LMFDB - Embedded newform 8649.2.a.bu.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8649,2,Mod(1,8649)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8649, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8649.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8649 = 3^{2} \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8649.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:	$69.0626127082$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 2883)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	8649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73783 q^{2} +1.02004 q^{4} -3.85240 q^{5} -2.44804 q^{7} +1.70300 q^{8} +6.69480 q^{10} +1.08295 q^{11} +4.06671 q^{13} +4.25427 q^{14} -4.99960 q^{16} -7.93311 q^{17} -1.66668 q^{19} -3.92959 q^{20} -1.88198 q^{22} +1.52487 q^{23} +9.84101 q^{25} -7.06724 q^{26} -2.49709 q^{28} -7.97442 q^{29} +5.28242 q^{32} +13.7864 q^{34} +9.43084 q^{35} -2.39623 q^{37} +2.89639 q^{38} -6.56066 q^{40} +3.81137 q^{41} +7.55291 q^{43} +1.10465 q^{44} -2.64995 q^{46} +5.06376 q^{47} -1.00709 q^{49} -17.1020 q^{50} +4.14820 q^{52} +3.29130 q^{53} -4.17197 q^{55} -4.16903 q^{56} +13.8581 q^{58} -14.2187 q^{59} -3.02518 q^{61} +0.819272 q^{64} -15.6666 q^{65} -9.93407 q^{67} -8.09206 q^{68} -16.3892 q^{70} -12.6838 q^{71} +5.17551 q^{73} +4.16423 q^{74} -1.70007 q^{76} -2.65111 q^{77} -1.53803 q^{79} +19.2605 q^{80} -6.62349 q^{82} +6.46074 q^{83} +30.5615 q^{85} -13.1256 q^{86} +1.84427 q^{88} +7.44199 q^{89} -9.95548 q^{91} +1.55542 q^{92} -8.79992 q^{94} +6.42071 q^{95} -7.34932 q^{97} +1.75015 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8} - 8 q^{10} - 16 q^{11} + 32 q^{13} + 24 q^{14} + 48 q^{16} - 32 q^{17} + 32 q^{19} + 24 q^{20} + 32 q^{22} - 32 q^{23} + 40 q^{25} - 16 q^{26} + 8 q^{28} - 48 q^{29}+ \cdots + 24 q^{98}+O(q^{100}) $ 24 * q + 32 * q^4 + 16 * q^7 + 24 * q^8 - 8 * q^10 - 16 * q^11 + 32 * q^13 + 24 * q^14 + 48 * q^16 - 32 * q^17 + 32 * q^19 + 24 * q^20 + 32 * q^22 - 32 * q^23 + 40 * q^25 - 16 * q^26 + 8 * q^28 - 48 * q^29 + 48 * q^32 + 48 * q^35 + 64 * q^37 + 24 * q^38 + 32 * q^43 - 48 * q^44 + 32 * q^46 + 48 * q^47 + 56 * q^49 + 24 * q^50 + 64 * q^52 - 80 * q^53 + 48 * q^56 + 32 * q^58 + 32 * q^61 + 56 * q^64 - 16 * q^65 - 16 * q^67 - 80 * q^68 + 8 * q^70 + 32 * q^73 + 56 * q^76 - 96 * q^77 + 32 * q^79 + 72 * q^80 + 8 * q^82 - 48 * q^83 + 96 * q^85 + 32 * q^86 + 96 * q^88 + 16 * q^89 - 32 * q^92 + 48 * q^94 + 48 * q^95 + 16 * q^97 + 24 * q^98

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 $

$2$	−1.73783	−1.22883	−0.614414	−	0.788984i	$-0.710608\pi$
$2$	−1.73783	−1.22883	−0.614414	+	0.788984i	$0.710608\pi$
$3$	0	0
$3$	0	0
$4$	1.02004	0.510019
$5$	−3.85240	−1.72285	−0.861423	−	0.507887i	$-0.830427\pi$
$5$	−3.85240	−1.72285	−0.861423	+	0.507887i	$0.830427\pi$
$6$	0	0
$7$	−2.44804	−0.925273	−0.462636	−	0.886548i	$-0.653096\pi$
$7$	−2.44804	−0.925273	−0.462636	+	0.886548i	$0.653096\pi$
$8$	1.70300	0.602103
$9$	0	0
$10$	6.69480	2.11708
$11$	1.08295	0.326522	0.163261	−	0.986583i	$-0.447799\pi$
$11$	1.08295	0.326522	0.163261	+	0.986583i	$0.447799\pi$
$12$	0	0
$13$	4.06671	1.12790	0.563952	−	0.825808i	$-0.309280\pi$
$13$	4.06671	1.12790	0.563952	+	0.825808i	$0.309280\pi$
$14$	4.25427	1.13700
$15$	0	0
$16$	−4.99960	−1.24990
$17$	−7.93311	−1.92406	−0.962030	−	0.272943i	$-0.912003\pi$
$17$	−7.93311	−1.92406	−0.962030	+	0.272943i	$0.912003\pi$
$18$	0	0
$19$	−1.66668	−0.382362	−0.191181	−	0.981555i	$-0.561232\pi$
$19$	−1.66668	−0.382362	−0.191181	+	0.981555i	$0.561232\pi$
$20$	−3.92959	−0.878684
$21$	0	0
$22$	−1.88198	−0.401240
$23$	1.52487	0.317957	0.158978	−	0.987282i	$-0.449180\pi$
$23$	1.52487	0.317957	0.158978	+	0.987282i	$0.449180\pi$
$24$	0	0
$25$	9.84101	1.96820
$26$	−7.06724	−1.38600
$27$	0	0
$28$	−2.49709	−0.471906
$29$	−7.97442	−1.48081	−0.740406	−	0.672160i	$-0.765367\pi$
$29$	−7.97442	−1.48081	−0.740406	+	0.672160i	$0.765367\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	5.28242	0.933809
$33$	0	0
$34$	13.7864	2.36434
$35$	9.43084	1.59410
$36$	0	0
$37$	−2.39623	−0.393938	−0.196969	−	0.980410i	$-0.563110\pi$
$37$	−2.39623	−0.393938	−0.196969	+	0.980410i	$0.563110\pi$
$38$	2.89639	0.469857
$39$	0	0
$40$	−6.56066	−1.03733
$41$	3.81137	0.595235	0.297618	−	0.954685i	$-0.403808\pi$
$41$	3.81137	0.595235	0.297618	+	0.954685i	$0.403808\pi$
$42$	0	0
$43$	7.55291	1.15181	0.575904	−	0.817517i	$-0.304650\pi$
$43$	7.55291	1.15181	0.575904	+	0.817517i	$0.304650\pi$
$44$	1.10465	0.166533
$45$	0	0
$46$	−2.64995	−0.390714
$47$	5.06376	0.738625	0.369312	−	0.929305i	$-0.379593\pi$
$47$	5.06376	0.738625	0.369312	+	0.929305i	$0.379593\pi$
$48$	0	0
$49$	−1.00709	−0.143870
$50$	−17.1020	−2.41858
$51$	0	0
$52$	4.14820	0.575252
$53$	3.29130	0.452095	0.226047	−	0.974116i	$-0.427420\pi$
$53$	3.29130	0.452095	0.226047	+	0.974116i	$0.427420\pi$
$54$	0	0
$55$	−4.17197	−0.562548
$56$	−4.16903	−0.557110
$57$	0	0
$58$	13.8581	1.81966
$59$	−14.2187	−1.85112	−0.925561	−	0.378599i	$-0.876406\pi$
$59$	−14.2187	−1.85112	−0.925561	+	0.378599i	$0.876406\pi$
$60$	0	0
$61$	−3.02518	−0.387334	−0.193667	−	0.981067i	$-0.562038\pi$
$61$	−3.02518	−0.387334	−0.193667	+	0.981067i	$0.562038\pi$
$62$	0	0
$63$	0	0
$64$	0.819272	0.102409
$65$	−15.6666	−1.94320
$66$	0	0
$67$	−9.93407	−1.21364	−0.606820	−	0.794839i	$-0.707555\pi$
$67$	−9.93407	−1.21364	−0.606820	+	0.794839i	$0.707555\pi$
$68$	−8.09206	−0.981307
$69$	0	0
$70$	−16.3892	−1.95888
$71$	−12.6838	−1.50530	−0.752648	−	0.658423i	$-0.771224\pi$
$71$	−12.6838	−1.50530	−0.752648	+	0.658423i	$0.771224\pi$
$72$	0	0
$73$	5.17551	0.605747	0.302874	−	0.953031i	$-0.402054\pi$
$73$	5.17551	0.605747	0.302874	+	0.953031i	$0.402054\pi$
$74$	4.16423	0.484082
$75$	0	0
$76$	−1.70007	−0.195011
$77$	−2.65111	−0.302122
$78$	0	0
$79$	−1.53803	−0.173042	−0.0865210	−	0.996250i	$-0.527575\pi$
$79$	−1.53803	−0.173042	−0.0865210	+	0.996250i	$0.527575\pi$
$80$	19.2605	2.15339
$81$	0	0
$82$	−6.62349	−0.731442
$83$	6.46074	0.709158	0.354579	−	0.935026i	$-0.384624\pi$
$83$	6.46074	0.709158	0.354579	+	0.935026i	$0.384624\pi$
$84$	0	0
$85$	30.5615	3.31486
$86$	−13.1256	−1.41537
$87$	0	0
$88$	1.84427	0.196600
$89$	7.44199	0.788850	0.394425	−	0.918928i	$-0.370944\pi$
$89$	7.44199	0.788850	0.394425	+	0.918928i	$0.370944\pi$
$90$	0	0
$91$	−9.95548	−1.04362
$92$	1.55542	0.162164
$93$	0	0
$94$	−8.79992	−0.907643
$95$	6.42071	0.658750
$96$	0	0
$97$	−7.34932	−0.746210	−0.373105	−	0.927789i	$-0.621707\pi$
$97$	−7.34932	−0.746210	−0.373105	+	0.927789i	$0.621707\pi$
$98$	1.75015	0.176792
$99$	0	0
$100$	10.0382	1.00382
$101$	−2.12363	−0.211310	−0.105655	−	0.994403i	$-0.533694\pi$
$101$	−2.12363	−0.211310	−0.105655	+	0.994403i	$0.533694\pi$
$102$	0	0
$103$	6.27988	0.618775	0.309388	−	0.950936i	$-0.399876\pi$
$103$	6.27988	0.618775	0.309388	+	0.950936i	$0.399876\pi$
$104$	6.92563	0.679114
$105$	0	0
$106$	−5.71971	−0.555547
$107$	11.8553	1.14610	0.573048	−	0.819522i	$-0.305761\pi$
$107$	11.8553	1.14610	0.573048	+	0.819522i	$0.305761\pi$
$108$	0	0
$109$	−18.7983	−1.80055	−0.900275	−	0.435322i	$-0.856635\pi$
$109$	−18.7983	−1.80055	−0.900275	+	0.435322i	$0.856635\pi$
$110$	7.25015	0.691275
$111$	0	0
$112$	12.2392	1.15650
$113$	−6.24648	−0.587620	−0.293810	−	0.955864i	$-0.594923\pi$
$113$	−6.24648	−0.587620	−0.293810	+	0.955864i	$0.594923\pi$
$114$	0	0
$115$	−5.87440	−0.547791
$116$	−8.13420	−0.755242
$117$	0	0
$118$	24.7097	2.27471
$119$	19.4206	1.78028
$120$	0	0
$121$	−9.82721	−0.893383
$122$	5.25723	0.475967
$123$	0	0
$124$	0	0
$125$	−18.6495	−1.66806
$126$	0	0
$127$	−5.25371	−0.466191	−0.233096	−	0.972454i	$-0.574886\pi$
$127$	−5.25371	−0.466191	−0.233096	+	0.972454i	$0.574886\pi$
$128$	−11.9886	−1.05965
$129$	0	0
$130$	27.2258	2.38786
$131$	1.24115	0.108440	0.0542200	−	0.998529i	$-0.482733\pi$
$131$	1.24115	0.108440	0.0542200	+	0.998529i	$0.482733\pi$
$132$	0	0
$133$	4.08009	0.353789
$134$	17.2637	1.49136
$135$	0	0
$136$	−13.5101	−1.15848
$137$	−6.76086	−0.577619	−0.288810	−	0.957387i	$-0.593259\pi$
$137$	−6.76086	−0.577619	−0.288810	+	0.957387i	$0.593259\pi$
$138$	0	0
$139$	−15.1040	−1.28110	−0.640552	−	0.767915i	$-0.721294\pi$
$139$	−15.1040	−1.28110	−0.640552	+	0.767915i	$0.721294\pi$
$140$	9.61981	0.813022
$141$	0	0
$142$	22.0423	1.84975
$143$	4.40406	0.368286
$144$	0	0
$145$	30.7207	2.55121
$146$	−8.99413	−0.744359
$147$	0	0
$148$	−2.44424	−0.200915
$149$	0.601270	0.0492580	0.0246290	−	0.999697i	$-0.492160\pi$
$149$	0.601270	0.0492580	0.0246290	+	0.999697i	$0.492160\pi$
$150$	0	0
$151$	−18.7150	−1.52300	−0.761502	−	0.648163i	$-0.775537\pi$
$151$	−18.7150	−1.52300	−0.761502	+	0.648163i	$0.775537\pi$
$152$	−2.83836	−0.230221
$153$	0	0
$154$	4.60717	0.371256
$155$	0	0
$156$	0	0
$157$	−1.66563	−0.132932	−0.0664658	−	0.997789i	$-0.521172\pi$
$157$	−1.66563	−0.132932	−0.0664658	+	0.997789i	$0.521172\pi$
$158$	2.67283	0.212639
$159$	0	0
$160$	−20.3500	−1.60881
$161$	−3.73294	−0.294197
$162$	0	0
$163$	−8.38889	−0.657068	−0.328534	−	0.944492i	$-0.606555\pi$
$163$	−8.38889	−0.657068	−0.328534	+	0.944492i	$0.606555\pi$
$164$	3.88774	0.303581
$165$	0	0
$166$	−11.2276	−0.871433
$167$	−17.2358	−1.33375	−0.666873	−	0.745171i	$-0.732368\pi$
$167$	−17.2358	−1.33375	−0.666873	+	0.745171i	$0.732368\pi$
$168$	0	0
$169$	3.53815	0.272165
$170$	−53.1106	−4.07340
$171$	0	0
$172$	7.70425	0.587444
$173$	−2.50350	−0.190337	−0.0951686	−	0.995461i	$-0.530339\pi$
$173$	−2.50350	−0.190337	−0.0951686	+	0.995461i	$0.530339\pi$
$174$	0	0
$175$	−24.0912	−1.82112
$176$	−5.41433	−0.408120
$177$	0	0
$178$	−12.9329	−0.969361
$179$	−6.22436	−0.465230	−0.232615	−	0.972569i	$-0.574728\pi$
$179$	−6.22436	−0.465230	−0.232615	+	0.972569i	$0.574728\pi$
$180$	0	0
$181$	13.1798	0.979649	0.489825	−	0.871821i	$-0.337061\pi$
$181$	13.1798	0.979649	0.489825	+	0.871821i	$0.337061\pi$
$182$	17.3009	1.28243
$183$	0	0
$184$	2.59686	0.191443
$185$	9.23124	0.678694
$186$	0	0
$187$	−8.59118	−0.628249
$188$	5.16522	0.376712
$189$	0	0
$190$	−11.1581	−0.809491
$191$	18.6607	1.35024	0.675119	−	0.737708i	$-0.264092\pi$
$191$	18.6607	1.35024	0.675119	+	0.737708i	$0.264092\pi$
$192$	0	0
$193$	−25.9315	−1.86659	−0.933294	−	0.359113i	$-0.883079\pi$
$193$	−25.9315	−1.86659	−0.933294	+	0.359113i	$0.883079\pi$
$194$	12.7718	0.916964
$195$	0	0
$196$	−1.02727	−0.0733764
$197$	3.41890	0.243586	0.121793	−	0.992556i	$-0.461136\pi$
$197$	3.41890	0.243586	0.121793	+	0.992556i	$0.461136\pi$
$198$	0	0
$199$	−10.4097	−0.737926	−0.368963	−	0.929444i	$-0.620287\pi$
$199$	−10.4097	−0.737926	−0.368963	+	0.929444i	$0.620287\pi$
$200$	16.7593	1.18506
$201$	0	0
$202$	3.69051	0.259663
$203$	19.5217	1.37016
$204$	0	0
$205$	−14.6829	−1.02550
$206$	−10.9133	−0.760369
$207$	0	0
$208$	−20.3319	−1.40977
$209$	−1.80493	−0.124850
$210$	0	0
$211$	−14.7553	−1.01580	−0.507900	−	0.861416i	$-0.669578\pi$
$211$	−14.7553	−1.01580	−0.507900	+	0.861416i	$0.669578\pi$
$212$	3.35725	0.230577
$213$	0	0
$214$	−20.6025	−1.40836
$215$	−29.0969	−1.98439
$216$	0	0
$217$	0	0
$218$	32.6681	2.21257
$219$	0	0
$220$	−4.25556	−0.286910
$221$	−32.2617	−2.17015
$222$	0	0
$223$	−11.4443	−0.766366	−0.383183	−	0.923673i	$-0.625172\pi$
$223$	−11.4443	−0.766366	−0.383183	+	0.923673i	$0.625172\pi$
$224$	−12.9316	−0.864028
$225$	0	0
$226$	10.8553	0.722083
$227$	−23.4894	−1.55904	−0.779522	−	0.626375i	$-0.784538\pi$
$227$	−23.4894	−1.55904	−0.779522	+	0.626375i	$0.784538\pi$
$228$	0	0
$229$	11.4045	0.753630	0.376815	−	0.926289i	$-0.377019\pi$
$229$	11.4045	0.753630	0.376815	+	0.926289i	$0.377019\pi$
$230$	10.2087	0.673141
$231$	0	0
$232$	−13.5805	−0.891602
$233$	−11.6370	−0.762365	−0.381182	−	0.924500i	$-0.624483\pi$
$233$	−11.6370	−0.762365	−0.381182	+	0.924500i	$0.624483\pi$
$234$	0	0
$235$	−19.5076	−1.27254
$236$	−14.5036	−0.944106
$237$	0	0
$238$	−33.7496	−2.18766
$239$	22.6157	1.46289	0.731445	−	0.681901i	$-0.238846\pi$
$239$	22.6157	1.46289	0.731445	+	0.681901i	$0.238846\pi$
$240$	0	0
$241$	23.0085	1.48211	0.741055	−	0.671444i	$-0.234326\pi$
$241$	23.0085	1.48211	0.741055	+	0.671444i	$0.234326\pi$
$242$	17.0780	1.09781
$243$	0	0
$244$	−3.08579	−0.197548
$245$	3.87972	0.247866
$246$	0	0
$247$	−6.77789	−0.431267
$248$	0	0
$249$	0	0
$250$	32.4096	2.04976
$251$	−20.7953	−1.31259	−0.656293	−	0.754506i	$-0.727876\pi$
$251$	−20.7953	−1.31259	−0.656293	+	0.754506i	$0.727876\pi$
$252$	0	0
$253$	1.65136	0.103820
$254$	9.13003	0.572869
$255$	0	0
$256$	19.1955	1.19972
$257$	0.0529016	0.00329991	0.00164995	−	0.999999i	$-0.499475\pi$
$257$	0.0529016	0.00329991	0.00164995	+	0.999999i	$0.499475\pi$
$258$	0	0
$259$	5.86607	0.364500
$260$	−15.9805	−0.991070
$261$	0	0
$262$	−2.15691	−0.133254
$263$	0.558965	0.0344672	0.0172336	−	0.999851i	$-0.494514\pi$
$263$	0.558965	0.0344672	0.0172336	+	0.999851i	$0.494514\pi$
$264$	0	0
$265$	−12.6794	−0.778890
$266$	−7.09049	−0.434746
$267$	0	0
$268$	−10.1331	−0.618979
$269$	12.3773	0.754660	0.377330	−	0.926079i	$-0.376842\pi$
$269$	12.3773	0.754660	0.377330	+	0.926079i	$0.376842\pi$
$270$	0	0
$271$	−2.94489	−0.178890	−0.0894448	−	0.995992i	$-0.528509\pi$
$271$	−2.94489	−0.178890	−0.0894448	+	0.995992i	$0.528509\pi$
$272$	39.6623	2.40488
$273$	0	0
$274$	11.7492	0.709795
$275$	10.6573	0.642662
$276$	0	0
$277$	1.43354	0.0861334	0.0430667	−	0.999072i	$-0.486287\pi$
$277$	1.43354	0.0861334	0.0430667	+	0.999072i	$0.486287\pi$
$278$	26.2481	1.57426
$279$	0	0
$280$	16.0608	0.959814
$281$	8.73944	0.521351	0.260676	−	0.965426i	$-0.416055\pi$
$281$	8.73944	0.521351	0.260676	+	0.965426i	$0.416055\pi$
$282$	0	0
$283$	−25.9054	−1.53992	−0.769958	−	0.638094i	$-0.779723\pi$
$283$	−25.9054	−1.53992	−0.769958	+	0.638094i	$0.779723\pi$
$284$	−12.9380	−0.767729
$285$	0	0
$286$	−7.65348	−0.452560
$287$	−9.33039	−0.550755
$288$	0	0
$289$	45.9342	2.70201
$290$	−53.3872	−3.13500
$291$	0	0
$292$	5.27921	0.308942
$293$	−9.05112	−0.528772	−0.264386	−	0.964417i	$-0.585169\pi$
$293$	−9.05112	−0.528772	−0.264386	+	0.964417i	$0.585169\pi$
$294$	0	0
$295$	54.7763	3.18920
$296$	−4.08079	−0.237191
$297$	0	0
$298$	−1.04490	−0.0605296
$299$	6.20120	0.358625
$300$	0	0
$301$	−18.4899	−1.06574
$302$	32.5234	1.87151
$303$	0	0
$304$	8.33271	0.477914
$305$	11.6542	0.667317
$306$	0	0
$307$	8.76298	0.500130	0.250065	−	0.968229i	$-0.419548\pi$
$307$	8.76298	0.500130	0.250065	+	0.968229i	$0.419548\pi$
$308$	−2.70423	−0.154088
$309$	0	0
$310$	0	0
$311$	0.722973	0.0409960	0.0204980	−	0.999790i	$-0.493475\pi$
$311$	0.722973	0.0409960	0.0204980	+	0.999790i	$0.493475\pi$
$312$	0	0
$313$	−9.55991	−0.540358	−0.270179	−	0.962810i	$-0.587083\pi$
$313$	−9.55991	−0.540358	−0.270179	+	0.962810i	$0.587083\pi$
$314$	2.89457	0.163350
$315$	0	0
$316$	−1.56885	−0.0882547
$317$	4.14299	0.232694	0.116347	−	0.993209i	$-0.462882\pi$
$317$	4.14299	0.232694	0.116347	+	0.993209i	$0.462882\pi$
$318$	0	0
$319$	−8.63592	−0.483518
$320$	−3.15617	−0.176435
$321$	0	0
$322$	6.48720	0.361517
$323$	13.2219	0.735687
$324$	0	0
$325$	40.0206	2.21994
$326$	14.5784	0.807424
$327$	0	0
$328$	6.49077	0.358393
$329$	−12.3963	−0.683429
$330$	0	0
$331$	5.62019	0.308914	0.154457	−	0.988000i	$-0.450637\pi$
$331$	5.62019	0.308914	0.154457	+	0.988000i	$0.450637\pi$
$332$	6.59019	0.361684
$333$	0	0
$334$	29.9528	1.63894
$335$	38.2700	2.09092
$336$	0	0
$337$	−21.7851	−1.18671	−0.593355	−	0.804941i	$-0.702197\pi$
$337$	−21.7851	−1.18671	−0.593355	+	0.804941i	$0.702197\pi$
$338$	−6.14869	−0.334445
$339$	0	0
$340$	31.1739	1.69064
$341$	0	0
$342$	0	0
$343$	19.6017	1.05839
$344$	12.8626	0.693507
$345$	0	0
$346$	4.35064	0.233892
$347$	9.59930	0.515317	0.257659	−	0.966236i	$-0.417049\pi$
$347$	9.59930	0.515317	0.257659	+	0.966236i	$0.417049\pi$
$348$	0	0
$349$	15.9285	0.852632	0.426316	−	0.904574i	$-0.359811\pi$
$349$	15.9285	0.852632	0.426316	+	0.904574i	$0.359811\pi$
$350$	41.8663	2.23785
$351$	0	0
$352$	5.72061	0.304910
$353$	26.0182	1.38481	0.692404	−	0.721510i	$-0.256552\pi$
$353$	26.0182	1.38481	0.692404	+	0.721510i	$0.256552\pi$
$354$	0	0
$355$	48.8633	2.59339
$356$	7.59111	0.402328
$357$	0	0
$358$	10.8168	0.571688
$359$	7.27753	0.384094	0.192047	−	0.981386i	$-0.438487\pi$
$359$	7.27753	0.384094	0.192047	+	0.981386i	$0.438487\pi$
$360$	0	0
$361$	−16.2222	−0.853800
$362$	−22.9043	−1.20382
$363$	0	0
$364$	−10.1550	−0.532265
$365$	−19.9381	−1.04361
$366$	0	0
$367$	1.05257	0.0549435	0.0274717	−	0.999623i	$-0.491254\pi$
$367$	1.05257	0.0549435	0.0274717	+	0.999623i	$0.491254\pi$
$368$	−7.62373	−0.397414
$369$	0	0
$370$	−16.0423	−0.833998
$371$	−8.05724	−0.418311
$372$	0	0
$373$	−13.8184	−0.715488	−0.357744	−	0.933820i	$-0.616454\pi$
$373$	−13.8184	−0.715488	−0.357744	+	0.933820i	$0.616454\pi$
$374$	14.9300	0.772010
$375$	0	0
$376$	8.62360	0.444728
$377$	−32.4297	−1.67021
$378$	0	0
$379$	27.9179	1.43405	0.717024	−	0.697049i	$-0.245504\pi$
$379$	27.9179	1.43405	0.717024	+	0.697049i	$0.245504\pi$
$380$	6.54936	0.335975
$381$	0	0
$382$	−32.4290	−1.65921
$383$	−32.4293	−1.65706	−0.828531	−	0.559943i	$-0.810823\pi$
$383$	−32.4293	−1.65706	−0.828531	+	0.559943i	$0.810823\pi$
$384$	0	0
$385$	10.2132	0.520511
$386$	45.0644	2.29372
$387$	0	0
$388$	−7.49657	−0.380581
$389$	−20.3874	−1.03368	−0.516840	−	0.856082i	$-0.672892\pi$
$389$	−20.3874	−1.03368	−0.516840	+	0.856082i	$0.672892\pi$
$390$	0	0
$391$	−12.0969	−0.611768
$392$	−1.71508	−0.0866246
$393$	0	0
$394$	−5.94145	−0.299326
$395$	5.92512	0.298125
$396$	0	0
$397$	−6.97908	−0.350270	−0.175135	−	0.984544i	$-0.556036\pi$
$397$	−6.97908	−0.350270	−0.175135	+	0.984544i	$0.556036\pi$
$398$	18.0903	0.906784
$399$	0	0
$400$	−49.2011	−2.46005
$401$	−12.5287	−0.625652	−0.312826	−	0.949810i	$-0.601276\pi$
$401$	−12.5287	−0.625652	−0.312826	+	0.949810i	$0.601276\pi$
$402$	0	0
$403$	0	0
$404$	−2.16619	−0.107772
$405$	0	0
$406$	−33.9253	−1.68369
$407$	−2.59500	−0.128629
$408$	0	0
$409$	15.2337	0.753258	0.376629	−	0.926364i	$-0.377083\pi$
$409$	15.2337	0.753258	0.376629	+	0.926364i	$0.377083\pi$
$410$	25.5164	1.26016
$411$	0	0
$412$	6.40572	0.315587
$413$	34.8081	1.71279
$414$	0	0
$415$	−24.8894	−1.22177
$416$	21.4821	1.05325
$417$	0	0
$418$	3.13665	0.153419
$419$	−0.695307	−0.0339679	−0.0169840	−	0.999856i	$-0.505406\pi$
$419$	−0.695307	−0.0339679	−0.0169840	+	0.999856i	$0.505406\pi$
$420$	0	0
$421$	30.0501	1.46455	0.732276	−	0.681008i	$-0.238458\pi$
$421$	30.0501	1.46455	0.732276	+	0.681008i	$0.238458\pi$
$422$	25.6422	1.24824
$423$	0	0
$424$	5.60510	0.272208
$425$	−78.0698	−3.78694
$426$	0	0
$427$	7.40576	0.358390
$428$	12.0929	0.584530
$429$	0	0
$430$	50.5653	2.43847
$431$	16.6803	0.803462	0.401731	−	0.915758i	$-0.368409\pi$
$431$	16.6803	0.803462	0.401731	+	0.915758i	$0.368409\pi$
$432$	0	0
$433$	6.72872	0.323362	0.161681	−	0.986843i	$-0.448308\pi$
$433$	6.72872	0.323362	0.161681	+	0.986843i	$0.448308\pi$
$434$	0	0
$435$	0	0
$436$	−19.1750	−0.918314
$437$	−2.54146	−0.121574
$438$	0	0
$439$	34.5522	1.64909	0.824543	−	0.565799i	$-0.191432\pi$
$439$	34.5522	1.64909	0.824543	+	0.565799i	$0.191432\pi$
$440$	−7.10488	−0.338712
$441$	0	0
$442$	56.0651	2.66675
$443$	22.7410	1.08046	0.540228	−	0.841518i	$-0.318338\pi$
$443$	22.7410	1.08046	0.540228	+	0.841518i	$0.318338\pi$
$444$	0	0
$445$	−28.6696	−1.35907
$446$	19.8882	0.941732
$447$	0	0
$448$	−2.00561	−0.0947563
$449$	36.8153	1.73742	0.868710	−	0.495321i	$-0.164950\pi$
$449$	36.8153	1.73742	0.868710	+	0.495321i	$0.164950\pi$
$450$	0	0
$451$	4.12753	0.194358
$452$	−6.37164	−0.299697
$453$	0	0
$454$	40.8204	1.91580
$455$	38.3525	1.79799
$456$	0	0
$457$	22.0731	1.03253	0.516267	−	0.856427i	$-0.327321\pi$
$457$	22.0731	1.03253	0.516267	+	0.856427i	$0.327321\pi$
$458$	−19.8190	−0.926082
$459$	0	0
$460$	−5.99211	−0.279384
$461$	23.5140	1.09515	0.547577	−	0.836755i	$-0.315550\pi$
$461$	23.5140	1.09515	0.547577	+	0.836755i	$0.315550\pi$
$462$	0	0
$463$	1.34434	0.0624768	0.0312384	−	0.999512i	$-0.490055\pi$
$463$	1.34434	0.0624768	0.0312384	+	0.999512i	$0.490055\pi$
$464$	39.8689	1.85087
$465$	0	0
$466$	20.2231	0.936816
$467$	−18.9658	−0.877631	−0.438816	−	0.898577i	$-0.644602\pi$
$467$	−18.9658	−0.877631	−0.438816	+	0.898577i	$0.644602\pi$
$468$	0	0
$469$	24.3190	1.12295
$470$	33.9009	1.56373
$471$	0	0
$472$	−24.2146	−1.11457
$473$	8.17945	0.376091
$474$	0	0
$475$	−16.4018	−0.752565
$476$	19.8097	0.907976
$477$	0	0
$478$	−39.3022	−1.79764
$479$	−24.2562	−1.10829	−0.554146	−	0.832420i	$-0.686955\pi$
$479$	−24.2562	−1.10829	−0.554146	+	0.832420i	$0.686955\pi$
$480$	0	0
$481$	−9.74477	−0.444323
$482$	−39.9848	−1.82126
$483$	0	0
$484$	−10.0241	−0.455642
$485$	28.3125	1.28561
$486$	0	0
$487$	−2.76752	−0.125408	−0.0627041	−	0.998032i	$-0.519972\pi$
$487$	−2.76752	−0.125408	−0.0627041	+	0.998032i	$0.519972\pi$
$488$	−5.15189	−0.233215
$489$	0	0
$490$	−6.74228	−0.304585
$491$	38.3241	1.72954	0.864772	−	0.502165i	$-0.167463\pi$
$491$	38.3241	1.72954	0.864772	+	0.502165i	$0.167463\pi$
$492$	0	0
$493$	63.2619	2.84917
$494$	11.7788	0.529953
$495$	0	0
$496$	0	0
$497$	31.0506	1.39281
$498$	0	0
$499$	−27.2331	−1.21912	−0.609560	−	0.792740i	$-0.708654\pi$
$499$	−27.2331	−1.21912	−0.609560	+	0.792740i	$0.708654\pi$
$500$	−19.0232	−0.850743
$501$	0	0
$502$	36.1385	1.61294
$503$	−6.29581	−0.280716	−0.140358	−	0.990101i	$-0.544825\pi$
$503$	−6.29581	−0.280716	−0.140358	+	0.990101i	$0.544825\pi$
$504$	0	0
$505$	8.18110	0.364054
$506$	−2.86977	−0.127577
$507$	0	0
$508$	−5.35898	−0.237766
$509$	−22.8095	−1.01101	−0.505507	−	0.862822i	$-0.668695\pi$
$509$	−22.8095	−1.01101	−0.505507	+	0.862822i	$0.668695\pi$
$510$	0	0
$511$	−12.6699	−0.560481
$512$	−9.38131	−0.414599
$513$	0	0
$514$	−0.0919337	−0.00405502
$515$	−24.1926	−1.06606
$516$	0	0
$517$	5.48381	0.241178
$518$	−10.1942	−0.447908
$519$	0	0
$520$	−26.6803	−1.17001
$521$	2.58247	0.113140	0.0565700	−	0.998399i	$-0.481984\pi$
$521$	2.58247	0.113140	0.0565700	+	0.998399i	$0.481984\pi$
$522$	0	0
$523$	22.3874	0.978934	0.489467	−	0.872022i	$-0.337191\pi$
$523$	22.3874	0.978934	0.489467	+	0.872022i	$0.337191\pi$
$524$	1.26602	0.0553065
$525$	0	0
$526$	−0.971383	−0.0423543
$527$	0	0
$528$	0	0
$529$	−20.6748	−0.898903
$530$	22.0346	0.957122
$531$	0	0
$532$	4.16184	0.180439
$533$	15.4997	0.671368
$534$	0	0
$535$	−45.6714	−1.97455
$536$	−16.9178	−0.730736
$537$	0	0
$538$	−21.5097	−0.927347
$539$	−1.09063	−0.0469768
$540$	0	0
$541$	13.0929	0.562907	0.281453	−	0.959575i	$-0.409184\pi$
$541$	13.0929	0.562907	0.281453	+	0.959575i	$0.409184\pi$
$542$	5.11771	0.219825
$543$	0	0
$544$	−41.9060	−1.79670
$545$	72.4186	3.10207
$546$	0	0
$547$	−5.04447	−0.215686	−0.107843	−	0.994168i	$-0.534394\pi$
$547$	−5.04447	−0.215686	−0.107843	+	0.994168i	$0.534394\pi$
$548$	−6.89633	−0.294597
$549$	0	0
$550$	−18.5206	−0.789721
$551$	13.2908	0.566206
$552$	0	0
$553$	3.76517	0.160111
$554$	−2.49125	−0.105843
$555$	0	0
$556$	−15.4066	−0.653387
$557$	−3.24936	−0.137680	−0.0688400	−	0.997628i	$-0.521930\pi$
$557$	−3.24936	−0.137680	−0.0688400	+	0.997628i	$0.521930\pi$
$558$	0	0
$559$	30.7155	1.29913
$560$	−47.1504	−1.99247
$561$	0	0
$562$	−15.1876	−0.640651
$563$	22.6069	0.952767	0.476383	−	0.879238i	$-0.341947\pi$
$563$	22.6069	0.952767	0.476383	+	0.879238i	$0.341947\pi$
$564$	0	0
$565$	24.0640	1.01238
$566$	45.0191	1.89229
$567$	0	0
$568$	−21.6007	−0.906343
$569$	15.4455	0.647509	0.323754	−	0.946141i	$-0.395055\pi$
$569$	15.4455	0.647509	0.323754	+	0.946141i	$0.395055\pi$
$570$	0	0
$571$	−7.03025	−0.294207	−0.147103	−	0.989121i	$-0.546995\pi$
$571$	−7.03025	−0.294207	−0.147103	+	0.989121i	$0.546995\pi$
$572$	4.49230	0.187833
$573$	0	0
$574$	16.2146	0.676784
$575$	15.0062	0.625803
$576$	0	0
$577$	−11.1732	−0.465144	−0.232572	−	0.972579i	$-0.574714\pi$
$577$	−11.1732	−0.465144	−0.232572	+	0.972579i	$0.574714\pi$
$578$	−79.8255	−3.32031
$579$	0	0
$580$	31.3362	1.30117
$581$	−15.8162	−0.656165
$582$	0	0
$583$	3.56432	0.147619
$584$	8.81391	0.364722
$585$	0	0
$586$	15.7293	0.649770
$587$	1.70100	0.0702076	0.0351038	−	0.999384i	$-0.488824\pi$
$587$	1.70100	0.0702076	0.0351038	+	0.999384i	$0.488824\pi$
$588$	0	0
$589$	0	0
$590$	−95.1916	−3.91898
$591$	0	0
$592$	11.9802	0.492382
$593$	17.1944	0.706090	0.353045	−	0.935606i	$-0.385146\pi$
$593$	17.1944	0.706090	0.353045	+	0.935606i	$0.385146\pi$
$594$	0	0
$595$	−74.8159	−3.06715
$596$	0.613318	0.0251225
$597$	0	0
$598$	−10.7766	−0.440688
$599$	23.6611	0.966764	0.483382	−	0.875409i	$-0.339408\pi$
$599$	23.6611	0.966764	0.483382	+	0.875409i	$0.339408\pi$
$600$	0	0
$601$	10.1358	0.413447	0.206723	−	0.978399i	$-0.433720\pi$
$601$	10.1358	0.413447	0.206723	+	0.978399i	$0.433720\pi$
$602$	32.1321	1.30961
$603$	0	0
$604$	−19.0900	−0.776760
$605$	37.8584	1.53916
$606$	0	0
$607$	1.41977	0.0576267	0.0288134	−	0.999585i	$-0.490827\pi$
$607$	1.41977	0.0576267	0.0288134	+	0.999585i	$0.490827\pi$
$608$	−8.80408	−0.357053
$609$	0	0
$610$	−20.2530	−0.820018
$611$	20.5928	0.833097
$612$	0	0
$613$	−26.0931	−1.05389	−0.526945	−	0.849900i	$-0.676663\pi$
$613$	−26.0931	−1.05389	−0.526945	+	0.849900i	$0.676663\pi$
$614$	−15.2285	−0.614573
$615$	0	0
$616$	−4.51486	−0.181909
$617$	14.0358	0.565061	0.282530	−	0.959258i	$-0.408826\pi$
$617$	14.0358	0.565061	0.282530	+	0.959258i	$0.408826\pi$
$618$	0	0
$619$	3.69898	0.148675	0.0743373	−	0.997233i	$-0.476316\pi$
$619$	3.69898	0.148675	0.0743373	+	0.997233i	$0.476316\pi$
$620$	0	0
$621$	0	0
$622$	−1.25640	−0.0503771
$623$	−18.2183	−0.729901
$624$	0	0
$625$	22.6404	0.905616
$626$	16.6135	0.664007
$627$	0	0
$628$	−1.69900	−0.0677976
$629$	19.0095	0.757960
$630$	0	0
$631$	21.7172	0.864549	0.432274	−	0.901742i	$-0.357711\pi$
$631$	21.7172	0.864549	0.432274	+	0.901742i	$0.357711\pi$
$632$	−2.61927	−0.104189
$633$	0	0
$634$	−7.19979	−0.285940
$635$	20.2394	0.803176
$636$	0	0
$637$	−4.09555	−0.162272
$638$	15.0077	0.594161
$639$	0	0
$640$	46.1849	1.82562
$641$	−3.40275	−0.134400	−0.0672002	−	0.997740i	$-0.521407\pi$
$641$	−3.40275	−0.134400	−0.0672002	+	0.997740i	$0.521407\pi$
$642$	0	0
$643$	−27.8988	−1.10022	−0.550111	−	0.835092i	$-0.685415\pi$
$643$	−27.8988	−1.10022	−0.550111	+	0.835092i	$0.685415\pi$
$644$	−3.80774	−0.150046
$645$	0	0
$646$	−22.9774	−0.904033
$647$	−17.2609	−0.678597	−0.339298	−	0.940679i	$-0.610190\pi$
$647$	−17.2609	−0.678597	−0.339298	+	0.940679i	$0.610190\pi$
$648$	0	0
$649$	−15.3982	−0.604433
$650$	−69.5487	−2.72793
$651$	0	0
$652$	−8.55698	−0.335117
$653$	13.7451	0.537886	0.268943	−	0.963156i	$-0.413326\pi$
$653$	13.7451	0.537886	0.268943	+	0.963156i	$0.413326\pi$
$654$	0	0
$655$	−4.78142	−0.186826
$656$	−19.0553	−0.743985
$657$	0	0
$658$	21.5426	0.839817
$659$	−30.9576	−1.20594	−0.602969	−	0.797765i	$-0.706016\pi$
$659$	−30.9576	−1.20594	−0.602969	+	0.797765i	$0.706016\pi$
$660$	0	0
$661$	−44.2197	−1.71995	−0.859973	−	0.510339i	$-0.829520\pi$
$661$	−44.2197	−1.71995	−0.859973	+	0.510339i	$0.829520\pi$
$662$	−9.76691	−0.379602
$663$	0	0
$664$	11.0027	0.426986
$665$	−15.7182	−0.609524
$666$	0	0
$667$	−12.1599	−0.470835
$668$	−17.5812	−0.680235
$669$	0	0
$670$	−66.5067	−2.56938
$671$	−3.27612	−0.126473
$672$	0	0
$673$	25.5392	0.984463	0.492231	−	0.870464i	$-0.336181\pi$
$673$	25.5392	0.984463	0.492231	+	0.870464i	$0.336181\pi$
$674$	37.8587	1.45826
$675$	0	0
$676$	3.60905	0.138809
$677$	6.44452	0.247683	0.123842	−	0.992302i	$-0.460479\pi$
$677$	6.44452	0.247683	0.123842	+	0.992302i	$0.460479\pi$
$678$	0	0
$679$	17.9914	0.690448
$680$	52.0464	1.99589
$681$	0	0
$682$	0	0
$683$	34.3784	1.31545	0.657726	−	0.753257i	$-0.271518\pi$
$683$	34.3784	1.31545	0.657726	+	0.753257i	$0.271518\pi$
$684$	0	0
$685$	26.0456	0.995149
$686$	−34.0643	−1.30058
$687$	0	0
$688$	−37.7615	−1.43965
$689$	13.3848	0.509919
$690$	0	0
$691$	25.2604	0.960952	0.480476	−	0.877008i	$-0.340464\pi$
$691$	25.2604	0.960952	0.480476	+	0.877008i	$0.340464\pi$
$692$	−2.55366	−0.0970756
$693$	0	0
$694$	−16.6819	−0.633236
$695$	58.1866	2.20715
$696$	0	0
$697$	−30.2360	−1.14527
$698$	−27.6809	−1.04774
$699$	0	0
$700$	−24.5739	−0.928807
$701$	16.0320	0.605521	0.302761	−	0.953067i	$-0.402092\pi$
$701$	16.0320	0.605521	0.302761	+	0.953067i	$0.402092\pi$
$702$	0	0
$703$	3.99374	0.150627
$704$	0.887233	0.0334388
$705$	0	0
$706$	−45.2151	−1.70169
$707$	5.19875	0.195519
$708$	0	0
$709$	−3.62098	−0.135989	−0.0679945	−	0.997686i	$-0.521660\pi$
$709$	−3.62098	−0.135989	−0.0679945	+	0.997686i	$0.521660\pi$
$710$	−84.9159	−3.18684
$711$	0	0
$712$	12.6737	0.474969
$713$	0	0
$714$	0	0
$715$	−16.9662	−0.634500
$716$	−6.34907	−0.237276
$717$	0	0
$718$	−12.6471	−0.471985
$719$	6.97181	0.260005	0.130002	−	0.991514i	$-0.458501\pi$
$719$	6.97181	0.260005	0.130002	+	0.991514i	$0.458501\pi$
$720$	0	0
$721$	−15.3734	−0.572536
$722$	28.1913	1.04917
$723$	0	0
$724$	13.4439	0.499639
$725$	−78.4763	−2.91454
$726$	0	0
$727$	−42.1636	−1.56376	−0.781881	−	0.623427i	$-0.785740\pi$
$727$	−42.1636	−1.56376	−0.781881	+	0.623427i	$0.785740\pi$
$728$	−16.9542	−0.628366
$729$	0	0
$730$	34.6490	1.28242
$731$	−59.9181	−2.21615
$732$	0	0
$733$	1.04801	0.0387091	0.0193545	−	0.999813i	$-0.493839\pi$
$733$	1.04801	0.0387091	0.0193545	+	0.999813i	$0.493839\pi$
$734$	−1.82918	−0.0675161
$735$	0	0
$736$	8.05499	0.296911
$737$	−10.7581	−0.396281
$738$	0	0
$739$	28.8950	1.06292	0.531459	−	0.847084i	$-0.321644\pi$
$739$	28.8950	1.06292	0.531459	+	0.847084i	$0.321644\pi$
$740$	9.41621	0.346147
$741$	0	0
$742$	14.0021	0.514033
$743$	−16.2261	−0.595277	−0.297638	−	0.954679i	$-0.596199\pi$
$743$	−16.2261	−0.595277	−0.297638	+	0.954679i	$0.596199\pi$
$744$	0	0
$745$	−2.31634	−0.0848640
$746$	24.0139	0.879212
$747$	0	0
$748$	−8.76332	−0.320419
$749$	−29.0223	−1.06045
$750$	0	0
$751$	−4.96932	−0.181333	−0.0906665	−	0.995881i	$-0.528900\pi$
$751$	−4.96932	−0.181333	−0.0906665	+	0.995881i	$0.528900\pi$
$752$	−25.3167	−0.923207
$753$	0	0
$754$	56.3571	2.05240
$755$	72.0976	2.62390
$756$	0	0
$757$	46.4717	1.68904	0.844521	−	0.535522i	$-0.179885\pi$
$757$	46.4717	1.68904	0.844521	+	0.535522i	$0.179885\pi$
$758$	−48.5165	−1.76220
$759$	0	0
$760$	10.9345	0.396636
$761$	−48.0573	−1.74208	−0.871038	−	0.491215i	$-0.836553\pi$
$761$	−48.0573	−1.74208	−0.871038	+	0.491215i	$0.836553\pi$
$762$	0	0
$763$	46.0190	1.66600
$764$	19.0346	0.688647
$765$	0	0
$766$	56.3565	2.03624
$767$	−57.8235	−2.08789
$768$	0	0
$769$	21.6466	0.780598	0.390299	−	0.920688i	$-0.372372\pi$
$769$	21.6466	0.780598	0.390299	+	0.920688i	$0.372372\pi$
$770$	−17.7487	−0.639618
$771$	0	0
$772$	−26.4511	−0.951995
$773$	42.0400	1.51207	0.756037	−	0.654529i	$-0.227133\pi$
$773$	42.0400	1.51207	0.756037	+	0.654529i	$0.227133\pi$
$774$	0	0
$775$	0	0
$776$	−12.5159	−0.449295
$777$	0	0
$778$	35.4297	1.27022
$779$	−6.35231	−0.227595
$780$	0	0
$781$	−13.7360	−0.491513
$782$	21.0224	0.751758
$783$	0	0
$784$	5.03505	0.179823
$785$	6.41667	0.229021
$786$	0	0
$787$	−3.83760	−0.136796	−0.0683979	−	0.997658i	$-0.521789\pi$
$787$	−3.83760	−0.136796	−0.0683979	+	0.997658i	$0.521789\pi$
$788$	3.48740	0.124234
$789$	0	0
$790$	−10.2968	−0.366344
$791$	15.2916	0.543708
$792$	0	0
$793$	−12.3025	−0.436875
$794$	12.1284	0.430422
$795$	0	0
$796$	−10.6183	−0.376356
$797$	−8.81463	−0.312230	−0.156115	−	0.987739i	$-0.549897\pi$
$797$	−8.81463	−0.312230	−0.156115	+	0.987739i	$0.549897\pi$
$798$	0	0
$799$	−40.1713	−1.42116
$800$	51.9843	1.83792
$801$	0	0
$802$	21.7727	0.768819
$803$	5.60483	0.197790
$804$	0	0
$805$	14.3808	0.506856
$806$	0	0
$807$	0	0
$808$	−3.61656	−0.127230
$809$	−17.1184	−0.601850	−0.300925	−	0.953648i	$-0.597295\pi$
$809$	−17.1184	−0.601850	−0.300925	+	0.953648i	$0.597295\pi$
$810$	0	0
$811$	39.7499	1.39581	0.697905	−	0.716191i	$-0.254116\pi$
$811$	39.7499	1.39581	0.697905	+	0.716191i	$0.254116\pi$
$812$	19.9129	0.698805
$813$	0	0
$814$	4.50966	0.158063
$815$	32.3174	1.13203
$816$	0	0
$817$	−12.5883	−0.440407
$818$	−26.4735	−0.925624
$819$	0	0
$820$	−14.9771	−0.523024
$821$	−24.1075	−0.841356	−0.420678	−	0.907210i	$-0.638208\pi$
$821$	−24.1075	−0.841356	−0.420678	+	0.907210i	$0.638208\pi$
$822$	0	0
$823$	14.0165	0.488584	0.244292	−	0.969702i	$-0.421444\pi$
$823$	14.0165	0.488584	0.244292	+	0.969702i	$0.421444\pi$
$824$	10.6947	0.372567
$825$	0	0
$826$	−60.4903	−2.10473
$827$	−42.4443	−1.47593	−0.737966	−	0.674837i	$-0.764214\pi$
$827$	−42.4443	−1.47593	−0.737966	+	0.674837i	$0.764214\pi$
$828$	0	0
$829$	27.8893	0.968634	0.484317	−	0.874893i	$-0.339068\pi$
$829$	27.8893	0.968634	0.484317	+	0.874893i	$0.339068\pi$
$830$	43.2534	1.50135
$831$	0	0
$832$	3.33175	0.115507
$833$	7.98936	0.276815
$834$	0	0
$835$	66.3992	2.29784
$836$	−1.84110	−0.0636756
$837$	0	0
$838$	1.20832	0.0417408
$839$	5.77939	0.199527	0.0997633	−	0.995011i	$-0.468191\pi$
$839$	5.77939	0.199527	0.0997633	+	0.995011i	$0.468191\pi$
$840$	0	0
$841$	34.5914	1.19281
$842$	−52.2218	−1.79968
$843$	0	0
$844$	−15.0510	−0.518077
$845$	−13.6304	−0.468899
$846$	0	0
$847$	24.0574	0.826623
$848$	−16.4552	−0.565073
$849$	0	0
$850$	135.672	4.65350
$851$	−3.65393	−0.125255
$852$	0	0
$853$	−17.0916	−0.585204	−0.292602	−	0.956234i	$-0.594521\pi$
$853$	−17.0916	−0.585204	−0.292602	+	0.956234i	$0.594521\pi$
$854$	−12.8699	−0.440399
$855$	0	0
$856$	20.1896	0.690068
$857$	−44.7670	−1.52921	−0.764606	−	0.644498i	$-0.777067\pi$
$857$	−44.7670	−1.52921	−0.764606	+	0.644498i	$0.777067\pi$
$858$	0	0
$859$	7.31104	0.249449	0.124725	−	0.992191i	$-0.460195\pi$
$859$	7.31104	0.249449	0.124725	+	0.992191i	$0.460195\pi$
$860$	−29.6799	−1.01208
$861$	0	0
$862$	−28.9875	−0.987317
$863$	−27.6370	−0.940775	−0.470388	−	0.882460i	$-0.655886\pi$
$863$	−27.6370	−0.940775	−0.470388	+	0.882460i	$0.655886\pi$
$864$	0	0
$865$	9.64448	0.327922
$866$	−11.6933	−0.397356
$867$	0	0
$868$	0	0
$869$	−1.66562	−0.0565021
$870$	0	0
$871$	−40.3990	−1.36887
$872$	−32.0136	−1.08412
$873$	0	0
$874$	4.41661	0.149394
$875$	45.6548	1.54341
$876$	0	0
$877$	−22.7150	−0.767031	−0.383515	−	0.923534i	$-0.625287\pi$
$877$	−22.7150	−0.767031	−0.383515	+	0.923534i	$0.625287\pi$
$878$	−60.0457	−2.02644
$879$	0	0
$880$	20.8582	0.703129
$881$	−43.7533	−1.47409	−0.737044	−	0.675845i	$-0.763779\pi$
$881$	−43.7533	−1.47409	−0.737044	+	0.675845i	$0.763779\pi$
$882$	0	0
$883$	−51.7405	−1.74121	−0.870603	−	0.491986i	$-0.836271\pi$
$883$	−51.7405	−1.74121	−0.870603	+	0.491986i	$0.836271\pi$
$884$	−32.9081	−1.10682
$885$	0	0
$886$	−39.5199	−1.32770
$887$	45.2793	1.52033	0.760165	−	0.649730i	$-0.225118\pi$
$887$	45.2793	1.52033	0.760165	+	0.649730i	$0.225118\pi$
$888$	0	0
$889$	12.8613	0.431354
$890$	49.8227	1.67006
$891$	0	0
$892$	−11.6736	−0.390861
$893$	−8.43964	−0.282422
$894$	0	0
$895$	23.9787	0.801520
$896$	29.3486	0.980467
$897$	0	0
$898$	−63.9785	−2.13499
$899$	0	0
$900$	0	0
$901$	−26.1102	−0.869858
$902$	−7.17292	−0.238832
$903$	0	0
$904$	−10.6378	−0.353807
$905$	−50.7740	−1.68779
$906$	0	0
$907$	10.9585	0.363872	0.181936	−	0.983310i	$-0.441764\pi$
$907$	10.9585	0.363872	0.181936	+	0.983310i	$0.441764\pi$
$908$	−23.9600	−0.795141
$909$	0	0
$910$	−66.6500	−2.20943
$911$	29.2094	0.967752	0.483876	−	0.875137i	$-0.339229\pi$
$911$	29.2094	0.967752	0.483876	+	0.875137i	$0.339229\pi$
$912$	0	0
$913$	6.99667	0.231556
$914$	−38.3591	−1.26881
$915$	0	0
$916$	11.6330	0.384365
$917$	−3.03840	−0.100337
$918$	0	0
$919$	23.3670	0.770806	0.385403	−	0.922748i	$-0.374062\pi$
$919$	23.3670	0.770806	0.385403	+	0.922748i	$0.374062\pi$
$920$	−10.0041	−0.329827
$921$	0	0
$922$	−40.8632	−1.34576
$923$	−51.5816	−1.69783
$924$	0	0
$925$	−23.5813	−0.775349
$926$	−2.33623	−0.0767733
$927$	0	0
$928$	−42.1242	−1.38280
$929$	−26.8470	−0.880821	−0.440411	−	0.897796i	$-0.645167\pi$
$929$	−26.8470	−0.880821	−0.440411	+	0.897796i	$0.645167\pi$
$930$	0	0
$931$	1.67849	0.0550104
$932$	−11.8702	−0.388820
$933$	0	0
$934$	32.9592	1.07846
$935$	33.0967	1.08238
$936$	0	0
$937$	18.4283	0.602026	0.301013	−	0.953620i	$-0.402675\pi$
$937$	18.4283	0.602026	0.301013	+	0.953620i	$0.402675\pi$
$938$	−42.2622	−1.37991
$939$	0	0
$940$	−19.8985	−0.649018
$941$	−20.7016	−0.674853	−0.337427	−	0.941352i	$-0.609556\pi$
$941$	−20.7016	−0.674853	−0.337427	+	0.941352i	$0.609556\pi$
$942$	0	0
$943$	5.81183	0.189259
$944$	71.0880	2.31372
$945$	0	0
$946$	−14.2144	−0.462152
$947$	−11.6183	−0.377543	−0.188771	−	0.982021i	$-0.560451\pi$
$947$	−11.6183	−0.377543	−0.188771	+	0.982021i	$0.560451\pi$
$948$	0	0
$949$	21.0473	0.683224
$950$	28.5034	0.924773
$951$	0	0
$952$	33.0733	1.07191
$953$	1.59717	0.0517374	0.0258687	−	0.999665i	$-0.491765\pi$
$953$	1.59717	0.0517374	0.0258687	+	0.999665i	$0.491765\pi$
$954$	0	0
$955$	−71.8884	−2.32625
$956$	23.0689	0.746101
$957$	0	0
$958$	42.1530	1.36190
$959$	16.5509	0.534455
$960$	0	0
$961$	0	0
$962$	16.9347	0.545997
$963$	0	0
$964$	23.4696	0.755904
$965$	99.8985	3.21585
$966$	0	0
$967$	24.1386	0.776245	0.388123	−	0.921608i	$-0.373124\pi$
$967$	24.1386	0.776245	0.388123	+	0.921608i	$0.373124\pi$
$968$	−16.7358	−0.537909
$969$	0	0
$970$	−49.2022	−1.57979
$971$	17.5140	0.562052	0.281026	−	0.959700i	$-0.409325\pi$
$971$	17.5140	0.562052	0.281026	+	0.959700i	$0.409325\pi$
$972$	0	0
$973$	36.9752	1.18537
$974$	4.80946	0.154105
$975$	0	0
$976$	15.1247	0.484129
$977$	−33.8011	−1.08139	−0.540697	−	0.841217i	$-0.681839\pi$
$977$	−33.8011	−1.08139	−0.540697	+	0.841217i	$0.681839\pi$
$978$	0	0
$979$	8.05932	0.257577
$980$	3.95746	0.126416
$981$	0	0
$982$	−66.6006	−2.12531
$983$	24.1804	0.771236	0.385618	−	0.922659i	$-0.373988\pi$
$983$	24.1804	0.771236	0.385618	+	0.922659i	$0.373988\pi$
$984$	0	0
$985$	−13.1710	−0.419662
$986$	−109.938	−3.50114
$987$	0	0
$988$	−6.91370	−0.219954
$989$	11.5172	0.366225
$990$	0	0
$991$	−32.1985	−1.02282	−0.511410	−	0.859337i	$-0.670877\pi$
$991$	−32.1985	−1.02282	−0.511410	+	0.859337i	$0.670877\pi$
$992$	0	0
$993$	0	0
$994$	−53.9605	−1.71152
$995$	40.1025	1.27133
$996$	0	0
$997$	33.3042	1.05475	0.527377	−	0.849631i	$-0.323175\pi$
$997$	33.3042	1.05475	0.527377	+	0.849631i	$0.323175\pi$
$998$	47.3263	1.49809
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.bu.1.6		24
3.2	odd	2		2883.2.a.v.1.19	yes	24
31.30	odd	2		8649.2.a.bv.1.6		24
93.92	even	2		2883.2.a.u.1.19	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2883.2.a.u.1.19	✓	24	93.92	even	2
2883.2.a.v.1.19	yes	24	3.2	odd	2
8649.2.a.bu.1.6		24	1.1	even	1	trivial
8649.2.a.bv.1.6		24	31.30	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:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73783 q^{2} +1.02004 q^{4} -3.85240 q^{5} -2.44804 q^{7} +1.70300 q^{8} +6.69480 q^{10} +1.08295 q^{11} +4.06671 q^{13} +4.25427 q^{14} -4.99960 q^{16} -7.93311 q^{17} -1.66668 q^{19} -3.92959 q^{20} -1.88198 q^{22} +1.52487 q^{23} +9.84101 q^{25} -7.06724 q^{26} -2.49709 q^{28} -7.97442 q^{29} +5.28242 q^{32} +13.7864 q^{34} +9.43084 q^{35} -2.39623 q^{37} +2.89639 q^{38} -6.56066 q^{40} +3.81137 q^{41} +7.55291 q^{43} +1.10465 q^{44} -2.64995 q^{46} +5.06376 q^{47} -1.00709 q^{49} -17.1020 q^{50} +4.14820 q^{52} +3.29130 q^{53} -4.17197 q^{55} -4.16903 q^{56} +13.8581 q^{58} -14.2187 q^{59} -3.02518 q^{61} +0.819272 q^{64} -15.6666 q^{65} -9.93407 q^{67} -8.09206 q^{68} -16.3892 q^{70} -12.6838 q^{71} +5.17551 q^{73} +4.16423 q^{74} -1.70007 q^{76} -2.65111 q^{77} -1.53803 q^{79} +19.2605 q^{80} -6.62349 q^{82} +6.46074 q^{83} +30.5615 q^{85} -13.1256 q^{86} +1.84427 q^{88} +7.44199 q^{89} -9.95548 q^{91} +1.55542 q^{92} -8.79992 q^{94} +6.42071 q^{95} -7.34932 q^{97} +1.75015 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8} - 8 q^{10} - 16 q^{11} + 32 q^{13} + 24 q^{14} + 48 q^{16} - 32 q^{17} + 32 q^{19} + 24 q^{20} + 32 q^{22} - 32 q^{23} + 40 q^{25} - 16 q^{26} + 8 q^{28} - 48 q^{29}+ \cdots + 24 q^{98}+O(q^{100}) \) 24 * q + 32 * q^4 + 16 * q^7 + 24 * q^8 - 8 * q^10 - 16 * q^11 + 32 * q^13 + 24 * q^14 + 48 * q^16 - 32 * q^17 + 32 * q^19 + 24 * q^20 + 32 * q^22 - 32 * q^23 + 40 * q^25 - 16 * q^26 + 8 * q^28 - 48 * q^29 + 48 * q^32 + 48 * q^35 + 64 * q^37 + 24 * q^38 + 32 * q^43 - 48 * q^44 + 32 * q^46 + 48 * q^47 + 56 * q^49 + 24 * q^50 + 64 * q^52 - 80 * q^53 + 48 * q^56 + 32 * q^58 + 32 * q^61 + 56 * q^64 - 16 * q^65 - 16 * q^67 - 80 * q^68 + 8 * q^70 + 32 * q^73 + 56 * q^76 - 96 * q^77 + 32 * q^79 + 72 * q^80 + 8 * q^82 - 48 * q^83 + 96 * q^85 + 32 * q^86 + 96 * q^88 + 16 * q^89 - 32 * q^92 + 48 * q^94 + 48 * q^95 + 16 * q^97 + 24 * q^98

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