LMFDB - Embedded newform 4725.2.a.cc.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4725,2,Mod(1,4725)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4725, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4725.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4725 = 3^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4725.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,0,6,0,0,5,-3,0,0,-9,0,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$37.7293149551$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.2161212.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 8x^{3} - x^{2} + 14x + 2 $ x^5 - 8x^3 - x^2 + 14x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 945)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.66057$ of defining polynomial
Character	$\chi$	$=$	4725.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.66057 q^{2} +0.757487 q^{4} +1.00000 q^{7} -2.06328 q^{8} -4.48133 q^{11} +1.64522 q^{13} +1.66057 q^{14} -4.94119 q^{16} +3.24698 q^{17} +6.66503 q^{19} -7.44156 q^{22} -8.79528 q^{23} +2.73200 q^{26} +0.757487 q^{28} -9.08678 q^{29} +0.903081 q^{31} -4.07862 q^{32} +5.39183 q^{34} +1.50779 q^{37} +11.0677 q^{38} +4.51466 q^{41} -4.51466 q^{43} -3.39455 q^{44} -14.6052 q^{46} -3.86256 q^{47} +1.00000 q^{49} +1.24623 q^{52} -5.81910 q^{53} -2.06328 q^{56} -15.0892 q^{58} -6.21734 q^{59} +4.54461 q^{61} +1.49963 q^{62} +3.10954 q^{64} -13.5681 q^{67} +2.45954 q^{68} -8.00000 q^{71} +0.170766 q^{73} +2.50378 q^{74} +5.04868 q^{76} -4.48133 q^{77} -7.37353 q^{79} +7.49691 q^{82} +1.56124 q^{83} -7.49691 q^{86} +9.24623 q^{88} -14.9851 q^{89} +1.64522 q^{91} -6.66231 q^{92} -6.41405 q^{94} -13.3399 q^{97} +1.66057 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 6 q^{4} + 5 q^{7} - 3 q^{8} - 9 q^{11} + 7 q^{13} - 4 q^{16} - 14 q^{17} - 3 q^{19} - 5 q^{22} - 2 q^{23} - 21 q^{26} + 6 q^{28} - 12 q^{29} - 6 q^{31} - 6 q^{32} + 7 q^{34} + 20 q^{38} - 7 q^{41}+ \cdots - 4 q^{97}+O(q^{100}) $ 5 * q + 6 * q^4 + 5 * q^7 - 3 * q^8 - 9 * q^11 + 7 * q^13 - 4 * q^16 - 14 * q^17 - 3 * q^19 - 5 * q^22 - 2 * q^23 - 21 * q^26 + 6 * q^28 - 12 * q^29 - 6 * q^31 - 6 * q^32 + 7 * q^34 + 20 * q^38 - 7 * q^41 + 7 * q^43 - 37 * q^44 - 5 * q^46 - 13 * q^47 + 5 * q^49 - q^52 + 5 * q^53 - 3 * q^56 - 34 * q^58 - 26 * q^59 + 2 * q^61 + 19 * q^62 - 21 * q^64 - 21 * q^67 - 29 * q^68 - 40 * q^71 - 9 * q^73 - 38 * q^74 + 2 * q^76 - 9 * q^77 + 8 * q^79 + 9 * q^82 - 17 * q^83 - 9 * q^86 + 39 * q^88 - 11 * q^89 + 7 * q^91 + 43 * q^92 + 17 * q^94 - 4 * q^97

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 $

$2$	1.66057	1.17420	0.587100	−	0.809515i	$-0.300270\pi$
$2$	1.66057	1.17420	0.587100	+	0.809515i	$0.300270\pi$
$3$	0	0
$3$	0	0
$4$	0.757487	0.378743
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.06328	−0.729479
$9$	0	0
$10$	0	0
$11$	−4.48133	−1.35117	−0.675586	−	0.737281i	$-0.736109\pi$
$11$	−4.48133	−1.35117	−0.675586	+	0.737281i	$0.736109\pi$
$12$	0	0
$13$	1.64522	0.456303	0.228151	−	0.973626i	$-0.426732\pi$
$13$	1.64522	0.456303	0.228151	+	0.973626i	$0.426732\pi$
$14$	1.66057	0.443806
$15$	0	0
$16$	−4.94119	−1.23530
$17$	3.24698	0.787508	0.393754	−	0.919216i	$-0.371176\pi$
$17$	3.24698	0.787508	0.393754	+	0.919216i	$0.371176\pi$
$18$	0	0
$19$	6.66503	1.52906	0.764532	−	0.644586i	$-0.222970\pi$
$19$	6.66503	1.52906	0.764532	+	0.644586i	$0.222970\pi$
$20$	0	0
$21$	0	0
$22$	−7.44156	−1.58655
$23$	−8.79528	−1.83394	−0.916972	−	0.398953i	$-0.869374\pi$
$23$	−8.79528	−1.83394	−0.916972	+	0.398953i	$0.869374\pi$
$24$	0	0
$25$	0	0
$26$	2.73200	0.535790
$27$	0	0
$28$	0.757487	0.143152
$29$	−9.08678	−1.68737	−0.843687	−	0.536836i	$-0.819619\pi$
$29$	−9.08678	−1.68737	−0.843687	+	0.536836i	$0.819619\pi$
$30$	0	0
$31$	0.903081	0.162198	0.0810991	−	0.996706i	$-0.474157\pi$
$31$	0.903081	0.162198	0.0810991	+	0.996706i	$0.474157\pi$
$32$	−4.07862	−0.721006
$33$	0	0
$34$	5.39183	0.924691
$35$	0	0
$36$	0	0
$37$	1.50779	0.247879	0.123939	−	0.992290i	$-0.460447\pi$
$37$	1.50779	0.247879	0.123939	+	0.992290i	$0.460447\pi$
$38$	11.0677	1.79542
$39$	0	0
$40$	0	0
$41$	4.51466	0.705072	0.352536	−	0.935798i	$-0.385319\pi$
$41$	4.51466	0.705072	0.352536	+	0.935798i	$0.385319\pi$
$42$	0	0
$43$	−4.51466	−0.688480	−0.344240	−	0.938882i	$-0.611863\pi$
$43$	−4.51466	−0.688480	−0.344240	+	0.938882i	$0.611863\pi$
$44$	−3.39455	−0.511748
$45$	0	0
$46$	−14.6052	−2.15341
$47$	−3.86256	−0.563413	−0.281706	−	0.959501i	$-0.590900\pi$
$47$	−3.86256	−0.563413	−0.281706	+	0.959501i	$0.590900\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	1.24623	0.172822
$53$	−5.81910	−0.799314	−0.399657	−	0.916665i	$-0.630871\pi$
$53$	−5.81910	−0.799314	−0.399657	+	0.916665i	$0.630871\pi$
$54$	0	0
$55$	0	0
$56$	−2.06328	−0.275717
$57$	0	0
$58$	−15.0892	−1.98131
$59$	−6.21734	−0.809429	−0.404714	−	0.914443i	$-0.632629\pi$
$59$	−6.21734	−0.809429	−0.404714	+	0.914443i	$0.632629\pi$
$60$	0	0
$61$	4.54461	0.581878	0.290939	−	0.956742i	$-0.406032\pi$
$61$	4.54461	0.581878	0.290939	+	0.956742i	$0.406032\pi$
$62$	1.49963	0.190453
$63$	0	0
$64$	3.10954	0.388693
$65$	0	0
$66$	0	0
$67$	−13.5681	−1.65761	−0.828805	−	0.559538i	$-0.810978\pi$
$67$	−13.5681	−1.65761	−0.828805	+	0.559538i	$0.810978\pi$
$68$	2.45954	0.298263
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	0.170766	0.0199867	0.00999333	−	0.999950i	$-0.496819\pi$
$73$	0.170766	0.0199867	0.00999333	+	0.999950i	$0.496819\pi$
$74$	2.50378	0.291059
$75$	0	0
$76$	5.04868	0.579123
$77$	−4.48133	−0.510695
$78$	0	0
$79$	−7.37353	−0.829587	−0.414794	−	0.909915i	$-0.636146\pi$
$79$	−7.37353	−0.829587	−0.414794	+	0.909915i	$0.636146\pi$
$80$	0	0
$81$	0	0
$82$	7.49691	0.827895
$83$	1.56124	0.171368	0.0856841	−	0.996322i	$-0.472692\pi$
$83$	1.56124	0.171368	0.0856841	+	0.996322i	$0.472692\pi$
$84$	0	0
$85$	0	0
$86$	−7.49691	−0.808412
$87$	0	0
$88$	9.24623	0.985652
$89$	−14.9851	−1.58842	−0.794210	−	0.607644i	$-0.792115\pi$
$89$	−14.9851	−1.58842	−0.794210	+	0.607644i	$0.792115\pi$
$90$	0	0
$91$	1.64522	0.172466
$92$	−6.66231	−0.694594
$93$	0	0
$94$	−6.41405	−0.661559
$95$	0	0
$96$	0	0
$97$	−13.3399	−1.35446	−0.677230	−	0.735771i	$-0.736820\pi$
$97$	−13.3399	−1.35446	−0.677230	+	0.735771i	$0.736820\pi$
$98$	1.66057	0.167743
$99$	0	0
$100$	0	0
$101$	−16.4042	−1.63228	−0.816141	−	0.577853i	$-0.803891\pi$
$101$	−16.4042	−1.63228	−0.816141	+	0.577853i	$0.803891\pi$
$102$	0	0
$103$	2.83611	0.279450	0.139725	−	0.990190i	$-0.455378\pi$
$103$	2.83611	0.279450	0.139725	+	0.990190i	$0.455378\pi$
$104$	−3.39455	−0.332863
$105$	0	0
$106$	−9.66301	−0.938554
$107$	13.3973	1.29517	0.647585	−	0.761993i	$-0.275779\pi$
$107$	13.3973	1.29517	0.647585	+	0.761993i	$0.275779\pi$
$108$	0	0
$109$	1.08103	0.103544	0.0517722	−	0.998659i	$-0.483513\pi$
$109$	1.08103	0.103544	0.0517722	+	0.998659i	$0.483513\pi$
$110$	0	0
$111$	0	0
$112$	−4.94119	−0.466898
$113$	20.1392	1.89453	0.947267	−	0.320444i	$-0.103832\pi$
$113$	20.1392	1.89453	0.947267	+	0.320444i	$0.103832\pi$
$114$	0	0
$115$	0	0
$116$	−6.88312	−0.639082
$117$	0	0
$118$	−10.3243	−0.950431
$119$	3.24698	0.297650
$120$	0	0
$121$	9.08234	0.825668
$122$	7.54664	0.683240
$123$	0	0
$124$	0.684072	0.0614315
$125$	0	0
$126$	0	0
$127$	−17.0012	−1.50862	−0.754308	−	0.656521i	$-0.772027\pi$
$127$	−17.0012	−1.50862	−0.754308	+	0.656521i	$0.772027\pi$
$128$	13.3209	1.17741
$129$	0	0
$130$	0	0
$131$	13.9723	1.22077	0.610385	−	0.792105i	$-0.291015\pi$
$131$	13.9723	1.22077	0.610385	+	0.792105i	$0.291015\pi$
$132$	0	0
$133$	6.66503	0.577932
$134$	−22.5308	−1.94636
$135$	0	0
$136$	−6.69942	−0.574470
$137$	−7.69498	−0.657427	−0.328713	−	0.944430i	$-0.606615\pi$
$137$	−7.69498	−0.657427	−0.328713	+	0.944430i	$0.606615\pi$
$138$	0	0
$139$	−15.9229	−1.35056	−0.675281	−	0.737561i	$-0.735978\pi$
$139$	−15.9229	−1.35056	−0.675281	+	0.737561i	$0.735978\pi$
$140$	0	0
$141$	0	0
$142$	−13.2845	−1.11481
$143$	−7.37279	−0.616544
$144$	0	0
$145$	0	0
$146$	0.283569	0.0234683
$147$	0	0
$148$	1.14213	0.0938824
$149$	16.8638	1.38154	0.690768	−	0.723076i	$-0.257273\pi$
$149$	16.8638	1.38154	0.690768	+	0.723076i	$0.257273\pi$
$150$	0	0
$151$	16.2668	1.32377	0.661886	−	0.749604i	$-0.269756\pi$
$151$	16.2668	1.32377	0.661886	+	0.749604i	$0.269756\pi$
$152$	−13.7518	−1.11542
$153$	0	0
$154$	−7.44156	−0.599658
$155$	0	0
$156$	0	0
$157$	14.2093	1.13402	0.567011	−	0.823710i	$-0.308100\pi$
$157$	14.2093	1.13402	0.567011	+	0.823710i	$0.308100\pi$
$158$	−12.2443	−0.974101
$159$	0	0
$160$	0	0
$161$	−8.79528	−0.693165
$162$	0	0
$163$	−15.8678	−1.24286	−0.621431	−	0.783469i	$-0.713448\pi$
$163$	−15.8678	−1.24286	−0.621431	+	0.783469i	$0.713448\pi$
$164$	3.41980	0.267041
$165$	0	0
$166$	2.59254	0.201220
$167$	3.74638	0.289904	0.144952	−	0.989439i	$-0.453697\pi$
$167$	3.74638	0.289904	0.144952	+	0.989439i	$0.453697\pi$
$168$	0	0
$169$	−10.2932	−0.791788
$170$	0	0
$171$	0	0
$172$	−3.41980	−0.260757
$173$	−11.7490	−0.893261	−0.446631	−	0.894718i	$-0.647376\pi$
$173$	−11.7490	−0.893261	−0.446631	+	0.894718i	$0.647376\pi$
$174$	0	0
$175$	0	0
$176$	22.1431	1.66910
$177$	0	0
$178$	−24.8838	−1.86512
$179$	−3.97300	−0.296956	−0.148478	−	0.988916i	$-0.547437\pi$
$179$	−3.97300	−0.296956	−0.148478	+	0.988916i	$0.547437\pi$
$180$	0	0
$181$	1.74420	0.129645	0.0648225	−	0.997897i	$-0.479352\pi$
$181$	1.74420	0.129645	0.0648225	+	0.997897i	$0.479352\pi$
$182$	2.73200	0.202510
$183$	0	0
$184$	18.1471	1.33782
$185$	0	0
$186$	0	0
$187$	−14.5508	−1.06406
$188$	−2.92584	−0.213389
$189$	0	0
$190$	0	0
$191$	−16.7458	−1.21168	−0.605840	−	0.795586i	$-0.707163\pi$
$191$	−16.7458	−1.21168	−0.605840	+	0.795586i	$0.707163\pi$
$192$	0	0
$193$	2.01612	0.145123	0.0725617	−	0.997364i	$-0.476883\pi$
$193$	2.01612	0.145123	0.0725617	+	0.997364i	$0.476883\pi$
$194$	−22.1518	−1.59041
$195$	0	0
$196$	0.757487	0.0541062
$197$	10.8371	0.772112	0.386056	−	0.922475i	$-0.373837\pi$
$197$	10.8371	0.772112	0.386056	+	0.922475i	$0.373837\pi$
$198$	0	0
$199$	−15.4934	−1.09830	−0.549151	−	0.835723i	$-0.685049\pi$
$199$	−15.4934	−1.09830	−0.549151	+	0.835723i	$0.685049\pi$
$200$	0	0
$201$	0	0
$202$	−27.2403	−1.91662
$203$	−9.08678	−0.637767
$204$	0	0
$205$	0	0
$206$	4.70955	0.328130
$207$	0	0
$208$	−8.12935	−0.563669
$209$	−29.8682	−2.06603
$210$	0	0
$211$	−19.4661	−1.34010	−0.670052	−	0.742314i	$-0.733728\pi$
$211$	−19.4661	−1.34010	−0.670052	+	0.742314i	$0.733728\pi$
$212$	−4.40789	−0.302735
$213$	0	0
$214$	22.2472	1.52079
$215$	0	0
$216$	0	0
$217$	0.903081	0.0613052
$218$	1.79513	0.121582
$219$	0	0
$220$	0	0
$221$	5.34200	0.359342
$222$	0	0
$223$	15.4468	1.03439	0.517197	−	0.855866i	$-0.326975\pi$
$223$	15.4468	1.03439	0.517197	+	0.855866i	$0.326975\pi$
$224$	−4.07862	−0.272514
$225$	0	0
$226$	33.4425	2.22456
$227$	−6.36402	−0.422395	−0.211197	−	0.977443i	$-0.567736\pi$
$227$	−6.36402	−0.422395	−0.211197	+	0.977443i	$0.567736\pi$
$228$	0	0
$229$	2.50892	0.165794	0.0828969	−	0.996558i	$-0.473583\pi$
$229$	2.50892	0.165794	0.0828969	+	0.996558i	$0.473583\pi$
$230$	0	0
$231$	0	0
$232$	18.7486	1.23090
$233$	−3.17651	−0.208100	−0.104050	−	0.994572i	$-0.533180\pi$
$233$	−3.17651	−0.208100	−0.104050	+	0.994572i	$0.533180\pi$
$234$	0	0
$235$	0	0
$236$	−4.70955	−0.306566
$237$	0	0
$238$	5.39183	0.349500
$239$	9.59511	0.620656	0.310328	−	0.950630i	$-0.399561\pi$
$239$	9.59511	0.620656	0.310328	+	0.950630i	$0.399561\pi$
$240$	0	0
$241$	13.5998	0.876040	0.438020	−	0.898965i	$-0.355680\pi$
$241$	13.5998	0.876040	0.438020	+	0.898965i	$0.355680\pi$
$242$	15.0819	0.969498
$243$	0	0
$244$	3.44248	0.220382
$245$	0	0
$246$	0	0
$247$	10.9655	0.697716
$248$	−1.86331	−0.118320
$249$	0	0
$250$	0	0
$251$	3.43312	0.216697	0.108348	−	0.994113i	$-0.465444\pi$
$251$	3.43312	0.216697	0.108348	+	0.994113i	$0.465444\pi$
$252$	0	0
$253$	39.4146	2.47797
$254$	−28.2317	−1.77141
$255$	0	0
$256$	15.9011	0.993819
$257$	9.36134	0.583945	0.291972	−	0.956427i	$-0.405689\pi$
$257$	9.36134	0.583945	0.291972	+	0.956427i	$0.405689\pi$
$258$	0	0
$259$	1.50779	0.0936893
$260$	0	0
$261$	0	0
$262$	23.2020	1.43343
$263$	4.12412	0.254304	0.127152	−	0.991883i	$-0.459416\pi$
$263$	4.12412	0.254304	0.127152	+	0.991883i	$0.459416\pi$
$264$	0	0
$265$	0	0
$266$	11.0677	0.678607
$267$	0	0
$268$	−10.2777	−0.627809
$269$	10.4705	0.638395	0.319197	−	0.947688i	$-0.396587\pi$
$269$	10.4705	0.638395	0.319197	+	0.947688i	$0.396587\pi$
$270$	0	0
$271$	16.7219	1.01578	0.507891	−	0.861422i	$-0.330425\pi$
$271$	16.7219	1.01578	0.507891	+	0.861422i	$0.330425\pi$
$272$	−16.0439	−0.972806
$273$	0	0
$274$	−12.7780	−0.771950
$275$	0	0
$276$	0	0
$277$	9.12011	0.547974	0.273987	−	0.961733i	$-0.411657\pi$
$277$	9.12011	0.547974	0.273987	+	0.961733i	$0.411657\pi$
$278$	−26.4410	−1.58583
$279$	0	0
$280$	0	0
$281$	−7.88720	−0.470511	−0.235255	−	0.971934i	$-0.575593\pi$
$281$	−7.88720	−0.470511	−0.235255	+	0.971934i	$0.575593\pi$
$282$	0	0
$283$	−1.88738	−0.112193	−0.0560965	−	0.998425i	$-0.517865\pi$
$283$	−1.88738	−0.112193	−0.0560965	+	0.998425i	$0.517865\pi$
$284$	−6.05990	−0.359589
$285$	0	0
$286$	−12.2430	−0.723945
$287$	4.51466	0.266492
$288$	0	0
$289$	−6.45713	−0.379831
$290$	0	0
$291$	0	0
$292$	0.129353	0.00756982
$293$	1.24698	0.0728492	0.0364246	−	0.999336i	$-0.488403\pi$
$293$	1.24698	0.0728492	0.0364246	+	0.999336i	$0.488403\pi$
$294$	0	0
$295$	0	0
$296$	−3.11098	−0.180822
$297$	0	0
$298$	28.0035	1.62220
$299$	−14.4702	−0.836833
$300$	0	0
$301$	−4.51466	−0.260221
$302$	27.0121	1.55437
$303$	0	0
$304$	−32.9332	−1.88885
$305$	0	0
$306$	0	0
$307$	28.1403	1.60605	0.803026	−	0.595944i	$-0.203222\pi$
$307$	28.1403	1.60605	0.803026	+	0.595944i	$0.203222\pi$
$308$	−3.39455	−0.193423
$309$	0	0
$310$	0	0
$311$	14.2692	0.809134	0.404567	−	0.914508i	$-0.367422\pi$
$311$	14.2692	0.809134	0.404567	+	0.914508i	$0.367422\pi$
$312$	0	0
$313$	22.0743	1.24772	0.623858	−	0.781538i	$-0.285564\pi$
$313$	22.0743	1.24772	0.623858	+	0.781538i	$0.285564\pi$
$314$	23.5954	1.33157
$315$	0	0
$316$	−5.58536	−0.314201
$317$	1.11016	0.0623530	0.0311765	−	0.999514i	$-0.490075\pi$
$317$	1.11016	0.0623530	0.0311765	+	0.999514i	$0.490075\pi$
$318$	0	0
$319$	40.7209	2.27993
$320$	0	0
$321$	0	0
$322$	−14.6052	−0.813914
$323$	21.6412	1.20415
$324$	0	0
$325$	0	0
$326$	−26.3496	−1.45937
$327$	0	0
$328$	−9.31500	−0.514335
$329$	−3.86256	−0.212950
$330$	0	0
$331$	−25.6641	−1.41063	−0.705314	−	0.708895i	$-0.749194\pi$
$331$	−25.6641	−1.41063	−0.705314	+	0.708895i	$0.749194\pi$
$332$	1.18262	0.0649046
$333$	0	0
$334$	6.22112	0.340404
$335$	0	0
$336$	0	0
$337$	−29.7195	−1.61892	−0.809462	−	0.587173i	$-0.800241\pi$
$337$	−29.7195	−1.61892	−0.809462	+	0.587173i	$0.800241\pi$
$338$	−17.0926	−0.929717
$339$	0	0
$340$	0	0
$341$	−4.04701	−0.219158
$342$	0	0
$343$	1.00000	0.0539949
$344$	9.31500	0.502231
$345$	0	0
$346$	−19.5100	−1.04887
$347$	4.76490	0.255793	0.127897	−	0.991787i	$-0.459177\pi$
$347$	4.76490	0.255793	0.127897	+	0.991787i	$0.459177\pi$
$348$	0	0
$349$	−28.4595	−1.52340	−0.761701	−	0.647929i	$-0.775635\pi$
$349$	−28.4595	−1.52340	−0.761701	+	0.647929i	$0.775635\pi$
$350$	0	0
$351$	0	0
$352$	18.2777	0.974203
$353$	2.24682	0.119586	0.0597931	−	0.998211i	$-0.480956\pi$
$353$	2.24682	0.119586	0.0597931	+	0.998211i	$0.480956\pi$
$354$	0	0
$355$	0	0
$356$	−11.3510	−0.601603
$357$	0	0
$358$	−6.59744	−0.348686
$359$	−16.7919	−0.886243	−0.443121	−	0.896462i	$-0.646129\pi$
$359$	−16.7919	−0.886243	−0.443121	+	0.896462i	$0.646129\pi$
$360$	0	0
$361$	25.4227	1.33804
$362$	2.89636	0.152229
$363$	0	0
$364$	1.24623	0.0653204
$365$	0	0
$366$	0	0
$367$	24.0771	1.25682	0.628408	−	0.777884i	$-0.283707\pi$
$367$	24.0771	1.25682	0.628408	+	0.777884i	$0.283707\pi$
$368$	43.4591	2.26546
$369$	0	0
$370$	0	0
$371$	−5.81910	−0.302112
$372$	0	0
$373$	4.89364	0.253383	0.126692	−	0.991942i	$-0.459564\pi$
$373$	4.89364	0.253383	0.126692	+	0.991942i	$0.459564\pi$
$374$	−24.1626	−1.24942
$375$	0	0
$376$	7.96954	0.410998
$377$	−14.9498	−0.769953
$378$	0	0
$379$	−16.4390	−0.844414	−0.422207	−	0.906499i	$-0.638745\pi$
$379$	−16.4390	−0.844414	−0.422207	+	0.906499i	$0.638745\pi$
$380$	0	0
$381$	0	0
$382$	−27.8075	−1.42275
$383$	24.9874	1.27680	0.638398	−	0.769706i	$-0.279597\pi$
$383$	24.9874	1.27680	0.638398	+	0.769706i	$0.279597\pi$
$384$	0	0
$385$	0	0
$386$	3.34790	0.170404
$387$	0	0
$388$	−10.1048	−0.512993
$389$	−0.217416	−0.0110234	−0.00551172	−	0.999985i	$-0.501754\pi$
$389$	−0.217416	−0.0110234	−0.00551172	+	0.999985i	$0.501754\pi$
$390$	0	0
$391$	−28.5581	−1.44424
$392$	−2.06328	−0.104211
$393$	0	0
$394$	17.9958	0.906613
$395$	0	0
$396$	0	0
$397$	31.7330	1.59263	0.796316	−	0.604880i	$-0.206779\pi$
$397$	31.7330	1.59263	0.796316	+	0.604880i	$0.206779\pi$
$398$	−25.7279	−1.28962
$399$	0	0
$400$	0	0
$401$	−11.3618	−0.567383	−0.283691	−	0.958916i	$-0.591559\pi$
$401$	−11.3618	−0.567383	−0.283691	+	0.958916i	$0.591559\pi$
$402$	0	0
$403$	1.48577	0.0740115
$404$	−12.4260	−0.618216
$405$	0	0
$406$	−15.0892	−0.748865
$407$	−6.75689	−0.334927
$408$	0	0
$409$	−5.53886	−0.273879	−0.136940	−	0.990579i	$-0.543727\pi$
$409$	−5.53886	−0.273879	−0.136940	+	0.990579i	$0.543727\pi$
$410$	0	0
$411$	0	0
$412$	2.14832	0.105840
$413$	−6.21734	−0.305935
$414$	0	0
$415$	0	0
$416$	−6.71024	−0.328997
$417$	0	0
$418$	−49.5982	−2.42593
$419$	−27.7747	−1.35688	−0.678440	−	0.734656i	$-0.737344\pi$
$419$	−27.7747	−1.35688	−0.678440	+	0.734656i	$0.737344\pi$
$420$	0	0
$421$	0.639289	0.0311570	0.0155785	−	0.999879i	$-0.495041\pi$
$421$	0.639289	0.0311570	0.0155785	+	0.999879i	$0.495041\pi$
$422$	−32.3248	−1.57355
$423$	0	0
$424$	12.0064	0.583083
$425$	0	0
$426$	0	0
$427$	4.54461	0.219929
$428$	10.1483	0.490537
$429$	0	0
$430$	0	0
$431$	11.0321	0.531399	0.265699	−	0.964056i	$-0.414397\pi$
$431$	11.0321	0.531399	0.265699	+	0.964056i	$0.414397\pi$
$432$	0	0
$433$	16.3489	0.785677	0.392838	−	0.919607i	$-0.371493\pi$
$433$	16.3489	0.785677	0.392838	+	0.919607i	$0.371493\pi$
$434$	1.49963	0.0719845
$435$	0	0
$436$	0.818870	0.0392167
$437$	−58.6208	−2.80422
$438$	0	0
$439$	31.1015	1.48440	0.742198	−	0.670181i	$-0.233784\pi$
$439$	31.1015	1.48440	0.742198	+	0.670181i	$0.233784\pi$
$440$	0	0
$441$	0	0
$442$	8.87076	0.421939
$443$	3.70111	0.175845	0.0879226	−	0.996127i	$-0.471977\pi$
$443$	3.70111	0.175845	0.0879226	+	0.996127i	$0.471977\pi$
$444$	0	0
$445$	0	0
$446$	25.6505	1.21458
$447$	0	0
$448$	3.10954	0.146912
$449$	17.3869	0.820539	0.410269	−	0.911964i	$-0.365435\pi$
$449$	17.3869	0.820539	0.410269	+	0.911964i	$0.365435\pi$
$450$	0	0
$451$	−20.2317	−0.952674
$452$	15.2552	0.717543
$453$	0	0
$454$	−10.5679	−0.495976
$455$	0	0
$456$	0	0
$457$	14.1806	0.663338	0.331669	−	0.943396i	$-0.392388\pi$
$457$	14.1806	0.663338	0.331669	+	0.943396i	$0.392388\pi$
$458$	4.16623	0.194675
$459$	0	0
$460$	0	0
$461$	−16.5364	−0.770178	−0.385089	−	0.922879i	$-0.625829\pi$
$461$	−16.5364	−0.770178	−0.385089	+	0.922879i	$0.625829\pi$
$462$	0	0
$463$	14.6216	0.679522	0.339761	−	0.940512i	$-0.389654\pi$
$463$	14.6216	0.679522	0.339761	+	0.940512i	$0.389654\pi$
$464$	44.8995	2.08441
$465$	0	0
$466$	−5.27482	−0.244351
$467$	−37.3100	−1.72650	−0.863250	−	0.504777i	$-0.831575\pi$
$467$	−37.3100	−1.72650	−0.863250	+	0.504777i	$0.831575\pi$
$468$	0	0
$469$	−13.5681	−0.626517
$470$	0	0
$471$	0	0
$472$	12.8281	0.590461
$473$	20.2317	0.930255
$474$	0	0
$475$	0	0
$476$	2.45954	0.112733
$477$	0	0
$478$	15.9333	0.728774
$479$	−8.94065	−0.408509	−0.204254	−	0.978918i	$-0.565477\pi$
$479$	−8.94065	−0.408509	−0.204254	+	0.978918i	$0.565477\pi$
$480$	0	0
$481$	2.48064	0.113108
$482$	22.5834	1.02865
$483$	0	0
$484$	6.87976	0.312716
$485$	0	0
$486$	0	0
$487$	24.4709	1.10888	0.554441	−	0.832223i	$-0.312932\pi$
$487$	24.4709	1.10888	0.554441	+	0.832223i	$0.312932\pi$
$488$	−9.37679	−0.424468
$489$	0	0
$490$	0	0
$491$	−27.8794	−1.25818	−0.629089	−	0.777333i	$-0.716572\pi$
$491$	−27.8794	−1.25818	−0.629089	+	0.777333i	$0.716572\pi$
$492$	0	0
$493$	−29.5046	−1.32882
$494$	18.2089	0.819257
$495$	0	0
$496$	−4.46229	−0.200363
$497$	−8.00000	−0.358849
$498$	0	0
$499$	2.26317	0.101314	0.0506568	−	0.998716i	$-0.483869\pi$
$499$	2.26317	0.101314	0.0506568	+	0.998716i	$0.483869\pi$
$500$	0	0
$501$	0	0
$502$	5.70093	0.254445
$503$	−14.1184	−0.629507	−0.314753	−	0.949173i	$-0.601922\pi$
$503$	−14.1184	−0.629507	−0.314753	+	0.949173i	$0.601922\pi$
$504$	0	0
$505$	0	0
$506$	65.4506	2.90963
$507$	0	0
$508$	−12.8782	−0.571378
$509$	37.5605	1.66484	0.832421	−	0.554144i	$-0.186954\pi$
$509$	37.5605	1.66484	0.832421	+	0.554144i	$0.186954\pi$
$510$	0	0
$511$	0.170766	0.00755425
$512$	−0.236841	−0.0104670
$513$	0	0
$514$	15.5451	0.685667
$515$	0	0
$516$	0	0
$517$	17.3094	0.761268
$518$	2.50378	0.110010
$519$	0	0
$520$	0	0
$521$	−17.6119	−0.771591	−0.385796	−	0.922584i	$-0.626073\pi$
$521$	−17.6119	−0.771591	−0.385796	+	0.922584i	$0.626073\pi$
$522$	0	0
$523$	3.80434	0.166352	0.0831762	−	0.996535i	$-0.473494\pi$
$523$	3.80434	0.166352	0.0831762	+	0.996535i	$0.473494\pi$
$524$	10.5839	0.462358
$525$	0	0
$526$	6.84838	0.298603
$527$	2.93229	0.127732
$528$	0	0
$529$	54.3570	2.36335
$530$	0	0
$531$	0	0
$532$	5.04868	0.218888
$533$	7.42763	0.321726
$534$	0	0
$535$	0	0
$536$	27.9948	1.20919
$537$	0	0
$538$	17.3869	0.749603
$539$	−4.48133	−0.193025
$540$	0	0
$541$	−23.4002	−1.00605	−0.503027	−	0.864271i	$-0.667780\pi$
$541$	−23.4002	−1.00605	−0.503027	+	0.864271i	$0.667780\pi$
$542$	27.7678	1.19273
$543$	0	0
$544$	−13.2432	−0.567798
$545$	0	0
$546$	0	0
$547$	10.6578	0.455693	0.227847	−	0.973697i	$-0.426832\pi$
$547$	10.6578	0.455693	0.227847	+	0.973697i	$0.426832\pi$
$548$	−5.82885	−0.248996
$549$	0	0
$550$	0	0
$551$	−60.5637	−2.58010
$552$	0	0
$553$	−7.37353	−0.313555
$554$	15.1446	0.643431
$555$	0	0
$556$	−12.0614	−0.511517
$557$	19.3186	0.818554	0.409277	−	0.912410i	$-0.365781\pi$
$557$	19.3186	0.818554	0.409277	+	0.912410i	$0.365781\pi$
$558$	0	0
$559$	−7.42763	−0.314155
$560$	0	0
$561$	0	0
$562$	−13.0972	−0.552473
$563$	2.49855	0.105301	0.0526506	−	0.998613i	$-0.483233\pi$
$563$	2.49855	0.105301	0.0526506	+	0.998613i	$0.483233\pi$
$564$	0	0
$565$	0	0
$566$	−3.13412	−0.131737
$567$	0	0
$568$	16.5062	0.692586
$569$	−20.9949	−0.880152	−0.440076	−	0.897961i	$-0.645049\pi$
$569$	−20.9949	−0.880152	−0.440076	+	0.897961i	$0.645049\pi$
$570$	0	0
$571$	−24.5019	−1.02537	−0.512686	−	0.858576i	$-0.671350\pi$
$571$	−24.5019	−1.02537	−0.512686	+	0.858576i	$0.671350\pi$
$572$	−5.58479	−0.233512
$573$	0	0
$574$	7.49691	0.312915
$575$	0	0
$576$	0	0
$577$	−2.55043	−0.106176	−0.0530879	−	0.998590i	$-0.516906\pi$
$577$	−2.55043	−0.106176	−0.0530879	+	0.998590i	$0.516906\pi$
$578$	−10.7225	−0.445998
$579$	0	0
$580$	0	0
$581$	1.56124	0.0647711
$582$	0	0
$583$	26.0773	1.08001
$584$	−0.352338	−0.0145798
$585$	0	0
$586$	2.07069	0.0855395
$587$	45.9024	1.89459	0.947297	−	0.320358i	$-0.103803\pi$
$587$	45.9024	1.89459	0.947297	+	0.320358i	$0.103803\pi$
$588$	0	0
$589$	6.01907	0.248011
$590$	0	0
$591$	0	0
$592$	−7.45026	−0.306204
$593$	−28.7110	−1.17902	−0.589509	−	0.807761i	$-0.700679\pi$
$593$	−28.7110	−1.17902	−0.589509	+	0.807761i	$0.700679\pi$
$594$	0	0
$595$	0	0
$596$	12.7741	0.523248
$597$	0	0
$598$	−24.0287	−0.982609
$599$	−42.5917	−1.74025	−0.870125	−	0.492831i	$-0.835962\pi$
$599$	−42.5917	−1.74025	−0.870125	+	0.492831i	$0.835962\pi$
$600$	0	0
$601$	13.1065	0.534624	0.267312	−	0.963610i	$-0.413865\pi$
$601$	13.1065	0.534624	0.267312	+	0.963610i	$0.413865\pi$
$602$	−7.49691	−0.305551
$603$	0	0
$604$	12.3219	0.501370
$605$	0	0
$606$	0	0
$607$	−3.52736	−0.143171	−0.0715856	−	0.997434i	$-0.522806\pi$
$607$	−3.52736	−0.143171	−0.0715856	+	0.997434i	$0.522806\pi$
$608$	−27.1842	−1.10246
$609$	0	0
$610$	0	0
$611$	−6.35478	−0.257087
$612$	0	0
$613$	−7.75038	−0.313035	−0.156517	−	0.987675i	$-0.550027\pi$
$613$	−7.75038	−0.313035	−0.156517	+	0.987675i	$0.550027\pi$
$614$	46.7289	1.88583
$615$	0	0
$616$	9.24623	0.372541
$617$	16.3117	0.656686	0.328343	−	0.944559i	$-0.393510\pi$
$617$	16.3117	0.656686	0.328343	+	0.944559i	$0.393510\pi$
$618$	0	0
$619$	−17.5794	−0.706577	−0.353288	−	0.935514i	$-0.614937\pi$
$619$	−17.5794	−0.706577	−0.353288	+	0.935514i	$0.614937\pi$
$620$	0	0
$621$	0	0
$622$	23.6950	0.950084
$623$	−14.9851	−0.600366
$624$	0	0
$625$	0	0
$626$	36.6559	1.46507
$627$	0	0
$628$	10.7633	0.429504
$629$	4.89575	0.195206
$630$	0	0
$631$	−42.2819	−1.68322	−0.841608	−	0.540089i	$-0.818391\pi$
$631$	−42.2819	−1.68322	−0.841608	+	0.540089i	$0.818391\pi$
$632$	15.2136	0.605166
$633$	0	0
$634$	1.84350	0.0732149
$635$	0	0
$636$	0	0
$637$	1.64522	0.0651861
$638$	67.6198	2.67709
$639$	0	0
$640$	0	0
$641$	9.59300	0.378901	0.189450	−	0.981890i	$-0.439329\pi$
$641$	9.59300	0.378901	0.189450	+	0.981890i	$0.439329\pi$
$642$	0	0
$643$	31.9586	1.26032	0.630162	−	0.776464i	$-0.282989\pi$
$643$	31.9586	1.26032	0.630162	+	0.776464i	$0.282989\pi$
$644$	−6.66231	−0.262532
$645$	0	0
$646$	35.9367	1.41391
$647$	10.0574	0.395396	0.197698	−	0.980263i	$-0.436653\pi$
$647$	10.0574	0.395396	0.197698	+	0.980263i	$0.436653\pi$
$648$	0	0
$649$	27.8620	1.09368
$650$	0	0
$651$	0	0
$652$	−12.0197	−0.470726
$653$	10.4588	0.409283	0.204641	−	0.978837i	$-0.434397\pi$
$653$	10.4588	0.409283	0.204641	+	0.978837i	$0.434397\pi$
$654$	0	0
$655$	0	0
$656$	−22.3078	−0.870973
$657$	0	0
$658$	−6.41405	−0.250046
$659$	−34.7947	−1.35541	−0.677704	−	0.735334i	$-0.737025\pi$
$659$	−34.7947	−1.35541	−0.677704	+	0.735334i	$0.737025\pi$
$660$	0	0
$661$	40.3945	1.57116	0.785581	−	0.618758i	$-0.212364\pi$
$661$	40.3945	1.57116	0.785581	+	0.618758i	$0.212364\pi$
$662$	−42.6171	−1.65636
$663$	0	0
$664$	−3.22127	−0.125009
$665$	0	0
$666$	0	0
$667$	79.9208	3.09455
$668$	2.83783	0.109799
$669$	0	0
$670$	0	0
$671$	−20.3659	−0.786217
$672$	0	0
$673$	−46.7268	−1.80119	−0.900593	−	0.434664i	$-0.856867\pi$
$673$	−46.7268	−1.80119	−0.900593	+	0.434664i	$0.856867\pi$
$674$	−49.3512	−1.90094
$675$	0	0
$676$	−7.79700	−0.299884
$677$	33.6084	1.29168	0.645838	−	0.763474i	$-0.276508\pi$
$677$	33.6084	1.29168	0.645838	+	0.763474i	$0.276508\pi$
$678$	0	0
$679$	−13.3399	−0.511938
$680$	0	0
$681$	0	0
$682$	−6.72033	−0.257335
$683$	8.44720	0.323223	0.161612	−	0.986854i	$-0.448331\pi$
$683$	8.44720	0.323223	0.161612	+	0.986854i	$0.448331\pi$
$684$	0	0
$685$	0	0
$686$	1.66057	0.0634008
$687$	0	0
$688$	22.3078	0.850477
$689$	−9.57371	−0.364729
$690$	0	0
$691$	−25.8453	−0.983201	−0.491601	−	0.870821i	$-0.663588\pi$
$691$	−25.8453	−0.983201	−0.491601	+	0.870821i	$0.663588\pi$
$692$	−8.89973	−0.338317
$693$	0	0
$694$	7.91244	0.300352
$695$	0	0
$696$	0	0
$697$	14.6590	0.555250
$698$	−47.2589	−1.78878
$699$	0	0
$700$	0	0
$701$	−12.5440	−0.473781	−0.236891	−	0.971536i	$-0.576128\pi$
$701$	−12.5440	−0.473781	−0.236891	+	0.971536i	$0.576128\pi$
$702$	0	0
$703$	10.0494	0.379022
$704$	−13.9349	−0.525191
$705$	0	0
$706$	3.73100	0.140418
$707$	−16.4042	−0.616944
$708$	0	0
$709$	−14.5033	−0.544685	−0.272342	−	0.962200i	$-0.587798\pi$
$709$	−14.5033	−0.544685	−0.272342	+	0.962200i	$0.587798\pi$
$710$	0	0
$711$	0	0
$712$	30.9185	1.15872
$713$	−7.94286	−0.297462
$714$	0	0
$715$	0	0
$716$	−3.00950	−0.112470
$717$	0	0
$718$	−27.8841	−1.04063
$719$	51.9206	1.93631	0.968155	−	0.250352i	$-0.0805462\pi$
$719$	51.9206	1.93631	0.968155	+	0.250352i	$0.0805462\pi$
$720$	0	0
$721$	2.83611	0.105622
$722$	42.2161	1.57112
$723$	0	0
$724$	1.32121	0.0491022
$725$	0	0
$726$	0	0
$727$	−26.2289	−0.972776	−0.486388	−	0.873743i	$-0.661686\pi$
$727$	−26.2289	−0.972776	−0.486388	+	0.873743i	$0.661686\pi$
$728$	−3.39455	−0.125810
$729$	0	0
$730$	0	0
$731$	−14.6590	−0.542183
$732$	0	0
$733$	48.4814	1.79070	0.895350	−	0.445363i	$-0.146925\pi$
$733$	48.4814	1.79070	0.895350	+	0.445363i	$0.146925\pi$
$734$	39.9817	1.47575
$735$	0	0
$736$	35.8726	1.32228
$737$	60.8032	2.23972
$738$	0	0
$739$	37.5127	1.37993	0.689964	−	0.723843i	$-0.257626\pi$
$739$	37.5127	1.37993	0.689964	+	0.723843i	$0.257626\pi$
$740$	0	0
$741$	0	0
$742$	−9.66301	−0.354740
$743$	−16.4544	−0.603652	−0.301826	−	0.953363i	$-0.597596\pi$
$743$	−16.4544	−0.603652	−0.301826	+	0.953363i	$0.597596\pi$
$744$	0	0
$745$	0	0
$746$	8.12622	0.297522
$747$	0	0
$748$	−11.0220	−0.403005
$749$	13.3973	0.489528
$750$	0	0
$751$	18.9510	0.691533	0.345767	−	0.938321i	$-0.387619\pi$
$751$	18.9510	0.691533	0.345767	+	0.938321i	$0.387619\pi$
$752$	19.0857	0.695982
$753$	0	0
$754$	−24.8251	−0.904078
$755$	0	0
$756$	0	0
$757$	−20.3544	−0.739792	−0.369896	−	0.929073i	$-0.620607\pi$
$757$	−20.3544	−0.739792	−0.369896	+	0.929073i	$0.620607\pi$
$758$	−27.2981	−0.991511
$759$	0	0
$760$	0	0
$761$	−2.63252	−0.0954288	−0.0477144	−	0.998861i	$-0.515194\pi$
$761$	−2.63252	−0.0954288	−0.0477144	+	0.998861i	$0.515194\pi$
$762$	0	0
$763$	1.08103	0.0391361
$764$	−12.6847	−0.458916
$765$	0	0
$766$	41.4933	1.49921
$767$	−10.2289	−0.369345
$768$	0	0
$769$	−15.9262	−0.574315	−0.287157	−	0.957883i	$-0.592710\pi$
$769$	−15.9262	−0.574315	−0.287157	+	0.957883i	$0.592710\pi$
$770$	0	0
$771$	0	0
$772$	1.52718	0.0549645
$773$	31.7879	1.14333	0.571666	−	0.820486i	$-0.306297\pi$
$773$	31.7879	1.14333	0.571666	+	0.820486i	$0.306297\pi$
$774$	0	0
$775$	0	0
$776$	27.5239	0.988051
$777$	0	0
$778$	−0.361034	−0.0129437
$779$	30.0904	1.07810
$780$	0	0
$781$	35.8507	1.28284
$782$	−47.4227	−1.69583
$783$	0	0
$784$	−4.94119	−0.176471
$785$	0	0
$786$	0	0
$787$	−47.1832	−1.68190	−0.840950	−	0.541112i	$-0.818003\pi$
$787$	−47.1832	−1.68190	−0.840950	+	0.541112i	$0.818003\pi$
$788$	8.20897	0.292432
$789$	0	0
$790$	0	0
$791$	20.1392	0.716067
$792$	0	0
$793$	7.47690	0.265512
$794$	52.6948	1.87007
$795$	0	0
$796$	−11.7361	−0.415974
$797$	−51.0233	−1.80734	−0.903669	−	0.428232i	$-0.859137\pi$
$797$	−51.0233	−1.80734	−0.903669	+	0.428232i	$0.859137\pi$
$798$	0	0
$799$	−12.5417	−0.443692
$800$	0	0
$801$	0	0
$802$	−18.8671	−0.666221
$803$	−0.765260	−0.0270054
$804$	0	0
$805$	0	0
$806$	2.46722	0.0869042
$807$	0	0
$808$	33.8465	1.19071
$809$	11.8135	0.415342	0.207671	−	0.978199i	$-0.433412\pi$
$809$	11.8135	0.415342	0.207671	+	0.978199i	$0.433412\pi$
$810$	0	0
$811$	−2.58613	−0.0908112	−0.0454056	−	0.998969i	$-0.514458\pi$
$811$	−2.58613	−0.0908112	−0.0454056	+	0.998969i	$0.514458\pi$
$812$	−6.88312	−0.241550
$813$	0	0
$814$	−11.2203	−0.393271
$815$	0	0
$816$	0	0
$817$	−30.0904	−1.05273
$818$	−9.19766	−0.321589
$819$	0	0
$820$	0	0
$821$	16.2587	0.567433	0.283716	−	0.958908i	$-0.408433\pi$
$821$	16.2587	0.567433	0.283716	+	0.958908i	$0.408433\pi$
$822$	0	0
$823$	−0.909326	−0.0316971	−0.0158486	−	0.999874i	$-0.505045\pi$
$823$	−0.909326	−0.0316971	−0.0158486	+	0.999874i	$0.505045\pi$
$824$	−5.85168	−0.203853
$825$	0	0
$826$	−10.3243	−0.359229
$827$	−9.62855	−0.334818	−0.167409	−	0.985888i	$-0.553540\pi$
$827$	−9.62855	−0.334818	−0.167409	+	0.985888i	$0.553540\pi$
$828$	0	0
$829$	7.01313	0.243576	0.121788	−	0.992556i	$-0.461137\pi$
$829$	7.01313	0.243576	0.121788	+	0.992556i	$0.461137\pi$
$830$	0	0
$831$	0	0
$832$	5.11589	0.177362
$833$	3.24698	0.112501
$834$	0	0
$835$	0	0
$836$	−22.6248	−0.782495
$837$	0	0
$838$	−46.1217	−1.59325
$839$	−6.61287	−0.228302	−0.114151	−	0.993463i	$-0.536415\pi$
$839$	−6.61287	−0.228302	−0.114151	+	0.993463i	$0.536415\pi$
$840$	0	0
$841$	53.5696	1.84723
$842$	1.06158	0.0365846
$843$	0	0
$844$	−14.7453	−0.507556
$845$	0	0
$846$	0	0
$847$	9.08234	0.312073
$848$	28.7532	0.987391
$849$	0	0
$850$	0	0
$851$	−13.2614	−0.454595
$852$	0	0
$853$	−27.2511	−0.933061	−0.466531	−	0.884505i	$-0.654496\pi$
$853$	−27.2511	−0.933061	−0.466531	+	0.884505i	$0.654496\pi$
$854$	7.54664	0.258241
$855$	0	0
$856$	−27.6425	−0.944800
$857$	10.2478	0.350057	0.175029	−	0.984563i	$-0.443998\pi$
$857$	10.2478	0.350057	0.175029	+	0.984563i	$0.443998\pi$
$858$	0	0
$859$	18.8887	0.644473	0.322236	−	0.946659i	$-0.395565\pi$
$859$	18.8887	0.644473	0.322236	+	0.946659i	$0.395565\pi$
$860$	0	0
$861$	0	0
$862$	18.3196	0.623968
$863$	−12.4734	−0.424600	−0.212300	−	0.977205i	$-0.568095\pi$
$863$	−12.4734	−0.424600	−0.212300	+	0.977205i	$0.568095\pi$
$864$	0	0
$865$	0	0
$866$	27.1484	0.922541
$867$	0	0
$868$	0.684072	0.0232189
$869$	33.0433	1.12092
$870$	0	0
$871$	−22.3226	−0.756371
$872$	−2.23047	−0.0755334
$873$	0	0
$874$	−97.3439	−3.29271
$875$	0	0
$876$	0	0
$877$	−41.5409	−1.40274	−0.701368	−	0.712799i	$-0.747427\pi$
$877$	−41.5409	−1.40274	−0.701368	+	0.712799i	$0.747427\pi$
$878$	51.6462	1.74298
$879$	0	0
$880$	0	0
$881$	43.4363	1.46341	0.731704	−	0.681623i	$-0.238725\pi$
$881$	43.4363	1.46341	0.731704	+	0.681623i	$0.238725\pi$
$882$	0	0
$883$	18.0849	0.608605	0.304302	−	0.952575i	$-0.401577\pi$
$883$	18.0849	0.608605	0.304302	+	0.952575i	$0.401577\pi$
$884$	4.04650	0.136098
$885$	0	0
$886$	6.14595	0.206477
$887$	41.8481	1.40512	0.702560	−	0.711624i	$-0.252040\pi$
$887$	41.8481	1.40512	0.702560	+	0.711624i	$0.252040\pi$
$888$	0	0
$889$	−17.0012	−0.570203
$890$	0	0
$891$	0	0
$892$	11.7007	0.391770
$893$	−25.7441	−0.861494
$894$	0	0
$895$	0	0
$896$	13.3209	0.445018
$897$	0	0
$898$	28.8721	0.963476
$899$	−8.20610	−0.273689
$900$	0	0
$901$	−18.8945	−0.629466
$902$	−33.5961	−1.11863
$903$	0	0
$904$	−41.5527	−1.38202
$905$	0	0
$906$	0	0
$907$	−19.3725	−0.643254	−0.321627	−	0.946866i	$-0.604230\pi$
$907$	−19.3725	−0.643254	−0.321627	+	0.946866i	$0.604230\pi$
$908$	−4.82066	−0.159979
$909$	0	0
$910$	0	0
$911$	−2.68932	−0.0891012	−0.0445506	−	0.999007i	$-0.514186\pi$
$911$	−2.68932	−0.0891012	−0.0445506	+	0.999007i	$0.514186\pi$
$912$	0	0
$913$	−6.99643	−0.231548
$914$	23.5478	0.778891
$915$	0	0
$916$	1.90047	0.0627933
$917$	13.9723	0.461407
$918$	0	0
$919$	41.8192	1.37949	0.689744	−	0.724053i	$-0.257723\pi$
$919$	41.8192	1.37949	0.689744	+	0.724053i	$0.257723\pi$
$920$	0	0
$921$	0	0
$922$	−27.4599	−0.904342
$923$	−13.1618	−0.433225
$924$	0	0
$925$	0	0
$926$	24.2801	0.797894
$927$	0	0
$928$	37.0616	1.21661
$929$	39.9284	1.31001	0.655004	−	0.755625i	$-0.272667\pi$
$929$	39.9284	1.31001	0.655004	+	0.755625i	$0.272667\pi$
$930$	0	0
$931$	6.66503	0.218438
$932$	−2.40617	−0.0788166
$933$	0	0
$934$	−61.9558	−2.02725
$935$	0	0
$936$	0	0
$937$	−18.1074	−0.591543	−0.295772	−	0.955259i	$-0.595577\pi$
$937$	−18.1074	−0.591543	−0.295772	+	0.955259i	$0.595577\pi$
$938$	−22.5308	−0.735656
$939$	0	0
$940$	0	0
$941$	52.9402	1.72580	0.862900	−	0.505375i	$-0.168646\pi$
$941$	52.9402	1.72580	0.862900	+	0.505375i	$0.168646\pi$
$942$	0	0
$943$	−39.7077	−1.29306
$944$	30.7210	0.999885
$945$	0	0
$946$	33.5961	1.09230
$947$	−48.4854	−1.57556	−0.787781	−	0.615955i	$-0.788770\pi$
$947$	−48.4854	−1.57556	−0.787781	+	0.615955i	$0.788770\pi$
$948$	0	0
$949$	0.280948	0.00911996
$950$	0	0
$951$	0	0
$952$	−6.69942	−0.217129
$953$	−9.76963	−0.316469	−0.158235	−	0.987402i	$-0.550580\pi$
$953$	−9.76963	−0.316469	−0.158235	+	0.987402i	$0.550580\pi$
$954$	0	0
$955$	0	0
$956$	7.26817	0.235070
$957$	0	0
$958$	−14.8466	−0.479671
$959$	−7.69498	−0.248484
$960$	0	0
$961$	−30.1844	−0.973692
$962$	4.11928	0.132811
$963$	0	0
$964$	10.3017	0.331795
$965$	0	0
$966$	0	0
$967$	40.0858	1.28907	0.644537	−	0.764573i	$-0.277050\pi$
$967$	40.0858	1.28907	0.644537	+	0.764573i	$0.277050\pi$
$968$	−18.7394	−0.602307
$969$	0	0
$970$	0	0
$971$	−36.2290	−1.16264	−0.581322	−	0.813674i	$-0.697464\pi$
$971$	−36.2290	−1.16264	−0.581322	+	0.813674i	$0.697464\pi$
$972$	0	0
$973$	−15.9229	−0.510464
$974$	40.6356	1.30205
$975$	0	0
$976$	−22.4558	−0.718792
$977$	−27.6379	−0.884214	−0.442107	−	0.896962i	$-0.645769\pi$
$977$	−27.6379	−0.884214	−0.442107	+	0.896962i	$0.645769\pi$
$978$	0	0
$979$	67.1533	2.14623
$980$	0	0
$981$	0	0
$982$	−46.2956	−1.47735
$983$	43.8409	1.39831	0.699154	−	0.714971i	$-0.253560\pi$
$983$	43.8409	1.39831	0.699154	+	0.714971i	$0.253560\pi$
$984$	0	0
$985$	0	0
$986$	−48.9944	−1.56030
$987$	0	0
$988$	8.30620	0.264255
$989$	39.7077	1.26263
$990$	0	0
$991$	27.6085	0.877012	0.438506	−	0.898728i	$-0.355508\pi$
$991$	27.6085	0.877012	0.438506	+	0.898728i	$0.355508\pi$
$992$	−3.68333	−0.116946
$993$	0	0
$994$	−13.2845	−0.421360
$995$	0	0
$996$	0	0
$997$	22.5255	0.713391	0.356696	−	0.934221i	$-0.383903\pi$
$997$	22.5255	0.713391	0.356696	+	0.934221i	$0.383903\pi$
$998$	3.75815	0.118962
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4725.2.a.cc.1.4		5
3.2	odd	2		4725.2.a.cd.1.2		5
5.2	odd	4		945.2.d.e.379.8	yes	10
5.3	odd	4		945.2.d.e.379.3	yes	10
5.4	even	2		4725.2.a.cb.1.2		5
15.2	even	4		945.2.d.d.379.3	✓	10
15.8	even	4		945.2.d.d.379.8	yes	10
15.14	odd	2		4725.2.a.ca.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
945.2.d.d.379.3	✓	10	15.2	even	4
945.2.d.d.379.8	yes	10	15.8	even	4
945.2.d.e.379.3	yes	10	5.3	odd	4
945.2.d.e.379.8	yes	10	5.2	odd	4
4725.2.a.ca.1.4		5	15.14	odd	2
4725.2.a.cb.1.2		5	5.4	even	2
4725.2.a.cc.1.4		5	1.1	even	1	trivial
4725.2.a.cd.1.2		5	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4725 = 3^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4725.a (trivial)

\(f(q)\)	\(=\)	\(q+1.66057 q^{2} +0.757487 q^{4} +1.00000 q^{7} -2.06328 q^{8} -4.48133 q^{11} +1.64522 q^{13} +1.66057 q^{14} -4.94119 q^{16} +3.24698 q^{17} +6.66503 q^{19} -7.44156 q^{22} -8.79528 q^{23} +2.73200 q^{26} +0.757487 q^{28} -9.08678 q^{29} +0.903081 q^{31} -4.07862 q^{32} +5.39183 q^{34} +1.50779 q^{37} +11.0677 q^{38} +4.51466 q^{41} -4.51466 q^{43} -3.39455 q^{44} -14.6052 q^{46} -3.86256 q^{47} +1.00000 q^{49} +1.24623 q^{52} -5.81910 q^{53} -2.06328 q^{56} -15.0892 q^{58} -6.21734 q^{59} +4.54461 q^{61} +1.49963 q^{62} +3.10954 q^{64} -13.5681 q^{67} +2.45954 q^{68} -8.00000 q^{71} +0.170766 q^{73} +2.50378 q^{74} +5.04868 q^{76} -4.48133 q^{77} -7.37353 q^{79} +7.49691 q^{82} +1.56124 q^{83} -7.49691 q^{86} +9.24623 q^{88} -14.9851 q^{89} +1.64522 q^{91} -6.66231 q^{92} -6.41405 q^{94} -13.3399 q^{97} +1.66057 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 6 q^{4} + 5 q^{7} - 3 q^{8} - 9 q^{11} + 7 q^{13} - 4 q^{16} - 14 q^{17} - 3 q^{19} - 5 q^{22} - 2 q^{23} - 21 q^{26} + 6 q^{28} - 12 q^{29} - 6 q^{31} - 6 q^{32} + 7 q^{34} + 20 q^{38} - 7 q^{41}+ \cdots - 4 q^{97}+O(q^{100}) \) 5 * q + 6 * q^4 + 5 * q^7 - 3 * q^8 - 9 * q^11 + 7 * q^13 - 4 * q^16 - 14 * q^17 - 3 * q^19 - 5 * q^22 - 2 * q^23 - 21 * q^26 + 6 * q^28 - 12 * q^29 - 6 * q^31 - 6 * q^32 + 7 * q^34 + 20 * q^38 - 7 * q^41 + 7 * q^43 - 37 * q^44 - 5 * q^46 - 13 * q^47 + 5 * q^49 - q^52 + 5 * q^53 - 3 * q^56 - 34 * q^58 - 26 * q^59 + 2 * q^61 + 19 * q^62 - 21 * q^64 - 21 * q^67 - 29 * q^68 - 40 * q^71 - 9 * q^73 - 38 * q^74 + 2 * q^76 - 9 * q^77 + 8 * q^79 + 9 * q^82 - 17 * q^83 - 9 * q^86 + 39 * q^88 - 11 * q^89 + 7 * q^91 + 43 * q^92 + 17 * q^94 - 4 * q^97

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