LMFDB - Embedded newform 8649.2.a.bu.1.8

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			1.8
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)

$f(q)$	$=$	$q-1.21162 q^{2} -0.531985 q^{4} -1.58010 q^{5} +4.11574 q^{7} +3.06780 q^{8} +O(q^{10})$	$q-1.21162 q^{2} -0.531985 q^{4} -1.58010 q^{5} +4.11574 q^{7} +3.06780 q^{8} +1.91448 q^{10} +1.03422 q^{11} +6.30039 q^{13} -4.98670 q^{14} -2.65302 q^{16} -3.15440 q^{17} -5.27176 q^{19} +0.840589 q^{20} -1.25308 q^{22} -5.66315 q^{23} -2.50329 q^{25} -7.63366 q^{26} -2.18951 q^{28} +6.35035 q^{29} -2.92114 q^{32} +3.82193 q^{34} -6.50329 q^{35} +4.52696 q^{37} +6.38735 q^{38} -4.84742 q^{40} +8.15535 q^{41} +5.10252 q^{43} -0.550190 q^{44} +6.86157 q^{46} -9.79654 q^{47} +9.93935 q^{49} +3.03302 q^{50} -3.35171 q^{52} +3.77956 q^{53} -1.63417 q^{55} +12.6263 q^{56} -7.69419 q^{58} +6.22906 q^{59} +5.15699 q^{61} +8.84535 q^{64} -9.95524 q^{65} -1.13241 q^{67} +1.67809 q^{68} +7.87949 q^{70} +13.8807 q^{71} -6.54260 q^{73} -5.48494 q^{74} +2.80449 q^{76} +4.25659 q^{77} -4.33699 q^{79} +4.19204 q^{80} -9.88116 q^{82} -9.51925 q^{83} +4.98427 q^{85} -6.18229 q^{86} +3.17278 q^{88} +4.37466 q^{89} +25.9308 q^{91} +3.01271 q^{92} +11.8697 q^{94} +8.32990 q^{95} +4.68445 q^{97} -12.0427 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8}+O(q^{10}) $ 24 * q + 32 * q^4 + 16 * q^7 + 24 * q^8	$ 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}+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

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 $

$2$	−1.21162	−0.856742	−0.428371	−	0.903603i	$-0.640912\pi$
$2$	−1.21162	−0.856742	−0.428371	+	0.903603i	$0.640912\pi$
$3$	0	0
$3$	0	0
$4$	−0.531985	−0.265992
$5$	−1.58010	−0.706642	−0.353321	−	0.935502i	$-0.614948\pi$
$5$	−1.58010	−0.706642	−0.353321	+	0.935502i	$0.614948\pi$
$6$	0	0
$7$	4.11574	1.55561	0.777803	−	0.628509i	$-0.216334\pi$
$7$	4.11574	1.55561	0.777803	+	0.628509i	$0.216334\pi$
$8$	3.06780	1.08463
$9$	0	0
$10$	1.91448	0.605410
$11$	1.03422	0.311829	0.155915	−	0.987771i	$-0.450168\pi$
$11$	1.03422	0.311829	0.155915	+	0.987771i	$0.450168\pi$
$12$	0	0
$13$	6.30039	1.74741	0.873707	−	0.486453i	$-0.161710\pi$
$13$	6.30039	1.74741	0.873707	+	0.486453i	$0.161710\pi$
$14$	−4.98670	−1.33275
$15$	0	0
$16$	−2.65302	−0.663256
$17$	−3.15440	−0.765055	−0.382527	−	0.923944i	$-0.624946\pi$
$17$	−3.15440	−0.765055	−0.382527	+	0.923944i	$0.624946\pi$
$18$	0	0
$19$	−5.27176	−1.20942	−0.604712	−	0.796444i	$-0.706712\pi$
$19$	−5.27176	−1.20942	−0.604712	+	0.796444i	$0.706712\pi$
$20$	0.840589	0.187961
$21$	0	0
$22$	−1.25308	−0.267157
$23$	−5.66315	−1.18085	−0.590424	−	0.807093i	$-0.701040\pi$
$23$	−5.66315	−1.18085	−0.590424	+	0.807093i	$0.701040\pi$
$24$	0	0
$25$	−2.50329	−0.500657
$26$	−7.63366	−1.49708
$27$	0	0
$28$	−2.18951	−0.413779
$29$	6.35035	1.17923	0.589615	−	0.807685i	$-0.299280\pi$
$29$	6.35035	1.17923	0.589615	+	0.807685i	$0.299280\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	−2.92114	−0.516390
$33$	0	0
$34$	3.82193	0.655455
$35$	−6.50329	−1.09926
$36$	0	0
$37$	4.52696	0.744228	0.372114	−	0.928187i	$-0.378633\pi$
$37$	4.52696	0.744228	0.372114	+	0.928187i	$0.378633\pi$
$38$	6.38735	1.03616
$39$	0	0
$40$	−4.84742	−0.766445
$41$	8.15535	1.27365	0.636826	−	0.771008i	$-0.280247\pi$
$41$	8.15535	1.27365	0.636826	+	0.771008i	$0.280247\pi$
$42$	0	0
$43$	5.10252	0.778126	0.389063	−	0.921211i	$-0.372799\pi$
$43$	5.10252	0.778126	0.389063	+	0.921211i	$0.372799\pi$
$44$	−0.550190	−0.0829442
$45$	0	0
$46$	6.86157	1.01168
$47$	−9.79654	−1.42897	−0.714486	−	0.699649i	$-0.753340\pi$
$47$	−9.79654	−1.42897	−0.714486	+	0.699649i	$0.753340\pi$
$48$	0	0
$49$	9.93935	1.41991
$50$	3.03302	0.428934
$51$	0	0
$52$	−3.35171	−0.464799
$53$	3.77956	0.519163	0.259581	−	0.965721i	$-0.416415\pi$
$53$	3.77956	0.519163	0.259581	+	0.965721i	$0.416415\pi$
$54$	0	0
$55$	−1.63417	−0.220352
$56$	12.6263	1.68725
$57$	0	0
$58$	−7.69419	−1.01030
$59$	6.22906	0.810955	0.405477	−	0.914105i	$-0.367105\pi$
$59$	6.22906	0.810955	0.405477	+	0.914105i	$0.367105\pi$
$60$	0	0
$61$	5.15699	0.660285	0.330143	−	0.943931i	$-0.392903\pi$
$61$	5.15699	0.660285	0.330143	+	0.943931i	$0.392903\pi$
$62$	0	0
$63$	0	0
$64$	8.84535	1.10567
$65$	−9.95524	−1.23480
$66$	0	0
$67$	−1.13241	−0.138346	−0.0691729	−	0.997605i	$-0.522036\pi$
$67$	−1.13241	−0.138346	−0.0691729	+	0.997605i	$0.522036\pi$
$68$	1.67809	0.203499
$69$	0	0
$70$	7.87949	0.941779
$71$	13.8807	1.64733	0.823667	−	0.567074i	$-0.191924\pi$
$71$	13.8807	1.64733	0.823667	+	0.567074i	$0.191924\pi$
$72$	0	0
$73$	−6.54260	−0.765753	−0.382877	−	0.923799i	$-0.625067\pi$
$73$	−6.54260	−0.765753	−0.382877	+	0.923799i	$0.625067\pi$
$74$	−5.48494	−0.637612
$75$	0	0
$76$	2.80449	0.321697
$77$	4.25659	0.485083
$78$	0	0
$79$	−4.33699	−0.487949	−0.243975	−	0.969782i	$-0.578451\pi$
$79$	−4.33699	−0.487949	−0.243975	+	0.969782i	$0.578451\pi$
$80$	4.19204	0.468684
$81$	0	0
$82$	−9.88116	−1.09119
$83$	−9.51925	−1.04487	−0.522436	−	0.852678i	$-0.674977\pi$
$83$	−9.51925	−1.04487	−0.522436	+	0.852678i	$0.674977\pi$
$84$	0	0
$85$	4.98427	0.540620
$86$	−6.18229	−0.666654
$87$	0	0
$88$	3.17278	0.338219
$89$	4.37466	0.463713	0.231857	−	0.972750i	$-0.425520\pi$
$89$	4.37466	0.463713	0.231857	+	0.972750i	$0.425520\pi$
$90$	0	0
$91$	25.9308	2.71828
$92$	3.01271	0.314097
$93$	0	0
$94$	11.8697	1.22426
$95$	8.32990	0.854630
$96$	0	0
$97$	4.68445	0.475633	0.237817	−	0.971310i	$-0.423568\pi$
$97$	4.68445	0.475633	0.237817	+	0.971310i	$0.423568\pi$
$98$	−12.0427	−1.21649
$99$	0	0
$100$	1.33171	0.133171
$101$	−11.7999	−1.17413	−0.587065	−	0.809540i	$-0.699717\pi$
$101$	−11.7999	−1.17413	−0.587065	+	0.809540i	$0.699717\pi$
$102$	0	0
$103$	6.82018	0.672012	0.336006	−	0.941860i	$-0.390924\pi$
$103$	6.82018	0.672012	0.336006	+	0.941860i	$0.390924\pi$
$104$	19.3283	1.89530
$105$	0	0
$106$	−4.57938	−0.444789
$107$	15.0633	1.45622	0.728111	−	0.685460i	$-0.240399\pi$
$107$	15.0633	1.45622	0.728111	+	0.685460i	$0.240399\pi$
$108$	0	0
$109$	7.33777	0.702830	0.351415	−	0.936220i	$-0.385701\pi$
$109$	7.33777	0.702830	0.351415	+	0.936220i	$0.385701\pi$
$110$	1.97999	0.188785
$111$	0	0
$112$	−10.9192	−1.03176
$113$	−13.1559	−1.23760	−0.618801	−	0.785548i	$-0.712381\pi$
$113$	−13.1559	−1.23760	−0.618801	+	0.785548i	$0.712381\pi$
$114$	0	0
$115$	8.94834	0.834437
$116$	−3.37829	−0.313666
$117$	0	0
$118$	−7.54723	−0.694779
$119$	−12.9827	−1.19012
$120$	0	0
$121$	−9.93039	−0.902762
$122$	−6.24830	−0.565694
$123$	0	0
$124$	0	0
$125$	11.8559	1.06043
$126$	0	0
$127$	9.70806	0.861451	0.430726	−	0.902483i	$-0.358258\pi$
$127$	9.70806	0.861451	0.430726	+	0.902483i	$0.358258\pi$
$128$	−4.87489	−0.430883
$129$	0	0
$130$	12.0619	1.05790
$131$	−0.564105	−0.0492860	−0.0246430	−	0.999696i	$-0.507845\pi$
$131$	−0.564105	−0.0492860	−0.0246430	+	0.999696i	$0.507845\pi$
$132$	0	0
$133$	−21.6972	−1.88139
$134$	1.37205	0.118527
$135$	0	0
$136$	−9.67706	−0.829801
$137$	−16.8898	−1.44299	−0.721495	−	0.692419i	$-0.756545\pi$
$137$	−16.8898	−1.44299	−0.721495	+	0.692419i	$0.756545\pi$
$138$	0	0
$139$	14.7007	1.24690	0.623451	−	0.781863i	$-0.285730\pi$
$139$	14.7007	1.24690	0.623451	+	0.781863i	$0.285730\pi$
$140$	3.45965	0.292394
$141$	0	0
$142$	−16.8181	−1.41134
$143$	6.51599	0.544895
$144$	0	0
$145$	−10.0342	−0.833293
$146$	7.92712	0.656053
$147$	0	0
$148$	−2.40827	−0.197959
$149$	13.6285	1.11649	0.558247	−	0.829675i	$-0.311474\pi$
$149$	13.6285	1.11649	0.558247	+	0.829675i	$0.311474\pi$
$150$	0	0
$151$	−2.25640	−0.183623	−0.0918117	−	0.995776i	$-0.529266\pi$
$151$	−2.25640	−0.183623	−0.0918117	+	0.995776i	$0.529266\pi$
$152$	−16.1727	−1.31178
$153$	0	0
$154$	−5.15735	−0.415591
$155$	0	0
$156$	0	0
$157$	1.79970	0.143631	0.0718157	−	0.997418i	$-0.477121\pi$
$157$	1.79970	0.143631	0.0718157	+	0.997418i	$0.477121\pi$
$158$	5.25477	0.418047
$159$	0	0
$160$	4.61570	0.364903
$161$	−23.3081	−1.83693
$162$	0	0
$163$	−10.9920	−0.860961	−0.430481	−	0.902600i	$-0.641656\pi$
$163$	−10.9920	−0.860961	−0.430481	+	0.902600i	$0.641656\pi$
$164$	−4.33852	−0.338782
$165$	0	0
$166$	11.5337	0.895187
$167$	15.0958	1.16815	0.584076	−	0.811699i	$-0.301457\pi$
$167$	15.0958	1.16815	0.584076	+	0.811699i	$0.301457\pi$
$168$	0	0
$169$	26.6949	2.05345
$170$	−6.03902	−0.463172
$171$	0	0
$172$	−2.71446	−0.206976
$173$	1.14351	0.0869394	0.0434697	−	0.999055i	$-0.486159\pi$
$173$	1.14351	0.0869394	0.0434697	+	0.999055i	$0.486159\pi$
$174$	0	0
$175$	−10.3029	−0.778825
$176$	−2.74381	−0.206823
$177$	0	0
$178$	−5.30042	−0.397283
$179$	−0.0355348	−0.00265600	−0.00132800	−	0.999999i	$-0.500423\pi$
$179$	−0.0355348	−0.00265600	−0.00132800	+	0.999999i	$0.500423\pi$
$180$	0	0
$181$	−19.6588	−1.46122	−0.730612	−	0.682793i	$-0.760765\pi$
$181$	−19.6588	−1.46122	−0.730612	+	0.682793i	$0.760765\pi$
$182$	−31.4182	−2.32887
$183$	0	0
$184$	−17.3734	−1.28078
$185$	−7.15305	−0.525903
$186$	0	0
$187$	−3.26235	−0.238566
$188$	5.21161	0.380096
$189$	0	0
$190$	−10.0926	−0.732197
$191$	1.51977	0.109967	0.0549833	−	0.998487i	$-0.482489\pi$
$191$	1.51977	0.109967	0.0549833	+	0.998487i	$0.482489\pi$
$192$	0	0
$193$	−21.4889	−1.54680	−0.773402	−	0.633916i	$-0.781447\pi$
$193$	−21.4889	−1.54680	−0.773402	+	0.633916i	$0.781447\pi$
$194$	−5.67575	−0.407495
$195$	0	0
$196$	−5.28758	−0.377684
$197$	−22.4989	−1.60298	−0.801489	−	0.598009i	$-0.795959\pi$
$197$	−22.4989	−1.60298	−0.801489	+	0.598009i	$0.795959\pi$
$198$	0	0
$199$	8.80907	0.624459	0.312229	−	0.950007i	$-0.398924\pi$
$199$	8.80907	0.624459	0.312229	+	0.950007i	$0.398924\pi$
$200$	−7.67957	−0.543027
$201$	0	0
$202$	14.2969	1.00593
$203$	26.1364	1.83442
$204$	0	0
$205$	−12.8863	−0.900016
$206$	−8.26345	−0.575742
$207$	0	0
$208$	−16.7151	−1.15898
$209$	−5.45216	−0.377134
$210$	0	0
$211$	10.7436	0.739622	0.369811	−	0.929107i	$-0.379422\pi$
$211$	10.7436	0.739622	0.369811	+	0.929107i	$0.379422\pi$
$212$	−2.01067	−0.138093
$213$	0	0
$214$	−18.2509	−1.24761
$215$	−8.06248	−0.549857
$216$	0	0
$217$	0	0
$218$	−8.89056	−0.602145
$219$	0	0
$220$	0.869355	0.0586119
$221$	−19.8740	−1.33687
$222$	0	0
$223$	−24.7256	−1.65575	−0.827875	−	0.560913i	$-0.810450\pi$
$223$	−24.7256	−1.65575	−0.827875	+	0.560913i	$0.810450\pi$
$224$	−12.0227	−0.803299
$225$	0	0
$226$	15.9399	1.06031
$227$	11.3724	0.754815	0.377408	−	0.926047i	$-0.376816\pi$
$227$	11.3724	0.754815	0.377408	+	0.926047i	$0.376816\pi$
$228$	0	0
$229$	−20.0617	−1.32572	−0.662858	−	0.748745i	$-0.730657\pi$
$229$	−20.0617	−1.32572	−0.662858	+	0.748745i	$0.730657\pi$
$230$	−10.8420	−0.714898
$231$	0	0
$232$	19.4816	1.27903
$233$	12.3086	0.806362	0.403181	−	0.915120i	$-0.367904\pi$
$233$	12.3086	0.806362	0.403181	+	0.915120i	$0.367904\pi$
$234$	0	0
$235$	15.4795	1.00977
$236$	−3.31377	−0.215708
$237$	0	0
$238$	15.7301	1.01963
$239$	16.6346	1.07600	0.538000	−	0.842945i	$-0.319180\pi$
$239$	16.6346	1.07600	0.538000	+	0.842945i	$0.319180\pi$
$240$	0	0
$241$	21.2856	1.37112	0.685562	−	0.728014i	$-0.259557\pi$
$241$	21.2856	1.37112	0.685562	+	0.728014i	$0.259557\pi$
$242$	12.0318	0.773435
$243$	0	0
$244$	−2.74344	−0.175631
$245$	−15.7052	−1.00337
$246$	0	0
$247$	−33.2141	−2.11336
$248$	0	0
$249$	0	0
$250$	−14.3649	−0.908513
$251$	7.71984	0.487272	0.243636	−	0.969867i	$-0.421660\pi$
$251$	7.71984	0.487272	0.243636	+	0.969867i	$0.421660\pi$
$252$	0	0
$253$	−5.85695	−0.368223
$254$	−11.7624	−0.738042
$255$	0	0
$256$	−11.7842	−0.736513
$257$	10.8569	0.677236	0.338618	−	0.940924i	$-0.390041\pi$
$257$	10.8569	0.677236	0.338618	+	0.940924i	$0.390041\pi$
$258$	0	0
$259$	18.6318	1.15772
$260$	5.29604	0.328446
$261$	0	0
$262$	0.683479	0.0422254
$263$	24.8050	1.52954	0.764771	−	0.644303i	$-0.222852\pi$
$263$	24.8050	1.52954	0.764771	+	0.644303i	$0.222852\pi$
$264$	0	0
$265$	−5.97209	−0.366862
$266$	26.2887	1.61186
$267$	0	0
$268$	0.602425	0.0367989
$269$	7.03948	0.429205	0.214602	−	0.976702i	$-0.431154\pi$
$269$	7.03948	0.429205	0.214602	+	0.976702i	$0.431154\pi$
$270$	0	0
$271$	−10.3229	−0.627071	−0.313535	−	0.949576i	$-0.601513\pi$
$271$	−10.3229	−0.627071	−0.313535	+	0.949576i	$0.601513\pi$
$272$	8.36870	0.507427
$273$	0	0
$274$	20.4639	1.23627
$275$	−2.58895	−0.156120
$276$	0	0
$277$	25.6150	1.53905	0.769527	−	0.638614i	$-0.220492\pi$
$277$	25.6150	1.53905	0.769527	+	0.638614i	$0.220492\pi$
$278$	−17.8117	−1.06827
$279$	0	0
$280$	−19.9507	−1.19229
$281$	−15.1662	−0.904739	−0.452370	−	0.891831i	$-0.649421\pi$
$281$	−15.1662	−0.904739	−0.452370	+	0.891831i	$0.649421\pi$
$282$	0	0
$283$	−4.24424	−0.252294	−0.126147	−	0.992012i	$-0.540261\pi$
$283$	−4.24424	−0.252294	−0.126147	+	0.992012i	$0.540261\pi$
$284$	−7.38431	−0.438178
$285$	0	0
$286$	−7.89489	−0.466834
$287$	33.5653	1.98130
$288$	0	0
$289$	−7.04975	−0.414691
$290$	12.1576	0.713918
$291$	0	0
$292$	3.48056	0.203685
$293$	16.0572	0.938071	0.469035	−	0.883179i	$-0.344602\pi$
$293$	16.0572	0.938071	0.469035	+	0.883179i	$0.344602\pi$
$294$	0	0
$295$	−9.84254	−0.573055
$296$	13.8878	0.807212
$297$	0	0
$298$	−16.5126	−0.956548
$299$	−35.6800	−2.06343
$300$	0	0
$301$	21.0006	1.21046
$302$	2.73389	0.157318
$303$	0	0
$304$	13.9861	0.802157
$305$	−8.14856	−0.466585
$306$	0	0
$307$	25.6485	1.46383	0.731917	−	0.681393i	$-0.238626\pi$
$307$	25.6485	1.46383	0.731917	+	0.681393i	$0.238626\pi$
$308$	−2.26444	−0.129028
$309$	0	0
$310$	0	0
$311$	−32.1626	−1.82377	−0.911887	−	0.410442i	$-0.865374\pi$
$311$	−32.1626	−1.82377	−0.911887	+	0.410442i	$0.865374\pi$
$312$	0	0
$313$	18.7995	1.06261	0.531305	−	0.847181i	$-0.321702\pi$
$313$	18.7995	1.06261	0.531305	+	0.847181i	$0.321702\pi$
$314$	−2.18054	−0.123055
$315$	0	0
$316$	2.30721	0.129791
$317$	8.58272	0.482054	0.241027	−	0.970518i	$-0.422516\pi$
$317$	8.58272	0.482054	0.241027	+	0.970518i	$0.422516\pi$
$318$	0	0
$319$	6.56766	0.367718
$320$	−13.9765	−0.781312
$321$	0	0
$322$	28.2405	1.57378
$323$	16.6292	0.925275
$324$	0	0
$325$	−15.7717	−0.874855
$326$	13.3181	0.737622
$327$	0	0
$328$	25.0189	1.38144
$329$	−40.3201	−2.22292
$330$	0	0
$331$	−16.2416	−0.892719	−0.446360	−	0.894854i	$-0.647280\pi$
$331$	−16.2416	−0.892719	−0.446360	+	0.894854i	$0.647280\pi$
$332$	5.06409	0.277928
$333$	0	0
$334$	−18.2904	−1.00080
$335$	1.78932	0.0977610
$336$	0	0
$337$	−16.9451	−0.923058	−0.461529	−	0.887125i	$-0.652699\pi$
$337$	−16.9451	−0.923058	−0.461529	+	0.887125i	$0.652699\pi$
$338$	−32.3440	−1.75928
$339$	0	0
$340$	−2.65155	−0.143801
$341$	0	0
$342$	0	0
$343$	12.0976	0.653209
$344$	15.6535	0.843979
$345$	0	0
$346$	−1.38549	−0.0744847
$347$	19.9422	1.07055	0.535276	−	0.844677i	$-0.320208\pi$
$347$	19.9422	1.07055	0.535276	+	0.844677i	$0.320208\pi$
$348$	0	0
$349$	15.3209	0.820109	0.410054	−	0.912061i	$-0.365510\pi$
$349$	15.3209	0.820109	0.410054	+	0.912061i	$0.365510\pi$
$350$	12.4831	0.667252
$351$	0	0
$352$	−3.02111	−0.161026
$353$	−16.9159	−0.900345	−0.450172	−	0.892942i	$-0.648637\pi$
$353$	−16.9159	−0.900345	−0.450172	+	0.892942i	$0.648637\pi$
$354$	0	0
$355$	−21.9329	−1.16408
$356$	−2.32725	−0.123344
$357$	0	0
$358$	0.0430546	0.00227551
$359$	23.0728	1.21773	0.608867	−	0.793272i	$-0.291624\pi$
$359$	23.0728	1.21773	0.608867	+	0.793272i	$0.291624\pi$
$360$	0	0
$361$	8.79141	0.462706
$362$	23.8189	1.25189
$363$	0	0
$364$	−13.7948	−0.723043
$365$	10.3380	0.541113
$366$	0	0
$367$	−18.9889	−0.991211	−0.495606	−	0.868548i	$-0.665054\pi$
$367$	−18.9889	−0.991211	−0.495606	+	0.868548i	$0.665054\pi$
$368$	15.0245	0.783204
$369$	0	0
$370$	8.66676	0.450563
$371$	15.5557	0.807613
$372$	0	0
$373$	10.9831	0.568683	0.284341	−	0.958723i	$-0.408225\pi$
$373$	10.9831	0.568683	0.284341	+	0.958723i	$0.408225\pi$
$374$	3.95272	0.204390
$375$	0	0
$376$	−30.0538	−1.54991
$377$	40.0097	2.06060
$378$	0	0
$379$	−6.34815	−0.326082	−0.163041	−	0.986619i	$-0.552130\pi$
$379$	−6.34815	−0.326082	−0.163041	+	0.986619i	$0.552130\pi$
$380$	−4.43138	−0.227325
$381$	0	0
$382$	−1.84138	−0.0942130
$383$	−0.745363	−0.0380863	−0.0190431	−	0.999819i	$-0.506062\pi$
$383$	−0.745363	−0.0380863	−0.0190431	+	0.999819i	$0.506062\pi$
$384$	0	0
$385$	−6.72583	−0.342780
$386$	26.0363	1.32521
$387$	0	0
$388$	−2.49205	−0.126515
$389$	4.21677	0.213799	0.106899	−	0.994270i	$-0.465908\pi$
$389$	4.21677	0.213799	0.106899	+	0.994270i	$0.465908\pi$
$390$	0	0
$391$	17.8638	0.903414
$392$	30.4919	1.54007
$393$	0	0
$394$	27.2600	1.37334
$395$	6.85287	0.344805
$396$	0	0
$397$	26.0337	1.30660	0.653298	−	0.757101i	$-0.273385\pi$
$397$	26.0337	1.30660	0.653298	+	0.757101i	$0.273385\pi$
$398$	−10.6732	−0.535000
$399$	0	0
$400$	6.64127	0.332064
$401$	5.33020	0.266177	0.133089	−	0.991104i	$-0.457510\pi$
$401$	5.33020	0.266177	0.133089	+	0.991104i	$0.457510\pi$
$402$	0	0
$403$	0	0
$404$	6.27734	0.312309
$405$	0	0
$406$	−31.6673	−1.57162
$407$	4.68188	0.232072
$408$	0	0
$409$	−8.16440	−0.403704	−0.201852	−	0.979416i	$-0.564696\pi$
$409$	−8.16440	−0.403704	−0.201852	+	0.979416i	$0.564696\pi$
$410$	15.6132	0.771082
$411$	0	0
$412$	−3.62823	−0.178750
$413$	25.6372	1.26153
$414$	0	0
$415$	15.0414	0.738351
$416$	−18.4043	−0.902347
$417$	0	0
$418$	6.60593	0.323107
$419$	1.27164	0.0621237	0.0310619	−	0.999517i	$-0.490111\pi$
$419$	1.27164	0.0621237	0.0310619	+	0.999517i	$0.490111\pi$
$420$	0	0
$421$	−20.6567	−1.00675	−0.503373	−	0.864069i	$-0.667908\pi$
$421$	−20.6567	−1.00675	−0.503373	+	0.864069i	$0.667908\pi$
$422$	−13.0172	−0.633666
$423$	0	0
$424$	11.5949	0.563099
$425$	7.89637	0.383030
$426$	0	0
$427$	21.2249	1.02714
$428$	−8.01343	−0.387344
$429$	0	0
$430$	9.76864	0.471086
$431$	27.1841	1.30941	0.654705	−	0.755884i	$-0.272793\pi$
$431$	27.1841	1.30941	0.654705	+	0.755884i	$0.272793\pi$
$432$	0	0
$433$	16.1720	0.777177	0.388589	−	0.921411i	$-0.372963\pi$
$433$	16.1720	0.777177	0.388589	+	0.921411i	$0.372963\pi$
$434$	0	0
$435$	0	0
$436$	−3.90358	−0.186948
$437$	29.8547	1.42815
$438$	0	0
$439$	7.60562	0.362996	0.181498	−	0.983391i	$-0.441905\pi$
$439$	7.60562	0.362996	0.181498	+	0.983391i	$0.441905\pi$
$440$	−5.01331	−0.239000
$441$	0	0
$442$	24.0796	1.14535
$443$	13.8515	0.658105	0.329053	−	0.944312i	$-0.393271\pi$
$443$	13.8515	0.658105	0.329053	+	0.944312i	$0.393271\pi$
$444$	0	0
$445$	−6.91240	−0.327679
$446$	29.9580	1.41855
$447$	0	0
$448$	36.4052	1.71998
$449$	13.8127	0.651861	0.325930	−	0.945394i	$-0.394323\pi$
$449$	13.8127	0.651861	0.325930	+	0.945394i	$0.394323\pi$
$450$	0	0
$451$	8.43443	0.397162
$452$	6.99873	0.329193
$453$	0	0
$454$	−13.7790	−0.646682
$455$	−40.9732	−1.92085
$456$	0	0
$457$	−13.3999	−0.626821	−0.313410	−	0.949618i	$-0.601472\pi$
$457$	−13.3999	−0.626821	−0.313410	+	0.949618i	$0.601472\pi$
$458$	24.3071	1.13580
$459$	0	0
$460$	−4.76038	−0.221954
$461$	1.65863	0.0772502	0.0386251	−	0.999254i	$-0.487702\pi$
$461$	1.65863	0.0772502	0.0386251	+	0.999254i	$0.487702\pi$
$462$	0	0
$463$	−33.5030	−1.55702	−0.778509	−	0.627633i	$-0.784024\pi$
$463$	−33.5030	−1.55702	−0.778509	+	0.627633i	$0.784024\pi$
$464$	−16.8476	−0.782131
$465$	0	0
$466$	−14.9133	−0.690845
$467$	2.54201	0.117630	0.0588151	−	0.998269i	$-0.481268\pi$
$467$	2.54201	0.117630	0.0588151	+	0.998269i	$0.481268\pi$
$468$	0	0
$469$	−4.66071	−0.215212
$470$	−18.7552	−0.865115
$471$	0	0
$472$	19.1095	0.879585
$473$	5.27713	0.242643
$474$	0	0
$475$	13.1967	0.605507
$476$	6.90660	0.316564
$477$	0	0
$478$	−20.1547	−0.921855
$479$	13.8704	0.633755	0.316877	−	0.948467i	$-0.397366\pi$
$479$	13.8704	0.633755	0.316877	+	0.948467i	$0.397366\pi$
$480$	0	0
$481$	28.5216	1.30047
$482$	−25.7900	−1.17470
$483$	0	0
$484$	5.28281	0.240128
$485$	−7.40189	−0.336103
$486$	0	0
$487$	−13.8012	−0.625390	−0.312695	−	0.949854i	$-0.601232\pi$
$487$	−13.8012	−0.625390	−0.312695	+	0.949854i	$0.601232\pi$
$488$	15.8206	0.716165
$489$	0	0
$490$	19.0286	0.859626
$491$	−6.49985	−0.293334	−0.146667	−	0.989186i	$-0.546855\pi$
$491$	−6.49985	−0.293334	−0.146667	+	0.989186i	$0.546855\pi$
$492$	0	0
$493$	−20.0315	−0.902175
$494$	40.2428	1.81061
$495$	0	0
$496$	0	0
$497$	57.1293	2.56260
$498$	0	0
$499$	2.76396	0.123732	0.0618659	−	0.998084i	$-0.480295\pi$
$499$	2.76396	0.123732	0.0618659	+	0.998084i	$0.480295\pi$
$500$	−6.30718	−0.282066
$501$	0	0
$502$	−9.35349	−0.417467
$503$	13.7691	0.613934	0.306967	−	0.951720i	$-0.400686\pi$
$503$	13.7691	0.613934	0.306967	+	0.951720i	$0.400686\pi$
$504$	0	0
$505$	18.6449	0.829689
$506$	7.09638	0.315472
$507$	0	0
$508$	−5.16454	−0.229139
$509$	2.98130	0.132144	0.0660719	−	0.997815i	$-0.478953\pi$
$509$	2.98130	0.132144	0.0660719	+	0.997815i	$0.478953\pi$
$510$	0	0
$511$	−26.9277	−1.19121
$512$	24.0277	1.06189
$513$	0	0
$514$	−13.1544	−0.580217
$515$	−10.7766	−0.474872
$516$	0	0
$517$	−10.1318	−0.445596
$518$	−22.5746	−0.991872
$519$	0	0
$520$	−30.5406	−1.33930
$521$	36.0596	1.57980	0.789899	−	0.613237i	$-0.210133\pi$
$521$	36.0596	1.57980	0.789899	+	0.613237i	$0.210133\pi$
$522$	0	0
$523$	3.24174	0.141751	0.0708756	−	0.997485i	$-0.477421\pi$
$523$	3.24174	0.141751	0.0708756	+	0.997485i	$0.477421\pi$
$524$	0.300095	0.0131097
$525$	0	0
$526$	−30.0541	−1.31042
$527$	0	0
$528$	0	0
$529$	9.07127	0.394403
$530$	7.23588	0.314307
$531$	0	0
$532$	11.5426	0.500434
$533$	51.3819	2.22560
$534$	0	0
$535$	−23.8015	−1.02903
$536$	−3.47400	−0.150054
$537$	0	0
$538$	−8.52915	−0.367718
$539$	10.2795	0.442769
$540$	0	0
$541$	−10.0578	−0.432419	−0.216209	−	0.976347i	$-0.569369\pi$
$541$	−10.0578	−0.432419	−0.216209	+	0.976347i	$0.569369\pi$
$542$	12.5074	0.537238
$543$	0	0
$544$	9.21446	0.395067
$545$	−11.5944	−0.496650
$546$	0	0
$547$	22.7733	0.973714	0.486857	−	0.873482i	$-0.338143\pi$
$547$	22.7733	0.973714	0.486857	+	0.873482i	$0.338143\pi$
$548$	8.98510	0.383824
$549$	0	0
$550$	3.13682	0.133754
$551$	−33.4775	−1.42619
$552$	0	0
$553$	−17.8499	−0.759056
$554$	−31.0355	−1.31857
$555$	0	0
$556$	−7.82057	−0.331666
$557$	−34.4919	−1.46147	−0.730734	−	0.682662i	$-0.760822\pi$
$557$	−34.4919	−1.46147	−0.730734	+	0.682662i	$0.760822\pi$
$558$	0	0
$559$	32.1478	1.35971
$560$	17.2534	0.729088
$561$	0	0
$562$	18.3756	0.775129
$563$	−18.4669	−0.778289	−0.389144	−	0.921177i	$-0.627229\pi$
$563$	−18.4669	−0.778289	−0.389144	+	0.921177i	$0.627229\pi$
$564$	0	0
$565$	20.7876	0.874541
$566$	5.14239	0.216151
$567$	0	0
$568$	42.5831	1.78675
$569$	22.3931	0.938766	0.469383	−	0.882995i	$-0.344476\pi$
$569$	22.3931	0.938766	0.469383	+	0.882995i	$0.344476\pi$
$570$	0	0
$571$	−6.62794	−0.277371	−0.138685	−	0.990336i	$-0.544288\pi$
$571$	−6.62794	−0.277371	−0.138685	+	0.990336i	$0.544288\pi$
$572$	−3.46641	−0.144938
$573$	0	0
$574$	−40.6683	−1.69746
$575$	14.1765	0.591200
$576$	0	0
$577$	35.9662	1.49729	0.748647	−	0.662969i	$-0.230704\pi$
$577$	35.9662	1.49729	0.748647	+	0.662969i	$0.230704\pi$
$578$	8.54160	0.355284
$579$	0	0
$580$	5.33803	0.221650
$581$	−39.1788	−1.62541
$582$	0	0
$583$	3.90890	0.161890
$584$	−20.0714	−0.830559
$585$	0	0
$586$	−19.4551	−0.803685
$587$	23.8767	0.985496	0.492748	−	0.870172i	$-0.335992\pi$
$587$	23.8767	0.985496	0.492748	+	0.870172i	$0.335992\pi$
$588$	0	0
$589$	0	0
$590$	11.9254	0.490960
$591$	0	0
$592$	−12.0101	−0.493613
$593$	15.4202	0.633233	0.316616	−	0.948554i	$-0.397453\pi$
$593$	15.4202	0.633233	0.316616	+	0.948554i	$0.397453\pi$
$594$	0	0
$595$	20.5140	0.840991
$596$	−7.25018	−0.296979
$597$	0	0
$598$	43.2305	1.76783
$599$	−11.5339	−0.471262	−0.235631	−	0.971843i	$-0.575716\pi$
$599$	−11.5339	−0.471262	−0.235631	+	0.971843i	$0.575716\pi$
$600$	0	0
$601$	2.88666	0.117749	0.0588747	−	0.998265i	$-0.481249\pi$
$601$	2.88666	0.117749	0.0588747	+	0.998265i	$0.481249\pi$
$602$	−25.4447	−1.03705
$603$	0	0
$604$	1.20037	0.0488424
$605$	15.6910	0.637930
$606$	0	0
$607$	20.7292	0.841370	0.420685	−	0.907207i	$-0.361790\pi$
$607$	20.7292	0.841370	0.420685	+	0.907207i	$0.361790\pi$
$608$	15.3996	0.624534
$609$	0	0
$610$	9.87293	0.399743
$611$	−61.7220	−2.49701
$612$	0	0
$613$	−38.8832	−1.57048	−0.785239	−	0.619193i	$-0.787460\pi$
$613$	−38.8832	−1.57048	−0.785239	+	0.619193i	$0.787460\pi$
$614$	−31.0761	−1.25413
$615$	0	0
$616$	13.0583	0.526136
$617$	26.6236	1.07182	0.535912	−	0.844274i	$-0.319968\pi$
$617$	26.6236	1.07182	0.535912	+	0.844274i	$0.319968\pi$
$618$	0	0
$619$	37.3450	1.50102	0.750510	−	0.660859i	$-0.229808\pi$
$619$	37.3450	1.50102	0.750510	+	0.660859i	$0.229808\pi$
$620$	0	0
$621$	0	0
$622$	38.9687	1.56250
$623$	18.0050	0.721355
$624$	0	0
$625$	−6.21714	−0.248685
$626$	−22.7778	−0.910383
$627$	0	0
$628$	−0.957411	−0.0382049
$629$	−14.2799	−0.569375
$630$	0	0
$631$	1.53031	0.0609208	0.0304604	−	0.999536i	$-0.490303\pi$
$631$	1.53031	0.0609208	0.0304604	+	0.999536i	$0.490303\pi$
$632$	−13.3050	−0.529244
$633$	0	0
$634$	−10.3990	−0.412996
$635$	−15.3397	−0.608738
$636$	0	0
$637$	62.6218	2.48116
$638$	−7.95749	−0.315040
$639$	0	0
$640$	7.70281	0.304480
$641$	−15.8558	−0.626268	−0.313134	−	0.949709i	$-0.601379\pi$
$641$	−15.8558	−0.626268	−0.313134	+	0.949709i	$0.601379\pi$
$642$	0	0
$643$	36.4143	1.43604	0.718020	−	0.696022i	$-0.245048\pi$
$643$	36.4143	1.43604	0.718020	+	0.696022i	$0.245048\pi$
$644$	12.3995	0.488610
$645$	0	0
$646$	−20.1483	−0.792723
$647$	−35.0613	−1.37840	−0.689201	−	0.724570i	$-0.742038\pi$
$647$	−35.0613	−1.37840	−0.689201	+	0.724570i	$0.742038\pi$
$648$	0	0
$649$	6.44222	0.252879
$650$	19.1092	0.749525
$651$	0	0
$652$	5.84758	0.229009
$653$	10.7222	0.419592	0.209796	−	0.977745i	$-0.432720\pi$
$653$	10.7222	0.419592	0.209796	+	0.977745i	$0.432720\pi$
$654$	0	0
$655$	0.891341	0.0348276
$656$	−21.6363	−0.844757
$657$	0	0
$658$	48.8525	1.90447
$659$	21.0146	0.818613	0.409307	−	0.912397i	$-0.365771\pi$
$659$	21.0146	0.818613	0.409307	+	0.912397i	$0.365771\pi$
$660$	0	0
$661$	39.8155	1.54864	0.774322	−	0.632792i	$-0.218091\pi$
$661$	39.8155	1.54864	0.774322	+	0.632792i	$0.218091\pi$
$662$	19.6786	0.764830
$663$	0	0
$664$	−29.2031	−1.13330
$665$	34.2837	1.32947
$666$	0	0
$667$	−35.9630	−1.39249
$668$	−8.03076	−0.310719
$669$	0	0
$670$	−2.16797	−0.0837560
$671$	5.33347	0.205896
$672$	0	0
$673$	−20.9166	−0.806274	−0.403137	−	0.915140i	$-0.632080\pi$
$673$	−20.9166	−0.806274	−0.403137	+	0.915140i	$0.632080\pi$
$674$	20.5310	0.790823
$675$	0	0
$676$	−14.2013	−0.546203
$677$	−18.8335	−0.723829	−0.361914	−	0.932211i	$-0.617877\pi$
$677$	−18.8335	−0.723829	−0.361914	+	0.932211i	$0.617877\pi$
$678$	0	0
$679$	19.2800	0.739898
$680$	15.2907	0.586372
$681$	0	0
$682$	0	0
$683$	−5.57551	−0.213341	−0.106670	−	0.994294i	$-0.534019\pi$
$683$	−5.57551	−0.213341	−0.106670	+	0.994294i	$0.534019\pi$
$684$	0	0
$685$	26.6875	1.01968
$686$	−14.6577	−0.559632
$687$	0	0
$688$	−13.5371	−0.516097
$689$	23.8127	0.907192
$690$	0	0
$691$	35.1574	1.33745	0.668726	−	0.743509i	$-0.266840\pi$
$691$	35.1574	1.33745	0.668726	+	0.743509i	$0.266840\pi$
$692$	−0.608329	−0.0231252
$693$	0	0
$694$	−24.1623	−0.917188
$695$	−23.2286	−0.881113
$696$	0	0
$697$	−25.7252	−0.974413
$698$	−18.5631	−0.702622
$699$	0	0
$700$	5.48098	0.207161
$701$	40.5354	1.53100	0.765500	−	0.643436i	$-0.222492\pi$
$701$	40.5354	1.53100	0.765500	+	0.643436i	$0.222492\pi$
$702$	0	0
$703$	−23.8650	−0.900087
$704$	9.14805	0.344780
$705$	0	0
$706$	20.4956	0.771364
$707$	−48.5652	−1.82648
$708$	0	0
$709$	−10.1325	−0.380535	−0.190268	−	0.981732i	$-0.560936\pi$
$709$	−10.1325	−0.380535	−0.190268	+	0.981732i	$0.560936\pi$
$710$	26.5742	0.997313
$711$	0	0
$712$	13.4206	0.502957
$713$	0	0
$714$	0	0
$715$	−10.2959	−0.385046
$716$	0.0189040	0.000706475 0
$717$	0	0
$718$	−27.9554	−1.04328
$719$	33.9133	1.26475	0.632377	−	0.774661i	$-0.282079\pi$
$719$	33.9133	1.26475	0.632377	+	0.774661i	$0.282079\pi$
$720$	0	0
$721$	28.0701	1.04539
$722$	−10.6518	−0.396420
$723$	0	0
$724$	10.4582	0.388674
$725$	−15.8967	−0.590390
$726$	0	0
$727$	−33.9219	−1.25809	−0.629047	−	0.777367i	$-0.716555\pi$
$727$	−33.9219	−1.25809	−0.629047	+	0.777367i	$0.716555\pi$
$728$	79.5503	2.94833
$729$	0	0
$730$	−12.5256	−0.463595
$731$	−16.0954	−0.595309
$732$	0	0
$733$	−37.6389	−1.39022	−0.695112	−	0.718901i	$-0.744645\pi$
$733$	−37.6389	−1.39022	−0.695112	+	0.718901i	$0.744645\pi$
$734$	23.0072	0.849213
$735$	0	0
$736$	16.5429	0.609779
$737$	−1.17116	−0.0431403
$738$	0	0
$739$	−26.4449	−0.972792	−0.486396	−	0.873739i	$-0.661689\pi$
$739$	−26.4449	−0.972792	−0.486396	+	0.873739i	$0.661689\pi$
$740$	3.80531	0.139886
$741$	0	0
$742$	−18.8476	−0.691916
$743$	−30.4100	−1.11563	−0.557817	−	0.829964i	$-0.688361\pi$
$743$	−30.4100	−1.11563	−0.557817	+	0.829964i	$0.688361\pi$
$744$	0	0
$745$	−21.5345	−0.788961
$746$	−13.3073	−0.487215
$747$	0	0
$748$	1.73552	0.0634569
$749$	61.9966	2.26531
$750$	0	0
$751$	−22.7132	−0.828817	−0.414408	−	0.910091i	$-0.636011\pi$
$751$	−22.7132	−0.828817	−0.414408	+	0.910091i	$0.636011\pi$
$752$	25.9904	0.947774
$753$	0	0
$754$	−48.4764	−1.76541
$755$	3.56534	0.129756
$756$	0	0
$757$	19.7651	0.718375	0.359188	−	0.933265i	$-0.383054\pi$
$757$	19.7651	0.718375	0.359188	+	0.933265i	$0.383054\pi$
$758$	7.69152	0.279369
$759$	0	0
$760$	25.5544	0.926956
$761$	−27.2378	−0.987368	−0.493684	−	0.869641i	$-0.664350\pi$
$761$	−27.2378	−0.987368	−0.493684	+	0.869641i	$0.664350\pi$
$762$	0	0
$763$	30.2004	1.09333
$764$	−0.808493	−0.0292503
$765$	0	0
$766$	0.903094	0.0326301
$767$	39.2455	1.41707
$768$	0	0
$769$	0.631295	0.0227651	0.0113825	−	0.999935i	$-0.496377\pi$
$769$	0.631295	0.0227651	0.0113825	+	0.999935i	$0.496377\pi$
$770$	8.14913	0.293674
$771$	0	0
$772$	11.4318	0.411438
$773$	6.71017	0.241348	0.120674	−	0.992692i	$-0.461494\pi$
$773$	6.71017	0.241348	0.120674	+	0.992692i	$0.461494\pi$
$774$	0	0
$775$	0	0
$776$	14.3709	0.515886
$777$	0	0
$778$	−5.10911	−0.183170
$779$	−42.9930	−1.54038
$780$	0	0
$781$	14.3557	0.513687
$782$	−21.6441	−0.773993
$783$	0	0
$784$	−26.3693	−0.941761
$785$	−2.84370	−0.101496
$786$	0	0
$787$	32.7300	1.16670	0.583350	−	0.812221i	$-0.301742\pi$
$787$	32.7300	1.16670	0.583350	+	0.812221i	$0.301742\pi$
$788$	11.9691	0.426380
$789$	0	0
$790$	−8.30305	−0.295409
$791$	−54.1463	−1.92522
$792$	0	0
$793$	32.4911	1.15379
$794$	−31.5429	−1.11942
$795$	0	0
$796$	−4.68629	−0.166101
$797$	−42.1989	−1.49476	−0.747380	−	0.664397i	$-0.768689\pi$
$797$	−42.1989	−1.49476	−0.747380	+	0.664397i	$0.768689\pi$
$798$	0	0
$799$	30.9022	1.09324
$800$	7.31246	0.258534
$801$	0	0
$802$	−6.45816	−0.228045
$803$	−6.76649	−0.238784
$804$	0	0
$805$	36.8291	1.29805
$806$	0	0
$807$	0	0
$808$	−36.1995	−1.27350
$809$	−31.6233	−1.11182	−0.555908	−	0.831244i	$-0.687629\pi$
$809$	−31.6233	−1.11182	−0.555908	+	0.831244i	$0.687629\pi$
$810$	0	0
$811$	−32.2709	−1.13318	−0.566592	−	0.823999i	$-0.691738\pi$
$811$	−32.2709	−1.13318	−0.566592	+	0.823999i	$0.691738\pi$
$812$	−13.9042	−0.487941
$813$	0	0
$814$	−5.67264	−0.198826
$815$	17.3685	0.608391
$816$	0	0
$817$	−26.8992	−0.941084
$818$	9.89212	0.345870
$819$	0	0
$820$	6.85530	0.239397
$821$	−26.0209	−0.908135	−0.454068	−	0.890967i	$-0.650028\pi$
$821$	−26.0209	−0.908135	−0.454068	+	0.890967i	$0.650028\pi$
$822$	0	0
$823$	−23.7776	−0.828835	−0.414417	−	0.910087i	$-0.636015\pi$
$823$	−23.7776	−0.828835	−0.414417	+	0.910087i	$0.636015\pi$
$824$	20.9229	0.728885
$825$	0	0
$826$	−31.0625	−1.08080
$827$	2.19619	0.0763689	0.0381845	−	0.999271i	$-0.487843\pi$
$827$	2.19619	0.0763689	0.0381845	+	0.999271i	$0.487843\pi$
$828$	0	0
$829$	5.94449	0.206461	0.103230	−	0.994657i	$-0.467082\pi$
$829$	5.94449	0.206461	0.103230	+	0.994657i	$0.467082\pi$
$830$	−18.2244	−0.632577
$831$	0	0
$832$	55.7292	1.93206
$833$	−31.3527	−1.08631
$834$	0	0
$835$	−23.8529	−0.825465
$836$	2.90047	0.100315
$837$	0	0
$838$	−1.54074	−0.0532240
$839$	−3.95918	−0.136686	−0.0683431	−	0.997662i	$-0.521771\pi$
$839$	−3.95918	−0.136686	−0.0683431	+	0.997662i	$0.521771\pi$
$840$	0	0
$841$	11.3269	0.390583
$842$	25.0280	0.862521
$843$	0	0
$844$	−5.71545	−0.196734
$845$	−42.1806	−1.45106
$846$	0	0
$847$	−40.8709	−1.40434
$848$	−10.0273	−0.344338
$849$	0	0
$850$	−9.56737	−0.328158
$851$	−25.6369	−0.878820
$852$	0	0
$853$	17.9236	0.613694	0.306847	−	0.951759i	$-0.400726\pi$
$853$	17.9236	0.613694	0.306847	+	0.951759i	$0.400726\pi$
$854$	−25.7164	−0.879997
$855$	0	0
$856$	46.2110	1.57946
$857$	27.5000	0.939382	0.469691	−	0.882831i	$-0.344365\pi$
$857$	27.5000	0.939382	0.469691	+	0.882831i	$0.344365\pi$
$858$	0	0
$859$	6.75092	0.230338	0.115169	−	0.993346i	$-0.463259\pi$
$859$	6.75092	0.230338	0.115169	+	0.993346i	$0.463259\pi$
$860$	4.28912	0.146258
$861$	0	0
$862$	−32.9367	−1.12183
$863$	17.3676	0.591199	0.295600	−	0.955312i	$-0.404481\pi$
$863$	17.3676	0.591199	0.295600	+	0.955312i	$0.404481\pi$
$864$	0	0
$865$	−1.80686	−0.0614350
$866$	−19.5943	−0.665841
$867$	0	0
$868$	0	0
$869$	−4.48540	−0.152157
$870$	0	0
$871$	−7.13462	−0.241747
$872$	22.5108	0.762311
$873$	0	0
$874$	−36.1725	−1.22355
$875$	48.7960	1.64961
$876$	0	0
$877$	0.0133353	0.000450301 0	0.000225151	−	1.00000i	$-0.499928\pi$
$877$	0.0133353	0.000450301 0	0.000225151		1.00000i	$0.499928\pi$
$878$	−9.21509	−0.310994
$879$	0	0
$880$	4.33550	0.146150
$881$	−1.01830	−0.0343074	−0.0171537	−	0.999853i	$-0.505460\pi$
$881$	−1.01830	−0.0343074	−0.0171537	+	0.999853i	$0.505460\pi$
$882$	0	0
$883$	33.9955	1.14404	0.572019	−	0.820240i	$-0.306160\pi$
$883$	33.9955	1.14404	0.572019	+	0.820240i	$0.306160\pi$
$884$	10.5726	0.355596
$885$	0	0
$886$	−16.7827	−0.563827
$887$	−49.7027	−1.66885	−0.834427	−	0.551118i	$-0.814201\pi$
$887$	−49.7027	−1.66885	−0.834427	+	0.551118i	$0.814201\pi$
$888$	0	0
$889$	39.9559	1.34008
$890$	8.37518	0.280737
$891$	0	0
$892$	13.1537	0.440417
$893$	51.6450	1.72823
$894$	0	0
$895$	0.0561486	0.00187684
$896$	−20.0638	−0.670284
$897$	0	0
$898$	−16.7357	−0.558477
$899$	0	0
$900$	0	0
$901$	−11.9223	−0.397188
$902$	−10.2193	−0.340265
$903$	0	0
$904$	−40.3596	−1.34234
$905$	31.0628	1.03256
$906$	0	0
$907$	−14.0366	−0.466077	−0.233039	−	0.972467i	$-0.574867\pi$
$907$	−14.0366	−0.466077	−0.233039	+	0.972467i	$0.574867\pi$
$908$	−6.04996	−0.200775
$909$	0	0
$910$	49.6438	1.64568
$911$	24.2175	0.802361	0.401180	−	0.915999i	$-0.368600\pi$
$911$	24.2175	0.802361	0.401180	+	0.915999i	$0.368600\pi$
$912$	0	0
$913$	−9.84500	−0.325822
$914$	16.2355	0.537024
$915$	0	0
$916$	10.6725	0.352630
$917$	−2.32171	−0.0766696
$918$	0	0
$919$	27.1293	0.894914	0.447457	−	0.894305i	$-0.352330\pi$
$919$	27.1293	0.894914	0.447457	+	0.894305i	$0.352330\pi$
$920$	27.4517	0.905055
$921$	0	0
$922$	−2.00963	−0.0661835
$923$	87.4537	2.87857
$924$	0	0
$925$	−11.3323	−0.372603
$926$	40.5928	1.33396
$927$	0	0
$928$	−18.5503	−0.608943
$929$	18.6995	0.613511	0.306756	−	0.951788i	$-0.400757\pi$
$929$	18.6995	0.613511	0.306756	+	0.951788i	$0.400757\pi$
$930$	0	0
$931$	−52.3978	−1.71727
$932$	−6.54798	−0.214486
$933$	0	0
$934$	−3.07994	−0.100779
$935$	5.15483	0.168581
$936$	0	0
$937$	59.7556	1.95213	0.976066	−	0.217475i	$-0.0697822\pi$
$937$	59.7556	1.95213	0.976066	+	0.217475i	$0.0697822\pi$
$938$	5.64699	0.184381
$939$	0	0
$940$	−8.23487	−0.268592
$941$	10.8785	0.354630	0.177315	−	0.984154i	$-0.443259\pi$
$941$	10.8785	0.354630	0.177315	+	0.984154i	$0.443259\pi$
$942$	0	0
$943$	−46.1850	−1.50399
$944$	−16.5258	−0.537870
$945$	0	0
$946$	−6.39386	−0.207882
$947$	6.14249	0.199604	0.0998020	−	0.995007i	$-0.468179\pi$
$947$	6.14249	0.199604	0.0998020	+	0.995007i	$0.468179\pi$
$948$	0	0
$949$	−41.2209	−1.33809
$950$	−15.9894	−0.518763
$951$	0	0
$952$	−39.8283	−1.29084
$953$	−21.5939	−0.699496	−0.349748	−	0.936844i	$-0.613733\pi$
$953$	−21.5939	−0.699496	−0.349748	+	0.936844i	$0.613733\pi$
$954$	0	0
$955$	−2.40138	−0.0777070
$956$	−8.84933	−0.286208
$957$	0	0
$958$	−16.8056	−0.542965
$959$	−69.5140	−2.24472
$960$	0	0
$961$	0	0
$962$	−34.5573	−1.11417
$963$	0	0
$964$	−11.3236	−0.364709
$965$	33.9546	1.09304
$966$	0	0
$967$	24.9912	0.803662	0.401831	−	0.915714i	$-0.368374\pi$
$967$	24.9912	0.803662	0.401831	+	0.915714i	$0.368374\pi$
$968$	−30.4644	−0.979163
$969$	0	0
$970$	8.96825	0.287953
$971$	−6.37825	−0.204688	−0.102344	−	0.994749i	$-0.532634\pi$
$971$	−6.37825	−0.204688	−0.102344	+	0.994749i	$0.532634\pi$
$972$	0	0
$973$	60.5045	1.93969
$974$	16.7217	0.535798
$975$	0	0
$976$	−13.6816	−0.437938
$977$	6.43724	0.205946	0.102973	−	0.994684i	$-0.467165\pi$
$977$	6.43724	0.205946	0.102973	+	0.994684i	$0.467165\pi$
$978$	0	0
$979$	4.52437	0.144599
$980$	8.35491	0.266888
$981$	0	0
$982$	7.87532	0.251312
$983$	42.2068	1.34619	0.673094	−	0.739557i	$-0.264965\pi$
$983$	42.2068	1.34619	0.673094	+	0.739557i	$0.264965\pi$
$984$	0	0
$985$	35.5505	1.13273
$986$	24.2706	0.772932
$987$	0	0
$988$	17.6694	0.562138
$989$	−28.8963	−0.918849
$990$	0	0
$991$	8.03041	0.255094	0.127547	−	0.991833i	$-0.459290\pi$
$991$	8.03041	0.255094	0.127547	+	0.991833i	$0.459290\pi$
$992$	0	0
$993$	0	0
$994$	−69.2189	−2.19549
$995$	−13.9192	−0.441269
$996$	0	0
$997$	26.3013	0.832971	0.416485	−	0.909142i	$-0.363262\pi$
$997$	26.3013	0.832971	0.416485	+	0.909142i	$0.363262\pi$
$998$	−3.34886	−0.106006
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.bu.1.8		24
3.2	odd	2		2883.2.a.v.1.17	yes	24
31.30	odd	2		8649.2.a.bv.1.8		24
93.92	even	2		2883.2.a.u.1.17	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2883.2.a.u.1.17	✓	24	93.92	even	2
2883.2.a.v.1.17	yes	24	3.2	odd	2
8649.2.a.bu.1.8		24	1.1	even	1	trivial
8649.2.a.bv.1.8		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)

\(f(q)\)	\(=\)	\(q-1.21162 q^{2} -0.531985 q^{4} -1.58010 q^{5} +4.11574 q^{7} +3.06780 q^{8} +O(q^{10})\)	\(q-1.21162 q^{2} -0.531985 q^{4} -1.58010 q^{5} +4.11574 q^{7} +3.06780 q^{8} +1.91448 q^{10} +1.03422 q^{11} +6.30039 q^{13} -4.98670 q^{14} -2.65302 q^{16} -3.15440 q^{17} -5.27176 q^{19} +0.840589 q^{20} -1.25308 q^{22} -5.66315 q^{23} -2.50329 q^{25} -7.63366 q^{26} -2.18951 q^{28} +6.35035 q^{29} -2.92114 q^{32} +3.82193 q^{34} -6.50329 q^{35} +4.52696 q^{37} +6.38735 q^{38} -4.84742 q^{40} +8.15535 q^{41} +5.10252 q^{43} -0.550190 q^{44} +6.86157 q^{46} -9.79654 q^{47} +9.93935 q^{49} +3.03302 q^{50} -3.35171 q^{52} +3.77956 q^{53} -1.63417 q^{55} +12.6263 q^{56} -7.69419 q^{58} +6.22906 q^{59} +5.15699 q^{61} +8.84535 q^{64} -9.95524 q^{65} -1.13241 q^{67} +1.67809 q^{68} +7.87949 q^{70} +13.8807 q^{71} -6.54260 q^{73} -5.48494 q^{74} +2.80449 q^{76} +4.25659 q^{77} -4.33699 q^{79} +4.19204 q^{80} -9.88116 q^{82} -9.51925 q^{83} +4.98427 q^{85} -6.18229 q^{86} +3.17278 q^{88} +4.37466 q^{89} +25.9308 q^{91} +3.01271 q^{92} +11.8697 q^{94} +8.32990 q^{95} +4.68445 q^{97} -12.0427 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8}+O(q^{10}) \) 24 * q + 32 * q^4 + 16 * q^7 + 24 * q^8	\( 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8} - 8 q^{10} - 16 q^{11} + 32 q^{13} + 24 q^{14} + 48 q^{16} - 32 q^{17} + 32 q^{19} + 24 q^{20} + 32 q^{22} - 32 q^{23} + 40 q^{25} - 16 q^{26} + 8 q^{28} - 48 q^{29} + 48 q^{32} + 48 q^{35} + 64 q^{37} + 24 q^{38} + 32 q^{43} - 48 q^{44} + 32 q^{46} + 48 q^{47} + 56 q^{49} + 24 q^{50} + 64 q^{52} - 80 q^{53} + 48 q^{56} + 32 q^{58} + 32 q^{61} + 56 q^{64} - 16 q^{65} - 16 q^{67} - 80 q^{68} + 8 q^{70} + 32 q^{73} + 56 q^{76} - 96 q^{77} + 32 q^{79} + 72 q^{80} + 8 q^{82} - 48 q^{83} + 96 q^{85} + 32 q^{86} + 96 q^{88} + 16 q^{89} - 32 q^{92} + 48 q^{94} + 48 q^{95} + 16 q^{97} + 24 q^{98}+O(q^{100}) \) 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

Introduction

L-functions

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