6	−0.0445497
$982$	− 335748.i	− 0.0111105i
$983$	− 1.81317e7i	− 0.598488i	−0.954177	−	0.299244i	$-0.903266\pi$
$983$	− 1.81317e7i	− 0.598488i	0.954177	−	0.299244i	$-0.0967344\pi$
$984$	−3.84538e6	−0.126605
$985$	0	0
$986$	−9.42035e6	−0.308585
$987$	9.70244e6i	0.317021i
$988$	2.58826e6i	0.0843558i
$989$	−9.43866e6	−0.306845
$990$	0	0
$991$	−2.02908e7	−0.656318	−0.328159	−	0.944623i	$-0.606428\pi$
$991$	−2.02908e7	−0.656318	−0.328159	+	0.944623i	$0.606428\pi$
$992$	1.96198e6i	0.0633018i
$993$	553140.i	0.0178017i
$994$	−1.24923e7	−0.401028
$995$	0	0
$996$	−1.40077e7	−0.447425
$997$	− 4.75390e7i	− 1.51465i	−0.653038	−	0.757325i	$-0.726506\pi$
$997$	− 4.75390e7i	− 1.51465i	0.653038	−	0.757325i	$-0.273494\pi$
$998$	2.84211e7i	0.903262i
$999$	−2.39768e7	−0.760112

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.6.c.g.99.2		2
5.2	odd	4		70.6.a.b.1.1	✓	1
5.3	odd	4		350.6.a.l.1.1		1
5.4	even	2	inner	350.6.c.g.99.1		2
15.2	even	4		630.6.a.i.1.1		1
20.7	even	4		560.6.a.e.1.1		1
35.27	even	4		490.6.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.6.a.b.1.1	✓	1	5.2	odd	4
350.6.a.l.1.1		1	5.3	odd	4
350.6.c.g.99.1		2	5.4	even	2	inner
350.6.c.g.99.2		2	1.1	even	1	trivial
490.6.a.g.1.1		1	35.27	even	4
560.6.a.e.1.1		1	20.7	even	4
630.6.a.i.1.1		1	15.2	even	4

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:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} +49.0000i q^{7} -64.0000i q^{8} +162.000 q^{9} +O(q^{10})\)	\(q+4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} +49.0000i q^{7} -64.0000i q^{8} +162.000 q^{9} -187.000 q^{11} +144.000i q^{12} +627.000i q^{13} -196.000 q^{14} +256.000 q^{16} -1813.00i q^{17} +648.000i q^{18} -258.000 q^{19} +441.000 q^{21} -748.000i q^{22} +2970.00i q^{23} -576.000 q^{24} -2508.00 q^{26} -3645.00i q^{27} -784.000i q^{28} -1299.00 q^{29} +1916.00 q^{31} +1024.00i q^{32} +1683.00i q^{33} +7252.00 q^{34} -2592.00 q^{36} -6578.00i q^{37} -1032.00i q^{38} +5643.00 q^{39} +6676.00 q^{41} +1764.00i q^{42} +3178.00i q^{43} +2992.00 q^{44} -11880.0 q^{46} +22001.0i q^{47} -2304.00i q^{48} -2401.00 q^{49} -16317.0 q^{51} -10032.0i q^{52} +26168.0i q^{53} +14580.0 q^{54} +3136.00 q^{56} +2322.00i q^{57} -5196.00i q^{58} -3932.00 q^{59} -48740.0 q^{61} +7664.00i q^{62} +7938.00i q^{63} -4096.00 q^{64} -6732.00 q^{66} +44832.0i q^{67} +29008.0i q^{68} +26730.0 q^{69} +63736.0 q^{71} -10368.0i q^{72} +60470.0i q^{73} +26312.0 q^{74} +4128.00 q^{76} -9163.00i q^{77} +22572.0i q^{78} +43721.0 q^{79} +6561.00 q^{81} +26704.0i q^{82} +97276.0i q^{83} -7056.00 q^{84} -12712.0 q^{86} +11691.0i q^{87} +11968.0i q^{88} -45560.0 q^{89} -30723.0 q^{91} -47520.0i q^{92} -17244.0i q^{93} -88004.0 q^{94} +9216.00 q^{96} +57295.0i q^{97} -9604.00i q^{98} -30294.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} + 72 q^{6} + 324 q^{9}+O(q^{10}) \) 2 * q - 32 * q^4 + 72 * q^6 + 324 * q^9	\( 2 q - 32 q^{4} + 72 q^{6} + 324 q^{9} - 374 q^{11} - 392 q^{14} + 512 q^{16} - 516 q^{19} + 882 q^{21} - 1152 q^{24} - 5016 q^{26} - 2598 q^{29} + 3832 q^{31} + 14504 q^{34} - 5184 q^{36} + 11286 q^{39} + 13352 q^{41} + 5984 q^{44} - 23760 q^{46} - 4802 q^{49} - 32634 q^{51} + 29160 q^{54} + 6272 q^{56} - 7864 q^{59} - 97480 q^{61} - 8192 q^{64} - 13464 q^{66} + 53460 q^{69} + 127472 q^{71} + 52624 q^{74} + 8256 q^{76} + 87442 q^{79} + 13122 q^{81} - 14112 q^{84} - 25424 q^{86} - 91120 q^{89} - 61446 q^{91} - 176008 q^{94} + 18432 q^{96} - 60588 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 + 72 * q^6 + 324 * q^9 - 374 * q^11 - 392 * q^14 + 512 * q^16 - 516 * q^19 + 882 * q^21 - 1152 * q^24 - 5016 * q^26 - 2598 * q^29 + 3832 * q^31 + 14504 * q^34 - 5184 * q^36 + 11286 * q^39 + 13352 * q^41 + 5984 * q^44 - 23760 * q^46 - 4802 * q^49 - 32634 * q^51 + 29160 * q^54 + 6272 * q^56 - 7864 * q^59 - 97480 * q^61 - 8192 * q^64 - 13464 * q^66 + 53460 * q^69 + 127472 * q^71 + 52624 * q^74 + 8256 * q^76 + 87442 * q^79 + 13122 * q^81 - 14112 * q^84 - 25424 * q^86 - 91120 * q^89 - 61446 * q^91 - 176008 * q^94 + 18432 * q^96 - 60588 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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