LMFDB - Embedded newform 2475.4.a.bx.1.8

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:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4 x^{9} - 49 x^{8} + 180 x^{7} + 753 x^{6} - 2420 x^{5} - 4059 x^{4} + 9796 x^{3} + 7226 x^{2} + \cdots - 2112 $ x^10 - 4x^9 - 49x^8 + 180x^7 + 753x^6 - 2420x^5 - 4059x^4 + 9796x^3 + 7226x^2 - 9120*x - 2112
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$3.67029$ 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)

$f(q)$	$=$	$q+3.67029 q^{2} +5.47106 q^{4} +21.0437 q^{7} -9.28196 q^{8} +O(q^{10})$	$q+3.67029 q^{2} +5.47106 q^{4} +21.0437 q^{7} -9.28196 q^{8} -11.0000 q^{11} +65.3141 q^{13} +77.2367 q^{14} -77.8360 q^{16} -58.8987 q^{17} -48.3479 q^{19} -40.3732 q^{22} +74.3859 q^{23} +239.722 q^{26} +115.132 q^{28} -17.8493 q^{29} +324.547 q^{31} -211.425 q^{32} -216.176 q^{34} +38.7634 q^{37} -177.451 q^{38} +372.242 q^{41} -459.275 q^{43} -60.1817 q^{44} +273.018 q^{46} +558.080 q^{47} +99.8391 q^{49} +357.337 q^{52} +363.938 q^{53} -195.327 q^{56} -65.5121 q^{58} +263.931 q^{59} -385.451 q^{61} +1191.18 q^{62} -153.305 q^{64} -367.513 q^{67} -322.238 q^{68} +647.416 q^{71} -759.614 q^{73} +142.273 q^{74} -264.514 q^{76} -231.481 q^{77} +707.603 q^{79} +1366.24 q^{82} +879.518 q^{83} -1685.67 q^{86} +102.102 q^{88} -380.167 q^{89} +1374.45 q^{91} +406.970 q^{92} +2048.32 q^{94} +149.441 q^{97} +366.439 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 4 q^{2} + 34 q^{4} - 2 q^{7} + 48 q^{8}+O(q^{10}) $ 10 * q + 4 * q^2 + 34 * q^4 - 2 * q^7 + 48 * q^8	$ 10 q + 4 q^{2} + 34 q^{4} - 2 q^{7} + 48 q^{8} - 110 q^{11} - 26 q^{13} + 72 q^{14} + 206 q^{16} + 148 q^{17} - 114 q^{19} - 44 q^{22} - 34 q^{23} + 100 q^{26} + 86 q^{28} - 38 q^{29} + 232 q^{31} + 448 q^{32} - 20 q^{34} - 754 q^{37} + 780 q^{38} + 160 q^{41} + 66 q^{43} - 374 q^{44} + 682 q^{46} + 450 q^{47} + 590 q^{49} - 200 q^{52} + 1068 q^{53} + 268 q^{56} + 138 q^{58} - 838 q^{59} - 566 q^{61} + 1230 q^{62} + 462 q^{64} - 430 q^{67} + 2234 q^{68} + 518 q^{71} + 184 q^{73} + 402 q^{74} + 386 q^{76} + 22 q^{77} + 956 q^{79} + 2180 q^{82} + 2094 q^{83} + 892 q^{86} - 528 q^{88} - 512 q^{89} - 858 q^{91} + 4476 q^{92} - 294 q^{94} + 1006 q^{97} + 2226 q^{98}+O(q^{100}) $ 10 * q + 4 * q^2 + 34 * q^4 - 2 * q^7 + 48 * q^8 - 110 * q^11 - 26 * q^13 + 72 * q^14 + 206 * q^16 + 148 * q^17 - 114 * q^19 - 44 * q^22 - 34 * q^23 + 100 * q^26 + 86 * q^28 - 38 * q^29 + 232 * q^31 + 448 * q^32 - 20 * q^34 - 754 * q^37 + 780 * q^38 + 160 * q^41 + 66 * q^43 - 374 * q^44 + 682 * q^46 + 450 * q^47 + 590 * q^49 - 200 * q^52 + 1068 * q^53 + 268 * q^56 + 138 * q^58 - 838 * q^59 - 566 * q^61 + 1230 * q^62 + 462 * q^64 - 430 * q^67 + 2234 * q^68 + 518 * q^71 + 184 * q^73 + 402 * q^74 + 386 * q^76 + 22 * q^77 + 956 * q^79 + 2180 * q^82 + 2094 * q^83 + 892 * q^86 - 528 * q^88 - 512 * q^89 - 858 * q^91 + 4476 * q^92 - 294 * q^94 + 1006 * q^97 + 2226 * 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 $

$2$	3.67029	1.29764	0.648822	−	0.760940i	$-0.275262\pi$
$2$	3.67029	1.29764	0.648822	+	0.760940i	$0.275262\pi$
$3$	0	0
$3$	0	0
$4$	5.47106	0.683882
$5$	0	0
$5$	0	0
$6$	0	0
$7$	21.0437	1.13626	0.568128	−	0.822940i	$-0.307668\pi$
$7$	21.0437	1.13626	0.568128	+	0.822940i	$0.307668\pi$
$8$	−9.28196	−0.410208
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	65.3141	1.39345	0.696726	−	0.717337i	$-0.254639\pi$
$13$	65.3141	1.39345	0.696726	+	0.717337i	$0.254639\pi$
$14$	77.2367	1.47446
$15$	0	0
$16$	−77.8360	−1.21619
$17$	−58.8987	−0.840296	−0.420148	−	0.907456i	$-0.638022\pi$
$17$	−58.8987	−0.840296	−0.420148	+	0.907456i	$0.638022\pi$
$18$	0	0
$19$	−48.3479	−0.583778	−0.291889	−	0.956452i	$-0.594284\pi$
$19$	−48.3479	−0.583778	−0.291889	+	0.956452i	$0.594284\pi$
$20$	0	0
$21$	0	0
$22$	−40.3732	−0.391255
$23$	74.3859	0.674371	0.337185	−	0.941438i	$-0.390525\pi$
$23$	74.3859	0.674371	0.337185	+	0.941438i	$0.390525\pi$
$24$	0	0
$25$	0	0
$26$	239.722	1.80821
$27$	0	0
$28$	115.132	0.777065
$29$	−17.8493	−0.114294	−0.0571470	−	0.998366i	$-0.518200\pi$
$29$	−17.8493	−0.114294	−0.0571470	+	0.998366i	$0.518200\pi$
$30$	0	0
$31$	324.547	1.88034	0.940168	−	0.340711i	$-0.110668\pi$
$31$	324.547	1.88034	0.940168	+	0.340711i	$0.110668\pi$
$32$	−211.425	−1.16797
$33$	0	0
$34$	−216.176	−1.09041
$35$	0	0
$36$	0	0
$37$	38.7634	0.172234	0.0861171	−	0.996285i	$-0.472554\pi$
$37$	38.7634	0.172234	0.0861171	+	0.996285i	$0.472554\pi$
$38$	−177.451	−0.757536
$39$	0	0
$40$	0	0
$41$	372.242	1.41791	0.708957	−	0.705252i	$-0.249166\pi$
$41$	372.242	1.41791	0.708957	+	0.705252i	$0.249166\pi$
$42$	0	0
$43$	−459.275	−1.62881	−0.814404	−	0.580299i	$-0.802936\pi$
$43$	−459.275	−1.62881	−0.814404	+	0.580299i	$0.802936\pi$
$44$	−60.1817	−0.206198
$45$	0	0
$46$	273.018	0.875094
$47$	558.080	1.73201	0.866003	−	0.500038i	$-0.166681\pi$
$47$	558.080	1.73201	0.866003	+	0.500038i	$0.166681\pi$
$48$	0	0
$49$	99.8391	0.291076
$50$	0	0
$51$	0	0
$52$	357.337	0.952958
$53$	363.938	0.943222	0.471611	−	0.881807i	$-0.343673\pi$
$53$	363.938	0.943222	0.471611	+	0.881807i	$0.343673\pi$
$54$	0	0
$55$	0	0
$56$	−195.327	−0.466101
$57$	0	0
$58$	−65.5121	−0.148313
$59$	263.931	0.582389	0.291194	−	0.956664i	$-0.405947\pi$
$59$	263.931	0.582389	0.291194	+	0.956664i	$0.405947\pi$
$60$	0	0
$61$	−385.451	−0.809048	−0.404524	−	0.914527i	$-0.632563\pi$
$61$	−385.451	−0.809048	−0.404524	+	0.914527i	$0.632563\pi$
$62$	1191.18	2.44001
$63$	0	0
$64$	−153.305	−0.299424
$65$	0	0
$66$	0	0
$67$	−367.513	−0.670132	−0.335066	−	0.942195i	$-0.608759\pi$
$67$	−367.513	−0.670132	−0.335066	+	0.942195i	$0.608759\pi$
$68$	−322.238	−0.574664
$69$	0	0
$70$	0	0
$71$	647.416	1.08217	0.541086	−	0.840967i	$-0.318013\pi$
$71$	647.416	1.08217	0.541086	+	0.840967i	$0.318013\pi$
$72$	0	0
$73$	−759.614	−1.21789	−0.608945	−	0.793212i	$-0.708407\pi$
$73$	−759.614	−1.21789	−0.608945	+	0.793212i	$0.708407\pi$
$74$	142.273	0.223499
$75$	0	0
$76$	−264.514	−0.399235
$77$	−231.481	−0.342594
$78$	0	0
$79$	707.603	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$79$	707.603	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$80$	0	0
$81$	0	0
$82$	1366.24	1.83995
$83$	879.518	1.16313	0.581564	−	0.813500i	$-0.302441\pi$
$83$	879.518	1.16313	0.581564	+	0.813500i	$0.302441\pi$
$84$	0	0
$85$	0	0
$86$	−1685.67	−2.11361
$87$	0	0
$88$	102.102	0.123682
$89$	−380.167	−0.452782	−0.226391	−	0.974037i	$-0.572693\pi$
$89$	−380.167	−0.452782	−0.226391	+	0.974037i	$0.572693\pi$
$90$	0	0
$91$	1374.45	1.58332
$92$	406.970	0.461190
$93$	0	0
$94$	2048.32	2.24753
$95$	0	0
$96$	0	0
$97$	149.441	0.156427	0.0782133	−	0.996937i	$-0.475078\pi$
$97$	149.441	0.156427	0.0782133	+	0.996937i	$0.475078\pi$
$98$	366.439	0.377713
$99$	0	0
$100$	0	0
$101$	1808.67	1.78188	0.890940	−	0.454122i	$-0.150047\pi$
$101$	1808.67	1.78188	0.890940	+	0.454122i	$0.150047\pi$
$102$	0	0
$103$	−1346.56	−1.28816	−0.644080	−	0.764959i	$-0.722759\pi$
$103$	−1346.56	−1.28816	−0.644080	+	0.764959i	$0.722759\pi$
$104$	−606.243	−0.571606
$105$	0	0
$106$	1335.76	1.22397
$107$	1611.32	1.45581	0.727906	−	0.685677i	$-0.240494\pi$
$107$	1611.32	1.45581	0.727906	+	0.685677i	$0.240494\pi$
$108$	0	0
$109$	−986.519	−0.866894	−0.433447	−	0.901179i	$-0.642703\pi$
$109$	−986.519	−0.866894	−0.433447	+	0.901179i	$0.642703\pi$
$110$	0	0
$111$	0	0
$112$	−1637.96	−1.38190
$113$	−589.981	−0.491157	−0.245578	−	0.969377i	$-0.578978\pi$
$113$	−589.981	−0.491157	−0.245578	+	0.969377i	$0.578978\pi$
$114$	0	0
$115$	0	0
$116$	−97.6545	−0.0781637
$117$	0	0
$118$	968.706	0.755734
$119$	−1239.45	−0.954791
$120$	0	0
$121$	121.000	0.0909091
$122$	−1414.72	−1.04986
$123$	0	0
$124$	1775.62	1.28593
$125$	0	0
$126$	0	0
$127$	2504.38	1.74982	0.874911	−	0.484284i	$-0.160920\pi$
$127$	2504.38	1.74982	0.874911	+	0.484284i	$0.160920\pi$
$128$	1128.73	0.779425
$129$	0	0
$130$	0	0
$131$	−1747.76	−1.16567	−0.582835	−	0.812590i	$-0.698057\pi$
$131$	−1747.76	−1.16567	−0.582835	+	0.812590i	$0.698057\pi$
$132$	0	0
$133$	−1017.42	−0.663320
$134$	−1348.88	−0.869593
$135$	0	0
$136$	546.695	0.344697
$137$	−1321.37	−0.824029	−0.412014	−	0.911177i	$-0.635175\pi$
$137$	−1321.37	−0.824029	−0.412014	+	0.911177i	$0.635175\pi$
$138$	0	0
$139$	−2765.95	−1.68781	−0.843904	−	0.536495i	$-0.819748\pi$
$139$	−2765.95	−1.68781	−0.843904	+	0.536495i	$0.819748\pi$
$140$	0	0
$141$	0	0
$142$	2376.21	1.40427
$143$	−718.455	−0.420142
$144$	0	0
$145$	0	0
$146$	−2788.01	−1.58039
$147$	0	0
$148$	212.077	0.117788
$149$	3142.37	1.72774	0.863870	−	0.503714i	$-0.168033\pi$
$149$	3142.37	1.72774	0.863870	+	0.503714i	$0.168033\pi$
$150$	0	0
$151$	2511.00	1.35326	0.676629	−	0.736324i	$-0.263440\pi$
$151$	2511.00	1.35326	0.676629	+	0.736324i	$0.263440\pi$
$152$	448.763	0.239470
$153$	0	0
$154$	−849.604	−0.444565
$155$	0	0
$156$	0	0
$157$	−1035.93	−0.526598	−0.263299	−	0.964714i	$-0.584811\pi$
$157$	−1035.93	−0.526598	−0.263299	+	0.964714i	$0.584811\pi$
$158$	2597.11	1.30769
$159$	0	0
$160$	0	0
$161$	1565.36	0.766258
$162$	0	0
$163$	329.221	0.158200	0.0790999	−	0.996867i	$-0.474795\pi$
$163$	329.221	0.158200	0.0790999	+	0.996867i	$0.474795\pi$
$164$	2036.56	0.969687
$165$	0	0
$166$	3228.09	1.50933
$167$	−1181.05	−0.547258	−0.273629	−	0.961835i	$-0.588224\pi$
$167$	−1181.05	−0.547258	−0.273629	+	0.961835i	$0.588224\pi$
$168$	0	0
$169$	2068.94	0.941710
$170$	0	0
$171$	0	0
$172$	−2512.72	−1.11391
$173$	2945.78	1.29459	0.647294	−	0.762241i	$-0.275901\pi$
$173$	2945.78	1.29459	0.647294	+	0.762241i	$0.275901\pi$
$174$	0	0
$175$	0	0
$176$	856.196	0.366694
$177$	0	0
$178$	−1395.32	−0.587550
$179$	4028.34	1.68208	0.841039	−	0.540974i	$-0.181944\pi$
$179$	4028.34	1.68208	0.841039	+	0.540974i	$0.181944\pi$
$180$	0	0
$181$	2045.58	0.840037	0.420018	−	0.907516i	$-0.362024\pi$
$181$	2045.58	0.840037	0.420018	+	0.907516i	$0.362024\pi$
$182$	5044.65	2.05458
$183$	0	0
$184$	−690.447	−0.276633
$185$	0	0
$186$	0	0
$187$	647.886	0.253359
$188$	3053.29	1.18449
$189$	0	0
$190$	0	0
$191$	−2094.10	−0.793318	−0.396659	−	0.917966i	$-0.629830\pi$
$191$	−2094.10	−0.793318	−0.396659	+	0.917966i	$0.629830\pi$
$192$	0	0
$193$	1714.38	0.639397	0.319698	−	0.947519i	$-0.396418\pi$
$193$	1714.38	0.639397	0.319698	+	0.947519i	$0.396418\pi$
$194$	548.491	0.202986
$195$	0	0
$196$	546.225	0.199062
$197$	−522.272	−0.188885	−0.0944425	−	0.995530i	$-0.530107\pi$
$197$	−522.272	−0.188885	−0.0944425	+	0.995530i	$0.530107\pi$
$198$	0	0
$199$	3327.80	1.18543	0.592717	−	0.805411i	$-0.298055\pi$
$199$	3327.80	1.18543	0.592717	+	0.805411i	$0.298055\pi$
$200$	0	0
$201$	0	0
$202$	6638.37	2.31225
$203$	−375.616	−0.129867
$204$	0	0
$205$	0	0
$206$	−4942.27	−1.67157
$207$	0	0
$208$	−5083.79	−1.69470
$209$	531.827	0.176016
$210$	0	0
$211$	4276.82	1.39539	0.697697	−	0.716393i	$-0.254208\pi$
$211$	4276.82	1.39539	0.697697	+	0.716393i	$0.254208\pi$
$212$	1991.13	0.645053
$213$	0	0
$214$	5914.01	1.88913
$215$	0	0
$216$	0	0
$217$	6829.69	2.13654
$218$	−3620.82	−1.12492
$219$	0	0
$220$	0	0
$221$	−3846.92	−1.17091
$222$	0	0
$223$	−2884.48	−0.866185	−0.433092	−	0.901350i	$-0.642578\pi$
$223$	−2884.48	−0.866185	−0.433092	+	0.901350i	$0.642578\pi$
$224$	−4449.18	−1.32711
$225$	0	0
$226$	−2165.40	−0.637347
$227$	5670.26	1.65792	0.828961	−	0.559306i	$-0.188932\pi$
$227$	5670.26	1.65792	0.828961	+	0.559306i	$0.188932\pi$
$228$	0	0
$229$	33.8426	0.00976585	0.00488292	−	0.999988i	$-0.498446\pi$
$229$	33.8426	0.00976585	0.00488292	+	0.999988i	$0.498446\pi$
$230$	0	0
$231$	0	0
$232$	165.676	0.0468844
$233$	5176.24	1.45539	0.727697	−	0.685899i	$-0.240591\pi$
$233$	5176.24	1.45539	0.727697	+	0.685899i	$0.240591\pi$
$234$	0	0
$235$	0	0
$236$	1443.98	0.398286
$237$	0	0
$238$	−4549.14	−1.23898
$239$	−240.063	−0.0649723	−0.0324861	−	0.999472i	$-0.510342\pi$
$239$	−240.063	−0.0649723	−0.0324861	+	0.999472i	$0.510342\pi$
$240$	0	0
$241$	2312.09	0.617985	0.308993	−	0.951064i	$-0.400008\pi$
$241$	2312.09	0.617985	0.308993	+	0.951064i	$0.400008\pi$
$242$	444.106	0.117968
$243$	0	0
$244$	−2108.82	−0.553293
$245$	0	0
$246$	0	0
$247$	−3157.80	−0.813466
$248$	−3012.43	−0.771330
$249$	0	0
$250$	0	0
$251$	−1530.57	−0.384894	−0.192447	−	0.981307i	$-0.561642\pi$
$251$	−1530.57	−0.384894	−0.192447	+	0.981307i	$0.561642\pi$
$252$	0	0
$253$	−818.245	−0.203331
$254$	9191.79	2.27065
$255$	0	0
$256$	5369.20	1.31084
$257$	−1580.87	−0.383704	−0.191852	−	0.981424i	$-0.561449\pi$
$257$	−1580.87	−0.383704	−0.191852	+	0.981424i	$0.561449\pi$
$258$	0	0
$259$	815.727	0.195702
$260$	0	0
$261$	0	0
$262$	−6414.81	−1.51263
$263$	−7620.96	−1.78680	−0.893400	−	0.449263i	$-0.851687\pi$
$263$	−7620.96	−1.78680	−0.893400	+	0.449263i	$0.851687\pi$
$264$	0	0
$265$	0	0
$266$	−3734.24	−0.860754
$267$	0	0
$268$	−2010.68	−0.458292
$269$	−1275.42	−0.289083	−0.144542	−	0.989499i	$-0.546171\pi$
$269$	−1275.42	−0.289083	−0.144542	+	0.989499i	$0.546171\pi$
$270$	0	0
$271$	−1325.32	−0.297075	−0.148537	−	0.988907i	$-0.547456\pi$
$271$	−1325.32	−0.297075	−0.148537	+	0.988907i	$0.547456\pi$
$272$	4584.44	1.02196
$273$	0	0
$274$	−4849.80	−1.06930
$275$	0	0
$276$	0	0
$277$	1478.50	0.320701	0.160351	−	0.987060i	$-0.448737\pi$
$277$	1478.50	0.320701	0.160351	+	0.987060i	$0.448737\pi$
$278$	−10151.9	−2.19017
$279$	0	0
$280$	0	0
$281$	−385.462	−0.0818318	−0.0409159	−	0.999163i	$-0.513028\pi$
$281$	−385.462	−0.0818318	−0.0409159	+	0.999163i	$0.513028\pi$
$282$	0	0
$283$	−4716.10	−0.990611	−0.495305	−	0.868719i	$-0.664944\pi$
$283$	−4716.10	−0.990611	−0.495305	+	0.868719i	$0.664944\pi$
$284$	3542.05	0.740078
$285$	0	0
$286$	−2636.94	−0.545195
$287$	7833.37	1.61111
$288$	0	0
$289$	−1443.94	−0.293902
$290$	0	0
$291$	0	0
$292$	−4155.89	−0.832894
$293$	5560.21	1.10864	0.554319	−	0.832304i	$-0.312979\pi$
$293$	5560.21	1.10864	0.554319	+	0.832304i	$0.312979\pi$
$294$	0	0
$295$	0	0
$296$	−359.800	−0.0706519
$297$	0	0
$298$	11533.4	2.24199
$299$	4858.45	0.939704
$300$	0	0
$301$	−9664.85	−1.85074
$302$	9216.10	1.75605
$303$	0	0
$304$	3763.21	0.709983
$305$	0	0
$306$	0	0
$307$	1284.57	0.238809	0.119404	−	0.992846i	$-0.461901\pi$
$307$	1284.57	0.238809	0.119404	+	0.992846i	$0.461901\pi$
$308$	−1266.45	−0.234294
$309$	0	0
$310$	0	0
$311$	−1630.37	−0.297266	−0.148633	−	0.988892i	$-0.547487\pi$
$311$	−1630.37	−0.297266	−0.148633	+	0.988892i	$0.547487\pi$
$312$	0	0
$313$	−3229.45	−0.583192	−0.291596	−	0.956542i	$-0.594186\pi$
$313$	−3229.45	−0.583192	−0.291596	+	0.956542i	$0.594186\pi$
$314$	−3802.15	−0.683337
$315$	0	0
$316$	3871.34	0.689176
$317$	−44.3705	−0.00786151	−0.00393075	−	0.999992i	$-0.501251\pi$
$317$	−44.3705	−0.00786151	−0.00393075	+	0.999992i	$0.501251\pi$
$318$	0	0
$319$	196.342	0.0344610
$320$	0	0
$321$	0	0
$322$	5745.32	0.994330
$323$	2847.63	0.490546
$324$	0	0
$325$	0	0
$326$	1208.34	0.205287
$327$	0	0
$328$	−3455.14	−0.581640
$329$	11744.1	1.96800
$330$	0	0
$331$	1040.13	0.172721	0.0863603	−	0.996264i	$-0.472476\pi$
$331$	1040.13	0.172721	0.0863603	+	0.996264i	$0.472476\pi$
$332$	4811.90	0.795443
$333$	0	0
$334$	−4334.79	−0.710147
$335$	0	0
$336$	0	0
$337$	3152.56	0.509587	0.254793	−	0.966996i	$-0.417993\pi$
$337$	3152.56	0.509587	0.254793	+	0.966996i	$0.417993\pi$
$338$	7593.60	1.22200
$339$	0	0
$340$	0	0
$341$	−3570.02	−0.566943
$342$	0	0
$343$	−5117.02	−0.805519
$344$	4262.97	0.668150
$345$	0	0
$346$	10811.9	1.67991
$347$	−5181.86	−0.801662	−0.400831	−	0.916152i	$-0.631279\pi$
$347$	−5181.86	−0.801662	−0.400831	+	0.916152i	$0.631279\pi$
$348$	0	0
$349$	−5468.78	−0.838789	−0.419394	−	0.907804i	$-0.637758\pi$
$349$	−5468.78	−0.838789	−0.419394	+	0.907804i	$0.637758\pi$
$350$	0	0
$351$	0	0
$352$	2325.68	0.352156
$353$	−5887.60	−0.887720	−0.443860	−	0.896096i	$-0.646391\pi$
$353$	−5887.60	−0.887720	−0.443860	+	0.896096i	$0.646391\pi$
$354$	0	0
$355$	0	0
$356$	−2079.91	−0.309650
$357$	0	0
$358$	14785.2	2.18274
$359$	−3930.72	−0.577871	−0.288935	−	0.957349i	$-0.593301\pi$
$359$	−3930.72	−0.577871	−0.288935	+	0.957349i	$0.593301\pi$
$360$	0	0
$361$	−4521.48	−0.659204
$362$	7507.88	1.09007
$363$	0	0
$364$	7519.72	1.08280
$365$	0	0
$366$	0	0
$367$	9860.59	1.40250	0.701251	−	0.712914i	$-0.252625\pi$
$367$	9860.59	1.40250	0.701251	+	0.712914i	$0.252625\pi$
$368$	−5789.90	−0.820161
$369$	0	0
$370$	0	0
$371$	7658.62	1.07174
$372$	0	0
$373$	2887.30	0.400801	0.200401	−	0.979714i	$-0.435776\pi$
$373$	2887.30	0.400801	0.200401	+	0.979714i	$0.435776\pi$
$374$	2377.93	0.328770
$375$	0	0
$376$	−5180.07	−0.710484
$377$	−1165.81	−0.159263
$378$	0	0
$379$	8563.50	1.16063	0.580313	−	0.814394i	$-0.302930\pi$
$379$	8563.50	1.16063	0.580313	+	0.814394i	$0.302930\pi$
$380$	0	0
$381$	0	0
$382$	−7685.96	−1.02944
$383$	5327.37	0.710746	0.355373	−	0.934725i	$-0.384354\pi$
$383$	5327.37	0.710746	0.355373	+	0.934725i	$0.384354\pi$
$384$	0	0
$385$	0	0
$386$	6292.27	0.829710
$387$	0	0
$388$	817.598	0.106977
$389$	−2468.21	−0.321705	−0.160852	−	0.986978i	$-0.551424\pi$
$389$	−2468.21	−0.321705	−0.160852	+	0.986978i	$0.551424\pi$
$390$	0	0
$391$	−4381.23	−0.566671
$392$	−926.702	−0.119402
$393$	0	0
$394$	−1916.89	−0.245106
$395$	0	0
$396$	0	0
$397$	7796.85	0.985675	0.492837	−	0.870122i	$-0.335960\pi$
$397$	7796.85	0.985675	0.492837	+	0.870122i	$0.335960\pi$
$398$	12214.0	1.53827
$399$	0	0
$400$	0	0
$401$	4914.79	0.612052	0.306026	−	0.952023i	$-0.401001\pi$
$401$	4914.79	0.612052	0.306026	+	0.952023i	$0.401001\pi$
$402$	0	0
$403$	21197.5	2.62016
$404$	9895.36	1.21860
$405$	0	0
$406$	−1378.62	−0.168522
$407$	−426.398	−0.0519306
$408$	0	0
$409$	−6911.42	−0.835569	−0.417785	−	0.908546i	$-0.637193\pi$
$409$	−6911.42	−0.835569	−0.417785	+	0.908546i	$0.637193\pi$
$410$	0	0
$411$	0	0
$412$	−7367.10	−0.880949
$413$	5554.11	0.661743
$414$	0	0
$415$	0	0
$416$	−13809.1	−1.62751
$417$	0	0
$418$	1951.96	0.228406
$419$	−6799.64	−0.792803	−0.396401	−	0.918077i	$-0.629741\pi$
$419$	−6799.64	−0.792803	−0.396401	+	0.918077i	$0.629741\pi$
$420$	0	0
$421$	3769.24	0.436345	0.218173	−	0.975910i	$-0.429990\pi$
$421$	3769.24	0.436345	0.218173	+	0.975910i	$0.429990\pi$
$422$	15697.2	1.81073
$423$	0	0
$424$	−3378.06	−0.386917
$425$	0	0
$426$	0	0
$427$	−8111.33	−0.919285
$428$	8815.61	0.995605
$429$	0	0
$430$	0	0
$431$	−10592.4	−1.18380	−0.591900	−	0.806011i	$-0.701622\pi$
$431$	−10592.4	−1.18380	−0.591900	+	0.806011i	$0.701622\pi$
$432$	0	0
$433$	−15929.7	−1.76798	−0.883989	−	0.467509i	$-0.845152\pi$
$433$	−15929.7	−1.76798	−0.883989	+	0.467509i	$0.845152\pi$
$434$	25067.0	2.77247
$435$	0	0
$436$	−5397.31	−0.592853
$437$	−3596.40	−0.393683
$438$	0	0
$439$	−14631.8	−1.59074	−0.795371	−	0.606122i	$-0.792724\pi$
$439$	−14631.8	−1.59074	−0.795371	+	0.606122i	$0.792724\pi$
$440$	0	0
$441$	0	0
$442$	−14119.3	−1.51943
$443$	1577.36	0.169170	0.0845852	−	0.996416i	$-0.473043\pi$
$443$	1577.36	0.169170	0.0845852	+	0.996416i	$0.473043\pi$
$444$	0	0
$445$	0	0
$446$	−10586.9	−1.12400
$447$	0	0
$448$	−3226.11	−0.340222
$449$	4498.25	0.472796	0.236398	−	0.971656i	$-0.424033\pi$
$449$	4498.25	0.472796	0.236398	+	0.971656i	$0.424033\pi$
$450$	0	0
$451$	−4094.67	−0.427517
$452$	−3227.82	−0.335894
$453$	0	0
$454$	20811.5	2.15139
$455$	0	0
$456$	0	0
$457$	−15877.5	−1.62520	−0.812600	−	0.582822i	$-0.801949\pi$
$457$	−15877.5	−1.62520	−0.812600	+	0.582822i	$0.801949\pi$
$458$	124.212	0.0126726
$459$	0	0
$460$	0	0
$461$	−17877.0	−1.80611	−0.903054	−	0.429527i	$-0.858680\pi$
$461$	−17877.0	−1.80611	−0.903054	+	0.429527i	$0.858680\pi$
$462$	0	0
$463$	−2226.70	−0.223506	−0.111753	−	0.993736i	$-0.535647\pi$
$463$	−2226.70	−0.223506	−0.111753	+	0.993736i	$0.535647\pi$
$464$	1389.32	0.139003
$465$	0	0
$466$	18998.3	1.88859
$467$	19540.6	1.93625	0.968125	−	0.250467i	$-0.0805841\pi$
$467$	19540.6	1.93625	0.968125	+	0.250467i	$0.0805841\pi$
$468$	0	0
$469$	−7733.85	−0.761441
$470$	0	0
$471$	0	0
$472$	−2449.80	−0.238901
$473$	5052.02	0.491104
$474$	0	0
$475$	0	0
$476$	−6781.10	−0.652965
$477$	0	0
$478$	−881.101	−0.0843110
$479$	3056.98	0.291602	0.145801	−	0.989314i	$-0.453424\pi$
$479$	3056.98	0.291602	0.145801	+	0.989314i	$0.453424\pi$
$480$	0	0
$481$	2531.80	0.240000
$482$	8486.03	0.801926
$483$	0	0
$484$	661.998	0.0621711
$485$	0	0
$486$	0	0
$487$	2753.54	0.256212	0.128106	−	0.991761i	$-0.459110\pi$
$487$	2753.54	0.256212	0.128106	+	0.991761i	$0.459110\pi$
$488$	3577.74	0.331878
$489$	0	0
$490$	0	0
$491$	7774.68	0.714595	0.357298	−	0.933991i	$-0.383698\pi$
$491$	7774.68	0.714595	0.357298	+	0.933991i	$0.383698\pi$
$492$	0	0
$493$	1051.30	0.0960409
$494$	−11590.1	−1.05559
$495$	0	0
$496$	−25261.5	−2.28684
$497$	13624.1	1.22962
$498$	0	0
$499$	603.250	0.0541186	0.0270593	−	0.999634i	$-0.491386\pi$
$499$	603.250	0.0541186	0.0270593	+	0.999634i	$0.491386\pi$
$500$	0	0
$501$	0	0
$502$	−5617.62	−0.499456
$503$	4927.64	0.436805	0.218402	−	0.975859i	$-0.429915\pi$
$503$	4927.64	0.436805	0.218402	+	0.975859i	$0.429915\pi$
$504$	0	0
$505$	0	0
$506$	−3003.20	−0.263851
$507$	0	0
$508$	13701.6	1.19667
$509$	−12663.0	−1.10271	−0.551353	−	0.834272i	$-0.685888\pi$
$509$	−12663.0	−1.10271	−0.551353	+	0.834272i	$0.685888\pi$
$510$	0	0
$511$	−15985.1	−1.38383
$512$	10676.7	0.921581
$513$	0	0
$514$	−5802.25	−0.497911
$515$	0	0
$516$	0	0
$517$	−6138.88	−0.522220
$518$	2993.96	0.253952
$519$	0	0
$520$	0	0
$521$	−12179.3	−1.02416	−0.512079	−	0.858939i	$-0.671124\pi$
$521$	−12179.3	−1.02416	−0.512079	+	0.858939i	$0.671124\pi$
$522$	0	0
$523$	1854.20	0.155025	0.0775127	−	0.996991i	$-0.475302\pi$
$523$	1854.20	0.155025	0.0775127	+	0.996991i	$0.475302\pi$
$524$	−9562.12	−0.797182
$525$	0	0
$526$	−27971.1	−2.31863
$527$	−19115.4	−1.58004
$528$	0	0
$529$	−6633.74	−0.545224
$530$	0	0
$531$	0	0
$532$	−5566.37	−0.453633
$533$	24312.7	1.97580
$534$	0	0
$535$	0	0
$536$	3411.24	0.274894
$537$	0	0
$538$	−4681.15	−0.375128
$539$	−1098.23	−0.0877627
$540$	0	0
$541$	−15546.5	−1.23548	−0.617741	−	0.786381i	$-0.711952\pi$
$541$	−15546.5	−1.23548	−0.617741	+	0.786381i	$0.711952\pi$
$542$	−4864.30	−0.385497
$543$	0	0
$544$	12452.7	0.981442
$545$	0	0
$546$	0	0
$547$	5579.72	0.436145	0.218073	−	0.975933i	$-0.430023\pi$
$547$	5579.72	0.436145	0.218073	+	0.975933i	$0.430023\pi$
$548$	−7229.27	−0.563539
$549$	0	0
$550$	0	0
$551$	862.976	0.0667223
$552$	0	0
$553$	14890.6	1.14505
$554$	5426.52	0.416157
$555$	0	0
$556$	−15132.7	−1.15426
$557$	11690.6	0.889314	0.444657	−	0.895701i	$-0.353326\pi$
$557$	11690.6	0.889314	0.444657	+	0.895701i	$0.353326\pi$
$558$	0	0
$559$	−29997.1	−2.26967
$560$	0	0
$561$	0	0
$562$	−1414.76	−0.106189
$563$	−15022.7	−1.12457	−0.562283	−	0.826945i	$-0.690077\pi$
$563$	−15022.7	−1.12457	−0.562283	+	0.826945i	$0.690077\pi$
$564$	0	0
$565$	0	0
$566$	−17309.5	−1.28546
$567$	0	0
$568$	−6009.29	−0.443916
$569$	−18726.0	−1.37967	−0.689836	−	0.723966i	$-0.742317\pi$
$569$	−18726.0	−1.37967	−0.689836	+	0.723966i	$0.742317\pi$
$570$	0	0
$571$	−6909.74	−0.506416	−0.253208	−	0.967412i	$-0.581486\pi$
$571$	−6909.74	−0.506416	−0.253208	+	0.967412i	$0.581486\pi$
$572$	−3930.71	−0.287328
$573$	0	0
$574$	28750.8	2.09065
$575$	0	0
$576$	0	0
$577$	−4108.29	−0.296413	−0.148206	−	0.988956i	$-0.547350\pi$
$577$	−4108.29	−0.296413	−0.148206	+	0.988956i	$0.547350\pi$
$578$	−5299.69	−0.381381
$579$	0	0
$580$	0	0
$581$	18508.4	1.32161
$582$	0	0
$583$	−4003.32	−0.284392
$584$	7050.70	0.499589
$585$	0	0
$586$	20407.6	1.43862
$587$	11358.3	0.798649	0.399324	−	0.916810i	$-0.369245\pi$
$587$	11358.3	0.798649	0.399324	+	0.916810i	$0.369245\pi$
$588$	0	0
$589$	−15691.2	−1.09770
$590$	0	0
$591$	0	0
$592$	−3017.19	−0.209469
$593$	5384.58	0.372880	0.186440	−	0.982466i	$-0.440305\pi$
$593$	5384.58	0.372880	0.186440	+	0.982466i	$0.440305\pi$
$594$	0	0
$595$	0	0
$596$	17192.1	1.18157
$597$	0	0
$598$	17831.9	1.21940
$599$	−11837.4	−0.807453	−0.403726	−	0.914880i	$-0.632285\pi$
$599$	−11837.4	−0.807453	−0.403726	+	0.914880i	$0.632285\pi$
$600$	0	0
$601$	−15988.5	−1.08517	−0.542583	−	0.840002i	$-0.682554\pi$
$601$	−15988.5	−1.08517	−0.542583	+	0.840002i	$0.682554\pi$
$602$	−35472.9	−2.40160
$603$	0	0
$604$	13737.8	0.925470
$605$	0	0
$606$	0	0
$607$	10111.5	0.676133	0.338067	−	0.941122i	$-0.390227\pi$
$607$	10111.5	0.676133	0.338067	+	0.941122i	$0.390227\pi$
$608$	10222.0	0.681835
$609$	0	0
$610$	0	0
$611$	36450.5	2.41347
$612$	0	0
$613$	18539.8	1.22156	0.610780	−	0.791800i	$-0.290856\pi$
$613$	18539.8	1.22156	0.610780	+	0.791800i	$0.290856\pi$
$614$	4714.75	0.309889
$615$	0	0
$616$	2148.60	0.140535
$617$	10846.1	0.707692	0.353846	−	0.935304i	$-0.384874\pi$
$617$	10846.1	0.707692	0.353846	+	0.935304i	$0.384874\pi$
$618$	0	0
$619$	−22761.2	−1.47795	−0.738974	−	0.673734i	$-0.764689\pi$
$619$	−22761.2	−1.47795	−0.738974	+	0.673734i	$0.764689\pi$
$620$	0	0
$621$	0	0
$622$	−5983.94	−0.385746
$623$	−8000.13	−0.514476
$624$	0	0
$625$	0	0
$626$	−11853.0	−0.756777
$627$	0	0
$628$	−5667.61	−0.360131
$629$	−2283.11	−0.144728
$630$	0	0
$631$	−14647.3	−0.924088	−0.462044	−	0.886857i	$-0.652884\pi$
$631$	−14647.3	−0.924088	−0.462044	+	0.886857i	$0.652884\pi$
$632$	−6567.94	−0.413384
$633$	0	0
$634$	−162.853	−0.0102014
$635$	0	0
$636$	0	0
$637$	6520.90	0.405600
$638$	720.633	0.0447181
$639$	0	0
$640$	0	0
$641$	−2393.83	−0.147505	−0.0737523	−	0.997277i	$-0.523497\pi$
$641$	−2393.83	−0.147505	−0.0737523	+	0.997277i	$0.523497\pi$
$642$	0	0
$643$	−24848.4	−1.52399	−0.761993	−	0.647585i	$-0.775779\pi$
$643$	−24848.4	−1.52399	−0.761993	+	0.647585i	$0.775779\pi$
$644$	8564.16	0.524030
$645$	0	0
$646$	10451.6	0.636555
$647$	8648.73	0.525528	0.262764	−	0.964860i	$-0.415366\pi$
$647$	8648.73	0.525528	0.262764	+	0.964860i	$0.415366\pi$
$648$	0	0
$649$	−2903.25	−0.175597
$650$	0	0
$651$	0	0
$652$	1801.19	0.108190
$653$	11712.1	0.701883	0.350941	−	0.936397i	$-0.385862\pi$
$653$	11712.1	0.701883	0.350941	+	0.936397i	$0.385862\pi$
$654$	0	0
$655$	0	0
$656$	−28973.8	−1.72445
$657$	0	0
$658$	43104.3	2.55377
$659$	−30097.0	−1.77908	−0.889539	−	0.456859i	$-0.848975\pi$
$659$	−30097.0	−1.77908	−0.889539	+	0.456859i	$0.848975\pi$
$660$	0	0
$661$	14719.1	0.866121	0.433061	−	0.901365i	$-0.357434\pi$
$661$	14719.1	0.866121	0.433061	+	0.901365i	$0.357434\pi$
$662$	3817.57	0.224130
$663$	0	0
$664$	−8163.65	−0.477125
$665$	0	0
$666$	0	0
$667$	−1327.73	−0.0770766
$668$	−6461.58	−0.374260
$669$	0	0
$670$	0	0
$671$	4239.96	0.243937
$672$	0	0
$673$	−16615.4	−0.951673	−0.475836	−	0.879534i	$-0.657855\pi$
$673$	−16615.4	−0.951673	−0.475836	+	0.879534i	$0.657855\pi$
$674$	11570.8	0.661263
$675$	0	0
$676$	11319.3	0.644019
$677$	5755.08	0.326714	0.163357	−	0.986567i	$-0.447768\pi$
$677$	5755.08	0.326714	0.163357	+	0.986567i	$0.447768\pi$
$678$	0	0
$679$	3144.79	0.177741
$680$	0	0
$681$	0	0
$682$	−13103.0	−0.735690
$683$	5821.51	0.326140	0.163070	−	0.986614i	$-0.447860\pi$
$683$	5821.51	0.326140	0.163070	+	0.986614i	$0.447860\pi$
$684$	0	0
$685$	0	0
$686$	−18781.0	−1.04528
$687$	0	0
$688$	35748.1	1.98093
$689$	23770.3	1.31433
$690$	0	0
$691$	6571.18	0.361765	0.180882	−	0.983505i	$-0.442105\pi$
$691$	6571.18	0.361765	0.180882	+	0.983505i	$0.442105\pi$
$692$	16116.5	0.885345
$693$	0	0
$694$	−19018.9	−1.04027
$695$	0	0
$696$	0	0
$697$	−21924.6	−1.19147
$698$	−20072.0	−1.08845
$699$	0	0
$700$	0	0
$701$	−2743.62	−0.147825	−0.0739124	−	0.997265i	$-0.523549\pi$
$701$	−2743.62	−0.147825	−0.0739124	+	0.997265i	$0.523549\pi$
$702$	0	0
$703$	−1874.13	−0.100546
$704$	1686.36	0.0902798
$705$	0	0
$706$	−21609.2	−1.15195
$707$	38061.3	2.02467
$708$	0	0
$709$	−24336.0	−1.28908	−0.644541	−	0.764570i	$-0.722951\pi$
$709$	−24336.0	−1.28908	−0.644541	+	0.764570i	$0.722951\pi$
$710$	0	0
$711$	0	0
$712$	3528.69	0.185735
$713$	24141.7	1.26804
$714$	0	0
$715$	0	0
$716$	22039.3	1.15034
$717$	0	0
$718$	−14426.9	−0.749871
$719$	2563.97	0.132990	0.0664950	−	0.997787i	$-0.478818\pi$
$719$	2563.97	0.132990	0.0664950	+	0.997787i	$0.478818\pi$
$720$	0	0
$721$	−28336.6	−1.46368
$722$	−16595.2	−0.855412
$723$	0	0
$724$	11191.5	0.574486
$725$	0	0
$726$	0	0
$727$	−27874.6	−1.42202	−0.711012	−	0.703180i	$-0.751763\pi$
$727$	−27874.6	−1.42202	−0.711012	+	0.703180i	$0.751763\pi$
$728$	−12757.6	−0.649490
$729$	0	0
$730$	0	0
$731$	27050.7	1.36868
$732$	0	0
$733$	26809.1	1.35091	0.675454	−	0.737402i	$-0.263948\pi$
$733$	26809.1	1.35091	0.675454	+	0.737402i	$0.263948\pi$
$734$	36191.3	1.81995
$735$	0	0
$736$	−15727.1	−0.787646
$737$	4042.64	0.202052
$738$	0	0
$739$	10016.5	0.498595	0.249297	−	0.968427i	$-0.419800\pi$
$739$	10016.5	0.498595	0.249297	+	0.968427i	$0.419800\pi$
$740$	0	0
$741$	0	0
$742$	28109.4	1.39074
$743$	−31647.9	−1.56265	−0.781324	−	0.624125i	$-0.785455\pi$
$743$	−31647.9	−1.56265	−0.781324	+	0.624125i	$0.785455\pi$
$744$	0	0
$745$	0	0
$746$	10597.3	0.520098
$747$	0	0
$748$	3544.62	0.173268
$749$	33908.1	1.65417
$750$	0	0
$751$	12461.0	0.605470	0.302735	−	0.953075i	$-0.402100\pi$
$751$	12461.0	0.605470	0.302735	+	0.953075i	$0.402100\pi$
$752$	−43438.7	−2.10644
$753$	0	0
$754$	−4278.87	−0.206667
$755$	0	0
$756$	0	0
$757$	8994.67	0.431858	0.215929	−	0.976409i	$-0.430722\pi$
$757$	8994.67	0.431858	0.215929	+	0.976409i	$0.430722\pi$
$758$	31430.6	1.50608
$759$	0	0
$760$	0	0
$761$	16238.0	0.773493	0.386747	−	0.922186i	$-0.373599\pi$
$761$	16238.0	0.773493	0.386747	+	0.922186i	$0.373599\pi$
$762$	0	0
$763$	−20760.1	−0.985013
$764$	−11456.9	−0.542536
$765$	0	0
$766$	19553.0	0.922296
$767$	17238.5	0.811531
$768$	0	0
$769$	38116.8	1.78742	0.893711	−	0.448643i	$-0.148092\pi$
$769$	38116.8	1.78742	0.893711	+	0.448643i	$0.148092\pi$
$770$	0	0
$771$	0	0
$772$	9379.46	0.437272
$773$	−2020.39	−0.0940082	−0.0470041	−	0.998895i	$-0.514967\pi$
$773$	−2020.39	−0.0940082	−0.0470041	+	0.998895i	$0.514967\pi$
$774$	0	0
$775$	0	0
$776$	−1387.10	−0.0641675
$777$	0	0
$778$	−9059.05	−0.417458
$779$	−17997.1	−0.827747
$780$	0	0
$781$	−7121.58	−0.326287
$782$	−16080.4	−0.735338
$783$	0	0
$784$	−7771.07	−0.354003
$785$	0	0
$786$	0	0
$787$	6512.11	0.294958	0.147479	−	0.989065i	$-0.452884\pi$
$787$	6512.11	0.294958	0.147479	+	0.989065i	$0.452884\pi$
$788$	−2857.38	−0.129175
$789$	0	0
$790$	0	0
$791$	−12415.4	−0.558080
$792$	0	0
$793$	−25175.4	−1.12737
$794$	28616.7	1.27906
$795$	0	0
$796$	18206.6	0.810697
$797$	−28502.2	−1.26675	−0.633375	−	0.773845i	$-0.718331\pi$
$797$	−28502.2	−1.26675	−0.633375	+	0.773845i	$0.718331\pi$
$798$	0	0
$799$	−32870.2	−1.45540
$800$	0	0
$801$	0	0
$802$	18038.7	0.794226
$803$	8355.75	0.367208
$804$	0	0
$805$	0	0
$806$	77801.2	3.40004
$807$	0	0
$808$	−16788.0	−0.730942
$809$	−8983.16	−0.390397	−0.195198	−	0.980764i	$-0.562535\pi$
$809$	−8983.16	−0.390397	−0.195198	+	0.980764i	$0.562535\pi$
$810$	0	0
$811$	32271.9	1.39731	0.698656	−	0.715458i	$-0.253782\pi$
$811$	32271.9	1.39731	0.698656	+	0.715458i	$0.253782\pi$
$812$	−2055.02	−0.0888139
$813$	0	0
$814$	−1565.00	−0.0673874
$815$	0	0
$816$	0	0
$817$	22205.0	0.950861
$818$	−25366.9	−1.08427
$819$	0	0
$820$	0	0
$821$	−43194.9	−1.83619	−0.918095	−	0.396359i	$-0.870273\pi$
$821$	−43194.9	−1.83619	−0.918095	+	0.396359i	$0.870273\pi$
$822$	0	0
$823$	−10766.5	−0.456009	−0.228004	−	0.973660i	$-0.573220\pi$
$823$	−10766.5	−0.456009	−0.228004	+	0.973660i	$0.573220\pi$
$824$	12498.7	0.528414
$825$	0	0
$826$	20385.2	0.858707
$827$	−9904.74	−0.416471	−0.208235	−	0.978079i	$-0.566772\pi$
$827$	−9904.74	−0.416471	−0.208235	+	0.978079i	$0.566772\pi$
$828$	0	0
$829$	23347.1	0.978141	0.489070	−	0.872244i	$-0.337336\pi$
$829$	23347.1	0.978141	0.489070	+	0.872244i	$0.337336\pi$
$830$	0	0
$831$	0	0
$832$	−10013.0	−0.417233
$833$	−5880.39	−0.244590
$834$	0	0
$835$	0	0
$836$	2909.66	0.120374
$837$	0	0
$838$	−24956.7	−1.02878
$839$	−9403.46	−0.386941	−0.193471	−	0.981106i	$-0.561974\pi$
$839$	−9403.46	−0.386941	−0.193471	+	0.981106i	$0.561974\pi$
$840$	0	0
$841$	−24070.4	−0.986937
$842$	13834.2	0.566221
$843$	0	0
$844$	23398.7	0.954286
$845$	0	0
$846$	0	0
$847$	2546.29	0.103296
$848$	−28327.5	−1.14713
$849$	0	0
$850$	0	0
$851$	2883.45	0.116150
$852$	0	0
$853$	−4698.34	−0.188591	−0.0942955	−	0.995544i	$-0.530060\pi$
$853$	−4698.34	−0.188591	−0.0942955	+	0.995544i	$0.530060\pi$
$854$	−29771.0	−1.19291
$855$	0	0
$856$	−14956.2	−0.597187
$857$	−27571.8	−1.09899	−0.549494	−	0.835498i	$-0.685180\pi$
$857$	−27571.8	−1.09899	−0.549494	+	0.835498i	$0.685180\pi$
$858$	0	0
$859$	7373.38	0.292871	0.146436	−	0.989220i	$-0.453220\pi$
$859$	7373.38	0.292871	0.146436	+	0.989220i	$0.453220\pi$
$860$	0	0
$861$	0	0
$862$	−38877.2	−1.53615
$863$	−47163.9	−1.86034	−0.930172	−	0.367125i	$-0.880342\pi$
$863$	−47163.9	−1.86034	−0.930172	+	0.367125i	$0.880342\pi$
$864$	0	0
$865$	0	0
$866$	−58466.8	−2.29421
$867$	0	0
$868$	37365.6	1.46114
$869$	−7783.63	−0.303845
$870$	0	0
$871$	−24003.8	−0.933797
$872$	9156.83	0.355607
$873$	0	0
$874$	−13199.9	−0.510860
$875$	0	0
$876$	0	0
$877$	28423.9	1.09442	0.547211	−	0.836995i	$-0.315690\pi$
$877$	28423.9	1.09442	0.547211	+	0.836995i	$0.315690\pi$
$878$	−53702.9	−2.06422
$879$	0	0
$880$	0	0
$881$	14187.0	0.542534	0.271267	−	0.962504i	$-0.412557\pi$
$881$	14187.0	0.542534	0.271267	+	0.962504i	$0.412557\pi$
$882$	0	0
$883$	−16463.0	−0.627433	−0.313716	−	0.949517i	$-0.601574\pi$
$883$	−16463.0	−0.627433	−0.313716	+	0.949517i	$0.601574\pi$
$884$	−21046.7	−0.800767
$885$	0	0
$886$	5789.36	0.219523
$887$	−36878.1	−1.39599	−0.697996	−	0.716102i	$-0.745925\pi$
$887$	−36878.1	−1.39599	−0.697996	+	0.716102i	$0.745925\pi$
$888$	0	0
$889$	52701.4	1.98824
$890$	0	0
$891$	0	0
$892$	−15781.2	−0.592369
$893$	−26982.0	−1.01111
$894$	0	0
$895$	0	0
$896$	23752.6	0.885625
$897$	0	0
$898$	16509.9	0.613522
$899$	−5792.94	−0.214911
$900$	0	0
$901$	−21435.5	−0.792586
$902$	−15028.6	−0.554766
$903$	0	0
$904$	5476.18	0.201477
$905$	0	0
$906$	0	0
$907$	−15527.4	−0.568443	−0.284221	−	0.958759i	$-0.591735\pi$
$907$	−15527.4	−0.568443	−0.284221	+	0.958759i	$0.591735\pi$
$908$	31022.3	1.13382
$909$	0	0
$910$	0	0
$911$	2542.54	0.0924676	0.0462338	−	0.998931i	$-0.485278\pi$
$911$	2542.54	0.0924676	0.0462338	+	0.998931i	$0.485278\pi$
$912$	0	0
$913$	−9674.70	−0.350696
$914$	−58275.0	−2.10893
$915$	0	0
$916$	185.155	0.00667869
$917$	−36779.5	−1.32450
$918$	0	0
$919$	−42147.5	−1.51286	−0.756430	−	0.654074i	$-0.773058\pi$
$919$	−42147.5	−1.51286	−0.756430	+	0.654074i	$0.773058\pi$
$920$	0	0
$921$	0	0
$922$	−65613.9	−2.34369
$923$	42285.4	1.50795
$924$	0	0
$925$	0	0
$926$	−8172.63	−0.290032
$927$	0	0
$928$	3773.79	0.133492
$929$	2571.44	0.0908140	0.0454070	−	0.998969i	$-0.485542\pi$
$929$	2571.44	0.0908140	0.0454070	+	0.998969i	$0.485542\pi$
$930$	0	0
$931$	−4827.01	−0.169924
$932$	28319.5	0.995319
$933$	0	0
$934$	71719.6	2.51257
$935$	0	0
$936$	0	0
$937$	16543.1	0.576775	0.288388	−	0.957514i	$-0.406881\pi$
$937$	16543.1	0.576775	0.288388	+	0.957514i	$0.406881\pi$
$938$	−28385.5	−0.988080
$939$	0	0
$940$	0	0
$941$	−22637.2	−0.784220	−0.392110	−	0.919918i	$-0.628255\pi$
$941$	−22637.2	−0.784220	−0.392110	+	0.919918i	$0.628255\pi$
$942$	0	0
$943$	27689.6	0.956200
$944$	−20543.4	−0.708294
$945$	0	0
$946$	18542.4	0.637278
$947$	21780.8	0.747392	0.373696	−	0.927551i	$-0.378090\pi$
$947$	21780.8	0.747392	0.373696	+	0.927551i	$0.378090\pi$
$948$	0	0
$949$	−49613.5	−1.69707
$950$	0	0
$951$	0	0
$952$	11504.5	0.391663
$953$	18898.7	0.642382	0.321191	−	0.947014i	$-0.395917\pi$
$953$	18898.7	0.642382	0.321191	+	0.947014i	$0.395917\pi$
$954$	0	0
$955$	0	0
$956$	−1313.40	−0.0444334
$957$	0	0
$958$	11220.0	0.378395
$959$	−27806.5	−0.936307
$960$	0	0
$961$	75540.0	2.53566
$962$	9292.45	0.311435
$963$	0	0
$964$	12649.6	0.422629
$965$	0	0
$966$	0	0
$967$	21045.5	0.699873	0.349937	−	0.936773i	$-0.386203\pi$
$967$	21045.5	0.699873	0.349937	+	0.936773i	$0.386203\pi$
$968$	−1123.12	−0.0372917
$969$	0	0
$970$	0	0
$971$	40895.7	1.35160	0.675800	−	0.737085i	$-0.263798\pi$
$971$	40895.7	1.35160	0.675800	+	0.737085i	$0.263798\pi$
$972$	0	0
$973$	−58206.0	−1.91778
$974$	10106.3	0.332472
$975$	0	0
$976$	30001.9	0.983953
$977$	−26029.8	−0.852371	−0.426186	−	0.904636i	$-0.640143\pi$
$977$	−26029.8	−0.852371	−0.426186	+	0.904636i	$0.640143\pi$
$978$	0	0
$979$	4181.83	0.136519
$980$	0	0
$981$	0	0
$982$	28535.4	0.927291
$983$	−34358.3	−1.11481	−0.557406	−	0.830240i	$-0.688203\pi$
$983$	−34358.3	−1.11481	−0.557406	+	0.830240i	$0.688203\pi$
$984$	0	0
$985$	0	0
$986$	3858.58	0.124627
$987$	0	0
$988$	−17276.5	−0.556315
$989$	−34163.5	−1.09842
$990$	0	0
$991$	−3179.29	−0.101911	−0.0509553	−	0.998701i	$-0.516227\pi$
$991$	−3179.29	−0.101911	−0.0509553	+	0.998701i	$0.516227\pi$
$992$	−68617.5	−2.19618
$993$	0	0
$994$	50004.3	1.59561
$995$	0	0
$996$	0	0
$997$	28378.0	0.901445	0.450722	−	0.892664i	$-0.351166\pi$
$997$	28378.0	0.901445	0.450722	+	0.892664i	$0.351166\pi$
$998$	2214.10	0.0702267
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.bx.1.8	yes	10
3.2	odd	2		2475.4.a.bu.1.3	✓	10
5.4	even	2		2475.4.a.bv.1.3	yes	10
15.14	odd	2		2475.4.a.by.1.8	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2475.4.a.bu.1.3	✓	10	3.2	odd	2
2475.4.a.bv.1.3	yes	10	5.4	even	2
2475.4.a.bx.1.8	yes	10	1.1	even	1	trivial
2475.4.a.by.1.8	yes	10	15.14	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+3.67029 q^{2} +5.47106 q^{4} +21.0437 q^{7} -9.28196 q^{8} +O(q^{10})\)	\(q+3.67029 q^{2} +5.47106 q^{4} +21.0437 q^{7} -9.28196 q^{8} -11.0000 q^{11} +65.3141 q^{13} +77.2367 q^{14} -77.8360 q^{16} -58.8987 q^{17} -48.3479 q^{19} -40.3732 q^{22} +74.3859 q^{23} +239.722 q^{26} +115.132 q^{28} -17.8493 q^{29} +324.547 q^{31} -211.425 q^{32} -216.176 q^{34} +38.7634 q^{37} -177.451 q^{38} +372.242 q^{41} -459.275 q^{43} -60.1817 q^{44} +273.018 q^{46} +558.080 q^{47} +99.8391 q^{49} +357.337 q^{52} +363.938 q^{53} -195.327 q^{56} -65.5121 q^{58} +263.931 q^{59} -385.451 q^{61} +1191.18 q^{62} -153.305 q^{64} -367.513 q^{67} -322.238 q^{68} +647.416 q^{71} -759.614 q^{73} +142.273 q^{74} -264.514 q^{76} -231.481 q^{77} +707.603 q^{79} +1366.24 q^{82} +879.518 q^{83} -1685.67 q^{86} +102.102 q^{88} -380.167 q^{89} +1374.45 q^{91} +406.970 q^{92} +2048.32 q^{94} +149.441 q^{97} +366.439 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{2} + 34 q^{4} - 2 q^{7} + 48 q^{8}+O(q^{10}) \) 10 * q + 4 * q^2 + 34 * q^4 - 2 * q^7 + 48 * q^8	\( 10 q + 4 q^{2} + 34 q^{4} - 2 q^{7} + 48 q^{8} - 110 q^{11} - 26 q^{13} + 72 q^{14} + 206 q^{16} + 148 q^{17} - 114 q^{19} - 44 q^{22} - 34 q^{23} + 100 q^{26} + 86 q^{28} - 38 q^{29} + 232 q^{31} + 448 q^{32} - 20 q^{34} - 754 q^{37} + 780 q^{38} + 160 q^{41} + 66 q^{43} - 374 q^{44} + 682 q^{46} + 450 q^{47} + 590 q^{49} - 200 q^{52} + 1068 q^{53} + 268 q^{56} + 138 q^{58} - 838 q^{59} - 566 q^{61} + 1230 q^{62} + 462 q^{64} - 430 q^{67} + 2234 q^{68} + 518 q^{71} + 184 q^{73} + 402 q^{74} + 386 q^{76} + 22 q^{77} + 956 q^{79} + 2180 q^{82} + 2094 q^{83} + 892 q^{86} - 528 q^{88} - 512 q^{89} - 858 q^{91} + 4476 q^{92} - 294 q^{94} + 1006 q^{97} + 2226 q^{98}+O(q^{100}) \) 10 * q + 4 * q^2 + 34 * q^4 - 2 * q^7 + 48 * q^8 - 110 * q^11 - 26 * q^13 + 72 * q^14 + 206 * q^16 + 148 * q^17 - 114 * q^19 - 44 * q^22 - 34 * q^23 + 100 * q^26 + 86 * q^28 - 38 * q^29 + 232 * q^31 + 448 * q^32 - 20 * q^34 - 754 * q^37 + 780 * q^38 + 160 * q^41 + 66 * q^43 - 374 * q^44 + 682 * q^46 + 450 * q^47 + 590 * q^49 - 200 * q^52 + 1068 * q^53 + 268 * q^56 + 138 * q^58 - 838 * q^59 - 566 * q^61 + 1230 * q^62 + 462 * q^64 - 430 * q^67 + 2234 * q^68 + 518 * q^71 + 184 * q^73 + 402 * q^74 + 386 * q^76 + 22 * q^77 + 956 * q^79 + 2180 * q^82 + 2094 * q^83 + 892 * q^86 - 528 * q^88 - 512 * q^89 - 858 * q^91 + 4476 * q^92 - 294 * q^94 + 1006 * q^97 + 2226 * q^98

Introduction

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