LMFDB - Embedded newform 8649.2.a.w.1.1

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:	$4$
Coefficient field:	$\Q(\zeta_{15})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 4x + 1 $ x^4 - x^3 - 4x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 93)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.82709$ of defining polynomial
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)

$f(q)$	$=$	$q-1.95630 q^{2} +1.82709 q^{4} +1.20906 q^{5} -1.40898 q^{7} +0.338261 q^{8} +O(q^{10})$	\(q-1.95630 q^{2} +1.82709 q^{4} +1.20906 q^{5} -1.40898 q^{7} +0.338261 q^{8} -2.36527 q^{10} -0.920147 q^{11} +6.20150 q^{13} +2.75638 q^{14} -4.31592 q^{16} +3.74724 q^{17} +0.408977 q^{19} +2.20906 q^{20} +1.80008 q^{22} -2.64505 q^{23} -3.53818 q^{25} -12.1320 q^{26} -2.57433 q^{28} +3.53976 q^{29} +7.76669 q^{32} -7.33070 q^{34} -1.70353 q^{35} -8.80284 q^{37} -0.800080 q^{38} +0.408977 q^{40} +8.17916 q^{41} +3.02392 q^{43} -1.68119 q^{44} +5.17449 q^{46} +8.67461 q^{47} -5.01478 q^{49} +6.92173 q^{50} +11.3307 q^{52} -12.7712 q^{53} -1.11251 q^{55} -0.476602 q^{56} -6.92482 q^{58} +9.55008 q^{59} +8.32624 q^{61} -6.56210 q^{64} +7.49797 q^{65} +2.47214 q^{67} +6.84655 q^{68} +3.33261 q^{70} +11.5652 q^{71} -5.77583 q^{73} +17.2210 q^{74} +0.747238 q^{76} +1.29647 q^{77} -14.3436 q^{79} -5.21819 q^{80} -16.0009 q^{82} +10.5862 q^{83} +4.53062 q^{85} -5.91568 q^{86} -0.311250 q^{88} +9.82709 q^{89} -8.73777 q^{91} -4.83274 q^{92} -16.9701 q^{94} +0.494477 q^{95} +11.8843 q^{97} +9.81040 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + q^{4} + 3 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) $ 4 * q + q^2 + q^4 + 3 * q^5 - 3 * q^7 - 3 * q^8	$ 4 q + q^{2} + q^{4} + 3 q^{5} - 3 q^{7} - 3 q^{8} + 2 q^{10} - 3 q^{11} + 6 q^{13} + 3 q^{14} - 9 q^{16} + 8 q^{17} - q^{19} + 7 q^{20} + 8 q^{22} + q^{23} - 9 q^{25} - 6 q^{26} + 3 q^{28} - 9 q^{29} - 8 q^{34} + 9 q^{35} + 4 q^{37} - 4 q^{38} - q^{40} + 10 q^{41} - 2 q^{43} + 13 q^{44} + 9 q^{46} - 3 q^{47} - 7 q^{49} + 9 q^{50} + 24 q^{52} - 30 q^{53} + 14 q^{55} + 6 q^{56} - 16 q^{58} - 8 q^{59} + 2 q^{61} - 7 q^{64} + 27 q^{65} - 8 q^{67} - 3 q^{68} + 21 q^{70} + 34 q^{71} + 9 q^{73} + 26 q^{74} - 4 q^{76} + 21 q^{77} + 6 q^{79} - 18 q^{80} + 5 q^{82} - 12 q^{83} - 4 q^{85} - 13 q^{86} - 4 q^{88} + 33 q^{89} + 18 q^{91} - q^{92} - 47 q^{94} - 12 q^{95} - 9 q^{97} + 17 q^{98}+O(q^{100}) $ 4 * q + q^2 + q^4 + 3 * q^5 - 3 * q^7 - 3 * q^8 + 2 * q^10 - 3 * q^11 + 6 * q^13 + 3 * q^14 - 9 * q^16 + 8 * q^17 - q^19 + 7 * q^20 + 8 * q^22 + q^23 - 9 * q^25 - 6 * q^26 + 3 * q^28 - 9 * q^29 - 8 * q^34 + 9 * q^35 + 4 * q^37 - 4 * q^38 - q^40 + 10 * q^41 - 2 * q^43 + 13 * q^44 + 9 * q^46 - 3 * q^47 - 7 * q^49 + 9 * q^50 + 24 * q^52 - 30 * q^53 + 14 * q^55 + 6 * q^56 - 16 * q^58 - 8 * q^59 + 2 * q^61 - 7 * q^64 + 27 * q^65 - 8 * q^67 - 3 * q^68 + 21 * q^70 + 34 * q^71 + 9 * q^73 + 26 * q^74 - 4 * q^76 + 21 * q^77 + 6 * q^79 - 18 * q^80 + 5 * q^82 - 12 * q^83 - 4 * q^85 - 13 * q^86 - 4 * q^88 + 33 * q^89 + 18 * q^91 - q^92 - 47 * q^94 - 12 * q^95 - 9 * q^97 + 17 * 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 $

$2$	−1.95630	−1.38331	−0.691655	−	0.722228i	$-0.743118\pi$
$2$	−1.95630	−1.38331	−0.691655	+	0.722228i	$0.743118\pi$
$3$	0	0
$3$	0	0
$4$	1.82709	0.913545
$5$	1.20906	0.540707	0.270353	−	0.962761i	$-0.412859\pi$
$5$	1.20906	0.540707	0.270353	+	0.962761i	$0.412859\pi$
$6$	0	0
$7$	−1.40898	−0.532543	−0.266272	−	0.963898i	$-0.585792\pi$
$7$	−1.40898	−0.532543	−0.266272	+	0.963898i	$0.585792\pi$
$8$	0.338261	0.119593
$9$	0	0
$10$	−2.36527	−0.747965
$11$	−0.920147	−0.277435	−0.138717	−	0.990332i	$-0.544298\pi$
$11$	−0.920147	−0.277435	−0.138717	+	0.990332i	$0.544298\pi$
$12$	0	0
$13$	6.20150	1.71999	0.859993	−	0.510305i	$-0.170468\pi$
$13$	6.20150	1.71999	0.859993	+	0.510305i	$0.170468\pi$
$14$	2.75638	0.736672
$15$	0	0
$16$	−4.31592	−1.07898
$17$	3.74724	0.908839	0.454419	−	0.890788i	$-0.349847\pi$
$17$	3.74724	0.908839	0.454419	+	0.890788i	$0.349847\pi$
$18$	0	0
$19$	0.408977	0.0938258	0.0469129	−	0.998899i	$-0.485062\pi$
$19$	0.408977	0.0938258	0.0469129	+	0.998899i	$0.485062\pi$
$20$	2.20906	0.493960
$21$	0	0
$22$	1.80008	0.383778
$23$	−2.64505	−0.551530	−0.275765	−	0.961225i	$-0.588931\pi$
$23$	−2.64505	−0.551530	−0.275765	+	0.961225i	$0.588931\pi$
$24$	0	0
$25$	−3.53818	−0.707636
$26$	−12.1320	−2.37927
$27$	0	0
$28$	−2.57433	−0.486502
$29$	3.53976	0.657317	0.328659	−	0.944449i	$-0.393403\pi$
$29$	3.53976	0.657317	0.328659	+	0.944449i	$0.393403\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	7.76669	1.37297
$33$	0	0
$34$	−7.33070	−1.25721
$35$	−1.70353	−0.287950
$36$	0	0
$37$	−8.80284	−1.44718	−0.723589	−	0.690231i	$-0.757509\pi$
$37$	−8.80284	−1.44718	−0.723589	+	0.690231i	$0.757509\pi$
$38$	−0.800080	−0.129790
$39$	0	0
$40$	0.408977	0.0646650
$41$	8.17916	1.27737	0.638685	−	0.769468i	$-0.279479\pi$
$41$	8.17916	1.27737	0.638685	+	0.769468i	$0.279479\pi$
$42$	0	0
$43$	3.02392	0.461144	0.230572	−	0.973055i	$-0.425940\pi$
$43$	3.02392	0.461144	0.230572	+	0.973055i	$0.425940\pi$
$44$	−1.68119	−0.253449
$45$	0	0
$46$	5.17449	0.762937
$47$	8.67461	1.26532	0.632661	−	0.774429i	$-0.281963\pi$
$47$	8.67461	1.26532	0.632661	+	0.774429i	$0.281963\pi$
$48$	0	0
$49$	−5.01478	−0.716398
$50$	6.92173	0.978880
$51$	0	0
$52$	11.3307	1.57129
$53$	−12.7712	−1.75425	−0.877127	−	0.480259i	$-0.840543\pi$
$53$	−12.7712	−1.75425	−0.877127	+	0.480259i	$0.840543\pi$
$54$	0	0
$55$	−1.11251	−0.150011
$56$	−0.476602	−0.0636887
$57$	0	0
$58$	−6.92482	−0.909273
$59$	9.55008	1.24331	0.621657	−	0.783289i	$-0.286460\pi$
$59$	9.55008	1.24331	0.621657	+	0.783289i	$0.286460\pi$
$60$	0	0
$61$	8.32624	1.06607	0.533033	−	0.846095i	$-0.321052\pi$
$61$	8.32624	1.06607	0.533033	+	0.846095i	$0.321052\pi$
$62$	0	0
$63$	0	0
$64$	−6.56210	−0.820263
$65$	7.49797	0.930008
$66$	0	0
$67$	2.47214	0.302019	0.151010	−	0.988532i	$-0.451748\pi$
$67$	2.47214	0.302019	0.151010	+	0.988532i	$0.451748\pi$
$68$	6.84655	0.830266
$69$	0	0
$70$	3.33261	0.398324
$71$	11.5652	1.37254	0.686268	−	0.727349i	$-0.259248\pi$
$71$	11.5652	1.37254	0.686268	+	0.727349i	$0.259248\pi$
$72$	0	0
$73$	−5.77583	−0.676010	−0.338005	−	0.941144i	$-0.609752\pi$
$73$	−5.77583	−0.676010	−0.338005	+	0.941144i	$0.609752\pi$
$74$	17.2210	2.00190
$75$	0	0
$76$	0.747238	0.0857141
$77$	1.29647	0.147746
$78$	0	0
$79$	−14.3436	−1.61378	−0.806889	−	0.590703i	$-0.798851\pi$
$79$	−14.3436	−1.61378	−0.806889	+	0.590703i	$0.798851\pi$
$80$	−5.21819	−0.583412
$81$	0	0
$82$	−16.0009	−1.76700
$83$	10.5862	1.16199	0.580995	−	0.813907i	$-0.302664\pi$
$83$	10.5862	1.16199	0.580995	+	0.813907i	$0.302664\pi$
$84$	0	0
$85$	4.53062	0.491415
$86$	−5.91568	−0.637904
$87$	0	0
$88$	−0.311250	−0.0331794
$89$	9.82709	1.04167	0.520835	−	0.853658i	$-0.325621\pi$
$89$	9.82709	1.04167	0.520835	+	0.853658i	$0.325621\pi$
$90$	0	0
$91$	−8.73777	−0.915967
$92$	−4.83274	−0.503848
$93$	0	0
$94$	−16.9701	−1.75033
$95$	0.494477	0.0507322
$96$	0	0
$97$	11.8843	1.20667	0.603333	−	0.797490i	$-0.293839\pi$
$97$	11.8843	1.20667	0.603333	+	0.797490i	$0.293839\pi$
$98$	9.81040	0.991000
$99$	0	0
$100$	−6.46458	−0.646458
$101$	−3.46202	−0.344484	−0.172242	−	0.985055i	$-0.555101\pi$
$101$	−3.46202	−0.344484	−0.172242	+	0.985055i	$0.555101\pi$
$102$	0	0
$103$	10.1084	0.996015	0.498007	−	0.867173i	$-0.334065\pi$
$103$	10.1084	0.996015	0.498007	+	0.867173i	$0.334065\pi$
$104$	2.09773	0.205699
$105$	0	0
$106$	24.9842	2.42668
$107$	−6.36994	−0.615806	−0.307903	−	0.951418i	$-0.599627\pi$
$107$	−6.36994	−0.615806	−0.307903	+	0.951418i	$0.599627\pi$
$108$	0	0
$109$	−8.05284	−0.771322	−0.385661	−	0.922640i	$-0.626027\pi$
$109$	−8.05284	−0.771322	−0.385661	+	0.922640i	$0.626027\pi$
$110$	2.17640	0.207512
$111$	0	0
$112$	6.08103	0.574604
$113$	9.05500	0.851823	0.425911	−	0.904765i	$-0.359954\pi$
$113$	9.05500	0.851823	0.425911	+	0.904765i	$0.359954\pi$
$114$	0	0
$115$	−3.19801	−0.298216
$116$	6.46747	0.600489
$117$	0	0
$118$	−18.6828	−1.71989
$119$	−5.27977	−0.483996
$120$	0	0
$121$	−10.1533	−0.923030
$122$	−16.2886	−1.47470
$123$	0	0
$124$	0	0
$125$	−10.3231	−0.923330
$126$	0	0
$127$	9.19861	0.816245	0.408122	−	0.912927i	$-0.366184\pi$
$127$	9.19861	0.816245	0.408122	+	0.912927i	$0.366184\pi$
$128$	−2.69598	−0.238293
$129$	0	0
$130$	−14.6682	−1.28649
$131$	−1.93026	−0.168648	−0.0843238	−	0.996438i	$-0.526873\pi$
$131$	−1.93026	−0.168648	−0.0843238	+	0.996438i	$0.526873\pi$
$132$	0	0
$133$	−0.576239	−0.0499663
$134$	−4.83623	−0.417786
$135$	0	0
$136$	1.26755	0.108691
$137$	14.7768	1.26247	0.631234	−	0.775593i	$-0.282549\pi$
$137$	14.7768	1.26247	0.631234	+	0.775593i	$0.282549\pi$
$138$	0	0
$139$	−10.0937	−0.856133	−0.428067	−	0.903747i	$-0.640805\pi$
$139$	−10.0937	−0.856133	−0.428067	+	0.903747i	$0.640805\pi$
$140$	−3.11251	−0.263055
$141$	0	0
$142$	−22.6249	−1.89864
$143$	−5.70629	−0.477184
$144$	0	0
$145$	4.27977	0.355416
$146$	11.2992	0.935131
$147$	0	0
$148$	−16.0836	−1.32206
$149$	−19.4190	−1.59086	−0.795432	−	0.606043i	$-0.792756\pi$
$149$	−19.4190	−1.59086	−0.795432	+	0.606043i	$0.792756\pi$
$150$	0	0
$151$	5.56771	0.453093	0.226547	−	0.974000i	$-0.427256\pi$
$151$	5.56771	0.453093	0.226547	+	0.974000i	$0.427256\pi$
$152$	0.138341	0.0112209
$153$	0	0
$154$	−2.53627	−0.204379
$155$	0	0
$156$	0	0
$157$	−10.9852	−0.876716	−0.438358	−	0.898801i	$-0.644440\pi$
$157$	−10.9852	−0.876716	−0.438358	+	0.898801i	$0.644440\pi$
$158$	28.0603	2.23236
$159$	0	0
$160$	9.39037	0.742374
$161$	3.72681	0.293714
$162$	0	0
$163$	15.5705	1.21958	0.609788	−	0.792564i	$-0.291255\pi$
$163$	15.5705	1.21958	0.609788	+	0.792564i	$0.291255\pi$
$164$	14.9441	1.16694
$165$	0	0
$166$	−20.7098	−1.60739
$167$	17.4583	1.35097	0.675483	−	0.737376i	$-0.263935\pi$
$167$	17.4583	1.35097	0.675483	+	0.737376i	$0.263935\pi$
$168$	0	0
$169$	25.4586	1.95835
$170$	−8.86324	−0.679779
$171$	0	0
$172$	5.52498	0.421276
$173$	−11.6457	−0.885406	−0.442703	−	0.896668i	$-0.645980\pi$
$173$	−11.6457	−0.885406	−0.442703	+	0.896668i	$0.645980\pi$
$174$	0	0
$175$	4.98522	0.376847
$176$	3.97128	0.299347
$177$	0	0
$178$	−19.2247	−1.44095
$179$	−13.7997	−1.03144	−0.515721	−	0.856757i	$-0.672476\pi$
$179$	−13.7997	−1.03144	−0.515721	+	0.856757i	$0.672476\pi$
$180$	0	0
$181$	−5.78846	−0.430253	−0.215126	−	0.976586i	$-0.569016\pi$
$181$	−5.78846	−0.430253	−0.215126	+	0.976586i	$0.569016\pi$
$182$	17.0937	1.26707
$183$	0	0
$184$	−0.894716	−0.0659593
$185$	−10.6431	−0.782499
$186$	0	0
$187$	−3.44801	−0.252144
$188$	15.8493	1.15593
$189$	0	0
$190$	−0.967342	−0.0701784
$191$	8.18514	0.592256	0.296128	−	0.955148i	$-0.404305\pi$
$191$	8.18514	0.592256	0.296128	+	0.955148i	$0.404305\pi$
$192$	0	0
$193$	−21.3686	−1.53815	−0.769074	−	0.639160i	$-0.779282\pi$
$193$	−21.3686	−1.53815	−0.769074	+	0.639160i	$0.779282\pi$
$194$	−23.2491	−1.66919
$195$	0	0
$196$	−9.16247	−0.654462
$197$	−8.04273	−0.573021	−0.286510	−	0.958077i	$-0.592495\pi$
$197$	−8.04273	−0.573021	−0.286510	+	0.958077i	$0.592495\pi$
$198$	0	0
$199$	12.9069	0.914949	0.457474	−	0.889223i	$-0.348754\pi$
$199$	12.9069	0.914949	0.457474	+	0.889223i	$0.348754\pi$
$200$	−1.19683	−0.0846286
$201$	0	0
$202$	6.77274	0.476528
$203$	−4.98744	−0.350050
$204$	0	0
$205$	9.88907	0.690683
$206$	−19.7751	−1.37780
$207$	0	0
$208$	−26.7652	−1.85583
$209$	−0.376319	−0.0260305
$210$	0	0
$211$	−10.9513	−0.753918	−0.376959	−	0.926230i	$-0.623030\pi$
$211$	−10.9513	−0.753918	−0.376959	+	0.926230i	$0.623030\pi$
$212$	−23.3341	−1.60259
$213$	0	0
$214$	12.4615	0.851850
$215$	3.65609	0.249343
$216$	0	0
$217$	0	0
$218$	15.7537	1.06698
$219$	0	0
$220$	−2.03266	−0.137042
$221$	23.2385	1.56319
$222$	0	0
$223$	28.6809	1.92061	0.960306	−	0.278948i	$-0.0899856\pi$
$223$	28.6809	1.92061	0.960306	+	0.278948i	$0.0899856\pi$
$224$	−10.9431	−0.731166
$225$	0	0
$226$	−17.7143	−1.17833
$227$	−7.74277	−0.513906	−0.256953	−	0.966424i	$-0.582719\pi$
$227$	−7.74277	−0.513906	−0.256953	+	0.966424i	$0.582719\pi$
$228$	0	0
$229$	14.5545	0.961791	0.480896	−	0.876778i	$-0.340312\pi$
$229$	14.5545	0.961791	0.480896	+	0.876778i	$0.340312\pi$
$230$	6.25625	0.412525
$231$	0	0
$232$	1.19736	0.0786108
$233$	2.67685	0.175366	0.0876832	−	0.996148i	$-0.472054\pi$
$233$	2.67685	0.175366	0.0876832	+	0.996148i	$0.472054\pi$
$234$	0	0
$235$	10.4881	0.684168
$236$	17.4489	1.13582
$237$	0	0
$238$	10.3288	0.669516
$239$	11.3447	0.733829	0.366914	−	0.930255i	$-0.380414\pi$
$239$	11.3447	0.733829	0.366914	+	0.930255i	$0.380414\pi$
$240$	0	0
$241$	3.78278	0.243670	0.121835	−	0.992550i	$-0.461122\pi$
$241$	3.78278	0.243670	0.121835	+	0.992550i	$0.461122\pi$
$242$	19.8629	1.27684
$243$	0	0
$244$	15.2128	0.973899
$245$	−6.06316	−0.387361
$246$	0	0
$247$	2.53627	0.161379
$248$	0	0
$249$	0	0
$250$	20.1951	1.27725
$251$	−8.91884	−0.562952	−0.281476	−	0.959568i	$-0.590824\pi$
$251$	−8.91884	−0.562952	−0.281476	+	0.959568i	$0.590824\pi$
$252$	0	0
$253$	2.43383	0.153014
$254$	−17.9952	−1.12912
$255$	0	0
$256$	18.3983	1.14990
$257$	−2.59378	−0.161796	−0.0808979	−	0.996722i	$-0.525779\pi$
$257$	−2.59378	−0.161796	−0.0808979	+	0.996722i	$0.525779\pi$
$258$	0	0
$259$	12.4030	0.770685
$260$	13.6995	0.849605
$261$	0	0
$262$	3.77616	0.233292
$263$	11.7482	0.724426	0.362213	−	0.932095i	$-0.382021\pi$
$263$	11.7482	0.724426	0.362213	+	0.932095i	$0.382021\pi$
$264$	0	0
$265$	−15.4411	−0.948537
$266$	1.12729	0.0691188
$267$	0	0
$268$	4.51682	0.275909
$269$	6.46767	0.394341	0.197170	−	0.980369i	$-0.436825\pi$
$269$	6.46767	0.394341	0.197170	+	0.980369i	$0.436825\pi$
$270$	0	0
$271$	30.5451	1.85548	0.927741	−	0.373225i	$-0.121748\pi$
$271$	30.5451	1.85548	0.927741	+	0.373225i	$0.121748\pi$
$272$	−16.1728	−0.980619
$273$	0	0
$274$	−28.9078	−1.74638
$275$	3.25565	0.196323
$276$	0	0
$277$	−6.75947	−0.406137	−0.203068	−	0.979165i	$-0.565091\pi$
$277$	−6.75947	−0.406137	−0.203068	+	0.979165i	$0.565091\pi$
$278$	19.7462	1.18430
$279$	0	0
$280$	−0.576239	−0.0344369
$281$	−18.8091	−1.12206	−0.561029	−	0.827796i	$-0.689595\pi$
$281$	−18.8091	−1.12206	−0.561029	+	0.827796i	$0.689595\pi$
$282$	0	0
$283$	−5.39419	−0.320652	−0.160326	−	0.987064i	$-0.551254\pi$
$283$	−5.39419	−0.320652	−0.160326	+	0.987064i	$0.551254\pi$
$284$	21.1307	1.25387
$285$	0	0
$286$	11.1632	0.660094
$287$	−11.5242	−0.680255
$288$	0	0
$289$	−2.95821	−0.174012
$290$	−8.37250	−0.491650
$291$	0	0
$292$	−10.5530	−0.617566
$293$	17.0420	0.995604	0.497802	−	0.867291i	$-0.334141\pi$
$293$	17.0420	0.995604	0.497802	+	0.867291i	$0.334141\pi$
$294$	0	0
$295$	11.5466	0.672268
$296$	−2.97766	−0.173073
$297$	0	0
$298$	37.9892	2.20066
$299$	−16.4032	−0.948624
$300$	0	0
$301$	−4.26063	−0.245579
$302$	−10.8921	−0.626769
$303$	0	0
$304$	−1.76511	−0.101236
$305$	10.0669	0.576429
$306$	0	0
$307$	−2.96190	−0.169045	−0.0845223	−	0.996422i	$-0.526936\pi$
$307$	−2.96190	−0.169045	−0.0845223	+	0.996422i	$0.526936\pi$
$308$	2.36876	0.134973
$309$	0	0
$310$	0	0
$311$	−25.0949	−1.42300	−0.711500	−	0.702686i	$-0.751984\pi$
$311$	−25.0949	−1.42300	−0.711500	+	0.702686i	$0.751984\pi$
$312$	0	0
$313$	−1.65885	−0.0937639	−0.0468819	−	0.998900i	$-0.514928\pi$
$313$	−1.65885	−0.0937639	−0.0468819	+	0.998900i	$0.514928\pi$
$314$	21.4903	1.21277
$315$	0	0
$316$	−26.2070	−1.47426
$317$	29.9057	1.67967	0.839835	−	0.542842i	$-0.182652\pi$
$317$	29.9057	1.67967	0.839835	+	0.542842i	$0.182652\pi$
$318$	0	0
$319$	−3.25710	−0.182363
$320$	−7.93395	−0.443522
$321$	0	0
$322$	−7.29074	−0.406297
$323$	1.53253	0.0852725
$324$	0	0
$325$	−21.9420	−1.21712
$326$	−30.4605	−1.68705
$327$	0	0
$328$	2.76669	0.152765
$329$	−12.2223	−0.673839
$330$	0	0
$331$	11.8467	0.651156	0.325578	−	0.945515i	$-0.394441\pi$
$331$	11.8467	0.651156	0.325578	+	0.945515i	$0.394441\pi$
$332$	19.3420	1.06153
$333$	0	0
$334$	−34.1536	−1.86880
$335$	2.98895	0.163304
$336$	0	0
$337$	−4.37343	−0.238236	−0.119118	−	0.992880i	$-0.538007\pi$
$337$	−4.37343	−0.238236	−0.119118	+	0.992880i	$0.538007\pi$
$338$	−49.8045	−2.70901
$339$	0	0
$340$	8.27786	0.448930
$341$	0	0
$342$	0	0
$343$	16.9286	0.914056
$344$	1.02287	0.0551497
$345$	0	0
$346$	22.7824	1.22479
$347$	−15.0162	−0.806114	−0.403057	−	0.915175i	$-0.632052\pi$
$347$	−15.0162	−0.806114	−0.403057	+	0.915175i	$0.632052\pi$
$348$	0	0
$349$	−4.01065	−0.214685	−0.107343	−	0.994222i	$-0.534234\pi$
$349$	−4.01065	−0.214685	−0.107343	+	0.994222i	$0.534234\pi$
$350$	−9.75255	−0.521296
$351$	0	0
$352$	−7.14650	−0.380910
$353$	4.81775	0.256423	0.128211	−	0.991747i	$-0.459076\pi$
$353$	4.81775	0.256423	0.128211	+	0.991747i	$0.459076\pi$
$354$	0	0
$355$	13.9830	0.742139
$356$	17.9550	0.951612
$357$	0	0
$358$	26.9964	1.42680
$359$	25.4227	1.34176	0.670878	−	0.741567i	$-0.265917\pi$
$359$	25.4227	1.34176	0.670878	+	0.741567i	$0.265917\pi$
$360$	0	0
$361$	−18.8327	−0.991197
$362$	11.3239	0.595172
$363$	0	0
$364$	−15.9647	−0.836778
$365$	−6.98331	−0.365523
$366$	0	0
$367$	−10.3288	−0.539159	−0.269579	−	0.962978i	$-0.586885\pi$
$367$	−10.3288	−0.539159	−0.269579	+	0.962978i	$0.586885\pi$
$368$	11.4158	0.595090
$369$	0	0
$370$	20.8211	1.08244
$371$	17.9943	0.934216
$372$	0	0
$373$	−1.70893	−0.0884852	−0.0442426	−	0.999021i	$-0.514087\pi$
$373$	−1.70893	−0.0884852	−0.0442426	+	0.999021i	$0.514087\pi$
$374$	6.74533	0.348793
$375$	0	0
$376$	2.93428	0.151324
$377$	21.9518	1.13058
$378$	0	0
$379$	0.437772	0.0224868	0.0112434	−	0.999937i	$-0.496421\pi$
$379$	0.437772	0.0224868	0.0112434	+	0.999937i	$0.496421\pi$
$380$	0.903454	0.0463462
$381$	0	0
$382$	−16.0125	−0.819273
$383$	−15.9713	−0.816094	−0.408047	−	0.912961i	$-0.633790\pi$
$383$	−15.9713	−0.816094	−0.408047	+	0.912961i	$0.633790\pi$
$384$	0	0
$385$	1.56750	0.0798873
$386$	41.8034	2.12774
$387$	0	0
$388$	21.7136	1.10234
$389$	22.7844	1.15521	0.577607	−	0.816315i	$-0.303987\pi$
$389$	22.7844	1.15521	0.577607	+	0.816315i	$0.303987\pi$
$390$	0	0
$391$	−9.91161	−0.501252
$392$	−1.69631	−0.0856764
$393$	0	0
$394$	15.7340	0.792665
$395$	−17.3422	−0.872581
$396$	0	0
$397$	−17.2017	−0.863331	−0.431665	−	0.902034i	$-0.642074\pi$
$397$	−17.2017	−0.863331	−0.431665	+	0.902034i	$0.642074\pi$
$398$	−25.2498	−1.26566
$399$	0	0
$400$	15.2705	0.763525
$401$	−0.182896	−0.00913340	−0.00456670	−	0.999990i	$-0.501454\pi$
$401$	−0.182896	−0.00913340	−0.00456670	+	0.999990i	$0.501454\pi$
$402$	0	0
$403$	0	0
$404$	−6.32543	−0.314702
$405$	0	0
$406$	9.75691	0.484227
$407$	8.09991	0.401498
$408$	0	0
$409$	34.9539	1.72836	0.864180	−	0.503182i	$-0.167838\pi$
$409$	34.9539	1.72836	0.864180	+	0.503182i	$0.167838\pi$
$410$	−19.3459	−0.955428
$411$	0	0
$412$	18.4690	0.909905
$413$	−13.4558	−0.662119
$414$	0	0
$415$	12.7994	0.628295
$416$	48.1651	2.36149
$417$	0	0
$418$	0.736191	0.0360083
$419$	−17.7031	−0.864855	−0.432427	−	0.901669i	$-0.642343\pi$
$419$	−17.7031	−0.864855	−0.432427	+	0.901669i	$0.642343\pi$
$420$	0	0
$421$	−7.86264	−0.383201	−0.191601	−	0.981473i	$-0.561368\pi$
$421$	−7.86264	−0.383201	−0.191601	+	0.981473i	$0.561368\pi$
$422$	21.4240	1.04290
$423$	0	0
$424$	−4.31999	−0.209797
$425$	−13.2584	−0.643127
$426$	0	0
$427$	−11.7315	−0.567726
$428$	−11.6385	−0.562566
$429$	0	0
$430$	−7.15240	−0.344919
$431$	−12.3149	−0.593190	−0.296595	−	0.955003i	$-0.595851\pi$
$431$	−12.3149	−0.593190	−0.296595	+	0.955003i	$0.595851\pi$
$432$	0	0
$433$	−36.0565	−1.73276	−0.866381	−	0.499383i	$-0.833560\pi$
$433$	−36.0565	−1.73276	−0.866381	+	0.499383i	$0.833560\pi$
$434$	0	0
$435$	0	0
$436$	−14.7133	−0.704638
$437$	−1.08176	−0.0517477
$438$	0	0
$439$	8.89378	0.424477	0.212239	−	0.977218i	$-0.431925\pi$
$439$	8.89378	0.424477	0.212239	+	0.977218i	$0.431925\pi$
$440$	−0.376319	−0.0179403
$441$	0	0
$442$	−45.4614	−2.16238
$443$	−3.09306	−0.146956	−0.0734778	−	0.997297i	$-0.523410\pi$
$443$	−3.09306	−0.146956	−0.0734778	+	0.997297i	$0.523410\pi$
$444$	0	0
$445$	11.8815	0.563238
$446$	−56.1082	−2.65680
$447$	0	0
$448$	9.24585	0.436825
$449$	8.94598	0.422187	0.211093	−	0.977466i	$-0.432298\pi$
$449$	8.94598	0.422187	0.211093	+	0.977466i	$0.432298\pi$
$450$	0	0
$451$	−7.52603	−0.354387
$452$	16.5443	0.778179
$453$	0	0
$454$	15.1471	0.710891
$455$	−10.5645	−0.495270
$456$	0	0
$457$	−8.30678	−0.388575	−0.194287	−	0.980945i	$-0.562239\pi$
$457$	−8.30678	−0.388575	−0.194287	+	0.980945i	$0.562239\pi$
$458$	−28.4730	−1.33046
$459$	0	0
$460$	−5.84306	−0.272434
$461$	−9.75191	−0.454192	−0.227096	−	0.973872i	$-0.572923\pi$
$461$	−9.75191	−0.454192	−0.227096	+	0.973872i	$0.572923\pi$
$462$	0	0
$463$	8.98115	0.417389	0.208695	−	0.977981i	$-0.433079\pi$
$463$	8.98115	0.417389	0.208695	+	0.977981i	$0.433079\pi$
$464$	−15.2773	−0.709232
$465$	0	0
$466$	−5.23671	−0.242586
$467$	−6.05029	−0.279974	−0.139987	−	0.990153i	$-0.544706\pi$
$467$	−6.05029	−0.279974	−0.139987	+	0.990153i	$0.544706\pi$
$468$	0	0
$469$	−3.48318	−0.160838
$470$	−20.5178	−0.946416
$471$	0	0
$472$	3.23042	0.148692
$473$	−2.78245	−0.127937
$474$	0	0
$475$	−1.44704	−0.0663945
$476$	−9.64662	−0.442152
$477$	0	0
$478$	−22.1936	−1.01511
$479$	−21.6597	−0.989657	−0.494829	−	0.868991i	$-0.664769\pi$
$479$	−21.6597	−0.989657	−0.494829	+	0.868991i	$0.664769\pi$
$480$	0	0
$481$	−54.5908	−2.48913
$482$	−7.40024	−0.337072
$483$	0	0
$484$	−18.5511	−0.843230
$485$	14.3688	0.652452
$486$	0	0
$487$	−0.541671	−0.0245455	−0.0122727	−	0.999925i	$-0.503907\pi$
$487$	−0.541671	−0.0245455	−0.0122727	+	0.999925i	$0.503907\pi$
$488$	2.81644	0.127494
$489$	0	0
$490$	11.8613	0.535840
$491$	39.8245	1.79726	0.898628	−	0.438712i	$-0.144565\pi$
$491$	39.8245	1.79726	0.898628	+	0.438712i	$0.144565\pi$
$492$	0	0
$493$	13.2643	0.597395
$494$	−4.96170	−0.223237
$495$	0	0
$496$	0	0
$497$	−16.2951	−0.730935
$498$	0	0
$499$	9.75577	0.436728	0.218364	−	0.975867i	$-0.429928\pi$
$499$	9.75577	0.436728	0.218364	+	0.975867i	$0.429928\pi$
$500$	−18.8613	−0.843504
$501$	0	0
$502$	17.4479	0.778737
$503$	44.2820	1.97444	0.987218	−	0.159373i	$-0.0509474\pi$
$503$	44.2820	1.97444	0.987218	+	0.159373i	$0.0509474\pi$
$504$	0	0
$505$	−4.18578	−0.186265
$506$	−4.76129	−0.211665
$507$	0	0
$508$	16.8067	0.745677
$509$	−16.0324	−0.710622	−0.355311	−	0.934748i	$-0.615625\pi$
$509$	−16.0324	−0.710622	−0.355311	+	0.934748i	$0.615625\pi$
$510$	0	0
$511$	8.13801	0.360004
$512$	−30.6006	−1.35237
$513$	0	0
$514$	5.07421	0.223814
$515$	12.2217	0.538552
$516$	0	0
$517$	−7.98192	−0.351045
$518$	−24.2639	−1.06610
$519$	0	0
$520$	2.53627	0.111223
$521$	−35.5205	−1.55618	−0.778091	−	0.628152i	$-0.783812\pi$
$521$	−35.5205	−1.55618	−0.778091	+	0.628152i	$0.783812\pi$
$522$	0	0
$523$	32.0980	1.40355	0.701773	−	0.712400i	$-0.252392\pi$
$523$	32.0980	1.40355	0.701773	+	0.712400i	$0.252392\pi$
$524$	−3.52676	−0.154067
$525$	0	0
$526$	−22.9830	−1.00211
$527$	0	0
$528$	0	0
$529$	−16.0037	−0.695815
$530$	30.2073	1.31212
$531$	0	0
$532$	−1.05284	−0.0456465
$533$	50.7231	2.19706
$534$	0	0
$535$	−7.70162	−0.332970
$536$	0.836228	0.0361195
$537$	0	0
$538$	−12.6527	−0.545495
$539$	4.61434	0.198754
$540$	0	0
$541$	−9.18074	−0.394711	−0.197355	−	0.980332i	$-0.563235\pi$
$541$	−9.18074	−0.394711	−0.197355	+	0.980332i	$0.563235\pi$
$542$	−59.7552	−2.56671
$543$	0	0
$544$	29.1036	1.24781
$545$	−9.73634	−0.417059
$546$	0	0
$547$	8.87677	0.379543	0.189772	−	0.981828i	$-0.439225\pi$
$547$	8.87677	0.379543	0.189772	+	0.981828i	$0.439225\pi$
$548$	26.9986	1.15332
$549$	0	0
$550$	−6.36901	−0.271575
$551$	1.44768	0.0616733
$552$	0	0
$553$	20.2098	0.859407
$554$	13.2235	0.561813
$555$	0	0
$556$	−18.4420	−0.782116
$557$	−25.8880	−1.09691	−0.548455	−	0.836180i	$-0.684784\pi$
$557$	−25.8880	−1.09691	−0.548455	+	0.836180i	$0.684784\pi$
$558$	0	0
$559$	18.7528	0.793161
$560$	7.35232	0.310692
$561$	0	0
$562$	36.7962	1.55215
$563$	44.8356	1.88959	0.944797	−	0.327656i	$-0.106259\pi$
$563$	44.8356	1.88959	0.944797	+	0.327656i	$0.106259\pi$
$564$	0	0
$565$	10.9480	0.460586
$566$	10.5526	0.443560
$567$	0	0
$568$	3.91206	0.164146
$569$	−5.92214	−0.248269	−0.124134	−	0.992265i	$-0.539615\pi$
$569$	−5.92214	−0.248269	−0.124134	+	0.992265i	$0.539615\pi$
$570$	0	0
$571$	−35.5370	−1.48718	−0.743588	−	0.668639i	$-0.766877\pi$
$571$	−35.5370	−1.48718	−0.743588	+	0.668639i	$0.766877\pi$
$572$	−10.4259	−0.435930
$573$	0	0
$574$	22.5448	0.941003
$575$	9.35865	0.390283
$576$	0	0
$577$	−41.6771	−1.73504	−0.867521	−	0.497401i	$-0.834288\pi$
$577$	−41.6771	−1.73504	−0.867521	+	0.497401i	$0.834288\pi$
$578$	5.78712	0.240713
$579$	0	0
$580$	7.81953	0.324688
$581$	−14.9158	−0.618810
$582$	0	0
$583$	11.7513	0.486691
$584$	−1.95374	−0.0808463
$585$	0	0
$586$	−33.3392	−1.37723
$587$	15.4357	0.637098	0.318549	−	0.947906i	$-0.396804\pi$
$587$	15.4357	0.637098	0.318549	+	0.947906i	$0.396804\pi$
$588$	0	0
$589$	0	0
$590$	−22.5885	−0.929955
$591$	0	0
$592$	37.9924	1.56148
$593$	−23.1887	−0.952247	−0.476124	−	0.879378i	$-0.657959\pi$
$593$	−23.1887	−0.952247	−0.476124	+	0.879378i	$0.657959\pi$
$594$	0	0
$595$	−6.38355	−0.261700
$596$	−35.4802	−1.45333
$597$	0	0
$598$	32.0896	1.31224
$599$	32.0574	1.30983	0.654916	−	0.755702i	$-0.272704\pi$
$599$	32.0574	1.30983	0.654916	+	0.755702i	$0.272704\pi$
$600$	0	0
$601$	−23.0257	−0.939240	−0.469620	−	0.882869i	$-0.655609\pi$
$601$	−23.0257	−0.939240	−0.469620	+	0.882869i	$0.655609\pi$
$602$	8.33506	0.339712
$603$	0	0
$604$	10.1727	0.413921
$605$	−12.2760	−0.499088
$606$	0	0
$607$	11.5223	0.467674	0.233837	−	0.972276i	$-0.424872\pi$
$607$	11.5223	0.467674	0.233837	+	0.972276i	$0.424872\pi$
$608$	3.17640	0.128820
$609$	0	0
$610$	−19.6938	−0.797379
$611$	53.7956	2.17634
$612$	0	0
$613$	12.4722	0.503748	0.251874	−	0.967760i	$-0.418953\pi$
$613$	12.4722	0.503748	0.251874	+	0.967760i	$0.418953\pi$
$614$	5.79435	0.233841
$615$	0	0
$616$	0.438544	0.0176695
$617$	27.5153	1.10772	0.553862	−	0.832608i	$-0.313153\pi$
$617$	27.5153	1.10772	0.553862	+	0.832608i	$0.313153\pi$
$618$	0	0
$619$	48.5927	1.95311	0.976553	−	0.215276i	$-0.0690651\pi$
$619$	48.5927	1.95311	0.976553	+	0.215276i	$0.0690651\pi$
$620$	0	0
$621$	0	0
$622$	49.0930	1.96845
$623$	−13.8461	−0.554734
$624$	0	0
$625$	5.20963	0.208385
$626$	3.24520	0.129704
$627$	0	0
$628$	−20.0710	−0.800920
$629$	−32.9863	−1.31525
$630$	0	0
$631$	3.12705	0.124486	0.0622429	−	0.998061i	$-0.480175\pi$
$631$	3.12705	0.124486	0.0622429	+	0.998061i	$0.480175\pi$
$632$	−4.85188	−0.192997
$633$	0	0
$634$	−58.5043	−2.32350
$635$	11.1216	0.441349
$636$	0	0
$637$	−31.0992	−1.23219
$638$	6.37185	0.252264
$639$	0	0
$640$	−3.25959	−0.128847
$641$	4.42002	0.174580	0.0872902	−	0.996183i	$-0.472179\pi$
$641$	4.42002	0.174580	0.0872902	+	0.996183i	$0.472179\pi$
$642$	0	0
$643$	7.71227	0.304142	0.152071	−	0.988370i	$-0.451406\pi$
$643$	7.71227	0.304142	0.152071	+	0.988370i	$0.451406\pi$
$644$	6.80922	0.268321
$645$	0	0
$646$	−2.99809	−0.117958
$647$	−23.3445	−0.917767	−0.458884	−	0.888496i	$-0.651751\pi$
$647$	−23.3445	−0.917767	−0.458884	+	0.888496i	$0.651751\pi$
$648$	0	0
$649$	−8.78748	−0.344939
$650$	42.9251	1.68366
$651$	0	0
$652$	28.4487	1.11414
$653$	47.6918	1.86632	0.933162	−	0.359456i	$-0.117038\pi$
$653$	47.6918	1.86632	0.933162	+	0.359456i	$0.117038\pi$
$654$	0	0
$655$	−2.33379	−0.0911889
$656$	−35.3006	−1.37826
$657$	0	0
$658$	23.9105	0.932128
$659$	34.5547	1.34606	0.673029	−	0.739616i	$-0.264993\pi$
$659$	34.5547	1.34606	0.673029	+	0.739616i	$0.264993\pi$
$660$	0	0
$661$	40.9673	1.59344	0.796722	−	0.604346i	$-0.206565\pi$
$661$	40.9673	1.59344	0.796722	+	0.604346i	$0.206565\pi$
$662$	−23.1757	−0.900751
$663$	0	0
$664$	3.58091	0.138966
$665$	−0.696706	−0.0270171
$666$	0	0
$667$	−9.36283	−0.362530
$668$	31.8980	1.23417
$669$	0	0
$670$	−5.84727	−0.225900
$671$	−7.66137	−0.295764
$672$	0	0
$673$	34.4795	1.32909	0.664544	−	0.747249i	$-0.268626\pi$
$673$	34.4795	1.32909	0.664544	+	0.747249i	$0.268626\pi$
$674$	8.55573	0.329554
$675$	0	0
$676$	46.5152	1.78905
$677$	0.862262	0.0331394	0.0165697	−	0.999863i	$-0.494725\pi$
$677$	0.862262	0.0331394	0.0165697	+	0.999863i	$0.494725\pi$
$678$	0	0
$679$	−16.7447	−0.642601
$680$	1.53253	0.0587700
$681$	0	0
$682$	0	0
$683$	−32.0304	−1.22561	−0.612804	−	0.790235i	$-0.709959\pi$
$683$	−32.0304	−1.22561	−0.612804	+	0.790235i	$0.709959\pi$
$684$	0	0
$685$	17.8660	0.682625
$686$	−33.1173	−1.26442
$687$	0	0
$688$	−13.0510	−0.497565
$689$	−79.2003	−3.01729
$690$	0	0
$691$	16.8108	0.639512	0.319756	−	0.947500i	$-0.396399\pi$
$691$	16.8108	0.639512	0.319756	+	0.947500i	$0.396399\pi$
$692$	−21.2777	−0.808858
$693$	0	0
$694$	29.3762	1.11511
$695$	−12.2038	−0.462917
$696$	0	0
$697$	30.6493	1.16092
$698$	7.84601	0.296976
$699$	0	0
$700$	9.10844	0.344267
$701$	34.3025	1.29559	0.647794	−	0.761816i	$-0.275692\pi$
$701$	34.3025	1.29559	0.647794	+	0.761816i	$0.275692\pi$
$702$	0	0
$703$	−3.60016	−0.135783
$704$	6.03810	0.227569
$705$	0	0
$706$	−9.42494	−0.354712
$707$	4.87791	0.183453
$708$	0	0
$709$	18.3125	0.687739	0.343869	−	0.939017i	$-0.388262\pi$
$709$	18.3125	0.687739	0.343869	+	0.939017i	$0.388262\pi$
$710$	−27.3548	−1.02661
$711$	0	0
$712$	3.32412	0.124577
$713$	0	0
$714$	0	0
$715$	−6.89923	−0.258017
$716$	−25.2134	−0.942269
$717$	0	0
$718$	−49.7342	−1.85606
$719$	28.7819	1.07338	0.536692	−	0.843778i	$-0.319674\pi$
$719$	28.7819	1.07338	0.536692	+	0.843778i	$0.319674\pi$
$720$	0	0
$721$	−14.2426	−0.530421
$722$	36.8424	1.37113
$723$	0	0
$724$	−10.5760	−0.393055
$725$	−12.5243	−0.465141
$726$	0	0
$727$	−20.3764	−0.755720	−0.377860	−	0.925863i	$-0.623340\pi$
$727$	−20.3764	−0.755720	−0.377860	+	0.925863i	$0.623340\pi$
$728$	−2.95565	−0.109544
$729$	0	0
$730$	13.6614	0.505631
$731$	11.3314	0.419105
$732$	0	0
$733$	−15.4226	−0.569646	−0.284823	−	0.958580i	$-0.591935\pi$
$733$	−15.4226	−0.569646	−0.284823	+	0.958580i	$0.591935\pi$
$734$	20.2062	0.745823
$735$	0	0
$736$	−20.5433	−0.757234
$737$	−2.27473	−0.0837907
$738$	0	0
$739$	31.7931	1.16953	0.584764	−	0.811204i	$-0.301187\pi$
$739$	31.7931	1.16953	0.584764	+	0.811204i	$0.301187\pi$
$740$	−19.4460	−0.714848
$741$	0	0
$742$	−35.2021	−1.29231
$743$	−3.69893	−0.135701	−0.0678503	−	0.997696i	$-0.521614\pi$
$743$	−3.69893	−0.135701	−0.0678503	+	0.997696i	$0.521614\pi$
$744$	0	0
$745$	−23.4786	−0.860191
$746$	3.34318	0.122402
$747$	0	0
$748$	−6.29983	−0.230345
$749$	8.97510	0.327943
$750$	0	0
$751$	29.1657	1.06427	0.532135	−	0.846659i	$-0.321390\pi$
$751$	29.1657	1.06427	0.532135	+	0.846659i	$0.321390\pi$
$752$	−37.4389	−1.36526
$753$	0	0
$754$	−42.9443	−1.56394
$755$	6.73167	0.244991
$756$	0	0
$757$	15.0512	0.547044	0.273522	−	0.961866i	$-0.411811\pi$
$757$	15.0512	0.547044	0.273522	+	0.961866i	$0.411811\pi$
$758$	−0.856411	−0.0311063
$759$	0	0
$760$	0.167262	0.00606724
$761$	8.66041	0.313939	0.156970	−	0.987603i	$-0.449827\pi$
$761$	8.66041	0.313939	0.156970	+	0.987603i	$0.449827\pi$
$762$	0	0
$763$	11.3463	0.410763
$764$	14.9550	0.541052
$765$	0	0
$766$	31.2445	1.12891
$767$	59.2248	2.13848
$768$	0	0
$769$	42.0435	1.51613	0.758063	−	0.652182i	$-0.226146\pi$
$769$	42.0435	1.51613	0.758063	+	0.652182i	$0.226146\pi$
$770$	−3.06650	−0.110509
$771$	0	0
$772$	−39.0424	−1.40517
$773$	8.87904	0.319357	0.159678	−	0.987169i	$-0.448954\pi$
$773$	8.87904	0.319357	0.159678	+	0.987169i	$0.448954\pi$
$774$	0	0
$775$	0	0
$776$	4.01999	0.144309
$777$	0	0
$778$	−44.5729	−1.59802
$779$	3.34509	0.119850
$780$	0	0
$781$	−10.6417	−0.380789
$782$	19.3900	0.693387
$783$	0	0
$784$	21.6434	0.772979
$785$	−13.2818	−0.474046
$786$	0	0
$787$	−17.2122	−0.613549	−0.306775	−	0.951782i	$-0.599250\pi$
$787$	−17.2122	−0.613549	−0.306775	+	0.951782i	$0.599250\pi$
$788$	−14.6948	−0.523480
$789$	0	0
$790$	33.9265	1.20705
$791$	−12.7583	−0.453632
$792$	0	0
$793$	51.6352	1.83362
$794$	33.6517	1.19425
$795$	0	0
$796$	23.5822	0.835847
$797$	−2.31941	−0.0821577	−0.0410789	−	0.999156i	$-0.513079\pi$
$797$	−2.31941	−0.0821577	−0.0410789	+	0.999156i	$0.513079\pi$
$798$	0	0
$799$	32.5058	1.14997
$800$	−27.4800	−0.971564
$801$	0	0
$802$	0.357799	0.0126343
$803$	5.31461	0.187549
$804$	0	0
$805$	4.50592	0.158813
$806$	0	0
$807$	0	0
$808$	−1.17107	−0.0411980
$809$	23.2869	0.818723	0.409361	−	0.912372i	$-0.365752\pi$
$809$	23.2869	0.818723	0.409361	+	0.912372i	$0.365752\pi$
$810$	0	0
$811$	27.7475	0.974347	0.487173	−	0.873305i	$-0.338028\pi$
$811$	27.7475	0.974347	0.487173	+	0.873305i	$0.338028\pi$
$812$	−9.11251	−0.319786
$813$	0	0
$814$	−15.8458	−0.555396
$815$	18.8256	0.659433
$816$	0	0
$817$	1.23671	0.0432671
$818$	−68.3802	−2.39086
$819$	0	0
$820$	18.0682	0.630970
$821$	−15.3949	−0.537286	−0.268643	−	0.963240i	$-0.586575\pi$
$821$	−15.3949	−0.537286	−0.268643	+	0.963240i	$0.586575\pi$
$822$	0	0
$823$	47.9826	1.67257	0.836284	−	0.548296i	$-0.184723\pi$
$823$	47.9826	1.67257	0.836284	+	0.548296i	$0.184723\pi$
$824$	3.41929	0.119117
$825$	0	0
$826$	26.3236	0.915915
$827$	−21.8207	−0.758779	−0.379389	−	0.925237i	$-0.623866\pi$
$827$	−21.8207	−0.758779	−0.379389	+	0.925237i	$0.623866\pi$
$828$	0	0
$829$	8.93997	0.310498	0.155249	−	0.987875i	$-0.450382\pi$
$829$	8.93997	0.310498	0.155249	+	0.987875i	$0.450382\pi$
$830$	−25.0393	−0.869127
$831$	0	0
$832$	−40.6949	−1.41084
$833$	−18.7916	−0.651090
$834$	0	0
$835$	21.1081	0.730476
$836$	−0.687569	−0.0237801
$837$	0	0
$838$	34.6326	1.19636
$839$	−39.6772	−1.36981	−0.684904	−	0.728633i	$-0.740156\pi$
$839$	−39.6772	−1.36981	−0.684904	+	0.728633i	$0.740156\pi$
$840$	0	0
$841$	−16.4701	−0.567934
$842$	15.3816	0.530086
$843$	0	0
$844$	−20.0090	−0.688739
$845$	30.7809	1.05890
$846$	0	0
$847$	14.3058	0.491553
$848$	55.1193	1.89280
$849$	0	0
$850$	25.9374	0.889644
$851$	23.2839	0.798162
$852$	0	0
$853$	−34.4794	−1.18055	−0.590277	−	0.807201i	$-0.700981\pi$
$853$	−34.4794	−1.18055	−0.590277	+	0.807201i	$0.700981\pi$
$854$	22.9502	0.785341
$855$	0	0
$856$	−2.15470	−0.0736463
$857$	30.2447	1.03314	0.516569	−	0.856245i	$-0.327209\pi$
$857$	30.2447	1.03314	0.516569	+	0.856245i	$0.327209\pi$
$858$	0	0
$859$	34.0433	1.16154	0.580771	−	0.814067i	$-0.302751\pi$
$859$	34.0433	1.16154	0.580771	+	0.814067i	$0.302751\pi$
$860$	6.68001	0.227787
$861$	0	0
$862$	24.0917	0.820565
$863$	−16.5521	−0.563440	−0.281720	−	0.959497i	$-0.590905\pi$
$863$	−16.5521	−0.563440	−0.281720	+	0.959497i	$0.590905\pi$
$864$	0	0
$865$	−14.0803	−0.478745
$866$	70.5371	2.39695
$867$	0	0
$868$	0	0
$869$	13.1982	0.447718
$870$	0	0
$871$	15.3310	0.519469
$872$	−2.72396	−0.0922451
$873$	0	0
$874$	2.11625	0.0715831
$875$	14.5451	0.491713
$876$	0	0
$877$	7.44459	0.251386	0.125693	−	0.992069i	$-0.459885\pi$
$877$	7.44459	0.251386	0.125693	+	0.992069i	$0.459885\pi$
$878$	−17.3989	−0.587183
$879$	0	0
$880$	4.80151	0.161859
$881$	7.80795	0.263057	0.131528	−	0.991312i	$-0.458012\pi$
$881$	7.80795	0.263057	0.131528	+	0.991312i	$0.458012\pi$
$882$	0	0
$883$	−22.1797	−0.746406	−0.373203	−	0.927750i	$-0.621740\pi$
$883$	−22.1797	−0.746406	−0.373203	+	0.927750i	$0.621740\pi$
$884$	42.4588	1.42805
$885$	0	0
$886$	6.05093	0.203285
$887$	−13.9035	−0.466833	−0.233417	−	0.972377i	$-0.574991\pi$
$887$	−13.9035	−0.466833	−0.233417	+	0.972377i	$0.574991\pi$
$888$	0	0
$889$	−12.9606	−0.434686
$890$	−23.2437	−0.779132
$891$	0	0
$892$	52.4025	1.75457
$893$	3.54772	0.118720
$894$	0	0
$895$	−16.6847	−0.557707
$896$	3.79857	0.126901
$897$	0	0
$898$	−17.5010	−0.584015
$899$	0	0
$900$	0	0
$901$	−47.8566	−1.59433
$902$	14.7231	0.490227
$903$	0	0
$904$	3.06295	0.101872
$905$	−6.99857	−0.232640
$906$	0	0
$907$	−30.5173	−1.01331	−0.506656	−	0.862148i	$-0.669118\pi$
$907$	−30.5173	−1.01331	−0.506656	+	0.862148i	$0.669118\pi$
$908$	−14.1467	−0.469476
$909$	0	0
$910$	20.6672	0.685111
$911$	56.9920	1.88823	0.944114	−	0.329618i	$-0.106920\pi$
$911$	56.9920	1.88823	0.944114	+	0.329618i	$0.106920\pi$
$912$	0	0
$913$	−9.74089	−0.322376
$914$	16.2505	0.537519
$915$	0	0
$916$	26.5925	0.878640
$917$	2.71969	0.0898122
$918$	0	0
$919$	28.9729	0.955728	0.477864	−	0.878434i	$-0.341411\pi$
$919$	28.9729	0.955728	0.477864	+	0.878434i	$0.341411\pi$
$920$	−1.08176	−0.0356647
$921$	0	0
$922$	19.0776	0.628288
$923$	71.7215	2.36074
$924$	0	0
$925$	31.1460	1.02408
$926$	−17.5698	−0.577379
$927$	0	0
$928$	27.4922	0.902477
$929$	29.2747	0.960472	0.480236	−	0.877139i	$-0.340551\pi$
$929$	29.2747	0.960472	0.480236	+	0.877139i	$0.340551\pi$
$930$	0	0
$931$	−2.05093	−0.0672166
$932$	4.89085	0.160205
$933$	0	0
$934$	11.8361	0.387290
$935$	−4.16884	−0.136336
$936$	0	0
$937$	42.4635	1.38722	0.693610	−	0.720350i	$-0.256019\pi$
$937$	42.4635	1.38722	0.693610	+	0.720350i	$0.256019\pi$
$938$	6.81413	0.222489
$939$	0	0
$940$	19.1627	0.625019
$941$	−1.58307	−0.0516065	−0.0258032	−	0.999667i	$-0.508214\pi$
$941$	−1.58307	−0.0516065	−0.0258032	+	0.999667i	$0.508214\pi$
$942$	0	0
$943$	−21.6342	−0.704508
$944$	−41.2174	−1.34151
$945$	0	0
$946$	5.44330	0.176977
$947$	38.1911	1.24104	0.620522	−	0.784189i	$-0.286921\pi$
$947$	38.1911	1.24104	0.620522	+	0.784189i	$0.286921\pi$
$948$	0	0
$949$	−35.8188	−1.16273
$950$	2.83083	0.0918442
$951$	0	0
$952$	−1.78594	−0.0578827
$953$	−0.297044	−0.00962219	−0.00481109	−	0.999988i	$-0.501531\pi$
$953$	−0.297044	−0.00962219	−0.00481109	+	0.999988i	$0.501531\pi$
$954$	0	0
$955$	9.89630	0.320237
$956$	20.7278	0.670386
$957$	0	0
$958$	42.3728	1.36900
$959$	−20.8202	−0.672319
$960$	0	0
$961$	0	0
$962$	106.796	3.44323
$963$	0	0
$964$	6.91149	0.222604
$965$	−25.8359	−0.831687
$966$	0	0
$967$	31.1353	1.00124	0.500622	−	0.865666i	$-0.333105\pi$
$967$	31.1353	1.00124	0.500622	+	0.865666i	$0.333105\pi$
$968$	−3.43448	−0.110388
$969$	0	0
$970$	−28.1095	−0.902543
$971$	37.1929	1.19358	0.596789	−	0.802398i	$-0.296443\pi$
$971$	37.1929	1.19358	0.596789	+	0.802398i	$0.296443\pi$
$972$	0	0
$973$	14.2217	0.455928
$974$	1.05967	0.0339540
$975$	0	0
$976$	−35.9354	−1.15026
$977$	32.5944	1.04279	0.521393	−	0.853317i	$-0.325413\pi$
$977$	32.5944	1.04279	0.521393	+	0.853317i	$0.325413\pi$
$978$	0	0
$979$	−9.04237	−0.288995
$980$	−11.0779	−0.353872
$981$	0	0
$982$	−77.9085	−2.48616
$983$	−3.11486	−0.0993487	−0.0496743	−	0.998765i	$-0.515818\pi$
$983$	−3.11486	−0.0993487	−0.0496743	+	0.998765i	$0.515818\pi$
$984$	0	0
$985$	−9.72412	−0.309836
$986$	−25.9489	−0.826383
$987$	0	0
$988$	4.63400	0.147427
$989$	−7.99841	−0.254335
$990$	0	0
$991$	−49.6191	−1.57620	−0.788102	−	0.615545i	$-0.788936\pi$
$991$	−49.6191	−1.57620	−0.788102	+	0.615545i	$0.788936\pi$
$992$	0	0
$993$	0	0
$994$	31.8780	1.01111
$995$	15.6052	0.494719
$996$	0	0
$997$	−29.1043	−0.921741	−0.460871	−	0.887467i	$-0.652463\pi$
$997$	−29.1043	−0.921741	−0.460871	+	0.887467i	$0.652463\pi$
$998$	−19.0852	−0.604130
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.w.1.1		4
3.2	odd	2		2883.2.a.k.1.4		4
31.2	even	5		279.2.i.b.190.1		8
31.16	even	5		279.2.i.b.163.1		8
31.30	odd	2		8649.2.a.x.1.1		4
93.2	odd	10		93.2.f.a.4.2	✓	8
93.47	odd	10		93.2.f.a.70.2	yes	8
93.92	even	2		2883.2.a.l.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.f.a.4.2	✓	8	93.2	odd	10
93.2.f.a.70.2	yes	8	93.47	odd	10
279.2.i.b.163.1		8	31.16	even	5
279.2.i.b.190.1		8	31.2	even	5
2883.2.a.k.1.4		4	3.2	odd	2
2883.2.a.l.1.4		4	93.92	even	2
8649.2.a.w.1.1		4	1.1	even	1	trivial
8649.2.a.x.1.1		4	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)

\(f(q)\)	\(=\)	\(q-1.95630 q^{2} +1.82709 q^{4} +1.20906 q^{5} -1.40898 q^{7} +0.338261 q^{8} +O(q^{10})\)	\(q-1.95630 q^{2} +1.82709 q^{4} +1.20906 q^{5} -1.40898 q^{7} +0.338261 q^{8} -2.36527 q^{10} -0.920147 q^{11} +6.20150 q^{13} +2.75638 q^{14} -4.31592 q^{16} +3.74724 q^{17} +0.408977 q^{19} +2.20906 q^{20} +1.80008 q^{22} -2.64505 q^{23} -3.53818 q^{25} -12.1320 q^{26} -2.57433 q^{28} +3.53976 q^{29} +7.76669 q^{32} -7.33070 q^{34} -1.70353 q^{35} -8.80284 q^{37} -0.800080 q^{38} +0.408977 q^{40} +8.17916 q^{41} +3.02392 q^{43} -1.68119 q^{44} +5.17449 q^{46} +8.67461 q^{47} -5.01478 q^{49} +6.92173 q^{50} +11.3307 q^{52} -12.7712 q^{53} -1.11251 q^{55} -0.476602 q^{56} -6.92482 q^{58} +9.55008 q^{59} +8.32624 q^{61} -6.56210 q^{64} +7.49797 q^{65} +2.47214 q^{67} +6.84655 q^{68} +3.33261 q^{70} +11.5652 q^{71} -5.77583 q^{73} +17.2210 q^{74} +0.747238 q^{76} +1.29647 q^{77} -14.3436 q^{79} -5.21819 q^{80} -16.0009 q^{82} +10.5862 q^{83} +4.53062 q^{85} -5.91568 q^{86} -0.311250 q^{88} +9.82709 q^{89} -8.73777 q^{91} -4.83274 q^{92} -16.9701 q^{94} +0.494477 q^{95} +11.8843 q^{97} +9.81040 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + q^{4} + 3 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 4 * q + q^2 + q^4 + 3 * q^5 - 3 * q^7 - 3 * q^8	\( 4 q + q^{2} + q^{4} + 3 q^{5} - 3 q^{7} - 3 q^{8} + 2 q^{10} - 3 q^{11} + 6 q^{13} + 3 q^{14} - 9 q^{16} + 8 q^{17} - q^{19} + 7 q^{20} + 8 q^{22} + q^{23} - 9 q^{25} - 6 q^{26} + 3 q^{28} - 9 q^{29} - 8 q^{34} + 9 q^{35} + 4 q^{37} - 4 q^{38} - q^{40} + 10 q^{41} - 2 q^{43} + 13 q^{44} + 9 q^{46} - 3 q^{47} - 7 q^{49} + 9 q^{50} + 24 q^{52} - 30 q^{53} + 14 q^{55} + 6 q^{56} - 16 q^{58} - 8 q^{59} + 2 q^{61} - 7 q^{64} + 27 q^{65} - 8 q^{67} - 3 q^{68} + 21 q^{70} + 34 q^{71} + 9 q^{73} + 26 q^{74} - 4 q^{76} + 21 q^{77} + 6 q^{79} - 18 q^{80} + 5 q^{82} - 12 q^{83} - 4 q^{85} - 13 q^{86} - 4 q^{88} + 33 q^{89} + 18 q^{91} - q^{92} - 47 q^{94} - 12 q^{95} - 9 q^{97} + 17 q^{98}+O(q^{100}) \) 4 * q + q^2 + q^4 + 3 * q^5 - 3 * q^7 - 3 * q^8 + 2 * q^10 - 3 * q^11 + 6 * q^13 + 3 * q^14 - 9 * q^16 + 8 * q^17 - q^19 + 7 * q^20 + 8 * q^22 + q^23 - 9 * q^25 - 6 * q^26 + 3 * q^28 - 9 * q^29 - 8 * q^34 + 9 * q^35 + 4 * q^37 - 4 * q^38 - q^40 + 10 * q^41 - 2 * q^43 + 13 * q^44 + 9 * q^46 - 3 * q^47 - 7 * q^49 + 9 * q^50 + 24 * q^52 - 30 * q^53 + 14 * q^55 + 6 * q^56 - 16 * q^58 - 8 * q^59 + 2 * q^61 - 7 * q^64 + 27 * q^65 - 8 * q^67 - 3 * q^68 + 21 * q^70 + 34 * q^71 + 9 * q^73 + 26 * q^74 - 4 * q^76 + 21 * q^77 + 6 * q^79 - 18 * q^80 + 5 * q^82 - 12 * q^83 - 4 * q^85 - 13 * q^86 - 4 * q^88 + 33 * q^89 + 18 * q^91 - q^92 - 47 * q^94 - 12 * q^95 - 9 * q^97 + 17 * q^98

Introduction

L-functions

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