LMFDB - Embedded newform 8649.2.a.bf.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8649,2,Mod(1,8649)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [8,-2,0,8,-3,0,-2,9,0,-13,18,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$69.0626127082$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.2051578125.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 9x^{6} + 19x^{5} + 14x^{4} - 28x^{3} - 11x^{2} + 6x + 1 $ x^8 - 2x^7 - 9x^6 + 19x^5 + 14x^4 - 28x^3 - 11x^2 + 6*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 31)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.143490$ 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)

$f(q)$	$=$	$q-1.23217 q^{2} -0.481752 q^{4} +1.54562 q^{5} -3.80376 q^{7} +3.05795 q^{8} -1.90447 q^{10} +3.76152 q^{11} -2.63506 q^{13} +4.68689 q^{14} -2.80441 q^{16} -3.77434 q^{17} +6.09831 q^{19} -0.744606 q^{20} -4.63485 q^{22} -0.909847 q^{23} -2.61105 q^{25} +3.24685 q^{26} +1.83247 q^{28} +6.80859 q^{29} -2.66037 q^{32} +4.65064 q^{34} -5.87917 q^{35} -1.81406 q^{37} -7.51417 q^{38} +4.72643 q^{40} -0.337145 q^{41} +3.88400 q^{43} -1.81212 q^{44} +1.12109 q^{46} +1.18915 q^{47} +7.46858 q^{49} +3.21726 q^{50} +1.26945 q^{52} +2.34413 q^{53} +5.81390 q^{55} -11.6317 q^{56} -8.38936 q^{58} -7.77883 q^{59} -2.72343 q^{61} +8.88686 q^{64} -4.07281 q^{65} -7.42118 q^{67} +1.81829 q^{68} +7.24416 q^{70} +5.09818 q^{71} +5.39174 q^{73} +2.23524 q^{74} -2.93787 q^{76} -14.3079 q^{77} -9.73723 q^{79} -4.33456 q^{80} +0.415420 q^{82} +8.39515 q^{83} -5.83370 q^{85} -4.78575 q^{86} +11.5025 q^{88} -5.09560 q^{89} +10.0231 q^{91} +0.438320 q^{92} -1.46523 q^{94} +9.42569 q^{95} +10.9142 q^{97} -9.20257 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 8 q^{4} - 3 q^{5} - 2 q^{7} + 9 q^{8} - 13 q^{10} + 18 q^{11} - 8 q^{13} + 9 q^{14} + 4 q^{16} + 14 q^{17} - 6 q^{19} + 7 q^{20} - 4 q^{22} + 22 q^{23} + 13 q^{25} + 9 q^{26} - 5 q^{28}+ \cdots - 10 q^{98}+O(q^{100}) $ 8 * q - 2 * q^2 + 8 * q^4 - 3 * q^5 - 2 * q^7 + 9 * q^8 - 13 * q^10 + 18 * q^11 - 8 * q^13 + 9 * q^14 + 4 * q^16 + 14 * q^17 - 6 * q^19 + 7 * q^20 - 4 * q^22 + 22 * q^23 + 13 * q^25 + 9 * q^26 - 5 * q^28 + 12 * q^29 + 21 * q^32 + 7 * q^34 + 12 * q^35 + 8 * q^37 - 7 * q^38 + 11 * q^40 + 22 * q^41 + 2 * q^43 + 4 * q^44 - 18 * q^46 + 18 * q^47 - 2 * q^49 - 27 * q^50 + q^52 + 6 * q^53 - 28 * q^55 - 30 * q^56 + 5 * q^58 + 4 * q^59 - 30 * q^61 + 9 * q^64 + 3 * q^65 - 13 * q^67 + 30 * q^68 - 39 * q^70 + q^71 - 2 * q^73 + 8 * q^74 - 33 * q^76 - 24 * q^77 - 8 * q^79 - 24 * q^80 - 19 * q^82 + 39 * q^83 + 26 * q^85 - 16 * q^86 + 17 * q^88 + 27 * q^89 - 16 * q^91 + 32 * q^92 + 22 * q^94 + 6 * q^95 + 34 * q^97 - 10 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.23217	−0.871277	−0.435639	−	0.900122i	$-0.643477\pi$
$2$	−1.23217	−0.871277	−0.435639	+	0.900122i	$0.643477\pi$
$3$	0	0
$3$	0	0
$4$	−0.481752	−0.240876
$5$	1.54562	0.691223	0.345612	−	0.938378i	$-0.387671\pi$
$5$	1.54562	0.691223	0.345612	+	0.938378i	$0.387671\pi$
$6$	0	0
$7$	−3.80376	−1.43769	−0.718843	−	0.695173i	$-0.755328\pi$
$7$	−3.80376	−1.43769	−0.718843	+	0.695173i	$0.755328\pi$
$8$	3.05795	1.08115
$9$	0	0
$10$	−1.90447	−0.602247
$11$	3.76152	1.13414	0.567071	−	0.823669i	$-0.308076\pi$
$11$	3.76152	1.13414	0.567071	+	0.823669i	$0.308076\pi$
$12$	0	0
$13$	−2.63506	−0.730835	−0.365418	−	0.930844i	$-0.619074\pi$
$13$	−2.63506	−0.730835	−0.365418	+	0.930844i	$0.619074\pi$
$14$	4.68689	1.25262
$15$	0	0
$16$	−2.80441	−0.701103
$17$	−3.77434	−0.915412	−0.457706	−	0.889104i	$-0.651329\pi$
$17$	−3.77434	−0.915412	−0.457706	+	0.889104i	$0.651329\pi$
$18$	0	0
$19$	6.09831	1.39905	0.699525	−	0.714608i	$-0.253395\pi$
$19$	6.09831	1.39905	0.699525	+	0.714608i	$0.253395\pi$
$20$	−0.744606	−0.166499
$21$	0	0
$22$	−4.63485	−0.988152
$23$	−0.909847	−0.189716	−0.0948581	−	0.995491i	$-0.530240\pi$
$23$	−0.909847	−0.189716	−0.0948581	+	0.995491i	$0.530240\pi$
$24$	0	0
$25$	−2.61105	−0.522210
$26$	3.24685	0.636760
$27$	0	0
$28$	1.83247	0.346304
$29$	6.80859	1.26432	0.632162	−	0.774836i	$-0.282168\pi$
$29$	6.80859	1.26432	0.632162	+	0.774836i	$0.282168\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	−2.66037	−0.470292
$33$	0	0
$34$	4.65064	0.797578
$35$	−5.87917	−0.993762
$36$	0	0
$37$	−1.81406	−0.298230	−0.149115	−	0.988820i	$-0.547643\pi$
$37$	−1.81406	−0.298230	−0.149115	+	0.988820i	$0.547643\pi$
$38$	−7.51417	−1.21896
$39$	0	0
$40$	4.72643	0.747314
$41$	−0.337145	−0.0526531	−0.0263266	−	0.999653i	$-0.508381\pi$
$41$	−0.337145	−0.0526531	−0.0263266	+	0.999653i	$0.508381\pi$
$42$	0	0
$43$	3.88400	0.592304	0.296152	−	0.955141i	$-0.404297\pi$
$43$	3.88400	0.592304	0.296152	+	0.955141i	$0.404297\pi$
$44$	−1.81212	−0.273187
$45$	0	0
$46$	1.12109	0.165295
$47$	1.18915	0.173455	0.0867275	−	0.996232i	$-0.472359\pi$
$47$	1.18915	0.173455	0.0867275	+	0.996232i	$0.472359\pi$
$48$	0	0
$49$	7.46858	1.06694
$50$	3.21726	0.454990
$51$	0	0
$52$	1.26945	0.176040
$53$	2.34413	0.321990	0.160995	−	0.986955i	$-0.448530\pi$
$53$	2.34413	0.321990	0.160995	+	0.986955i	$0.448530\pi$
$54$	0	0
$55$	5.81390	0.783946
$56$	−11.6317	−1.55435
$57$	0	0
$58$	−8.38936	−1.10158
$59$	−7.77883	−1.01272	−0.506359	−	0.862323i	$-0.669009\pi$
$59$	−7.77883	−1.01272	−0.506359	+	0.862323i	$0.669009\pi$
$60$	0	0
$61$	−2.72343	−0.348700	−0.174350	−	0.984684i	$-0.555782\pi$
$61$	−2.72343	−0.348700	−0.174350	+	0.984684i	$0.555782\pi$
$62$	0	0
$63$	0	0
$64$	8.88686	1.11086
$65$	−4.07281	−0.505170
$66$	0	0
$67$	−7.42118	−0.906642	−0.453321	−	0.891347i	$-0.649761\pi$
$67$	−7.42118	−0.906642	−0.453321	+	0.891347i	$0.649761\pi$
$68$	1.81829	0.220501
$69$	0	0
$70$	7.24416	0.865842
$71$	5.09818	0.605043	0.302521	−	0.953143i	$-0.402172\pi$
$71$	5.09818	0.605043	0.302521	+	0.953143i	$0.402172\pi$
$72$	0	0
$73$	5.39174	0.631056	0.315528	−	0.948916i	$-0.397818\pi$
$73$	5.39174	0.631056	0.315528	+	0.948916i	$0.397818\pi$
$74$	2.23524	0.259841
$75$	0	0
$76$	−2.93787	−0.336997
$77$	−14.3079	−1.63054
$78$	0	0
$79$	−9.73723	−1.09552	−0.547762	−	0.836634i	$-0.684520\pi$
$79$	−9.73723	−1.09552	−0.547762	+	0.836634i	$0.684520\pi$
$80$	−4.33456	−0.484619
$81$	0	0
$82$	0.415420	0.0458755
$83$	8.39515	0.921487	0.460743	−	0.887533i	$-0.347583\pi$
$83$	8.39515	0.921487	0.460743	+	0.887533i	$0.347583\pi$
$84$	0	0
$85$	−5.83370	−0.632754
$86$	−4.78575	−0.516061
$87$	0	0
$88$	11.5025	1.22617
$89$	−5.09560	−0.540132	−0.270066	−	0.962842i	$-0.587046\pi$
$89$	−5.09560	−0.540132	−0.270066	+	0.962842i	$0.587046\pi$
$90$	0	0
$91$	10.0231	1.05071
$92$	0.438320	0.0456980
$93$	0	0
$94$	−1.46523	−0.151127
$95$	9.42569	0.967056
$96$	0	0
$97$	10.9142	1.10817	0.554086	−	0.832460i	$-0.313068\pi$
$97$	10.9142	1.10817	0.554086	+	0.832460i	$0.313068\pi$
$98$	−9.20257	−0.929600
$99$	0	0
$100$	1.25788	0.125788
$101$	0.398727	0.0396748	0.0198374	−	0.999803i	$-0.493685\pi$
$101$	0.398727	0.0396748	0.0198374	+	0.999803i	$0.493685\pi$
$102$	0	0
$103$	3.27680	0.322873	0.161436	−	0.986883i	$-0.448387\pi$
$103$	3.27680	0.322873	0.161436	+	0.986883i	$0.448387\pi$
$104$	−8.05788	−0.790140
$105$	0	0
$106$	−2.88837	−0.280543
$107$	−3.15685	−0.305184	−0.152592	−	0.988289i	$-0.548762\pi$
$107$	−3.15685	−0.305184	−0.152592	+	0.988289i	$0.548762\pi$
$108$	0	0
$109$	6.95622	0.666285	0.333142	−	0.942877i	$-0.391891\pi$
$109$	6.95622	0.666285	0.333142	+	0.942877i	$0.391891\pi$
$110$	−7.16372	−0.683034
$111$	0	0
$112$	10.6673	1.00797
$113$	15.2468	1.43430	0.717148	−	0.696920i	$-0.245447\pi$
$113$	15.2468	1.43430	0.717148	+	0.696920i	$0.245447\pi$
$114$	0	0
$115$	−1.40628	−0.131136
$116$	−3.28005	−0.304545
$117$	0	0
$118$	9.58486	0.882358
$119$	14.3567	1.31607
$120$	0	0
$121$	3.14907	0.286279
$122$	3.35574	0.303814
$123$	0	0
$124$	0	0
$125$	−11.7638	−1.05219
$126$	0	0
$127$	−20.2264	−1.79480	−0.897400	−	0.441218i	$-0.854547\pi$
$127$	−20.2264	−1.79480	−0.897400	+	0.441218i	$0.854547\pi$
$128$	−5.62940	−0.497573
$129$	0	0
$130$	5.01841	0.440143
$131$	−12.6513	−1.10535	−0.552674	−	0.833398i	$-0.686392\pi$
$131$	−12.6513	−1.10535	−0.552674	+	0.833398i	$0.686392\pi$
$132$	0	0
$133$	−23.1965	−2.01139
$134$	9.14417	0.789936
$135$	0	0
$136$	−11.5417	−0.989695
$137$	9.87801	0.843936	0.421968	−	0.906611i	$-0.361340\pi$
$137$	9.87801	0.843936	0.421968	+	0.906611i	$0.361340\pi$
$138$	0	0
$139$	−15.4990	−1.31461	−0.657303	−	0.753626i	$-0.728303\pi$
$139$	−15.4990	−1.31461	−0.657303	+	0.753626i	$0.728303\pi$
$140$	2.83230	0.239373
$141$	0	0
$142$	−6.28184	−0.527160
$143$	−9.91185	−0.828871
$144$	0	0
$145$	10.5235	0.873930
$146$	−6.64356	−0.549825
$147$	0	0
$148$	0.873928	0.0718364
$149$	15.2432	1.24878	0.624388	−	0.781115i	$-0.285349\pi$
$149$	15.2432	1.24878	0.624388	+	0.781115i	$0.285349\pi$
$150$	0	0
$151$	−0.261828	−0.0213073	−0.0106536	−	0.999943i	$-0.503391\pi$
$151$	−0.261828	−0.0213073	−0.0106536	+	0.999943i	$0.503391\pi$
$152$	18.6483	1.51258
$153$	0	0
$154$	17.6298	1.42065
$155$	0	0
$156$	0	0
$157$	15.9265	1.27108	0.635538	−	0.772070i	$-0.280778\pi$
$157$	15.9265	1.27108	0.635538	+	0.772070i	$0.280778\pi$
$158$	11.9979	0.954505
$159$	0	0
$160$	−4.11193	−0.325077
$161$	3.46084	0.272752
$162$	0	0
$163$	−0.133175	−0.0104311	−0.00521555	−	0.999986i	$-0.501660\pi$
$163$	−0.133175	−0.0104311	−0.00521555	+	0.999986i	$0.501660\pi$
$164$	0.162420	0.0126829
$165$	0	0
$166$	−10.3443	−0.802871
$167$	3.94878	0.305566	0.152783	−	0.988260i	$-0.451176\pi$
$167$	3.94878	0.305566	0.152783	+	0.988260i	$0.451176\pi$
$168$	0	0
$169$	−6.05644	−0.465880
$170$	7.18813	0.551304
$171$	0	0
$172$	−1.87112	−0.142672
$173$	14.4569	1.09914	0.549570	−	0.835447i	$-0.314791\pi$
$173$	14.4569	1.09914	0.549570	+	0.835447i	$0.314791\pi$
$174$	0	0
$175$	9.93180	0.750774
$176$	−10.5489	−0.795151
$177$	0	0
$178$	6.27865	0.470605
$179$	17.7199	1.32445	0.662223	−	0.749307i	$-0.269613\pi$
$179$	17.7199	1.32445	0.662223	+	0.749307i	$0.269613\pi$
$180$	0	0
$181$	−12.0592	−0.896350	−0.448175	−	0.893946i	$-0.647926\pi$
$181$	−12.0592	−0.896350	−0.448175	+	0.893946i	$0.647926\pi$
$182$	−12.3502	−0.915461
$183$	0	0
$184$	−2.78226	−0.205111
$185$	−2.80386	−0.206144
$186$	0	0
$187$	−14.1973	−1.03821
$188$	−0.572874	−0.0417811
$189$	0	0
$190$	−11.6141	−0.842574
$191$	−5.07152	−0.366962	−0.183481	−	0.983023i	$-0.558737\pi$
$191$	−5.07152	−0.366962	−0.183481	+	0.983023i	$0.558737\pi$
$192$	0	0
$193$	−20.0417	−1.44264	−0.721318	−	0.692604i	$-0.756463\pi$
$193$	−20.0417	−1.44264	−0.721318	+	0.692604i	$0.756463\pi$
$194$	−13.4482	−0.965525
$195$	0	0
$196$	−3.59800	−0.257000
$197$	−21.7084	−1.54666	−0.773328	−	0.634006i	$-0.781410\pi$
$197$	−21.7084	−1.54666	−0.773328	+	0.634006i	$0.781410\pi$
$198$	0	0
$199$	15.1655	1.07505	0.537527	−	0.843246i	$-0.319359\pi$
$199$	15.1655	1.07505	0.537527	+	0.843246i	$0.319359\pi$
$200$	−7.98445	−0.564586
$201$	0	0
$202$	−0.491300	−0.0345678
$203$	−25.8982	−1.81770
$204$	0	0
$205$	−0.521098	−0.0363951
$206$	−4.03759	−0.281312
$207$	0	0
$208$	7.38980	0.512391
$209$	22.9390	1.58672
$210$	0	0
$211$	6.30441	0.434013	0.217007	−	0.976170i	$-0.430371\pi$
$211$	6.30441	0.434013	0.217007	+	0.976170i	$0.430371\pi$
$212$	−1.12929	−0.0775597
$213$	0	0
$214$	3.88978	0.265900
$215$	6.00319	0.409414
$216$	0	0
$217$	0	0
$218$	−8.57126	−0.580519
$219$	0	0
$220$	−2.80085	−0.188834
$221$	9.94562	0.669015
$222$	0	0
$223$	7.38911	0.494811	0.247406	−	0.968912i	$-0.420422\pi$
$223$	7.38911	0.494811	0.247406	+	0.968912i	$0.420422\pi$
$224$	10.1194	0.676132
$225$	0	0
$226$	−18.7867	−1.24967
$227$	−19.2398	−1.27699	−0.638495	−	0.769626i	$-0.720443\pi$
$227$	−19.2398	−1.27699	−0.638495	+	0.769626i	$0.720443\pi$
$228$	0	0
$229$	−6.08710	−0.402247	−0.201124	−	0.979566i	$-0.564459\pi$
$229$	−6.08710	−0.402247	−0.201124	+	0.979566i	$0.564459\pi$
$230$	1.73278	0.114256
$231$	0	0
$232$	20.8203	1.36692
$233$	−17.5280	−1.14830	−0.574150	−	0.818750i	$-0.694667\pi$
$233$	−17.5280	−1.14830	−0.574150	+	0.818750i	$0.694667\pi$
$234$	0	0
$235$	1.83797	0.119896
$236$	3.74746	0.243939
$237$	0	0
$238$	−17.6899	−1.14667
$239$	27.9120	1.80548	0.902740	−	0.430187i	$-0.141552\pi$
$239$	27.9120	1.80548	0.902740	+	0.430187i	$0.141552\pi$
$240$	0	0
$241$	22.7497	1.46544	0.732720	−	0.680531i	$-0.238251\pi$
$241$	22.7497	1.46544	0.732720	+	0.680531i	$0.238251\pi$
$242$	−3.88019	−0.249428
$243$	0	0
$244$	1.31202	0.0839933
$245$	11.5436	0.737494
$246$	0	0
$247$	−16.0694	−1.02247
$248$	0	0
$249$	0	0
$250$	14.4950	0.916747
$251$	16.2345	1.02471	0.512356	−	0.858773i	$-0.328773\pi$
$251$	16.2345	1.02471	0.512356	+	0.858773i	$0.328773\pi$
$252$	0	0
$253$	−3.42241	−0.215165
$254$	24.9224	1.56377
$255$	0	0
$256$	−10.8373	−0.677333
$257$	−23.3521	−1.45666	−0.728332	−	0.685224i	$-0.759704\pi$
$257$	−23.3521	−1.45666	−0.728332	+	0.685224i	$0.759704\pi$
$258$	0	0
$259$	6.90026	0.428761
$260$	1.96208	0.121683
$261$	0	0
$262$	15.5886	0.963064
$263$	10.3944	0.640944	0.320472	−	0.947258i	$-0.396159\pi$
$263$	10.3944	0.640944	0.320472	+	0.947258i	$0.396159\pi$
$264$	0	0
$265$	3.62313	0.222567
$266$	28.5821	1.75248
$267$	0	0
$268$	3.57517	0.218388
$269$	4.41805	0.269373	0.134686	−	0.990888i	$-0.456997\pi$
$269$	4.41805	0.269373	0.134686	+	0.990888i	$0.456997\pi$
$270$	0	0
$271$	6.15286	0.373760	0.186880	−	0.982383i	$-0.440162\pi$
$271$	6.15286	0.373760	0.186880	+	0.982383i	$0.440162\pi$
$272$	10.5848	0.641798
$273$	0	0
$274$	−12.1714	−0.735302
$275$	−9.82153	−0.592261
$276$	0	0
$277$	22.0474	1.32470	0.662350	−	0.749194i	$-0.269559\pi$
$277$	22.0474	1.32470	0.662350	+	0.749194i	$0.269559\pi$
$278$	19.0974	1.14539
$279$	0	0
$280$	−17.9782	−1.07440
$281$	30.7736	1.83580	0.917899	−	0.396815i	$-0.129885\pi$
$281$	30.7736	1.83580	0.917899	+	0.396815i	$0.129885\pi$
$282$	0	0
$283$	−21.3453	−1.26884	−0.634422	−	0.772987i	$-0.718762\pi$
$283$	−21.3453	−1.26884	−0.634422	+	0.772987i	$0.718762\pi$
$284$	−2.45606	−0.145740
$285$	0	0
$286$	12.2131	0.722176
$287$	1.28242	0.0756987
$288$	0	0
$289$	−2.75436	−0.162021
$290$	−12.9668	−0.761436
$291$	0	0
$292$	−2.59748	−0.152006
$293$	1.79903	0.105100	0.0525501	−	0.998618i	$-0.483265\pi$
$293$	1.79903	0.105100	0.0525501	+	0.998618i	$0.483265\pi$
$294$	0	0
$295$	−12.0231	−0.700014
$296$	−5.54731	−0.322431
$297$	0	0
$298$	−18.7823	−1.08803
$299$	2.39750	0.138651
$300$	0	0
$301$	−14.7738	−0.851546
$302$	0.322617	0.0185645
$303$	0	0
$304$	−17.1022	−0.980878
$305$	−4.20940	−0.241030
$306$	0	0
$307$	26.7694	1.52781	0.763905	−	0.645328i	$-0.223279\pi$
$307$	26.7694	1.52781	0.763905	+	0.645328i	$0.223279\pi$
$308$	6.89287	0.392758
$309$	0	0
$310$	0	0
$311$	−4.18114	−0.237090	−0.118545	−	0.992949i	$-0.537823\pi$
$311$	−4.18114	−0.237090	−0.118545	+	0.992949i	$0.537823\pi$
$312$	0	0
$313$	−11.3428	−0.641135	−0.320567	−	0.947226i	$-0.603874\pi$
$313$	−11.3428	−0.641135	−0.320567	+	0.947226i	$0.603874\pi$
$314$	−19.6242	−1.10746
$315$	0	0
$316$	4.69092	0.263885
$317$	−25.8813	−1.45364	−0.726820	−	0.686828i	$-0.759003\pi$
$317$	−25.8813	−1.45364	−0.726820	+	0.686828i	$0.759003\pi$
$318$	0	0
$319$	25.6107	1.43392
$320$	13.7357	0.767851
$321$	0	0
$322$	−4.26435	−0.237643
$323$	−23.0171	−1.28071
$324$	0	0
$325$	6.88028	0.381649
$326$	0.164095	0.00908838
$327$	0	0
$328$	−1.03097	−0.0569258
$329$	−4.52323	−0.249374
$330$	0	0
$331$	0.765788	0.0420915	0.0210457	−	0.999779i	$-0.493300\pi$
$331$	0.765788	0.0420915	0.0210457	+	0.999779i	$0.493300\pi$
$332$	−4.04437	−0.221964
$333$	0	0
$334$	−4.86558	−0.266232
$335$	−11.4703	−0.626692
$336$	0	0
$337$	2.39434	0.130428	0.0652140	−	0.997871i	$-0.479227\pi$
$337$	2.39434	0.130428	0.0652140	+	0.997871i	$0.479227\pi$
$338$	7.46258	0.405911
$339$	0	0
$340$	2.81040	0.152415
$341$	0	0
$342$	0	0
$343$	−1.78235	−0.0962378
$344$	11.8770	0.640367
$345$	0	0
$346$	−17.8134	−0.957656
$347$	−5.65780	−0.303727	−0.151863	−	0.988402i	$-0.548527\pi$
$347$	−5.65780	−0.303727	−0.151863	+	0.988402i	$0.548527\pi$
$348$	0	0
$349$	29.0676	1.55595	0.777977	−	0.628293i	$-0.216246\pi$
$349$	29.0676	1.55595	0.777977	+	0.628293i	$0.216246\pi$
$350$	−12.2377	−0.654132
$351$	0	0
$352$	−10.0071	−0.533378
$353$	19.2752	1.02592	0.512958	−	0.858414i	$-0.328550\pi$
$353$	19.2752	1.02592	0.512958	+	0.858414i	$0.328550\pi$
$354$	0	0
$355$	7.87986	0.418220
$356$	2.45481	0.130105
$357$	0	0
$358$	−21.8340	−1.15396
$359$	−10.3443	−0.545950	−0.272975	−	0.962021i	$-0.588008\pi$
$359$	−10.3443	−0.545950	−0.272975	+	0.962021i	$0.588008\pi$
$360$	0	0
$361$	18.1894	0.957339
$362$	14.8590	0.780970
$363$	0	0
$364$	−4.82867	−0.253091
$365$	8.33360	0.436201
$366$	0	0
$367$	−0.136564	−0.00712857	−0.00356428	−	0.999994i	$-0.501135\pi$
$367$	−0.136564	−0.00712857	−0.00356428	+	0.999994i	$0.501135\pi$
$368$	2.55159	0.133011
$369$	0	0
$370$	3.45484	0.179608
$371$	−8.91649	−0.462921
$372$	0	0
$373$	−7.36393	−0.381290	−0.190645	−	0.981659i	$-0.561058\pi$
$373$	−7.36393	−0.381290	−0.190645	+	0.981659i	$0.561058\pi$
$374$	17.4935	0.904566
$375$	0	0
$376$	3.63635	0.187530
$377$	−17.9411	−0.924012
$378$	0	0
$379$	−4.60813	−0.236704	−0.118352	−	0.992972i	$-0.537761\pi$
$379$	−4.60813	−0.236704	−0.118352	+	0.992972i	$0.537761\pi$
$380$	−4.54084	−0.232940
$381$	0	0
$382$	6.24899	0.319726
$383$	32.2948	1.65019	0.825093	−	0.564997i	$-0.191123\pi$
$383$	32.2948	1.65019	0.825093	+	0.564997i	$0.191123\pi$
$384$	0	0
$385$	−22.1147	−1.12707
$386$	24.6949	1.25694
$387$	0	0
$388$	−5.25794	−0.266932
$389$	35.2713	1.78832	0.894162	−	0.447743i	$-0.147772\pi$
$389$	35.2713	1.78832	0.894162	+	0.447743i	$0.147772\pi$
$390$	0	0
$391$	3.43407	0.173668
$392$	22.8385	1.15352
$393$	0	0
$394$	26.7484	1.34757
$395$	−15.0501	−0.757252
$396$	0	0
$397$	−4.03402	−0.202462	−0.101231	−	0.994863i	$-0.532278\pi$
$397$	−4.03402	−0.202462	−0.101231	+	0.994863i	$0.532278\pi$
$398$	−18.6865	−0.936671
$399$	0	0
$400$	7.32246	0.366123
$401$	−24.8832	−1.24261	−0.621304	−	0.783570i	$-0.713397\pi$
$401$	−24.8832	−1.24261	−0.621304	+	0.783570i	$0.713397\pi$
$402$	0	0
$403$	0	0
$404$	−0.192087	−0.00955671
$405$	0	0
$406$	31.9111	1.58372
$407$	−6.82365	−0.338236
$408$	0	0
$409$	3.50722	0.173421	0.0867105	−	0.996234i	$-0.472364\pi$
$409$	3.50722	0.173421	0.0867105	+	0.996234i	$0.472364\pi$
$410$	0.642083	0.0317102
$411$	0	0
$412$	−1.57860	−0.0777723
$413$	29.5888	1.45597
$414$	0	0
$415$	12.9757	0.636953
$416$	7.01025	0.343706
$417$	0	0
$418$	−28.2647	−1.38247
$419$	−4.12031	−0.201290	−0.100645	−	0.994922i	$-0.532091\pi$
$419$	−4.12031	−0.201290	−0.100645	+	0.994922i	$0.532091\pi$
$420$	0	0
$421$	27.6014	1.34521	0.672605	−	0.740002i	$-0.265175\pi$
$421$	27.6014	1.34521	0.672605	+	0.740002i	$0.265175\pi$
$422$	−7.76812	−0.378146
$423$	0	0
$424$	7.16821	0.348119
$425$	9.85499	0.478037
$426$	0	0
$427$	10.3593	0.501321
$428$	1.52082	0.0735114
$429$	0	0
$430$	−7.39697	−0.356713
$431$	14.8739	0.716452	0.358226	−	0.933635i	$-0.383382\pi$
$431$	14.8739	0.716452	0.358226	+	0.933635i	$0.383382\pi$
$432$	0	0
$433$	9.26195	0.445101	0.222550	−	0.974921i	$-0.428562\pi$
$433$	9.26195	0.445101	0.222550	+	0.974921i	$0.428562\pi$
$434$	0	0
$435$	0	0
$436$	−3.35117	−0.160492
$437$	−5.54853	−0.265422
$438$	0	0
$439$	17.7182	0.845643	0.422821	−	0.906213i	$-0.361040\pi$
$439$	17.7182	0.845643	0.422821	+	0.906213i	$0.361040\pi$
$440$	17.7786	0.847561
$441$	0	0
$442$	−12.2547	−0.582898
$443$	18.4502	0.876597	0.438299	−	0.898829i	$-0.355581\pi$
$443$	18.4502	0.876597	0.438299	+	0.898829i	$0.355581\pi$
$444$	0	0
$445$	−7.87587	−0.373352
$446$	−9.10465	−0.431118
$447$	0	0
$448$	−33.8035	−1.59706
$449$	−22.7102	−1.07176	−0.535881	−	0.844294i	$-0.680020\pi$
$449$	−22.7102	−1.07176	−0.535881	+	0.844294i	$0.680020\pi$
$450$	0	0
$451$	−1.26818	−0.0597162
$452$	−7.34516	−0.345487
$453$	0	0
$454$	23.7067	1.11261
$455$	15.4920	0.726276
$456$	0	0
$457$	−16.1619	−0.756020	−0.378010	−	0.925802i	$-0.623392\pi$
$457$	−16.1619	−0.756020	−0.378010	+	0.925802i	$0.623392\pi$
$458$	7.50036	0.350469
$459$	0	0
$460$	0.677478	0.0315876
$461$	−21.5388	−1.00316	−0.501580	−	0.865111i	$-0.667248\pi$
$461$	−21.5388	−1.00316	−0.501580	+	0.865111i	$0.667248\pi$
$462$	0	0
$463$	−17.5098	−0.813750	−0.406875	−	0.913484i	$-0.633381\pi$
$463$	−17.5098	−0.813750	−0.406875	+	0.913484i	$0.633381\pi$
$464$	−19.0941	−0.886421
$465$	0	0
$466$	21.5976	1.00049
$467$	11.8066	0.546346	0.273173	−	0.961965i	$-0.411927\pi$
$467$	11.8066	0.546346	0.273173	+	0.961965i	$0.411927\pi$
$468$	0	0
$469$	28.2284	1.30347
$470$	−2.26470	−0.104463
$471$	0	0
$472$	−23.7872	−1.09490
$473$	14.6097	0.671757
$474$	0	0
$475$	−15.9230	−0.730598
$476$	−6.91635	−0.317010
$477$	0	0
$478$	−34.3924	−1.57307
$479$	15.8087	0.722319	0.361159	−	0.932504i	$-0.382381\pi$
$479$	15.8087	0.722319	0.361159	+	0.932504i	$0.382381\pi$
$480$	0	0
$481$	4.78017	0.217957
$482$	−28.0316	−1.27680
$483$	0	0
$484$	−1.51707	−0.0689576
$485$	16.8693	0.765994
$486$	0	0
$487$	12.7916	0.579645	0.289822	−	0.957080i	$-0.406404\pi$
$487$	12.7916	0.579645	0.289822	+	0.957080i	$0.406404\pi$
$488$	−8.32811	−0.376996
$489$	0	0
$490$	−14.2237	−0.642561
$491$	18.5282	0.836166	0.418083	−	0.908409i	$-0.362702\pi$
$491$	18.5282	0.836166	0.418083	+	0.908409i	$0.362702\pi$
$492$	0	0
$493$	−25.6979	−1.15738
$494$	19.8003	0.890859
$495$	0	0
$496$	0	0
$497$	−19.3922	−0.869861
$498$	0	0
$499$	40.8873	1.83036	0.915182	−	0.403040i	$-0.132046\pi$
$499$	40.8873	1.83036	0.915182	+	0.403040i	$0.132046\pi$
$500$	5.66723	0.253446
$501$	0	0
$502$	−20.0037	−0.892809
$503$	−20.0771	−0.895194	−0.447597	−	0.894235i	$-0.647720\pi$
$503$	−20.0771	−0.895194	−0.447597	+	0.894235i	$0.647720\pi$
$504$	0	0
$505$	0.616282	0.0274242
$506$	4.21700	0.187469
$507$	0	0
$508$	9.74408	0.432324
$509$	31.9112	1.41444	0.707220	−	0.706994i	$-0.249949\pi$
$509$	31.9112	1.41444	0.707220	+	0.706994i	$0.249949\pi$
$510$	0	0
$511$	−20.5089	−0.907260
$512$	24.6123	1.08772
$513$	0	0
$514$	28.7738	1.26916
$515$	5.06470	0.223177
$516$	0	0
$517$	4.47301	0.196723
$518$	−8.50231	−0.373570
$519$	0	0
$520$	−12.4544	−0.546163
$521$	−2.10756	−0.0923339	−0.0461670	−	0.998934i	$-0.514701\pi$
$521$	−2.10756	−0.0923339	−0.0461670	+	0.998934i	$0.514701\pi$
$522$	0	0
$523$	4.90777	0.214602	0.107301	−	0.994227i	$-0.465779\pi$
$523$	4.90777	0.214602	0.107301	+	0.994227i	$0.465779\pi$
$524$	6.09477	0.266251
$525$	0	0
$526$	−12.8076	−0.558440
$527$	0	0
$528$	0	0
$529$	−22.1722	−0.964008
$530$	−4.46432	−0.193918
$531$	0	0
$532$	11.1750	0.484496
$533$	0.888398	0.0384808
$534$	0	0
$535$	−4.87929	−0.210950
$536$	−22.6936	−0.980213
$537$	0	0
$538$	−5.44379	−0.234699
$539$	28.0932	1.21006
$540$	0	0
$541$	−12.4370	−0.534709	−0.267355	−	0.963598i	$-0.586150\pi$
$541$	−12.4370	−0.534709	−0.267355	+	0.963598i	$0.586150\pi$
$542$	−7.58139	−0.325648
$543$	0	0
$544$	10.0411	0.430511
$545$	10.7517	0.460552
$546$	0	0
$547$	−13.9650	−0.597101	−0.298550	−	0.954394i	$-0.596503\pi$
$547$	−13.9650	−0.597101	−0.298550	+	0.954394i	$0.596503\pi$
$548$	−4.75875	−0.203284
$549$	0	0
$550$	12.1018	0.516023
$551$	41.5209	1.76885
$552$	0	0
$553$	37.0381	1.57502
$554$	−27.1662	−1.15418
$555$	0	0
$556$	7.46666	0.316657
$557$	37.2207	1.57709	0.788546	−	0.614976i	$-0.210834\pi$
$557$	37.2207	1.57709	0.788546	+	0.614976i	$0.210834\pi$
$558$	0	0
$559$	−10.2346	−0.432876
$560$	16.4876	0.696730
$561$	0	0
$562$	−37.9183	−1.59949
$563$	30.0267	1.26548	0.632738	−	0.774366i	$-0.281931\pi$
$563$	30.0267	1.26548	0.632738	+	0.774366i	$0.281931\pi$
$564$	0	0
$565$	23.5658	0.991420
$566$	26.3010	1.10551
$567$	0	0
$568$	15.5900	0.654140
$569$	40.3713	1.69245	0.846227	−	0.532823i	$-0.178869\pi$
$569$	40.3713	1.69245	0.846227	+	0.532823i	$0.178869\pi$
$570$	0	0
$571$	42.6015	1.78282	0.891410	−	0.453199i	$-0.149717\pi$
$571$	42.6015	1.78282	0.891410	+	0.453199i	$0.149717\pi$
$572$	4.77505	0.199655
$573$	0	0
$574$	−1.58016	−0.0659545
$575$	2.37566	0.0990717
$576$	0	0
$577$	−6.53777	−0.272171	−0.136085	−	0.990697i	$-0.543452\pi$
$577$	−6.53777	−0.272171	−0.136085	+	0.990697i	$0.543452\pi$
$578$	3.39385	0.141165
$579$	0	0
$580$	−5.06972	−0.210509
$581$	−31.9331	−1.32481
$582$	0	0
$583$	8.81749	0.365183
$584$	16.4877	0.682264
$585$	0	0
$586$	−2.21671	−0.0915715
$587$	21.3777	0.882352	0.441176	−	0.897421i	$-0.354561\pi$
$587$	21.3777	0.882352	0.441176	+	0.897421i	$0.354561\pi$
$588$	0	0
$589$	0	0
$590$	14.8146	0.609907
$591$	0	0
$592$	5.08738	0.209090
$593$	−25.9906	−1.06731	−0.533653	−	0.845704i	$-0.679181\pi$
$593$	−25.9906	−1.06731	−0.533653	+	0.845704i	$0.679181\pi$
$594$	0	0
$595$	22.1900	0.909701
$596$	−7.34346	−0.300800
$597$	0	0
$598$	−2.95414	−0.120804
$599$	9.18796	0.375410	0.187705	−	0.982225i	$-0.439895\pi$
$599$	9.18796	0.375410	0.187705	+	0.982225i	$0.439895\pi$
$600$	0	0
$601$	33.3190	1.35911	0.679556	−	0.733624i	$-0.262172\pi$
$601$	33.3190	1.35911	0.679556	+	0.733624i	$0.262172\pi$
$602$	18.2038	0.741933
$603$	0	0
$604$	0.126136	0.00513240
$605$	4.86727	0.197883
$606$	0	0
$607$	−48.7008	−1.97670	−0.988352	−	0.152188i	$-0.951368\pi$
$607$	−48.7008	−1.97670	−0.988352	+	0.152188i	$0.951368\pi$
$608$	−16.2238	−0.657961
$609$	0	0
$610$	5.18671	0.210004
$611$	−3.13348	−0.126767
$612$	0	0
$613$	20.9140	0.844709	0.422354	−	0.906431i	$-0.361204\pi$
$613$	20.9140	0.844709	0.422354	+	0.906431i	$0.361204\pi$
$614$	−32.9845	−1.33115
$615$	0	0
$616$	−43.7529	−1.76285
$617$	−45.3840	−1.82709	−0.913545	−	0.406738i	$-0.866666\pi$
$617$	−45.3840	−1.82709	−0.913545	+	0.406738i	$0.866666\pi$
$618$	0	0
$619$	−10.2462	−0.411832	−0.205916	−	0.978570i	$-0.566017\pi$
$619$	−10.2462	−0.411832	−0.205916	+	0.978570i	$0.566017\pi$
$620$	0	0
$621$	0	0
$622$	5.15188	0.206572
$623$	19.3824	0.776540
$624$	0	0
$625$	−5.12716	−0.205086
$626$	13.9763	0.558606
$627$	0	0
$628$	−7.67263	−0.306171
$629$	6.84689	0.273003
$630$	0	0
$631$	−10.6770	−0.425043	−0.212522	−	0.977156i	$-0.568168\pi$
$631$	−10.6770	−0.425043	−0.212522	+	0.977156i	$0.568168\pi$
$632$	−29.7759	−1.18442
$633$	0	0
$634$	31.8903	1.26652
$635$	−31.2623	−1.24061
$636$	0	0
$637$	−19.6802	−0.779757
$638$	−31.5568	−1.24934
$639$	0	0
$640$	−8.70093	−0.343934
$641$	−23.8621	−0.942496	−0.471248	−	0.882001i	$-0.656196\pi$
$641$	−23.8621	−0.942496	−0.471248	+	0.882001i	$0.656196\pi$
$642$	0	0
$643$	−7.93926	−0.313094	−0.156547	−	0.987671i	$-0.550036\pi$
$643$	−7.93926	−0.313094	−0.156547	+	0.987671i	$0.550036\pi$
$644$	−1.66726	−0.0656994
$645$	0	0
$646$	28.3610	1.11585
$647$	−2.47761	−0.0974050	−0.0487025	−	0.998813i	$-0.515509\pi$
$647$	−2.47761	−0.0974050	−0.0487025	+	0.998813i	$0.515509\pi$
$648$	0	0
$649$	−29.2603	−1.14857
$650$	−8.47769	−0.332523
$651$	0	0
$652$	0.0641574	0.00251260
$653$	−22.4940	−0.880260	−0.440130	−	0.897934i	$-0.645068\pi$
$653$	−22.4940	−0.880260	−0.440130	+	0.897934i	$0.645068\pi$
$654$	0	0
$655$	−19.5541	−0.764042
$656$	0.945493	0.0369153
$657$	0	0
$658$	5.57340	0.217274
$659$	11.9596	0.465881	0.232941	−	0.972491i	$-0.425165\pi$
$659$	11.9596	0.465881	0.232941	+	0.972491i	$0.425165\pi$
$660$	0	0
$661$	23.8360	0.927112	0.463556	−	0.886068i	$-0.346573\pi$
$661$	23.8360	0.927112	0.463556	+	0.886068i	$0.346573\pi$
$662$	−0.943582	−0.0366734
$663$	0	0
$664$	25.6719	0.996263
$665$	−35.8531	−1.39032
$666$	0	0
$667$	−6.19478	−0.239863
$668$	−1.90233	−0.0736034
$669$	0	0
$670$	14.1334	0.546022
$671$	−10.2443	−0.395475
$672$	0	0
$673$	3.49412	0.134688	0.0673442	−	0.997730i	$-0.478547\pi$
$673$	3.49412	0.134688	0.0673442	+	0.997730i	$0.478547\pi$
$674$	−2.95024	−0.113639
$675$	0	0
$676$	2.91770	0.112219
$677$	19.2457	0.739674	0.369837	−	0.929097i	$-0.379414\pi$
$677$	19.2457	0.739674	0.369837	+	0.929097i	$0.379414\pi$
$678$	0	0
$679$	−41.5151	−1.59320
$680$	−17.8391	−0.684100
$681$	0	0
$682$	0	0
$683$	39.8738	1.52573	0.762865	−	0.646558i	$-0.223792\pi$
$683$	39.8738	1.52573	0.762865	+	0.646558i	$0.223792\pi$
$684$	0	0
$685$	15.2677	0.583348
$686$	2.19616	0.0838498
$687$	0	0
$688$	−10.8923	−0.415266
$689$	−6.17692	−0.235322
$690$	0	0
$691$	38.4104	1.46120	0.730601	−	0.682804i	$-0.239240\pi$
$691$	38.4104	1.46120	0.730601	+	0.682804i	$0.239240\pi$
$692$	−6.96465	−0.264756
$693$	0	0
$694$	6.97139	0.264630
$695$	−23.9556	−0.908687
$696$	0	0
$697$	1.27250	0.0481993
$698$	−35.8163	−1.35567
$699$	0	0
$700$	−4.78466	−0.180843
$701$	35.4255	1.33800	0.669001	−	0.743262i	$-0.266722\pi$
$701$	35.4255	1.33800	0.669001	+	0.743262i	$0.266722\pi$
$702$	0	0
$703$	−11.0627	−0.417239
$704$	33.4281	1.25987
$705$	0	0
$706$	−23.7504	−0.893857
$707$	−1.51666	−0.0570399
$708$	0	0
$709$	−25.8093	−0.969288	−0.484644	−	0.874711i	$-0.661051\pi$
$709$	−25.8093	−0.969288	−0.484644	+	0.874711i	$0.661051\pi$
$710$	−9.70935	−0.364385
$711$	0	0
$712$	−15.5821	−0.583962
$713$	0	0
$714$	0	0
$715$	−15.3200	−0.572935
$716$	−8.53658	−0.319027
$717$	0	0
$718$	12.7459	0.475674
$719$	39.0467	1.45620	0.728099	−	0.685472i	$-0.240404\pi$
$719$	39.0467	1.45620	0.728099	+	0.685472i	$0.240404\pi$
$720$	0	0
$721$	−12.4642	−0.464190
$722$	−22.4125	−0.834108
$723$	0	0
$724$	5.80952	0.215909
$725$	−17.7776	−0.660243
$726$	0	0
$727$	24.9836	0.926592	0.463296	−	0.886204i	$-0.346667\pi$
$727$	24.9836	0.926592	0.463296	+	0.886204i	$0.346667\pi$
$728$	30.6502	1.13597
$729$	0	0
$730$	−10.2684	−0.380052
$731$	−14.6595	−0.542202
$732$	0	0
$733$	−16.9490	−0.626027	−0.313014	−	0.949749i	$-0.601339\pi$
$733$	−16.9490	−0.626027	−0.313014	+	0.949749i	$0.601339\pi$
$734$	0.168270	0.00621096
$735$	0	0
$736$	2.42053	0.0892220
$737$	−27.9150	−1.02826
$738$	0	0
$739$	14.3821	0.529055	0.264528	−	0.964378i	$-0.414784\pi$
$739$	14.3821	0.529055	0.264528	+	0.964378i	$0.414784\pi$
$740$	1.35076	0.0496550
$741$	0	0
$742$	10.9866	0.403333
$743$	27.7705	1.01880	0.509400	−	0.860530i	$-0.329867\pi$
$743$	27.7705	1.01880	0.509400	+	0.860530i	$0.329867\pi$
$744$	0	0
$745$	23.5603	0.863183
$746$	9.07363	0.332209
$747$	0	0
$748$	6.83956	0.250079
$749$	12.0079	0.438758
$750$	0	0
$751$	−43.5702	−1.58990	−0.794950	−	0.606676i	$-0.792503\pi$
$751$	−43.5702	−1.58990	−0.794950	+	0.606676i	$0.792503\pi$
$752$	−3.33486	−0.121610
$753$	0	0
$754$	22.1065	0.805071
$755$	−0.404687	−0.0147281
$756$	0	0
$757$	25.2171	0.916531	0.458266	−	0.888815i	$-0.348471\pi$
$757$	25.2171	0.916531	0.458266	+	0.888815i	$0.348471\pi$
$758$	5.67802	0.206235
$759$	0	0
$760$	28.8233	1.04553
$761$	2.61593	0.0948274	0.0474137	−	0.998875i	$-0.484902\pi$
$761$	2.61593	0.0948274	0.0474137	+	0.998875i	$0.484902\pi$
$762$	0	0
$763$	−26.4598	−0.957908
$764$	2.44321	0.0883923
$765$	0	0
$766$	−39.7927	−1.43777
$767$	20.4977	0.740130
$768$	0	0
$769$	11.3507	0.409316	0.204658	−	0.978834i	$-0.434392\pi$
$769$	11.3507	0.409316	0.204658	+	0.978834i	$0.434392\pi$
$770$	27.2491	0.981988
$771$	0	0
$772$	9.65514	0.347496
$773$	11.4174	0.410655	0.205327	−	0.978693i	$-0.434174\pi$
$773$	11.4174	0.410655	0.205327	+	0.978693i	$0.434174\pi$
$774$	0	0
$775$	0	0
$776$	33.3751	1.19810
$777$	0	0
$778$	−43.4603	−1.55813
$779$	−2.05601	−0.0736644
$780$	0	0
$781$	19.1769	0.686204
$782$	−4.23137	−0.151313
$783$	0	0
$784$	−20.9450	−0.748035
$785$	24.6164	0.878597
$786$	0	0
$787$	−27.0892	−0.965626	−0.482813	−	0.875723i	$-0.660385\pi$
$787$	−27.0892	−0.965626	−0.482813	+	0.875723i	$0.660385\pi$
$788$	10.4580	0.372552
$789$	0	0
$790$	18.5443	0.659776
$791$	−57.9951	−2.06207
$792$	0	0
$793$	7.17642	0.254842
$794$	4.97060	0.176400
$795$	0	0
$796$	−7.30601	−0.258955
$797$	−48.6558	−1.72348	−0.861739	−	0.507352i	$-0.830624\pi$
$797$	−48.6558	−1.72348	−0.861739	+	0.507352i	$0.830624\pi$
$798$	0	0
$799$	−4.48825	−0.158783
$800$	6.94637	0.245591
$801$	0	0
$802$	30.6604	1.08266
$803$	20.2812	0.715707
$804$	0	0
$805$	5.34915	0.188533
$806$	0	0
$807$	0	0
$808$	1.21929	0.0428943
$809$	18.8091	0.661294	0.330647	−	0.943755i	$-0.392733\pi$
$809$	18.8091	0.661294	0.330647	+	0.943755i	$0.392733\pi$
$810$	0	0
$811$	−39.5385	−1.38839	−0.694193	−	0.719789i	$-0.744239\pi$
$811$	−39.5385	−1.38839	−0.694193	+	0.719789i	$0.744239\pi$
$812$	12.4765	0.437840
$813$	0	0
$814$	8.40791	0.294697
$815$	−0.205839	−0.00721022
$816$	0	0
$817$	23.6858	0.828662
$818$	−4.32150	−0.151098
$819$	0	0
$820$	0.251040	0.00876670
$821$	14.6367	0.510825	0.255412	−	0.966832i	$-0.417789\pi$
$821$	14.6367	0.510825	0.255412	+	0.966832i	$0.417789\pi$
$822$	0	0
$823$	41.7867	1.45659	0.728296	−	0.685262i	$-0.240312\pi$
$823$	41.7867	1.45659	0.728296	+	0.685262i	$0.240312\pi$
$824$	10.0203	0.349073
$825$	0	0
$826$	−36.4585	−1.26855
$827$	9.40613	0.327083	0.163542	−	0.986536i	$-0.447708\pi$
$827$	9.40613	0.327083	0.163542	+	0.986536i	$0.447708\pi$
$828$	0	0
$829$	40.3307	1.40074	0.700372	−	0.713778i	$-0.253017\pi$
$829$	40.3307	1.40074	0.700372	+	0.713778i	$0.253017\pi$
$830$	−15.9883	−0.554963
$831$	0	0
$832$	−23.4174	−0.811854
$833$	−28.1889	−0.976689
$834$	0	0
$835$	6.10332	0.211214
$836$	−11.0509	−0.382203
$837$	0	0
$838$	5.07693	0.175380
$839$	−7.14965	−0.246833	−0.123417	−	0.992355i	$-0.539385\pi$
$839$	−7.14965	−0.246833	−0.123417	+	0.992355i	$0.539385\pi$
$840$	0	0
$841$	17.3569	0.598514
$842$	−34.0097	−1.17205
$843$	0	0
$844$	−3.03716	−0.104543
$845$	−9.36097	−0.322027
$846$	0	0
$847$	−11.9783	−0.411579
$848$	−6.57389	−0.225748
$849$	0	0
$850$	−12.1430	−0.416503
$851$	1.65052	0.0565791
$852$	0	0
$853$	1.12667	0.0385765	0.0192882	−	0.999814i	$-0.493860\pi$
$853$	1.12667	0.0385765	0.0192882	+	0.999814i	$0.493860\pi$
$854$	−12.7644	−0.436789
$855$	0	0
$856$	−9.65346	−0.329949
$857$	23.8540	0.814838	0.407419	−	0.913241i	$-0.366429\pi$
$857$	23.8540	0.814838	0.407419	+	0.913241i	$0.366429\pi$
$858$	0	0
$859$	32.4657	1.10771	0.553857	−	0.832612i	$-0.313155\pi$
$859$	32.4657	1.10771	0.553857	+	0.832612i	$0.313155\pi$
$860$	−2.89205	−0.0986180
$861$	0	0
$862$	−18.3273	−0.624229
$863$	23.8418	0.811585	0.405793	−	0.913965i	$-0.366996\pi$
$863$	23.8418	0.811585	0.405793	+	0.913965i	$0.366996\pi$
$864$	0	0
$865$	22.3450	0.759752
$866$	−11.4123	−0.387806
$867$	0	0
$868$	0	0
$869$	−36.6268	−1.24248
$870$	0	0
$871$	19.5553	0.662605
$872$	21.2717	0.720352
$873$	0	0
$874$	6.83675	0.231256
$875$	44.7467	1.51271
$876$	0	0
$877$	29.2746	0.988534	0.494267	−	0.869310i	$-0.335436\pi$
$877$	29.2746	0.988534	0.494267	+	0.869310i	$0.335436\pi$
$878$	−21.8319	−0.736789
$879$	0	0
$880$	−16.3046	−0.549627
$881$	−9.03008	−0.304231	−0.152116	−	0.988363i	$-0.548609\pi$
$881$	−9.03008	−0.304231	−0.152116	+	0.988363i	$0.548609\pi$
$882$	0	0
$883$	21.4076	0.720422	0.360211	−	0.932871i	$-0.382705\pi$
$883$	21.4076	0.720422	0.360211	+	0.932871i	$0.382705\pi$
$884$	−4.79132	−0.161150
$885$	0	0
$886$	−22.7339	−0.763759
$887$	−24.5950	−0.825820	−0.412910	−	0.910772i	$-0.635488\pi$
$887$	−24.5950	−0.825820	−0.412910	+	0.910772i	$0.635488\pi$
$888$	0	0
$889$	76.9362	2.58036
$890$	9.70443	0.325293
$891$	0	0
$892$	−3.55971	−0.119188
$893$	7.25179	0.242672
$894$	0	0
$895$	27.3883	0.915488
$896$	21.4129	0.715354
$897$	0	0
$898$	27.9829	0.933801
$899$	0	0
$900$	0	0
$901$	−8.84753	−0.294754
$902$	1.56261	0.0520293
$903$	0	0
$904$	46.6238	1.55069
$905$	−18.6389	−0.619578
$906$	0	0
$907$	−19.7663	−0.656329	−0.328165	−	0.944621i	$-0.606430\pi$
$907$	−19.7663	−0.656329	−0.328165	+	0.944621i	$0.606430\pi$
$908$	9.26880	0.307596
$909$	0	0
$910$	−19.0888	−0.632788
$911$	35.6216	1.18020	0.590098	−	0.807332i	$-0.299089\pi$
$911$	35.6216	1.18020	0.590098	+	0.807332i	$0.299089\pi$
$912$	0	0
$913$	31.5785	1.04510
$914$	19.9142	0.658703
$915$	0	0
$916$	2.93247	0.0968916
$917$	48.1224	1.58914
$918$	0	0
$919$	−17.1563	−0.565935	−0.282968	−	0.959129i	$-0.591319\pi$
$919$	−17.1563	−0.565935	−0.282968	+	0.959129i	$0.591319\pi$
$920$	−4.30033	−0.141778
$921$	0	0
$922$	26.5395	0.874030
$923$	−13.4340	−0.442186
$924$	0	0
$925$	4.73661	0.155739
$926$	21.5751	0.709002
$927$	0	0
$928$	−18.1134	−0.594601
$929$	−45.5222	−1.49353	−0.746767	−	0.665086i	$-0.768395\pi$
$929$	−45.5222	−1.49353	−0.746767	+	0.665086i	$0.768395\pi$
$930$	0	0
$931$	45.5457	1.49270
$932$	8.44416	0.276598
$933$	0	0
$934$	−14.5478	−0.476018
$935$	−21.9436	−0.717633
$936$	0	0
$937$	25.6175	0.836887	0.418443	−	0.908243i	$-0.362576\pi$
$937$	25.6175	0.836887	0.418443	+	0.908243i	$0.362576\pi$
$938$	−34.7822	−1.13568
$939$	0	0
$940$	−0.885446	−0.0288801
$941$	−12.1238	−0.395226	−0.197613	−	0.980280i	$-0.563319\pi$
$941$	−12.1238	−0.395226	−0.197613	+	0.980280i	$0.563319\pi$
$942$	0	0
$943$	0.306750	0.00998916
$944$	21.8151	0.710019
$945$	0	0
$946$	−18.0017	−0.585286
$947$	27.2011	0.883916	0.441958	−	0.897036i	$-0.354284\pi$
$947$	27.2011	0.883916	0.441958	+	0.897036i	$0.354284\pi$
$948$	0	0
$949$	−14.2076	−0.461198
$950$	19.6199	0.636553
$951$	0	0
$952$	43.9019	1.42287
$953$	−24.8927	−0.806353	−0.403177	−	0.915122i	$-0.632094\pi$
$953$	−24.8927	−0.806353	−0.403177	+	0.915122i	$0.632094\pi$
$954$	0	0
$955$	−7.83866	−0.253653
$956$	−13.4467	−0.434896
$957$	0	0
$958$	−19.4791	−0.629340
$959$	−37.5736	−1.21331
$960$	0	0
$961$	0	0
$962$	−5.89000	−0.189901
$963$	0	0
$964$	−10.9597	−0.352989
$965$	−30.9770	−0.997184
$966$	0	0
$967$	52.1998	1.67863	0.839316	−	0.543643i	$-0.182956\pi$
$967$	52.1998	1.67863	0.839316	+	0.543643i	$0.182956\pi$
$968$	9.62967	0.309509
$969$	0	0
$970$	−20.7858	−0.667393
$971$	19.9608	0.640574	0.320287	−	0.947321i	$-0.396221\pi$
$971$	19.9608	0.640574	0.320287	+	0.947321i	$0.396221\pi$
$972$	0	0
$973$	58.9544	1.88999
$974$	−15.7615	−0.505031
$975$	0	0
$976$	7.63763	0.244475
$977$	60.2506	1.92759	0.963794	−	0.266648i	$-0.0859162\pi$
$977$	60.2506	1.92759	0.963794	+	0.266648i	$0.0859162\pi$
$978$	0	0
$979$	−19.1672	−0.612587
$980$	−5.56115	−0.177644
$981$	0	0
$982$	−22.8299	−0.728532
$983$	11.5387	0.368028	0.184014	−	0.982924i	$-0.441091\pi$
$983$	11.5387	0.368028	0.184014	+	0.982924i	$0.441091\pi$
$984$	0	0
$985$	−33.5529	−1.06909
$986$	31.6643	1.00840
$987$	0	0
$988$	7.74148	0.246289
$989$	−3.53384	−0.112370
$990$	0	0
$991$	37.3423	1.18622	0.593109	−	0.805122i	$-0.297900\pi$
$991$	37.3423	1.18622	0.593109	+	0.805122i	$0.297900\pi$
$992$	0	0
$993$	0	0
$994$	23.8946	0.757890
$995$	23.4402	0.743103
$996$	0	0
$997$	20.8024	0.658820	0.329410	−	0.944187i	$-0.393150\pi$
$997$	20.8024	0.658820	0.329410	+	0.944187i	$0.393150\pi$
$998$	−50.3801	−1.59476
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.bf.1.4		8
3.2	odd	2		961.2.a.i.1.5		8
31.10	even	15		279.2.y.c.100.1		16
31.28	even	15		279.2.y.c.226.1		16
31.30	odd	2		8649.2.a.be.1.4		8
93.2	odd	10		961.2.d.o.531.3		16
93.5	odd	6		961.2.c.j.521.5		16
93.8	odd	10		961.2.d.p.374.2		16
93.11	even	30		961.2.g.m.338.1		16
93.14	odd	30		961.2.g.s.816.1		16
93.17	even	30		961.2.g.m.816.1		16
93.20	odd	30		961.2.g.s.338.1		16
93.23	even	10		961.2.d.q.374.2		16
93.26	even	6		961.2.c.i.521.5		16
93.29	even	10		961.2.d.n.531.3		16
93.35	odd	10		961.2.d.p.388.2		16
93.38	odd	30		961.2.g.t.235.1		16
93.41	odd	30		31.2.g.a.7.2	✓	16
93.44	even	30		961.2.g.j.448.2		16
93.47	odd	10		961.2.d.o.628.3		16
93.50	odd	30		961.2.g.k.547.2		16
93.53	even	30		961.2.g.n.732.1		16
93.56	odd	6		961.2.c.j.439.5		16
93.59	odd	30		31.2.g.a.9.2	yes	16
93.65	even	30		961.2.g.l.846.2		16
93.68	even	6		961.2.c.i.439.5		16
93.71	odd	30		961.2.g.t.732.1		16
93.74	even	30		961.2.g.j.547.2		16
93.77	even	10		961.2.d.n.628.3		16
93.80	odd	30		961.2.g.k.448.2		16
93.83	even	30		961.2.g.l.844.2		16
93.86	even	30		961.2.g.n.235.1		16
93.89	even	10		961.2.d.q.388.2		16
93.92	even	2		961.2.a.j.1.5		8
372.59	even	30		496.2.bg.c.257.2		16
372.227	even	30		496.2.bg.c.193.2		16
465.59	odd	30		775.2.bl.a.226.1		16
465.134	odd	30		775.2.bl.a.751.1		16
465.152	even	60		775.2.ck.a.474.3		32
465.227	even	60		775.2.ck.a.224.2		32
465.338	even	60		775.2.ck.a.474.2		32
465.413	even	60		775.2.ck.a.224.3		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.2.g.a.7.2	✓	16	93.41	odd	30
31.2.g.a.9.2	yes	16	93.59	odd	30
279.2.y.c.100.1		16	31.10	even	15
279.2.y.c.226.1		16	31.28	even	15
496.2.bg.c.193.2		16	372.227	even	30
496.2.bg.c.257.2		16	372.59	even	30
775.2.bl.a.226.1		16	465.59	odd	30
775.2.bl.a.751.1		16	465.134	odd	30
775.2.ck.a.224.2		32	465.227	even	60
775.2.ck.a.224.3		32	465.413	even	60
775.2.ck.a.474.2		32	465.338	even	60
775.2.ck.a.474.3		32	465.152	even	60
961.2.a.i.1.5		8	3.2	odd	2
961.2.a.j.1.5		8	93.92	even	2
961.2.c.i.439.5		16	93.68	even	6
961.2.c.i.521.5		16	93.26	even	6
961.2.c.j.439.5		16	93.56	odd	6
961.2.c.j.521.5		16	93.5	odd	6
961.2.d.n.531.3		16	93.29	even	10
961.2.d.n.628.3		16	93.77	even	10
961.2.d.o.531.3		16	93.2	odd	10
961.2.d.o.628.3		16	93.47	odd	10
961.2.d.p.374.2		16	93.8	odd	10
961.2.d.p.388.2		16	93.35	odd	10
961.2.d.q.374.2		16	93.23	even	10
961.2.d.q.388.2		16	93.89	even	10
961.2.g.j.448.2		16	93.44	even	30
961.2.g.j.547.2		16	93.74	even	30
961.2.g.k.448.2		16	93.80	odd	30
961.2.g.k.547.2		16	93.50	odd	30
961.2.g.l.844.2		16	93.83	even	30
961.2.g.l.846.2		16	93.65	even	30
961.2.g.m.338.1		16	93.11	even	30
961.2.g.m.816.1		16	93.17	even	30
961.2.g.n.235.1		16	93.86	even	30
961.2.g.n.732.1		16	93.53	even	30
961.2.g.s.338.1		16	93.20	odd	30
961.2.g.s.816.1		16	93.14	odd	30
961.2.g.t.235.1		16	93.38	odd	30
961.2.g.t.732.1		16	93.71	odd	30
8649.2.a.be.1.4		8	31.30	odd	2
8649.2.a.bf.1.4		8	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

\(f(q)\)	\(=\)	\(q-1.23217 q^{2} -0.481752 q^{4} +1.54562 q^{5} -3.80376 q^{7} +3.05795 q^{8} -1.90447 q^{10} +3.76152 q^{11} -2.63506 q^{13} +4.68689 q^{14} -2.80441 q^{16} -3.77434 q^{17} +6.09831 q^{19} -0.744606 q^{20} -4.63485 q^{22} -0.909847 q^{23} -2.61105 q^{25} +3.24685 q^{26} +1.83247 q^{28} +6.80859 q^{29} -2.66037 q^{32} +4.65064 q^{34} -5.87917 q^{35} -1.81406 q^{37} -7.51417 q^{38} +4.72643 q^{40} -0.337145 q^{41} +3.88400 q^{43} -1.81212 q^{44} +1.12109 q^{46} +1.18915 q^{47} +7.46858 q^{49} +3.21726 q^{50} +1.26945 q^{52} +2.34413 q^{53} +5.81390 q^{55} -11.6317 q^{56} -8.38936 q^{58} -7.77883 q^{59} -2.72343 q^{61} +8.88686 q^{64} -4.07281 q^{65} -7.42118 q^{67} +1.81829 q^{68} +7.24416 q^{70} +5.09818 q^{71} +5.39174 q^{73} +2.23524 q^{74} -2.93787 q^{76} -14.3079 q^{77} -9.73723 q^{79} -4.33456 q^{80} +0.415420 q^{82} +8.39515 q^{83} -5.83370 q^{85} -4.78575 q^{86} +11.5025 q^{88} -5.09560 q^{89} +10.0231 q^{91} +0.438320 q^{92} -1.46523 q^{94} +9.42569 q^{95} +10.9142 q^{97} -9.20257 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 8 q^{4} - 3 q^{5} - 2 q^{7} + 9 q^{8} - 13 q^{10} + 18 q^{11} - 8 q^{13} + 9 q^{14} + 4 q^{16} + 14 q^{17} - 6 q^{19} + 7 q^{20} - 4 q^{22} + 22 q^{23} + 13 q^{25} + 9 q^{26} - 5 q^{28}+ \cdots - 10 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 + 8 * q^4 - 3 * q^5 - 2 * q^7 + 9 * q^8 - 13 * q^10 + 18 * q^11 - 8 * q^13 + 9 * q^14 + 4 * q^16 + 14 * q^17 - 6 * q^19 + 7 * q^20 - 4 * q^22 + 22 * q^23 + 13 * q^25 + 9 * q^26 - 5 * q^28 + 12 * q^29 + 21 * q^32 + 7 * q^34 + 12 * q^35 + 8 * q^37 - 7 * q^38 + 11 * q^40 + 22 * q^41 + 2 * q^43 + 4 * q^44 - 18 * q^46 + 18 * q^47 - 2 * q^49 - 27 * q^50 + q^52 + 6 * q^53 - 28 * q^55 - 30 * q^56 + 5 * q^58 + 4 * q^59 - 30 * q^61 + 9 * q^64 + 3 * q^65 - 13 * q^67 + 30 * q^68 - 39 * q^70 + q^71 - 2 * q^73 + 8 * q^74 - 33 * q^76 - 24 * q^77 - 8 * q^79 - 24 * q^80 - 19 * q^82 + 39 * q^83 + 26 * q^85 - 16 * q^86 + 17 * q^88 + 27 * q^89 - 16 * q^91 + 32 * q^92 + 22 * q^94 + 6 * q^95 + 34 * q^97 - 10 * q^98

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