LMFDB - Embedded newform 8649.2.a.bu.1.3

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			1.3
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)

$f(q)$	$=$	$q-2.39335 q^{2} +3.72812 q^{4} +3.81642 q^{5} +3.08197 q^{7} -4.13599 q^{8} +O(q^{10})$	$q-2.39335 q^{2} +3.72812 q^{4} +3.81642 q^{5} +3.08197 q^{7} -4.13599 q^{8} -9.13404 q^{10} -5.10289 q^{11} -2.96369 q^{13} -7.37623 q^{14} +2.44263 q^{16} -2.76477 q^{17} -0.941501 q^{19} +14.2281 q^{20} +12.2130 q^{22} -1.69550 q^{23} +9.56510 q^{25} +7.09315 q^{26} +11.4899 q^{28} -5.12553 q^{29} +2.42592 q^{32} +6.61705 q^{34} +11.7621 q^{35} +3.49060 q^{37} +2.25334 q^{38} -15.7847 q^{40} +3.59114 q^{41} +7.63468 q^{43} -19.0242 q^{44} +4.05792 q^{46} -8.51490 q^{47} +2.49854 q^{49} -22.8926 q^{50} -11.0490 q^{52} -3.59928 q^{53} -19.4748 q^{55} -12.7470 q^{56} +12.2672 q^{58} +13.2472 q^{59} +1.40700 q^{61} -10.6913 q^{64} -11.3107 q^{65} -0.347230 q^{67} -10.3074 q^{68} -28.1508 q^{70} +1.48005 q^{71} +8.48652 q^{73} -8.35422 q^{74} -3.51003 q^{76} -15.7269 q^{77} +1.72215 q^{79} +9.32211 q^{80} -8.59486 q^{82} +4.35067 q^{83} -10.5515 q^{85} -18.2725 q^{86} +21.1055 q^{88} +15.0354 q^{89} -9.13402 q^{91} -6.32102 q^{92} +20.3791 q^{94} -3.59317 q^{95} -17.0577 q^{97} -5.97987 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.39335	−1.69235	−0.846177	−	0.532903i	$-0.821101\pi$
$2$	−2.39335	−1.69235	−0.846177	+	0.532903i	$0.821101\pi$
$3$	0	0
$3$	0	0
$4$	3.72812	1.86406
$5$	3.81642	1.70676	0.853379	−	0.521292i	$-0.174550\pi$
$5$	3.81642	1.70676	0.853379	+	0.521292i	$0.174550\pi$
$6$	0	0
$7$	3.08197	1.16488	0.582438	−	0.812875i	$-0.302099\pi$
$7$	3.08197	1.16488	0.582438	+	0.812875i	$0.302099\pi$
$8$	−4.13599	−1.46229
$9$	0	0
$10$	−9.13404	−2.88844
$11$	−5.10289	−1.53858	−0.769289	−	0.638901i	$-0.779389\pi$
$11$	−5.10289	−1.53858	−0.769289	+	0.638901i	$0.779389\pi$
$12$	0	0
$13$	−2.96369	−0.821981	−0.410990	−	0.911640i	$-0.634817\pi$
$13$	−2.96369	−0.821981	−0.410990	+	0.911640i	$0.634817\pi$
$14$	−7.37623	−1.97138
$15$	0	0
$16$	2.44263	0.610657
$17$	−2.76477	−0.670554	−0.335277	−	0.942120i	$-0.608830\pi$
$17$	−2.76477	−0.670554	−0.335277	+	0.942120i	$0.608830\pi$
$18$	0	0
$19$	−0.941501	−0.215995	−0.107998	−	0.994151i	$-0.534444\pi$
$19$	−0.941501	−0.215995	−0.107998	+	0.994151i	$0.534444\pi$
$20$	14.2281	3.18150
$21$	0	0
$22$	12.2130	2.60382
$23$	−1.69550	−0.353536	−0.176768	−	0.984253i	$-0.556564\pi$
$23$	−1.69550	−0.353536	−0.176768	+	0.984253i	$0.556564\pi$
$24$	0	0
$25$	9.56510	1.91302
$26$	7.09315	1.39108
$27$	0	0
$28$	11.4899	2.17140
$29$	−5.12553	−0.951787	−0.475894	−	0.879503i	$-0.657875\pi$
$29$	−5.12553	−0.951787	−0.475894	+	0.879503i	$0.657875\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	2.42592	0.428845
$33$	0	0
$34$	6.61705	1.13481
$35$	11.7621	1.98816
$36$	0	0
$37$	3.49060	0.573851	0.286925	−	0.957953i	$-0.407367\pi$
$37$	3.49060	0.573851	0.286925	+	0.957953i	$0.407367\pi$
$38$	2.25334	0.365540
$39$	0	0
$40$	−15.7847	−2.49578
$41$	3.59114	0.560842	0.280421	−	0.959877i	$-0.409526\pi$
$41$	3.59114	0.560842	0.280421	+	0.959877i	$0.409526\pi$
$42$	0	0
$43$	7.63468	1.16428	0.582139	−	0.813089i	$-0.302216\pi$
$43$	7.63468	1.16428	0.582139	+	0.813089i	$0.302216\pi$
$44$	−19.0242	−2.86800
$45$	0	0
$46$	4.05792	0.598308
$47$	−8.51490	−1.24203	−0.621013	−	0.783800i	$-0.713278\pi$
$47$	−8.51490	−1.24203	−0.621013	+	0.783800i	$0.713278\pi$
$48$	0	0
$49$	2.49854	0.356934
$50$	−22.8926	−3.23751
$51$	0	0
$52$	−11.0490	−1.53222
$53$	−3.59928	−0.494400	−0.247200	−	0.968965i	$-0.579510\pi$
$53$	−3.59928	−0.494400	−0.247200	+	0.968965i	$0.579510\pi$
$54$	0	0
$55$	−19.4748	−2.62598
$56$	−12.7470	−1.70339
$57$	0	0
$58$	12.2672	1.61076
$59$	13.2472	1.72463	0.862317	−	0.506369i	$-0.169012\pi$
$59$	13.2472	1.72463	0.862317	+	0.506369i	$0.169012\pi$
$60$	0	0
$61$	1.40700	0.180148	0.0900740	−	0.995935i	$-0.471290\pi$
$61$	1.40700	0.180148	0.0900740	+	0.995935i	$0.471290\pi$
$62$	0	0
$63$	0	0
$64$	−10.6913	−1.33642
$65$	−11.3107	−1.40292
$66$	0	0
$67$	−0.347230	−0.0424209	−0.0212105	−	0.999775i	$-0.506752\pi$
$67$	−0.347230	−0.0424209	−0.0212105	+	0.999775i	$0.506752\pi$
$68$	−10.3074	−1.24995
$69$	0	0
$70$	−28.1508	−3.36467
$71$	1.48005	0.175649	0.0878247	−	0.996136i	$-0.472008\pi$
$71$	1.48005	0.175649	0.0878247	+	0.996136i	$0.472008\pi$
$72$	0	0
$73$	8.48652	0.993272	0.496636	−	0.867959i	$-0.334568\pi$
$73$	8.48652	0.993272	0.496636	+	0.867959i	$0.334568\pi$
$74$	−8.35422	−0.971158
$75$	0	0
$76$	−3.51003	−0.402628
$77$	−15.7269	−1.79225
$78$	0	0
$79$	1.72215	0.193757	0.0968785	−	0.995296i	$-0.469114\pi$
$79$	1.72215	0.193757	0.0968785	+	0.995296i	$0.469114\pi$
$80$	9.32211	1.04224
$81$	0	0
$82$	−8.59486	−0.949143
$83$	4.35067	0.477549	0.238774	−	0.971075i	$-0.423254\pi$
$83$	4.35067	0.477549	0.238774	+	0.971075i	$0.423254\pi$
$84$	0	0
$85$	−10.5515	−1.14447
$86$	−18.2725	−1.97037
$87$	0	0
$88$	21.1055	2.24985
$89$	15.0354	1.59375	0.796875	−	0.604144i	$-0.206485\pi$
$89$	15.0354	1.59375	0.796875	+	0.604144i	$0.206485\pi$
$90$	0	0
$91$	−9.13402	−0.957505
$92$	−6.32102	−0.659012
$93$	0	0
$94$	20.3791	2.10195
$95$	−3.59317	−0.368651
$96$	0	0
$97$	−17.0577	−1.73195	−0.865976	−	0.500086i	$-0.833302\pi$
$97$	−17.0577	−1.73195	−0.865976	+	0.500086i	$0.833302\pi$
$98$	−5.97987	−0.604058
$99$	0	0
$100$	35.6598	3.56598
$101$	−12.9798	−1.29154	−0.645770	−	0.763532i	$-0.723464\pi$
$101$	−12.9798	−1.29154	−0.645770	+	0.763532i	$0.723464\pi$
$102$	0	0
$103$	5.43306	0.535335	0.267667	−	0.963511i	$-0.413747\pi$
$103$	5.43306	0.535335	0.267667	+	0.963511i	$0.413747\pi$
$104$	12.2578	1.20198
$105$	0	0
$106$	8.61434	0.836699
$107$	10.5068	1.01573	0.507865	−	0.861437i	$-0.330435\pi$
$107$	10.5068	1.01573	0.507865	+	0.861437i	$0.330435\pi$
$108$	0	0
$109$	−0.658227	−0.0630468	−0.0315234	−	0.999503i	$-0.510036\pi$
$109$	−0.658227	−0.0630468	−0.0315234	+	0.999503i	$0.510036\pi$
$110$	46.6100	4.44408
$111$	0	0
$112$	7.52811	0.711340
$113$	13.1798	1.23985	0.619925	−	0.784661i	$-0.287163\pi$
$113$	13.1798	1.23985	0.619925	+	0.784661i	$0.287163\pi$
$114$	0	0
$115$	−6.47075	−0.603400
$116$	−19.1086	−1.77419
$117$	0	0
$118$	−31.7051	−2.91869
$119$	−8.52093	−0.781112
$120$	0	0
$121$	15.0395	1.36722
$122$	−3.36744	−0.304874
$123$	0	0
$124$	0	0
$125$	17.4224	1.55830
$126$	0	0
$127$	−8.97813	−0.796680	−0.398340	−	0.917238i	$-0.630414\pi$
$127$	−8.97813	−0.796680	−0.398340	+	0.917238i	$0.630414\pi$
$128$	20.7362	1.83284
$129$	0	0
$130$	27.0705	2.37424
$131$	1.03984	0.0908516	0.0454258	−	0.998968i	$-0.485536\pi$
$131$	1.03984	0.0908516	0.0454258	+	0.998968i	$0.485536\pi$
$132$	0	0
$133$	−2.90168	−0.251607
$134$	0.831043	0.0717912
$135$	0	0
$136$	11.4350	0.980547
$137$	17.3846	1.48527	0.742633	−	0.669699i	$-0.233577\pi$
$137$	17.3846	1.48527	0.742633	+	0.669699i	$0.233577\pi$
$138$	0	0
$139$	2.64255	0.224138	0.112069	−	0.993700i	$-0.464252\pi$
$139$	2.64255	0.224138	0.112069	+	0.993700i	$0.464252\pi$
$140$	43.8505	3.70605
$141$	0	0
$142$	−3.54227	−0.297261
$143$	15.1234	1.26468
$144$	0	0
$145$	−19.5612	−1.62447
$146$	−20.3112	−1.68097
$147$	0	0
$148$	13.0134	1.06969
$149$	16.4449	1.34722	0.673611	−	0.739086i	$-0.264742\pi$
$149$	16.4449	1.34722	0.673611	+	0.739086i	$0.264742\pi$
$150$	0	0
$151$	20.5048	1.66866	0.834329	−	0.551267i	$-0.185856\pi$
$151$	20.5048	1.66866	0.834329	+	0.551267i	$0.185856\pi$
$152$	3.89404	0.315848
$153$	0	0
$154$	37.6401	3.03312
$155$	0	0
$156$	0	0
$157$	24.2469	1.93512	0.967558	−	0.252647i	$-0.0813012\pi$
$157$	24.2469	1.93512	0.967558	+	0.252647i	$0.0813012\pi$
$158$	−4.12171	−0.327905
$159$	0	0
$160$	9.25832	0.731935
$161$	−5.22548	−0.411825
$162$	0	0
$163$	21.0271	1.64697	0.823485	−	0.567338i	$-0.192027\pi$
$163$	21.0271	1.64697	0.823485	+	0.567338i	$0.192027\pi$
$164$	13.3882	1.04544
$165$	0	0
$166$	−10.4127	−0.808181
$167$	22.7279	1.75874	0.879369	−	0.476141i	$-0.157965\pi$
$167$	22.7279	1.75874	0.879369	+	0.476141i	$0.157965\pi$
$168$	0	0
$169$	−4.21651	−0.324347
$170$	25.2535	1.93685
$171$	0	0
$172$	28.4630	2.17028
$173$	8.24863	0.627132	0.313566	−	0.949566i	$-0.398476\pi$
$173$	8.24863	0.627132	0.313566	+	0.949566i	$0.398476\pi$
$174$	0	0
$175$	29.4793	2.22843
$176$	−12.4645	−0.939544
$177$	0	0
$178$	−35.9850	−2.69719
$179$	−18.5066	−1.38325	−0.691624	−	0.722258i	$-0.743104\pi$
$179$	−18.5066	−1.38325	−0.691624	+	0.722258i	$0.743104\pi$
$180$	0	0
$181$	−7.55355	−0.561451	−0.280726	−	0.959788i	$-0.590575\pi$
$181$	−7.55355	−0.561451	−0.280726	+	0.959788i	$0.590575\pi$
$182$	21.8609	1.62044
$183$	0	0
$184$	7.01257	0.516973
$185$	13.3216	0.979424
$186$	0	0
$187$	14.1083	1.03170
$188$	−31.7445	−2.31521
$189$	0	0
$190$	8.59970	0.623888
$191$	−10.7554	−0.778234	−0.389117	−	0.921188i	$-0.627220\pi$
$191$	−10.7554	−0.778234	−0.389117	+	0.921188i	$0.627220\pi$
$192$	0	0
$193$	−12.5417	−0.902774	−0.451387	−	0.892328i	$-0.649071\pi$
$193$	−12.5417	−0.902774	−0.451387	+	0.892328i	$0.649071\pi$
$194$	40.8251	2.93107
$195$	0	0
$196$	9.31484	0.665346
$197$	13.2494	0.943980	0.471990	−	0.881604i	$-0.343536\pi$
$197$	13.2494	0.943980	0.471990	+	0.881604i	$0.343536\pi$
$198$	0	0
$199$	−8.63547	−0.612152	−0.306076	−	0.952007i	$-0.599016\pi$
$199$	−8.63547	−0.612152	−0.306076	+	0.952007i	$0.599016\pi$
$200$	−39.5612	−2.79740
$201$	0	0
$202$	31.0652	2.18574
$203$	−15.7967	−1.10871
$204$	0	0
$205$	13.7053	0.957221
$206$	−13.0032	−0.905976
$207$	0	0
$208$	−7.23921	−0.501949
$209$	4.80437	0.332325
$210$	0	0
$211$	17.7220	1.22003	0.610017	−	0.792388i	$-0.291163\pi$
$211$	17.7220	1.22003	0.610017	+	0.792388i	$0.291163\pi$
$212$	−13.4186	−0.921590
$213$	0	0
$214$	−25.1464	−1.71897
$215$	29.1372	1.98714
$216$	0	0
$217$	0	0
$218$	1.57537	0.106697
$219$	0	0
$220$	−72.6043	−4.89498
$221$	8.19392	0.551183
$222$	0	0
$223$	−7.60638	−0.509361	−0.254680	−	0.967025i	$-0.581970\pi$
$223$	−7.60638	−0.509361	−0.254680	+	0.967025i	$0.581970\pi$
$224$	7.47660	0.499551
$225$	0	0
$226$	−31.5438	−2.09826
$227$	10.2803	0.682328	0.341164	−	0.940004i	$-0.389179\pi$
$227$	10.2803	0.682328	0.341164	+	0.940004i	$0.389179\pi$
$228$	0	0
$229$	−1.89362	−0.125134	−0.0625668	−	0.998041i	$-0.519929\pi$
$229$	−1.89362	−0.125134	−0.0625668	+	0.998041i	$0.519929\pi$
$230$	15.4868	1.02117
$231$	0	0
$232$	21.1991	1.39179
$233$	10.7724	0.705725	0.352862	−	0.935675i	$-0.385208\pi$
$233$	10.7724	0.705725	0.352862	+	0.935675i	$0.385208\pi$
$234$	0	0
$235$	−32.4965	−2.11984
$236$	49.3870	3.21482
$237$	0	0
$238$	20.3935	1.32192
$239$	13.1113	0.848102	0.424051	−	0.905638i	$-0.360608\pi$
$239$	13.1113	0.848102	0.424051	+	0.905638i	$0.360608\pi$
$240$	0	0
$241$	13.3470	0.859754	0.429877	−	0.902887i	$-0.358557\pi$
$241$	13.3470	0.859754	0.429877	+	0.902887i	$0.358557\pi$
$242$	−35.9947	−2.31383
$243$	0	0
$244$	5.24547	0.335807
$245$	9.53548	0.609199
$246$	0	0
$247$	2.79032	0.177544
$248$	0	0
$249$	0	0
$250$	−41.6978	−2.63720
$251$	−10.4907	−0.662165	−0.331082	−	0.943602i	$-0.607414\pi$
$251$	−10.4907	−0.662165	−0.331082	+	0.943602i	$0.607414\pi$
$252$	0	0
$253$	8.65194	0.543943
$254$	21.4878	1.34826
$255$	0	0
$256$	−28.2464	−1.76540
$257$	−10.2725	−0.640782	−0.320391	−	0.947285i	$-0.603814\pi$
$257$	−10.2725	−0.640782	−0.320391	+	0.947285i	$0.603814\pi$
$258$	0	0
$259$	10.7579	0.668464
$260$	−42.1677	−2.61513
$261$	0	0
$262$	−2.48871	−0.153753
$263$	9.79214	0.603809	0.301905	−	0.953338i	$-0.402378\pi$
$263$	9.79214	0.603809	0.301905	+	0.953338i	$0.402378\pi$
$264$	0	0
$265$	−13.7364	−0.843820
$266$	6.94473	0.425808
$267$	0	0
$268$	−1.29452	−0.0790751
$269$	18.0649	1.10144	0.550718	−	0.834692i	$-0.314354\pi$
$269$	18.0649	1.10144	0.550718	+	0.834692i	$0.314354\pi$
$270$	0	0
$271$	19.0649	1.15811	0.579054	−	0.815289i	$-0.303422\pi$
$271$	19.0649	1.15811	0.579054	+	0.815289i	$0.303422\pi$
$272$	−6.75330	−0.409479
$273$	0	0
$274$	−41.6074	−2.51359
$275$	−48.8096	−2.94333
$276$	0	0
$277$	11.1429	0.669511	0.334755	−	0.942305i	$-0.391346\pi$
$277$	11.1429	0.669511	0.334755	+	0.942305i	$0.391346\pi$
$278$	−6.32454	−0.379321
$279$	0	0
$280$	−48.6480	−2.90727
$281$	10.4579	0.623869	0.311934	−	0.950104i	$-0.399023\pi$
$281$	10.4579	0.623869	0.311934	+	0.950104i	$0.399023\pi$
$282$	0	0
$283$	−10.2033	−0.606521	−0.303260	−	0.952908i	$-0.598075\pi$
$283$	−10.2033	−0.606521	−0.303260	+	0.952908i	$0.598075\pi$
$284$	5.51780	0.327421
$285$	0	0
$286$	−36.1956	−2.14029
$287$	11.0678	0.653311
$288$	0	0
$289$	−9.35607	−0.550357
$290$	46.8168	2.74918
$291$	0	0
$292$	31.6387	1.85152
$293$	−3.15756	−0.184467	−0.0922334	−	0.995737i	$-0.529401\pi$
$293$	−3.15756	−0.184467	−0.0922334	+	0.995737i	$0.529401\pi$
$294$	0	0
$295$	50.5568	2.94353
$296$	−14.4371	−0.839138
$297$	0	0
$298$	−39.3585	−2.27998
$299$	5.02494	0.290600
$300$	0	0
$301$	23.5299	1.35624
$302$	−49.0751	−2.82396
$303$	0	0
$304$	−2.29974	−0.131899
$305$	5.36971	0.307469
$306$	0	0
$307$	7.37176	0.420728	0.210364	−	0.977623i	$-0.432535\pi$
$307$	7.37176	0.420728	0.210364	+	0.977623i	$0.432535\pi$
$308$	−58.6319	−3.34086
$309$	0	0
$310$	0	0
$311$	17.2437	0.977803	0.488902	−	0.872339i	$-0.337398\pi$
$311$	17.2437	0.977803	0.488902	+	0.872339i	$0.337398\pi$
$312$	0	0
$313$	−31.0607	−1.75565	−0.877827	−	0.478977i	$-0.841008\pi$
$313$	−31.0607	−1.75565	−0.877827	+	0.478977i	$0.841008\pi$
$314$	−58.0314	−3.27490
$315$	0	0
$316$	6.42038	0.361175
$317$	−18.9263	−1.06301	−0.531504	−	0.847055i	$-0.678373\pi$
$317$	−18.9263	−1.06301	−0.531504	+	0.847055i	$0.678373\pi$
$318$	0	0
$319$	26.1550	1.46440
$320$	−40.8026	−2.28094
$321$	0	0
$322$	12.5064	0.696954
$323$	2.60303	0.144836
$324$	0	0
$325$	−28.3480	−1.57247
$326$	−50.3252	−2.78725
$327$	0	0
$328$	−14.8529	−0.820116
$329$	−26.2427	−1.44680
$330$	0	0
$331$	1.91233	0.105111	0.0525555	−	0.998618i	$-0.483263\pi$
$331$	1.91233	0.105111	0.0525555	+	0.998618i	$0.483263\pi$
$332$	16.2198	0.890179
$333$	0	0
$334$	−54.3958	−2.97641
$335$	−1.32518	−0.0724022
$336$	0	0
$337$	−17.3061	−0.942724	−0.471362	−	0.881940i	$-0.656237\pi$
$337$	−17.3061	−0.942724	−0.471362	+	0.881940i	$0.656237\pi$
$338$	10.0916	0.548910
$339$	0	0
$340$	−39.3373	−2.13337
$341$	0	0
$342$	0	0
$343$	−13.8734	−0.749092
$344$	−31.5770	−1.70252
$345$	0	0
$346$	−19.7418	−1.06133
$347$	25.4967	1.36874	0.684368	−	0.729137i	$-0.260078\pi$
$347$	25.4967	1.36874	0.684368	+	0.729137i	$0.260078\pi$
$348$	0	0
$349$	−3.55625	−0.190361	−0.0951807	−	0.995460i	$-0.530343\pi$
$349$	−3.55625	−0.190361	−0.0951807	+	0.995460i	$0.530343\pi$
$350$	−70.5544	−3.77129
$351$	0	0
$352$	−12.3792	−0.659812
$353$	36.3702	1.93579	0.967894	−	0.251358i	$-0.0808771\pi$
$353$	36.3702	1.93579	0.967894	+	0.251358i	$0.0808771\pi$
$354$	0	0
$355$	5.64849	0.299791
$356$	56.0538	2.97085
$357$	0	0
$358$	44.2927	2.34094
$359$	6.85109	0.361587	0.180793	−	0.983521i	$-0.442133\pi$
$359$	6.85109	0.361587	0.180793	+	0.983521i	$0.442133\pi$
$360$	0	0
$361$	−18.1136	−0.953346
$362$	18.0783	0.950174
$363$	0	0
$364$	−34.0527	−1.78485
$365$	32.3882	1.69527
$366$	0	0
$367$	16.4919	0.860870	0.430435	−	0.902622i	$-0.358360\pi$
$367$	16.4919	0.860870	0.430435	+	0.902622i	$0.358360\pi$
$368$	−4.14148	−0.215889
$369$	0	0
$370$	−31.8832	−1.65753
$371$	−11.0929	−0.575914
$372$	0	0
$373$	−16.8765	−0.873834	−0.436917	−	0.899502i	$-0.643930\pi$
$373$	−16.8765	−0.873834	−0.436917	+	0.899502i	$0.643930\pi$
$374$	−33.7661	−1.74600
$375$	0	0
$376$	35.2175	1.81621
$377$	15.1905	0.782351
$378$	0	0
$379$	−32.0192	−1.64472	−0.822359	−	0.568970i	$-0.807342\pi$
$379$	−32.0192	−1.64472	−0.822359	+	0.568970i	$0.807342\pi$
$380$	−13.3958	−0.687188
$381$	0	0
$382$	25.7414	1.31705
$383$	−6.00912	−0.307052	−0.153526	−	0.988145i	$-0.549063\pi$
$383$	−6.00912	−0.307052	−0.153526	+	0.988145i	$0.549063\pi$
$384$	0	0
$385$	−60.0207	−3.05894
$386$	30.0167	1.52781
$387$	0	0
$388$	−63.5933	−3.22846
$389$	−6.99388	−0.354604	−0.177302	−	0.984157i	$-0.556737\pi$
$389$	−6.99388	−0.354604	−0.177302	+	0.984157i	$0.556737\pi$
$390$	0	0
$391$	4.68766	0.237065
$392$	−10.3339	−0.521942
$393$	0	0
$394$	−31.7104	−1.59755
$395$	6.57246	0.330696
$396$	0	0
$397$	5.58662	0.280384	0.140192	−	0.990124i	$-0.455228\pi$
$397$	5.58662	0.280384	0.140192	+	0.990124i	$0.455228\pi$
$398$	20.6677	1.03598
$399$	0	0
$400$	23.3640	1.16820
$401$	11.6794	0.583243	0.291622	−	0.956534i	$-0.405805\pi$
$401$	11.6794	0.583243	0.291622	+	0.956534i	$0.405805\pi$
$402$	0	0
$403$	0	0
$404$	−48.3903	−2.40751
$405$	0	0
$406$	37.8071	1.87633
$407$	−17.8121	−0.882914
$408$	0	0
$409$	−8.34003	−0.412388	−0.206194	−	0.978511i	$-0.566108\pi$
$409$	−8.34003	−0.412388	−0.206194	+	0.978511i	$0.566108\pi$
$410$	−32.8016	−1.61996
$411$	0	0
$412$	20.2551	0.997896
$413$	40.8274	2.00898
$414$	0	0
$415$	16.6040	0.815059
$416$	−7.18967	−0.352503
$417$	0	0
$418$	−11.4985	−0.562412
$419$	7.77430	0.379799	0.189900	−	0.981803i	$-0.439184\pi$
$419$	7.77430	0.379799	0.189900	+	0.981803i	$0.439184\pi$
$420$	0	0
$421$	19.0984	0.930801	0.465401	−	0.885100i	$-0.345910\pi$
$421$	19.0984	0.930801	0.465401	+	0.885100i	$0.345910\pi$
$422$	−42.4150	−2.06473
$423$	0	0
$424$	14.8866	0.722957
$425$	−26.4453	−1.28278
$426$	0	0
$427$	4.33633	0.209850
$428$	39.1706	1.89338
$429$	0	0
$430$	−69.7355	−3.36294
$431$	−1.11397	−0.0536582	−0.0268291	−	0.999640i	$-0.508541\pi$
$431$	−1.11397	−0.0536582	−0.0268291	+	0.999640i	$0.508541\pi$
$432$	0	0
$433$	25.4923	1.22508	0.612541	−	0.790439i	$-0.290147\pi$
$433$	25.4923	1.22508	0.612541	+	0.790439i	$0.290147\pi$
$434$	0	0
$435$	0	0
$436$	−2.45395	−0.117523
$437$	1.59631	0.0763621
$438$	0	0
$439$	34.3287	1.63842	0.819209	−	0.573495i	$-0.194413\pi$
$439$	34.3287	1.63842	0.819209	+	0.573495i	$0.194413\pi$
$440$	80.5475	3.83995
$441$	0	0
$442$	−19.6109	−0.932796
$443$	−19.4247	−0.922897	−0.461448	−	0.887167i	$-0.652670\pi$
$443$	−19.4247	−0.922897	−0.461448	+	0.887167i	$0.652670\pi$
$444$	0	0
$445$	57.3815	2.72015
$446$	18.2047	0.862019
$447$	0	0
$448$	−32.9503	−1.55676
$449$	21.3206	1.00618	0.503090	−	0.864234i	$-0.332196\pi$
$449$	21.3206	1.00618	0.503090	+	0.864234i	$0.332196\pi$
$450$	0	0
$451$	−18.3252	−0.862900
$452$	49.1358	2.31115
$453$	0	0
$454$	−24.6044	−1.15474
$455$	−34.8593	−1.63423
$456$	0	0
$457$	−34.7495	−1.62551	−0.812757	−	0.582603i	$-0.802034\pi$
$457$	−34.7495	−1.62551	−0.812757	+	0.582603i	$0.802034\pi$
$458$	4.53208	0.211770
$459$	0	0
$460$	−24.1237	−1.12477
$461$	−1.56859	−0.0730564	−0.0365282	−	0.999333i	$-0.511630\pi$
$461$	−1.56859	−0.0730564	−0.0365282	+	0.999333i	$0.511630\pi$
$462$	0	0
$463$	18.0988	0.841122	0.420561	−	0.907264i	$-0.361833\pi$
$463$	18.0988	0.841122	0.420561	+	0.907264i	$0.361833\pi$
$464$	−12.5198	−0.581216
$465$	0	0
$466$	−25.7822	−1.19434
$467$	1.63829	0.0758110	0.0379055	−	0.999281i	$-0.487931\pi$
$467$	1.63829	0.0758110	0.0379055	+	0.999281i	$0.487931\pi$
$468$	0	0
$469$	−1.07015	−0.0494151
$470$	77.7754	3.58751
$471$	0	0
$472$	−54.7901	−2.52192
$473$	−38.9589	−1.79133
$474$	0	0
$475$	−9.00555	−0.413203
$476$	−31.7670	−1.45604
$477$	0	0
$478$	−31.3800	−1.43529
$479$	−25.1353	−1.14846	−0.574231	−	0.818694i	$-0.694699\pi$
$479$	−25.1353	−1.14846	−0.574231	+	0.818694i	$0.694699\pi$
$480$	0	0
$481$	−10.3451	−0.471694
$482$	−31.9440	−1.45501
$483$	0	0
$484$	56.0689	2.54859
$485$	−65.0996	−2.95602
$486$	0	0
$487$	−12.0354	−0.545374	−0.272687	−	0.962103i	$-0.587912\pi$
$487$	−12.0354	−0.545374	−0.272687	+	0.962103i	$0.587912\pi$
$488$	−5.81934	−0.263429
$489$	0	0
$490$	−22.8217	−1.03098
$491$	−7.63531	−0.344577	−0.172288	−	0.985047i	$-0.555116\pi$
$491$	−7.63531	−0.344577	−0.172288	+	0.985047i	$0.555116\pi$
$492$	0	0
$493$	14.1709	0.638225
$494$	−6.67821	−0.300467
$495$	0	0
$496$	0	0
$497$	4.56147	0.204610
$498$	0	0
$499$	20.0518	0.897641	0.448820	−	0.893622i	$-0.351844\pi$
$499$	20.0518	0.897641	0.448820	+	0.893622i	$0.351844\pi$
$500$	64.9526	2.90477
$501$	0	0
$502$	25.1078	1.12062
$503$	−9.06226	−0.404066	−0.202033	−	0.979379i	$-0.564755\pi$
$503$	−9.06226	−0.404066	−0.202033	+	0.979379i	$0.564755\pi$
$504$	0	0
$505$	−49.5365	−2.20435
$506$	−20.7071	−0.920544
$507$	0	0
$508$	−33.4715	−1.48506
$509$	−0.209697	−0.00929466	−0.00464733	−	0.999989i	$-0.501479\pi$
$509$	−0.209697	−0.00929466	−0.00464733	+	0.999989i	$0.501479\pi$
$510$	0	0
$511$	26.1552	1.15704
$512$	26.1310	1.15484
$513$	0	0
$514$	24.5857	1.08443
$515$	20.7348	0.913687
$516$	0	0
$517$	43.4506	1.91095
$518$	−25.7474	−1.13128
$519$	0	0
$520$	46.7810	2.05148
$521$	−13.9311	−0.610334	−0.305167	−	0.952299i	$-0.598712\pi$
$521$	−13.9311	−0.610334	−0.305167	+	0.952299i	$0.598712\pi$
$522$	0	0
$523$	13.5934	0.594396	0.297198	−	0.954816i	$-0.403948\pi$
$523$	13.5934	0.594396	0.297198	+	0.954816i	$0.403948\pi$
$524$	3.87666	0.169353
$525$	0	0
$526$	−23.4360	−1.02186
$527$	0	0
$528$	0	0
$529$	−20.1253	−0.875012
$530$	32.8760	1.42804
$531$	0	0
$532$	−10.8178	−0.469011
$533$	−10.6430	−0.461002
$534$	0	0
$535$	40.0984	1.73360
$536$	1.43614	0.0620318
$537$	0	0
$538$	−43.2356	−1.86402
$539$	−12.7498	−0.549171
$540$	0	0
$541$	−27.9843	−1.20314	−0.601570	−	0.798820i	$-0.705458\pi$
$541$	−27.9843	−1.20314	−0.601570	+	0.798820i	$0.705458\pi$
$542$	−45.6289	−1.95993
$543$	0	0
$544$	−6.70709	−0.287564
$545$	−2.51208	−0.107606
$546$	0	0
$547$	9.01058	0.385265	0.192632	−	0.981271i	$-0.438298\pi$
$547$	9.01058	0.385265	0.192632	+	0.981271i	$0.438298\pi$
$548$	64.8118	2.76862
$549$	0	0
$550$	116.818	4.98116
$551$	4.82569	0.205581
$552$	0	0
$553$	5.30762	0.225703
$554$	−26.6688	−1.13305
$555$	0	0
$556$	9.85173	0.417807
$557$	−6.90131	−0.292418	−0.146209	−	0.989254i	$-0.546707\pi$
$557$	−6.90131	−0.292418	−0.146209	+	0.989254i	$0.546707\pi$
$558$	0	0
$559$	−22.6269	−0.957015
$560$	28.7305	1.21408
$561$	0	0
$562$	−25.0295	−1.05581
$563$	−31.2560	−1.31728	−0.658641	−	0.752458i	$-0.728868\pi$
$563$	−31.2560	−1.31728	−0.658641	+	0.752458i	$0.728868\pi$
$564$	0	0
$565$	50.2996	2.11612
$566$	24.4200	1.02645
$567$	0	0
$568$	−6.12147	−0.256851
$569$	−37.5583	−1.57452	−0.787262	−	0.616619i	$-0.788502\pi$
$569$	−37.5583	−1.57452	−0.787262	+	0.616619i	$0.788502\pi$
$570$	0	0
$571$	30.7216	1.28566	0.642830	−	0.766009i	$-0.277760\pi$
$571$	30.7216	1.28566	0.642830	+	0.766009i	$0.277760\pi$
$572$	56.3818	2.35744
$573$	0	0
$574$	−26.4891	−1.10563
$575$	−16.2176	−0.676321
$576$	0	0
$577$	−44.4950	−1.85235	−0.926175	−	0.377094i	$-0.876923\pi$
$577$	−44.4950	−1.85235	−0.926175	+	0.377094i	$0.876923\pi$
$578$	22.3923	0.931398
$579$	0	0
$580$	−72.9265	−3.02811
$581$	13.4086	0.556284
$582$	0	0
$583$	18.3667	0.760673
$584$	−35.1001	−1.45245
$585$	0	0
$586$	7.55715	0.312183
$587$	−20.8388	−0.860109	−0.430054	−	0.902803i	$-0.641506\pi$
$587$	−20.8388	−0.860109	−0.430054	+	0.902803i	$0.641506\pi$
$588$	0	0
$589$	0	0
$590$	−121.000	−4.98150
$591$	0	0
$592$	8.52624	0.350426
$593$	12.4620	0.511751	0.255876	−	0.966710i	$-0.417636\pi$
$593$	12.4620	0.511751	0.255876	+	0.966710i	$0.417636\pi$
$594$	0	0
$595$	−32.5195	−1.33317
$596$	61.3087	2.51130
$597$	0	0
$598$	−12.0264	−0.491798
$599$	−6.51238	−0.266089	−0.133044	−	0.991110i	$-0.542475\pi$
$599$	−6.51238	−0.266089	−0.133044	+	0.991110i	$0.542475\pi$
$600$	0	0
$601$	18.0953	0.738121	0.369061	−	0.929405i	$-0.379679\pi$
$601$	18.0953	0.738121	0.369061	+	0.929405i	$0.379679\pi$
$602$	−56.3152	−2.29524
$603$	0	0
$604$	76.4443	3.11048
$605$	57.3970	2.33352
$606$	0	0
$607$	11.3830	0.462020	0.231010	−	0.972951i	$-0.425797\pi$
$607$	11.3830	0.462020	0.231010	+	0.972951i	$0.425797\pi$
$608$	−2.28400	−0.0926285
$609$	0	0
$610$	−12.8516	−0.520346
$611$	25.2356	1.02092
$612$	0	0
$613$	32.2276	1.30166	0.650830	−	0.759223i	$-0.274421\pi$
$613$	32.2276	1.30166	0.650830	+	0.759223i	$0.274421\pi$
$614$	−17.6432	−0.712021
$615$	0	0
$616$	65.0465	2.62080
$617$	−9.82738	−0.395635	−0.197818	−	0.980239i	$-0.563385\pi$
$617$	−9.82738	−0.395635	−0.197818	+	0.980239i	$0.563385\pi$
$618$	0	0
$619$	31.7958	1.27798	0.638990	−	0.769215i	$-0.279353\pi$
$619$	31.7958	1.27798	0.638990	+	0.769215i	$0.279353\pi$
$620$	0	0
$621$	0	0
$622$	−41.2703	−1.65479
$623$	46.3387	1.85652
$624$	0	0
$625$	18.6656	0.746625
$626$	74.3391	2.97119
$627$	0	0
$628$	90.3955	3.60717
$629$	−9.65068	−0.384798
$630$	0	0
$631$	43.7125	1.74017	0.870083	−	0.492905i	$-0.164065\pi$
$631$	43.7125	1.74017	0.870083	+	0.492905i	$0.164065\pi$
$632$	−7.12280	−0.283330
$633$	0	0
$634$	45.2973	1.79899
$635$	−34.2644	−1.35974
$636$	0	0
$637$	−7.40490	−0.293393
$638$	−62.5981	−2.47828
$639$	0	0
$640$	79.1383	3.12821
$641$	21.1881	0.836881	0.418440	−	0.908244i	$-0.362577\pi$
$641$	21.1881	0.836881	0.418440	+	0.908244i	$0.362577\pi$
$642$	0	0
$643$	11.9734	0.472184	0.236092	−	0.971731i	$-0.424133\pi$
$643$	11.9734	0.472184	0.236092	+	0.971731i	$0.424133\pi$
$644$	−19.4812	−0.767667
$645$	0	0
$646$	−6.22996	−0.245114
$647$	−33.8451	−1.33059	−0.665294	−	0.746582i	$-0.731694\pi$
$647$	−33.8451	−1.33059	−0.665294	+	0.746582i	$0.731694\pi$
$648$	0	0
$649$	−67.5988	−2.65349
$650$	67.8467	2.66117
$651$	0	0
$652$	78.3915	3.07005
$653$	−29.1132	−1.13929	−0.569644	−	0.821892i	$-0.692919\pi$
$653$	−29.1132	−1.13929	−0.569644	+	0.821892i	$0.692919\pi$
$654$	0	0
$655$	3.96849	0.155062
$656$	8.77183	0.342482
$657$	0	0
$658$	62.8078	2.44850
$659$	49.4293	1.92549	0.962745	−	0.270410i	$-0.0871591\pi$
$659$	49.4293	1.92549	0.962745	+	0.270410i	$0.0871591\pi$
$660$	0	0
$661$	27.8737	1.08416	0.542080	−	0.840327i	$-0.317637\pi$
$661$	27.8737	1.08416	0.542080	+	0.840327i	$0.317637\pi$
$662$	−4.57686	−0.177885
$663$	0	0
$664$	−17.9943	−0.698316
$665$	−11.0740	−0.429433
$666$	0	0
$667$	8.69034	0.336491
$668$	84.7323	3.27839
$669$	0	0
$670$	3.17161	0.122530
$671$	−7.17977	−0.277172
$672$	0	0
$673$	41.4533	1.59791	0.798954	−	0.601393i	$-0.205387\pi$
$673$	41.4533	1.59791	0.798954	+	0.601393i	$0.205387\pi$
$674$	41.4196	1.59542
$675$	0	0
$676$	−15.7197	−0.604603
$677$	−18.3685	−0.705960	−0.352980	−	0.935631i	$-0.614832\pi$
$677$	−18.3685	−0.705960	−0.352980	+	0.935631i	$0.614832\pi$
$678$	0	0
$679$	−52.5714	−2.01751
$680$	43.6410	1.67356
$681$	0	0
$682$	0	0
$683$	−16.3776	−0.626672	−0.313336	−	0.949642i	$-0.601447\pi$
$683$	−16.3776	−0.626672	−0.313336	+	0.949642i	$0.601447\pi$
$684$	0	0
$685$	66.3470	2.53499
$686$	33.2038	1.26773
$687$	0	0
$688$	18.6487	0.710975
$689$	10.6672	0.406387
$690$	0	0
$691$	24.9679	0.949822	0.474911	−	0.880034i	$-0.342480\pi$
$691$	24.9679	0.949822	0.474911	+	0.880034i	$0.342480\pi$
$692$	30.7519	1.16901
$693$	0	0
$694$	−61.0226	−2.31638
$695$	10.0851	0.382549
$696$	0	0
$697$	−9.92867	−0.376075
$698$	8.51134	0.322159
$699$	0	0
$700$	109.902	4.15392
$701$	30.8777	1.16623	0.583117	−	0.812388i	$-0.301833\pi$
$701$	30.8777	1.16623	0.583117	+	0.812388i	$0.301833\pi$
$702$	0	0
$703$	−3.28640	−0.123949
$704$	54.5566	2.05618
$705$	0	0
$706$	−87.0465	−3.27604
$707$	−40.0034	−1.50448
$708$	0	0
$709$	−21.2282	−0.797241	−0.398620	−	0.917116i	$-0.630511\pi$
$709$	−21.2282	−0.797241	−0.398620	+	0.917116i	$0.630511\pi$
$710$	−13.5188	−0.507352
$711$	0	0
$712$	−62.1863	−2.33053
$713$	0	0
$714$	0	0
$715$	57.7173	2.15851
$716$	−68.9947	−2.57845
$717$	0	0
$718$	−16.3971	−0.611933
$719$	−45.5834	−1.69997	−0.849987	−	0.526804i	$-0.823390\pi$
$719$	−45.5834	−1.69997	−0.849987	+	0.526804i	$0.823390\pi$
$720$	0	0
$721$	16.7445	0.623598
$722$	43.3521	1.61340
$723$	0	0
$724$	−28.1605	−1.04658
$725$	−49.0262	−1.82079
$726$	0	0
$727$	10.5498	0.391269	0.195635	−	0.980677i	$-0.437323\pi$
$727$	10.5498	0.391269	0.195635	+	0.980677i	$0.437323\pi$
$728$	37.7782	1.40015
$729$	0	0
$730$	−77.5161	−2.86900
$731$	−21.1081	−0.780712
$732$	0	0
$733$	−31.4319	−1.16096	−0.580482	−	0.814273i	$-0.697136\pi$
$733$	−31.4319	−1.16096	−0.580482	+	0.814273i	$0.697136\pi$
$734$	−39.4708	−1.45690
$735$	0	0
$736$	−4.11314	−0.151612
$737$	1.77188	0.0652679
$738$	0	0
$739$	1.17168	0.0431009	0.0215504	−	0.999768i	$-0.493140\pi$
$739$	1.17168	0.0431009	0.0215504	+	0.999768i	$0.493140\pi$
$740$	49.6645	1.82570
$741$	0	0
$742$	26.5491	0.974650
$743$	−13.4137	−0.492099	−0.246050	−	0.969257i	$-0.579133\pi$
$743$	−13.4137	−0.492099	−0.246050	+	0.969257i	$0.579133\pi$
$744$	0	0
$745$	62.7609	2.29938
$746$	40.3914	1.47884
$747$	0	0
$748$	52.5974	1.92315
$749$	32.3816	1.18320
$750$	0	0
$751$	1.29409	0.0472218	0.0236109	−	0.999721i	$-0.492484\pi$
$751$	1.29409	0.0472218	0.0236109	+	0.999721i	$0.492484\pi$
$752$	−20.7987	−0.758452
$753$	0	0
$754$	−36.3562	−1.32401
$755$	78.2550	2.84799
$756$	0	0
$757$	41.6223	1.51279	0.756395	−	0.654116i	$-0.226959\pi$
$757$	41.6223	1.51279	0.756395	+	0.654116i	$0.226959\pi$
$758$	76.6332	2.78344
$759$	0	0
$760$	14.8613	0.539076
$761$	11.3577	0.411717	0.205858	−	0.978582i	$-0.434001\pi$
$761$	11.3577	0.411717	0.205858	+	0.978582i	$0.434001\pi$
$762$	0	0
$763$	−2.02864	−0.0734416
$764$	−40.0974	−1.45067
$765$	0	0
$766$	14.3819	0.519640
$767$	−39.2606	−1.41762
$768$	0	0
$769$	5.37400	0.193791	0.0968957	−	0.995295i	$-0.469109\pi$
$769$	5.37400	0.193791	0.0968957	+	0.995295i	$0.469109\pi$
$770$	143.650	5.17680
$771$	0	0
$772$	−46.7571	−1.68282
$773$	−37.0370	−1.33213	−0.666064	−	0.745894i	$-0.732022\pi$
$773$	−37.0370	−1.33213	−0.666064	+	0.745894i	$0.732022\pi$
$774$	0	0
$775$	0	0
$776$	70.5506	2.53262
$777$	0	0
$778$	16.7388	0.600115
$779$	−3.38106	−0.121139
$780$	0	0
$781$	−7.55252	−0.270250
$782$	−11.2192	−0.401198
$783$	0	0
$784$	6.10300	0.217964
$785$	92.5366	3.30277
$786$	0	0
$787$	−30.2676	−1.07892	−0.539461	−	0.842011i	$-0.681372\pi$
$787$	−30.2676	−1.07892	−0.539461	+	0.842011i	$0.681372\pi$
$788$	49.3953	1.75963
$789$	0	0
$790$	−15.7302	−0.559655
$791$	40.6197	1.44427
$792$	0	0
$793$	−4.16992	−0.148078
$794$	−13.3707	−0.474509
$795$	0	0
$796$	−32.1940	−1.14109
$797$	8.88635	0.314771	0.157385	−	0.987537i	$-0.449694\pi$
$797$	8.88635	0.314771	0.157385	+	0.987537i	$0.449694\pi$
$798$	0	0
$799$	23.5417	0.832845
$800$	23.2041	0.820389
$801$	0	0
$802$	−27.9530	−0.987053
$803$	−43.3057	−1.52823
$804$	0	0
$805$	−19.9426	−0.702886
$806$	0	0
$807$	0	0
$808$	53.6844	1.88861
$809$	−34.0004	−1.19539	−0.597696	−	0.801723i	$-0.703917\pi$
$809$	−34.0004	−1.19539	−0.597696	+	0.801723i	$0.703917\pi$
$810$	0	0
$811$	−5.81257	−0.204107	−0.102053	−	0.994779i	$-0.532541\pi$
$811$	−5.81257	−0.204107	−0.102053	+	0.994779i	$0.532541\pi$
$812$	−58.8921	−2.06671
$813$	0	0
$814$	42.6306	1.49420
$815$	80.2484	2.81098
$816$	0	0
$817$	−7.18806	−0.251478
$818$	19.9606	0.697906
$819$	0	0
$820$	51.0951	1.78432
$821$	−31.6055	−1.10304	−0.551519	−	0.834162i	$-0.685952\pi$
$821$	−31.6055	−1.10304	−0.551519	+	0.834162i	$0.685952\pi$
$822$	0	0
$823$	−18.7070	−0.652085	−0.326042	−	0.945355i	$-0.605715\pi$
$823$	−18.7070	−0.652085	−0.326042	+	0.945355i	$0.605715\pi$
$824$	−22.4711	−0.782817
$825$	0	0
$826$	−97.7141	−3.39991
$827$	−19.8511	−0.690292	−0.345146	−	0.938549i	$-0.612171\pi$
$827$	−19.8511	−0.690292	−0.345146	+	0.938549i	$0.612171\pi$
$828$	0	0
$829$	−35.8877	−1.24643	−0.623216	−	0.782050i	$-0.714174\pi$
$829$	−35.8877	−1.24643	−0.623216	+	0.782050i	$0.714174\pi$
$830$	−39.7392	−1.37937
$831$	0	0
$832$	31.6858	1.09851
$833$	−6.90787	−0.239344
$834$	0	0
$835$	86.7393	3.00174
$836$	17.9113	0.619474
$837$	0	0
$838$	−18.6066	−0.642755
$839$	22.2305	0.767482	0.383741	−	0.923441i	$-0.374635\pi$
$839$	22.2305	0.767482	0.383741	+	0.923441i	$0.374635\pi$
$840$	0	0
$841$	−2.72893	−0.0941009
$842$	−45.7092	−1.57524
$843$	0	0
$844$	66.0698	2.27421
$845$	−16.0920	−0.553582
$846$	0	0
$847$	46.3512	1.59264
$848$	−8.79172	−0.301909
$849$	0	0
$850$	63.2927	2.17092
$851$	−5.91831	−0.202877
$852$	0	0
$853$	−43.8647	−1.50190	−0.750949	−	0.660360i	$-0.770404\pi$
$853$	−43.8647	−1.50190	−0.750949	+	0.660360i	$0.770404\pi$
$854$	−10.3784	−0.355140
$855$	0	0
$856$	−43.4560	−1.48530
$857$	39.9377	1.36424	0.682122	−	0.731238i	$-0.261057\pi$
$857$	39.9377	1.36424	0.682122	+	0.731238i	$0.261057\pi$
$858$	0	0
$859$	−19.5694	−0.667699	−0.333849	−	0.942626i	$-0.608348\pi$
$859$	−19.5694	−0.667699	−0.333849	+	0.942626i	$0.608348\pi$
$860$	108.627	3.70415
$861$	0	0
$862$	2.66613	0.0908086
$863$	31.0832	1.05808	0.529042	−	0.848595i	$-0.322551\pi$
$863$	31.0832	1.05808	0.529042	+	0.848595i	$0.322551\pi$
$864$	0	0
$865$	31.4803	1.07036
$866$	−61.0120	−2.07327
$867$	0	0
$868$	0	0
$869$	−8.78794	−0.298110
$870$	0	0
$871$	1.02908	0.0348692
$872$	2.72242	0.0921928
$873$	0	0
$874$	−3.82054	−0.129232
$875$	53.6952	1.81523
$876$	0	0
$877$	36.9864	1.24894	0.624471	−	0.781048i	$-0.285315\pi$
$877$	36.9864	1.24894	0.624471	+	0.781048i	$0.285315\pi$
$878$	−82.1605	−2.77278
$879$	0	0
$880$	−47.5697	−1.60357
$881$	−47.2972	−1.59348	−0.796741	−	0.604321i	$-0.793445\pi$
$881$	−47.2972	−1.59348	−0.796741	+	0.604321i	$0.793445\pi$
$882$	0	0
$883$	−54.8384	−1.84546	−0.922729	−	0.385450i	$-0.874046\pi$
$883$	−54.8384	−1.84546	−0.922729	+	0.385450i	$0.874046\pi$
$884$	30.5479	1.02744
$885$	0	0
$886$	46.4901	1.56187
$887$	−7.32248	−0.245865	−0.122932	−	0.992415i	$-0.539230\pi$
$887$	−7.32248	−0.245865	−0.122932	+	0.992415i	$0.539230\pi$
$888$	0	0
$889$	−27.6703	−0.928033
$890$	−137.334	−4.60345
$891$	0	0
$892$	−28.3575	−0.949479
$893$	8.01678	0.268271
$894$	0	0
$895$	−70.6290	−2.36087
$896$	63.9084	2.13503
$897$	0	0
$898$	−51.0276	−1.70281
$899$	0	0
$900$	0	0
$901$	9.95118	0.331522
$902$	43.8586	1.46033
$903$	0	0
$904$	−54.5114	−1.81302
$905$	−28.8276	−0.958261
$906$	0	0
$907$	6.97833	0.231712	0.115856	−	0.993266i	$-0.463039\pi$
$907$	6.97833	0.231712	0.115856	+	0.993266i	$0.463039\pi$
$908$	38.3262	1.27190
$909$	0	0
$910$	83.4304	2.76569
$911$	−6.46515	−0.214200	−0.107100	−	0.994248i	$-0.534156\pi$
$911$	−6.46515	−0.214200	−0.107100	+	0.994248i	$0.534156\pi$
$912$	0	0
$913$	−22.2010	−0.734746
$914$	83.1677	2.75094
$915$	0	0
$916$	−7.05963	−0.233257
$917$	3.20477	0.105831
$918$	0	0
$919$	23.5545	0.776992	0.388496	−	0.921450i	$-0.372995\pi$
$919$	23.5545	0.776992	0.388496	+	0.921450i	$0.372995\pi$
$920$	26.7629	0.882348
$921$	0	0
$922$	3.75418	0.123637
$923$	−4.38641	−0.144381
$924$	0	0
$925$	33.3879	1.09779
$926$	−43.3167	−1.42348
$927$	0	0
$928$	−12.4341	−0.408169
$929$	18.9289	0.621036	0.310518	−	0.950568i	$-0.399498\pi$
$929$	18.9289	0.621036	0.310518	+	0.950568i	$0.399498\pi$
$930$	0	0
$931$	−2.35237	−0.0770960
$932$	40.1609	1.31551
$933$	0	0
$934$	−3.92100	−0.128299
$935$	53.8432	1.76086
$936$	0	0
$937$	−36.8198	−1.20285	−0.601425	−	0.798929i	$-0.705400\pi$
$937$	−36.8198	−1.20285	−0.601425	+	0.798929i	$0.705400\pi$
$938$	2.56125	0.0836278
$939$	0	0
$940$	−121.151	−3.95150
$941$	−15.6184	−0.509145	−0.254572	−	0.967054i	$-0.581935\pi$
$941$	−15.6184	−0.509145	−0.254572	+	0.967054i	$0.581935\pi$
$942$	0	0
$943$	−6.08878	−0.198278
$944$	32.3579	1.05316
$945$	0	0
$946$	93.2423	3.03157
$947$	−8.23472	−0.267592	−0.133796	−	0.991009i	$-0.542717\pi$
$947$	−8.23472	−0.267592	−0.133796	+	0.991009i	$0.542717\pi$
$948$	0	0
$949$	−25.1514	−0.816450
$950$	21.5534	0.699285
$951$	0	0
$952$	35.2425	1.14221
$953$	−59.9174	−1.94092	−0.970458	−	0.241272i	$-0.922435\pi$
$953$	−59.9174	−1.94092	−0.970458	+	0.241272i	$0.922435\pi$
$954$	0	0
$955$	−41.0472	−1.32826
$956$	48.8806	1.58091
$957$	0	0
$958$	60.1575	1.94360
$959$	53.5788	1.73015
$960$	0	0
$961$	0	0
$962$	24.7593	0.798273
$963$	0	0
$964$	49.7591	1.60263
$965$	−47.8646	−1.54082
$966$	0	0
$967$	−9.29744	−0.298985	−0.149493	−	0.988763i	$-0.547764\pi$
$967$	−9.29744	−0.298985	−0.149493	+	0.988763i	$0.547764\pi$
$968$	−62.2031	−1.99928
$969$	0	0
$970$	155.806	5.00263
$971$	−7.97360	−0.255885	−0.127942	−	0.991782i	$-0.540837\pi$
$971$	−7.97360	−0.255885	−0.127942	+	0.991782i	$0.540837\pi$
$972$	0	0
$973$	8.14425	0.261093
$974$	28.8048	0.922965
$975$	0	0
$976$	3.43678	0.110009
$977$	−35.6908	−1.14185	−0.570925	−	0.821002i	$-0.693415\pi$
$977$	−35.6908	−1.14185	−0.570925	+	0.821002i	$0.693415\pi$
$978$	0	0
$979$	−76.7240	−2.45211
$980$	35.5494	1.13558
$981$	0	0
$982$	18.2740	0.583146
$983$	53.1289	1.69455	0.847274	−	0.531156i	$-0.178242\pi$
$983$	53.1289	1.69455	0.847274	+	0.531156i	$0.178242\pi$
$984$	0	0
$985$	50.5653	1.61114
$986$	−33.9159	−1.08010
$987$	0	0
$988$	10.4026	0.330952
$989$	−12.9446	−0.411614
$990$	0	0
$991$	−52.7050	−1.67423	−0.837115	−	0.547026i	$-0.815760\pi$
$991$	−52.7050	−1.67423	−0.837115	+	0.547026i	$0.815760\pi$
$992$	0	0
$993$	0	0
$994$	−10.9172	−0.346272
$995$	−32.9566	−1.04479
$996$	0	0
$997$	−21.9522	−0.695234	−0.347617	−	0.937637i	$-0.613009\pi$
$997$	−21.9522	−0.695234	−0.347617	+	0.937637i	$0.613009\pi$
$998$	−47.9909	−1.51912
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.bu.1.3		24
3.2	odd	2		2883.2.a.v.1.22	yes	24
31.30	odd	2		8649.2.a.bv.1.3		24
93.92	even	2		2883.2.a.u.1.22	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2883.2.a.u.1.22	✓	24	93.92	even	2
2883.2.a.v.1.22	yes	24	3.2	odd	2
8649.2.a.bu.1.3		24	1.1	even	1	trivial
8649.2.a.bv.1.3		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)

\(f(q)\)	\(=\)	\(q-2.39335 q^{2} +3.72812 q^{4} +3.81642 q^{5} +3.08197 q^{7} -4.13599 q^{8} +O(q^{10})\)	\(q-2.39335 q^{2} +3.72812 q^{4} +3.81642 q^{5} +3.08197 q^{7} -4.13599 q^{8} -9.13404 q^{10} -5.10289 q^{11} -2.96369 q^{13} -7.37623 q^{14} +2.44263 q^{16} -2.76477 q^{17} -0.941501 q^{19} +14.2281 q^{20} +12.2130 q^{22} -1.69550 q^{23} +9.56510 q^{25} +7.09315 q^{26} +11.4899 q^{28} -5.12553 q^{29} +2.42592 q^{32} +6.61705 q^{34} +11.7621 q^{35} +3.49060 q^{37} +2.25334 q^{38} -15.7847 q^{40} +3.59114 q^{41} +7.63468 q^{43} -19.0242 q^{44} +4.05792 q^{46} -8.51490 q^{47} +2.49854 q^{49} -22.8926 q^{50} -11.0490 q^{52} -3.59928 q^{53} -19.4748 q^{55} -12.7470 q^{56} +12.2672 q^{58} +13.2472 q^{59} +1.40700 q^{61} -10.6913 q^{64} -11.3107 q^{65} -0.347230 q^{67} -10.3074 q^{68} -28.1508 q^{70} +1.48005 q^{71} +8.48652 q^{73} -8.35422 q^{74} -3.51003 q^{76} -15.7269 q^{77} +1.72215 q^{79} +9.32211 q^{80} -8.59486 q^{82} +4.35067 q^{83} -10.5515 q^{85} -18.2725 q^{86} +21.1055 q^{88} +15.0354 q^{89} -9.13402 q^{91} -6.32102 q^{92} +20.3791 q^{94} -3.59317 q^{95} -17.0577 q^{97} -5.97987 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8}+O(q^{10}) \) 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

Modular forms

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