LMFDB - Embedded newform 5850.2.e.h.5149.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5850,2,Mod(5149,5850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5850, 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("5850.5149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5850.e (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:	$46.7124851824$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5149.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	5850.5149
Dual form			5850.2.e.h.5149.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{7} -1.00000i q^{8} -4.00000 q^{11} +1.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -4.00000i q^{17} +2.00000 q^{19} -4.00000i q^{22} +2.00000i q^{23} -1.00000 q^{26} +2.00000i q^{28} +8.00000 q^{29} +4.00000 q^{31} +1.00000i q^{32} +4.00000 q^{34} +6.00000i q^{37} +2.00000i q^{38} -10.0000 q^{41} -4.00000i q^{43} +4.00000 q^{44} -2.00000 q^{46} +3.00000 q^{49} -1.00000i q^{52} +6.00000i q^{53} -2.00000 q^{56} +8.00000i q^{58} -12.0000 q^{59} -2.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -8.00000i q^{67} +4.00000i q^{68} -6.00000 q^{74} -2.00000 q^{76} +8.00000i q^{77} +8.00000 q^{79} -10.0000i q^{82} -12.0000i q^{83} +4.00000 q^{86} +4.00000i q^{88} -10.0000 q^{89} +2.00000 q^{91} -2.00000i q^{92} -8.00000i q^{97} +3.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} - 8 q^{11} + 4 q^{14} + 2 q^{16} + 4 q^{19} - 2 q^{26} + 16 q^{29} + 8 q^{31} + 8 q^{34} - 20 q^{41} + 8 q^{44} - 4 q^{46} + 6 q^{49} - 4 q^{56} - 24 q^{59} - 4 q^{61} - 2 q^{64} - 12 q^{74} - 4 q^{76} + 16 q^{79} + 8 q^{86} - 20 q^{89} + 4 q^{91}+O(q^{100}) $ 2 * q - 2 * q^4 - 8 * q^11 + 4 * q^14 + 2 * q^16 + 4 * q^19 - 2 * q^26 + 16 * q^29 + 8 * q^31 + 8 * q^34 - 20 * q^41 + 8 * q^44 - 4 * q^46 + 6 * q^49 - 4 * q^56 - 24 * q^59 - 4 * q^61 - 2 * q^64 - 12 * q^74 - 4 * q^76 + 16 * q^79 + 8 * q^86 - 20 * q^89 + 4 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5850\mathbb{Z}\right)^\times$.

$n$	$2251$	$3251$	$3277$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	− 4.00000i	− 0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$	− 4.00000i	− 0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	− 4.00000i	− 0.852803i
$23$	2.00000i	0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$	2.00000i	0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$	0	0
$25$	0	0
$26$	−1.00000	−0.196116
$27$	0	0
$28$	2.00000i	0.377964i
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	4.00000	0.685994
$35$	0	0
$36$	0	0
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	2.00000i	0.324443i
$39$	0	0
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−2.00000	−0.294884
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	− 1.00000i	− 0.138675i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	0	0
$56$	−2.00000	−0.267261
$57$	0	0
$58$	8.00000i	1.05045i
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	4.00000i	0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	4.00000i	0.485071i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	−2.00000	−0.229416
$77$	8.00000i	0.911685i
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	0	0
$82$	− 10.0000i	− 1.10432i
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	4.00000i	0.426401i
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	− 2.00000i	− 0.208514i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 8.00000i	− 0.812277i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	− 8.00000i	− 0.812277i	0.913812	−	0.406138i	$-0.133125\pi$
$98$	3.00000i	0.303046i
$99$	0	0
$100$	0	0
$101$	−20.0000	−1.99007	−0.995037	−	0.0995037i	$-0.968274\pi$
$101$	−20.0000	−1.99007	−0.995037	+	0.0995037i	$0.968274\pi$
$102$	0	0
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−6.00000	−0.582772
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	0	0
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	0	0
$112$	− 2.00000i	− 0.188982i
$113$	− 16.0000i	− 1.50515i	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	− 16.0000i	− 1.50515i	0.658505	−	0.752577i	$-0.271189\pi$
$114$	0	0
$115$	0	0
$116$	−8.00000	−0.742781
$117$	0	0
$118$	− 12.0000i	− 1.10469i
$119$	−8.00000	−0.733359
$120$	0	0
$121$	5.00000	0.454545
$122$	− 2.00000i	− 0.181071i
$123$	0	0
$124$	−4.00000	−0.359211
$125$	0	0
$126$	0	0
$127$	20.0000i	1.77471i	0.461084	+	0.887357i	$0.347461\pi$
$127$	20.0000i	1.77471i	−0.461084	+	0.887357i	$0.652539\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−14.0000	−1.22319	−0.611593	−	0.791173i	$-0.709471\pi$
$131$	−14.0000	−1.22319	−0.611593	+	0.791173i	$0.709471\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	8.00000	0.691095
$135$	0	0
$136$	−4.00000	−0.342997
$137$	2.00000i	0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$	2.00000i	0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 4.00000i	− 0.334497i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	− 6.00000i	− 0.493197i
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	− 2.00000i	− 0.162221i
$153$	0	0
$154$	−8.00000	−0.644658
$155$	0	0
$156$	0	0
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	8.00000i	0.636446i
$159$	0	0
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	− 12.0000i	− 0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$	− 12.0000i	− 0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	12.0000	0.931381
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	4.00000i	0.304997i
$173$	14.0000i	1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$	14.0000i	1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$	0	0
$175$	0	0
$176$	−4.00000	−0.301511
$177$	0	0
$178$	− 10.0000i	− 0.749532i
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	2.00000i	0.148250i
$183$	0	0
$184$	2.00000	0.147442
$185$	0	0
$186$	0	0
$187$	16.0000i	1.17004i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	− 12.0000i	− 0.863779i	−0.901927	−	0.431889i	$-0.857847\pi$
$193$	− 12.0000i	− 0.863779i	0.901927	−	0.431889i	$-0.142153\pi$
$194$	8.00000	0.574367
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 10.0000i	− 0.712470i	−0.934396	−	0.356235i	$-0.884060\pi$
$197$	− 10.0000i	− 0.712470i	0.934396	−	0.356235i	$-0.115940\pi$
$198$	0	0
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	0	0
$201$	0	0
$202$	− 20.0000i	− 1.40720i
$203$	− 16.0000i	− 1.12298i
$204$	0	0
$205$	0	0
$206$	4.00000	0.278693
$207$	0	0
$208$	1.00000i	0.0693375i
$209$	−8.00000	−0.553372
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	− 6.00000i	− 0.412082i
$213$	0	0
$214$	4.00000	0.273434
$215$	0	0
$216$	0	0
$217$	− 8.00000i	− 0.543075i
$218$	− 4.00000i	− 0.270914i
$219$	0	0
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	22.0000i	1.47323i	0.676313	+	0.736614i	$0.263577\pi$
$223$	22.0000i	1.47323i	−0.676313	+	0.736614i	$0.736423\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	16.0000	1.06430
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	−28.0000	−1.85029	−0.925146	−	0.379611i	$-0.876058\pi$
$229$	−28.0000	−1.85029	−0.925146	+	0.379611i	$0.876058\pi$
$230$	0	0
$231$	0	0
$232$	− 8.00000i	− 0.525226i
$233$	28.0000i	1.83434i	0.398495	+	0.917170i	$0.369533\pi$
$233$	28.0000i	1.83434i	−0.398495	+	0.917170i	$0.630467\pi$
$234$	0	0
$235$	0	0
$236$	12.0000	0.781133
$237$	0	0
$238$	− 8.00000i	− 0.518563i
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	5.00000i	0.321412i
$243$	0	0
$244$	2.00000	0.128037
$245$	0	0
$246$	0	0
$247$	2.00000i	0.127257i
$248$	− 4.00000i	− 0.254000i
$249$	0	0
$250$	0	0
$251$	−10.0000	−0.631194	−0.315597	−	0.948893i	$-0.602205\pi$
$251$	−10.0000	−0.631194	−0.315597	+	0.948893i	$0.602205\pi$
$252$	0	0
$253$	− 8.00000i	− 0.502956i
$254$	−20.0000	−1.25491
$255$	0	0
$256$	1.00000	0.0625000
$257$	8.00000i	0.499026i	0.968371	+	0.249513i	$0.0802706\pi$
$257$	8.00000i	0.499026i	−0.968371	+	0.249513i	$0.919729\pi$
$258$	0	0
$259$	12.0000	0.745644
$260$	0	0
$261$	0	0
$262$	− 14.0000i	− 0.864923i
$263$	− 18.0000i	− 1.10993i	−0.831875	−	0.554964i	$-0.812732\pi$
$263$	− 18.0000i	− 1.10993i	0.831875	−	0.554964i	$-0.187268\pi$
$264$	0	0
$265$	0	0
$266$	4.00000	0.245256
$267$	0	0
$268$	8.00000i	0.488678i
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	− 4.00000i	− 0.242536i
$273$	0	0
$274$	−2.00000	−0.120824
$275$	0	0
$276$	0	0
$277$	22.0000i	1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	16.0000i	0.959616i
$279$	0	0
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	− 28.0000i	− 1.66443i	−0.554455	−	0.832214i	$-0.687073\pi$
$283$	− 28.0000i	− 1.66443i	0.554455	−	0.832214i	$-0.312927\pi$
$284$	0	0
$285$	0	0
$286$	4.00000	0.236525
$287$	20.0000i	1.18056i
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	0	0
$295$	0	0
$296$	6.00000	0.348743
$297$	0	0
$298$	− 22.0000i	− 1.27443i
$299$	−2.00000	−0.115663
$300$	0	0
$301$	−8.00000	−0.461112
$302$	− 4.00000i	− 0.230174i
$303$	0	0
$304$	2.00000	0.114708
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	− 8.00000i	− 0.455842i
$309$	0	0
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	0	0
$313$	10.0000i	0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$	10.0000i	0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	−8.00000	−0.450035
$317$	2.00000i	0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$	2.00000i	0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$	0	0
$319$	−32.0000	−1.79166
$320$	0	0
$321$	0	0
$322$	4.00000i	0.222911i
$323$	− 8.00000i	− 0.445132i
$324$	0	0
$325$	0	0
$326$	12.0000	0.664619
$327$	0	0
$328$	10.0000i	0.552158i
$329$	0	0
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	12.0000i	0.658586i
$333$	0	0
$334$	−12.0000	−0.656611
$335$	0	0
$336$	0	0
$337$	− 26.0000i	− 1.41631i	−0.706057	−	0.708155i	$-0.749528\pi$
$337$	− 26.0000i	− 1.41631i	0.706057	−	0.708155i	$-0.250472\pi$
$338$	− 1.00000i	− 0.0543928i
$339$	0	0
$340$	0	0
$341$	−16.0000	−0.866449
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−14.0000	−0.752645
$347$	− 16.0000i	− 0.858925i	−0.903085	−	0.429463i	$-0.858703\pi$
$347$	− 16.0000i	− 0.858925i	0.903085	−	0.429463i	$-0.141297\pi$
$348$	0	0
$349$	28.0000	1.49881	0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000	1.49881	0.749403	+	0.662114i	$0.230341\pi$
$350$	0	0
$351$	0	0
$352$	− 4.00000i	− 0.213201i
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	0	0
$355$	0	0
$356$	10.0000	0.529999
$357$	0	0
$358$	2.00000i	0.105703i
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	− 22.0000i	− 1.15629i
$363$	0	0
$364$	−2.00000	−0.104828
$365$	0	0
$366$	0	0
$367$	− 4.00000i	− 0.208798i	−0.994535	−	0.104399i	$-0.966708\pi$
$367$	− 4.00000i	− 0.208798i	0.994535	−	0.104399i	$-0.0332919\pi$
$368$	2.00000i	0.104257i
$369$	0	0
$370$	0	0
$371$	12.0000	0.623009
$372$	0	0
$373$	− 14.0000i	− 0.724893i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	− 14.0000i	− 0.724893i	0.932005	−	0.362446i	$-0.118058\pi$
$374$	−16.0000	−0.827340
$375$	0	0
$376$	0	0
$377$	8.00000i	0.412021i
$378$	0	0
$379$	14.0000	0.719132	0.359566	−	0.933120i	$-0.382925\pi$
$379$	14.0000	0.719132	0.359566	+	0.933120i	$0.382925\pi$
$380$	0	0
$381$	0	0
$382$	8.00000i	0.409316i
$383$	− 20.0000i	− 1.02195i	−0.859595	−	0.510976i	$-0.829284\pi$
$383$	− 20.0000i	− 1.02195i	0.859595	−	0.510976i	$-0.170716\pi$
$384$	0	0
$385$	0	0
$386$	12.0000	0.610784
$387$	0	0
$388$	8.00000i	0.406138i
$389$	20.0000	1.01404	0.507020	−	0.861934i	$-0.330747\pi$
$389$	20.0000	1.01404	0.507020	+	0.861934i	$0.330747\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	− 3.00000i	− 0.151523i
$393$	0	0
$394$	10.0000	0.503793
$395$	0	0
$396$	0	0
$397$	6.00000i	0.301131i	0.988600	+	0.150566i	$0.0481095\pi$
$397$	6.00000i	0.301131i	−0.988600	+	0.150566i	$0.951890\pi$
$398$	− 24.0000i	− 1.20301i
$399$	0	0
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	4.00000i	0.199254i
$404$	20.0000	0.995037
$405$	0	0
$406$	16.0000	0.794067
$407$	− 24.0000i	− 1.18964i
$408$	0	0
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	0	0
$412$	4.00000i	0.197066i
$413$	24.0000i	1.18096i
$414$	0	0
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	− 8.00000i	− 0.391293i
$419$	14.0000	0.683945	0.341972	−	0.939710i	$-0.388905\pi$
$419$	14.0000	0.683945	0.341972	+	0.939710i	$0.388905\pi$
$420$	0	0
$421$	−16.0000	−0.779792	−0.389896	−	0.920859i	$-0.627489\pi$
$421$	−16.0000	−0.779792	−0.389896	+	0.920859i	$0.627489\pi$
$422$	− 4.00000i	− 0.194717i
$423$	0	0
$424$	6.00000	0.291386
$425$	0	0
$426$	0	0
$427$	4.00000i	0.193574i
$428$	4.00000i	0.193347i
$429$	0	0
$430$	0	0
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	0	0
$433$	26.0000i	1.24948i	0.780833	+	0.624740i	$0.214795\pi$
$433$	26.0000i	1.24948i	−0.780833	+	0.624740i	$0.785205\pi$
$434$	8.00000	0.384012
$435$	0	0
$436$	4.00000	0.191565
$437$	4.00000i	0.191346i
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	0	0
$442$	4.00000i	0.190261i
$443$	− 16.0000i	− 0.760183i	−0.924949	−	0.380091i	$-0.875893\pi$
$443$	− 16.0000i	− 0.760183i	0.924949	−	0.380091i	$-0.124107\pi$
$444$	0	0
$445$	0	0
$446$	−22.0000	−1.04173
$447$	0	0
$448$	2.00000i	0.0944911i
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	40.0000	1.88353
$452$	16.0000i	0.752577i
$453$	0	0
$454$	12.0000	0.563188
$455$	0	0
$456$	0	0
$457$	− 16.0000i	− 0.748448i	−0.927338	−	0.374224i	$-0.877909\pi$
$457$	− 16.0000i	− 0.748448i	0.927338	−	0.374224i	$-0.122091\pi$
$458$	− 28.0000i	− 1.30835i
$459$	0	0
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	− 2.00000i	− 0.0929479i	−0.998920	−	0.0464739i	$-0.985202\pi$
$463$	− 2.00000i	− 0.0929479i	0.998920	−	0.0464739i	$-0.0147984\pi$
$464$	8.00000	0.371391
$465$	0	0
$466$	−28.0000	−1.29707
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	0	0
$472$	12.0000i	0.552345i
$473$	16.0000i	0.735681i
$474$	0	0
$475$	0	0
$476$	8.00000	0.366679
$477$	0	0
$478$	− 8.00000i	− 0.365911i
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	− 22.0000i	− 1.00207i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	0	0
$487$	38.0000i	1.72194i	0.508652	+	0.860972i	$0.330144\pi$
$487$	38.0000i	1.72194i	−0.508652	+	0.860972i	$0.669856\pi$
$488$	2.00000i	0.0905357i
$489$	0	0
$490$	0	0
$491$	−38.0000	−1.71492	−0.857458	−	0.514554i	$-0.827958\pi$
$491$	−38.0000	−1.71492	−0.857458	+	0.514554i	$0.827958\pi$
$492$	0	0
$493$	− 32.0000i	− 1.44121i
$494$	−2.00000	−0.0899843
$495$	0	0
$496$	4.00000	0.179605
$497$	0	0
$498$	0	0
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	0	0
$501$	0	0
$502$	− 10.0000i	− 0.446322i
$503$	− 42.0000i	− 1.87269i	−0.351085	−	0.936344i	$-0.614187\pi$
$503$	− 42.0000i	− 1.87269i	0.351085	−	0.936344i	$-0.385813\pi$
$504$	0	0
$505$	0	0
$506$	8.00000	0.355643
$507$	0	0
$508$	− 20.0000i	− 0.887357i
$509$	38.0000	1.68432	0.842160	−	0.539227i	$-0.181284\pi$
$509$	38.0000	1.68432	0.842160	+	0.539227i	$0.181284\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	−8.00000	−0.352865
$515$	0	0
$516$	0	0
$517$	0	0
$518$	12.0000i	0.527250i
$519$	0	0
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	14.0000	0.611593
$525$	0	0
$526$	18.0000	0.784837
$527$	− 16.0000i	− 0.696971i
$528$	0	0
$529$	19.0000	0.826087
$530$	0	0
$531$	0	0
$532$	4.00000i	0.173422i
$533$	− 10.0000i	− 0.433148i
$534$	0	0
$535$	0	0
$536$	−8.00000	−0.345547
$537$	0	0
$538$	− 4.00000i	− 0.172452i
$539$	−12.0000	−0.516877
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	− 28.0000i	− 1.20270i
$543$	0	0
$544$	4.00000	0.171499
$545$	0	0
$546$	0	0
$547$	− 12.0000i	− 0.513083i	−0.966533	−	0.256541i	$-0.917417\pi$
$547$	− 12.0000i	− 0.513083i	0.966533	−	0.256541i	$-0.0825830\pi$
$548$	− 2.00000i	− 0.0854358i
$549$	0	0
$550$	0	0
$551$	16.0000	0.681623
$552$	0	0
$553$	− 16.0000i	− 0.680389i
$554$	−22.0000	−0.934690
$555$	0	0
$556$	−16.0000	−0.678551
$557$	10.0000i	0.423714i	0.977301	+	0.211857i	$0.0679510\pi$
$557$	10.0000i	0.423714i	−0.977301	+	0.211857i	$0.932049\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	0	0
$562$	10.0000i	0.421825i
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	0	0
$565$	0	0
$566$	28.0000	1.17693
$567$	0	0
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	4.00000i	0.167248i
$573$	0	0
$574$	−20.0000	−0.834784
$575$	0	0
$576$	0	0
$577$	− 4.00000i	− 0.166522i	−0.996528	−	0.0832611i	$-0.973466\pi$
$577$	− 4.00000i	− 0.166522i	0.996528	−	0.0832611i	$-0.0265335\pi$
$578$	1.00000i	0.0415945i
$579$	0	0
$580$	0	0
$581$	−24.0000	−0.995688
$582$	0	0
$583$	− 24.0000i	− 0.993978i
$584$	0	0
$585$	0	0
$586$	6.00000	0.247858
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	0	0
$592$	6.00000i	0.246598i
$593$	34.0000i	1.39621i	0.715994	+	0.698106i	$0.245974\pi$
$593$	34.0000i	1.39621i	−0.715994	+	0.698106i	$0.754026\pi$
$594$	0	0
$595$	0	0
$596$	22.0000	0.901155
$597$	0	0
$598$	− 2.00000i	− 0.0817861i
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−42.0000	−1.71322	−0.856608	−	0.515968i	$-0.827432\pi$
$601$	−42.0000	−1.71322	−0.856608	+	0.515968i	$0.827432\pi$
$602$	− 8.00000i	− 0.326056i
$603$	0	0
$604$	4.00000	0.162758
$605$	0	0
$606$	0	0
$607$	− 12.0000i	− 0.487065i	−0.969893	−	0.243532i	$-0.921694\pi$
$607$	− 12.0000i	− 0.487065i	0.969893	−	0.243532i	$-0.0783062\pi$
$608$	2.00000i	0.0811107i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	2.00000i	0.0807792i	0.999184	+	0.0403896i	$0.0128599\pi$
$613$	2.00000i	0.0807792i	−0.999184	+	0.0403896i	$0.987140\pi$
$614$	0	0
$615$	0	0
$616$	8.00000	0.322329
$617$	− 38.0000i	− 1.52982i	−0.644136	−	0.764911i	$-0.722783\pi$
$617$	− 38.0000i	− 1.52982i	0.644136	−	0.764911i	$-0.277217\pi$
$618$	0	0
$619$	−26.0000	−1.04503	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	0	0
$621$	0	0
$622$	16.0000i	0.641542i
$623$	20.0000i	0.801283i
$624$	0	0
$625$	0	0
$626$	−10.0000	−0.399680
$627$	0	0
$628$	10.0000i	0.399043i
$629$	24.0000	0.956943
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	− 8.00000i	− 0.318223i
$633$	0	0
$634$	−2.00000	−0.0794301
$635$	0	0
$636$	0	0
$637$	3.00000i	0.118864i
$638$	− 32.0000i	− 1.26689i
$639$	0	0
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	40.0000i	1.57745i	0.614749	+	0.788723i	$0.289257\pi$
$643$	40.0000i	1.57745i	−0.614749	+	0.788723i	$0.710743\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	8.00000	0.314756
$647$	18.0000i	0.707653i	0.935311	+	0.353827i	$0.115120\pi$
$647$	18.0000i	0.707653i	−0.935311	+	0.353827i	$0.884880\pi$
$648$	0	0
$649$	48.0000	1.88416
$650$	0	0
$651$	0	0
$652$	12.0000i	0.469956i
$653$	− 38.0000i	− 1.48705i	−0.668705	−	0.743527i	$-0.733151\pi$
$653$	− 38.0000i	− 1.48705i	0.668705	−	0.743527i	$-0.266849\pi$
$654$	0	0
$655$	0	0
$656$	−10.0000	−0.390434
$657$	0	0
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	−16.0000	−0.622328	−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000	−0.622328	−0.311164	+	0.950356i	$0.600719\pi$
$662$	− 10.0000i	− 0.388661i
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	0	0
$667$	16.0000i	0.619522i
$668$	− 12.0000i	− 0.464294i
$669$	0	0
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	46.0000i	1.77317i	0.462566	+	0.886585i	$0.346929\pi$
$673$	46.0000i	1.77317i	−0.462566	+	0.886585i	$0.653071\pi$
$674$	26.0000	1.00148
$675$	0	0
$676$	1.00000	0.0384615
$677$	22.0000i	0.845529i	0.906240	+	0.422764i	$0.138940\pi$
$677$	22.0000i	0.845529i	−0.906240	+	0.422764i	$0.861060\pi$
$678$	0	0
$679$	−16.0000	−0.614024
$680$	0	0
$681$	0	0
$682$	− 16.0000i	− 0.612672i
$683$	36.0000i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$	36.0000i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	0	0
$685$	0	0
$686$	20.0000	0.763604
$687$	0	0
$688$	− 4.00000i	− 0.152499i
$689$	−6.00000	−0.228582
$690$	0	0
$691$	−18.0000	−0.684752	−0.342376	−	0.939563i	$-0.611232\pi$
$691$	−18.0000	−0.684752	−0.342376	+	0.939563i	$0.611232\pi$
$692$	− 14.0000i	− 0.532200i
$693$	0	0
$694$	16.0000	0.607352
$695$	0	0
$696$	0	0
$697$	40.0000i	1.51511i
$698$	28.0000i	1.05982i
$699$	0	0
$700$	0	0
$701$	20.0000	0.755390	0.377695	−	0.925930i	$-0.376717\pi$
$701$	20.0000	0.755390	0.377695	+	0.925930i	$0.376717\pi$
$702$	0	0
$703$	12.0000i	0.452589i
$704$	4.00000	0.150756
$705$	0	0
$706$	−6.00000	−0.225813
$707$	40.0000i	1.50435i
$708$	0	0
$709$	−4.00000	−0.150223	−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000	−0.150223	−0.0751116	+	0.997175i	$0.523931\pi$
$710$	0	0
$711$	0	0
$712$	10.0000i	0.374766i
$713$	8.00000i	0.299602i
$714$	0	0
$715$	0	0
$716$	−2.00000	−0.0747435
$717$	0	0
$718$	24.0000i	0.895672i
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	− 15.0000i	− 0.558242i
$723$	0	0
$724$	22.0000	0.817624
$725$	0	0
$726$	0	0
$727$	8.00000i	0.296704i	0.988935	+	0.148352i	$0.0473968\pi$
$727$	8.00000i	0.296704i	−0.988935	+	0.148352i	$0.952603\pi$
$728$	− 2.00000i	− 0.0741249i
$729$	0	0
$730$	0	0
$731$	−16.0000	−0.591781
$732$	0	0
$733$	− 6.00000i	− 0.221615i	−0.993842	−	0.110808i	$-0.964656\pi$
$733$	− 6.00000i	− 0.221615i	0.993842	−	0.110808i	$-0.0353437\pi$
$734$	4.00000	0.147643
$735$	0	0
$736$	−2.00000	−0.0737210
$737$	32.0000i	1.17874i
$738$	0	0
$739$	−50.0000	−1.83928	−0.919640	−	0.392763i	$-0.871519\pi$
$739$	−50.0000	−1.83928	−0.919640	+	0.392763i	$0.871519\pi$
$740$	0	0
$741$	0	0
$742$	12.0000i	0.440534i
$743$	36.0000i	1.32071i	0.750953	+	0.660356i	$0.229595\pi$
$743$	36.0000i	1.32071i	−0.750953	+	0.660356i	$0.770405\pi$
$744$	0	0
$745$	0	0
$746$	14.0000	0.512576
$747$	0	0
$748$	− 16.0000i	− 0.585018i
$749$	−8.00000	−0.292314
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	0	0
$753$	0	0
$754$	−8.00000	−0.291343
$755$	0	0
$756$	0	0
$757$	− 38.0000i	− 1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$	− 38.0000i	− 1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	14.0000i	0.508503i
$759$	0	0
$760$	0	0
$761$	46.0000	1.66750	0.833749	−	0.552143i	$-0.186190\pi$
$761$	46.0000	1.66750	0.833749	+	0.552143i	$0.186190\pi$
$762$	0	0
$763$	8.00000i	0.289619i
$764$	−8.00000	−0.289430
$765$	0	0
$766$	20.0000	0.722629
$767$	− 12.0000i	− 0.433295i
$768$	0	0
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	0	0
$772$	12.0000i	0.431889i
$773$	26.0000i	0.935155i	0.883952	+	0.467578i	$0.154873\pi$
$773$	26.0000i	0.935155i	−0.883952	+	0.467578i	$0.845127\pi$
$774$	0	0
$775$	0	0
$776$	−8.00000	−0.287183
$777$	0	0
$778$	20.0000i	0.717035i
$779$	−20.0000	−0.716574
$780$	0	0
$781$	0	0
$782$	8.00000i	0.286079i
$783$	0	0
$784$	3.00000	0.107143
$785$	0	0
$786$	0	0
$787$	− 28.0000i	− 0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$	− 28.0000i	− 0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$	10.0000i	0.356235i
$789$	0	0
$790$	0	0
$791$	−32.0000	−1.13779
$792$	0	0
$793$	− 2.00000i	− 0.0710221i
$794$	−6.00000	−0.212932
$795$	0	0
$796$	24.0000	0.850657
$797$	42.0000i	1.48772i	0.668338	+	0.743858i	$0.267006\pi$
$797$	42.0000i	1.48772i	−0.668338	+	0.743858i	$0.732994\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	30.0000i	1.05934i
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−4.00000	−0.140894
$807$	0	0
$808$	20.0000i	0.703598i
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	−34.0000	−1.19390	−0.596951	−	0.802278i	$-0.703621\pi$
$811$	−34.0000	−1.19390	−0.596951	+	0.802278i	$0.703621\pi$
$812$	16.0000i	0.561490i
$813$	0	0
$814$	24.0000	0.841200
$815$	0	0
$816$	0	0
$817$	− 8.00000i	− 0.279885i
$818$	18.0000i	0.629355i
$819$	0	0
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	0	0
$823$	20.0000i	0.697156i	0.937280	+	0.348578i	$0.113335\pi$
$823$	20.0000i	0.697156i	−0.937280	+	0.348578i	$0.886665\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	−24.0000	−0.835067
$827$	44.0000i	1.53003i	0.644013	+	0.765015i	$0.277268\pi$
$827$	44.0000i	1.53003i	−0.644013	+	0.765015i	$0.722732\pi$
$828$	0	0
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	0	0
$831$	0	0
$832$	− 1.00000i	− 0.0346688i
$833$	− 12.0000i	− 0.415775i
$834$	0	0
$835$	0	0
$836$	8.00000	0.276686
$837$	0	0
$838$	14.0000i	0.483622i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	− 16.0000i	− 0.551396i
$843$	0	0
$844$	4.00000	0.137686
$845$	0	0
$846$	0	0
$847$	− 10.0000i	− 0.343604i
$848$	6.00000i	0.206041i
$849$	0	0
$850$	0	0
$851$	−12.0000	−0.411355
$852$	0	0
$853$	10.0000i	0.342393i	0.985237	+	0.171197i	$0.0547634\pi$
$853$	10.0000i	0.342393i	−0.985237	+	0.171197i	$0.945237\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	−4.00000	−0.136717
$857$	16.0000i	0.546550i	0.961936	+	0.273275i	$0.0881068\pi$
$857$	16.0000i	0.546550i	−0.961936	+	0.273275i	$0.911893\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	0	0
$861$	0	0
$862$	− 16.0000i	− 0.544962i
$863$	− 36.0000i	− 1.22545i	−0.790295	−	0.612727i	$-0.790072\pi$
$863$	− 36.0000i	− 1.22545i	0.790295	−	0.612727i	$-0.209928\pi$
$864$	0	0
$865$	0	0
$866$	−26.0000	−0.883516
$867$	0	0
$868$	8.00000i	0.271538i
$869$	−32.0000	−1.08553
$870$	0	0
$871$	8.00000	0.271070
$872$	4.00000i	0.135457i
$873$	0	0
$874$	−4.00000	−0.135302
$875$	0	0
$876$	0	0
$877$	− 22.0000i	− 0.742887i	−0.928456	−	0.371444i	$-0.878863\pi$
$877$	− 22.0000i	− 0.742887i	0.928456	−	0.371444i	$-0.121137\pi$
$878$	− 8.00000i	− 0.269987i
$879$	0	0
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	− 44.0000i	− 1.48072i	−0.672212	−	0.740359i	$-0.734656\pi$
$883$	− 44.0000i	− 1.48072i	0.672212	−	0.740359i	$-0.265344\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	16.0000	0.537531
$887$	6.00000i	0.201460i	0.994914	+	0.100730i	$0.0321179\pi$
$887$	6.00000i	0.201460i	−0.994914	+	0.100730i	$0.967882\pi$
$888$	0	0
$889$	40.0000	1.34156
$890$	0	0
$891$	0	0
$892$	− 22.0000i	− 0.736614i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	− 18.0000i	− 0.600668i
$899$	32.0000	1.06726
$900$	0	0
$901$	24.0000	0.799556
$902$	40.0000i	1.33185i
$903$	0	0
$904$	−16.0000	−0.532152
$905$	0	0
$906$	0	0
$907$	− 44.0000i	− 1.46100i	−0.682915	−	0.730498i	$-0.739288\pi$
$907$	− 44.0000i	− 1.46100i	0.682915	−	0.730498i	$-0.260712\pi$
$908$	12.0000i	0.398234i
$909$	0	0
$910$	0	0
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	0	0
$913$	48.0000i	1.58857i
$914$	16.0000	0.529233
$915$	0	0
$916$	28.0000	0.925146
$917$	28.0000i	0.924641i
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	0	0
$922$	− 30.0000i	− 0.987997i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	2.00000	0.0657241
$927$	0	0
$928$	8.00000i	0.262613i
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	− 28.0000i	− 0.917170i
$933$	0	0
$934$	12.0000	0.392652
$935$	0	0
$936$	0	0
$937$	10.0000i	0.326686i	0.986569	+	0.163343i	$0.0522277\pi$
$937$	10.0000i	0.326686i	−0.986569	+	0.163343i	$0.947772\pi$
$938$	− 16.0000i	− 0.522419i
$939$	0	0
$940$	0	0
$941$	38.0000	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$942$	0	0
$943$	− 20.0000i	− 0.651290i
$944$	−12.0000	−0.390567
$945$	0	0
$946$	−16.0000	−0.520205
$947$	4.00000i	0.129983i	0.997886	+	0.0649913i	$0.0207020\pi$
$947$	4.00000i	0.129983i	−0.997886	+	0.0649913i	$0.979298\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	8.00000i	0.259281i
$953$	− 36.0000i	− 1.16615i	−0.812417	−	0.583077i	$-0.801849\pi$
$953$	− 36.0000i	− 1.16615i	0.812417	−	0.583077i	$-0.198151\pi$
$954$	0	0
$955$	0	0
$956$	8.00000	0.258738
$957$	0	0
$958$	24.0000i	0.775405i
$959$	4.00000	0.129167
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 6.00000i	− 0.193448i
$963$	0	0
$964$	22.0000	0.708572
$965$	0	0
$966$	0	0
$967$	14.0000i	0.450210i	0.974335	+	0.225105i	$0.0722725\pi$
$967$	14.0000i	0.450210i	−0.974335	+	0.225105i	$0.927728\pi$
$968$	− 5.00000i	− 0.160706i
$969$	0	0
$970$	0	0
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	0	0
$973$	− 32.0000i	− 1.02587i
$974$	−38.0000	−1.21760
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	0	0
$979$	40.0000	1.27841
$980$	0	0
$981$	0	0
$982$	− 38.0000i	− 1.21263i
$983$	− 12.0000i	− 0.382741i	−0.981518	−	0.191370i	$-0.938707\pi$
$983$	− 12.0000i	− 0.382741i	0.981518	−	0.191370i	$-0.0612931\pi$
$984$	0	0
$985$	0	0
$986$	32.0000	1.01909
$987$	0	0
$988$	− 2.00000i	− 0.0636285i
$989$	8.00000	0.254385
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	4.00000i	0.127000i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 26.0000i	− 0.823428i	−0.911313	−	0.411714i	$-0.864930\pi$
$997$	− 26.0000i	− 0.823428i	0.911313	−	0.411714i	$-0.135070\pi$
$998$	14.0000i	0.443162i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5850.2.e.h.5149.2		2
3.2	odd	2		1950.2.e.m.1249.1		2
5.2	odd	4		5850.2.a.s.1.1		1
5.3	odd	4		1170.2.a.j.1.1		1
5.4	even	2	inner	5850.2.e.h.5149.1		2
15.2	even	4		1950.2.a.ba.1.1		1
15.8	even	4		390.2.a.b.1.1	✓	1
15.14	odd	2		1950.2.e.m.1249.2		2
20.3	even	4		9360.2.a.v.1.1		1
60.23	odd	4		3120.2.a.y.1.1		1
195.8	odd	4		5070.2.b.f.1351.1		2
195.38	even	4		5070.2.a.n.1.1		1
195.83	odd	4		5070.2.b.f.1351.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.b.1.1	✓	1	15.8	even	4
1170.2.a.j.1.1		1	5.3	odd	4
1950.2.a.ba.1.1		1	15.2	even	4
1950.2.e.m.1249.1		2	3.2	odd	2
1950.2.e.m.1249.2		2	15.14	odd	2
3120.2.a.y.1.1		1	60.23	odd	4
5070.2.a.n.1.1		1	195.38	even	4
5070.2.b.f.1351.1		2	195.8	odd	4
5070.2.b.f.1351.2		2	195.83	odd	4
5850.2.a.s.1.1		1	5.2	odd	4
5850.2.e.h.5149.1		2	5.4	even	2	inner
5850.2.e.h.5149.2		2	1.1	even	1	trivial
9360.2.a.v.1.1		1	20.3	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:	\( N \)	\(=\)	\( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5850.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{7} -1.00000i q^{8} -4.00000 q^{11} +1.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -4.00000i q^{17} +2.00000 q^{19} -4.00000i q^{22} +2.00000i q^{23} -1.00000 q^{26} +2.00000i q^{28} +8.00000 q^{29} +4.00000 q^{31} +1.00000i q^{32} +4.00000 q^{34} +6.00000i q^{37} +2.00000i q^{38} -10.0000 q^{41} -4.00000i q^{43} +4.00000 q^{44} -2.00000 q^{46} +3.00000 q^{49} -1.00000i q^{52} +6.00000i q^{53} -2.00000 q^{56} +8.00000i q^{58} -12.0000 q^{59} -2.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -8.00000i q^{67} +4.00000i q^{68} -6.00000 q^{74} -2.00000 q^{76} +8.00000i q^{77} +8.00000 q^{79} -10.0000i q^{82} -12.0000i q^{83} +4.00000 q^{86} +4.00000i q^{88} -10.0000 q^{89} +2.00000 q^{91} -2.00000i q^{92} -8.00000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} - 8 q^{11} + 4 q^{14} + 2 q^{16} + 4 q^{19} - 2 q^{26} + 16 q^{29} + 8 q^{31} + 8 q^{34} - 20 q^{41} + 8 q^{44} - 4 q^{46} + 6 q^{49} - 4 q^{56} - 24 q^{59} - 4 q^{61} - 2 q^{64} - 12 q^{74} - 4 q^{76} + 16 q^{79} + 8 q^{86} - 20 q^{89} + 4 q^{91}+O(q^{100}) \) 2 * q - 2 * q^4 - 8 * q^11 + 4 * q^14 + 2 * q^16 + 4 * q^19 - 2 * q^26 + 16 * q^29 + 8 * q^31 + 8 * q^34 - 20 * q^41 + 8 * q^44 - 4 * q^46 + 6 * q^49 - 4 * q^56 - 24 * q^59 - 4 * q^61 - 2 * q^64 - 12 * q^74 - 4 * q^76 + 16 * q^79 + 8 * q^86 - 20 * q^89 + 4 * q^91

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