LMFDB - Embedded newform 475.2.a.j.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(1,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	475.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:	$3.79289409601$
Analytic rank:	$0$
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	$\chi$	$=$	475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.906968 q^{2} -3.21789 q^{3} -1.17741 q^{4} -2.91852 q^{6} -2.59637 q^{7} -2.88181 q^{8} +7.35482 q^{9} +0.741113 q^{11} +3.78878 q^{12} +3.78878 q^{13} -2.35482 q^{14} -0.258887 q^{16} +3.16725 q^{17} +6.67058 q^{18} -1.00000 q^{19} +8.35482 q^{21} +0.672165 q^{22} -0.570885 q^{23} +9.27334 q^{24} +3.43630 q^{26} -14.0133 q^{27} +3.05699 q^{28} +6.00000 q^{29} +5.83705 q^{31} +5.52881 q^{32} -2.38482 q^{33} +2.87259 q^{34} -8.65964 q^{36} +1.40396 q^{37} -0.906968 q^{38} -12.1919 q^{39} -3.83705 q^{41} +7.57755 q^{42} +2.59637 q^{43} -0.872594 q^{44} -0.517774 q^{46} -5.08247 q^{47} +0.833070 q^{48} -0.258887 q^{49} -10.1919 q^{51} -4.46094 q^{52} +0.160905 q^{53} -12.7096 q^{54} +7.48223 q^{56} +3.21789 q^{57} +5.44181 q^{58} +8.35482 q^{59} -8.57816 q^{61} +5.29401 q^{62} -19.0958 q^{63} +5.53223 q^{64} -2.16295 q^{66} +14.8464 q^{67} -3.72915 q^{68} +1.83705 q^{69} +3.64518 q^{71} -21.1952 q^{72} +10.8461 q^{73} +1.27334 q^{74} +1.17741 q^{76} -1.92420 q^{77} -11.0576 q^{78} +1.83705 q^{79} +23.0289 q^{81} -3.48008 q^{82} +4.19876 q^{83} -9.83705 q^{84} +2.35482 q^{86} -19.3073 q^{87} -2.13574 q^{88} -16.9015 q^{89} -9.83705 q^{91} +0.672165 q^{92} -18.7830 q^{93} -4.60963 q^{94} -17.7911 q^{96} +3.78878 q^{97} -0.234802 q^{98} +5.45075 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 8 q^{4} + 14 q^{9} + 2 q^{11} + 16 q^{14} - 4 q^{16} - 6 q^{19} + 20 q^{21} + 8 q^{24} + 8 q^{26} + 36 q^{29} - 8 q^{34} - 32 q^{36} - 8 q^{39} + 12 q^{41} + 20 q^{44} - 8 q^{46} - 4 q^{49} + 4 q^{51}+ \cdots - 30 q^{99}+O(q^{100}) $ 6 * q + 8 * q^4 + 14 * q^9 + 2 * q^11 + 16 * q^14 - 4 * q^16 - 6 * q^19 + 20 * q^21 + 8 * q^24 + 8 * q^26 + 36 * q^29 - 8 * q^34 - 32 * q^36 - 8 * q^39 + 12 * q^41 + 20 * q^44 - 8 * q^46 - 4 * q^49 + 4 * q^51 - 16 * q^54 + 40 * q^56 + 20 * q^59 - 14 * q^61 - 12 * q^64 - 48 * q^66 - 24 * q^69 + 52 * q^71 - 40 * q^74 - 8 * q^76 - 24 * q^79 + 38 * q^81 - 24 * q^84 - 16 * q^86 + 24 * q^89 - 24 * q^91 - 48 * q^94 - 64 * q^96 - 30 * q^99

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	$0.396095\pi$
$3$	−3.21789	−1.85785	−0.928925	−	0.370268i	$-0.879266\pi$
$3$	−3.21789	−1.85785	−0.928925	+	0.370268i	$0.879266\pi$
$4$	−1.17741	−0.588705
$5$	0	0
$5$	0	0
$6$	−2.91852	−1.19148
$7$	−2.59637	−0.981334	−0.490667	−	0.871347i	$-0.663247\pi$
$7$	−2.59637	−0.981334	−0.490667	+	0.871347i	$0.663247\pi$
$8$	−2.88181	−1.01887
$9$	7.35482	2.45161
$10$	0	0
$11$	0.741113	0.223454	0.111727	−	0.993739i	$-0.464362\pi$
$11$	0.741113	0.223454	0.111727	+	0.993739i	$0.464362\pi$
$12$	3.78878	1.09373
$13$	3.78878	1.05082	0.525409	−	0.850850i	$-0.323912\pi$
$13$	3.78878	1.05082	0.525409	+	0.850850i	$0.323912\pi$
$14$	−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	$0.374517\pi$
$18$	6.67058	1.57227
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	8.35482	1.82317
$22$	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	$0.518957\pi$
$24$	9.27334	1.89291
$25$	0	0
$26$	3.43630	0.673913
$27$	−14.0133	−2.69687
$28$	3.05699	0.577716
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	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
$33$	−2.38482	−0.415144
$34$	2.87259	0.492646
$35$	0	0
$36$	−8.65964	−1.44327
$37$	1.40396	0.230809	0.115404	−	0.993319i	$-0.463184\pi$
$37$	1.40396	0.230809	0.115404	+	0.993319i	$0.463184\pi$
$38$	−0.906968	−0.147130
$39$	−12.1919	−1.95226
$40$	0	0
$41$	−3.83705	−0.599246	−0.299623	−	0.954058i	$-0.596861\pi$
$41$	−3.83705	−0.599246	−0.299623	+	0.954058i	$0.596861\pi$
$42$	7.57755	1.16924
$43$	2.59637	0.395942	0.197971	−	0.980208i	$-0.436565\pi$
$43$	2.59637	0.395942	0.197971	+	0.980208i	$0.436565\pi$
$44$	−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	$0.620874\pi$
$48$	0.833070	0.120243
$49$	−0.258887	−0.0369839
$50$	0	0
$51$	−10.1919	−1.42715
$52$	−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	$0.496482\pi$
$54$	−12.7096	−1.72956
$55$	0	0
$56$	7.48223	0.999854
$57$	3.21789	0.426220
$58$	5.44181	0.714544
$59$	8.35482	1.08770	0.543852	−	0.839181i	$-0.316965\pi$
$59$	8.35482	1.08770	0.543852	+	0.839181i	$0.316965\pi$
$60$	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
$63$	−19.0958	−2.40584
$64$	5.53223	0.691529
$65$	0	0
$66$	−2.16295	−0.266241
$67$	14.8464	1.81378	0.906888	−	0.421371i	$-0.138451\pi$
$67$	14.8464	1.81378	0.906888	+	0.421371i	$0.138451\pi$
$68$	−3.72915	−0.452226
$69$	1.83705	0.221154
$70$	0	0
$71$	3.64518	0.432603	0.216302	−	0.976327i	$-0.430601\pi$
$71$	3.64518	0.432603	0.216302	+	0.976327i	$0.430601\pi$
$72$	−21.1952	−2.49788
$73$	10.8461	1.26944	0.634719	−	0.772743i	$-0.281116\pi$
$73$	10.8461	1.26944	0.634719	+	0.772743i	$0.281116\pi$
$74$	1.27334	0.148023
$75$	0	0
$76$	1.17741	0.135058
$77$	−1.92420	−0.219283
$78$	−11.0576	−1.25203
$79$	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
$81$	23.0289	2.55877
$82$	−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	$0.425985\pi$
$84$	−9.83705	−1.07331
$85$	0	0
$86$	2.35482	0.253927
$87$	−19.3073	−2.06996
$88$	−2.13574	−0.227671
$89$	−16.9015	−1.79156	−0.895778	−	0.444502i	$-0.853381\pi$
$89$	−16.9015	−1.79156	−0.895778	+	0.444502i	$0.853381\pi$
$90$	0	0
$91$	−9.83705	−1.03120
$92$	0.672165	0.0700781
$93$	−18.7830	−1.94770
$94$	−4.60963	−0.475447
$95$	0	0
$96$	−17.7911	−1.81580
$97$	3.78878	0.384692	0.192346	−	0.981327i	$-0.438390\pi$
$97$	3.78878	0.384692	0.192346	+	0.981327i	$0.438390\pi$
$98$	−0.234802	−0.0237186
$99$	5.45075	0.547821
$100$	0	0
$101$	8.35482	0.831336	0.415668	−	0.909517i	$-0.363548\pi$
$101$	8.35482	0.831336	0.415668	+	0.909517i	$0.363548\pi$
$102$	−9.24369	−0.915262
$103$	−2.07612	−0.204566	−0.102283	−	0.994755i	$-0.532615\pi$
$103$	−2.07612	−0.204566	−0.102283	+	0.994755i	$0.532615\pi$
$104$	−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	$0.411086\pi$
$108$	16.4994	1.58766
$109$	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
$111$	−4.51777	−0.428808
$112$	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	$0.441428\pi$
$114$	2.91852	0.273345
$115$	0	0
$116$	−7.06446	−0.655918
$117$	27.8658	2.57619
$118$	7.57755	0.697570
$119$	−8.22334	−0.753832
$120$	0	0
$121$	−10.4508	−0.950068
$122$	−7.78011	−0.704378
$123$	12.3472	1.11331
$124$	−6.87259	−0.617177
$125$	0	0
$126$	−17.3193	−1.54292
$127$	14.4233	1.27986	0.639931	−	0.768432i	$-0.278963\pi$
$127$	14.4233	1.27986	0.639931	+	0.768432i	$0.278963\pi$
$128$	−6.04007	−0.533872
$129$	−8.35482	−0.735601
$130$	0	0
$131$	9.96853	0.870954	0.435477	−	0.900200i	$-0.356580\pi$
$131$	9.96853	0.870954	0.435477	+	0.900200i	$0.356580\pi$
$132$	2.80791	0.244397
$133$	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	$0.363940\pi$
$138$	1.66614	0.141831
$139$	−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
$141$	16.3548	1.37732
$142$	3.30606	0.277438
$143$	2.80791	0.234809
$144$	−1.90407	−0.158672
$145$	0	0
$146$	9.83705	0.814120
$147$	0.833070	0.0687105
$148$	−1.65303	−0.135878
$149$	15.0959	1.23671	0.618353	−	0.785900i	$-0.287800\pi$
$149$	15.0959	1.23671	0.618353	+	0.785900i	$0.287800\pi$
$150$	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
$153$	23.2946	1.88325
$154$	−1.74519	−0.140631
$155$	0	0
$156$	14.3548	1.14931
$157$	7.57755	0.604754	0.302377	−	0.953188i	$-0.402220\pi$
$157$	7.57755	0.604754	0.302377	+	0.953188i	$0.402220\pi$
$158$	1.66614	0.132551
$159$	−0.517774	−0.0410622
$160$	0	0
$161$	1.48223	0.116816
$162$	20.8865	1.64100
$163$	−19.6757	−1.54112	−0.770559	−	0.637369i	$-0.780023\pi$
$163$	−19.6757	−1.54112	−0.770559	+	0.637369i	$0.780023\pi$
$164$	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	$0.637160\pi$
$168$	−24.0770	−1.85758
$169$	1.35482	0.104217
$170$	0	0
$171$	−7.35482	−0.562437
$172$	−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	$0.217703\pi$
$174$	−17.5111	−1.32752
$175$	0	0
$176$	−0.191865	−0.0144623
$177$	−26.8849	−2.02079
$178$	−15.3291	−1.14897
$179$	25.0645	1.87341	0.936703	−	0.350126i	$-0.113861\pi$
$179$	25.0645	1.87341	0.936703	+	0.350126i	$0.113861\pi$
$180$	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
$183$	27.6036	2.04051
$184$	1.64518	0.121284
$185$	0	0
$186$	−17.0355	−1.24911
$187$	2.34729	0.171651
$188$	5.98414	0.436439
$189$	36.3837	2.64653
$190$	0	0
$191$	11.4508	0.828547	0.414274	−	0.910152i	$-0.364036\pi$
$191$	11.4508	0.828547	0.414274	+	0.910152i	$0.364036\pi$
$192$	−17.8021	−1.28476
$193$	3.78878	0.272722	0.136361	−	0.990659i	$-0.456459\pi$
$193$	3.78878	0.272722	0.136361	+	0.990659i	$0.456459\pi$
$194$	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	$0.474078\pi$
$198$	4.94366	0.351330
$199$	−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
$201$	−47.7741	−3.36972
$202$	7.57755	0.533155
$203$	−15.5782	−1.09337
$204$	12.0000	0.840168
$205$	0	0
$206$	−1.88297	−0.131193
$207$	−4.19876	−0.291834
$208$	−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
$213$	−11.7298	−0.803712
$214$	5.17334	0.353642
$215$	0	0
$216$	40.3837	2.74776
$217$	−15.1551	−1.02880
$218$	1.49213	0.101059
$219$	−34.9015	−2.35843
$220$	0	0
$221$	12.0000	0.807207
$222$	−4.09748	−0.275005
$223$	−4.03785	−0.270394	−0.135197	−	0.990819i	$-0.543167\pi$
$223$	−4.03785	−0.270394	−0.135197	+	0.990819i	$0.543167\pi$
$224$	−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	$0.621431\pi$
$228$	−3.78878	−0.250918
$229$	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
$231$	6.19186	0.407395
$232$	−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	$0.522193\pi$
$234$	25.2733	1.65217
$235$	0	0
$236$	−9.83705	−0.640337
$237$	−5.91141	−0.383987
$238$	−7.45830	−0.483450
$239$	−14.4152	−0.932442	−0.466221	−	0.884668i	$-0.654385\pi$
$239$	−14.4152	−0.932442	−0.466221	+	0.884668i	$0.654385\pi$
$240$	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
$243$	−32.0645	−2.05694
$244$	10.1000	0.646587
$245$	0	0
$246$	11.1985	0.713990
$247$	−3.78878	−0.241074
$248$	−16.8212	−1.06815
$249$	−13.5111	−0.856233
$250$	0	0
$251$	12.9330	0.816322	0.408161	−	0.912910i	$-0.366170\pi$
$251$	12.9330	0.816322	0.408161	+	0.912910i	$0.366170\pi$
$252$	22.4836	1.41633
$253$	−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	$0.611941\pi$
$258$	−7.57755	−0.471758
$259$	−3.64518	−0.226501
$260$	0	0
$261$	44.1289	2.73151
$262$	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	$0.314466\pi$
$264$	6.87259	0.422979
$265$	0	0
$266$	2.35482	0.144383
$267$	54.3872	3.32844
$268$	−17.4803	−1.06778
$269$	24.9934	1.52387	0.761936	−	0.647652i	$-0.224249\pi$
$269$	24.9934	1.52387	0.761936	+	0.647652i	$0.224249\pi$
$270$	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
$273$	31.6545	1.91582
$274$	8.80150	0.531718
$275$	0	0
$276$	−2.16295	−0.130195
$277$	−21.2315	−1.27568	−0.637840	−	0.770169i	$-0.720172\pi$
$277$	−21.2315	−1.27568	−0.637840	+	0.770169i	$0.720172\pi$
$278$	−12.1994	−0.731671
$279$	42.9304	2.57018
$280$	0	0
$281$	3.83705	0.228899	0.114449	−	0.993429i	$-0.463490\pi$
$281$	3.83705	0.228899	0.114449	+	0.993429i	$0.463490\pi$
$282$	14.8333	0.883310
$283$	0.211545	0.0125751	0.00628753	−	0.999980i	$-0.497999\pi$
$283$	0.211545	0.0125751	0.00628753	+	0.999980i	$0.497999\pi$
$284$	−4.29187	−0.254676
$285$	0	0
$286$	2.54668	0.150589
$287$	9.96237	0.588060
$288$	40.6634	2.39612
$289$	−6.96853	−0.409913
$290$	0	0
$291$	−12.1919	−0.714700
$292$	−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	$0.644308\pi$
$294$	0.755568	0.0440656
$295$	0	0
$296$	−4.04593	−0.235165
$297$	−10.3855	−0.602626
$298$	13.6915	0.793129
$299$	−2.16295	−0.125087
$300$	0	0
$301$	−6.74111	−0.388551
$302$	12.8716	0.740675
$303$	−26.8849	−1.54450
$304$	0.258887	0.0148482
$305$	0	0
$306$	21.1274	1.20777
$307$	1.65303	0.0943434	0.0471717	−	0.998887i	$-0.484979\pi$
$307$	1.65303	0.0943434	0.0471717	+	0.998887i	$0.484979\pi$
$308$	2.26557	0.129093
$309$	6.68073	0.380053
$310$	0	0
$311$	−0.741113	−0.0420247	−0.0210123	−	0.999779i	$-0.506689\pi$
$311$	−0.741113	−0.0420247	−0.0210123	+	0.999779i	$0.506689\pi$
$312$	35.1346	1.98911
$313$	−26.8849	−1.51962	−0.759812	−	0.650143i	$-0.774709\pi$
$313$	−26.8849	−1.51962	−0.759812	+	0.650143i	$0.774709\pi$
$314$	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	$0.573611\pi$
$318$	−0.469604	−0.0263341
$319$	4.44668	0.248966
$320$	0	0
$321$	−18.3548	−1.02447
$322$	1.34433	0.0749166
$323$	−3.16725	−0.176231
$324$	−27.1145	−1.50636
$325$	0	0
$326$	−17.8452	−0.988354
$327$	−5.29401	−0.292759
$328$	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
$333$	10.3258	0.565852
$334$	−9.79112	−0.535746
$335$	0	0
$336$	−2.16295	−0.117999
$337$	−9.90275	−0.539437	−0.269718	−	0.962939i	$-0.586931\pi$
$337$	−9.90275	−0.539437	−0.269718	+	0.962939i	$0.586931\pi$
$338$	1.22878	0.0668367
$339$	−12.5178	−0.679872
$340$	0	0
$341$	4.32591	0.234261
$342$	−6.67058	−0.360704
$343$	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	$0.693496\pi$
$348$	22.7327	1.21860
$349$	−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
$351$	−53.0934	−2.83391
$352$	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	$0.280854\pi$
$354$	−24.3837	−1.29598
$355$	0	0
$356$	19.9000	1.05470
$357$	26.4618	1.40051
$358$	22.7327	1.20146
$359$	2.22334	0.117343	0.0586717	−	0.998277i	$-0.481313\pi$
$359$	2.22334	0.117343	0.0586717	+	0.998277i	$0.481313\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−17.6127	−0.925701
$363$	33.6294	1.76508
$364$	11.5822	0.607074
$365$	0	0
$366$	25.0355	1.30863
$367$	−4.52057	−0.235972	−0.117986	−	0.993015i	$-0.537644\pi$
$367$	−4.52057	−0.235972	−0.117986	+	0.993015i	$0.537644\pi$
$368$	0.147795	0.00770433
$369$	−28.2208	−1.46911
$370$	0	0
$371$	−0.417768	−0.0216894
$372$	22.1153	1.14662
$373$	15.5186	0.803521	0.401760	−	0.915745i	$-0.368398\pi$
$373$	15.5186	0.803521	0.401760	+	0.915745i	$0.368398\pi$
$374$	2.12892	0.110084
$375$	0	0
$376$	14.6467	0.755345
$377$	22.7327	1.17079
$378$	32.9989	1.69728
$379$	−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
$381$	−46.4126	−2.37779
$382$	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	$0.613865\pi$
$384$	19.4363	0.991854
$385$	0	0
$386$	3.43630	0.174903
$387$	19.0958	0.970694
$388$	−4.46094	−0.226470
$389$	12.7411	0.646000	0.323000	−	0.946399i	$-0.395309\pi$
$389$	12.7411	0.646000	0.323000	+	0.946399i	$0.395309\pi$
$390$	0	0
$391$	−1.80814	−0.0914413
$392$	0.746063	0.0376819
$393$	−32.0776	−1.61810
$394$	2.07110	0.104340
$395$	0	0
$396$	−6.41777	−0.322505
$397$	−38.6522	−1.93990	−0.969950	−	0.243306i	$-0.921768\pi$
$397$	−38.6522	−1.93990	−0.969950	+	0.243306i	$0.921768\pi$
$398$	−17.6412	−0.884274
$399$	−8.35482	−0.418264
$400$	0	0
$401$	31.8660	1.59131	0.795655	−	0.605750i	$-0.207127\pi$
$401$	31.8660	1.59131	0.795655	+	0.605750i	$0.207127\pi$
$402$	−43.3296	−2.16108
$403$	22.1153	1.10164
$404$	−9.83705	−0.489411
$405$	0	0
$406$	−14.1289	−0.701206
$407$	1.04049	0.0515751
$408$	29.3710	1.45408
$409$	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
$411$	−31.2274	−1.54033
$412$	2.44444	0.120429
$413$	−21.6922	−1.06740
$414$	−3.80814	−0.187160
$415$	0	0
$416$	20.9474	1.02703
$417$	43.2830	2.11958
$418$	−0.672165	−0.0328767
$419$	−25.7452	−1.25773	−0.628867	−	0.777513i	$-0.716481\pi$
$419$	−25.7452	−1.25773	−0.628867	+	0.777513i	$0.716481\pi$
$420$	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
$423$	−37.3806	−1.81751
$424$	−0.463697	−0.0225191
$425$	0	0
$426$	−10.6385	−0.515439
$427$	22.2720	1.07782
$428$	−6.71593	−0.324627
$429$	−9.03555	−0.436240
$430$	0	0
$431$	1.74519	0.0840627	0.0420314	−	0.999116i	$-0.486617\pi$
$431$	1.74519	0.0840627	0.0420314	+	0.999116i	$0.486617\pi$
$432$	3.62787	0.174546
$433$	−18.5208	−0.890052	−0.445026	−	0.895518i	$-0.646806\pi$
$433$	−18.5208	−0.890052	−0.445026	+	0.895518i	$0.646806\pi$
$434$	−13.7452	−0.659790
$435$	0	0
$436$	−1.93705	−0.0927679
$437$	0.570885	0.0273091
$438$	−31.6545	−1.51251
$439$	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
$441$	−1.90407	−0.0906699
$442$	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	$0.410042\pi$
$444$	5.31927	0.252441
$445$	0	0
$446$	−3.66220	−0.173410
$447$	−48.5771	−2.29762
$448$	−14.3637	−0.678620
$449$	−7.06446	−0.333392	−0.166696	−	0.986008i	$-0.553310\pi$
$449$	−7.06446	−0.333392	−0.166696	+	0.986008i	$0.553310\pi$
$450$	0	0
$451$	−2.84368	−0.133904
$452$	−4.58019	−0.215434
$453$	−45.6679	−2.14566
$454$	−10.1748	−0.477529
$455$	0	0
$456$	−9.27334	−0.434264
$457$	−34.5000	−1.61384	−0.806920	−	0.590660i	$-0.798867\pi$
$457$	−34.5000	−1.61384	−0.806920	+	0.590660i	$0.798867\pi$
$458$	14.6307	0.683649
$459$	−44.3837	−2.07166
$460$	0	0
$461$	8.03147	0.374063	0.187032	−	0.982354i	$-0.440113\pi$
$461$	8.03147	0.374063	0.187032	+	0.982354i	$0.440113\pi$
$462$	5.61582	0.261272
$463$	−25.3290	−1.17714	−0.588570	−	0.808447i	$-0.700309\pi$
$463$	−25.3290	−1.17714	−0.588570	+	0.808447i	$0.700309\pi$
$464$	−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	$0.286388\pi$
$468$	−32.8094	−1.51662
$469$	−38.5467	−1.77992
$470$	0	0
$471$	−24.3837	−1.12354
$472$	−24.0770	−1.10823
$473$	1.92420	0.0884748
$474$	−5.36146	−0.246260
$475$	0	0
$476$	9.68224	0.443785
$477$	1.18343	0.0541854
$478$	−13.0741	−0.597996
$479$	−28.9015	−1.32054	−0.660272	−	0.751027i	$-0.729559\pi$
$479$	−28.9015	−1.32054	−0.660272	+	0.751027i	$0.729559\pi$
$480$	0	0
$481$	5.31927	0.242538
$482$	−0.147795	−0.00673187
$483$	−4.76964	−0.217026
$484$	12.3048	0.559310
$485$	0	0
$486$	−29.0815	−1.31916
$487$	−17.7294	−0.803395	−0.401697	−	0.915773i	$-0.631580\pi$
$487$	−17.7294	−0.803395	−0.401697	+	0.915773i	$0.631580\pi$
$488$	24.7206	1.11905
$489$	63.3141	2.86316
$490$	0	0
$491$	−35.1645	−1.58695	−0.793475	−	0.608603i	$-0.791730\pi$
$491$	−35.1645	−1.58695	−0.793475	+	0.608603i	$0.791730\pi$
$492$	−14.5377	−0.655410
$493$	19.0035	0.855875
$494$	−3.43630	−0.154606
$495$	0	0
$496$	−1.51114	−0.0678520
$497$	−9.46422	−0.424528
$498$	−12.2542	−0.549122
$499$	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
$501$	34.7385	1.55200
$502$	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	$0.537988\pi$
$504$	55.0304	2.45125
$505$	0	0
$506$	−0.383729	−0.0170588
$507$	−4.35966	−0.193619
$508$	−16.9821	−0.753461
$509$	36.1919	1.60418	0.802088	−	0.597206i	$-0.203722\pi$
$509$	36.1919	1.60418	0.802088	+	0.597206i	$0.203722\pi$
$510$	0	0
$511$	−28.1604	−1.24574
$512$	−2.92346	−0.129200
$513$	14.0133	0.618704
$514$	−10.0170	−0.441832
$515$	0	0
$516$	9.83705	0.433052
$517$	−3.76668	−0.165658
$518$	−3.30606	−0.145260
$519$	−65.6111	−2.88001
$520$	0	0
$521$	−2.77259	−0.121469	−0.0607346	−	0.998154i	$-0.519344\pi$
$521$	−2.77259	−0.121469	−0.0607346	+	0.998154i	$0.519344\pi$
$522$	40.0235	1.75178
$523$	−20.5373	−0.898033	−0.449016	−	0.893524i	$-0.648226\pi$
$523$	−20.5373	−0.898033	−0.449016	+	0.893524i	$0.648226\pi$
$524$	−11.7370	−0.512735
$525$	0	0
$526$	16.1919	0.705999
$527$	18.4874	0.805323
$528$	0.617399	0.0268688
$529$	−22.6741	−0.985830
$530$	0	0
$531$	61.4482	2.66662
$532$	−3.05699	−0.132537
$533$	−14.5377	−0.629698
$534$	49.3274	2.13461
$535$	0	0
$536$	−42.7845	−1.84801
$537$	−80.6547	−3.48051
$538$	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
$543$	62.4891	2.68166
$544$	17.5111	0.750784
$545$	0	0
$546$	28.7096	1.22866
$547$	43.0756	1.84178	0.920890	−	0.389822i	$-0.127463\pi$
$547$	43.0756	1.84178	0.920890	+	0.389822i	$0.127463\pi$
$548$	−11.4260	−0.488092
$549$	−63.0908	−2.69265
$550$	0	0
$551$	−6.00000	−0.255609
$552$	−5.29401	−0.225328
$553$	−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	$0.172512\pi$
$558$	38.9365	1.64831
$559$	9.83705	0.416063
$560$	0	0
$561$	−7.55332	−0.318902
$562$	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	$0.363609\pi$
$564$	−19.2563	−0.810837
$565$	0	0
$566$	0.191865	0.00806467
$567$	−59.7915	−2.51101
$568$	−10.5047	−0.440768
$569$	18.6807	0.783137	0.391568	−	0.920149i	$-0.371933\pi$
$569$	18.6807	0.783137	0.391568	+	0.920149i	$0.371933\pi$
$570$	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
$573$	−36.8473	−1.53932
$574$	9.03555	0.377137
$575$	0	0
$576$	40.6885	1.69536
$577$	−0.156779	−0.00652679	−0.00326339	−	0.999995i	$-0.501039\pi$
$577$	−0.156779	−0.00652679	−0.00326339	+	0.999995i	$0.501039\pi$
$578$	−6.32023	−0.262887
$579$	−12.1919	−0.506677
$580$	0	0
$581$	−10.9015	−0.452271
$582$	−11.0576	−0.458353
$583$	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	$0.277775\pi$
$588$	−0.980865	−0.0404502
$589$	−5.83705	−0.240511
$590$	0	0
$591$	−7.34818	−0.302264
$592$	−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	$0.298009\pi$
$594$	−9.41928	−0.386478
$595$	0	0
$596$	−17.7741	−0.728055
$597$	62.5904	2.56165
$598$	−1.96173	−0.0802211
$599$	25.3274	1.03485	0.517425	−	0.855728i	$-0.326891\pi$
$599$	25.3274	1.03485	0.517425	+	0.855728i	$0.326891\pi$
$600$	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
$603$	109.193	4.44667
$604$	−16.7096	−0.679906
$605$	0	0
$606$	−24.3837	−0.990521
$607$	−2.49921	−0.101440	−0.0507199	−	0.998713i	$-0.516152\pi$
$607$	−2.49921	−0.101440	−0.0507199	+	0.998713i	$0.516152\pi$
$608$	−5.52881	−0.224223
$609$	50.1289	2.03133
$610$	0	0
$611$	−19.2563	−0.779027
$612$	−27.4272	−1.10868
$613$	−0.883711	−0.0356927	−0.0178464	−	0.999841i	$-0.505681\pi$
$613$	−0.883711	−0.0356927	−0.0178464	+	0.999841i	$0.505681\pi$
$614$	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	$0.701650\pi$
$618$	6.05921	0.243737
$619$	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
$621$	8.00000	0.321029
$622$	−0.672165	−0.0269514
$623$	43.8825	1.75811
$624$	3.15632	0.126354
$625$	0	0
$626$	−24.3837	−0.974570
$627$	2.38482	0.0952405
$628$	−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
$633$	−36.1286	−1.43598
$634$	−7.40226	−0.293981
$635$	0	0
$636$	0.609632	0.0241735
$637$	−0.980865	−0.0388633
$638$	4.03299	0.159668
$639$	26.8096	1.06057
$640$	0	0
$641$	−32.6675	−1.29029	−0.645143	−	0.764062i	$-0.723202\pi$
$641$	−32.6675	−1.29029	−0.645143	+	0.764062i	$0.723202\pi$
$642$	−16.6472	−0.657014
$643$	31.8661	1.25668	0.628338	−	0.777941i	$-0.283736\pi$
$643$	31.8661	1.25668	0.628338	+	0.777941i	$0.283736\pi$
$644$	−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	$0.362763\pi$
$648$	−66.3649	−2.60706
$649$	6.19186	0.243052
$650$	0	0
$651$	48.7675	1.91135
$652$	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	$0.419202\pi$
$654$	−4.80150	−0.187753
$655$	0	0
$656$	0.993361	0.0387842
$657$	79.7710	3.11216
$658$	11.9683	0.466573
$659$	20.3548	0.792911	0.396456	−	0.918054i	$-0.370240\pi$
$659$	20.3548	0.792911	0.396456	+	0.918054i	$0.370240\pi$
$660$	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
$663$	−38.6147	−1.49967
$664$	−12.1000	−0.469571
$665$	0	0
$666$	9.36520	0.362894
$667$	−3.42531	−0.132629
$668$	12.7107	0.491790
$669$	12.9934	0.502352
$670$	0	0
$671$	−6.35738	−0.245424
$672$	46.1922	1.78190
$673$	21.2094	0.817564	0.408782	−	0.912632i	$-0.365954\pi$
$673$	21.2094	0.817564	0.408782	+	0.912632i	$0.365954\pi$
$674$	−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	$0.430544\pi$
$678$	−11.3532	−0.436018
$679$	−9.83705	−0.377511
$680$	0	0
$681$	36.1000	1.38336
$682$	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	$0.575993\pi$
$684$	8.65964	0.331109
$685$	0	0
$686$	17.0934	0.652628
$687$	−51.9093	−1.98046
$688$	−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
$693$	−14.1521	−0.537595
$694$	−19.2985	−0.732561
$695$	0	0
$696$	55.6401	2.10903
$697$	−12.1529	−0.460323
$698$	−14.8881	−0.563521
$699$	6.84368	0.258852
$700$	0	0
$701$	−16.0289	−0.605404	−0.302702	−	0.953085i	$-0.597889\pi$
$701$	−16.0289	−0.605404	−0.302702	+	0.953085i	$0.597889\pi$
$702$	−48.1540	−1.81745
$703$	−1.40396	−0.0529512
$704$	4.10001	0.154525
$705$	0	0
$706$	21.6533	0.814934
$707$	−21.6922	−0.815818
$708$	31.6545	1.18965
$709$	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
$711$	13.5111	0.506707
$712$	48.7069	1.82537
$713$	−3.33228	−0.124795
$714$	24.0000	0.898177
$715$	0	0
$716$	−29.5111	−1.10288
$717$	46.3865	1.73234
$718$	2.01650	0.0752550
$719$	11.2589	0.419886	0.209943	−	0.977714i	$-0.432672\pi$
$719$	11.2589	0.419886	0.209943	+	0.977714i	$0.432672\pi$
$720$	0	0
$721$	5.39037	0.200748
$722$	0.906968	0.0337538
$723$	0.524371	0.0195016
$724$	22.8644	0.849750
$725$	0	0
$726$	30.5008	1.13199
$727$	48.9829	1.81668	0.908338	−	0.418237i	$-0.137352\pi$
$727$	48.9829	1.81668	0.908338	+	0.418237i	$0.137352\pi$
$728$	28.3485	1.05066
$729$	34.0934	1.26272
$730$	0	0
$731$	8.22334	0.304151
$732$	−32.5007	−1.20126
$733$	−35.9260	−1.32696	−0.663479	−	0.748195i	$-0.730921\pi$
$733$	−35.9260	−1.32696	−0.663479	+	0.748195i	$0.730921\pi$
$734$	−4.10001	−0.151334
$735$	0	0
$736$	−3.15632	−0.116343
$737$	11.0029	0.405296
$738$	−25.5953	−0.942177
$739$	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
$741$	12.1919	0.447879
$742$	−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	$0.425982\pi$
$744$	54.1289	1.98446
$745$	0	0
$746$	14.0748	0.515316
$747$	30.8811	1.12988
$748$	−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
$753$	−41.6169	−1.51660
$754$	20.6178	0.750855
$755$	0	0
$756$	−42.8386	−1.55802
$757$	−15.7350	−0.571897	−0.285949	−	0.958245i	$-0.592309\pi$
$757$	−15.7350	−0.571897	−0.285949	+	0.958245i	$0.592309\pi$
$758$	−17.1431	−0.622664
$759$	1.36146	0.0494178
$760$	0	0
$761$	16.9619	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$761$	16.9619	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$762$	−42.0948	−1.52493
$763$	−4.27149	−0.154638
$764$	−13.4822	−0.487770
$765$	0	0
$766$	−12.4297	−0.449102
$767$	31.6545	1.14298
$768$	53.2323	1.92086
$769$	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
$771$	35.5400	1.27994
$772$	−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	$0.763559\pi$
$774$	17.3193	0.622528
$775$	0	0
$776$	−10.9185	−0.391952
$777$	11.7298	0.420804
$778$	11.5558	0.414295
$779$	3.83705	0.137476
$780$	0	0
$781$	2.70149	0.0966669
$782$	−1.63992	−0.0586434
$783$	−84.0800	−3.00477
$784$	0.0670225	0.00239366
$785$	0	0
$786$	−29.0934	−1.03773
$787$	−28.5379	−1.01727	−0.508634	−	0.860983i	$-0.669849\pi$
$787$	−28.5379	−1.01727	−0.508634	+	0.860983i	$0.669849\pi$
$788$	−2.68866	−0.0957796
$789$	−57.4482	−2.04521
$790$	0	0
$791$	−10.1000	−0.359115
$792$	−15.7080	−0.558160
$793$	−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	$0.700636\pi$
$798$	−7.57755	−0.268242
$799$	−16.0974	−0.569487
$800$	0	0
$801$	−124.308	−4.39219
$802$	28.9014	1.02054
$803$	8.03817	0.283661
$804$	56.2497	1.98377
$805$	0	0
$806$	20.0578	0.706507
$807$	−80.4259	−2.83113
$808$	−24.0770	−0.847025
$809$	−34.4234	−1.21026	−0.605130	−	0.796126i	$-0.706879\pi$
$809$	−34.4234	−1.21026	−0.605130	+	0.796126i	$0.706879\pi$
$810$	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
$813$	76.7980	2.69342
$814$	0.943690	0.0330763
$815$	0	0
$816$	2.63854	0.0923674
$817$	−2.59637	−0.0908353
$818$	10.0351	0.350869
$819$	−72.3497	−2.52810
$820$	0	0
$821$	−31.3167	−1.09296	−0.546480	−	0.837472i	$-0.684033\pi$
$821$	−31.3167	−1.09296	−0.546480	+	0.837472i	$0.684033\pi$
$822$	−28.3223	−0.987852
$823$	13.4800	0.469882	0.234941	−	0.972010i	$-0.424510\pi$
$823$	13.4800	0.469882	0.234941	+	0.972010i	$0.424510\pi$
$824$	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	$0.589665\pi$
$828$	4.94366	0.171804
$829$	−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
$831$	68.3208	2.37002
$832$	20.9604	0.726670
$833$	−0.819960	−0.0284099
$834$	39.2563	1.35934
$835$	0	0
$836$	0.872594	0.0301793
$837$	−81.7965	−2.82730
$838$	−23.3501	−0.806614
$839$	−18.9934	−0.655724	−0.327862	−	0.944726i	$-0.606328\pi$
$839$	−18.9934	−0.655724	−0.327862	+	0.944726i	$0.606328\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	24.8946	0.857925
$843$	−12.3472	−0.425260
$844$	−13.2193	−0.455026
$845$	0	0
$846$	−33.9030	−1.16561
$847$	27.1340	0.932334
$848$	−0.0416562	−0.00143048
$849$	−0.680729	−0.0233626
$850$	0	0
$851$	−0.801497	−0.0274750
$852$	13.8108	0.473149
$853$	44.1210	1.51067	0.755337	−	0.655337i	$-0.227473\pi$
$853$	44.1210	1.51067	0.755337	+	0.655337i	$0.227473\pi$
$854$	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	$0.313240\pi$
$858$	−8.19495	−0.279771
$859$	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
$861$	−32.0578	−1.09253
$862$	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	$0.595895\pi$
$864$	−77.4771	−2.63582
$865$	0	0
$866$	−16.7978	−0.570811
$867$	22.4240	0.761557
$868$	17.8438	0.605657
$869$	1.36146	0.0461843
$870$	0	0
$871$	56.2497	1.90595
$872$	−4.74109	−0.160554
$873$	27.8658	0.943113
$874$	0.517774	0.0175140
$875$	0	0
$876$	41.0934	1.38842
$877$	23.2904	0.786462	0.393231	−	0.919440i	$-0.371357\pi$
$877$	23.2904	0.786462	0.393231	+	0.919440i	$0.371357\pi$
$878$	26.7086	0.901370
$879$	48.2497	1.62742
$880$	0	0
$881$	28.9619	0.975751	0.487875	−	0.872913i	$-0.337772\pi$
$881$	28.9619	0.975751	0.487875	+	0.872913i	$0.337772\pi$
$882$	−1.72693	−0.0581487
$883$	−37.3627	−1.25735	−0.628677	−	0.777667i	$-0.716403\pi$
$883$	−37.3627	−1.25735	−0.628677	+	0.777667i	$0.716403\pi$
$884$	−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	$0.626325\pi$
$888$	13.0194	0.436901
$889$	−37.4482	−1.25597
$890$	0	0
$891$	17.0670	0.571767
$892$	4.75420	0.159183
$893$	5.08247	0.170078
$894$	−44.0578	−1.47351
$895$	0	0
$896$	15.6822	0.523907
$897$	6.96015	0.232393
$898$	−6.40723	−0.213812
$899$	35.0223	1.16806
$900$	0	0
$901$	0.509626	0.0169781
$902$	−2.57913	−0.0858756
$903$	21.6922	0.721870
$904$	−11.2104	−0.372852
$905$	0	0
$906$	−41.4193	−1.37606
$907$	−35.2693	−1.17110	−0.585549	−	0.810637i	$-0.699121\pi$
$907$	−35.2693	−1.17110	−0.585549	+	0.810637i	$0.699121\pi$
$908$	13.2088	0.438350
$909$	61.4482	2.03811
$910$	0	0
$911$	−4.97260	−0.164750	−0.0823748	−	0.996601i	$-0.526250\pi$
$911$	−4.97260	−0.164750	−0.0823748	+	0.996601i	$0.526250\pi$
$912$	−0.833070	−0.0275857
$913$	3.11175	0.102984
$914$	−31.2904	−1.03499
$915$	0	0
$916$	−18.9934	−0.627558
$917$	−25.8819	−0.854697
$918$	−40.2546	−1.32860
$919$	−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
$921$	−5.31927	−0.175276
$922$	7.28429	0.239895
$923$	13.8108	0.454587
$924$	−7.29036	−0.239835
$925$	0	0
$926$	−22.9726	−0.754926
$927$	−15.2695	−0.501516
$928$	33.1729	1.08895
$929$	32.5126	1.06671	0.533353	−	0.845893i	$-0.320932\pi$
$929$	32.5126	1.06671	0.533353	+	0.845893i	$0.320932\pi$
$930$	0	0
$931$	0.258887	0.00848468
$932$	2.50407	0.0820235
$933$	2.38482	0.0780755
$934$	24.3756	0.797593
$935$	0	0
$936$	−80.3038	−2.62481
$937$	−0.385560	−0.0125957	−0.00629785	−	0.999980i	$-0.502005\pi$
$937$	−0.385560	−0.0125957	−0.00629785	+	0.999980i	$0.502005\pi$
$938$	−34.9606	−1.14150
$939$	86.5126	2.82323
$940$	0	0
$941$	45.3482	1.47831	0.739154	−	0.673536i	$-0.235225\pi$
$941$	45.3482	1.47831	0.739154	+	0.673536i	$0.235225\pi$
$942$	−22.1153	−0.720554
$943$	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	$0.450084\pi$
$948$	6.96015	0.226055
$949$	41.0934	1.33395
$950$	0	0
$951$	26.2630	0.851635
$952$	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	$0.920842\pi$
$954$	1.07333	0.0347503
$955$	0	0
$956$	16.9726	0.548933
$957$	−14.3089	−0.462542
$958$	−26.2127	−0.846895
$959$	−25.1959	−0.813619
$960$	0	0
$961$	3.07110	0.0990676
$962$	4.82441	0.155545
$963$	41.9518	1.35188
$964$	0.191865	0.00617954
$965$	0	0
$966$	−4.32591	−0.139184
$967$	−0.368324	−0.0118445	−0.00592225	−	0.999982i	$-0.501885\pi$
$967$	−0.368324	−0.0118445	−0.00592225	+	0.999982i	$0.501885\pi$
$968$	30.1171	0.967999
$969$	10.1919	0.327410
$970$	0	0
$971$	−45.8370	−1.47098	−0.735490	−	0.677535i	$-0.763048\pi$
$971$	−45.8370	−1.47098	−0.735490	+	0.677535i	$0.763048\pi$
$972$	37.7531	1.21093
$973$	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	$0.346257\pi$
$978$	57.4239	1.83621
$979$	−12.5259	−0.400330
$980$	0	0
$981$	12.1000	0.386323
$982$	−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	$0.556212\pi$
$984$	−35.5822	−1.13432
$985$	0	0
$986$	17.2356	0.548892
$987$	−42.4631	−1.35161
$988$	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
$993$	−25.7431	−0.816933
$994$	−8.58374	−0.272260
$995$	0	0
$996$	15.9081	0.504069
$997$	20.3854	0.645611	0.322805	−	0.946465i	$-0.395374\pi$
$997$	20.3854	0.645611	0.322805	+	0.946465i	$0.395374\pi$
$998$	19.4551	0.615842
$999$	−19.6741	−0.622461

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.a.j.1.4		6
3.2	odd	2		4275.2.a.br.1.3		6
4.3	odd	2		7600.2.a.ck.1.6		6
5.2	odd	4		95.2.b.b.39.4	yes	6
5.3	odd	4		95.2.b.b.39.3	✓	6
5.4	even	2	inner	475.2.a.j.1.3		6
15.2	even	4		855.2.c.d.514.3		6
15.8	even	4		855.2.c.d.514.4		6
15.14	odd	2		4275.2.a.br.1.4		6
19.18	odd	2		9025.2.a.bx.1.3		6
20.3	even	4		1520.2.d.h.609.6		6
20.7	even	4		1520.2.d.h.609.1		6
20.19	odd	2		7600.2.a.ck.1.1		6
95.18	even	4		1805.2.b.e.1084.4		6
95.37	even	4		1805.2.b.e.1084.3		6
95.94	odd	2		9025.2.a.bx.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.b.b.39.3	✓	6	5.3	odd	4
95.2.b.b.39.4	yes	6	5.2	odd	4
475.2.a.j.1.3		6	5.4	even	2	inner
475.2.a.j.1.4		6	1.1	even	1	trivial
855.2.c.d.514.3		6	15.2	even	4
855.2.c.d.514.4		6	15.8	even	4
1520.2.d.h.609.1		6	20.7	even	4
1520.2.d.h.609.6		6	20.3	even	4
1805.2.b.e.1084.3		6	95.37	even	4
1805.2.b.e.1084.4		6	95.18	even	4
4275.2.a.br.1.3		6	3.2	odd	2
4275.2.a.br.1.4		6	15.14	odd	2
7600.2.a.ck.1.1		6	20.19	odd	2
7600.2.a.ck.1.6		6	4.3	odd	2
9025.2.a.bx.1.3		6	19.18	odd	2
9025.2.a.bx.1.4		6	95.94	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.a (trivial)

\(f(q)\)	\(=\)	\(q+0.906968 q^{2} -3.21789 q^{3} -1.17741 q^{4} -2.91852 q^{6} -2.59637 q^{7} -2.88181 q^{8} +7.35482 q^{9} +0.741113 q^{11} +3.78878 q^{12} +3.78878 q^{13} -2.35482 q^{14} -0.258887 q^{16} +3.16725 q^{17} +6.67058 q^{18} -1.00000 q^{19} +8.35482 q^{21} +0.672165 q^{22} -0.570885 q^{23} +9.27334 q^{24} +3.43630 q^{26} -14.0133 q^{27} +3.05699 q^{28} +6.00000 q^{29} +5.83705 q^{31} +5.52881 q^{32} -2.38482 q^{33} +2.87259 q^{34} -8.65964 q^{36} +1.40396 q^{37} -0.906968 q^{38} -12.1919 q^{39} -3.83705 q^{41} +7.57755 q^{42} +2.59637 q^{43} -0.872594 q^{44} -0.517774 q^{46} -5.08247 q^{47} +0.833070 q^{48} -0.258887 q^{49} -10.1919 q^{51} -4.46094 q^{52} +0.160905 q^{53} -12.7096 q^{54} +7.48223 q^{56} +3.21789 q^{57} +5.44181 q^{58} +8.35482 q^{59} -8.57816 q^{61} +5.29401 q^{62} -19.0958 q^{63} +5.53223 q^{64} -2.16295 q^{66} +14.8464 q^{67} -3.72915 q^{68} +1.83705 q^{69} +3.64518 q^{71} -21.1952 q^{72} +10.8461 q^{73} +1.27334 q^{74} +1.17741 q^{76} -1.92420 q^{77} -11.0576 q^{78} +1.83705 q^{79} +23.0289 q^{81} -3.48008 q^{82} +4.19876 q^{83} -9.83705 q^{84} +2.35482 q^{86} -19.3073 q^{87} -2.13574 q^{88} -16.9015 q^{89} -9.83705 q^{91} +0.672165 q^{92} -18.7830 q^{93} -4.60963 q^{94} -17.7911 q^{96} +3.78878 q^{97} -0.234802 q^{98} +5.45075 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{4} + 14 q^{9} + 2 q^{11} + 16 q^{14} - 4 q^{16} - 6 q^{19} + 20 q^{21} + 8 q^{24} + 8 q^{26} + 36 q^{29} - 8 q^{34} - 32 q^{36} - 8 q^{39} + 12 q^{41} + 20 q^{44} - 8 q^{46} - 4 q^{49} + 4 q^{51}+ \cdots - 30 q^{99}+O(q^{100}) \) 6 * q + 8 * q^4 + 14 * q^9 + 2 * q^11 + 16 * q^14 - 4 * q^16 - 6 * q^19 + 20 * q^21 + 8 * q^24 + 8 * q^26 + 36 * q^29 - 8 * q^34 - 32 * q^36 - 8 * q^39 + 12 * q^41 + 20 * q^44 - 8 * q^46 - 4 * q^49 + 4 * q^51 - 16 * q^54 + 40 * q^56 + 20 * q^59 - 14 * q^61 - 12 * q^64 - 48 * q^66 - 24 * q^69 + 52 * q^71 - 40 * q^74 - 8 * q^76 - 24 * q^79 + 38 * q^81 - 24 * q^84 - 16 * q^86 + 24 * q^89 - 24 * q^91 - 48 * q^94 - 64 * q^96 - 30 * q^99

Introduction

L-functions

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