LMFDB - Embedded newform 3700.2.d.i.149.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3700,2,Mod(149,3700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3700, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3700.149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3700 = 2^{2} \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3700.d (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:	$29.5446487479$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 23x^{6} + 139x^{4} + 273x^{2} + 144 $ x^8 + 23x^6 + 139x^4 + 273*x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 740)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			149.6
Root			$1.51951i$ of defining polynomial
Character	$\chi$	$=$	3700.149
Dual form			3700.2.d.i.149.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.51951i q^{3} -0.519509i q^{7} -3.34793 q^{9} +O(q^{10})$	$q+2.51951i q^{3} -0.519509i q^{7} -3.34793 q^{9} +6.16241 q^{11} -6.51034i q^{13} +2.51034i q^{17} -6.68192 q^{19} +1.30891 q^{21} -3.03902i q^{23} -0.876607i q^{27} +3.03902 q^{29} -4.33876 q^{31} +15.5263i q^{33} -1.00000i q^{37} +16.4029 q^{39} +12.1624 q^{41} -2.34316i q^{43} -9.54019i q^{47} +6.73011 q^{49} -6.32482 q^{51} -9.20143i q^{53} -16.8352i q^{57} +9.37778 q^{59} +2.00000 q^{61} +1.73928i q^{63} -0.357097i q^{67} +7.65684 q^{69} -2.87661 q^{71} -10.2404i q^{73} -3.20143i q^{77} -0.681922 q^{79} -7.83516 q^{81} +6.17635i q^{83} +7.65684i q^{87} +11.0207 q^{89} -3.38218 q^{91} -10.9315i q^{93} -6.34316i q^{97} -20.6313 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 26 q^{9}+O(q^{10}) $ 8 * q - 26 * q^9	$ 8 q - 26 q^{9} + 10 q^{11} + 38 q^{21} - 4 q^{29} - 8 q^{31} - 4 q^{39} + 58 q^{41} - 2 q^{49} + 28 q^{51} + 20 q^{59} + 16 q^{61} + 88 q^{69} - 34 q^{71} + 48 q^{79} + 56 q^{81} + 8 q^{89} + 28 q^{91} + 44 q^{99}+O(q^{100}) $ 8 * q - 26 * q^9 + 10 * q^11 + 38 * q^21 - 4 * q^29 - 8 * q^31 - 4 * q^39 + 58 * q^41 - 2 * q^49 + 28 * q^51 + 20 * q^59 + 16 * q^61 + 88 * q^69 - 34 * q^71 + 48 * q^79 + 56 * q^81 + 8 * q^89 + 28 * q^91 + 44 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1777$	$1851$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.51951i	1.45464i	0.686299	+	0.727320i	$0.259234\pi$
$3$	2.51951i	1.45464i	−0.686299	+	0.727320i	$0.740766\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.519509i	− 0.196356i	−0.995169	−	0.0981780i	$-0.968699\pi$
$7$	− 0.519509i	− 0.196356i	0.995169	−	0.0981780i	$-0.0313015\pi$
$8$	0	0
$9$	−3.34793	−1.11598
$10$	0	0
$11$	6.16241	1.85804	0.929019	−	0.370033i	$-0.120654\pi$
$11$	6.16241	1.85804	0.929019	+	0.370033i	$0.120654\pi$
$12$	0	0
$13$	− 6.51034i	− 1.80564i	−0.430015	−	0.902822i	$-0.641492\pi$
$13$	− 6.51034i	− 1.80564i	0.430015	−	0.902822i	$-0.358508\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.51034i	0.608847i	0.952537	+	0.304423i	$0.0984638\pi$
$17$	2.51034i	0.608847i	−0.952537	+	0.304423i	$0.901536\pi$
$18$	0	0
$19$	−6.68192	−1.53294	−0.766469	−	0.642281i	$-0.777988\pi$
$19$	−6.68192	−1.53294	−0.766469	+	0.642281i	$0.777988\pi$
$20$	0	0
$21$	1.30891	0.285627
$22$	0	0
$23$	− 3.03902i	− 0.633679i	−0.948479	−	0.316840i	$-0.897378\pi$
$23$	− 3.03902i	− 0.633679i	0.948479	−	0.316840i	$-0.102622\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 0.876607i	− 0.168703i
$28$	0	0
$29$	3.03902	0.564332	0.282166	−	0.959366i	$-0.408947\pi$
$29$	3.03902	0.564332	0.282166	+	0.959366i	$0.408947\pi$
$30$	0	0
$31$	−4.33876	−0.779264	−0.389632	−	0.920971i	$-0.627398\pi$
$31$	−4.33876	−0.779264	−0.389632	+	0.920971i	$0.627398\pi$
$32$	0	0
$33$	15.5263i	2.70277i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 1.00000i	− 0.164399i
$37$	− 1.00000i	− 0.164399i
$38$	0	0
$39$	16.4029	2.62656
$40$	0	0
$41$	12.1624	1.89945	0.949725	−	0.313086i	$-0.101363\pi$
$41$	12.1624	1.89945	0.949725	+	0.313086i	$0.101363\pi$
$42$	0	0
$43$	− 2.34316i	− 0.357329i	−0.983910	−	0.178665i	$-0.942822\pi$
$43$	− 2.34316i	− 0.357329i	0.983910	−	0.178665i	$-0.0571777\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 9.54019i	− 1.39158i	−0.718245	−	0.695790i	$-0.755055\pi$
$47$	− 9.54019i	− 1.39158i	0.718245	−	0.695790i	$-0.244945\pi$
$48$	0	0
$49$	6.73011	0.961444
$50$	0	0
$51$	−6.32482	−0.885653
$52$	0	0
$53$	− 9.20143i	− 1.26391i	−0.775004	−	0.631957i	$-0.782252\pi$
$53$	− 9.20143i	− 1.26391i	0.775004	−	0.631957i	$-0.217748\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 16.8352i	− 2.22987i
$58$	0	0
$59$	9.37778	1.22088	0.610441	−	0.792062i	$-0.290992\pi$
$59$	9.37778	1.22088	0.610441	+	0.792062i	$0.290992\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	1.73928i	0.219129i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 0.357097i	− 0.0436264i	−0.999762	−	0.0218132i	$-0.993056\pi$
$67$	− 0.357097i	− 0.0436264i	0.999762	−	0.0218132i	$-0.00694390\pi$
$68$	0	0
$69$	7.65684	0.921775
$70$	0	0
$71$	−2.87661	−0.341390	−0.170695	−	0.985324i	$-0.554601\pi$
$71$	−2.87661	−0.341390	−0.170695	+	0.985324i	$0.554601\pi$
$72$	0	0
$73$	− 10.2404i	− 1.19855i	−0.800542	−	0.599277i	$-0.795455\pi$
$73$	− 10.2404i	− 1.19855i	0.800542	−	0.599277i	$-0.204545\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.20143i	− 0.364837i
$78$	0	0
$79$	−0.681922	−0.0767222	−0.0383611	−	0.999264i	$-0.512214\pi$
$79$	−0.681922	−0.0767222	−0.0383611	+	0.999264i	$0.512214\pi$
$80$	0	0
$81$	−7.83516	−0.870574
$82$	0	0
$83$	6.17635i	0.677942i	0.940797	+	0.338971i	$0.110079\pi$
$83$	6.17635i	0.677942i	−0.940797	+	0.338971i	$0.889921\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.65684i	0.820899i
$88$	0	0
$89$	11.0207	1.16819	0.584095	−	0.811685i	$-0.301450\pi$
$89$	11.0207	1.16819	0.584095	+	0.811685i	$0.301450\pi$
$90$	0	0
$91$	−3.38218	−0.354549
$92$	0	0
$93$	− 10.9315i	− 1.13355i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 6.34316i	− 0.644051i	−0.946731	−	0.322025i	$-0.895636\pi$
$97$	− 6.34316i	− 0.644051i	0.946731	−	0.322025i	$-0.104364\pi$
$98$	0	0
$99$	−20.6313	−2.07352
$100$	0	0
$101$	19.4466	1.93501	0.967507	−	0.252845i	$-0.0813664\pi$
$101$	19.4466	1.93501	0.967507	+	0.252845i	$0.0813664\pi$
$102$	0	0
$103$	2.00000i	0.197066i	0.995134	+	0.0985329i	$0.0314150\pi$
$103$	2.00000i	0.197066i	−0.995134	+	0.0985329i	$0.968585\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.0251i	1.45253i	0.687415	+	0.726265i	$0.258745\pi$
$107$	15.0251i	1.45253i	−0.687415	+	0.726265i	$0.741255\pi$
$108$	0	0
$109$	−15.7070	−1.50446	−0.752229	−	0.658902i	$-0.771021\pi$
$109$	−15.7070	−1.50446	−0.752229	+	0.658902i	$0.771021\pi$
$110$	0	0
$111$	2.51951	0.239141
$112$	0	0
$113$	− 2.18552i	− 0.205596i	−0.994702	−	0.102798i	$-0.967220\pi$
$113$	− 2.18552i	− 0.205596i	0.994702	−	0.102798i	$-0.0327795\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	21.7961i	2.01505i
$118$	0	0
$119$	1.30414	0.119551
$120$	0	0
$121$	26.9753	2.45230
$122$	0	0
$123$	30.6433i	2.76301i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2.17635i	− 0.193119i	−0.995327	−	0.0965597i	$-0.969216\pi$
$127$	− 2.17635i	− 0.193119i	0.995327	−	0.0965597i	$-0.0307839\pi$
$128$	0	0
$129$	5.90362	0.519785
$130$	0	0
$131$	−12.7416	−1.11324	−0.556620	−	0.830767i	$-0.687902\pi$
$131$	−12.7416	−1.11324	−0.556620	+	0.830767i	$0.687902\pi$
$132$	0	0
$133$	3.47132i	0.301002i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.26513i	0.364394i	0.983262	+	0.182197i	$0.0583209\pi$
$137$	4.26513i	0.364394i	−0.983262	+	0.182197i	$0.941679\pi$
$138$	0	0
$139$	11.6290	0.986356	0.493178	−	0.869928i	$-0.335835\pi$
$139$	11.6290	0.986356	0.493178	+	0.869928i	$0.335835\pi$
$140$	0	0
$141$	24.0366	2.02425
$142$	0	0
$143$	− 40.1194i	− 3.35495i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	16.9566i	1.39855i
$148$	0	0
$149$	17.3591	1.42211	0.711056	−	0.703136i	$-0.248217\pi$
$149$	17.3591	1.42211	0.711056	+	0.703136i	$0.248217\pi$
$150$	0	0
$151$	0.343164	0.0279263	0.0139631	−	0.999903i	$-0.495555\pi$
$151$	0.343164	0.0279263	0.0139631	+	0.999903i	$0.495555\pi$
$152$	0	0
$153$	− 8.40444i	− 0.679458i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.16241i	0.332197i	0.986109	+	0.166098i	$0.0531169\pi$
$157$	4.16241i	0.332197i	−0.986109	+	0.166098i	$0.946883\pi$
$158$	0	0
$159$	23.1831	1.83854
$160$	0	0
$161$	−1.57880	−0.124427
$162$	0	0
$163$	19.3455i	1.51526i	0.652686	+	0.757628i	$0.273642\pi$
$163$	19.3455i	1.51526i	−0.652686	+	0.757628i	$0.726358\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.979321i	0.0757821i	0.999282	+	0.0378911i	$0.0120640\pi$
$167$	0.979321i	0.0757821i	−0.999282	+	0.0378911i	$0.987936\pi$
$168$	0	0
$169$	−29.3845	−2.26035
$170$	0	0
$171$	22.3706	1.71072
$172$	0	0
$173$	− 22.9363i	− 1.74381i	−0.489671	−	0.871907i	$-0.662883\pi$
$173$	− 22.9363i	− 1.74381i	0.489671	−	0.871907i	$-0.337117\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	23.6274i	1.77594i
$178$	0	0
$179$	16.7600	1.25270	0.626349	−	0.779543i	$-0.284548\pi$
$179$	16.7600	1.25270	0.626349	+	0.779543i	$0.284548\pi$
$180$	0	0
$181$	−11.4466	−0.850822	−0.425411	−	0.905000i	$-0.639871\pi$
$181$	−11.4466	−0.850822	−0.425411	+	0.905000i	$0.639871\pi$
$182$	0	0
$183$	5.03902i	0.372495i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	15.4697i	1.13126i
$188$	0	0
$189$	−0.455405	−0.0331259
$190$	0	0
$191$	−10.4351	−0.755060	−0.377530	−	0.925997i	$-0.623226\pi$
$191$	−10.4351	−0.755060	−0.377530	+	0.925997i	$0.623226\pi$
$192$	0	0
$193$	20.4029i	1.46863i	0.678809	+	0.734315i	$0.262497\pi$
$193$	20.4029i	1.46863i	−0.678809	+	0.734315i	$0.737503\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.1051i	0.791202i	0.918422	+	0.395601i	$0.129464\pi$
$197$	11.1051i	0.791202i	−0.918422	+	0.395601i	$0.870536\pi$
$198$	0	0
$199$	0.622223	0.0441083	0.0220541	−	0.999757i	$-0.492979\pi$
$199$	0.622223	0.0441083	0.0220541	+	0.999757i	$0.492979\pi$
$200$	0	0
$201$	0.899710	0.0634606
$202$	0	0
$203$	− 1.57880i	− 0.110810i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	10.1744i	0.707171i
$208$	0	0
$209$	−41.1768	−2.84826
$210$	0	0
$211$	−18.3001	−1.25983	−0.629917	−	0.776662i	$-0.716911\pi$
$211$	−18.3001	−1.25983	−0.629917	+	0.776662i	$0.716911\pi$
$212$	0	0
$213$	− 7.24764i	− 0.496600i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.25403i	0.153013i
$218$	0	0
$219$	25.8009	1.74346
$220$	0	0
$221$	16.3432	1.09936
$222$	0	0
$223$	− 0.784636i	− 0.0525431i	−0.999655	−	0.0262715i	$-0.991637\pi$
$223$	− 0.784636i	− 0.0525431i	0.999655	−	0.0262715i	$-0.00836345\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 6.32482i	− 0.419793i	−0.977724	−	0.209897i	$-0.932687\pi$
$227$	− 6.32482i	− 0.419793i	0.977724	−	0.209897i	$-0.0673128\pi$
$228$	0	0
$229$	−2.08437	−0.137739	−0.0688697	−	0.997626i	$-0.521939\pi$
$229$	−2.08437	−0.137739	−0.0688697	+	0.997626i	$0.521939\pi$
$230$	0	0
$231$	8.06604	0.530706
$232$	0	0
$233$	26.4626i	1.73362i	0.498639	+	0.866810i	$0.333833\pi$
$233$	26.4626i	1.73362i	−0.498639	+	0.866810i	$0.666167\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1.71811i	− 0.111603i
$238$	0	0
$239$	−18.6636	−1.20725	−0.603623	−	0.797270i	$-0.706277\pi$
$239$	−18.6636	−1.20725	−0.603623	+	0.797270i	$0.706277\pi$
$240$	0	0
$241$	−0.0596981	−0.00384549	−0.00192274	−	0.999998i	$-0.500612\pi$
$241$	−0.0596981	−0.00384549	−0.00192274	+	0.999998i	$0.500612\pi$
$242$	0	0
$243$	− 22.3706i	− 1.43507i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	43.5016i	2.76794i
$248$	0	0
$249$	−15.5614	−0.986161
$250$	0	0
$251$	3.05295	0.192701	0.0963503	−	0.995347i	$-0.469283\pi$
$251$	3.05295	0.192701	0.0963503	+	0.995347i	$0.469283\pi$
$252$	0	0
$253$	− 18.7277i	− 1.17740i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20.9132i	1.30453i	0.757991	+	0.652265i	$0.226181\pi$
$257$	20.9132i	1.30453i	−0.757991	+	0.652265i	$0.773819\pi$
$258$	0	0
$259$	−0.519509	−0.0322807
$260$	0	0
$261$	−10.1744	−0.629780
$262$	0	0
$263$	− 7.91122i	− 0.487827i	−0.969797	−	0.243913i	$-0.921569\pi$
$263$	− 7.91122i	− 0.487827i	0.969797	−	0.243913i	$-0.0784312\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	27.7667i	1.69929i
$268$	0	0
$269$	0.606717	0.0369922	0.0184961	−	0.999829i	$-0.494112\pi$
$269$	0.606717	0.0369922	0.0184961	+	0.999829i	$0.494112\pi$
$270$	0	0
$271$	20.5931	1.25094	0.625472	−	0.780247i	$-0.284906\pi$
$271$	20.5931	1.25094	0.625472	+	0.780247i	$0.284906\pi$
$272$	0	0
$273$	− 8.52144i	− 0.515741i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 21.2842i	− 1.27885i	−0.768855	−	0.639423i	$-0.779173\pi$
$277$	− 21.2842i	− 1.27885i	0.768855	−	0.639423i	$-0.220827\pi$
$278$	0	0
$279$	14.5258	0.869640
$280$	0	0
$281$	−10.3432	−0.617021	−0.308511	−	0.951221i	$-0.599831\pi$
$281$	−10.3432	−0.617021	−0.308511	+	0.951221i	$0.599831\pi$
$282$	0	0
$283$	1.67518i	0.0995789i	0.998760	+	0.0497894i	$0.0158550\pi$
$283$	1.67518i	0.0995789i	−0.998760	+	0.0497894i	$0.984145\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 6.31849i	− 0.372969i
$288$	0	0
$289$	10.6982	0.629306
$290$	0	0
$291$	15.9817	0.936862
$292$	0	0
$293$	9.96332i	0.582063i	0.956713	+	0.291032i	$0.0939985\pi$
$293$	9.96332i	0.582063i	−0.956713	+	0.291032i	$0.906001\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 5.40201i	− 0.313456i
$298$	0	0
$299$	−19.7850	−1.14420
$300$	0	0
$301$	−1.21730	−0.0701637
$302$	0	0
$303$	48.9960i	2.81475i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 26.2265i	− 1.49683i	−0.663233	−	0.748413i	$-0.730816\pi$
$307$	− 26.2265i	− 1.49683i	0.663233	−	0.748413i	$-0.269184\pi$
$308$	0	0
$309$	−5.03902	−0.286660
$310$	0	0
$311$	1.74881	0.0991658	0.0495829	−	0.998770i	$-0.484211\pi$
$311$	1.74881	0.0991658	0.0495829	+	0.998770i	$0.484211\pi$
$312$	0	0
$313$	− 21.7762i	− 1.23087i	−0.788189	−	0.615433i	$-0.788981\pi$
$313$	− 21.7762i	− 1.23087i	0.788189	−	0.615433i	$-0.211019\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 4.42120i	− 0.248319i	−0.992262	−	0.124160i	$-0.960376\pi$
$317$	− 4.42120i	− 0.248319i	0.992262	−	0.124160i	$-0.0396235\pi$
$318$	0	0
$319$	18.7277	1.04855
$320$	0	0
$321$	−37.8558	−2.11291
$322$	0	0
$323$	− 16.7739i	− 0.933324i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 39.5740i	− 2.18844i
$328$	0	0
$329$	−4.95622	−0.273245
$330$	0	0
$331$	32.4168	1.78179	0.890894	−	0.454211i	$-0.150079\pi$
$331$	32.4168	1.78179	0.890894	+	0.454211i	$0.150079\pi$
$332$	0	0
$333$	3.34793i	0.183465i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 29.8789i	− 1.62761i	−0.581138	−	0.813805i	$-0.697392\pi$
$337$	− 29.8789i	− 1.62761i	0.581138	−	0.813805i	$-0.302608\pi$
$338$	0	0
$339$	5.50643	0.299068
$340$	0	0
$341$	−26.7372	−1.44790
$342$	0	0
$343$	− 7.13292i	− 0.385142i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.88294i	− 0.154765i	−0.997001	−	0.0773823i	$-0.975344\pi$
$347$	− 2.88294i	− 0.154765i	0.997001	−	0.0773823i	$-0.0246562\pi$
$348$	0	0
$349$	−31.5494	−1.68880	−0.844399	−	0.535714i	$-0.820042\pi$
$349$	−31.5494	−1.68880	−0.844399	+	0.535714i	$0.820042\pi$
$350$	0	0
$351$	−5.70701	−0.304617
$352$	0	0
$353$	− 2.68633i	− 0.142979i	−0.997441	−	0.0714894i	$-0.977225\pi$
$353$	− 2.68633i	− 0.142979i	0.997441	−	0.0714894i	$-0.0227752\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.28581i	0.173903i
$358$	0	0
$359$	−21.9570	−1.15885	−0.579423	−	0.815027i	$-0.696722\pi$
$359$	−21.9570	−1.15885	−0.579423	+	0.815027i	$0.696722\pi$
$360$	0	0
$361$	25.6481	1.34990
$362$	0	0
$363$	67.9646i	3.56722i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	25.3499i	1.32325i	0.749833	+	0.661627i	$0.230134\pi$
$367$	25.3499i	1.32325i	−0.749833	+	0.661627i	$0.769866\pi$
$368$	0	0
$369$	−40.7189	−2.11974
$370$	0	0
$371$	−4.78023	−0.248177
$372$	0	0
$373$	23.6322i	1.22363i	0.791002	+	0.611813i	$0.209560\pi$
$373$	23.6322i	1.22363i	−0.791002	+	0.611813i	$0.790440\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 19.7850i	− 1.01898i
$378$	0	0
$379$	18.8216	0.966800	0.483400	−	0.875400i	$-0.339402\pi$
$379$	18.8216	0.966800	0.483400	+	0.875400i	$0.339402\pi$
$380$	0	0
$381$	5.48332	0.280919
$382$	0	0
$383$	11.0485i	0.564554i	0.959333	+	0.282277i	$0.0910898\pi$
$383$	11.0485i	0.564554i	−0.959333	+	0.282277i	$0.908910\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.84474i	0.398771i
$388$	0	0
$389$	−29.4235	−1.49183	−0.745916	−	0.666040i	$-0.767988\pi$
$389$	−29.4235	−1.49183	−0.745916	+	0.666040i	$0.767988\pi$
$390$	0	0
$391$	7.62897	0.385814
$392$	0	0
$393$	− 32.1026i	− 1.61936i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 17.2014i	− 0.863315i	−0.902038	−	0.431658i	$-0.857929\pi$
$397$	− 17.2014i	− 0.863315i	0.902038	−	0.431658i	$-0.142071\pi$
$398$	0	0
$399$	−8.74603	−0.437849
$400$	0	0
$401$	18.6496	0.931319	0.465660	−	0.884964i	$-0.345817\pi$
$401$	18.6496	0.931319	0.465660	+	0.884964i	$0.345817\pi$
$402$	0	0
$403$	28.2468i	1.40707i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 6.16241i	− 0.305459i
$408$	0	0
$409$	−11.0390	−0.545844	−0.272922	−	0.962036i	$-0.587990\pi$
$409$	−11.0390	−0.545844	−0.272922	+	0.962036i	$0.587990\pi$
$410$	0	0
$411$	−10.7460	−0.530062
$412$	0	0
$413$	− 4.87184i	− 0.239728i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	29.2993i	1.43479i
$418$	0	0
$419$	9.95698	0.486430	0.243215	−	0.969972i	$-0.421798\pi$
$419$	9.95698	0.486430	0.243215	+	0.969972i	$0.421798\pi$
$420$	0	0
$421$	22.8153	1.11195	0.555974	−	0.831200i	$-0.312346\pi$
$421$	22.8153	1.11195	0.555974	+	0.831200i	$0.312346\pi$
$422$	0	0
$423$	31.9399i	1.55297i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1.03902i	− 0.0502816i
$428$	0	0
$429$	101.081	4.88025
$430$	0	0
$431$	8.67239	0.417735	0.208867	−	0.977944i	$-0.433022\pi$
$431$	8.67239	0.417735	0.208867	+	0.977944i	$0.433022\pi$
$432$	0	0
$433$	− 5.38852i	− 0.258956i	−0.991582	−	0.129478i	$-0.958670\pi$
$433$	− 5.38852i	− 0.258956i	0.991582	−	0.129478i	$-0.0413301\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	20.3065i	0.971391i
$438$	0	0
$439$	−0.603884	−0.0288218	−0.0144109	−	0.999896i	$-0.504587\pi$
$439$	−0.603884	−0.0288218	−0.0144109	+	0.999896i	$0.504587\pi$
$440$	0	0
$441$	−22.5319	−1.07295
$442$	0	0
$443$	27.2655i	1.29542i	0.761885	+	0.647712i	$0.224274\pi$
$443$	27.2655i	1.29542i	−0.761885	+	0.647712i	$0.775726\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	43.7364i	2.06866i
$448$	0	0
$449$	−9.28581	−0.438224	−0.219112	−	0.975700i	$-0.570316\pi$
$449$	−9.28581	−0.438224	−0.219112	+	0.975700i	$0.570316\pi$
$450$	0	0
$451$	74.9498	3.52925
$452$	0	0
$453$	0.864604i	0.0406227i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 12.2651i	− 0.573738i	−0.957970	−	0.286869i	$-0.907385\pi$
$457$	− 12.2651i	− 0.573738i	0.957970	−	0.286869i	$-0.0926145\pi$
$458$	0	0
$459$	2.20058	0.102714
$460$	0	0
$461$	1.81291	0.0844357	0.0422178	−	0.999108i	$-0.486558\pi$
$461$	1.81291	0.0844357	0.0422178	+	0.999108i	$0.486558\pi$
$462$	0	0
$463$	− 23.7349i	− 1.10305i	−0.834157	−	0.551527i	$-0.814046\pi$
$463$	− 23.7349i	− 1.10305i	0.834157	−	0.551527i	$-0.185954\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6.00000i	− 0.277647i	−0.990317	−	0.138823i	$-0.955668\pi$
$467$	− 6.00000i	− 0.277647i	0.990317	−	0.138823i	$-0.0443321\pi$
$468$	0	0
$469$	−0.185515	−0.00856630
$470$	0	0
$471$	−10.4872	−0.483226
$472$	0	0
$473$	− 14.4395i	− 0.663931i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	30.8057i	1.41050i
$478$	0	0
$479$	9.62456	0.439758	0.219879	−	0.975527i	$-0.429434\pi$
$479$	9.62456	0.439758	0.219879	+	0.975527i	$0.429434\pi$
$480$	0	0
$481$	−6.51034	−0.296846
$482$	0	0
$483$	− 3.97780i	− 0.180996i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	31.7165i	1.43721i	0.695417	+	0.718607i	$0.255220\pi$
$487$	31.7165i	1.43721i	−0.695417	+	0.718607i	$0.744780\pi$
$488$	0	0
$489$	−48.7412	−2.20415
$490$	0	0
$491$	−5.02068	−0.226580	−0.113290	−	0.993562i	$-0.536139\pi$
$491$	−5.02068	−0.226580	−0.113290	+	0.993562i	$0.536139\pi$
$492$	0	0
$493$	7.62897i	0.343591i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.49442i	0.0670341i
$498$	0	0
$499$	−34.8794	−1.56141	−0.780707	−	0.624897i	$-0.785141\pi$
$499$	−34.8794	−1.56141	−0.780707	+	0.624897i	$0.785141\pi$
$500$	0	0
$501$	−2.46741	−0.110236
$502$	0	0
$503$	20.2158i	0.901377i	0.892681	+	0.450688i	$0.148821\pi$
$503$	20.2158i	0.901377i	−0.892681	+	0.450688i	$0.851179\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 74.0346i	− 3.28799i
$508$	0	0
$509$	−4.18075	−0.185309	−0.0926543	−	0.995698i	$-0.529535\pi$
$509$	−4.18075	−0.185309	−0.0926543	+	0.995698i	$0.529535\pi$
$510$	0	0
$511$	−5.32001	−0.235343
$512$	0	0
$513$	5.85742i	0.258611i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 58.7906i	− 2.58561i
$518$	0	0
$519$	57.7882	2.53662
$520$	0	0
$521$	−6.26036	−0.274271	−0.137136	−	0.990552i	$-0.543790\pi$
$521$	−6.26036	−0.274271	−0.137136	+	0.990552i	$0.543790\pi$
$522$	0	0
$523$	30.3432i	1.32681i	0.748259	+	0.663407i	$0.230890\pi$
$523$	30.3432i	1.32681i	−0.748259	+	0.663407i	$0.769110\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 10.8918i	− 0.474452i
$528$	0	0
$529$	13.7644	0.598451
$530$	0	0
$531$	−31.3961	−1.36248
$532$	0	0
$533$	− 79.1814i	− 3.42973i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	42.2269i	1.82222i
$538$	0	0
$539$	41.4737	1.78640
$540$	0	0
$541$	−16.0502	−0.690051	−0.345025	−	0.938593i	$-0.612130\pi$
$541$	−16.0502	−0.690051	−0.345025	+	0.938593i	$0.612130\pi$
$542$	0	0
$543$	− 28.8399i	− 1.23764i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 21.0804i	− 0.901332i	−0.892693	−	0.450666i	$-0.851187\pi$
$547$	− 21.0804i	− 0.901332i	0.892693	−	0.450666i	$-0.148813\pi$
$548$	0	0
$549$	−6.69586	−0.285772
$550$	0	0
$551$	−20.3065	−0.865085
$552$	0	0
$553$	0.354265i	0.0150649i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 0.959408i	− 0.0406514i	−0.999793	−	0.0203257i	$-0.993530\pi$
$557$	− 0.959408i	− 0.0406514i	0.999793	−	0.0203257i	$-0.00647032\pi$
$558$	0	0
$559$	−15.2548	−0.645209
$560$	0	0
$561$	−38.9762	−1.64558
$562$	0	0
$563$	− 41.3957i	− 1.74462i	−0.488954	−	0.872310i	$-0.662621\pi$
$563$	− 41.3957i	− 1.74462i	0.488954	−	0.872310i	$-0.337379\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.07044i	0.170942i
$568$	0	0
$569$	26.4626	1.10937	0.554684	−	0.832061i	$-0.312839\pi$
$569$	26.4626	1.10937	0.554684	+	0.832061i	$0.312839\pi$
$570$	0	0
$571$	17.4482	0.730185	0.365093	−	0.930971i	$-0.381037\pi$
$571$	17.4482	0.730185	0.365093	+	0.930971i	$0.381037\pi$
$572$	0	0
$573$	− 26.2914i	− 1.09834i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 30.2954i	− 1.26121i	−0.776103	−	0.630607i	$-0.782806\pi$
$577$	− 30.2954i	− 1.26121i	0.776103	−	0.630607i	$-0.217194\pi$
$578$	0	0
$579$	−51.4052	−2.13633
$580$	0	0
$581$	3.20867	0.133118
$582$	0	0
$583$	− 56.7030i	− 2.34840i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 21.2858i	− 0.878559i	−0.898350	−	0.439280i	$-0.855234\pi$
$587$	− 21.2858i	− 0.878559i	0.898350	−	0.439280i	$-0.144766\pi$
$588$	0	0
$589$	28.9912	1.19456
$590$	0	0
$591$	−27.9793	−1.15091
$592$	0	0
$593$	− 17.5629i	− 0.721223i	−0.932716	−	0.360612i	$-0.882568\pi$
$593$	− 17.5629i	− 0.721223i	0.932716	−	0.360612i	$-0.117432\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.56770i	0.0641616i
$598$	0	0
$599$	12.8901	0.526675	0.263338	−	0.964704i	$-0.415177\pi$
$599$	12.8901	0.526675	0.263338	+	0.964704i	$0.415177\pi$
$600$	0	0
$601$	−13.8424	−0.564641	−0.282321	−	0.959320i	$-0.591104\pi$
$601$	−13.8424	−0.564641	−0.282321	+	0.959320i	$0.591104\pi$
$602$	0	0
$603$	1.19554i	0.0486860i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 24.7460i	− 1.00441i	−0.864749	−	0.502205i	$-0.832522\pi$
$607$	− 24.7460i	− 1.00441i	0.864749	−	0.502205i	$-0.167478\pi$
$608$	0	0
$609$	3.97780	0.161189
$610$	0	0
$611$	−62.1099	−2.51270
$612$	0	0
$613$	− 11.9753i	− 0.483679i	−0.970316	−	0.241839i	$-0.922249\pi$
$613$	− 11.9753i	− 0.483679i	0.970316	−	0.241839i	$-0.0777507\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	25.5167i	1.02726i	0.858010	+	0.513632i	$0.171700\pi$
$617$	25.5167i	1.02726i	−0.858010	+	0.513632i	$0.828300\pi$
$618$	0	0
$619$	24.0660	0.967296	0.483648	−	0.875263i	$-0.339312\pi$
$619$	24.0660	0.967296	0.483648	+	0.875263i	$0.339312\pi$
$620$	0	0
$621$	−2.66402	−0.106904
$622$	0	0
$623$	− 5.72535i	− 0.229381i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 103.745i	− 4.14318i
$628$	0	0
$629$	2.51034	0.100094
$630$	0	0
$631$	−24.5545	−0.977500	−0.488750	−	0.872424i	$-0.662547\pi$
$631$	−24.5545	−0.977500	−0.488750	+	0.872424i	$0.662547\pi$
$632$	0	0
$633$	− 46.1074i	− 1.83260i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 43.8153i	− 1.73603i
$638$	0	0
$639$	9.63067	0.380983
$640$	0	0
$641$	−5.20300	−0.205506	−0.102753	−	0.994707i	$-0.532765\pi$
$641$	−5.20300	−0.205506	−0.102753	+	0.994707i	$0.532765\pi$
$642$	0	0
$643$	− 0.0875650i	− 0.00345323i	−0.999999	−	0.00172661i	$-0.999450\pi$
$643$	− 0.0875650i	− 0.00345323i	0.999999	−	0.00172661i	$-0.000549599\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.81606i	0.228653i	0.993443	+	0.114326i	$0.0364710\pi$
$647$	5.81606i	0.228653i	−0.993443	+	0.114326i	$0.963529\pi$
$648$	0	0
$649$	57.7897	2.26845
$650$	0	0
$651$	−5.67904	−0.222579
$652$	0	0
$653$	12.5716i	0.491965i	0.969274	+	0.245983i	$0.0791106\pi$
$653$	12.5716i	0.491965i	−0.969274	+	0.245983i	$0.920889\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	34.2843i	1.33756i
$658$	0	0
$659$	−21.7730	−0.848157	−0.424079	−	0.905625i	$-0.639402\pi$
$659$	−21.7730	−0.848157	−0.424079	+	0.905625i	$0.639402\pi$
$660$	0	0
$661$	−9.29462	−0.361519	−0.180759	−	0.983527i	$-0.557856\pi$
$661$	−9.29462	−0.361519	−0.180759	+	0.983527i	$0.557856\pi$
$662$	0	0
$663$	41.1768i	1.59917i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 9.23563i	− 0.357605i
$668$	0	0
$669$	1.97690	0.0764312
$670$	0	0
$671$	12.3248	0.475795
$672$	0	0
$673$	34.2221i	1.31917i	0.751632	+	0.659583i	$0.229267\pi$
$673$	34.2221i	1.31917i	−0.751632	+	0.659583i	$0.770733\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 15.4761i	− 0.594794i	−0.954754	−	0.297397i	$-0.903881\pi$
$677$	− 15.4761i	− 0.594794i	0.954754	−	0.297397i	$-0.0961185\pi$
$678$	0	0
$679$	−3.29533	−0.126463
$680$	0	0
$681$	15.9355	0.610648
$682$	0	0
$683$	12.3750i	0.473516i	0.971569	+	0.236758i	$0.0760849\pi$
$683$	12.3750i	0.473516i	−0.971569	+	0.236758i	$0.923915\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 5.25160i	− 0.200361i
$688$	0	0
$689$	−59.9044	−2.28218
$690$	0	0
$691$	1.69819	0.0646024	0.0323012	−	0.999478i	$-0.489716\pi$
$691$	1.69819	0.0646024	0.0323012	+	0.999478i	$0.489716\pi$
$692$	0	0
$693$	10.7182i	0.407149i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	30.5318i	1.15647i
$698$	0	0
$699$	−66.6727	−2.52179
$700$	0	0
$701$	33.3869	1.26100	0.630502	−	0.776187i	$-0.282849\pi$
$701$	33.3869	1.26100	0.630502	+	0.776187i	$0.282849\pi$
$702$	0	0
$703$	6.68192i	0.252013i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 10.1027i	− 0.379952i
$708$	0	0
$709$	19.9403	0.748874	0.374437	−	0.927252i	$-0.377836\pi$
$709$	19.9403	0.748874	0.374437	+	0.927252i	$0.377836\pi$
$710$	0	0
$711$	2.28302	0.0856201
$712$	0	0
$713$	13.1856i	0.493803i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 47.0231i	− 1.75611i
$718$	0	0
$719$	−43.8694	−1.63605	−0.818027	−	0.575180i	$-0.804932\pi$
$719$	−43.8694	−1.63605	−0.818027	+	0.575180i	$0.804932\pi$
$720$	0	0
$721$	1.03902	0.0386951
$722$	0	0
$723$	− 0.150410i	− 0.00559380i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	41.4052i	1.53563i	0.640669	+	0.767817i	$0.278657\pi$
$727$	41.4052i	1.53563i	−0.640669	+	0.767817i	$0.721343\pi$
$728$	0	0
$729$	32.8574	1.21694
$730$	0	0
$731$	5.88214	0.217559
$732$	0	0
$733$	25.6919i	0.948950i	0.880269	+	0.474475i	$0.157362\pi$
$733$	25.6919i	0.948950i	−0.880269	+	0.474475i	$0.842638\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2.20058i	− 0.0810594i
$738$	0	0
$739$	46.4195	1.70757	0.853785	−	0.520625i	$-0.174301\pi$
$739$	46.4195	1.70757	0.853785	+	0.520625i	$0.174301\pi$
$740$	0	0
$741$	−109.603	−4.02635
$742$	0	0
$743$	9.86501i	0.361912i	0.983491	+	0.180956i	$0.0579192\pi$
$743$	9.86501i	0.361912i	−0.983491	+	0.180956i	$0.942081\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 20.6780i	− 0.756567i
$748$	0	0
$749$	7.80567	0.285213
$750$	0	0
$751$	37.5860	1.37153	0.685765	−	0.727823i	$-0.259468\pi$
$751$	37.5860	1.37153	0.685765	+	0.727823i	$0.259468\pi$
$752$	0	0
$753$	7.69194i	0.280310i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 47.1672i	− 1.71432i	−0.515049	−	0.857161i	$-0.672226\pi$
$757$	− 47.1672i	− 1.71432i	0.515049	−	0.857161i	$-0.327774\pi$
$758$	0	0
$759$	47.1846	1.71269
$760$	0	0
$761$	−30.0167	−1.08810	−0.544052	−	0.839052i	$-0.683111\pi$
$761$	−30.0167	−1.08810	−0.544052	+	0.839052i	$0.683111\pi$
$762$	0	0
$763$	8.15994i	0.295410i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 61.0525i	− 2.20448i
$768$	0	0
$769$	32.3662	1.16715	0.583577	−	0.812058i	$-0.301653\pi$
$769$	32.3662	1.16715	0.583577	+	0.812058i	$0.301653\pi$
$770$	0	0
$771$	−52.6910	−1.89762
$772$	0	0
$773$	− 40.6338i	− 1.46150i	−0.682648	−	0.730748i	$-0.739172\pi$
$773$	− 40.6338i	− 1.46150i	0.682648	−	0.730748i	$-0.260828\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 1.30891i	− 0.0469568i
$778$	0	0
$779$	−81.2683	−2.91174
$780$	0	0
$781$	−17.7268	−0.634316
$782$	0	0
$783$	− 2.66402i	− 0.0952044i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	13.7089i	0.488671i	0.969691	+	0.244335i	$0.0785698\pi$
$787$	13.7089i	0.488671i	−0.969691	+	0.244335i	$0.921430\pi$
$788$	0	0
$789$	19.9324	0.709612
$790$	0	0
$791$	−1.13540	−0.0403700
$792$	0	0
$793$	− 13.0207i	− 0.462378i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 18.3321i	− 0.649355i	−0.945825	−	0.324677i	$-0.894744\pi$
$797$	− 18.3321i	− 0.649355i	0.945825	−	0.324677i	$-0.105256\pi$
$798$	0	0
$799$	23.9491	0.847259
$800$	0	0
$801$	−36.8964	−1.30367
$802$	0	0
$803$	− 63.1059i	− 2.22696i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.52863i	0.0538103i
$808$	0	0
$809$	−18.3248	−0.644267	−0.322133	−	0.946694i	$-0.604400\pi$
$809$	−18.3248	−0.644267	−0.322133	+	0.946694i	$0.604400\pi$
$810$	0	0
$811$	−9.45775	−0.332106	−0.166053	−	0.986117i	$-0.553102\pi$
$811$	−9.45775	−0.332106	−0.166053	+	0.986117i	$0.553102\pi$
$812$	0	0
$813$	51.8846i	1.81967i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	15.6568i	0.547763i
$818$	0	0
$819$	11.3233	0.395668
$820$	0	0
$821$	−32.4689	−1.13317	−0.566586	−	0.824003i	$-0.691736\pi$
$821$	−32.4689	−1.13317	−0.566586	+	0.824003i	$0.691736\pi$
$822$	0	0
$823$	32.0370i	1.11674i	0.829593	+	0.558369i	$0.188573\pi$
$823$	32.0370i	1.11674i	−0.829593	+	0.558369i	$0.811427\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 2.20543i	− 0.0766903i	−0.999265	−	0.0383451i	$-0.987791\pi$
$827$	− 2.20543i	− 0.0766903i	0.999265	−	0.0383451i	$-0.0122086\pi$
$828$	0	0
$829$	−26.9745	−0.936862	−0.468431	−	0.883500i	$-0.655181\pi$
$829$	−26.9745	−0.936862	−0.468431	+	0.883500i	$0.655181\pi$
$830$	0	0
$831$	53.6258	1.86026
$832$	0	0
$833$	16.8949i	0.585372i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.80338i	0.131464i
$838$	0	0
$839$	20.8424	0.719560	0.359780	−	0.933037i	$-0.382852\pi$
$839$	20.8424	0.719560	0.359780	+	0.933037i	$0.382852\pi$
$840$	0	0
$841$	−19.7644	−0.681530
$842$	0	0
$843$	− 26.0597i	− 0.897544i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 14.0139i	− 0.481524i
$848$	0	0
$849$	−4.22062	−0.144851
$850$	0	0
$851$	−3.03902	−0.104176
$852$	0	0
$853$	− 48.7093i	− 1.66778i	−0.551933	−	0.833888i	$-0.686110\pi$
$853$	− 48.7093i	− 1.66778i	0.551933	−	0.833888i	$-0.313890\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.4212i	0.970850i	0.874278	+	0.485425i	$0.161335\pi$
$857$	28.4212i	0.970850i	−0.874278	+	0.485425i	$0.838665\pi$
$858$	0	0
$859$	−16.2926	−0.555895	−0.277947	−	0.960596i	$-0.589654\pi$
$859$	−16.2926	−0.555895	−0.277947	+	0.960596i	$0.589654\pi$
$860$	0	0
$861$	15.9195	0.542535
$862$	0	0
$863$	28.1333i	0.957670i	0.877905	+	0.478835i	$0.158941\pi$
$863$	28.1333i	0.957670i	−0.877905	+	0.478835i	$0.841059\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	26.9542i	0.915413i
$868$	0	0
$869$	−4.20228	−0.142553
$870$	0	0
$871$	−2.32482	−0.0787737
$872$	0	0
$873$	21.2365i	0.718745i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	25.0899i	0.847226i	0.905843	+	0.423613i	$0.139238\pi$
$877$	25.0899i	0.847226i	−0.905843	+	0.423613i	$0.860762\pi$
$878$	0	0
$879$	−25.1027	−0.846692
$880$	0	0
$881$	29.2197	0.984436	0.492218	−	0.870472i	$-0.336186\pi$
$881$	29.2197	0.984436	0.492218	+	0.870472i	$0.336186\pi$
$882$	0	0
$883$	26.9745i	0.907763i	0.891062	+	0.453882i	$0.149961\pi$
$883$	26.9745i	0.907763i	−0.891062	+	0.453882i	$0.850039\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	31.2839i	1.05041i	0.850976	+	0.525205i	$0.176011\pi$
$887$	31.2839i	1.05041i	−0.850976	+	0.525205i	$0.823989\pi$
$888$	0	0
$889$	−1.13063	−0.0379202
$890$	0	0
$891$	−48.2835	−1.61756
$892$	0	0
$893$	63.7468i	2.13321i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 49.8486i	− 1.66440i
$898$	0	0
$899$	−13.1856	−0.439763
$900$	0	0
$901$	23.0987	0.769530
$902$	0	0
$903$	− 3.06699i	− 0.102063i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 17.4697i	− 0.580073i	−0.957016	−	0.290037i	$-0.906332\pi$
$907$	− 17.4697i	− 0.580073i	0.957016	−	0.290037i	$-0.0936675\pi$
$908$	0	0
$909$	−65.1060	−2.15943
$910$	0	0
$911$	−37.6659	−1.24793	−0.623964	−	0.781453i	$-0.714479\pi$
$911$	−37.6659	−1.24793	−0.623964	+	0.781453i	$0.714479\pi$
$912$	0	0
$913$	38.0612i	1.25964i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.61939i	0.218592i
$918$	0	0
$919$	43.6119	1.43862	0.719312	−	0.694687i	$-0.244457\pi$
$919$	43.6119	1.43862	0.719312	+	0.694687i	$0.244457\pi$
$920$	0	0
$921$	66.0780	2.17734
$922$	0	0
$923$	18.7277i	0.616429i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 6.69586i	− 0.219921i
$928$	0	0
$929$	−22.8367	−0.749249	−0.374625	−	0.927177i	$-0.622228\pi$
$929$	−22.8367	−0.749249	−0.374625	+	0.927177i	$0.622228\pi$
$930$	0	0
$931$	−44.9701	−1.47383
$932$	0	0
$933$	4.40614i	0.144250i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	14.2309i	0.464904i	0.972608	+	0.232452i	$0.0746749\pi$
$937$	14.2309i	0.464904i	−0.972608	+	0.232452i	$0.925325\pi$
$938$	0	0
$939$	54.8654	1.79047
$940$	0	0
$941$	16.4140	0.535080	0.267540	−	0.963547i	$-0.413789\pi$
$941$	16.4140	0.535080	0.267540	+	0.963547i	$0.413789\pi$
$942$	0	0
$943$	− 36.9618i	− 1.20364i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	33.9506i	1.10325i	0.834093	+	0.551624i	$0.185992\pi$
$947$	33.9506i	1.10325i	−0.834093	+	0.551624i	$0.814008\pi$
$948$	0	0
$949$	−66.6688	−2.16416
$950$	0	0
$951$	11.1393	0.361215
$952$	0	0
$953$	− 41.5358i	− 1.34548i	−0.739881	−	0.672738i	$-0.765118\pi$
$953$	− 41.5358i	− 1.34548i	0.739881	−	0.672738i	$-0.234882\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	47.1846i	1.52526i
$958$	0	0
$959$	2.21577	0.0715510
$960$	0	0
$961$	−12.1752	−0.392748
$962$	0	0
$963$	− 50.3029i	− 1.62099i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	16.2341i	0.522054i	0.965332	+	0.261027i	$0.0840611\pi$
$967$	16.2341i	0.522054i	−0.965332	+	0.261027i	$0.915939\pi$
$968$	0	0
$969$	42.2620	1.35765
$970$	0	0
$971$	30.7277	0.986098	0.493049	−	0.870001i	$-0.335882\pi$
$971$	30.7277	0.986098	0.493049	+	0.870001i	$0.335882\pi$
$972$	0	0
$973$	− 6.04136i	− 0.193677i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20.8654i	0.667544i	0.942654	+	0.333772i	$0.108322\pi$
$977$	20.8654i	0.667544i	−0.942654	+	0.333772i	$0.891678\pi$
$978$	0	0
$979$	67.9140	2.17054
$980$	0	0
$981$	52.5859	1.67894
$982$	0	0
$983$	− 20.2360i	− 0.645430i	−0.946496	−	0.322715i	$-0.895405\pi$
$983$	− 20.2360i	− 0.645430i	0.946496	−	0.322715i	$-0.104595\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 12.4872i	− 0.397473i
$988$	0	0
$989$	−7.12092	−0.226432
$990$	0	0
$991$	−9.33642	−0.296581	−0.148291	−	0.988944i	$-0.547377\pi$
$991$	−9.33642	−0.296581	−0.148291	+	0.988944i	$0.547377\pi$
$992$	0	0
$993$	81.6744i	2.59186i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	13.9506i	0.441821i	0.975294	+	0.220911i	$0.0709029\pi$
$997$	13.9506i	0.441821i	−0.975294	+	0.220911i	$0.929097\pi$
$998$	0	0
$999$	−0.876607	−0.0277346

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3700.2.d.i.149.6		8
5.2	odd	4		740.2.a.f.1.3	✓	4
5.3	odd	4		3700.2.a.j.1.2		4
5.4	even	2	inner	3700.2.d.i.149.3		8
15.2	even	4		6660.2.a.r.1.3		4
20.7	even	4		2960.2.a.v.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
740.2.a.f.1.3	✓	4	5.2	odd	4
2960.2.a.v.1.2		4	20.7	even	4
3700.2.a.j.1.2		4	5.3	odd	4
3700.2.d.i.149.3		8	5.4	even	2	inner
3700.2.d.i.149.6		8	1.1	even	1	trivial
6660.2.a.r.1.3		4	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:	\( N \)	\(=\)	\( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3700.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.51951i q^{3} -0.519509i q^{7} -3.34793 q^{9} +O(q^{10})\)	\(q+2.51951i q^{3} -0.519509i q^{7} -3.34793 q^{9} +6.16241 q^{11} -6.51034i q^{13} +2.51034i q^{17} -6.68192 q^{19} +1.30891 q^{21} -3.03902i q^{23} -0.876607i q^{27} +3.03902 q^{29} -4.33876 q^{31} +15.5263i q^{33} -1.00000i q^{37} +16.4029 q^{39} +12.1624 q^{41} -2.34316i q^{43} -9.54019i q^{47} +6.73011 q^{49} -6.32482 q^{51} -9.20143i q^{53} -16.8352i q^{57} +9.37778 q^{59} +2.00000 q^{61} +1.73928i q^{63} -0.357097i q^{67} +7.65684 q^{69} -2.87661 q^{71} -10.2404i q^{73} -3.20143i q^{77} -0.681922 q^{79} -7.83516 q^{81} +6.17635i q^{83} +7.65684i q^{87} +11.0207 q^{89} -3.38218 q^{91} -10.9315i q^{93} -6.34316i q^{97} -20.6313 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 26 q^{9}+O(q^{10}) \) 8 * q - 26 * q^9	\( 8 q - 26 q^{9} + 10 q^{11} + 38 q^{21} - 4 q^{29} - 8 q^{31} - 4 q^{39} + 58 q^{41} - 2 q^{49} + 28 q^{51} + 20 q^{59} + 16 q^{61} + 88 q^{69} - 34 q^{71} + 48 q^{79} + 56 q^{81} + 8 q^{89} + 28 q^{91} + 44 q^{99}+O(q^{100}) \) 8 * q - 26 * q^9 + 10 * q^11 + 38 * q^21 - 4 * q^29 - 8 * q^31 - 4 * q^39 + 58 * q^41 - 2 * q^49 + 28 * q^51 + 20 * q^59 + 16 * q^61 + 88 * q^69 - 34 * q^71 + 48 * q^79 + 56 * q^81 + 8 * q^89 + 28 * q^91 + 44 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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