LMFDB - Embedded newform 2475.4.a.bv.1.6

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:	$10$
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			1.6
Root			$-0.207736$ 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)

$f(q)$	$=$	$q+0.207736 q^{2} -7.95685 q^{4} -31.6389 q^{7} -3.31481 q^{8} +O(q^{10})$	$q+0.207736 q^{2} -7.95685 q^{4} -31.6389 q^{7} -3.31481 q^{8} -11.0000 q^{11} +0.254633 q^{13} -6.57254 q^{14} +62.9662 q^{16} -117.599 q^{17} -17.3405 q^{19} -2.28509 q^{22} +178.365 q^{23} +0.0528964 q^{26} +251.746 q^{28} +254.919 q^{29} +48.5542 q^{31} +39.5988 q^{32} -24.4295 q^{34} +147.517 q^{37} -3.60225 q^{38} +469.046 q^{41} -355.130 q^{43} +87.5253 q^{44} +37.0529 q^{46} -266.283 q^{47} +658.023 q^{49} -2.02608 q^{52} +283.583 q^{53} +104.877 q^{56} +52.9558 q^{58} +508.261 q^{59} -605.417 q^{61} +10.0864 q^{62} -495.503 q^{64} +351.877 q^{67} +935.716 q^{68} -419.836 q^{71} -942.799 q^{73} +30.6446 q^{74} +137.976 q^{76} +348.028 q^{77} -1070.08 q^{79} +97.4377 q^{82} +1163.87 q^{83} -73.7733 q^{86} +36.4629 q^{88} -372.618 q^{89} -8.05632 q^{91} -1419.23 q^{92} -55.3166 q^{94} -524.502 q^{97} +136.695 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.207736	0.0734457	0.0367229	−	0.999325i	$-0.488308\pi$
$2$	0.207736	0.0734457	0.0367229	+	0.999325i	$0.488308\pi$
$3$	0	0
$3$	0	0
$4$	−7.95685	−0.994606
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−31.6389	−1.70834	−0.854171	−	0.519992i	$-0.825935\pi$
$7$	−31.6389	−1.70834	−0.854171	+	0.519992i	$0.825935\pi$
$8$	−3.31481	−0.146495
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	0.254633	0.00543250	0.00271625	−	0.999996i	$-0.499135\pi$
$13$	0.254633	0.00543250	0.00271625	+	0.999996i	$0.499135\pi$
$14$	−6.57254	−0.125470
$15$	0	0
$16$	62.9662	0.983846
$17$	−117.599	−1.67776	−0.838880	−	0.544316i	$-0.816789\pi$
$17$	−117.599	−1.67776	−0.838880	+	0.544316i	$0.816789\pi$
$18$	0	0
$19$	−17.3405	−0.209378	−0.104689	−	0.994505i	$-0.533385\pi$
$19$	−17.3405	−0.209378	−0.104689	+	0.994505i	$0.533385\pi$
$20$	0	0
$21$	0	0
$22$	−2.28509	−0.0221447
$23$	178.365	1.61703	0.808517	−	0.588473i	$-0.200271\pi$
$23$	178.365	1.61703	0.808517	+	0.588473i	$0.200271\pi$
$24$	0	0
$25$	0	0
$26$	0.0528964	0.000398994 0
$27$	0	0
$28$	251.746	1.69913
$29$	254.919	1.63232	0.816160	−	0.577826i	$-0.196099\pi$
$29$	254.919	1.63232	0.816160	+	0.577826i	$0.196099\pi$
$30$	0	0
$31$	48.5542	0.281309	0.140655	−	0.990059i	$-0.455079\pi$
$31$	48.5542	0.281309	0.140655	+	0.990059i	$0.455079\pi$
$32$	39.5988	0.218755
$33$	0	0
$34$	−24.4295	−0.123224
$35$	0	0
$36$	0	0
$37$	147.517	0.655449	0.327725	−	0.944773i	$-0.393718\pi$
$37$	147.517	0.655449	0.327725	+	0.944773i	$0.393718\pi$
$38$	−3.60225	−0.0153779
$39$	0	0
$40$	0	0
$41$	469.046	1.78665	0.893325	−	0.449410i	$-0.148366\pi$
$41$	469.046	1.78665	0.893325	+	0.449410i	$0.148366\pi$
$42$	0	0
$43$	−355.130	−1.25946	−0.629731	−	0.776814i	$-0.716835\pi$
$43$	−355.130	−1.25946	−0.629731	+	0.776814i	$0.716835\pi$
$44$	87.5253	0.299885
$45$	0	0
$46$	37.0529	0.118764
$47$	−266.283	−0.826413	−0.413206	−	0.910637i	$-0.635591\pi$
$47$	−266.283	−0.826413	−0.413206	+	0.910637i	$0.635591\pi$
$48$	0	0
$49$	658.023	1.91843
$50$	0	0
$51$	0	0
$52$	−2.02608	−0.00540319
$53$	283.583	0.734964	0.367482	−	0.930031i	$-0.380220\pi$
$53$	283.583	0.734964	0.367482	+	0.930031i	$0.380220\pi$
$54$	0	0
$55$	0	0
$56$	104.877	0.250264
$57$	0	0
$58$	52.9558	0.119887
$59$	508.261	1.12152	0.560762	−	0.827977i	$-0.310508\pi$
$59$	508.261	1.12152	0.560762	+	0.827977i	$0.310508\pi$
$60$	0	0
$61$	−605.417	−1.27075	−0.635375	−	0.772204i	$-0.719154\pi$
$61$	−605.417	−1.27075	−0.635375	+	0.772204i	$0.719154\pi$
$62$	10.0864	0.0206610
$63$	0	0
$64$	−495.503	−0.967780
$65$	0	0
$66$	0	0
$67$	351.877	0.641621	0.320811	−	0.947143i	$-0.396045\pi$
$67$	351.877	0.641621	0.320811	+	0.947143i	$0.396045\pi$
$68$	935.716	1.66871
$69$	0	0
$70$	0	0
$71$	−419.836	−0.701766	−0.350883	−	0.936419i	$-0.614119\pi$
$71$	−419.836	−0.701766	−0.350883	+	0.936419i	$0.614119\pi$
$72$	0	0
$73$	−942.799	−1.51159	−0.755797	−	0.654807i	$-0.772750\pi$
$73$	−942.799	−1.51159	−0.755797	+	0.654807i	$0.772750\pi$
$74$	30.6446	0.0481400
$75$	0	0
$76$	137.976	0.208249
$77$	348.028	0.515085
$78$	0	0
$79$	−1070.08	−1.52397	−0.761983	−	0.647597i	$-0.775774\pi$
$79$	−1070.08	−1.52397	−0.761983	+	0.647597i	$0.775774\pi$
$80$	0	0
$81$	0	0
$82$	97.4377	0.131222
$83$	1163.87	1.53917	0.769584	−	0.638545i	$-0.220463\pi$
$83$	1163.87	1.53917	0.769584	+	0.638545i	$0.220463\pi$
$84$	0	0
$85$	0	0
$86$	−73.7733	−0.0925020
$87$	0	0
$88$	36.4629	0.0441700
$89$	−372.618	−0.443791	−0.221895	−	0.975070i	$-0.571224\pi$
$89$	−372.618	−0.443791	−0.221895	+	0.975070i	$0.571224\pi$
$90$	0	0
$91$	−8.05632	−0.00928057
$92$	−1419.23	−1.60831
$93$	0	0
$94$	−55.3166	−0.0606965
$95$	0	0
$96$	0	0
$97$	−524.502	−0.549022	−0.274511	−	0.961584i	$-0.588516\pi$
$97$	−524.502	−0.549022	−0.274511	+	0.961584i	$0.588516\pi$
$98$	136.695	0.140901
$99$	0	0
$100$	0	0
$101$	340.997	0.335946	0.167973	−	0.985792i	$-0.446278\pi$
$101$	340.997	0.335946	0.167973	+	0.985792i	$0.446278\pi$
$102$	0	0
$103$	1787.40	1.70988	0.854938	−	0.518729i	$-0.173595\pi$
$103$	1787.40	1.70988	0.854938	+	0.518729i	$0.173595\pi$
$104$	−0.844060	−0.000795835 0
$105$	0	0
$106$	58.9103	0.0539800
$107$	936.154	0.845808	0.422904	−	0.906174i	$-0.361011\pi$
$107$	936.154	0.845808	0.422904	+	0.906174i	$0.361011\pi$
$108$	0	0
$109$	1168.71	1.02699	0.513494	−	0.858093i	$-0.328351\pi$
$109$	1168.71	1.02699	0.513494	+	0.858093i	$0.328351\pi$
$110$	0	0
$111$	0	0
$112$	−1992.18	−1.68075
$113$	−1840.82	−1.53247	−0.766237	−	0.642558i	$-0.777873\pi$
$113$	−1840.82	−1.53247	−0.766237	+	0.642558i	$0.777873\pi$
$114$	0	0
$115$	0	0
$116$	−2028.35	−1.62351
$117$	0	0
$118$	105.584	0.0823712
$119$	3720.70	2.86619
$120$	0	0
$121$	121.000	0.0909091
$122$	−125.767	−0.0933311
$123$	0	0
$124$	−386.338	−0.279792
$125$	0	0
$126$	0	0
$127$	1615.59	1.12882	0.564410	−	0.825494i	$-0.309104\pi$
$127$	1615.59	1.12882	0.564410	+	0.825494i	$0.309104\pi$
$128$	−419.724	−0.289834
$129$	0	0
$130$	0	0
$131$	−2375.70	−1.58447	−0.792236	−	0.610215i	$-0.791083\pi$
$131$	−2375.70	−1.58447	−0.792236	+	0.610215i	$0.791083\pi$
$132$	0	0
$133$	548.636	0.357690
$134$	73.0975	0.0471244
$135$	0	0
$136$	389.818	0.245784
$137$	807.295	0.503444	0.251722	−	0.967800i	$-0.419003\pi$
$137$	807.295	0.503444	0.251722	+	0.967800i	$0.419003\pi$
$138$	0	0
$139$	2409.45	1.47026	0.735131	−	0.677925i	$-0.237120\pi$
$139$	2409.45	1.47026	0.735131	+	0.677925i	$0.237120\pi$
$140$	0	0
$141$	0	0
$142$	−87.2151	−0.0515417
$143$	−2.80096	−0.00163796
$144$	0	0
$145$	0	0
$146$	−195.853	−0.111020
$147$	0	0
$148$	−1173.77	−0.651914
$149$	710.292	0.390533	0.195266	−	0.980750i	$-0.437443\pi$
$149$	710.292	0.390533	0.195266	+	0.980750i	$0.437443\pi$
$150$	0	0
$151$	−201.621	−0.108660	−0.0543300	−	0.998523i	$-0.517302\pi$
$151$	−201.621	−0.108660	−0.0543300	+	0.998523i	$0.517302\pi$
$152$	57.4805	0.0306729
$153$	0	0
$154$	72.2980	0.0378308
$155$	0	0
$156$	0	0
$157$	−1078.10	−0.548037	−0.274019	−	0.961724i	$-0.588353\pi$
$157$	−1078.10	−0.548037	−0.274019	+	0.961724i	$0.588353\pi$
$158$	−222.294	−0.111929
$159$	0	0
$160$	0	0
$161$	−5643.29	−2.76245
$162$	0	0
$163$	−673.385	−0.323580	−0.161790	−	0.986825i	$-0.551727\pi$
$163$	−673.385	−0.323580	−0.161790	+	0.986825i	$0.551727\pi$
$164$	−3732.13	−1.77701
$165$	0	0
$166$	241.777	0.113045
$167$	−2816.25	−1.30496	−0.652479	−	0.757807i	$-0.726271\pi$
$167$	−2816.25	−1.30496	−0.652479	+	0.757807i	$0.726271\pi$
$168$	0	0
$169$	−2196.94	−0.999970
$170$	0	0
$171$	0	0
$172$	2825.72	1.25267
$173$	−2265.71	−0.995717	−0.497859	−	0.867258i	$-0.665880\pi$
$173$	−2265.71	−0.995717	−0.497859	+	0.867258i	$0.665880\pi$
$174$	0	0
$175$	0	0
$176$	−692.628	−0.296641
$177$	0	0
$178$	−77.4061	−0.0325945
$179$	502.841	0.209967	0.104983	−	0.994474i	$-0.466521\pi$
$179$	502.841	0.209967	0.104983	+	0.994474i	$0.466521\pi$
$180$	0	0
$181$	3966.01	1.62868	0.814340	−	0.580388i	$-0.197099\pi$
$181$	3966.01	1.62868	0.814340	+	0.580388i	$0.197099\pi$
$182$	−1.67359	−0.000681618 0
$183$	0	0
$184$	−591.247	−0.236888
$185$	0	0
$186$	0	0
$187$	1293.59	0.505864
$188$	2118.77	0.821955
$189$	0	0
$190$	0	0
$191$	1642.53	0.622247	0.311124	−	0.950369i	$-0.399295\pi$
$191$	1642.53	0.622247	0.311124	+	0.950369i	$0.399295\pi$
$192$	0	0
$193$	−2658.66	−0.991579	−0.495789	−	0.868443i	$-0.665121\pi$
$193$	−2658.66	−0.991579	−0.495789	+	0.868443i	$0.665121\pi$
$194$	−108.958	−0.0403233
$195$	0	0
$196$	−5235.79	−1.90809
$197$	−3376.29	−1.22107	−0.610535	−	0.791989i	$-0.709046\pi$
$197$	−3376.29	−1.22107	−0.610535	+	0.791989i	$0.709046\pi$
$198$	0	0
$199$	−2536.55	−0.903574	−0.451787	−	0.892126i	$-0.649213\pi$
$199$	−2536.55	−0.903574	−0.451787	+	0.892126i	$0.649213\pi$
$200$	0	0
$201$	0	0
$202$	70.8374	0.0246738
$203$	−8065.37	−2.78856
$204$	0	0
$205$	0	0
$206$	371.306	0.125583
$207$	0	0
$208$	16.0333	0.00534474
$209$	190.746	0.0631299
$210$	0	0
$211$	−2875.09	−0.938055	−0.469027	−	0.883184i	$-0.655395\pi$
$211$	−2875.09	−0.938055	−0.469027	+	0.883184i	$0.655395\pi$
$212$	−2256.42	−0.730999
$213$	0	0
$214$	194.473	0.0621210
$215$	0	0
$216$	0	0
$217$	−1536.20	−0.480573
$218$	242.782	0.0754278
$219$	0	0
$220$	0	0
$221$	−29.9446	−0.00911443
$222$	0	0
$223$	3933.01	1.18105	0.590524	−	0.807020i	$-0.298921\pi$
$223$	3933.01	1.18105	0.590524	+	0.807020i	$0.298921\pi$
$224$	−1252.86	−0.373708
$225$	0	0
$226$	−382.404	−0.112554
$227$	2820.31	0.824628	0.412314	−	0.911042i	$-0.364721\pi$
$227$	2820.31	0.824628	0.412314	+	0.911042i	$0.364721\pi$
$228$	0	0
$229$	−1292.46	−0.372962	−0.186481	−	0.982459i	$-0.559708\pi$
$229$	−1292.46	−0.372962	−0.186481	+	0.982459i	$0.559708\pi$
$230$	0	0
$231$	0	0
$232$	−845.008	−0.239127
$233$	318.169	0.0894591	0.0447295	−	0.998999i	$-0.485757\pi$
$233$	318.169	0.0894591	0.0447295	+	0.998999i	$0.485757\pi$
$234$	0	0
$235$	0	0
$236$	−4044.15	−1.11547
$237$	0	0
$238$	772.924	0.210509
$239$	5914.81	1.60083	0.800413	−	0.599450i	$-0.204614\pi$
$239$	5914.81	1.60083	0.800413	+	0.599450i	$0.204614\pi$
$240$	0	0
$241$	−907.387	−0.242531	−0.121265	−	0.992620i	$-0.538695\pi$
$241$	−907.387	−0.242531	−0.121265	+	0.992620i	$0.538695\pi$
$242$	25.1360	0.00667688
$243$	0	0
$244$	4817.21	1.26389
$245$	0	0
$246$	0	0
$247$	−4.41547	−0.00113745
$248$	−160.948	−0.0412105
$249$	0	0
$250$	0	0
$251$	7197.14	1.80988	0.904939	−	0.425541i	$-0.139916\pi$
$251$	7197.14	1.80988	0.904939	+	0.425541i	$0.139916\pi$
$252$	0	0
$253$	−1962.02	−0.487554
$254$	335.615	0.0829070
$255$	0	0
$256$	3876.83	0.946493
$257$	−2342.91	−0.568664	−0.284332	−	0.958726i	$-0.591772\pi$
$257$	−2342.91	−0.568664	−0.284332	+	0.958726i	$0.591772\pi$
$258$	0	0
$259$	−4667.28	−1.11973
$260$	0	0
$261$	0	0
$262$	−493.518	−0.116373
$263$	−1749.19	−0.410113	−0.205057	−	0.978750i	$-0.565738\pi$
$263$	−1749.19	−0.410113	−0.205057	+	0.978750i	$0.565738\pi$
$264$	0	0
$265$	0	0
$266$	113.971	0.0262708
$267$	0	0
$268$	−2799.83	−0.638160
$269$	1336.09	0.302835	0.151417	−	0.988470i	$-0.451616\pi$
$269$	1336.09	0.302835	0.151417	+	0.988470i	$0.451616\pi$
$270$	0	0
$271$	−6262.35	−1.40373	−0.701864	−	0.712311i	$-0.747649\pi$
$271$	−6262.35	−1.40373	−0.701864	+	0.712311i	$0.747649\pi$
$272$	−7404.75	−1.65066
$273$	0	0
$274$	167.704	0.0369758
$275$	0	0
$276$	0	0
$277$	183.932	0.0398967	0.0199483	−	0.999801i	$-0.493650\pi$
$277$	183.932	0.0398967	0.0199483	+	0.999801i	$0.493650\pi$
$278$	500.528	0.107985
$279$	0	0
$280$	0	0
$281$	3.36918	0.000715261 0	0.000357630	−	1.00000i	$-0.499886\pi$
$281$	3.36918	0.000715261 0	0.000357630		1.00000i	$0.499886\pi$
$282$	0	0
$283$	4234.13	0.889374	0.444687	−	0.895686i	$-0.353315\pi$
$283$	4234.13	0.889374	0.444687	+	0.895686i	$0.353315\pi$
$284$	3340.57	0.697981
$285$	0	0
$286$	−0.581861	−0.000120301 0
$287$	−14840.1	−3.05221
$288$	0	0
$289$	8916.50	1.81488
$290$	0	0
$291$	0	0
$292$	7501.71	1.50344
$293$	−7346.30	−1.46476	−0.732381	−	0.680895i	$-0.761591\pi$
$293$	−7346.30	−1.46476	−0.732381	+	0.680895i	$0.761591\pi$
$294$	0	0
$295$	0	0
$296$	−488.990	−0.0960202
$297$	0	0
$298$	147.553	0.0286829
$299$	45.4177	0.00878453
$300$	0	0
$301$	11235.9	2.15159
$302$	−41.8838	−0.00798061
$303$	0	0
$304$	−1091.87	−0.205996
$305$	0	0
$306$	0	0
$307$	−5220.64	−0.970547	−0.485273	−	0.874363i	$-0.661280\pi$
$307$	−5220.64	−0.970547	−0.485273	+	0.874363i	$0.661280\pi$
$308$	−2769.21	−0.512306
$309$	0	0
$310$	0	0
$311$	4231.75	0.771578	0.385789	−	0.922587i	$-0.373929\pi$
$311$	4231.75	0.771578	0.385789	+	0.922587i	$0.373929\pi$
$312$	0	0
$313$	2438.03	0.440273	0.220136	−	0.975469i	$-0.429350\pi$
$313$	2438.03	0.440273	0.220136	+	0.975469i	$0.429350\pi$
$314$	−223.960	−0.0402510
$315$	0	0
$316$	8514.45	1.51575
$317$	−9183.21	−1.62707	−0.813534	−	0.581518i	$-0.802459\pi$
$317$	−9183.21	−1.62707	−0.813534	+	0.581518i	$0.802459\pi$
$318$	0	0
$319$	−2804.11	−0.492163
$320$	0	0
$321$	0	0
$322$	−1172.31	−0.202890
$323$	2039.23	0.351287
$324$	0	0
$325$	0	0
$326$	−139.886	−0.0237656
$327$	0	0
$328$	−1554.80	−0.261736
$329$	8424.92	1.41180
$330$	0	0
$331$	−224.822	−0.0373333	−0.0186667	−	0.999826i	$-0.505942\pi$
$331$	−224.822	−0.0373333	−0.0186667	+	0.999826i	$0.505942\pi$
$332$	−9260.71	−1.53087
$333$	0	0
$334$	−585.037	−0.0958436
$335$	0	0
$336$	0	0
$337$	−6825.59	−1.10330	−0.551652	−	0.834074i	$-0.686002\pi$
$337$	−6825.59	−1.10330	−0.551652	+	0.834074i	$0.686002\pi$
$338$	−456.382	−0.0734436
$339$	0	0
$340$	0	0
$341$	−534.096	−0.0848180
$342$	0	0
$343$	−9966.99	−1.56900
$344$	1177.19	0.184505
$345$	0	0
$346$	−470.670	−0.0731312
$347$	−9125.46	−1.41176	−0.705879	−	0.708332i	$-0.749448\pi$
$347$	−9125.46	−1.41176	−0.705879	+	0.708332i	$0.749448\pi$
$348$	0	0
$349$	9733.00	1.49282	0.746412	−	0.665484i	$-0.231775\pi$
$349$	9733.00	1.49282	0.746412	+	0.665484i	$0.231775\pi$
$350$	0	0
$351$	0	0
$352$	−435.587	−0.0659570
$353$	−4247.34	−0.640406	−0.320203	−	0.947349i	$-0.603751\pi$
$353$	−4247.34	−0.640406	−0.320203	+	0.947349i	$0.603751\pi$
$354$	0	0
$355$	0	0
$356$	2964.86	0.441397
$357$	0	0
$358$	104.458	0.0154212
$359$	−4113.57	−0.604752	−0.302376	−	0.953189i	$-0.597780\pi$
$359$	−4113.57	−0.604752	−0.302376	+	0.953189i	$0.597780\pi$
$360$	0	0
$361$	−6558.31	−0.956161
$362$	823.883	0.119620
$363$	0	0
$364$	64.1029	0.00923051
$365$	0	0
$366$	0	0
$367$	2213.98	0.314901	0.157451	−	0.987527i	$-0.449672\pi$
$367$	2213.98	0.314901	0.157451	+	0.987527i	$0.449672\pi$
$368$	11231.0	1.59091
$369$	0	0
$370$	0	0
$371$	−8972.26	−1.25557
$372$	0	0
$373$	1310.35	0.181897	0.0909483	−	0.995856i	$-0.471010\pi$
$373$	1310.35	0.181897	0.0909483	+	0.995856i	$0.471010\pi$
$374$	268.725	0.0371535
$375$	0	0
$376$	882.678	0.121066
$377$	64.9108	0.00886758
$378$	0	0
$379$	−7479.09	−1.01365	−0.506827	−	0.862048i	$-0.669182\pi$
$379$	−7479.09	−1.01365	−0.506827	+	0.862048i	$0.669182\pi$
$380$	0	0
$381$	0	0
$382$	341.212	0.0457014
$383$	−1829.27	−0.244050	−0.122025	−	0.992527i	$-0.538939\pi$
$383$	−1829.27	−0.244050	−0.122025	+	0.992527i	$0.538939\pi$
$384$	0	0
$385$	0	0
$386$	−552.299	−0.0728272
$387$	0	0
$388$	4173.38	0.546061
$389$	13433.7	1.75094	0.875470	−	0.483272i	$-0.160552\pi$
$389$	13433.7	1.75094	0.875470	+	0.483272i	$0.160552\pi$
$390$	0	0
$391$	−20975.6	−2.71299
$392$	−2181.22	−0.281041
$393$	0	0
$394$	−701.377	−0.0896824
$395$	0	0
$396$	0	0
$397$	4120.45	0.520905	0.260453	−	0.965487i	$-0.416128\pi$
$397$	4120.45	0.520905	0.260453	+	0.965487i	$0.416128\pi$
$398$	−526.932	−0.0663636
$399$	0	0
$400$	0	0
$401$	−8686.06	−1.08170	−0.540849	−	0.841120i	$-0.681897\pi$
$401$	−8686.06	−1.08170	−0.540849	+	0.841120i	$0.681897\pi$
$402$	0	0
$403$	12.3635	0.00152821
$404$	−2713.26	−0.334133
$405$	0	0
$406$	−1675.47	−0.204808
$407$	−1622.69	−0.197625
$408$	0	0
$409$	−7230.83	−0.874184	−0.437092	−	0.899417i	$-0.643992\pi$
$409$	−7230.83	−0.874184	−0.437092	+	0.899417i	$0.643992\pi$
$410$	0	0
$411$	0	0
$412$	−14222.0	−1.70065
$413$	−16080.8	−1.91595
$414$	0	0
$415$	0	0
$416$	10.0832	0.00118838
$417$	0	0
$418$	39.6247	0.00463662
$419$	8861.11	1.03316	0.516579	−	0.856239i	$-0.327205\pi$
$419$	8861.11	1.03316	0.516579	+	0.856239i	$0.327205\pi$
$420$	0	0
$421$	15509.1	1.79540	0.897702	−	0.440602i	$-0.145235\pi$
$421$	15509.1	1.79540	0.897702	+	0.440602i	$0.145235\pi$
$422$	−597.260	−0.0688961
$423$	0	0
$424$	−940.023	−0.107669
$425$	0	0
$426$	0	0
$427$	19154.8	2.17087
$428$	−7448.84	−0.841246
$429$	0	0
$430$	0	0
$431$	−2229.91	−0.249213	−0.124607	−	0.992206i	$-0.539767\pi$
$431$	−2229.91	−0.249213	−0.124607	+	0.992206i	$0.539767\pi$
$432$	0	0
$433$	8922.37	0.990258	0.495129	−	0.868819i	$-0.335121\pi$
$433$	8922.37	0.990258	0.495129	+	0.868819i	$0.335121\pi$
$434$	−319.125	−0.0352960
$435$	0	0
$436$	−9299.21	−1.02145
$437$	−3092.95	−0.338572
$438$	0	0
$439$	−1988.01	−0.216133	−0.108066	−	0.994144i	$-0.534466\pi$
$439$	−1988.01	−0.216133	−0.108066	+	0.994144i	$0.534466\pi$
$440$	0	0
$441$	0	0
$442$	−6.22056	−0.000669416 0
$443$	1533.65	0.164483	0.0822413	−	0.996612i	$-0.473792\pi$
$443$	1533.65	0.164483	0.0822413	+	0.996612i	$0.473792\pi$
$444$	0	0
$445$	0	0
$446$	817.027	0.0867429
$447$	0	0
$448$	15677.2	1.65330
$449$	−12496.7	−1.31348	−0.656742	−	0.754116i	$-0.728066\pi$
$449$	−12496.7	−1.31348	−0.656742	+	0.754116i	$0.728066\pi$
$450$	0	0
$451$	−5159.51	−0.538695
$452$	14647.1	1.52421
$453$	0	0
$454$	585.880	0.0605654
$455$	0	0
$456$	0	0
$457$	13640.8	1.39625	0.698127	−	0.715974i	$-0.254017\pi$
$457$	13640.8	1.39625	0.698127	+	0.715974i	$0.254017\pi$
$458$	−268.491	−0.0273925
$459$	0	0
$460$	0	0
$461$	−4647.16	−0.469501	−0.234750	−	0.972056i	$-0.575427\pi$
$461$	−4647.16	−0.469501	−0.234750	+	0.972056i	$0.575427\pi$
$462$	0	0
$463$	−4887.00	−0.490536	−0.245268	−	0.969455i	$-0.578876\pi$
$463$	−4887.00	−0.490536	−0.245268	+	0.969455i	$0.578876\pi$
$464$	16051.3	1.60595
$465$	0	0
$466$	66.0952	0.00657039
$467$	−10720.4	−1.06227	−0.531136	−	0.847287i	$-0.678235\pi$
$467$	−10720.4	−1.06227	−0.531136	+	0.847287i	$0.678235\pi$
$468$	0	0
$469$	−11133.0	−1.09611
$470$	0	0
$471$	0	0
$472$	−1684.79	−0.164298
$473$	3906.43	0.379742
$474$	0	0
$475$	0	0
$476$	−29605.1	−2.85073
$477$	0	0
$478$	1228.72	0.117574
$479$	−6456.26	−0.615854	−0.307927	−	0.951410i	$-0.599635\pi$
$479$	−6456.26	−0.615854	−0.307927	+	0.951410i	$0.599635\pi$
$480$	0	0
$481$	37.5627	0.00356073
$482$	−188.497	−0.0178128
$483$	0	0
$484$	−962.778	−0.0904187
$485$	0	0
$486$	0	0
$487$	−19458.3	−1.81056	−0.905278	−	0.424820i	$-0.860337\pi$
$487$	−19458.3	−1.81056	−0.905278	+	0.424820i	$0.860337\pi$
$488$	2006.84	0.186159
$489$	0	0
$490$	0	0
$491$	−5892.02	−0.541555	−0.270777	−	0.962642i	$-0.587281\pi$
$491$	−5892.02	−0.541555	−0.270777	+	0.962642i	$0.587281\pi$
$492$	0	0
$493$	−29978.2	−2.73864
$494$	−0.917251	−8.35407e−5 0
$495$	0	0
$496$	3057.27	0.276765
$497$	13283.2	1.19886
$498$	0	0
$499$	15553.1	1.39530	0.697648	−	0.716440i	$-0.254230\pi$
$499$	15553.1	1.39530	0.697648	+	0.716440i	$0.254230\pi$
$500$	0	0
$501$	0	0
$502$	1495.10	0.132928
$503$	12964.1	1.14918	0.574592	−	0.818440i	$-0.305161\pi$
$503$	12964.1	1.14918	0.574592	+	0.818440i	$0.305161\pi$
$504$	0	0
$505$	0	0
$506$	−407.582	−0.0358088
$507$	0	0
$508$	−12855.0	−1.12273
$509$	11276.5	0.981965	0.490983	−	0.871169i	$-0.336638\pi$
$509$	11276.5	0.981965	0.490983	+	0.871169i	$0.336638\pi$
$510$	0	0
$511$	29829.2	2.58232
$512$	4163.15	0.359350
$513$	0	0
$514$	−486.706	−0.0417659
$515$	0	0
$516$	0	0
$517$	2929.12	0.249173
$518$	−969.561	−0.0822395
$519$	0	0
$520$	0	0
$521$	−19639.9	−1.65152	−0.825759	−	0.564023i	$-0.809253\pi$
$521$	−19639.9	−1.65152	−0.825759	+	0.564023i	$0.809253\pi$
$522$	0	0
$523$	−4175.40	−0.349097	−0.174548	−	0.984649i	$-0.555847\pi$
$523$	−4175.40	−0.349097	−0.174548	+	0.984649i	$0.555847\pi$
$524$	18903.1	1.57592
$525$	0	0
$526$	−363.370	−0.0301210
$527$	−5709.92	−0.471970
$528$	0	0
$529$	19647.2	1.61480
$530$	0	0
$531$	0	0
$532$	−4365.41	−0.355760
$533$	119.435	0.00970598
$534$	0	0
$535$	0	0
$536$	−1166.41	−0.0939945
$537$	0	0
$538$	277.553	0.0222419
$539$	−7238.25	−0.578429
$540$	0	0
$541$	11904.7	0.946068	0.473034	−	0.881044i	$-0.343159\pi$
$541$	11904.7	0.946068	0.473034	+	0.881044i	$0.343159\pi$
$542$	−1300.91	−0.103098
$543$	0	0
$544$	−4656.78	−0.367018
$545$	0	0
$546$	0	0
$547$	−1584.81	−0.123878	−0.0619392	−	0.998080i	$-0.519728\pi$
$547$	−1584.81	−0.123878	−0.0619392	+	0.998080i	$0.519728\pi$
$548$	−6423.52	−0.500729
$549$	0	0
$550$	0	0
$551$	−4420.43	−0.341772
$552$	0	0
$553$	33856.2	2.60346
$554$	38.2092	0.00293024
$555$	0	0
$556$	−19171.6	−1.46233
$557$	−21202.0	−1.61285	−0.806424	−	0.591338i	$-0.798600\pi$
$557$	−21202.0	−1.61285	−0.806424	+	0.591338i	$0.798600\pi$
$558$	0	0
$559$	−90.4279	−0.00684202
$560$	0	0
$561$	0	0
$562$	0.699899	5.25329e−5 0
$563$	−11126.0	−0.832867	−0.416433	−	0.909166i	$-0.636720\pi$
$563$	−11126.0	−0.832867	−0.416433	+	0.909166i	$0.636720\pi$
$564$	0	0
$565$	0	0
$566$	879.581	0.0653207
$567$	0	0
$568$	1391.68	0.102805
$569$	−16907.3	−1.24568	−0.622841	−	0.782349i	$-0.714022\pi$
$569$	−16907.3	−1.24568	−0.622841	+	0.782349i	$0.714022\pi$
$570$	0	0
$571$	10875.1	0.797037	0.398519	−	0.917160i	$-0.369524\pi$
$571$	10875.1	0.797037	0.398519	+	0.917160i	$0.369524\pi$
$572$	22.2868	0.00162912
$573$	0	0
$574$	−3082.83	−0.224172
$575$	0	0
$576$	0	0
$577$	−8978.69	−0.647812	−0.323906	−	0.946089i	$-0.604996\pi$
$577$	−8978.69	−0.647812	−0.323906	+	0.946089i	$0.604996\pi$
$578$	1852.28	0.133295
$579$	0	0
$580$	0	0
$581$	−36823.5	−2.62943
$582$	0	0
$583$	−3119.41	−0.221600
$584$	3125.20	0.221441
$585$	0	0
$586$	−1526.09	−0.107581
$587$	−820.314	−0.0576797	−0.0288398	−	0.999584i	$-0.509181\pi$
$587$	−820.314	−0.0576797	−0.0288398	+	0.999584i	$0.509181\pi$
$588$	0	0
$589$	−841.955	−0.0589001
$590$	0	0
$591$	0	0
$592$	9288.57	0.644861
$593$	10199.1	0.706281	0.353141	−	0.935570i	$-0.385114\pi$
$593$	10199.1	0.706281	0.353141	+	0.935570i	$0.385114\pi$
$594$	0	0
$595$	0	0
$596$	−5651.68	−0.388426
$597$	0	0
$598$	9.43489	0.000645186 0
$599$	9647.48	0.658072	0.329036	−	0.944317i	$-0.393276\pi$
$599$	9647.48	0.658072	0.329036	+	0.944317i	$0.393276\pi$
$600$	0	0
$601$	−11840.0	−0.803597	−0.401798	−	0.915728i	$-0.631615\pi$
$601$	−11840.0	−0.803597	−0.401798	+	0.915728i	$0.631615\pi$
$602$	2334.11	0.158025
$603$	0	0
$604$	1604.26	0.108074
$605$	0	0
$606$	0	0
$607$	−14317.8	−0.957402	−0.478701	−	0.877978i	$-0.658892\pi$
$607$	−14317.8	−0.957402	−0.478701	+	0.877978i	$0.658892\pi$
$608$	−686.664	−0.0458025
$609$	0	0
$610$	0	0
$611$	−67.8045	−0.00448949
$612$	0	0
$613$	22627.1	1.49087	0.745433	−	0.666581i	$-0.232243\pi$
$613$	22627.1	1.49087	0.745433	+	0.666581i	$0.232243\pi$
$614$	−1084.51	−0.0712825
$615$	0	0
$616$	−1153.65	−0.0754575
$617$	7831.84	0.511018	0.255509	−	0.966807i	$-0.417757\pi$
$617$	7831.84	0.511018	0.255509	+	0.966807i	$0.417757\pi$
$618$	0	0
$619$	578.127	0.0375394	0.0187697	−	0.999824i	$-0.494025\pi$
$619$	578.127	0.0375394	0.0187697	+	0.999824i	$0.494025\pi$
$620$	0	0
$621$	0	0
$622$	879.087	0.0566691
$623$	11789.2	0.758147
$624$	0	0
$625$	0	0
$626$	506.465	0.0323361
$627$	0	0
$628$	8578.28	0.545081
$629$	−17347.8	−1.09969
$630$	0	0
$631$	18946.0	1.19529	0.597645	−	0.801761i	$-0.296103\pi$
$631$	18946.0	1.19529	0.597645	+	0.801761i	$0.296103\pi$
$632$	3547.11	0.223254
$633$	0	0
$634$	−1907.68	−0.119501
$635$	0	0
$636$	0	0
$637$	167.554	0.0104219
$638$	−582.514	−0.0361473
$639$	0	0
$640$	0	0
$641$	28510.4	1.75677	0.878387	−	0.477949i	$-0.158620\pi$
$641$	28510.4	1.75677	0.878387	+	0.477949i	$0.158620\pi$
$642$	0	0
$643$	−16683.0	−1.02319	−0.511596	−	0.859226i	$-0.670945\pi$
$643$	−16683.0	−1.02319	−0.511596	+	0.859226i	$0.670945\pi$
$644$	44902.8	2.74755
$645$	0	0
$646$	423.620	0.0258005
$647$	−20533.1	−1.24767	−0.623834	−	0.781557i	$-0.714426\pi$
$647$	−20533.1	−1.24767	−0.623834	+	0.781557i	$0.714426\pi$
$648$	0	0
$649$	−5590.87	−0.338152
$650$	0	0
$651$	0	0
$652$	5358.02	0.321835
$653$	−19000.9	−1.13869	−0.569344	−	0.822099i	$-0.692803\pi$
$653$	−19000.9	−1.13869	−0.569344	+	0.822099i	$0.692803\pi$
$654$	0	0
$655$	0	0
$656$	29534.0	1.75779
$657$	0	0
$658$	1750.16	0.103690
$659$	21809.6	1.28920	0.644598	−	0.764522i	$-0.277025\pi$
$659$	21809.6	1.28920	0.644598	+	0.764522i	$0.277025\pi$
$660$	0	0
$661$	−20272.8	−1.19292	−0.596461	−	0.802642i	$-0.703427\pi$
$661$	−20272.8	−1.19292	−0.596461	+	0.802642i	$0.703427\pi$
$662$	−46.7036	−0.00274197
$663$	0	0
$664$	−3858.00	−0.225481
$665$	0	0
$666$	0	0
$667$	45468.8	2.63952
$668$	22408.5	1.29792
$669$	0	0
$670$	0	0
$671$	6659.59	0.383145
$672$	0	0
$673$	1757.51	0.100664	0.0503322	−	0.998733i	$-0.483972\pi$
$673$	1757.51	0.100664	0.0503322	+	0.998733i	$0.483972\pi$
$674$	−1417.92	−0.0810330
$675$	0	0
$676$	17480.7	0.994576
$677$	5107.84	0.289971	0.144985	−	0.989434i	$-0.453686\pi$
$677$	5107.84	0.289971	0.144985	+	0.989434i	$0.453686\pi$
$678$	0	0
$679$	16594.7	0.937918
$680$	0	0
$681$	0	0
$682$	−110.951	−0.00622952
$683$	−9379.62	−0.525478	−0.262739	−	0.964867i	$-0.584626\pi$
$683$	−9379.62	−0.525478	−0.262739	+	0.964867i	$0.584626\pi$
$684$	0	0
$685$	0	0
$686$	−2070.50	−0.115236
$687$	0	0
$688$	−22361.2	−1.23912
$689$	72.2095	0.00399269
$690$	0	0
$691$	−28560.3	−1.57233	−0.786167	−	0.618014i	$-0.787938\pi$
$691$	−28560.3	−1.57233	−0.786167	+	0.618014i	$0.787938\pi$
$692$	18027.9	0.990346
$693$	0	0
$694$	−1895.68	−0.103688
$695$	0	0
$696$	0	0
$697$	−55159.3	−2.99757
$698$	2021.89	0.109642
$699$	0	0
$700$	0	0
$701$	4454.76	0.240020	0.120010	−	0.992773i	$-0.461707\pi$
$701$	4454.76	0.240020	0.120010	+	0.992773i	$0.461707\pi$
$702$	0	0
$703$	−2558.02	−0.137237
$704$	5450.54	0.291797
$705$	0	0
$706$	−882.325	−0.0470351
$707$	−10788.8	−0.573910
$708$	0	0
$709$	4691.66	0.248518	0.124259	−	0.992250i	$-0.460345\pi$
$709$	4691.66	0.248518	0.124259	+	0.992250i	$0.460345\pi$
$710$	0	0
$711$	0	0
$712$	1235.16	0.0650133
$713$	8660.39	0.454887
$714$	0	0
$715$	0	0
$716$	−4001.03	−0.208834
$717$	0	0
$718$	−854.536	−0.0444164
$719$	−13711.6	−0.711205	−0.355602	−	0.934637i	$-0.615724\pi$
$719$	−13711.6	−0.711205	−0.355602	+	0.934637i	$0.615724\pi$
$720$	0	0
$721$	−56551.3	−2.92106
$722$	−1362.40	−0.0702259
$723$	0	0
$724$	−31556.9	−1.61990
$725$	0	0
$726$	0	0
$727$	−7468.37	−0.380999	−0.190500	−	0.981687i	$-0.561011\pi$
$727$	−7468.37	−0.380999	−0.190500	+	0.981687i	$0.561011\pi$
$728$	26.7052	0.00135956
$729$	0	0
$730$	0	0
$731$	41762.9	2.11307
$732$	0	0
$733$	8068.26	0.406559	0.203280	−	0.979121i	$-0.434840\pi$
$733$	8068.26	0.406559	0.203280	+	0.979121i	$0.434840\pi$
$734$	459.923	0.0231281
$735$	0	0
$736$	7063.06	0.353733
$737$	−3870.65	−0.193456
$738$	0	0
$739$	−28950.1	−1.44107	−0.720533	−	0.693421i	$-0.756103\pi$
$739$	−28950.1	−1.44107	−0.720533	+	0.693421i	$0.756103\pi$
$740$	0	0
$741$	0	0
$742$	−1863.86	−0.0922162
$743$	36353.4	1.79499	0.897495	−	0.441025i	$-0.145385\pi$
$743$	36353.4	1.79499	0.897495	+	0.441025i	$0.145385\pi$
$744$	0	0
$745$	0	0
$746$	272.207	0.0133595
$747$	0	0
$748$	−10292.9	−0.503135
$749$	−29618.9	−1.44493
$750$	0	0
$751$	29066.4	1.41232	0.706158	−	0.708055i	$-0.250427\pi$
$751$	29066.4	1.41232	0.706158	+	0.708055i	$0.250427\pi$
$752$	−16766.8	−0.813063
$753$	0	0
$754$	13.4843	0.000651286 0
$755$	0	0
$756$	0	0
$757$	17429.9	0.836858	0.418429	−	0.908249i	$-0.362581\pi$
$757$	17429.9	0.836858	0.418429	+	0.908249i	$0.362581\pi$
$758$	−1553.68	−0.0744486
$759$	0	0
$760$	0	0
$761$	−4266.56	−0.203236	−0.101618	−	0.994823i	$-0.532402\pi$
$761$	−4266.56	−0.203236	−0.101618	+	0.994823i	$0.532402\pi$
$762$	0	0
$763$	−36976.6	−1.75445
$764$	−13069.4	−0.618891
$765$	0	0
$766$	−380.004	−0.0179244
$767$	129.420	0.00609268
$768$	0	0
$769$	13815.6	0.647860	0.323930	−	0.946081i	$-0.394996\pi$
$769$	13815.6	0.647860	0.323930	+	0.946081i	$0.394996\pi$
$770$	0	0
$771$	0	0
$772$	21154.6	0.986230
$773$	−5401.85	−0.251347	−0.125673	−	0.992072i	$-0.540109\pi$
$773$	−5401.85	−0.251347	−0.125673	+	0.992072i	$0.540109\pi$
$774$	0	0
$775$	0	0
$776$	1738.63	0.0804291
$777$	0	0
$778$	2790.66	0.128599
$779$	−8133.50	−0.374086
$780$	0	0
$781$	4618.20	0.211590
$782$	−4357.38	−0.199258
$783$	0	0
$784$	41433.2	1.88744
$785$	0	0
$786$	0	0
$787$	9169.94	0.415341	0.207670	−	0.978199i	$-0.433412\pi$
$787$	9169.94	0.415341	0.207670	+	0.978199i	$0.433412\pi$
$788$	26864.6	1.21448
$789$	0	0
$790$	0	0
$791$	58241.5	2.61799
$792$	0	0
$793$	−154.159	−0.00690334
$794$	855.965	0.0382583
$795$	0	0
$796$	20182.9	0.898699
$797$	−23741.7	−1.05518	−0.527588	−	0.849500i	$-0.676904\pi$
$797$	−23741.7	−1.05518	−0.527588	+	0.849500i	$0.676904\pi$
$798$	0	0
$799$	31314.6	1.38652
$800$	0	0
$801$	0	0
$802$	−1804.41	−0.0794461
$803$	10370.8	0.455762
$804$	0	0
$805$	0	0
$806$	2.56834	0.000112241 0
$807$	0	0
$808$	−1130.34	−0.0492144
$809$	−41400.4	−1.79921	−0.899605	−	0.436704i	$-0.856146\pi$
$809$	−41400.4	−1.79921	−0.899605	+	0.436704i	$0.856146\pi$
$810$	0	0
$811$	19019.0	0.823487	0.411743	−	0.911300i	$-0.364920\pi$
$811$	19019.0	0.823487	0.411743	+	0.911300i	$0.364920\pi$
$812$	64174.9	2.77352
$813$	0	0
$814$	−337.090	−0.0145147
$815$	0	0
$816$	0	0
$817$	6158.14	0.263704
$818$	−1502.10	−0.0642051
$819$	0	0
$820$	0	0
$821$	12230.5	0.519912	0.259956	−	0.965621i	$-0.416292\pi$
$821$	12230.5	0.519912	0.259956	+	0.965621i	$0.416292\pi$
$822$	0	0
$823$	27169.7	1.15076	0.575381	−	0.817886i	$-0.304854\pi$
$823$	27169.7	1.15076	0.575381	+	0.817886i	$0.304854\pi$
$824$	−5924.88	−0.250489
$825$	0	0
$826$	−3340.57	−0.140718
$827$	8407.80	0.353528	0.176764	−	0.984253i	$-0.443437\pi$
$827$	8407.80	0.353528	0.176764	+	0.984253i	$0.443437\pi$
$828$	0	0
$829$	−21518.6	−0.901536	−0.450768	−	0.892641i	$-0.648850\pi$
$829$	−21518.6	−0.901536	−0.450768	+	0.892641i	$0.648850\pi$
$830$	0	0
$831$	0	0
$832$	−126.171	−0.00525746
$833$	−77382.7	−3.21867
$834$	0	0
$835$	0	0
$836$	−1517.73	−0.0627894
$837$	0	0
$838$	1840.77	0.0758811
$839$	−16858.8	−0.693719	−0.346859	−	0.937917i	$-0.612752\pi$
$839$	−16858.8	−0.693719	−0.346859	+	0.937917i	$0.612752\pi$
$840$	0	0
$841$	40594.7	1.66447
$842$	3221.79	0.131865
$843$	0	0
$844$	22876.7	0.932995
$845$	0	0
$846$	0	0
$847$	−3828.31	−0.155304
$848$	17856.1	0.723091
$849$	0	0
$850$	0	0
$851$	26311.9	1.05988
$852$	0	0
$853$	1485.47	0.0596268	0.0298134	−	0.999555i	$-0.490509\pi$
$853$	1485.47	0.0596268	0.0298134	+	0.999555i	$0.490509\pi$
$854$	3979.13	0.159441
$855$	0	0
$856$	−3103.17	−0.123907
$857$	−33557.0	−1.33755	−0.668777	−	0.743463i	$-0.733182\pi$
$857$	−33557.0	−1.33755	−0.668777	+	0.743463i	$0.733182\pi$
$858$	0	0
$859$	−22792.7	−0.905330	−0.452665	−	0.891681i	$-0.649527\pi$
$859$	−22792.7	−0.905330	−0.452665	+	0.891681i	$0.649527\pi$
$860$	0	0
$861$	0	0
$862$	−463.232	−0.0183036
$863$	−7401.25	−0.291937	−0.145968	−	0.989289i	$-0.546630\pi$
$863$	−7401.25	−0.291937	−0.145968	+	0.989289i	$0.546630\pi$
$864$	0	0
$865$	0	0
$866$	1853.50	0.0727302
$867$	0	0
$868$	12223.3	0.477980
$869$	11770.9	0.459493
$870$	0	0
$871$	89.5995	0.00348561
$872$	−3874.03	−0.150449
$873$	0	0
$874$	−642.517	−0.0248666
$875$	0	0
$876$	0	0
$877$	22513.9	0.866864	0.433432	−	0.901186i	$-0.357302\pi$
$877$	22513.9	0.866864	0.433432	+	0.901186i	$0.357302\pi$
$878$	−412.980	−0.0158740
$879$	0	0
$880$	0	0
$881$	−19905.8	−0.761229	−0.380615	−	0.924734i	$-0.624288\pi$
$881$	−19905.8	−0.761229	−0.380615	+	0.924734i	$0.624288\pi$
$882$	0	0
$883$	34812.3	1.32676	0.663379	−	0.748283i	$-0.269122\pi$
$883$	34812.3	1.32676	0.663379	+	0.748283i	$0.269122\pi$
$884$	238.264	0.00906526
$885$	0	0
$886$	318.594	0.0120805
$887$	20010.7	0.757490	0.378745	−	0.925501i	$-0.376356\pi$
$887$	20010.7	0.757490	0.378745	+	0.925501i	$0.376356\pi$
$888$	0	0
$889$	−51115.5	−1.92841
$890$	0	0
$891$	0	0
$892$	−31294.3	−1.17468
$893$	4617.49	0.173033
$894$	0	0
$895$	0	0
$896$	13279.6	0.495135
$897$	0	0
$898$	−2596.01	−0.0964697
$899$	12377.4	0.459187
$900$	0	0
$901$	−33349.0	−1.23309
$902$	−1071.81	−0.0395649
$903$	0	0
$904$	6101.96	0.224500
$905$	0	0
$906$	0	0
$907$	−27826.7	−1.01871	−0.509356	−	0.860556i	$-0.670116\pi$
$907$	−27826.7	−1.01871	−0.509356	+	0.860556i	$0.670116\pi$
$908$	−22440.8	−0.820180
$909$	0	0
$910$	0	0
$911$	17716.2	0.644306	0.322153	−	0.946688i	$-0.395593\pi$
$911$	17716.2	0.644306	0.322153	+	0.946688i	$0.395593\pi$
$912$	0	0
$913$	−12802.5	−0.464077
$914$	2833.67	0.102549
$915$	0	0
$916$	10283.9	0.370951
$917$	75164.6	2.70682
$918$	0	0
$919$	21228.9	0.762000	0.381000	−	0.924575i	$-0.375580\pi$
$919$	21228.9	0.762000	0.381000	+	0.924575i	$0.375580\pi$
$920$	0	0
$921$	0	0
$922$	−965.382	−0.0344828
$923$	−106.904	−0.00381234
$924$	0	0
$925$	0	0
$926$	−1015.20	−0.0360278
$927$	0	0
$928$	10094.5	0.357077
$929$	−3940.84	−0.139176	−0.0695882	−	0.997576i	$-0.522169\pi$
$929$	−3940.84	−0.139176	−0.0695882	+	0.997576i	$0.522169\pi$
$930$	0	0
$931$	−11410.5	−0.401678
$932$	−2531.62	−0.0889765
$933$	0	0
$934$	−2227.01	−0.0780193
$935$	0	0
$936$	0	0
$937$	−13736.2	−0.478912	−0.239456	−	0.970907i	$-0.576969\pi$
$937$	−13736.2	−0.478912	−0.239456	+	0.970907i	$0.576969\pi$
$938$	−2312.73	−0.0805045
$939$	0	0
$940$	0	0
$941$	−10339.6	−0.358196	−0.179098	−	0.983831i	$-0.557318\pi$
$941$	−10339.6	−0.358196	−0.179098	+	0.983831i	$0.557318\pi$
$942$	0	0
$943$	83661.6	2.88907
$944$	32003.2	1.10341
$945$	0	0
$946$	811.506	0.0278904
$947$	−3000.03	−0.102944	−0.0514720	−	0.998674i	$-0.516391\pi$
$947$	−3000.03	−0.102944	−0.0514720	+	0.998674i	$0.516391\pi$
$948$	0	0
$949$	−240.068	−0.00821173
$950$	0	0
$951$	0	0
$952$	−12333.4	−0.419883
$953$	−19151.8	−0.650983	−0.325492	−	0.945545i	$-0.605530\pi$
$953$	−19151.8	−0.650983	−0.325492	+	0.945545i	$0.605530\pi$
$954$	0	0
$955$	0	0
$956$	−47063.2	−1.59219
$957$	0	0
$958$	−1341.20	−0.0452319
$959$	−25542.0	−0.860055
$960$	0	0
$961$	−27433.5	−0.920865
$962$	7.80311	0.000261520 0
$963$	0	0
$964$	7219.93	0.241222
$965$	0	0
$966$	0	0
$967$	−433.840	−0.0144274	−0.00721372	−	0.999974i	$-0.502296\pi$
$967$	−433.840	−0.0144274	−0.00721372	+	0.999974i	$0.502296\pi$
$968$	−401.092	−0.0133178
$969$	0	0
$970$	0	0
$971$	42679.5	1.41056	0.705279	−	0.708930i	$-0.250822\pi$
$971$	42679.5	1.41056	0.705279	+	0.708930i	$0.250822\pi$
$972$	0	0
$973$	−76232.3	−2.51171
$974$	−4042.19	−0.132978
$975$	0	0
$976$	−38120.8	−1.25022
$977$	28979.7	0.948969	0.474484	−	0.880264i	$-0.342635\pi$
$977$	28979.7	0.948969	0.474484	+	0.880264i	$0.342635\pi$
$978$	0	0
$979$	4098.79	0.133808
$980$	0	0
$981$	0	0
$982$	−1223.98	−0.0397749
$983$	−37541.0	−1.21808	−0.609040	−	0.793139i	$-0.708445\pi$
$983$	−37541.0	−1.21808	−0.609040	+	0.793139i	$0.708445\pi$
$984$	0	0
$985$	0	0
$986$	−6227.55	−0.201141
$987$	0	0
$988$	35.1332	0.00113131
$989$	−63342.9	−2.03659
$990$	0	0
$991$	17890.3	0.573466	0.286733	−	0.958011i	$-0.407431\pi$
$991$	17890.3	0.573466	0.286733	+	0.958011i	$0.407431\pi$
$992$	1922.69	0.0615377
$993$	0	0
$994$	2759.39	0.0880509
$995$	0	0
$996$	0	0
$997$	13532.7	0.429875	0.214938	−	0.976628i	$-0.431045\pi$
$997$	13532.7	0.429875	0.214938	+	0.976628i	$0.431045\pi$
$998$	3230.94	0.102479
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.bv.1.6	yes	10
3.2	odd	2		2475.4.a.by.1.5	yes	10
5.4	even	2		2475.4.a.bx.1.5	yes	10
15.14	odd	2		2475.4.a.bu.1.6	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2475.4.a.bu.1.6	✓	10	15.14	odd	2
2475.4.a.bv.1.6	yes	10	1.1	even	1	trivial
2475.4.a.bx.1.5	yes	10	5.4	even	2
2475.4.a.by.1.5	yes	10	3.2	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)

\(f(q)\)	\(=\)	\(q+0.207736 q^{2} -7.95685 q^{4} -31.6389 q^{7} -3.31481 q^{8} +O(q^{10})\)	\(q+0.207736 q^{2} -7.95685 q^{4} -31.6389 q^{7} -3.31481 q^{8} -11.0000 q^{11} +0.254633 q^{13} -6.57254 q^{14} +62.9662 q^{16} -117.599 q^{17} -17.3405 q^{19} -2.28509 q^{22} +178.365 q^{23} +0.0528964 q^{26} +251.746 q^{28} +254.919 q^{29} +48.5542 q^{31} +39.5988 q^{32} -24.4295 q^{34} +147.517 q^{37} -3.60225 q^{38} +469.046 q^{41} -355.130 q^{43} +87.5253 q^{44} +37.0529 q^{46} -266.283 q^{47} +658.023 q^{49} -2.02608 q^{52} +283.583 q^{53} +104.877 q^{56} +52.9558 q^{58} +508.261 q^{59} -605.417 q^{61} +10.0864 q^{62} -495.503 q^{64} +351.877 q^{67} +935.716 q^{68} -419.836 q^{71} -942.799 q^{73} +30.6446 q^{74} +137.976 q^{76} +348.028 q^{77} -1070.08 q^{79} +97.4377 q^{82} +1163.87 q^{83} -73.7733 q^{86} +36.4629 q^{88} -372.618 q^{89} -8.05632 q^{91} -1419.23 q^{92} -55.3166 q^{94} -524.502 q^{97} +136.695 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 4 q^{2} + 34 q^{4} + 2 q^{7} - 48 q^{8}+O(q^{10}) \) 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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