LMFDB - Embedded newform 6561.2.a.c.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6561,2,Mod(1,6561)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6561, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6561.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6561 = 3^{8} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6561.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:	$52.3898487662$
Analytic rank:	$1$
Dimension:	$72$
Twist minimal:	no (minimal twist has level 81)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	6561.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.91259 q^{2} +1.65799 q^{4} +2.48473 q^{5} -4.55307 q^{7} +0.654119 q^{8} +O(q^{10})$	\(q-1.91259 q^{2} +1.65799 q^{4} +2.48473 q^{5} -4.55307 q^{7} +0.654119 q^{8} -4.75227 q^{10} +0.317801 q^{11} -3.40235 q^{13} +8.70815 q^{14} -4.56705 q^{16} +5.09977 q^{17} -1.24484 q^{19} +4.11967 q^{20} -0.607822 q^{22} -1.90780 q^{23} +1.17390 q^{25} +6.50729 q^{26} -7.54896 q^{28} -3.46559 q^{29} +6.78656 q^{31} +7.42664 q^{32} -9.75376 q^{34} -11.3132 q^{35} +8.53717 q^{37} +2.38087 q^{38} +1.62531 q^{40} -6.35091 q^{41} -6.09082 q^{43} +0.526911 q^{44} +3.64883 q^{46} +4.07772 q^{47} +13.7304 q^{49} -2.24519 q^{50} -5.64107 q^{52} +7.76272 q^{53} +0.789650 q^{55} -2.97825 q^{56} +6.62825 q^{58} +3.96901 q^{59} +2.45912 q^{61} -12.9799 q^{62} -5.07001 q^{64} -8.45393 q^{65} -3.44535 q^{67} +8.45538 q^{68} +21.6374 q^{70} -2.82097 q^{71} -4.22982 q^{73} -16.3281 q^{74} -2.06394 q^{76} -1.44697 q^{77} -7.24456 q^{79} -11.3479 q^{80} +12.1467 q^{82} -12.0814 q^{83} +12.6716 q^{85} +11.6492 q^{86} +0.207879 q^{88} +3.73584 q^{89} +15.4911 q^{91} -3.16312 q^{92} -7.79901 q^{94} -3.09310 q^{95} +14.6254 q^{97} -26.2607 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8}+O(q^{10}) $ 72 * q - 9 * q^2 + 63 * q^4 - 18 * q^5 - 27 * q^8	$ 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8} - 36 q^{11} - 36 q^{14} + 45 q^{16} - 36 q^{17} - 54 q^{20} - 54 q^{23} + 36 q^{25} - 45 q^{26} + 9 q^{28} - 54 q^{29} - 63 q^{32} - 72 q^{35} - 54 q^{38} - 72 q^{41} - 90 q^{44} - 90 q^{47} + 18 q^{49} - 45 q^{50} - 45 q^{53} + 9 q^{55} - 108 q^{56} + 18 q^{58} - 108 q^{59} - 72 q^{62} + 9 q^{64} - 72 q^{65} - 108 q^{68} - 126 q^{71} - 90 q^{74} - 72 q^{77} - 144 q^{80} - 18 q^{82} - 108 q^{83} - 90 q^{86} - 108 q^{89} - 72 q^{92} - 144 q^{95} - 81 q^{98}+O(q^{100}) $ 72 * q - 9 * q^2 + 63 * q^4 - 18 * q^5 - 27 * q^8 - 36 * q^11 - 36 * q^14 + 45 * q^16 - 36 * q^17 - 54 * q^20 - 54 * q^23 + 36 * q^25 - 45 * q^26 + 9 * q^28 - 54 * q^29 - 63 * q^32 - 72 * q^35 - 54 * q^38 - 72 * q^41 - 90 * q^44 - 90 * q^47 + 18 * q^49 - 45 * q^50 - 45 * q^53 + 9 * q^55 - 108 * q^56 + 18 * q^58 - 108 * q^59 - 72 * q^62 + 9 * q^64 - 72 * q^65 - 108 * q^68 - 126 * q^71 - 90 * q^74 - 72 * q^77 - 144 * q^80 - 18 * q^82 - 108 * q^83 - 90 * q^86 - 108 * q^89 - 72 * q^92 - 144 * q^95 - 81 * 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 $

$2$	−1.91259	−1.35240	−0.676202	−	0.736716i	$-0.736375\pi$
$2$	−1.91259	−1.35240	−0.676202	+	0.736716i	$0.736375\pi$
$3$	0	0
$3$	0	0
$4$	1.65799	0.828996
$5$	2.48473	1.11121	0.555603	−	0.831448i	$-0.312487\pi$
$5$	2.48473	1.11121	0.555603	+	0.831448i	$0.312487\pi$
$6$	0	0
$7$	−4.55307	−1.72090	−0.860449	−	0.509536i	$-0.829817\pi$
$7$	−4.55307	−1.72090	−0.860449	+	0.509536i	$0.829817\pi$
$8$	0.654119	0.231266
$9$	0	0
$10$	−4.75227	−1.50280
$11$	0.317801	0.0958205	0.0479103	−	0.998852i	$-0.484744\pi$
$11$	0.317801	0.0958205	0.0479103	+	0.998852i	$0.484744\pi$
$12$	0	0
$13$	−3.40235	−0.943642	−0.471821	−	0.881694i	$-0.656403\pi$
$13$	−3.40235	−0.943642	−0.471821	+	0.881694i	$0.656403\pi$
$14$	8.70815	2.32735
$15$	0	0
$16$	−4.56705	−1.14176
$17$	5.09977	1.23688	0.618438	−	0.785834i	$-0.287766\pi$
$17$	5.09977	1.23688	0.618438	+	0.785834i	$0.287766\pi$
$18$	0	0
$19$	−1.24484	−0.285586	−0.142793	−	0.989753i	$-0.545608\pi$
$19$	−1.24484	−0.285586	−0.142793	+	0.989753i	$0.545608\pi$
$20$	4.11967	0.921186
$21$	0	0
$22$	−0.607822	−0.129588
$23$	−1.90780	−0.397803	−0.198902	−	0.980019i	$-0.563737\pi$
$23$	−1.90780	−0.397803	−0.198902	+	0.980019i	$0.563737\pi$
$24$	0	0
$25$	1.17390	0.234780
$26$	6.50729	1.27618
$27$	0	0
$28$	−7.54896	−1.42662
$29$	−3.46559	−0.643544	−0.321772	−	0.946817i	$-0.604278\pi$
$29$	−3.46559	−0.643544	−0.321772	+	0.946817i	$0.604278\pi$
$30$	0	0
$31$	6.78656	1.21890	0.609451	−	0.792824i	$-0.291390\pi$
$31$	6.78656	1.21890	0.609451	+	0.792824i	$0.291390\pi$
$32$	7.42664	1.31286
$33$	0	0
$34$	−9.75376	−1.67276
$35$	−11.3132	−1.91227
$36$	0	0
$37$	8.53717	1.40350	0.701751	−	0.712422i	$-0.252402\pi$
$37$	8.53717	1.40350	0.701751	+	0.712422i	$0.252402\pi$
$38$	2.38087	0.386228
$39$	0	0
$40$	1.62531	0.256984
$41$	−6.35091	−0.991845	−0.495923	−	0.868367i	$-0.665170\pi$
$41$	−6.35091	−0.991845	−0.495923	+	0.868367i	$0.665170\pi$
$42$	0	0
$43$	−6.09082	−0.928842	−0.464421	−	0.885615i	$-0.653738\pi$
$43$	−6.09082	−0.928842	−0.464421	+	0.885615i	$0.653738\pi$
$44$	0.526911	0.0794349
$45$	0	0
$46$	3.64883	0.537991
$47$	4.07772	0.594797	0.297399	−	0.954753i	$-0.403881\pi$
$47$	4.07772	0.594797	0.297399	+	0.954753i	$0.403881\pi$
$48$	0	0
$49$	13.7304	1.96149
$50$	−2.24519	−0.317517
$51$	0	0
$52$	−5.64107	−0.782276
$53$	7.76272	1.06629	0.533146	−	0.846024i	$-0.321010\pi$
$53$	7.76272	1.06629	0.533146	+	0.846024i	$0.321010\pi$
$54$	0	0
$55$	0.789650	0.106476
$56$	−2.97825	−0.397985
$57$	0	0
$58$	6.62825	0.870332
$59$	3.96901	0.516721	0.258361	−	0.966049i	$-0.416818\pi$
$59$	3.96901	0.516721	0.258361	+	0.966049i	$0.416818\pi$
$60$	0	0
$61$	2.45912	0.314858	0.157429	−	0.987530i	$-0.449679\pi$
$61$	2.45912	0.314858	0.157429	+	0.987530i	$0.449679\pi$
$62$	−12.9799	−1.64845
$63$	0	0
$64$	−5.07001	−0.633751
$65$	−8.45393	−1.04858
$66$	0	0
$67$	−3.44535	−0.420916	−0.210458	−	0.977603i	$-0.567496\pi$
$67$	−3.44535	−0.420916	−0.210458	+	0.977603i	$0.567496\pi$
$68$	8.45538	1.02537
$69$	0	0
$70$	21.6374	2.58617
$71$	−2.82097	−0.334788	−0.167394	−	0.985890i	$-0.553535\pi$
$71$	−2.82097	−0.334788	−0.167394	+	0.985890i	$0.553535\pi$
$72$	0	0
$73$	−4.22982	−0.495063	−0.247531	−	0.968880i	$-0.579619\pi$
$73$	−4.22982	−0.495063	−0.247531	+	0.968880i	$0.579619\pi$
$74$	−16.3281	−1.89810
$75$	0	0
$76$	−2.06394	−0.236750
$77$	−1.44697	−0.164897
$78$	0	0
$79$	−7.24456	−0.815076	−0.407538	−	0.913188i	$-0.633613\pi$
$79$	−7.24456	−0.815076	−0.407538	+	0.913188i	$0.633613\pi$
$80$	−11.3479	−1.26873
$81$	0	0
$82$	12.1467	1.34138
$83$	−12.0814	−1.32611	−0.663053	−	0.748572i	$-0.730740\pi$
$83$	−12.0814	−1.32611	−0.663053	+	0.748572i	$0.730740\pi$
$84$	0	0
$85$	12.6716	1.37442
$86$	11.6492	1.25617
$87$	0	0
$88$	0.207879	0.0221600
$89$	3.73584	0.395998	0.197999	−	0.980202i	$-0.436556\pi$
$89$	3.73584	0.395998	0.197999	+	0.980202i	$0.436556\pi$
$90$	0	0
$91$	15.4911	1.62391
$92$	−3.16312	−0.329778
$93$	0	0
$94$	−7.79901	−0.804406
$95$	−3.09310	−0.317345
$96$	0	0
$97$	14.6254	1.48499	0.742493	−	0.669854i	$-0.233643\pi$
$97$	14.6254	1.48499	0.742493	+	0.669854i	$0.233643\pi$
$98$	−26.2607	−2.65273
$99$	0	0
$100$	1.94632	0.194632
$101$	17.7329	1.76449	0.882244	−	0.470792i	$-0.156032\pi$
$101$	17.7329	1.76449	0.882244	+	0.470792i	$0.156032\pi$
$102$	0	0
$103$	14.2535	1.40444	0.702219	−	0.711961i	$-0.252193\pi$
$103$	14.2535	1.40444	0.702219	+	0.711961i	$0.252193\pi$
$104$	−2.22554	−0.218232
$105$	0	0
$106$	−14.8469	−1.44206
$107$	−3.40986	−0.329644	−0.164822	−	0.986323i	$-0.552705\pi$
$107$	−3.40986	−0.329644	−0.164822	+	0.986323i	$0.552705\pi$
$108$	0	0
$109$	0.173590	0.0166269	0.00831345	−	0.999965i	$-0.497354\pi$
$109$	0.173590	0.0166269	0.00831345	+	0.999965i	$0.497354\pi$
$110$	−1.51028	−0.143999
$111$	0	0
$112$	20.7941	1.96486
$113$	−1.99830	−0.187984	−0.0939920	−	0.995573i	$-0.529963\pi$
$113$	−1.99830	−0.187984	−0.0939920	+	0.995573i	$0.529963\pi$
$114$	0	0
$115$	−4.74037	−0.442042
$116$	−5.74593	−0.533496
$117$	0	0
$118$	−7.59109	−0.698816
$119$	−23.2196	−2.12854
$120$	0	0
$121$	−10.8990	−0.990818
$122$	−4.70328	−0.425815
$123$	0	0
$124$	11.2521	1.01047
$125$	−9.50684	−0.850317
$126$	0	0
$127$	−3.18255	−0.282405	−0.141203	−	0.989981i	$-0.545097\pi$
$127$	−3.18255	−0.282405	−0.141203	+	0.989981i	$0.545097\pi$
$128$	−5.15644	−0.455769
$129$	0	0
$130$	16.1689	1.41811
$131$	−11.3410	−0.990867	−0.495434	−	0.868646i	$-0.664991\pi$
$131$	−11.3410	−0.990867	−0.495434	+	0.868646i	$0.664991\pi$
$132$	0	0
$133$	5.66785	0.491465
$134$	6.58953	0.569249
$135$	0	0
$136$	3.33585	0.286047
$137$	9.70862	0.829463	0.414732	−	0.909944i	$-0.363875\pi$
$137$	9.70862	0.829463	0.414732	+	0.909944i	$0.363875\pi$
$138$	0	0
$139$	7.10074	0.602277	0.301139	−	0.953580i	$-0.402633\pi$
$139$	7.10074	0.602277	0.301139	+	0.953580i	$0.402633\pi$
$140$	−18.7571	−1.58527
$141$	0	0
$142$	5.39535	0.452768
$143$	−1.08127	−0.0904203
$144$	0	0
$145$	−8.61107	−0.715111
$146$	8.08990	0.669525
$147$	0	0
$148$	14.1546	1.16350
$149$	−21.2557	−1.74134	−0.870668	−	0.491872i	$-0.836313\pi$
$149$	−21.2557	−1.74134	−0.870668	+	0.491872i	$0.836313\pi$
$150$	0	0
$151$	−20.1683	−1.64128	−0.820638	−	0.571448i	$-0.806382\pi$
$151$	−20.1683	−1.64128	−0.820638	+	0.571448i	$0.806382\pi$
$152$	−0.814274	−0.0660463
$153$	0	0
$154$	2.76746	0.223008
$155$	16.8628	1.35445
$156$	0	0
$157$	−8.08270	−0.645069	−0.322535	−	0.946558i	$-0.604535\pi$
$157$	−8.08270	−0.645069	−0.322535	+	0.946558i	$0.604535\pi$
$158$	13.8559	1.10231
$159$	0	0
$160$	18.4532	1.45885
$161$	8.68634	0.684579
$162$	0	0
$163$	−9.17644	−0.718754	−0.359377	−	0.933192i	$-0.617011\pi$
$163$	−9.17644	−0.718754	−0.359377	+	0.933192i	$0.617011\pi$
$164$	−10.5298	−0.822236
$165$	0	0
$166$	23.1067	1.79343
$167$	−16.3516	−1.26533	−0.632664	−	0.774426i	$-0.718039\pi$
$167$	−16.3516	−1.26533	−0.632664	+	0.774426i	$0.718039\pi$
$168$	0	0
$169$	−1.42402	−0.109540
$170$	−24.2355	−1.85878
$171$	0	0
$172$	−10.0985	−0.770007
$173$	11.8039	0.897434	0.448717	−	0.893674i	$-0.351881\pi$
$173$	11.8039	0.897434	0.448717	+	0.893674i	$0.351881\pi$
$174$	0	0
$175$	−5.34485	−0.404033
$176$	−1.45141	−0.109404
$177$	0	0
$178$	−7.14512	−0.535549
$179$	−6.32575	−0.472809	−0.236404	−	0.971655i	$-0.575969\pi$
$179$	−6.32575	−0.472809	−0.236404	+	0.971655i	$0.575969\pi$
$180$	0	0
$181$	−17.1488	−1.27466	−0.637330	−	0.770591i	$-0.719961\pi$
$181$	−17.1488	−1.27466	−0.637330	+	0.770591i	$0.719961\pi$
$182$	−29.6282	−2.19618
$183$	0	0
$184$	−1.24793	−0.0919984
$185$	21.2126	1.55958
$186$	0	0
$187$	1.62071	0.118518
$188$	6.76084	0.493085
$189$	0	0
$190$	5.91583	0.429179
$191$	7.69091	0.556494	0.278247	−	0.960509i	$-0.410247\pi$
$191$	7.69091	0.556494	0.278247	+	0.960509i	$0.410247\pi$
$192$	0	0
$193$	9.50992	0.684539	0.342269	−	0.939602i	$-0.388804\pi$
$193$	9.50992	0.684539	0.342269	+	0.939602i	$0.388804\pi$
$194$	−27.9724	−2.00830
$195$	0	0
$196$	22.7650	1.62607
$197$	−8.80030	−0.626996	−0.313498	−	0.949589i	$-0.601501\pi$
$197$	−8.80030	−0.626996	−0.313498	+	0.949589i	$0.601501\pi$
$198$	0	0
$199$	2.02333	0.143430	0.0717151	−	0.997425i	$-0.477153\pi$
$199$	2.02333	0.143430	0.0717151	+	0.997425i	$0.477153\pi$
$200$	0.767870	0.0542966
$201$	0	0
$202$	−33.9157	−2.38630
$203$	15.7791	1.10747
$204$	0	0
$205$	−15.7803	−1.10215
$206$	−27.2610	−1.89937
$207$	0	0
$208$	15.5387	1.07741
$209$	−0.395612	−0.0273650
$210$	0	0
$211$	−25.8502	−1.77960	−0.889802	−	0.456347i	$-0.849157\pi$
$211$	−25.8502	−1.77960	−0.889802	+	0.456347i	$0.849157\pi$
$212$	12.8705	0.883952
$213$	0	0
$214$	6.52166	0.445812
$215$	−15.1341	−1.03214
$216$	0	0
$217$	−30.8997	−2.09761
$218$	−0.332006	−0.0224863
$219$	0	0
$220$	1.30923	0.0882686
$221$	−17.3512	−1.16717
$222$	0	0
$223$	−8.60277	−0.576084	−0.288042	−	0.957618i	$-0.593004\pi$
$223$	−8.60277	−0.576084	−0.288042	+	0.957618i	$0.593004\pi$
$224$	−33.8140	−2.25929
$225$	0	0
$226$	3.82192	0.254230
$227$	18.4664	1.22566	0.612830	−	0.790215i	$-0.290031\pi$
$227$	18.4664	1.22566	0.612830	+	0.790215i	$0.290031\pi$
$228$	0	0
$229$	−0.116898	−0.00772481	−0.00386240	−	0.999993i	$-0.501229\pi$
$229$	−0.116898	−0.00772481	−0.00386240	+	0.999993i	$0.501229\pi$
$230$	9.06638	0.597819
$231$	0	0
$232$	−2.26691	−0.148830
$233$	−11.3718	−0.744993	−0.372496	−	0.928034i	$-0.621498\pi$
$233$	−11.3718	−0.744993	−0.372496	+	0.928034i	$0.621498\pi$
$234$	0	0
$235$	10.1321	0.660943
$236$	6.58059	0.428360
$237$	0	0
$238$	44.4095	2.87864
$239$	6.00715	0.388570	0.194285	−	0.980945i	$-0.437761\pi$
$239$	6.00715	0.388570	0.194285	+	0.980945i	$0.437761\pi$
$240$	0	0
$241$	18.7989	1.21094	0.605472	−	0.795867i	$-0.292985\pi$
$241$	18.7989	1.21094	0.605472	+	0.795867i	$0.292985\pi$
$242$	20.8453	1.33999
$243$	0	0
$244$	4.07720	0.261016
$245$	34.1165	2.17962
$246$	0	0
$247$	4.23539	0.269491
$248$	4.43922	0.281890
$249$	0	0
$250$	18.1827	1.14997
$251$	−17.6564	−1.11446	−0.557231	−	0.830358i	$-0.688136\pi$
$251$	−17.6564	−1.11446	−0.557231	+	0.830358i	$0.688136\pi$
$252$	0	0
$253$	−0.606300	−0.0381177
$254$	6.08690	0.381926
$255$	0	0
$256$	20.0022	1.25014
$257$	−13.2603	−0.827153	−0.413576	−	0.910469i	$-0.635721\pi$
$257$	−13.2603	−0.827153	−0.413576	+	0.910469i	$0.635721\pi$
$258$	0	0
$259$	−38.8703	−2.41528
$260$	−14.0166	−0.869270
$261$	0	0
$262$	21.6907	1.34005
$263$	2.31332	0.142646	0.0713228	−	0.997453i	$-0.477278\pi$
$263$	2.31332	0.142646	0.0713228	+	0.997453i	$0.477278\pi$
$264$	0	0
$265$	19.2883	1.18487
$266$	−10.8403	−0.664659
$267$	0	0
$268$	−5.71236	−0.348938
$269$	18.3134	1.11659	0.558294	−	0.829643i	$-0.311456\pi$
$269$	18.3134	1.11659	0.558294	+	0.829643i	$0.311456\pi$
$270$	0	0
$271$	9.15164	0.555923	0.277961	−	0.960592i	$-0.410341\pi$
$271$	9.15164	0.555923	0.277961	+	0.960592i	$0.410341\pi$
$272$	−23.2909	−1.41222
$273$	0	0
$274$	−18.5686	−1.12177
$275$	0.373066	0.0224967
$276$	0	0
$277$	−16.6390	−0.999739	−0.499869	−	0.866101i	$-0.666619\pi$
$277$	−16.6390	−0.999739	−0.499869	+	0.866101i	$0.666619\pi$
$278$	−13.5808	−0.814522
$279$	0	0
$280$	−7.40015	−0.442244
$281$	6.87767	0.410287	0.205144	−	0.978732i	$-0.434234\pi$
$281$	6.87767	0.410287	0.205144	+	0.978732i	$0.434234\pi$
$282$	0	0
$283$	22.4289	1.33326	0.666631	−	0.745388i	$-0.267736\pi$
$283$	22.4289	1.33326	0.666631	+	0.745388i	$0.267736\pi$
$284$	−4.67715	−0.277538
$285$	0	0
$286$	2.06802	0.122285
$287$	28.9161	1.70687
$288$	0	0
$289$	9.00763	0.529861
$290$	16.4694	0.967118
$291$	0	0
$292$	−7.01301	−0.410405
$293$	−12.6529	−0.739190	−0.369595	−	0.929193i	$-0.620504\pi$
$293$	−12.6529	−0.739190	−0.369595	+	0.929193i	$0.620504\pi$
$294$	0	0
$295$	9.86194	0.574184
$296$	5.58432	0.324582
$297$	0	0
$298$	40.6534	2.35499
$299$	6.49100	0.375384
$300$	0	0
$301$	27.7319	1.59844
$302$	38.5737	2.21967
$303$	0	0
$304$	5.68525	0.326071
$305$	6.11026	0.349872
$306$	0	0
$307$	0.844202	0.0481811	0.0240906	−	0.999710i	$-0.492331\pi$
$307$	0.844202	0.0481811	0.0240906	+	0.999710i	$0.492331\pi$
$308$	−2.39906	−0.136699
$309$	0	0
$310$	−32.2516	−1.83177
$311$	−5.42802	−0.307795	−0.153897	−	0.988087i	$-0.549183\pi$
$311$	−5.42802	−0.307795	−0.153897	+	0.988087i	$0.549183\pi$
$312$	0	0
$313$	−21.3920	−1.20915	−0.604574	−	0.796549i	$-0.706657\pi$
$313$	−21.3920	−1.20915	−0.604574	+	0.796549i	$0.706657\pi$
$314$	15.4589	0.872394
$315$	0	0
$316$	−12.0114	−0.675695
$317$	3.98930	0.224061	0.112031	−	0.993705i	$-0.464264\pi$
$317$	3.98930	0.224061	0.112031	+	0.993705i	$0.464264\pi$
$318$	0	0
$319$	−1.10137	−0.0616647
$320$	−12.5976	−0.704229
$321$	0	0
$322$	−16.6134	−0.925828
$323$	−6.34840	−0.353235
$324$	0	0
$325$	−3.99402	−0.221548
$326$	17.5507	0.972046
$327$	0	0
$328$	−4.15425	−0.229380
$329$	−18.5662	−1.02359
$330$	0	0
$331$	7.62895	0.419325	0.209662	−	0.977774i	$-0.432763\pi$
$331$	7.62895	0.419325	0.209662	+	0.977774i	$0.432763\pi$
$332$	−20.0309	−1.09934
$333$	0	0
$334$	31.2740	1.71123
$335$	−8.56077	−0.467725
$336$	0	0
$337$	−5.35197	−0.291540	−0.145770	−	0.989318i	$-0.546566\pi$
$337$	−5.35197	−0.291540	−0.145770	+	0.989318i	$0.546566\pi$
$338$	2.72357	0.148142
$339$	0	0
$340$	21.0094	1.13939
$341$	2.15677	0.116796
$342$	0	0
$343$	−30.6442	−1.65463
$344$	−3.98412	−0.214809
$345$	0	0
$346$	−22.5760	−1.21369
$347$	13.2310	0.710280	0.355140	−	0.934813i	$-0.384433\pi$
$347$	13.2310	0.710280	0.355140	+	0.934813i	$0.384433\pi$
$348$	0	0
$349$	18.1331	0.970641	0.485321	−	0.874336i	$-0.338703\pi$
$349$	18.1331	0.970641	0.485321	+	0.874336i	$0.338703\pi$
$350$	10.2225	0.546415
$351$	0	0
$352$	2.36019	0.125799
$353$	−7.13365	−0.379686	−0.189843	−	0.981814i	$-0.560798\pi$
$353$	−7.13365	−0.379686	−0.189843	+	0.981814i	$0.560798\pi$
$354$	0	0
$355$	−7.00936	−0.372018
$356$	6.19399	0.328281
$357$	0	0
$358$	12.0986	0.639428
$359$	−13.4077	−0.707630	−0.353815	−	0.935315i	$-0.615116\pi$
$359$	−13.4077	−0.707630	−0.353815	+	0.935315i	$0.615116\pi$
$360$	0	0
$361$	−17.4504	−0.918440
$362$	32.7986	1.72386
$363$	0	0
$364$	25.6842	1.34622
$365$	−10.5100	−0.550117
$366$	0	0
$367$	18.2648	0.953417	0.476709	−	0.879061i	$-0.341830\pi$
$367$	18.2648	0.953417	0.476709	+	0.879061i	$0.341830\pi$
$368$	8.71300	0.454197
$369$	0	0
$370$	−40.5709	−2.10918
$371$	−35.3442	−1.83498
$372$	0	0
$373$	32.0643	1.66023	0.830113	−	0.557595i	$-0.188276\pi$
$373$	32.0643	1.66023	0.830113	+	0.557595i	$0.188276\pi$
$374$	−3.09975	−0.160284
$375$	0	0
$376$	2.66732	0.137556
$377$	11.7912	0.607275
$378$	0	0
$379$	−14.0025	−0.719259	−0.359629	−	0.933095i	$-0.617097\pi$
$379$	−14.0025	−0.719259	−0.359629	+	0.933095i	$0.617097\pi$
$380$	−5.12834	−0.263078
$381$	0	0
$382$	−14.7095	−0.752605
$383$	6.96459	0.355874	0.177937	−	0.984042i	$-0.443058\pi$
$383$	6.96459	0.355874	0.177937	+	0.984042i	$0.443058\pi$
$384$	0	0
$385$	−3.59533	−0.183235
$386$	−18.1886	−0.925773
$387$	0	0
$388$	24.2488	1.23105
$389$	−7.41351	−0.375880	−0.187940	−	0.982181i	$-0.560181\pi$
$389$	−7.41351	−0.375880	−0.187940	+	0.982181i	$0.560181\pi$
$390$	0	0
$391$	−9.72933	−0.492033
$392$	8.98134	0.453626
$393$	0	0
$394$	16.8314	0.847951
$395$	−18.0008	−0.905718
$396$	0	0
$397$	10.6682	0.535422	0.267711	−	0.963499i	$-0.413733\pi$
$397$	10.6682	0.535422	0.267711	+	0.963499i	$0.413733\pi$
$398$	−3.86980	−0.193976
$399$	0	0
$400$	−5.36126	−0.268063
$401$	−23.3362	−1.16535	−0.582677	−	0.812704i	$-0.697995\pi$
$401$	−23.3362	−1.16535	−0.582677	+	0.812704i	$0.697995\pi$
$402$	0	0
$403$	−23.0902	−1.15021
$404$	29.4010	1.46275
$405$	0	0
$406$	−30.1789	−1.49775
$407$	2.71312	0.134484
$408$	0	0
$409$	9.30429	0.460067	0.230034	−	0.973183i	$-0.426116\pi$
$409$	9.30429	0.460067	0.230034	+	0.973183i	$0.426116\pi$
$410$	30.1812	1.49055
$411$	0	0
$412$	23.6322	1.16427
$413$	−18.0712	−0.889225
$414$	0	0
$415$	−30.0191	−1.47358
$416$	−25.2680	−1.23887
$417$	0	0
$418$	0.756642	0.0370086
$419$	−2.85835	−0.139639	−0.0698197	−	0.997560i	$-0.522242\pi$
$419$	−2.85835	−0.139639	−0.0698197	+	0.997560i	$0.522242\pi$
$420$	0	0
$421$	16.5162	0.804950	0.402475	−	0.915431i	$-0.368150\pi$
$421$	16.5162	0.804950	0.402475	+	0.915431i	$0.368150\pi$
$422$	49.4408	2.40674
$423$	0	0
$424$	5.07774	0.246597
$425$	5.98662	0.290394
$426$	0	0
$427$	−11.1965	−0.541839
$428$	−5.65353	−0.273274
$429$	0	0
$430$	28.9453	1.39586
$431$	−23.7722	−1.14507	−0.572534	−	0.819881i	$-0.694040\pi$
$431$	−23.7722	−1.14507	−0.572534	+	0.819881i	$0.694040\pi$
$432$	0	0
$433$	−2.93302	−0.140952	−0.0704761	−	0.997513i	$-0.522452\pi$
$433$	−2.93302	−0.140952	−0.0704761	+	0.997513i	$0.522452\pi$
$434$	59.0984	2.83681
$435$	0	0
$436$	0.287811	0.0137836
$437$	2.37491	0.113607
$438$	0	0
$439$	33.5957	1.60343	0.801717	−	0.597704i	$-0.203920\pi$
$439$	33.5957	1.60343	0.801717	+	0.597704i	$0.203920\pi$
$440$	0.516525	0.0246244
$441$	0	0
$442$	33.1857	1.57848
$443$	24.1189	1.14592	0.572962	−	0.819582i	$-0.305794\pi$
$443$	24.1189	1.14592	0.572962	+	0.819582i	$0.305794\pi$
$444$	0	0
$445$	9.28256	0.440036
$446$	16.4535	0.779098
$447$	0	0
$448$	23.0841	1.09062
$449$	−27.9463	−1.31887	−0.659435	−	0.751762i	$-0.729204\pi$
$449$	−27.9463	−1.31887	−0.659435	+	0.751762i	$0.729204\pi$
$450$	0	0
$451$	−2.01832	−0.0950392
$452$	−3.31316	−0.155838
$453$	0	0
$454$	−35.3187	−1.65759
$455$	38.4913	1.80450
$456$	0	0
$457$	−14.2905	−0.668481	−0.334240	−	0.942488i	$-0.608480\pi$
$457$	−14.2905	−0.668481	−0.334240	+	0.942488i	$0.608480\pi$
$458$	0.223577	0.0104471
$459$	0	0
$460$	−7.85950	−0.366451
$461$	2.45670	0.114420	0.0572099	−	0.998362i	$-0.481780\pi$
$461$	2.45670	0.114420	0.0572099	+	0.998362i	$0.481780\pi$
$462$	0	0
$463$	−38.6985	−1.79847	−0.899236	−	0.437465i	$-0.855876\pi$
$463$	−38.6985	−1.79847	−0.899236	+	0.437465i	$0.855876\pi$
$464$	15.8275	0.734774
$465$	0	0
$466$	21.7496	1.00753
$467$	−26.6365	−1.23259	−0.616295	−	0.787516i	$-0.711367\pi$
$467$	−26.6365	−1.23259	−0.616295	+	0.787516i	$0.711367\pi$
$468$	0	0
$469$	15.6869	0.724354
$470$	−19.3785	−0.893861
$471$	0	0
$472$	2.59620	0.119500
$473$	−1.93567	−0.0890021
$474$	0	0
$475$	−1.46132	−0.0670500
$476$	−38.4979	−1.76455
$477$	0	0
$478$	−11.4892	−0.525504
$479$	−35.5689	−1.62518	−0.812592	−	0.582833i	$-0.801944\pi$
$479$	−35.5689	−1.62518	−0.812592	+	0.582833i	$0.801944\pi$
$480$	0	0
$481$	−29.0464	−1.32440
$482$	−35.9545	−1.63768
$483$	0	0
$484$	−18.0705	−0.821385
$485$	36.3403	1.65013
$486$	0	0
$487$	1.87640	0.0850280	0.0425140	−	0.999096i	$-0.486463\pi$
$487$	1.87640	0.0850280	0.0425140	+	0.999096i	$0.486463\pi$
$488$	1.60856	0.0728159
$489$	0	0
$490$	−65.2508	−2.94773
$491$	−18.0842	−0.816130	−0.408065	−	0.912953i	$-0.633796\pi$
$491$	−18.0842	−0.816130	−0.408065	+	0.912953i	$0.633796\pi$
$492$	0	0
$493$	−17.6737	−0.795984
$494$	−8.10055	−0.364461
$495$	0	0
$496$	−30.9945	−1.39170
$497$	12.8441	0.576135
$498$	0	0
$499$	2.73770	0.122556	0.0612781	−	0.998121i	$-0.480482\pi$
$499$	2.73770	0.122556	0.0612781	+	0.998121i	$0.480482\pi$
$500$	−15.7623	−0.704910
$501$	0	0
$502$	33.7694	1.50720
$503$	−13.5127	−0.602503	−0.301252	−	0.953545i	$-0.597404\pi$
$503$	−13.5127	−0.602503	−0.301252	+	0.953545i	$0.597404\pi$
$504$	0	0
$505$	44.0615	1.96071
$506$	1.15960	0.0515506
$507$	0	0
$508$	−5.27664	−0.234113
$509$	−41.2343	−1.82768	−0.913839	−	0.406077i	$-0.866896\pi$
$509$	−41.2343	−1.82768	−0.913839	+	0.406077i	$0.866896\pi$
$510$	0	0
$511$	19.2586	0.851952
$512$	−27.9430	−1.23492
$513$	0	0
$514$	25.3614	1.11864
$515$	35.4161	1.56062
$516$	0	0
$517$	1.29590	0.0569938
$518$	74.3429	3.26644
$519$	0	0
$520$	−5.52987	−0.242501
$521$	12.2459	0.536502	0.268251	−	0.963349i	$-0.413554\pi$
$521$	12.2459	0.536502	0.268251	+	0.963349i	$0.413554\pi$
$522$	0	0
$523$	−4.60384	−0.201312	−0.100656	−	0.994921i	$-0.532094\pi$
$523$	−4.60384	−0.201312	−0.100656	+	0.994921i	$0.532094\pi$
$524$	−18.8033	−0.821426
$525$	0	0
$526$	−4.42443	−0.192915
$527$	34.6099	1.50763
$528$	0	0
$529$	−19.3603	−0.841752
$530$	−36.8905	−1.60242
$531$	0	0
$532$	9.39726	0.407423
$533$	21.6080	0.935947
$534$	0	0
$535$	−8.47260	−0.366302
$536$	−2.25367	−0.0973435
$537$	0	0
$538$	−35.0260	−1.51008
$539$	4.36355	0.187951
$540$	0	0
$541$	29.1406	1.25285	0.626427	−	0.779480i	$-0.284517\pi$
$541$	29.1406	1.25285	0.626427	+	0.779480i	$0.284517\pi$
$542$	−17.5033	−0.751832
$543$	0	0
$544$	37.8741	1.62384
$545$	0.431325	0.0184759
$546$	0	0
$547$	−29.5683	−1.26425	−0.632124	−	0.774868i	$-0.717817\pi$
$547$	−29.5683	−1.26425	−0.632124	+	0.774868i	$0.717817\pi$
$548$	16.0968	0.687622
$549$	0	0
$550$	−0.713522	−0.0304247
$551$	4.31411	0.183787
$552$	0	0
$553$	32.9850	1.40266
$554$	31.8235	1.35205
$555$	0	0
$556$	11.7730	0.499286
$557$	−34.8999	−1.47875	−0.739377	−	0.673291i	$-0.764880\pi$
$557$	−34.8999	−1.47875	−0.739377	+	0.673291i	$0.764880\pi$
$558$	0	0
$559$	20.7231	0.876494
$560$	51.6677	2.18336
$561$	0	0
$562$	−13.1541	−0.554874
$563$	40.6269	1.71222	0.856109	−	0.516796i	$-0.172875\pi$
$563$	40.6269	1.71222	0.856109	+	0.516796i	$0.172875\pi$
$564$	0	0
$565$	−4.96523	−0.208889
$566$	−42.8973	−1.80311
$567$	0	0
$568$	−1.84525	−0.0774249
$569$	−30.2015	−1.26611	−0.633056	−	0.774106i	$-0.718200\pi$
$569$	−30.2015	−1.26611	−0.633056	+	0.774106i	$0.718200\pi$
$570$	0	0
$571$	−39.1109	−1.63674	−0.818371	−	0.574691i	$-0.805122\pi$
$571$	−39.1109	−1.63674	−0.818371	+	0.574691i	$0.805122\pi$
$572$	−1.79274	−0.0749581
$573$	0	0
$574$	−55.3047	−2.30837
$575$	−2.23957	−0.0933963
$576$	0	0
$577$	−19.7648	−0.822819	−0.411409	−	0.911451i	$-0.634963\pi$
$577$	−19.7648	−0.822819	−0.411409	+	0.911451i	$0.634963\pi$
$578$	−17.2279	−0.716586
$579$	0	0
$580$	−14.2771	−0.592824
$581$	55.0075	2.28209
$582$	0	0
$583$	2.46700	0.102173
$584$	−2.76680	−0.114491
$585$	0	0
$586$	24.1998	0.999683
$587$	−13.7458	−0.567349	−0.283675	−	0.958921i	$-0.591554\pi$
$587$	−13.7458	−0.567349	−0.283675	+	0.958921i	$0.591554\pi$
$588$	0	0
$589$	−8.44819	−0.348102
$590$	−18.8618	−0.776529
$591$	0	0
$592$	−38.9896	−1.60246
$593$	21.5632	0.885494	0.442747	−	0.896647i	$-0.354004\pi$
$593$	21.5632	0.885494	0.442747	+	0.896647i	$0.354004\pi$
$594$	0	0
$595$	−57.6945	−2.36524
$596$	−35.2418	−1.44356
$597$	0	0
$598$	−12.4146	−0.507671
$599$	−16.5582	−0.676551	−0.338276	−	0.941047i	$-0.609844\pi$
$599$	−16.5582	−0.676551	−0.338276	+	0.941047i	$0.609844\pi$
$600$	0	0
$601$	−18.5087	−0.754984	−0.377492	−	0.926013i	$-0.623213\pi$
$601$	−18.5087	−0.754984	−0.377492	+	0.926013i	$0.623213\pi$
$602$	−53.0398	−2.16174
$603$	0	0
$604$	−33.4390	−1.36061
$605$	−27.0811	−1.10100
$606$	0	0
$607$	21.8623	0.887362	0.443681	−	0.896185i	$-0.353672\pi$
$607$	21.8623	0.887362	0.443681	+	0.896185i	$0.353672\pi$
$608$	−9.24499	−0.374934
$609$	0	0
$610$	−11.6864	−0.473169
$611$	−13.8738	−0.561276
$612$	0	0
$613$	−14.5904	−0.589301	−0.294650	−	0.955605i	$-0.595203\pi$
$613$	−14.5904	−0.589301	−0.294650	+	0.955605i	$0.595203\pi$
$614$	−1.61461	−0.0651603
$615$	0	0
$616$	−0.946489	−0.0381351
$617$	43.6084	1.75561	0.877804	−	0.479019i	$-0.159008\pi$
$617$	43.6084	1.75561	0.877804	+	0.479019i	$0.159008\pi$
$618$	0	0
$619$	6.48658	0.260717	0.130359	−	0.991467i	$-0.458387\pi$
$619$	6.48658	0.260717	0.130359	+	0.991467i	$0.458387\pi$
$620$	27.9584	1.12284
$621$	0	0
$622$	10.3816	0.416263
$623$	−17.0095	−0.681472
$624$	0	0
$625$	−29.4915	−1.17966
$626$	40.9141	1.63526
$627$	0	0
$628$	−13.4011	−0.534760
$629$	43.5376	1.73596
$630$	0	0
$631$	−17.1800	−0.683923	−0.341962	−	0.939714i	$-0.611091\pi$
$631$	−17.1800	−0.683923	−0.341962	+	0.939714i	$0.611091\pi$
$632$	−4.73880	−0.188499
$633$	0	0
$634$	−7.62989	−0.303022
$635$	−7.90778	−0.313811
$636$	0	0
$637$	−46.7158	−1.85095
$638$	2.10646	0.0833956
$639$	0	0
$640$	−12.8124	−0.506453
$641$	−32.1627	−1.27035	−0.635176	−	0.772367i	$-0.719072\pi$
$641$	−32.1627	−1.27035	−0.635176	+	0.772367i	$0.719072\pi$
$642$	0	0
$643$	20.2283	0.797727	0.398863	−	0.917010i	$-0.369405\pi$
$643$	20.2283	0.797727	0.398863	+	0.917010i	$0.369405\pi$
$644$	14.4019	0.567514
$645$	0	0
$646$	12.1419	0.477716
$647$	−47.4222	−1.86436	−0.932179	−	0.361999i	$-0.882094\pi$
$647$	−47.4222	−1.86436	−0.932179	+	0.361999i	$0.882094\pi$
$648$	0	0
$649$	1.26136	0.0495125
$650$	7.63891	0.299623
$651$	0	0
$652$	−15.2145	−0.595845
$653$	3.64019	0.142451	0.0712257	−	0.997460i	$-0.477309\pi$
$653$	3.64019	0.142451	0.0712257	+	0.997460i	$0.477309\pi$
$654$	0	0
$655$	−28.1794	−1.10106
$656$	29.0049	1.13245
$657$	0	0
$658$	35.5094	1.38430
$659$	14.7090	0.572981	0.286490	−	0.958083i	$-0.407511\pi$
$659$	14.7090	0.572981	0.286490	+	0.958083i	$0.407511\pi$
$660$	0	0
$661$	−2.25606	−0.0877505	−0.0438752	−	0.999037i	$-0.513970\pi$
$661$	−2.25606	−0.0877505	−0.0438752	+	0.999037i	$0.513970\pi$
$662$	−14.5910	−0.567097
$663$	0	0
$664$	−7.90267	−0.306683
$665$	14.0831	0.546119
$666$	0	0
$667$	6.61165	0.256004
$668$	−27.1109	−1.04895
$669$	0	0
$670$	16.3732	0.632553
$671$	0.781510	0.0301699
$672$	0	0
$673$	39.5982	1.52640	0.763200	−	0.646162i	$-0.223627\pi$
$673$	39.5982	1.52640	0.763200	+	0.646162i	$0.223627\pi$
$674$	10.2361	0.394280
$675$	0	0
$676$	−2.36102	−0.0908083
$677$	−23.5482	−0.905032	−0.452516	−	0.891756i	$-0.649473\pi$
$677$	−23.5482	−0.905032	−0.452516	+	0.891756i	$0.649473\pi$
$678$	0	0
$679$	−66.5905	−2.55551
$680$	8.28871	0.317857
$681$	0	0
$682$	−4.12502	−0.157955
$683$	17.9368	0.686333	0.343167	−	0.939275i	$-0.388500\pi$
$683$	17.9368	0.686333	0.343167	+	0.939275i	$0.388500\pi$
$684$	0	0
$685$	24.1233	0.921705
$686$	58.6097	2.23773
$687$	0	0
$688$	27.8171	1.06052
$689$	−26.4115	−1.00620
$690$	0	0
$691$	−44.4428	−1.69068	−0.845342	−	0.534226i	$-0.820603\pi$
$691$	−44.4428	−1.69068	−0.845342	+	0.534226i	$0.820603\pi$
$692$	19.5708	0.743970
$693$	0	0
$694$	−25.3055	−0.960585
$695$	17.6434	0.669254
$696$	0	0
$697$	−32.3882	−1.22679
$698$	−34.6811	−1.31270
$699$	0	0
$700$	−8.86172	−0.334942
$701$	−30.2178	−1.14131	−0.570656	−	0.821189i	$-0.693311\pi$
$701$	−30.2178	−1.14131	−0.570656	+	0.821189i	$0.693311\pi$
$702$	0	0
$703$	−10.6274	−0.400821
$704$	−1.61125	−0.0607264
$705$	0	0
$706$	13.6437	0.513489
$707$	−80.7391	−3.03651
$708$	0	0
$709$	−44.6710	−1.67766	−0.838828	−	0.544396i	$-0.816759\pi$
$709$	−44.6710	−1.67766	−0.838828	+	0.544396i	$0.816759\pi$
$710$	13.4060	0.503119
$711$	0	0
$712$	2.44368	0.0915808
$713$	−12.9474	−0.484884
$714$	0	0
$715$	−2.68667	−0.100476
$716$	−10.4880	−0.391957
$717$	0	0
$718$	25.6434	0.957002
$719$	47.5278	1.77249	0.886243	−	0.463220i	$-0.153306\pi$
$719$	47.5278	1.77249	0.886243	+	0.463220i	$0.153306\pi$
$720$	0	0
$721$	−64.8971	−2.41689
$722$	33.3754	1.24210
$723$	0	0
$724$	−28.4326	−1.05669
$725$	−4.06826	−0.151091
$726$	0	0
$727$	−0.276450	−0.0102530	−0.00512648	−	0.999987i	$-0.501632\pi$
$727$	−0.276450	−0.0102530	−0.00512648	+	0.999987i	$0.501632\pi$
$728$	10.1330	0.375555
$729$	0	0
$730$	20.1012	0.743980
$731$	−31.0618	−1.14886
$732$	0	0
$733$	−11.6799	−0.431406	−0.215703	−	0.976459i	$-0.569204\pi$
$733$	−11.6799	−0.431406	−0.215703	+	0.976459i	$0.569204\pi$
$734$	−34.9331	−1.28941
$735$	0	0
$736$	−14.1685	−0.522259
$737$	−1.09493	−0.0403324
$738$	0	0
$739$	11.8553	0.436105	0.218053	−	0.975937i	$-0.430030\pi$
$739$	11.8553	0.436105	0.218053	+	0.975937i	$0.430030\pi$
$740$	35.1703	1.29289
$741$	0	0
$742$	67.5989	2.48163
$743$	−38.9102	−1.42748	−0.713739	−	0.700412i	$-0.753000\pi$
$743$	−38.9102	−1.42748	−0.713739	+	0.700412i	$0.753000\pi$
$744$	0	0
$745$	−52.8148	−1.93498
$746$	−61.3258	−2.24530
$747$	0	0
$748$	2.68713	0.0982511
$749$	15.5253	0.567284
$750$	0	0
$751$	−40.8016	−1.48887	−0.744436	−	0.667694i	$-0.767282\pi$
$751$	−40.8016	−1.48887	−0.744436	+	0.667694i	$0.767282\pi$
$752$	−18.6232	−0.679117
$753$	0	0
$754$	−22.5516	−0.821281
$755$	−50.1130	−1.82380
$756$	0	0
$757$	28.2347	1.02621	0.513104	−	0.858326i	$-0.328495\pi$
$757$	28.2347	1.02621	0.513104	+	0.858326i	$0.328495\pi$
$758$	26.7810	0.972728
$759$	0	0
$760$	−2.02325	−0.0733911
$761$	−31.1104	−1.12775	−0.563876	−	0.825860i	$-0.690690\pi$
$761$	−31.1104	−1.12775	−0.563876	+	0.825860i	$0.690690\pi$
$762$	0	0
$763$	−0.790367	−0.0286132
$764$	12.7515	0.461332
$765$	0	0
$766$	−13.3204	−0.481285
$767$	−13.5040	−0.487600
$768$	0	0
$769$	−15.8429	−0.571311	−0.285655	−	0.958332i	$-0.592211\pi$
$769$	−15.8429	−0.571311	−0.285655	+	0.958332i	$0.592211\pi$
$770$	6.87639	0.247808
$771$	0	0
$772$	15.7674	0.567480
$773$	32.9880	1.18649	0.593247	−	0.805020i	$-0.297846\pi$
$773$	32.9880	1.18649	0.593247	+	0.805020i	$0.297846\pi$
$774$	0	0
$775$	7.96675	0.286174
$776$	9.56676	0.343427
$777$	0	0
$778$	14.1790	0.508341
$779$	7.90588	0.283257
$780$	0	0
$781$	−0.896506	−0.0320795
$782$	18.6082	0.665428
$783$	0	0
$784$	−62.7076	−2.23956
$785$	−20.0833	−0.716805
$786$	0	0
$787$	−32.6162	−1.16264	−0.581321	−	0.813675i	$-0.697464\pi$
$787$	−32.6162	−1.16264	−0.581321	+	0.813675i	$0.697464\pi$
$788$	−14.5908	−0.519777
$789$	0	0
$790$	34.4281	1.22490
$791$	9.09838	0.323501
$792$	0	0
$793$	−8.36678	−0.297113
$794$	−20.4039	−0.724107
$795$	0	0
$796$	3.35467	0.118903
$797$	28.7305	1.01769	0.508843	−	0.860859i	$-0.330073\pi$
$797$	28.7305	1.01769	0.508843	+	0.860859i	$0.330073\pi$
$798$	0	0
$799$	20.7955	0.735690
$800$	8.71813	0.308233
$801$	0	0
$802$	44.6325	1.57603
$803$	−1.34424	−0.0474372
$804$	0	0
$805$	21.5832	0.760709
$806$	44.1621	1.55554
$807$	0	0
$808$	11.5994	0.408066
$809$	36.8884	1.29693	0.648464	−	0.761246i	$-0.275412\pi$
$809$	36.8884	1.29693	0.648464	+	0.761246i	$0.275412\pi$
$810$	0	0
$811$	52.6316	1.84815	0.924073	−	0.382217i	$-0.124839\pi$
$811$	52.6316	1.84815	0.924073	+	0.382217i	$0.124839\pi$
$812$	26.1616	0.918092
$813$	0	0
$814$	−5.18908	−0.181877
$815$	−22.8010	−0.798684
$816$	0	0
$817$	7.58211	0.265265
$818$	−17.7953	−0.622197
$819$	0	0
$820$	−26.1637	−0.913674
$821$	−44.3254	−1.54697	−0.773484	−	0.633816i	$-0.781488\pi$
$821$	−44.3254	−1.54697	−0.773484	+	0.633816i	$0.781488\pi$
$822$	0	0
$823$	−8.49072	−0.295968	−0.147984	−	0.988990i	$-0.547278\pi$
$823$	−8.49072	−0.295968	−0.147984	+	0.988990i	$0.547278\pi$
$824$	9.32347	0.324798
$825$	0	0
$826$	34.5627	1.20259
$827$	14.2670	0.496113	0.248057	−	0.968746i	$-0.420208\pi$
$827$	14.2670	0.496113	0.248057	+	0.968746i	$0.420208\pi$
$828$	0	0
$829$	−5.56424	−0.193254	−0.0966270	−	0.995321i	$-0.530805\pi$
$829$	−5.56424	−0.193254	−0.0966270	+	0.995321i	$0.530805\pi$
$830$	57.4141	1.99287
$831$	0	0
$832$	17.2499	0.598034
$833$	70.0221	2.42612
$834$	0	0
$835$	−40.6295	−1.40604
$836$	−0.655921	−0.0226855
$837$	0	0
$838$	5.46684	0.188849
$839$	−55.1833	−1.90514	−0.952570	−	0.304318i	$-0.901571\pi$
$839$	−55.1833	−1.90514	−0.952570	+	0.304318i	$0.901571\pi$
$840$	0	0
$841$	−16.9897	−0.585851
$842$	−31.5887	−1.08862
$843$	0	0
$844$	−42.8595	−1.47529
$845$	−3.53831	−0.121722
$846$	0	0
$847$	49.6239	1.70510
$848$	−35.4527	−1.21745
$849$	0	0
$850$	−11.4499	−0.392730
$851$	−16.2872	−0.558318
$852$	0	0
$853$	0.412827	0.0141349	0.00706746	−	0.999975i	$-0.497750\pi$
$853$	0.412827	0.0141349	0.00706746	+	0.999975i	$0.497750\pi$
$854$	21.4144	0.732785
$855$	0	0
$856$	−2.23045	−0.0762354
$857$	20.1714	0.689041	0.344521	−	0.938779i	$-0.388041\pi$
$857$	20.1714	0.689041	0.344521	+	0.938779i	$0.388041\pi$
$858$	0	0
$859$	−19.1978	−0.655019	−0.327510	−	0.944848i	$-0.606209\pi$
$859$	−19.1978	−0.655019	−0.327510	+	0.944848i	$0.606209\pi$
$860$	−25.0922	−0.855637
$861$	0	0
$862$	45.4665	1.54860
$863$	−9.67483	−0.329335	−0.164668	−	0.986349i	$-0.552655\pi$
$863$	−9.67483	−0.329335	−0.164668	+	0.986349i	$0.552655\pi$
$864$	0	0
$865$	29.3296	0.997235
$866$	5.60967	0.190624
$867$	0	0
$868$	−51.2315	−1.73891
$869$	−2.30233	−0.0781010
$870$	0	0
$871$	11.7223	0.397194
$872$	0.113548	0.00384523
$873$	0	0
$874$	−4.54222	−0.153643
$875$	43.2853	1.46331
$876$	0	0
$877$	−20.5086	−0.692528	−0.346264	−	0.938137i	$-0.612550\pi$
$877$	−20.5086	−0.692528	−0.346264	+	0.938137i	$0.612550\pi$
$878$	−64.2547	−2.16849
$879$	0	0
$880$	−3.60637	−0.121571
$881$	−49.1410	−1.65560	−0.827801	−	0.561021i	$-0.810409\pi$
$881$	−49.1410	−1.65560	−0.827801	+	0.561021i	$0.810409\pi$
$882$	0	0
$883$	40.7250	1.37050	0.685252	−	0.728306i	$-0.259692\pi$
$883$	40.7250	1.37050	0.685252	+	0.728306i	$0.259692\pi$
$884$	−28.7682	−0.967578
$885$	0	0
$886$	−46.1295	−1.54975
$887$	−56.0239	−1.88110	−0.940550	−	0.339656i	$-0.889689\pi$
$887$	−56.0239	−1.88110	−0.940550	+	0.339656i	$0.889689\pi$
$888$	0	0
$889$	14.4903	0.485991
$890$	−17.7537	−0.595106
$891$	0	0
$892$	−14.2633	−0.477571
$893$	−5.07612	−0.169866
$894$	0	0
$895$	−15.7178	−0.525388
$896$	23.4776	0.784332
$897$	0	0
$898$	53.4498	1.78364
$899$	−23.5194	−0.784417
$900$	0	0
$901$	39.5881	1.31887
$902$	3.86022	0.128531
$903$	0	0
$904$	−1.30712	−0.0434743
$905$	−42.6102	−1.41641
$906$	0	0
$907$	−5.43210	−0.180370	−0.0901849	−	0.995925i	$-0.528746\pi$
$907$	−5.43210	−0.180370	−0.0901849	+	0.995925i	$0.528746\pi$
$908$	30.6172	1.01607
$909$	0	0
$910$	−73.6181	−2.44042
$911$	−51.0288	−1.69066	−0.845329	−	0.534246i	$-0.820596\pi$
$911$	−51.0288	−1.69066	−0.845329	+	0.534246i	$0.820596\pi$
$912$	0	0
$913$	−3.83948	−0.127068
$914$	27.3318	0.904056
$915$	0	0
$916$	−0.193815	−0.00640384
$917$	51.6363	1.70518
$918$	0	0
$919$	49.1948	1.62279	0.811393	−	0.584501i	$-0.198710\pi$
$919$	49.1948	1.62279	0.811393	+	0.584501i	$0.198710\pi$
$920$	−3.10076	−0.102229
$921$	0	0
$922$	−4.69865	−0.154742
$923$	9.59793	0.315920
$924$	0	0
$925$	10.0218	0.329514
$926$	74.0143	2.43226
$927$	0	0
$928$	−25.7377	−0.844881
$929$	19.0079	0.623628	0.311814	−	0.950143i	$-0.399063\pi$
$929$	19.0079	0.623628	0.311814	+	0.950143i	$0.399063\pi$
$930$	0	0
$931$	−17.0922	−0.560175
$932$	−18.8544	−0.617596
$933$	0	0
$934$	50.9446	1.66696
$935$	4.02703	0.131698
$936$	0	0
$937$	11.0181	0.359945	0.179972	−	0.983672i	$-0.442399\pi$
$937$	11.0181	0.359945	0.179972	+	0.983672i	$0.442399\pi$
$938$	−30.0026	−0.979619
$939$	0	0
$940$	16.7989	0.547919
$941$	−35.4734	−1.15640	−0.578201	−	0.815895i	$-0.696245\pi$
$941$	−35.4734	−1.15640	−0.578201	+	0.815895i	$0.696245\pi$
$942$	0	0
$943$	12.1163	0.394560
$944$	−18.1267	−0.589973
$945$	0	0
$946$	3.70214	0.120367
$947$	11.7753	0.382646	0.191323	−	0.981527i	$-0.438722\pi$
$947$	11.7753	0.382646	0.191323	+	0.981527i	$0.438722\pi$
$948$	0	0
$949$	14.3913	0.467162
$950$	2.79490	0.0906786
$951$	0	0
$952$	−15.1884	−0.492258
$953$	−34.8027	−1.12737	−0.563685	−	0.825990i	$-0.690617\pi$
$953$	−34.8027	−1.12737	−0.563685	+	0.825990i	$0.690617\pi$
$954$	0	0
$955$	19.1099	0.618380
$956$	9.95981	0.322123
$957$	0	0
$958$	68.0286	2.19791
$959$	−44.2040	−1.42742
$960$	0	0
$961$	15.0574	0.485723
$962$	55.5539	1.79113
$963$	0	0
$964$	31.1684	1.00387
$965$	23.6296	0.760664
$966$	0	0
$967$	−20.2757	−0.652023	−0.326011	−	0.945366i	$-0.605705\pi$
$967$	−20.2757	−0.652023	−0.326011	+	0.945366i	$0.605705\pi$
$968$	−7.12924	−0.229142
$969$	0	0
$970$	−69.5040	−2.23164
$971$	17.1843	0.551471	0.275735	−	0.961234i	$-0.411079\pi$
$971$	17.1843	0.551471	0.275735	+	0.961234i	$0.411079\pi$
$972$	0	0
$973$	−32.3302	−1.03646
$974$	−3.58879	−0.114992
$975$	0	0
$976$	−11.2309	−0.359493
$977$	38.1786	1.22144	0.610721	−	0.791846i	$-0.290880\pi$
$977$	38.1786	1.22144	0.610721	+	0.791846i	$0.290880\pi$
$978$	0	0
$979$	1.18725	0.0379447
$980$	56.5649	1.80690
$981$	0	0
$982$	34.5877	1.10374
$983$	−5.16551	−0.164754	−0.0823770	−	0.996601i	$-0.526251\pi$
$983$	−5.16551	−0.164754	−0.0823770	+	0.996601i	$0.526251\pi$
$984$	0	0
$985$	−21.8664	−0.696722
$986$	33.8025	1.07649
$987$	0	0
$988$	7.02224	0.223407
$989$	11.6201	0.369497
$990$	0	0
$991$	−39.1040	−1.24218	−0.621090	−	0.783739i	$-0.713310\pi$
$991$	−39.1040	−1.24218	−0.621090	+	0.783739i	$0.713310\pi$
$992$	50.4013	1.60024
$993$	0	0
$994$	−24.5654	−0.779168
$995$	5.02744	0.159381
$996$	0	0
$997$	−24.9615	−0.790539	−0.395269	−	0.918565i	$-0.629349\pi$
$997$	−24.9615	−0.790539	−0.395269	+	0.918565i	$0.629349\pi$
$998$	−5.23609	−0.165746
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.c.1.15		72
3.2	odd	2		6561.2.a.d.1.58		72
81.4	even	27		81.2.g.a.16.3	✓	144
81.7	even	27		729.2.g.c.352.6		144
81.20	odd	54		243.2.g.a.118.6		144
81.23	odd	54		729.2.g.b.379.3		144
81.31	even	27		729.2.g.d.622.6		144
81.34	even	27		729.2.g.d.109.6		144
81.47	odd	54		729.2.g.a.109.3		144
81.50	odd	54		729.2.g.a.622.3		144
81.58	even	27		729.2.g.c.379.6		144
81.61	even	27		81.2.g.a.76.3	yes	144
81.74	odd	54		729.2.g.b.352.3		144
81.77	odd	54		243.2.g.a.208.6		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.16.3	✓	144	81.4	even	27
81.2.g.a.76.3	yes	144	81.61	even	27
243.2.g.a.118.6		144	81.20	odd	54
243.2.g.a.208.6		144	81.77	odd	54
729.2.g.a.109.3		144	81.47	odd	54
729.2.g.a.622.3		144	81.50	odd	54
729.2.g.b.352.3		144	81.74	odd	54
729.2.g.b.379.3		144	81.23	odd	54
729.2.g.c.352.6		144	81.7	even	27
729.2.g.c.379.6		144	81.58	even	27
729.2.g.d.109.6		144	81.34	even	27
729.2.g.d.622.6		144	81.31	even	27
6561.2.a.c.1.15		72	1.1	even	1	trivial
6561.2.a.d.1.58		72	3.2	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:	\( N \)	\(=\)	\( 6561 = 3^{8} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6561.a (trivial)

\(f(q)\)	\(=\)	\(q-1.91259 q^{2} +1.65799 q^{4} +2.48473 q^{5} -4.55307 q^{7} +0.654119 q^{8} +O(q^{10})\)	\(q-1.91259 q^{2} +1.65799 q^{4} +2.48473 q^{5} -4.55307 q^{7} +0.654119 q^{8} -4.75227 q^{10} +0.317801 q^{11} -3.40235 q^{13} +8.70815 q^{14} -4.56705 q^{16} +5.09977 q^{17} -1.24484 q^{19} +4.11967 q^{20} -0.607822 q^{22} -1.90780 q^{23} +1.17390 q^{25} +6.50729 q^{26} -7.54896 q^{28} -3.46559 q^{29} +6.78656 q^{31} +7.42664 q^{32} -9.75376 q^{34} -11.3132 q^{35} +8.53717 q^{37} +2.38087 q^{38} +1.62531 q^{40} -6.35091 q^{41} -6.09082 q^{43} +0.526911 q^{44} +3.64883 q^{46} +4.07772 q^{47} +13.7304 q^{49} -2.24519 q^{50} -5.64107 q^{52} +7.76272 q^{53} +0.789650 q^{55} -2.97825 q^{56} +6.62825 q^{58} +3.96901 q^{59} +2.45912 q^{61} -12.9799 q^{62} -5.07001 q^{64} -8.45393 q^{65} -3.44535 q^{67} +8.45538 q^{68} +21.6374 q^{70} -2.82097 q^{71} -4.22982 q^{73} -16.3281 q^{74} -2.06394 q^{76} -1.44697 q^{77} -7.24456 q^{79} -11.3479 q^{80} +12.1467 q^{82} -12.0814 q^{83} +12.6716 q^{85} +11.6492 q^{86} +0.207879 q^{88} +3.73584 q^{89} +15.4911 q^{91} -3.16312 q^{92} -7.79901 q^{94} -3.09310 q^{95} +14.6254 q^{97} -26.2607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8}+O(q^{10}) \) 72 * q - 9 * q^2 + 63 * q^4 - 18 * q^5 - 27 * q^8	\( 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8} - 36 q^{11} - 36 q^{14} + 45 q^{16} - 36 q^{17} - 54 q^{20} - 54 q^{23} + 36 q^{25} - 45 q^{26} + 9 q^{28} - 54 q^{29} - 63 q^{32} - 72 q^{35} - 54 q^{38} - 72 q^{41} - 90 q^{44} - 90 q^{47} + 18 q^{49} - 45 q^{50} - 45 q^{53} + 9 q^{55} - 108 q^{56} + 18 q^{58} - 108 q^{59} - 72 q^{62} + 9 q^{64} - 72 q^{65} - 108 q^{68} - 126 q^{71} - 90 q^{74} - 72 q^{77} - 144 q^{80} - 18 q^{82} - 108 q^{83} - 90 q^{86} - 108 q^{89} - 72 q^{92} - 144 q^{95} - 81 q^{98}+O(q^{100}) \) 72 * q - 9 * q^2 + 63 * q^4 - 18 * q^5 - 27 * q^8 - 36 * q^11 - 36 * q^14 + 45 * q^16 - 36 * q^17 - 54 * q^20 - 54 * q^23 + 36 * q^25 - 45 * q^26 + 9 * q^28 - 54 * q^29 - 63 * q^32 - 72 * q^35 - 54 * q^38 - 72 * q^41 - 90 * q^44 - 90 * q^47 + 18 * q^49 - 45 * q^50 - 45 * q^53 + 9 * q^55 - 108 * q^56 + 18 * q^58 - 108 * q^59 - 72 * q^62 + 9 * q^64 - 72 * q^65 - 108 * q^68 - 126 * q^71 - 90 * q^74 - 72 * q^77 - 144 * q^80 - 18 * q^82 - 108 * q^83 - 90 * q^86 - 108 * q^89 - 72 * q^92 - 144 * q^95 - 81 * q^98

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