LMFDB - Embedded newform 6498.2.a.bu.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6498,2,Mod(1,6498)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6498.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6498, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6498 = 2 \cdot 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6498.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,0,3,6,0,3,3,0,6,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$51.8867912334$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	6498.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +0.120615 q^{5} -4.29086 q^{7} +1.00000 q^{8} +0.120615 q^{10} -2.57398 q^{11} -0.815207 q^{13} -4.29086 q^{14} +1.00000 q^{16} +0.467911 q^{17} +0.120615 q^{20} -2.57398 q^{22} -5.55438 q^{23} -4.98545 q^{25} -0.815207 q^{26} -4.29086 q^{28} +2.34730 q^{29} -5.34730 q^{31} +1.00000 q^{32} +0.467911 q^{34} -0.517541 q^{35} +8.51754 q^{37} +0.120615 q^{40} +3.83750 q^{41} +9.33275 q^{43} -2.57398 q^{44} -5.55438 q^{46} +9.47565 q^{47} +11.4115 q^{49} -4.98545 q^{50} -0.815207 q^{52} +12.7588 q^{53} -0.310460 q^{55} -4.29086 q^{56} +2.34730 q^{58} +15.0496 q^{59} -1.71688 q^{61} -5.34730 q^{62} +1.00000 q^{64} -0.0983261 q^{65} -11.4561 q^{67} +0.467911 q^{68} -0.517541 q^{70} +13.3327 q^{71} +2.28312 q^{73} +8.51754 q^{74} +11.0446 q^{77} -4.85710 q^{79} +0.120615 q^{80} +3.83750 q^{82} -3.24897 q^{83} +0.0564370 q^{85} +9.33275 q^{86} -2.57398 q^{88} +3.43107 q^{89} +3.49794 q^{91} -5.55438 q^{92} +9.47565 q^{94} -3.10101 q^{97} +11.4115 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 3 q^{7} + 3 q^{8} + 6 q^{10} - 6 q^{13} + 3 q^{14} + 3 q^{16} + 6 q^{17} + 6 q^{20} - 6 q^{23} + 3 q^{25} - 6 q^{26} + 3 q^{28} + 6 q^{29} - 15 q^{31} + 3 q^{32} + 6 q^{34}+ \cdots + 24 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 + 3 * q^7 + 3 * q^8 + 6 * q^10 - 6 * q^13 + 3 * q^14 + 3 * q^16 + 6 * q^17 + 6 * q^20 - 6 * q^23 + 3 * q^25 - 6 * q^26 + 3 * q^28 + 6 * q^29 - 15 * q^31 + 3 * q^32 + 6 * q^34 + 21 * q^35 + 3 * q^37 + 6 * q^40 + 9 * q^41 + 9 * q^43 - 6 * q^46 + 9 * q^47 + 24 * q^49 + 3 * q^50 - 6 * q^52 + 27 * q^53 + 12 * q^55 + 3 * q^56 + 6 * q^58 + 18 * q^59 + 3 * q^61 - 15 * q^62 + 3 * q^64 - 12 * q^65 - 12 * q^67 + 6 * q^68 + 21 * q^70 + 21 * q^71 + 15 * q^73 + 3 * q^74 + 21 * q^77 - 15 * q^79 + 6 * q^80 + 9 * q^82 + 3 * q^83 + 15 * q^85 + 9 * q^86 + 3 * q^89 - 15 * q^91 - 6 * q^92 + 9 * q^94 - 12 * q^97 + 24 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0.120615	0.0539406	0.0269703	−	0.999636i	$-0.491414\pi$
$5$	0.120615	0.0539406	0.0269703	+	0.999636i	$0.491414\pi$
$6$	0	0
$7$	−4.29086	−1.62179	−0.810896	−	0.585190i	$-0.801020\pi$
$7$	−4.29086	−1.62179	−0.810896	+	0.585190i	$0.801020\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0.120615	0.0381417
$11$	−2.57398	−0.776084	−0.388042	−	0.921642i	$-0.626848\pi$
$11$	−2.57398	−0.776084	−0.388042	+	0.921642i	$0.626848\pi$
$12$	0	0
$13$	−0.815207	−0.226098	−0.113049	−	0.993589i	$-0.536062\pi$
$13$	−0.815207	−0.226098	−0.113049	+	0.993589i	$0.536062\pi$
$14$	−4.29086	−1.14678
$15$	0	0
$16$	1.00000	0.250000
$17$	0.467911	0.113485	0.0567426	−	0.998389i	$-0.481929\pi$
$17$	0.467911	0.113485	0.0567426	+	0.998389i	$0.481929\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0.120615	0.0269703
$21$	0	0
$22$	−2.57398	−0.548774
$23$	−5.55438	−1.15817	−0.579084	−	0.815268i	$-0.696590\pi$
$23$	−5.55438	−1.15817	−0.579084	+	0.815268i	$0.696590\pi$
$24$	0	0
$25$	−4.98545	−0.997090
$26$	−0.815207	−0.159875
$27$	0	0
$28$	−4.29086	−0.810896
$29$	2.34730	0.435882	0.217941	−	0.975962i	$-0.430066\pi$
$29$	2.34730	0.435882	0.217941	+	0.975962i	$0.430066\pi$
$30$	0	0
$31$	−5.34730	−0.960403	−0.480201	−	0.877158i	$-0.659436\pi$
$31$	−5.34730	−0.960403	−0.480201	+	0.877158i	$0.659436\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0.467911	0.0802461
$35$	−0.517541	−0.0874804
$36$	0	0
$37$	8.51754	1.40028	0.700138	−	0.714008i	$-0.253122\pi$
$37$	8.51754	1.40028	0.700138	+	0.714008i	$0.253122\pi$
$38$	0	0
$39$	0	0
$40$	0.120615	0.0190709
$41$	3.83750	0.599316	0.299658	−	0.954047i	$-0.403127\pi$
$41$	3.83750	0.599316	0.299658	+	0.954047i	$0.403127\pi$
$42$	0	0
$43$	9.33275	1.42323	0.711615	−	0.702569i	$-0.247964\pi$
$43$	9.33275	1.42323	0.711615	+	0.702569i	$0.247964\pi$
$44$	−2.57398	−0.388042
$45$	0	0
$46$	−5.55438	−0.818948
$47$	9.47565	1.38217	0.691083	−	0.722775i	$-0.257134\pi$
$47$	9.47565	1.38217	0.691083	+	0.722775i	$0.257134\pi$
$48$	0	0
$49$	11.4115	1.63021
$50$	−4.98545	−0.705049
$51$	0	0
$52$	−0.815207	−0.113049
$53$	12.7588	1.75255	0.876276	−	0.481810i	$-0.160020\pi$
$53$	12.7588	1.75255	0.876276	+	0.481810i	$0.160020\pi$
$54$	0	0
$55$	−0.310460	−0.0418624
$56$	−4.29086	−0.573390
$57$	0	0
$58$	2.34730	0.308215
$59$	15.0496	1.95929	0.979647	−	0.200726i	$-0.0643300\pi$
$59$	15.0496	1.95929	0.979647	+	0.200726i	$0.0643300\pi$
$60$	0	0
$61$	−1.71688	−0.219824	−0.109912	−	0.993941i	$-0.535057\pi$
$61$	−1.71688	−0.219824	−0.109912	+	0.993941i	$0.535057\pi$
$62$	−5.34730	−0.679107
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.0983261	−0.0121958
$66$	0	0
$67$	−11.4561	−1.39958	−0.699790	−	0.714349i	$-0.746723\pi$
$67$	−11.4561	−1.39958	−0.699790	+	0.714349i	$0.746723\pi$
$68$	0.467911	0.0567426
$69$	0	0
$70$	−0.517541	−0.0618580
$71$	13.3327	1.58231	0.791153	−	0.611618i	$-0.209481\pi$
$71$	13.3327	1.58231	0.791153	+	0.611618i	$0.209481\pi$
$72$	0	0
$73$	2.28312	0.267219	0.133609	−	0.991034i	$-0.457343\pi$
$73$	2.28312	0.267219	0.133609	+	0.991034i	$0.457343\pi$
$74$	8.51754	0.990144
$75$	0	0
$76$	0	0
$77$	11.0446	1.25865
$78$	0	0
$79$	−4.85710	−0.546466	−0.273233	−	0.961948i	$-0.588093\pi$
$79$	−4.85710	−0.546466	−0.273233	+	0.961948i	$0.588093\pi$
$80$	0.120615	0.0134851
$81$	0	0
$82$	3.83750	0.423781
$83$	−3.24897	−0.356621	−0.178310	−	0.983974i	$-0.557063\pi$
$83$	−3.24897	−0.356621	−0.178310	+	0.983974i	$0.557063\pi$
$84$	0	0
$85$	0.0564370	0.00612145
$86$	9.33275	1.00638
$87$	0	0
$88$	−2.57398	−0.274387
$89$	3.43107	0.363693	0.181847	−	0.983327i	$-0.441793\pi$
$89$	3.43107	0.363693	0.181847	+	0.983327i	$0.441793\pi$
$90$	0	0
$91$	3.49794	0.366684
$92$	−5.55438	−0.579084
$93$	0	0
$94$	9.47565	0.977339
$95$	0	0
$96$	0	0
$97$	−3.10101	−0.314860	−0.157430	−	0.987530i	$-0.550321\pi$
$97$	−3.10101	−0.314860	−0.157430	+	0.987530i	$0.550321\pi$
$98$	11.4115	1.15273
$99$	0	0
$100$	−4.98545	−0.498545
$101$	16.7297	1.66466	0.832332	−	0.554277i	$-0.187005\pi$
$101$	16.7297	1.66466	0.832332	+	0.554277i	$0.187005\pi$
$102$	0	0
$103$	10.3969	1.02444	0.512220	−	0.858854i	$-0.328823\pi$
$103$	10.3969	1.02444	0.512220	+	0.858854i	$0.328823\pi$
$104$	−0.815207	−0.0799377
$105$	0	0
$106$	12.7588	1.23924
$107$	−9.11381	−0.881065	−0.440533	−	0.897737i	$-0.645210\pi$
$107$	−9.11381	−0.881065	−0.440533	+	0.897737i	$0.645210\pi$
$108$	0	0
$109$	−3.82976	−0.366824	−0.183412	−	0.983036i	$-0.558714\pi$
$109$	−3.82976	−0.366824	−0.183412	+	0.983036i	$0.558714\pi$
$110$	−0.310460	−0.0296012
$111$	0	0
$112$	−4.29086	−0.405448
$113$	5.41147	0.509069	0.254534	−	0.967064i	$-0.418078\pi$
$113$	5.41147	0.509069	0.254534	+	0.967064i	$0.418078\pi$
$114$	0	0
$115$	−0.669940	−0.0624722
$116$	2.34730	0.217941
$117$	0	0
$118$	15.0496	1.38543
$119$	−2.00774	−0.184049
$120$	0	0
$121$	−4.37464	−0.397694
$122$	−1.71688	−0.155439
$123$	0	0
$124$	−5.34730	−0.480201
$125$	−1.20439	−0.107724
$126$	0	0
$127$	−4.24123	−0.376348	−0.188174	−	0.982136i	$-0.560257\pi$
$127$	−4.24123	−0.376348	−0.188174	+	0.982136i	$0.560257\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−0.0983261	−0.00862377
$131$	−21.0428	−1.83852	−0.919260	−	0.393651i	$-0.871212\pi$
$131$	−21.0428	−1.83852	−0.919260	+	0.393651i	$0.871212\pi$
$132$	0	0
$133$	0	0
$134$	−11.4561	−0.989652
$135$	0	0
$136$	0.467911	0.0401230
$137$	12.3209	1.05264	0.526322	−	0.850285i	$-0.323571\pi$
$137$	12.3209	1.05264	0.526322	+	0.850285i	$0.323571\pi$
$138$	0	0
$139$	−5.58853	−0.474013	−0.237006	−	0.971508i	$-0.576166\pi$
$139$	−5.58853	−0.474013	−0.237006	+	0.971508i	$0.576166\pi$
$140$	−0.517541	−0.0437402
$141$	0	0
$142$	13.3327	1.11886
$143$	2.09833	0.175471
$144$	0	0
$145$	0.283119	0.0235117
$146$	2.28312	0.188952
$147$	0	0
$148$	8.51754	0.700138
$149$	−11.0942	−0.908873	−0.454436	−	0.890779i	$-0.650159\pi$
$149$	−11.0942	−0.908873	−0.454436	+	0.890779i	$0.650159\pi$
$150$	0	0
$151$	−12.5963	−1.02507	−0.512535	−	0.858666i	$-0.671293\pi$
$151$	−12.5963	−1.02507	−0.512535	+	0.858666i	$0.671293\pi$
$152$	0	0
$153$	0	0
$154$	11.0446	0.889997
$155$	−0.644963	−0.0518047
$156$	0	0
$157$	−3.97771	−0.317456	−0.158728	−	0.987322i	$-0.550739\pi$
$157$	−3.97771	−0.317456	−0.158728	+	0.987322i	$0.550739\pi$
$158$	−4.85710	−0.386410
$159$	0	0
$160$	0.120615	0.00953543
$161$	23.8331	1.87831
$162$	0	0
$163$	19.2003	1.50388	0.751941	−	0.659231i	$-0.229118\pi$
$163$	19.2003	1.50388	0.751941	+	0.659231i	$0.229118\pi$
$164$	3.83750	0.299658
$165$	0	0
$166$	−3.24897	−0.252169
$167$	−15.3746	−1.18973	−0.594863	−	0.803827i	$-0.702794\pi$
$167$	−15.3746	−1.18973	−0.594863	+	0.803827i	$0.702794\pi$
$168$	0	0
$169$	−12.3354	−0.948880
$170$	0.0564370	0.00432852
$171$	0	0
$172$	9.33275	0.711615
$173$	−6.59358	−0.501300	−0.250650	−	0.968078i	$-0.580644\pi$
$173$	−6.59358	−0.501300	−0.250650	+	0.968078i	$0.580644\pi$
$174$	0	0
$175$	21.3919	1.61707
$176$	−2.57398	−0.194021
$177$	0	0
$178$	3.43107	0.257170
$179$	1.24123	0.0927738	0.0463869	−	0.998924i	$-0.485229\pi$
$179$	1.24123	0.0927738	0.0463869	+	0.998924i	$0.485229\pi$
$180$	0	0
$181$	−4.03003	−0.299550	−0.149775	−	0.988720i	$-0.547855\pi$
$181$	−4.03003	−0.299550	−0.149775	+	0.988720i	$0.547855\pi$
$182$	3.49794	0.259285
$183$	0	0
$184$	−5.55438	−0.409474
$185$	1.02734	0.0755316
$186$	0	0
$187$	−1.20439	−0.0880739
$188$	9.47565	0.691083
$189$	0	0
$190$	0	0
$191$	24.0847	1.74271	0.871354	−	0.490654i	$-0.163242\pi$
$191$	24.0847	1.74271	0.871354	+	0.490654i	$0.163242\pi$
$192$	0	0
$193$	9.85978	0.709723	0.354861	−	0.934919i	$-0.384528\pi$
$193$	9.85978	0.709723	0.354861	+	0.934919i	$0.384528\pi$
$194$	−3.10101	−0.222640
$195$	0	0
$196$	11.4115	0.815105
$197$	6.45605	0.459975	0.229987	−	0.973194i	$-0.426132\pi$
$197$	6.45605	0.459975	0.229987	+	0.973194i	$0.426132\pi$
$198$	0	0
$199$	24.7716	1.75601	0.878005	−	0.478652i	$-0.158874\pi$
$199$	24.7716	1.75601	0.878005	+	0.478652i	$0.158874\pi$
$200$	−4.98545	−0.352525
$201$	0	0
$202$	16.7297	1.17710
$203$	−10.0719	−0.706910
$204$	0	0
$205$	0.462859	0.0323275
$206$	10.3969	0.724388
$207$	0	0
$208$	−0.815207	−0.0565245
$209$	0	0
$210$	0	0
$211$	1.38413	0.0952876	0.0476438	−	0.998864i	$-0.484829\pi$
$211$	1.38413	0.0952876	0.0476438	+	0.998864i	$0.484829\pi$
$212$	12.7588	0.876276
$213$	0	0
$214$	−9.11381	−0.623007
$215$	1.12567	0.0767699
$216$	0	0
$217$	22.9445	1.55757
$218$	−3.82976	−0.259384
$219$	0	0
$220$	−0.310460	−0.0209312
$221$	−0.381445	−0.0256587
$222$	0	0
$223$	−22.3131	−1.49420	−0.747099	−	0.664712i	$-0.768554\pi$
$223$	−22.3131	−1.49420	−0.747099	+	0.664712i	$0.768554\pi$
$224$	−4.29086	−0.286695
$225$	0	0
$226$	5.41147	0.359966
$227$	−20.7665	−1.37832	−0.689161	−	0.724608i	$-0.742021\pi$
$227$	−20.7665	−1.37832	−0.689161	+	0.724608i	$0.742021\pi$
$228$	0	0
$229$	5.51754	0.364609	0.182305	−	0.983242i	$-0.441644\pi$
$229$	5.51754	0.364609	0.182305	+	0.983242i	$0.441644\pi$
$230$	−0.669940	−0.0441745
$231$	0	0
$232$	2.34730	0.154108
$233$	15.0865	0.988347	0.494174	−	0.869363i	$-0.335471\pi$
$233$	15.0865	0.988347	0.494174	+	0.869363i	$0.335471\pi$
$234$	0	0
$235$	1.14290	0.0745548
$236$	15.0496	0.979647
$237$	0	0
$238$	−2.00774	−0.130143
$239$	28.9513	1.87270	0.936352	−	0.351062i	$-0.114179\pi$
$239$	28.9513	1.87270	0.936352	+	0.351062i	$0.114179\pi$
$240$	0	0
$241$	19.3696	1.24770	0.623852	−	0.781542i	$-0.285567\pi$
$241$	19.3696	1.24770	0.623852	+	0.781542i	$0.285567\pi$
$242$	−4.37464	−0.281212
$243$	0	0
$244$	−1.71688	−0.109912
$245$	1.37639	0.0879345
$246$	0	0
$247$	0	0
$248$	−5.34730	−0.339554
$249$	0	0
$250$	−1.20439	−0.0761725
$251$	4.34224	0.274080	0.137040	−	0.990566i	$-0.456241\pi$
$251$	4.34224	0.274080	0.137040	+	0.990566i	$0.456241\pi$
$252$	0	0
$253$	14.2968	0.898835
$254$	−4.24123	−0.266118
$255$	0	0
$256$	1.00000	0.0625000
$257$	17.7665	1.10824	0.554122	−	0.832435i	$-0.313054\pi$
$257$	17.7665	1.10824	0.554122	+	0.832435i	$0.313054\pi$
$258$	0	0
$259$	−36.5476	−2.27096
$260$	−0.0983261	−0.00609792
$261$	0	0
$262$	−21.0428	−1.30003
$263$	2.31315	0.142635	0.0713174	−	0.997454i	$-0.477280\pi$
$263$	2.31315	0.142635	0.0713174	+	0.997454i	$0.477280\pi$
$264$	0	0
$265$	1.53890	0.0945336
$266$	0	0
$267$	0	0
$268$	−11.4561	−0.699790
$269$	−2.81790	−0.171810	−0.0859051	−	0.996303i	$-0.527378\pi$
$269$	−2.81790	−0.171810	−0.0859051	+	0.996303i	$0.527378\pi$
$270$	0	0
$271$	−6.68954	−0.406361	−0.203180	−	0.979141i	$-0.565128\pi$
$271$	−6.68954	−0.406361	−0.203180	+	0.979141i	$0.565128\pi$
$272$	0.467911	0.0283713
$273$	0	0
$274$	12.3209	0.744332
$275$	12.8324	0.773825
$276$	0	0
$277$	8.25166	0.495794	0.247897	−	0.968786i	$-0.420261\pi$
$277$	8.25166	0.495794	0.247897	+	0.968786i	$0.420261\pi$
$278$	−5.58853	−0.335178
$279$	0	0
$280$	−0.517541	−0.0309290
$281$	27.9368	1.66657	0.833284	−	0.552846i	$-0.186458\pi$
$281$	27.9368	1.66657	0.833284	+	0.552846i	$0.186458\pi$
$282$	0	0
$283$	2.42602	0.144212	0.0721060	−	0.997397i	$-0.477028\pi$
$283$	2.42602	0.144212	0.0721060	+	0.997397i	$0.477028\pi$
$284$	13.3327	0.791153
$285$	0	0
$286$	2.09833	0.124077
$287$	−16.4662	−0.971966
$288$	0	0
$289$	−16.7811	−0.987121
$290$	0.283119	0.0166253
$291$	0	0
$292$	2.28312	0.133609
$293$	−14.1010	−0.823790	−0.411895	−	0.911231i	$-0.635133\pi$
$293$	−14.1010	−0.823790	−0.411895	+	0.911231i	$0.635133\pi$
$294$	0	0
$295$	1.81521	0.105685
$296$	8.51754	0.495072
$297$	0	0
$298$	−11.0942	−0.642670
$299$	4.52797	0.261859
$300$	0	0
$301$	−40.0455	−2.30818
$302$	−12.5963	−0.724834
$303$	0	0
$304$	0	0
$305$	−0.207081	−0.0118574
$306$	0	0
$307$	20.2071	1.15328	0.576640	−	0.816999i	$-0.304364\pi$
$307$	20.2071	1.15328	0.576640	+	0.816999i	$0.304364\pi$
$308$	11.0446	0.629323
$309$	0	0
$310$	−0.644963	−0.0366314
$311$	−9.61856	−0.545418	−0.272709	−	0.962097i	$-0.587920\pi$
$311$	−9.61856	−0.545418	−0.272709	+	0.962097i	$0.587920\pi$
$312$	0	0
$313$	24.8881	1.40676	0.703378	−	0.710816i	$-0.251674\pi$
$313$	24.8881	1.40676	0.703378	+	0.710816i	$0.251674\pi$
$314$	−3.97771	−0.224475
$315$	0	0
$316$	−4.85710	−0.273233
$317$	17.2635	0.969616	0.484808	−	0.874621i	$-0.338890\pi$
$317$	17.2635	0.969616	0.484808	+	0.874621i	$0.338890\pi$
$318$	0	0
$319$	−6.04189	−0.338281
$320$	0.120615	0.00674257
$321$	0	0
$322$	23.8331	1.32816
$323$	0	0
$324$	0	0
$325$	4.06418	0.225440
$326$	19.2003	1.06340
$327$	0	0
$328$	3.83750	0.211890
$329$	−40.6587	−2.24159
$330$	0	0
$331$	−9.99050	−0.549128	−0.274564	−	0.961569i	$-0.588533\pi$
$331$	−9.99050	−0.549128	−0.274564	+	0.961569i	$0.588533\pi$
$332$	−3.24897	−0.178310
$333$	0	0
$334$	−15.3746	−0.841263
$335$	−1.38177	−0.0754941
$336$	0	0
$337$	−6.76382	−0.368449	−0.184224	−	0.982884i	$-0.558977\pi$
$337$	−6.76382	−0.368449	−0.184224	+	0.982884i	$0.558977\pi$
$338$	−12.3354	−0.670959
$339$	0	0
$340$	0.0564370	0.00306073
$341$	13.7638	0.745353
$342$	0	0
$343$	−18.9290	−1.02207
$344$	9.33275	0.503188
$345$	0	0
$346$	−6.59358	−0.354473
$347$	−24.6013	−1.32067	−0.660334	−	0.750972i	$-0.729585\pi$
$347$	−24.6013	−1.32067	−0.660334	+	0.750972i	$0.729585\pi$
$348$	0	0
$349$	−6.83481	−0.365859	−0.182929	−	0.983126i	$-0.558558\pi$
$349$	−6.83481	−0.365859	−0.182929	+	0.983126i	$0.558558\pi$
$350$	21.3919	1.14344
$351$	0	0
$352$	−2.57398	−0.137193
$353$	5.08378	0.270582	0.135291	−	0.990806i	$-0.456803\pi$
$353$	5.08378	0.270582	0.135291	+	0.990806i	$0.456803\pi$
$354$	0	0
$355$	1.60813	0.0853505
$356$	3.43107	0.181847
$357$	0	0
$358$	1.24123	0.0656010
$359$	2.86753	0.151342	0.0756711	−	0.997133i	$-0.475890\pi$
$359$	2.86753	0.151342	0.0756711	+	0.997133i	$0.475890\pi$
$360$	0	0
$361$	0	0
$362$	−4.03003	−0.211814
$363$	0	0
$364$	3.49794	0.183342
$365$	0.275378	0.0144139
$366$	0	0
$367$	22.1429	1.15585	0.577925	−	0.816090i	$-0.303863\pi$
$367$	22.1429	1.15585	0.577925	+	0.816090i	$0.303863\pi$
$368$	−5.55438	−0.289542
$369$	0	0
$370$	1.02734	0.0534089
$371$	−54.7461	−2.84228
$372$	0	0
$373$	−17.6732	−0.915086	−0.457543	−	0.889188i	$-0.651270\pi$
$373$	−17.6732	−0.915086	−0.457543	+	0.889188i	$0.651270\pi$
$374$	−1.20439	−0.0622777
$375$	0	0
$376$	9.47565	0.488669
$377$	−1.91353	−0.0985520
$378$	0	0
$379$	9.75970	0.501322	0.250661	−	0.968075i	$-0.419352\pi$
$379$	9.75970	0.501322	0.250661	+	0.968075i	$0.419352\pi$
$380$	0	0
$381$	0	0
$382$	24.0847	1.23228
$383$	−22.3155	−1.14027	−0.570135	−	0.821551i	$-0.693109\pi$
$383$	−22.3155	−1.14027	−0.570135	+	0.821551i	$0.693109\pi$
$384$	0	0
$385$	1.33214	0.0678921
$386$	9.85978	0.501850
$387$	0	0
$388$	−3.10101	−0.157430
$389$	−7.02229	−0.356044	−0.178022	−	0.984026i	$-0.556970\pi$
$389$	−7.02229	−0.356044	−0.178022	+	0.984026i	$0.556970\pi$
$390$	0	0
$391$	−2.59896	−0.131435
$392$	11.4115	0.576366
$393$	0	0
$394$	6.45605	0.325251
$395$	−0.585838	−0.0294767
$396$	0	0
$397$	−19.9786	−1.00270	−0.501350	−	0.865245i	$-0.667163\pi$
$397$	−19.9786	−1.00270	−0.501350	+	0.865245i	$0.667163\pi$
$398$	24.7716	1.24169
$399$	0	0
$400$	−4.98545	−0.249273
$401$	−1.18984	−0.0594180	−0.0297090	−	0.999559i	$-0.509458\pi$
$401$	−1.18984	−0.0594180	−0.0297090	+	0.999559i	$0.509458\pi$
$402$	0	0
$403$	4.35916	0.217145
$404$	16.7297	0.832332
$405$	0	0
$406$	−10.0719	−0.499861
$407$	−21.9240	−1.08673
$408$	0	0
$409$	−9.48845	−0.469173	−0.234587	−	0.972095i	$-0.575374\pi$
$409$	−9.48845	−0.469173	−0.234587	+	0.972095i	$0.575374\pi$
$410$	0.462859	0.0228590
$411$	0	0
$412$	10.3969	0.512220
$413$	−64.5758	−3.17757
$414$	0	0
$415$	−0.391874	−0.0192363
$416$	−0.815207	−0.0399688
$417$	0	0
$418$	0	0
$419$	−4.72638	−0.230899	−0.115449	−	0.993313i	$-0.536831\pi$
$419$	−4.72638	−0.230899	−0.115449	+	0.993313i	$0.536831\pi$
$420$	0	0
$421$	−1.38682	−0.0675895	−0.0337948	−	0.999429i	$-0.510759\pi$
$421$	−1.38682	−0.0675895	−0.0337948	+	0.999429i	$0.510759\pi$
$422$	1.38413	0.0673785
$423$	0	0
$424$	12.7588	0.619621
$425$	−2.33275	−0.113155
$426$	0	0
$427$	7.36690	0.356509
$428$	−9.11381	−0.440533
$429$	0	0
$430$	1.12567	0.0542845
$431$	2.71183	0.130624	0.0653121	−	0.997865i	$-0.479196\pi$
$431$	2.71183	0.130624	0.0653121	+	0.997865i	$0.479196\pi$
$432$	0	0
$433$	20.2412	0.972731	0.486366	−	0.873755i	$-0.338322\pi$
$433$	20.2412	0.972731	0.486366	+	0.873755i	$0.338322\pi$
$434$	22.9445	1.10137
$435$	0	0
$436$	−3.82976	−0.183412
$437$	0	0
$438$	0	0
$439$	18.2003	0.868652	0.434326	−	0.900756i	$-0.356987\pi$
$439$	18.2003	0.868652	0.434326	+	0.900756i	$0.356987\pi$
$440$	−0.310460	−0.0148006
$441$	0	0
$442$	−0.381445	−0.0181435
$443$	35.7425	1.69818	0.849088	−	0.528252i	$-0.177152\pi$
$443$	35.7425	1.69818	0.849088	+	0.528252i	$0.177152\pi$
$444$	0	0
$445$	0.413838	0.0196178
$446$	−22.3131	−1.05656
$447$	0	0
$448$	−4.29086	−0.202724
$449$	−18.9905	−0.896217	−0.448109	−	0.893979i	$-0.647902\pi$
$449$	−18.9905	−0.896217	−0.448109	+	0.893979i	$0.647902\pi$
$450$	0	0
$451$	−9.87763	−0.465119
$452$	5.41147	0.254534
$453$	0	0
$454$	−20.7665	−0.974621
$455$	0.421903	0.0197791
$456$	0	0
$457$	25.9632	1.21451	0.607253	−	0.794509i	$-0.292272\pi$
$457$	25.9632	1.21451	0.607253	+	0.794509i	$0.292272\pi$
$458$	5.51754	0.257818
$459$	0	0
$460$	−0.669940	−0.0312361
$461$	29.2344	1.36158	0.680791	−	0.732477i	$-0.261636\pi$
$461$	29.2344	1.36158	0.680791	+	0.732477i	$0.261636\pi$
$462$	0	0
$463$	−21.1908	−0.984819	−0.492410	−	0.870364i	$-0.663884\pi$
$463$	−21.1908	−0.984819	−0.492410	+	0.870364i	$0.663884\pi$
$464$	2.34730	0.108970
$465$	0	0
$466$	15.0865	0.698867
$467$	2.14527	0.0992711	0.0496356	−	0.998767i	$-0.484194\pi$
$467$	2.14527	0.0992711	0.0496356	+	0.998767i	$0.484194\pi$
$468$	0	0
$469$	49.1563	2.26983
$470$	1.14290	0.0527182
$471$	0	0
$472$	15.0496	0.692715
$473$	−24.0223	−1.10455
$474$	0	0
$475$	0	0
$476$	−2.00774	−0.0920246
$477$	0	0
$478$	28.9513	1.32420
$479$	3.39693	0.155210	0.0776048	−	0.996984i	$-0.475273\pi$
$479$	3.39693	0.155210	0.0776048	+	0.996984i	$0.475273\pi$
$480$	0	0
$481$	−6.94356	−0.316599
$482$	19.3696	0.882260
$483$	0	0
$484$	−4.37464	−0.198847
$485$	−0.374028	−0.0169837
$486$	0	0
$487$	−19.7861	−0.896594	−0.448297	−	0.893885i	$-0.647969\pi$
$487$	−19.7861	−0.896594	−0.448297	+	0.893885i	$0.647969\pi$
$488$	−1.71688	−0.0777196
$489$	0	0
$490$	1.37639	0.0621791
$491$	−25.1438	−1.13473	−0.567363	−	0.823468i	$-0.692036\pi$
$491$	−25.1438	−1.13473	−0.567363	+	0.823468i	$0.692036\pi$
$492$	0	0
$493$	1.09833	0.0494661
$494$	0	0
$495$	0	0
$496$	−5.34730	−0.240101
$497$	−57.2089	−2.56617
$498$	0	0
$499$	35.1584	1.57391	0.786953	−	0.617013i	$-0.211658\pi$
$499$	35.1584	1.57391	0.786953	+	0.617013i	$0.211658\pi$
$500$	−1.20439	−0.0538621
$501$	0	0
$502$	4.34224	0.193804
$503$	−20.3455	−0.907163	−0.453581	−	0.891215i	$-0.649854\pi$
$503$	−20.3455	−0.907163	−0.453581	+	0.891215i	$0.649854\pi$
$504$	0	0
$505$	2.01785	0.0897930
$506$	14.2968	0.635572
$507$	0	0
$508$	−4.24123	−0.188174
$509$	−13.1652	−0.583537	−0.291768	−	0.956489i	$-0.594244\pi$
$509$	−13.1652	−0.583537	−0.291768	+	0.956489i	$0.594244\pi$
$510$	0	0
$511$	−9.79654	−0.433373
$512$	1.00000	0.0441942
$513$	0	0
$514$	17.7665	0.783647
$515$	1.25402	0.0552588
$516$	0	0
$517$	−24.3901	−1.07268
$518$	−36.5476	−1.60581
$519$	0	0
$520$	−0.0983261	−0.00431188
$521$	20.4911	0.897733	0.448866	−	0.893599i	$-0.351828\pi$
$521$	20.4911	0.897733	0.448866	+	0.893599i	$0.351828\pi$
$522$	0	0
$523$	42.9195	1.87674	0.938370	−	0.345633i	$-0.112336\pi$
$523$	42.9195	1.87674	0.938370	+	0.345633i	$0.112336\pi$
$524$	−21.0428	−0.919260
$525$	0	0
$526$	2.31315	0.100858
$527$	−2.50206	−0.108991
$528$	0	0
$529$	7.85111	0.341353
$530$	1.53890	0.0668454
$531$	0	0
$532$	0	0
$533$	−3.12836	−0.135504
$534$	0	0
$535$	−1.09926	−0.0475251
$536$	−11.4561	−0.494826
$537$	0	0
$538$	−2.81790	−0.121488
$539$	−29.3729	−1.26518
$540$	0	0
$541$	−14.5767	−0.626700	−0.313350	−	0.949638i	$-0.601451\pi$
$541$	−14.5767	−0.626700	−0.313350	+	0.949638i	$0.601451\pi$
$542$	−6.68954	−0.287340
$543$	0	0
$544$	0.467911	0.0200615
$545$	−0.461925	−0.0197867
$546$	0	0
$547$	−18.5689	−0.793950	−0.396975	−	0.917829i	$-0.629940\pi$
$547$	−18.5689	−0.793950	−0.396975	+	0.917829i	$0.629940\pi$
$548$	12.3209	0.526322
$549$	0	0
$550$	12.8324	0.547177
$551$	0	0
$552$	0	0
$553$	20.8411	0.886254
$554$	8.25166	0.350579
$555$	0	0
$556$	−5.58853	−0.237006
$557$	−15.5381	−0.658369	−0.329185	−	0.944266i	$-0.606774\pi$
$557$	−15.5381	−0.658369	−0.329185	+	0.944266i	$0.606774\pi$
$558$	0	0
$559$	−7.60813	−0.321789
$560$	−0.517541	−0.0218701
$561$	0	0
$562$	27.9368	1.17844
$563$	−12.1557	−0.512302	−0.256151	−	0.966637i	$-0.582454\pi$
$563$	−12.1557	−0.512302	−0.256151	+	0.966637i	$0.582454\pi$
$564$	0	0
$565$	0.652704	0.0274594
$566$	2.42602	0.101973
$567$	0	0
$568$	13.3327	0.559430
$569$	13.6709	0.573113	0.286556	−	0.958063i	$-0.407489\pi$
$569$	13.6709	0.573113	0.286556	+	0.958063i	$0.407489\pi$
$570$	0	0
$571$	28.2003	1.18014	0.590072	−	0.807350i	$-0.299099\pi$
$571$	28.2003	1.18014	0.590072	+	0.807350i	$0.299099\pi$
$572$	2.09833	0.0877354
$573$	0	0
$574$	−16.4662	−0.687284
$575$	27.6911	1.15480
$576$	0	0
$577$	20.2189	0.841726	0.420863	−	0.907124i	$-0.361727\pi$
$577$	20.2189	0.841726	0.420863	+	0.907124i	$0.361727\pi$
$578$	−16.7811	−0.698000
$579$	0	0
$580$	0.283119	0.0117559
$581$	13.9409	0.578365
$582$	0	0
$583$	−32.8408	−1.36013
$584$	2.28312	0.0944761
$585$	0	0
$586$	−14.1010	−0.582508
$587$	5.20708	0.214919	0.107460	−	0.994209i	$-0.465728\pi$
$587$	5.20708	0.214919	0.107460	+	0.994209i	$0.465728\pi$
$588$	0	0
$589$	0	0
$590$	1.81521	0.0747309
$591$	0	0
$592$	8.51754	0.350069
$593$	43.5449	1.78817	0.894087	−	0.447893i	$-0.147826\pi$
$593$	43.5449	1.78817	0.894087	+	0.447893i	$0.147826\pi$
$594$	0	0
$595$	−0.242163	−0.00992772
$596$	−11.0942	−0.454436
$597$	0	0
$598$	4.52797	0.185162
$599$	−32.7246	−1.33709	−0.668546	−	0.743671i	$-0.733083\pi$
$599$	−32.7246	−1.33709	−0.668546	+	0.743671i	$0.733083\pi$
$600$	0	0
$601$	20.7733	0.847361	0.423681	−	0.905812i	$-0.360738\pi$
$601$	20.7733	0.847361	0.423681	+	0.905812i	$0.360738\pi$
$602$	−40.0455	−1.63213
$603$	0	0
$604$	−12.5963	−0.512535
$605$	−0.527646	−0.0214519
$606$	0	0
$607$	23.0419	0.935241	0.467621	−	0.883929i	$-0.345111\pi$
$607$	23.0419	0.935241	0.467621	+	0.883929i	$0.345111\pi$
$608$	0	0
$609$	0	0
$610$	−0.207081	−0.00838447
$611$	−7.72462	−0.312505
$612$	0	0
$613$	21.2003	0.856271	0.428136	−	0.903715i	$-0.359171\pi$
$613$	21.2003	0.856271	0.428136	+	0.903715i	$0.359171\pi$
$614$	20.2071	0.815491
$615$	0	0
$616$	11.0446	0.444999
$617$	20.4825	0.824593	0.412296	−	0.911050i	$-0.364727\pi$
$617$	20.4825	0.824593	0.412296	+	0.911050i	$0.364727\pi$
$618$	0	0
$619$	−15.7452	−0.632851	−0.316426	−	0.948617i	$-0.602483\pi$
$619$	−15.7452	−0.632851	−0.316426	+	0.948617i	$0.602483\pi$
$620$	−0.644963	−0.0259023
$621$	0	0
$622$	−9.61856	−0.385669
$623$	−14.7223	−0.589835
$624$	0	0
$625$	24.7820	0.991280
$626$	24.8881	0.994727
$627$	0	0
$628$	−3.97771	−0.158728
$629$	3.98545	0.158910
$630$	0	0
$631$	−21.1530	−0.842088	−0.421044	−	0.907040i	$-0.638336\pi$
$631$	−21.1530	−0.842088	−0.421044	+	0.907040i	$0.638336\pi$
$632$	−4.85710	−0.193205
$633$	0	0
$634$	17.2635	0.685622
$635$	−0.511555	−0.0203004
$636$	0	0
$637$	−9.30272	−0.368587
$638$	−6.04189	−0.239201
$639$	0	0
$640$	0.120615	0.00476772
$641$	8.22668	0.324934	0.162467	−	0.986714i	$-0.448055\pi$
$641$	8.22668	0.324934	0.162467	+	0.986714i	$0.448055\pi$
$642$	0	0
$643$	−4.27631	−0.168641	−0.0843206	−	0.996439i	$-0.526872\pi$
$643$	−4.27631	−0.168641	−0.0843206	+	0.996439i	$0.526872\pi$
$644$	23.8331	0.939154
$645$	0	0
$646$	0	0
$647$	−0.947682	−0.0372572	−0.0186286	−	0.999826i	$-0.505930\pi$
$647$	−0.947682	−0.0372572	−0.0186286	+	0.999826i	$0.505930\pi$
$648$	0	0
$649$	−38.7374	−1.52058
$650$	4.06418	0.159410
$651$	0	0
$652$	19.2003	0.751941
$653$	32.8435	1.28526	0.642632	−	0.766175i	$-0.277842\pi$
$653$	32.8435	1.28526	0.642632	+	0.766175i	$0.277842\pi$
$654$	0	0
$655$	−2.53807	−0.0991708
$656$	3.83750	0.149829
$657$	0	0
$658$	−40.6587	−1.58504
$659$	4.74422	0.184809	0.0924043	−	0.995722i	$-0.470545\pi$
$659$	4.74422	0.184809	0.0924043	+	0.995722i	$0.470545\pi$
$660$	0	0
$661$	−39.4056	−1.53270	−0.766350	−	0.642423i	$-0.777929\pi$
$661$	−39.4056	−1.53270	−0.766350	+	0.642423i	$0.777929\pi$
$662$	−9.99050	−0.388292
$663$	0	0
$664$	−3.24897	−0.126085
$665$	0	0
$666$	0	0
$667$	−13.0378	−0.504824
$668$	−15.3746	−0.594863
$669$	0	0
$670$	−1.38177	−0.0533824
$671$	4.41921	0.170602
$672$	0	0
$673$	−17.1429	−0.660810	−0.330405	−	0.943839i	$-0.607185\pi$
$673$	−17.1429	−0.660810	−0.330405	+	0.943839i	$0.607185\pi$
$674$	−6.76382	−0.260533
$675$	0	0
$676$	−12.3354	−0.474440
$677$	7.13011	0.274032	0.137016	−	0.990569i	$-0.456249\pi$
$677$	7.13011	0.274032	0.137016	+	0.990569i	$0.456249\pi$
$678$	0	0
$679$	13.3060	0.510638
$680$	0.0564370	0.00216426
$681$	0	0
$682$	13.7638	0.527044
$683$	−26.0000	−0.994862	−0.497431	−	0.867503i	$-0.665723\pi$
$683$	−26.0000	−0.994862	−0.497431	+	0.867503i	$0.665723\pi$
$684$	0	0
$685$	1.48608	0.0567802
$686$	−18.9290	−0.722713
$687$	0	0
$688$	9.33275	0.355808
$689$	−10.4010	−0.396248
$690$	0	0
$691$	39.3715	1.49776	0.748880	−	0.662705i	$-0.230592\pi$
$691$	39.3715	1.49776	0.748880	+	0.662705i	$0.230592\pi$
$692$	−6.59358	−0.250650
$693$	0	0
$694$	−24.6013	−0.933853
$695$	−0.674059	−0.0255685
$696$	0	0
$697$	1.79561	0.0680135
$698$	−6.83481	−0.258701
$699$	0	0
$700$	21.3919	0.808537
$701$	−5.63404	−0.212795	−0.106397	−	0.994324i	$-0.533932\pi$
$701$	−5.63404	−0.212795	−0.106397	+	0.994324i	$0.533932\pi$
$702$	0	0
$703$	0	0
$704$	−2.57398	−0.0970104
$705$	0	0
$706$	5.08378	0.191331
$707$	−71.7847	−2.69974
$708$	0	0
$709$	30.6664	1.15170	0.575851	−	0.817555i	$-0.304671\pi$
$709$	30.6664	1.15170	0.575851	+	0.817555i	$0.304671\pi$
$710$	1.60813	0.0603519
$711$	0	0
$712$	3.43107	0.128585
$713$	29.7009	1.11231
$714$	0	0
$715$	0.253089	0.00946500
$716$	1.24123	0.0463869
$717$	0	0
$718$	2.86753	0.107015
$719$	−6.33956	−0.236426	−0.118213	−	0.992988i	$-0.537716\pi$
$719$	−6.33956	−0.236426	−0.118213	+	0.992988i	$0.537716\pi$
$720$	0	0
$721$	−44.6117	−1.66143
$722$	0	0
$723$	0	0
$724$	−4.03003	−0.149775
$725$	−11.7023	−0.434614
$726$	0	0
$727$	−23.3414	−0.865685	−0.432843	−	0.901469i	$-0.642489\pi$
$727$	−23.3414	−0.865685	−0.432843	+	0.901469i	$0.642489\pi$
$728$	3.49794	0.129642
$729$	0	0
$730$	0.275378	0.0101922
$731$	4.36690	0.161516
$732$	0	0
$733$	1.58946	0.0587080	0.0293540	−	0.999569i	$-0.490655\pi$
$733$	1.58946	0.0587080	0.0293540	+	0.999569i	$0.490655\pi$
$734$	22.1429	0.817309
$735$	0	0
$736$	−5.55438	−0.204737
$737$	29.4876	1.08619
$738$	0	0
$739$	3.65951	0.134617	0.0673086	−	0.997732i	$-0.478559\pi$
$739$	3.65951	0.134617	0.0673086	+	0.997732i	$0.478559\pi$
$740$	1.02734	0.0377658
$741$	0	0
$742$	−54.7461	−2.00979
$743$	−18.3800	−0.674297	−0.337149	−	0.941451i	$-0.609463\pi$
$743$	−18.3800	−0.674297	−0.337149	+	0.941451i	$0.609463\pi$
$744$	0	0
$745$	−1.33813	−0.0490251
$746$	−17.6732	−0.647063
$747$	0	0
$748$	−1.20439	−0.0440370
$749$	39.1061	1.42890
$750$	0	0
$751$	−30.6655	−1.11900	−0.559500	−	0.828830i	$-0.689007\pi$
$751$	−30.6655	−1.11900	−0.559500	+	0.828830i	$0.689007\pi$
$752$	9.47565	0.345541
$753$	0	0
$754$	−1.91353	−0.0696868
$755$	−1.51930	−0.0552928
$756$	0	0
$757$	−7.78787	−0.283055	−0.141527	−	0.989934i	$-0.545201\pi$
$757$	−7.78787	−0.283055	−0.141527	+	0.989934i	$0.545201\pi$
$758$	9.75970	0.354488
$759$	0	0
$760$	0	0
$761$	−28.3969	−1.02939	−0.514694	−	0.857374i	$-0.672094\pi$
$761$	−28.3969	−1.02939	−0.514694	+	0.857374i	$0.672094\pi$
$762$	0	0
$763$	16.4329	0.594912
$764$	24.0847	0.871354
$765$	0	0
$766$	−22.3155	−0.806292
$767$	−12.2686	−0.442992
$768$	0	0
$769$	35.2344	1.27059	0.635293	−	0.772271i	$-0.280879\pi$
$769$	35.2344	1.27059	0.635293	+	0.772271i	$0.280879\pi$
$770$	1.33214	0.0480070
$771$	0	0
$772$	9.85978	0.354861
$773$	−20.1034	−0.723068	−0.361534	−	0.932359i	$-0.617747\pi$
$773$	−20.1034	−0.723068	−0.361534	+	0.932359i	$0.617747\pi$
$774$	0	0
$775$	26.6587	0.957608
$776$	−3.10101	−0.111320
$777$	0	0
$778$	−7.02229	−0.251761
$779$	0	0
$780$	0	0
$781$	−34.3182	−1.22800
$782$	−2.59896	−0.0929384
$783$	0	0
$784$	11.4115	0.407553
$785$	−0.479771	−0.0171238
$786$	0	0
$787$	23.8530	0.850267	0.425133	−	0.905131i	$-0.360227\pi$
$787$	23.8530	0.850267	0.425133	+	0.905131i	$0.360227\pi$
$788$	6.45605	0.229987
$789$	0	0
$790$	−0.585838	−0.0208432
$791$	−23.2199	−0.825604
$792$	0	0
$793$	1.39961	0.0497018
$794$	−19.9786	−0.709016
$795$	0	0
$796$	24.7716	0.878005
$797$	−28.5262	−1.01045	−0.505225	−	0.862988i	$-0.668591\pi$
$797$	−28.5262	−1.01045	−0.505225	+	0.862988i	$0.668591\pi$
$798$	0	0
$799$	4.43376	0.156855
$800$	−4.98545	−0.176262
$801$	0	0
$802$	−1.18984	−0.0420149
$803$	−5.87670	−0.207384
$804$	0	0
$805$	2.87462	0.101317
$806$	4.35916	0.153545
$807$	0	0
$808$	16.7297	0.588548
$809$	−24.9290	−0.876457	−0.438229	−	0.898863i	$-0.644394\pi$
$809$	−24.9290	−0.876457	−0.438229	+	0.898863i	$0.644394\pi$
$810$	0	0
$811$	−18.5243	−0.650478	−0.325239	−	0.945632i	$-0.605445\pi$
$811$	−18.5243	−0.650478	−0.325239	+	0.945632i	$0.605445\pi$
$812$	−10.0719	−0.353455
$813$	0	0
$814$	−21.9240	−0.768434
$815$	2.31584	0.0811202
$816$	0	0
$817$	0	0
$818$	−9.48845	−0.331756
$819$	0	0
$820$	0.462859	0.0161637
$821$	−35.5613	−1.24110	−0.620549	−	0.784168i	$-0.713090\pi$
$821$	−35.5613	−1.24110	−0.620549	+	0.784168i	$0.713090\pi$
$822$	0	0
$823$	−18.4953	−0.644704	−0.322352	−	0.946620i	$-0.604473\pi$
$823$	−18.4953	−0.644704	−0.322352	+	0.946620i	$0.604473\pi$
$824$	10.3969	0.362194
$825$	0	0
$826$	−64.5758	−2.24688
$827$	31.5945	1.09865	0.549324	−	0.835609i	$-0.314885\pi$
$827$	31.5945	1.09865	0.549324	+	0.835609i	$0.314885\pi$
$828$	0	0
$829$	41.1438	1.42898	0.714492	−	0.699643i	$-0.246658\pi$
$829$	41.1438	1.42898	0.714492	+	0.699643i	$0.246658\pi$
$830$	−0.391874	−0.0136021
$831$	0	0
$832$	−0.815207	−0.0282622
$833$	5.33956	0.185005
$834$	0	0
$835$	−1.85441	−0.0641744
$836$	0	0
$837$	0	0
$838$	−4.72638	−0.163270
$839$	19.9804	0.689800	0.344900	−	0.938639i	$-0.387913\pi$
$839$	19.9804	0.689800	0.344900	+	0.938639i	$0.387913\pi$
$840$	0	0
$841$	−23.4902	−0.810007
$842$	−1.38682	−0.0477930
$843$	0	0
$844$	1.38413	0.0476438
$845$	−1.48784	−0.0511831
$846$	0	0
$847$	18.7710	0.644978
$848$	12.7588	0.438138
$849$	0	0
$850$	−2.33275	−0.0800126
$851$	−47.3096	−1.62175
$852$	0	0
$853$	−18.9358	−0.648350	−0.324175	−	0.945997i	$-0.605087\pi$
$853$	−18.9358	−0.648350	−0.324175	+	0.945997i	$0.605087\pi$
$854$	7.36690	0.252090
$855$	0	0
$856$	−9.11381	−0.311504
$857$	−21.3402	−0.728966	−0.364483	−	0.931210i	$-0.618754\pi$
$857$	−21.3402	−0.728966	−0.364483	+	0.931210i	$0.618754\pi$
$858$	0	0
$859$	45.8367	1.56393	0.781964	−	0.623324i	$-0.214218\pi$
$859$	45.8367	1.56393	0.781964	+	0.623324i	$0.214218\pi$
$860$	1.12567	0.0383849
$861$	0	0
$862$	2.71183	0.0923653
$863$	15.2594	0.519436	0.259718	−	0.965685i	$-0.416370\pi$
$863$	15.2594	0.519436	0.259718	+	0.965685i	$0.416370\pi$
$864$	0	0
$865$	−0.795283	−0.0270404
$866$	20.2412	0.687825
$867$	0	0
$868$	22.9445	0.778787
$869$	12.5021	0.424103
$870$	0	0
$871$	9.33906	0.316442
$872$	−3.82976	−0.129692
$873$	0	0
$874$	0	0
$875$	5.16788	0.174706
$876$	0	0
$877$	−22.3806	−0.755740	−0.377870	−	0.925859i	$-0.623343\pi$
$877$	−22.3806	−0.755740	−0.377870	+	0.925859i	$0.623343\pi$
$878$	18.2003	0.614229
$879$	0	0
$880$	−0.310460	−0.0104656
$881$	27.0104	0.910004	0.455002	−	0.890490i	$-0.349638\pi$
$881$	27.0104	0.910004	0.455002	+	0.890490i	$0.349638\pi$
$882$	0	0
$883$	17.4142	0.586033	0.293017	−	0.956107i	$-0.405341\pi$
$883$	17.4142	0.586033	0.293017	+	0.956107i	$0.405341\pi$
$884$	−0.381445	−0.0128294
$885$	0	0
$886$	35.7425	1.20079
$887$	58.5768	1.96682	0.983408	−	0.181408i	$-0.0580655\pi$
$887$	58.5768	1.96682	0.983408	+	0.181408i	$0.0580655\pi$
$888$	0	0
$889$	18.1985	0.610359
$890$	0.413838	0.0138719
$891$	0	0
$892$	−22.3131	−0.747099
$893$	0	0
$894$	0	0
$895$	0.149711	0.00500427
$896$	−4.29086	−0.143348
$897$	0	0
$898$	−18.9905	−0.633721
$899$	−12.5517	−0.418622
$900$	0	0
$901$	5.96997	0.198889
$902$	−9.87763	−0.328889
$903$	0	0
$904$	5.41147	0.179983
$905$	−0.486081	−0.0161579
$906$	0	0
$907$	9.86753	0.327646	0.163823	−	0.986490i	$-0.447617\pi$
$907$	9.86753	0.327646	0.163823	+	0.986490i	$0.447617\pi$
$908$	−20.7665	−0.689161
$909$	0	0
$910$	0.421903	0.0139860
$911$	−13.9813	−0.463222	−0.231611	−	0.972808i	$-0.574400\pi$
$911$	−13.9813	−0.463222	−0.231611	+	0.972808i	$0.574400\pi$
$912$	0	0
$913$	8.36278	0.276768
$914$	25.9632	0.858785
$915$	0	0
$916$	5.51754	0.182305
$917$	90.2918	2.98170
$918$	0	0
$919$	−12.7246	−0.419747	−0.209873	−	0.977729i	$-0.567305\pi$
$919$	−12.7246	−0.419747	−0.209873	+	0.977729i	$0.567305\pi$
$920$	−0.669940	−0.0220873
$921$	0	0
$922$	29.2344	0.962784
$923$	−10.8690	−0.357756
$924$	0	0
$925$	−42.4638	−1.39620
$926$	−21.1908	−0.696372
$927$	0	0
$928$	2.34730	0.0770538
$929$	−20.9463	−0.687224	−0.343612	−	0.939112i	$-0.611651\pi$
$929$	−20.9463	−0.687224	−0.343612	+	0.939112i	$0.611651\pi$
$930$	0	0
$931$	0	0
$932$	15.0865	0.494174
$933$	0	0
$934$	2.14527	0.0701953
$935$	−0.145268	−0.00475076
$936$	0	0
$937$	−9.72462	−0.317690	−0.158845	−	0.987304i	$-0.550777\pi$
$937$	−9.72462	−0.317690	−0.158845	+	0.987304i	$0.550777\pi$
$938$	49.1563	1.60501
$939$	0	0
$940$	1.14290	0.0372774
$941$	−23.1857	−0.755833	−0.377917	−	0.925840i	$-0.623359\pi$
$941$	−23.1857	−0.755833	−0.377917	+	0.925840i	$0.623359\pi$
$942$	0	0
$943$	−21.3149	−0.694109
$944$	15.0496	0.489824
$945$	0	0
$946$	−24.0223	−0.781032
$947$	−27.5212	−0.894318	−0.447159	−	0.894455i	$-0.647564\pi$
$947$	−27.5212	−0.894318	−0.447159	+	0.894455i	$0.647564\pi$
$948$	0	0
$949$	−1.86122	−0.0604176
$950$	0	0
$951$	0	0
$952$	−2.00774	−0.0650713
$953$	50.4243	1.63340	0.816701	−	0.577061i	$-0.195800\pi$
$953$	50.4243	1.63340	0.816701	+	0.577061i	$0.195800\pi$
$954$	0	0
$955$	2.90497	0.0940027
$956$	28.9513	0.936352
$957$	0	0
$958$	3.39693	0.109750
$959$	−52.8672	−1.70717
$960$	0	0
$961$	−2.40642	−0.0776265
$962$	−6.94356	−0.223869
$963$	0	0
$964$	19.3696	0.623852
$965$	1.18924	0.0382828
$966$	0	0
$967$	31.8640	1.02468	0.512339	−	0.858783i	$-0.328779\pi$
$967$	31.8640	1.02468	0.512339	+	0.858783i	$0.328779\pi$
$968$	−4.37464	−0.140606
$969$	0	0
$970$	−0.374028	−0.0120093
$971$	−2.24123	−0.0719245	−0.0359622	−	0.999353i	$-0.511450\pi$
$971$	−2.24123	−0.0719245	−0.0359622	+	0.999353i	$0.511450\pi$
$972$	0	0
$973$	23.9796	0.768750
$974$	−19.7861	−0.633988
$975$	0	0
$976$	−1.71688	−0.0549560
$977$	−1.46791	−0.0469626	−0.0234813	−	0.999724i	$-0.507475\pi$
$977$	−1.46791	−0.0469626	−0.0234813	+	0.999724i	$0.507475\pi$
$978$	0	0
$979$	−8.83151	−0.282256
$980$	1.37639	0.0439672
$981$	0	0
$982$	−25.1438	−0.802372
$983$	30.0188	0.957450	0.478725	−	0.877965i	$-0.341099\pi$
$983$	30.0188	0.957450	0.478725	+	0.877965i	$0.341099\pi$
$984$	0	0
$985$	0.778695	0.0248113
$986$	1.09833	0.0349778
$987$	0	0
$988$	0	0
$989$	−51.8376	−1.64834
$990$	0	0
$991$	−54.7351	−1.73872	−0.869358	−	0.494183i	$-0.835467\pi$
$991$	−54.7351	−1.73872	−0.869358	+	0.494183i	$0.835467\pi$
$992$	−5.34730	−0.169777
$993$	0	0
$994$	−57.2089	−1.81456
$995$	2.98782	0.0947201
$996$	0	0
$997$	−48.6219	−1.53987	−0.769935	−	0.638123i	$-0.779711\pi$
$997$	−48.6219	−1.53987	−0.769935	+	0.638123i	$0.779711\pi$
$998$	35.1584	1.11292
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6498.2.a.bu.1.1		3
3.2	odd	2		2166.2.a.p.1.3		3
19.2	odd	18		342.2.u.b.289.1		6
19.10	odd	18		342.2.u.b.271.1		6
19.18	odd	2		6498.2.a.bp.1.1		3
57.2	even	18		114.2.i.c.61.1	yes	6
57.29	even	18		114.2.i.c.43.1	✓	6
57.56	even	2		2166.2.a.r.1.3		3
228.59	odd	18		912.2.bo.d.289.1		6
228.143	odd	18		912.2.bo.d.385.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.i.c.43.1	✓	6	57.29	even	18
114.2.i.c.61.1	yes	6	57.2	even	18
342.2.u.b.271.1		6	19.10	odd	18
342.2.u.b.289.1		6	19.2	odd	18
912.2.bo.d.289.1		6	228.59	odd	18
912.2.bo.d.385.1		6	228.143	odd	18
2166.2.a.p.1.3		3	3.2	odd	2
2166.2.a.r.1.3		3	57.56	even	2
6498.2.a.bp.1.1		3	19.18	odd	2
6498.2.a.bu.1.1		3	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:	\( N \)	\(=\)	\( 6498 = 2 \cdot 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6498.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.120615 q^{5} -4.29086 q^{7} +1.00000 q^{8} +0.120615 q^{10} -2.57398 q^{11} -0.815207 q^{13} -4.29086 q^{14} +1.00000 q^{16} +0.467911 q^{17} +0.120615 q^{20} -2.57398 q^{22} -5.55438 q^{23} -4.98545 q^{25} -0.815207 q^{26} -4.29086 q^{28} +2.34730 q^{29} -5.34730 q^{31} +1.00000 q^{32} +0.467911 q^{34} -0.517541 q^{35} +8.51754 q^{37} +0.120615 q^{40} +3.83750 q^{41} +9.33275 q^{43} -2.57398 q^{44} -5.55438 q^{46} +9.47565 q^{47} +11.4115 q^{49} -4.98545 q^{50} -0.815207 q^{52} +12.7588 q^{53} -0.310460 q^{55} -4.29086 q^{56} +2.34730 q^{58} +15.0496 q^{59} -1.71688 q^{61} -5.34730 q^{62} +1.00000 q^{64} -0.0983261 q^{65} -11.4561 q^{67} +0.467911 q^{68} -0.517541 q^{70} +13.3327 q^{71} +2.28312 q^{73} +8.51754 q^{74} +11.0446 q^{77} -4.85710 q^{79} +0.120615 q^{80} +3.83750 q^{82} -3.24897 q^{83} +0.0564370 q^{85} +9.33275 q^{86} -2.57398 q^{88} +3.43107 q^{89} +3.49794 q^{91} -5.55438 q^{92} +9.47565 q^{94} -3.10101 q^{97} +11.4115 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 3 q^{7} + 3 q^{8} + 6 q^{10} - 6 q^{13} + 3 q^{14} + 3 q^{16} + 6 q^{17} + 6 q^{20} - 6 q^{23} + 3 q^{25} - 6 q^{26} + 3 q^{28} + 6 q^{29} - 15 q^{31} + 3 q^{32} + 6 q^{34}+ \cdots + 24 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 + 3 * q^7 + 3 * q^8 + 6 * q^10 - 6 * q^13 + 3 * q^14 + 3 * q^16 + 6 * q^17 + 6 * q^20 - 6 * q^23 + 3 * q^25 - 6 * q^26 + 3 * q^28 + 6 * q^29 - 15 * q^31 + 3 * q^32 + 6 * q^34 + 21 * q^35 + 3 * q^37 + 6 * q^40 + 9 * q^41 + 9 * q^43 - 6 * q^46 + 9 * q^47 + 24 * q^49 + 3 * q^50 - 6 * q^52 + 27 * q^53 + 12 * q^55 + 3 * q^56 + 6 * q^58 + 18 * q^59 + 3 * q^61 - 15 * q^62 + 3 * q^64 - 12 * q^65 - 12 * q^67 + 6 * q^68 + 21 * q^70 + 21 * q^71 + 15 * q^73 + 3 * q^74 + 21 * q^77 - 15 * q^79 + 6 * q^80 + 9 * q^82 + 3 * q^83 + 15 * q^85 + 9 * q^86 + 3 * q^89 - 15 * q^91 - 6 * q^92 + 9 * q^94 - 12 * q^97 + 24 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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