LMFDB - Embedded newform 2475.4.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2475.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:	$146.029727264$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.1539480.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 25x^{2} + 9x + 96 $ x^4 - x^3 - 25x^2 + 9x + 96
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.03085$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.03085 q^{2} -3.87566 q^{4} -27.2109 q^{7} +24.1176 q^{8} +O(q^{10})$	$q-2.03085 q^{2} -3.87566 q^{4} -27.2109 q^{7} +24.1176 q^{8} -11.0000 q^{11} -14.8309 q^{13} +55.2611 q^{14} -17.9739 q^{16} +91.7661 q^{17} -12.7627 q^{19} +22.3393 q^{22} -10.2751 q^{23} +30.1193 q^{26} +105.460 q^{28} -153.965 q^{29} -115.601 q^{31} -156.439 q^{32} -186.363 q^{34} -201.704 q^{37} +25.9191 q^{38} -398.269 q^{41} +438.107 q^{43} +42.6323 q^{44} +20.8672 q^{46} +372.988 q^{47} +397.431 q^{49} +57.4797 q^{52} +454.843 q^{53} -656.262 q^{56} +312.679 q^{58} +766.019 q^{59} -357.517 q^{61} +234.768 q^{62} +461.495 q^{64} +947.404 q^{67} -355.654 q^{68} -568.207 q^{71} +503.547 q^{73} +409.631 q^{74} +49.4639 q^{76} +299.319 q^{77} -386.931 q^{79} +808.823 q^{82} +754.462 q^{83} -889.727 q^{86} -265.294 q^{88} +848.909 q^{89} +403.562 q^{91} +39.8229 q^{92} -757.482 q^{94} -823.953 q^{97} -807.121 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + 19 q^{4} - 9 q^{7} + 33 q^{8}+O(q^{10}) $ 4 * q + q^2 + 19 * q^4 - 9 * q^7 + 33 * q^8	$ 4 q + q^{2} + 19 q^{4} - 9 q^{7} + 33 q^{8} - 44 q^{11} - 70 q^{13} + 49 q^{14} - 37 q^{16} + 103 q^{17} - 205 q^{19} - 11 q^{22} - 56 q^{23} + 86 q^{26} + 551 q^{28} + 79 q^{29} + 49 q^{31} + 225 q^{32} - 939 q^{34} - 289 q^{37} + 145 q^{38} - 736 q^{41} + 152 q^{43} - 209 q^{44} - 334 q^{46} + 412 q^{47} + 37 q^{49} - 1598 q^{52} + 1685 q^{53} + 257 q^{56} - 609 q^{58} + 842 q^{59} - 1097 q^{61} + 1359 q^{62} - 165 q^{64} + 122 q^{67} - 757 q^{68} + 521 q^{71} + 590 q^{73} - 3257 q^{74} - 2825 q^{76} + 99 q^{77} - 1118 q^{79} + 402 q^{82} - 122 q^{83} - 3452 q^{86} - 363 q^{88} + 181 q^{89} - 2190 q^{91} + 430 q^{92} + 1034 q^{94} - 1474 q^{97} - 872 q^{98}+O(q^{100}) $ 4 * q + q^2 + 19 * q^4 - 9 * q^7 + 33 * q^8 - 44 * q^11 - 70 * q^13 + 49 * q^14 - 37 * q^16 + 103 * q^17 - 205 * q^19 - 11 * q^22 - 56 * q^23 + 86 * q^26 + 551 * q^28 + 79 * q^29 + 49 * q^31 + 225 * q^32 - 939 * q^34 - 289 * q^37 + 145 * q^38 - 736 * q^41 + 152 * q^43 - 209 * q^44 - 334 * q^46 + 412 * q^47 + 37 * q^49 - 1598 * q^52 + 1685 * q^53 + 257 * q^56 - 609 * q^58 + 842 * q^59 - 1097 * q^61 + 1359 * q^62 - 165 * q^64 + 122 * q^67 - 757 * q^68 + 521 * q^71 + 590 * q^73 - 3257 * q^74 - 2825 * q^76 + 99 * q^77 - 1118 * q^79 + 402 * q^82 - 122 * q^83 - 3452 * q^86 - 363 * q^88 + 181 * q^89 - 2190 * q^91 + 430 * q^92 + 1034 * q^94 - 1474 * q^97 - 872 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−2.03085	−0.718013	−0.359006	−	0.933335i	$-0.616884\pi$
$2$	−2.03085	−0.718013	−0.359006	+	0.933335i	$0.616884\pi$
$3$	0	0
$3$	0	0
$4$	−3.87566	−0.484458
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−27.2109	−1.46925	−0.734624	−	0.678474i	$-0.762641\pi$
$7$	−27.2109	−1.46925	−0.734624	+	0.678474i	$0.762641\pi$
$8$	24.1176	1.06586
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−14.8309	−0.316412	−0.158206	−	0.987406i	$-0.550571\pi$
$13$	−14.8309	−0.316412	−0.158206	+	0.987406i	$0.550571\pi$
$14$	55.2611	1.05494
$15$	0	0
$16$	−17.9739	−0.280843
$17$	91.7661	1.30921	0.654604	−	0.755972i	$-0.272835\pi$
$17$	91.7661	1.30921	0.654604	+	0.755972i	$0.272835\pi$
$18$	0	0
$19$	−12.7627	−0.154103	−0.0770517	−	0.997027i	$-0.524551\pi$
$19$	−12.7627	−0.154103	−0.0770517	+	0.997027i	$0.524551\pi$
$20$	0	0
$21$	0	0
$22$	22.3393	0.216489
$23$	−10.2751	−0.0931527	−0.0465763	−	0.998915i	$-0.514831\pi$
$23$	−10.2751	−0.0931527	−0.0465763	+	0.998915i	$0.514831\pi$
$24$	0	0
$25$	0	0
$26$	30.1193	0.227188
$27$	0	0
$28$	105.460	0.711789
$29$	−153.965	−0.985883	−0.492941	−	0.870063i	$-0.664078\pi$
$29$	−153.965	−0.985883	−0.492941	+	0.870063i	$0.664078\pi$
$30$	0	0
$31$	−115.601	−0.669760	−0.334880	−	0.942261i	$-0.608696\pi$
$31$	−115.601	−0.669760	−0.334880	+	0.942261i	$0.608696\pi$
$32$	−156.439	−0.864211
$33$	0	0
$34$	−186.363	−0.940028
$35$	0	0
$36$	0	0
$37$	−201.704	−0.896217	−0.448108	−	0.893979i	$-0.647902\pi$
$37$	−201.704	−0.896217	−0.448108	+	0.893979i	$0.647902\pi$
$38$	25.9191	0.110648
$39$	0	0
$40$	0	0
$41$	−398.269	−1.51705	−0.758527	−	0.651642i	$-0.774080\pi$
$41$	−398.269	−1.51705	−0.758527	+	0.651642i	$0.774080\pi$
$42$	0	0
$43$	438.107	1.55374	0.776868	−	0.629664i	$-0.216807\pi$
$43$	438.107	1.55374	0.776868	+	0.629664i	$0.216807\pi$
$44$	42.6323	0.146070
$45$	0	0
$46$	20.8672	0.0668848
$47$	372.988	1.15757	0.578786	−	0.815479i	$-0.303527\pi$
$47$	372.988	1.15757	0.578786	+	0.815479i	$0.303527\pi$
$48$	0	0
$49$	397.431	1.15869
$50$	0	0
$51$	0	0
$52$	57.4797	0.153288
$53$	454.843	1.17882	0.589410	−	0.807834i	$-0.299360\pi$
$53$	454.843	1.17882	0.589410	+	0.807834i	$0.299360\pi$
$54$	0	0
$55$	0	0
$56$	−656.262	−1.56601
$57$	0	0
$58$	312.679	0.707876
$59$	766.019	1.69029	0.845145	−	0.534537i	$-0.179514\pi$
$59$	766.019	1.69029	0.845145	+	0.534537i	$0.179514\pi$
$60$	0	0
$61$	−357.517	−0.750417	−0.375208	−	0.926941i	$-0.622429\pi$
$61$	−357.517	−0.750417	−0.375208	+	0.926941i	$0.622429\pi$
$62$	234.768	0.480896
$63$	0	0
$64$	461.495	0.901357
$65$	0	0
$66$	0	0
$67$	947.404	1.72752	0.863760	−	0.503904i	$-0.168104\pi$
$67$	947.404	1.72752	0.863760	+	0.503904i	$0.168104\pi$
$68$	−355.654	−0.634256
$69$	0	0
$70$	0	0
$71$	−568.207	−0.949772	−0.474886	−	0.880047i	$-0.657511\pi$
$71$	−568.207	−0.949772	−0.474886	+	0.880047i	$0.657511\pi$
$72$	0	0
$73$	503.547	0.807338	0.403669	−	0.914905i	$-0.367735\pi$
$73$	503.547	0.807338	0.403669	+	0.914905i	$0.367735\pi$
$74$	409.631	0.643495
$75$	0	0
$76$	49.4639	0.0746566
$77$	299.319	0.442995
$78$	0	0
$79$	−386.931	−0.551053	−0.275526	−	0.961294i	$-0.588852\pi$
$79$	−386.931	−0.551053	−0.275526	+	0.961294i	$0.588852\pi$
$80$	0	0
$81$	0	0
$82$	808.823	1.08926
$83$	754.462	0.997747	0.498874	−	0.866675i	$-0.333747\pi$
$83$	754.462	0.997747	0.498874	+	0.866675i	$0.333747\pi$
$84$	0	0
$85$	0	0
$86$	−889.727	−1.11560
$87$	0	0
$88$	−265.294	−0.321369
$89$	848.909	1.01106	0.505529	−	0.862810i	$-0.331297\pi$
$89$	848.909	1.01106	0.505529	+	0.862810i	$0.331297\pi$
$90$	0	0
$91$	403.562	0.464888
$92$	39.8229	0.0451285
$93$	0	0
$94$	−757.482	−0.831152
$95$	0	0
$96$	0	0
$97$	−823.953	−0.862472	−0.431236	−	0.902239i	$-0.641922\pi$
$97$	−823.953	−0.862472	−0.431236	+	0.902239i	$0.641922\pi$
$98$	−807.121	−0.831954
$99$	0	0
$100$	0	0
$101$	1328.83	1.30914	0.654572	−	0.755999i	$-0.272849\pi$
$101$	1328.83	1.30914	0.654572	+	0.755999i	$0.272849\pi$
$102$	0	0
$103$	−1752.94	−1.67692	−0.838459	−	0.544964i	$-0.816543\pi$
$103$	−1752.94	−1.67692	−0.838459	+	0.544964i	$0.816543\pi$
$104$	−357.687	−0.337251
$105$	0	0
$106$	−923.716	−0.846408
$107$	−1175.53	−1.06208	−0.531040	−	0.847346i	$-0.678199\pi$
$107$	−1175.53	−1.06208	−0.531040	+	0.847346i	$0.678199\pi$
$108$	0	0
$109$	2131.94	1.87342	0.936711	−	0.350104i	$-0.113854\pi$
$109$	2131.94	1.87342	0.936711	+	0.350104i	$0.113854\pi$
$110$	0	0
$111$	0	0
$112$	489.086	0.412628
$113$	83.9161	0.0698599	0.0349299	−	0.999390i	$-0.488879\pi$
$113$	83.9161	0.0698599	0.0349299	+	0.999390i	$0.488879\pi$
$114$	0	0
$115$	0	0
$116$	596.717	0.477618
$117$	0	0
$118$	−1555.67	−1.21365
$119$	−2497.03	−1.92355
$120$	0	0
$121$	121.000	0.0909091
$122$	726.063	0.538809
$123$	0	0
$124$	448.030	0.324470
$125$	0	0
$126$	0	0
$127$	550.609	0.384714	0.192357	−	0.981325i	$-0.438387\pi$
$127$	550.609	0.384714	0.192357	+	0.981325i	$0.438387\pi$
$128$	314.286	0.217025
$129$	0	0
$130$	0	0
$131$	−593.759	−0.396007	−0.198004	−	0.980201i	$-0.563446\pi$
$131$	−593.759	−0.396007	−0.198004	+	0.980201i	$0.563446\pi$
$132$	0	0
$133$	347.284	0.226416
$134$	−1924.03	−1.24038
$135$	0	0
$136$	2213.18	1.39543
$137$	−242.900	−0.151477	−0.0757385	−	0.997128i	$-0.524131\pi$
$137$	−242.900	−0.151477	−0.0757385	+	0.997128i	$0.524131\pi$
$138$	0	0
$139$	−47.7927	−0.0291635	−0.0145817	−	0.999894i	$-0.504642\pi$
$139$	−47.7927	−0.0291635	−0.0145817	+	0.999894i	$0.504642\pi$
$140$	0	0
$141$	0	0
$142$	1153.94	0.681948
$143$	163.140	0.0954019
$144$	0	0
$145$	0	0
$146$	−1022.63	−0.579679
$147$	0	0
$148$	781.739	0.434179
$149$	323.326	0.177771	0.0888856	−	0.996042i	$-0.471669\pi$
$149$	323.326	0.177771	0.0888856	+	0.996042i	$0.471669\pi$
$150$	0	0
$151$	370.894	0.199887	0.0999435	−	0.994993i	$-0.468134\pi$
$151$	370.894	0.199887	0.0999435	+	0.994993i	$0.468134\pi$
$152$	−307.806	−0.164253
$153$	0	0
$154$	−607.872	−0.318076
$155$	0	0
$156$	0	0
$157$	3875.01	1.96981	0.984903	−	0.173105i	$-0.0553801\pi$
$157$	3875.01	1.96981	0.984903	+	0.173105i	$0.0553801\pi$
$158$	785.798	0.395663
$159$	0	0
$160$	0	0
$161$	279.595	0.136864
$162$	0	0
$163$	201.884	0.0970110	0.0485055	−	0.998823i	$-0.484554\pi$
$163$	201.884	0.0970110	0.0485055	+	0.998823i	$0.484554\pi$
$164$	1543.56	0.734948
$165$	0	0
$166$	−1532.20	−0.716395
$167$	−1039.35	−0.481599	−0.240799	−	0.970575i	$-0.577410\pi$
$167$	−1039.35	−0.481599	−0.240799	+	0.970575i	$0.577410\pi$
$168$	0	0
$169$	−1977.04	−0.899883
$170$	0	0
$171$	0	0
$172$	−1697.95	−0.752719
$173$	−208.810	−0.0917660	−0.0458830	−	0.998947i	$-0.514610\pi$
$173$	−208.810	−0.0917660	−0.0458830	+	0.998947i	$0.514610\pi$
$174$	0	0
$175$	0	0
$176$	197.713	0.0846773
$177$	0	0
$178$	−1724.00	−0.725952
$179$	3609.03	1.50699	0.753495	−	0.657453i	$-0.228366\pi$
$179$	3609.03	1.50699	0.753495	+	0.657453i	$0.228366\pi$
$180$	0	0
$181$	1679.68	0.689776	0.344888	−	0.938644i	$-0.387917\pi$
$181$	1679.68	0.689776	0.344888	+	0.938644i	$0.387917\pi$
$182$	−819.573	−0.333795
$183$	0	0
$184$	−247.812	−0.0992876
$185$	0	0
$186$	0	0
$187$	−1009.43	−0.394741
$188$	−1445.58	−0.560795
$189$	0	0
$190$	0	0
$191$	−3562.66	−1.34966	−0.674830	−	0.737973i	$-0.735783\pi$
$191$	−3562.66	−1.34966	−0.674830	+	0.737973i	$0.735783\pi$
$192$	0	0
$193$	−4855.85	−1.81105	−0.905523	−	0.424296i	$-0.860522\pi$
$193$	−4855.85	−1.81105	−0.905523	+	0.424296i	$0.860522\pi$
$194$	1673.32	0.619266
$195$	0	0
$196$	−1540.31	−0.561336
$197$	2308.61	0.834934	0.417467	−	0.908692i	$-0.362918\pi$
$197$	2308.61	0.834934	0.417467	+	0.908692i	$0.362918\pi$
$198$	0	0
$199$	−4004.50	−1.42649	−0.713244	−	0.700915i	$-0.752775\pi$
$199$	−4004.50	−1.42649	−0.713244	+	0.700915i	$0.752775\pi$
$200$	0	0
$201$	0	0
$202$	−2698.65	−0.939983
$203$	4189.52	1.44851
$204$	0	0
$205$	0	0
$206$	3559.96	1.20405
$207$	0	0
$208$	266.570	0.0888621
$209$	140.390	0.0464639
$210$	0	0
$211$	−3187.65	−1.04003	−0.520017	−	0.854156i	$-0.674074\pi$
$211$	−3187.65	−1.04003	−0.520017	+	0.854156i	$0.674074\pi$
$212$	−1762.82	−0.571089
$213$	0	0
$214$	2387.32	0.762588
$215$	0	0
$216$	0	0
$217$	3145.60	0.984043
$218$	−4329.65	−1.34514
$219$	0	0
$220$	0	0
$221$	−1360.98	−0.414250
$222$	0	0
$223$	−3381.16	−1.01533	−0.507666	−	0.861554i	$-0.669492\pi$
$223$	−3381.16	−1.01533	−0.507666	+	0.861554i	$0.669492\pi$
$224$	4256.84	1.26974
$225$	0	0
$226$	−170.421	−0.0501603
$227$	1812.18	0.529862	0.264931	−	0.964267i	$-0.414651\pi$
$227$	1812.18	0.529862	0.264931	+	0.964267i	$0.414651\pi$
$228$	0	0
$229$	−3616.43	−1.04358	−0.521792	−	0.853073i	$-0.674736\pi$
$229$	−3616.43	−1.04358	−0.521792	+	0.853073i	$0.674736\pi$
$230$	0	0
$231$	0	0
$232$	−3713.27	−1.05081
$233$	−401.957	−0.113017	−0.0565087	−	0.998402i	$-0.517997\pi$
$233$	−401.957	−0.113017	−0.0565087	+	0.998402i	$0.517997\pi$
$234$	0	0
$235$	0	0
$236$	−2968.83	−0.818874
$237$	0	0
$238$	5071.09	1.38113
$239$	3529.88	0.955352	0.477676	−	0.878536i	$-0.341479\pi$
$239$	3529.88	0.955352	0.477676	+	0.878536i	$0.341479\pi$
$240$	0	0
$241$	327.477	0.0875297	0.0437648	−	0.999042i	$-0.486065\pi$
$241$	327.477	0.0875297	0.0437648	+	0.999042i	$0.486065\pi$
$242$	−245.732	−0.0652739
$243$	0	0
$244$	1385.62	0.363545
$245$	0	0
$246$	0	0
$247$	189.283	0.0487602
$248$	−2788.02	−0.713870
$249$	0	0
$250$	0	0
$251$	2997.53	0.753795	0.376897	−	0.926255i	$-0.376991\pi$
$251$	2997.53	0.753795	0.376897	+	0.926255i	$0.376991\pi$
$252$	0	0
$253$	113.026	0.0280866
$254$	−1118.20	−0.276229
$255$	0	0
$256$	−4330.22	−1.05718
$257$	−3694.40	−0.896694	−0.448347	−	0.893860i	$-0.647987\pi$
$257$	−3694.40	−0.896694	−0.448347	+	0.893860i	$0.647987\pi$
$258$	0	0
$259$	5488.55	1.31676
$260$	0	0
$261$	0	0
$262$	1205.83	0.284338
$263$	1783.85	0.418238	0.209119	−	0.977890i	$-0.432940\pi$
$263$	1783.85	0.418238	0.209119	+	0.977890i	$0.432940\pi$
$264$	0	0
$265$	0	0
$266$	−705.280	−0.162570
$267$	0	0
$268$	−3671.82	−0.836910
$269$	2406.63	0.545482	0.272741	−	0.962087i	$-0.412070\pi$
$269$	2406.63	0.545482	0.272741	+	0.962087i	$0.412070\pi$
$270$	0	0
$271$	−7331.47	−1.64338	−0.821688	−	0.569937i	$-0.806967\pi$
$271$	−7331.47	−1.64338	−0.821688	+	0.569937i	$0.806967\pi$
$272$	−1649.40	−0.367682
$273$	0	0
$274$	493.293	0.108762
$275$	0	0
$276$	0	0
$277$	−3280.80	−0.711639	−0.355820	−	0.934555i	$-0.615798\pi$
$277$	−3280.80	−0.711639	−0.355820	+	0.934555i	$0.615798\pi$
$278$	97.0596	0.0209397
$279$	0	0
$280$	0	0
$281$	3987.45	0.846516	0.423258	−	0.906009i	$-0.360886\pi$
$281$	3987.45	0.846516	0.423258	+	0.906009i	$0.360886\pi$
$282$	0	0
$283$	6007.11	1.26179	0.630893	−	0.775870i	$-0.282689\pi$
$283$	6007.11	1.26179	0.630893	+	0.775870i	$0.282689\pi$
$284$	2202.18	0.460124
$285$	0	0
$286$	−331.313	−0.0684997
$287$	10837.2	2.22893
$288$	0	0
$289$	3508.02	0.714027
$290$	0	0
$291$	0	0
$292$	−1951.58	−0.391121
$293$	−7671.29	−1.52956	−0.764781	−	0.644290i	$-0.777153\pi$
$293$	−7671.29	−1.52956	−0.764781	+	0.644290i	$0.777153\pi$
$294$	0	0
$295$	0	0
$296$	−4864.64	−0.955241
$297$	0	0
$298$	−656.625	−0.127642
$299$	152.390	0.0294746
$300$	0	0
$301$	−11921.3	−2.28282
$302$	−753.229	−0.143521
$303$	0	0
$304$	229.396	0.0432788
$305$	0	0
$306$	0	0
$307$	6365.60	1.18340	0.591700	−	0.806158i	$-0.298457\pi$
$307$	6365.60	1.18340	0.591700	+	0.806158i	$0.298457\pi$
$308$	−1160.06	−0.214612
$309$	0	0
$310$	0	0
$311$	−3950.06	−0.720217	−0.360108	−	0.932910i	$-0.617260\pi$
$311$	−3950.06	−0.720217	−0.360108	+	0.932910i	$0.617260\pi$
$312$	0	0
$313$	138.061	0.0249318	0.0124659	−	0.999922i	$-0.496032\pi$
$313$	138.061	0.0249318	0.0124659	+	0.999922i	$0.496032\pi$
$314$	−7869.55	−1.41435
$315$	0	0
$316$	1499.61	0.266962
$317$	4816.40	0.853362	0.426681	−	0.904402i	$-0.359683\pi$
$317$	4816.40	0.853362	0.426681	+	0.904402i	$0.359683\pi$
$318$	0	0
$319$	1693.62	0.297255
$320$	0	0
$321$	0	0
$322$	−567.814	−0.0982703
$323$	−1171.18	−0.201753
$324$	0	0
$325$	0	0
$326$	−409.996	−0.0696552
$327$	0	0
$328$	−9605.31	−1.61697
$329$	−10149.3	−1.70076
$330$	0	0
$331$	6885.32	1.14336	0.571679	−	0.820478i	$-0.306292\pi$
$331$	6885.32	1.14336	0.571679	+	0.820478i	$0.306292\pi$
$332$	−2924.04	−0.483366
$333$	0	0
$334$	2110.75	0.345794
$335$	0	0
$336$	0	0
$337$	−7944.37	−1.28415	−0.642073	−	0.766643i	$-0.721926\pi$
$337$	−7944.37	−1.28415	−0.642073	+	0.766643i	$0.721926\pi$
$338$	4015.07	0.646128
$339$	0	0
$340$	0	0
$341$	1271.61	0.201940
$342$	0	0
$343$	−1481.11	−0.233155
$344$	10566.1	1.65606
$345$	0	0
$346$	424.061	0.0658891
$347$	−10587.3	−1.63791	−0.818957	−	0.573855i	$-0.805447\pi$
$347$	−10587.3	−1.63791	−0.818957	+	0.573855i	$0.805447\pi$
$348$	0	0
$349$	−4931.86	−0.756437	−0.378219	−	0.925716i	$-0.623463\pi$
$349$	−4931.86	−0.756437	−0.378219	+	0.925716i	$0.623463\pi$
$350$	0	0
$351$	0	0
$352$	1720.83	0.260569
$353$	783.519	0.118137	0.0590687	−	0.998254i	$-0.481187\pi$
$353$	783.519	0.118137	0.0590687	+	0.998254i	$0.481187\pi$
$354$	0	0
$355$	0	0
$356$	−3290.08	−0.489815
$357$	0	0
$358$	−7329.38	−1.08204
$359$	−973.712	−0.143149	−0.0715746	−	0.997435i	$-0.522802\pi$
$359$	−973.712	−0.143149	−0.0715746	+	0.997435i	$0.522802\pi$
$360$	0	0
$361$	−6696.11	−0.976252
$362$	−3411.17	−0.495268
$363$	0	0
$364$	−1564.07	−0.225219
$365$	0	0
$366$	0	0
$367$	−8373.15	−1.19094	−0.595470	−	0.803377i	$-0.703034\pi$
$367$	−8373.15	−1.19094	−0.595470	+	0.803377i	$0.703034\pi$
$368$	184.684	0.0261613
$369$	0	0
$370$	0	0
$371$	−12376.7	−1.73198
$372$	0	0
$373$	−3367.38	−0.467444	−0.233722	−	0.972303i	$-0.575091\pi$
$373$	−3367.38	−0.467444	−0.233722	+	0.972303i	$0.575091\pi$
$374$	2049.99	0.283429
$375$	0	0
$376$	8995.60	1.23381
$377$	2283.44	0.311945
$378$	0	0
$379$	12549.8	1.70090	0.850448	−	0.526059i	$-0.176331\pi$
$379$	12549.8	1.70090	0.850448	+	0.526059i	$0.176331\pi$
$380$	0	0
$381$	0	0
$382$	7235.22	0.969073
$383$	6138.95	0.819022	0.409511	−	0.912305i	$-0.365699\pi$
$383$	6138.95	0.819022	0.409511	+	0.912305i	$0.365699\pi$
$384$	0	0
$385$	0	0
$386$	9861.49	1.30035
$387$	0	0
$388$	3193.36	0.417831
$389$	−9586.25	−1.24947	−0.624733	−	0.780839i	$-0.714792\pi$
$389$	−9586.25	−1.24947	−0.624733	+	0.780839i	$0.714792\pi$
$390$	0	0
$391$	−942.908	−0.121956
$392$	9585.09	1.23500
$393$	0	0
$394$	−4688.44	−0.599493
$395$	0	0
$396$	0	0
$397$	3406.35	0.430629	0.215315	−	0.976545i	$-0.430922\pi$
$397$	3406.35	0.430629	0.215315	+	0.976545i	$0.430922\pi$
$398$	8132.52	1.02424
$399$	0	0
$400$	0	0
$401$	−6168.72	−0.768207	−0.384104	−	0.923290i	$-0.625489\pi$
$401$	−6168.72	−0.768207	−0.384104	+	0.923290i	$0.625489\pi$
$402$	0	0
$403$	1714.47	0.211920
$404$	−5150.10	−0.634226
$405$	0	0
$406$	−8508.27	−1.04005
$407$	2218.75	0.270219
$408$	0	0
$409$	−8625.82	−1.04283	−0.521417	−	0.853302i	$-0.674597\pi$
$409$	−8625.82	−1.04283	−0.521417	+	0.853302i	$0.674597\pi$
$410$	0	0
$411$	0	0
$412$	6793.82	0.812396
$413$	−20844.0	−2.48346
$414$	0	0
$415$	0	0
$416$	2320.13	0.273447
$417$	0	0
$418$	−285.110	−0.0333617
$419$	−1704.58	−0.198745	−0.0993724	−	0.995050i	$-0.531683\pi$
$419$	−1704.58	−0.198745	−0.0993724	+	0.995050i	$0.531683\pi$
$420$	0	0
$421$	−4880.19	−0.564954	−0.282477	−	0.959274i	$-0.591156\pi$
$421$	−4880.19	−0.564954	−0.282477	+	0.959274i	$0.591156\pi$
$422$	6473.63	0.746757
$423$	0	0
$424$	10969.7	1.25646
$425$	0	0
$426$	0	0
$427$	9728.36	1.10255
$428$	4555.95	0.514533
$429$	0	0
$430$	0	0
$431$	3706.55	0.414241	0.207121	−	0.978315i	$-0.433591\pi$
$431$	3706.55	0.414241	0.207121	+	0.978315i	$0.433591\pi$
$432$	0	0
$433$	478.409	0.0530967	0.0265484	−	0.999648i	$-0.491548\pi$
$433$	478.409	0.0530967	0.0265484	+	0.999648i	$0.491548\pi$
$434$	−6388.23	−0.706555
$435$	0	0
$436$	−8262.68	−0.907594
$437$	131.138	0.0143551
$438$	0	0
$439$	−4353.30	−0.473284	−0.236642	−	0.971597i	$-0.576047\pi$
$439$	−4353.30	−0.473284	−0.236642	+	0.971597i	$0.576047\pi$
$440$	0	0
$441$	0	0
$442$	2763.93	0.297436
$443$	−1807.76	−0.193881	−0.0969405	−	0.995290i	$-0.530906\pi$
$443$	−1807.76	−0.193881	−0.0969405	+	0.995290i	$0.530906\pi$
$444$	0	0
$445$	0	0
$446$	6866.61	0.729022
$447$	0	0
$448$	−12557.7	−1.32432
$449$	−14011.3	−1.47268	−0.736341	−	0.676610i	$-0.763448\pi$
$449$	−14011.3	−1.47268	−0.736341	+	0.676610i	$0.763448\pi$
$450$	0	0
$451$	4380.96	0.457409
$452$	−325.231	−0.0338442
$453$	0	0
$454$	−3680.26	−0.380448
$455$	0	0
$456$	0	0
$457$	10025.1	1.02616	0.513079	−	0.858341i	$-0.328505\pi$
$457$	10025.1	1.02616	0.513079	+	0.858341i	$0.328505\pi$
$458$	7344.42	0.749306
$459$	0	0
$460$	0	0
$461$	−9366.05	−0.946249	−0.473124	−	0.880996i	$-0.656874\pi$
$461$	−9366.05	−0.946249	−0.473124	+	0.880996i	$0.656874\pi$
$462$	0	0
$463$	2431.75	0.244088	0.122044	−	0.992525i	$-0.461055\pi$
$463$	2431.75	0.244088	0.122044	+	0.992525i	$0.461055\pi$
$464$	2767.36	0.276878
$465$	0	0
$466$	816.312	0.0811479
$467$	61.7268	0.00611643	0.00305822	−	0.999995i	$-0.499027\pi$
$467$	61.7268	0.00611643	0.00305822	+	0.999995i	$0.499027\pi$
$468$	0	0
$469$	−25779.7	−2.53815
$470$	0	0
$471$	0	0
$472$	18474.6	1.80161
$473$	−4819.17	−0.468469
$474$	0	0
$475$	0	0
$476$	9677.66	0.931880
$477$	0	0
$478$	−7168.65	−0.685955
$479$	−15444.7	−1.47325	−0.736624	−	0.676302i	$-0.763581\pi$
$479$	−15444.7	−1.47325	−0.736624	+	0.676302i	$0.763581\pi$
$480$	0	0
$481$	2991.46	0.283574
$482$	−665.056	−0.0628474
$483$	0	0
$484$	−468.955	−0.0440416
$485$	0	0
$486$	0	0
$487$	−11763.9	−1.09461	−0.547304	−	0.836934i	$-0.684346\pi$
$487$	−11763.9	−1.09461	−0.547304	+	0.836934i	$0.684346\pi$
$488$	−8622.48	−0.799839
$489$	0	0
$490$	0	0
$491$	−10134.6	−0.931499	−0.465749	−	0.884917i	$-0.654215\pi$
$491$	−10134.6	−0.931499	−0.465749	+	0.884917i	$0.654215\pi$
$492$	0	0
$493$	−14128.8	−1.29073
$494$	−384.404	−0.0350104
$495$	0	0
$496$	2077.81	0.188097
$497$	15461.4	1.39545
$498$	0	0
$499$	15229.5	1.36627	0.683134	−	0.730293i	$-0.260616\pi$
$499$	15229.5	1.36627	0.683134	+	0.730293i	$0.260616\pi$
$500$	0	0
$501$	0	0
$502$	−6087.53	−0.541234
$503$	2000.27	0.177311	0.0886557	−	0.996062i	$-0.471743\pi$
$503$	2000.27	0.177311	0.0886557	+	0.996062i	$0.471743\pi$
$504$	0	0
$505$	0	0
$506$	−229.539	−0.0201665
$507$	0	0
$508$	−2133.98	−0.186378
$509$	18499.8	1.61098	0.805490	−	0.592609i	$-0.201902\pi$
$509$	18499.8	1.61098	0.805490	+	0.592609i	$0.201902\pi$
$510$	0	0
$511$	−13701.9	−1.18618
$512$	6279.74	0.542046
$513$	0	0
$514$	7502.76	0.643838
$515$	0	0
$516$	0	0
$517$	−4102.87	−0.349021
$518$	−11146.4	−0.945454
$519$	0	0
$520$	0	0
$521$	15555.0	1.30802	0.654010	−	0.756486i	$-0.273085\pi$
$521$	15555.0	1.30802	0.654010	+	0.756486i	$0.273085\pi$
$522$	0	0
$523$	−3123.96	−0.261188	−0.130594	−	0.991436i	$-0.541688\pi$
$523$	−3123.96	−0.261188	−0.130594	+	0.991436i	$0.541688\pi$
$524$	2301.21	0.191849
$525$	0	0
$526$	−3622.72	−0.300301
$527$	−10608.3	−0.876855
$528$	0	0
$529$	−12061.4	−0.991323
$530$	0	0
$531$	0	0
$532$	−1345.96	−0.109689
$533$	5906.70	0.480014
$534$	0	0
$535$	0	0
$536$	22849.1	1.84129
$537$	0	0
$538$	−4887.49	−0.391663
$539$	−4371.74	−0.349358
$540$	0	0
$541$	−21119.0	−1.67833	−0.839166	−	0.543876i	$-0.816956\pi$
$541$	−21119.0	−1.67833	−0.839166	+	0.543876i	$0.816956\pi$
$542$	14889.1	1.17996
$543$	0	0
$544$	−14355.8	−1.13143
$545$	0	0
$546$	0	0
$547$	1394.41	0.108996	0.0544979	−	0.998514i	$-0.482644\pi$
$547$	1394.41	0.108996	0.0544979	+	0.998514i	$0.482644\pi$
$548$	941.399	0.0733842
$549$	0	0
$550$	0	0
$551$	1965.01	0.151928
$552$	0	0
$553$	10528.7	0.809633
$554$	6662.80	0.510966
$555$	0	0
$556$	185.228	0.0141285
$557$	833.454	0.0634014	0.0317007	−	0.999497i	$-0.489908\pi$
$557$	833.454	0.0634014	0.0317007	+	0.999497i	$0.489908\pi$
$558$	0	0
$559$	−6497.52	−0.491621
$560$	0	0
$561$	0	0
$562$	−8097.89	−0.607810
$563$	−19864.5	−1.48701	−0.743507	−	0.668729i	$-0.766839\pi$
$563$	−19864.5	−1.48701	−0.743507	+	0.668729i	$0.766839\pi$
$564$	0	0
$565$	0	0
$566$	−12199.5	−0.905979
$567$	0	0
$568$	−13703.8	−1.01232
$569$	−2584.14	−0.190391	−0.0951957	−	0.995459i	$-0.530348\pi$
$569$	−2584.14	−0.190391	−0.0951957	+	0.995459i	$0.530348\pi$
$570$	0	0
$571$	4580.32	0.335693	0.167846	−	0.985813i	$-0.446319\pi$
$571$	4580.32	0.335693	0.167846	+	0.985813i	$0.446319\pi$
$572$	−632.276	−0.0462182
$573$	0	0
$574$	−22008.8	−1.60040
$575$	0	0
$576$	0	0
$577$	−17278.3	−1.24663	−0.623314	−	0.781972i	$-0.714214\pi$
$577$	−17278.3	−1.24663	−0.623314	+	0.781972i	$0.714214\pi$
$578$	−7124.24	−0.512681
$579$	0	0
$580$	0	0
$581$	−20529.6	−1.46594
$582$	0	0
$583$	−5003.27	−0.355428
$584$	12144.4	0.860509
$585$	0	0
$586$	15579.2	1.09825
$587$	−22148.1	−1.55733	−0.778663	−	0.627442i	$-0.784102\pi$
$587$	−22148.1	−1.55733	−0.778663	+	0.627442i	$0.784102\pi$
$588$	0	0
$589$	1475.38	0.103212
$590$	0	0
$591$	0	0
$592$	3625.42	0.251696
$593$	−3639.71	−0.252049	−0.126024	−	0.992027i	$-0.540222\pi$
$593$	−3639.71	−0.252049	−0.126024	+	0.992027i	$0.540222\pi$
$594$	0	0
$595$	0	0
$596$	−1253.10	−0.0861226
$597$	0	0
$598$	−309.480	−0.0211632
$599$	−8532.55	−0.582021	−0.291011	−	0.956720i	$-0.593991\pi$
$599$	−8532.55	−0.582021	−0.291011	+	0.956720i	$0.593991\pi$
$600$	0	0
$601$	12658.8	0.859172	0.429586	−	0.903026i	$-0.358660\pi$
$601$	12658.8	0.859172	0.429586	+	0.903026i	$0.358660\pi$
$602$	24210.2	1.63910
$603$	0	0
$604$	−1437.46	−0.0968368
$605$	0	0
$606$	0	0
$607$	14445.0	0.965906	0.482953	−	0.875646i	$-0.339564\pi$
$607$	14445.0	0.965906	0.482953	+	0.875646i	$0.339564\pi$
$608$	1996.58	0.133178
$609$	0	0
$610$	0	0
$611$	−5531.76	−0.366270
$612$	0	0
$613$	−7680.41	−0.506050	−0.253025	−	0.967460i	$-0.581426\pi$
$613$	−7680.41	−0.506050	−0.253025	+	0.967460i	$0.581426\pi$
$614$	−12927.5	−0.849696
$615$	0	0
$616$	7218.88	0.472170
$617$	21182.8	1.38215	0.691074	−	0.722784i	$-0.257138\pi$
$617$	21182.8	1.38215	0.691074	+	0.722784i	$0.257138\pi$
$618$	0	0
$619$	19612.3	1.27348	0.636740	−	0.771078i	$-0.280282\pi$
$619$	19612.3	1.27348	0.636740	+	0.771078i	$0.280282\pi$
$620$	0	0
$621$	0	0
$622$	8021.97	0.517125
$623$	−23099.5	−1.48549
$624$	0	0
$625$	0	0
$626$	−280.380	−0.0179013
$627$	0	0
$628$	−15018.2	−0.954288
$629$	−18509.6	−1.17333
$630$	0	0
$631$	−1779.35	−0.112258	−0.0561289	−	0.998424i	$-0.517876\pi$
$631$	−1779.35	−0.112258	−0.0561289	+	0.998424i	$0.517876\pi$
$632$	−9331.87	−0.587345
$633$	0	0
$634$	−9781.36	−0.612725
$635$	0	0
$636$	0	0
$637$	−5894.26	−0.366624
$638$	−3439.47	−0.213433
$639$	0	0
$640$	0	0
$641$	−26938.9	−1.65994	−0.829971	−	0.557806i	$-0.811643\pi$
$641$	−26938.9	−1.65994	−0.829971	+	0.557806i	$0.811643\pi$
$642$	0	0
$643$	11554.8	0.708671	0.354336	−	0.935118i	$-0.384707\pi$
$643$	11554.8	0.708671	0.354336	+	0.935118i	$0.384707\pi$
$644$	−1083.62	−0.0663050
$645$	0	0
$646$	2378.49	0.144862
$647$	−23430.5	−1.42372	−0.711860	−	0.702321i	$-0.752147\pi$
$647$	−23430.5	−1.42372	−0.711860	+	0.702321i	$0.752147\pi$
$648$	0	0
$649$	−8426.20	−0.509642
$650$	0	0
$651$	0	0
$652$	−782.435	−0.0469978
$653$	6963.49	0.417308	0.208654	−	0.977990i	$-0.433092\pi$
$653$	6963.49	0.417308	0.208654	+	0.977990i	$0.433092\pi$
$654$	0	0
$655$	0	0
$656$	7158.47	0.426054
$657$	0	0
$658$	20611.7	1.22117
$659$	9811.30	0.579960	0.289980	−	0.957033i	$-0.406351\pi$
$659$	9811.30	0.579960	0.289980	+	0.957033i	$0.406351\pi$
$660$	0	0
$661$	−31077.0	−1.82867	−0.914337	−	0.404954i	$-0.867288\pi$
$661$	−31077.0	−1.82867	−0.914337	+	0.404954i	$0.867288\pi$
$662$	−13983.0	−0.820945
$663$	0	0
$664$	18195.9	1.06346
$665$	0	0
$666$	0	0
$667$	1582.01	0.0918376
$668$	4028.15	0.233314
$669$	0	0
$670$	0	0
$671$	3932.69	0.226259
$672$	0	0
$673$	13034.1	0.746550	0.373275	−	0.927721i	$-0.378235\pi$
$673$	13034.1	0.746550	0.373275	+	0.927721i	$0.378235\pi$
$674$	16133.8	0.922034
$675$	0	0
$676$	7662.35	0.435956
$677$	−26895.0	−1.52682	−0.763410	−	0.645914i	$-0.776477\pi$
$677$	−26895.0	−1.52682	−0.763410	+	0.645914i	$0.776477\pi$
$678$	0	0
$679$	22420.5	1.26719
$680$	0	0
$681$	0	0
$682$	−2582.45	−0.144996
$683$	10858.7	0.608339	0.304169	−	0.952618i	$-0.401621\pi$
$683$	10858.7	0.608339	0.304169	+	0.952618i	$0.401621\pi$
$684$	0	0
$685$	0	0
$686$	3007.90	0.167408
$687$	0	0
$688$	−7874.50	−0.436355
$689$	−6745.74	−0.372993
$690$	0	0
$691$	10663.6	0.587065	0.293533	−	0.955949i	$-0.405169\pi$
$691$	10663.6	0.587065	0.293533	+	0.955949i	$0.405169\pi$
$692$	809.276	0.0444567
$693$	0	0
$694$	21501.2	1.17604
$695$	0	0
$696$	0	0
$697$	−36547.6	−1.98614
$698$	10015.9	0.543131
$699$	0	0
$700$	0	0
$701$	6606.95	0.355979	0.177989	−	0.984032i	$-0.443041\pi$
$701$	6606.95	0.355979	0.177989	+	0.984032i	$0.443041\pi$
$702$	0	0
$703$	2574.29	0.138110
$704$	−5076.44	−0.271769
$705$	0	0
$706$	−1591.21	−0.0848242
$707$	−36158.6	−1.92346
$708$	0	0
$709$	8059.91	0.426934	0.213467	−	0.976950i	$-0.431524\pi$
$709$	8059.91	0.426934	0.213467	+	0.976950i	$0.431524\pi$
$710$	0	0
$711$	0	0
$712$	20473.7	1.07765
$713$	1187.81	0.0623899
$714$	0	0
$715$	0	0
$716$	−13987.4	−0.730073
$717$	0	0
$718$	1977.46	0.102783
$719$	29768.8	1.54408	0.772038	−	0.635577i	$-0.219238\pi$
$719$	29768.8	1.54408	0.772038	+	0.635577i	$0.219238\pi$
$720$	0	0
$721$	47699.1	2.46381
$722$	13598.8	0.700961
$723$	0	0
$724$	−6509.86	−0.334167
$725$	0	0
$726$	0	0
$727$	−4419.98	−0.225486	−0.112743	−	0.993624i	$-0.535964\pi$
$727$	−4419.98	−0.225486	−0.112743	+	0.993624i	$0.535964\pi$
$728$	9732.97	0.495505
$729$	0	0
$730$	0	0
$731$	40203.3	2.03416
$732$	0	0
$733$	1202.63	0.0606007	0.0303003	−	0.999541i	$-0.490354\pi$
$733$	1202.63	0.0606007	0.0303003	+	0.999541i	$0.490354\pi$
$734$	17004.6	0.855110
$735$	0	0
$736$	1607.43	0.0805035
$737$	−10421.4	−0.520867
$738$	0	0
$739$	19392.8	0.965327	0.482663	−	0.875806i	$-0.339669\pi$
$739$	19392.8	0.965327	0.482663	+	0.875806i	$0.339669\pi$
$740$	0	0
$741$	0	0
$742$	25135.1	1.24358
$743$	2255.15	0.111350	0.0556752	−	0.998449i	$-0.482269\pi$
$743$	2255.15	0.111350	0.0556752	+	0.998449i	$0.482269\pi$
$744$	0	0
$745$	0	0
$746$	6838.64	0.335631
$747$	0	0
$748$	3912.20	0.191235
$749$	31987.1	1.56046
$750$	0	0
$751$	−26643.6	−1.29459	−0.647295	−	0.762239i	$-0.724100\pi$
$751$	−26643.6	−1.29459	−0.647295	+	0.762239i	$0.724100\pi$
$752$	−6704.07	−0.325096
$753$	0	0
$754$	−4637.32	−0.223981
$755$	0	0
$756$	0	0
$757$	−9048.83	−0.434459	−0.217229	−	0.976121i	$-0.569702\pi$
$757$	−9048.83	−0.434459	−0.217229	+	0.976121i	$0.569702\pi$
$758$	−25486.7	−1.22126
$759$	0	0
$760$	0	0
$761$	2439.38	0.116199	0.0580994	−	0.998311i	$-0.481496\pi$
$761$	2439.38	0.116199	0.0580994	+	0.998311i	$0.481496\pi$
$762$	0	0
$763$	−58011.9	−2.75252
$764$	13807.7	0.653853
$765$	0	0
$766$	−12467.3	−0.588068
$767$	−11360.8	−0.534828
$768$	0	0
$769$	−25729.3	−1.20653	−0.603265	−	0.797540i	$-0.706134\pi$
$769$	−25729.3	−1.20653	−0.603265	+	0.797540i	$0.706134\pi$
$770$	0	0
$771$	0	0
$772$	18819.7	0.877376
$773$	−2384.74	−0.110961	−0.0554807	−	0.998460i	$-0.517669\pi$
$773$	−2384.74	−0.110961	−0.0554807	+	0.998460i	$0.517669\pi$
$774$	0	0
$775$	0	0
$776$	−19871.8	−0.919274
$777$	0	0
$778$	19468.2	0.897132
$779$	5082.99	0.233783
$780$	0	0
$781$	6250.28	0.286367
$782$	1914.90	0.0875661
$783$	0	0
$784$	−7143.40	−0.325410
$785$	0	0
$786$	0	0
$787$	−308.809	−0.0139871	−0.00699354	−	0.999976i	$-0.502226\pi$
$787$	−308.809	−0.0139871	−0.00699354	+	0.999976i	$0.502226\pi$
$788$	−8947.41	−0.404490
$789$	0	0
$790$	0	0
$791$	−2283.43	−0.102641
$792$	0	0
$793$	5302.31	0.237441
$794$	−6917.77	−0.309197
$795$	0	0
$796$	15520.1	0.691074
$797$	−12395.9	−0.550921	−0.275460	−	0.961312i	$-0.588830\pi$
$797$	−12395.9	−0.550921	−0.275460	+	0.961312i	$0.588830\pi$
$798$	0	0
$799$	34227.7	1.51550
$800$	0	0
$801$	0	0
$802$	12527.7	0.551583
$803$	−5539.01	−0.243422
$804$	0	0
$805$	0	0
$806$	−3481.82	−0.152161
$807$	0	0
$808$	32048.3	1.39536
$809$	−9408.60	−0.408886	−0.204443	−	0.978878i	$-0.565538\pi$
$809$	−9408.60	−0.408886	−0.204443	+	0.978878i	$0.565538\pi$
$810$	0	0
$811$	5489.40	0.237680	0.118840	−	0.992913i	$-0.462082\pi$
$811$	5489.40	0.237680	0.118840	+	0.992913i	$0.462082\pi$
$812$	−16237.2	−0.701740
$813$	0	0
$814$	−4505.94	−0.194021
$815$	0	0
$816$	0	0
$817$	−5591.42	−0.239436
$818$	17517.7	0.748769
$819$	0	0
$820$	0	0
$821$	−1365.73	−0.0580565	−0.0290282	−	0.999579i	$-0.509241\pi$
$821$	−1365.73	−0.0580565	−0.0290282	+	0.999579i	$0.509241\pi$
$822$	0	0
$823$	43252.1	1.83192	0.915962	−	0.401264i	$-0.131429\pi$
$823$	43252.1	1.83192	0.915962	+	0.401264i	$0.131429\pi$
$824$	−42276.9	−1.78736
$825$	0	0
$826$	42331.0	1.78315
$827$	−3670.83	−0.154350	−0.0771750	−	0.997018i	$-0.524590\pi$
$827$	−3670.83	−0.154350	−0.0771750	+	0.997018i	$0.524590\pi$
$828$	0	0
$829$	8734.48	0.365936	0.182968	−	0.983119i	$-0.441430\pi$
$829$	8734.48	0.365936	0.182968	+	0.983119i	$0.441430\pi$
$830$	0	0
$831$	0	0
$832$	−6844.39	−0.285200
$833$	36470.7	1.51697
$834$	0	0
$835$	0	0
$836$	−544.103	−0.0225098
$837$	0	0
$838$	3461.73	0.142701
$839$	−19284.1	−0.793515	−0.396758	−	0.917923i	$-0.629865\pi$
$839$	−19284.1	−0.793515	−0.396758	+	0.917923i	$0.629865\pi$
$840$	0	0
$841$	−683.762	−0.0280357
$842$	9910.91	0.405644
$843$	0	0
$844$	12354.3	0.503852
$845$	0	0
$846$	0	0
$847$	−3292.51	−0.133568
$848$	−8175.32	−0.331063
$849$	0	0
$850$	0	0
$851$	2072.54	0.0834850
$852$	0	0
$853$	−15460.2	−0.620569	−0.310285	−	0.950644i	$-0.600424\pi$
$853$	−15460.2	−0.620569	−0.310285	+	0.950644i	$0.600424\pi$
$854$	−19756.8	−0.791644
$855$	0	0
$856$	−28351.0	−1.13203
$857$	14479.2	0.577130	0.288565	−	0.957460i	$-0.406822\pi$
$857$	14479.2	0.577130	0.288565	+	0.957460i	$0.406822\pi$
$858$	0	0
$859$	−5772.69	−0.229292	−0.114646	−	0.993406i	$-0.536573\pi$
$859$	−5772.69	−0.229292	−0.114646	+	0.993406i	$0.536573\pi$
$860$	0	0
$861$	0	0
$862$	−7527.43	−0.297431
$863$	−4210.06	−0.166063	−0.0830314	−	0.996547i	$-0.526460\pi$
$863$	−4210.06	−0.166063	−0.0830314	+	0.996547i	$0.526460\pi$
$864$	0	0
$865$	0	0
$866$	−971.576	−0.0381241
$867$	0	0
$868$	−12191.3	−0.476727
$869$	4256.24	0.166149
$870$	0	0
$871$	−14050.9	−0.546608
$872$	51417.4	1.99680
$873$	0	0
$874$	−266.322	−0.0103072
$875$	0	0
$876$	0	0
$877$	−2685.99	−0.103420	−0.0517101	−	0.998662i	$-0.516467\pi$
$877$	−2685.99	−0.103420	−0.0517101	+	0.998662i	$0.516467\pi$
$878$	8840.88	0.339824
$879$	0	0
$880$	0	0
$881$	6740.65	0.257773	0.128887	−	0.991659i	$-0.458860\pi$
$881$	6740.65	0.257773	0.128887	+	0.991659i	$0.458860\pi$
$882$	0	0
$883$	16250.8	0.619348	0.309674	−	0.950843i	$-0.399780\pi$
$883$	16250.8	0.619348	0.309674	+	0.950843i	$0.399780\pi$
$884$	5274.68	0.200686
$885$	0	0
$886$	3671.28	0.139209
$887$	35779.6	1.35441	0.677204	−	0.735795i	$-0.263191\pi$
$887$	35779.6	1.35441	0.677204	+	0.735795i	$0.263191\pi$
$888$	0	0
$889$	−14982.5	−0.565240
$890$	0	0
$891$	0	0
$892$	13104.2	0.491886
$893$	−4760.33	−0.178386
$894$	0	0
$895$	0	0
$896$	−8551.98	−0.318864
$897$	0	0
$898$	28454.8	1.05740
$899$	17798.5	0.660304
$900$	0	0
$901$	41739.2	1.54332
$902$	−8897.06	−0.328425
$903$	0	0
$904$	2023.86	0.0744608
$905$	0	0
$906$	0	0
$907$	−10164.2	−0.372101	−0.186050	−	0.982540i	$-0.559569\pi$
$907$	−10164.2	−0.372101	−0.186050	+	0.982540i	$0.559569\pi$
$908$	−7023.40	−0.256696
$909$	0	0
$910$	0	0
$911$	26746.3	0.972717	0.486358	−	0.873759i	$-0.338325\pi$
$911$	26746.3	0.972717	0.486358	+	0.873759i	$0.338325\pi$
$912$	0	0
$913$	−8299.09	−0.300832
$914$	−20359.4	−0.736795
$915$	0	0
$916$	14016.1	0.505572
$917$	16156.7	0.581833
$918$	0	0
$919$	−20170.5	−0.724010	−0.362005	−	0.932176i	$-0.617908\pi$
$919$	−20170.5	−0.724010	−0.362005	+	0.932176i	$0.617908\pi$
$920$	0	0
$921$	0	0
$922$	19021.0	0.679419
$923$	8427.04	0.300519
$924$	0	0
$925$	0	0
$926$	−4938.50	−0.175258
$927$	0	0
$928$	24086.1	0.852010
$929$	15842.7	0.559506	0.279753	−	0.960072i	$-0.409747\pi$
$929$	15842.7	0.559506	0.279753	+	0.960072i	$0.409747\pi$
$930$	0	0
$931$	−5072.29	−0.178558
$932$	1557.85	0.0547522
$933$	0	0
$934$	−125.358	−0.00439168
$935$	0	0
$936$	0	0
$937$	−2061.07	−0.0718593	−0.0359296	−	0.999354i	$-0.511439\pi$
$937$	−2061.07	−0.0718593	−0.0359296	+	0.999354i	$0.511439\pi$
$938$	52354.5	1.82243
$939$	0	0
$940$	0	0
$941$	27936.9	0.967818	0.483909	−	0.875118i	$-0.339217\pi$
$941$	27936.9	0.967818	0.483909	+	0.875118i	$0.339217\pi$
$942$	0	0
$943$	4092.26	0.141318
$944$	−13768.4	−0.474706
$945$	0	0
$946$	9787.00	0.336367
$947$	−53763.9	−1.84487	−0.922436	−	0.386151i	$-0.873804\pi$
$947$	−53763.9	−1.84487	−0.922436	+	0.386151i	$0.873804\pi$
$948$	0	0
$949$	−7468.06	−0.255452
$950$	0	0
$951$	0	0
$952$	−60222.6	−2.05024
$953$	−15951.3	−0.542198	−0.271099	−	0.962551i	$-0.587387\pi$
$953$	−15951.3	−0.542198	−0.271099	+	0.962551i	$0.587387\pi$
$954$	0	0
$955$	0	0
$956$	−13680.6	−0.462828
$957$	0	0
$958$	31365.8	1.05781
$959$	6609.52	0.222557
$960$	0	0
$961$	−16427.4	−0.551422
$962$	−6075.20	−0.203610
$963$	0	0
$964$	−1269.19	−0.0424044
$965$	0	0
$966$	0	0
$967$	−20328.9	−0.676042	−0.338021	−	0.941139i	$-0.609757\pi$
$967$	−20328.9	−0.676042	−0.338021	+	0.941139i	$0.609757\pi$
$968$	2918.24	0.0968963
$969$	0	0
$970$	0	0
$971$	−34000.9	−1.12373	−0.561865	−	0.827229i	$-0.689916\pi$
$971$	−34000.9	−1.12373	−0.561865	+	0.827229i	$0.689916\pi$
$972$	0	0
$973$	1300.48	0.0428484
$974$	23890.7	0.785943
$975$	0	0
$976$	6426.00	0.210749
$977$	7698.19	0.252085	0.126042	−	0.992025i	$-0.459772\pi$
$977$	7698.19	0.252085	0.126042	+	0.992025i	$0.459772\pi$
$978$	0	0
$979$	−9337.99	−0.304845
$980$	0	0
$981$	0	0
$982$	20581.7	0.668828
$983$	17353.3	0.563057	0.281528	−	0.959553i	$-0.409159\pi$
$983$	17353.3	0.563057	0.281528	+	0.959553i	$0.409159\pi$
$984$	0	0
$985$	0	0
$986$	28693.4	0.926758
$987$	0	0
$988$	−733.595	−0.0236222
$989$	−4501.60	−0.144735
$990$	0	0
$991$	42007.7	1.34654	0.673268	−	0.739399i	$-0.264890\pi$
$991$	42007.7	1.34654	0.673268	+	0.739399i	$0.264890\pi$
$992$	18084.5	0.578814
$993$	0	0
$994$	−31399.8	−1.00195
$995$	0	0
$996$	0	0
$997$	21798.3	0.692438	0.346219	−	0.938154i	$-0.387465\pi$
$997$	21798.3	0.692438	0.346219	+	0.938154i	$0.387465\pi$
$998$	−30928.9	−0.980998
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.bc.1.2		4
3.2	odd	2		275.4.a.e.1.3		4
5.4	even	2		495.4.a.n.1.3		4
15.2	even	4		275.4.b.e.199.5		8
15.8	even	4		275.4.b.e.199.4		8
15.14	odd	2		55.4.a.d.1.2	✓	4
60.59	even	2		880.4.a.z.1.4		4
165.164	even	2		605.4.a.j.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.4.a.d.1.2	✓	4	15.14	odd	2
275.4.a.e.1.3		4	3.2	odd	2
275.4.b.e.199.4		8	15.8	even	4
275.4.b.e.199.5		8	15.2	even	4
495.4.a.n.1.3		4	5.4	even	2
605.4.a.j.1.3		4	165.164	even	2
880.4.a.z.1.4		4	60.59	even	2
2475.4.a.bc.1.2		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q-2.03085 q^{2} -3.87566 q^{4} -27.2109 q^{7} +24.1176 q^{8} +O(q^{10})\)	\(q-2.03085 q^{2} -3.87566 q^{4} -27.2109 q^{7} +24.1176 q^{8} -11.0000 q^{11} -14.8309 q^{13} +55.2611 q^{14} -17.9739 q^{16} +91.7661 q^{17} -12.7627 q^{19} +22.3393 q^{22} -10.2751 q^{23} +30.1193 q^{26} +105.460 q^{28} -153.965 q^{29} -115.601 q^{31} -156.439 q^{32} -186.363 q^{34} -201.704 q^{37} +25.9191 q^{38} -398.269 q^{41} +438.107 q^{43} +42.6323 q^{44} +20.8672 q^{46} +372.988 q^{47} +397.431 q^{49} +57.4797 q^{52} +454.843 q^{53} -656.262 q^{56} +312.679 q^{58} +766.019 q^{59} -357.517 q^{61} +234.768 q^{62} +461.495 q^{64} +947.404 q^{67} -355.654 q^{68} -568.207 q^{71} +503.547 q^{73} +409.631 q^{74} +49.4639 q^{76} +299.319 q^{77} -386.931 q^{79} +808.823 q^{82} +754.462 q^{83} -889.727 q^{86} -265.294 q^{88} +848.909 q^{89} +403.562 q^{91} +39.8229 q^{92} -757.482 q^{94} -823.953 q^{97} -807.121 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 19 q^{4} - 9 q^{7} + 33 q^{8}+O(q^{10}) \) 4 * q + q^2 + 19 * q^4 - 9 * q^7 + 33 * q^8	\( 4 q + q^{2} + 19 q^{4} - 9 q^{7} + 33 q^{8} - 44 q^{11} - 70 q^{13} + 49 q^{14} - 37 q^{16} + 103 q^{17} - 205 q^{19} - 11 q^{22} - 56 q^{23} + 86 q^{26} + 551 q^{28} + 79 q^{29} + 49 q^{31} + 225 q^{32} - 939 q^{34} - 289 q^{37} + 145 q^{38} - 736 q^{41} + 152 q^{43} - 209 q^{44} - 334 q^{46} + 412 q^{47} + 37 q^{49} - 1598 q^{52} + 1685 q^{53} + 257 q^{56} - 609 q^{58} + 842 q^{59} - 1097 q^{61} + 1359 q^{62} - 165 q^{64} + 122 q^{67} - 757 q^{68} + 521 q^{71} + 590 q^{73} - 3257 q^{74} - 2825 q^{76} + 99 q^{77} - 1118 q^{79} + 402 q^{82} - 122 q^{83} - 3452 q^{86} - 363 q^{88} + 181 q^{89} - 2190 q^{91} + 430 q^{92} + 1034 q^{94} - 1474 q^{97} - 872 q^{98}+O(q^{100}) \) 4 * q + q^2 + 19 * q^4 - 9 * q^7 + 33 * q^8 - 44 * q^11 - 70 * q^13 + 49 * q^14 - 37 * q^16 + 103 * q^17 - 205 * q^19 - 11 * q^22 - 56 * q^23 + 86 * q^26 + 551 * q^28 + 79 * q^29 + 49 * q^31 + 225 * q^32 - 939 * q^34 - 289 * q^37 + 145 * q^38 - 736 * q^41 + 152 * q^43 - 209 * q^44 - 334 * q^46 + 412 * q^47 + 37 * q^49 - 1598 * q^52 + 1685 * q^53 + 257 * q^56 - 609 * q^58 + 842 * q^59 - 1097 * q^61 + 1359 * q^62 - 165 * q^64 + 122 * q^67 - 757 * q^68 + 521 * q^71 + 590 * q^73 - 3257 * q^74 - 2825 * q^76 + 99 * q^77 - 1118 * q^79 + 402 * q^82 - 122 * q^83 - 3452 * q^86 - 363 * q^88 + 181 * q^89 - 2190 * q^91 + 430 * q^92 + 1034 * q^94 - 1474 * q^97 - 872 * q^98

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