LMFDB - Embedded newform 1170.2.b.d.181.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1170,2,Mod(181,1170)]

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, 1]))

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

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

chi := DirichletCharacter("1170.181");

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.b (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:	$9.34249703649$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.2
Root			$1.30278i$ of defining polynomial
Character	$\chi$	$=$	1170.181
Dual form			1170.2.b.d.181.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} +2.60555i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} +2.60555i q^{7} +1.00000i q^{8} -1.00000 q^{10} -3.60555 q^{13} +2.60555 q^{14} +1.00000 q^{16} +2.60555 q^{17} +2.60555i q^{19} +1.00000i q^{20} +8.60555 q^{23} -1.00000 q^{25} +3.60555i q^{26} -2.60555i q^{28} +2.60555 q^{29} +6.00000i q^{31} -1.00000i q^{32} -2.60555i q^{34} +2.60555 q^{35} +5.21110i q^{37} +2.60555 q^{38} +1.00000 q^{40} +11.2111i q^{41} +8.00000 q^{43} -8.60555i q^{46} -5.21110i q^{47} +0.211103 q^{49} +1.00000i q^{50} +3.60555 q^{52} -6.00000 q^{53} -2.60555 q^{56} -2.60555i q^{58} -5.21110i q^{59} +3.21110 q^{61} +6.00000 q^{62} -1.00000 q^{64} +3.60555i q^{65} +11.2111i q^{67} -2.60555 q^{68} -2.60555i q^{70} +5.21110i q^{71} -8.60555i q^{73} +5.21110 q^{74} -2.60555i q^{76} +14.4222 q^{79} -1.00000i q^{80} +11.2111 q^{82} -17.2111i q^{83} -2.60555i q^{85} -8.00000i q^{86} -0.788897i q^{89} -9.39445i q^{91} -8.60555 q^{92} -5.21110 q^{94} +2.60555 q^{95} +8.60555i q^{97} -0.211103i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} - 4 q^{10} - 4 q^{14} + 4 q^{16} - 4 q^{17} + 20 q^{23} - 4 q^{25} - 4 q^{29} - 4 q^{35} - 4 q^{38} + 4 q^{40} + 32 q^{43} - 28 q^{49} - 24 q^{53} + 4 q^{56} - 16 q^{61} + 24 q^{62} - 4 q^{64} + 4 q^{68} - 8 q^{74} + 16 q^{82} - 20 q^{92} + 8 q^{94} - 4 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^10 - 4 * q^14 + 4 * q^16 - 4 * q^17 + 20 * q^23 - 4 * q^25 - 4 * q^29 - 4 * q^35 - 4 * q^38 + 4 * q^40 + 32 * q^43 - 28 * q^49 - 24 * q^53 + 4 * q^56 - 16 * q^61 + 24 * q^62 - 4 * q^64 + 4 * q^68 - 8 * q^74 + 16 * q^82 - 20 * q^92 + 8 * q^94 - 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times$.

$n$	$911$	$937$	$1081$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	2.60555i	0.984806i	0.870367	+	0.492403i	$0.163881\pi$
$7$	2.60555i	0.984806i	−0.870367	+	0.492403i	$0.836119\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	−1.00000	−0.316228
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	−3.60555	−1.00000
$13$	−3.60555	−1.00000
$14$	2.60555	0.696363
$15$	0	0
$16$	1.00000	0.250000
$17$	2.60555	0.631939	0.315970	−	0.948769i	$-0.397670\pi$
$17$	2.60555	0.631939	0.315970	+	0.948769i	$0.397670\pi$
$18$	0	0
$19$	2.60555i	0.597754i	0.954292	+	0.298877i	$0.0966121\pi$
$19$	2.60555i	0.597754i	−0.954292	+	0.298877i	$0.903388\pi$
$20$	1.00000i	0.223607i
$21$	0	0
$22$	0	0
$23$	8.60555	1.79438	0.897191	−	0.441643i	$-0.145604\pi$
$23$	8.60555	1.79438	0.897191	+	0.441643i	$0.145604\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	3.60555i	0.707107i
$27$	0	0
$28$	− 2.60555i	− 0.492403i
$29$	2.60555	0.483839	0.241919	−	0.970296i	$-0.422223\pi$
$29$	2.60555	0.483839	0.241919	+	0.970296i	$0.422223\pi$
$30$	0	0
$31$	6.00000i	1.07763i	0.842424	+	0.538816i	$0.181128\pi$
$31$	6.00000i	1.07763i	−0.842424	+	0.538816i	$0.818872\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	− 2.60555i	− 0.446848i
$35$	2.60555	0.440419
$36$	0	0
$37$	5.21110i	0.856700i	0.903613	+	0.428350i	$0.140905\pi$
$37$	5.21110i	0.856700i	−0.903613	+	0.428350i	$0.859095\pi$
$38$	2.60555	0.422676
$39$	0	0
$40$	1.00000	0.158114
$41$	11.2111i	1.75088i	0.483327	+	0.875440i	$0.339428\pi$
$41$	11.2111i	1.75088i	−0.483327	+	0.875440i	$0.660572\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	0	0
$46$	− 8.60555i	− 1.26882i
$47$	− 5.21110i	− 0.760117i	−0.924962	−	0.380059i	$-0.875904\pi$
$47$	− 5.21110i	− 0.760117i	0.924962	−	0.380059i	$-0.124096\pi$
$48$	0	0
$49$	0.211103	0.0301575
$50$	1.00000i	0.141421i
$51$	0	0
$52$	3.60555	0.500000
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	−2.60555	−0.348181
$57$	0	0
$58$	− 2.60555i	− 0.342126i
$59$	− 5.21110i	− 0.678428i	−0.940709	−	0.339214i	$-0.889839\pi$
$59$	− 5.21110i	− 0.678428i	0.940709	−	0.339214i	$-0.110161\pi$
$60$	0	0
$61$	3.21110	0.411140	0.205570	−	0.978642i	$-0.434095\pi$
$61$	3.21110	0.411140	0.205570	+	0.978642i	$0.434095\pi$
$62$	6.00000	0.762001
$63$	0	0
$64$	−1.00000	−0.125000
$65$	3.60555i	0.447214i
$66$	0	0
$67$	11.2111i	1.36965i	0.728706	+	0.684827i	$0.240122\pi$
$67$	11.2111i	1.36965i	−0.728706	+	0.684827i	$0.759878\pi$
$68$	−2.60555	−0.315970
$69$	0	0
$70$	− 2.60555i	− 0.311423i
$71$	5.21110i	0.618444i	0.950990	+	0.309222i	$0.100069\pi$
$71$	5.21110i	0.618444i	−0.950990	+	0.309222i	$0.899931\pi$
$72$	0	0
$73$	− 8.60555i	− 1.00720i	−0.863936	−	0.503602i	$-0.832008\pi$
$73$	− 8.60555i	− 1.00720i	0.863936	−	0.503602i	$-0.167992\pi$
$74$	5.21110	0.605778
$75$	0	0
$76$	− 2.60555i	− 0.298877i
$77$	0	0
$78$	0	0
$79$	14.4222	1.62262	0.811312	−	0.584613i	$-0.198754\pi$
$79$	14.4222	1.62262	0.811312	+	0.584613i	$0.198754\pi$
$80$	− 1.00000i	− 0.111803i
$81$	0	0
$82$	11.2111	1.23806
$83$	− 17.2111i	− 1.88916i	−0.328276	−	0.944582i	$-0.606467\pi$
$83$	− 17.2111i	− 1.88916i	0.328276	−	0.944582i	$-0.393533\pi$
$84$	0	0
$85$	− 2.60555i	− 0.282612i
$86$	− 8.00000i	− 0.862662i
$87$	0	0
$88$	0	0
$89$	− 0.788897i	− 0.0836230i	−0.999126	−	0.0418115i	$-0.986687\pi$
$89$	− 0.788897i	− 0.0836230i	0.999126	−	0.0418115i	$-0.0133129\pi$
$90$	0	0
$91$	− 9.39445i	− 0.984806i
$92$	−8.60555	−0.897191
$93$	0	0
$94$	−5.21110	−0.537484
$95$	2.60555	0.267324
$96$	0	0
$97$	8.60555i	0.873761i	0.899519	+	0.436881i	$0.143917\pi$
$97$	8.60555i	0.873761i	−0.899519	+	0.436881i	$0.856083\pi$
$98$	− 0.211103i	− 0.0213246i
$99$	0	0
$100$	1.00000	0.100000
$101$	14.6056	1.45331	0.726653	−	0.687004i	$-0.241075\pi$
$101$	14.6056	1.45331	0.726653	+	0.687004i	$0.241075\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	− 3.60555i	− 0.353553i
$105$	0	0
$106$	6.00000i	0.582772i
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	8.60555i	0.824262i	0.911125	+	0.412131i	$0.135215\pi$
$109$	8.60555i	0.824262i	−0.911125	+	0.412131i	$0.864785\pi$
$110$	0	0
$111$	0	0
$112$	2.60555i	0.246201i
$113$	−7.81665	−0.735329	−0.367664	−	0.929959i	$-0.619843\pi$
$113$	−7.81665	−0.735329	−0.367664	+	0.929959i	$0.619843\pi$
$114$	0	0
$115$	− 8.60555i	− 0.802472i
$116$	−2.60555	−0.241919
$117$	0	0
$118$	−5.21110	−0.479721
$119$	6.78890i	0.622337i
$120$	0	0
$121$	11.0000	1.00000
$122$	− 3.21110i	− 0.290720i
$123$	0	0
$124$	− 6.00000i	− 0.538816i
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	−13.2111	−1.17230	−0.586148	−	0.810204i	$-0.699356\pi$
$127$	−13.2111	−1.17230	−0.586148	+	0.810204i	$0.699356\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	3.60555	0.316228
$131$	−15.3944	−1.34502	−0.672510	−	0.740088i	$-0.734784\pi$
$131$	−15.3944	−1.34502	−0.672510	+	0.740088i	$0.734784\pi$
$132$	0	0
$133$	−6.78890	−0.588672
$134$	11.2111	0.968492
$135$	0	0
$136$	2.60555i	0.223424i
$137$	11.2111i	0.957829i	0.877862	+	0.478915i	$0.158970\pi$
$137$	11.2111i	0.957829i	−0.877862	+	0.478915i	$0.841030\pi$
$138$	0	0
$139$	2.78890	0.236551	0.118276	−	0.992981i	$-0.462263\pi$
$139$	2.78890	0.236551	0.118276	+	0.992981i	$0.462263\pi$
$140$	−2.60555	−0.220209
$141$	0	0
$142$	5.21110	0.437306
$143$	0	0
$144$	0	0
$145$	− 2.60555i	− 0.216379i
$146$	−8.60555	−0.712200
$147$	0	0
$148$	− 5.21110i	− 0.428350i
$149$	− 0.788897i	− 0.0646290i	−0.999478	−	0.0323145i	$-0.989712\pi$
$149$	− 0.788897i	− 0.0646290i	0.999478	−	0.0323145i	$-0.0102878\pi$
$150$	0	0
$151$	6.00000i	0.488273i	0.969741	+	0.244137i	$0.0785045\pi$
$151$	6.00000i	0.488273i	−0.969741	+	0.244137i	$0.921495\pi$
$152$	−2.60555	−0.211338
$153$	0	0
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	−8.42221	−0.672165	−0.336083	−	0.941833i	$-0.609102\pi$
$157$	−8.42221	−0.672165	−0.336083	+	0.941833i	$0.609102\pi$
$158$	− 14.4222i	− 1.14737i
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	22.4222i	1.76712i
$162$	0	0
$163$	− 4.42221i	− 0.346374i	−0.984889	−	0.173187i	$-0.944594\pi$
$163$	− 4.42221i	− 0.346374i	0.984889	−	0.173187i	$-0.0554065\pi$
$164$	− 11.2111i	− 0.875440i
$165$	0	0
$166$	−17.2111	−1.33584
$167$	5.21110i	0.403247i	0.979463	+	0.201624i	$0.0646218\pi$
$167$	5.21110i	0.403247i	−0.979463	+	0.201624i	$0.935378\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	−2.60555	−0.199837
$171$	0	0
$172$	−8.00000	−0.609994
$173$	−16.4222	−1.24856	−0.624279	−	0.781202i	$-0.714607\pi$
$173$	−16.4222	−1.24856	−0.624279	+	0.781202i	$0.714607\pi$
$174$	0	0
$175$	− 2.60555i	− 0.196961i
$176$	0	0
$177$	0	0
$178$	−0.788897	−0.0591304
$179$	−1.81665	−0.135783	−0.0678915	−	0.997693i	$-0.521627\pi$
$179$	−1.81665	−0.135783	−0.0678915	+	0.997693i	$0.521627\pi$
$180$	0	0
$181$	−20.4222	−1.51797	−0.758985	−	0.651108i	$-0.774305\pi$
$181$	−20.4222	−1.51797	−0.758985	+	0.651108i	$0.774305\pi$
$182$	−9.39445	−0.696363
$183$	0	0
$184$	8.60555i	0.634410i
$185$	5.21110	0.383128
$186$	0	0
$187$	0	0
$188$	5.21110i	0.380059i
$189$	0	0
$190$	− 2.60555i	− 0.189027i
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	− 13.8167i	− 0.994545i	−0.867595	−	0.497272i	$-0.834335\pi$
$193$	− 13.8167i	− 0.994545i	0.867595	−	0.497272i	$-0.165665\pi$
$194$	8.60555	0.617843
$195$	0	0
$196$	−0.211103	−0.0150788
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	0	0
$199$	−6.42221	−0.455258	−0.227629	−	0.973748i	$-0.573097\pi$
$199$	−6.42221	−0.455258	−0.227629	+	0.973748i	$0.573097\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	0	0
$202$	− 14.6056i	− 1.02764i
$203$	6.78890i	0.476487i
$204$	0	0
$205$	11.2111	0.783017
$206$	4.00000i	0.278693i
$207$	0	0
$208$	−3.60555	−0.250000
$209$	0	0
$210$	0	0
$211$	−2.78890	−0.191996	−0.0959978	−	0.995382i	$-0.530604\pi$
$211$	−2.78890	−0.191996	−0.0959978	+	0.995382i	$0.530604\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	0	0
$215$	− 8.00000i	− 0.545595i
$216$	0	0
$217$	−15.6333	−1.06126
$218$	8.60555	0.582841
$219$	0	0
$220$	0	0
$221$	−9.39445	−0.631939
$222$	0	0
$223$	− 19.8167i	− 1.32702i	−0.748167	−	0.663511i	$-0.769066\pi$
$223$	− 19.8167i	− 1.32702i	0.748167	−	0.663511i	$-0.230934\pi$
$224$	2.60555	0.174091
$225$	0	0
$226$	7.81665i	0.519956i
$227$	− 24.0000i	− 1.59294i	−0.604681	−	0.796468i	$-0.706699\pi$
$227$	− 24.0000i	− 1.59294i	0.604681	−	0.796468i	$-0.293301\pi$
$228$	0	0
$229$	1.81665i	0.120048i	0.998197	+	0.0600239i	$0.0191177\pi$
$229$	1.81665i	0.120048i	−0.998197	+	0.0600239i	$0.980882\pi$
$230$	−8.60555	−0.567433
$231$	0	0
$232$	2.60555i	0.171063i
$233$	−19.8167	−1.29823	−0.649116	−	0.760689i	$-0.724861\pi$
$233$	−19.8167	−1.29823	−0.649116	+	0.760689i	$0.724861\pi$
$234$	0	0
$235$	−5.21110	−0.339935
$236$	5.21110i	0.339214i
$237$	0	0
$238$	6.78890	0.440059
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	22.4222i	1.44434i	0.691715	+	0.722171i	$0.256855\pi$
$241$	22.4222i	1.44434i	−0.691715	+	0.722171i	$0.743145\pi$
$242$	− 11.0000i	− 0.707107i
$243$	0	0
$244$	−3.21110	−0.205570
$245$	− 0.211103i	− 0.0134868i
$246$	0	0
$247$	− 9.39445i	− 0.597754i
$248$	−6.00000	−0.381000
$249$	0	0
$250$	1.00000	0.0632456
$251$	20.6056	1.30061	0.650305	−	0.759673i	$-0.274641\pi$
$251$	20.6056	1.30061	0.650305	+	0.759673i	$0.274641\pi$
$252$	0	0
$253$	0	0
$254$	13.2111i	0.828938i
$255$	0	0
$256$	1.00000	0.0625000
$257$	21.3944	1.33455	0.667275	−	0.744812i	$-0.267461\pi$
$257$	21.3944	1.33455	0.667275	+	0.744812i	$0.267461\pi$
$258$	0	0
$259$	−13.5778	−0.843683
$260$	− 3.60555i	− 0.223607i
$261$	0	0
$262$	15.3944i	0.951072i
$263$	−13.8167	−0.851971	−0.425986	−	0.904730i	$-0.640073\pi$
$263$	−13.8167	−0.851971	−0.425986	+	0.904730i	$0.640073\pi$
$264$	0	0
$265$	6.00000i	0.368577i
$266$	6.78890i	0.416254i
$267$	0	0
$268$	− 11.2111i	− 0.684827i
$269$	−4.18335	−0.255063	−0.127532	−	0.991835i	$-0.540705\pi$
$269$	−4.18335	−0.255063	−0.127532	+	0.991835i	$0.540705\pi$
$270$	0	0
$271$	28.4222i	1.72653i	0.504754	+	0.863263i	$0.331583\pi$
$271$	28.4222i	1.72653i	−0.504754	+	0.863263i	$0.668417\pi$
$272$	2.60555	0.157985
$273$	0	0
$274$	11.2111	0.677287
$275$	0	0
$276$	0	0
$277$	12.4222	0.746378	0.373189	−	0.927755i	$-0.378264\pi$
$277$	12.4222	0.746378	0.373189	+	0.927755i	$0.378264\pi$
$278$	− 2.78890i	− 0.167267i
$279$	0	0
$280$	2.60555i	0.155711i
$281$	− 12.7889i	− 0.762922i	−0.924385	−	0.381461i	$-0.875421\pi$
$281$	− 12.7889i	− 0.762922i	0.924385	−	0.381461i	$-0.124579\pi$
$282$	0	0
$283$	−18.4222	−1.09509	−0.547543	−	0.836777i	$-0.684437\pi$
$283$	−18.4222	−1.09509	−0.547543	+	0.836777i	$0.684437\pi$
$284$	− 5.21110i	− 0.309222i
$285$	0	0
$286$	0	0
$287$	−29.2111	−1.72428
$288$	0	0
$289$	−10.2111	−0.600653
$290$	−2.60555	−0.153003
$291$	0	0
$292$	8.60555i	0.503602i
$293$	− 18.0000i	− 1.05157i	−0.850617	−	0.525786i	$-0.823771\pi$
$293$	− 18.0000i	− 1.05157i	0.850617	−	0.525786i	$-0.176229\pi$
$294$	0	0
$295$	−5.21110	−0.303402
$296$	−5.21110	−0.302889
$297$	0	0
$298$	−0.788897	−0.0456996
$299$	−31.0278	−1.79438
$300$	0	0
$301$	20.8444i	1.20145i
$302$	6.00000	0.345261
$303$	0	0
$304$	2.60555i	0.149439i
$305$	− 3.21110i	− 0.183867i
$306$	0	0
$307$	− 23.2111i	− 1.32473i	−0.749182	−	0.662364i	$-0.769553\pi$
$307$	− 23.2111i	− 1.32473i	0.749182	−	0.662364i	$-0.230447\pi$
$308$	0	0
$309$	0	0
$310$	− 6.00000i	− 0.340777i
$311$	−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
$313$	32.4222	1.83261	0.916306	−	0.400480i	$-0.131156\pi$
$313$	32.4222	1.83261	0.916306	+	0.400480i	$0.131156\pi$
$314$	8.42221i	0.475293i
$315$	0	0
$316$	−14.4222	−0.811312
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	0	0
$319$	0	0
$320$	1.00000i	0.0559017i
$321$	0	0
$322$	22.4222	1.24954
$323$	6.78890i	0.377744i
$324$	0	0
$325$	3.60555	0.200000
$326$	−4.42221	−0.244923
$327$	0	0
$328$	−11.2111	−0.619030
$329$	13.5778	0.748568
$330$	0	0
$331$	− 9.39445i	− 0.516366i	−0.966096	−	0.258183i	$-0.916876\pi$
$331$	− 9.39445i	− 0.516366i	0.966096	−	0.258183i	$-0.0831237\pi$
$332$	17.2111i	0.944582i
$333$	0	0
$334$	5.21110	0.285139
$335$	11.2111	0.612528
$336$	0	0
$337$	−29.6333	−1.61423	−0.807115	−	0.590395i	$-0.798972\pi$
$337$	−29.6333	−1.61423	−0.807115	+	0.590395i	$0.798972\pi$
$338$	− 13.0000i	− 0.707107i
$339$	0	0
$340$	2.60555i	0.141306i
$341$	0	0
$342$	0	0
$343$	18.7889i	1.01451i
$344$	8.00000i	0.431331i
$345$	0	0
$346$	16.4222i	0.882863i
$347$	−15.6333	−0.839240	−0.419620	−	0.907700i	$-0.637837\pi$
$347$	−15.6333	−0.839240	−0.419620	+	0.907700i	$0.637837\pi$
$348$	0	0
$349$	13.8167i	0.739589i	0.929114	+	0.369794i	$0.120572\pi$
$349$	13.8167i	0.739589i	−0.929114	+	0.369794i	$0.879428\pi$
$350$	−2.60555	−0.139273
$351$	0	0
$352$	0	0
$353$	23.2111i	1.23540i	0.786413	+	0.617701i	$0.211936\pi$
$353$	23.2111i	1.23540i	−0.786413	+	0.617701i	$0.788064\pi$
$354$	0	0
$355$	5.21110	0.276577
$356$	0.788897i	0.0418115i
$357$	0	0
$358$	1.81665i	0.0960131i
$359$	27.6333i	1.45843i	0.684285	+	0.729215i	$0.260115\pi$
$359$	27.6333i	1.45843i	−0.684285	+	0.729215i	$0.739885\pi$
$360$	0	0
$361$	12.2111	0.642690
$362$	20.4222i	1.07337i
$363$	0	0
$364$	9.39445i	0.492403i
$365$	−8.60555	−0.450435
$366$	0	0
$367$	23.6333	1.23365	0.616824	−	0.787101i	$-0.288419\pi$
$367$	23.6333	1.23365	0.616824	+	0.787101i	$0.288419\pi$
$368$	8.60555	0.448595
$369$	0	0
$370$	− 5.21110i	− 0.270912i
$371$	− 15.6333i	− 0.811641i
$372$	0	0
$373$	8.42221	0.436085	0.218043	−	0.975939i	$-0.430033\pi$
$373$	8.42221	0.436085	0.218043	+	0.975939i	$0.430033\pi$
$374$	0	0
$375$	0	0
$376$	5.21110	0.268742
$377$	−9.39445	−0.483839
$378$	0	0
$379$	− 1.02776i	− 0.0527923i	−0.999652	−	0.0263961i	$-0.991597\pi$
$379$	− 1.02776i	− 0.0527923i	0.999652	−	0.0263961i	$-0.00840313\pi$
$380$	−2.60555	−0.133662
$381$	0	0
$382$	− 12.0000i	− 0.613973i
$383$	− 15.6333i	− 0.798825i	−0.916771	−	0.399412i	$-0.869214\pi$
$383$	− 15.6333i	− 0.798825i	0.916771	−	0.399412i	$-0.130786\pi$
$384$	0	0
$385$	0	0
$386$	−13.8167	−0.703249
$387$	0	0
$388$	− 8.60555i	− 0.436881i
$389$	2.60555	0.132107	0.0660533	−	0.997816i	$-0.478959\pi$
$389$	2.60555	0.132107	0.0660533	+	0.997816i	$0.478959\pi$
$390$	0	0
$391$	22.4222	1.13394
$392$	0.211103i	0.0106623i
$393$	0	0
$394$	6.00000	0.302276
$395$	− 14.4222i	− 0.725660i
$396$	0	0
$397$	39.6333i	1.98914i	0.104076	+	0.994569i	$0.466811\pi$
$397$	39.6333i	1.98914i	−0.104076	+	0.994569i	$0.533189\pi$
$398$	6.42221i	0.321916i
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	− 23.2111i	− 1.15911i	−0.814934	−	0.579554i	$-0.803227\pi$
$401$	− 23.2111i	− 1.15911i	0.814934	−	0.579554i	$-0.196773\pi$
$402$	0	0
$403$	− 21.6333i	− 1.07763i
$404$	−14.6056	−0.726653
$405$	0	0
$406$	6.78890	0.336927
$407$	0	0
$408$	0	0
$409$	− 29.2111i	− 1.44440i	−0.691686	−	0.722198i	$-0.743132\pi$
$409$	− 29.2111i	− 1.44440i	0.691686	−	0.722198i	$-0.256868\pi$
$410$	− 11.2111i	− 0.553677i
$411$	0	0
$412$	4.00000	0.197066
$413$	13.5778	0.668120
$414$	0	0
$415$	−17.2111	−0.844860
$416$	3.60555i	0.176777i
$417$	0	0
$418$	0	0
$419$	−25.8167	−1.26123	−0.630613	−	0.776097i	$-0.717196\pi$
$419$	−25.8167	−1.26123	−0.630613	+	0.776097i	$0.717196\pi$
$420$	0	0
$421$	1.81665i	0.0885383i	0.999020	+	0.0442691i	$0.0140959\pi$
$421$	1.81665i	0.0885383i	−0.999020	+	0.0442691i	$0.985904\pi$
$422$	2.78890i	0.135761i
$423$	0	0
$424$	− 6.00000i	− 0.291386i
$425$	−2.60555	−0.126388
$426$	0	0
$427$	8.36669i	0.404893i
$428$	0	0
$429$	0	0
$430$	−8.00000	−0.385794
$431$	− 12.0000i	− 0.578020i	−0.957326	−	0.289010i	$-0.906674\pi$
$431$	− 12.0000i	− 0.578020i	0.957326	−	0.289010i	$-0.0933260\pi$
$432$	0	0
$433$	4.78890	0.230140	0.115070	−	0.993357i	$-0.463291\pi$
$433$	4.78890	0.230140	0.115070	+	0.993357i	$0.463291\pi$
$434$	15.6333i	0.750423i
$435$	0	0
$436$	− 8.60555i	− 0.412131i
$437$	22.4222i	1.07260i
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	0	0
$442$	9.39445i	0.446848i
$443$	27.6333	1.31290	0.656449	−	0.754370i	$-0.272058\pi$
$443$	27.6333	1.31290	0.656449	+	0.754370i	$0.272058\pi$
$444$	0	0
$445$	−0.788897	−0.0373973
$446$	−19.8167	−0.938346
$447$	0	0
$448$	− 2.60555i	− 0.123101i
$449$	9.63331i	0.454624i	0.973822	+	0.227312i	$0.0729937\pi$
$449$	9.63331i	0.454624i	−0.973822	+	0.227312i	$0.927006\pi$
$450$	0	0
$451$	0	0
$452$	7.81665	0.367664
$453$	0	0
$454$	−24.0000	−1.12638
$455$	−9.39445	−0.440419
$456$	0	0
$457$	− 12.2389i	− 0.572510i	−0.958154	−	0.286255i	$-0.907590\pi$
$457$	− 12.2389i	− 0.572510i	0.958154	−	0.286255i	$-0.0924104\pi$
$458$	1.81665	0.0848867
$459$	0	0
$460$	8.60555i	0.401236i
$461$	− 9.63331i	− 0.448668i	−0.974512	−	0.224334i	$-0.927979\pi$
$461$	− 9.63331i	− 0.448668i	0.974512	−	0.224334i	$-0.0720206\pi$
$462$	0	0
$463$	− 38.6056i	− 1.79415i	−0.441876	−	0.897076i	$-0.645687\pi$
$463$	− 38.6056i	− 1.79415i	0.441876	−	0.897076i	$-0.354313\pi$
$464$	2.60555	0.120960
$465$	0	0
$466$	19.8167i	0.917989i
$467$	1.57779	0.0730116	0.0365058	−	0.999333i	$-0.488377\pi$
$467$	1.57779	0.0730116	0.0365058	+	0.999333i	$0.488377\pi$
$468$	0	0
$469$	−29.2111	−1.34884
$470$	5.21110i	0.240370i
$471$	0	0
$472$	5.21110	0.239860
$473$	0	0
$474$	0	0
$475$	− 2.60555i	− 0.119551i
$476$	− 6.78890i	− 0.311169i
$477$	0	0
$478$	0	0
$479$	− 34.4222i	− 1.57279i	−0.617724	−	0.786395i	$-0.711945\pi$
$479$	− 34.4222i	− 1.57279i	0.617724	−	0.786395i	$-0.288055\pi$
$480$	0	0
$481$	− 18.7889i	− 0.856700i
$482$	22.4222	1.02130
$483$	0	0
$484$	−11.0000	−0.500000
$485$	8.60555	0.390758
$486$	0	0
$487$	− 37.0278i	− 1.67789i	−0.544218	−	0.838944i	$-0.683174\pi$
$487$	− 37.0278i	− 1.67789i	0.544218	−	0.838944i	$-0.316826\pi$
$488$	3.21110i	0.145360i
$489$	0	0
$490$	−0.211103	−0.00953664
$491$	13.8167	0.623537	0.311768	−	0.950158i	$-0.399079\pi$
$491$	13.8167	0.623537	0.311768	+	0.950158i	$0.399079\pi$
$492$	0	0
$493$	6.78890	0.305757
$494$	−9.39445	−0.422676
$495$	0	0
$496$	6.00000i	0.269408i
$497$	−13.5778	−0.609047
$498$	0	0
$499$	− 13.0278i	− 0.583202i	−0.956540	−	0.291601i	$-0.905812\pi$
$499$	− 13.0278i	− 0.583202i	0.956540	−	0.291601i	$-0.0941880\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	0	0
$502$	− 20.6056i	− 0.919671i
$503$	41.4500	1.84816	0.924081	−	0.382196i	$-0.124832\pi$
$503$	41.4500	1.84816	0.924081	+	0.382196i	$0.124832\pi$
$504$	0	0
$505$	− 14.6056i	− 0.649939i
$506$	0	0
$507$	0	0
$508$	13.2111	0.586148
$509$	− 9.63331i	− 0.426989i	−0.976944	−	0.213494i	$-0.931515\pi$
$509$	− 9.63331i	− 0.426989i	0.976944	−	0.213494i	$-0.0684845\pi$
$510$	0	0
$511$	22.4222	0.991900
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	− 21.3944i	− 0.943669i
$515$	4.00000i	0.176261i
$516$	0	0
$517$	0	0
$518$	13.5778i	0.596574i
$519$	0	0
$520$	−3.60555	−0.158114
$521$	21.6333	0.947772	0.473886	−	0.880586i	$-0.342851\pi$
$521$	21.6333	0.947772	0.473886	+	0.880586i	$0.342851\pi$
$522$	0	0
$523$	−24.8444	−1.08637	−0.543185	−	0.839613i	$-0.682782\pi$
$523$	−24.8444	−1.08637	−0.543185	+	0.839613i	$0.682782\pi$
$524$	15.3944	0.672510
$525$	0	0
$526$	13.8167i	0.602435i
$527$	15.6333i	0.680998i
$528$	0	0
$529$	51.0555	2.21980
$530$	6.00000	0.260623
$531$	0	0
$532$	6.78890	0.294336
$533$	− 40.4222i	− 1.75088i
$534$	0	0
$535$	0	0
$536$	−11.2111	−0.484246
$537$	0	0
$538$	4.18335i	0.180357i
$539$	0	0
$540$	0	0
$541$	43.0278i	1.84991i	0.380079	+	0.924954i	$0.375897\pi$
$541$	43.0278i	1.84991i	−0.380079	+	0.924954i	$0.624103\pi$
$542$	28.4222	1.22084
$543$	0	0
$544$	− 2.60555i	− 0.111712i
$545$	8.60555	0.368621
$546$	0	0
$547$	−14.4222	−0.616649	−0.308324	−	0.951281i	$-0.599768\pi$
$547$	−14.4222	−0.616649	−0.308324	+	0.951281i	$0.599768\pi$
$548$	− 11.2111i	− 0.478915i
$549$	0	0
$550$	0	0
$551$	6.78890i	0.289217i
$552$	0	0
$553$	37.5778i	1.59797i
$554$	− 12.4222i	− 0.527769i
$555$	0	0
$556$	−2.78890	−0.118276
$557$	− 40.4222i	− 1.71274i	−0.516360	−	0.856372i	$-0.672713\pi$
$557$	− 40.4222i	− 1.71274i	0.516360	−	0.856372i	$-0.327287\pi$
$558$	0	0
$559$	−28.8444	−1.21999
$560$	2.60555	0.110105
$561$	0	0
$562$	−12.7889	−0.539467
$563$	38.0555	1.60385	0.801924	−	0.597426i	$-0.203810\pi$
$563$	38.0555	1.60385	0.801924	+	0.597426i	$0.203810\pi$
$564$	0	0
$565$	7.81665i	0.328849i
$566$	18.4222i	0.774343i
$567$	0	0
$568$	−5.21110	−0.218653
$569$	9.63331	0.403849	0.201925	−	0.979401i	$-0.435280\pi$
$569$	9.63331	0.403849	0.201925	+	0.979401i	$0.435280\pi$
$570$	0	0
$571$	42.0555	1.75997	0.879984	−	0.475003i	$-0.157553\pi$
$571$	42.0555	1.75997	0.879984	+	0.475003i	$0.157553\pi$
$572$	0	0
$573$	0	0
$574$	29.2111i	1.21925i
$575$	−8.60555	−0.358876
$576$	0	0
$577$	− 44.6056i	− 1.85695i	−0.371391	−	0.928477i	$-0.621119\pi$
$577$	− 44.6056i	− 1.85695i	0.371391	−	0.928477i	$-0.378881\pi$
$578$	10.2111i	0.424726i
$579$	0	0
$580$	2.60555i	0.108190i
$581$	44.8444	1.86046
$582$	0	0
$583$	0	0
$584$	8.60555	0.356100
$585$	0	0
$586$	−18.0000	−0.743573
$587$	− 22.4222i	− 0.925463i	−0.886498	−	0.462732i	$-0.846869\pi$
$587$	− 22.4222i	− 0.925463i	0.886498	−	0.462732i	$-0.153131\pi$
$588$	0	0
$589$	−15.6333	−0.644159
$590$	5.21110i	0.214538i
$591$	0	0
$592$	5.21110i	0.214175i
$593$	− 4.42221i	− 0.181598i	−0.995869	−	0.0907991i	$-0.971058\pi$
$593$	− 4.42221i	− 0.181598i	0.995869	−	0.0907991i	$-0.0289421\pi$
$594$	0	0
$595$	6.78890	0.278318
$596$	0.788897i	0.0323145i
$597$	0	0
$598$	31.0278i	1.26882i
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−41.6333	−1.69826	−0.849129	−	0.528185i	$-0.822873\pi$
$601$	−41.6333	−1.69826	−0.849129	+	0.528185i	$0.822873\pi$
$602$	20.8444	0.849555
$603$	0	0
$604$	− 6.00000i	− 0.244137i
$605$	− 11.0000i	− 0.447214i
$606$	0	0
$607$	2.78890	0.113198	0.0565989	−	0.998397i	$-0.481974\pi$
$607$	2.78890	0.113198	0.0565989	+	0.998397i	$0.481974\pi$
$608$	2.60555	0.105669
$609$	0	0
$610$	−3.21110	−0.130014
$611$	18.7889i	0.760117i
$612$	0	0
$613$	− 18.7889i	− 0.758876i	−0.925217	−	0.379438i	$-0.876117\pi$
$613$	− 18.7889i	− 0.758876i	0.925217	−	0.379438i	$-0.123883\pi$
$614$	−23.2111	−0.936724
$615$	0	0
$616$	0	0
$617$	16.4222i	0.661133i	0.943783	+	0.330567i	$0.107240\pi$
$617$	16.4222i	0.661133i	−0.943783	+	0.330567i	$0.892760\pi$
$618$	0	0
$619$	4.18335i	0.168143i	0.996460	+	0.0840714i	$0.0267924\pi$
$619$	4.18335i	0.168143i	−0.996460	+	0.0840714i	$0.973208\pi$
$620$	−6.00000	−0.240966
$621$	0	0
$622$	12.0000i	0.481156i
$623$	2.05551	0.0823524
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 32.4222i	− 1.29585i
$627$	0	0
$628$	8.42221	0.336083
$629$	13.5778i	0.541382i
$630$	0	0
$631$	− 11.2111i	− 0.446307i	−0.974783	−	0.223153i	$-0.928365\pi$
$631$	− 11.2111i	− 0.446307i	0.974783	−	0.223153i	$-0.0716351\pi$
$632$	14.4222i	0.573685i
$633$	0	0
$634$	18.0000	0.714871
$635$	13.2111i	0.524267i
$636$	0	0
$637$	−0.761141	−0.0301575
$638$	0	0
$639$	0	0
$640$	1.00000	0.0395285
$641$	28.4222	1.12261	0.561305	−	0.827609i	$-0.310300\pi$
$641$	28.4222	1.12261	0.561305	+	0.827609i	$0.310300\pi$
$642$	0	0
$643$	− 33.6333i	− 1.32637i	−0.748456	−	0.663184i	$-0.769205\pi$
$643$	− 33.6333i	− 1.32637i	0.748456	−	0.663184i	$-0.230795\pi$
$644$	− 22.4222i	− 0.883559i
$645$	0	0
$646$	6.78890	0.267106
$647$	27.3944	1.07699	0.538493	−	0.842630i	$-0.318994\pi$
$647$	27.3944	1.07699	0.538493	+	0.842630i	$0.318994\pi$
$648$	0	0
$649$	0	0
$650$	− 3.60555i	− 0.141421i
$651$	0	0
$652$	4.42221i	0.173187i
$653$	−24.7889	−0.970065	−0.485032	−	0.874496i	$-0.661192\pi$
$653$	−24.7889	−0.970065	−0.485032	+	0.874496i	$0.661192\pi$
$654$	0	0
$655$	15.3944i	0.601511i
$656$	11.2111i	0.437720i
$657$	0	0
$658$	− 13.5778i	− 0.529318i
$659$	−24.2389	−0.944212	−0.472106	−	0.881542i	$-0.656506\pi$
$659$	−24.2389	−0.944212	−0.472106	+	0.881542i	$0.656506\pi$
$660$	0	0
$661$	− 0.238859i	− 0.00929054i	−0.999989	−	0.00464527i	$-0.998521\pi$
$661$	− 0.238859i	− 0.00929054i	0.999989	−	0.00464527i	$-0.00147864\pi$
$662$	−9.39445	−0.365126
$663$	0	0
$664$	17.2111	0.667920
$665$	6.78890i	0.263262i
$666$	0	0
$667$	22.4222	0.868191
$668$	− 5.21110i	− 0.201624i
$669$	0	0
$670$	− 11.2111i	− 0.433123i
$671$	0	0
$672$	0	0
$673$	5.63331	0.217148	0.108574	−	0.994088i	$-0.465372\pi$
$673$	5.63331	0.217148	0.108574	+	0.994088i	$0.465372\pi$
$674$	29.6333i	1.14143i
$675$	0	0
$676$	−13.0000	−0.500000
$677$	−44.0555	−1.69319	−0.846595	−	0.532237i	$-0.821352\pi$
$677$	−44.0555	−1.69319	−0.846595	+	0.532237i	$0.821352\pi$
$678$	0	0
$679$	−22.4222	−0.860485
$680$	2.60555	0.0999183
$681$	0	0
$682$	0	0
$683$	5.21110i	0.199397i	0.995018	+	0.0996986i	$0.0317879\pi$
$683$	5.21110i	0.199397i	−0.995018	+	0.0996986i	$0.968212\pi$
$684$	0	0
$685$	11.2111	0.428354
$686$	18.7889	0.717363
$687$	0	0
$688$	8.00000	0.304997
$689$	21.6333	0.824163
$690$	0	0
$691$	− 30.2389i	− 1.15034i	−0.818034	−	0.575170i	$-0.804936\pi$
$691$	− 30.2389i	− 1.15034i	0.818034	−	0.575170i	$-0.195064\pi$
$692$	16.4222	0.624279
$693$	0	0
$694$	15.6333i	0.593432i
$695$	− 2.78890i	− 0.105789i
$696$	0	0
$697$	29.2111i	1.10645i
$698$	13.8167	0.522968
$699$	0	0
$700$	2.60555i	0.0984806i
$701$	−10.9722	−0.414416	−0.207208	−	0.978297i	$-0.566438\pi$
$701$	−10.9722	−0.414416	−0.207208	+	0.978297i	$0.566438\pi$
$702$	0	0
$703$	−13.5778	−0.512096
$704$	0	0
$705$	0	0
$706$	23.2111	0.873561
$707$	38.0555i	1.43122i
$708$	0	0
$709$	− 8.60555i	− 0.323188i	−0.986857	−	0.161594i	$-0.948336\pi$
$709$	− 8.60555i	− 0.323188i	0.986857	−	0.161594i	$-0.0516635\pi$
$710$	− 5.21110i	− 0.195569i
$711$	0	0
$712$	0.788897	0.0295652
$713$	51.6333i	1.93368i
$714$	0	0
$715$	0	0
$716$	1.81665	0.0678915
$717$	0	0
$718$	27.6333	1.03127
$719$	8.36669	0.312025	0.156012	−	0.987755i	$-0.450136\pi$
$719$	8.36669	0.312025	0.156012	+	0.987755i	$0.450136\pi$
$720$	0	0
$721$	− 10.4222i	− 0.388143i
$722$	− 12.2111i	− 0.454450i
$723$	0	0
$724$	20.4222	0.758985
$725$	−2.60555	−0.0967677
$726$	0	0
$727$	14.4222	0.534890	0.267445	−	0.963573i	$-0.413821\pi$
$727$	14.4222	0.534890	0.267445	+	0.963573i	$0.413821\pi$
$728$	9.39445	0.348181
$729$	0	0
$730$	8.60555i	0.318506i
$731$	20.8444	0.770958
$732$	0	0
$733$	38.0555i	1.40561i	0.711381	+	0.702806i	$0.248070\pi$
$733$	38.0555i	1.40561i	−0.711381	+	0.702806i	$0.751930\pi$
$734$	− 23.6333i	− 0.872321i
$735$	0	0
$736$	− 8.60555i	− 0.317205i
$737$	0	0
$738$	0	0
$739$	− 30.2389i	− 1.11235i	−0.831064	−	0.556177i	$-0.812268\pi$
$739$	− 30.2389i	− 1.11235i	0.831064	−	0.556177i	$-0.187732\pi$
$740$	−5.21110	−0.191564
$741$	0	0
$742$	−15.6333	−0.573917
$743$	20.8444i	0.764707i	0.924016	+	0.382354i	$0.124886\pi$
$743$	20.8444i	0.764707i	−0.924016	+	0.382354i	$0.875114\pi$
$744$	0	0
$745$	−0.788897	−0.0289030
$746$	− 8.42221i	− 0.308359i
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	18.4222	0.672236	0.336118	−	0.941820i	$-0.390886\pi$
$751$	18.4222	0.672236	0.336118	+	0.941820i	$0.390886\pi$
$752$	− 5.21110i	− 0.190029i
$753$	0	0
$754$	9.39445i	0.342126i
$755$	6.00000	0.218362
$756$	0	0
$757$	27.2111	0.989004	0.494502	−	0.869176i	$-0.335350\pi$
$757$	27.2111	0.989004	0.494502	+	0.869176i	$0.335350\pi$
$758$	−1.02776	−0.0373298
$759$	0	0
$760$	2.60555i	0.0945133i
$761$	9.63331i	0.349207i	0.984639	+	0.174604i	$0.0558644\pi$
$761$	9.63331i	0.349207i	−0.984639	+	0.174604i	$0.944136\pi$
$762$	0	0
$763$	−22.4222	−0.811738
$764$	−12.0000	−0.434145
$765$	0	0
$766$	−15.6333	−0.564854
$767$	18.7889i	0.678428i
$768$	0	0
$769$	44.8444i	1.61713i	0.588406	+	0.808565i	$0.299756\pi$
$769$	44.8444i	1.61713i	−0.588406	+	0.808565i	$0.700244\pi$
$770$	0	0
$771$	0	0
$772$	13.8167i	0.497272i
$773$	− 30.0000i	− 1.07903i	−0.841978	−	0.539513i	$-0.818609\pi$
$773$	− 30.0000i	− 1.07903i	0.841978	−	0.539513i	$-0.181391\pi$
$774$	0	0
$775$	− 6.00000i	− 0.215526i
$776$	−8.60555	−0.308921
$777$	0	0
$778$	− 2.60555i	− 0.0934135i
$779$	−29.2111	−1.04660
$780$	0	0
$781$	0	0
$782$	− 22.4222i	− 0.801816i
$783$	0	0
$784$	0.211103	0.00753938
$785$	8.42221i	0.300601i
$786$	0	0
$787$	− 37.2666i	− 1.32841i	−0.747550	−	0.664206i	$-0.768770\pi$
$787$	− 37.2666i	− 1.32841i	0.747550	−	0.664206i	$-0.231230\pi$
$788$	− 6.00000i	− 0.213741i
$789$	0	0
$790$	−14.4222	−0.513119
$791$	− 20.3667i	− 0.724156i
$792$	0	0
$793$	−11.5778	−0.411140
$794$	39.6333	1.40653
$795$	0	0
$796$	6.42221	0.227629
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	0	0
$799$	− 13.5778i	− 0.480348i
$800$	1.00000i	0.0353553i
$801$	0	0
$802$	−23.2111	−0.819613
$803$	0	0
$804$	0	0
$805$	22.4222	0.790279
$806$	−21.6333	−0.762001
$807$	0	0
$808$	14.6056i	0.513822i
$809$	−50.8444	−1.78759	−0.893797	−	0.448471i	$-0.851969\pi$
$809$	−50.8444	−1.78759	−0.893797	+	0.448471i	$0.851969\pi$
$810$	0	0
$811$	− 18.2389i	− 0.640453i	−0.947341	−	0.320226i	$-0.896241\pi$
$811$	− 18.2389i	− 0.640453i	0.947341	−	0.320226i	$-0.103759\pi$
$812$	− 6.78890i	− 0.238244i
$813$	0	0
$814$	0	0
$815$	−4.42221	−0.154903
$816$	0	0
$817$	20.8444i	0.729254i
$818$	−29.2111	−1.02134
$819$	0	0
$820$	−11.2111	−0.391509
$821$	− 11.2111i	− 0.391270i	−0.980677	−	0.195635i	$-0.937323\pi$
$821$	− 11.2111i	− 0.391270i	0.980677	−	0.195635i	$-0.0626768\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	− 4.00000i	− 0.139347i
$825$	0	0
$826$	− 13.5778i	− 0.472432i
$827$	− 15.6333i	− 0.543623i	−0.962350	−	0.271812i	$-0.912377\pi$
$827$	− 15.6333i	− 0.543623i	0.962350	−	0.271812i	$-0.0876228\pi$
$828$	0	0
$829$	−10.8444	−0.376642	−0.188321	−	0.982108i	$-0.560305\pi$
$829$	−10.8444	−0.376642	−0.188321	+	0.982108i	$0.560305\pi$
$830$	17.2111i	0.597406i
$831$	0	0
$832$	3.60555	0.125000
$833$	0.550039	0.0190577
$834$	0	0
$835$	5.21110	0.180338
$836$	0	0
$837$	0	0
$838$	25.8167i	0.891822i
$839$	− 10.4222i	− 0.359814i	−0.983684	−	0.179907i	$-0.942420\pi$
$839$	− 10.4222i	− 0.359814i	0.983684	−	0.179907i	$-0.0575798\pi$
$840$	0	0
$841$	−22.2111	−0.765900
$842$	1.81665	0.0626060
$843$	0	0
$844$	2.78890	0.0959978
$845$	− 13.0000i	− 0.447214i
$846$	0	0
$847$	28.6611i	0.984806i
$848$	−6.00000	−0.206041
$849$	0	0
$850$	2.60555i	0.0893697i
$851$	44.8444i	1.53725i
$852$	0	0
$853$	− 29.2111i	− 1.00017i	−0.865977	−	0.500085i	$-0.833302\pi$
$853$	− 29.2111i	− 1.00017i	0.865977	−	0.500085i	$-0.166698\pi$
$854$	8.36669	0.286302
$855$	0	0
$856$	0	0
$857$	−13.0278	−0.445020	−0.222510	−	0.974930i	$-0.571425\pi$
$857$	−13.0278	−0.445020	−0.222510	+	0.974930i	$0.571425\pi$
$858$	0	0
$859$	10.7889	0.368112	0.184056	−	0.982916i	$-0.441077\pi$
$859$	10.7889	0.368112	0.184056	+	0.982916i	$0.441077\pi$
$860$	8.00000i	0.272798i
$861$	0	0
$862$	−12.0000	−0.408722
$863$	− 8.36669i	− 0.284806i	−0.989809	−	0.142403i	$-0.954517\pi$
$863$	− 8.36669i	− 0.284806i	0.989809	−	0.142403i	$-0.0454828\pi$
$864$	0	0
$865$	16.4222i	0.558372i
$866$	− 4.78890i	− 0.162733i
$867$	0	0
$868$	15.6333	0.530629
$869$	0	0
$870$	0	0
$871$	− 40.4222i	− 1.36965i
$872$	−8.60555	−0.291421
$873$	0	0
$874$	22.4222	0.758442
$875$	−2.60555	−0.0880837
$876$	0	0
$877$	32.8444i	1.10908i	0.832158	+	0.554538i	$0.187105\pi$
$877$	32.8444i	1.10908i	−0.832158	+	0.554538i	$0.812895\pi$
$878$	− 8.00000i	− 0.269987i
$879$	0	0
$880$	0	0
$881$	−24.7889	−0.835159	−0.417580	−	0.908640i	$-0.637122\pi$
$881$	−24.7889	−0.835159	−0.417580	+	0.908640i	$0.637122\pi$
$882$	0	0
$883$	38.4222	1.29301	0.646505	−	0.762910i	$-0.276230\pi$
$883$	38.4222	1.29301	0.646505	+	0.762910i	$0.276230\pi$
$884$	9.39445	0.315970
$885$	0	0
$886$	− 27.6333i	− 0.928359i
$887$	−43.0278	−1.44473	−0.722365	−	0.691512i	$-0.756945\pi$
$887$	−43.0278	−1.44473	−0.722365	+	0.691512i	$0.756945\pi$
$888$	0	0
$889$	− 34.4222i	− 1.15448i
$890$	0.788897i	0.0264439i
$891$	0	0
$892$	19.8167i	0.663511i
$893$	13.5778	0.454364
$894$	0	0
$895$	1.81665i	0.0607240i
$896$	−2.60555	−0.0870454
$897$	0	0
$898$	9.63331	0.321468
$899$	15.6333i	0.521400i
$900$	0	0
$901$	−15.6333	−0.520821
$902$	0	0
$903$	0	0
$904$	− 7.81665i	− 0.259978i
$905$	20.4222i	0.678857i
$906$	0	0
$907$	50.4222	1.67424	0.837121	−	0.547018i	$-0.184237\pi$
$907$	50.4222	1.67424	0.837121	+	0.547018i	$0.184237\pi$
$908$	24.0000i	0.796468i
$909$	0	0
$910$	9.39445i	0.311423i
$911$	−15.6333	−0.517955	−0.258977	−	0.965883i	$-0.583385\pi$
$911$	−15.6333	−0.517955	−0.258977	+	0.965883i	$0.583385\pi$
$912$	0	0
$913$	0	0
$914$	−12.2389	−0.404825
$915$	0	0
$916$	− 1.81665i	− 0.0600239i
$917$	− 40.1110i	− 1.32458i
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	8.60555	0.283717
$921$	0	0
$922$	−9.63331	−0.317256
$923$	− 18.7889i	− 0.618444i
$924$	0	0
$925$	− 5.21110i	− 0.171340i
$926$	−38.6056	−1.26866
$927$	0	0
$928$	− 2.60555i	− 0.0855314i
$929$	− 24.7889i	− 0.813297i	−0.913585	−	0.406649i	$-0.866697\pi$
$929$	− 24.7889i	− 0.813297i	0.913585	−	0.406649i	$-0.133303\pi$
$930$	0	0
$931$	0.550039i	0.0180268i
$932$	19.8167	0.649116
$933$	0	0
$934$	− 1.57779i	− 0.0516270i
$935$	0	0
$936$	0	0
$937$	−53.6333	−1.75212	−0.876062	−	0.482199i	$-0.839838\pi$
$937$	−53.6333	−1.75212	−0.876062	+	0.482199i	$0.839838\pi$
$938$	29.2111i	0.953776i
$939$	0	0
$940$	5.21110	0.169967
$941$	54.0000i	1.76035i	0.474650	+	0.880175i	$0.342575\pi$
$941$	54.0000i	1.76035i	−0.474650	+	0.880175i	$0.657425\pi$
$942$	0	0
$943$	96.4777i	3.14175i
$944$	− 5.21110i	− 0.169607i
$945$	0	0
$946$	0	0
$947$	− 27.6333i	− 0.897962i	−0.893541	−	0.448981i	$-0.851787\pi$
$947$	− 27.6333i	− 0.897962i	0.893541	−	0.448981i	$-0.148213\pi$
$948$	0	0
$949$	31.0278i	1.00720i
$950$	−2.60555	−0.0845352
$951$	0	0
$952$	−6.78890	−0.220029
$953$	−30.2389	−0.979533	−0.489766	−	0.871854i	$-0.662918\pi$
$953$	−30.2389	−0.979533	−0.489766	+	0.871854i	$0.662918\pi$
$954$	0	0
$955$	− 12.0000i	− 0.388311i
$956$	0	0
$957$	0	0
$958$	−34.4222	−1.11213
$959$	−29.2111	−0.943276
$960$	0	0
$961$	−5.00000	−0.161290
$962$	−18.7889	−0.605778
$963$	0	0
$964$	− 22.4222i	− 0.722171i
$965$	−13.8167	−0.444774
$966$	0	0
$967$	42.2389i	1.35831i	0.733995	+	0.679155i	$0.237654\pi$
$967$	42.2389i	1.35831i	−0.733995	+	0.679155i	$0.762346\pi$
$968$	11.0000i	0.353553i
$969$	0	0
$970$	− 8.60555i	− 0.276308i
$971$	16.9722	0.544665	0.272333	−	0.962203i	$-0.412205\pi$
$971$	16.9722	0.544665	0.272333	+	0.962203i	$0.412205\pi$
$972$	0	0
$973$	7.26662i	0.232957i
$974$	−37.0278	−1.18645
$975$	0	0
$976$	3.21110	0.102785
$977$	38.8444i	1.24274i	0.783516	+	0.621371i	$0.213424\pi$
$977$	38.8444i	1.24274i	−0.783516	+	0.621371i	$0.786576\pi$
$978$	0	0
$979$	0	0
$980$	0.211103i	0.00674342i
$981$	0	0
$982$	− 13.8167i	− 0.440907i
$983$	13.5778i	0.433064i	0.976275	+	0.216532i	$0.0694746\pi$
$983$	13.5778i	0.433064i	−0.976275	+	0.216532i	$0.930525\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	− 6.78890i	− 0.216203i
$987$	0	0
$988$	9.39445i	0.298877i
$989$	68.8444	2.18912
$990$	0	0
$991$	6.42221	0.204008	0.102004	−	0.994784i	$-0.467475\pi$
$991$	6.42221	0.204008	0.102004	+	0.994784i	$0.467475\pi$
$992$	6.00000	0.190500
$993$	0	0
$994$	13.5778i	0.430662i
$995$	6.42221i	0.203598i
$996$	0	0
$997$	12.4222	0.393415	0.196708	−	0.980462i	$-0.436975\pi$
$997$	12.4222	0.393415	0.196708	+	0.980462i	$0.436975\pi$
$998$	−13.0278	−0.412386
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1170.2.b.d.181.2		4
3.2	odd	2		390.2.b.c.181.4	yes	4
12.11	even	2		3120.2.g.q.961.3		4
13.12	even	2	inner	1170.2.b.d.181.3		4
15.2	even	4		1950.2.f.m.649.3		4
15.8	even	4		1950.2.f.n.649.2		4
15.14	odd	2		1950.2.b.k.1351.1		4
39.5	even	4		5070.2.a.bf.1.1		2
39.8	even	4		5070.2.a.z.1.2		2
39.38	odd	2		390.2.b.c.181.1	✓	4
156.155	even	2		3120.2.g.q.961.2		4
195.38	even	4		1950.2.f.m.649.1		4
195.77	even	4		1950.2.f.n.649.4		4
195.194	odd	2		1950.2.b.k.1351.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.b.c.181.1	✓	4	39.38	odd	2
390.2.b.c.181.4	yes	4	3.2	odd	2
1170.2.b.d.181.2		4	1.1	even	1	trivial
1170.2.b.d.181.3		4	13.12	even	2	inner
1950.2.b.k.1351.1		4	15.14	odd	2
1950.2.b.k.1351.4		4	195.194	odd	2
1950.2.f.m.649.1		4	195.38	even	4
1950.2.f.m.649.3		4	15.2	even	4
1950.2.f.n.649.2		4	15.8	even	4
1950.2.f.n.649.4		4	195.77	even	4
3120.2.g.q.961.2		4	156.155	even	2
3120.2.g.q.961.3		4	12.11	even	2
5070.2.a.z.1.2		2	39.8	even	4
5070.2.a.bf.1.1		2	39.5	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:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} +2.60555i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} +2.60555i q^{7} +1.00000i q^{8} -1.00000 q^{10} -3.60555 q^{13} +2.60555 q^{14} +1.00000 q^{16} +2.60555 q^{17} +2.60555i q^{19} +1.00000i q^{20} +8.60555 q^{23} -1.00000 q^{25} +3.60555i q^{26} -2.60555i q^{28} +2.60555 q^{29} +6.00000i q^{31} -1.00000i q^{32} -2.60555i q^{34} +2.60555 q^{35} +5.21110i q^{37} +2.60555 q^{38} +1.00000 q^{40} +11.2111i q^{41} +8.00000 q^{43} -8.60555i q^{46} -5.21110i q^{47} +0.211103 q^{49} +1.00000i q^{50} +3.60555 q^{52} -6.00000 q^{53} -2.60555 q^{56} -2.60555i q^{58} -5.21110i q^{59} +3.21110 q^{61} +6.00000 q^{62} -1.00000 q^{64} +3.60555i q^{65} +11.2111i q^{67} -2.60555 q^{68} -2.60555i q^{70} +5.21110i q^{71} -8.60555i q^{73} +5.21110 q^{74} -2.60555i q^{76} +14.4222 q^{79} -1.00000i q^{80} +11.2111 q^{82} -17.2111i q^{83} -2.60555i q^{85} -8.00000i q^{86} -0.788897i q^{89} -9.39445i q^{91} -8.60555 q^{92} -5.21110 q^{94} +2.60555 q^{95} +8.60555i q^{97} -0.211103i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} - 4 q^{10} - 4 q^{14} + 4 q^{16} - 4 q^{17} + 20 q^{23} - 4 q^{25} - 4 q^{29} - 4 q^{35} - 4 q^{38} + 4 q^{40} + 32 q^{43} - 28 q^{49} - 24 q^{53} + 4 q^{56} - 16 q^{61} + 24 q^{62} - 4 q^{64} + 4 q^{68} - 8 q^{74} + 16 q^{82} - 20 q^{92} + 8 q^{94} - 4 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^10 - 4 * q^14 + 4 * q^16 - 4 * q^17 + 20 * q^23 - 4 * q^25 - 4 * q^29 - 4 * q^35 - 4 * q^38 + 4 * q^40 + 32 * q^43 - 28 * q^49 - 24 * q^53 + 4 * q^56 - 16 * q^61 + 24 * q^62 - 4 * q^64 + 4 * q^68 - 8 * q^74 + 16 * q^82 - 20 * q^92 + 8 * q^94 - 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more