LMFDB - Embedded newform 8046.2.a.o.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, 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("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 29 x^{10} + 76 x^{9} + 320 x^{8} - 724 x^{7} - 1643 x^{6} + 3265 x^{5} + 3921 x^{4} + \cdots + 423 $ x^12 - 3x^11 - 29x^10 + 76x^9 + 320x^8 - 724x^7 - 1643x^6 + 3265x^5 + 3921x^4 - 6927x^3 - 3639x^2 + 5508*x + 423
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$1.84210$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.84210 q^{5} +2.73011 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.84210 q^{5} +2.73011 q^{7} +1.00000 q^{8} +1.84210 q^{10} -3.84740 q^{11} -1.70279 q^{13} +2.73011 q^{14} +1.00000 q^{16} +1.33871 q^{17} +4.04876 q^{19} +1.84210 q^{20} -3.84740 q^{22} -3.24440 q^{23} -1.60665 q^{25} -1.70279 q^{26} +2.73011 q^{28} +1.77539 q^{29} -3.60491 q^{31} +1.00000 q^{32} +1.33871 q^{34} +5.02915 q^{35} -4.85467 q^{37} +4.04876 q^{38} +1.84210 q^{40} +11.5668 q^{41} +7.48373 q^{43} -3.84740 q^{44} -3.24440 q^{46} +7.47502 q^{47} +0.453494 q^{49} -1.60665 q^{50} -1.70279 q^{52} +8.47922 q^{53} -7.08731 q^{55} +2.73011 q^{56} +1.77539 q^{58} +9.23848 q^{59} +7.09661 q^{61} -3.60491 q^{62} +1.00000 q^{64} -3.13671 q^{65} -10.8507 q^{67} +1.33871 q^{68} +5.02915 q^{70} +7.94712 q^{71} +11.5681 q^{73} -4.85467 q^{74} +4.04876 q^{76} -10.5038 q^{77} +8.02055 q^{79} +1.84210 q^{80} +11.5668 q^{82} +6.08371 q^{83} +2.46604 q^{85} +7.48373 q^{86} -3.84740 q^{88} -8.43503 q^{89} -4.64879 q^{91} -3.24440 q^{92} +7.47502 q^{94} +7.45825 q^{95} +7.75492 q^{97} +0.453494 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{4} + 3 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q + 12 * q^2 + 12 * q^4 + 3 * q^5 + 6 * q^7 + 12 * q^8	$ 12 q + 12 q^{2} + 12 q^{4} + 3 q^{5} + 6 q^{7} + 12 q^{8} + 3 q^{10} + 10 q^{11} + 5 q^{13} + 6 q^{14} + 12 q^{16} + 8 q^{17} + 2 q^{19} + 3 q^{20} + 10 q^{22} + 9 q^{23} + 7 q^{25} + 5 q^{26} + 6 q^{28} + 19 q^{29} + 10 q^{31} + 12 q^{32} + 8 q^{34} + 20 q^{35} + 11 q^{37} + 2 q^{38} + 3 q^{40} + 8 q^{41} + 13 q^{43} + 10 q^{44} + 9 q^{46} + 11 q^{47} + 2 q^{49} + 7 q^{50} + 5 q^{52} + 24 q^{53} + 3 q^{55} + 6 q^{56} + 19 q^{58} + 10 q^{59} + 10 q^{62} + 12 q^{64} + 28 q^{65} + 21 q^{67} + 8 q^{68} + 20 q^{70} + 37 q^{71} - 2 q^{73} + 11 q^{74} + 2 q^{76} + 2 q^{77} + 7 q^{79} + 3 q^{80} + 8 q^{82} + 22 q^{83} + 15 q^{85} + 13 q^{86} + 10 q^{88} + 40 q^{89} + q^{91} + 9 q^{92} + 11 q^{94} + 11 q^{95} + 7 q^{97} + 2 q^{98}+O(q^{100}) $ 12 * q + 12 * q^2 + 12 * q^4 + 3 * q^5 + 6 * q^7 + 12 * q^8 + 3 * q^10 + 10 * q^11 + 5 * q^13 + 6 * q^14 + 12 * q^16 + 8 * q^17 + 2 * q^19 + 3 * q^20 + 10 * q^22 + 9 * q^23 + 7 * q^25 + 5 * q^26 + 6 * q^28 + 19 * q^29 + 10 * q^31 + 12 * q^32 + 8 * q^34 + 20 * q^35 + 11 * q^37 + 2 * q^38 + 3 * q^40 + 8 * q^41 + 13 * q^43 + 10 * q^44 + 9 * q^46 + 11 * q^47 + 2 * q^49 + 7 * q^50 + 5 * q^52 + 24 * q^53 + 3 * q^55 + 6 * q^56 + 19 * q^58 + 10 * q^59 + 10 * q^62 + 12 * q^64 + 28 * q^65 + 21 * q^67 + 8 * q^68 + 20 * q^70 + 37 * q^71 - 2 * q^73 + 11 * q^74 + 2 * q^76 + 2 * q^77 + 7 * q^79 + 3 * q^80 + 8 * q^82 + 22 * q^83 + 15 * q^85 + 13 * q^86 + 10 * q^88 + 40 * q^89 + q^91 + 9 * q^92 + 11 * q^94 + 11 * q^95 + 7 * q^97 + 2 * 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
$5$	1.84210	0.823814	0.411907	−	0.911226i	$-0.364863\pi$
$5$	1.84210	0.823814	0.411907	+	0.911226i	$0.364863\pi$
$6$	0	0
$7$	2.73011	1.03188	0.515942	−	0.856623i	$-0.327442\pi$
$7$	2.73011	1.03188	0.515942	+	0.856623i	$0.327442\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.84210	0.582525
$11$	−3.84740	−1.16003	−0.580017	−	0.814604i	$-0.696954\pi$
$11$	−3.84740	−1.16003	−0.580017	+	0.814604i	$0.696954\pi$
$12$	0	0
$13$	−1.70279	−0.472268	−0.236134	−	0.971721i	$-0.575880\pi$
$13$	−1.70279	−0.472268	−0.236134	+	0.971721i	$0.575880\pi$
$14$	2.73011	0.729652
$15$	0	0
$16$	1.00000	0.250000
$17$	1.33871	0.324684	0.162342	−	0.986735i	$-0.448095\pi$
$17$	1.33871	0.324684	0.162342	+	0.986735i	$0.448095\pi$
$18$	0	0
$19$	4.04876	0.928850	0.464425	−	0.885612i	$-0.346261\pi$
$19$	4.04876	0.928850	0.464425	+	0.885612i	$0.346261\pi$
$20$	1.84210	0.411907
$21$	0	0
$22$	−3.84740	−0.820269
$23$	−3.24440	−0.676503	−0.338252	−	0.941056i	$-0.609835\pi$
$23$	−3.24440	−0.676503	−0.338252	+	0.941056i	$0.609835\pi$
$24$	0	0
$25$	−1.60665	−0.321330
$26$	−1.70279	−0.333944
$27$	0	0
$28$	2.73011	0.515942
$29$	1.77539	0.329682	0.164841	−	0.986320i	$-0.447289\pi$
$29$	1.77539	0.329682	0.164841	+	0.986320i	$0.447289\pi$
$30$	0	0
$31$	−3.60491	−0.647462	−0.323731	−	0.946149i	$-0.604937\pi$
$31$	−3.60491	−0.647462	−0.323731	+	0.946149i	$0.604937\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.33871	0.229586
$35$	5.02915	0.850081
$36$	0	0
$37$	−4.85467	−0.798103	−0.399051	−	0.916929i	$-0.630661\pi$
$37$	−4.85467	−0.798103	−0.399051	+	0.916929i	$0.630661\pi$
$38$	4.04876	0.656796
$39$	0	0
$40$	1.84210	0.291262
$41$	11.5668	1.80643	0.903213	−	0.429192i	$-0.141202\pi$
$41$	11.5668	1.80643	0.903213	+	0.429192i	$0.141202\pi$
$42$	0	0
$43$	7.48373	1.14126	0.570629	−	0.821208i	$-0.306699\pi$
$43$	7.48373	1.14126	0.570629	+	0.821208i	$0.306699\pi$
$44$	−3.84740	−0.580017
$45$	0	0
$46$	−3.24440	−0.478360
$47$	7.47502	1.09034	0.545172	−	0.838324i	$-0.316464\pi$
$47$	7.47502	1.09034	0.545172	+	0.838324i	$0.316464\pi$
$48$	0	0
$49$	0.453494	0.0647849
$50$	−1.60665	−0.227215
$51$	0	0
$52$	−1.70279	−0.236134
$53$	8.47922	1.16471	0.582355	−	0.812934i	$-0.302131\pi$
$53$	8.47922	1.16471	0.582355	+	0.812934i	$0.302131\pi$
$54$	0	0
$55$	−7.08731	−0.955653
$56$	2.73011	0.364826
$57$	0	0
$58$	1.77539	0.233120
$59$	9.23848	1.20275	0.601374	−	0.798968i	$-0.294620\pi$
$59$	9.23848	1.20275	0.601374	+	0.798968i	$0.294620\pi$
$60$	0	0
$61$	7.09661	0.908628	0.454314	−	0.890842i	$-0.349884\pi$
$61$	7.09661	0.908628	0.454314	+	0.890842i	$0.349884\pi$
$62$	−3.60491	−0.457825
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.13671	−0.389061
$66$	0	0
$67$	−10.8507	−1.32563	−0.662815	−	0.748783i	$-0.730638\pi$
$67$	−10.8507	−1.32563	−0.662815	+	0.748783i	$0.730638\pi$
$68$	1.33871	0.162342
$69$	0	0
$70$	5.02915	0.601098
$71$	7.94712	0.943150	0.471575	−	0.881826i	$-0.343686\pi$
$71$	7.94712	0.943150	0.471575	+	0.881826i	$0.343686\pi$
$72$	0	0
$73$	11.5681	1.35394	0.676971	−	0.736010i	$-0.263292\pi$
$73$	11.5681	1.35394	0.676971	+	0.736010i	$0.263292\pi$
$74$	−4.85467	−0.564344
$75$	0	0
$76$	4.04876	0.464425
$77$	−10.5038	−1.19702
$78$	0	0
$79$	8.02055	0.902382	0.451191	−	0.892427i	$-0.350999\pi$
$79$	8.02055	0.902382	0.451191	+	0.892427i	$0.350999\pi$
$80$	1.84210	0.205954
$81$	0	0
$82$	11.5668	1.27734
$83$	6.08371	0.667774	0.333887	−	0.942613i	$-0.391640\pi$
$83$	6.08371	0.667774	0.333887	+	0.942613i	$0.391640\pi$
$84$	0	0
$85$	2.46604	0.267480
$86$	7.48373	0.806992
$87$	0	0
$88$	−3.84740	−0.410134
$89$	−8.43503	−0.894112	−0.447056	−	0.894506i	$-0.647527\pi$
$89$	−8.43503	−0.894112	−0.447056	+	0.894506i	$0.647527\pi$
$90$	0	0
$91$	−4.64879	−0.487326
$92$	−3.24440	−0.338252
$93$	0	0
$94$	7.47502	0.770989
$95$	7.45825	0.765200
$96$	0	0
$97$	7.75492	0.787393	0.393696	−	0.919241i	$-0.371196\pi$
$97$	7.75492	0.787393	0.393696	+	0.919241i	$0.371196\pi$
$98$	0.453494	0.0458098
$99$	0	0
$100$	−1.60665	−0.160665
$101$	3.84379	0.382471	0.191236	−	0.981544i	$-0.438751\pi$
$101$	3.84379	0.382471	0.191236	+	0.981544i	$0.438751\pi$
$102$	0	0
$103$	8.63407	0.850740	0.425370	−	0.905020i	$-0.360144\pi$
$103$	8.63407	0.850740	0.425370	+	0.905020i	$0.360144\pi$
$104$	−1.70279	−0.166972
$105$	0	0
$106$	8.47922	0.823575
$107$	9.38555	0.907335	0.453668	−	0.891171i	$-0.350115\pi$
$107$	9.38555	0.907335	0.453668	+	0.891171i	$0.350115\pi$
$108$	0	0
$109$	−3.26109	−0.312355	−0.156178	−	0.987729i	$-0.549917\pi$
$109$	−3.26109	−0.312355	−0.156178	+	0.987729i	$0.549917\pi$
$110$	−7.08731	−0.675749
$111$	0	0
$112$	2.73011	0.257971
$113$	0.402240	0.0378396	0.0189198	−	0.999821i	$-0.493977\pi$
$113$	0.402240	0.0378396	0.0189198	+	0.999821i	$0.493977\pi$
$114$	0	0
$115$	−5.97652	−0.557313
$116$	1.77539	0.164841
$117$	0	0
$118$	9.23848	0.850471
$119$	3.65482	0.335037
$120$	0	0
$121$	3.80249	0.345681
$122$	7.09661	0.642497
$123$	0	0
$124$	−3.60491	−0.323731
$125$	−12.1701	−1.08853
$126$	0	0
$127$	−14.3340	−1.27194	−0.635969	−	0.771715i	$-0.719399\pi$
$127$	−14.3340	−1.27194	−0.635969	+	0.771715i	$0.719399\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−3.13671	−0.275108
$131$	14.9878	1.30949	0.654744	−	0.755850i	$-0.272776\pi$
$131$	14.9878	1.30949	0.654744	+	0.755850i	$0.272776\pi$
$132$	0	0
$133$	11.0536	0.958466
$134$	−10.8507	−0.937362
$135$	0	0
$136$	1.33871	0.114793
$137$	2.93206	0.250503	0.125252	−	0.992125i	$-0.460026\pi$
$137$	2.93206	0.250503	0.125252	+	0.992125i	$0.460026\pi$
$138$	0	0
$139$	−1.09705	−0.0930503	−0.0465251	−	0.998917i	$-0.514815\pi$
$139$	−1.09705	−0.0930503	−0.0465251	+	0.998917i	$0.514815\pi$
$140$	5.02915	0.425040
$141$	0	0
$142$	7.94712	0.666908
$143$	6.55130	0.547847
$144$	0	0
$145$	3.27046	0.271597
$146$	11.5681	0.957382
$147$	0	0
$148$	−4.85467	−0.399051
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	1.39102	0.113199	0.0565997	−	0.998397i	$-0.481974\pi$
$151$	1.39102	0.113199	0.0565997	+	0.998397i	$0.481974\pi$
$152$	4.04876	0.328398
$153$	0	0
$154$	−10.5038	−0.846422
$155$	−6.64063	−0.533388
$156$	0	0
$157$	−2.97780	−0.237654	−0.118827	−	0.992915i	$-0.537913\pi$
$157$	−2.97780	−0.237654	−0.118827	+	0.992915i	$0.537913\pi$
$158$	8.02055	0.638080
$159$	0	0
$160$	1.84210	0.145631
$161$	−8.85755	−0.698073
$162$	0	0
$163$	−3.73847	−0.292820	−0.146410	−	0.989224i	$-0.546772\pi$
$163$	−3.73847	−0.292820	−0.146410	+	0.989224i	$0.546772\pi$
$164$	11.5668	0.903213
$165$	0	0
$166$	6.08371	0.472188
$167$	−12.0610	−0.933312	−0.466656	−	0.884439i	$-0.654541\pi$
$167$	−12.0610	−0.933312	−0.466656	+	0.884439i	$0.654541\pi$
$168$	0	0
$169$	−10.1005	−0.776963
$170$	2.46604	0.189137
$171$	0	0
$172$	7.48373	0.570629
$173$	−7.80209	−0.593182	−0.296591	−	0.955005i	$-0.595850\pi$
$173$	−7.80209	−0.593182	−0.296591	+	0.955005i	$0.595850\pi$
$174$	0	0
$175$	−4.38633	−0.331575
$176$	−3.84740	−0.290009
$177$	0	0
$178$	−8.43503	−0.632232
$179$	−6.68488	−0.499651	−0.249826	−	0.968291i	$-0.580373\pi$
$179$	−6.68488	−0.499651	−0.249826	+	0.968291i	$0.580373\pi$
$180$	0	0
$181$	7.53147	0.559810	0.279905	−	0.960028i	$-0.409697\pi$
$181$	7.53147	0.559810	0.279905	+	0.960028i	$0.409697\pi$
$182$	−4.64879	−0.344591
$183$	0	0
$184$	−3.24440	−0.239180
$185$	−8.94281	−0.657488
$186$	0	0
$187$	−5.15054	−0.376645
$188$	7.47502	0.545172
$189$	0	0
$190$	7.45825	0.541078
$191$	16.4428	1.18976	0.594881	−	0.803814i	$-0.297199\pi$
$191$	16.4428	1.18976	0.594881	+	0.803814i	$0.297199\pi$
$192$	0	0
$193$	−4.21567	−0.303450	−0.151725	−	0.988423i	$-0.548483\pi$
$193$	−4.21567	−0.303450	−0.151725	+	0.988423i	$0.548483\pi$
$194$	7.75492	0.556771
$195$	0	0
$196$	0.453494	0.0323925
$197$	−3.69511	−0.263266	−0.131633	−	0.991299i	$-0.542022\pi$
$197$	−3.69511	−0.263266	−0.131633	+	0.991299i	$0.542022\pi$
$198$	0	0
$199$	−1.88171	−0.133391	−0.0666954	−	0.997773i	$-0.521246\pi$
$199$	−1.88171	−0.133391	−0.0666954	+	0.997773i	$0.521246\pi$
$200$	−1.60665	−0.113607
$201$	0	0
$202$	3.84379	0.270448
$203$	4.84701	0.340194
$204$	0	0
$205$	21.3072	1.48816
$206$	8.63407	0.601564
$207$	0	0
$208$	−1.70279	−0.118067
$209$	−15.5772	−1.07750
$210$	0	0
$211$	−5.87565	−0.404497	−0.202248	−	0.979334i	$-0.564825\pi$
$211$	−5.87565	−0.404497	−0.202248	+	0.979334i	$0.564825\pi$
$212$	8.47922	0.582355
$213$	0	0
$214$	9.38555	0.641583
$215$	13.7858	0.940185
$216$	0	0
$217$	−9.84181	−0.668106
$218$	−3.26109	−0.220869
$219$	0	0
$220$	−7.08731	−0.477827
$221$	−2.27953	−0.153338
$222$	0	0
$223$	−2.59470	−0.173754	−0.0868769	−	0.996219i	$-0.527689\pi$
$223$	−2.59470	−0.173754	−0.0868769	+	0.996219i	$0.527689\pi$
$224$	2.73011	0.182413
$225$	0	0
$226$	0.402240	0.0267566
$227$	−7.65667	−0.508191	−0.254096	−	0.967179i	$-0.581778\pi$
$227$	−7.65667	−0.508191	−0.254096	+	0.967179i	$0.581778\pi$
$228$	0	0
$229$	17.4892	1.15572	0.577859	−	0.816137i	$-0.303888\pi$
$229$	17.4892	1.15572	0.577859	+	0.816137i	$0.303888\pi$
$230$	−5.97652	−0.394080
$231$	0	0
$232$	1.77539	0.116560
$233$	−24.6338	−1.61382	−0.806909	−	0.590676i	$-0.798861\pi$
$233$	−24.6338	−1.61382	−0.806909	+	0.590676i	$0.798861\pi$
$234$	0	0
$235$	13.7698	0.898240
$236$	9.23848	0.601374
$237$	0	0
$238$	3.65482	0.236907
$239$	−12.6170	−0.816124	−0.408062	−	0.912954i	$-0.633795\pi$
$239$	−12.6170	−0.816124	−0.408062	+	0.912954i	$0.633795\pi$
$240$	0	0
$241$	−20.8387	−1.34234	−0.671170	−	0.741304i	$-0.734208\pi$
$241$	−20.8387	−1.34234	−0.671170	+	0.741304i	$0.734208\pi$
$242$	3.80249	0.244433
$243$	0	0
$244$	7.09661	0.454314
$245$	0.835384	0.0533707
$246$	0	0
$247$	−6.89418	−0.438666
$248$	−3.60491	−0.228912
$249$	0	0
$250$	−12.1701	−0.769707
$251$	1.32351	0.0835390	0.0417695	−	0.999127i	$-0.486700\pi$
$251$	1.32351	0.0835390	0.0417695	+	0.999127i	$0.486700\pi$
$252$	0	0
$253$	12.4825	0.784767
$254$	−14.3340	−0.899396
$255$	0	0
$256$	1.00000	0.0625000
$257$	24.7362	1.54300	0.771501	−	0.636229i	$-0.219506\pi$
$257$	24.7362	1.54300	0.771501	+	0.636229i	$0.219506\pi$
$258$	0	0
$259$	−13.2538	−0.823550
$260$	−3.13671	−0.194531
$261$	0	0
$262$	14.9878	0.925948
$263$	9.78048	0.603091	0.301545	−	0.953452i	$-0.402498\pi$
$263$	9.78048	0.603091	0.301545	+	0.953452i	$0.402498\pi$
$264$	0	0
$265$	15.6196	0.959505
$266$	11.0536	0.677738
$267$	0	0
$268$	−10.8507	−0.662815
$269$	24.7843	1.51113	0.755564	−	0.655075i	$-0.227363\pi$
$269$	24.7843	1.51113	0.755564	+	0.655075i	$0.227363\pi$
$270$	0	0
$271$	−8.26372	−0.501985	−0.250993	−	0.967989i	$-0.580757\pi$
$271$	−8.26372	−0.501985	−0.250993	+	0.967989i	$0.580757\pi$
$272$	1.33871	0.0811711
$273$	0	0
$274$	2.93206	0.177132
$275$	6.18143	0.372754
$276$	0	0
$277$	8.98709	0.539982	0.269991	−	0.962863i	$-0.412979\pi$
$277$	8.98709	0.539982	0.269991	+	0.962863i	$0.412979\pi$
$278$	−1.09705	−0.0657965
$279$	0	0
$280$	5.02915	0.300549
$281$	−17.6404	−1.05234	−0.526168	−	0.850380i	$-0.676372\pi$
$281$	−17.6404	−1.05234	−0.526168	+	0.850380i	$0.676372\pi$
$282$	0	0
$283$	−6.56481	−0.390238	−0.195119	−	0.980780i	$-0.562509\pi$
$283$	−6.56481	−0.390238	−0.195119	+	0.980780i	$0.562509\pi$
$284$	7.94712	0.471575
$285$	0	0
$286$	6.55130	0.387387
$287$	31.5786	1.86402
$288$	0	0
$289$	−15.2079	−0.894580
$290$	3.27046	0.192048
$291$	0	0
$292$	11.5681	0.676971
$293$	−23.6982	−1.38446	−0.692232	−	0.721675i	$-0.743372\pi$
$293$	−23.6982	−1.38446	−0.692232	+	0.721675i	$0.743372\pi$
$294$	0	0
$295$	17.0182	0.990840
$296$	−4.85467	−0.282172
$297$	0	0
$298$	1.00000	0.0579284
$299$	5.52451	0.319491
$300$	0	0
$301$	20.4314	1.17765
$302$	1.39102	0.0800441
$303$	0	0
$304$	4.04876	0.232213
$305$	13.0727	0.748541
$306$	0	0
$307$	15.3247	0.874628	0.437314	−	0.899309i	$-0.355930\pi$
$307$	15.3247	0.874628	0.437314	+	0.899309i	$0.355930\pi$
$308$	−10.5038	−0.598511
$309$	0	0
$310$	−6.64063	−0.377162
$311$	15.5470	0.881590	0.440795	−	0.897608i	$-0.354697\pi$
$311$	15.5470	0.881590	0.440795	+	0.897608i	$0.354697\pi$
$312$	0	0
$313$	−24.0093	−1.35709	−0.678543	−	0.734561i	$-0.737388\pi$
$313$	−24.0093	−1.35709	−0.678543	+	0.734561i	$0.737388\pi$
$314$	−2.97780	−0.168047
$315$	0	0
$316$	8.02055	0.451191
$317$	−9.26814	−0.520551	−0.260275	−	0.965534i	$-0.583813\pi$
$317$	−9.26814	−0.520551	−0.260275	+	0.965534i	$0.583813\pi$
$318$	0	0
$319$	−6.83065	−0.382443
$320$	1.84210	0.102977
$321$	0	0
$322$	−8.85755	−0.493612
$323$	5.42011	0.301583
$324$	0	0
$325$	2.73578	0.151754
$326$	−3.73847	−0.207055
$327$	0	0
$328$	11.5668	0.638668
$329$	20.4076	1.12511
$330$	0	0
$331$	−31.4534	−1.72884	−0.864419	−	0.502772i	$-0.832313\pi$
$331$	−31.4534	−1.72884	−0.864419	+	0.502772i	$0.832313\pi$
$332$	6.08371	0.333887
$333$	0	0
$334$	−12.0610	−0.659951
$335$	−19.9882	−1.09207
$336$	0	0
$337$	−6.35364	−0.346105	−0.173052	−	0.984913i	$-0.555363\pi$
$337$	−6.35364	−0.346105	−0.173052	+	0.984913i	$0.555363\pi$
$338$	−10.1005	−0.549396
$339$	0	0
$340$	2.46604	0.133740
$341$	13.8696	0.751078
$342$	0	0
$343$	−17.8727	−0.965034
$344$	7.48373	0.403496
$345$	0	0
$346$	−7.80209	−0.419443
$347$	−35.0246	−1.88022	−0.940109	−	0.340874i	$-0.889277\pi$
$347$	−35.0246	−1.88022	−0.940109	+	0.340874i	$0.889277\pi$
$348$	0	0
$349$	−4.88382	−0.261425	−0.130713	−	0.991420i	$-0.541726\pi$
$349$	−4.88382	−0.261425	−0.130713	+	0.991420i	$0.541726\pi$
$350$	−4.38633	−0.234459
$351$	0	0
$352$	−3.84740	−0.205067
$353$	20.5830	1.09552	0.547761	−	0.836635i	$-0.315480\pi$
$353$	20.5830	1.09552	0.547761	+	0.836635i	$0.315480\pi$
$354$	0	0
$355$	14.6394	0.776980
$356$	−8.43503	−0.447056
$357$	0	0
$358$	−6.68488	−0.353307
$359$	5.57294	0.294128	0.147064	−	0.989127i	$-0.453018\pi$
$359$	5.57294	0.294128	0.147064	+	0.989127i	$0.453018\pi$
$360$	0	0
$361$	−2.60751	−0.137238
$362$	7.53147	0.395845
$363$	0	0
$364$	−4.64879	−0.243663
$365$	21.3096	1.11540
$366$	0	0
$367$	28.6963	1.49794	0.748968	−	0.662607i	$-0.230550\pi$
$367$	28.6963	1.49794	0.748968	+	0.662607i	$0.230550\pi$
$368$	−3.24440	−0.169126
$369$	0	0
$370$	−8.94281	−0.464915
$371$	23.1492	1.20185
$372$	0	0
$373$	−24.1074	−1.24823	−0.624117	−	0.781331i	$-0.714541\pi$
$373$	−24.1074	−1.24823	−0.624117	+	0.781331i	$0.714541\pi$
$374$	−5.15054	−0.266328
$375$	0	0
$376$	7.47502	0.385495
$377$	−3.02311	−0.155698
$378$	0	0
$379$	−6.75620	−0.347043	−0.173521	−	0.984830i	$-0.555515\pi$
$379$	−6.75620	−0.347043	−0.173521	+	0.984830i	$0.555515\pi$
$380$	7.45825	0.382600
$381$	0	0
$382$	16.4428	0.841289
$383$	−37.2939	−1.90563	−0.952815	−	0.303553i	$-0.901827\pi$
$383$	−37.2939	−1.90563	−0.952815	+	0.303553i	$0.901827\pi$
$384$	0	0
$385$	−19.3491	−0.986123
$386$	−4.21567	−0.214572
$387$	0	0
$388$	7.75492	0.393696
$389$	−24.6656	−1.25059	−0.625297	−	0.780386i	$-0.715022\pi$
$389$	−24.6656	−1.25059	−0.625297	+	0.780386i	$0.715022\pi$
$390$	0	0
$391$	−4.34330	−0.219650
$392$	0.453494	0.0229049
$393$	0	0
$394$	−3.69511	−0.186157
$395$	14.7747	0.743395
$396$	0	0
$397$	−35.3952	−1.77644	−0.888218	−	0.459422i	$-0.848056\pi$
$397$	−35.3952	−1.77644	−0.888218	+	0.459422i	$0.848056\pi$
$398$	−1.88171	−0.0943215
$399$	0	0
$400$	−1.60665	−0.0803325
$401$	−14.3746	−0.717832	−0.358916	−	0.933370i	$-0.616854\pi$
$401$	−14.3746	−0.717832	−0.358916	+	0.933370i	$0.616854\pi$
$402$	0	0
$403$	6.13840	0.305775
$404$	3.84379	0.191236
$405$	0	0
$406$	4.84701	0.240553
$407$	18.6779	0.925827
$408$	0	0
$409$	24.8501	1.22876	0.614379	−	0.789011i	$-0.289407\pi$
$409$	24.8501	1.22876	0.614379	+	0.789011i	$0.289407\pi$
$410$	21.3072	1.05229
$411$	0	0
$412$	8.63407	0.425370
$413$	25.2220	1.24110
$414$	0	0
$415$	11.2068	0.550122
$416$	−1.70279	−0.0834860
$417$	0	0
$418$	−15.5772	−0.761907
$419$	0.643871	0.0314552	0.0157276	−	0.999876i	$-0.494994\pi$
$419$	0.643871	0.0314552	0.0157276	+	0.999876i	$0.494994\pi$
$420$	0	0
$421$	−0.0549889	−0.00268000	−0.00134000	−	0.999999i	$-0.500427\pi$
$421$	−0.0549889	−0.00268000	−0.00134000	+	0.999999i	$0.500427\pi$
$422$	−5.87565	−0.286022
$423$	0	0
$424$	8.47922	0.411787
$425$	−2.15084	−0.104331
$426$	0	0
$427$	19.3745	0.937599
$428$	9.38555	0.453668
$429$	0	0
$430$	13.7858	0.664811
$431$	−19.6080	−0.944482	−0.472241	−	0.881469i	$-0.656555\pi$
$431$	−19.6080	−0.944482	−0.472241	+	0.881469i	$0.656555\pi$
$432$	0	0
$433$	−24.8137	−1.19247	−0.596235	−	0.802810i	$-0.703337\pi$
$433$	−24.8137	−1.19247	−0.596235	+	0.802810i	$0.703337\pi$
$434$	−9.84181	−0.472422
$435$	0	0
$436$	−3.26109	−0.156178
$437$	−13.1358	−0.628370
$438$	0	0
$439$	22.6370	1.08041	0.540204	−	0.841534i	$-0.318347\pi$
$439$	22.6370	1.08041	0.540204	+	0.841534i	$0.318347\pi$
$440$	−7.08731	−0.337874
$441$	0	0
$442$	−2.27953	−0.108426
$443$	−13.4672	−0.639846	−0.319923	−	0.947443i	$-0.603657\pi$
$443$	−13.4672	−0.639846	−0.319923	+	0.947443i	$0.603657\pi$
$444$	0	0
$445$	−15.5382	−0.736582
$446$	−2.59470	−0.122863
$447$	0	0
$448$	2.73011	0.128986
$449$	16.2157	0.765266	0.382633	−	0.923900i	$-0.375017\pi$
$449$	16.2157	0.765266	0.382633	+	0.923900i	$0.375017\pi$
$450$	0	0
$451$	−44.5020	−2.09552
$452$	0.402240	0.0189198
$453$	0	0
$454$	−7.65667	−0.359345
$455$	−8.56356	−0.401466
$456$	0	0
$457$	22.8894	1.07072	0.535361	−	0.844624i	$-0.320176\pi$
$457$	22.8894	1.07072	0.535361	+	0.844624i	$0.320176\pi$
$458$	17.4892	0.817216
$459$	0	0
$460$	−5.97652	−0.278656
$461$	8.83698	0.411579	0.205790	−	0.978596i	$-0.434024\pi$
$461$	8.83698	0.411579	0.205790	+	0.978596i	$0.434024\pi$
$462$	0	0
$463$	−27.1889	−1.26357	−0.631787	−	0.775142i	$-0.717678\pi$
$463$	−27.1889	−1.26357	−0.631787	+	0.775142i	$0.717678\pi$
$464$	1.77539	0.0824205
$465$	0	0
$466$	−24.6338	−1.14114
$467$	19.2357	0.890120	0.445060	−	0.895501i	$-0.353182\pi$
$467$	19.2357	0.890120	0.445060	+	0.895501i	$0.353182\pi$
$468$	0	0
$469$	−29.6237	−1.36790
$470$	13.7698	0.635152
$471$	0	0
$472$	9.23848	0.425235
$473$	−28.7929	−1.32390
$474$	0	0
$475$	−6.50495	−0.298468
$476$	3.65482	0.167518
$477$	0	0
$478$	−12.6170	−0.577087
$479$	−17.8607	−0.816076	−0.408038	−	0.912965i	$-0.633787\pi$
$479$	−17.8607	−0.816076	−0.408038	+	0.912965i	$0.633787\pi$
$480$	0	0
$481$	8.26647	0.376918
$482$	−20.8387	−0.949177
$483$	0	0
$484$	3.80249	0.172841
$485$	14.2854	0.648665
$486$	0	0
$487$	−18.9480	−0.858614	−0.429307	−	0.903159i	$-0.641242\pi$
$487$	−18.9480	−0.858614	−0.429307	+	0.903159i	$0.641242\pi$
$488$	7.09661	0.321249
$489$	0	0
$490$	0.835384	0.0377388
$491$	17.6159	0.794994	0.397497	−	0.917603i	$-0.369879\pi$
$491$	17.6159	0.794994	0.397497	+	0.917603i	$0.369879\pi$
$492$	0	0
$493$	2.37673	0.107043
$494$	−6.89418	−0.310184
$495$	0	0
$496$	−3.60491	−0.161865
$497$	21.6965	0.973221
$498$	0	0
$499$	−37.0676	−1.65937	−0.829686	−	0.558230i	$-0.811481\pi$
$499$	−37.0676	−1.65937	−0.829686	+	0.558230i	$0.811481\pi$
$500$	−12.1701	−0.544265
$501$	0	0
$502$	1.32351	0.0590710
$503$	37.4833	1.67130	0.835648	−	0.549265i	$-0.185092\pi$
$503$	37.4833	1.67130	0.835648	+	0.549265i	$0.185092\pi$
$504$	0	0
$505$	7.08066	0.315085
$506$	12.4825	0.554914
$507$	0	0
$508$	−14.3340	−0.635969
$509$	18.5165	0.820728	0.410364	−	0.911922i	$-0.365402\pi$
$509$	18.5165	0.820728	0.410364	+	0.911922i	$0.365402\pi$
$510$	0	0
$511$	31.5821	1.39711
$512$	1.00000	0.0441942
$513$	0	0
$514$	24.7362	1.09107
$515$	15.9049	0.700852
$516$	0	0
$517$	−28.7594	−1.26484
$518$	−13.2538	−0.582338
$519$	0	0
$520$	−3.13671	−0.137554
$521$	11.1319	0.487696	0.243848	−	0.969813i	$-0.421590\pi$
$521$	11.1319	0.487696	0.243848	+	0.969813i	$0.421590\pi$
$522$	0	0
$523$	−31.4707	−1.37612	−0.688059	−	0.725654i	$-0.741537\pi$
$523$	−31.4707	−1.37612	−0.688059	+	0.725654i	$0.741537\pi$
$524$	14.9878	0.654744
$525$	0	0
$526$	9.78048	0.426449
$527$	−4.82593	−0.210221
$528$	0	0
$529$	−12.4739	−0.542343
$530$	15.6196	0.678473
$531$	0	0
$532$	11.0536	0.479233
$533$	−19.6957	−0.853118
$534$	0	0
$535$	17.2892	0.747476
$536$	−10.8507	−0.468681
$537$	0	0
$538$	24.7843	1.06853
$539$	−1.74477	−0.0751528
$540$	0	0
$541$	−16.5938	−0.713424	−0.356712	−	0.934214i	$-0.616102\pi$
$541$	−16.5938	−0.713424	−0.356712	+	0.934214i	$0.616102\pi$
$542$	−8.26372	−0.354957
$543$	0	0
$544$	1.33871	0.0573966
$545$	−6.00726	−0.257323
$546$	0	0
$547$	38.3320	1.63896	0.819480	−	0.573108i	$-0.194262\pi$
$547$	38.3320	1.63896	0.819480	+	0.573108i	$0.194262\pi$
$548$	2.93206	0.125252
$549$	0	0
$550$	6.18143	0.263577
$551$	7.18814	0.306225
$552$	0	0
$553$	21.8970	0.931154
$554$	8.98709	0.381825
$555$	0	0
$556$	−1.09705	−0.0465251
$557$	1.54484	0.0654572	0.0327286	−	0.999464i	$-0.489580\pi$
$557$	1.54484	0.0654572	0.0327286	+	0.999464i	$0.489580\pi$
$558$	0	0
$559$	−12.7432	−0.538980
$560$	5.02915	0.212520
$561$	0	0
$562$	−17.6404	−0.744114
$563$	24.4870	1.03200	0.516001	−	0.856588i	$-0.327420\pi$
$563$	24.4870	1.03200	0.516001	+	0.856588i	$0.327420\pi$
$564$	0	0
$565$	0.740969	0.0311728
$566$	−6.56481	−0.275940
$567$	0	0
$568$	7.94712	0.333454
$569$	9.18879	0.385214	0.192607	−	0.981276i	$-0.438306\pi$
$569$	9.18879	0.385214	0.192607	+	0.981276i	$0.438306\pi$
$570$	0	0
$571$	−10.2322	−0.428204	−0.214102	−	0.976811i	$-0.568682\pi$
$571$	−10.2322	−0.428204	−0.214102	+	0.976811i	$0.568682\pi$
$572$	6.55130	0.273924
$573$	0	0
$574$	31.5786	1.31806
$575$	5.21261	0.217381
$576$	0	0
$577$	10.9927	0.457631	0.228815	−	0.973470i	$-0.426515\pi$
$577$	10.9927	0.457631	0.228815	+	0.973470i	$0.426515\pi$
$578$	−15.2079	−0.632564
$579$	0	0
$580$	3.27046	0.135798
$581$	16.6092	0.689065
$582$	0	0
$583$	−32.6230	−1.35111
$584$	11.5681	0.478691
$585$	0	0
$586$	−23.6982	−0.978963
$587$	17.3288	0.715235	0.357618	−	0.933868i	$-0.383589\pi$
$587$	17.3288	0.715235	0.357618	+	0.933868i	$0.383589\pi$
$588$	0	0
$589$	−14.5954	−0.601395
$590$	17.0182	0.700630
$591$	0	0
$592$	−4.85467	−0.199526
$593$	−8.16220	−0.335181	−0.167591	−	0.985857i	$-0.553599\pi$
$593$	−8.16220	−0.335181	−0.167591	+	0.985857i	$0.553599\pi$
$594$	0	0
$595$	6.73256	0.276008
$596$	1.00000	0.0409616
$597$	0	0
$598$	5.52451	0.225914
$599$	29.5063	1.20559	0.602797	−	0.797895i	$-0.294053\pi$
$599$	29.5063	1.20559	0.602797	+	0.797895i	$0.294053\pi$
$600$	0	0
$601$	−3.32303	−0.135549	−0.0677746	−	0.997701i	$-0.521590\pi$
$601$	−3.32303	−0.135549	−0.0677746	+	0.997701i	$0.521590\pi$
$602$	20.4314	0.832722
$603$	0	0
$604$	1.39102	0.0565997
$605$	7.00459	0.284777
$606$	0	0
$607$	41.4666	1.68308	0.841539	−	0.540197i	$-0.181650\pi$
$607$	41.4666	1.68308	0.841539	+	0.540197i	$0.181650\pi$
$608$	4.04876	0.164199
$609$	0	0
$610$	13.0727	0.529298
$611$	−12.7284	−0.514934
$612$	0	0
$613$	−10.7240	−0.433139	−0.216569	−	0.976267i	$-0.569487\pi$
$613$	−10.7240	−0.433139	−0.216569	+	0.976267i	$0.569487\pi$
$614$	15.3247	0.618456
$615$	0	0
$616$	−10.5038	−0.423211
$617$	−3.34030	−0.134476	−0.0672378	−	0.997737i	$-0.521419\pi$
$617$	−3.34030	−0.134476	−0.0672378	+	0.997737i	$0.521419\pi$
$618$	0	0
$619$	7.84424	0.315287	0.157643	−	0.987496i	$-0.449610\pi$
$619$	7.84424	0.315287	0.157643	+	0.987496i	$0.449610\pi$
$620$	−6.64063	−0.266694
$621$	0	0
$622$	15.5470	0.623378
$623$	−23.0286	−0.922620
$624$	0	0
$625$	−14.3854	−0.575417
$626$	−24.0093	−0.959605
$627$	0	0
$628$	−2.97780	−0.118827
$629$	−6.49898	−0.259131
$630$	0	0
$631$	13.6879	0.544907	0.272454	−	0.962169i	$-0.412165\pi$
$631$	13.6879	0.544907	0.272454	+	0.962169i	$0.412165\pi$
$632$	8.02055	0.319040
$633$	0	0
$634$	−9.26814	−0.368085
$635$	−26.4047	−1.04784
$636$	0	0
$637$	−0.772204	−0.0305958
$638$	−6.83065	−0.270428
$639$	0	0
$640$	1.84210	0.0728156
$641$	31.6955	1.25190	0.625948	−	0.779865i	$-0.284712\pi$
$641$	31.6955	1.25190	0.625948	+	0.779865i	$0.284712\pi$
$642$	0	0
$643$	37.2213	1.46787	0.733933	−	0.679222i	$-0.237682\pi$
$643$	37.2213	1.46787	0.733933	+	0.679222i	$0.237682\pi$
$644$	−8.85755	−0.349036
$645$	0	0
$646$	5.42011	0.213251
$647$	24.2301	0.952582	0.476291	−	0.879288i	$-0.341981\pi$
$647$	24.2301	0.952582	0.476291	+	0.879288i	$0.341981\pi$
$648$	0	0
$649$	−35.5441	−1.39523
$650$	2.73578	0.107306
$651$	0	0
$652$	−3.73847	−0.146410
$653$	0.185511	0.00725959	0.00362980	−	0.999993i	$-0.498845\pi$
$653$	0.185511	0.00725959	0.00362980	+	0.999993i	$0.498845\pi$
$654$	0	0
$655$	27.6091	1.07878
$656$	11.5668	0.451607
$657$	0	0
$658$	20.4076	0.795572
$659$	33.2302	1.29447	0.647233	−	0.762292i	$-0.275926\pi$
$659$	33.2302	1.29447	0.647233	+	0.762292i	$0.275926\pi$
$660$	0	0
$661$	7.77760	0.302514	0.151257	−	0.988495i	$-0.451668\pi$
$661$	7.77760	0.302514	0.151257	+	0.988495i	$0.451668\pi$
$662$	−31.4534	−1.22247
$663$	0	0
$664$	6.08371	0.236094
$665$	20.3618	0.789598
$666$	0	0
$667$	−5.76007	−0.223031
$668$	−12.0610	−0.466656
$669$	0	0
$670$	−19.9882	−0.772212
$671$	−27.3035	−1.05404
$672$	0	0
$673$	2.63951	0.101746	0.0508729	−	0.998705i	$-0.483800\pi$
$673$	2.63951	0.101746	0.0508729	+	0.998705i	$0.483800\pi$
$674$	−6.35364	−0.244733
$675$	0	0
$676$	−10.1005	−0.388481
$677$	−31.8665	−1.22473	−0.612364	−	0.790576i	$-0.709781\pi$
$677$	−31.8665	−1.22473	−0.612364	+	0.790576i	$0.709781\pi$
$678$	0	0
$679$	21.1718	0.812498
$680$	2.46604	0.0945683
$681$	0	0
$682$	13.8696	0.531093
$683$	21.6399	0.828029	0.414014	−	0.910270i	$-0.364126\pi$
$683$	21.6399	0.828029	0.414014	+	0.910270i	$0.364126\pi$
$684$	0	0
$685$	5.40117	0.206368
$686$	−17.8727	−0.682382
$687$	0	0
$688$	7.48373	0.285315
$689$	−14.4383	−0.550056
$690$	0	0
$691$	24.4428	0.929849	0.464924	−	0.885350i	$-0.346082\pi$
$691$	24.4428	0.929849	0.464924	+	0.885350i	$0.346082\pi$
$692$	−7.80209	−0.296591
$693$	0	0
$694$	−35.0246	−1.32952
$695$	−2.02087	−0.0766561
$696$	0	0
$697$	15.4845	0.586518
$698$	−4.88382	−0.184855
$699$	0	0
$700$	−4.38633	−0.165788
$701$	−18.9570	−0.715994	−0.357997	−	0.933723i	$-0.616540\pi$
$701$	−18.9570	−0.715994	−0.357997	+	0.933723i	$0.616540\pi$
$702$	0	0
$703$	−19.6554	−0.741318
$704$	−3.84740	−0.145004
$705$	0	0
$706$	20.5830	0.774651
$707$	10.4940	0.394666
$708$	0	0
$709$	6.74360	0.253261	0.126631	−	0.991950i	$-0.459584\pi$
$709$	6.74360	0.253261	0.126631	+	0.991950i	$0.459584\pi$
$710$	14.6394	0.549408
$711$	0	0
$712$	−8.43503	−0.316116
$713$	11.6958	0.438010
$714$	0	0
$715$	12.0682	0.451324
$716$	−6.68488	−0.249826
$717$	0	0
$718$	5.57294	0.207980
$719$	12.7405	0.475141	0.237571	−	0.971370i	$-0.423649\pi$
$719$	12.7405	0.475141	0.237571	+	0.971370i	$0.423649\pi$
$720$	0	0
$721$	23.5719	0.877865
$722$	−2.60751	−0.0970416
$723$	0	0
$724$	7.53147	0.279905
$725$	−2.85244	−0.105937
$726$	0	0
$727$	−8.65308	−0.320925	−0.160462	−	0.987042i	$-0.551299\pi$
$727$	−8.65308	−0.320925	−0.160462	+	0.987042i	$0.551299\pi$
$728$	−4.64879	−0.172296
$729$	0	0
$730$	21.3096	0.788705
$731$	10.0185	0.370549
$732$	0	0
$733$	39.6643	1.46503	0.732516	−	0.680749i	$-0.238346\pi$
$733$	39.6643	1.46503	0.732516	+	0.680749i	$0.238346\pi$
$734$	28.6963	1.05920
$735$	0	0
$736$	−3.24440	−0.119590
$737$	41.7472	1.53778
$738$	0	0
$739$	−10.6187	−0.390616	−0.195308	−	0.980742i	$-0.562571\pi$
$739$	−10.6187	−0.390616	−0.195308	+	0.980742i	$0.562571\pi$
$740$	−8.94281	−0.328744
$741$	0	0
$742$	23.1492	0.849834
$743$	15.7928	0.579381	0.289690	−	0.957120i	$-0.406448\pi$
$743$	15.7928	0.579381	0.289690	+	0.957120i	$0.406448\pi$
$744$	0	0
$745$	1.84210	0.0674895
$746$	−24.1074	−0.882635
$747$	0	0
$748$	−5.15054	−0.188323
$749$	25.6236	0.936265
$750$	0	0
$751$	14.8473	0.541785	0.270893	−	0.962610i	$-0.412681\pi$
$751$	14.8473	0.541785	0.270893	+	0.962610i	$0.412681\pi$
$752$	7.47502	0.272586
$753$	0	0
$754$	−3.02311	−0.110095
$755$	2.56240	0.0932554
$756$	0	0
$757$	19.0177	0.691211	0.345605	−	0.938380i	$-0.387674\pi$
$757$	19.0177	0.691211	0.345605	+	0.938380i	$0.387674\pi$
$758$	−6.75620	−0.245396
$759$	0	0
$760$	7.45825	0.270539
$761$	−0.927512	−0.0336223	−0.0168111	−	0.999859i	$-0.505351\pi$
$761$	−0.927512	−0.0336223	−0.0168111	+	0.999859i	$0.505351\pi$
$762$	0	0
$763$	−8.90312	−0.322315
$764$	16.4428	0.594881
$765$	0	0
$766$	−37.2939	−1.34748
$767$	−15.7312	−0.568019
$768$	0	0
$769$	−23.1561	−0.835029	−0.417515	−	0.908670i	$-0.637099\pi$
$769$	−23.1561	−0.835029	−0.417515	+	0.908670i	$0.637099\pi$
$770$	−19.3491	−0.697295
$771$	0	0
$772$	−4.21567	−0.151725
$773$	13.5151	0.486106	0.243053	−	0.970013i	$-0.421851\pi$
$773$	13.5151	0.486106	0.243053	+	0.970013i	$0.421851\pi$
$774$	0	0
$775$	5.79184	0.208049
$776$	7.75492	0.278385
$777$	0	0
$778$	−24.6656	−0.884304
$779$	46.8311	1.67790
$780$	0	0
$781$	−30.5758	−1.09409
$782$	−4.34330	−0.155316
$783$	0	0
$784$	0.453494	0.0161962
$785$	−5.48542	−0.195783
$786$	0	0
$787$	−9.06810	−0.323243	−0.161621	−	0.986853i	$-0.551672\pi$
$787$	−9.06810	−0.323243	−0.161621	+	0.986853i	$0.551672\pi$
$788$	−3.69511	−0.131633
$789$	0	0
$790$	14.7747	0.525660
$791$	1.09816	0.0390461
$792$	0	0
$793$	−12.0840	−0.429116
$794$	−35.3952	−1.25613
$795$	0	0
$796$	−1.88171	−0.0666954
$797$	−36.6337	−1.29763	−0.648816	−	0.760946i	$-0.724735\pi$
$797$	−36.6337	−1.29763	−0.648816	+	0.760946i	$0.724735\pi$
$798$	0	0
$799$	10.0069	0.354017
$800$	−1.60665	−0.0568037
$801$	0	0
$802$	−14.3746	−0.507584
$803$	−44.5071	−1.57062
$804$	0	0
$805$	−16.3165	−0.575082
$806$	6.13840	0.216216
$807$	0	0
$808$	3.84379	0.135224
$809$	−29.2181	−1.02725	−0.513627	−	0.858014i	$-0.671698\pi$
$809$	−29.2181	−1.02725	−0.513627	+	0.858014i	$0.671698\pi$
$810$	0	0
$811$	−7.81028	−0.274256	−0.137128	−	0.990553i	$-0.543787\pi$
$811$	−7.81028	−0.274256	−0.137128	+	0.990553i	$0.543787\pi$
$812$	4.84701	0.170097
$813$	0	0
$814$	18.6779	0.654659
$815$	−6.88665	−0.241229
$816$	0	0
$817$	30.2999	1.06006
$818$	24.8501	0.868863
$819$	0	0
$820$	21.3072	0.744080
$821$	23.0808	0.805525	0.402762	−	0.915305i	$-0.368050\pi$
$821$	23.0808	0.805525	0.402762	+	0.915305i	$0.368050\pi$
$822$	0	0
$823$	−19.4399	−0.677631	−0.338815	−	0.940853i	$-0.610026\pi$
$823$	−19.4399	−0.677631	−0.338815	+	0.940853i	$0.610026\pi$
$824$	8.63407	0.300782
$825$	0	0
$826$	25.2220	0.877587
$827$	19.4077	0.674872	0.337436	−	0.941349i	$-0.390440\pi$
$827$	19.4077	0.674872	0.337436	+	0.941349i	$0.390440\pi$
$828$	0	0
$829$	39.9376	1.38709	0.693546	−	0.720413i	$-0.256048\pi$
$829$	39.9376	1.38709	0.693546	+	0.720413i	$0.256048\pi$
$830$	11.2068	0.388995
$831$	0	0
$832$	−1.70279	−0.0590335
$833$	0.607096	0.0210346
$834$	0	0
$835$	−22.2177	−0.768875
$836$	−15.5772	−0.538749
$837$	0	0
$838$	0.643871	0.0222422
$839$	37.2494	1.28599	0.642996	−	0.765870i	$-0.277691\pi$
$839$	37.2494	1.28599	0.642996	+	0.765870i	$0.277691\pi$
$840$	0	0
$841$	−25.8480	−0.891310
$842$	−0.0549889	−0.00189504
$843$	0	0
$844$	−5.87565	−0.202248
$845$	−18.6062	−0.640073
$846$	0	0
$847$	10.3812	0.356703
$848$	8.47922	0.291178
$849$	0	0
$850$	−2.15084	−0.0737731
$851$	15.7505	0.539919
$852$	0	0
$853$	−12.1494	−0.415987	−0.207993	−	0.978130i	$-0.566693\pi$
$853$	−12.1494	−0.415987	−0.207993	+	0.978130i	$0.566693\pi$
$854$	19.3745	0.662983
$855$	0	0
$856$	9.38555	0.320792
$857$	8.78731	0.300169	0.150084	−	0.988673i	$-0.452045\pi$
$857$	8.78731	0.300169	0.150084	+	0.988673i	$0.452045\pi$
$858$	0	0
$859$	−15.1402	−0.516576	−0.258288	−	0.966068i	$-0.583158\pi$
$859$	−15.1402	−0.516576	−0.258288	+	0.966068i	$0.583158\pi$
$860$	13.7858	0.470092
$861$	0	0
$862$	−19.6080	−0.667850
$863$	−6.37600	−0.217042	−0.108521	−	0.994094i	$-0.534611\pi$
$863$	−6.37600	−0.217042	−0.108521	+	0.994094i	$0.534611\pi$
$864$	0	0
$865$	−14.3723	−0.488672
$866$	−24.8137	−0.843204
$867$	0	0
$868$	−9.84181	−0.334053
$869$	−30.8583	−1.04679
$870$	0	0
$871$	18.4765	0.626052
$872$	−3.26109	−0.110434
$873$	0	0
$874$	−13.1358	−0.444325
$875$	−33.2258	−1.12324
$876$	0	0
$877$	4.76734	0.160982	0.0804909	−	0.996755i	$-0.474351\pi$
$877$	4.76734	0.160982	0.0804909	+	0.996755i	$0.474351\pi$
$878$	22.6370	0.763963
$879$	0	0
$880$	−7.08731	−0.238913
$881$	−17.5273	−0.590510	−0.295255	−	0.955418i	$-0.595405\pi$
$881$	−17.5273	−0.590510	−0.295255	+	0.955418i	$0.595405\pi$
$882$	0	0
$883$	−27.3672	−0.920978	−0.460489	−	0.887665i	$-0.652326\pi$
$883$	−27.3672	−0.920978	−0.460489	+	0.887665i	$0.652326\pi$
$884$	−2.27953	−0.0766690
$885$	0	0
$886$	−13.4672	−0.452440
$887$	−33.7793	−1.13420	−0.567098	−	0.823650i	$-0.691934\pi$
$887$	−33.7793	−1.13420	−0.567098	+	0.823650i	$0.691934\pi$
$888$	0	0
$889$	−39.1334	−1.31249
$890$	−15.5382	−0.520842
$891$	0	0
$892$	−2.59470	−0.0868769
$893$	30.2646	1.01277
$894$	0	0
$895$	−12.3143	−0.411620
$896$	2.73011	0.0912065
$897$	0	0
$898$	16.2157	0.541125
$899$	−6.40014	−0.213457
$900$	0	0
$901$	11.3512	0.378163
$902$	−44.5020	−1.48176
$903$	0	0
$904$	0.402240	0.0133783
$905$	13.8738	0.461179
$906$	0	0
$907$	44.3120	1.47136	0.735679	−	0.677331i	$-0.236863\pi$
$907$	44.3120	1.47136	0.735679	+	0.677331i	$0.236863\pi$
$908$	−7.65667	−0.254096
$909$	0	0
$910$	−8.56356	−0.283879
$911$	6.55341	0.217124	0.108562	−	0.994090i	$-0.465375\pi$
$911$	6.55341	0.217124	0.108562	+	0.994090i	$0.465375\pi$
$912$	0	0
$913$	−23.4065	−0.774641
$914$	22.8894	0.757114
$915$	0	0
$916$	17.4892	0.577859
$917$	40.9183	1.35124
$918$	0	0
$919$	−22.8805	−0.754758	−0.377379	−	0.926059i	$-0.623175\pi$
$919$	−22.8805	−0.754758	−0.377379	+	0.926059i	$0.623175\pi$
$920$	−5.97652	−0.197040
$921$	0	0
$922$	8.83698	0.291030
$923$	−13.5323	−0.445419
$924$	0	0
$925$	7.79976	0.256455
$926$	−27.1889	−0.893482
$927$	0	0
$928$	1.77539	0.0582801
$929$	−10.7381	−0.352307	−0.176153	−	0.984363i	$-0.556365\pi$
$929$	−10.7381	−0.352307	−0.176153	+	0.984363i	$0.556365\pi$
$930$	0	0
$931$	1.83609	0.0601755
$932$	−24.6338	−0.806909
$933$	0	0
$934$	19.2357	0.629410
$935$	−9.48784	−0.310286
$936$	0	0
$937$	−8.18867	−0.267512	−0.133756	−	0.991014i	$-0.542704\pi$
$937$	−8.18867	−0.267512	−0.133756	+	0.991014i	$0.542704\pi$
$938$	−29.6237	−0.967249
$939$	0	0
$940$	13.7698	0.449120
$941$	−39.3933	−1.28418	−0.642092	−	0.766627i	$-0.721933\pi$
$941$	−39.3933	−1.28418	−0.642092	+	0.766627i	$0.721933\pi$
$942$	0	0
$943$	−37.5272	−1.22205
$944$	9.23848	0.300687
$945$	0	0
$946$	−28.7929	−0.936139
$947$	−7.65920	−0.248891	−0.124445	−	0.992226i	$-0.539715\pi$
$947$	−7.65920	−0.248891	−0.124445	+	0.992226i	$0.539715\pi$
$948$	0	0
$949$	−19.6980	−0.639424
$950$	−6.50495	−0.211048
$951$	0	0
$952$	3.65482	0.118453
$953$	0.707314	0.0229122	0.0114561	−	0.999934i	$-0.496353\pi$
$953$	0.707314	0.0229122	0.0114561	+	0.999934i	$0.496353\pi$
$954$	0	0
$955$	30.2894	0.980143
$956$	−12.6170	−0.408062
$957$	0	0
$958$	−17.8607	−0.577053
$959$	8.00485	0.258490
$960$	0	0
$961$	−18.0046	−0.580793
$962$	8.26647	0.266522
$963$	0	0
$964$	−20.8387	−0.671170
$965$	−7.76570	−0.249987
$966$	0	0
$967$	−44.2449	−1.42282	−0.711409	−	0.702778i	$-0.751943\pi$
$967$	−44.2449	−1.42282	−0.711409	+	0.702778i	$0.751943\pi$
$968$	3.80249	0.122217
$969$	0	0
$970$	14.2854	0.458676
$971$	−36.4341	−1.16922	−0.584612	−	0.811313i	$-0.698753\pi$
$971$	−36.4341	−1.16922	−0.584612	+	0.811313i	$0.698753\pi$
$972$	0	0
$973$	−2.99506	−0.0960171
$974$	−18.9480	−0.607132
$975$	0	0
$976$	7.09661	0.227157
$977$	41.2830	1.32076	0.660379	−	0.750932i	$-0.270395\pi$
$977$	41.2830	1.32076	0.660379	+	0.750932i	$0.270395\pi$
$978$	0	0
$979$	32.4530	1.03720
$980$	0.835384	0.0266854
$981$	0	0
$982$	17.6159	0.562146
$983$	29.7978	0.950403	0.475201	−	0.879877i	$-0.342375\pi$
$983$	29.7978	0.950403	0.475201	+	0.879877i	$0.342375\pi$
$984$	0	0
$985$	−6.80678	−0.216882
$986$	2.37673	0.0756905
$987$	0	0
$988$	−6.89418	−0.219333
$989$	−24.2802	−0.772065
$990$	0	0
$991$	−49.3414	−1.56738	−0.783690	−	0.621152i	$-0.786665\pi$
$991$	−49.3414	−1.56738	−0.783690	+	0.621152i	$0.786665\pi$
$992$	−3.60491	−0.114456
$993$	0	0
$994$	21.6965	0.688171
$995$	−3.46630	−0.109889
$996$	0	0
$997$	41.9804	1.32953	0.664766	−	0.747051i	$-0.268531\pi$
$997$	41.9804	1.32953	0.664766	+	0.747051i	$0.268531\pi$
$998$	−37.0676	−1.17335
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.o.1.9	yes	12
3.2	odd	2		8046.2.a.j.1.4	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.j.1.4	✓	12	3.2	odd	2
8046.2.a.o.1.9	yes	12	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.84210 q^{5} +2.73011 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.84210 q^{5} +2.73011 q^{7} +1.00000 q^{8} +1.84210 q^{10} -3.84740 q^{11} -1.70279 q^{13} +2.73011 q^{14} +1.00000 q^{16} +1.33871 q^{17} +4.04876 q^{19} +1.84210 q^{20} -3.84740 q^{22} -3.24440 q^{23} -1.60665 q^{25} -1.70279 q^{26} +2.73011 q^{28} +1.77539 q^{29} -3.60491 q^{31} +1.00000 q^{32} +1.33871 q^{34} +5.02915 q^{35} -4.85467 q^{37} +4.04876 q^{38} +1.84210 q^{40} +11.5668 q^{41} +7.48373 q^{43} -3.84740 q^{44} -3.24440 q^{46} +7.47502 q^{47} +0.453494 q^{49} -1.60665 q^{50} -1.70279 q^{52} +8.47922 q^{53} -7.08731 q^{55} +2.73011 q^{56} +1.77539 q^{58} +9.23848 q^{59} +7.09661 q^{61} -3.60491 q^{62} +1.00000 q^{64} -3.13671 q^{65} -10.8507 q^{67} +1.33871 q^{68} +5.02915 q^{70} +7.94712 q^{71} +11.5681 q^{73} -4.85467 q^{74} +4.04876 q^{76} -10.5038 q^{77} +8.02055 q^{79} +1.84210 q^{80} +11.5668 q^{82} +6.08371 q^{83} +2.46604 q^{85} +7.48373 q^{86} -3.84740 q^{88} -8.43503 q^{89} -4.64879 q^{91} -3.24440 q^{92} +7.47502 q^{94} +7.45825 q^{95} +7.75492 q^{97} +0.453494 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} + 3 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^4 + 3 * q^5 + 6 * q^7 + 12 * q^8	\( 12 q + 12 q^{2} + 12 q^{4} + 3 q^{5} + 6 q^{7} + 12 q^{8} + 3 q^{10} + 10 q^{11} + 5 q^{13} + 6 q^{14} + 12 q^{16} + 8 q^{17} + 2 q^{19} + 3 q^{20} + 10 q^{22} + 9 q^{23} + 7 q^{25} + 5 q^{26} + 6 q^{28} + 19 q^{29} + 10 q^{31} + 12 q^{32} + 8 q^{34} + 20 q^{35} + 11 q^{37} + 2 q^{38} + 3 q^{40} + 8 q^{41} + 13 q^{43} + 10 q^{44} + 9 q^{46} + 11 q^{47} + 2 q^{49} + 7 q^{50} + 5 q^{52} + 24 q^{53} + 3 q^{55} + 6 q^{56} + 19 q^{58} + 10 q^{59} + 10 q^{62} + 12 q^{64} + 28 q^{65} + 21 q^{67} + 8 q^{68} + 20 q^{70} + 37 q^{71} - 2 q^{73} + 11 q^{74} + 2 q^{76} + 2 q^{77} + 7 q^{79} + 3 q^{80} + 8 q^{82} + 22 q^{83} + 15 q^{85} + 13 q^{86} + 10 q^{88} + 40 q^{89} + q^{91} + 9 q^{92} + 11 q^{94} + 11 q^{95} + 7 q^{97} + 2 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 + 3 * q^5 + 6 * q^7 + 12 * q^8 + 3 * q^10 + 10 * q^11 + 5 * q^13 + 6 * q^14 + 12 * q^16 + 8 * q^17 + 2 * q^19 + 3 * q^20 + 10 * q^22 + 9 * q^23 + 7 * q^25 + 5 * q^26 + 6 * q^28 + 19 * q^29 + 10 * q^31 + 12 * q^32 + 8 * q^34 + 20 * q^35 + 11 * q^37 + 2 * q^38 + 3 * q^40 + 8 * q^41 + 13 * q^43 + 10 * q^44 + 9 * q^46 + 11 * q^47 + 2 * q^49 + 7 * q^50 + 5 * q^52 + 24 * q^53 + 3 * q^55 + 6 * q^56 + 19 * q^58 + 10 * q^59 + 10 * q^62 + 12 * q^64 + 28 * q^65 + 21 * q^67 + 8 * q^68 + 20 * q^70 + 37 * q^71 - 2 * q^73 + 11 * q^74 + 2 * q^76 + 2 * q^77 + 7 * q^79 + 3 * q^80 + 8 * q^82 + 22 * q^83 + 15 * q^85 + 13 * q^86 + 10 * q^88 + 40 * q^89 + q^91 + 9 * q^92 + 11 * q^94 + 11 * q^95 + 7 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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