LMFDB - Embedded newform 8649.2.a.bh.1.6

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,-1,0,13,-3,0,-1,-9,0,6,5,0,-4] 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:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 14x^{6} + 10x^{5} + 62x^{4} - 31x^{3} - 82x^{2} + 42x + 9 $ x^8 - x^7 - 14x^6 + 10x^5 + 62x^4 - 31x^3 - 82x^2 + 42x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.6
Root			$-1.67120$ 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)

$f(q)$	$=$	\(q+1.67120 q^{2} +0.792899 q^{4} -3.89560 q^{5} +4.51364 q^{7} -2.01730 q^{8} -6.51032 q^{10} -0.335097 q^{11} -1.74603 q^{13} +7.54317 q^{14} -4.95711 q^{16} +3.47039 q^{17} +0.211855 q^{19} -3.08882 q^{20} -0.560013 q^{22} -2.74271 q^{23} +10.1757 q^{25} -2.91796 q^{26} +3.57886 q^{28} +7.23800 q^{29} -4.24970 q^{32} +5.79970 q^{34} -17.5833 q^{35} -7.41580 q^{37} +0.354051 q^{38} +7.85861 q^{40} -6.51032 q^{41} +3.05753 q^{43} -0.265698 q^{44} -4.58361 q^{46} -3.33460 q^{47} +13.3729 q^{49} +17.0056 q^{50} -1.38442 q^{52} -12.1365 q^{53} +1.30540 q^{55} -9.10537 q^{56} +12.0961 q^{58} +1.37161 q^{59} +3.58978 q^{61} +2.81214 q^{64} +6.80184 q^{65} -1.37225 q^{67} +2.75167 q^{68} -29.3852 q^{70} +1.73742 q^{71} +13.0174 q^{73} -12.3933 q^{74} +0.167979 q^{76} -1.51250 q^{77} +8.50919 q^{79} +19.3109 q^{80} -10.8800 q^{82} +7.95742 q^{83} -13.5193 q^{85} +5.10973 q^{86} +0.675992 q^{88} +8.05400 q^{89} -7.88094 q^{91} -2.17469 q^{92} -5.57278 q^{94} -0.825302 q^{95} -3.44476 q^{97} +22.3488 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 13 q^{4} - 3 q^{5} - q^{7} - 9 q^{8} + 6 q^{10} + 5 q^{11} - 4 q^{13} - 3 q^{14} + 19 q^{16} - 2 q^{17} - 7 q^{19} - q^{20} + 9 q^{23} + 15 q^{25} + 8 q^{26} - 3 q^{28} + q^{29} - 54 q^{32}+ \cdots + 41 q^{98}+O(q^{100}) $ 8 * q - q^2 + 13 * q^4 - 3 * q^5 - q^7 - 9 * q^8 + 6 * q^10 + 5 * q^11 - 4 * q^13 - 3 * q^14 + 19 * q^16 - 2 * q^17 - 7 * q^19 - q^20 + 9 * q^23 + 15 * q^25 + 8 * q^26 - 3 * q^28 + q^29 - 54 * q^32 + 48 * q^34 - 51 * q^35 - 14 * q^37 + 16 * q^38 + 59 * q^40 + 6 * q^41 + 36 * q^43 + 27 * q^44 - 3 * q^46 - 35 * q^47 + 3 * q^49 + 37 * q^50 - 46 * q^52 - 8 * q^53 + 6 * q^55 - 52 * q^56 + 52 * q^58 + 28 * q^59 + 17 * q^64 + 3 * q^65 + 30 * q^67 + 15 * q^68 - 35 * q^70 + 6 * q^71 - q^73 + 30 * q^74 - 34 * q^76 - q^77 + 36 * q^79 - 38 * q^80 - 7 * q^82 + 22 * q^83 + 8 * q^85 + 3 * q^86 - 16 * q^88 + q^89 - 6 * q^91 + 35 * q^92 + 37 * q^94 - 50 * q^95 + 29 * q^97 + 41 * 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 $

$2$	1.67120	1.18171	0.590857	−	0.806776i	$-0.298790\pi$
$2$	1.67120	1.18171	0.590857	+	0.806776i	$0.298790\pi$
$3$	0	0
$3$	0	0
$4$	0.792899	0.396449
$5$	−3.89560	−1.74217	−0.871083	−	0.491136i	$-0.836582\pi$
$5$	−3.89560	−1.74217	−0.871083	+	0.491136i	$0.836582\pi$
$6$	0	0
$7$	4.51364	1.70599	0.852997	−	0.521916i	$-0.174783\pi$
$7$	4.51364	1.70599	0.852997	+	0.521916i	$0.174783\pi$
$8$	−2.01730	−0.713225
$9$	0	0
$10$	−6.51032	−2.05874
$11$	−0.335097	−0.101035	−0.0505177	−	0.998723i	$-0.516087\pi$
$11$	−0.335097	−0.101035	−0.0505177	+	0.998723i	$0.516087\pi$
$12$	0	0
$13$	−1.74603	−0.484261	−0.242131	−	0.970244i	$-0.577846\pi$
$13$	−1.74603	−0.484261	−0.242131	+	0.970244i	$0.577846\pi$
$14$	7.54317	2.01600
$15$	0	0
$16$	−4.95711	−1.23928
$17$	3.47039	0.841693	0.420847	−	0.907132i	$-0.361733\pi$
$17$	3.47039	0.841693	0.420847	+	0.907132i	$0.361733\pi$
$18$	0	0
$19$	0.211855	0.0486028	0.0243014	−	0.999705i	$-0.492264\pi$
$19$	0.211855	0.0486028	0.0243014	+	0.999705i	$0.492264\pi$
$20$	−3.08882	−0.690681
$21$	0	0
$22$	−0.560013	−0.119395
$23$	−2.74271	−0.571895	−0.285947	−	0.958245i	$-0.592308\pi$
$23$	−2.74271	−0.571895	−0.285947	+	0.958245i	$0.592308\pi$
$24$	0	0
$25$	10.1757	2.03514
$26$	−2.91796	−0.572259
$27$	0	0
$28$	3.57886	0.676340
$29$	7.23800	1.34406	0.672031	−	0.740523i	$-0.265422\pi$
$29$	7.23800	1.34406	0.672031	+	0.740523i	$0.265422\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	−4.24970	−0.751247
$33$	0	0
$34$	5.79970	0.994641
$35$	−17.5833	−2.97212
$36$	0	0
$37$	−7.41580	−1.21915	−0.609575	−	0.792729i	$-0.708660\pi$
$37$	−7.41580	−1.21915	−0.609575	+	0.792729i	$0.708660\pi$
$38$	0.354051	0.0574347
$39$	0	0
$40$	7.85861	1.24256
$41$	−6.51032	−1.01674	−0.508370	−	0.861139i	$-0.669752\pi$
$41$	−6.51032	−1.01674	−0.508370	+	0.861139i	$0.669752\pi$
$42$	0	0
$43$	3.05753	0.466269	0.233134	−	0.972445i	$-0.425102\pi$
$43$	3.05753	0.466269	0.233134	+	0.972445i	$0.425102\pi$
$44$	−0.265698	−0.0400555
$45$	0	0
$46$	−4.58361	−0.675816
$47$	−3.33460	−0.486402	−0.243201	−	0.969976i	$-0.578197\pi$
$47$	−3.33460	−0.486402	−0.243201	+	0.969976i	$0.578197\pi$
$48$	0	0
$49$	13.3729	1.91042
$50$	17.0056	2.40496
$51$	0	0
$52$	−1.38442	−0.191985
$53$	−12.1365	−1.66708	−0.833541	−	0.552458i	$-0.813690\pi$
$53$	−12.1365	−1.66708	−0.833541	+	0.552458i	$0.813690\pi$
$54$	0	0
$55$	1.30540	0.176021
$56$	−9.10537	−1.21676
$57$	0	0
$58$	12.0961	1.58830
$59$	1.37161	0.178568	0.0892842	−	0.996006i	$-0.471542\pi$
$59$	1.37161	0.178568	0.0892842	+	0.996006i	$0.471542\pi$
$60$	0	0
$61$	3.58978	0.459624	0.229812	−	0.973235i	$-0.426189\pi$
$61$	3.58978	0.459624	0.229812	+	0.973235i	$0.426189\pi$
$62$	0	0
$63$	0	0
$64$	2.81214	0.351517
$65$	6.80184	0.843664
$66$	0	0
$67$	−1.37225	−0.167647	−0.0838233	−	0.996481i	$-0.526713\pi$
$67$	−1.37225	−0.167647	−0.0838233	+	0.996481i	$0.526713\pi$
$68$	2.75167	0.333689
$69$	0	0
$70$	−29.3852	−3.51220
$71$	1.73742	0.206194	0.103097	−	0.994671i	$-0.467125\pi$
$71$	1.73742	0.206194	0.103097	+	0.994671i	$0.467125\pi$
$72$	0	0
$73$	13.0174	1.52357	0.761784	−	0.647832i	$-0.224324\pi$
$73$	13.0174	1.52357	0.761784	+	0.647832i	$0.224324\pi$
$74$	−12.3933	−1.44069
$75$	0	0
$76$	0.167979	0.0192686
$77$	−1.51250	−0.172366
$78$	0	0
$79$	8.50919	0.957358	0.478679	−	0.877990i	$-0.341116\pi$
$79$	8.50919	0.957358	0.478679	+	0.877990i	$0.341116\pi$
$80$	19.3109	2.15903
$81$	0	0
$82$	−10.8800	−1.20150
$83$	7.95742	0.873440	0.436720	−	0.899597i	$-0.356140\pi$
$83$	7.95742	0.873440	0.436720	+	0.899597i	$0.356140\pi$
$84$	0	0
$85$	−13.5193	−1.46637
$86$	5.10973	0.550997
$87$	0	0
$88$	0.675992	0.0720610
$89$	8.05400	0.853722	0.426861	−	0.904317i	$-0.359619\pi$
$89$	8.05400	0.853722	0.426861	+	0.904317i	$0.359619\pi$
$90$	0	0
$91$	−7.88094	−0.826147
$92$	−2.17469	−0.226727
$93$	0	0
$94$	−5.57278	−0.574788
$95$	−0.825302	−0.0846742
$96$	0	0
$97$	−3.44476	−0.349763	−0.174881	−	0.984589i	$-0.555954\pi$
$97$	−3.44476	−0.349763	−0.174881	+	0.984589i	$0.555954\pi$
$98$	22.3488	2.25757
$99$	0	0
$100$	8.06831	0.806831
$101$	0.632314	0.0629176	0.0314588	−	0.999505i	$-0.489985\pi$
$101$	0.632314	0.0629176	0.0314588	+	0.999505i	$0.489985\pi$
$102$	0	0
$103$	1.08793	0.107197	0.0535985	−	0.998563i	$-0.482931\pi$
$103$	1.08793	0.107197	0.0535985	+	0.998563i	$0.482931\pi$
$104$	3.52227	0.345387
$105$	0	0
$106$	−20.2825	−1.97001
$107$	7.97759	0.771223	0.385611	−	0.922661i	$-0.373991\pi$
$107$	7.97759	0.771223	0.385611	+	0.922661i	$0.373991\pi$
$108$	0	0
$109$	0.219454	0.0210198	0.0105099	−	0.999945i	$-0.496655\pi$
$109$	0.219454	0.0210198	0.0105099	+	0.999945i	$0.496655\pi$
$110$	2.18159	0.208006
$111$	0	0
$112$	−22.3746	−2.11420
$113$	11.9252	1.12183	0.560916	−	0.827873i	$-0.310449\pi$
$113$	11.9252	1.12183	0.560916	+	0.827873i	$0.310449\pi$
$114$	0	0
$115$	10.6845	0.996336
$116$	5.73900	0.532853
$117$	0	0
$118$	2.29223	0.211017
$119$	15.6641	1.43592
$120$	0	0
$121$	−10.8877	−0.989792
$122$	5.99922	0.543144
$123$	0	0
$124$	0	0
$125$	−20.1625	−1.80339
$126$	0	0
$127$	12.9672	1.15065	0.575324	−	0.817925i	$-0.304876\pi$
$127$	12.9672	1.15065	0.575324	+	0.817925i	$0.304876\pi$
$128$	13.1990	1.16664
$129$	0	0
$130$	11.3672	0.996970
$131$	−13.2669	−1.15914	−0.579568	−	0.814924i	$-0.696779\pi$
$131$	−13.2669	−1.15914	−0.579568	+	0.814924i	$0.696779\pi$
$132$	0	0
$133$	0.956236	0.0829161
$134$	−2.29330	−0.198111
$135$	0	0
$136$	−7.00083	−0.600316
$137$	10.5121	0.898109	0.449054	−	0.893504i	$-0.351761\pi$
$137$	10.5121	0.898109	0.449054	+	0.893504i	$0.351761\pi$
$138$	0	0
$139$	23.4907	1.99245	0.996227	−	0.0867830i	$-0.0276587\pi$
$139$	23.4907	1.99245	0.996227	+	0.0867830i	$0.0276587\pi$
$140$	−13.9418	−1.17830
$141$	0	0
$142$	2.90357	0.243662
$143$	0.585089	0.0489276
$144$	0	0
$145$	−28.1963	−2.34158
$146$	21.7546	1.80042
$147$	0	0
$148$	−5.87998	−0.483331
$149$	10.6943	0.876113	0.438056	−	0.898948i	$-0.355667\pi$
$149$	10.6943	0.876113	0.438056	+	0.898948i	$0.355667\pi$
$150$	0	0
$151$	2.81490	0.229073	0.114537	−	0.993419i	$-0.463462\pi$
$151$	2.81490	0.229073	0.114537	+	0.993419i	$0.463462\pi$
$152$	−0.427376	−0.0346647
$153$	0	0
$154$	−2.52769	−0.203687
$155$	0	0
$156$	0	0
$157$	14.6205	1.16684	0.583420	−	0.812171i	$-0.301714\pi$
$157$	14.6205	1.16684	0.583420	+	0.812171i	$0.301714\pi$
$158$	14.2205	1.13132
$159$	0	0
$160$	16.5551	1.30880
$161$	−12.3796	−0.975649
$162$	0	0
$163$	−17.2452	−1.35075	−0.675374	−	0.737476i	$-0.736018\pi$
$163$	−17.2452	−1.35075	−0.675374	+	0.737476i	$0.736018\pi$
$164$	−5.16202	−0.403086
$165$	0	0
$166$	13.2984	1.03216
$167$	0.408108	0.0315804	0.0157902	−	0.999875i	$-0.494974\pi$
$167$	0.408108	0.0315804	0.0157902	+	0.999875i	$0.494974\pi$
$168$	0	0
$169$	−9.95138	−0.765491
$170$	−22.5933	−1.73283
$171$	0	0
$172$	2.42431	0.184852
$173$	−3.74370	−0.284628	−0.142314	−	0.989822i	$-0.545454\pi$
$173$	−3.74370	−0.284628	−0.142314	+	0.989822i	$0.545454\pi$
$174$	0	0
$175$	45.9295	3.47194
$176$	1.66111	0.125211
$177$	0	0
$178$	13.4598	1.00886
$179$	−1.83178	−0.136913	−0.0684567	−	0.997654i	$-0.521808\pi$
$179$	−1.83178	−0.136913	−0.0684567	+	0.997654i	$0.521808\pi$
$180$	0	0
$181$	−11.5508	−0.858565	−0.429283	−	0.903170i	$-0.641234\pi$
$181$	−11.5508	−0.858565	−0.429283	+	0.903170i	$0.641234\pi$
$182$	−13.1706	−0.976270
$183$	0	0
$184$	5.53288	0.407889
$185$	28.8890	2.12396
$186$	0	0
$187$	−1.16292	−0.0850409
$188$	−2.64400	−0.192834
$189$	0	0
$190$	−1.37924	−0.100061
$191$	8.89041	0.643287	0.321644	−	0.946861i	$-0.395765\pi$
$191$	8.89041	0.643287	0.321644	+	0.946861i	$0.395765\pi$
$192$	0	0
$193$	−11.3286	−0.815449	−0.407725	−	0.913105i	$-0.633678\pi$
$193$	−11.3286	−0.815449	−0.407725	+	0.913105i	$0.633678\pi$
$194$	−5.75688	−0.413320
$195$	0	0
$196$	10.6034	0.757383
$197$	13.9315	0.992579	0.496289	−	0.868157i	$-0.334696\pi$
$197$	13.9315	0.992579	0.496289	+	0.868157i	$0.334696\pi$
$198$	0	0
$199$	0.446722	0.0316673	0.0158336	−	0.999875i	$-0.494960\pi$
$199$	0.446722	0.0316673	0.0158336	+	0.999875i	$0.494960\pi$
$200$	−20.5275	−1.45151
$201$	0	0
$202$	1.05672	0.0743506
$203$	32.6697	2.29296
$204$	0	0
$205$	25.3616	1.77133
$206$	1.81815	0.126676
$207$	0	0
$208$	8.65526	0.600134
$209$	−0.0709919	−0.00491061
$210$	0	0
$211$	20.4011	1.40447	0.702234	−	0.711946i	$-0.252186\pi$
$211$	20.4011	1.40447	0.702234	+	0.711946i	$0.252186\pi$
$212$	−9.62304	−0.660913
$213$	0	0
$214$	13.3321	0.911365
$215$	−11.9109	−0.812318
$216$	0	0
$217$	0	0
$218$	0.366750	0.0248394
$219$	0	0
$220$	1.03505	0.0697833
$221$	−6.05940	−0.407599
$222$	0	0
$223$	20.4372	1.36857	0.684287	−	0.729213i	$-0.260114\pi$
$223$	20.4372	1.36857	0.684287	+	0.729213i	$0.260114\pi$
$224$	−19.1816	−1.28162
$225$	0	0
$226$	19.9294	1.32568
$227$	12.1282	0.804976	0.402488	−	0.915425i	$-0.368146\pi$
$227$	12.1282	0.804976	0.402488	+	0.915425i	$0.368146\pi$
$228$	0	0
$229$	23.5742	1.55783	0.778915	−	0.627130i	$-0.215770\pi$
$229$	23.5742	1.55783	0.778915	+	0.627130i	$0.215770\pi$
$230$	17.8559	1.17738
$231$	0	0
$232$	−14.6012	−0.958618
$233$	−7.68952	−0.503757	−0.251879	−	0.967759i	$-0.581048\pi$
$233$	−7.68952	−0.503757	−0.251879	+	0.967759i	$0.581048\pi$
$234$	0	0
$235$	12.9903	0.847393
$236$	1.08755	0.0707934
$237$	0	0
$238$	26.1777	1.69685
$239$	22.3543	1.44598	0.722990	−	0.690859i	$-0.242767\pi$
$239$	22.3543	1.44598	0.722990	+	0.690859i	$0.242767\pi$
$240$	0	0
$241$	8.78690	0.566014	0.283007	−	0.959118i	$-0.408668\pi$
$241$	8.78690	0.566014	0.283007	+	0.959118i	$0.408668\pi$
$242$	−18.1955	−1.16965
$243$	0	0
$244$	2.84633	0.182218
$245$	−52.0955	−3.32826
$246$	0	0
$247$	−0.369905	−0.0235365
$248$	0	0
$249$	0	0
$250$	−33.6955	−2.13109
$251$	−9.27250	−0.585275	−0.292637	−	0.956223i	$-0.594533\pi$
$251$	−9.27250	−0.585275	−0.292637	+	0.956223i	$0.594533\pi$
$252$	0	0
$253$	0.919074	0.0577817
$254$	21.6707	1.35974
$255$	0	0
$256$	16.4339	1.02712
$257$	2.03509	0.126946	0.0634729	−	0.997984i	$-0.479782\pi$
$257$	2.03509	0.126946	0.0634729	+	0.997984i	$0.479782\pi$
$258$	0	0
$259$	−33.4722	−2.07986
$260$	5.39317	0.334470
$261$	0	0
$262$	−22.1716	−1.36977
$263$	29.6069	1.82564	0.912819	−	0.408364i	$-0.133900\pi$
$263$	29.6069	1.82564	0.912819	+	0.408364i	$0.133900\pi$
$264$	0	0
$265$	47.2791	2.90433
$266$	1.59806	0.0979832
$267$	0	0
$268$	−1.08805	−0.0664634
$269$	20.6451	1.25875	0.629376	−	0.777101i	$-0.283311\pi$
$269$	20.6451	1.25875	0.629376	+	0.777101i	$0.283311\pi$
$270$	0	0
$271$	6.46297	0.392598	0.196299	−	0.980544i	$-0.437108\pi$
$271$	6.46297	0.392598	0.196299	+	0.980544i	$0.437108\pi$
$272$	−17.2031	−1.04309
$273$	0	0
$274$	17.5678	1.06131
$275$	−3.40985	−0.205622
$276$	0	0
$277$	9.45301	0.567976	0.283988	−	0.958828i	$-0.408342\pi$
$277$	9.45301	0.567976	0.283988	+	0.958828i	$0.408342\pi$
$278$	39.2576	2.35451
$279$	0	0
$280$	35.4709	2.11979
$281$	2.27165	0.135515	0.0677575	−	0.997702i	$-0.478416\pi$
$281$	2.27165	0.135515	0.0677575	+	0.997702i	$0.478416\pi$
$282$	0	0
$283$	−27.1242	−1.61237	−0.806183	−	0.591667i	$-0.798470\pi$
$283$	−27.1242	−1.61237	−0.806183	+	0.591667i	$0.798470\pi$
$284$	1.37760	0.0817454
$285$	0	0
$286$	0.977799	0.0578184
$287$	−29.3852	−1.73455
$288$	0	0
$289$	−4.95640	−0.291553
$290$	−47.1216	−2.76708
$291$	0	0
$292$	10.3215	0.604017
$293$	−9.17316	−0.535902	−0.267951	−	0.963433i	$-0.586347\pi$
$293$	−9.17316	−0.535902	−0.267951	+	0.963433i	$0.586347\pi$
$294$	0	0
$295$	−5.34325	−0.311096
$296$	14.9599	0.869527
$297$	0	0
$298$	17.8723	1.03532
$299$	4.78885	0.276947
$300$	0	0
$301$	13.8006	0.795452
$302$	4.70425	0.270699
$303$	0	0
$304$	−1.05019	−0.0602324
$305$	−13.9843	−0.800741
$306$	0	0
$307$	−9.06925	−0.517609	−0.258805	−	0.965930i	$-0.583329\pi$
$307$	−9.06925	−0.517609	−0.258805	+	0.965930i	$0.583329\pi$
$308$	−1.19926	−0.0683344
$309$	0	0
$310$	0	0
$311$	11.8965	0.674587	0.337293	−	0.941400i	$-0.390489\pi$
$311$	11.8965	0.674587	0.337293	+	0.941400i	$0.390489\pi$
$312$	0	0
$313$	−15.6841	−0.886516	−0.443258	−	0.896394i	$-0.646177\pi$
$313$	−15.6841	−0.886516	−0.443258	+	0.896394i	$0.646177\pi$
$314$	24.4337	1.37887
$315$	0	0
$316$	6.74692	0.379544
$317$	15.0304	0.844194	0.422097	−	0.906551i	$-0.361294\pi$
$317$	15.0304	0.844194	0.422097	+	0.906551i	$0.361294\pi$
$318$	0	0
$319$	−2.42543	−0.135798
$320$	−10.9550	−0.612401
$321$	0	0
$322$	−20.6887	−1.15294
$323$	0.735219	0.0409087
$324$	0	0
$325$	−17.7671	−0.985541
$326$	−28.8201	−1.59620
$327$	0	0
$328$	13.1333	0.725164
$329$	−15.0512	−0.829799
$330$	0	0
$331$	13.5963	0.747318	0.373659	−	0.927566i	$-0.378103\pi$
$331$	13.5963	0.747318	0.373659	+	0.927566i	$0.378103\pi$
$332$	6.30943	0.346275
$333$	0	0
$334$	0.682029	0.0373190
$335$	5.34573	0.292068
$336$	0	0
$337$	−10.3562	−0.564136	−0.282068	−	0.959394i	$-0.591020\pi$
$337$	−10.3562	−0.564136	−0.282068	+	0.959394i	$0.591020\pi$
$338$	−16.6307	−0.904592
$339$	0	0
$340$	−10.7194	−0.581341
$341$	0	0
$342$	0	0
$343$	28.7650	1.55316
$344$	−6.16796	−0.332554
$345$	0	0
$346$	−6.25645	−0.336349
$347$	2.52600	0.135603	0.0678013	−	0.997699i	$-0.478402\pi$
$347$	2.52600	0.135603	0.0678013	+	0.997699i	$0.478402\pi$
$348$	0	0
$349$	34.8988	1.86809	0.934046	−	0.357153i	$-0.116252\pi$
$349$	34.8988	1.86809	0.934046	+	0.357153i	$0.116252\pi$
$350$	76.7572	4.10284
$351$	0	0
$352$	1.42406	0.0759027
$353$	−9.14039	−0.486494	−0.243247	−	0.969964i	$-0.578213\pi$
$353$	−9.14039	−0.486494	−0.243247	+	0.969964i	$0.578213\pi$
$354$	0	0
$355$	−6.76830	−0.359224
$356$	6.38601	0.338458
$357$	0	0
$358$	−3.06126	−0.161793
$359$	4.71581	0.248891	0.124445	−	0.992226i	$-0.460285\pi$
$359$	4.71581	0.248891	0.124445	+	0.992226i	$0.460285\pi$
$360$	0	0
$361$	−18.9551	−0.997638
$362$	−19.3037	−1.01458
$363$	0	0
$364$	−6.24879	−0.327526
$365$	−50.7105	−2.65431
$366$	0	0
$367$	−24.4353	−1.27551	−0.637756	−	0.770238i	$-0.720137\pi$
$367$	−24.4353	−1.27551	−0.637756	+	0.770238i	$0.720137\pi$
$368$	13.5959	0.708736
$369$	0	0
$370$	48.2792	2.50992
$371$	−54.7799	−2.84403
$372$	0	0
$373$	−3.54248	−0.183422	−0.0917112	−	0.995786i	$-0.529234\pi$
$373$	−3.54248	−0.183422	−0.0917112	+	0.995786i	$0.529234\pi$
$374$	−1.94346	−0.100494
$375$	0	0
$376$	6.72691	0.346914
$377$	−12.6378	−0.650877
$378$	0	0
$379$	24.1617	1.24111	0.620553	−	0.784165i	$-0.286908\pi$
$379$	24.1617	1.24111	0.620553	+	0.784165i	$0.286908\pi$
$380$	−0.654381	−0.0335690
$381$	0	0
$382$	14.8576	0.760182
$383$	−1.18780	−0.0606935	−0.0303468	−	0.999539i	$-0.509661\pi$
$383$	−1.18780	−0.0606935	−0.0303468	+	0.999539i	$0.509661\pi$
$384$	0	0
$385$	5.89212	0.300290
$386$	−18.9323	−0.963628
$387$	0	0
$388$	−2.73135	−0.138663
$389$	−18.2656	−0.926102	−0.463051	−	0.886332i	$-0.653245\pi$
$389$	−18.2656	−0.926102	−0.463051	+	0.886332i	$0.653245\pi$
$390$	0	0
$391$	−9.51827	−0.481360
$392$	−26.9772	−1.36256
$393$	0	0
$394$	23.2823	1.17294
$395$	−33.1484	−1.66788
$396$	0	0
$397$	−2.65218	−0.133109	−0.0665546	−	0.997783i	$-0.521201\pi$
$397$	−2.65218	−0.133109	−0.0665546	+	0.997783i	$0.521201\pi$
$398$	0.746560	0.0374217
$399$	0	0
$400$	−50.4421	−2.52211
$401$	16.6033	0.829131	0.414566	−	0.910019i	$-0.363934\pi$
$401$	16.6033	0.829131	0.414566	+	0.910019i	$0.363934\pi$
$402$	0	0
$403$	0	0
$404$	0.501361	0.0249436
$405$	0	0
$406$	54.5975	2.70963
$407$	2.48501	0.123177
$408$	0	0
$409$	25.6723	1.26942	0.634708	−	0.772752i	$-0.281120\pi$
$409$	25.6723	1.26942	0.634708	+	0.772752i	$0.281120\pi$
$410$	42.3842	2.09321
$411$	0	0
$412$	0.862619	0.0424982
$413$	6.19095	0.304637
$414$	0	0
$415$	−30.9989	−1.52168
$416$	7.42010	0.363800
$417$	0	0
$418$	−0.118641	−0.00580294
$419$	−31.1576	−1.52215	−0.761075	−	0.648664i	$-0.775328\pi$
$419$	−31.1576	−1.52215	−0.761075	+	0.648664i	$0.775328\pi$
$420$	0	0
$421$	−30.5518	−1.48900	−0.744502	−	0.667620i	$-0.767313\pi$
$421$	−30.5518	−1.48900	−0.744502	+	0.667620i	$0.767313\pi$
$422$	34.0942	1.65968
$423$	0	0
$424$	24.4831	1.18900
$425$	35.3137	1.71297
$426$	0	0
$427$	16.2029	0.784115
$428$	6.32542	0.305751
$429$	0	0
$430$	−19.9055	−0.959928
$431$	36.6717	1.76641	0.883206	−	0.468986i	$-0.155380\pi$
$431$	36.6717	1.76641	0.883206	+	0.468986i	$0.155380\pi$
$432$	0	0
$433$	−13.7562	−0.661083	−0.330541	−	0.943791i	$-0.607231\pi$
$433$	−13.7562	−0.661083	−0.330541	+	0.943791i	$0.607231\pi$
$434$	0	0
$435$	0	0
$436$	0.174004	0.00833330
$437$	−0.581057	−0.0277957
$438$	0	0
$439$	−5.74749	−0.274313	−0.137156	−	0.990549i	$-0.543796\pi$
$439$	−5.74749	−0.274313	−0.137156	+	0.990549i	$0.543796\pi$
$440$	−2.63340	−0.125542
$441$	0	0
$442$	−10.1265	−0.481666
$443$	−32.0745	−1.52390	−0.761952	−	0.647633i	$-0.775759\pi$
$443$	−32.0745	−1.52390	−0.761952	+	0.647633i	$0.775759\pi$
$444$	0	0
$445$	−31.3752	−1.48733
$446$	34.1545	1.61726
$447$	0	0
$448$	12.6930	0.599686
$449$	22.5738	1.06532	0.532662	−	0.846328i	$-0.321192\pi$
$449$	22.5738	1.06532	0.532662	+	0.846328i	$0.321192\pi$
$450$	0	0
$451$	2.18159	0.102727
$452$	9.45550	0.444749
$453$	0	0
$454$	20.2686	0.951252
$455$	30.7010	1.43929
$456$	0	0
$457$	8.28399	0.387509	0.193754	−	0.981050i	$-0.437934\pi$
$457$	8.28399	0.387509	0.193754	+	0.981050i	$0.437934\pi$
$458$	39.3972	1.84091
$459$	0	0
$460$	8.47173	0.394997
$461$	15.7988	0.735822	0.367911	−	0.929861i	$-0.380073\pi$
$461$	15.7988	0.735822	0.367911	+	0.929861i	$0.380073\pi$
$462$	0	0
$463$	−14.2339	−0.661506	−0.330753	−	0.943717i	$-0.607303\pi$
$463$	−14.2339	−0.661506	−0.330753	+	0.943717i	$0.607303\pi$
$464$	−35.8795	−1.66567
$465$	0	0
$466$	−12.8507	−0.595298
$467$	−26.2440	−1.21443	−0.607213	−	0.794539i	$-0.707713\pi$
$467$	−26.2440	−1.21443	−0.607213	+	0.794539i	$0.707713\pi$
$468$	0	0
$469$	−6.19382	−0.286004
$470$	21.7093	1.00138
$471$	0	0
$472$	−2.76695	−0.127359
$473$	−1.02457	−0.0471097
$474$	0	0
$475$	2.15577	0.0989137
$476$	12.4200	0.569271
$477$	0	0
$478$	37.3584	1.70874
$479$	41.7540	1.90779	0.953895	−	0.300141i	$-0.0970338\pi$
$479$	41.7540	1.90779	0.953895	+	0.300141i	$0.0970338\pi$
$480$	0	0
$481$	12.9482	0.590387
$482$	14.6846	0.668867
$483$	0	0
$484$	−8.63285	−0.392402
$485$	13.4194	0.609345
$486$	0	0
$487$	−34.5409	−1.56520	−0.782598	−	0.622527i	$-0.786106\pi$
$487$	−34.5409	−1.56520	−0.782598	+	0.622527i	$0.786106\pi$
$488$	−7.24167	−0.327815
$489$	0	0
$490$	−87.0619	−3.93305
$491$	−13.4986	−0.609183	−0.304592	−	0.952483i	$-0.598520\pi$
$491$	−13.4986	−0.609183	−0.304592	+	0.952483i	$0.598520\pi$
$492$	0	0
$493$	25.1187	1.13129
$494$	−0.618184	−0.0278134
$495$	0	0
$496$	0	0
$497$	7.84208	0.351766
$498$	0	0
$499$	−37.7824	−1.69137	−0.845687	−	0.533679i	$-0.820809\pi$
$499$	−37.7824	−1.69137	−0.845687	+	0.533679i	$0.820809\pi$
$500$	−15.9868	−0.714953
$501$	0	0
$502$	−15.4962	−0.691628
$503$	−5.37870	−0.239824	−0.119912	−	0.992785i	$-0.538261\pi$
$503$	−5.37870	−0.239824	−0.119912	+	0.992785i	$0.538261\pi$
$504$	0	0
$505$	−2.46324	−0.109613
$506$	1.53595	0.0682814
$507$	0	0
$508$	10.2816	0.456174
$509$	18.4656	0.818475	0.409237	−	0.912428i	$-0.365795\pi$
$509$	18.4656	0.818475	0.409237	+	0.912428i	$0.365795\pi$
$510$	0	0
$511$	58.7556	2.59920
$512$	1.06623	0.0471210
$513$	0	0
$514$	3.40104	0.150014
$515$	−4.23815	−0.186755
$516$	0	0
$517$	1.11742	0.0491439
$518$	−55.9387	−2.45780
$519$	0	0
$520$	−13.7214	−0.601722
$521$	20.3286	0.890610	0.445305	−	0.895379i	$-0.353095\pi$
$521$	20.3286	0.890610	0.445305	+	0.895379i	$0.353095\pi$
$522$	0	0
$523$	−5.37216	−0.234908	−0.117454	−	0.993078i	$-0.537473\pi$
$523$	−5.37216	−0.234908	−0.117454	+	0.993078i	$0.537473\pi$
$524$	−10.5193	−0.459539
$525$	0	0
$526$	49.4789	2.15738
$527$	0	0
$528$	0	0
$529$	−15.4775	−0.672936
$530$	79.0127	3.43209
$531$	0	0
$532$	0.758198	0.0328721
$533$	11.3672	0.492368
$534$	0	0
$535$	−31.0775	−1.34360
$536$	2.76824	0.119570
$537$	0	0
$538$	34.5020	1.48748
$539$	−4.48122	−0.193020
$540$	0	0
$541$	29.3978	1.26391	0.631954	−	0.775006i	$-0.282253\pi$
$541$	29.3978	1.26391	0.631954	+	0.775006i	$0.282253\pi$
$542$	10.8009	0.463938
$543$	0	0
$544$	−14.7481	−0.632320
$545$	−0.854904	−0.0366200
$546$	0	0
$547$	−37.7150	−1.61258	−0.806288	−	0.591523i	$-0.798527\pi$
$547$	−37.7150	−1.61258	−0.806288	+	0.591523i	$0.798527\pi$
$548$	8.33502	0.356055
$549$	0	0
$550$	−5.69853	−0.242986
$551$	1.53340	0.0653252
$552$	0	0
$553$	38.4074	1.63325
$554$	15.7978	0.671186
$555$	0	0
$556$	18.6257	0.789907
$557$	−2.12741	−0.0901411	−0.0450706	−	0.998984i	$-0.514351\pi$
$557$	−2.12741	−0.0901411	−0.0450706	+	0.998984i	$0.514351\pi$
$558$	0	0
$559$	−5.33854	−0.225796
$560$	87.1625	3.68329
$561$	0	0
$562$	3.79637	0.160140
$563$	15.4591	0.651523	0.325761	−	0.945452i	$-0.394379\pi$
$563$	15.4591	0.651523	0.325761	+	0.945452i	$0.394379\pi$
$564$	0	0
$565$	−46.4559	−1.95442
$566$	−45.3299	−1.90536
$567$	0	0
$568$	−3.50491	−0.147063
$569$	−43.4928	−1.82331	−0.911656	−	0.410954i	$-0.865196\pi$
$569$	−43.4928	−1.82331	−0.911656	+	0.410954i	$0.865196\pi$
$570$	0	0
$571$	9.60615	0.402005	0.201002	−	0.979591i	$-0.435580\pi$
$571$	9.60615	0.402005	0.201002	+	0.979591i	$0.435580\pi$
$572$	0.463916	0.0193973
$573$	0	0
$574$	−49.1085	−2.04975
$575$	−27.9090	−1.16389
$576$	0	0
$577$	45.8418	1.90842	0.954210	−	0.299139i	$-0.0966993\pi$
$577$	45.8418	1.90842	0.954210	+	0.299139i	$0.0966993\pi$
$578$	−8.28312	−0.344532
$579$	0	0
$580$	−22.3568	−0.928318
$581$	35.9169	1.49008
$582$	0	0
$583$	4.06691	0.168434
$584$	−26.2600	−1.08665
$585$	0	0
$586$	−15.3302	−0.633283
$587$	0.304600	0.0125722	0.00628610	−	0.999980i	$-0.497999\pi$
$587$	0.304600	0.0125722	0.00628610	+	0.999980i	$0.497999\pi$
$588$	0	0
$589$	0	0
$590$	−8.92962	−0.367627
$591$	0	0
$592$	36.7609	1.51086
$593$	−32.9188	−1.35181	−0.675906	−	0.736988i	$-0.736247\pi$
$593$	−32.9188	−1.35181	−0.675906	+	0.736988i	$0.736247\pi$
$594$	0	0
$595$	−61.0210	−2.50162
$596$	8.47951	0.347334
$597$	0	0
$598$	8.00312	0.327272
$599$	−0.464587	−0.0189825	−0.00949125	−	0.999955i	$-0.503021\pi$
$599$	−0.464587	−0.0189825	−0.00949125	+	0.999955i	$0.503021\pi$
$600$	0	0
$601$	4.23305	0.172670	0.0863349	−	0.996266i	$-0.472484\pi$
$601$	4.23305	0.172670	0.0863349	+	0.996266i	$0.472484\pi$
$602$	23.0635	0.939997
$603$	0	0
$604$	2.23193	0.0908160
$605$	42.4142	1.72438
$606$	0	0
$607$	−16.3205	−0.662430	−0.331215	−	0.943555i	$-0.607459\pi$
$607$	−16.3205	−0.662430	−0.331215	+	0.943555i	$0.607459\pi$
$608$	−0.900319	−0.0365128
$609$	0	0
$610$	−23.3706	−0.946247
$611$	5.82232	0.235546
$612$	0	0
$613$	−19.5044	−0.787773	−0.393887	−	0.919159i	$-0.628870\pi$
$613$	−19.5044	−0.787773	−0.393887	+	0.919159i	$0.628870\pi$
$614$	−15.1565	−0.611666
$615$	0	0
$616$	3.05118	0.122936
$617$	3.39753	0.136779	0.0683897	−	0.997659i	$-0.478214\pi$
$617$	3.39753	0.136779	0.0683897	+	0.997659i	$0.478214\pi$
$618$	0	0
$619$	−26.4142	−1.06168	−0.530839	−	0.847473i	$-0.678123\pi$
$619$	−26.4142	−1.06168	−0.530839	+	0.847473i	$0.678123\pi$
$620$	0	0
$621$	0	0
$622$	19.8813	0.797169
$623$	36.3528	1.45644
$624$	0	0
$625$	27.6666	1.10666
$626$	−26.2112	−1.04761
$627$	0	0
$628$	11.5925	0.462593
$629$	−25.7357	−1.02615
$630$	0	0
$631$	−27.5230	−1.09567	−0.547836	−	0.836586i	$-0.684548\pi$
$631$	−27.5230	−1.09567	−0.547836	+	0.836586i	$0.684548\pi$
$632$	−17.1656	−0.682811
$633$	0	0
$634$	25.1188	0.997596
$635$	−50.5149	−2.00462
$636$	0	0
$637$	−23.3495	−0.925141
$638$	−4.05337	−0.160474
$639$	0	0
$640$	−51.4182	−2.03248
$641$	8.91667	0.352187	0.176094	−	0.984373i	$-0.443654\pi$
$641$	8.91667	0.352187	0.176094	+	0.984373i	$0.443654\pi$
$642$	0	0
$643$	−12.6584	−0.499199	−0.249600	−	0.968349i	$-0.580299\pi$
$643$	−12.6584	−0.499199	−0.249600	+	0.968349i	$0.580299\pi$
$644$	−9.81577	−0.386795
$645$	0	0
$646$	1.22870	0.0483424
$647$	29.0256	1.14111	0.570556	−	0.821259i	$-0.306728\pi$
$647$	29.0256	1.14111	0.570556	+	0.821259i	$0.306728\pi$
$648$	0	0
$649$	−0.459622	−0.0180418
$650$	−29.6923	−1.16463
$651$	0	0
$652$	−13.6737	−0.535503
$653$	−32.4706	−1.27067	−0.635337	−	0.772235i	$-0.719139\pi$
$653$	−32.4706	−1.27067	−0.635337	+	0.772235i	$0.719139\pi$
$654$	0	0
$655$	51.6826	2.01941
$656$	32.2724	1.26002
$657$	0	0
$658$	−25.1535	−0.980586
$659$	−9.23568	−0.359771	−0.179886	−	0.983688i	$-0.557573\pi$
$659$	−9.23568	−0.359771	−0.179886	+	0.983688i	$0.557573\pi$
$660$	0	0
$661$	−21.1259	−0.821704	−0.410852	−	0.911702i	$-0.634769\pi$
$661$	−21.1259	−0.821704	−0.410852	+	0.911702i	$0.634769\pi$
$662$	22.7220	0.883116
$663$	0	0
$664$	−16.0525	−0.622959
$665$	−3.72511	−0.144454
$666$	0	0
$667$	−19.8517	−0.768662
$668$	0.323588	0.0125200
$669$	0	0
$670$	8.93377	0.345141
$671$	−1.20292	−0.0464383
$672$	0	0
$673$	40.4793	1.56036	0.780181	−	0.625554i	$-0.215127\pi$
$673$	40.4793	1.56036	0.780181	+	0.625554i	$0.215127\pi$
$674$	−17.3072	−0.666648
$675$	0	0
$676$	−7.89044	−0.303478
$677$	−16.5546	−0.636245	−0.318123	−	0.948050i	$-0.603052\pi$
$677$	−16.5546	−0.636245	−0.318123	+	0.948050i	$0.603052\pi$
$678$	0	0
$679$	−15.5484	−0.596693
$680$	27.2724	1.04585
$681$	0	0
$682$	0	0
$683$	15.8493	0.606455	0.303228	−	0.952918i	$-0.401936\pi$
$683$	15.8493	0.606455	0.303228	+	0.952918i	$0.401936\pi$
$684$	0	0
$685$	−40.9509	−1.56465
$686$	48.0719	1.83540
$687$	0	0
$688$	−15.1565	−0.577836
$689$	21.1907	0.807303
$690$	0	0
$691$	0.273282	0.0103961	0.00519807	−	0.999986i	$-0.498345\pi$
$691$	0.273282	0.0103961	0.00519807	+	0.999986i	$0.498345\pi$
$692$	−2.96837	−0.112841
$693$	0	0
$694$	4.22144	0.160244
$695$	−91.5104	−3.47119
$696$	0	0
$697$	−22.5933	−0.855784
$698$	58.3228	2.20755
$699$	0	0
$700$	36.4174	1.37645
$701$	22.4179	0.846711	0.423356	−	0.905964i	$-0.360852\pi$
$701$	22.4179	0.846711	0.423356	+	0.905964i	$0.360852\pi$
$702$	0	0
$703$	−1.57107	−0.0592541
$704$	−0.942338	−0.0355157
$705$	0	0
$706$	−15.2754	−0.574897
$707$	2.85403	0.107337
$708$	0	0
$709$	14.8070	0.556090	0.278045	−	0.960568i	$-0.410314\pi$
$709$	14.8070	0.556090	0.278045	+	0.960568i	$0.410314\pi$
$710$	−11.3112	−0.424500
$711$	0	0
$712$	−16.2474	−0.608896
$713$	0	0
$714$	0	0
$715$	−2.27927	−0.0852400
$716$	−1.45241	−0.0542793
$717$	0	0
$718$	7.88105	0.294118
$719$	25.1086	0.936392	0.468196	−	0.883625i	$-0.344904\pi$
$719$	25.1086	0.936392	0.468196	+	0.883625i	$0.344904\pi$
$720$	0	0
$721$	4.91052	0.182877
$722$	−31.6777	−1.17892
$723$	0	0
$724$	−9.15863	−0.340378
$725$	73.6518	2.73536
$726$	0	0
$727$	1.32303	0.0490686	0.0245343	−	0.999699i	$-0.492190\pi$
$727$	1.32303	0.0490686	0.0245343	+	0.999699i	$0.492190\pi$
$728$	15.8983	0.589228
$729$	0	0
$730$	−84.7472	−3.13663
$731$	10.6108	0.392455
$732$	0	0
$733$	21.3453	0.788408	0.394204	−	0.919023i	$-0.371020\pi$
$733$	21.3453	0.788408	0.394204	+	0.919023i	$0.371020\pi$
$734$	−40.8362	−1.50729
$735$	0	0
$736$	11.6557	0.429635
$737$	0.459836	0.0169383
$738$	0	0
$739$	−22.1603	−0.815178	−0.407589	−	0.913165i	$-0.633630\pi$
$739$	−22.1603	−0.815178	−0.407589	+	0.913165i	$0.633630\pi$
$740$	22.9060	0.842043
$741$	0	0
$742$	−91.5480	−3.36083
$743$	29.6842	1.08901	0.544503	−	0.838759i	$-0.316718\pi$
$743$	29.6842	1.08901	0.544503	+	0.838759i	$0.316718\pi$
$744$	0	0
$745$	−41.6608	−1.52633
$746$	−5.92018	−0.216753
$747$	0	0
$748$	−0.922075	−0.0337144
$749$	36.0079	1.31570
$750$	0	0
$751$	15.5930	0.568996	0.284498	−	0.958677i	$-0.408173\pi$
$751$	15.5930	0.568996	0.284498	+	0.958677i	$0.408173\pi$
$752$	16.5300	0.602787
$753$	0	0
$754$	−21.1202	−0.769151
$755$	−10.9657	−0.399084
$756$	0	0
$757$	21.2045	0.770691	0.385345	−	0.922772i	$-0.374082\pi$
$757$	21.2045	0.770691	0.385345	+	0.922772i	$0.374082\pi$
$758$	40.3790	1.46663
$759$	0	0
$760$	1.66488	0.0603917
$761$	24.4271	0.885481	0.442740	−	0.896650i	$-0.354006\pi$
$761$	24.4271	0.885481	0.442740	+	0.896650i	$0.354006\pi$
$762$	0	0
$763$	0.990533	0.0358597
$764$	7.04920	0.255031
$765$	0	0
$766$	−1.98504	−0.0717224
$767$	−2.39487	−0.0864738
$768$	0	0
$769$	23.9568	0.863903	0.431952	−	0.901897i	$-0.357825\pi$
$769$	23.9568	0.863903	0.431952	+	0.901897i	$0.357825\pi$
$770$	9.84689	0.354857
$771$	0	0
$772$	−8.98242	−0.323284
$773$	−45.1977	−1.62565	−0.812825	−	0.582508i	$-0.802071\pi$
$773$	−45.1977	−1.62565	−0.812825	+	0.582508i	$0.802071\pi$
$774$	0	0
$775$	0	0
$776$	6.94914	0.249459
$777$	0	0
$778$	−30.5254	−1.09439
$779$	−1.37924	−0.0494165
$780$	0	0
$781$	−0.582204	−0.0208329
$782$	−15.9069	−0.568830
$783$	0	0
$784$	−66.2910	−2.36753
$785$	−56.9555	−2.03283
$786$	0	0
$787$	−19.8996	−0.709346	−0.354673	−	0.934990i	$-0.615408\pi$
$787$	−19.8996	−0.709346	−0.354673	+	0.934990i	$0.615408\pi$
$788$	11.0463	0.393507
$789$	0	0
$790$	−55.3975	−1.97095
$791$	53.8261	1.91384
$792$	0	0
$793$	−6.26785	−0.222578
$794$	−4.43232	−0.157297
$795$	0	0
$796$	0.354205	0.0125545
$797$	41.1645	1.45812	0.729060	−	0.684449i	$-0.239957\pi$
$797$	41.1645	1.45812	0.729060	+	0.684449i	$0.239957\pi$
$798$	0	0
$799$	−11.5724	−0.409401
$800$	−43.2437	−1.52890
$801$	0	0
$802$	27.7474	0.979796
$803$	−4.36208	−0.153934
$804$	0	0
$805$	48.2260	1.69974
$806$	0	0
$807$	0	0
$808$	−1.27557	−0.0448744
$809$	−44.0715	−1.54947	−0.774736	−	0.632285i	$-0.782117\pi$
$809$	−44.0715	−1.54947	−0.774736	+	0.632285i	$0.782117\pi$
$810$	0	0
$811$	24.0093	0.843079	0.421539	−	0.906810i	$-0.361490\pi$
$811$	24.0093	0.843079	0.421539	+	0.906810i	$0.361490\pi$
$812$	25.9037	0.909043
$813$	0	0
$814$	4.15294	0.145561
$815$	67.1804	2.35323
$816$	0	0
$817$	0.647752	0.0226620
$818$	42.9035	1.50009
$819$	0	0
$820$	20.1092	0.702243
$821$	−41.8591	−1.46089	−0.730447	−	0.682970i	$-0.760688\pi$
$821$	−41.8591	−1.46089	−0.730447	+	0.682970i	$0.760688\pi$
$822$	0	0
$823$	41.7357	1.45481	0.727407	−	0.686206i	$-0.240725\pi$
$823$	41.7357	1.45481	0.727407	+	0.686206i	$0.240725\pi$
$824$	−2.19469	−0.0764555
$825$	0	0
$826$	10.3463	0.359994
$827$	−18.0321	−0.627039	−0.313520	−	0.949582i	$-0.601508\pi$
$827$	−18.0321	−0.627039	−0.313520	+	0.949582i	$0.601508\pi$
$828$	0	0
$829$	12.6733	0.440162	0.220081	−	0.975482i	$-0.429368\pi$
$829$	12.6733	0.440162	0.220081	+	0.975482i	$0.429368\pi$
$830$	−51.8053	−1.79819
$831$	0	0
$832$	−4.91007	−0.170226
$833$	46.4092	1.60798
$834$	0	0
$835$	−1.58983	−0.0550182
$836$	−0.0562894	−0.00194681
$837$	0	0
$838$	−52.0705	−1.79875
$839$	43.0991	1.48795	0.743974	−	0.668209i	$-0.232939\pi$
$839$	43.0991	1.48795	0.743974	+	0.668209i	$0.232939\pi$
$840$	0	0
$841$	23.3886	0.806503
$842$	−51.0581	−1.75958
$843$	0	0
$844$	16.1760	0.556800
$845$	38.7666	1.33361
$846$	0	0
$847$	−49.1432	−1.68858
$848$	60.1621	2.06598
$849$	0	0
$850$	59.0161	2.02424
$851$	20.3394	0.697225
$852$	0	0
$853$	44.8847	1.53682	0.768411	−	0.639956i	$-0.221048\pi$
$853$	44.8847	1.53682	0.768411	+	0.639956i	$0.221048\pi$
$854$	27.0783	0.926600
$855$	0	0
$856$	−16.0932	−0.550055
$857$	−14.2215	−0.485796	−0.242898	−	0.970052i	$-0.578098\pi$
$857$	−14.2215	−0.485796	−0.242898	+	0.970052i	$0.578098\pi$
$858$	0	0
$859$	−23.5553	−0.803698	−0.401849	−	0.915706i	$-0.631632\pi$
$859$	−23.5553	−0.803698	−0.401849	+	0.915706i	$0.631632\pi$
$860$	−9.44415	−0.322043
$861$	0	0
$862$	61.2856	2.08739
$863$	−33.9831	−1.15680	−0.578400	−	0.815754i	$-0.696323\pi$
$863$	−33.9831	−1.15680	−0.578400	+	0.815754i	$0.696323\pi$
$864$	0	0
$865$	14.5840	0.495869
$866$	−22.9894	−0.781211
$867$	0	0
$868$	0	0
$869$	−2.85140	−0.0967272
$870$	0	0
$871$	2.39598	0.0811848
$872$	−0.442704	−0.0149919
$873$	0	0
$874$	−0.971060	−0.0328466
$875$	−91.0062	−3.07657
$876$	0	0
$877$	−42.3591	−1.43036	−0.715182	−	0.698938i	$-0.753656\pi$
$877$	−42.3591	−1.43036	−0.715182	+	0.698938i	$0.753656\pi$
$878$	−9.60519	−0.324160
$879$	0	0
$880$	−6.47103	−0.218138
$881$	33.4197	1.12594	0.562970	−	0.826478i	$-0.309659\pi$
$881$	33.4197	1.12594	0.562970	+	0.826478i	$0.309659\pi$
$882$	0	0
$883$	−16.1834	−0.544616	−0.272308	−	0.962210i	$-0.587787\pi$
$883$	−16.1834	−0.544616	−0.272308	+	0.962210i	$0.587787\pi$
$884$	−4.80449	−0.161593
$885$	0	0
$886$	−53.6028	−1.80082
$887$	43.0207	1.44449	0.722247	−	0.691635i	$-0.243109\pi$
$887$	43.0207	1.44449	0.722247	+	0.691635i	$0.243109\pi$
$888$	0	0
$889$	58.5290	1.96300
$890$	−52.4341	−1.75759
$891$	0	0
$892$	16.2046	0.542570
$893$	−0.706452	−0.0236405
$894$	0	0
$895$	7.13588	0.238526
$896$	59.5756	1.99028
$897$	0	0
$898$	37.7253	1.25891
$899$	0	0
$900$	0	0
$901$	−42.1185	−1.40317
$902$	3.64586	0.121394
$903$	0	0
$904$	−24.0568	−0.800117
$905$	44.9974	1.49576
$906$	0	0
$907$	18.4721	0.613354	0.306677	−	0.951814i	$-0.400783\pi$
$907$	18.4721	0.613354	0.306677	+	0.951814i	$0.400783\pi$
$908$	9.61642	0.319132
$909$	0	0
$910$	51.3074	1.70082
$911$	−20.8824	−0.691864	−0.345932	−	0.938260i	$-0.612437\pi$
$911$	−20.8824	−0.691864	−0.345932	+	0.938260i	$0.612437\pi$
$912$	0	0
$913$	−2.66651	−0.0882485
$914$	13.8442	0.457925
$915$	0	0
$916$	18.6920	0.617601
$917$	−59.8820	−1.97748
$918$	0	0
$919$	−14.9218	−0.492225	−0.246112	−	0.969241i	$-0.579153\pi$
$919$	−14.9218	−0.492225	−0.246112	+	0.969241i	$0.579153\pi$
$920$	−21.5539	−0.710611
$921$	0	0
$922$	26.4028	0.869531
$923$	−3.03359	−0.0998518
$924$	0	0
$925$	−75.4610	−2.48114
$926$	−23.7877	−0.781712
$927$	0	0
$928$	−30.7593	−1.00972
$929$	19.8663	0.651793	0.325897	−	0.945405i	$-0.394334\pi$
$929$	19.8663	0.651793	0.325897	+	0.945405i	$0.394334\pi$
$930$	0	0
$931$	2.83312	0.0928516
$932$	−6.09701	−0.199714
$933$	0	0
$934$	−43.8588	−1.43511
$935$	4.53026	0.148155
$936$	0	0
$937$	10.1006	0.329973	0.164986	−	0.986296i	$-0.447242\pi$
$937$	10.1006	0.329973	0.164986	+	0.986296i	$0.447242\pi$
$938$	−10.3511	−0.337975
$939$	0	0
$940$	10.3000	0.335949
$941$	12.6891	0.413651	0.206826	−	0.978378i	$-0.433687\pi$
$941$	12.6891	0.413651	0.206826	+	0.978378i	$0.433687\pi$
$942$	0	0
$943$	17.8559	0.581469
$944$	−6.79922	−0.221296
$945$	0	0
$946$	−1.71226	−0.0556702
$947$	−16.6123	−0.539828	−0.269914	−	0.962884i	$-0.586995\pi$
$947$	−16.6123	−0.539828	−0.269914	+	0.962884i	$0.586995\pi$
$948$	0	0
$949$	−22.7287	−0.737805
$950$	3.60272	0.116888
$951$	0	0
$952$	−31.5992	−1.02414
$953$	55.2584	1.79000	0.894998	−	0.446070i	$-0.147177\pi$
$953$	55.2584	1.79000	0.894998	+	0.446070i	$0.147177\pi$
$954$	0	0
$955$	−34.6335	−1.12071
$956$	17.7247	0.573258
$957$	0	0
$958$	69.7792	2.25446
$959$	47.4478	1.53217
$960$	0	0
$961$	0	0
$962$	21.6390	0.697669
$963$	0	0
$964$	6.96712	0.224396
$965$	44.1316	1.42065
$966$	0	0
$967$	−44.8302	−1.44164	−0.720821	−	0.693121i	$-0.756235\pi$
$967$	−44.8302	−1.44164	−0.720821	+	0.693121i	$0.756235\pi$
$968$	21.9638	0.705944
$969$	0	0
$970$	22.4265	0.720072
$971$	60.8757	1.95360	0.976798	−	0.214165i	$-0.0687030\pi$
$971$	60.8757	1.95360	0.976798	+	0.214165i	$0.0687030\pi$
$972$	0	0
$973$	106.028	3.39912
$974$	−57.7246	−1.84962
$975$	0	0
$976$	−17.7949	−0.569601
$977$	−28.7783	−0.920699	−0.460349	−	0.887738i	$-0.652276\pi$
$977$	−28.7783	−0.920699	−0.460349	+	0.887738i	$0.652276\pi$
$978$	0	0
$979$	−2.69887	−0.0862562
$980$	−41.3065	−1.31949
$981$	0	0
$982$	−22.5588	−0.719881
$983$	−31.6935	−1.01086	−0.505432	−	0.862866i	$-0.668667\pi$
$983$	−31.6935	−1.01086	−0.505432	+	0.862866i	$0.668667\pi$
$984$	0	0
$985$	−54.2716	−1.72924
$986$	41.9782	1.33686
$987$	0	0
$988$	−0.293297	−0.00933102
$989$	−8.38592	−0.266657
$990$	0	0
$991$	−2.33148	−0.0740618	−0.0370309	−	0.999314i	$-0.511790\pi$
$991$	−2.33148	−0.0740618	−0.0370309	+	0.999314i	$0.511790\pi$
$992$	0	0
$993$	0	0
$994$	13.1057	0.415686
$995$	−1.74025	−0.0551696
$996$	0	0
$997$	22.7279	0.719800	0.359900	−	0.932991i	$-0.382811\pi$
$997$	22.7279	0.719800	0.359900	+	0.932991i	$0.382811\pi$
$998$	−63.1419	−1.99872
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.bh.1.6		8
3.2	odd	2		2883.2.a.p.1.3		8
31.2	even	5		279.2.i.c.190.3		16
31.16	even	5		279.2.i.c.163.3		16
31.30	odd	2		8649.2.a.bg.1.6		8
93.2	odd	10		93.2.f.b.4.2	✓	16
93.47	odd	10		93.2.f.b.70.2	yes	16
93.92	even	2		2883.2.a.o.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.f.b.4.2	✓	16	93.2	odd	10
93.2.f.b.70.2	yes	16	93.47	odd	10
279.2.i.c.163.3		16	31.16	even	5
279.2.i.c.190.3		16	31.2	even	5
2883.2.a.o.1.3		8	93.92	even	2
2883.2.a.p.1.3		8	3.2	odd	2
8649.2.a.bg.1.6		8	31.30	odd	2
8649.2.a.bh.1.6		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)

\(f(q)\)	\(=\)	\(q+1.67120 q^{2} +0.792899 q^{4} -3.89560 q^{5} +4.51364 q^{7} -2.01730 q^{8} -6.51032 q^{10} -0.335097 q^{11} -1.74603 q^{13} +7.54317 q^{14} -4.95711 q^{16} +3.47039 q^{17} +0.211855 q^{19} -3.08882 q^{20} -0.560013 q^{22} -2.74271 q^{23} +10.1757 q^{25} -2.91796 q^{26} +3.57886 q^{28} +7.23800 q^{29} -4.24970 q^{32} +5.79970 q^{34} -17.5833 q^{35} -7.41580 q^{37} +0.354051 q^{38} +7.85861 q^{40} -6.51032 q^{41} +3.05753 q^{43} -0.265698 q^{44} -4.58361 q^{46} -3.33460 q^{47} +13.3729 q^{49} +17.0056 q^{50} -1.38442 q^{52} -12.1365 q^{53} +1.30540 q^{55} -9.10537 q^{56} +12.0961 q^{58} +1.37161 q^{59} +3.58978 q^{61} +2.81214 q^{64} +6.80184 q^{65} -1.37225 q^{67} +2.75167 q^{68} -29.3852 q^{70} +1.73742 q^{71} +13.0174 q^{73} -12.3933 q^{74} +0.167979 q^{76} -1.51250 q^{77} +8.50919 q^{79} +19.3109 q^{80} -10.8800 q^{82} +7.95742 q^{83} -13.5193 q^{85} +5.10973 q^{86} +0.675992 q^{88} +8.05400 q^{89} -7.88094 q^{91} -2.17469 q^{92} -5.57278 q^{94} -0.825302 q^{95} -3.44476 q^{97} +22.3488 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 13 q^{4} - 3 q^{5} - q^{7} - 9 q^{8} + 6 q^{10} + 5 q^{11} - 4 q^{13} - 3 q^{14} + 19 q^{16} - 2 q^{17} - 7 q^{19} - q^{20} + 9 q^{23} + 15 q^{25} + 8 q^{26} - 3 q^{28} + q^{29} - 54 q^{32}+ \cdots + 41 q^{98}+O(q^{100}) \) 8 * q - q^2 + 13 * q^4 - 3 * q^5 - q^7 - 9 * q^8 + 6 * q^10 + 5 * q^11 - 4 * q^13 - 3 * q^14 + 19 * q^16 - 2 * q^17 - 7 * q^19 - q^20 + 9 * q^23 + 15 * q^25 + 8 * q^26 - 3 * q^28 + q^29 - 54 * q^32 + 48 * q^34 - 51 * q^35 - 14 * q^37 + 16 * q^38 + 59 * q^40 + 6 * q^41 + 36 * q^43 + 27 * q^44 - 3 * q^46 - 35 * q^47 + 3 * q^49 + 37 * q^50 - 46 * q^52 - 8 * q^53 + 6 * q^55 - 52 * q^56 + 52 * q^58 + 28 * q^59 + 17 * q^64 + 3 * q^65 + 30 * q^67 + 15 * q^68 - 35 * q^70 + 6 * q^71 - q^73 + 30 * q^74 - 34 * q^76 - q^77 + 36 * q^79 - 38 * q^80 - 7 * q^82 + 22 * q^83 + 8 * q^85 + 3 * q^86 - 16 * q^88 + q^89 - 6 * q^91 + 35 * q^92 + 37 * q^94 - 50 * q^95 + 29 * q^97 + 41 * q^98

Introduction

L-functions

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Database

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