LMFDB - Embedded newform 6498.2.a.bp.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6498,2,Mod(1,6498)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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)

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

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−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	$-0.463938\pi$
$13$	0.815207	0.226098	0.113049	+	0.993589i	$0.463938\pi$
$14$	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	$-0.569934\pi$
$29$	−2.34730	−0.435882	−0.217941	+	0.975962i	$0.569934\pi$
$30$	0	0
$31$	5.34730	0.960403	0.480201	−	0.877158i	$-0.340564\pi$
$31$	5.34730	0.960403	0.480201	+	0.877158i	$0.340564\pi$
$32$	−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	$-0.746878\pi$
$37$	−8.51754	−1.40028	−0.700138	+	0.714008i	$0.746878\pi$
$38$	0	0
$39$	0	0
$40$	−0.120615	−0.0190709
$41$	−3.83750	−0.599316	−0.299658	−	0.954047i	$-0.596873\pi$
$41$	−3.83750	−0.599316	−0.299658	+	0.954047i	$0.596873\pi$
$42$	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	$-0.839980\pi$
$53$	−12.7588	−1.75255	−0.876276	+	0.481810i	$0.839980\pi$
$54$	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	$-0.935670\pi$
$59$	−15.0496	−1.95929	−0.979647	+	0.200726i	$0.935670\pi$
$60$	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	$-0.253277\pi$
$67$	11.4561	1.39958	0.699790	+	0.714349i	$0.253277\pi$
$68$	0.467911	0.0567426
$69$	0	0
$70$	0.517541	0.0618580
$71$	−13.3327	−1.58231	−0.791153	−	0.611618i	$-0.790519\pi$
$71$	−13.3327	−1.58231	−0.791153	+	0.611618i	$0.790519\pi$
$72$	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	$-0.411907\pi$
$79$	4.85710	0.546466	0.273233	+	0.961948i	$0.411907\pi$
$80$	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	$-0.558207\pi$
$89$	−3.43107	−0.363693	−0.181847	+	0.983327i	$0.558207\pi$
$90$	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	$-0.449679\pi$
$97$	3.10101	0.314860	0.157430	+	0.987530i	$0.449679\pi$
$98$	−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	$-0.671177\pi$
$103$	−10.3969	−1.02444	−0.512220	+	0.858854i	$0.671177\pi$
$104$	−0.815207	−0.0799377
$105$	0	0
$106$	12.7588	1.23924
$107$	9.11381	0.881065	0.440533	−	0.897737i	$-0.354790\pi$
$107$	9.11381	0.881065	0.440533	+	0.897737i	$0.354790\pi$
$108$	0	0
$109$	3.82976	0.366824	0.183412	−	0.983036i	$-0.441286\pi$
$109$	3.82976	0.366824	0.183412	+	0.983036i	$0.441286\pi$
$110$	0.310460	0.0296012
$111$	0	0
$112$	−4.29086	−0.405448
$113$	−5.41147	−0.509069	−0.254534	−	0.967064i	$-0.581922\pi$
$113$	−5.41147	−0.509069	−0.254534	+	0.967064i	$0.581922\pi$
$114$	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	$-0.439743\pi$
$127$	4.24123	0.376348	0.188174	+	0.982136i	$0.439743\pi$
$128$	−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	$-0.328707\pi$
$151$	12.5963	1.02507	0.512535	+	0.858666i	$0.328707\pi$
$152$	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	$-0.297206\pi$
$167$	15.3746	1.18973	0.594863	+	0.803827i	$0.297206\pi$
$168$	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	$-0.419356\pi$
$173$	6.59358	0.501300	0.250650	+	0.968078i	$0.419356\pi$
$174$	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	$-0.514771\pi$
$179$	−1.24123	−0.0927738	−0.0463869	+	0.998924i	$0.514771\pi$
$180$	0	0
$181$	4.03003	0.299550	0.149775	−	0.988720i	$-0.452145\pi$
$181$	4.03003	0.299550	0.149775	+	0.988720i	$0.452145\pi$
$182$	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	$-0.615472\pi$
$193$	−9.85978	−0.709723	−0.354861	+	0.934919i	$0.615472\pi$
$194$	−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	$-0.515171\pi$
$211$	−1.38413	−0.0952876	−0.0476438	+	0.998864i	$0.515171\pi$
$212$	−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	$-0.231446\pi$
$223$	22.3131	1.49420	0.747099	+	0.664712i	$0.231446\pi$
$224$	4.29086	0.286695
$225$	0	0
$226$	5.41147	0.359966
$227$	20.7665	1.37832	0.689161	−	0.724608i	$-0.257979\pi$
$227$	20.7665	1.37832	0.689161	+	0.724608i	$0.257979\pi$
$228$	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	$-0.714433\pi$
$241$	−19.3696	−1.24770	−0.623852	+	0.781542i	$0.714433\pi$
$242$	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	$-0.686946\pi$
$257$	−17.7665	−1.10824	−0.554122	+	0.832435i	$0.686946\pi$
$258$	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	$-0.472622\pi$
$269$	2.81790	0.171810	0.0859051	+	0.996303i	$0.472622\pi$
$270$	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	$-0.813542\pi$
$281$	−27.9368	−1.66657	−0.833284	+	0.552846i	$0.813542\pi$
$282$	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	$-0.364867\pi$
$293$	14.1010	0.823790	0.411895	+	0.911231i	$0.364867\pi$
$294$	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	$-0.695636\pi$
$307$	−20.2071	−1.15328	−0.576640	+	0.816999i	$0.695636\pi$
$308$	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	$-0.661110\pi$
$317$	−17.2635	−0.969616	−0.484808	+	0.874621i	$0.661110\pi$
$318$	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	$-0.411467\pi$
$331$	9.99050	0.549128	0.274564	+	0.961569i	$0.411467\pi$
$332$	−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	$-0.441023\pi$
$337$	6.76382	0.368449	0.184224	+	0.982884i	$0.441023\pi$
$338$	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	$-0.348730\pi$
$373$	17.6732	0.915086	0.457543	+	0.889188i	$0.348730\pi$
$374$	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	$-0.580648\pi$
$379$	−9.75970	−0.501322	−0.250661	+	0.968075i	$0.580648\pi$
$380$	0	0
$381$	0	0
$382$	−24.0847	−1.23228
$383$	22.3155	1.14027	0.570135	−	0.821551i	$-0.306891\pi$
$383$	22.3155	1.14027	0.570135	+	0.821551i	$0.306891\pi$
$384$	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	$-0.490542\pi$
$401$	1.18984	0.0594180	0.0297090	+	0.999559i	$0.490542\pi$
$402$	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	$-0.424626\pi$
$409$	9.48845	0.469173	0.234587	+	0.972095i	$0.424626\pi$
$410$	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	$-0.489241\pi$
$421$	1.38682	0.0675895	0.0337948	+	0.999429i	$0.489241\pi$
$422$	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	$-0.520804\pi$
$431$	−2.71183	−0.130624	−0.0653121	+	0.997865i	$0.520804\pi$
$432$	0	0
$433$	−20.2412	−0.972731	−0.486366	−	0.873755i	$-0.661678\pi$
$433$	−20.2412	−0.972731	−0.486366	+	0.873755i	$0.661678\pi$
$434$	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	$-0.643013\pi$
$439$	−18.2003	−0.868652	−0.434326	+	0.900756i	$0.643013\pi$
$440$	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	$-0.352098\pi$
$449$	18.9905	0.896217	0.448109	+	0.893979i	$0.352098\pi$
$450$	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	$-0.352031\pi$
$487$	19.7861	0.896594	0.448297	+	0.893885i	$0.352031\pi$
$488$	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	$-0.405756\pi$
$509$	13.1652	0.583537	0.291768	+	0.956489i	$0.405756\pi$
$510$	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	$-0.648172\pi$
$521$	−20.4911	−0.897733	−0.448866	+	0.893599i	$0.648172\pi$
$522$	0	0
$523$	−42.9195	−1.87674	−0.938370	−	0.345633i	$-0.887664\pi$
$523$	−42.9195	−1.87674	−0.938370	+	0.345633i	$0.887664\pi$
$524$	−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	$-0.370060\pi$
$547$	18.5689	0.793950	0.396975	+	0.917829i	$0.370060\pi$
$548$	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	$-0.417546\pi$
$563$	12.1557	0.512302	0.256151	+	0.966637i	$0.417546\pi$
$564$	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	$-0.592511\pi$
$569$	−13.6709	−0.573113	−0.286556	+	0.958063i	$0.592511\pi$
$570$	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	$-0.266917\pi$
$599$	32.7246	1.33709	0.668546	+	0.743671i	$0.266917\pi$
$600$	0	0
$601$	−20.7733	−0.847361	−0.423681	−	0.905812i	$-0.639262\pi$
$601$	−20.7733	−0.847361	−0.423681	+	0.905812i	$0.639262\pi$
$602$	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	$-0.654889\pi$
$607$	−23.0419	−0.935241	−0.467621	+	0.883929i	$0.654889\pi$
$608$	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	$-0.551945\pi$
$641$	−8.22668	−0.324934	−0.162467	+	0.986714i	$0.551945\pi$
$642$	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	$-0.529455\pi$
$659$	−4.74422	−0.184809	−0.0924043	+	0.995722i	$0.529455\pi$
$660$	0	0
$661$	39.4056	1.53270	0.766350	−	0.642423i	$-0.222071\pi$
$661$	39.4056	1.53270	0.766350	+	0.642423i	$0.222071\pi$
$662$	−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	$-0.392815\pi$
$673$	17.1429	0.660810	0.330405	+	0.943839i	$0.392815\pi$
$674$	−6.76382	−0.260533
$675$	0	0
$676$	−12.3354	−0.474440
$677$	−7.13011	−0.274032	−0.137016	−	0.990569i	$-0.543751\pi$
$677$	−7.13011	−0.274032	−0.137016	+	0.990569i	$0.543751\pi$
$678$	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	$-0.334277\pi$
$683$	26.0000	0.994862	0.497431	+	0.867503i	$0.334277\pi$
$684$	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	$-0.390537\pi$
$743$	18.3800	0.674297	0.337149	+	0.941451i	$0.390537\pi$
$744$	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	$-0.310993\pi$
$751$	30.6655	1.11900	0.559500	+	0.828830i	$0.310993\pi$
$752$	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	$-0.382253\pi$
$773$	20.1034	0.723068	0.361534	+	0.932359i	$0.382253\pi$
$774$	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	$-0.639773\pi$
$787$	−23.8530	−0.850267	−0.425133	+	0.905131i	$0.639773\pi$
$788$	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	$-0.331409\pi$
$797$	28.5262	1.01045	0.505225	+	0.862988i	$0.331409\pi$
$798$	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	$-0.394555\pi$
$811$	18.5243	0.650478	0.325239	+	0.945632i	$0.394555\pi$
$812$	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	$-0.685115\pi$
$827$	−31.5945	−1.09865	−0.549324	+	0.835609i	$0.685115\pi$
$828$	0	0
$829$	−41.1438	−1.42898	−0.714492	−	0.699643i	$-0.753342\pi$
$829$	−41.1438	−1.42898	−0.714492	+	0.699643i	$0.753342\pi$
$830$	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	$-0.612087\pi$
$839$	−19.9804	−0.689800	−0.344900	+	0.938639i	$0.612087\pi$
$840$	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	$-0.381246\pi$
$857$	21.3402	0.728966	0.364483	+	0.931210i	$0.381246\pi$
$858$	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	$-0.583630\pi$
$863$	−15.2594	−0.519436	−0.259718	+	0.965685i	$0.583630\pi$
$864$	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	$-0.376657\pi$
$877$	22.3806	0.755740	0.377870	+	0.925859i	$0.376657\pi$
$878$	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	$-0.941934\pi$
$887$	−58.5768	−1.96682	−0.983408	+	0.181408i	$0.941934\pi$
$888$	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	$-0.552383\pi$
$907$	−9.86753	−0.327646	−0.163823	+	0.986490i	$0.552383\pi$
$908$	20.7665	0.689161
$909$	0	0
$910$	0.421903	0.0139860
$911$	13.9813	0.463222	0.231611	−	0.972808i	$-0.425600\pi$
$911$	13.9813	0.463222	0.231611	+	0.972808i	$0.425600\pi$
$912$	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	$-0.376641\pi$
$941$	23.1857	0.755833	0.377917	+	0.925840i	$0.376641\pi$
$942$	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	$-0.804200\pi$
$953$	−50.4243	−1.63340	−0.816701	+	0.577061i	$0.804200\pi$
$954$	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	$-0.488550\pi$
$971$	2.24123	0.0719245	0.0359622	+	0.999353i	$0.488550\pi$
$972$	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	$-0.492525\pi$
$977$	1.46791	0.0469626	0.0234813	+	0.999724i	$0.492525\pi$
$978$	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	$-0.658901\pi$
$983$	−30.0188	−0.957450	−0.478725	+	0.877965i	$0.658901\pi$
$984$	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	$-0.164533\pi$
$991$	54.7351	1.73872	0.869358	+	0.494183i	$0.164533\pi$
$992$	−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.bp.1.1		3
3.2	odd	2		2166.2.a.r.1.3		3
19.9	even	9		342.2.u.b.271.1		6
19.17	even	9		342.2.u.b.289.1		6
19.18	odd	2		6498.2.a.bu.1.1		3
57.17	odd	18		114.2.i.c.61.1	yes	6
57.47	odd	18		114.2.i.c.43.1	✓	6
57.56	even	2		2166.2.a.p.1.3		3
228.47	even	18		912.2.bo.d.385.1		6
228.131	even	18		912.2.bo.d.289.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.i.c.43.1	✓	6	57.47	odd	18
114.2.i.c.61.1	yes	6	57.17	odd	18
342.2.u.b.271.1		6	19.9	even	9
342.2.u.b.289.1		6	19.17	even	9
912.2.bo.d.289.1		6	228.131	even	18
912.2.bo.d.385.1		6	228.47	even	18
2166.2.a.p.1.3		3	57.56	even	2
2166.2.a.r.1.3		3	3.2	odd	2
6498.2.a.bp.1.1		3	1.1	even	1	trivial
6498.2.a.bu.1.1		3	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

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

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