LMFDB - Embedded newform 1170.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1170,2,Mod(1,1170)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1170, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1170.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1170.a (trivial)

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:	yes
Analytic conductor:	$9.34249703649$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	1170.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.60555 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.60555 q^{7} +1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{13} -2.60555 q^{14} +1.00000 q^{16} +4.60555 q^{17} +6.60555 q^{19} +1.00000 q^{20} +4.60555 q^{23} +1.00000 q^{25} +1.00000 q^{26} -2.60555 q^{28} -4.60555 q^{29} +2.00000 q^{31} +1.00000 q^{32} +4.60555 q^{34} -2.60555 q^{35} +11.2111 q^{37} +6.60555 q^{38} +1.00000 q^{40} +3.21110 q^{41} +5.21110 q^{43} +4.60555 q^{46} -9.21110 q^{47} -0.211103 q^{49} +1.00000 q^{50} +1.00000 q^{52} -2.60555 q^{56} -4.60555 q^{58} -9.21110 q^{59} -7.21110 q^{61} +2.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} -7.21110 q^{67} +4.60555 q^{68} -2.60555 q^{70} -12.0000 q^{71} +6.60555 q^{73} +11.2111 q^{74} +6.60555 q^{76} -1.21110 q^{79} +1.00000 q^{80} +3.21110 q^{82} +4.60555 q^{85} +5.21110 q^{86} +3.21110 q^{89} -2.60555 q^{91} +4.60555 q^{92} -9.21110 q^{94} +6.60555 q^{95} +6.60555 q^{97} -0.211103 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} + 2 q^{8} + 2 q^{10} + 2 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} + 6 q^{19} + 2 q^{20} + 2 q^{23} + 2 q^{25} + 2 q^{26} + 2 q^{28} - 2 q^{29} + 4 q^{31} + 2 q^{32} + 2 q^{34} + 2 q^{35} + 8 q^{37} + 6 q^{38} + 2 q^{40} - 8 q^{41} - 4 q^{43} + 2 q^{46} - 4 q^{47} + 14 q^{49} + 2 q^{50} + 2 q^{52} + 2 q^{56} - 2 q^{58} - 4 q^{59} + 4 q^{62} + 2 q^{64} + 2 q^{65} + 2 q^{68} + 2 q^{70} - 24 q^{71} + 6 q^{73} + 8 q^{74} + 6 q^{76} + 12 q^{79} + 2 q^{80} - 8 q^{82} + 2 q^{85} - 4 q^{86} - 8 q^{89} + 2 q^{91} + 2 q^{92} - 4 q^{94} + 6 q^{95} + 6 q^{97} + 14 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^7 + 2 * q^8 + 2 * q^10 + 2 * q^13 + 2 * q^14 + 2 * q^16 + 2 * q^17 + 6 * q^19 + 2 * q^20 + 2 * q^23 + 2 * q^25 + 2 * q^26 + 2 * q^28 - 2 * q^29 + 4 * q^31 + 2 * q^32 + 2 * q^34 + 2 * q^35 + 8 * q^37 + 6 * q^38 + 2 * q^40 - 8 * q^41 - 4 * q^43 + 2 * q^46 - 4 * q^47 + 14 * q^49 + 2 * q^50 + 2 * q^52 + 2 * q^56 - 2 * q^58 - 4 * q^59 + 4 * q^62 + 2 * q^64 + 2 * q^65 + 2 * q^68 + 2 * q^70 - 24 * q^71 + 6 * q^73 + 8 * q^74 + 6 * q^76 + 12 * q^79 + 2 * q^80 - 8 * q^82 + 2 * q^85 - 4 * q^86 - 8 * q^89 + 2 * q^91 + 2 * q^92 - 4 * q^94 + 6 * q^95 + 6 * q^97 + 14 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.60555	−0.984806	−0.492403	−	0.870367i	$-0.663881\pi$
$7$	−2.60555	−0.984806	−0.492403	+	0.870367i	$0.663881\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−2.60555	−0.696363
$15$	0	0
$16$	1.00000	0.250000
$17$	4.60555	1.11701	0.558505	−	0.829501i	$-0.311375\pi$
$17$	4.60555	1.11701	0.558505	+	0.829501i	$0.311375\pi$
$18$	0	0
$19$	6.60555	1.51542	0.757709	−	0.652593i	$-0.226319\pi$
$19$	6.60555	1.51542	0.757709	+	0.652593i	$0.226319\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	0	0
$23$	4.60555	0.960324	0.480162	−	0.877180i	$-0.340578\pi$
$23$	4.60555	0.960324	0.480162	+	0.877180i	$0.340578\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	0	0
$28$	−2.60555	−0.492403
$29$	−4.60555	−0.855229	−0.427615	−	0.903961i	$-0.640646\pi$
$29$	−4.60555	−0.855229	−0.427615	+	0.903961i	$0.640646\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.60555	0.789846
$35$	−2.60555	−0.440419
$36$	0	0
$37$	11.2111	1.84309	0.921547	−	0.388267i	$-0.126926\pi$
$37$	11.2111	1.84309	0.921547	+	0.388267i	$0.126926\pi$
$38$	6.60555	1.07156
$39$	0	0
$40$	1.00000	0.158114
$41$	3.21110	0.501490	0.250745	−	0.968053i	$-0.419324\pi$
$41$	3.21110	0.501490	0.250745	+	0.968053i	$0.419324\pi$
$42$	0	0
$43$	5.21110	0.794686	0.397343	−	0.917670i	$-0.369932\pi$
$43$	5.21110	0.794686	0.397343	+	0.917670i	$0.369932\pi$
$44$	0	0
$45$	0	0
$46$	4.60555	0.679051
$47$	−9.21110	−1.34358	−0.671789	−	0.740743i	$-0.734474\pi$
$47$	−9.21110	−1.34358	−0.671789	+	0.740743i	$0.734474\pi$
$48$	0	0
$49$	−0.211103	−0.0301575
$50$	1.00000	0.141421
$51$	0	0
$52$	1.00000	0.138675
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	−2.60555	−0.348181
$57$	0	0
$58$	−4.60555	−0.604739
$59$	−9.21110	−1.19918	−0.599592	−	0.800306i	$-0.704670\pi$
$59$	−9.21110	−1.19918	−0.599592	+	0.800306i	$0.704670\pi$
$60$	0	0
$61$	−7.21110	−0.923287	−0.461644	−	0.887066i	$-0.652740\pi$
$61$	−7.21110	−0.923287	−0.461644	+	0.887066i	$0.652740\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	0	0
$67$	−7.21110	−0.880976	−0.440488	−	0.897758i	$-0.645195\pi$
$67$	−7.21110	−0.880976	−0.440488	+	0.897758i	$0.645195\pi$
$68$	4.60555	0.558505
$69$	0	0
$70$	−2.60555	−0.311423
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	6.60555	0.773121	0.386561	−	0.922264i	$-0.373663\pi$
$73$	6.60555	0.773121	0.386561	+	0.922264i	$0.373663\pi$
$74$	11.2111	1.30326
$75$	0	0
$76$	6.60555	0.757709
$77$	0	0
$78$	0	0
$79$	−1.21110	−0.136260	−0.0681298	−	0.997676i	$-0.521703\pi$
$79$	−1.21110	−0.136260	−0.0681298	+	0.997676i	$0.521703\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	3.21110	0.354607
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	4.60555	0.499542
$86$	5.21110	0.561928
$87$	0	0
$88$	0	0
$89$	3.21110	0.340376	0.170188	−	0.985412i	$-0.445562\pi$
$89$	3.21110	0.340376	0.170188	+	0.985412i	$0.445562\pi$
$90$	0	0
$91$	−2.60555	−0.273136
$92$	4.60555	0.480162
$93$	0	0
$94$	−9.21110	−0.950053
$95$	6.60555	0.677715
$96$	0	0
$97$	6.60555	0.670692	0.335346	−	0.942095i	$-0.391147\pi$
$97$	6.60555	0.670692	0.335346	+	0.942095i	$0.391147\pi$
$98$	−0.211103	−0.0213246
$99$	0	0
$100$	1.00000	0.100000
$101$	−13.8167	−1.37481	−0.687404	−	0.726275i	$-0.741250\pi$
$101$	−13.8167	−1.37481	−0.687404	+	0.726275i	$0.741250\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	18.6056	1.78209	0.891044	−	0.453916i	$-0.149974\pi$
$109$	18.6056	1.78209	0.891044	+	0.453916i	$0.149974\pi$
$110$	0	0
$111$	0	0
$112$	−2.60555	−0.246201
$113$	7.39445	0.695611	0.347806	−	0.937567i	$-0.386927\pi$
$113$	7.39445	0.695611	0.347806	+	0.937567i	$0.386927\pi$
$114$	0	0
$115$	4.60555	0.429470
$116$	−4.60555	−0.427615
$117$	0	0
$118$	−9.21110	−0.847951
$119$	−12.0000	−1.10004
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−7.21110	−0.652863
$123$	0	0
$124$	2.00000	0.179605
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.21110	−0.107468	−0.0537340	−	0.998555i	$-0.517112\pi$
$127$	−1.21110	−0.107468	−0.0537340	+	0.998555i	$0.517112\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	1.00000	0.0877058
$131$	−16.6056	−1.45083	−0.725417	−	0.688310i	$-0.758353\pi$
$131$	−16.6056	−1.45083	−0.725417	+	0.688310i	$0.758353\pi$
$132$	0	0
$133$	−17.2111	−1.49239
$134$	−7.21110	−0.622944
$135$	0	0
$136$	4.60555	0.394923
$137$	21.6333	1.84826	0.924129	−	0.382080i	$-0.124792\pi$
$137$	21.6333	1.84826	0.924129	+	0.382080i	$0.124792\pi$
$138$	0	0
$139$	14.4222	1.22328	0.611638	−	0.791138i	$-0.290511\pi$
$139$	14.4222	1.22328	0.611638	+	0.791138i	$0.290511\pi$
$140$	−2.60555	−0.220209
$141$	0	0
$142$	−12.0000	−1.00702
$143$	0	0
$144$	0	0
$145$	−4.60555	−0.382470
$146$	6.60555	0.546679
$147$	0	0
$148$	11.2111	0.921547
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	8.42221	0.685389	0.342695	−	0.939447i	$-0.388660\pi$
$151$	8.42221	0.685389	0.342695	+	0.939447i	$0.388660\pi$
$152$	6.60555	0.535781
$153$	0	0
$154$	0	0
$155$	2.00000	0.160644
$156$	0	0
$157$	11.2111	0.894743	0.447372	−	0.894348i	$-0.352360\pi$
$157$	11.2111	0.894743	0.447372	+	0.894348i	$0.352360\pi$
$158$	−1.21110	−0.0963501
$159$	0	0
$160$	1.00000	0.0790569
$161$	−12.0000	−0.945732
$162$	0	0
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	3.21110	0.250745
$165$	0	0
$166$	0	0
$167$	−18.4222	−1.42555	−0.712777	−	0.701391i	$-0.752563\pi$
$167$	−18.4222	−1.42555	−0.712777	+	0.701391i	$0.752563\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	4.60555	0.353230
$171$	0	0
$172$	5.21110	0.397343
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	0	0
$175$	−2.60555	−0.196961
$176$	0	0
$177$	0	0
$178$	3.21110	0.240682
$179$	11.0278	0.824253	0.412127	−	0.911127i	$-0.364786\pi$
$179$	11.0278	0.824253	0.412127	+	0.911127i	$0.364786\pi$
$180$	0	0
$181$	−19.2111	−1.42795	−0.713975	−	0.700171i	$-0.753107\pi$
$181$	−19.2111	−1.42795	−0.713975	+	0.700171i	$0.753107\pi$
$182$	−2.60555	−0.193136
$183$	0	0
$184$	4.60555	0.339526
$185$	11.2111	0.824257
$186$	0	0
$187$	0	0
$188$	−9.21110	−0.671789
$189$	0	0
$190$	6.60555	0.479217
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	−14.6056	−1.05133	−0.525665	−	0.850691i	$-0.676184\pi$
$193$	−14.6056	−1.05133	−0.525665	+	0.850691i	$0.676184\pi$
$194$	6.60555	0.474251
$195$	0	0
$196$	−0.211103	−0.0150788
$197$	−24.4222	−1.74001	−0.870005	−	0.493043i	$-0.835885\pi$
$197$	−24.4222	−1.74001	−0.870005	+	0.493043i	$0.835885\pi$
$198$	0	0
$199$	5.21110	0.369405	0.184703	−	0.982794i	$-0.440868\pi$
$199$	5.21110	0.369405	0.184703	+	0.982794i	$0.440868\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−13.8167	−0.972136
$203$	12.0000	0.842235
$204$	0	0
$205$	3.21110	0.224273
$206$	−4.00000	−0.278693
$207$	0	0
$208$	1.00000	0.0693375
$209$	0	0
$210$	0	0
$211$	−25.2111	−1.73560	−0.867802	−	0.496910i	$-0.834468\pi$
$211$	−25.2111	−1.73560	−0.867802	+	0.496910i	$0.834468\pi$
$212$	0	0
$213$	0	0
$214$	−12.0000	−0.820303
$215$	5.21110	0.355394
$216$	0	0
$217$	−5.21110	−0.353753
$218$	18.6056	1.26013
$219$	0	0
$220$	0	0
$221$	4.60555	0.309803
$222$	0	0
$223$	−17.3944	−1.16482	−0.582409	−	0.812896i	$-0.697890\pi$
$223$	−17.3944	−1.16482	−0.582409	+	0.812896i	$0.697890\pi$
$224$	−2.60555	−0.174091
$225$	0	0
$226$	7.39445	0.491871
$227$	6.42221	0.426257	0.213128	−	0.977024i	$-0.431635\pi$
$227$	6.42221	0.426257	0.213128	+	0.977024i	$0.431635\pi$
$228$	0	0
$229$	0.183346	0.0121159	0.00605793	−	0.999982i	$-0.498072\pi$
$229$	0.183346	0.0121159	0.00605793	+	0.999982i	$0.498072\pi$
$230$	4.60555	0.303681
$231$	0	0
$232$	−4.60555	−0.302369
$233$	−1.81665	−0.119013	−0.0595065	−	0.998228i	$-0.518953\pi$
$233$	−1.81665	−0.119013	−0.0595065	+	0.998228i	$0.518953\pi$
$234$	0	0
$235$	−9.21110	−0.600866
$236$	−9.21110	−0.599592
$237$	0	0
$238$	−12.0000	−0.777844
$239$	18.4222	1.19163	0.595817	−	0.803120i	$-0.296828\pi$
$239$	18.4222	1.19163	0.595817	+	0.803120i	$0.296828\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−11.0000	−0.707107
$243$	0	0
$244$	−7.21110	−0.461644
$245$	−0.211103	−0.0134868
$246$	0	0
$247$	6.60555	0.420301
$248$	2.00000	0.127000
$249$	0	0
$250$	1.00000	0.0632456
$251$	4.60555	0.290700	0.145350	−	0.989380i	$-0.453569\pi$
$251$	4.60555	0.290700	0.145350	+	0.989380i	$0.453569\pi$
$252$	0	0
$253$	0	0
$254$	−1.21110	−0.0759913
$255$	0	0
$256$	1.00000	0.0625000
$257$	−11.0278	−0.687893	−0.343946	−	0.938989i	$-0.611764\pi$
$257$	−11.0278	−0.687893	−0.343946	+	0.938989i	$0.611764\pi$
$258$	0	0
$259$	−29.2111	−1.81509
$260$	1.00000	0.0620174
$261$	0	0
$262$	−16.6056	−1.02589
$263$	−11.0278	−0.680001	−0.340000	−	0.940425i	$-0.610427\pi$
$263$	−11.0278	−0.680001	−0.340000	+	0.940425i	$0.610427\pi$
$264$	0	0
$265$	0	0
$266$	−17.2111	−1.05528
$267$	0	0
$268$	−7.21110	−0.440488
$269$	10.1833	0.620890	0.310445	−	0.950591i	$-0.399522\pi$
$269$	10.1833	0.620890	0.310445	+	0.950591i	$0.399522\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	4.60555	0.279253
$273$	0	0
$274$	21.6333	1.30692
$275$	0	0
$276$	0	0
$277$	−31.2111	−1.87529	−0.937647	−	0.347590i	$-0.887000\pi$
$277$	−31.2111	−1.87529	−0.937647	+	0.347590i	$0.887000\pi$
$278$	14.4222	0.864986
$279$	0	0
$280$	−2.60555	−0.155711
$281$	−3.21110	−0.191558	−0.0957792	−	0.995403i	$-0.530534\pi$
$281$	−3.21110	−0.191558	−0.0957792	+	0.995403i	$0.530534\pi$
$282$	0	0
$283$	−1.21110	−0.0719926	−0.0359963	−	0.999352i	$-0.511460\pi$
$283$	−1.21110	−0.0719926	−0.0359963	+	0.999352i	$0.511460\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	0	0
$287$	−8.36669	−0.493870
$288$	0	0
$289$	4.21110	0.247712
$290$	−4.60555	−0.270447
$291$	0	0
$292$	6.60555	0.386561
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	0	0
$295$	−9.21110	−0.536291
$296$	11.2111	0.651632
$297$	0	0
$298$	−6.00000	−0.347571
$299$	4.60555	0.266346
$300$	0	0
$301$	−13.5778	−0.782611
$302$	8.42221	0.484643
$303$	0	0
$304$	6.60555	0.378854
$305$	−7.21110	−0.412907
$306$	0	0
$307$	4.78890	0.273317	0.136658	−	0.990618i	$-0.456364\pi$
$307$	4.78890	0.273317	0.136658	+	0.990618i	$0.456364\pi$
$308$	0	0
$309$	0	0
$310$	2.00000	0.113592
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	11.2111	0.633689	0.316844	−	0.948478i	$-0.397377\pi$
$313$	11.2111	0.633689	0.316844	+	0.948478i	$0.397377\pi$
$314$	11.2111	0.632679
$315$	0	0
$316$	−1.21110	−0.0681298
$317$	−12.4222	−0.697701	−0.348850	−	0.937178i	$-0.613428\pi$
$317$	−12.4222	−0.697701	−0.348850	+	0.937178i	$0.613428\pi$
$318$	0	0
$319$	0	0
$320$	1.00000	0.0559017
$321$	0	0
$322$	−12.0000	−0.668734
$323$	30.4222	1.69274
$324$	0	0
$325$	1.00000	0.0554700
$326$	2.00000	0.110770
$327$	0	0
$328$	3.21110	0.177303
$329$	24.0000	1.32316
$330$	0	0
$331$	−5.39445	−0.296506	−0.148253	−	0.988949i	$-0.547365\pi$
$331$	−5.39445	−0.296506	−0.148253	+	0.988949i	$0.547365\pi$
$332$	0	0
$333$	0	0
$334$	−18.4222	−1.00802
$335$	−7.21110	−0.393985
$336$	0	0
$337$	26.8444	1.46231	0.731154	−	0.682212i	$-0.238982\pi$
$337$	26.8444	1.46231	0.731154	+	0.682212i	$0.238982\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	4.60555	0.249771
$341$	0	0
$342$	0	0
$343$	18.7889	1.01451
$344$	5.21110	0.280964
$345$	0	0
$346$	12.0000	0.645124
$347$	14.7889	0.793910	0.396955	−	0.917838i	$-0.370067\pi$
$347$	14.7889	0.793910	0.396955	+	0.917838i	$0.370067\pi$
$348$	0	0
$349$	−11.8167	−0.632531	−0.316265	−	0.948671i	$-0.602429\pi$
$349$	−11.8167	−0.632531	−0.316265	+	0.948671i	$0.602429\pi$
$350$	−2.60555	−0.139273
$351$	0	0
$352$	0	0
$353$	21.6333	1.15142	0.575712	−	0.817652i	$-0.304725\pi$
$353$	21.6333	1.15142	0.575712	+	0.817652i	$0.304725\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	3.21110	0.170188
$357$	0	0
$358$	11.0278	0.582835
$359$	−30.4222	−1.60562	−0.802811	−	0.596233i	$-0.796663\pi$
$359$	−30.4222	−1.60562	−0.802811	+	0.596233i	$0.796663\pi$
$360$	0	0
$361$	24.6333	1.29649
$362$	−19.2111	−1.00971
$363$	0	0
$364$	−2.60555	−0.136568
$365$	6.60555	0.345750
$366$	0	0
$367$	−6.78890	−0.354378	−0.177189	−	0.984177i	$-0.556700\pi$
$367$	−6.78890	−0.354378	−0.177189	+	0.984177i	$0.556700\pi$
$368$	4.60555	0.240081
$369$	0	0
$370$	11.2111	0.582837
$371$	0	0
$372$	0	0
$373$	−0.788897	−0.0408476	−0.0204238	−	0.999791i	$-0.506502\pi$
$373$	−0.788897	−0.0408476	−0.0204238	+	0.999791i	$0.506502\pi$
$374$	0	0
$375$	0	0
$376$	−9.21110	−0.475026
$377$	−4.60555	−0.237198
$378$	0	0
$379$	−26.6056	−1.36664	−0.683318	−	0.730121i	$-0.739464\pi$
$379$	−26.6056	−1.36664	−0.683318	+	0.730121i	$0.739464\pi$
$380$	6.60555	0.338858
$381$	0	0
$382$	−12.0000	−0.613973
$383$	18.4222	0.941331	0.470665	−	0.882312i	$-0.344014\pi$
$383$	18.4222	0.941331	0.470665	+	0.882312i	$0.344014\pi$
$384$	0	0
$385$	0	0
$386$	−14.6056	−0.743403
$387$	0	0
$388$	6.60555	0.335346
$389$	−1.81665	−0.0921080	−0.0460540	−	0.998939i	$-0.514665\pi$
$389$	−1.81665	−0.0921080	−0.0460540	+	0.998939i	$0.514665\pi$
$390$	0	0
$391$	21.2111	1.07269
$392$	−0.211103	−0.0106623
$393$	0	0
$394$	−24.4222	−1.23037
$395$	−1.21110	−0.0609372
$396$	0	0
$397$	35.2111	1.76719	0.883597	−	0.468248i	$-0.155114\pi$
$397$	35.2111	1.76719	0.883597	+	0.468248i	$0.155114\pi$
$398$	5.21110	0.261209
$399$	0	0
$400$	1.00000	0.0500000
$401$	33.6333	1.67957	0.839784	−	0.542921i	$-0.182682\pi$
$401$	33.6333	1.67957	0.839784	+	0.542921i	$0.182682\pi$
$402$	0	0
$403$	2.00000	0.0996271
$404$	−13.8167	−0.687404
$405$	0	0
$406$	12.0000	0.595550
$407$	0	0
$408$	0	0
$409$	−13.6333	−0.674124	−0.337062	−	0.941483i	$-0.609433\pi$
$409$	−13.6333	−0.674124	−0.337062	+	0.941483i	$0.609433\pi$
$410$	3.21110	0.158585
$411$	0	0
$412$	−4.00000	−0.197066
$413$	24.0000	1.18096
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	0	0
$418$	0	0
$419$	−29.4500	−1.43872	−0.719362	−	0.694635i	$-0.755566\pi$
$419$	−29.4500	−1.43872	−0.719362	+	0.694635i	$0.755566\pi$
$420$	0	0
$421$	−9.02776	−0.439986	−0.219993	−	0.975501i	$-0.570603\pi$
$421$	−9.02776	−0.439986	−0.219993	+	0.975501i	$0.570603\pi$
$422$	−25.2111	−1.22726
$423$	0	0
$424$	0	0
$425$	4.60555	0.223402
$426$	0	0
$427$	18.7889	0.909258
$428$	−12.0000	−0.580042
$429$	0	0
$430$	5.21110	0.251302
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	−22.8444	−1.09783	−0.548916	−	0.835877i	$-0.684959\pi$
$433$	−22.8444	−1.09783	−0.548916	+	0.835877i	$0.684959\pi$
$434$	−5.21110	−0.250141
$435$	0	0
$436$	18.6056	0.891044
$437$	30.4222	1.45529
$438$	0	0
$439$	32.8444	1.56758	0.783789	−	0.621027i	$-0.213284\pi$
$439$	32.8444	1.56758	0.783789	+	0.621027i	$0.213284\pi$
$440$	0	0
$441$	0	0
$442$	4.60555	0.219064
$443$	−39.6333	−1.88304	−0.941518	−	0.336964i	$-0.890600\pi$
$443$	−39.6333	−1.88304	−0.941518	+	0.336964i	$0.890600\pi$
$444$	0	0
$445$	3.21110	0.152221
$446$	−17.3944	−0.823651
$447$	0	0
$448$	−2.60555	−0.123101
$449$	−9.63331	−0.454624	−0.227312	−	0.973822i	$-0.572994\pi$
$449$	−9.63331	−0.454624	−0.227312	+	0.973822i	$0.572994\pi$
$450$	0	0
$451$	0	0
$452$	7.39445	0.347806
$453$	0	0
$454$	6.42221	0.301409
$455$	−2.60555	−0.122150
$456$	0	0
$457$	−17.3944	−0.813678	−0.406839	−	0.913500i	$-0.633369\pi$
$457$	−17.3944	−0.813678	−0.406839	+	0.913500i	$0.633369\pi$
$458$	0.183346	0.00856720
$459$	0	0
$460$	4.60555	0.214735
$461$	−3.21110	−0.149556	−0.0747780	−	0.997200i	$-0.523825\pi$
$461$	−3.21110	−0.149556	−0.0747780	+	0.997200i	$0.523825\pi$
$462$	0	0
$463$	12.1833	0.566208	0.283104	−	0.959089i	$-0.408636\pi$
$463$	12.1833	0.566208	0.283104	+	0.959089i	$0.408636\pi$
$464$	−4.60555	−0.213807
$465$	0	0
$466$	−1.81665	−0.0841549
$467$	−5.57779	−0.258110	−0.129055	−	0.991637i	$-0.541194\pi$
$467$	−5.57779	−0.258110	−0.129055	+	0.991637i	$0.541194\pi$
$468$	0	0
$469$	18.7889	0.867591
$470$	−9.21110	−0.424876
$471$	0	0
$472$	−9.21110	−0.423975
$473$	0	0
$474$	0	0
$475$	6.60555	0.303083
$476$	−12.0000	−0.550019
$477$	0	0
$478$	18.4222	0.842612
$479$	18.4222	0.841732	0.420866	−	0.907123i	$-0.361726\pi$
$479$	18.4222	0.841732	0.420866	+	0.907123i	$0.361726\pi$
$480$	0	0
$481$	11.2111	0.511182
$482$	−10.0000	−0.455488
$483$	0	0
$484$	−11.0000	−0.500000
$485$	6.60555	0.299943
$486$	0	0
$487$	2.97224	0.134685	0.0673426	−	0.997730i	$-0.478548\pi$
$487$	2.97224	0.134685	0.0673426	+	0.997730i	$0.478548\pi$
$488$	−7.21110	−0.326431
$489$	0	0
$490$	−0.211103	−0.00953664
$491$	32.2389	1.45492	0.727460	−	0.686150i	$-0.240701\pi$
$491$	32.2389	1.45492	0.727460	+	0.686150i	$0.240701\pi$
$492$	0	0
$493$	−21.2111	−0.955300
$494$	6.60555	0.297198
$495$	0	0
$496$	2.00000	0.0898027
$497$	31.2666	1.40250
$498$	0	0
$499$	27.8167	1.24524	0.622622	−	0.782523i	$-0.286067\pi$
$499$	27.8167	1.24524	0.622622	+	0.782523i	$0.286067\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	4.60555	0.205556
$503$	20.2389	0.902406	0.451203	−	0.892421i	$-0.350995\pi$
$503$	20.2389	0.902406	0.451203	+	0.892421i	$0.350995\pi$
$504$	0	0
$505$	−13.8167	−0.614833
$506$	0	0
$507$	0	0
$508$	−1.21110	−0.0537340
$509$	9.63331	0.426989	0.213494	−	0.976944i	$-0.431515\pi$
$509$	9.63331	0.426989	0.213494	+	0.976944i	$0.431515\pi$
$510$	0	0
$511$	−17.2111	−0.761374
$512$	1.00000	0.0441942
$513$	0	0
$514$	−11.0278	−0.486413
$515$	−4.00000	−0.176261
$516$	0	0
$517$	0	0
$518$	−29.2111	−1.28346
$519$	0	0
$520$	1.00000	0.0438529
$521$	−21.2111	−0.929275	−0.464638	−	0.885501i	$-0.653815\pi$
$521$	−21.2111	−0.929275	−0.464638	+	0.885501i	$0.653815\pi$
$522$	0	0
$523$	38.4222	1.68009	0.840043	−	0.542520i	$-0.182530\pi$
$523$	38.4222	1.68009	0.840043	+	0.542520i	$0.182530\pi$
$524$	−16.6056	−0.725417
$525$	0	0
$526$	−11.0278	−0.480833
$527$	9.21110	0.401242
$528$	0	0
$529$	−1.78890	−0.0777781
$530$	0	0
$531$	0	0
$532$	−17.2111	−0.746196
$533$	3.21110	0.139088
$534$	0	0
$535$	−12.0000	−0.518805
$536$	−7.21110	−0.311472
$537$	0	0
$538$	10.1833	0.439035
$539$	0	0
$540$	0	0
$541$	−9.02776	−0.388134	−0.194067	−	0.980988i	$-0.562168\pi$
$541$	−9.02776	−0.388134	−0.194067	+	0.980988i	$0.562168\pi$
$542$	2.00000	0.0859074
$543$	0	0
$544$	4.60555	0.197461
$545$	18.6056	0.796974
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	21.6333	0.924129
$549$	0	0
$550$	0	0
$551$	−30.4222	−1.29603
$552$	0	0
$553$	3.15559	0.134189
$554$	−31.2111	−1.32603
$555$	0	0
$556$	14.4222	0.611638
$557$	−18.8444	−0.798463	−0.399232	−	0.916850i	$-0.630723\pi$
$557$	−18.8444	−0.798463	−0.399232	+	0.916850i	$0.630723\pi$
$558$	0	0
$559$	5.21110	0.220406
$560$	−2.60555	−0.110105
$561$	0	0
$562$	−3.21110	−0.135452
$563$	−15.6333	−0.658865	−0.329433	−	0.944179i	$-0.606857\pi$
$563$	−15.6333	−0.658865	−0.329433	+	0.944179i	$0.606857\pi$
$564$	0	0
$565$	7.39445	0.311087
$566$	−1.21110	−0.0509064
$567$	0	0
$568$	−12.0000	−0.503509
$569$	−26.7889	−1.12305	−0.561525	−	0.827460i	$-0.689785\pi$
$569$	−26.7889	−1.12305	−0.561525	+	0.827460i	$0.689785\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	0	0
$574$	−8.36669	−0.349219
$575$	4.60555	0.192065
$576$	0	0
$577$	27.8167	1.15802	0.579011	−	0.815320i	$-0.303439\pi$
$577$	27.8167	1.15802	0.579011	+	0.815320i	$0.303439\pi$
$578$	4.21110	0.175159
$579$	0	0
$580$	−4.60555	−0.191235
$581$	0	0
$582$	0	0
$583$	0	0
$584$	6.60555	0.273340
$585$	0	0
$586$	30.0000	1.23929
$587$	−6.42221	−0.265073	−0.132536	−	0.991178i	$-0.542312\pi$
$587$	−6.42221	−0.265073	−0.132536	+	0.991178i	$0.542312\pi$
$588$	0	0
$589$	13.2111	0.544354
$590$	−9.21110	−0.379215
$591$	0	0
$592$	11.2111	0.460773
$593$	20.7889	0.853698	0.426849	−	0.904323i	$-0.359624\pi$
$593$	20.7889	0.853698	0.426849	+	0.904323i	$0.359624\pi$
$594$	0	0
$595$	−12.0000	−0.491952
$596$	−6.00000	−0.245770
$597$	0	0
$598$	4.60555	0.188335
$599$	−36.8444	−1.50542	−0.752711	−	0.658351i	$-0.771254\pi$
$599$	−36.8444	−1.50542	−0.752711	+	0.658351i	$0.771254\pi$
$600$	0	0
$601$	17.6333	0.719278	0.359639	−	0.933092i	$-0.382900\pi$
$601$	17.6333	0.719278	0.359639	+	0.933092i	$0.382900\pi$
$602$	−13.5778	−0.553390
$603$	0	0
$604$	8.42221	0.342695
$605$	−11.0000	−0.447214
$606$	0	0
$607$	41.2111	1.67271	0.836354	−	0.548190i	$-0.184683\pi$
$607$	41.2111	1.67271	0.836354	+	0.548190i	$0.184683\pi$
$608$	6.60555	0.267890
$609$	0	0
$610$	−7.21110	−0.291969
$611$	−9.21110	−0.372641
$612$	0	0
$613$	5.63331	0.227527	0.113764	−	0.993508i	$-0.463709\pi$
$613$	5.63331	0.227527	0.113764	+	0.993508i	$0.463709\pi$
$614$	4.78890	0.193264
$615$	0	0
$616$	0	0
$617$	−18.8444	−0.758647	−0.379324	−	0.925264i	$-0.623843\pi$
$617$	−18.8444	−0.758647	−0.379324	+	0.925264i	$0.623843\pi$
$618$	0	0
$619$	−2.60555	−0.104726	−0.0523630	−	0.998628i	$-0.516675\pi$
$619$	−2.60555	−0.104726	−0.0523630	+	0.998628i	$0.516675\pi$
$620$	2.00000	0.0803219
$621$	0	0
$622$	−12.0000	−0.481156
$623$	−8.36669	−0.335204
$624$	0	0
$625$	1.00000	0.0400000
$626$	11.2111	0.448086
$627$	0	0
$628$	11.2111	0.447372
$629$	51.6333	2.05875
$630$	0	0
$631$	23.2111	0.924019	0.462010	−	0.886875i	$-0.347129\pi$
$631$	23.2111	0.924019	0.462010	+	0.886875i	$0.347129\pi$
$632$	−1.21110	−0.0481751
$633$	0	0
$634$	−12.4222	−0.493349
$635$	−1.21110	−0.0480611
$636$	0	0
$637$	−0.211103	−0.00836419
$638$	0	0
$639$	0	0
$640$	1.00000	0.0395285
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	0	0
$643$	48.0555	1.89512	0.947562	−	0.319571i	$-0.103539\pi$
$643$	48.0555	1.89512	0.947562	+	0.319571i	$0.103539\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	30.4222	1.19695
$647$	−35.0278	−1.37708	−0.688542	−	0.725197i	$-0.741749\pi$
$647$	−35.0278	−1.37708	−0.688542	+	0.725197i	$0.741749\pi$
$648$	0	0
$649$	0	0
$650$	1.00000	0.0392232
$651$	0	0
$652$	2.00000	0.0783260
$653$	−2.78890	−0.109138	−0.0545690	−	0.998510i	$-0.517378\pi$
$653$	−2.78890	−0.109138	−0.0545690	+	0.998510i	$0.517378\pi$
$654$	0	0
$655$	−16.6056	−0.648833
$656$	3.21110	0.125372
$657$	0	0
$658$	24.0000	0.935617
$659$	13.8167	0.538220	0.269110	−	0.963109i	$-0.413270\pi$
$659$	13.8167	0.538220	0.269110	+	0.963109i	$0.413270\pi$
$660$	0	0
$661$	−21.0278	−0.817885	−0.408942	−	0.912560i	$-0.634102\pi$
$661$	−21.0278	−0.817885	−0.408942	+	0.912560i	$0.634102\pi$
$662$	−5.39445	−0.209661
$663$	0	0
$664$	0	0
$665$	−17.2111	−0.667418
$666$	0	0
$667$	−21.2111	−0.821297
$668$	−18.4222	−0.712777
$669$	0	0
$670$	−7.21110	−0.278589
$671$	0	0
$672$	0	0
$673$	−19.2111	−0.740534	−0.370267	−	0.928925i	$-0.620734\pi$
$673$	−19.2111	−0.740534	−0.370267	+	0.928925i	$0.620734\pi$
$674$	26.8444	1.03401
$675$	0	0
$676$	1.00000	0.0384615
$677$	9.21110	0.354011	0.177006	−	0.984210i	$-0.443359\pi$
$677$	9.21110	0.354011	0.177006	+	0.984210i	$0.443359\pi$
$678$	0	0
$679$	−17.2111	−0.660501
$680$	4.60555	0.176615
$681$	0	0
$682$	0	0
$683$	5.57779	0.213428	0.106714	−	0.994290i	$-0.465967\pi$
$683$	5.57779	0.213428	0.106714	+	0.994290i	$0.465967\pi$
$684$	0	0
$685$	21.6333	0.826566
$686$	18.7889	0.717363
$687$	0	0
$688$	5.21110	0.198671
$689$	0	0
$690$	0	0
$691$	−23.8167	−0.906028	−0.453014	−	0.891503i	$-0.649651\pi$
$691$	−23.8167	−0.906028	−0.453014	+	0.891503i	$0.649651\pi$
$692$	12.0000	0.456172
$693$	0	0
$694$	14.7889	0.561379
$695$	14.4222	0.547065
$696$	0	0
$697$	14.7889	0.560169
$698$	−11.8167	−0.447267
$699$	0	0
$700$	−2.60555	−0.0984806
$701$	16.6056	0.627183	0.313592	−	0.949558i	$-0.398468\pi$
$701$	16.6056	0.627183	0.313592	+	0.949558i	$0.398468\pi$
$702$	0	0
$703$	74.0555	2.79306
$704$	0	0
$705$	0	0
$706$	21.6333	0.814180
$707$	36.0000	1.35392
$708$	0	0
$709$	31.4500	1.18113	0.590564	−	0.806991i	$-0.298905\pi$
$709$	31.4500	1.18113	0.590564	+	0.806991i	$0.298905\pi$
$710$	−12.0000	−0.450352
$711$	0	0
$712$	3.21110	0.120341
$713$	9.21110	0.344959
$714$	0	0
$715$	0	0
$716$	11.0278	0.412127
$717$	0	0
$718$	−30.4222	−1.13535
$719$	21.2111	0.791041	0.395520	−	0.918457i	$-0.370564\pi$
$719$	21.2111	0.791041	0.395520	+	0.918457i	$0.370564\pi$
$720$	0	0
$721$	10.4222	0.388143
$722$	24.6333	0.916757
$723$	0	0
$724$	−19.2111	−0.713975
$725$	−4.60555	−0.171046
$726$	0	0
$727$	−43.6333	−1.61827	−0.809135	−	0.587623i	$-0.800064\pi$
$727$	−43.6333	−1.61827	−0.809135	+	0.587623i	$0.800064\pi$
$728$	−2.60555	−0.0965682
$729$	0	0
$730$	6.60555	0.244482
$731$	24.0000	0.887672
$732$	0	0
$733$	41.6333	1.53776	0.768881	−	0.639392i	$-0.220814\pi$
$733$	41.6333	1.53776	0.768881	+	0.639392i	$0.220814\pi$
$734$	−6.78890	−0.250583
$735$	0	0
$736$	4.60555	0.169763
$737$	0	0
$738$	0	0
$739$	−26.6056	−0.978701	−0.489351	−	0.872087i	$-0.662766\pi$
$739$	−26.6056	−0.978701	−0.489351	+	0.872087i	$0.662766\pi$
$740$	11.2111	0.412128
$741$	0	0
$742$	0	0
$743$	−46.0555	−1.68961	−0.844806	−	0.535072i	$-0.820284\pi$
$743$	−46.0555	−1.68961	−0.844806	+	0.535072i	$0.820284\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	−0.788897	−0.0288836
$747$	0	0
$748$	0	0
$749$	31.2666	1.14246
$750$	0	0
$751$	−35.2666	−1.28690	−0.643449	−	0.765489i	$-0.722497\pi$
$751$	−35.2666	−1.28690	−0.643449	+	0.765489i	$0.722497\pi$
$752$	−9.21110	−0.335894
$753$	0	0
$754$	−4.60555	−0.167724
$755$	8.42221	0.306515
$756$	0	0
$757$	−34.8444	−1.26644	−0.633221	−	0.773971i	$-0.718268\pi$
$757$	−34.8444	−1.26644	−0.633221	+	0.773971i	$0.718268\pi$
$758$	−26.6056	−0.966357
$759$	0	0
$760$	6.60555	0.239609
$761$	−9.63331	−0.349207	−0.174604	−	0.984639i	$-0.555864\pi$
$761$	−9.63331	−0.349207	−0.174604	+	0.984639i	$0.555864\pi$
$762$	0	0
$763$	−48.4777	−1.75501
$764$	−12.0000	−0.434145
$765$	0	0
$766$	18.4222	0.665621
$767$	−9.21110	−0.332594
$768$	0	0
$769$	−34.8444	−1.25652	−0.628261	−	0.778003i	$-0.716233\pi$
$769$	−34.8444	−1.25652	−0.628261	+	0.778003i	$0.716233\pi$
$770$	0	0
$771$	0	0
$772$	−14.6056	−0.525665
$773$	−12.4222	−0.446796	−0.223398	−	0.974727i	$-0.571715\pi$
$773$	−12.4222	−0.446796	−0.223398	+	0.974727i	$0.571715\pi$
$774$	0	0
$775$	2.00000	0.0718421
$776$	6.60555	0.237125
$777$	0	0
$778$	−1.81665	−0.0651302
$779$	21.2111	0.759967
$780$	0	0
$781$	0	0
$782$	21.2111	0.758507
$783$	0	0
$784$	−0.211103	−0.00753938
$785$	11.2111	0.400141
$786$	0	0
$787$	38.0000	1.35455	0.677277	−	0.735728i	$-0.263160\pi$
$787$	38.0000	1.35455	0.677277	+	0.735728i	$0.263160\pi$
$788$	−24.4222	−0.870005
$789$	0	0
$790$	−1.21110	−0.0430891
$791$	−19.2666	−0.685042
$792$	0	0
$793$	−7.21110	−0.256074
$794$	35.2111	1.24960
$795$	0	0
$796$	5.21110	0.184703
$797$	42.4222	1.50267	0.751336	−	0.659920i	$-0.229410\pi$
$797$	42.4222	1.50267	0.751336	+	0.659920i	$0.229410\pi$
$798$	0	0
$799$	−42.4222	−1.50079
$800$	1.00000	0.0353553
$801$	0	0
$802$	33.6333	1.18763
$803$	0	0
$804$	0	0
$805$	−12.0000	−0.422944
$806$	2.00000	0.0704470
$807$	0	0
$808$	−13.8167	−0.486068
$809$	6.42221	0.225793	0.112896	−	0.993607i	$-0.463987\pi$
$809$	6.42221	0.225793	0.112896	+	0.993607i	$0.463987\pi$
$810$	0	0
$811$	49.0278	1.72160	0.860799	−	0.508946i	$-0.169965\pi$
$811$	49.0278	1.72160	0.860799	+	0.508946i	$0.169965\pi$
$812$	12.0000	0.421117
$813$	0	0
$814$	0	0
$815$	2.00000	0.0700569
$816$	0	0
$817$	34.4222	1.20428
$818$	−13.6333	−0.476677
$819$	0	0
$820$	3.21110	0.112137
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	24.0000	0.835067
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	0	0
$832$	1.00000	0.0346688
$833$	−0.972244	−0.0336862
$834$	0	0
$835$	−18.4222	−0.637527
$836$	0	0
$837$	0	0
$838$	−29.4500	−1.01733
$839$	−42.4222	−1.46458	−0.732289	−	0.680994i	$-0.761548\pi$
$839$	−42.4222	−1.46458	−0.732289	+	0.680994i	$0.761548\pi$
$840$	0	0
$841$	−7.78890	−0.268583
$842$	−9.02776	−0.311117
$843$	0	0
$844$	−25.2111	−0.867802
$845$	1.00000	0.0344010
$846$	0	0
$847$	28.6611	0.984806
$848$	0	0
$849$	0	0
$850$	4.60555	0.157969
$851$	51.6333	1.76997
$852$	0	0
$853$	24.0555	0.823645	0.411823	−	0.911264i	$-0.364892\pi$
$853$	24.0555	0.823645	0.411823	+	0.911264i	$0.364892\pi$
$854$	18.7889	0.642943
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−29.4500	−1.00599	−0.502996	−	0.864289i	$-0.667769\pi$
$857$	−29.4500	−1.00599	−0.502996	+	0.864289i	$0.667769\pi$
$858$	0	0
$859$	4.36669	0.148990	0.0744948	−	0.997221i	$-0.476266\pi$
$859$	4.36669	0.148990	0.0744948	+	0.997221i	$0.476266\pi$
$860$	5.21110	0.177697
$861$	0	0
$862$	−24.0000	−0.817443
$863$	−42.4222	−1.44407	−0.722034	−	0.691857i	$-0.756793\pi$
$863$	−42.4222	−1.44407	−0.722034	+	0.691857i	$0.756793\pi$
$864$	0	0
$865$	12.0000	0.408012
$866$	−22.8444	−0.776285
$867$	0	0
$868$	−5.21110	−0.176876
$869$	0	0
$870$	0	0
$871$	−7.21110	−0.244339
$872$	18.6056	0.630063
$873$	0	0
$874$	30.4222	1.02905
$875$	−2.60555	−0.0880837
$876$	0	0
$877$	26.0000	0.877958	0.438979	−	0.898497i	$-0.355340\pi$
$877$	26.0000	0.877958	0.438979	+	0.898497i	$0.355340\pi$
$878$	32.8444	1.10845
$879$	0	0
$880$	0	0
$881$	26.7889	0.902541	0.451270	−	0.892387i	$-0.350971\pi$
$881$	26.7889	0.902541	0.451270	+	0.892387i	$0.350971\pi$
$882$	0	0
$883$	14.4222	0.485346	0.242673	−	0.970108i	$-0.421976\pi$
$883$	14.4222	0.485346	0.242673	+	0.970108i	$0.421976\pi$
$884$	4.60555	0.154901
$885$	0	0
$886$	−39.6333	−1.33151
$887$	4.60555	0.154639	0.0773196	−	0.997006i	$-0.475364\pi$
$887$	4.60555	0.154639	0.0773196	+	0.997006i	$0.475364\pi$
$888$	0	0
$889$	3.15559	0.105835
$890$	3.21110	0.107636
$891$	0	0
$892$	−17.3944	−0.582409
$893$	−60.8444	−2.03608
$894$	0	0
$895$	11.0278	0.368617
$896$	−2.60555	−0.0870454
$897$	0	0
$898$	−9.63331	−0.321468
$899$	−9.21110	−0.307207
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	7.39445	0.245936
$905$	−19.2111	−0.638599
$906$	0	0
$907$	−12.3667	−0.410629	−0.205315	−	0.978696i	$-0.565822\pi$
$907$	−12.3667	−0.410629	−0.205315	+	0.978696i	$0.565822\pi$
$908$	6.42221	0.213128
$909$	0	0
$910$	−2.60555	−0.0863732
$911$	15.6333	0.517955	0.258977	−	0.965883i	$-0.416615\pi$
$911$	15.6333	0.517955	0.258977	+	0.965883i	$0.416615\pi$
$912$	0	0
$913$	0	0
$914$	−17.3944	−0.575357
$915$	0	0
$916$	0.183346	0.00605793
$917$	43.2666	1.42879
$918$	0	0
$919$	−52.0000	−1.71532	−0.857661	−	0.514216i	$-0.828083\pi$
$919$	−52.0000	−1.71532	−0.857661	+	0.514216i	$0.828083\pi$
$920$	4.60555	0.151841
$921$	0	0
$922$	−3.21110	−0.105752
$923$	−12.0000	−0.394985
$924$	0	0
$925$	11.2111	0.368619
$926$	12.1833	0.400370
$927$	0	0
$928$	−4.60555	−0.151185
$929$	−40.0555	−1.31418	−0.657089	−	0.753813i	$-0.728213\pi$
$929$	−40.0555	−1.31418	−0.657089	+	0.753813i	$0.728213\pi$
$930$	0	0
$931$	−1.39445	−0.0457012
$932$	−1.81665	−0.0595065
$933$	0	0
$934$	−5.57779	−0.182511
$935$	0	0
$936$	0	0
$937$	−13.6333	−0.445381	−0.222690	−	0.974889i	$-0.571484\pi$
$937$	−13.6333	−0.445381	−0.222690	+	0.974889i	$0.571484\pi$
$938$	18.7889	0.613479
$939$	0	0
$940$	−9.21110	−0.300433
$941$	21.6333	0.705226	0.352613	−	0.935769i	$-0.385293\pi$
$941$	21.6333	0.705226	0.352613	+	0.935769i	$0.385293\pi$
$942$	0	0
$943$	14.7889	0.481593
$944$	−9.21110	−0.299796
$945$	0	0
$946$	0	0
$947$	42.4222	1.37854	0.689268	−	0.724506i	$-0.257932\pi$
$947$	42.4222	1.37854	0.689268	+	0.724506i	$0.257932\pi$
$948$	0	0
$949$	6.60555	0.214425
$950$	6.60555	0.214312
$951$	0	0
$952$	−12.0000	−0.388922
$953$	16.6056	0.537907	0.268953	−	0.963153i	$-0.413322\pi$
$953$	16.6056	0.537907	0.268953	+	0.963153i	$0.413322\pi$
$954$	0	0
$955$	−12.0000	−0.388311
$956$	18.4222	0.595817
$957$	0	0
$958$	18.4222	0.595194
$959$	−56.3667	−1.82018
$960$	0	0
$961$	−27.0000	−0.870968
$962$	11.2111	0.361460
$963$	0	0
$964$	−10.0000	−0.322078
$965$	−14.6056	−0.470169
$966$	0	0
$967$	−26.6056	−0.855577	−0.427788	−	0.903879i	$-0.640707\pi$
$967$	−26.6056	−0.855577	−0.427788	+	0.903879i	$0.640707\pi$
$968$	−11.0000	−0.353553
$969$	0	0
$970$	6.60555	0.212091
$971$	−19.3944	−0.622397	−0.311199	−	0.950345i	$-0.600730\pi$
$971$	−19.3944	−0.622397	−0.311199	+	0.950345i	$0.600730\pi$
$972$	0	0
$973$	−37.5778	−1.20469
$974$	2.97224	0.0952368
$975$	0	0
$976$	−7.21110	−0.230822
$977$	57.6333	1.84385	0.921926	−	0.387365i	$-0.126615\pi$
$977$	57.6333	1.84385	0.921926	+	0.387365i	$0.126615\pi$
$978$	0	0
$979$	0	0
$980$	−0.211103	−0.00674342
$981$	0	0
$982$	32.2389	1.02878
$983$	27.6333	0.881366	0.440683	−	0.897663i	$-0.354736\pi$
$983$	27.6333	0.881366	0.440683	+	0.897663i	$0.354736\pi$
$984$	0	0
$985$	−24.4222	−0.778156
$986$	−21.2111	−0.675499
$987$	0	0
$988$	6.60555	0.210151
$989$	24.0000	0.763156
$990$	0	0
$991$	−31.6333	−1.00487	−0.502433	−	0.864616i	$-0.667561\pi$
$991$	−31.6333	−1.00487	−0.502433	+	0.864616i	$0.667561\pi$
$992$	2.00000	0.0635001
$993$	0	0
$994$	31.2666	0.991717
$995$	5.21110	0.165203
$996$	0	0
$997$	4.78890	0.151666	0.0758330	−	0.997121i	$-0.475838\pi$
$997$	4.78890	0.151666	0.0758330	+	0.997121i	$0.475838\pi$
$998$	27.8167	0.880521
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1170.2.a.q.1.1	yes	2
3.2	odd	2		1170.2.a.p.1.1	✓	2
4.3	odd	2		9360.2.a.cn.1.2		2
5.2	odd	4		5850.2.e.bj.5149.3		4
5.3	odd	4		5850.2.e.bj.5149.2		4
5.4	even	2		5850.2.a.ce.1.2		2
12.11	even	2		9360.2.a.cf.1.2		2
15.2	even	4		5850.2.e.bl.5149.1		4
15.8	even	4		5850.2.e.bl.5149.4		4
15.14	odd	2		5850.2.a.ck.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1170.2.a.p.1.1	✓	2	3.2	odd	2
1170.2.a.q.1.1	yes	2	1.1	even	1	trivial
5850.2.a.ce.1.2		2	5.4	even	2
5850.2.a.ck.1.2		2	15.14	odd	2
5850.2.e.bj.5149.2		4	5.3	odd	4
5850.2.e.bj.5149.3		4	5.2	odd	4
5850.2.e.bl.5149.1		4	15.2	even	4
5850.2.e.bl.5149.4		4	15.8	even	4
9360.2.a.cf.1.2		2	12.11	even	2
9360.2.a.cn.1.2		2	4.3	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:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.60555 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.60555 q^{7} +1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{13} -2.60555 q^{14} +1.00000 q^{16} +4.60555 q^{17} +6.60555 q^{19} +1.00000 q^{20} +4.60555 q^{23} +1.00000 q^{25} +1.00000 q^{26} -2.60555 q^{28} -4.60555 q^{29} +2.00000 q^{31} +1.00000 q^{32} +4.60555 q^{34} -2.60555 q^{35} +11.2111 q^{37} +6.60555 q^{38} +1.00000 q^{40} +3.21110 q^{41} +5.21110 q^{43} +4.60555 q^{46} -9.21110 q^{47} -0.211103 q^{49} +1.00000 q^{50} +1.00000 q^{52} -2.60555 q^{56} -4.60555 q^{58} -9.21110 q^{59} -7.21110 q^{61} +2.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} -7.21110 q^{67} +4.60555 q^{68} -2.60555 q^{70} -12.0000 q^{71} +6.60555 q^{73} +11.2111 q^{74} +6.60555 q^{76} -1.21110 q^{79} +1.00000 q^{80} +3.21110 q^{82} +4.60555 q^{85} +5.21110 q^{86} +3.21110 q^{89} -2.60555 q^{91} +4.60555 q^{92} -9.21110 q^{94} +6.60555 q^{95} +6.60555 q^{97} -0.211103 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} + 2 q^{8} + 2 q^{10} + 2 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} + 6 q^{19} + 2 q^{20} + 2 q^{23} + 2 q^{25} + 2 q^{26} + 2 q^{28} - 2 q^{29} + 4 q^{31} + 2 q^{32} + 2 q^{34} + 2 q^{35} + 8 q^{37} + 6 q^{38} + 2 q^{40} - 8 q^{41} - 4 q^{43} + 2 q^{46} - 4 q^{47} + 14 q^{49} + 2 q^{50} + 2 q^{52} + 2 q^{56} - 2 q^{58} - 4 q^{59} + 4 q^{62} + 2 q^{64} + 2 q^{65} + 2 q^{68} + 2 q^{70} - 24 q^{71} + 6 q^{73} + 8 q^{74} + 6 q^{76} + 12 q^{79} + 2 q^{80} - 8 q^{82} + 2 q^{85} - 4 q^{86} - 8 q^{89} + 2 q^{91} + 2 q^{92} - 4 q^{94} + 6 q^{95} + 6 q^{97} + 14 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^7 + 2 * q^8 + 2 * q^10 + 2 * q^13 + 2 * q^14 + 2 * q^16 + 2 * q^17 + 6 * q^19 + 2 * q^20 + 2 * q^23 + 2 * q^25 + 2 * q^26 + 2 * q^28 - 2 * q^29 + 4 * q^31 + 2 * q^32 + 2 * q^34 + 2 * q^35 + 8 * q^37 + 6 * q^38 + 2 * q^40 - 8 * q^41 - 4 * q^43 + 2 * q^46 - 4 * q^47 + 14 * q^49 + 2 * q^50 + 2 * q^52 + 2 * q^56 - 2 * q^58 - 4 * q^59 + 4 * q^62 + 2 * q^64 + 2 * q^65 + 2 * q^68 + 2 * q^70 - 24 * q^71 + 6 * q^73 + 8 * q^74 + 6 * q^76 + 12 * q^79 + 2 * q^80 - 8 * q^82 + 2 * q^85 - 4 * q^86 - 8 * q^89 + 2 * q^91 + 2 * q^92 - 4 * q^94 + 6 * q^95 + 6 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more