LMFDB - Embedded newform 8649.2.a.bf.1.8

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			1.8
Root			$2.28064$ 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)

$f(q)$	$=$	$q+2.69016 q^{2} +5.23694 q^{4} -0.608384 q^{5} -1.72688 q^{7} +8.70786 q^{8} -1.63665 q^{10} +1.33739 q^{11} +3.67067 q^{13} -4.64557 q^{14} +12.9516 q^{16} +2.76282 q^{17} -2.56768 q^{19} -3.18607 q^{20} +3.59779 q^{22} +0.539261 q^{23} -4.62987 q^{25} +9.87466 q^{26} -9.04356 q^{28} +8.13087 q^{29} +17.4262 q^{32} +7.43241 q^{34} +1.05061 q^{35} +7.74498 q^{37} -6.90745 q^{38} -5.29772 q^{40} +0.104052 q^{41} -3.00470 q^{43} +7.00384 q^{44} +1.45070 q^{46} +6.72498 q^{47} -4.01789 q^{49} -12.4551 q^{50} +19.2230 q^{52} -2.79875 q^{53} -0.813648 q^{55} -15.0374 q^{56} +21.8733 q^{58} -0.466233 q^{59} -5.11468 q^{61} +20.9759 q^{64} -2.23317 q^{65} +8.29847 q^{67} +14.4687 q^{68} +2.82629 q^{70} -4.75871 q^{71} +7.50619 q^{73} +20.8352 q^{74} -13.4468 q^{76} -2.30951 q^{77} -9.69896 q^{79} -7.87956 q^{80} +0.279916 q^{82} +16.6846 q^{83} -1.68085 q^{85} -8.08312 q^{86} +11.6458 q^{88} +15.3163 q^{89} -6.33879 q^{91} +2.82407 q^{92} +18.0912 q^{94} +1.56213 q^{95} -1.27918 q^{97} -10.8087 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.69016	1.90223	0.951114	−	0.308841i	$-0.0999412\pi$
$2$	2.69016	1.90223	0.951114	+	0.308841i	$0.0999412\pi$
$3$	0	0
$3$	0	0
$4$	5.23694	2.61847
$5$	−0.608384	−0.272077	−0.136039	−	0.990704i	$-0.543437\pi$
$5$	−0.608384	−0.272077	−0.136039	+	0.990704i	$0.543437\pi$
$6$	0	0
$7$	−1.72688	−0.652699	−0.326349	−	0.945249i	$-0.605819\pi$
$7$	−1.72688	−0.652699	−0.326349	+	0.945249i	$0.605819\pi$
$8$	8.70786	3.07869
$9$	0	0
$10$	−1.63665	−0.517553
$11$	1.33739	0.403239	0.201619	−	0.979464i	$-0.435380\pi$
$11$	1.33739	0.403239	0.201619	+	0.979464i	$0.435380\pi$
$12$	0	0
$13$	3.67067	1.01806	0.509030	−	0.860749i	$-0.330004\pi$
$13$	3.67067	1.01806	0.509030	+	0.860749i	$0.330004\pi$
$14$	−4.64557	−1.24158
$15$	0	0
$16$	12.9516	3.23791
$17$	2.76282	0.670082	0.335041	−	0.942204i	$-0.391250\pi$
$17$	2.76282	0.670082	0.335041	+	0.942204i	$0.391250\pi$
$18$	0	0
$19$	−2.56768	−0.589066	−0.294533	−	0.955641i	$-0.595164\pi$
$19$	−2.56768	−0.589066	−0.294533	+	0.955641i	$0.595164\pi$
$20$	−3.18607	−0.712426
$21$	0	0
$22$	3.59779	0.767052
$23$	0.539261	0.112444	0.0562218	−	0.998418i	$-0.482095\pi$
$23$	0.539261	0.112444	0.0562218	+	0.998418i	$0.482095\pi$
$24$	0	0
$25$	−4.62987	−0.925974
$26$	9.87466	1.93658
$27$	0	0
$28$	−9.04356	−1.70907
$29$	8.13087	1.50986	0.754932	−	0.655803i	$-0.227670\pi$
$29$	8.13087	1.50986	0.754932	+	0.655803i	$0.227670\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	17.4262	3.08054
$33$	0	0
$34$	7.43241	1.27465
$35$	1.05061	0.177585
$36$	0	0
$37$	7.74498	1.27327	0.636633	−	0.771167i	$-0.280327\pi$
$37$	7.74498	1.27327	0.636633	+	0.771167i	$0.280327\pi$
$38$	−6.90745	−1.12054
$39$	0	0
$40$	−5.29772	−0.837643
$41$	0.104052	0.0162502	0.00812508	−	0.999967i	$-0.497414\pi$
$41$	0.104052	0.0162502	0.00812508	+	0.999967i	$0.497414\pi$
$42$	0	0
$43$	−3.00470	−0.458213	−0.229106	−	0.973401i	$-0.573580\pi$
$43$	−3.00470	−0.458213	−0.229106	+	0.973401i	$0.573580\pi$
$44$	7.00384	1.05587
$45$	0	0
$46$	1.45070	0.213893
$47$	6.72498	0.980939	0.490470	−	0.871458i	$-0.336825\pi$
$47$	6.72498	0.980939	0.490470	+	0.871458i	$0.336825\pi$
$48$	0	0
$49$	−4.01789	−0.573984
$50$	−12.4551	−1.76141
$51$	0	0
$52$	19.2230	2.66576
$53$	−2.79875	−0.384437	−0.192219	−	0.981352i	$-0.561568\pi$
$53$	−2.79875	−0.384437	−0.192219	+	0.981352i	$0.561568\pi$
$54$	0	0
$55$	−0.813648	−0.109712
$56$	−15.0374	−2.00946
$57$	0	0
$58$	21.8733	2.87210
$59$	−0.466233	−0.0606983	−0.0303492	−	0.999539i	$-0.509662\pi$
$59$	−0.466233	−0.0606983	−0.0303492	+	0.999539i	$0.509662\pi$
$60$	0	0
$61$	−5.11468	−0.654867	−0.327434	−	0.944874i	$-0.606184\pi$
$61$	−5.11468	−0.654867	−0.327434	+	0.944874i	$0.606184\pi$
$62$	0	0
$63$	0	0
$64$	20.9759	2.62198
$65$	−2.23317	−0.276991
$66$	0	0
$67$	8.29847	1.01382	0.506910	−	0.861999i	$-0.330788\pi$
$67$	8.29847	1.01382	0.506910	+	0.861999i	$0.330788\pi$
$68$	14.4687	1.75459
$69$	0	0
$70$	2.82629	0.337806
$71$	−4.75871	−0.564755	−0.282377	−	0.959303i	$-0.591123\pi$
$71$	−4.75871	−0.564755	−0.282377	+	0.959303i	$0.591123\pi$
$72$	0	0
$73$	7.50619	0.878533	0.439267	−	0.898357i	$-0.355238\pi$
$73$	7.50619	0.878533	0.439267	+	0.898357i	$0.355238\pi$
$74$	20.8352	2.42204
$75$	0	0
$76$	−13.4468	−1.54245
$77$	−2.30951	−0.263194
$78$	0	0
$79$	−9.69896	−1.09122	−0.545609	−	0.838040i	$-0.683702\pi$
$79$	−9.69896	−1.09122	−0.545609	+	0.838040i	$0.683702\pi$
$80$	−7.87956	−0.880962
$81$	0	0
$82$	0.279916	0.0309115
$83$	16.6846	1.83137	0.915686	−	0.401895i	$-0.131648\pi$
$83$	16.6846	1.83137	0.915686	+	0.401895i	$0.131648\pi$
$84$	0	0
$85$	−1.68085	−0.182314
$86$	−8.08312	−0.871625
$87$	0	0
$88$	11.6458	1.24145
$89$	15.3163	1.62352	0.811761	−	0.583990i	$-0.198509\pi$
$89$	15.3163	1.62352	0.811761	+	0.583990i	$0.198509\pi$
$90$	0	0
$91$	−6.33879	−0.664486
$92$	2.82407	0.294430
$93$	0	0
$94$	18.0912	1.86597
$95$	1.56213	0.160271
$96$	0	0
$97$	−1.27918	−0.129881	−0.0649404	−	0.997889i	$-0.520686\pi$
$97$	−1.27918	−0.129881	−0.0649404	+	0.997889i	$0.520686\pi$
$98$	−10.8087	−1.09185
$99$	0	0
$100$	−24.2463	−2.42463
$101$	−8.12024	−0.807994	−0.403997	−	0.914760i	$-0.632379\pi$
$101$	−8.12024	−0.807994	−0.403997	+	0.914760i	$0.632379\pi$
$102$	0	0
$103$	15.1525	1.49302	0.746511	−	0.665374i	$-0.231728\pi$
$103$	15.1525	1.49302	0.746511	+	0.665374i	$0.231728\pi$
$104$	31.9636	3.13429
$105$	0	0
$106$	−7.52906	−0.731287
$107$	12.3756	1.19640	0.598198	−	0.801348i	$-0.295884\pi$
$107$	12.3756	1.19640	0.598198	+	0.801348i	$0.295884\pi$
$108$	0	0
$109$	7.20958	0.690553	0.345276	−	0.938501i	$-0.387785\pi$
$109$	7.20958	0.690553	0.345276	+	0.938501i	$0.387785\pi$
$110$	−2.18884	−0.208698
$111$	0	0
$112$	−22.3659	−2.11338
$113$	−16.6718	−1.56835	−0.784176	−	0.620539i	$-0.786914\pi$
$113$	−16.6718	−1.56835	−0.784176	+	0.620539i	$0.786914\pi$
$114$	0	0
$115$	−0.328077	−0.0305934
$116$	42.5808	3.95353
$117$	0	0
$118$	−1.25424	−0.115462
$119$	−4.77105	−0.437362
$120$	0	0
$121$	−9.21138	−0.837398
$122$	−13.7593	−1.24571
$123$	0	0
$124$	0	0
$125$	5.85866	0.524014
$126$	0	0
$127$	−12.9094	−1.14553	−0.572764	−	0.819721i	$-0.694129\pi$
$127$	−12.9094	−1.14553	−0.572764	+	0.819721i	$0.694129\pi$
$128$	21.5760	1.90707
$129$	0	0
$130$	−6.00758	−0.526900
$131$	2.43778	0.212990	0.106495	−	0.994313i	$-0.466037\pi$
$131$	2.43778	0.212990	0.106495	+	0.994313i	$0.466037\pi$
$132$	0	0
$133$	4.43407	0.384482
$134$	22.3242	1.92851
$135$	0	0
$136$	24.0583	2.06298
$137$	−8.14382	−0.695774	−0.347887	−	0.937536i	$-0.613101\pi$
$137$	−8.14382	−0.695774	−0.347887	+	0.937536i	$0.613101\pi$
$138$	0	0
$139$	−1.35336	−0.114791	−0.0573954	−	0.998352i	$-0.518280\pi$
$139$	−1.35336	−0.114791	−0.0573954	+	0.998352i	$0.518280\pi$
$140$	5.50195	0.465000
$141$	0	0
$142$	−12.8017	−1.07429
$143$	4.90912	0.410521
$144$	0	0
$145$	−4.94669	−0.410800
$146$	20.1928	1.67117
$147$	0	0
$148$	40.5600	3.33401
$149$	9.00385	0.737624	0.368812	−	0.929504i	$-0.379765\pi$
$149$	9.00385	0.737624	0.368812	+	0.929504i	$0.379765\pi$
$150$	0	0
$151$	−3.20841	−0.261097	−0.130548	−	0.991442i	$-0.541674\pi$
$151$	−3.20841	−0.261097	−0.130548	+	0.991442i	$0.541674\pi$
$152$	−22.3590	−1.81355
$153$	0	0
$154$	−6.21295	−0.500654
$155$	0	0
$156$	0	0
$157$	−3.32677	−0.265505	−0.132753	−	0.991149i	$-0.542382\pi$
$157$	−3.32677	−0.265505	−0.132753	+	0.991149i	$0.542382\pi$
$158$	−26.0917	−2.07574
$159$	0	0
$160$	−10.6018	−0.838146
$161$	−0.931238	−0.0733918
$162$	0	0
$163$	1.74035	0.136314	0.0681572	−	0.997675i	$-0.478288\pi$
$163$	1.74035	0.136314	0.0681572	+	0.997675i	$0.478288\pi$
$164$	0.544913	0.0425506
$165$	0	0
$166$	44.8841	3.48368
$167$	14.0165	1.08463	0.542314	−	0.840176i	$-0.317548\pi$
$167$	14.0165	1.08463	0.542314	+	0.840176i	$0.317548\pi$
$168$	0	0
$169$	0.473782	0.0364447
$170$	−4.52176	−0.346803
$171$	0	0
$172$	−15.7354	−1.19982
$173$	0.0119809	0.000910890 0	0.000455445	−	1.00000i	$-0.499855\pi$
$173$	0.0119809	0.000910890 0	0.000455445		1.00000i	$0.499855\pi$
$174$	0	0
$175$	7.99522	0.604382
$176$	17.3214	1.30565
$177$	0	0
$178$	41.2032	3.08831
$179$	9.97800	0.745791	0.372896	−	0.927873i	$-0.378365\pi$
$179$	9.97800	0.745791	0.372896	+	0.927873i	$0.378365\pi$
$180$	0	0
$181$	−18.6032	−1.38276	−0.691381	−	0.722490i	$-0.742997\pi$
$181$	−18.6032	−1.38276	−0.691381	+	0.722490i	$0.742997\pi$
$182$	−17.0523	−1.26400
$183$	0	0
$184$	4.69581	0.346180
$185$	−4.71192	−0.346427
$186$	0	0
$187$	3.69497	0.270203
$188$	35.2183	2.56856
$189$	0	0
$190$	4.20238	0.304873
$191$	−1.19812	−0.0866928	−0.0433464	−	0.999060i	$-0.513802\pi$
$191$	−1.19812	−0.0866928	−0.0433464	+	0.999060i	$0.513802\pi$
$192$	0	0
$193$	12.7266	0.916078	0.458039	−	0.888932i	$-0.348552\pi$
$193$	12.7266	0.916078	0.458039	+	0.888932i	$0.348552\pi$
$194$	−3.44118	−0.247063
$195$	0	0
$196$	−21.0414	−1.50296
$197$	2.96860	0.211504	0.105752	−	0.994393i	$-0.466275\pi$
$197$	2.96860	0.211504	0.105752	+	0.994393i	$0.466275\pi$
$198$	0	0
$199$	−13.1841	−0.934593	−0.467296	−	0.884101i	$-0.654772\pi$
$199$	−13.1841	−0.934593	−0.467296	+	0.884101i	$0.654772\pi$
$200$	−40.3163	−2.85079
$201$	0	0
$202$	−21.8447	−1.53699
$203$	−14.0410	−0.985487
$204$	0	0
$205$	−0.0633035	−0.00442131
$206$	40.7626	2.84007
$207$	0	0
$208$	47.5411	3.29638
$209$	−3.43399	−0.237534
$210$	0	0
$211$	18.3323	1.26205	0.631023	−	0.775764i	$-0.282635\pi$
$211$	18.3323	1.26205	0.631023	+	0.775764i	$0.282635\pi$
$212$	−14.6569	−1.00664
$213$	0	0
$214$	33.2923	2.27582
$215$	1.82801	0.124669
$216$	0	0
$217$	0	0
$218$	19.3949	1.31359
$219$	0	0
$220$	−4.26102	−0.287278
$221$	10.1414	0.682183
$222$	0	0
$223$	9.39601	0.629203	0.314602	−	0.949224i	$-0.398129\pi$
$223$	9.39601	0.629203	0.314602	+	0.949224i	$0.398129\pi$
$224$	−30.0929	−2.01067
$225$	0	0
$226$	−44.8497	−2.98336
$227$	−8.45942	−0.561472	−0.280736	−	0.959785i	$-0.590578\pi$
$227$	−8.45942	−0.561472	−0.280736	+	0.959785i	$0.590578\pi$
$228$	0	0
$229$	22.8434	1.50954	0.754768	−	0.655991i	$-0.227749\pi$
$229$	22.8434	1.50954	0.754768	+	0.655991i	$0.227749\pi$
$230$	−0.882579	−0.0581956
$231$	0	0
$232$	70.8025	4.64841
$233$	−16.2535	−1.06480	−0.532400	−	0.846493i	$-0.678710\pi$
$233$	−16.2535	−1.06480	−0.532400	+	0.846493i	$0.678710\pi$
$234$	0	0
$235$	−4.09137	−0.266891
$236$	−2.44163	−0.158937
$237$	0	0
$238$	−12.8349	−0.831962
$239$	−18.2730	−1.18198	−0.590991	−	0.806678i	$-0.701263\pi$
$239$	−18.2730	−1.18198	−0.590991	+	0.806678i	$0.701263\pi$
$240$	0	0
$241$	−18.1715	−1.17053	−0.585266	−	0.810841i	$-0.699010\pi$
$241$	−18.1715	−1.17053	−0.585266	+	0.810841i	$0.699010\pi$
$242$	−24.7800	−1.59292
$243$	0	0
$244$	−26.7852	−1.71475
$245$	2.44442	0.156168
$246$	0	0
$247$	−9.42508	−0.599704
$248$	0	0
$249$	0	0
$250$	15.7607	0.996794
$251$	22.9302	1.44734	0.723671	−	0.690145i	$-0.242453\pi$
$251$	22.9302	1.44734	0.723671	+	0.690145i	$0.242453\pi$
$252$	0	0
$253$	0.721203	0.0453417
$254$	−34.7284	−2.17905
$255$	0	0
$256$	16.0910	1.00569
$257$	−0.106193	−0.00662414	−0.00331207	−	0.999995i	$-0.501054\pi$
$257$	−0.106193	−0.00662414	−0.00331207	+	0.999995i	$0.501054\pi$
$258$	0	0
$259$	−13.3746	−0.831060
$260$	−11.6950	−0.725292
$261$	0	0
$262$	6.55802	0.405156
$263$	7.29731	0.449971	0.224986	−	0.974362i	$-0.427766\pi$
$263$	7.29731	0.449971	0.224986	+	0.974362i	$0.427766\pi$
$264$	0	0
$265$	1.70271	0.104597
$266$	11.9283	0.731373
$267$	0	0
$268$	43.4586	2.65465
$269$	−16.2194	−0.988912	−0.494456	−	0.869203i	$-0.664633\pi$
$269$	−16.2194	−0.988912	−0.494456	+	0.869203i	$0.664633\pi$
$270$	0	0
$271$	−5.14564	−0.312575	−0.156288	−	0.987712i	$-0.549953\pi$
$271$	−5.14564	−0.312575	−0.156288	+	0.987712i	$0.549953\pi$
$272$	35.7830	2.16966
$273$	0	0
$274$	−21.9082	−1.32352
$275$	−6.19195	−0.373389
$276$	0	0
$277$	−16.9228	−1.01679	−0.508397	−	0.861123i	$-0.669762\pi$
$277$	−16.9228	−1.01679	−0.508397	+	0.861123i	$0.669762\pi$
$278$	−3.64076	−0.218358
$279$	0	0
$280$	9.14853	0.546729
$281$	−1.90667	−0.113742	−0.0568711	−	0.998382i	$-0.518112\pi$
$281$	−1.90667	−0.113742	−0.0568711	+	0.998382i	$0.518112\pi$
$282$	0	0
$283$	−15.3098	−0.910074	−0.455037	−	0.890473i	$-0.650374\pi$
$283$	−15.3098	−0.910074	−0.455037	+	0.890473i	$0.650374\pi$
$284$	−24.9211	−1.47879
$285$	0	0
$286$	13.2063	0.780905
$287$	−0.179685	−0.0106065
$288$	0	0
$289$	−9.36683	−0.550990
$290$	−13.3074	−0.781435
$291$	0	0
$292$	39.3094	2.30041
$293$	−26.7937	−1.56530	−0.782651	−	0.622460i	$-0.786133\pi$
$293$	−26.7937	−1.56530	−0.782651	+	0.622460i	$0.786133\pi$
$294$	0	0
$295$	0.283648	0.0165146
$296$	67.4422	3.92000
$297$	0	0
$298$	24.2218	1.40313
$299$	1.97945	0.114474
$300$	0	0
$301$	5.18876	0.299075
$302$	−8.63112	−0.496665
$303$	0	0
$304$	−33.2556	−1.90734
$305$	3.11169	0.178175
$306$	0	0
$307$	8.95126	0.510875	0.255438	−	0.966826i	$-0.417780\pi$
$307$	8.95126	0.510875	0.255438	+	0.966826i	$0.417780\pi$
$308$	−12.0948	−0.689164
$309$	0	0
$310$	0	0
$311$	−20.6556	−1.17127	−0.585637	−	0.810574i	$-0.699156\pi$
$311$	−20.6556	−1.17127	−0.585637	+	0.810574i	$0.699156\pi$
$312$	0	0
$313$	0.451321	0.0255102	0.0127551	−	0.999919i	$-0.495940\pi$
$313$	0.451321	0.0255102	0.0127551	+	0.999919i	$0.495940\pi$
$314$	−8.94953	−0.505051
$315$	0	0
$316$	−50.7928	−2.85732
$317$	5.58084	0.313451	0.156726	−	0.987642i	$-0.449906\pi$
$317$	5.58084	0.313451	0.156726	+	0.987642i	$0.449906\pi$
$318$	0	0
$319$	10.8742	0.608836
$320$	−12.7614	−0.713383
$321$	0	0
$322$	−2.50517	−0.139608
$323$	−7.09403	−0.394722
$324$	0	0
$325$	−16.9947	−0.942696
$326$	4.68180	0.259301
$327$	0	0
$328$	0.906069	0.0500293
$329$	−11.6132	−0.640258
$330$	0	0
$331$	24.2055	1.33045	0.665226	−	0.746642i	$-0.268335\pi$
$331$	24.2055	1.33045	0.665226	+	0.746642i	$0.268335\pi$
$332$	87.3761	4.79539
$333$	0	0
$334$	37.7065	2.06321
$335$	−5.04865	−0.275837
$336$	0	0
$337$	11.0360	0.601168	0.300584	−	0.953755i	$-0.402818\pi$
$337$	11.0360	0.601168	0.300584	+	0.953755i	$0.402818\pi$
$338$	1.27455	0.0693262
$339$	0	0
$340$	−8.80253	−0.477384
$341$	0	0
$342$	0	0
$343$	19.0266	1.02734
$344$	−26.1645	−1.41070
$345$	0	0
$346$	0.0322304	0.00173272
$347$	18.5674	0.996749	0.498375	−	0.866962i	$-0.333930\pi$
$347$	18.5674	0.996749	0.498375	+	0.866962i	$0.333930\pi$
$348$	0	0
$349$	−32.1746	−1.72227	−0.861134	−	0.508378i	$-0.830245\pi$
$349$	−32.1746	−1.72227	−0.861134	+	0.508378i	$0.830245\pi$
$350$	21.5084	1.14967
$351$	0	0
$352$	23.3056	1.24219
$353$	−32.2134	−1.71455	−0.857273	−	0.514863i	$-0.827843\pi$
$353$	−32.2134	−1.71455	−0.857273	+	0.514863i	$0.827843\pi$
$354$	0	0
$355$	2.89512	0.153657
$356$	80.2104	4.25114
$357$	0	0
$358$	26.8424	1.41866
$359$	−21.1623	−1.11690	−0.558451	−	0.829538i	$-0.688604\pi$
$359$	−21.1623	−1.11690	−0.558451	+	0.829538i	$0.688604\pi$
$360$	0	0
$361$	−12.4070	−0.653002
$362$	−50.0454	−2.63033
$363$	0	0
$364$	−33.1959	−1.73994
$365$	−4.56664	−0.239029
$366$	0	0
$367$	12.9964	0.678408	0.339204	−	0.940713i	$-0.389842\pi$
$367$	12.9964	0.678408	0.339204	+	0.940713i	$0.389842\pi$
$368$	6.98431	0.364082
$369$	0	0
$370$	−12.6758	−0.658983
$371$	4.83309	0.250922
$372$	0	0
$373$	−4.42592	−0.229166	−0.114583	−	0.993414i	$-0.536553\pi$
$373$	−4.42592	−0.229166	−0.114583	+	0.993414i	$0.536553\pi$
$374$	9.94005	0.513988
$375$	0	0
$376$	58.5602	3.02001
$377$	29.8457	1.53713
$378$	0	0
$379$	−14.9557	−0.768225	−0.384112	−	0.923286i	$-0.625492\pi$
$379$	−14.9557	−0.768225	−0.384112	+	0.923286i	$0.625492\pi$
$380$	8.18079	0.419666
$381$	0	0
$382$	−3.22312	−0.164909
$383$	11.6503	0.595305	0.297652	−	0.954674i	$-0.403796\pi$
$383$	11.6503	0.595305	0.297652	+	0.954674i	$0.403796\pi$
$384$	0	0
$385$	1.40507	0.0716091
$386$	34.2364	1.74259
$387$	0	0
$388$	−6.69897	−0.340089
$389$	−14.4019	−0.730207	−0.365103	−	0.930967i	$-0.618966\pi$
$389$	−14.4019	−0.730207	−0.365103	+	0.930967i	$0.618966\pi$
$390$	0	0
$391$	1.48988	0.0753465
$392$	−34.9872	−1.76712
$393$	0	0
$394$	7.98598	0.402328
$395$	5.90069	0.296896
$396$	0	0
$397$	15.7901	0.792484	0.396242	−	0.918146i	$-0.370314\pi$
$397$	15.7901	0.792484	0.396242	+	0.918146i	$0.370314\pi$
$398$	−35.4672	−1.77781
$399$	0	0
$400$	−59.9644	−2.99822
$401$	−31.0232	−1.54923	−0.774613	−	0.632436i	$-0.782055\pi$
$401$	−31.0232	−1.54923	−0.774613	+	0.632436i	$0.782055\pi$
$402$	0	0
$403$	0	0
$404$	−42.5252	−2.11571
$405$	0	0
$406$	−37.7725	−1.87462
$407$	10.3581	0.513431
$408$	0	0
$409$	6.58582	0.325648	0.162824	−	0.986655i	$-0.447940\pi$
$409$	6.58582	0.325648	0.162824	+	0.986655i	$0.447940\pi$
$410$	−0.170296	−0.00841033
$411$	0	0
$412$	79.3527	3.90943
$413$	0.805127	0.0396177
$414$	0	0
$415$	−10.1506	−0.498275
$416$	63.9657	3.13618
$417$	0	0
$418$	−9.23797	−0.451844
$419$	14.6720	0.716773	0.358387	−	0.933573i	$-0.383327\pi$
$419$	14.6720	0.716773	0.358387	+	0.933573i	$0.383327\pi$
$420$	0	0
$421$	−26.7159	−1.30205	−0.651026	−	0.759056i	$-0.725661\pi$
$421$	−26.7159	−1.30205	−0.651026	+	0.759056i	$0.725661\pi$
$422$	49.3167	2.40070
$423$	0	0
$424$	−24.3711	−1.18356
$425$	−12.7915	−0.620479
$426$	0	0
$427$	8.83243	0.427431
$428$	64.8103	3.13272
$429$	0	0
$430$	4.91764	0.237150
$431$	−3.98801	−0.192096	−0.0960478	−	0.995377i	$-0.530620\pi$
$431$	−3.98801	−0.192096	−0.0960478	+	0.995377i	$0.530620\pi$
$432$	0	0
$433$	18.0766	0.868704	0.434352	−	0.900743i	$-0.356977\pi$
$433$	18.0766	0.868704	0.434352	+	0.900743i	$0.356977\pi$
$434$	0	0
$435$	0	0
$436$	37.7561	1.80819
$437$	−1.38465	−0.0662367
$438$	0	0
$439$	22.9086	1.09337	0.546684	−	0.837339i	$-0.315890\pi$
$439$	22.9086	1.09337	0.546684	+	0.837339i	$0.315890\pi$
$440$	−7.08513	−0.337771
$441$	0	0
$442$	27.2819	1.29767
$443$	21.4913	1.02108	0.510542	−	0.859853i	$-0.329445\pi$
$443$	21.4913	1.02108	0.510542	+	0.859853i	$0.329445\pi$
$444$	0	0
$445$	−9.31817	−0.441724
$446$	25.2767	1.19689
$447$	0	0
$448$	−36.2228	−1.71137
$449$	−18.7996	−0.887209	−0.443604	−	0.896223i	$-0.646300\pi$
$449$	−18.7996	−0.887209	−0.443604	+	0.896223i	$0.646300\pi$
$450$	0	0
$451$	0.139158	0.00655270
$452$	−87.3092	−4.10668
$453$	0	0
$454$	−22.7572	−1.06805
$455$	3.85642	0.180792
$456$	0	0
$457$	4.10599	0.192070	0.0960351	−	0.995378i	$-0.469384\pi$
$457$	4.10599	0.192070	0.0960351	+	0.995378i	$0.469384\pi$
$458$	61.4524	2.87148
$459$	0	0
$460$	−1.71812	−0.0801078
$461$	22.8750	1.06540	0.532698	−	0.846305i	$-0.321178\pi$
$461$	22.8750	1.06540	0.532698	+	0.846305i	$0.321178\pi$
$462$	0	0
$463$	−16.5670	−0.769932	−0.384966	−	0.922931i	$-0.625787\pi$
$463$	−16.5670	−0.769932	−0.384966	+	0.922931i	$0.625787\pi$
$464$	105.308	4.88880
$465$	0	0
$466$	−43.7244	−2.02549
$467$	−38.1462	−1.76520	−0.882599	−	0.470127i	$-0.844208\pi$
$467$	−38.1462	−1.76520	−0.882599	+	0.470127i	$0.844208\pi$
$468$	0	0
$469$	−14.3305	−0.661719
$470$	−11.0064	−0.507688
$471$	0	0
$472$	−4.05989	−0.186872
$473$	−4.01847	−0.184769
$474$	0	0
$475$	11.8880	0.545459
$476$	−24.9857	−1.14522
$477$	0	0
$478$	−49.1572	−2.24840
$479$	33.6848	1.53910	0.769548	−	0.638589i	$-0.220482\pi$
$479$	33.6848	1.53910	0.769548	+	0.638589i	$0.220482\pi$
$480$	0	0
$481$	28.4292	1.29626
$482$	−48.8843	−2.22662
$483$	0	0
$484$	−48.2394	−2.19270
$485$	0.778230	0.0353376
$486$	0	0
$487$	−20.6767	−0.936950	−0.468475	−	0.883477i	$-0.655196\pi$
$487$	−20.6767	−0.936950	−0.468475	+	0.883477i	$0.655196\pi$
$488$	−44.5379	−2.01614
$489$	0	0
$490$	6.57587	0.297067
$491$	−24.7327	−1.11617	−0.558087	−	0.829783i	$-0.688464\pi$
$491$	−24.7327	−1.11617	−0.558087	+	0.829783i	$0.688464\pi$
$492$	0	0
$493$	22.4641	1.01173
$494$	−25.3549	−1.14077
$495$	0	0
$496$	0	0
$497$	8.21771	0.368615
$498$	0	0
$499$	−24.5602	−1.09946	−0.549732	−	0.835341i	$-0.685270\pi$
$499$	−24.5602	−1.09946	−0.549732	+	0.835341i	$0.685270\pi$
$500$	30.6814	1.37211
$501$	0	0
$502$	61.6858	2.75317
$503$	−14.7391	−0.657185	−0.328593	−	0.944472i	$-0.606574\pi$
$503$	−14.7391	−0.657185	−0.328593	+	0.944472i	$0.606574\pi$
$504$	0	0
$505$	4.94022	0.219837
$506$	1.94015	0.0862501
$507$	0	0
$508$	−67.6059	−2.99953
$509$	−15.5063	−0.687304	−0.343652	−	0.939097i	$-0.611664\pi$
$509$	−15.5063	−0.687304	−0.343652	+	0.939097i	$0.611664\pi$
$510$	0	0
$511$	−12.9623	−0.573418
$512$	0.135395	0.00598366
$513$	0	0
$514$	−0.285676	−0.0126006
$515$	−9.21854	−0.406217
$516$	0	0
$517$	8.99394	0.395553
$518$	−35.9799	−1.58086
$519$	0	0
$520$	−19.4462	−0.852771
$521$	−29.4146	−1.28867	−0.644337	−	0.764741i	$-0.722867\pi$
$521$	−29.4146	−1.28867	−0.644337	+	0.764741i	$0.722867\pi$
$522$	0	0
$523$	45.0260	1.96885	0.984424	−	0.175810i	$-0.0562543\pi$
$523$	45.0260	1.96885	0.984424	+	0.175810i	$0.0562543\pi$
$524$	12.7665	0.557708
$525$	0	0
$526$	19.6309	0.855947
$527$	0	0
$528$	0	0
$529$	−22.7092	−0.987356
$530$	4.58056	0.198967
$531$	0	0
$532$	23.2209	1.00676
$533$	0.381940	0.0165436
$534$	0	0
$535$	−7.52912	−0.325512
$536$	72.2619	3.12124
$537$	0	0
$538$	−43.6326	−1.88114
$539$	−5.37349	−0.231453
$540$	0	0
$541$	37.7787	1.62423	0.812117	−	0.583495i	$-0.198315\pi$
$541$	37.7787	1.62423	0.812117	+	0.583495i	$0.198315\pi$
$542$	−13.8426	−0.594589
$543$	0	0
$544$	48.1454	2.06422
$545$	−4.38619	−0.187884
$546$	0	0
$547$	−18.2088	−0.778550	−0.389275	−	0.921122i	$-0.627274\pi$
$547$	−18.2088	−0.778550	−0.389275	+	0.921122i	$0.627274\pi$
$548$	−42.6487	−1.82186
$549$	0	0
$550$	−16.6573	−0.710270
$551$	−20.8774	−0.889409
$552$	0	0
$553$	16.7489	0.712237
$554$	−45.5250	−1.93417
$555$	0	0
$556$	−7.08748	−0.300576
$557$	9.19760	0.389715	0.194857	−	0.980832i	$-0.437576\pi$
$557$	9.19760	0.389715	0.194857	+	0.980832i	$0.437576\pi$
$558$	0	0
$559$	−11.0293	−0.466488
$560$	13.6071	0.575003
$561$	0	0
$562$	−5.12923	−0.216364
$563$	36.1618	1.52404	0.762019	−	0.647555i	$-0.224208\pi$
$563$	36.1618	1.52404	0.762019	+	0.647555i	$0.224208\pi$
$564$	0	0
$565$	10.1429	0.426713
$566$	−41.1858	−1.73117
$567$	0	0
$568$	−41.4382	−1.73871
$569$	−41.4998	−1.73976	−0.869880	−	0.493263i	$-0.835804\pi$
$569$	−41.4998	−1.73976	−0.869880	+	0.493263i	$0.835804\pi$
$570$	0	0
$571$	−32.3931	−1.35561	−0.677805	−	0.735242i	$-0.737069\pi$
$571$	−32.3931	−1.35561	−0.677805	+	0.735242i	$0.737069\pi$
$572$	25.7087	1.07494
$573$	0	0
$574$	−0.483381	−0.0201759
$575$	−2.49671	−0.104120
$576$	0	0
$577$	40.9825	1.70613	0.853063	−	0.521809i	$-0.174743\pi$
$577$	40.9825	1.70613	0.853063	+	0.521809i	$0.174743\pi$
$578$	−25.1982	−1.04811
$579$	0	0
$580$	−25.9055	−1.07567
$581$	−28.8123	−1.19533
$582$	0	0
$583$	−3.74302	−0.155020
$584$	65.3629	2.70474
$585$	0	0
$586$	−72.0791	−2.97756
$587$	−25.5314	−1.05379	−0.526897	−	0.849929i	$-0.676644\pi$
$587$	−25.5314	−1.05379	−0.526897	+	0.849929i	$0.676644\pi$
$588$	0	0
$589$	0	0
$590$	0.763058	0.0314146
$591$	0	0
$592$	100.310	4.12272
$593$	−44.5318	−1.82870	−0.914350	−	0.404925i	$-0.867298\pi$
$593$	−44.5318	−1.82870	−0.914350	+	0.404925i	$0.867298\pi$
$594$	0	0
$595$	2.90263	0.118996
$596$	47.1526	1.93145
$597$	0	0
$598$	5.32502	0.217756
$599$	−18.9800	−0.775500	−0.387750	−	0.921765i	$-0.626748\pi$
$599$	−18.9800	−0.775500	−0.387750	+	0.921765i	$0.626748\pi$
$600$	0	0
$601$	−12.8398	−0.523747	−0.261873	−	0.965102i	$-0.584340\pi$
$601$	−12.8398	−0.523747	−0.261873	+	0.965102i	$0.584340\pi$
$602$	13.9586	0.568909
$603$	0	0
$604$	−16.8022	−0.683673
$605$	5.60405	0.227837
$606$	0	0
$607$	−38.1393	−1.54803	−0.774013	−	0.633170i	$-0.781754\pi$
$607$	−38.1393	−1.54803	−0.774013	+	0.633170i	$0.781754\pi$
$608$	−44.7448	−1.81464
$609$	0	0
$610$	8.37092	0.338929
$611$	24.6852	0.998654
$612$	0	0
$613$	−18.5424	−0.748920	−0.374460	−	0.927243i	$-0.622172\pi$
$613$	−18.5424	−0.748920	−0.374460	+	0.927243i	$0.622172\pi$
$614$	24.0803	0.971801
$615$	0	0
$616$	−20.1109	−0.810293
$617$	−8.13555	−0.327525	−0.163762	−	0.986500i	$-0.552363\pi$
$617$	−8.13555	−0.327525	−0.163762	+	0.986500i	$0.552363\pi$
$618$	0	0
$619$	−5.36063	−0.215462	−0.107731	−	0.994180i	$-0.534359\pi$
$619$	−5.36063	−0.215462	−0.107731	+	0.994180i	$0.534359\pi$
$620$	0	0
$621$	0	0
$622$	−55.5669	−2.22803
$623$	−26.4494	−1.05967
$624$	0	0
$625$	19.5850	0.783401
$626$	1.21412	0.0485261
$627$	0	0
$628$	−17.4221	−0.695217
$629$	21.3980	0.853193
$630$	0	0
$631$	29.0470	1.15634	0.578172	−	0.815915i	$-0.303766\pi$
$631$	29.0470	1.15634	0.578172	+	0.815915i	$0.303766\pi$
$632$	−84.4572	−3.35953
$633$	0	0
$634$	15.0133	0.596256
$635$	7.85389	0.311672
$636$	0	0
$637$	−14.7483	−0.584350
$638$	29.2532	1.15814
$639$	0	0
$640$	−13.1265	−0.518870
$641$	27.6055	1.09035	0.545177	−	0.838321i	$-0.316463\pi$
$641$	27.6055	1.09035	0.545177	+	0.838321i	$0.316463\pi$
$642$	0	0
$643$	−4.22904	−0.166777	−0.0833885	−	0.996517i	$-0.526574\pi$
$643$	−4.22904	−0.166777	−0.0833885	+	0.996517i	$0.526574\pi$
$644$	−4.87683	−0.192174
$645$	0	0
$646$	−19.0840	−0.750852
$647$	26.2809	1.03321	0.516604	−	0.856225i	$-0.327196\pi$
$647$	26.2809	1.03321	0.516604	+	0.856225i	$0.327196\pi$
$648$	0	0
$649$	−0.623536	−0.0244759
$650$	−45.7184	−1.79322
$651$	0	0
$652$	9.11408	0.356935
$653$	13.3517	0.522494	0.261247	−	0.965272i	$-0.415866\pi$
$653$	13.3517	0.522494	0.261247	+	0.965272i	$0.415866\pi$
$654$	0	0
$655$	−1.48311	−0.0579498
$656$	1.34764	0.0526166
$657$	0	0
$658$	−31.2414	−1.21792
$659$	15.4158	0.600515	0.300258	−	0.953858i	$-0.402927\pi$
$659$	15.4158	0.600515	0.300258	+	0.953858i	$0.402927\pi$
$660$	0	0
$661$	−4.72306	−0.183706	−0.0918530	−	0.995773i	$-0.529279\pi$
$661$	−4.72306	−0.183706	−0.0918530	+	0.995773i	$0.529279\pi$
$662$	65.1164	2.53082
$663$	0	0
$664$	145.287	5.63823
$665$	−2.69762	−0.104609
$666$	0	0
$667$	4.38466	0.169775
$668$	73.4034	2.84006
$669$	0	0
$670$	−13.5817	−0.524705
$671$	−6.84033	−0.264068
$672$	0	0
$673$	−31.0730	−1.19778	−0.598888	−	0.800833i	$-0.704391\pi$
$673$	−31.0730	−1.19778	−0.598888	+	0.800833i	$0.704391\pi$
$674$	29.6885	1.14356
$675$	0	0
$676$	2.48116	0.0954294
$677$	9.03995	0.347434	0.173717	−	0.984796i	$-0.444422\pi$
$677$	9.03995	0.347434	0.173717	+	0.984796i	$0.444422\pi$
$678$	0	0
$679$	2.20898	0.0847730
$680$	−14.6367	−0.561290
$681$	0	0
$682$	0	0
$683$	−7.13535	−0.273027	−0.136513	−	0.990638i	$-0.543590\pi$
$683$	−7.13535	−0.273027	−0.136513	+	0.990638i	$0.543590\pi$
$684$	0	0
$685$	4.95457	0.189304
$686$	51.1844	1.95423
$687$	0	0
$688$	−38.9158	−1.48365
$689$	−10.2733	−0.391380
$690$	0	0
$691$	36.4754	1.38759	0.693795	−	0.720172i	$-0.255937\pi$
$691$	36.4754	1.38759	0.693795	+	0.720172i	$0.255937\pi$
$692$	0.0627431	0.00238514
$693$	0	0
$694$	49.9491	1.89604
$695$	0.823364	0.0312320
$696$	0	0
$697$	0.287476	0.0108889
$698$	−86.5547	−3.27615
$699$	0	0
$700$	41.8705	1.58256
$701$	1.61444	0.0609766	0.0304883	−	0.999535i	$-0.490294\pi$
$701$	1.61444	0.0609766	0.0304883	+	0.999535i	$0.490294\pi$
$702$	0	0
$703$	−19.8866	−0.750037
$704$	28.0530	1.05729
$705$	0	0
$706$	−86.6590	−3.26145
$707$	14.0227	0.527377
$708$	0	0
$709$	11.4880	0.431439	0.215720	−	0.976455i	$-0.430790\pi$
$709$	11.4880	0.431439	0.215720	+	0.976455i	$0.430790\pi$
$710$	7.78833	0.292291
$711$	0	0
$712$	133.372	4.99833
$713$	0	0
$714$	0	0
$715$	−2.98663	−0.111694
$716$	52.2542	1.95283
$717$	0	0
$718$	−56.9298	−2.12460
$719$	14.1710	0.528489	0.264244	−	0.964456i	$-0.414877\pi$
$719$	14.1710	0.528489	0.264244	+	0.964456i	$0.414877\pi$
$720$	0	0
$721$	−26.1666	−0.974493
$722$	−33.3768	−1.24216
$723$	0	0
$724$	−97.4236	−3.62072
$725$	−37.6449	−1.39809
$726$	0	0
$727$	−26.9477	−0.999436	−0.499718	−	0.866188i	$-0.666563\pi$
$727$	−26.9477	−0.999436	−0.499718	+	0.866188i	$0.666563\pi$
$728$	−55.1974	−2.04575
$729$	0	0
$730$	−12.2850	−0.454688
$731$	−8.30145	−0.307040
$732$	0	0
$733$	−28.4235	−1.04985	−0.524924	−	0.851149i	$-0.675906\pi$
$733$	−28.4235	−1.04985	−0.524924	+	0.851149i	$0.675906\pi$
$734$	34.9624	1.29049
$735$	0	0
$736$	9.39725	0.346387
$737$	11.0983	0.408811
$738$	0	0
$739$	−0.697306	−0.0256508	−0.0128254	−	0.999918i	$-0.504083\pi$
$739$	−0.697306	−0.0256508	−0.0128254	+	0.999918i	$0.504083\pi$
$740$	−24.6760	−0.907108
$741$	0	0
$742$	13.0018	0.477310
$743$	18.1815	0.667015	0.333508	−	0.942747i	$-0.391768\pi$
$743$	18.1815	0.667015	0.333508	+	0.942747i	$0.391768\pi$
$744$	0	0
$745$	−5.47779	−0.200691
$746$	−11.9064	−0.435925
$747$	0	0
$748$	19.3503	0.707519
$749$	−21.3712	−0.780886
$750$	0	0
$751$	−37.0660	−1.35256	−0.676279	−	0.736646i	$-0.736408\pi$
$751$	−37.0660	−1.35256	−0.676279	+	0.736646i	$0.736408\pi$
$752$	87.0995	3.17619
$753$	0	0
$754$	80.2895	2.92397
$755$	1.95194	0.0710385
$756$	0	0
$757$	6.40201	0.232685	0.116342	−	0.993209i	$-0.462883\pi$
$757$	6.40201	0.232685	0.116342	+	0.993209i	$0.462883\pi$
$758$	−40.2333	−1.46134
$759$	0	0
$760$	13.6028	0.493427
$761$	21.3388	0.773532	0.386766	−	0.922178i	$-0.373592\pi$
$761$	21.3388	0.773532	0.386766	+	0.922178i	$0.373592\pi$
$762$	0	0
$763$	−12.4501	−0.450723
$764$	−6.27447	−0.227002
$765$	0	0
$766$	31.3412	1.13241
$767$	−1.71138	−0.0617945
$768$	0	0
$769$	−12.2566	−0.441983	−0.220991	−	0.975276i	$-0.570929\pi$
$769$	−12.2566	−0.441983	−0.220991	+	0.975276i	$0.570929\pi$
$770$	3.77986	0.136217
$771$	0	0
$772$	66.6482	2.39872
$773$	34.7035	1.24820	0.624098	−	0.781346i	$-0.285466\pi$
$773$	34.7035	1.24820	0.624098	+	0.781346i	$0.285466\pi$
$774$	0	0
$775$	0	0
$776$	−11.1389	−0.399863
$777$	0	0
$778$	−38.7434	−1.38902
$779$	−0.267172	−0.00957242
$780$	0	0
$781$	−6.36426	−0.227731
$782$	4.00801	0.143326
$783$	0	0
$784$	−52.0382	−1.85851
$785$	2.02395	0.0722380
$786$	0	0
$787$	−20.1182	−0.717136	−0.358568	−	0.933504i	$-0.616735\pi$
$787$	−20.1182	−0.717136	−0.358568	+	0.933504i	$0.616735\pi$
$788$	15.5463	0.553816
$789$	0	0
$790$	15.8738	0.564763
$791$	28.7902	1.02366
$792$	0	0
$793$	−18.7743	−0.666694
$794$	42.4779	1.50748
$795$	0	0
$796$	−69.0441	−2.44720
$797$	−26.2019	−0.928121	−0.464060	−	0.885804i	$-0.653608\pi$
$797$	−26.2019	−0.928121	−0.464060	+	0.885804i	$0.653608\pi$
$798$	0	0
$799$	18.5799	0.657310
$800$	−80.6809	−2.85250
$801$	0	0
$802$	−83.4573	−2.94698
$803$	10.0387	0.354259
$804$	0	0
$805$	0.566550	0.0199683
$806$	0	0
$807$	0	0
$808$	−70.7099	−2.48757
$809$	−36.4501	−1.28152	−0.640759	−	0.767742i	$-0.721380\pi$
$809$	−36.4501	−1.28152	−0.640759	+	0.767742i	$0.721380\pi$
$810$	0	0
$811$	−19.1064	−0.670918	−0.335459	−	0.942055i	$-0.608891\pi$
$811$	−19.1064	−0.670918	−0.335459	+	0.942055i	$0.608891\pi$
$812$	−73.5320	−2.58047
$813$	0	0
$814$	27.8648	0.976662
$815$	−1.05880	−0.0370881
$816$	0	0
$817$	7.71511	0.269917
$818$	17.7169	0.619456
$819$	0	0
$820$	−0.331516	−0.0115770
$821$	−39.4598	−1.37716	−0.688578	−	0.725162i	$-0.741765\pi$
$821$	−39.4598	−1.37716	−0.688578	+	0.725162i	$0.741765\pi$
$822$	0	0
$823$	−31.5880	−1.10109	−0.550544	−	0.834806i	$-0.685580\pi$
$823$	−31.5880	−1.10109	−0.550544	+	0.834806i	$0.685580\pi$
$824$	131.946	4.59656
$825$	0	0
$826$	2.16592	0.0753619
$827$	−19.8298	−0.689551	−0.344776	−	0.938685i	$-0.612045\pi$
$827$	−19.8298	−0.689551	−0.344776	+	0.938685i	$0.612045\pi$
$828$	0	0
$829$	8.99987	0.312578	0.156289	−	0.987711i	$-0.450047\pi$
$829$	8.99987	0.312578	0.156289	+	0.987711i	$0.450047\pi$
$830$	−27.3068	−0.947832
$831$	0	0
$832$	76.9954	2.66933
$833$	−11.1007	−0.384617
$834$	0	0
$835$	−8.52740	−0.295103
$836$	−17.9836	−0.621976
$837$	0	0
$838$	39.4699	1.36347
$839$	31.8452	1.09942	0.549709	−	0.835356i	$-0.314739\pi$
$839$	31.8452	1.09942	0.549709	+	0.835356i	$0.314739\pi$
$840$	0	0
$841$	37.1110	1.27969
$842$	−71.8698	−2.47680
$843$	0	0
$844$	96.0050	3.30463
$845$	−0.288241	−0.00991579
$846$	0	0
$847$	15.9069	0.546569
$848$	−36.2483	−1.24477
$849$	0	0
$850$	−34.4111	−1.18029
$851$	4.17656	0.143171
$852$	0	0
$853$	−49.0917	−1.68087	−0.840434	−	0.541914i	$-0.817700\pi$
$853$	−49.0917	−1.68087	−0.840434	+	0.541914i	$0.817700\pi$
$854$	23.7606	0.813071
$855$	0	0
$856$	107.765	3.68334
$857$	−19.2405	−0.657242	−0.328621	−	0.944462i	$-0.606584\pi$
$857$	−19.2405	−0.657242	−0.328621	+	0.944462i	$0.606584\pi$
$858$	0	0
$859$	4.64215	0.158388	0.0791940	−	0.996859i	$-0.474765\pi$
$859$	4.64215	0.158388	0.0791940	+	0.996859i	$0.474765\pi$
$860$	9.57319	0.326443
$861$	0	0
$862$	−10.7284	−0.365410
$863$	38.5304	1.31159	0.655795	−	0.754939i	$-0.272333\pi$
$863$	38.5304	1.31159	0.655795	+	0.754939i	$0.272333\pi$
$864$	0	0
$865$	−0.00728897	−0.000247833 0
$866$	48.6287	1.65247
$867$	0	0
$868$	0	0
$869$	−12.9713	−0.440022
$870$	0	0
$871$	30.4609	1.03213
$872$	62.7800	2.12600
$873$	0	0
$874$	−3.72492	−0.125997
$875$	−10.1172	−0.342023
$876$	0	0
$877$	−9.25834	−0.312632	−0.156316	−	0.987707i	$-0.549962\pi$
$877$	−9.25834	−0.312632	−0.156316	+	0.987707i	$0.549962\pi$
$878$	61.6277	2.07983
$879$	0	0
$880$	−10.5381	−0.355238
$881$	17.8791	0.602361	0.301181	−	0.953567i	$-0.402619\pi$
$881$	17.8791	0.602361	0.301181	+	0.953567i	$0.402619\pi$
$882$	0	0
$883$	−48.3156	−1.62595	−0.812975	−	0.582299i	$-0.802153\pi$
$883$	−48.3156	−1.62595	−0.812975	+	0.582299i	$0.802153\pi$
$884$	53.1098	1.78628
$885$	0	0
$886$	57.8150	1.94233
$887$	−23.5122	−0.789463	−0.394732	−	0.918797i	$-0.629162\pi$
$887$	−23.5122	−0.789463	−0.394732	+	0.918797i	$0.629162\pi$
$888$	0	0
$889$	22.2930	0.747684
$890$	−25.0673	−0.840259
$891$	0	0
$892$	49.2063	1.64755
$893$	−17.2676	−0.577838
$894$	0	0
$895$	−6.07046	−0.202913
$896$	−37.2591	−1.24474
$897$	0	0
$898$	−50.5739	−1.68767
$899$	0	0
$900$	0	0
$901$	−7.73243	−0.257605
$902$	0.374357	0.0124647
$903$	0	0
$904$	−145.176	−4.82847
$905$	11.3179	0.376219
$906$	0	0
$907$	−12.5821	−0.417783	−0.208892	−	0.977939i	$-0.566986\pi$
$907$	−12.5821	−0.417783	−0.208892	+	0.977939i	$0.566986\pi$
$908$	−44.3015	−1.47020
$909$	0	0
$910$	10.3744	0.343907
$911$	47.2845	1.56660	0.783302	−	0.621641i	$-0.213534\pi$
$911$	47.2845	1.56660	0.783302	+	0.621641i	$0.213534\pi$
$912$	0	0
$913$	22.3138	0.738480
$914$	11.0458	0.365361
$915$	0	0
$916$	119.630	3.95267
$917$	−4.20976	−0.139018
$918$	0	0
$919$	−23.1319	−0.763050	−0.381525	−	0.924359i	$-0.624601\pi$
$919$	−23.1319	−0.763050	−0.381525	+	0.924359i	$0.624601\pi$
$920$	−2.85685	−0.0941877
$921$	0	0
$922$	61.5374	2.02663
$923$	−17.4676	−0.574954
$924$	0	0
$925$	−35.8582	−1.17901
$926$	−44.5677	−1.46459
$927$	0	0
$928$	141.690	4.65120
$929$	−1.68694	−0.0553468	−0.0276734	−	0.999617i	$-0.508810\pi$
$929$	−1.68694	−0.0553468	−0.0276734	+	0.999617i	$0.508810\pi$
$930$	0	0
$931$	10.3166	0.338114
$932$	−85.1184	−2.78815
$933$	0	0
$934$	−102.619	−3.35781
$935$	−2.24796	−0.0735162
$936$	0	0
$937$	19.7267	0.644442	0.322221	−	0.946665i	$-0.395571\pi$
$937$	19.7267	0.644442	0.322221	+	0.946665i	$0.395571\pi$
$938$	−38.5511	−1.25874
$939$	0	0
$940$	−21.4262	−0.698847
$941$	8.82326	0.287630	0.143815	−	0.989605i	$-0.454063\pi$
$941$	8.82326	0.287630	0.143815	+	0.989605i	$0.454063\pi$
$942$	0	0
$943$	0.0561111	0.00182723
$944$	−6.03847	−0.196536
$945$	0	0
$946$	−10.8103	−0.351473
$947$	−15.5612	−0.505670	−0.252835	−	0.967509i	$-0.581363\pi$
$947$	−15.5612	−0.505670	−0.252835	+	0.967509i	$0.581363\pi$
$948$	0	0
$949$	27.5527	0.894399
$950$	31.9806	1.03759
$951$	0	0
$952$	−41.5457	−1.34650
$953$	−9.65172	−0.312650	−0.156325	−	0.987706i	$-0.549965\pi$
$953$	−9.65172	−0.312650	−0.156325	+	0.987706i	$0.549965\pi$
$954$	0	0
$955$	0.728916	0.0235872
$956$	−95.6945	−3.09498
$957$	0	0
$958$	90.6172	2.92771
$959$	14.0634	0.454131
$960$	0	0
$961$	0	0
$962$	76.4790	2.46578
$963$	0	0
$964$	−95.1632	−3.06500
$965$	−7.74263	−0.249244
$966$	0	0
$967$	28.3976	0.913204	0.456602	−	0.889671i	$-0.349066\pi$
$967$	28.3976	0.913204	0.456602	+	0.889671i	$0.349066\pi$
$968$	−80.2114	−2.57809
$969$	0	0
$970$	2.09356	0.0672202
$971$	22.1271	0.710092	0.355046	−	0.934849i	$-0.384465\pi$
$971$	22.1271	0.710092	0.355046	+	0.934849i	$0.384465\pi$
$972$	0	0
$973$	2.33709	0.0749238
$974$	−55.6235	−1.78229
$975$	0	0
$976$	−66.2434	−2.12040
$977$	19.9123	0.637051	0.318525	−	0.947914i	$-0.396812\pi$
$977$	19.9123	0.637051	0.318525	+	0.947914i	$0.396812\pi$
$978$	0	0
$979$	20.4839	0.654667
$980$	12.8013	0.408921
$981$	0	0
$982$	−66.5349	−2.12322
$983$	24.9304	0.795156	0.397578	−	0.917568i	$-0.369851\pi$
$983$	24.9304	0.795156	0.397578	+	0.917568i	$0.369851\pi$
$984$	0	0
$985$	−1.80605	−0.0575454
$986$	60.4320	1.92455
$987$	0	0
$988$	−49.3586	−1.57031
$989$	−1.62032	−0.0515231
$990$	0	0
$991$	−42.0512	−1.33580	−0.667900	−	0.744251i	$-0.732807\pi$
$991$	−42.0512	−1.33580	−0.667900	+	0.744251i	$0.732807\pi$
$992$	0	0
$993$	0	0
$994$	22.1069	0.701189
$995$	8.02096	0.254282
$996$	0	0
$997$	14.4647	0.458100	0.229050	−	0.973415i	$-0.426438\pi$
$997$	14.4647	0.458100	0.229050	+	0.973415i	$0.426438\pi$
$998$	−66.0707	−2.09143
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.bf.1.8		8
3.2	odd	2		961.2.a.i.1.1		8
31.18	even	15		279.2.y.c.262.2		16
31.19	even	15		279.2.y.c.82.2		16
31.30	odd	2		8649.2.a.be.1.8		8
93.2	odd	10		961.2.d.o.531.1		16
93.5	odd	6		961.2.c.j.521.1		16
93.8	odd	10		961.2.d.p.374.4		16
93.11	even	30		961.2.g.n.338.2		16
93.14	odd	30		961.2.g.t.816.2		16
93.17	even	30		961.2.g.n.816.2		16
93.20	odd	30		961.2.g.t.338.2		16
93.23	even	10		961.2.d.q.374.4		16
93.26	even	6		961.2.c.i.521.1		16
93.29	even	10		961.2.d.n.531.1		16
93.35	odd	10		961.2.d.p.388.4		16
93.38	odd	30		961.2.g.s.235.2		16
93.41	odd	30		961.2.g.k.844.1		16
93.44	even	30		961.2.g.l.448.1		16
93.47	odd	10		961.2.d.o.628.1		16
93.50	odd	30		31.2.g.a.20.1	yes	16
93.53	even	30		961.2.g.m.732.2		16
93.56	odd	6		961.2.c.j.439.1		16
93.59	odd	30		961.2.g.k.846.1		16
93.65	even	30		961.2.g.j.846.1		16
93.68	even	6		961.2.c.i.439.1		16
93.71	odd	30		961.2.g.s.732.2		16
93.74	even	30		961.2.g.l.547.1		16
93.77	even	10		961.2.d.n.628.1		16
93.80	odd	30		31.2.g.a.14.1	✓	16
93.83	even	30		961.2.g.j.844.1		16
93.86	even	30		961.2.g.m.235.2		16
93.89	even	10		961.2.d.q.388.4		16
93.92	even	2		961.2.a.j.1.1		8
372.143	even	30		496.2.bg.c.113.1		16
372.359	even	30		496.2.bg.c.417.1		16
465.143	even	60		775.2.ck.a.299.1		32
465.173	even	60		775.2.ck.a.324.4		32
465.329	odd	30		775.2.bl.a.51.2		16
465.359	odd	30		775.2.bl.a.76.2		16
465.422	even	60		775.2.ck.a.299.4		32
465.452	even	60		775.2.ck.a.324.1		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.2.g.a.14.1	✓	16	93.80	odd	30
31.2.g.a.20.1	yes	16	93.50	odd	30
279.2.y.c.82.2		16	31.19	even	15
279.2.y.c.262.2		16	31.18	even	15
496.2.bg.c.113.1		16	372.143	even	30
496.2.bg.c.417.1		16	372.359	even	30
775.2.bl.a.51.2		16	465.329	odd	30
775.2.bl.a.76.2		16	465.359	odd	30
775.2.ck.a.299.1		32	465.143	even	60
775.2.ck.a.299.4		32	465.422	even	60
775.2.ck.a.324.1		32	465.452	even	60
775.2.ck.a.324.4		32	465.173	even	60
961.2.a.i.1.1		8	3.2	odd	2
961.2.a.j.1.1		8	93.92	even	2
961.2.c.i.439.1		16	93.68	even	6
961.2.c.i.521.1		16	93.26	even	6
961.2.c.j.439.1		16	93.56	odd	6
961.2.c.j.521.1		16	93.5	odd	6
961.2.d.n.531.1		16	93.29	even	10
961.2.d.n.628.1		16	93.77	even	10
961.2.d.o.531.1		16	93.2	odd	10
961.2.d.o.628.1		16	93.47	odd	10
961.2.d.p.374.4		16	93.8	odd	10
961.2.d.p.388.4		16	93.35	odd	10
961.2.d.q.374.4		16	93.23	even	10
961.2.d.q.388.4		16	93.89	even	10
961.2.g.j.844.1		16	93.83	even	30
961.2.g.j.846.1		16	93.65	even	30
961.2.g.k.844.1		16	93.41	odd	30
961.2.g.k.846.1		16	93.59	odd	30
961.2.g.l.448.1		16	93.44	even	30
961.2.g.l.547.1		16	93.74	even	30
961.2.g.m.235.2		16	93.86	even	30
961.2.g.m.732.2		16	93.53	even	30
961.2.g.n.338.2		16	93.11	even	30
961.2.g.n.816.2		16	93.17	even	30
961.2.g.s.235.2		16	93.38	odd	30
961.2.g.s.732.2		16	93.71	odd	30
961.2.g.t.338.2		16	93.20	odd	30
961.2.g.t.816.2		16	93.14	odd	30
8649.2.a.be.1.8		8	31.30	odd	2
8649.2.a.bf.1.8		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)

\(f(q)\)	\(=\)	\(q+2.69016 q^{2} +5.23694 q^{4} -0.608384 q^{5} -1.72688 q^{7} +8.70786 q^{8} -1.63665 q^{10} +1.33739 q^{11} +3.67067 q^{13} -4.64557 q^{14} +12.9516 q^{16} +2.76282 q^{17} -2.56768 q^{19} -3.18607 q^{20} +3.59779 q^{22} +0.539261 q^{23} -4.62987 q^{25} +9.87466 q^{26} -9.04356 q^{28} +8.13087 q^{29} +17.4262 q^{32} +7.43241 q^{34} +1.05061 q^{35} +7.74498 q^{37} -6.90745 q^{38} -5.29772 q^{40} +0.104052 q^{41} -3.00470 q^{43} +7.00384 q^{44} +1.45070 q^{46} +6.72498 q^{47} -4.01789 q^{49} -12.4551 q^{50} +19.2230 q^{52} -2.79875 q^{53} -0.813648 q^{55} -15.0374 q^{56} +21.8733 q^{58} -0.466233 q^{59} -5.11468 q^{61} +20.9759 q^{64} -2.23317 q^{65} +8.29847 q^{67} +14.4687 q^{68} +2.82629 q^{70} -4.75871 q^{71} +7.50619 q^{73} +20.8352 q^{74} -13.4468 q^{76} -2.30951 q^{77} -9.69896 q^{79} -7.87956 q^{80} +0.279916 q^{82} +16.6846 q^{83} -1.68085 q^{85} -8.08312 q^{86} +11.6458 q^{88} +15.3163 q^{89} -6.33879 q^{91} +2.82407 q^{92} +18.0912 q^{94} +1.56213 q^{95} -1.27918 q^{97} -10.8087 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 8 q^{4} - 3 q^{5} - 2 q^{7} + 9 q^{8} - 13 q^{10} + 18 q^{11} - 8 q^{13} + 9 q^{14} + 4 q^{16} + 14 q^{17} - 6 q^{19} + 7 q^{20} - 4 q^{22} + 22 q^{23} + 13 q^{25} + 9 q^{26} - 5 q^{28}+ \cdots - 10 q^{98}+O(q^{100}) \) 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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