LMFDB - Embedded newform 363.3.c.e.241.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,3,Mod(241,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("363.241");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	363.c (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:	$9.89103359628$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 77 x^{12} + 88 x^{11} - 577 x^{10} + 578 x^{9} + 1520 x^{8} + \cdots + 83521 $ x^16 - 2x^15 + 3x^14 - 4x^13 + 77x^12 + 88x^11 - 577x^10 + 578x^9 + 1520x^8 + 1868x^7 - 1619x^6 - 16804x^5 + 32427x^4 + 43316x^3 - 71672x^2 + 83521
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 11^{4} $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			241.5
Root			$1.60675 - 1.36085i$ of defining polynomial
Character	$\chi$	$=$	363.241
Dual form			363.3.c.e.241.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.21517i q^{2} +1.73205 q^{3} -0.906963 q^{4} +8.69502 q^{5} -3.83678i q^{6} +6.67189i q^{7} -6.85159i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q-2.21517i q^{2} +1.73205 q^{3} -0.906963 q^{4} +8.69502 q^{5} -3.83678i q^{6} +6.67189i q^{7} -6.85159i q^{8} +3.00000 q^{9} -19.2609i q^{10} -1.57091 q^{12} +11.1547i q^{13} +14.7794 q^{14} +15.0602 q^{15} -18.8053 q^{16} -7.82687i q^{17} -6.64550i q^{18} +8.44217i q^{19} -7.88606 q^{20} +11.5561i q^{21} +9.30611 q^{23} -11.8673i q^{24} +50.6034 q^{25} +24.7094 q^{26} +5.19615 q^{27} -6.05116i q^{28} +7.17852i q^{29} -33.3609i q^{30} -27.4243 q^{31} +14.2504i q^{32} -17.3378 q^{34} +58.0122i q^{35} -2.72089 q^{36} -52.9416 q^{37} +18.7008 q^{38} +19.3205i q^{39} -59.5747i q^{40} -47.8535i q^{41} +25.5986 q^{42} -45.8381i q^{43} +26.0851 q^{45} -20.6146i q^{46} -16.0778 q^{47} -32.5717 q^{48} +4.48587 q^{49} -112.095i q^{50} -13.5565i q^{51} -10.1169i q^{52} -54.9322 q^{53} -11.5103i q^{54} +45.7131 q^{56} +14.6223i q^{57} +15.9016 q^{58} -71.2113 q^{59} -13.6591 q^{60} -1.30131i q^{61} +60.7494i q^{62} +20.0157i q^{63} -43.6540 q^{64} +96.9901i q^{65} -28.9406 q^{67} +7.09868i q^{68} +16.1187 q^{69} +128.507 q^{70} +22.0521 q^{71} -20.5548i q^{72} -125.227i q^{73} +117.274i q^{74} +87.6477 q^{75} -7.65674i q^{76} +42.7980 q^{78} -42.8764i q^{79} -163.512 q^{80} +9.00000 q^{81} -106.004 q^{82} +133.214i q^{83} -10.4809i q^{84} -68.0548i q^{85} -101.539 q^{86} +12.4336i q^{87} -9.48441 q^{89} -57.7828i q^{90} -74.4227 q^{91} -8.44030 q^{92} -47.5003 q^{93} +35.6151i q^{94} +73.4049i q^{95} +24.6825i q^{96} +156.020 q^{97} -9.93695i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 20 q^{4} - 4 q^{5} + 48 q^{9}+O(q^{10}) $ 16 * q - 20 * q^4 - 4 * q^5 + 48 * q^9	$ 16 q - 20 q^{4} - 4 q^{5} + 48 q^{9} - 24 q^{12} - 52 q^{14} + 36 q^{15} - 44 q^{16} - 108 q^{20} + 132 q^{23} + 88 q^{25} - 4 q^{26} + 40 q^{31} - 368 q^{34} - 60 q^{36} - 16 q^{37} + 280 q^{38} + 36 q^{42} - 12 q^{45} + 80 q^{47} + 144 q^{48} - 140 q^{49} - 128 q^{53} + 524 q^{56} + 140 q^{58} - 220 q^{59} - 384 q^{60} - 8 q^{64} + 36 q^{67} - 180 q^{69} - 100 q^{70} + 644 q^{71} + 312 q^{75} - 312 q^{78} + 264 q^{80} + 144 q^{81} - 476 q^{82} + 76 q^{86} + 76 q^{89} - 624 q^{91} + 120 q^{92} - 336 q^{93} + 216 q^{97}+O(q^{100}) $ 16 * q - 20 * q^4 - 4 * q^5 + 48 * q^9 - 24 * q^12 - 52 * q^14 + 36 * q^15 - 44 * q^16 - 108 * q^20 + 132 * q^23 + 88 * q^25 - 4 * q^26 + 40 * q^31 - 368 * q^34 - 60 * q^36 - 16 * q^37 + 280 * q^38 + 36 * q^42 - 12 * q^45 + 80 * q^47 + 144 * q^48 - 140 * q^49 - 128 * q^53 + 524 * q^56 + 140 * q^58 - 220 * q^59 - 384 * q^60 - 8 * q^64 + 36 * q^67 - 180 * q^69 - 100 * q^70 + 644 * q^71 + 312 * q^75 - 312 * q^78 + 264 * q^80 + 144 * q^81 - 476 * q^82 + 76 * q^86 + 76 * q^89 - 624 * q^91 + 120 * q^92 - 336 * q^93 + 216 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/363\mathbb{Z}\right)^\times$.

$n$	$122$	$244$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.21517i	− 1.10758i	−0.832655	−	0.553792i	$-0.813180\pi$
$2$	− 2.21517i	− 1.10758i	0.832655	−	0.553792i	$-0.186820\pi$
$3$	1.73205	0.577350
$3$	1.73205	0.577350
$4$	−0.906963	−0.226741
$5$	8.69502	1.73900	0.869502	−	0.493929i	$-0.164440\pi$
$5$	8.69502	1.73900	0.869502	+	0.493929i	$0.164440\pi$
$6$	− 3.83678i	− 0.639464i
$7$	6.67189i	0.953127i	0.879140	+	0.476564i	$0.158118\pi$
$7$	6.67189i	0.953127i	−0.879140	+	0.476564i	$0.841882\pi$
$8$	− 6.85159i	− 0.856449i
$9$	3.00000	0.333333
$10$	− 19.2609i	− 1.92609i
$11$	0	0
$11$	0	0
$12$	−1.57091	−0.130909
$13$	11.1547i	0.858051i	0.903292	+	0.429026i	$0.141143\pi$
$13$	11.1547i	0.858051i	−0.903292	+	0.429026i	$0.858857\pi$
$14$	14.7794	1.05567
$15$	15.0602	1.00401
$16$	−18.8053	−1.17533
$17$	− 7.82687i	− 0.460404i	−0.973143	−	0.230202i	$-0.926061\pi$
$17$	− 7.82687i	− 0.460404i	0.973143	−	0.230202i	$-0.0739387\pi$
$18$	− 6.64550i	− 0.369194i
$19$	8.44217i	0.444325i	0.975010	+	0.222162i	$0.0713115\pi$
$19$	8.44217i	0.444325i	−0.975010	+	0.222162i	$0.928688\pi$
$20$	−7.88606	−0.394303
$21$	11.5561i	0.550288i
$22$	0	0
$23$	9.30611	0.404613	0.202307	−	0.979322i	$-0.435156\pi$
$23$	9.30611	0.404613	0.202307	+	0.979322i	$0.435156\pi$
$24$	− 11.8673i	− 0.494471i
$25$	50.6034	2.02414
$26$	24.7094	0.950363
$27$	5.19615	0.192450
$28$	− 6.05116i	− 0.216113i
$29$	7.17852i	0.247535i	0.992311	+	0.123768i	$0.0394978\pi$
$29$	7.17852i	0.247535i	−0.992311	+	0.123768i	$0.960502\pi$
$30$	− 33.3609i	− 1.11203i
$31$	−27.4243	−0.884655	−0.442328	−	0.896854i	$-0.645847\pi$
$31$	−27.4243	−0.884655	−0.442328	+	0.896854i	$0.645847\pi$
$32$	14.2504i	0.445326i
$33$	0	0
$34$	−17.3378	−0.509936
$35$	58.0122i	1.65749i
$36$	−2.72089	−0.0755802
$37$	−52.9416	−1.43085	−0.715427	−	0.698688i	$-0.753768\pi$
$37$	−52.9416	−1.43085	−0.715427	+	0.698688i	$0.753768\pi$
$38$	18.7008	0.492127
$39$	19.3205i	0.495396i
$40$	− 59.5747i	− 1.48937i
$41$	− 47.8535i	− 1.16716i	−0.812056	−	0.583580i	$-0.801652\pi$
$41$	− 47.8535i	− 1.16716i	0.812056	−	0.583580i	$-0.198348\pi$
$42$	25.5986	0.609490
$43$	− 45.8381i	− 1.06600i	−0.846114	−	0.533002i	$-0.821064\pi$
$43$	− 45.8381i	− 1.06600i	0.846114	−	0.533002i	$-0.178936\pi$
$44$	0	0
$45$	26.0851	0.579668
$46$	− 20.6146i	− 0.448143i
$47$	−16.0778	−0.342082	−0.171041	−	0.985264i	$-0.554713\pi$
$47$	−16.0778	−0.342082	−0.171041	+	0.985264i	$0.554713\pi$
$48$	−32.5717	−0.678577
$49$	4.48587	0.0915484
$50$	− 112.095i	− 2.24190i
$51$	− 13.5565i	− 0.265815i
$52$	− 10.1169i	− 0.194555i
$53$	−54.9322	−1.03646	−0.518228	−	0.855243i	$-0.673408\pi$
$53$	−54.9322	−1.03646	−0.518228	+	0.855243i	$0.673408\pi$
$54$	− 11.5103i	− 0.213155i
$55$	0	0
$56$	45.7131	0.816305
$57$	14.6223i	0.256531i
$58$	15.9016	0.274166
$59$	−71.2113	−1.20697	−0.603486	−	0.797374i	$-0.706222\pi$
$59$	−71.2113	−1.20697	−0.603486	+	0.797374i	$0.706222\pi$
$60$	−13.6591	−0.227651
$61$	− 1.30131i	− 0.0213330i	−0.999943	−	0.0106665i	$-0.996605\pi$
$61$	− 1.30131i	− 0.0213330i	0.999943	−	0.0106665i	$-0.00339531\pi$
$62$	60.7494i	0.979830i
$63$	20.0157i	0.317709i
$64$	−43.6540	−0.682094
$65$	96.9901i	1.49216i
$66$	0	0
$67$	−28.9406	−0.431949	−0.215975	−	0.976399i	$-0.569293\pi$
$67$	−28.9406	−0.431949	−0.215975	+	0.976399i	$0.569293\pi$
$68$	7.09868i	0.104392i
$69$	16.1187	0.233604
$70$	128.507	1.83581
$71$	22.0521	0.310593	0.155296	−	0.987868i	$-0.450367\pi$
$71$	22.0521	0.310593	0.155296	+	0.987868i	$0.450367\pi$
$72$	− 20.5548i	− 0.285483i
$73$	− 125.227i	− 1.71544i	−0.514120	−	0.857718i	$-0.671881\pi$
$73$	− 125.227i	− 1.71544i	0.514120	−	0.857718i	$-0.328119\pi$
$74$	117.274i	1.58479i
$75$	87.6477	1.16864
$76$	− 7.65674i	− 0.100747i
$77$	0	0
$78$	42.7980	0.548693
$79$	− 42.8764i	− 0.542739i	−0.962475	−	0.271370i	$-0.912523\pi$
$79$	− 42.8764i	− 0.542739i	0.962475	−	0.271370i	$-0.0874765\pi$
$80$	−163.512	−2.04390
$81$	9.00000	0.111111
$82$	−106.004	−1.29273
$83$	133.214i	1.60499i	0.596661	+	0.802493i	$0.296494\pi$
$83$	133.214i	1.60499i	−0.596661	+	0.802493i	$0.703506\pi$
$84$	− 10.4809i	− 0.124773i
$85$	− 68.0548i	− 0.800645i
$86$	−101.539	−1.18069
$87$	12.4336i	0.142915i
$88$	0	0
$89$	−9.48441	−0.106566	−0.0532832	−	0.998579i	$-0.516969\pi$
$89$	−9.48441	−0.106566	−0.0532832	+	0.998579i	$0.516969\pi$
$90$	− 57.7828i	− 0.642031i
$91$	−74.4227	−0.817832
$92$	−8.44030	−0.0917423
$93$	−47.5003	−0.510756
$94$	35.6151i	0.378884i
$95$	73.4049i	0.772683i
$96$	24.6825i	0.257109i
$97$	156.020	1.60845	0.804226	−	0.594324i	$-0.202580\pi$
$97$	156.020	1.60845	0.804226	+	0.594324i	$0.202580\pi$
$98$	− 9.93695i	− 0.101397i
$99$	0	0
$100$	−45.8954	−0.458954
$101$	55.0762i	0.545309i	0.962112	+	0.272655i	$0.0879016\pi$
$101$	55.0762i	0.545309i	−0.962112	+	0.272655i	$0.912098\pi$
$102$	−30.0300	−0.294412
$103$	−81.0177	−0.786579	−0.393290	−	0.919415i	$-0.628663\pi$
$103$	−81.0177	−0.786579	−0.393290	+	0.919415i	$0.628663\pi$
$104$	76.4272	0.734877
$105$	100.480i	0.956954i
$106$	121.684i	1.14796i
$107$	78.1987i	0.730829i	0.930845	+	0.365415i	$0.119073\pi$
$107$	78.1987i	0.730829i	−0.930845	+	0.365415i	$0.880927\pi$
$108$	−4.71272	−0.0436363
$109$	2.67841i	0.0245725i	0.999925	+	0.0122863i	$0.00391094\pi$
$109$	2.67841i	0.0245725i	−0.999925	+	0.0122863i	$0.996089\pi$
$110$	0	0
$111$	−91.6975	−0.826104
$112$	− 125.467i	− 1.12024i
$113$	184.206	1.63014	0.815069	−	0.579364i	$-0.196699\pi$
$113$	184.206	1.63014	0.815069	+	0.579364i	$0.196699\pi$
$114$	32.3908	0.284130
$115$	80.9168	0.703624
$116$	− 6.51066i	− 0.0561263i
$117$	33.4640i	0.286017i
$118$	157.745i	1.33682i
$119$	52.2200	0.438824
$120$	− 103.186i	− 0.859887i
$121$	0	0
$122$	−2.88262	−0.0236280
$123$	− 82.8847i	− 0.673860i
$124$	24.8728	0.200587
$125$	222.622	1.78098
$126$	44.3381	0.351889
$127$	119.107i	0.937849i	0.883238	+	0.468924i	$0.155358\pi$
$127$	119.107i	0.937849i	−0.883238	+	0.468924i	$0.844642\pi$
$128$	153.703i	1.20080i
$129$	− 79.3940i	− 0.615457i
$130$	214.849	1.65269
$131$	17.9999i	0.137403i	0.997637	+	0.0687017i	$0.0218857\pi$
$131$	17.9999i	0.137403i	−0.997637	+	0.0687017i	$0.978114\pi$
$132$	0	0
$133$	−56.3253	−0.423498
$134$	64.1083i	0.478420i
$135$	45.1807	0.334672
$136$	−53.6265	−0.394313
$137$	92.5606	0.675625	0.337812	−	0.941213i	$-0.390313\pi$
$137$	92.5606	0.675625	0.337812	+	0.941213i	$0.390313\pi$
$138$	− 35.7055i	− 0.258735i
$139$	216.817i	1.55983i	0.625883	+	0.779917i	$0.284739\pi$
$139$	216.817i	1.55983i	−0.625883	+	0.779917i	$0.715261\pi$
$140$	− 52.6150i	− 0.375821i
$141$	−27.8476	−0.197501
$142$	− 48.8490i	− 0.344007i
$143$	0	0
$144$	−56.4158	−0.391776
$145$	62.4174i	0.430465i
$146$	−277.398	−1.89999
$147$	7.76975	0.0528555
$148$	48.0161	0.324433
$149$	− 140.721i	− 0.944438i	−0.881481	−	0.472219i	$-0.843453\pi$
$149$	− 140.721i	− 0.944438i	0.881481	−	0.472219i	$-0.156547\pi$
$150$	− 194.154i	− 1.29436i
$151$	11.5088i	0.0762172i	0.999274	+	0.0381086i	$0.0121333\pi$
$151$	11.5088i	0.0762172i	−0.999274	+	0.0381086i	$0.987867\pi$
$152$	57.8423	0.380542
$153$	− 23.4806i	− 0.153468i
$154$	0	0
$155$	−238.455	−1.53842
$156$	− 17.5229i	− 0.112327i
$157$	−149.729	−0.953690	−0.476845	−	0.878987i	$-0.658220\pi$
$157$	−149.729	−0.953690	−0.476845	+	0.878987i	$0.658220\pi$
$158$	−94.9783	−0.601129
$159$	−95.1453	−0.598398
$160$	123.908i	0.774424i
$161$	62.0893i	0.385648i
$162$	− 19.9365i	− 0.123065i
$163$	−54.8173	−0.336302	−0.168151	−	0.985761i	$-0.553780\pi$
$163$	−54.8173	−0.336302	−0.168151	+	0.985761i	$0.553780\pi$
$164$	43.4014i	0.264643i
$165$	0	0
$166$	295.091	1.77766
$167$	42.9417i	0.257136i	0.991701	+	0.128568i	$0.0410381\pi$
$167$	42.9417i	0.257136i	−0.991701	+	0.128568i	$0.958962\pi$
$168$	79.1774	0.471294
$169$	44.5734	0.263748
$170$	−150.753	−0.886781
$171$	25.3265i	0.148108i
$172$	41.5735i	0.241706i
$173$	16.2431i	0.0938907i	0.998897	+	0.0469453i	$0.0149487\pi$
$173$	16.2431i	0.0938907i	−0.998897	+	0.0469453i	$0.985051\pi$
$174$	27.5424	0.158290
$175$	337.620i	1.92926i
$176$	0	0
$177$	−123.342	−0.696845
$178$	21.0095i	0.118031i
$179$	130.353	0.728228	0.364114	−	0.931354i	$-0.381372\pi$
$179$	130.353	0.728228	0.364114	+	0.931354i	$0.381372\pi$
$180$	−23.6582	−0.131434
$181$	−101.541	−0.560998	−0.280499	−	0.959854i	$-0.590500\pi$
$181$	−101.541	−0.560998	−0.280499	+	0.959854i	$0.590500\pi$
$182$	164.859i	0.905817i
$183$	− 2.25394i	− 0.0123166i
$184$	− 63.7617i	− 0.346531i
$185$	−460.328	−2.48826
$186$	105.221i	0.565705i
$187$	0	0
$188$	14.5820	0.0775639
$189$	34.6682i	0.183429i
$190$	162.604	0.855811
$191$	−161.775	−0.846987	−0.423494	−	0.905899i	$-0.639196\pi$
$191$	−161.775	−0.846987	−0.423494	+	0.905899i	$0.639196\pi$
$192$	−75.6109	−0.393807
$193$	− 34.1916i	− 0.177159i	−0.996069	−	0.0885793i	$-0.971767\pi$
$193$	− 34.1916i	− 0.177159i	0.996069	−	0.0885793i	$-0.0282327\pi$
$194$	− 345.610i	− 1.78149i
$195$	167.992i	0.861496i
$196$	−4.06852	−0.0207577
$197$	− 61.8792i	− 0.314107i	−0.987590	−	0.157054i	$-0.949800\pi$
$197$	− 61.8792i	− 0.314107i	0.987590	−	0.157054i	$-0.0501996\pi$
$198$	0	0
$199$	238.301	1.19749	0.598747	−	0.800938i	$-0.295666\pi$
$199$	238.301	1.19749	0.598747	+	0.800938i	$0.295666\pi$
$200$	− 346.714i	− 1.73357i
$201$	−50.1266	−0.249386
$202$	122.003	0.603975
$203$	−47.8943	−0.235933
$204$	12.2953i	0.0602710i
$205$	− 416.087i	− 2.02969i
$206$	179.468i	0.871202i
$207$	27.9183	0.134871
$208$	− 209.767i	− 1.00849i
$209$	0	0
$210$	222.580	1.05991
$211$	188.443i	0.893096i	0.894760	+	0.446548i	$0.147347\pi$
$211$	188.443i	0.893096i	−0.894760	+	0.446548i	$0.852653\pi$
$212$	49.8214	0.235007
$213$	38.1953	0.179321
$214$	173.223	0.809454
$215$	− 398.564i	− 1.85378i
$216$	− 35.6019i	− 0.164824i
$217$	− 182.972i	− 0.843189i
$218$	5.93311	0.0272161
$219$	− 216.899i	− 0.990408i
$220$	0	0
$221$	87.3062	0.395050
$222$	203.125i	0.914979i
$223$	−258.263	−1.15813	−0.579065	−	0.815281i	$-0.696582\pi$
$223$	−258.263	−1.15813	−0.579065	+	0.815281i	$0.696582\pi$
$224$	−95.0774	−0.424453
$225$	151.810	0.674712
$226$	− 408.046i	− 1.80551i
$227$	− 44.3088i	− 0.195193i	−0.995226	−	0.0975965i	$-0.968885\pi$
$227$	− 44.3088i	− 0.195193i	0.995226	−	0.0975965i	$-0.0311155\pi$
$228$	− 13.2619i	− 0.0581661i
$229$	−3.83877	−0.0167632	−0.00838160	−	0.999965i	$-0.502668\pi$
$229$	−3.83877	−0.0167632	−0.00838160	+	0.999965i	$0.502668\pi$
$230$	− 179.244i	− 0.779323i
$231$	0	0
$232$	49.1843	0.212001
$233$	241.342i	1.03580i	0.855440	+	0.517902i	$0.173287\pi$
$233$	241.342i	1.03580i	−0.855440	+	0.517902i	$0.826713\pi$
$234$	74.1283	0.316788
$235$	−139.797	−0.594882
$236$	64.5860	0.273670
$237$	− 74.2641i	− 0.313351i
$238$	− 115.676i	− 0.486034i
$239$	− 463.393i	− 1.93888i	−0.245323	−	0.969441i	$-0.578894\pi$
$239$	− 463.393i	− 1.93888i	0.245323	−	0.969441i	$-0.421106\pi$
$240$	−283.211	−1.18005
$241$	447.213i	1.85565i	0.373011	+	0.927827i	$0.378325\pi$
$241$	447.213i	1.85565i	−0.373011	+	0.927827i	$0.621675\pi$
$242$	0	0
$243$	15.5885	0.0641500
$244$	1.18024i	0.00483705i
$245$	39.0047	0.159203
$246$	−183.604	−0.746356
$247$	−94.1696	−0.381254
$248$	187.900i	0.757662i
$249$	230.733i	0.926639i
$250$	− 493.145i	− 1.97258i
$251$	380.631	1.51646	0.758230	−	0.651987i	$-0.226064\pi$
$251$	380.631	1.51646	0.758230	+	0.651987i	$0.226064\pi$
$252$	− 18.1535i	− 0.0720376i
$253$	0	0
$254$	263.841	1.03875
$255$	− 117.874i	− 0.462253i
$256$	165.861	0.647894
$257$	3.33118	0.0129618	0.00648089	−	0.999979i	$-0.497937\pi$
$257$	3.33118	0.0129618	0.00648089	+	0.999979i	$0.497937\pi$
$258$	−175.871	−0.681670
$259$	− 353.220i	− 1.36379i
$260$	− 87.9664i	− 0.338332i
$261$	21.5356i	0.0825118i
$262$	39.8727	0.152186
$263$	− 379.793i	− 1.44408i	−0.691850	−	0.722041i	$-0.743204\pi$
$263$	− 379.793i	− 1.44408i	0.691850	−	0.722041i	$-0.256796\pi$
$264$	0	0
$265$	−477.636	−1.80240
$266$	124.770i	0.469060i
$267$	−16.4275	−0.0615261
$268$	26.2481	0.0979405
$269$	−187.216	−0.695972	−0.347986	−	0.937500i	$-0.613134\pi$
$269$	−187.216	−0.695972	−0.347986	+	0.937500i	$0.613134\pi$
$270$	− 100.083i	− 0.370677i
$271$	305.158i	1.12604i	0.826442	+	0.563022i	$0.190361\pi$
$271$	305.158i	1.12604i	−0.826442	+	0.563022i	$0.809639\pi$
$272$	147.186i	0.541127i
$273$	−128.904	−0.472176
$274$	− 205.037i	− 0.748311i
$275$	0	0
$276$	−14.6190	−0.0529675
$277$	− 480.172i	− 1.73347i	−0.498765	−	0.866737i	$-0.666213\pi$
$277$	− 480.172i	− 1.73347i	0.498765	−	0.866737i	$-0.333787\pi$
$278$	480.285	1.72765
$279$	−82.2730	−0.294885
$280$	397.476	1.41956
$281$	− 9.05822i	− 0.0322357i	−0.999870	−	0.0161178i	$-0.994869\pi$
$281$	− 9.05822i	− 0.0322357i	0.999870	−	0.0161178i	$-0.00513069\pi$
$282$	61.6872i	0.218749i
$283$	266.704i	0.942416i	0.882022	+	0.471208i	$0.156182\pi$
$283$	266.704i	0.942416i	−0.882022	+	0.471208i	$0.843818\pi$
$284$	−20.0004	−0.0704240
$285$	127.141i	0.446109i
$286$	0	0
$287$	319.274	1.11245
$288$	42.7513i	0.148442i
$289$	227.740	0.788028
$290$	138.265	0.476776
$291$	270.234	0.928640
$292$	113.576i	0.388959i
$293$	− 170.234i	− 0.581004i	−0.956874	−	0.290502i	$-0.906178\pi$
$293$	− 170.234i	− 0.581004i	0.956874	−	0.290502i	$-0.0938223\pi$
$294$	− 17.2113i	− 0.0585418i
$295$	−619.184	−2.09893
$296$	362.734i	1.22545i
$297$	0	0
$298$	−311.721	−1.04604
$299$	103.807i	0.347179i
$300$	−79.4932	−0.264977
$301$	305.827	1.01604
$302$	25.4939	0.0844169
$303$	95.3948i	0.314834i
$304$	− 158.757i	− 0.522228i
$305$	− 11.3149i	− 0.0370981i
$306$	−52.0135	−0.169979
$307$	− 228.869i	− 0.745501i	−0.927932	−	0.372750i	$-0.878415\pi$
$307$	− 228.869i	− 0.745501i	0.927932	−	0.372750i	$-0.121585\pi$
$308$	0	0
$309$	−140.327	−0.454132
$310$	528.218i	1.70393i
$311$	282.560	0.908555	0.454277	−	0.890860i	$-0.349898\pi$
$311$	282.560	0.908555	0.454277	+	0.890860i	$0.349898\pi$
$312$	132.376	0.424282
$313$	−350.176	−1.11877	−0.559386	−	0.828907i	$-0.688963\pi$
$313$	−350.176	−1.11877	−0.559386	+	0.828907i	$0.688963\pi$
$314$	331.675i	1.05629i
$315$	174.037i	0.552498i
$316$	38.8873i	0.123061i
$317$	374.415	1.18112	0.590559	−	0.806994i	$-0.298907\pi$
$317$	374.415	1.18112	0.590559	+	0.806994i	$0.298907\pi$
$318$	210.763i	0.662776i
$319$	0	0
$320$	−379.572	−1.18616
$321$	135.444i	0.421944i
$322$	137.538	0.427137
$323$	66.0758	0.204569
$324$	−8.16267	−0.0251934
$325$	564.464i	1.73681i
$326$	121.429i	0.372483i
$327$	4.63913i	0.0141870i
$328$	−327.873	−0.999612
$329$	− 107.270i	− 0.326048i
$330$	0	0
$331$	262.925	0.794334	0.397167	−	0.917746i	$-0.369993\pi$
$331$	262.925	0.794334	0.397167	+	0.917746i	$0.369993\pi$
$332$	− 120.820i	− 0.363916i
$333$	−158.825	−0.476951
$334$	95.1230	0.284799
$335$	−251.639	−0.751162
$336$	− 217.315i	− 0.646770i
$337$	31.6019i	0.0937743i	0.998900	+	0.0468872i	$0.0149301\pi$
$337$	31.6019i	0.0937743i	−0.998900	+	0.0468872i	$0.985070\pi$
$338$	− 98.7375i	− 0.292123i
$339$	319.053	0.941161
$340$	61.7232i	0.181539i
$341$	0	0
$342$	56.1025	0.164042
$343$	356.852i	1.04038i
$344$	−314.064	−0.912977
$345$	140.152	0.406238
$346$	35.9811	0.103992
$347$	413.522i	1.19170i	0.803094	+	0.595852i	$0.203186\pi$
$347$	413.522i	1.19170i	−0.803094	+	0.595852i	$0.796814\pi$
$348$	− 11.2768i	− 0.0324046i
$349$	367.213i	1.05219i	0.850427	+	0.526094i	$0.176344\pi$
$349$	367.213i	1.05219i	−0.850427	+	0.526094i	$0.823656\pi$
$350$	747.885	2.13682
$351$	57.9614i	0.165132i
$352$	0	0
$353$	−415.577	−1.17727	−0.588636	−	0.808398i	$-0.700335\pi$
$353$	−415.577	−1.17727	−0.588636	+	0.808398i	$0.700335\pi$
$354$	273.222i	0.771814i
$355$	191.743	0.540122
$356$	8.60201	0.0241629
$357$	90.4478	0.253355
$358$	− 288.753i	− 0.806574i
$359$	− 154.752i	− 0.431065i	−0.976497	−	0.215532i	$-0.930851\pi$
$359$	− 154.752i	− 0.431065i	0.976497	−	0.215532i	$-0.0691487\pi$
$360$	− 178.724i	− 0.496456i
$361$	289.730	0.802575
$362$	224.929i	0.621352i
$363$	0	0
$364$	67.4987	0.185436
$365$	− 1088.85i	− 2.98315i
$366$	−4.99284	−0.0136416
$367$	−108.936	−0.296829	−0.148414	−	0.988925i	$-0.547417\pi$
$367$	−108.936	−0.296829	−0.148414	+	0.988925i	$0.547417\pi$
$368$	−175.004	−0.475554
$369$	− 143.561i	− 0.389053i
$370$	1019.70i	2.75596i
$371$	− 366.501i	− 0.987875i
$372$	43.0810	0.115809
$373$	− 721.229i	− 1.93359i	−0.255555	−	0.966795i	$-0.582258\pi$
$373$	− 721.229i	− 1.93359i	0.255555	−	0.966795i	$-0.417742\pi$
$374$	0	0
$375$	385.593	1.02825
$376$	110.159i	0.292976i
$377$	−80.0741	−0.212398
$378$	76.7958	0.203163
$379$	−12.9375	−0.0341358	−0.0170679	−	0.999854i	$-0.505433\pi$
$379$	−12.9375	−0.0341358	−0.0170679	+	0.999854i	$0.505433\pi$
$380$	− 66.5755i	− 0.175199i
$381$	206.299i	0.541467i
$382$	358.358i	0.938109i
$383$	−99.6126	−0.260085	−0.130043	−	0.991508i	$-0.541511\pi$
$383$	−99.6126	−0.260085	−0.130043	+	0.991508i	$0.541511\pi$
$384$	266.221i	0.693283i
$385$	0	0
$386$	−75.7401	−0.196218
$387$	− 137.514i	− 0.355334i
$388$	−141.504	−0.364701
$389$	202.122	0.519595	0.259797	−	0.965663i	$-0.416344\pi$
$389$	202.122	0.519595	0.259797	+	0.965663i	$0.416344\pi$
$390$	372.130	0.954179
$391$	− 72.8377i	− 0.186286i
$392$	− 30.7353i	− 0.0784065i
$393$	31.1767i	0.0793299i
$394$	−137.073	−0.347900
$395$	− 372.811i	− 0.943826i
$396$	0	0
$397$	332.729	0.838108	0.419054	−	0.907961i	$-0.362362\pi$
$397$	332.729	0.838108	0.419054	+	0.907961i	$0.362362\pi$
$398$	− 527.877i	− 1.32632i
$399$	−97.5582	−0.244507
$400$	−951.611	−2.37903
$401$	656.880	1.63811	0.819053	−	0.573718i	$-0.194500\pi$
$401$	656.880	1.63811	0.819053	+	0.573718i	$0.194500\pi$
$402$	111.039i	0.276216i
$403$	− 305.909i	− 0.759080i
$404$	− 49.9521i	− 0.123644i
$405$	78.2552	0.193223
$406$	106.094i	0.261315i
$407$	0	0
$408$	−92.8839	−0.227657
$409$	− 469.056i	− 1.14684i	−0.819263	−	0.573418i	$-0.805617\pi$
$409$	− 469.056i	− 1.14684i	0.819263	−	0.573418i	$-0.194383\pi$
$410$	−921.703	−2.24806
$411$	160.320	0.390072
$412$	73.4800	0.178350
$413$	− 475.114i	− 1.15040i
$414$	− 61.8437i	− 0.149381i
$415$	1158.30i	2.79108i
$416$	−158.959	−0.382113
$417$	375.538i	0.900570i
$418$	0	0
$419$	242.229	0.578112	0.289056	−	0.957312i	$-0.406659\pi$
$419$	242.229	0.578112	0.289056	+	0.957312i	$0.406659\pi$
$420$	− 91.1318i	− 0.216980i
$421$	−737.569	−1.75195	−0.875973	−	0.482360i	$-0.839780\pi$
$421$	−737.569	−1.75195	−0.875973	+	0.482360i	$0.839780\pi$
$422$	417.433	0.989178
$423$	−48.2335	−0.114027
$424$	376.373i	0.887672i
$425$	− 396.066i	− 0.931921i
$426$	− 84.6090i	− 0.198613i
$427$	8.68220	0.0203330
$428$	− 70.9233i	− 0.165709i
$429$	0	0
$430$	−882.885	−2.05322
$431$	491.440i	1.14023i	0.821564	+	0.570116i	$0.193102\pi$
$431$	491.440i	1.14023i	−0.821564	+	0.570116i	$0.806898\pi$
$432$	−97.7150	−0.226192
$433$	−402.664	−0.929939	−0.464970	−	0.885327i	$-0.653935\pi$
$433$	−402.664	−0.929939	−0.464970	+	0.885327i	$0.653935\pi$
$434$	−405.314	−0.933902
$435$	108.110i	0.248529i
$436$	− 2.42921i	− 0.00557159i
$437$	78.5638i	0.179780i
$438$	−480.468	−1.09696
$439$	− 68.4030i	− 0.155815i	−0.996961	−	0.0779077i	$-0.975176\pi$
$439$	− 68.4030i	− 0.155815i	0.996961	−	0.0779077i	$-0.0248240\pi$
$440$	0	0
$441$	13.4576	0.0305161
$442$	− 193.398i	− 0.437551i
$443$	313.441	0.707542	0.353771	−	0.935332i	$-0.384899\pi$
$443$	313.441	0.707542	0.353771	+	0.935332i	$0.384899\pi$
$444$	83.1662	0.187311
$445$	−82.4671	−0.185319
$446$	572.096i	1.28273i
$447$	− 243.736i	− 0.545272i
$448$	− 291.255i	− 0.650122i
$449$	329.111	0.732987	0.366494	−	0.930421i	$-0.380558\pi$
$449$	329.111	0.732987	0.366494	+	0.930421i	$0.380558\pi$
$450$	− 336.285i	− 0.747300i
$451$	0	0
$452$	−167.068	−0.369619
$453$	19.9338i	0.0440040i
$454$	−98.1514	−0.216193
$455$	−647.107	−1.42221
$456$	100.186	0.219706
$457$	88.5487i	0.193761i	0.995296	+	0.0968804i	$0.0308864\pi$
$457$	88.5487i	0.193761i	−0.995296	+	0.0968804i	$0.969114\pi$
$458$	8.50352i	0.0185666i
$459$	− 40.6696i	− 0.0886048i
$460$	−73.3886	−0.159540
$461$	607.310i	1.31737i	0.752417	+	0.658687i	$0.228888\pi$
$461$	607.310i	1.31737i	−0.752417	+	0.658687i	$0.771112\pi$
$462$	0	0
$463$	−40.7126	−0.0879321	−0.0439660	−	0.999033i	$-0.513999\pi$
$463$	−40.7126	−0.0879321	−0.0439660	+	0.999033i	$0.513999\pi$
$464$	− 134.994i	− 0.290936i
$465$	−413.016	−0.888207
$466$	534.614	1.14724
$467$	−447.800	−0.958887	−0.479443	−	0.877573i	$-0.659161\pi$
$467$	−447.800	−0.958887	−0.479443	+	0.877573i	$0.659161\pi$
$468$	− 30.3506i	− 0.0648517i
$469$	− 193.089i	− 0.411703i
$470$	309.674i	0.658881i
$471$	−259.339	−0.550613
$472$	487.911i	1.03371i
$473$	0	0
$474$	−164.507	−0.347062
$475$	427.203i	0.899374i
$476$	−47.3616	−0.0994992
$477$	−164.797	−0.345485
$478$	−1026.49	−2.14747
$479$	58.4924i	0.122114i	0.998134	+	0.0610568i	$0.0194471\pi$
$479$	58.4924i	0.122114i	−0.998134	+	0.0610568i	$0.980553\pi$
$480$	214.615i	0.447114i
$481$	− 590.546i	− 1.22775i
$482$	990.650	2.05529
$483$	107.542i	0.222654i
$484$	0	0
$485$	1356.60	2.79710
$486$	− 34.5310i	− 0.0710515i
$487$	−653.283	−1.34144	−0.670722	−	0.741709i	$-0.734015\pi$
$487$	−653.283	−1.34144	−0.670722	+	0.741709i	$0.734015\pi$
$488$	−8.91605	−0.0182706
$489$	−94.9463	−0.194164
$490$	− 86.4020i	− 0.176331i
$491$	294.767i	0.600339i	0.953886	+	0.300170i	$0.0970433\pi$
$491$	294.767i	0.600339i	−0.953886	+	0.300170i	$0.902957\pi$
$492$	75.1734i	0.152791i
$493$	56.1854	0.113966
$494$	208.601i	0.422270i
$495$	0	0
$496$	515.722	1.03976
$497$	147.129i	0.296034i
$498$	511.112	1.02633
$499$	41.3213	0.0828083	0.0414042	−	0.999142i	$-0.486817\pi$
$499$	41.3213	0.0828083	0.0414042	+	0.999142i	$0.486817\pi$
$500$	−201.910	−0.403820
$501$	74.3772i	0.148458i
$502$	− 843.162i	− 1.67961i
$503$	612.314i	1.21733i	0.793429	+	0.608663i	$0.208294\pi$
$503$	612.314i	1.21733i	−0.793429	+	0.608663i	$0.791706\pi$
$504$	137.139	0.272102
$505$	478.889i	0.948295i
$506$	0	0
$507$	77.2034	0.152275
$508$	− 108.025i	− 0.212648i
$509$	−299.983	−0.589358	−0.294679	−	0.955596i	$-0.595213\pi$
$509$	−299.983	−0.589358	−0.294679	+	0.955596i	$0.595213\pi$
$510$	−261.111	−0.511983
$511$	835.500	1.63503
$512$	247.401i	0.483205i
$513$	43.8668i	0.0855104i
$514$	− 7.37911i	− 0.0143562i
$515$	−704.450	−1.36786
$516$	72.0074i	0.139549i
$517$	0	0
$518$	−782.442	−1.51051
$519$	28.1339i	0.0542078i
$520$	664.537	1.27795
$521$	635.801	1.22035	0.610174	−	0.792268i	$-0.291100\pi$
$521$	635.801	1.22035	0.610174	+	0.792268i	$0.291100\pi$
$522$	47.7049	0.0913887
$523$	8.23200i	0.0157400i	0.999969	+	0.00786998i	$0.00250512\pi$
$523$	8.23200i	0.0157400i	−0.999969	+	0.00786998i	$0.997495\pi$
$524$	− 16.3252i	− 0.0311550i
$525$	584.776i	1.11386i
$526$	−841.306	−1.59944
$527$	214.647i	0.407299i
$528$	0	0
$529$	−442.396	−0.836288
$530$	1058.04i	1.99631i
$531$	−213.634	−0.402324
$532$	51.0849	0.0960243
$533$	533.790	1.00148
$534$	36.3896i	0.0681453i
$535$	679.939i	1.27091i
$536$	198.289i	0.369943i
$537$	225.778	0.420443
$538$	414.716i	0.770847i
$539$	0	0
$540$	−40.9772	−0.0758837
$541$	205.864i	0.380526i	0.981733	+	0.190263i	$0.0609340\pi$
$541$	205.864i	0.380526i	−0.981733	+	0.190263i	$0.939066\pi$
$542$	675.975	1.24719
$543$	−175.874	−0.323892
$544$	111.536	0.205030
$545$	23.2888i	0.0427317i
$546$	285.544i	0.522974i
$547$	− 932.049i	− 1.70393i	−0.523600	−	0.851964i	$-0.675411\pi$
$547$	− 932.049i	− 1.70393i	0.523600	−	0.851964i	$-0.324589\pi$
$548$	−83.9491	−0.153192
$549$	− 3.90393i	− 0.00711099i
$550$	0	0
$551$	−60.6023	−0.109986
$552$	− 110.438i	− 0.200070i
$553$	286.067	0.517299
$554$	−1063.66	−1.91997
$555$	−797.312	−1.43660
$556$	− 196.645i	− 0.353678i
$557$	− 745.673i	− 1.33873i	−0.742933	−	0.669366i	$-0.766566\pi$
$557$	− 745.673i	− 1.33873i	0.742933	−	0.669366i	$-0.233434\pi$
$558$	182.248i	0.326610i
$559$	511.309	0.914686
$560$	− 1090.94i	− 1.94810i
$561$	0	0
$562$	−20.0655	−0.0357037
$563$	− 766.304i	− 1.36111i	−0.732698	−	0.680554i	$-0.761739\pi$
$563$	− 766.304i	− 1.36111i	0.732698	−	0.680554i	$-0.238261\pi$
$564$	25.2568	0.0447815
$565$	1601.67	2.83482
$566$	590.793	1.04380
$567$	60.0470i	0.105903i
$568$	− 151.092i	− 0.266007i
$569$	− 886.251i	− 1.55756i	−0.627297	−	0.778780i	$-0.715839\pi$
$569$	− 886.251i	− 1.55756i	0.627297	−	0.778780i	$-0.284161\pi$
$570$	281.638	0.494103
$571$	− 504.852i	− 0.884154i	−0.896977	−	0.442077i	$-0.854242\pi$
$571$	− 504.852i	− 0.884154i	0.896977	−	0.442077i	$-0.145758\pi$
$572$	0	0
$573$	−280.202	−0.489008
$574$	− 707.244i	− 1.23213i
$575$	470.921	0.818993
$576$	−130.962	−0.227365
$577$	865.114	1.49933	0.749666	−	0.661817i	$-0.230214\pi$
$577$	865.114	1.49933	0.749666	+	0.661817i	$0.230214\pi$
$578$	− 504.482i	− 0.872807i
$579$	− 59.2216i	− 0.102283i
$580$	− 56.6103i	− 0.0976040i
$581$	−888.788	−1.52976
$582$	− 598.614i	− 1.02855i
$583$	0	0
$584$	−858.004	−1.46918
$585$	290.970i	0.497385i
$586$	−377.097	−0.643511
$587$	−733.629	−1.24979	−0.624897	−	0.780708i	$-0.714859\pi$
$587$	−733.629	−1.24979	−0.624897	+	0.780708i	$0.714859\pi$
$588$	−7.04688	−0.0119845
$589$	− 231.521i	− 0.393074i
$590$	1371.60i	2.32474i
$591$	− 107.178i	− 0.181350i
$592$	995.581	1.68172
$593$	− 1126.19i	− 1.89915i	−0.313544	−	0.949574i	$-0.601516\pi$
$593$	− 1126.19i	− 1.89915i	0.313544	−	0.949574i	$-0.398484\pi$
$594$	0	0
$595$	454.054	0.763117
$596$	127.629i	0.214143i
$597$	412.750	0.691373
$598$	229.949	0.384530
$599$	−36.7240	−0.0613089	−0.0306545	−	0.999530i	$-0.509759\pi$
$599$	−36.7240	−0.0613089	−0.0306545	+	0.999530i	$0.509759\pi$
$600$	− 600.526i	− 1.00088i
$601$	642.204i	1.06856i	0.845308	+	0.534279i	$0.179417\pi$
$601$	642.204i	1.06856i	−0.845308	+	0.534279i	$0.820583\pi$
$602$	− 677.458i	− 1.12535i
$603$	−86.8218	−0.143983
$604$	− 10.4381i	− 0.0172815i
$605$	0	0
$606$	211.315	0.348705
$607$	− 660.993i	− 1.08895i	−0.838777	−	0.544475i	$-0.816729\pi$
$607$	− 660.993i	− 1.08895i	0.838777	−	0.544475i	$-0.183271\pi$
$608$	−120.305	−0.197869
$609$	−82.9554	−0.136216
$610$	−25.0644	−0.0410892
$611$	− 179.343i	− 0.293524i
$612$	21.2960i	0.0347975i
$613$	− 729.719i	− 1.19041i	−0.803575	−	0.595203i	$-0.797072\pi$
$613$	− 729.719i	− 1.19041i	0.803575	−	0.595203i	$-0.202928\pi$
$614$	−506.982	−0.825704
$615$	− 720.685i	− 1.17184i
$616$	0	0
$617$	−329.848	−0.534600	−0.267300	−	0.963613i	$-0.586132\pi$
$617$	−329.848	−0.534600	−0.267300	+	0.963613i	$0.586132\pi$
$618$	310.847i	0.502989i
$619$	1033.26	1.66924	0.834620	−	0.550826i	$-0.185687\pi$
$619$	1033.26	1.66924	0.834620	+	0.550826i	$0.185687\pi$
$620$	216.270	0.348822
$621$	48.3560	0.0778679
$622$	− 625.919i	− 1.00630i
$623$	− 63.2789i	− 0.101571i
$624$	− 363.326i	− 0.582254i
$625$	670.620	1.07299
$626$	775.697i	1.23913i
$627$	0	0
$628$	135.799	0.216240
$629$	414.367i	0.658771i
$630$	385.520	0.611937
$631$	−854.749	−1.35459	−0.677297	−	0.735709i	$-0.736849\pi$
$631$	−854.749	−1.35459	−0.677297	+	0.735709i	$0.736849\pi$
$632$	−293.771	−0.464828
$633$	326.393i	0.515629i
$634$	− 829.391i	− 1.30819i
$635$	1035.64i	1.63092i
$636$	86.2933	0.135681
$637$	50.0384i	0.0785532i
$638$	0	0
$639$	66.1562	0.103531
$640$	1336.45i	2.08820i
$641$	452.979	0.706676	0.353338	−	0.935496i	$-0.385047\pi$
$641$	452.979	0.706676	0.353338	+	0.935496i	$0.385047\pi$
$642$	300.031	0.467339
$643$	−265.255	−0.412527	−0.206264	−	0.978496i	$-0.566130\pi$
$643$	−265.255	−0.412527	−0.206264	+	0.978496i	$0.566130\pi$
$644$	− 56.3127i	− 0.0874421i
$645$	− 690.332i	− 1.07028i
$646$	− 146.369i	− 0.226577i
$647$	−342.818	−0.529859	−0.264929	−	0.964268i	$-0.585349\pi$
$647$	−342.818	−0.529859	−0.264929	+	0.964268i	$0.585349\pi$
$648$	− 61.6643i	− 0.0951610i
$649$	0	0
$650$	1250.38	1.92366
$651$	− 316.917i	− 0.486816i
$652$	49.7172	0.0762534
$653$	543.167	0.831802	0.415901	−	0.909410i	$-0.363466\pi$
$653$	543.167	0.831802	0.415901	+	0.909410i	$0.363466\pi$
$654$	10.2765	0.0157132
$655$	156.509i	0.238945i
$656$	899.899i	1.37180i
$657$	− 375.681i	− 0.571812i
$658$	−237.620	−0.361125
$659$	309.878i	0.470224i	0.971968	+	0.235112i	$0.0755457\pi$
$659$	309.878i	0.470224i	−0.971968	+	0.235112i	$0.924454\pi$
$660$	0	0
$661$	389.483	0.589234	0.294617	−	0.955615i	$-0.404808\pi$
$661$	389.483	0.589234	0.294617	+	0.955615i	$0.404808\pi$
$662$	− 582.422i	− 0.879791i
$663$	151.219	0.228083
$664$	912.727	1.37459
$665$	−489.749	−0.736465
$666$	351.823i	0.528263i
$667$	66.8041i	0.100156i
$668$	− 38.9465i	− 0.0583032i
$669$	−447.325	−0.668647
$670$	557.423i	0.831974i
$671$	0	0
$672$	−164.679	−0.245058
$673$	− 721.948i	− 1.07273i	−0.843986	−	0.536365i	$-0.819797\pi$
$673$	− 721.948i	− 1.07273i	0.843986	−	0.536365i	$-0.180203\pi$
$674$	70.0036	0.103863
$675$	262.943	0.389545
$676$	−40.4264	−0.0598024
$677$	− 846.044i	− 1.24970i	−0.780746	−	0.624848i	$-0.785161\pi$
$677$	− 846.044i	− 1.24970i	0.780746	−	0.624848i	$-0.214839\pi$
$678$	− 706.757i	− 1.04241i
$679$	1040.95i	1.53306i
$680$	−466.284	−0.685712
$681$	− 76.7451i	− 0.112695i
$682$	0	0
$683$	49.2192	0.0720632	0.0360316	−	0.999351i	$-0.488528\pi$
$683$	49.2192	0.0720632	0.0360316	+	0.999351i	$0.488528\pi$
$684$	− 22.9702i	− 0.0335822i
$685$	804.817	1.17491
$686$	790.486	1.15231
$687$	−6.64895	−0.00967824
$688$	861.999i	1.25290i
$689$	− 612.750i	− 0.889333i
$690$	− 310.460i	− 0.449942i
$691$	705.693	1.02126	0.510632	−	0.859799i	$-0.329411\pi$
$691$	705.693	1.02126	0.510632	+	0.859799i	$0.329411\pi$
$692$	− 14.7319i	− 0.0212888i
$693$	0	0
$694$	916.019	1.31991
$695$	1885.23i	2.71256i
$696$	85.1898	0.122399
$697$	−374.543	−0.537365
$698$	813.439	1.16538
$699$	418.017i	0.598022i
$700$	− 306.209i	− 0.437442i
$701$	56.9789i	0.0812823i	0.999174	+	0.0406412i	$0.0129400\pi$
$701$	56.9789i	0.0812823i	−0.999174	+	0.0406412i	$0.987060\pi$
$702$	128.394	0.182898
$703$	− 446.942i	− 0.635764i
$704$	0	0
$705$	−242.136	−0.343455
$706$	920.573i	1.30393i
$707$	−367.463	−0.519749
$708$	111.866	0.158003
$709$	481.533	0.679172	0.339586	−	0.940575i	$-0.389713\pi$
$709$	481.533	0.679172	0.339586	+	0.940575i	$0.389713\pi$
$710$	− 424.743i	− 0.598230i
$711$	− 128.629i	− 0.180913i
$712$	64.9833i	0.0912687i
$713$	−255.214	−0.357943
$714$	− 200.357i	− 0.280612i
$715$	0	0
$716$	−118.225	−0.165119
$717$	− 802.620i	− 1.11941i
$718$	−342.802	−0.477440
$719$	987.683	1.37369	0.686845	−	0.726804i	$-0.258995\pi$
$719$	987.683	1.37369	0.686845	+	0.726804i	$0.258995\pi$
$720$	−490.537	−0.681301
$721$	− 540.541i	− 0.749710i
$722$	− 641.800i	− 0.888919i
$723$	774.595i	1.07136i
$724$	92.0936	0.127201
$725$	363.258i	0.501045i
$726$	0	0
$727$	−146.472	−0.201474	−0.100737	−	0.994913i	$-0.532120\pi$
$727$	−146.472	−0.201474	−0.100737	+	0.994913i	$0.532120\pi$
$728$	509.914i	0.700432i
$729$	27.0000	0.0370370
$730$	−2411.99	−3.30409
$731$	−358.769	−0.490792
$732$	2.04424i	0.00279267i
$733$	140.454i	0.191615i	0.995400	+	0.0958076i	$0.0305433\pi$
$733$	140.454i	0.191615i	−0.995400	+	0.0958076i	$0.969457\pi$
$734$	241.312i	0.328763i
$735$	67.5582	0.0919159
$736$	132.616i	0.180185i
$737$	0	0
$738$	−318.011	−0.430909
$739$	− 662.223i	− 0.896107i	−0.894007	−	0.448053i	$-0.852117\pi$
$739$	− 662.223i	− 0.896107i	0.894007	−	0.448053i	$-0.147883\pi$
$740$	417.501	0.564190
$741$	−163.107	−0.220117
$742$	−811.862	−1.09415
$743$	191.960i	0.258358i	0.991621	+	0.129179i	$0.0412341\pi$
$743$	191.960i	0.258358i	−0.991621	+	0.129179i	$0.958766\pi$
$744$	325.453i	0.437437i
$745$	− 1223.57i	− 1.64238i
$746$	−1597.64	−2.14161
$747$	399.642i	0.534995i
$748$	0	0
$749$	−521.733	−0.696573
$750$	− 854.153i	− 1.13887i
$751$	−1199.85	−1.59767	−0.798833	−	0.601552i	$-0.794549\pi$
$751$	−1199.85	−1.59767	−0.798833	+	0.601552i	$0.794549\pi$
$752$	302.348	0.402059
$753$	659.273	0.875529
$754$	177.377i	0.235249i
$755$	100.069i	0.132542i
$756$	− 31.4427i	− 0.0415909i
$757$	−572.112	−0.755762	−0.377881	−	0.925854i	$-0.623347\pi$
$757$	−572.112	−0.755762	−0.377881	+	0.925854i	$0.623347\pi$
$758$	28.6586i	0.0378082i
$759$	0	0
$760$	502.940	0.661764
$761$	71.9549i	0.0945531i	0.998882	+	0.0472765i	$0.0150542\pi$
$761$	71.9549i	0.0945531i	−0.998882	+	0.0472765i	$0.984946\pi$
$762$	456.987	0.599720
$763$	−17.8700	−0.0234207
$764$	146.724	0.192047
$765$	− 204.164i	− 0.266882i
$766$	220.658i	0.288066i
$767$	− 794.338i	− 1.03564i
$768$	287.280	0.374062
$769$	− 87.6070i	− 0.113923i	−0.998376	−	0.0569616i	$-0.981859\pi$
$769$	− 87.6070i	− 0.113923i	0.998376	−	0.0569616i	$-0.0181413\pi$
$770$	0	0
$771$	5.76976	0.00748348
$772$	31.0105i	0.0401691i
$773$	−767.814	−0.993291	−0.496646	−	0.867953i	$-0.665435\pi$
$773$	−767.814	−0.993291	−0.496646	+	0.867953i	$0.665435\pi$
$774$	−304.617	−0.393562
$775$	−1387.76	−1.79066
$776$	− 1068.98i	− 1.37756i
$777$	− 611.796i	− 0.787382i
$778$	− 447.735i	− 0.575495i
$779$	403.988	0.518598
$780$	− 152.362i	− 0.195336i
$781$	0	0
$782$	−161.348	−0.206327
$783$	37.3007i	0.0476382i
$784$	−84.3580	−0.107599
$785$	−1301.90	−1.65847
$786$	69.0615	0.0878645
$787$	313.510i	0.398361i	0.979963	+	0.199181i	$0.0638280\pi$
$787$	313.510i	0.398361i	−0.979963	+	0.199181i	$0.936172\pi$
$788$	56.1221i	0.0712210i
$789$	− 657.822i	− 0.833741i
$790$	−825.839	−1.04537
$791$	1229.00i	1.55373i
$792$	0	0
$793$	14.5157	0.0183048
$794$	− 737.050i	− 0.928274i
$795$	−827.291	−1.04062
$796$	−216.130	−0.271521
$797$	−671.426	−0.842442	−0.421221	−	0.906958i	$-0.638398\pi$
$797$	−671.426	−0.842442	−0.421221	+	0.906958i	$0.638398\pi$
$798$	216.108i	0.270812i
$799$	125.839i	0.157496i
$800$	721.121i	0.901401i
$801$	−28.4532	−0.0355221
$802$	− 1455.10i	− 1.81434i
$803$	0	0
$804$	45.4630	0.0565460
$805$	539.868i	0.670644i
$806$	−677.640	−0.840744
$807$	−324.268	−0.401820
$808$	377.360	0.467029
$809$	− 1059.92i	− 1.31016i	−0.755562	−	0.655078i	$-0.772636\pi$
$809$	− 1059.92i	− 1.31016i	0.755562	−	0.655078i	$-0.227364\pi$
$810$	− 173.348i	− 0.214010i
$811$	868.931i	1.07143i	0.844398	+	0.535716i	$0.179958\pi$
$811$	868.931i	1.07143i	−0.844398	+	0.535716i	$0.820042\pi$
$812$	43.4384	0.0534956
$813$	528.549i	0.650122i
$814$	0	0
$815$	−476.637	−0.584831
$816$	254.934i	0.312420i
$817$	386.974	0.473652
$818$	−1039.04	−1.27022
$819$	−223.268	−0.272611
$820$	377.376i	0.460215i
$821$	749.652i	0.913096i	0.889699	+	0.456548i	$0.150914\pi$
$821$	749.652i	0.913096i	−0.889699	+	0.456548i	$0.849086\pi$
$822$	− 355.135i	− 0.432038i
$823$	1184.79	1.43960	0.719800	−	0.694182i	$-0.244234\pi$
$823$	1184.79	1.43960	0.719800	+	0.694182i	$0.244234\pi$
$824$	555.100i	0.673665i
$825$	0	0
$826$	−1052.46	−1.27416
$827$	− 100.344i	− 0.121335i	−0.998158	−	0.0606674i	$-0.980677\pi$
$827$	− 100.344i	− 0.121335i	0.998158	−	0.0606674i	$-0.0193229\pi$
$828$	−25.3209	−0.0305808
$829$	249.143	0.300535	0.150267	−	0.988645i	$-0.451987\pi$
$829$	249.143	0.300535	0.150267	+	0.988645i	$0.451987\pi$
$830$	2565.82	3.09135
$831$	− 831.683i	− 1.00082i
$832$	− 486.946i	− 0.585271i
$833$	− 35.1103i	− 0.0421492i
$834$	831.879	0.997456
$835$	373.379i	0.447161i
$836$	0	0
$837$	−142.501	−0.170252
$838$	− 536.577i	− 0.640307i
$839$	−121.681	−0.145030	−0.0725152	−	0.997367i	$-0.523103\pi$
$839$	−121.681	−0.145030	−0.0725152	+	0.997367i	$0.523103\pi$
$840$	688.449	0.819582
$841$	789.469	0.938726
$842$	1633.84i	1.94043i
$843$	− 15.6893i	− 0.0186113i
$844$	− 170.911i	− 0.202501i
$845$	387.567	0.458659
$846$	106.845i	0.126295i
$847$	0	0
$848$	1033.01	1.21818
$849$	461.944i	0.544104i
$850$	−877.353	−1.03218
$851$	−492.680	−0.578942
$852$	−34.6417	−0.0406593
$853$	144.122i	0.168959i	0.996425	+	0.0844795i	$0.0269228\pi$
$853$	144.122i	0.168959i	−0.996425	+	0.0844795i	$0.973077\pi$
$854$	− 19.2325i	− 0.0225205i
$855$	220.215i	0.257561i
$856$	535.786	0.625918
$857$	− 527.163i	− 0.615126i	−0.951528	−	0.307563i	$-0.900487\pi$
$857$	− 527.163i	− 0.615126i	0.951528	−	0.307563i	$-0.0995134\pi$
$858$	0	0
$859$	122.027	0.142057	0.0710284	−	0.997474i	$-0.477372\pi$
$859$	122.027	0.142057	0.0710284	+	0.997474i	$0.477372\pi$
$860$	361.482i	0.420328i
$861$	552.998	0.642274
$862$	1088.62	1.26290
$863$	845.250	0.979432	0.489716	−	0.871882i	$-0.337100\pi$
$863$	845.250	0.979432	0.489716	+	0.871882i	$0.337100\pi$
$864$	74.0474i	0.0857031i
$865$	141.234i	0.163276i
$866$	891.967i	1.02998i
$867$	394.457	0.454968
$868$	165.949i	0.191185i
$869$	0	0
$870$	239.482	0.275267
$871$	− 322.823i	− 0.370635i
$872$	18.3513	0.0210451
$873$	468.059	0.536151
$874$	174.032	0.199121
$875$	1485.31i	1.69750i
$876$	196.720i	0.224566i
$877$	914.615i	1.04289i	0.853285	+	0.521445i	$0.174607\pi$
$877$	914.615i	1.04289i	−0.853285	+	0.521445i	$0.825393\pi$
$878$	−151.524	−0.172579
$879$	− 294.854i	− 0.335443i
$880$	0	0
$881$	618.978	0.702586	0.351293	−	0.936266i	$-0.385742\pi$
$881$	618.978	0.702586	0.351293	+	0.936266i	$0.385742\pi$
$882$	− 29.8108i	− 0.0337991i
$883$	−58.5805	−0.0663425	−0.0331713	−	0.999450i	$-0.510561\pi$
$883$	−58.5805	−0.0663425	−0.0331713	+	0.999450i	$0.510561\pi$
$884$	−79.1835	−0.0895740
$885$	−1072.46	−1.21182
$886$	− 694.324i	− 0.783661i
$887$	1060.58i	1.19569i	0.801612	+	0.597845i	$0.203976\pi$
$887$	1060.58i	1.19569i	−0.801612	+	0.597845i	$0.796024\pi$
$888$	628.274i	0.707516i
$889$	−794.667	−0.893889
$890$	182.678i	0.205257i
$891$	0	0
$892$	234.235	0.262595
$893$	− 135.732i	− 0.151995i
$894$	−539.917	−0.603934
$895$	1133.42	1.26639
$896$	−1025.49	−1.14452
$897$	179.798i	0.200444i
$898$	− 729.036i	− 0.811845i
$899$	− 196.866i	− 0.218983i
$900$	−137.686	−0.152985
$901$	429.947i	0.477189i
$902$	0	0
$903$	529.708	0.586609
$904$	− 1262.10i	− 1.39613i
$905$	−882.898	−0.975578
$906$	44.1567	0.0487381
$907$	−1248.22	−1.37621	−0.688105	−	0.725611i	$-0.741557\pi$
$907$	−1248.22	−1.37621	−0.688105	+	0.725611i	$0.741557\pi$
$908$	40.1865i	0.0442582i
$909$	165.229i	0.181770i
$910$	1433.45i	1.57522i
$911$	−619.331	−0.679837	−0.339918	−	0.940455i	$-0.610399\pi$
$911$	−619.331	−0.679837	−0.339918	+	0.940455i	$0.610399\pi$
$912$	− 274.976i	− 0.301509i
$913$	0	0
$914$	196.150	0.214606
$915$	− 19.5980i	− 0.0214186i
$916$	3.48162	0.00380090
$917$	−120.093	−0.130963
$918$	−90.0900	−0.0981372
$919$	1704.20i	1.85441i	0.374558	+	0.927204i	$0.377795\pi$
$919$	1704.20i	1.85441i	−0.374558	+	0.927204i	$0.622205\pi$
$920$	− 554.409i	− 0.602618i
$921$	− 396.412i	− 0.430415i
$922$	1345.29	1.45910
$923$	245.984i	0.266504i
$924$	0	0
$925$	−2679.02	−2.89624
$926$	90.1851i	0.0973921i
$927$	−243.053	−0.262193
$928$	−102.297	−0.110234
$929$	−505.092	−0.543694	−0.271847	−	0.962340i	$-0.587635\pi$
$929$	−505.092	−0.543694	−0.271847	+	0.962340i	$0.587635\pi$
$930$	914.900i	0.983763i
$931$	37.8705i	0.0406772i
$932$	− 218.889i	− 0.234859i
$933$	489.409	0.524554
$934$	991.952i	1.06205i
$935$	0	0
$936$	229.282	0.244959
$937$	1305.95i	1.39376i	0.717189	+	0.696878i	$0.245428\pi$
$937$	1305.95i	1.39376i	−0.717189	+	0.696878i	$0.754572\pi$
$938$	−427.723	−0.455995
$939$	−606.522	−0.645923
$940$	126.791	0.134884
$941$	− 31.8347i	− 0.0338307i	−0.999857	−	0.0169154i	$-0.994615\pi$
$941$	− 31.8347i	− 0.0338307i	0.999857	−	0.0169154i	$-0.00538459\pi$
$942$	574.479i	0.609850i
$943$	− 445.330i	− 0.472248i
$944$	1339.15	1.41859
$945$	301.440i	0.318985i
$946$	0	0
$947$	−749.175	−0.791103	−0.395552	−	0.918444i	$-0.629447\pi$
$947$	−749.175	−0.791103	−0.395552	+	0.918444i	$0.629447\pi$
$948$	67.3548i	0.0710493i
$949$	1396.86	1.47193
$950$	946.325	0.996132
$951$	648.505	0.681919
$952$	− 357.790i	− 0.375830i
$953$	1452.72i	1.52437i	0.647361	+	0.762183i	$0.275873\pi$
$953$	1452.72i	1.52437i	−0.647361	+	0.762183i	$0.724127\pi$
$954$	365.052i	0.382654i
$955$	−1406.63	−1.47291
$956$	420.280i	0.439624i
$957$	0	0
$958$	129.571	0.135251
$959$	617.554i	0.643957i
$960$	−657.439	−0.684832
$961$	−208.907	−0.217385
$962$	−1308.16	−1.35983
$963$	234.596i	0.243610i
$964$	− 405.605i	− 0.420752i
$965$	− 297.297i	− 0.308079i
$966$	238.223	0.246608
$967$	− 950.193i	− 0.982619i	−0.870985	−	0.491310i	$-0.836518\pi$
$967$	− 950.193i	− 0.982619i	0.870985	−	0.491310i	$-0.163482\pi$
$968$	0	0
$969$	114.447	0.118108
$970$	− 3005.09i	− 3.09803i
$971$	−659.055	−0.678738	−0.339369	−	0.940653i	$-0.610214\pi$
$971$	−659.055	−0.678738	−0.339369	+	0.940653i	$0.610214\pi$
$972$	−14.1382	−0.0145454
$973$	−1446.58	−1.48672
$974$	1447.13i	1.48576i
$975$	977.681i	1.00275i
$976$	24.4715i	0.0250733i
$977$	206.627	0.211491	0.105746	−	0.994393i	$-0.466277\pi$
$977$	206.627	0.211491	0.105746	+	0.994393i	$0.466277\pi$
$978$	210.322i	0.215053i
$979$	0	0
$980$	−35.3758	−0.0360978
$981$	8.03522i	0.00819084i
$982$	652.957	0.664926
$983$	228.441	0.232391	0.116196	−	0.993226i	$-0.462930\pi$
$983$	228.441	0.232391	0.116196	+	0.993226i	$0.462930\pi$
$984$	−567.892	−0.577126
$985$	− 538.041i	− 0.546234i
$986$	− 124.460i	− 0.126227i
$987$	− 185.796i	− 0.188244i
$988$	85.4084	0.0864457
$989$	− 426.575i	− 0.431319i
$990$	0	0
$991$	−1872.78	−1.88979	−0.944895	−	0.327373i	$-0.893837\pi$
$991$	−1872.78	−1.88979	−0.944895	+	0.327373i	$0.893837\pi$
$992$	− 390.809i	− 0.393960i
$993$	455.399	0.458609
$994$	325.915	0.327883
$995$	2072.03	2.08245
$996$	− 209.266i	− 0.210107i
$997$	− 1536.35i	− 1.54097i	−0.637459	−	0.770484i	$-0.720015\pi$
$997$	− 1536.35i	− 1.54097i	0.637459	−	0.770484i	$-0.279985\pi$
$998$	− 91.5337i	− 0.0917171i
$999$	−275.093	−0.275368

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.3.c.e.241.5		16
3.2	odd	2		1089.3.c.m.604.12		16
11.2	odd	10		33.3.g.a.7.3	✓	16
11.3	even	5		363.3.g.a.112.2		16
11.4	even	5		363.3.g.g.94.3		16
11.5	even	5		33.3.g.a.19.3	yes	16
11.6	odd	10		363.3.g.f.118.2		16
11.7	odd	10		363.3.g.a.94.2		16
11.8	odd	10		363.3.g.g.112.3		16
11.9	even	5		363.3.g.f.40.2		16
11.10	odd	2	inner	363.3.c.e.241.12		16
33.2	even	10		99.3.k.c.73.2		16
33.5	odd	10		99.3.k.c.19.2		16
33.32	even	2		1089.3.c.m.604.5		16
44.27	odd	10		528.3.bf.b.481.1		16
44.35	even	10		528.3.bf.b.337.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.g.a.7.3	✓	16	11.2	odd	10
33.3.g.a.19.3	yes	16	11.5	even	5
99.3.k.c.19.2		16	33.5	odd	10
99.3.k.c.73.2		16	33.2	even	10
363.3.c.e.241.5		16	1.1	even	1	trivial
363.3.c.e.241.12		16	11.10	odd	2	inner
363.3.g.a.94.2		16	11.7	odd	10
363.3.g.a.112.2		16	11.3	even	5
363.3.g.f.40.2		16	11.9	even	5
363.3.g.f.118.2		16	11.6	odd	10
363.3.g.g.94.3		16	11.4	even	5
363.3.g.g.112.3		16	11.8	odd	10
528.3.bf.b.337.1		16	44.35	even	10
528.3.bf.b.481.1		16	44.27	odd	10
1089.3.c.m.604.5		16	33.32	even	2
1089.3.c.m.604.12		16	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:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	363.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.21517i q^{2} +1.73205 q^{3} -0.906963 q^{4} +8.69502 q^{5} -3.83678i q^{6} +6.67189i q^{7} -6.85159i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-2.21517i q^{2} +1.73205 q^{3} -0.906963 q^{4} +8.69502 q^{5} -3.83678i q^{6} +6.67189i q^{7} -6.85159i q^{8} +3.00000 q^{9} -19.2609i q^{10} -1.57091 q^{12} +11.1547i q^{13} +14.7794 q^{14} +15.0602 q^{15} -18.8053 q^{16} -7.82687i q^{17} -6.64550i q^{18} +8.44217i q^{19} -7.88606 q^{20} +11.5561i q^{21} +9.30611 q^{23} -11.8673i q^{24} +50.6034 q^{25} +24.7094 q^{26} +5.19615 q^{27} -6.05116i q^{28} +7.17852i q^{29} -33.3609i q^{30} -27.4243 q^{31} +14.2504i q^{32} -17.3378 q^{34} +58.0122i q^{35} -2.72089 q^{36} -52.9416 q^{37} +18.7008 q^{38} +19.3205i q^{39} -59.5747i q^{40} -47.8535i q^{41} +25.5986 q^{42} -45.8381i q^{43} +26.0851 q^{45} -20.6146i q^{46} -16.0778 q^{47} -32.5717 q^{48} +4.48587 q^{49} -112.095i q^{50} -13.5565i q^{51} -10.1169i q^{52} -54.9322 q^{53} -11.5103i q^{54} +45.7131 q^{56} +14.6223i q^{57} +15.9016 q^{58} -71.2113 q^{59} -13.6591 q^{60} -1.30131i q^{61} +60.7494i q^{62} +20.0157i q^{63} -43.6540 q^{64} +96.9901i q^{65} -28.9406 q^{67} +7.09868i q^{68} +16.1187 q^{69} +128.507 q^{70} +22.0521 q^{71} -20.5548i q^{72} -125.227i q^{73} +117.274i q^{74} +87.6477 q^{75} -7.65674i q^{76} +42.7980 q^{78} -42.8764i q^{79} -163.512 q^{80} +9.00000 q^{81} -106.004 q^{82} +133.214i q^{83} -10.4809i q^{84} -68.0548i q^{85} -101.539 q^{86} +12.4336i q^{87} -9.48441 q^{89} -57.7828i q^{90} -74.4227 q^{91} -8.44030 q^{92} -47.5003 q^{93} +35.6151i q^{94} +73.4049i q^{95} +24.6825i q^{96} +156.020 q^{97} -9.93695i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 20 q^{4} - 4 q^{5} + 48 q^{9}+O(q^{10}) \) 16 * q - 20 * q^4 - 4 * q^5 + 48 * q^9	\( 16 q - 20 q^{4} - 4 q^{5} + 48 q^{9} - 24 q^{12} - 52 q^{14} + 36 q^{15} - 44 q^{16} - 108 q^{20} + 132 q^{23} + 88 q^{25} - 4 q^{26} + 40 q^{31} - 368 q^{34} - 60 q^{36} - 16 q^{37} + 280 q^{38} + 36 q^{42} - 12 q^{45} + 80 q^{47} + 144 q^{48} - 140 q^{49} - 128 q^{53} + 524 q^{56} + 140 q^{58} - 220 q^{59} - 384 q^{60} - 8 q^{64} + 36 q^{67} - 180 q^{69} - 100 q^{70} + 644 q^{71} + 312 q^{75} - 312 q^{78} + 264 q^{80} + 144 q^{81} - 476 q^{82} + 76 q^{86} + 76 q^{89} - 624 q^{91} + 120 q^{92} - 336 q^{93} + 216 q^{97}+O(q^{100}) \) 16 * q - 20 * q^4 - 4 * q^5 + 48 * q^9 - 24 * q^12 - 52 * q^14 + 36 * q^15 - 44 * q^16 - 108 * q^20 + 132 * q^23 + 88 * q^25 - 4 * q^26 + 40 * q^31 - 368 * q^34 - 60 * q^36 - 16 * q^37 + 280 * q^38 + 36 * q^42 - 12 * q^45 + 80 * q^47 + 144 * q^48 - 140 * q^49 - 128 * q^53 + 524 * q^56 + 140 * q^58 - 220 * q^59 - 384 * q^60 - 8 * q^64 + 36 * q^67 - 180 * q^69 - 100 * q^70 + 644 * q^71 + 312 * q^75 - 312 * q^78 + 264 * q^80 + 144 * q^81 - 476 * q^82 + 76 * q^86 + 76 * q^89 - 624 * q^91 + 120 * q^92 - 336 * q^93 + 216 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more