LMFDB - Embedded newform 2475.4.a.by.1.4

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			1.4
Root			$-1.29917$ 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)

$f(q)$	$=$	$q-1.29917 q^{2} -6.31217 q^{4} +16.4390 q^{7} +18.5939 q^{8} +O(q^{10})$	$q-1.29917 q^{2} -6.31217 q^{4} +16.4390 q^{7} +18.5939 q^{8} +11.0000 q^{11} +4.28526 q^{13} -21.3570 q^{14} +26.3409 q^{16} +72.6873 q^{17} +133.612 q^{19} -14.2908 q^{22} +116.750 q^{23} -5.56726 q^{26} -103.766 q^{28} +144.653 q^{29} +235.345 q^{31} -182.972 q^{32} -94.4328 q^{34} +75.2732 q^{37} -173.585 q^{38} +355.585 q^{41} -2.86919 q^{43} -69.4339 q^{44} -151.677 q^{46} -83.0358 q^{47} -72.7595 q^{49} -27.0493 q^{52} -213.608 q^{53} +305.665 q^{56} -187.928 q^{58} -477.946 q^{59} -294.389 q^{61} -305.752 q^{62} +26.9842 q^{64} +1004.34 q^{67} -458.814 q^{68} +1023.22 q^{71} -76.8447 q^{73} -97.7923 q^{74} -843.384 q^{76} +180.829 q^{77} +456.683 q^{79} -461.963 q^{82} +504.319 q^{83} +3.72755 q^{86} +204.533 q^{88} -849.405 q^{89} +70.4454 q^{91} -736.943 q^{92} +107.877 q^{94} -306.403 q^{97} +94.5266 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.29917	−0.459324	−0.229662	−	0.973270i	$-0.573762\pi$
$2$	−1.29917	−0.459324	−0.229662	+	0.973270i	$0.573762\pi$
$3$	0	0
$3$	0	0
$4$	−6.31217	−0.789021
$5$	0	0
$5$	0	0
$6$	0	0
$7$	16.4390	0.887622	0.443811	−	0.896120i	$-0.353626\pi$
$7$	16.4390	0.887622	0.443811	+	0.896120i	$0.353626\pi$
$8$	18.5939	0.821741
$9$	0	0
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	4.28526	0.0914245	0.0457122	−	0.998955i	$-0.485444\pi$
$13$	4.28526	0.0914245	0.0457122	+	0.998955i	$0.485444\pi$
$14$	−21.3570	−0.407706
$15$	0	0
$16$	26.3409	0.411576
$17$	72.6873	1.03701	0.518507	−	0.855073i	$-0.326488\pi$
$17$	72.6873	1.03701	0.518507	+	0.855073i	$0.326488\pi$
$18$	0	0
$19$	133.612	1.61330	0.806652	−	0.591026i	$-0.201277\pi$
$19$	133.612	1.61330	0.806652	+	0.591026i	$0.201277\pi$
$20$	0	0
$21$	0	0
$22$	−14.2908	−0.138491
$23$	116.750	1.05843	0.529216	−	0.848487i	$-0.322486\pi$
$23$	116.750	1.05843	0.529216	+	0.848487i	$0.322486\pi$
$24$	0	0
$25$	0	0
$26$	−5.56726	−0.0419935
$27$	0	0
$28$	−103.766	−0.700353
$29$	144.653	0.926253	0.463127	−	0.886292i	$-0.346727\pi$
$29$	144.653	0.926253	0.463127	+	0.886292i	$0.346727\pi$
$30$	0	0
$31$	235.345	1.36352	0.681762	−	0.731574i	$-0.261214\pi$
$31$	235.345	1.36352	0.681762	+	0.731574i	$0.261214\pi$
$32$	−182.972	−1.01079
$33$	0	0
$34$	−94.4328	−0.476326
$35$	0	0
$36$	0	0
$37$	75.2732	0.334455	0.167227	−	0.985918i	$-0.446519\pi$
$37$	75.2732	0.334455	0.167227	+	0.985918i	$0.446519\pi$
$38$	−173.585	−0.741030
$39$	0	0
$40$	0	0
$41$	355.585	1.35446	0.677232	−	0.735770i	$-0.263179\pi$
$41$	355.585	1.35446	0.677232	+	0.735770i	$0.263179\pi$
$42$	0	0
$43$	−2.86919	−0.0101755	−0.00508776	−	0.999987i	$-0.501619\pi$
$43$	−2.86919	−0.0101755	−0.00508776	+	0.999987i	$0.501619\pi$
$44$	−69.4339	−0.237899
$45$	0	0
$46$	−151.677	−0.486164
$47$	−83.0358	−0.257702	−0.128851	−	0.991664i	$-0.541129\pi$
$47$	−83.0358	−0.257702	−0.128851	+	0.991664i	$0.541129\pi$
$48$	0	0
$49$	−72.7595	−0.212127
$50$	0	0
$51$	0	0
$52$	−27.0493	−0.0721359
$53$	−213.608	−0.553610	−0.276805	−	0.960926i	$-0.589276\pi$
$53$	−213.608	−0.553610	−0.276805	+	0.960926i	$0.589276\pi$
$54$	0	0
$55$	0	0
$56$	305.665	0.729395
$57$	0	0
$58$	−187.928	−0.425451
$59$	−477.946	−1.05463	−0.527315	−	0.849670i	$-0.676801\pi$
$59$	−477.946	−1.05463	−0.527315	+	0.849670i	$0.676801\pi$
$60$	0	0
$61$	−294.389	−0.617912	−0.308956	−	0.951076i	$-0.599980\pi$
$61$	−294.389	−0.617912	−0.308956	+	0.951076i	$0.599980\pi$
$62$	−305.752	−0.626300
$63$	0	0
$64$	26.9842	0.0527034
$65$	0	0
$66$	0	0
$67$	1004.34	1.83133	0.915667	−	0.401937i	$-0.131663\pi$
$67$	1004.34	1.83133	0.915667	+	0.401937i	$0.131663\pi$
$68$	−458.814	−0.818227
$69$	0	0
$70$	0	0
$71$	1023.22	1.71033	0.855164	−	0.518357i	$-0.173456\pi$
$71$	1023.22	1.71033	0.855164	+	0.518357i	$0.173456\pi$
$72$	0	0
$73$	−76.8447	−0.123205	−0.0616027	−	0.998101i	$-0.519621\pi$
$73$	−76.8447	−0.123205	−0.0616027	+	0.998101i	$0.519621\pi$
$74$	−97.7923	−0.153623
$75$	0	0
$76$	−843.384	−1.27293
$77$	180.829	0.267628
$78$	0	0
$79$	456.683	0.650391	0.325196	−	0.945647i	$-0.394570\pi$
$79$	456.683	0.650391	0.325196	+	0.945647i	$0.394570\pi$
$80$	0	0
$81$	0	0
$82$	−461.963	−0.622138
$83$	504.319	0.666943	0.333471	−	0.942760i	$-0.391780\pi$
$83$	504.319	0.666943	0.333471	+	0.942760i	$0.391780\pi$
$84$	0	0
$85$	0	0
$86$	3.72755	0.00467386
$87$	0	0
$88$	204.533	0.247764
$89$	−849.405	−1.01165	−0.505825	−	0.862636i	$-0.668812\pi$
$89$	−849.405	−1.01165	−0.505825	+	0.862636i	$0.668812\pi$
$90$	0	0
$91$	70.4454	0.0811504
$92$	−736.943	−0.835126
$93$	0	0
$94$	107.877	0.118369
$95$	0	0
$96$	0	0
$97$	−306.403	−0.320727	−0.160363	−	0.987058i	$-0.551267\pi$
$97$	−306.403	−0.320727	−0.160363	+	0.987058i	$0.551267\pi$
$98$	94.5266	0.0974350
$99$	0	0
$100$	0	0
$101$	−588.634	−0.579913	−0.289957	−	0.957040i	$-0.593641\pi$
$101$	−588.634	−0.579913	−0.289957	+	0.957040i	$0.593641\pi$
$102$	0	0
$103$	−1302.57	−1.24608	−0.623039	−	0.782191i	$-0.714102\pi$
$103$	−1302.57	−1.24608	−0.623039	+	0.782191i	$0.714102\pi$
$104$	79.6796	0.0751272
$105$	0	0
$106$	277.512	0.254286
$107$	1900.93	1.71748	0.858738	−	0.512414i	$-0.171249\pi$
$107$	1900.93	1.71748	0.858738	+	0.512414i	$0.171249\pi$
$108$	0	0
$109$	512.079	0.449984	0.224992	−	0.974361i	$-0.427764\pi$
$109$	512.079	0.449984	0.224992	+	0.974361i	$0.427764\pi$
$110$	0	0
$111$	0	0
$112$	433.017	0.365324
$113$	−1552.38	−1.29235	−0.646174	−	0.763190i	$-0.723632\pi$
$113$	−1552.38	−1.29235	−0.646174	+	0.763190i	$0.723632\pi$
$114$	0	0
$115$	0	0
$116$	−913.073	−0.730834
$117$	0	0
$118$	620.930	0.484417
$119$	1194.91	0.920477
$120$	0	0
$121$	121.000	0.0909091
$122$	382.460	0.283822
$123$	0	0
$124$	−1485.54	−1.07585
$125$	0	0
$126$	0	0
$127$	−835.326	−0.583647	−0.291823	−	0.956472i	$-0.594262\pi$
$127$	−835.326	−0.583647	−0.291823	+	0.956472i	$0.594262\pi$
$128$	1428.72	0.986580
$129$	0	0
$130$	0	0
$131$	1569.32	1.04666	0.523329	−	0.852131i	$-0.324690\pi$
$131$	1569.32	1.04666	0.523329	+	0.852131i	$0.324690\pi$
$132$	0	0
$133$	2196.45	1.43201
$134$	−1304.80	−0.841176
$135$	0	0
$136$	1351.54	0.852157
$137$	−494.205	−0.308195	−0.154098	−	0.988056i	$-0.549247\pi$
$137$	−494.205	−0.308195	−0.154098	+	0.988056i	$0.549247\pi$
$138$	0	0
$139$	−2336.57	−1.42579	−0.712897	−	0.701269i	$-0.752617\pi$
$139$	−2336.57	−1.42579	−0.712897	+	0.701269i	$0.752617\pi$
$140$	0	0
$141$	0	0
$142$	−1329.33	−0.785595
$143$	47.1379	0.0275655
$144$	0	0
$145$	0	0
$146$	99.8340	0.0565912
$147$	0	0
$148$	−475.137	−0.263892
$149$	−1283.86	−0.705890	−0.352945	−	0.935644i	$-0.614820\pi$
$149$	−1283.86	−0.705890	−0.352945	+	0.935644i	$0.614820\pi$
$150$	0	0
$151$	−3544.32	−1.91015	−0.955076	−	0.296362i	$-0.904227\pi$
$151$	−3544.32	−1.91015	−0.955076	+	0.296362i	$0.904227\pi$
$152$	2484.37	1.32572
$153$	0	0
$154$	−234.927	−0.122928
$155$	0	0
$156$	0	0
$157$	1309.42	0.665624	0.332812	−	0.942993i	$-0.392003\pi$
$157$	1309.42	0.665624	0.332812	+	0.942993i	$0.392003\pi$
$158$	−593.307	−0.298740
$159$	0	0
$160$	0	0
$161$	1919.24	0.939488
$162$	0	0
$163$	−3414.52	−1.64077	−0.820385	−	0.571812i	$-0.806241\pi$
$163$	−3414.52	−1.64077	−0.820385	+	0.571812i	$0.806241\pi$
$164$	−2244.51	−1.06870
$165$	0	0
$166$	−655.194	−0.306343
$167$	3735.79	1.73104	0.865522	−	0.500872i	$-0.166987\pi$
$167$	3735.79	1.73104	0.865522	+	0.500872i	$0.166987\pi$
$168$	0	0
$169$	−2178.64	−0.991642
$170$	0	0
$171$	0	0
$172$	18.1108	0.00802870
$173$	−2425.53	−1.06595	−0.532975	−	0.846131i	$-0.678926\pi$
$173$	−2425.53	−1.06595	−0.532975	+	0.846131i	$0.678926\pi$
$174$	0	0
$175$	0	0
$176$	289.749	0.124095
$177$	0	0
$178$	1103.52	0.464675
$179$	−3368.99	−1.40676	−0.703380	−	0.710814i	$-0.748327\pi$
$179$	−3368.99	−1.40676	−0.703380	+	0.710814i	$0.748327\pi$
$180$	0	0
$181$	−2230.19	−0.915849	−0.457924	−	0.888991i	$-0.651407\pi$
$181$	−2230.19	−0.915849	−0.457924	+	0.888991i	$0.651407\pi$
$182$	−91.5202	−0.0372743
$183$	0	0
$184$	2170.83	0.869757
$185$	0	0
$186$	0	0
$187$	799.560	0.312672
$188$	524.136	0.203333
$189$	0	0
$190$	0	0
$191$	−2691.12	−1.01949	−0.509746	−	0.860325i	$-0.670261\pi$
$191$	−2691.12	−1.01949	−0.509746	+	0.860325i	$0.670261\pi$
$192$	0	0
$193$	1089.18	0.406221	0.203111	−	0.979156i	$-0.434895\pi$
$193$	1089.18	0.406221	0.203111	+	0.979156i	$0.434895\pi$
$194$	398.068	0.147317
$195$	0	0
$196$	459.270	0.167373
$197$	108.355	0.0391877	0.0195938	−	0.999808i	$-0.493763\pi$
$197$	108.355	0.0391877	0.0195938	+	0.999808i	$0.493763\pi$
$198$	0	0
$199$	2078.20	0.740300	0.370150	−	0.928972i	$-0.379306\pi$
$199$	2078.20	0.740300	0.370150	+	0.928972i	$0.379306\pi$
$200$	0	0
$201$	0	0
$202$	764.733	0.266368
$203$	2377.95	0.822163
$204$	0	0
$205$	0	0
$206$	1692.25	0.572354
$207$	0	0
$208$	112.877	0.0376281
$209$	1469.74	0.486430
$210$	0	0
$211$	−1130.84	−0.368959	−0.184479	−	0.982836i	$-0.559060\pi$
$211$	−1130.84	−0.368959	−0.184479	+	0.982836i	$0.559060\pi$
$212$	1348.33	0.436810
$213$	0	0
$214$	−2469.62	−0.788879
$215$	0	0
$216$	0	0
$217$	3868.84	1.21029
$218$	−665.275	−0.206689
$219$	0	0
$220$	0	0
$221$	311.484	0.0948085
$222$	0	0
$223$	−4231.76	−1.27076	−0.635381	−	0.772199i	$-0.719157\pi$
$223$	−4231.76	−1.27076	−0.635381	+	0.772199i	$0.719157\pi$
$224$	−3007.88	−0.897197
$225$	0	0
$226$	2016.79	0.593606
$227$	1467.83	0.429177	0.214589	−	0.976705i	$-0.431159\pi$
$227$	1467.83	0.429177	0.214589	+	0.976705i	$0.431159\pi$
$228$	0	0
$229$	−63.2356	−0.0182477	−0.00912385	−	0.999958i	$-0.502904\pi$
$229$	−63.2356	−0.0182477	−0.00912385	+	0.999958i	$0.502904\pi$
$230$	0	0
$231$	0	0
$232$	2689.66	0.761140
$233$	2715.78	0.763591	0.381796	−	0.924247i	$-0.375306\pi$
$233$	2715.78	0.763591	0.381796	+	0.924247i	$0.375306\pi$
$234$	0	0
$235$	0	0
$236$	3016.87	0.832126
$237$	0	0
$238$	−1552.38	−0.422798
$239$	4354.46	1.17852	0.589261	−	0.807943i	$-0.299419\pi$
$239$	4354.46	1.17852	0.589261	+	0.807943i	$0.299419\pi$
$240$	0	0
$241$	19.1263	0.00511218	0.00255609	−	0.999997i	$-0.499186\pi$
$241$	19.1263	0.00511218	0.00255609	+	0.999997i	$0.499186\pi$
$242$	−157.199	−0.0417567
$243$	0	0
$244$	1858.23	0.487546
$245$	0	0
$246$	0	0
$247$	572.564	0.147496
$248$	4375.98	1.12046
$249$	0	0
$250$	0	0
$251$	−1322.79	−0.332644	−0.166322	−	0.986071i	$-0.553189\pi$
$251$	−1322.79	−0.332644	−0.166322	+	0.986071i	$0.553189\pi$
$252$	0	0
$253$	1284.24	0.319130
$254$	1085.23	0.268083
$255$	0	0
$256$	−2072.02	−0.505863
$257$	7061.45	1.71393	0.856967	−	0.515371i	$-0.172346\pi$
$257$	7061.45	1.71393	0.856967	+	0.515371i	$0.172346\pi$
$258$	0	0
$259$	1237.42	0.296870
$260$	0	0
$261$	0	0
$262$	−2038.81	−0.480755
$263$	4160.06	0.975362	0.487681	−	0.873022i	$-0.337843\pi$
$263$	4160.06	0.975362	0.487681	+	0.873022i	$0.337843\pi$
$264$	0	0
$265$	0	0
$266$	−2853.56	−0.657755
$267$	0	0
$268$	−6339.55	−1.44496
$269$	1770.52	0.401303	0.200652	−	0.979663i	$-0.435694\pi$
$269$	1770.52	0.401303	0.200652	+	0.979663i	$0.435694\pi$
$270$	0	0
$271$	2045.62	0.458533	0.229267	−	0.973364i	$-0.426367\pi$
$271$	2045.62	0.458533	0.229267	+	0.973364i	$0.426367\pi$
$272$	1914.64	0.426810
$273$	0	0
$274$	642.053	0.141562
$275$	0	0
$276$	0	0
$277$	−3291.00	−0.713852	−0.356926	−	0.934133i	$-0.616175\pi$
$277$	−3291.00	−0.713852	−0.356926	+	0.934133i	$0.616175\pi$
$278$	3035.59	0.654901
$279$	0	0
$280$	0	0
$281$	5172.09	1.09801	0.549005	−	0.835819i	$-0.315007\pi$
$281$	5172.09	1.09801	0.549005	+	0.835819i	$0.315007\pi$
$282$	0	0
$283$	7373.40	1.54877	0.774387	−	0.632712i	$-0.218058\pi$
$283$	7373.40	1.54877	0.774387	+	0.632712i	$0.218058\pi$
$284$	−6458.71	−1.34949
$285$	0	0
$286$	−61.2399	−0.0126615
$287$	5845.46	1.20225
$288$	0	0
$289$	370.440	0.0753999
$290$	0	0
$291$	0	0
$292$	485.057	0.0972117
$293$	8270.22	1.64898	0.824490	−	0.565877i	$-0.191462\pi$
$293$	8270.22	1.64898	0.824490	+	0.565877i	$0.191462\pi$
$294$	0	0
$295$	0	0
$296$	1399.62	0.274835
$297$	0	0
$298$	1667.94	0.324232
$299$	500.302	0.0967667
$300$	0	0
$301$	−47.1666	−0.00903202
$302$	4604.66	0.877379
$303$	0	0
$304$	3519.47	0.663997
$305$	0	0
$306$	0	0
$307$	5660.81	1.05238	0.526188	−	0.850368i	$-0.323621\pi$
$307$	5660.81	1.05238	0.526188	+	0.850368i	$0.323621\pi$
$308$	−1141.42	−0.211164
$309$	0	0
$310$	0	0
$311$	7756.92	1.41432	0.707161	−	0.707052i	$-0.249975\pi$
$311$	7756.92	1.41432	0.707161	+	0.707052i	$0.249975\pi$
$312$	0	0
$313$	−4681.02	−0.845325	−0.422663	−	0.906287i	$-0.638904\pi$
$313$	−4681.02	−0.845325	−0.422663	+	0.906287i	$0.638904\pi$
$314$	−1701.15	−0.305737
$315$	0	0
$316$	−2882.66	−0.513172
$317$	3387.47	0.600187	0.300094	−	0.953910i	$-0.402982\pi$
$317$	3387.47	0.600187	0.300094	+	0.953910i	$0.402982\pi$
$318$	0	0
$319$	1591.18	0.279276
$320$	0	0
$321$	0	0
$322$	−2493.42	−0.431530
$323$	9711.92	1.67302
$324$	0	0
$325$	0	0
$326$	4436.02	0.753645
$327$	0	0
$328$	6611.70	1.11302
$329$	−1365.02	−0.228742
$330$	0	0
$331$	−5684.94	−0.944026	−0.472013	−	0.881592i	$-0.656472\pi$
$331$	−5684.94	−0.944026	−0.472013	+	0.881592i	$0.656472\pi$
$332$	−3183.35	−0.526232
$333$	0	0
$334$	−4853.41	−0.795110
$335$	0	0
$336$	0	0
$337$	−6527.04	−1.05505	−0.527523	−	0.849541i	$-0.676879\pi$
$337$	−6527.04	−1.05505	−0.527523	+	0.849541i	$0.676879\pi$
$338$	2830.41	0.455485
$339$	0	0
$340$	0	0
$341$	2588.80	0.411118
$342$	0	0
$343$	−6834.67	−1.07591
$344$	−53.3494	−0.00836164
$345$	0	0
$346$	3151.16	0.489617
$347$	7720.15	1.19435	0.597175	−	0.802111i	$-0.296290\pi$
$347$	7720.15	1.19435	0.597175	+	0.802111i	$0.296290\pi$
$348$	0	0
$349$	2049.69	0.314377	0.157188	−	0.987569i	$-0.449757\pi$
$349$	2049.69	0.314377	0.157188	+	0.987569i	$0.449757\pi$
$350$	0	0
$351$	0	0
$352$	−2012.69	−0.304764
$353$	−6718.33	−1.01298	−0.506488	−	0.862247i	$-0.669057\pi$
$353$	−6718.33	−1.01298	−0.506488	+	0.862247i	$0.669057\pi$
$354$	0	0
$355$	0	0
$356$	5361.59	0.798213
$357$	0	0
$358$	4376.87	0.646159
$359$	−7086.78	−1.04186	−0.520928	−	0.853601i	$-0.674414\pi$
$359$	−7086.78	−1.04186	−0.520928	+	0.853601i	$0.674414\pi$
$360$	0	0
$361$	10993.3	1.60275
$362$	2897.38	0.420672
$363$	0	0
$364$	−444.663	−0.0640294
$365$	0	0
$366$	0	0
$367$	4522.84	0.643298	0.321649	−	0.946859i	$-0.395763\pi$
$367$	4522.84	0.643298	0.321649	+	0.946859i	$0.395763\pi$
$368$	3075.28	0.435625
$369$	0	0
$370$	0	0
$371$	−3511.50	−0.491397
$372$	0	0
$373$	−3342.40	−0.463976	−0.231988	−	0.972719i	$-0.574523\pi$
$373$	−3342.40	−0.463976	−0.231988	+	0.972719i	$0.574523\pi$
$374$	−1038.76	−0.143618
$375$	0	0
$376$	−1543.96	−0.211765
$377$	619.875	0.0846822
$378$	0	0
$379$	−899.769	−0.121947	−0.0609736	−	0.998139i	$-0.519421\pi$
$379$	−899.769	−0.121947	−0.0609736	+	0.998139i	$0.519421\pi$
$380$	0	0
$381$	0	0
$382$	3496.21	0.468277
$383$	916.550	0.122281	0.0611404	−	0.998129i	$-0.480526\pi$
$383$	916.550	0.122281	0.0611404	+	0.998129i	$0.480526\pi$
$384$	0	0
$385$	0	0
$386$	−1415.02	−0.186587
$387$	0	0
$388$	1934.07	0.253060
$389$	−9875.11	−1.28712	−0.643558	−	0.765398i	$-0.722542\pi$
$389$	−9875.11	−1.28712	−0.643558	+	0.765398i	$0.722542\pi$
$390$	0	0
$391$	8486.20	1.09761
$392$	−1352.88	−0.174313
$393$	0	0
$394$	−140.771	−0.0179999
$395$	0	0
$396$	0	0
$397$	−10077.7	−1.27401	−0.637006	−	0.770858i	$-0.719828\pi$
$397$	−10077.7	−1.27401	−0.637006	+	0.770858i	$0.719828\pi$
$398$	−2699.93	−0.340038
$399$	0	0
$400$	0	0
$401$	5372.45	0.669046	0.334523	−	0.942388i	$-0.391425\pi$
$401$	5372.45	0.669046	0.334523	+	0.942388i	$0.391425\pi$
$402$	0	0
$403$	1008.52	0.124659
$404$	3715.56	0.457564
$405$	0	0
$406$	−3089.34	−0.377639
$407$	828.005	0.100842
$408$	0	0
$409$	−5952.05	−0.719584	−0.359792	−	0.933033i	$-0.617152\pi$
$409$	−5952.05	−0.719584	−0.359792	+	0.933033i	$0.617152\pi$
$410$	0	0
$411$	0	0
$412$	8222.04	0.983182
$413$	−7856.94	−0.936114
$414$	0	0
$415$	0	0
$416$	−784.084	−0.0924107
$417$	0	0
$418$	−1909.43	−0.223429
$419$	5393.13	0.628810	0.314405	−	0.949289i	$-0.398195\pi$
$419$	5393.13	0.628810	0.314405	+	0.949289i	$0.398195\pi$
$420$	0	0
$421$	−10963.9	−1.26924	−0.634620	−	0.772824i	$-0.718843\pi$
$421$	−10963.9	−1.26924	−0.634620	+	0.772824i	$0.718843\pi$
$422$	1469.15	0.169472
$423$	0	0
$424$	−3971.80	−0.454924
$425$	0	0
$426$	0	0
$427$	−4839.46	−0.548473
$428$	−11999.0	−1.35513
$429$	0	0
$430$	0	0
$431$	15391.2	1.72012	0.860058	−	0.510196i	$-0.170427\pi$
$431$	15391.2	1.72012	0.860058	+	0.510196i	$0.170427\pi$
$432$	0	0
$433$	2998.44	0.332785	0.166392	−	0.986060i	$-0.446788\pi$
$433$	2998.44	0.332785	0.166392	+	0.986060i	$0.446788\pi$
$434$	−5026.26	−0.555918
$435$	0	0
$436$	−3232.33	−0.355047
$437$	15599.2	1.70757
$438$	0	0
$439$	−5831.19	−0.633958	−0.316979	−	0.948433i	$-0.602668\pi$
$439$	−5831.19	−0.633958	−0.316979	+	0.948433i	$0.602668\pi$
$440$	0	0
$441$	0	0
$442$	−404.669	−0.0435479
$443$	16101.1	1.72683	0.863415	−	0.504494i	$-0.168321\pi$
$443$	16101.1	1.72683	0.863415	+	0.504494i	$0.168321\pi$
$444$	0	0
$445$	0	0
$446$	5497.76	0.583692
$447$	0	0
$448$	443.592	0.0467807
$449$	−15892.2	−1.67038	−0.835189	−	0.549963i	$-0.814642\pi$
$449$	−15892.2	−1.67038	−0.835189	+	0.549963i	$0.814642\pi$
$450$	0	0
$451$	3911.43	0.408386
$452$	9798.86	1.01969
$453$	0	0
$454$	−1906.95	−0.197131
$455$	0	0
$456$	0	0
$457$	−1290.47	−0.132092	−0.0660458	−	0.997817i	$-0.521038\pi$
$457$	−1290.47	−0.132092	−0.0660458	+	0.997817i	$0.521038\pi$
$458$	82.1534	0.00838161
$459$	0	0
$460$	0	0
$461$	−805.732	−0.0814027	−0.0407014	−	0.999171i	$-0.512959\pi$
$461$	−805.732	−0.0814027	−0.0407014	+	0.999171i	$0.512959\pi$
$462$	0	0
$463$	−4326.61	−0.434286	−0.217143	−	0.976140i	$-0.569674\pi$
$463$	−4326.61	−0.434286	−0.217143	+	0.976140i	$0.569674\pi$
$464$	3810.28	0.381223
$465$	0	0
$466$	−3528.25	−0.350736
$467$	−111.263	−0.0110249	−0.00551247	−	0.999985i	$-0.501755\pi$
$467$	−111.263	−0.0110249	−0.00551247	+	0.999985i	$0.501755\pi$
$468$	0	0
$469$	16510.3	1.62553
$470$	0	0
$471$	0	0
$472$	−8886.86	−0.866633
$473$	−31.5611	−0.00306804
$474$	0	0
$475$	0	0
$476$	−7542.45	−0.726276
$477$	0	0
$478$	−5657.16	−0.541323
$479$	778.887	0.0742970	0.0371485	−	0.999310i	$-0.488173\pi$
$479$	778.887	0.0742970	0.0371485	+	0.999310i	$0.488173\pi$
$480$	0	0
$481$	322.565	0.0305774
$482$	−24.8482	−0.00234815
$483$	0	0
$484$	−763.773	−0.0717292
$485$	0	0
$486$	0	0
$487$	10375.6	0.965427	0.482713	−	0.875778i	$-0.339651\pi$
$487$	10375.6	0.965427	0.482713	+	0.875778i	$0.339651\pi$
$488$	−5473.83	−0.507764
$489$	0	0
$490$	0	0
$491$	−10721.9	−0.985488	−0.492744	−	0.870174i	$-0.664006\pi$
$491$	−10721.9	−0.985488	−0.492744	+	0.870174i	$0.664006\pi$
$492$	0	0
$493$	10514.4	0.960538
$494$	−743.856	−0.0677483
$495$	0	0
$496$	6199.19	0.561194
$497$	16820.6	1.51813
$498$	0	0
$499$	2650.84	0.237812	0.118906	−	0.992906i	$-0.462061\pi$
$499$	2650.84	0.237812	0.118906	+	0.992906i	$0.462061\pi$
$500$	0	0
$501$	0	0
$502$	1718.52	0.152791
$503$	4081.05	0.361759	0.180880	−	0.983505i	$-0.442106\pi$
$503$	4081.05	0.361759	0.180880	+	0.983505i	$0.442106\pi$
$504$	0	0
$505$	0	0
$506$	−1668.45	−0.146584
$507$	0	0
$508$	5272.72	0.460510
$509$	−15502.7	−1.34999	−0.674994	−	0.737823i	$-0.735854\pi$
$509$	−15502.7	−1.34999	−0.674994	+	0.737823i	$0.735854\pi$
$510$	0	0
$511$	−1263.25	−0.109360
$512$	−8737.87	−0.754224
$513$	0	0
$514$	−9173.99	−0.787251
$515$	0	0
$516$	0	0
$517$	−913.394	−0.0777002
$518$	−1607.61	−0.136359
$519$	0	0
$520$	0	0
$521$	−16290.8	−1.36989	−0.684945	−	0.728594i	$-0.740174\pi$
$521$	−16290.8	−1.36989	−0.684945	+	0.728594i	$0.740174\pi$
$522$	0	0
$523$	21829.9	1.82515	0.912575	−	0.408910i	$-0.134091\pi$
$523$	21829.9	1.82515	0.912575	+	0.408910i	$0.134091\pi$
$524$	−9905.82	−0.825835
$525$	0	0
$526$	−5404.60	−0.448008
$527$	17106.6	1.41399
$528$	0	0
$529$	1463.45	0.120280
$530$	0	0
$531$	0	0
$532$	−13864.4	−1.12988
$533$	1523.77	0.123831
$534$	0	0
$535$	0	0
$536$	18674.5	1.50488
$537$	0	0
$538$	−2300.20	−0.184328
$539$	−800.355	−0.0639587
$540$	0	0
$541$	20431.2	1.62367	0.811835	−	0.583888i	$-0.198469\pi$
$541$	20431.2	1.62367	0.811835	+	0.583888i	$0.198469\pi$
$542$	−2657.60	−0.210615
$543$	0	0
$544$	−13299.7	−1.04820
$545$	0	0
$546$	0	0
$547$	10728.3	0.838589	0.419294	−	0.907850i	$-0.362278\pi$
$547$	10728.3	0.838589	0.419294	+	0.907850i	$0.362278\pi$
$548$	3119.50	0.243173
$549$	0	0
$550$	0	0
$551$	19327.4	1.49433
$552$	0	0
$553$	7507.42	0.577302
$554$	4275.55	0.327889
$555$	0	0
$556$	14748.8	1.12498
$557$	7755.35	0.589954	0.294977	−	0.955504i	$-0.404688\pi$
$557$	7755.35	0.589954	0.294977	+	0.955504i	$0.404688\pi$
$558$	0	0
$559$	−12.2952	−0.000930292 0
$560$	0	0
$561$	0	0
$562$	−6719.39	−0.504343
$563$	−2213.68	−0.165711	−0.0828556	−	0.996562i	$-0.526404\pi$
$563$	−2213.68	−0.165711	−0.0828556	+	0.996562i	$0.526404\pi$
$564$	0	0
$565$	0	0
$566$	−9579.27	−0.711390
$567$	0	0
$568$	19025.5	1.40545
$569$	−4643.97	−0.342154	−0.171077	−	0.985258i	$-0.554725\pi$
$569$	−4643.97	−0.342154	−0.171077	+	0.985258i	$0.554725\pi$
$570$	0	0
$571$	12883.4	0.944224	0.472112	−	0.881539i	$-0.343492\pi$
$571$	12883.4	0.944224	0.472112	+	0.881539i	$0.343492\pi$
$572$	−297.542	−0.0217498
$573$	0	0
$574$	−7594.21	−0.552223
$575$	0	0
$576$	0	0
$577$	−8662.43	−0.624995	−0.312497	−	0.949919i	$-0.601166\pi$
$577$	−8662.43	−0.624995	−0.312497	+	0.949919i	$0.601166\pi$
$578$	−481.262	−0.0346330
$579$	0	0
$580$	0	0
$581$	8290.50	0.591993
$582$	0	0
$583$	−2349.69	−0.166920
$584$	−1428.84	−0.101243
$585$	0	0
$586$	−10744.4	−0.757416
$587$	−20547.5	−1.44478	−0.722391	−	0.691485i	$-0.756957\pi$
$587$	−20547.5	−1.44478	−0.722391	+	0.691485i	$0.756957\pi$
$588$	0	0
$589$	31445.0	2.19978
$590$	0	0
$591$	0	0
$592$	1982.76	0.137654
$593$	−2360.70	−0.163478	−0.0817388	−	0.996654i	$-0.526047\pi$
$593$	−2360.70	−0.163478	−0.0817388	+	0.996654i	$0.526047\pi$
$594$	0	0
$595$	0	0
$596$	8103.92	0.556962
$597$	0	0
$598$	−649.975	−0.0444473
$599$	−11752.1	−0.801630	−0.400815	−	0.916159i	$-0.631273\pi$
$599$	−11752.1	−0.801630	−0.400815	+	0.916159i	$0.631273\pi$
$600$	0	0
$601$	6667.12	0.452508	0.226254	−	0.974068i	$-0.427352\pi$
$601$	6667.12	0.452508	0.226254	+	0.974068i	$0.427352\pi$
$602$	61.2772	0.00414863
$603$	0	0
$604$	22372.4	1.50715
$605$	0	0
$606$	0	0
$607$	−16579.2	−1.10862	−0.554308	−	0.832312i	$-0.687017\pi$
$607$	−16579.2	−1.10862	−0.554308	+	0.832312i	$0.687017\pi$
$608$	−24447.3	−1.63071
$609$	0	0
$610$	0	0
$611$	−355.830	−0.0235603
$612$	0	0
$613$	−23230.3	−1.53061	−0.765305	−	0.643668i	$-0.777412\pi$
$613$	−23230.3	−1.53061	−0.765305	+	0.643668i	$0.777412\pi$
$614$	−7354.33	−0.483382
$615$	0	0
$616$	3362.31	0.219921
$617$	3455.91	0.225494	0.112747	−	0.993624i	$-0.464035\pi$
$617$	3455.91	0.225494	0.112747	+	0.993624i	$0.464035\pi$
$618$	0	0
$619$	11286.3	0.732850	0.366425	−	0.930448i	$-0.380582\pi$
$619$	11286.3	0.732850	0.366425	+	0.930448i	$0.380582\pi$
$620$	0	0
$621$	0	0
$622$	−10077.5	−0.649633
$623$	−13963.4	−0.897962
$624$	0	0
$625$	0	0
$626$	6081.41	0.388278
$627$	0	0
$628$	−8265.27	−0.525191
$629$	5471.40	0.346835
$630$	0	0
$631$	−13976.1	−0.881742	−0.440871	−	0.897571i	$-0.645330\pi$
$631$	−13976.1	−0.881742	−0.440871	+	0.897571i	$0.645330\pi$
$632$	8491.51	0.534453
$633$	0	0
$634$	−4400.89	−0.275681
$635$	0	0
$636$	0	0
$637$	−311.794	−0.0193936
$638$	−2067.21	−0.128278
$639$	0	0
$640$	0	0
$641$	973.071	0.0599594	0.0299797	−	0.999551i	$-0.490456\pi$
$641$	973.071	0.0599594	0.0299797	+	0.999551i	$0.490456\pi$
$642$	0	0
$643$	−24227.2	−1.48589	−0.742947	−	0.669351i	$-0.766572\pi$
$643$	−24227.2	−1.48589	−0.742947	+	0.669351i	$0.766572\pi$
$644$	−12114.6	−0.741276
$645$	0	0
$646$	−12617.4	−0.768459
$647$	−25850.8	−1.57079	−0.785393	−	0.618998i	$-0.787539\pi$
$647$	−25850.8	−1.57079	−0.785393	+	0.618998i	$0.787539\pi$
$648$	0	0
$649$	−5257.40	−0.317983
$650$	0	0
$651$	0	0
$652$	21553.0	1.29460
$653$	9955.72	0.596627	0.298314	−	0.954468i	$-0.403576\pi$
$653$	9955.72	0.596627	0.298314	+	0.954468i	$0.403576\pi$
$654$	0	0
$655$	0	0
$656$	9366.40	0.557464
$657$	0	0
$658$	1773.39	0.105067
$659$	17797.8	1.05205	0.526027	−	0.850468i	$-0.323681\pi$
$659$	17797.8	1.05205	0.526027	+	0.850468i	$0.323681\pi$
$660$	0	0
$661$	10703.0	0.629801	0.314901	−	0.949125i	$-0.398029\pi$
$661$	10703.0	0.629801	0.314901	+	0.949125i	$0.398029\pi$
$662$	7385.67	0.433614
$663$	0	0
$664$	9377.25	0.548054
$665$	0	0
$666$	0	0
$667$	16888.1	0.980377
$668$	−23581.0	−1.36583
$669$	0	0
$670$	0	0
$671$	−3238.28	−0.186308
$672$	0	0
$673$	−30467.9	−1.74510	−0.872551	−	0.488524i	$-0.837536\pi$
$673$	−30467.9	−1.74510	−0.872551	+	0.488524i	$0.837536\pi$
$674$	8479.70	0.484608
$675$	0	0
$676$	13751.9	0.782426
$677$	1486.08	0.0843642	0.0421821	−	0.999110i	$-0.486569\pi$
$677$	1486.08	0.0843642	0.0421821	+	0.999110i	$0.486569\pi$
$678$	0	0
$679$	−5036.95	−0.284684
$680$	0	0
$681$	0	0
$682$	−3363.28	−0.188836
$683$	12122.7	0.679154	0.339577	−	0.940578i	$-0.389716\pi$
$683$	12122.7	0.679154	0.339577	+	0.940578i	$0.389716\pi$
$684$	0	0
$685$	0	0
$686$	8879.36	0.494192
$687$	0	0
$688$	−75.5769	−0.00418800
$689$	−915.367	−0.0506135
$690$	0	0
$691$	25011.9	1.37699	0.688493	−	0.725243i	$-0.258273\pi$
$691$	25011.9	1.37699	0.688493	+	0.725243i	$0.258273\pi$
$692$	15310.3	0.841058
$693$	0	0
$694$	−10029.7	−0.548594
$695$	0	0
$696$	0	0
$697$	25846.5	1.40460
$698$	−2662.89	−0.144401
$699$	0	0
$700$	0	0
$701$	−9699.17	−0.522586	−0.261293	−	0.965260i	$-0.584149\pi$
$701$	−9699.17	−0.522586	−0.261293	+	0.965260i	$0.584149\pi$
$702$	0	0
$703$	10057.4	0.539578
$704$	296.826	0.0158907
$705$	0	0
$706$	8728.22	0.465285
$707$	−9676.55	−0.514744
$708$	0	0
$709$	−4659.20	−0.246798	−0.123399	−	0.992357i	$-0.539380\pi$
$709$	−4659.20	−0.246798	−0.123399	+	0.992357i	$0.539380\pi$
$710$	0	0
$711$	0	0
$712$	−15793.7	−0.831314
$713$	27476.4	1.44320
$714$	0	0
$715$	0	0
$716$	21265.6	1.10996
$717$	0	0
$718$	9206.90	0.478550
$719$	−25990.6	−1.34810	−0.674051	−	0.738685i	$-0.735447\pi$
$719$	−25990.6	−1.34810	−0.674051	+	0.738685i	$0.735447\pi$
$720$	0	0
$721$	−21412.9	−1.10605
$722$	−14282.1	−0.736183
$723$	0	0
$724$	14077.3	0.722624
$725$	0	0
$726$	0	0
$727$	6713.33	0.342481	0.171240	−	0.985229i	$-0.445222\pi$
$727$	6713.33	0.342481	0.171240	+	0.985229i	$0.445222\pi$
$728$	1309.85	0.0666846
$729$	0	0
$730$	0	0
$731$	−208.554	−0.0105522
$732$	0	0
$733$	27390.9	1.38023	0.690114	−	0.723701i	$-0.257561\pi$
$733$	27390.9	1.38023	0.690114	+	0.723701i	$0.257561\pi$
$734$	−5875.91	−0.295482
$735$	0	0
$736$	−21361.9	−1.06985
$737$	11047.7	0.552168
$738$	0	0
$739$	6126.07	0.304941	0.152470	−	0.988308i	$-0.451277\pi$
$739$	6126.07	0.304941	0.152470	+	0.988308i	$0.451277\pi$
$740$	0	0
$741$	0	0
$742$	4562.02	0.225710
$743$	24446.6	1.20708	0.603538	−	0.797334i	$-0.293757\pi$
$743$	24446.6	1.20708	0.603538	+	0.797334i	$0.293757\pi$
$744$	0	0
$745$	0	0
$746$	4342.33	0.213115
$747$	0	0
$748$	−5046.96	−0.246705
$749$	31249.4	1.52447
$750$	0	0
$751$	−22814.3	−1.10853	−0.554265	−	0.832340i	$-0.687001\pi$
$751$	−22814.3	−1.10853	−0.554265	+	0.832340i	$0.687001\pi$
$752$	−2187.23	−0.106064
$753$	0	0
$754$	−805.320	−0.0388966
$755$	0	0
$756$	0	0
$757$	7619.36	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$757$	7619.36	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$758$	1168.95	0.0560133
$759$	0	0
$760$	0	0
$761$	−17564.1	−0.836660	−0.418330	−	0.908295i	$-0.637384\pi$
$761$	−17564.1	−0.836660	−0.418330	+	0.908295i	$0.637384\pi$
$762$	0	0
$763$	8418.06	0.399416
$764$	16986.8	0.804400
$765$	0	0
$766$	−1190.75	−0.0561665
$767$	−2048.12	−0.0964191
$768$	0	0
$769$	−23406.8	−1.09762	−0.548810	−	0.835947i	$-0.684919\pi$
$769$	−23406.8	−1.09762	−0.548810	+	0.835947i	$0.684919\pi$
$770$	0	0
$771$	0	0
$772$	−6875.07	−0.320517
$773$	29880.7	1.39034	0.695170	−	0.718845i	$-0.255329\pi$
$773$	29880.7	1.39034	0.695170	+	0.718845i	$0.255329\pi$
$774$	0	0
$775$	0	0
$776$	−5697.21	−0.263554
$777$	0	0
$778$	12829.4	0.591203
$779$	47510.5	2.18516
$780$	0	0
$781$	11255.4	0.515684
$782$	−11025.0	−0.504159
$783$	0	0
$784$	−1916.55	−0.0873063
$785$	0	0
$786$	0	0
$787$	−41231.8	−1.86754	−0.933770	−	0.357875i	$-0.883502\pi$
$787$	−41231.8	−1.86754	−0.933770	+	0.357875i	$0.883502\pi$
$788$	−683.955	−0.0309199
$789$	0	0
$790$	0	0
$791$	−25519.5	−1.14712
$792$	0	0
$793$	−1261.53	−0.0564923
$794$	13092.5	0.585185
$795$	0	0
$796$	−13118.0	−0.584113
$797$	−25257.5	−1.12254	−0.561270	−	0.827632i	$-0.689687\pi$
$797$	−25257.5	−1.12254	−0.561270	+	0.827632i	$0.689687\pi$
$798$	0	0
$799$	−6035.64	−0.267241
$800$	0	0
$801$	0	0
$802$	−6979.70	−0.307309
$803$	−845.292	−0.0371478
$804$	0	0
$805$	0	0
$806$	−1310.23	−0.0572591
$807$	0	0
$808$	−10945.0	−0.476539
$809$	24273.8	1.05491	0.527455	−	0.849583i	$-0.323146\pi$
$809$	24273.8	1.05491	0.527455	+	0.849583i	$0.323146\pi$
$810$	0	0
$811$	6741.86	0.291910	0.145955	−	0.989291i	$-0.453375\pi$
$811$	6741.86	0.291910	0.145955	+	0.989291i	$0.453375\pi$
$812$	−15010.0	−0.648704
$813$	0	0
$814$	−1075.72	−0.0463192
$815$	0	0
$816$	0	0
$817$	−383.360	−0.0164162
$818$	7732.70	0.330522
$819$	0	0
$820$	0	0
$821$	−42424.0	−1.80342	−0.901710	−	0.432341i	$-0.857688\pi$
$821$	−42424.0	−1.80342	−0.901710	+	0.432341i	$0.857688\pi$
$822$	0	0
$823$	16941.4	0.717544	0.358772	−	0.933425i	$-0.383195\pi$
$823$	16941.4	0.717544	0.358772	+	0.933425i	$0.383195\pi$
$824$	−24219.8	−1.02395
$825$	0	0
$826$	10207.5	0.429980
$827$	−24563.9	−1.03285	−0.516427	−	0.856332i	$-0.672738\pi$
$827$	−24563.9	−1.03285	−0.516427	+	0.856332i	$0.672738\pi$
$828$	0	0
$829$	1626.98	0.0681634	0.0340817	−	0.999419i	$-0.489149\pi$
$829$	1626.98	0.0681634	0.0340817	+	0.999419i	$0.489149\pi$
$830$	0	0
$831$	0	0
$832$	115.634	0.00481838
$833$	−5288.69	−0.219979
$834$	0	0
$835$	0	0
$836$	−9277.23	−0.383803
$837$	0	0
$838$	−7006.56	−0.288828
$839$	−1228.51	−0.0505515	−0.0252758	−	0.999681i	$-0.508046\pi$
$839$	−1228.51	−0.0505515	−0.0252758	+	0.999681i	$0.508046\pi$
$840$	0	0
$841$	−3464.57	−0.142055
$842$	14244.0	0.582993
$843$	0	0
$844$	7138.06	0.291116
$845$	0	0
$846$	0	0
$847$	1989.12	0.0806929
$848$	−5626.62	−0.227852
$849$	0	0
$850$	0	0
$851$	8788.11	0.353998
$852$	0	0
$853$	13253.5	0.531993	0.265997	−	0.963974i	$-0.414299\pi$
$853$	13253.5	0.531993	0.265997	+	0.963974i	$0.414299\pi$
$854$	6287.25	0.251927
$855$	0	0
$856$	35345.7	1.41132
$857$	−43231.0	−1.72316	−0.861578	−	0.507626i	$-0.830523\pi$
$857$	−43231.0	−1.72316	−0.861578	+	0.507626i	$0.830523\pi$
$858$	0	0
$859$	−676.318	−0.0268634	−0.0134317	−	0.999910i	$-0.504276\pi$
$859$	−676.318	−0.0268634	−0.0134317	+	0.999910i	$0.504276\pi$
$860$	0	0
$861$	0	0
$862$	−19995.8	−0.790091
$863$	3044.63	0.120093	0.0600466	−	0.998196i	$-0.480875\pi$
$863$	3044.63	0.120093	0.0600466	+	0.998196i	$0.480875\pi$
$864$	0	0
$865$	0	0
$866$	−3895.47	−0.152856
$867$	0	0
$868$	−24420.8	−0.954948
$869$	5023.52	0.196100
$870$	0	0
$871$	4303.85	0.167429
$872$	9521.53	0.369770
$873$	0	0
$874$	−20265.9	−0.784331
$875$	0	0
$876$	0	0
$877$	32314.0	1.24420	0.622102	−	0.782936i	$-0.286279\pi$
$877$	32314.0	1.24420	0.622102	+	0.782936i	$0.286279\pi$
$878$	7575.68	0.291192
$879$	0	0
$880$	0	0
$881$	−10298.1	−0.393814	−0.196907	−	0.980422i	$-0.563090\pi$
$881$	−10298.1	−0.393814	−0.196907	+	0.980422i	$0.563090\pi$
$882$	0	0
$883$	2233.80	0.0851339	0.0425670	−	0.999094i	$-0.486446\pi$
$883$	2233.80	0.0851339	0.0425670	+	0.999094i	$0.486446\pi$
$884$	−1966.14	−0.0748060
$885$	0	0
$886$	−20918.0	−0.793175
$887$	26821.0	1.01529	0.507645	−	0.861566i	$-0.330516\pi$
$887$	26821.0	1.01529	0.507645	+	0.861566i	$0.330516\pi$
$888$	0	0
$889$	−13731.9	−0.518058
$890$	0	0
$891$	0	0
$892$	26711.6	1.00266
$893$	−11094.6	−0.415753
$894$	0	0
$895$	0	0
$896$	23486.7	0.875710
$897$	0	0
$898$	20646.6	0.767245
$899$	34043.3	1.26297
$900$	0	0
$901$	−15526.6	−0.574102
$902$	−5081.60	−0.187582
$903$	0	0
$904$	−28864.7	−1.06197
$905$	0	0
$906$	0	0
$907$	−40663.3	−1.48865	−0.744324	−	0.667819i	$-0.767228\pi$
$907$	−40663.3	−1.48865	−0.744324	+	0.667819i	$0.767228\pi$
$908$	−9265.18	−0.338630
$909$	0	0
$910$	0	0
$911$	−7433.95	−0.270360	−0.135180	−	0.990821i	$-0.543161\pi$
$911$	−7433.95	−0.270360	−0.135180	+	0.990821i	$0.543161\pi$
$912$	0	0
$913$	5547.51	0.201091
$914$	1676.54	0.0606729
$915$	0	0
$916$	399.154	0.0143978
$917$	25798.0	0.929037
$918$	0	0
$919$	40068.5	1.43823	0.719117	−	0.694889i	$-0.244546\pi$
$919$	40068.5	1.43823	0.719117	+	0.694889i	$0.244546\pi$
$920$	0	0
$921$	0	0
$922$	1046.78	0.0373902
$923$	4384.75	0.156366
$924$	0	0
$925$	0	0
$926$	5620.98	0.199478
$927$	0	0
$928$	−26467.4	−0.936245
$929$	13790.9	0.487045	0.243522	−	0.969895i	$-0.421697\pi$
$929$	13790.9	0.487045	0.243522	+	0.969895i	$0.421697\pi$
$930$	0	0
$931$	−9721.58	−0.342225
$932$	−17142.5	−0.602490
$933$	0	0
$934$	144.549	0.00506402
$935$	0	0
$936$	0	0
$937$	20498.3	0.714674	0.357337	−	0.933975i	$-0.383685\pi$
$937$	20498.3	0.714674	0.357337	+	0.933975i	$0.383685\pi$
$938$	−21449.6	−0.746647
$939$	0	0
$940$	0	0
$941$	−19843.6	−0.687440	−0.343720	−	0.939072i	$-0.611687\pi$
$941$	−19843.6	−0.687440	−0.343720	+	0.939072i	$0.611687\pi$
$942$	0	0
$943$	41514.3	1.43361
$944$	−12589.5	−0.434061
$945$	0	0
$946$	41.0031	0.00140922
$947$	−35045.2	−1.20255	−0.601276	−	0.799042i	$-0.705341\pi$
$947$	−35045.2	−1.20255	−0.601276	+	0.799042i	$0.705341\pi$
$948$	0	0
$949$	−329.300	−0.0112640
$950$	0	0
$951$	0	0
$952$	22217.9	0.756394
$953$	−2356.19	−0.0800885	−0.0400443	−	0.999198i	$-0.512750\pi$
$953$	−2356.19	−0.0800885	−0.0400443	+	0.999198i	$0.512750\pi$
$954$	0	0
$955$	0	0
$956$	−27486.1	−0.929878
$957$	0	0
$958$	−1011.90	−0.0341264
$959$	−8124.23	−0.273561
$960$	0	0
$961$	25596.4	0.859198
$962$	−419.066	−0.0140449
$963$	0	0
$964$	−120.729	−0.00403362
$965$	0	0
$966$	0	0
$967$	−5720.37	−0.190232	−0.0951162	−	0.995466i	$-0.530322\pi$
$967$	−5720.37	−0.190232	−0.0951162	+	0.995466i	$0.530322\pi$
$968$	2249.86	0.0747037
$969$	0	0
$970$	0	0
$971$	−12491.1	−0.412829	−0.206415	−	0.978465i	$-0.566180\pi$
$971$	−12491.1	−0.412829	−0.206415	+	0.978465i	$0.566180\pi$
$972$	0	0
$973$	−38410.8	−1.26557
$974$	−13479.6	−0.443444
$975$	0	0
$976$	−7754.46	−0.254318
$977$	59099.0	1.93526	0.967628	−	0.252380i	$-0.0812132\pi$
$977$	59099.0	1.93526	0.967628	+	0.252380i	$0.0812132\pi$
$978$	0	0
$979$	−9343.46	−0.305024
$980$	0	0
$981$	0	0
$982$	13929.6	0.452658
$983$	−42365.3	−1.37461	−0.687305	−	0.726369i	$-0.741207\pi$
$983$	−42365.3	−1.37461	−0.687305	+	0.726369i	$0.741207\pi$
$984$	0	0
$985$	0	0
$986$	−13660.0	−0.441199
$987$	0	0
$988$	−3614.12	−0.116377
$989$	−334.977	−0.0107701
$990$	0	0
$991$	−51165.3	−1.64008	−0.820040	−	0.572306i	$-0.806049\pi$
$991$	−51165.3	−1.64008	−0.820040	+	0.572306i	$0.806049\pi$
$992$	−43061.6	−1.37823
$993$	0	0
$994$	−21852.8	−0.697312
$995$	0	0
$996$	0	0
$997$	238.109	0.00756369	0.00378184	−	0.999993i	$-0.498796\pi$
$997$	238.109	0.00756369	0.00378184	+	0.999993i	$0.498796\pi$
$998$	−3443.88	−0.109233
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.by.1.4	yes	10
3.2	odd	2		2475.4.a.bv.1.7	yes	10
5.4	even	2		2475.4.a.bu.1.7	✓	10
15.14	odd	2		2475.4.a.bx.1.4	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2475.4.a.bu.1.7	✓	10	5.4	even	2
2475.4.a.bv.1.7	yes	10	3.2	odd	2
2475.4.a.bx.1.4	yes	10	15.14	odd	2
2475.4.a.by.1.4	yes	10	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)

\(f(q)\)	\(=\)	\(q-1.29917 q^{2} -6.31217 q^{4} +16.4390 q^{7} +18.5939 q^{8} +O(q^{10})\)	\(q-1.29917 q^{2} -6.31217 q^{4} +16.4390 q^{7} +18.5939 q^{8} +11.0000 q^{11} +4.28526 q^{13} -21.3570 q^{14} +26.3409 q^{16} +72.6873 q^{17} +133.612 q^{19} -14.2908 q^{22} +116.750 q^{23} -5.56726 q^{26} -103.766 q^{28} +144.653 q^{29} +235.345 q^{31} -182.972 q^{32} -94.4328 q^{34} +75.2732 q^{37} -173.585 q^{38} +355.585 q^{41} -2.86919 q^{43} -69.4339 q^{44} -151.677 q^{46} -83.0358 q^{47} -72.7595 q^{49} -27.0493 q^{52} -213.608 q^{53} +305.665 q^{56} -187.928 q^{58} -477.946 q^{59} -294.389 q^{61} -305.752 q^{62} +26.9842 q^{64} +1004.34 q^{67} -458.814 q^{68} +1023.22 q^{71} -76.8447 q^{73} -97.7923 q^{74} -843.384 q^{76} +180.829 q^{77} +456.683 q^{79} -461.963 q^{82} +504.319 q^{83} +3.72755 q^{86} +204.533 q^{88} -849.405 q^{89} +70.4454 q^{91} -736.943 q^{92} +107.877 q^{94} -306.403 q^{97} +94.5266 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{2} + 34 q^{4} + 2 q^{7} + 48 q^{8}+O(q^{10}) \) 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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