LMFDB - Embedded newform 7200.2.d.r.2449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7200,2,Mod(2449,7200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("7200.2449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7200.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$57.4922894553$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.399424.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 $ x^6 - 2x^5 + 3x^4 - 6x^3 + 6x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2449.2
Root			$1.40680 + 0.144584i$ of defining polynomial
Character	$\chi$	$=$	7200.2449
Dual form			7200.2.d.r.2449.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.62721i q^{7} +O(q^{10})$	$q-3.62721i q^{7} +6.20555i q^{11} +0.578337 q^{13} +1.42166i q^{17} -5.62721i q^{19} -5.62721i q^{23} +2.00000i q^{29} +2.57834 q^{31} -7.83276 q^{37} -5.25443 q^{41} -7.25443 q^{43} -6.78389i q^{47} -6.15667 q^{49} -2.00000 q^{53} +2.20555i q^{59} +12.4111i q^{61} -4.00000 q^{67} +8.41110 q^{71} -6.00000i q^{73} +22.5089 q^{77} +5.42166 q^{79} -3.25443 q^{83} -13.2544 q^{89} -2.09775i q^{91} -4.84333i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q + 12 q^{31} + 8 q^{37} + 20 q^{41} + 8 q^{43} - 30 q^{49} - 12 q^{53} - 24 q^{67} - 8 q^{71} + 32 q^{77} + 36 q^{79} + 32 q^{83} - 28 q^{89}+O(q^{100}) $ 6 * q + 12 * q^31 + 8 * q^37 + 20 * q^41 + 8 * q^43 - 30 * q^49 - 12 * q^53 - 24 * q^67 - 8 * q^71 + 32 * q^77 + 36 * q^79 + 32 * q^83 - 28 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$6401$	$6751$
$\chi(n)$	$-1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.62721i	− 1.37096i	−0.728093	−	0.685479i	$-0.759593\pi$
$7$	− 3.62721i	− 1.37096i	0.728093	−	0.685479i	$-0.240407\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.20555i	1.87104i	0.353269	+	0.935522i	$0.385070\pi$
$11$	6.20555i	1.87104i	−0.353269	+	0.935522i	$0.614930\pi$
$12$	0	0
$13$	0.578337	0.160402	0.0802009	−	0.996779i	$-0.474444\pi$
$13$	0.578337	0.160402	0.0802009	+	0.996779i	$0.474444\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.42166i	0.344804i	0.985027	+	0.172402i	$0.0551528\pi$
$17$	1.42166i	0.344804i	−0.985027	+	0.172402i	$0.944847\pi$
$18$	0	0
$19$	− 5.62721i	− 1.29097i	−0.763772	−	0.645486i	$-0.776655\pi$
$19$	− 5.62721i	− 1.29097i	0.763772	−	0.645486i	$-0.223345\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 5.62721i	− 1.17336i	−0.809821	−	0.586678i	$-0.800436\pi$
$23$	− 5.62721i	− 1.17336i	0.809821	−	0.586678i	$-0.199564\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.00000i	0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$	2.00000i	0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$	0	0
$31$	2.57834	0.463083	0.231542	−	0.972825i	$-0.425623\pi$
$31$	2.57834	0.463083	0.231542	+	0.972825i	$0.425623\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.83276	−1.28770	−0.643849	−	0.765152i	$-0.722664\pi$
$37$	−7.83276	−1.28770	−0.643849	+	0.765152i	$0.722664\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.25443	−0.820603	−0.410302	−	0.911950i	$-0.634577\pi$
$41$	−5.25443	−0.820603	−0.410302	+	0.911950i	$0.634577\pi$
$42$	0	0
$43$	−7.25443	−1.10629	−0.553145	−	0.833085i	$-0.686572\pi$
$43$	−7.25443	−1.10629	−0.553145	+	0.833085i	$0.686572\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 6.78389i	− 0.989532i	−0.869026	−	0.494766i	$-0.835254\pi$
$47$	− 6.78389i	− 0.989532i	0.869026	−	0.494766i	$-0.164746\pi$
$48$	0	0
$49$	−6.15667	−0.879525
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.20555i	0.287138i	0.989640	+	0.143569i	$0.0458579\pi$
$59$	2.20555i	0.287138i	−0.989640	+	0.143569i	$0.954142\pi$
$60$	0	0
$61$	12.4111i	1.58908i	0.607213	+	0.794539i	$0.292288\pi$
$61$	12.4111i	1.58908i	−0.607213	+	0.794539i	$0.707712\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.41110	0.998214	0.499107	−	0.866540i	$-0.333661\pi$
$71$	8.41110	0.998214	0.499107	+	0.866540i	$0.333661\pi$
$72$	0	0
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	22.5089	2.56512
$78$	0	0
$79$	5.42166	0.609985	0.304992	−	0.952355i	$-0.401346\pi$
$79$	5.42166	0.609985	0.304992	+	0.952355i	$0.401346\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.25443	−0.357220	−0.178610	−	0.983920i	$-0.557160\pi$
$83$	−3.25443	−0.357220	−0.178610	+	0.983920i	$0.557160\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−13.2544	−1.40497	−0.702483	−	0.711700i	$-0.747925\pi$
$89$	−13.2544	−1.40497	−0.702483	+	0.711700i	$0.747925\pi$
$90$	0	0
$91$	− 2.09775i	− 0.219904i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 4.84333i	− 0.491765i	−0.969300	−	0.245883i	$-0.920922\pi$
$97$	− 4.84333i	− 0.491765i	0.969300	−	0.245883i	$-0.0790778\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.00000i	0.199007i	0.995037	+	0.0995037i	$0.0317255\pi$
$101$	2.00000i	0.199007i	−0.995037	+	0.0995037i	$0.968274\pi$
$102$	0	0
$103$	− 2.47054i	− 0.243429i	−0.992565	−	0.121715i	$-0.961161\pi$
$103$	− 2.47054i	− 0.243429i	0.992565	−	0.121715i	$-0.0388393\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.0978	1.36288	0.681441	−	0.731873i	$-0.261354\pi$
$107$	14.0978	1.36288	0.681441	+	0.731873i	$0.261354\pi$
$108$	0	0
$109$	− 7.25443i	− 0.694848i	−0.937708	−	0.347424i	$-0.887056\pi$
$109$	− 7.25443i	− 0.694848i	0.937708	−	0.347424i	$-0.112944\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.08719i	0.854851i	0.904051	+	0.427425i	$0.140579\pi$
$113$	9.08719i	0.854851i	−0.904051	+	0.427425i	$0.859421\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.15667	0.472712
$120$	0	0
$121$	−27.5089	−2.50080
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 10.4705i	− 0.929110i	−0.885544	−	0.464555i	$-0.846214\pi$
$127$	− 10.4705i	− 0.929110i	0.885544	−	0.464555i	$-0.153786\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 13.4600i	− 1.17600i	−0.808860	−	0.588002i	$-0.799915\pi$
$131$	− 13.4600i	− 1.17600i	0.808860	−	0.588002i	$-0.200085\pi$
$132$	0	0
$133$	−20.4111	−1.76987
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 10.5783i	− 0.903768i	−0.892077	−	0.451884i	$-0.850752\pi$
$137$	− 10.5783i	− 0.903768i	0.892077	−	0.451884i	$-0.149248\pi$
$138$	0	0
$139$	− 12.4705i	− 1.05774i	−0.848704	−	0.528869i	$-0.822616\pi$
$139$	− 12.4705i	− 1.05774i	0.848704	−	0.528869i	$-0.177384\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.58890i	0.300119i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.00000i	0.163846i	0.996639	+	0.0819232i	$0.0261062\pi$
$149$	2.00000i	0.163846i	−0.996639	+	0.0819232i	$0.973894\pi$
$150$	0	0
$151$	−12.6761	−1.03157	−0.515783	−	0.856719i	$-0.672499\pi$
$151$	−12.6761	−1.03157	−0.515783	+	0.856719i	$0.672499\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−1.32391	−0.105660	−0.0528298	−	0.998604i	$-0.516824\pi$
$157$	−1.32391	−0.105660	−0.0528298	+	0.998604i	$0.516824\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−20.4111	−1.60862
$162$	0	0
$163$	15.2544	1.19482	0.597409	−	0.801936i	$-0.296197\pi$
$163$	15.2544	1.19482	0.597409	+	0.801936i	$0.296197\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 10.7839i	− 0.834482i	−0.908796	−	0.417241i	$-0.862997\pi$
$167$	− 10.7839i	− 0.834482i	0.908796	−	0.417241i	$-0.137003\pi$
$168$	0	0
$169$	−12.6655	−0.974271
$170$	0	0
$171$	0	0
$172$	0	0
$173$	13.6655	1.03897	0.519485	−	0.854479i	$-0.326124\pi$
$173$	13.6655	1.03897	0.519485	+	0.854479i	$0.326124\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.04888i	0.676345i	0.941084	+	0.338172i	$0.109809\pi$
$179$	9.04888i	0.676345i	−0.941084	+	0.338172i	$0.890191\pi$
$180$	0	0
$181$	− 23.2544i	− 1.72849i	−0.503073	−	0.864244i	$-0.667797\pi$
$181$	− 23.2544i	− 1.72849i	0.503073	−	0.864244i	$-0.332203\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−8.82220	−0.645143
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	25.6655i	1.84745i	0.383062	+	0.923723i	$0.374869\pi$
$193$	25.6655i	1.84745i	−0.383062	+	0.923723i	$0.625131\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.1567	−1.07987	−0.539934	−	0.841707i	$-0.681551\pi$
$197$	−15.1567	−1.07987	−0.539934	+	0.841707i	$0.681551\pi$
$198$	0	0
$199$	−20.6761	−1.46569	−0.732845	−	0.680396i	$-0.761808\pi$
$199$	−20.6761	−1.46569	−0.732845	+	0.680396i	$0.761808\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7.25443	0.509161
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	34.9200	2.41546
$210$	0	0
$211$	2.03831i	0.140323i	0.997536	+	0.0701616i	$0.0223515\pi$
$211$	2.03831i	0.140323i	−0.997536	+	0.0701616i	$0.977648\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 9.35218i	− 0.634867i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.822200i	0.0553072i
$222$	0	0
$223$	7.21611i	0.483227i	0.970373	+	0.241613i	$0.0776766\pi$
$223$	7.21611i	0.483227i	−0.970373	+	0.241613i	$0.922323\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.15667	0.0767712	0.0383856	−	0.999263i	$-0.487778\pi$
$227$	1.15667	0.0767712	0.0383856	+	0.999263i	$0.487778\pi$
$228$	0	0
$229$	− 14.0978i	− 0.931606i	−0.884889	−	0.465803i	$-0.845766\pi$
$229$	− 14.0978i	− 0.931606i	0.884889	−	0.465803i	$-0.154234\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.5783i	0.955059i	0.878616	+	0.477529i	$0.158468\pi$
$233$	14.5783i	0.955059i	−0.878616	+	0.477529i	$0.841532\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−19.2544	−1.24547	−0.622733	−	0.782435i	$-0.713978\pi$
$239$	−19.2544	−1.24547	−0.622733	+	0.782435i	$0.713978\pi$
$240$	0	0
$241$	−13.6655	−0.880274	−0.440137	−	0.897931i	$-0.645070\pi$
$241$	−13.6655	−0.880274	−0.440137	+	0.897931i	$0.645070\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 3.25443i	− 0.207074i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 7.14663i	− 0.451091i	−0.974233	−	0.225546i	$-0.927584\pi$
$251$	− 7.14663i	− 0.451091i	0.974233	−	0.225546i	$-0.0724165\pi$
$252$	0	0
$253$	34.9200	2.19540
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.73501i	0.482497i	0.970463	+	0.241248i	$0.0775568\pi$
$257$	7.73501i	0.482497i	−0.970463	+	0.241248i	$0.922443\pi$
$258$	0	0
$259$	28.4111i	1.76538i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 18.7839i	− 1.15826i	−0.815234	−	0.579132i	$-0.803392\pi$
$263$	− 18.7839i	− 1.15826i	0.815234	−	0.579132i	$-0.196608\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8.50885i	0.518794i	0.965771	+	0.259397i	$0.0835238\pi$
$269$	8.50885i	0.518794i	−0.965771	+	0.259397i	$0.916476\pi$
$270$	0	0
$271$	−30.9894	−1.88247	−0.941237	−	0.337746i	$-0.890335\pi$
$271$	−30.9894	−1.88247	−0.941237	+	0.337746i	$0.890335\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	9.51941	0.571966	0.285983	−	0.958235i	$-0.407680\pi$
$277$	9.51941	0.571966	0.285983	+	0.958235i	$0.407680\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−13.6655	−0.815217	−0.407608	−	0.913157i	$-0.633637\pi$
$281$	−13.6655	−0.815217	−0.407608	+	0.913157i	$0.633637\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	19.0589i	1.12501i
$288$	0	0
$289$	14.9789	0.881110
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−4.31335	−0.251989	−0.125994	−	0.992031i	$-0.540212\pi$
$293$	−4.31335	−0.251989	−0.125994	+	0.992031i	$0.540212\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 3.25443i	− 0.188208i
$300$	0	0
$301$	26.3133i	1.51668i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−25.5678	−1.45923	−0.729615	−	0.683858i	$-0.760301\pi$
$307$	−25.5678	−1.45923	−0.729615	+	0.683858i	$0.760301\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−20.0766	−1.13844	−0.569221	−	0.822185i	$-0.692755\pi$
$311$	−20.0766	−1.13844	−0.569221	+	0.822185i	$0.692755\pi$
$312$	0	0
$313$	7.15667i	0.404519i	0.979332	+	0.202260i	$0.0648285\pi$
$313$	7.15667i	0.404519i	−0.979332	+	0.202260i	$0.935172\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.1744	−1.35777	−0.678884	−	0.734245i	$-0.737536\pi$
$317$	−24.1744	−1.35777	−0.678884	+	0.734245i	$0.737536\pi$
$318$	0	0
$319$	−12.4111	−0.694888
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8.00000	0.445132
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−24.6066	−1.35661
$330$	0	0
$331$	27.1950i	1.49477i	0.664390	+	0.747386i	$0.268691\pi$
$331$	27.1950i	1.49477i	−0.664390	+	0.747386i	$0.731309\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 22.8222i	− 1.24320i	−0.783333	−	0.621602i	$-0.786482\pi$
$337$	− 22.8222i	− 1.24320i	0.783333	−	0.621602i	$-0.213518\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	16.0000i	0.866449i
$342$	0	0
$343$	− 3.05892i	− 0.165166i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−23.6655	−1.27043	−0.635216	−	0.772335i	$-0.719089\pi$
$347$	−23.6655	−1.27043	−0.635216	+	0.772335i	$0.719089\pi$
$348$	0	0
$349$	34.9200i	1.86922i	0.355671	+	0.934611i	$0.384252\pi$
$349$	34.9200i	1.86922i	−0.355671	+	0.934611i	$0.615748\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.9305i	0.847896i	0.905687	+	0.423948i	$0.139356\pi$
$353$	15.9305i	0.847896i	−0.905687	+	0.423948i	$0.860644\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8.41110	−0.443921	−0.221960	−	0.975056i	$-0.571246\pi$
$359$	−8.41110	−0.443921	−0.221960	+	0.975056i	$0.571246\pi$
$360$	0	0
$361$	−12.6655	−0.666607
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 24.4494i	− 1.27625i	−0.769933	−	0.638124i	$-0.779711\pi$
$367$	− 24.4494i	− 1.27625i	0.769933	−	0.638124i	$-0.220289\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7.25443i	0.376631i
$372$	0	0
$373$	0.167237	0.00865920	0.00432960	−	0.999991i	$-0.498622\pi$
$373$	0.167237	0.00865920	0.00432960	+	0.999991i	$0.498622\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.15667i	0.0595718i
$378$	0	0
$379$	7.72496i	0.396805i	0.980121	+	0.198402i	$0.0635753\pi$
$379$	7.72496i	0.396805i	−0.980121	+	0.198402i	$0.936425\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 1.62721i	− 0.0831467i	−0.999135	−	0.0415734i	$-0.986763\pi$
$383$	− 1.62721i	− 0.0831467i	0.999135	−	0.0415734i	$-0.0132370\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	12.3133i	0.624312i	0.950031	+	0.312156i	$0.101051\pi$
$389$	12.3133i	0.624312i	−0.950031	+	0.312156i	$0.898949\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−19.0872	−0.957959	−0.478979	−	0.877826i	$-0.658993\pi$
$397$	−19.0872	−0.957959	−0.478979	+	0.877826i	$0.658993\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.4111	0.719656	0.359828	−	0.933019i	$-0.382835\pi$
$401$	14.4111	0.719656	0.359828	+	0.933019i	$0.382835\pi$
$402$	0	0
$403$	1.49115	0.0742794
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 48.6066i	− 2.40934i
$408$	0	0
$409$	8.31335	0.411069	0.205534	−	0.978650i	$-0.434107\pi$
$409$	8.31335	0.411069	0.205534	+	0.978650i	$0.434107\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	7.36222i	0.359668i	0.983697	+	0.179834i	$0.0575561\pi$
$419$	7.36222i	0.359668i	−0.983697	+	0.179834i	$0.942444\pi$
$420$	0	0
$421$	− 30.0978i	− 1.46687i	−0.679757	−	0.733437i	$-0.737915\pi$
$421$	− 30.0978i	− 1.46687i	0.679757	−	0.733437i	$-0.262085\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	45.0177	2.17856
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.41110	0.405148	0.202574	−	0.979267i	$-0.435069\pi$
$431$	8.41110	0.405148	0.202574	+	0.979267i	$0.435069\pi$
$432$	0	0
$433$	− 4.31335i	− 0.207286i	−0.994615	−	0.103643i	$-0.966950\pi$
$433$	− 4.31335i	− 0.207286i	0.994615	−	0.103643i	$-0.0330500\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−31.6655	−1.51477
$438$	0	0
$439$	9.83276	0.469292	0.234646	−	0.972081i	$-0.424607\pi$
$439$	9.83276	0.469292	0.234646	+	0.972081i	$0.424607\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−21.3522	−1.01447	−0.507236	−	0.861807i	$-0.669333\pi$
$443$	−21.3522	−1.01447	−0.507236	+	0.861807i	$0.669333\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−20.3133	−0.958646	−0.479323	−	0.877639i	$-0.659118\pi$
$449$	−20.3133	−0.958646	−0.479323	+	0.877639i	$0.659118\pi$
$450$	0	0
$451$	− 32.6066i	− 1.53539i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.35218i	0.156808i	0.996922	+	0.0784041i	$0.0249825\pi$
$457$	3.35218i	0.156808i	−0.996922	+	0.0784041i	$0.975018\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 28.5089i	− 1.32779i	−0.747826	−	0.663895i	$-0.768902\pi$
$461$	− 28.5089i	− 1.32779i	0.747826	−	0.663895i	$-0.231098\pi$
$462$	0	0
$463$	− 23.6272i	− 1.09805i	−0.835806	−	0.549025i	$-0.814999\pi$
$463$	− 23.6272i	− 1.09805i	0.835806	−	0.549025i	$-0.185001\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−29.5678	−1.36823	−0.684117	−	0.729373i	$-0.739812\pi$
$467$	−29.5678	−1.36823	−0.684117	+	0.729373i	$0.739812\pi$
$468$	0	0
$469$	14.5089i	0.669957i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 45.0177i	− 2.06992i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	22.0978	1.00967	0.504836	−	0.863215i	$-0.331553\pi$
$479$	22.0978	1.00967	0.504836	+	0.863215i	$0.331553\pi$
$480$	0	0
$481$	−4.52998	−0.206549
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.03831i	0.182993i	0.995805	+	0.0914967i	$0.0291651\pi$
$487$	4.03831i	0.182993i	−0.995805	+	0.0914967i	$0.970835\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	18.2056i	0.821605i	0.911724	+	0.410802i	$0.134751\pi$
$491$	18.2056i	0.821605i	−0.911724	+	0.410802i	$0.865249\pi$
$492$	0	0
$493$	−2.84333	−0.128057
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 30.5089i	− 1.36851i
$498$	0	0
$499$	0.0594386i	0.00266084i	0.999999	+	0.00133042i	$0.000423486\pi$
$499$	0.0594386i	0.00266084i	−0.999999	+	0.00133042i	$0.999577\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2.03831i	0.0908839i	0.998967	+	0.0454419i	$0.0144696\pi$
$503$	2.03831i	0.0908839i	−0.998967	+	0.0454419i	$0.985530\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 40.7044i	− 1.80419i	−0.431539	−	0.902094i	$-0.642029\pi$
$509$	− 40.7044i	− 1.80419i	0.431539	−	0.902094i	$-0.357971\pi$
$510$	0	0
$511$	−21.7633	−0.962751
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	42.0978	1.85146
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	35.3311	1.54492	0.772460	−	0.635064i	$-0.219026\pi$
$523$	35.3311	1.54492	0.772460	+	0.635064i	$0.219026\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.66553i	0.159673i
$528$	0	0
$529$	−8.66553	−0.376762
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3.03883	−0.131626
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 38.2056i	− 1.64563i
$540$	0	0
$541$	3.05892i	0.131513i	0.997836	+	0.0657567i	$0.0209461\pi$
$541$	3.05892i	0.131513i	−0.997836	+	0.0657567i	$0.979054\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	32.0766	1.37150	0.685749	−	0.727838i	$-0.259475\pi$
$547$	32.0766	1.37150	0.685749	+	0.727838i	$0.259475\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	11.2544	0.479455
$552$	0	0
$553$	− 19.6655i	− 0.836263i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	33.6655	1.42645	0.713227	−	0.700933i	$-0.247233\pi$
$557$	33.6655	1.42645	0.713227	+	0.700933i	$0.247233\pi$
$558$	0	0
$559$	−4.19550	−0.177451
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5.35218	0.225567	0.112784	−	0.993620i	$-0.464023\pi$
$563$	5.35218	0.225567	0.112784	+	0.993620i	$0.464023\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.58890	0.234299	0.117149	−	0.993114i	$-0.462624\pi$
$569$	5.58890	0.234299	0.117149	+	0.993114i	$0.462624\pi$
$570$	0	0
$571$	10.3728i	0.434088i	0.976162	+	0.217044i	$0.0696414\pi$
$571$	10.3728i	0.434088i	−0.976162	+	0.217044i	$0.930359\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 21.6655i	− 0.901948i	−0.892537	−	0.450974i	$-0.851077\pi$
$577$	− 21.6655i	− 0.901948i	0.892537	−	0.450974i	$-0.148923\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	11.8045i	0.489733i
$582$	0	0
$583$	− 12.4111i	− 0.514015i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.90225	0.0785142	0.0392571	−	0.999229i	$-0.487501\pi$
$587$	1.90225	0.0785142	0.0392571	+	0.999229i	$0.487501\pi$
$588$	0	0
$589$	− 14.5089i	− 0.597827i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.57834i	0.105880i	0.998598	+	0.0529398i	$0.0168591\pi$
$593$	2.57834i	0.105880i	−0.998598	+	0.0529398i	$0.983141\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−26.7244	−1.09193	−0.545966	−	0.837808i	$-0.683837\pi$
$599$	−26.7244	−1.09193	−0.545966	+	0.837808i	$0.683837\pi$
$600$	0	0
$601$	33.3311	1.35960	0.679801	−	0.733397i	$-0.262066\pi$
$601$	33.3311	1.35960	0.679801	+	0.733397i	$0.262066\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 21.9406i	− 0.890540i	−0.895396	−	0.445270i	$-0.853108\pi$
$607$	− 21.9406i	− 0.890540i	0.895396	−	0.445270i	$-0.146892\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 3.92337i	− 0.158723i
$612$	0	0
$613$	3.42166	0.138200	0.0690998	−	0.997610i	$-0.477987\pi$
$613$	3.42166	0.138200	0.0690998	+	0.997610i	$0.477987\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	19.7350i	0.794502i	0.917710	+	0.397251i	$0.130036\pi$
$617$	19.7350i	0.794502i	−0.917710	+	0.397251i	$0.869964\pi$
$618$	0	0
$619$	− 20.4705i	− 0.822780i	−0.911459	−	0.411390i	$-0.865043\pi$
$619$	− 20.4705i	− 0.822780i	0.911459	−	0.411390i	$-0.134957\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	48.0766i	1.92615i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 11.1355i	− 0.444003i
$630$	0	0
$631$	−1.08719	−0.0432803	−0.0216402	−	0.999766i	$-0.506889\pi$
$631$	−1.08719	−0.0432803	−0.0216402	+	0.999766i	$0.506889\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−3.56063	−0.141077
$638$	0	0
$639$	0	0
$640$	0	0
$641$	27.9789	1.10510	0.552550	−	0.833480i	$-0.313655\pi$
$641$	27.9789	1.10510	0.552550	+	0.833480i	$0.313655\pi$
$642$	0	0
$643$	−4.94108	−0.194857	−0.0974285	−	0.995243i	$-0.531062\pi$
$643$	−4.94108	−0.194857	−0.0974285	+	0.995243i	$0.531062\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	49.3694i	1.94091i	0.241282	+	0.970455i	$0.422432\pi$
$647$	49.3694i	1.94091i	−0.241282	+	0.970455i	$0.577568\pi$
$648$	0	0
$649$	−13.6867	−0.537248
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−40.1744	−1.57214	−0.786072	−	0.618134i	$-0.787889\pi$
$653$	−40.1744	−1.57214	−0.786072	+	0.618134i	$0.787889\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 21.1255i	− 0.822933i	−0.911425	−	0.411466i	$-0.865017\pi$
$659$	− 21.1255i	− 0.822933i	0.911425	−	0.411466i	$-0.134983\pi$
$660$	0	0
$661$	10.9200i	0.424737i	0.977190	+	0.212368i	$0.0681177\pi$
$661$	10.9200i	0.424737i	−0.977190	+	0.212368i	$0.931882\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	11.2544	0.435773
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−77.0177	−2.97324
$672$	0	0
$673$	− 18.0000i	− 0.693849i	−0.937893	−	0.346925i	$-0.887226\pi$
$673$	− 18.0000i	− 0.693849i	0.937893	−	0.346925i	$-0.112774\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	30.4877	1.17174	0.585869	−	0.810406i	$-0.300753\pi$
$677$	30.4877	1.17174	0.585869	+	0.810406i	$0.300753\pi$
$678$	0	0
$679$	−17.5678	−0.674189
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−35.2544	−1.34897	−0.674487	−	0.738287i	$-0.735635\pi$
$683$	−35.2544	−1.34897	−0.674487	+	0.738287i	$0.735635\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.15667	−0.0440658
$690$	0	0
$691$	− 28.1361i	− 1.07035i	−0.844742	−	0.535173i	$-0.820246\pi$
$691$	− 28.1361i	− 1.07035i	0.844742	−	0.535173i	$-0.179754\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 7.47002i	− 0.282947i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 34.8222i	− 1.31522i	−0.753360	−	0.657608i	$-0.771568\pi$
$701$	− 34.8222i	− 1.31522i	0.753360	−	0.657608i	$-0.228432\pi$
$702$	0	0
$703$	44.0766i	1.66238i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.25443	0.272831
$708$	0	0
$709$	− 7.58890i	− 0.285007i	−0.989794	−	0.142504i	$-0.954485\pi$
$709$	− 7.58890i	− 0.285007i	0.989794	−	0.142504i	$-0.0455152\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 14.5089i	− 0.543361i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	3.66553	0.136701	0.0683505	−	0.997661i	$-0.478226\pi$
$719$	3.66553	0.136701	0.0683505	+	0.997661i	$0.478226\pi$
$720$	0	0
$721$	−8.96117	−0.333731
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 36.1149i	− 1.33943i	−0.742619	−	0.669714i	$-0.766416\pi$
$727$	− 36.1149i	− 1.33943i	0.742619	−	0.669714i	$-0.233584\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 10.3133i	− 0.381453i
$732$	0	0
$733$	−34.0071	−1.25608	−0.628041	−	0.778180i	$-0.716143\pi$
$733$	−34.0071	−1.25608	−0.628041	+	0.778180i	$0.716143\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 24.8222i	− 0.914338i
$738$	0	0
$739$	52.0172i	1.91348i	0.290939	+	0.956742i	$0.406032\pi$
$739$	52.0172i	1.91348i	−0.290939	+	0.956742i	$0.593968\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 23.3139i	− 0.855303i	−0.903944	−	0.427651i	$-0.859341\pi$
$743$	− 23.3139i	− 0.855303i	0.903944	−	0.427651i	$-0.140659\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 51.1355i	− 1.86845i
$750$	0	0
$751$	−11.1083	−0.405348	−0.202674	−	0.979246i	$-0.564963\pi$
$751$	−11.1083	−0.405348	−0.202674	+	0.979246i	$0.564963\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	13.3239	0.484266	0.242133	−	0.970243i	$-0.422153\pi$
$757$	13.3239	0.484266	0.242133	+	0.970243i	$0.422153\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	17.1355	0.621163	0.310582	−	0.950547i	$-0.399476\pi$
$761$	17.1355	0.621163	0.310582	+	0.950547i	$0.399476\pi$
$762$	0	0
$763$	−26.3133	−0.952607
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.27555i	0.0460575i
$768$	0	0
$769$	−5.47002	−0.197254	−0.0986270	−	0.995124i	$-0.531445\pi$
$769$	−5.47002	−0.197254	−0.0986270	+	0.995124i	$0.531445\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−3.15667	−0.113538	−0.0567688	−	0.998387i	$-0.518080\pi$
$773$	−3.15667	−0.113538	−0.0567688	+	0.998387i	$0.518080\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	29.5678i	1.05938i
$780$	0	0
$781$	52.1955i	1.86770i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−31.1355	−1.10986	−0.554931	−	0.831896i	$-0.687255\pi$
$787$	−31.1355	−1.10986	−0.554931	+	0.831896i	$0.687255\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	32.9612	1.17196
$792$	0	0
$793$	7.17780i	0.254891i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	10.0000	0.354218	0.177109	−	0.984191i	$-0.443325\pi$
$797$	10.0000	0.354218	0.177109	+	0.984191i	$0.443325\pi$
$798$	0	0
$799$	9.64440	0.341194
$800$	0	0
$801$	0	0
$802$	0	0
$803$	37.2333	1.31393
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	29.0388	1.02095	0.510475	−	0.859892i	$-0.329469\pi$
$809$	29.0388	1.02095	0.510475	+	0.859892i	$0.329469\pi$
$810$	0	0
$811$	− 2.58838i	− 0.0908904i	−0.998967	−	0.0454452i	$-0.985529\pi$
$811$	− 2.58838i	− 0.0908904i	0.998967	−	0.0454452i	$-0.0144706\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	40.8222i	1.42819i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	38.1955i	1.33303i	0.745491	+	0.666516i	$0.232215\pi$
$821$	38.1955i	1.33303i	−0.745491	+	0.666516i	$0.767785\pi$
$822$	0	0
$823$	18.3517i	0.639699i	0.947468	+	0.319849i	$0.103632\pi$
$823$	18.3517i	0.639699i	−0.947468	+	0.319849i	$0.896368\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	0	0
$829$	24.7456i	0.859449i	0.902960	+	0.429725i	$0.141389\pi$
$829$	24.7456i	0.859449i	−0.902960	+	0.429725i	$0.858611\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 8.75272i	− 0.303264i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	53.4288	1.84457	0.922284	−	0.386514i	$-0.126321\pi$
$839$	53.4288	1.84457	0.922284	+	0.386514i	$0.126321\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	99.7805i	3.42850i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	44.0766i	1.51093i
$852$	0	0
$853$	29.0661	0.995203	0.497602	−	0.867406i	$-0.334214\pi$
$853$	29.0661	0.995203	0.497602	+	0.867406i	$0.334214\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 10.2439i	− 0.349924i	−0.984575	−	0.174962i	$-0.944020\pi$
$857$	− 10.2439i	− 0.349924i	0.984575	−	0.174962i	$-0.0559802\pi$
$858$	0	0
$859$	− 10.9794i	− 0.374612i	−0.982302	−	0.187306i	$-0.940024\pi$
$859$	− 10.9794i	− 0.374612i	0.982302	−	0.187306i	$-0.0599756\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	38.8605i	1.32283i	0.750021	+	0.661414i	$0.230043\pi$
$863$	38.8605i	1.32283i	−0.750021	+	0.661414i	$0.769957\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	33.6444i	1.14131i
$870$	0	0
$871$	−2.31335	−0.0783848
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−22.3416	−0.754423	−0.377211	−	0.926127i	$-0.623117\pi$
$877$	−22.3416	−0.754423	−0.377211	+	0.926127i	$0.623117\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−9.88112	−0.332903	−0.166452	−	0.986050i	$-0.553231\pi$
$881$	−9.88112	−0.332903	−0.166452	+	0.986050i	$0.553231\pi$
$882$	0	0
$883$	10.6277	0.357652	0.178826	−	0.983881i	$-0.442770\pi$
$883$	10.6277	0.357652	0.178826	+	0.983881i	$0.442770\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 11.6061i	− 0.389694i	−0.980834	−	0.194847i	$-0.937579\pi$
$887$	− 11.6061i	− 0.389694i	0.980834	−	0.194847i	$-0.0624211\pi$
$888$	0	0
$889$	−37.9789	−1.27377
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−38.1744	−1.27746
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5.15667i	0.171985i
$900$	0	0
$901$	− 2.84333i	− 0.0947249i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−0.195504	−0.00649159	−0.00324580	−	0.999995i	$-0.501033\pi$
$907$	−0.195504	−0.00649159	−0.00324580	+	0.999995i	$0.501033\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−7.88112	−0.261113	−0.130557	−	0.991441i	$-0.541676\pi$
$911$	−7.88112	−0.261113	−0.130557	+	0.991441i	$0.541676\pi$
$912$	0	0
$913$	− 20.1955i	− 0.668374i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−48.8222	−1.61225
$918$	0	0
$919$	9.75614	0.321825	0.160913	−	0.986969i	$-0.448556\pi$
$919$	9.75614	0.321825	0.160913	+	0.986969i	$0.448556\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.86445	0.160115
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	6.82220	0.223829	0.111915	−	0.993718i	$-0.464302\pi$
$929$	6.82220	0.223829	0.111915	+	0.993718i	$0.464302\pi$
$930$	0	0
$931$	34.6449i	1.13544i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 57.5266i	− 1.87931i	−0.342123	−	0.939655i	$-0.611146\pi$
$937$	− 57.5266i	− 1.87931i	0.342123	−	0.939655i	$-0.388854\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0.508852i	0.0165881i	0.999966	+	0.00829405i	$0.00264011\pi$
$941$	0.508852i	0.0165881i	−0.999966	+	0.00829405i	$0.997360\pi$
$942$	0	0
$943$	29.5678i	0.962859i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.68665	0.0548088	0.0274044	−	0.999624i	$-0.491276\pi$
$947$	1.68665	0.0548088	0.0274044	+	0.999624i	$0.491276\pi$
$948$	0	0
$949$	− 3.47002i	− 0.112642i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 9.22616i	− 0.298865i	−0.988772	−	0.149432i	$-0.952255\pi$
$953$	− 9.22616i	− 0.298865i	0.988772	−	0.149432i	$-0.0477446\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−38.3699	−1.23903
$960$	0	0
$961$	−24.3522	−0.785554
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	12.2338i	0.393413i	0.980462	+	0.196707i	$0.0630246\pi$
$967$	12.2338i	0.393413i	−0.980462	+	0.196707i	$0.936975\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	33.2444i	1.06686i	0.845843	+	0.533431i	$0.179098\pi$
$971$	33.2444i	1.06686i	−0.845843	+	0.533431i	$0.820902\pi$
$972$	0	0
$973$	−45.2333	−1.45011
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7.93051i	0.253720i	0.991921	+	0.126860i	$0.0404898\pi$
$977$	7.93051i	0.253720i	−0.991921	+	0.126860i	$0.959510\pi$
$978$	0	0
$979$	− 82.2510i	− 2.62875i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	41.8993i	1.33638i	0.743990	+	0.668191i	$0.232931\pi$
$983$	41.8993i	1.33638i	−0.743990	+	0.668191i	$0.767069\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	40.8222i	1.29807i
$990$	0	0
$991$	35.1849	1.11769	0.558843	−	0.829273i	$-0.311245\pi$
$991$	35.1849	1.11769	0.558843	+	0.829273i	$0.311245\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	8.04836	0.254894	0.127447	−	0.991845i	$-0.459322\pi$
$997$	8.04836	0.254894	0.127447	+	0.991845i	$0.459322\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7200.2.d.r.2449.2		6
3.2	odd	2		2400.2.d.e.49.2		6
4.3	odd	2		1800.2.d.r.1549.4		6
5.2	odd	4		1440.2.k.f.721.6		6
5.3	odd	4		7200.2.k.p.3601.2		6
5.4	even	2		7200.2.d.q.2449.5		6
8.3	odd	2		1800.2.d.q.1549.4		6
8.5	even	2		7200.2.d.q.2449.2		6
12.11	even	2		600.2.d.f.349.3		6
15.2	even	4		480.2.k.b.241.6		6
15.8	even	4		2400.2.k.c.1201.1		6
15.14	odd	2		2400.2.d.f.49.5		6
20.3	even	4		1800.2.k.p.901.6		6
20.7	even	4		360.2.k.f.181.1		6
20.19	odd	2		1800.2.d.q.1549.3		6
24.5	odd	2		2400.2.d.f.49.2		6
24.11	even	2		600.2.d.e.349.3		6
40.3	even	4		1800.2.k.p.901.5		6
40.13	odd	4		7200.2.k.p.3601.1		6
40.19	odd	2		1800.2.d.r.1549.3		6
40.27	even	4		360.2.k.f.181.2		6
40.29	even	2	inner	7200.2.d.r.2449.5		6
40.37	odd	4		1440.2.k.f.721.3		6
60.23	odd	4		600.2.k.c.301.1		6
60.47	odd	4		120.2.k.b.61.6	yes	6
60.59	even	2		600.2.d.e.349.4		6
120.29	odd	2		2400.2.d.e.49.5		6
120.53	even	4		2400.2.k.c.1201.4		6
120.59	even	2		600.2.d.f.349.4		6
120.77	even	4		480.2.k.b.241.3		6
120.83	odd	4		600.2.k.c.301.2		6
120.107	odd	4		120.2.k.b.61.5	✓	6
240.77	even	4		3840.2.a.bo.1.1		3
240.107	odd	4		3840.2.a.bp.1.3		3
240.197	even	4		3840.2.a.br.1.1		3
240.227	odd	4		3840.2.a.bq.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.k.b.61.5	✓	6	120.107	odd	4
120.2.k.b.61.6	yes	6	60.47	odd	4
360.2.k.f.181.1		6	20.7	even	4
360.2.k.f.181.2		6	40.27	even	4
480.2.k.b.241.3		6	120.77	even	4
480.2.k.b.241.6		6	15.2	even	4
600.2.d.e.349.3		6	24.11	even	2
600.2.d.e.349.4		6	60.59	even	2
600.2.d.f.349.3		6	12.11	even	2
600.2.d.f.349.4		6	120.59	even	2
600.2.k.c.301.1		6	60.23	odd	4
600.2.k.c.301.2		6	120.83	odd	4
1440.2.k.f.721.3		6	40.37	odd	4
1440.2.k.f.721.6		6	5.2	odd	4
1800.2.d.q.1549.3		6	20.19	odd	2
1800.2.d.q.1549.4		6	8.3	odd	2
1800.2.d.r.1549.3		6	40.19	odd	2
1800.2.d.r.1549.4		6	4.3	odd	2
1800.2.k.p.901.5		6	40.3	even	4
1800.2.k.p.901.6		6	20.3	even	4
2400.2.d.e.49.2		6	3.2	odd	2
2400.2.d.e.49.5		6	120.29	odd	2
2400.2.d.f.49.2		6	24.5	odd	2
2400.2.d.f.49.5		6	15.14	odd	2
2400.2.k.c.1201.1		6	15.8	even	4
2400.2.k.c.1201.4		6	120.53	even	4
3840.2.a.bo.1.1		3	240.77	even	4
3840.2.a.bp.1.3		3	240.107	odd	4
3840.2.a.bq.1.3		3	240.227	odd	4
3840.2.a.br.1.1		3	240.197	even	4
7200.2.d.q.2449.2		6	8.5	even	2
7200.2.d.q.2449.5		6	5.4	even	2
7200.2.d.r.2449.2		6	1.1	even	1	trivial
7200.2.d.r.2449.5		6	40.29	even	2	inner
7200.2.k.p.3601.1		6	40.13	odd	4
7200.2.k.p.3601.2		6	5.3	odd	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:	\( N \)	\(=\)	\( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7200.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.62721i q^{7} +O(q^{10})\)	\(q-3.62721i q^{7} +6.20555i q^{11} +0.578337 q^{13} +1.42166i q^{17} -5.62721i q^{19} -5.62721i q^{23} +2.00000i q^{29} +2.57834 q^{31} -7.83276 q^{37} -5.25443 q^{41} -7.25443 q^{43} -6.78389i q^{47} -6.15667 q^{49} -2.00000 q^{53} +2.20555i q^{59} +12.4111i q^{61} -4.00000 q^{67} +8.41110 q^{71} -6.00000i q^{73} +22.5089 q^{77} +5.42166 q^{79} -3.25443 q^{83} -13.2544 q^{89} -2.09775i q^{91} -4.84333i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q + 12 q^{31} + 8 q^{37} + 20 q^{41} + 8 q^{43} - 30 q^{49} - 12 q^{53} - 24 q^{67} - 8 q^{71} + 32 q^{77} + 36 q^{79} + 32 q^{83} - 28 q^{89}+O(q^{100}) \) 6 * q + 12 * q^31 + 8 * q^37 + 20 * q^41 + 8 * q^43 - 30 * q^49 - 12 * q^53 - 24 * q^67 - 8 * q^71 + 32 * q^77 + 36 * q^79 + 32 * q^83 - 28 * q^89

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