LMFDB - Embedded newform 4275.2.a.br.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4275,2,Mod(1,4275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4275, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4275 = 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4275.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$34.1360468641$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.66064384.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} + 13x^{2} - 1 $ x^6 - 9x^4 + 13x^2 - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.285442$ of defining polynomial
Character	$\chi$	$=$	4275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.906968 q^{2} -1.17741 q^{4} +2.59637 q^{7} -2.88181 q^{8} -0.741113 q^{11} -3.78878 q^{13} +2.35482 q^{14} -0.258887 q^{16} +3.16725 q^{17} -1.00000 q^{19} -0.672165 q^{22} -0.570885 q^{23} -3.43630 q^{26} -3.05699 q^{28} -6.00000 q^{29} +5.83705 q^{31} +5.52881 q^{32} +2.87259 q^{34} -1.40396 q^{37} -0.906968 q^{38} +3.83705 q^{41} -2.59637 q^{43} +0.872594 q^{44} -0.517774 q^{46} -5.08247 q^{47} -0.258887 q^{49} +4.46094 q^{52} +0.160905 q^{53} -7.48223 q^{56} -5.44181 q^{58} -8.35482 q^{59} -8.57816 q^{61} +5.29401 q^{62} +5.53223 q^{64} -14.8464 q^{67} -3.72915 q^{68} -3.64518 q^{71} -10.8461 q^{73} -1.27334 q^{74} +1.17741 q^{76} -1.92420 q^{77} +1.83705 q^{79} +3.48008 q^{82} +4.19876 q^{83} -2.35482 q^{86} +2.13574 q^{88} +16.9015 q^{89} -9.83705 q^{91} +0.672165 q^{92} -4.60963 q^{94} -3.78878 q^{97} -0.234802 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 8 q^{4} - 2 q^{11} - 16 q^{14} - 4 q^{16} - 6 q^{19} - 8 q^{26} - 36 q^{29} - 8 q^{34} - 12 q^{41} - 20 q^{44} - 8 q^{46} - 4 q^{49} - 40 q^{56} - 20 q^{59} - 14 q^{61} - 12 q^{64} - 52 q^{71} + 40 q^{74}+ \cdots - 48 q^{94}+O(q^{100}) $ 6 * q + 8 * q^4 - 2 * q^11 - 16 * q^14 - 4 * q^16 - 6 * q^19 - 8 * q^26 - 36 * q^29 - 8 * q^34 - 12 * q^41 - 20 * q^44 - 8 * q^46 - 4 * q^49 - 40 * q^56 - 20 * q^59 - 14 * q^61 - 12 * q^64 - 52 * q^71 + 40 * q^74 - 8 * q^76 - 24 * q^79 + 16 * q^86 - 24 * q^89 - 24 * q^91 - 48 * q^94

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 $

$2$	0.906968	0.641323	0.320661	−	0.947194i	$-0.396095\pi$
$2$	0.906968	0.641323	0.320661	+	0.947194i	$0.396095\pi$
$3$	0	0
$3$	0	0
$4$	−1.17741	−0.588705
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.59637	0.981334	0.490667	−	0.871347i	$-0.336753\pi$
$7$	2.59637	0.981334	0.490667	+	0.871347i	$0.336753\pi$
$8$	−2.88181	−1.01887
$9$	0	0
$10$	0	0
$11$	−0.741113	−0.223454	−0.111727	−	0.993739i	$-0.535638\pi$
$11$	−0.741113	−0.223454	−0.111727	+	0.993739i	$0.535638\pi$
$12$	0	0
$13$	−3.78878	−1.05082	−0.525409	−	0.850850i	$-0.676088\pi$
$13$	−3.78878	−1.05082	−0.525409	+	0.850850i	$0.676088\pi$
$14$	2.35482	0.629352
$15$	0	0
$16$	−0.258887	−0.0647218
$17$	3.16725	0.768171	0.384086	−	0.923298i	$-0.374517\pi$
$17$	3.16725	0.768171	0.384086	+	0.923298i	$0.374517\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	−0.672165	−0.143306
$23$	−0.570885	−0.119038	−0.0595189	−	0.998227i	$-0.518957\pi$
$23$	−0.570885	−0.119038	−0.0595189	+	0.998227i	$0.518957\pi$
$24$	0	0
$25$	0	0
$26$	−3.43630	−0.673913
$27$	0	0
$28$	−3.05699	−0.577716
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	5.83705	1.04836	0.524182	−	0.851606i	$-0.324371\pi$
$31$	5.83705	1.04836	0.524182	+	0.851606i	$0.324371\pi$
$32$	5.52881	0.977365
$33$	0	0
$34$	2.87259	0.492646
$35$	0	0
$36$	0	0
$37$	−1.40396	−0.230809	−0.115404	−	0.993319i	$-0.536816\pi$
$37$	−1.40396	−0.230809	−0.115404	+	0.993319i	$0.536816\pi$
$38$	−0.906968	−0.147130
$39$	0	0
$40$	0	0
$41$	3.83705	0.599246	0.299623	−	0.954058i	$-0.403139\pi$
$41$	3.83705	0.599246	0.299623	+	0.954058i	$0.403139\pi$
$42$	0	0
$43$	−2.59637	−0.395942	−0.197971	−	0.980208i	$-0.563435\pi$
$43$	−2.59637	−0.395942	−0.197971	+	0.980208i	$0.563435\pi$
$44$	0.872594	0.131548
$45$	0	0
$46$	−0.517774	−0.0763416
$47$	−5.08247	−0.741354	−0.370677	−	0.928762i	$-0.620874\pi$
$47$	−5.08247	−0.741354	−0.370677	+	0.928762i	$0.620874\pi$
$48$	0	0
$49$	−0.258887	−0.0369839
$50$	0	0
$51$	0	0
$52$	4.46094	0.618621
$53$	0.160905	0.0221020	0.0110510	−	0.999939i	$-0.496482\pi$
$53$	0.160905	0.0221020	0.0110510	+	0.999939i	$0.496482\pi$
$54$	0	0
$55$	0	0
$56$	−7.48223	−0.999854
$57$	0	0
$58$	−5.44181	−0.714544
$59$	−8.35482	−1.08770	−0.543852	−	0.839181i	$-0.683035\pi$
$59$	−8.35482	−1.08770	−0.543852	+	0.839181i	$0.683035\pi$
$60$	0	0
$61$	−8.57816	−1.09832	−0.549160	−	0.835717i	$-0.685052\pi$
$61$	−8.57816	−1.09832	−0.549160	+	0.835717i	$0.685052\pi$
$62$	5.29401	0.672340
$63$	0	0
$64$	5.53223	0.691529
$65$	0	0
$66$	0	0
$67$	−14.8464	−1.81378	−0.906888	−	0.421371i	$-0.861549\pi$
$67$	−14.8464	−1.81378	−0.906888	+	0.421371i	$0.861549\pi$
$68$	−3.72915	−0.452226
$69$	0	0
$70$	0	0
$71$	−3.64518	−0.432603	−0.216302	−	0.976327i	$-0.569399\pi$
$71$	−3.64518	−0.432603	−0.216302	+	0.976327i	$0.569399\pi$
$72$	0	0
$73$	−10.8461	−1.26944	−0.634719	−	0.772743i	$-0.718884\pi$
$73$	−10.8461	−1.26944	−0.634719	+	0.772743i	$0.718884\pi$
$74$	−1.27334	−0.148023
$75$	0	0
$76$	1.17741	0.135058
$77$	−1.92420	−0.219283
$78$	0	0
$79$	1.83705	0.206684	0.103342	−	0.994646i	$-0.467046\pi$
$79$	1.83705	0.206684	0.103342	+	0.994646i	$0.467046\pi$
$80$	0	0
$81$	0	0
$82$	3.48008	0.384310
$83$	4.19876	0.460873	0.230437	−	0.973087i	$-0.425985\pi$
$83$	4.19876	0.460873	0.230437	+	0.973087i	$0.425985\pi$
$84$	0	0
$85$	0	0
$86$	−2.35482	−0.253927
$87$	0	0
$88$	2.13574	0.227671
$89$	16.9015	1.79156	0.895778	−	0.444502i	$-0.146619\pi$
$89$	16.9015	1.79156	0.895778	+	0.444502i	$0.146619\pi$
$90$	0	0
$91$	−9.83705	−1.03120
$92$	0.672165	0.0700781
$93$	0	0
$94$	−4.60963	−0.475447
$95$	0	0
$96$	0	0
$97$	−3.78878	−0.384692	−0.192346	−	0.981327i	$-0.561610\pi$
$97$	−3.78878	−0.384692	−0.192346	+	0.981327i	$0.561610\pi$
$98$	−0.234802	−0.0237186
$99$	0	0
$100$	0	0
$101$	−8.35482	−0.831336	−0.415668	−	0.909517i	$-0.636452\pi$
$101$	−8.35482	−0.831336	−0.415668	+	0.909517i	$0.636452\pi$
$102$	0	0
$103$	2.07612	0.204566	0.102283	−	0.994755i	$-0.467385\pi$
$103$	2.07612	0.204566	0.102283	+	0.994755i	$0.467385\pi$
$104$	10.9185	1.07065
$105$	0	0
$106$	0.145935	0.0141745
$107$	5.70399	0.551426	0.275713	−	0.961240i	$-0.411086\pi$
$107$	5.70399	0.551426	0.275713	+	0.961240i	$0.411086\pi$
$108$	0	0
$109$	1.64518	0.157580	0.0787899	−	0.996891i	$-0.474894\pi$
$109$	1.64518	0.157580	0.0787899	+	0.996891i	$0.474894\pi$
$110$	0	0
$111$	0	0
$112$	−0.672165	−0.0635137
$113$	3.89006	0.365946	0.182973	−	0.983118i	$-0.441428\pi$
$113$	3.89006	0.365946	0.182973	+	0.983118i	$0.441428\pi$
$114$	0	0
$115$	0	0
$116$	7.06446	0.655918
$117$	0	0
$118$	−7.57755	−0.697570
$119$	8.22334	0.753832
$120$	0	0
$121$	−10.4508	−0.950068
$122$	−7.78011	−0.704378
$123$	0	0
$124$	−6.87259	−0.617177
$125$	0	0
$126$	0	0
$127$	−14.4233	−1.27986	−0.639931	−	0.768432i	$-0.721037\pi$
$127$	−14.4233	−1.27986	−0.639931	+	0.768432i	$0.721037\pi$
$128$	−6.04007	−0.533872
$129$	0	0
$130$	0	0
$131$	−9.96853	−0.870954	−0.435477	−	0.900200i	$-0.643420\pi$
$131$	−9.96853	−0.870954	−0.435477	+	0.900200i	$0.643420\pi$
$132$	0	0
$133$	−2.59637	−0.225133
$134$	−13.4652	−1.16322
$135$	0	0
$136$	−9.12741	−0.782669
$137$	9.70431	0.829095	0.414548	−	0.910028i	$-0.363940\pi$
$137$	9.70431	0.829095	0.414548	+	0.910028i	$0.363940\pi$
$138$	0	0
$139$	−13.4508	−1.14088	−0.570439	−	0.821340i	$-0.693227\pi$
$139$	−13.4508	−1.14088	−0.570439	+	0.821340i	$0.693227\pi$
$140$	0	0
$141$	0	0
$142$	−3.30606	−0.277438
$143$	2.80791	0.234809
$144$	0	0
$145$	0	0
$146$	−9.83705	−0.814120
$147$	0	0
$148$	1.65303	0.135878
$149$	−15.0959	−1.23671	−0.618353	−	0.785900i	$-0.712200\pi$
$149$	−15.0959	−1.23671	−0.618353	+	0.785900i	$0.712200\pi$
$150$	0	0
$151$	14.1919	1.15492	0.577459	−	0.816420i	$-0.304044\pi$
$151$	14.1919	1.15492	0.577459	+	0.816420i	$0.304044\pi$
$152$	2.88181	0.233745
$153$	0	0
$154$	−1.74519	−0.140631
$155$	0	0
$156$	0	0
$157$	−7.57755	−0.604754	−0.302377	−	0.953188i	$-0.597780\pi$
$157$	−7.57755	−0.604754	−0.302377	+	0.953188i	$0.597780\pi$
$158$	1.66614	0.132551
$159$	0	0
$160$	0	0
$161$	−1.48223	−0.116816
$162$	0	0
$163$	19.6757	1.54112	0.770559	−	0.637369i	$-0.219977\pi$
$163$	19.6757	1.54112	0.770559	+	0.637369i	$0.219977\pi$
$164$	−4.51777	−0.352779
$165$	0	0
$166$	3.80814	0.295569
$167$	−10.7954	−0.835376	−0.417688	−	0.908590i	$-0.637160\pi$
$167$	−10.7954	−0.835376	−0.417688	+	0.908590i	$0.637160\pi$
$168$	0	0
$169$	1.35482	0.104217
$170$	0	0
$171$	0	0
$172$	3.05699	0.233093
$173$	20.3895	1.55018	0.775092	−	0.631848i	$-0.217703\pi$
$173$	20.3895	1.55018	0.775092	+	0.631848i	$0.217703\pi$
$174$	0	0
$175$	0	0
$176$	0.191865	0.0144623
$177$	0	0
$178$	15.3291	1.14897
$179$	−25.0645	−1.87341	−0.936703	−	0.350126i	$-0.886139\pi$
$179$	−25.0645	−1.87341	−0.936703	+	0.350126i	$0.886139\pi$
$180$	0	0
$181$	−19.4193	−1.44342	−0.721712	−	0.692194i	$-0.756644\pi$
$181$	−19.4193	−1.44342	−0.721712	+	0.692194i	$0.756644\pi$
$182$	−8.92188	−0.661334
$183$	0	0
$184$	1.64518	0.121284
$185$	0	0
$186$	0	0
$187$	−2.34729	−0.171651
$188$	5.98414	0.436439
$189$	0	0
$190$	0	0
$191$	−11.4508	−0.828547	−0.414274	−	0.910152i	$-0.635964\pi$
$191$	−11.4508	−0.828547	−0.414274	+	0.910152i	$0.635964\pi$
$192$	0	0
$193$	−3.78878	−0.272722	−0.136361	−	0.990659i	$-0.543541\pi$
$193$	−3.78878	−0.272722	−0.136361	+	0.990659i	$0.543541\pi$
$194$	−3.43630	−0.246712
$195$	0	0
$196$	0.304816	0.0217726
$197$	2.28354	0.162695	0.0813477	−	0.996686i	$-0.474078\pi$
$197$	2.28354	0.162695	0.0813477	+	0.996686i	$0.474078\pi$
$198$	0	0
$199$	−19.4508	−1.37883	−0.689414	−	0.724368i	$-0.742132\pi$
$199$	−19.4508	−1.37883	−0.689414	+	0.724368i	$0.742132\pi$
$200$	0	0
$201$	0	0
$202$	−7.57755	−0.533155
$203$	−15.5782	−1.09337
$204$	0	0
$205$	0	0
$206$	1.88297	0.131193
$207$	0	0
$208$	0.980865	0.0680108
$209$	0.741113	0.0512639
$210$	0	0
$211$	11.2274	0.772927	0.386463	−	0.922305i	$-0.373697\pi$
$211$	11.2274	0.772927	0.386463	+	0.922305i	$0.373697\pi$
$212$	−0.189451	−0.0130115
$213$	0	0
$214$	5.17334	0.353642
$215$	0	0
$216$	0	0
$217$	15.1551	1.02880
$218$	1.49213	0.101059
$219$	0	0
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	4.03785	0.270394	0.135197	−	0.990819i	$-0.456833\pi$
$223$	4.03785	0.270394	0.135197	+	0.990819i	$0.456833\pi$
$224$	14.3548	0.959122
$225$	0	0
$226$	3.52815	0.234689
$227$	−11.2185	−0.744600	−0.372300	−	0.928112i	$-0.621431\pi$
$227$	−11.2185	−0.744600	−0.372300	+	0.928112i	$0.621431\pi$
$228$	0	0
$229$	16.1315	1.06600	0.532999	−	0.846116i	$-0.321065\pi$
$229$	16.1315	1.06600	0.532999	+	0.846116i	$0.321065\pi$
$230$	0	0
$231$	0	0
$232$	17.2908	1.13520
$233$	−2.12676	−0.139329	−0.0696644	−	0.997570i	$-0.522193\pi$
$233$	−2.12676	−0.139329	−0.0696644	+	0.997570i	$0.522193\pi$
$234$	0	0
$235$	0	0
$236$	9.83705	0.640337
$237$	0	0
$238$	7.45830	0.483450
$239$	14.4152	0.932442	0.466221	−	0.884668i	$-0.345615\pi$
$239$	14.4152	0.932442	0.466221	+	0.884668i	$0.345615\pi$
$240$	0	0
$241$	−0.162955	−0.0104968	−0.00524842	−	0.999986i	$-0.501671\pi$
$241$	−0.162955	−0.0104968	−0.00524842	+	0.999986i	$0.501671\pi$
$242$	−9.47849	−0.609301
$243$	0	0
$244$	10.1000	0.646587
$245$	0	0
$246$	0	0
$247$	3.78878	0.241074
$248$	−16.8212	−1.06815
$249$	0	0
$250$	0	0
$251$	−12.9330	−0.816322	−0.408161	−	0.912910i	$-0.633830\pi$
$251$	−12.9330	−0.816322	−0.408161	+	0.912910i	$0.633830\pi$
$252$	0	0
$253$	0.423090	0.0265995
$254$	−13.0815	−0.820805
$255$	0	0
$256$	−16.5426	−1.03391
$257$	−11.0445	−0.688938	−0.344469	−	0.938798i	$-0.611941\pi$
$257$	−11.0445	−0.688938	−0.344469	+	0.938798i	$0.611941\pi$
$258$	0	0
$259$	−3.64518	−0.226501
$260$	0	0
$261$	0	0
$262$	−9.04113	−0.558563
$263$	17.8527	1.10085	0.550424	−	0.834885i	$-0.314466\pi$
$263$	17.8527	1.10085	0.550424	+	0.834885i	$0.314466\pi$
$264$	0	0
$265$	0	0
$266$	−2.35482	−0.144383
$267$	0	0
$268$	17.4803	1.06778
$269$	−24.9934	−1.52387	−0.761936	−	0.647652i	$-0.775751\pi$
$269$	−24.9934	−1.52387	−0.761936	+	0.647652i	$0.775751\pi$
$270$	0	0
$271$	−23.8660	−1.44975	−0.724877	−	0.688879i	$-0.758103\pi$
$271$	−23.8660	−1.44975	−0.724877	+	0.688879i	$0.758103\pi$
$272$	−0.819960	−0.0497174
$273$	0	0
$274$	8.80150	0.531718
$275$	0	0
$276$	0	0
$277$	21.2315	1.27568	0.637840	−	0.770169i	$-0.279828\pi$
$277$	21.2315	1.27568	0.637840	+	0.770169i	$0.279828\pi$
$278$	−12.1994	−0.731671
$279$	0	0
$280$	0	0
$281$	−3.83705	−0.228899	−0.114449	−	0.993429i	$-0.536510\pi$
$281$	−3.83705	−0.228899	−0.114449	+	0.993429i	$0.536510\pi$
$282$	0	0
$283$	−0.211545	−0.0125751	−0.00628753	−	0.999980i	$-0.502001\pi$
$283$	−0.211545	−0.0125751	−0.00628753	+	0.999980i	$0.502001\pi$
$284$	4.29187	0.254676
$285$	0	0
$286$	2.54668	0.150589
$287$	9.96237	0.588060
$288$	0	0
$289$	−6.96853	−0.409913
$290$	0	0
$291$	0	0
$292$	12.7703	0.747324
$293$	−14.9942	−0.875970	−0.437985	−	0.898982i	$-0.644308\pi$
$293$	−14.9942	−0.875970	−0.437985	+	0.898982i	$0.644308\pi$
$294$	0	0
$295$	0	0
$296$	4.04593	0.235165
$297$	0	0
$298$	−13.6915	−0.793129
$299$	2.16295	0.125087
$300$	0	0
$301$	−6.74111	−0.388551
$302$	12.8716	0.740675
$303$	0	0
$304$	0.258887	0.0148482
$305$	0	0
$306$	0	0
$307$	−1.65303	−0.0943434	−0.0471717	−	0.998887i	$-0.515021\pi$
$307$	−1.65303	−0.0943434	−0.0471717	+	0.998887i	$0.515021\pi$
$308$	2.26557	0.129093
$309$	0	0
$310$	0	0
$311$	0.741113	0.0420247	0.0210123	−	0.999779i	$-0.493311\pi$
$311$	0.741113	0.0420247	0.0210123	+	0.999779i	$0.493311\pi$
$312$	0	0
$313$	26.8849	1.51962	0.759812	−	0.650143i	$-0.225291\pi$
$313$	26.8849	1.51962	0.759812	+	0.650143i	$0.225291\pi$
$314$	−6.87259	−0.387843
$315$	0	0
$316$	−2.16295	−0.121676
$317$	−8.16155	−0.458398	−0.229199	−	0.973380i	$-0.573611\pi$
$317$	−8.16155	−0.458398	−0.229199	+	0.973380i	$0.573611\pi$
$318$	0	0
$319$	4.44668	0.248966
$320$	0	0
$321$	0	0
$322$	−1.34433	−0.0749166
$323$	−3.16725	−0.176231
$324$	0	0
$325$	0	0
$326$	17.8452	0.988354
$327$	0	0
$328$	−11.0576	−0.610555
$329$	−13.1959	−0.727516
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	−4.94366	−0.271318
$333$	0	0
$334$	−9.79112	−0.535746
$335$	0	0
$336$	0	0
$337$	9.90275	0.539437	0.269718	−	0.962939i	$-0.413069\pi$
$337$	9.90275	0.539437	0.269718	+	0.962939i	$0.413069\pi$
$338$	1.22878	0.0668367
$339$	0	0
$340$	0	0
$341$	−4.32591	−0.234261
$342$	0	0
$343$	−18.8467	−1.01763
$344$	7.48223	0.403415
$345$	0	0
$346$	18.4926	0.994169
$347$	−21.2781	−1.14227	−0.571133	−	0.820858i	$-0.693496\pi$
$347$	−21.2781	−1.14227	−0.571133	+	0.820858i	$0.693496\pi$
$348$	0	0
$349$	−16.4152	−0.878686	−0.439343	−	0.898319i	$-0.644789\pi$
$349$	−16.4152	−0.878686	−0.439343	+	0.898319i	$0.644789\pi$
$350$	0	0
$351$	0	0
$352$	−4.09748	−0.218396
$353$	23.8744	1.27071	0.635354	−	0.772221i	$-0.280854\pi$
$353$	23.8744	1.27071	0.635354	+	0.772221i	$0.280854\pi$
$354$	0	0
$355$	0	0
$356$	−19.9000	−1.05470
$357$	0	0
$358$	−22.7327	−1.20146
$359$	−2.22334	−0.117343	−0.0586717	−	0.998277i	$-0.518687\pi$
$359$	−2.22334	−0.117343	−0.0586717	+	0.998277i	$0.518687\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−17.6127	−0.925701
$363$	0	0
$364$	11.5822	0.607074
$365$	0	0
$366$	0	0
$367$	4.52057	0.235972	0.117986	−	0.993015i	$-0.462356\pi$
$367$	4.52057	0.235972	0.117986	+	0.993015i	$0.462356\pi$
$368$	0.147795	0.00770433
$369$	0	0
$370$	0	0
$371$	0.417768	0.0216894
$372$	0	0
$373$	−15.5186	−0.803521	−0.401760	−	0.915745i	$-0.631602\pi$
$373$	−15.5186	−0.803521	−0.401760	+	0.915745i	$0.631602\pi$
$374$	−2.12892	−0.110084
$375$	0	0
$376$	14.6467	0.755345
$377$	22.7327	1.17079
$378$	0	0
$379$	−18.9015	−0.970905	−0.485453	−	0.874263i	$-0.661345\pi$
$379$	−18.9015	−0.970905	−0.485453	+	0.874263i	$0.661345\pi$
$380$	0	0
$381$	0	0
$382$	−10.3855	−0.531366
$383$	−13.7046	−0.700274	−0.350137	−	0.936699i	$-0.613865\pi$
$383$	−13.7046	−0.700274	−0.350137	+	0.936699i	$0.613865\pi$
$384$	0	0
$385$	0	0
$386$	−3.43630	−0.174903
$387$	0	0
$388$	4.46094	0.226470
$389$	−12.7411	−0.646000	−0.323000	−	0.946399i	$-0.604691\pi$
$389$	−12.7411	−0.646000	−0.323000	+	0.946399i	$0.604691\pi$
$390$	0	0
$391$	−1.80814	−0.0914413
$392$	0.746063	0.0376819
$393$	0	0
$394$	2.07110	0.104340
$395$	0	0
$396$	0	0
$397$	38.6522	1.93990	0.969950	−	0.243306i	$-0.0782319\pi$
$397$	38.6522	1.93990	0.969950	+	0.243306i	$0.0782319\pi$
$398$	−17.6412	−0.884274
$399$	0	0
$400$	0	0
$401$	−31.8660	−1.59131	−0.795655	−	0.605750i	$-0.792873\pi$
$401$	−31.8660	−1.59131	−0.795655	+	0.605750i	$0.792873\pi$
$402$	0	0
$403$	−22.1153	−1.10164
$404$	9.83705	0.489411
$405$	0	0
$406$	−14.1289	−0.701206
$407$	1.04049	0.0515751
$408$	0	0
$409$	11.0645	0.547102	0.273551	−	0.961857i	$-0.411802\pi$
$409$	11.0645	0.547102	0.273551	+	0.961857i	$0.411802\pi$
$410$	0	0
$411$	0	0
$412$	−2.44444	−0.120429
$413$	−21.6922	−1.06740
$414$	0	0
$415$	0	0
$416$	−20.9474	−1.02703
$417$	0	0
$418$	0.672165	0.0328767
$419$	25.7452	1.25773	0.628867	−	0.777513i	$-0.283519\pi$
$419$	25.7452	1.25773	0.628867	+	0.777513i	$0.283519\pi$
$420$	0	0
$421$	27.4482	1.33774	0.668871	−	0.743378i	$-0.266778\pi$
$421$	27.4482	1.33774	0.668871	+	0.743378i	$0.266778\pi$
$422$	10.1829	0.495696
$423$	0	0
$424$	−0.463697	−0.0225191
$425$	0	0
$426$	0	0
$427$	−22.2720	−1.07782
$428$	−6.71593	−0.324627
$429$	0	0
$430$	0	0
$431$	−1.74519	−0.0840627	−0.0420314	−	0.999116i	$-0.513383\pi$
$431$	−1.74519	−0.0840627	−0.0420314	+	0.999116i	$0.513383\pi$
$432$	0	0
$433$	18.5208	0.890052	0.445026	−	0.895518i	$-0.353194\pi$
$433$	18.5208	0.890052	0.445026	+	0.895518i	$0.353194\pi$
$434$	13.7452	0.659790
$435$	0	0
$436$	−1.93705	−0.0927679
$437$	0.570885	0.0273091
$438$	0	0
$439$	29.4482	1.40549	0.702743	−	0.711444i	$-0.251959\pi$
$439$	29.4482	1.40549	0.702743	+	0.711444i	$0.251959\pi$
$440$	0	0
$441$	0	0
$442$	−10.8836	−0.517681
$443$	11.7388	0.557726	0.278863	−	0.960331i	$-0.410042\pi$
$443$	11.7388	0.557726	0.278863	+	0.960331i	$0.410042\pi$
$444$	0	0
$445$	0	0
$446$	3.66220	0.173410
$447$	0	0
$448$	14.3637	0.678620
$449$	7.06446	0.333392	0.166696	−	0.986008i	$-0.446690\pi$
$449$	7.06446	0.333392	0.166696	+	0.986008i	$0.446690\pi$
$450$	0	0
$451$	−2.84368	−0.133904
$452$	−4.58019	−0.215434
$453$	0	0
$454$	−10.1748	−0.477529
$455$	0	0
$456$	0	0
$457$	34.5000	1.61384	0.806920	−	0.590660i	$-0.201133\pi$
$457$	34.5000	1.61384	0.806920	+	0.590660i	$0.201133\pi$
$458$	14.6307	0.683649
$459$	0	0
$460$	0	0
$461$	−8.03147	−0.374063	−0.187032	−	0.982354i	$-0.559887\pi$
$461$	−8.03147	−0.374063	−0.187032	+	0.982354i	$0.559887\pi$
$462$	0	0
$463$	25.3290	1.17714	0.588570	−	0.808447i	$-0.299691\pi$
$463$	25.3290	1.17714	0.588570	+	0.808447i	$0.299691\pi$
$464$	1.55332	0.0721112
$465$	0	0
$466$	−1.92890	−0.0893547
$467$	26.8759	1.24367	0.621834	−	0.783149i	$-0.286388\pi$
$467$	26.8759	1.24367	0.621834	+	0.783149i	$0.286388\pi$
$468$	0	0
$469$	−38.5467	−1.77992
$470$	0	0
$471$	0	0
$472$	24.0770	1.10823
$473$	1.92420	0.0884748
$474$	0	0
$475$	0	0
$476$	−9.68224	−0.443785
$477$	0	0
$478$	13.0741	0.597996
$479$	28.9015	1.32054	0.660272	−	0.751027i	$-0.270441\pi$
$479$	28.9015	1.32054	0.660272	+	0.751027i	$0.270441\pi$
$480$	0	0
$481$	5.31927	0.242538
$482$	−0.147795	−0.00673187
$483$	0	0
$484$	12.3048	0.559310
$485$	0	0
$486$	0	0
$487$	17.7294	0.803395	0.401697	−	0.915773i	$-0.368420\pi$
$487$	17.7294	0.803395	0.401697	+	0.915773i	$0.368420\pi$
$488$	24.7206	1.11905
$489$	0	0
$490$	0	0
$491$	35.1645	1.58695	0.793475	−	0.608603i	$-0.208270\pi$
$491$	35.1645	1.58695	0.793475	+	0.608603i	$0.208270\pi$
$492$	0	0
$493$	−19.0035	−0.855875
$494$	3.43630	0.154606
$495$	0	0
$496$	−1.51114	−0.0678520
$497$	−9.46422	−0.424528
$498$	0	0
$499$	21.4508	0.960268	0.480134	−	0.877195i	$-0.340588\pi$
$499$	21.4508	0.960268	0.480134	+	0.877195i	$0.340588\pi$
$500$	0	0
$501$	0	0
$502$	−11.7298	−0.523526
$503$	−5.34053	−0.238122	−0.119061	−	0.992887i	$-0.537988\pi$
$503$	−5.34053	−0.238122	−0.119061	+	0.992887i	$0.537988\pi$
$504$	0	0
$505$	0	0
$506$	0.383729	0.0170588
$507$	0	0
$508$	16.9821	0.753461
$509$	−36.1919	−1.60418	−0.802088	−	0.597206i	$-0.796278\pi$
$509$	−36.1919	−1.60418	−0.802088	+	0.597206i	$0.796278\pi$
$510$	0	0
$511$	−28.1604	−1.24574
$512$	−2.92346	−0.129200
$513$	0	0
$514$	−10.0170	−0.441832
$515$	0	0
$516$	0	0
$517$	3.76668	0.165658
$518$	−3.30606	−0.145260
$519$	0	0
$520$	0	0
$521$	2.77259	0.121469	0.0607346	−	0.998154i	$-0.480656\pi$
$521$	2.77259	0.121469	0.0607346	+	0.998154i	$0.480656\pi$
$522$	0	0
$523$	20.5373	0.898033	0.449016	−	0.893524i	$-0.351774\pi$
$523$	20.5373	0.898033	0.449016	+	0.893524i	$0.351774\pi$
$524$	11.7370	0.512735
$525$	0	0
$526$	16.1919	0.705999
$527$	18.4874	0.805323
$528$	0	0
$529$	−22.6741	−0.985830
$530$	0	0
$531$	0	0
$532$	3.05699	0.132537
$533$	−14.5377	−0.629698
$534$	0	0
$535$	0	0
$536$	42.7845	1.84801
$537$	0	0
$538$	−22.6682	−0.977294
$539$	0.191865	0.00826419
$540$	0	0
$541$	35.4797	1.52539	0.762695	−	0.646758i	$-0.223876\pi$
$541$	35.4797	1.52539	0.762695	+	0.646758i	$0.223876\pi$
$542$	−21.6456	−0.929760
$543$	0	0
$544$	17.5111	0.750784
$545$	0	0
$546$	0	0
$547$	−43.0756	−1.84178	−0.920890	−	0.389822i	$-0.872537\pi$
$547$	−43.0756	−1.84178	−0.920890	+	0.389822i	$0.872537\pi$
$548$	−11.4260	−0.488092
$549$	0	0
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	4.76964	0.202826
$554$	19.2563	0.818123
$555$	0	0
$556$	15.8370	0.671640
$557$	40.4376	1.71340	0.856698	−	0.515818i	$-0.172512\pi$
$557$	40.4376	1.71340	0.856698	+	0.515818i	$0.172512\pi$
$558$	0	0
$559$	9.83705	0.416063
$560$	0	0
$561$	0	0
$562$	−3.48008	−0.146798
$563$	19.7173	0.830986	0.415493	−	0.909596i	$-0.363609\pi$
$563$	19.7173	0.830986	0.415493	+	0.909596i	$0.363609\pi$
$564$	0	0
$565$	0	0
$566$	−0.191865	−0.00806467
$567$	0	0
$568$	10.5047	0.440768
$569$	−18.6807	−0.783137	−0.391568	−	0.920149i	$-0.628067\pi$
$569$	−18.6807	−0.783137	−0.391568	+	0.920149i	$0.628067\pi$
$570$	0	0
$571$	−29.9371	−1.25283	−0.626413	−	0.779491i	$-0.715478\pi$
$571$	−29.9371	−1.25283	−0.626413	+	0.779491i	$0.715478\pi$
$572$	−3.30606	−0.138233
$573$	0	0
$574$	9.03555	0.377137
$575$	0	0
$576$	0	0
$577$	0.156779	0.00652679	0.00326339	−	0.999995i	$-0.498961\pi$
$577$	0.156779	0.00652679	0.00326339	+	0.999995i	$0.498961\pi$
$578$	−6.32023	−0.262887
$579$	0	0
$580$	0	0
$581$	10.9015	0.452271
$582$	0	0
$583$	−0.119249	−0.00493877
$584$	31.2563	1.29340
$585$	0	0
$586$	−13.5993	−0.561780
$587$	31.1474	1.28559	0.642795	−	0.766038i	$-0.277775\pi$
$587$	31.1474	1.28559	0.642795	+	0.766038i	$0.277775\pi$
$588$	0	0
$589$	−5.83705	−0.240511
$590$	0	0
$591$	0	0
$592$	0.363466	0.0149384
$593$	28.8728	1.18567	0.592833	−	0.805326i	$-0.298009\pi$
$593$	28.8728	1.18567	0.592833	+	0.805326i	$0.298009\pi$
$594$	0	0
$595$	0	0
$596$	17.7741	0.728055
$597$	0	0
$598$	1.96173	0.0802211
$599$	−25.3274	−1.03485	−0.517425	−	0.855728i	$-0.673109\pi$
$599$	−25.3274	−1.03485	−0.517425	+	0.855728i	$0.673109\pi$
$600$	0	0
$601$	−19.8370	−0.809170	−0.404585	−	0.914500i	$-0.632584\pi$
$601$	−19.8370	−0.809170	−0.404585	+	0.914500i	$0.632584\pi$
$602$	−6.11397	−0.249187
$603$	0	0
$604$	−16.7096	−0.679906
$605$	0	0
$606$	0	0
$607$	2.49921	0.101440	0.0507199	−	0.998713i	$-0.483848\pi$
$607$	2.49921	0.101440	0.0507199	+	0.998713i	$0.483848\pi$
$608$	−5.52881	−0.224223
$609$	0	0
$610$	0	0
$611$	19.2563	0.779027
$612$	0	0
$613$	0.883711	0.0356927	0.0178464	−	0.999841i	$-0.494319\pi$
$613$	0.883711	0.0356927	0.0178464	+	0.999841i	$0.494319\pi$
$614$	−1.49925	−0.0605046
$615$	0	0
$616$	5.54517	0.223421
$617$	−29.4085	−1.18394	−0.591971	−	0.805959i	$-0.701650\pi$
$617$	−29.4085	−1.18394	−0.591971	+	0.805959i	$0.701650\pi$
$618$	0	0
$619$	30.3208	1.21870	0.609348	−	0.792903i	$-0.291431\pi$
$619$	30.3208	1.21870	0.609348	+	0.792903i	$0.291431\pi$
$620$	0	0
$621$	0	0
$622$	0.672165	0.0269514
$623$	43.8825	1.75811
$624$	0	0
$625$	0	0
$626$	24.3837	0.974570
$627$	0	0
$628$	8.92188	0.356022
$629$	−4.44668	−0.177301
$630$	0	0
$631$	17.7767	0.707678	0.353839	−	0.935306i	$-0.384876\pi$
$631$	17.7767	0.707678	0.353839	+	0.935306i	$0.384876\pi$
$632$	−5.29401	−0.210584
$633$	0	0
$634$	−7.40226	−0.293981
$635$	0	0
$636$	0	0
$637$	0.980865	0.0388633
$638$	4.03299	0.159668
$639$	0	0
$640$	0	0
$641$	32.6675	1.29029	0.645143	−	0.764062i	$-0.276798\pi$
$641$	32.6675	1.29029	0.645143	+	0.764062i	$0.276798\pi$
$642$	0	0
$643$	−31.8661	−1.25668	−0.628338	−	0.777941i	$-0.716264\pi$
$643$	−31.8661	−1.25668	−0.628338	+	0.777941i	$0.716264\pi$
$644$	1.74519	0.0687700
$645$	0	0
$646$	−2.87259	−0.113021
$647$	21.2601	0.835820	0.417910	−	0.908488i	$-0.362763\pi$
$647$	21.2601	0.835820	0.417910	+	0.908488i	$0.362763\pi$
$648$	0	0
$649$	6.19186	0.243052
$650$	0	0
$651$	0	0
$652$	−23.1663	−0.907263
$653$	12.8340	0.502234	0.251117	−	0.967957i	$-0.419202\pi$
$653$	12.8340	0.502234	0.251117	+	0.967957i	$0.419202\pi$
$654$	0	0
$655$	0	0
$656$	−0.993361	−0.0387842
$657$	0	0
$658$	−11.9683	−0.466573
$659$	−20.3548	−0.792911	−0.396456	−	0.918054i	$-0.629760\pi$
$659$	−20.3548	−0.792911	−0.396456	+	0.918054i	$0.629760\pi$
$660$	0	0
$661$	−30.7385	−1.19559	−0.597795	−	0.801649i	$-0.703957\pi$
$661$	−30.7385	−1.19559	−0.597795	+	0.801649i	$0.703957\pi$
$662$	7.25574	0.282002
$663$	0	0
$664$	−12.1000	−0.469571
$665$	0	0
$666$	0	0
$667$	3.42531	0.132629
$668$	12.7107	0.491790
$669$	0	0
$670$	0	0
$671$	6.35738	0.245424
$672$	0	0
$673$	−21.2094	−0.817564	−0.408782	−	0.912632i	$-0.634046\pi$
$673$	−21.2094	−0.817564	−0.408782	+	0.912632i	$0.634046\pi$
$674$	8.98147	0.345953
$675$	0	0
$676$	−1.59518	−0.0613530
$677$	11.2650	0.432951	0.216475	−	0.976288i	$-0.430544\pi$
$677$	11.2650	0.432951	0.216475	+	0.976288i	$0.430544\pi$
$678$	0	0
$679$	−9.83705	−0.377511
$680$	0	0
$681$	0	0
$682$	−3.92346	−0.150237
$683$	−12.3603	−0.472954	−0.236477	−	0.971637i	$-0.575993\pi$
$683$	−12.3603	−0.472954	−0.236477	+	0.971637i	$0.575993\pi$
$684$	0	0
$685$	0	0
$686$	−17.0934	−0.652628
$687$	0	0
$688$	0.672165	0.0256261
$689$	−0.609632	−0.0232251
$690$	0	0
$691$	22.7493	0.865423	0.432711	−	0.901533i	$-0.357557\pi$
$691$	22.7493	0.865423	0.432711	+	0.901533i	$0.357557\pi$
$692$	−24.0068	−0.912601
$693$	0	0
$694$	−19.2985	−0.732561
$695$	0	0
$696$	0	0
$697$	12.1529	0.460323
$698$	−14.8881	−0.563521
$699$	0	0
$700$	0	0
$701$	16.0289	0.605404	0.302702	−	0.953085i	$-0.402111\pi$
$701$	16.0289	0.605404	0.302702	+	0.953085i	$0.402111\pi$
$702$	0	0
$703$	1.40396	0.0529512
$704$	−4.10001	−0.154525
$705$	0	0
$706$	21.6533	0.814934
$707$	−21.6922	−0.815818
$708$	0	0
$709$	31.4193	1.17998	0.589988	−	0.807412i	$-0.299133\pi$
$709$	31.4193	1.17998	0.589988	+	0.807412i	$0.299133\pi$
$710$	0	0
$711$	0	0
$712$	−48.7069	−1.82537
$713$	−3.33228	−0.124795
$714$	0	0
$715$	0	0
$716$	29.5111	1.10288
$717$	0	0
$718$	−2.01650	−0.0752550
$719$	−11.2589	−0.419886	−0.209943	−	0.977714i	$-0.567328\pi$
$719$	−11.2589	−0.419886	−0.209943	+	0.977714i	$0.567328\pi$
$720$	0	0
$721$	5.39037	0.200748
$722$	0.906968	0.0337538
$723$	0	0
$724$	22.8644	0.849750
$725$	0	0
$726$	0	0
$727$	−48.9829	−1.81668	−0.908338	−	0.418237i	$-0.862648\pi$
$727$	−48.9829	−1.81668	−0.908338	+	0.418237i	$0.862648\pi$
$728$	28.3485	1.05066
$729$	0	0
$730$	0	0
$731$	−8.22334	−0.304151
$732$	0	0
$733$	35.9260	1.32696	0.663479	−	0.748195i	$-0.269079\pi$
$733$	35.9260	1.32696	0.663479	+	0.748195i	$0.269079\pi$
$734$	4.10001	0.151334
$735$	0	0
$736$	−3.15632	−0.116343
$737$	11.0029	0.405296
$738$	0	0
$739$	14.3523	0.527956	0.263978	−	0.964529i	$-0.414965\pi$
$739$	14.3523	0.527956	0.263978	+	0.964529i	$0.414965\pi$
$740$	0	0
$741$	0	0
$742$	0.378902	0.0139099
$743$	12.5629	0.460887	0.230443	−	0.973086i	$-0.425982\pi$
$743$	12.5629	0.460887	0.230443	+	0.973086i	$0.425982\pi$
$744$	0	0
$745$	0	0
$746$	−14.0748	−0.515316
$747$	0	0
$748$	2.76372	0.101052
$749$	14.8096	0.541133
$750$	0	0
$751$	26.4548	0.965350	0.482675	−	0.875799i	$-0.339665\pi$
$751$	26.4548	0.965350	0.482675	+	0.875799i	$0.339665\pi$
$752$	1.31578	0.0479817
$753$	0	0
$754$	20.6178	0.750855
$755$	0	0
$756$	0	0
$757$	15.7350	0.571897	0.285949	−	0.958245i	$-0.407691\pi$
$757$	15.7350	0.571897	0.285949	+	0.958245i	$0.407691\pi$
$758$	−17.1431	−0.622664
$759$	0	0
$760$	0	0
$761$	−16.9619	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$761$	−16.9619	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$762$	0	0
$763$	4.27149	0.154638
$764$	13.4822	0.487770
$765$	0	0
$766$	−12.4297	−0.449102
$767$	31.6545	1.14298
$768$	0	0
$769$	41.9974	1.51447	0.757233	−	0.653145i	$-0.226551\pi$
$769$	41.9974	1.51447	0.757233	+	0.653145i	$0.226551\pi$
$770$	0	0
$771$	0	0
$772$	4.46094	0.160553
$773$	−40.9579	−1.47315	−0.736576	−	0.676355i	$-0.763559\pi$
$773$	−40.9579	−1.47315	−0.736576	+	0.676355i	$0.763559\pi$
$774$	0	0
$775$	0	0
$776$	10.9185	0.391952
$777$	0	0
$778$	−11.5558	−0.414295
$779$	−3.83705	−0.137476
$780$	0	0
$781$	2.70149	0.0966669
$782$	−1.63992	−0.0586434
$783$	0	0
$784$	0.0670225	0.00239366
$785$	0	0
$786$	0	0
$787$	28.5379	1.01727	0.508634	−	0.860983i	$-0.330151\pi$
$787$	28.5379	1.01727	0.508634	+	0.860983i	$0.330151\pi$
$788$	−2.68866	−0.0957796
$789$	0	0
$790$	0	0
$791$	10.1000	0.359115
$792$	0	0
$793$	32.5007	1.15413
$794$	35.0563	1.24410
$795$	0	0
$796$	22.9015	0.811722
$797$	−33.2790	−1.17880	−0.589402	−	0.807840i	$-0.700636\pi$
$797$	−33.2790	−1.17880	−0.589402	+	0.807840i	$0.700636\pi$
$798$	0	0
$799$	−16.0974	−0.569487
$800$	0	0
$801$	0	0
$802$	−28.9014	−1.02054
$803$	8.03817	0.283661
$804$	0	0
$805$	0	0
$806$	−20.0578	−0.706507
$807$	0	0
$808$	24.0770	0.847025
$809$	34.4234	1.21026	0.605130	−	0.796126i	$-0.293121\pi$
$809$	34.4234	1.21026	0.605130	+	0.796126i	$0.293121\pi$
$810$	0	0
$811$	−11.2563	−0.395263	−0.197631	−	0.980276i	$-0.563325\pi$
$811$	−11.2563	−0.395263	−0.197631	+	0.980276i	$0.563325\pi$
$812$	18.3419	0.643675
$813$	0	0
$814$	0.943690	0.0330763
$815$	0	0
$816$	0	0
$817$	2.59637	0.0908353
$818$	10.0351	0.350869
$819$	0	0
$820$	0	0
$821$	31.3167	1.09296	0.546480	−	0.837472i	$-0.315967\pi$
$821$	31.3167	1.09296	0.546480	+	0.837472i	$0.315967\pi$
$822$	0	0
$823$	−13.4800	−0.469882	−0.234941	−	0.972010i	$-0.575490\pi$
$823$	−13.4800	−0.469882	−0.234941	+	0.972010i	$0.575490\pi$
$824$	−5.98298	−0.208427
$825$	0	0
$826$	−19.6741	−0.684549
$827$	−15.9882	−0.555963	−0.277982	−	0.960586i	$-0.589665\pi$
$827$	−15.9882	−0.555963	−0.277982	+	0.960586i	$0.589665\pi$
$828$	0	0
$829$	−48.4837	−1.68391	−0.841955	−	0.539548i	$-0.818595\pi$
$829$	−48.4837	−1.68391	−0.841955	+	0.539548i	$0.818595\pi$
$830$	0	0
$831$	0	0
$832$	−20.9604	−0.726670
$833$	−0.819960	−0.0284099
$834$	0	0
$835$	0	0
$836$	−0.872594	−0.0301793
$837$	0	0
$838$	23.3501	0.806614
$839$	18.9934	0.655724	0.327862	−	0.944726i	$-0.393672\pi$
$839$	18.9934	0.655724	0.327862	+	0.944726i	$0.393672\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	24.8946	0.857925
$843$	0	0
$844$	−13.2193	−0.455026
$845$	0	0
$846$	0	0
$847$	−27.1340	−0.932334
$848$	−0.0416562	−0.00143048
$849$	0	0
$850$	0	0
$851$	0.801497	0.0274750
$852$	0	0
$853$	−44.1210	−1.51067	−0.755337	−	0.655337i	$-0.772527\pi$
$853$	−44.1210	−1.51067	−0.755337	+	0.655337i	$0.772527\pi$
$854$	−20.2000	−0.691230
$855$	0	0
$856$	−16.4378	−0.561833
$857$	32.4149	1.10727	0.553635	−	0.832759i	$-0.313240\pi$
$857$	32.4149	1.10727	0.553635	+	0.832759i	$0.313240\pi$
$858$	0	0
$859$	6.29444	0.214763	0.107382	−	0.994218i	$-0.465753\pi$
$859$	6.29444	0.214763	0.107382	+	0.994218i	$0.465753\pi$
$860$	0	0
$861$	0	0
$862$	−1.58283	−0.0539113
$863$	−17.4338	−0.593453	−0.296726	−	0.954963i	$-0.595895\pi$
$863$	−17.4338	−0.593453	−0.296726	+	0.954963i	$0.595895\pi$
$864$	0	0
$865$	0	0
$866$	16.7978	0.570811
$867$	0	0
$868$	−17.8438	−0.605657
$869$	−1.36146	−0.0461843
$870$	0	0
$871$	56.2497	1.90595
$872$	−4.74109	−0.160554
$873$	0	0
$874$	0.517774	0.0175140
$875$	0	0
$876$	0	0
$877$	−23.2904	−0.786462	−0.393231	−	0.919440i	$-0.628643\pi$
$877$	−23.2904	−0.786462	−0.393231	+	0.919440i	$0.628643\pi$
$878$	26.7086	0.901370
$879$	0	0
$880$	0	0
$881$	−28.9619	−0.975751	−0.487875	−	0.872913i	$-0.662228\pi$
$881$	−28.9619	−0.975751	−0.487875	+	0.872913i	$0.662228\pi$
$882$	0	0
$883$	37.3627	1.25735	0.628677	−	0.777667i	$-0.283597\pi$
$883$	37.3627	1.25735	0.628677	+	0.777667i	$0.283597\pi$
$884$	14.1289	0.475207
$885$	0	0
$886$	10.6467	0.357683
$887$	−23.0234	−0.773050	−0.386525	−	0.922279i	$-0.626325\pi$
$887$	−23.0234	−0.773050	−0.386525	+	0.922279i	$0.626325\pi$
$888$	0	0
$889$	−37.4482	−1.25597
$890$	0	0
$891$	0	0
$892$	−4.75420	−0.159183
$893$	5.08247	0.170078
$894$	0	0
$895$	0	0
$896$	−15.6822	−0.523907
$897$	0	0
$898$	6.40723	0.213812
$899$	−35.0223	−1.16806
$900$	0	0
$901$	0.509626	0.0169781
$902$	−2.57913	−0.0858756
$903$	0	0
$904$	−11.2104	−0.372852
$905$	0	0
$906$	0	0
$907$	35.2693	1.17110	0.585549	−	0.810637i	$-0.300879\pi$
$907$	35.2693	1.17110	0.585549	+	0.810637i	$0.300879\pi$
$908$	13.2088	0.438350
$909$	0	0
$910$	0	0
$911$	4.97260	0.164750	0.0823748	−	0.996601i	$-0.473750\pi$
$911$	4.97260	0.164750	0.0823748	+	0.996601i	$0.473750\pi$
$912$	0	0
$913$	−3.11175	−0.102984
$914$	31.2904	1.03499
$915$	0	0
$916$	−18.9934	−0.627558
$917$	−25.8819	−0.854697
$918$	0	0
$919$	−48.7096	−1.60678	−0.803391	−	0.595451i	$-0.796973\pi$
$919$	−48.7096	−1.60678	−0.803391	+	0.595451i	$0.796973\pi$
$920$	0	0
$921$	0	0
$922$	−7.28429	−0.239895
$923$	13.8108	0.454587
$924$	0	0
$925$	0	0
$926$	22.9726	0.754926
$927$	0	0
$928$	−33.1729	−1.08895
$929$	−32.5126	−1.06671	−0.533353	−	0.845893i	$-0.679068\pi$
$929$	−32.5126	−1.06671	−0.533353	+	0.845893i	$0.679068\pi$
$930$	0	0
$931$	0.258887	0.00848468
$932$	2.50407	0.0820235
$933$	0	0
$934$	24.3756	0.797593
$935$	0	0
$936$	0	0
$937$	0.385560	0.0125957	0.00629785	−	0.999980i	$-0.497995\pi$
$937$	0.385560	0.0125957	0.00629785	+	0.999980i	$0.497995\pi$
$938$	−34.9606	−1.14150
$939$	0	0
$940$	0	0
$941$	−45.3482	−1.47831	−0.739154	−	0.673536i	$-0.764775\pi$
$941$	−45.3482	−1.47831	−0.739154	+	0.673536i	$0.764775\pi$
$942$	0	0
$943$	−2.19051	−0.0713329
$944$	2.16295	0.0703982
$945$	0	0
$946$	1.74519	0.0567409
$947$	9.61202	0.312349	0.156174	−	0.987730i	$-0.450084\pi$
$947$	9.61202	0.312349	0.156174	+	0.987730i	$0.450084\pi$
$948$	0	0
$949$	41.0934	1.33395
$950$	0	0
$951$	0	0
$952$	−23.6981	−0.768059
$953$	−59.8421	−1.93848	−0.969238	−	0.246126i	$-0.920842\pi$
$953$	−59.8421	−1.93848	−0.969238	+	0.246126i	$0.920842\pi$
$954$	0	0
$955$	0	0
$956$	−16.9726	−0.548933
$957$	0	0
$958$	26.2127	0.846895
$959$	25.1959	0.813619
$960$	0	0
$961$	3.07110	0.0990676
$962$	4.82441	0.155545
$963$	0	0
$964$	0.191865	0.00617954
$965$	0	0
$966$	0	0
$967$	0.368324	0.0118445	0.00592225	−	0.999982i	$-0.498115\pi$
$967$	0.368324	0.0118445	0.00592225	+	0.999982i	$0.498115\pi$
$968$	30.1171	0.967999
$969$	0	0
$970$	0	0
$971$	45.8370	1.47098	0.735490	−	0.677535i	$-0.236952\pi$
$971$	45.8370	1.47098	0.735490	+	0.677535i	$0.236952\pi$
$972$	0	0
$973$	−34.9231	−1.11958
$974$	16.0800	0.515235
$975$	0	0
$976$	2.22077	0.0710853
$977$	29.0337	0.928872	0.464436	−	0.885607i	$-0.346257\pi$
$977$	29.0337	0.928872	0.464436	+	0.885607i	$0.346257\pi$
$978$	0	0
$979$	−12.5259	−0.400330
$980$	0	0
$981$	0	0
$982$	31.8930	1.01775
$983$	−11.0160	−0.351355	−0.175677	−	0.984448i	$-0.556212\pi$
$983$	−11.0160	−0.351355	−0.175677	+	0.984448i	$0.556212\pi$
$984$	0	0
$985$	0	0
$986$	−17.2356	−0.548892
$987$	0	0
$988$	−4.46094	−0.141921
$989$	1.48223	0.0471320
$990$	0	0
$991$	−25.6822	−0.815823	−0.407912	−	0.913021i	$-0.633743\pi$
$991$	−25.6822	−0.815823	−0.407912	+	0.913021i	$0.633743\pi$
$992$	32.2719	1.02463
$993$	0	0
$994$	−8.58374	−0.272260
$995$	0	0
$996$	0	0
$997$	−20.3854	−0.645611	−0.322805	−	0.946465i	$-0.604626\pi$
$997$	−20.3854	−0.645611	−0.322805	+	0.946465i	$0.604626\pi$
$998$	19.4551	0.615842
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.br.1.4		6
3.2	odd	2		475.2.a.j.1.3		6
5.2	odd	4		855.2.c.d.514.4		6
5.3	odd	4		855.2.c.d.514.3		6
5.4	even	2	inner	4275.2.a.br.1.3		6
12.11	even	2		7600.2.a.ck.1.1		6
15.2	even	4		95.2.b.b.39.3	✓	6
15.8	even	4		95.2.b.b.39.4	yes	6
15.14	odd	2		475.2.a.j.1.4		6
57.56	even	2		9025.2.a.bx.1.4		6
60.23	odd	4		1520.2.d.h.609.1		6
60.47	odd	4		1520.2.d.h.609.6		6
60.59	even	2		7600.2.a.ck.1.6		6
285.113	odd	4		1805.2.b.e.1084.3		6
285.227	odd	4		1805.2.b.e.1084.4		6
285.284	even	2		9025.2.a.bx.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.b.b.39.3	✓	6	15.2	even	4
95.2.b.b.39.4	yes	6	15.8	even	4
475.2.a.j.1.3		6	3.2	odd	2
475.2.a.j.1.4		6	15.14	odd	2
855.2.c.d.514.3		6	5.3	odd	4
855.2.c.d.514.4		6	5.2	odd	4
1520.2.d.h.609.1		6	60.23	odd	4
1520.2.d.h.609.6		6	60.47	odd	4
1805.2.b.e.1084.3		6	285.113	odd	4
1805.2.b.e.1084.4		6	285.227	odd	4
4275.2.a.br.1.3		6	5.4	even	2	inner
4275.2.a.br.1.4		6	1.1	even	1	trivial
7600.2.a.ck.1.1		6	12.11	even	2
7600.2.a.ck.1.6		6	60.59	even	2
9025.2.a.bx.1.3		6	285.284	even	2
9025.2.a.bx.1.4		6	57.56	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4275.a (trivial)

\(f(q)\)	\(=\)	\(q+0.906968 q^{2} -1.17741 q^{4} +2.59637 q^{7} -2.88181 q^{8} -0.741113 q^{11} -3.78878 q^{13} +2.35482 q^{14} -0.258887 q^{16} +3.16725 q^{17} -1.00000 q^{19} -0.672165 q^{22} -0.570885 q^{23} -3.43630 q^{26} -3.05699 q^{28} -6.00000 q^{29} +5.83705 q^{31} +5.52881 q^{32} +2.87259 q^{34} -1.40396 q^{37} -0.906968 q^{38} +3.83705 q^{41} -2.59637 q^{43} +0.872594 q^{44} -0.517774 q^{46} -5.08247 q^{47} -0.258887 q^{49} +4.46094 q^{52} +0.160905 q^{53} -7.48223 q^{56} -5.44181 q^{58} -8.35482 q^{59} -8.57816 q^{61} +5.29401 q^{62} +5.53223 q^{64} -14.8464 q^{67} -3.72915 q^{68} -3.64518 q^{71} -10.8461 q^{73} -1.27334 q^{74} +1.17741 q^{76} -1.92420 q^{77} +1.83705 q^{79} +3.48008 q^{82} +4.19876 q^{83} -2.35482 q^{86} +2.13574 q^{88} +16.9015 q^{89} -9.83705 q^{91} +0.672165 q^{92} -4.60963 q^{94} -3.78878 q^{97} -0.234802 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{4} - 2 q^{11} - 16 q^{14} - 4 q^{16} - 6 q^{19} - 8 q^{26} - 36 q^{29} - 8 q^{34} - 12 q^{41} - 20 q^{44} - 8 q^{46} - 4 q^{49} - 40 q^{56} - 20 q^{59} - 14 q^{61} - 12 q^{64} - 52 q^{71} + 40 q^{74}+ \cdots - 48 q^{94}+O(q^{100}) \) 6 * q + 8 * q^4 - 2 * q^11 - 16 * q^14 - 4 * q^16 - 6 * q^19 - 8 * q^26 - 36 * q^29 - 8 * q^34 - 12 * q^41 - 20 * q^44 - 8 * q^46 - 4 * q^49 - 40 * q^56 - 20 * q^59 - 14 * q^61 - 12 * q^64 - 52 * q^71 + 40 * q^74 - 8 * q^76 - 24 * q^79 + 16 * q^86 - 24 * q^89 - 24 * q^91 - 48 * q^94

Introduction

L-functions

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