LMFDB - Embedded newform 8046.2.a.p.1.3

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} - 5 x^{11} - 23 x^{10} + 142 x^{9} + 104 x^{8} - 1302 x^{7} + 607 x^{6} + 4323 x^{5} - 4461 x^{4} + \cdots - 553 $ x^12 - 5x^11 - 23x^10 + 142x^9 + 104x^8 - 1302x^7 + 607x^6 + 4323x^5 - 4461x^4 - 3333x^3 + 4805x^2 - 224*x - 553
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.38357$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -2.38357 q^{5} +4.24377 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -2.38357 q^{5} +4.24377 q^{7} +1.00000 q^{8} -2.38357 q^{10} +3.02184 q^{11} -4.77318 q^{13} +4.24377 q^{14} +1.00000 q^{16} -3.59582 q^{17} -0.186813 q^{19} -2.38357 q^{20} +3.02184 q^{22} +1.80017 q^{23} +0.681408 q^{25} -4.77318 q^{26} +4.24377 q^{28} +9.75888 q^{29} -2.75394 q^{31} +1.00000 q^{32} -3.59582 q^{34} -10.1153 q^{35} -10.7593 q^{37} -0.186813 q^{38} -2.38357 q^{40} +1.84886 q^{41} +8.66820 q^{43} +3.02184 q^{44} +1.80017 q^{46} +9.97430 q^{47} +11.0096 q^{49} +0.681408 q^{50} -4.77318 q^{52} -1.29170 q^{53} -7.20278 q^{55} +4.24377 q^{56} +9.75888 q^{58} +5.85085 q^{59} +7.73588 q^{61} -2.75394 q^{62} +1.00000 q^{64} +11.3772 q^{65} +11.8732 q^{67} -3.59582 q^{68} -10.1153 q^{70} +9.41056 q^{71} +1.40310 q^{73} -10.7593 q^{74} -0.186813 q^{76} +12.8240 q^{77} -9.05695 q^{79} -2.38357 q^{80} +1.84886 q^{82} +5.97278 q^{83} +8.57089 q^{85} +8.66820 q^{86} +3.02184 q^{88} +0.434476 q^{89} -20.2563 q^{91} +1.80017 q^{92} +9.97430 q^{94} +0.445281 q^{95} -7.48855 q^{97} +11.0096 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8	$ 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8} + 5 q^{10} + 6 q^{11} + 3 q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} + 8 q^{19} + 5 q^{20} + 6 q^{22} + 11 q^{23} + 11 q^{25} + 3 q^{26} + 6 q^{28} + 29 q^{29} + 2 q^{31} + 12 q^{32} + 6 q^{34} + 4 q^{35} + 5 q^{37} + 8 q^{38} + 5 q^{40} + 22 q^{41} + 9 q^{43} + 6 q^{44} + 11 q^{46} + 15 q^{47} + 14 q^{49} + 11 q^{50} + 3 q^{52} + 12 q^{53} + 13 q^{55} + 6 q^{56} + 29 q^{58} + 34 q^{59} - 4 q^{61} + 2 q^{62} + 12 q^{64} + 12 q^{65} + q^{67} + 6 q^{68} + 4 q^{70} + 21 q^{71} - 2 q^{73} + 5 q^{74} + 8 q^{76} + 34 q^{77} + 9 q^{79} + 5 q^{80} + 22 q^{82} + 10 q^{83} + 5 q^{85} + 9 q^{86} + 6 q^{88} - 2 q^{89} + 17 q^{91} + 11 q^{92} + 15 q^{94} + 69 q^{95} - 13 q^{97} + 14 q^{98}+O(q^{100}) $ 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8 + 5 * q^10 + 6 * q^11 + 3 * q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 + 8 * q^19 + 5 * q^20 + 6 * q^22 + 11 * q^23 + 11 * q^25 + 3 * q^26 + 6 * q^28 + 29 * q^29 + 2 * q^31 + 12 * q^32 + 6 * q^34 + 4 * q^35 + 5 * q^37 + 8 * q^38 + 5 * q^40 + 22 * q^41 + 9 * q^43 + 6 * q^44 + 11 * q^46 + 15 * q^47 + 14 * q^49 + 11 * q^50 + 3 * q^52 + 12 * q^53 + 13 * q^55 + 6 * q^56 + 29 * q^58 + 34 * q^59 - 4 * q^61 + 2 * q^62 + 12 * q^64 + 12 * q^65 + q^67 + 6 * q^68 + 4 * q^70 + 21 * q^71 - 2 * q^73 + 5 * q^74 + 8 * q^76 + 34 * q^77 + 9 * q^79 + 5 * q^80 + 22 * q^82 + 10 * q^83 + 5 * q^85 + 9 * q^86 + 6 * q^88 - 2 * q^89 + 17 * q^91 + 11 * q^92 + 15 * q^94 + 69 * q^95 - 13 * q^97 + 14 * 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
$5$	−2.38357	−1.06597	−0.532983	−	0.846126i	$-0.678929\pi$
$5$	−2.38357	−1.06597	−0.532983	+	0.846126i	$0.678929\pi$
$6$	0	0
$7$	4.24377	1.60400	0.801998	−	0.597327i	$-0.203771\pi$
$7$	4.24377	1.60400	0.801998	+	0.597327i	$0.203771\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−2.38357	−0.753751
$11$	3.02184	0.911120	0.455560	−	0.890205i	$-0.349439\pi$
$11$	3.02184	0.911120	0.455560	+	0.890205i	$0.349439\pi$
$12$	0	0
$13$	−4.77318	−1.32384	−0.661921	−	0.749574i	$-0.730259\pi$
$13$	−4.77318	−1.32384	−0.661921	+	0.749574i	$0.730259\pi$
$14$	4.24377	1.13420
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.59582	−0.872114	−0.436057	−	0.899919i	$-0.643625\pi$
$17$	−3.59582	−0.872114	−0.436057	+	0.899919i	$0.643625\pi$
$18$	0	0
$19$	−0.186813	−0.0428577	−0.0214289	−	0.999770i	$-0.506822\pi$
$19$	−0.186813	−0.0428577	−0.0214289	+	0.999770i	$0.506822\pi$
$20$	−2.38357	−0.532983
$21$	0	0
$22$	3.02184	0.644259
$23$	1.80017	0.375362	0.187681	−	0.982230i	$-0.439903\pi$
$23$	1.80017	0.375362	0.187681	+	0.982230i	$0.439903\pi$
$24$	0	0
$25$	0.681408	0.136282
$26$	−4.77318	−0.936098
$27$	0	0
$28$	4.24377	0.801998
$29$	9.75888	1.81218	0.906089	−	0.423086i	$-0.139053\pi$
$29$	9.75888	1.81218	0.906089	+	0.423086i	$0.139053\pi$
$30$	0	0
$31$	−2.75394	−0.494622	−0.247311	−	0.968936i	$-0.579547\pi$
$31$	−2.75394	−0.494622	−0.247311	+	0.968936i	$0.579547\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.59582	−0.616678
$35$	−10.1153	−1.70980
$36$	0	0
$37$	−10.7593	−1.76882	−0.884410	−	0.466711i	$-0.845439\pi$
$37$	−10.7593	−1.76882	−0.884410	+	0.466711i	$0.845439\pi$
$38$	−0.186813	−0.0303050
$39$	0	0
$40$	−2.38357	−0.376876
$41$	1.84886	0.288743	0.144371	−	0.989524i	$-0.453884\pi$
$41$	1.84886	0.288743	0.144371	+	0.989524i	$0.453884\pi$
$42$	0	0
$43$	8.66820	1.32189	0.660944	−	0.750435i	$-0.270156\pi$
$43$	8.66820	1.32189	0.660944	+	0.750435i	$0.270156\pi$
$44$	3.02184	0.455560
$45$	0	0
$46$	1.80017	0.265421
$47$	9.97430	1.45490	0.727450	−	0.686160i	$-0.240705\pi$
$47$	9.97430	1.45490	0.727450	+	0.686160i	$0.240705\pi$
$48$	0	0
$49$	11.0096	1.57280
$50$	0.681408	0.0963656
$51$	0	0
$52$	−4.77318	−0.661921
$53$	−1.29170	−0.177429	−0.0887147	−	0.996057i	$-0.528276\pi$
$53$	−1.29170	−0.177429	−0.0887147	+	0.996057i	$0.528276\pi$
$54$	0	0
$55$	−7.20278	−0.971223
$56$	4.24377	0.567098
$57$	0	0
$58$	9.75888	1.28140
$59$	5.85085	0.761715	0.380858	−	0.924634i	$-0.375629\pi$
$59$	5.85085	0.761715	0.380858	+	0.924634i	$0.375629\pi$
$60$	0	0
$61$	7.73588	0.990478	0.495239	−	0.868757i	$-0.335081\pi$
$61$	7.73588	0.990478	0.495239	+	0.868757i	$0.335081\pi$
$62$	−2.75394	−0.349751
$63$	0	0
$64$	1.00000	0.125000
$65$	11.3772	1.41117
$66$	0	0
$67$	11.8732	1.45054	0.725268	−	0.688466i	$-0.241716\pi$
$67$	11.8732	1.45054	0.725268	+	0.688466i	$0.241716\pi$
$68$	−3.59582	−0.436057
$69$	0	0
$70$	−10.1153	−1.20901
$71$	9.41056	1.11683	0.558414	−	0.829562i	$-0.311410\pi$
$71$	9.41056	1.11683	0.558414	+	0.829562i	$0.311410\pi$
$72$	0	0
$73$	1.40310	0.164221	0.0821103	−	0.996623i	$-0.473834\pi$
$73$	1.40310	0.164221	0.0821103	+	0.996623i	$0.473834\pi$
$74$	−10.7593	−1.25074
$75$	0	0
$76$	−0.186813	−0.0214289
$77$	12.8240	1.46143
$78$	0	0
$79$	−9.05695	−1.01899	−0.509493	−	0.860475i	$-0.670167\pi$
$79$	−9.05695	−1.01899	−0.509493	+	0.860475i	$0.670167\pi$
$80$	−2.38357	−0.266491
$81$	0	0
$82$	1.84886	0.204172
$83$	5.97278	0.655598	0.327799	−	0.944748i	$-0.393693\pi$
$83$	5.97278	0.655598	0.327799	+	0.944748i	$0.393693\pi$
$84$	0	0
$85$	8.57089	0.929643
$86$	8.66820	0.934716
$87$	0	0
$88$	3.02184	0.322130
$89$	0.434476	0.0460544	0.0230272	−	0.999735i	$-0.492670\pi$
$89$	0.434476	0.0460544	0.0230272	+	0.999735i	$0.492670\pi$
$90$	0	0
$91$	−20.2563	−2.12344
$92$	1.80017	0.187681
$93$	0	0
$94$	9.97430	1.02877
$95$	0.445281	0.0456849
$96$	0	0
$97$	−7.48855	−0.760347	−0.380174	−	0.924915i	$-0.624136\pi$
$97$	−7.48855	−0.760347	−0.380174	+	0.924915i	$0.624136\pi$
$98$	11.0096	1.11214
$99$	0	0
$100$	0.681408	0.0681408
$101$	−7.20086	−0.716512	−0.358256	−	0.933623i	$-0.616629\pi$
$101$	−7.20086	−0.716512	−0.358256	+	0.933623i	$0.616629\pi$
$102$	0	0
$103$	−15.1707	−1.49481	−0.747406	−	0.664368i	$-0.768701\pi$
$103$	−15.1707	−1.49481	−0.747406	+	0.664368i	$0.768701\pi$
$104$	−4.77318	−0.468049
$105$	0	0
$106$	−1.29170	−0.125461
$107$	2.90215	0.280562	0.140281	−	0.990112i	$-0.455199\pi$
$107$	2.90215	0.280562	0.140281	+	0.990112i	$0.455199\pi$
$108$	0	0
$109$	16.0307	1.53546	0.767730	−	0.640774i	$-0.221386\pi$
$109$	16.0307	1.53546	0.767730	+	0.640774i	$0.221386\pi$
$110$	−7.20278	−0.686758
$111$	0	0
$112$	4.24377	0.400999
$113$	3.16837	0.298055	0.149028	−	0.988833i	$-0.452386\pi$
$113$	3.16837	0.298055	0.149028	+	0.988833i	$0.452386\pi$
$114$	0	0
$115$	−4.29084	−0.400123
$116$	9.75888	0.906089
$117$	0	0
$118$	5.85085	0.538614
$119$	−15.2598	−1.39887
$120$	0	0
$121$	−1.86845	−0.169859
$122$	7.73588	0.700374
$123$	0	0
$124$	−2.75394	−0.247311
$125$	10.2937	0.920694
$126$	0	0
$127$	−0.901656	−0.0800090	−0.0400045	−	0.999199i	$-0.512737\pi$
$127$	−0.901656	−0.0800090	−0.0400045	+	0.999199i	$0.512737\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	11.3772	0.997847
$131$	7.20947	0.629894	0.314947	−	0.949109i	$-0.398013\pi$
$131$	7.20947	0.629894	0.314947	+	0.949109i	$0.398013\pi$
$132$	0	0
$133$	−0.792790	−0.0687436
$134$	11.8732	1.02568
$135$	0	0
$136$	−3.59582	−0.308339
$137$	7.21427	0.616356	0.308178	−	0.951329i	$-0.400281\pi$
$137$	7.21427	0.616356	0.308178	+	0.951329i	$0.400281\pi$
$138$	0	0
$139$	−18.1470	−1.53920	−0.769602	−	0.638524i	$-0.779545\pi$
$139$	−18.1470	−1.53920	−0.769602	+	0.638524i	$0.779545\pi$
$140$	−10.1153	−0.854901
$141$	0	0
$142$	9.41056	0.789717
$143$	−14.4238	−1.20618
$144$	0	0
$145$	−23.2610	−1.93172
$146$	1.40310	0.116121
$147$	0	0
$148$	−10.7593	−0.884410
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−0.270277	−0.0219949	−0.0109974	−	0.999940i	$-0.503501\pi$
$151$	−0.270277	−0.0219949	−0.0109974	+	0.999940i	$0.503501\pi$
$152$	−0.186813	−0.0151525
$153$	0	0
$154$	12.8240	1.03339
$155$	6.56421	0.527250
$156$	0	0
$157$	0.523007	0.0417405	0.0208702	−	0.999782i	$-0.493356\pi$
$157$	0.523007	0.0417405	0.0208702	+	0.999782i	$0.493356\pi$
$158$	−9.05695	−0.720533
$159$	0	0
$160$	−2.38357	−0.188438
$161$	7.63952	0.602079
$162$	0	0
$163$	−17.5841	−1.37729	−0.688645	−	0.725099i	$-0.741794\pi$
$163$	−17.5841	−1.37729	−0.688645	+	0.725099i	$0.741794\pi$
$164$	1.84886	0.144371
$165$	0	0
$166$	5.97278	0.463578
$167$	10.7254	0.829959	0.414979	−	0.909831i	$-0.363789\pi$
$167$	10.7254	0.829959	0.414979	+	0.909831i	$0.363789\pi$
$168$	0	0
$169$	9.78325	0.752557
$170$	8.57089	0.657357
$171$	0	0
$172$	8.66820	0.660944
$173$	−1.33631	−0.101598	−0.0507988	−	0.998709i	$-0.516177\pi$
$173$	−1.33631	−0.101598	−0.0507988	+	0.998709i	$0.516177\pi$
$174$	0	0
$175$	2.89174	0.218595
$176$	3.02184	0.227780
$177$	0	0
$178$	0.434476	0.0325653
$179$	−10.3669	−0.774858	−0.387429	−	0.921900i	$-0.626637\pi$
$179$	−10.3669	−0.774858	−0.387429	+	0.921900i	$0.626637\pi$
$180$	0	0
$181$	7.10722	0.528276	0.264138	−	0.964485i	$-0.414913\pi$
$181$	7.10722	0.528276	0.264138	+	0.964485i	$0.414913\pi$
$182$	−20.2563	−1.50150
$183$	0	0
$184$	1.80017	0.132710
$185$	25.6456	1.88550
$186$	0	0
$187$	−10.8660	−0.794601
$188$	9.97430	0.727450
$189$	0	0
$190$	0.445281	0.0323041
$191$	15.8013	1.14334	0.571669	−	0.820484i	$-0.306296\pi$
$191$	15.8013	1.14334	0.571669	+	0.820484i	$0.306296\pi$
$192$	0	0
$193$	3.48510	0.250863	0.125431	−	0.992102i	$-0.459969\pi$
$193$	3.48510	0.250863	0.125431	+	0.992102i	$0.459969\pi$
$194$	−7.48855	−0.537647
$195$	0	0
$196$	11.0096	0.786400
$197$	19.3760	1.38048	0.690241	−	0.723580i	$-0.257504\pi$
$197$	19.3760	1.38048	0.690241	+	0.723580i	$0.257504\pi$
$198$	0	0
$199$	12.1663	0.862448	0.431224	−	0.902245i	$-0.358082\pi$
$199$	12.1663	0.862448	0.431224	+	0.902245i	$0.358082\pi$
$200$	0.681408	0.0481828
$201$	0	0
$202$	−7.20086	−0.506651
$203$	41.4145	2.90673
$204$	0	0
$205$	−4.40688	−0.307790
$206$	−15.1707	−1.05699
$207$	0	0
$208$	−4.77318	−0.330960
$209$	−0.564519	−0.0390486
$210$	0	0
$211$	2.39138	0.164629	0.0823146	−	0.996606i	$-0.473769\pi$
$211$	2.39138	0.164629	0.0823146	+	0.996606i	$0.473769\pi$
$212$	−1.29170	−0.0887147
$213$	0	0
$214$	2.90215	0.198387
$215$	−20.6613	−1.40909
$216$	0	0
$217$	−11.6871	−0.793371
$218$	16.0307	1.08573
$219$	0	0
$220$	−7.20278	−0.485611
$221$	17.1635	1.15454
$222$	0	0
$223$	13.6534	0.914302	0.457151	−	0.889389i	$-0.348870\pi$
$223$	13.6534	0.914302	0.457151	+	0.889389i	$0.348870\pi$
$224$	4.24377	0.283549
$225$	0	0
$226$	3.16837	0.210757
$227$	−3.02417	−0.200721	−0.100361	−	0.994951i	$-0.532000\pi$
$227$	−3.02417	−0.200721	−0.100361	+	0.994951i	$0.532000\pi$
$228$	0	0
$229$	5.88592	0.388953	0.194476	−	0.980907i	$-0.437699\pi$
$229$	5.88592	0.388953	0.194476	+	0.980907i	$0.437699\pi$
$230$	−4.29084	−0.282929
$231$	0	0
$232$	9.75888	0.640702
$233$	4.86587	0.318774	0.159387	−	0.987216i	$-0.449048\pi$
$233$	4.86587	0.318774	0.159387	+	0.987216i	$0.449048\pi$
$234$	0	0
$235$	−23.7744	−1.55087
$236$	5.85085	0.380858
$237$	0	0
$238$	−15.2598	−0.989148
$239$	15.8490	1.02519	0.512593	−	0.858632i	$-0.328685\pi$
$239$	15.8490	1.02519	0.512593	+	0.858632i	$0.328685\pi$
$240$	0	0
$241$	29.1733	1.87922	0.939608	−	0.342253i	$-0.111190\pi$
$241$	29.1733	1.87922	0.939608	+	0.342253i	$0.111190\pi$
$242$	−1.86845	−0.120109
$243$	0	0
$244$	7.73588	0.495239
$245$	−26.2422	−1.67655
$246$	0	0
$247$	0.891690	0.0567369
$248$	−2.75394	−0.174875
$249$	0	0
$250$	10.2937	0.651029
$251$	−14.7903	−0.933555	−0.466778	−	0.884375i	$-0.654585\pi$
$251$	−14.7903	−0.933555	−0.466778	+	0.884375i	$0.654585\pi$
$252$	0	0
$253$	5.43984	0.342000
$254$	−0.901656	−0.0565749
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.10154	0.380604	0.190302	−	0.981726i	$-0.439053\pi$
$257$	6.10154	0.380604	0.190302	+	0.981726i	$0.439053\pi$
$258$	0	0
$259$	−45.6601	−2.83718
$260$	11.3772	0.705585
$261$	0	0
$262$	7.20947	0.445402
$263$	−24.9970	−1.54138	−0.770691	−	0.637209i	$-0.780089\pi$
$263$	−24.9970	−1.54138	−0.770691	+	0.637209i	$0.780089\pi$
$264$	0	0
$265$	3.07887	0.189133
$266$	−0.792790	−0.0486091
$267$	0	0
$268$	11.8732	0.725268
$269$	−5.06247	−0.308664	−0.154332	−	0.988019i	$-0.549323\pi$
$269$	−5.06247	−0.308664	−0.154332	+	0.988019i	$0.549323\pi$
$270$	0	0
$271$	25.8216	1.56855	0.784274	−	0.620414i	$-0.213035\pi$
$271$	25.8216	1.56855	0.784274	+	0.620414i	$0.213035\pi$
$272$	−3.59582	−0.218029
$273$	0	0
$274$	7.21427	0.435830
$275$	2.05911	0.124169
$276$	0	0
$277$	29.6146	1.77937	0.889685	−	0.456575i	$-0.150924\pi$
$277$	29.6146	1.77937	0.889685	+	0.456575i	$0.150924\pi$
$278$	−18.1470	−1.08838
$279$	0	0
$280$	−10.1153	−0.604507
$281$	−5.08230	−0.303185	−0.151592	−	0.988443i	$-0.548440\pi$
$281$	−5.08230	−0.303185	−0.151592	+	0.988443i	$0.548440\pi$
$282$	0	0
$283$	1.96090	0.116563	0.0582817	−	0.998300i	$-0.481438\pi$
$283$	1.96090	0.116563	0.0582817	+	0.998300i	$0.481438\pi$
$284$	9.41056	0.558414
$285$	0	0
$286$	−14.4238	−0.852898
$287$	7.84613	0.463142
$288$	0	0
$289$	−4.07008	−0.239416
$290$	−23.2610	−1.36593
$291$	0	0
$292$	1.40310	0.0821103
$293$	−12.9104	−0.754234	−0.377117	−	0.926166i	$-0.623085\pi$
$293$	−12.9104	−0.754234	−0.377117	+	0.926166i	$0.623085\pi$
$294$	0	0
$295$	−13.9459	−0.811962
$296$	−10.7593	−0.625372
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	−8.59255	−0.496920
$300$	0	0
$301$	36.7858	2.12030
$302$	−0.270277	−0.0155527
$303$	0	0
$304$	−0.186813	−0.0107144
$305$	−18.4390	−1.05581
$306$	0	0
$307$	11.8181	0.674496	0.337248	−	0.941416i	$-0.390504\pi$
$307$	11.8181	0.674496	0.337248	+	0.941416i	$0.390504\pi$
$308$	12.8240	0.730716
$309$	0	0
$310$	6.56421	0.372822
$311$	−17.7340	−1.00560	−0.502801	−	0.864402i	$-0.667697\pi$
$311$	−17.7340	−1.00560	−0.502801	+	0.864402i	$0.667697\pi$
$312$	0	0
$313$	−26.9984	−1.52604	−0.763020	−	0.646375i	$-0.776284\pi$
$313$	−26.9984	−1.52604	−0.763020	+	0.646375i	$0.776284\pi$
$314$	0.523007	0.0295150
$315$	0	0
$316$	−9.05695	−0.509493
$317$	8.15122	0.457818	0.228909	−	0.973448i	$-0.426484\pi$
$317$	8.15122	0.457818	0.228909	+	0.973448i	$0.426484\pi$
$318$	0	0
$319$	29.4898	1.65111
$320$	−2.38357	−0.133246
$321$	0	0
$322$	7.63952	0.425734
$323$	0.671744	0.0373769
$324$	0	0
$325$	−3.25248	−0.180415
$326$	−17.5841	−0.973891
$327$	0	0
$328$	1.84886	0.102086
$329$	42.3286	2.33365
$330$	0	0
$331$	25.3921	1.39568	0.697839	−	0.716255i	$-0.254145\pi$
$331$	25.3921	1.39568	0.697839	+	0.716255i	$0.254145\pi$
$332$	5.97278	0.327799
$333$	0	0
$334$	10.7254	0.586870
$335$	−28.3005	−1.54622
$336$	0	0
$337$	6.97919	0.380180	0.190090	−	0.981767i	$-0.439122\pi$
$337$	6.97919	0.380180	0.190090	+	0.981767i	$0.439122\pi$
$338$	9.78325	0.532138
$339$	0	0
$340$	8.57089	0.464822
$341$	−8.32198	−0.450660
$342$	0	0
$343$	17.0158	0.918769
$344$	8.66820	0.467358
$345$	0	0
$346$	−1.33631	−0.0718403
$347$	−14.6682	−0.787428	−0.393714	−	0.919233i	$-0.628810\pi$
$347$	−14.6682	−0.787428	−0.393714	+	0.919233i	$0.628810\pi$
$348$	0	0
$349$	−15.9194	−0.852144	−0.426072	−	0.904689i	$-0.640103\pi$
$349$	−15.9194	−0.852144	−0.426072	+	0.904689i	$0.640103\pi$
$350$	2.89174	0.154570
$351$	0	0
$352$	3.02184	0.161065
$353$	29.0530	1.54634	0.773168	−	0.634202i	$-0.218671\pi$
$353$	29.0530	1.54634	0.773168	+	0.634202i	$0.218671\pi$
$354$	0	0
$355$	−22.4307	−1.19050
$356$	0.434476	0.0230272
$357$	0	0
$358$	−10.3669	−0.547907
$359$	−16.0434	−0.846737	−0.423368	−	0.905958i	$-0.639152\pi$
$359$	−16.0434	−0.846737	−0.423368	+	0.905958i	$0.639152\pi$
$360$	0	0
$361$	−18.9651	−0.998163
$362$	7.10722	0.373547
$363$	0	0
$364$	−20.2563	−1.06172
$365$	−3.34439	−0.175053
$366$	0	0
$367$	8.55330	0.446479	0.223239	−	0.974764i	$-0.428337\pi$
$367$	8.55330	0.446479	0.223239	+	0.974764i	$0.428337\pi$
$368$	1.80017	0.0938405
$369$	0	0
$370$	25.6456	1.33325
$371$	−5.48170	−0.284596
$372$	0	0
$373$	−19.1127	−0.989621	−0.494810	−	0.869001i	$-0.664762\pi$
$373$	−19.1127	−0.989621	−0.494810	+	0.869001i	$0.664762\pi$
$374$	−10.8660	−0.561868
$375$	0	0
$376$	9.97430	0.514385
$377$	−46.5809	−2.39904
$378$	0	0
$379$	32.7673	1.68314	0.841570	−	0.540147i	$-0.181632\pi$
$379$	32.7673	1.68314	0.841570	+	0.540147i	$0.181632\pi$
$380$	0.445281	0.0228424
$381$	0	0
$382$	15.8013	0.808463
$383$	−25.0882	−1.28195	−0.640973	−	0.767564i	$-0.721469\pi$
$383$	−25.0882	−1.28195	−0.640973	+	0.767564i	$0.721469\pi$
$384$	0	0
$385$	−30.5670	−1.55784
$386$	3.48510	0.177387
$387$	0	0
$388$	−7.48855	−0.380174
$389$	15.7312	0.797606	0.398803	−	0.917037i	$-0.369426\pi$
$389$	15.7312	0.797606	0.398803	+	0.917037i	$0.369426\pi$
$390$	0	0
$391$	−6.47310	−0.327359
$392$	11.0096	0.556069
$393$	0	0
$394$	19.3760	0.976148
$395$	21.5879	1.08620
$396$	0	0
$397$	26.3693	1.32344	0.661718	−	0.749753i	$-0.269828\pi$
$397$	26.3693	1.32344	0.661718	+	0.749753i	$0.269828\pi$
$398$	12.1663	0.609843
$399$	0	0
$400$	0.681408	0.0340704
$401$	14.1978	0.709004	0.354502	−	0.935055i	$-0.384650\pi$
$401$	14.1978	0.709004	0.354502	+	0.935055i	$0.384650\pi$
$402$	0	0
$403$	13.1450	0.654801
$404$	−7.20086	−0.358256
$405$	0	0
$406$	41.4145	2.05537
$407$	−32.5130	−1.61161
$408$	0	0
$409$	−34.4325	−1.70258	−0.851290	−	0.524696i	$-0.824179\pi$
$409$	−34.4325	−1.70258	−0.851290	+	0.524696i	$0.824179\pi$
$410$	−4.40688	−0.217640
$411$	0	0
$412$	−15.1707	−0.747406
$413$	24.8297	1.22179
$414$	0	0
$415$	−14.2365	−0.698844
$416$	−4.77318	−0.234024
$417$	0	0
$418$	−0.564519	−0.0276115
$419$	−28.5257	−1.39357	−0.696785	−	0.717280i	$-0.745387\pi$
$419$	−28.5257	−1.39357	−0.696785	+	0.717280i	$0.745387\pi$
$420$	0	0
$421$	16.4084	0.799696	0.399848	−	0.916582i	$-0.369063\pi$
$421$	16.4084	0.799696	0.399848	+	0.916582i	$0.369063\pi$
$422$	2.39138	0.116410
$423$	0	0
$424$	−1.29170	−0.0627307
$425$	−2.45022	−0.118853
$426$	0	0
$427$	32.8293	1.58872
$428$	2.90215	0.140281
$429$	0	0
$430$	−20.6613	−0.996374
$431$	11.7732	0.567097	0.283549	−	0.958958i	$-0.408488\pi$
$431$	11.7732	0.567097	0.283549	+	0.958958i	$0.408488\pi$
$432$	0	0
$433$	3.98250	0.191387	0.0956934	−	0.995411i	$-0.469493\pi$
$433$	3.98250	0.191387	0.0956934	+	0.995411i	$0.469493\pi$
$434$	−11.6871	−0.560998
$435$	0	0
$436$	16.0307	0.767730
$437$	−0.336295	−0.0160872
$438$	0	0
$439$	13.3979	0.639446	0.319723	−	0.947511i	$-0.396410\pi$
$439$	13.3979	0.639446	0.319723	+	0.947511i	$0.396410\pi$
$440$	−7.20278	−0.343379
$441$	0	0
$442$	17.1635	0.816384
$443$	28.2871	1.34396	0.671981	−	0.740568i	$-0.265444\pi$
$443$	28.2871	1.34396	0.671981	+	0.740568i	$0.265444\pi$
$444$	0	0
$445$	−1.03560	−0.0490923
$446$	13.6534	0.646509
$447$	0	0
$448$	4.24377	0.200499
$449$	15.9872	0.754485	0.377242	−	0.926115i	$-0.376872\pi$
$449$	15.9872	0.754485	0.377242	+	0.926115i	$0.376872\pi$
$450$	0	0
$451$	5.58696	0.263080
$452$	3.16837	0.149028
$453$	0	0
$454$	−3.02417	−0.141931
$455$	48.2823	2.26351
$456$	0	0
$457$	−3.00001	−0.140335	−0.0701673	−	0.997535i	$-0.522353\pi$
$457$	−3.00001	−0.140335	−0.0701673	+	0.997535i	$0.522353\pi$
$458$	5.88592	0.275031
$459$	0	0
$460$	−4.29084	−0.200061
$461$	14.2661	0.664437	0.332219	−	0.943202i	$-0.392203\pi$
$461$	14.2661	0.664437	0.332219	+	0.943202i	$0.392203\pi$
$462$	0	0
$463$	−13.8671	−0.644461	−0.322230	−	0.946661i	$-0.604433\pi$
$463$	−13.8671	−0.644461	−0.322230	+	0.946661i	$0.604433\pi$
$464$	9.75888	0.453045
$465$	0	0
$466$	4.86587	0.225407
$467$	−31.8022	−1.47163	−0.735816	−	0.677182i	$-0.763201\pi$
$467$	−31.8022	−1.47163	−0.735816	+	0.677182i	$0.763201\pi$
$468$	0	0
$469$	50.3869	2.32665
$470$	−23.7744	−1.09663
$471$	0	0
$472$	5.85085	0.269307
$473$	26.1939	1.20440
$474$	0	0
$475$	−0.127296	−0.00584072
$476$	−15.2598	−0.699434
$477$	0	0
$478$	15.8490	0.724916
$479$	27.0793	1.23729	0.618643	−	0.785672i	$-0.287683\pi$
$479$	27.0793	1.23729	0.618643	+	0.785672i	$0.287683\pi$
$480$	0	0
$481$	51.3561	2.34164
$482$	29.1733	1.32881
$483$	0	0
$484$	−1.86845	−0.0849297
$485$	17.8495	0.810503
$486$	0	0
$487$	−36.9878	−1.67608	−0.838038	−	0.545612i	$-0.816297\pi$
$487$	−36.9878	−1.67608	−0.838038	+	0.545612i	$0.816297\pi$
$488$	7.73588	0.350187
$489$	0	0
$490$	−26.2422	−1.18550
$491$	−6.03716	−0.272453	−0.136227	−	0.990678i	$-0.543498\pi$
$491$	−6.03716	−0.272453	−0.136227	+	0.990678i	$0.543498\pi$
$492$	0	0
$493$	−35.0912	−1.58043
$494$	0.891690	0.0401190
$495$	0	0
$496$	−2.75394	−0.123656
$497$	39.9363	1.79139
$498$	0	0
$499$	−22.7307	−1.01756	−0.508782	−	0.860895i	$-0.669904\pi$
$499$	−22.7307	−1.01756	−0.508782	+	0.860895i	$0.669904\pi$
$500$	10.2937	0.460347
$501$	0	0
$502$	−14.7903	−0.660123
$503$	−42.6176	−1.90022	−0.950112	−	0.311908i	$-0.899032\pi$
$503$	−42.6176	−1.90022	−0.950112	+	0.311908i	$0.899032\pi$
$504$	0	0
$505$	17.1638	0.763777
$506$	5.43984	0.241830
$507$	0	0
$508$	−0.901656	−0.0400045
$509$	17.8076	0.789309	0.394655	−	0.918830i	$-0.370864\pi$
$509$	17.8076	0.789309	0.394655	+	0.918830i	$0.370864\pi$
$510$	0	0
$511$	5.95444	0.263409
$512$	1.00000	0.0441942
$513$	0	0
$514$	6.10154	0.269127
$515$	36.1604	1.59342
$516$	0	0
$517$	30.1408	1.32559
$518$	−45.6601	−2.00619
$519$	0	0
$520$	11.3772	0.498924
$521$	45.0643	1.97430	0.987151	−	0.159789i	$-0.0510812\pi$
$521$	45.0643	1.97430	0.987151	+	0.159789i	$0.0510812\pi$
$522$	0	0
$523$	5.64776	0.246959	0.123480	−	0.992347i	$-0.460595\pi$
$523$	5.64776	0.246959	0.123480	+	0.992347i	$0.460595\pi$
$524$	7.20947	0.314947
$525$	0	0
$526$	−24.9970	−1.08992
$527$	9.90267	0.431367
$528$	0	0
$529$	−19.7594	−0.859103
$530$	3.07887	0.133738
$531$	0	0
$532$	−0.792790	−0.0343718
$533$	−8.82492	−0.382250
$534$	0	0
$535$	−6.91748	−0.299069
$536$	11.8732	0.512842
$537$	0	0
$538$	−5.06247	−0.218259
$539$	33.2693	1.43301
$540$	0	0
$541$	32.4953	1.39708	0.698542	−	0.715569i	$-0.253833\pi$
$541$	32.4953	1.39708	0.698542	+	0.715569i	$0.253833\pi$
$542$	25.8216	1.10913
$543$	0	0
$544$	−3.59582	−0.154169
$545$	−38.2102	−1.63675
$546$	0	0
$547$	8.40831	0.359514	0.179757	−	0.983711i	$-0.442469\pi$
$547$	8.40831	0.359514	0.179757	+	0.983711i	$0.442469\pi$
$548$	7.21427	0.308178
$549$	0	0
$550$	2.05911	0.0878007
$551$	−1.82308	−0.0776659
$552$	0	0
$553$	−38.4356	−1.63445
$554$	29.6146	1.25820
$555$	0	0
$556$	−18.1470	−0.769602
$557$	−17.0870	−0.724000	−0.362000	−	0.932178i	$-0.617906\pi$
$557$	−17.0870	−0.724000	−0.362000	+	0.932178i	$0.617906\pi$
$558$	0	0
$559$	−41.3749	−1.74997
$560$	−10.1153	−0.427451
$561$	0	0
$562$	−5.08230	−0.214384
$563$	23.3666	0.984784	0.492392	−	0.870374i	$-0.336123\pi$
$563$	23.3666	0.984784	0.492392	+	0.870374i	$0.336123\pi$
$564$	0	0
$565$	−7.55204	−0.317717
$566$	1.96090	0.0824228
$567$	0	0
$568$	9.41056	0.394858
$569$	26.9656	1.13046	0.565228	−	0.824935i	$-0.308788\pi$
$569$	26.9656	1.13046	0.565228	+	0.824935i	$0.308788\pi$
$570$	0	0
$571$	−20.9519	−0.876811	−0.438406	−	0.898777i	$-0.644457\pi$
$571$	−20.9519	−0.876811	−0.438406	+	0.898777i	$0.644457\pi$
$572$	−14.4238	−0.603090
$573$	0	0
$574$	7.84613	0.327491
$575$	1.22665	0.0511549
$576$	0	0
$577$	0.938325	0.0390630	0.0195315	−	0.999809i	$-0.493783\pi$
$577$	0.938325	0.0390630	0.0195315	+	0.999809i	$0.493783\pi$
$578$	−4.07008	−0.169293
$579$	0	0
$580$	−23.2610	−0.965860
$581$	25.3471	1.05158
$582$	0	0
$583$	−3.90333	−0.161659
$584$	1.40310	0.0580607
$585$	0	0
$586$	−12.9104	−0.533324
$587$	−4.21974	−0.174167	−0.0870836	−	0.996201i	$-0.527755\pi$
$587$	−4.21974	−0.174167	−0.0870836	+	0.996201i	$0.527755\pi$
$588$	0	0
$589$	0.514471	0.0211984
$590$	−13.9459	−0.574144
$591$	0	0
$592$	−10.7593	−0.442205
$593$	−3.04491	−0.125039	−0.0625197	−	0.998044i	$-0.519914\pi$
$593$	−3.04491	−0.125039	−0.0625197	+	0.998044i	$0.519914\pi$
$594$	0	0
$595$	36.3729	1.49114
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	−8.59255	−0.351375
$599$	−15.6959	−0.641318	−0.320659	−	0.947195i	$-0.603904\pi$
$599$	−15.6959	−0.641318	−0.320659	+	0.947195i	$0.603904\pi$
$600$	0	0
$601$	−42.8088	−1.74621	−0.873104	−	0.487533i	$-0.837897\pi$
$601$	−42.8088	−1.74621	−0.873104	+	0.487533i	$0.837897\pi$
$602$	36.7858	1.49928
$603$	0	0
$604$	−0.270277	−0.0109974
$605$	4.45359	0.181064
$606$	0	0
$607$	−40.6762	−1.65100	−0.825498	−	0.564405i	$-0.809105\pi$
$607$	−40.6762	−1.65100	−0.825498	+	0.564405i	$0.809105\pi$
$608$	−0.186813	−0.00757625
$609$	0	0
$610$	−18.4390	−0.746574
$611$	−47.6091	−1.92606
$612$	0	0
$613$	18.6084	0.751586	0.375793	−	0.926704i	$-0.377370\pi$
$613$	18.6084	0.751586	0.375793	+	0.926704i	$0.377370\pi$
$614$	11.8181	0.476940
$615$	0	0
$616$	12.8240	0.516695
$617$	10.9754	0.441854	0.220927	−	0.975290i	$-0.429092\pi$
$617$	10.9754	0.441854	0.220927	+	0.975290i	$0.429092\pi$
$618$	0	0
$619$	−7.49427	−0.301220	−0.150610	−	0.988593i	$-0.548124\pi$
$619$	−7.49427	−0.301220	−0.150610	+	0.988593i	$0.548124\pi$
$620$	6.56421	0.263625
$621$	0	0
$622$	−17.7340	−0.711068
$623$	1.84382	0.0738710
$624$	0	0
$625$	−27.9427	−1.11771
$626$	−26.9984	−1.07907
$627$	0	0
$628$	0.523007	0.0208702
$629$	38.6885	1.54261
$630$	0	0
$631$	18.4962	0.736324	0.368162	−	0.929762i	$-0.379987\pi$
$631$	18.4962	0.736324	0.368162	+	0.929762i	$0.379987\pi$
$632$	−9.05695	−0.360266
$633$	0	0
$634$	8.15122	0.323726
$635$	2.14916	0.0852868
$636$	0	0
$637$	−52.5508	−2.08214
$638$	29.4898	1.16751
$639$	0	0
$640$	−2.38357	−0.0942189
$641$	−48.5557	−1.91783	−0.958917	−	0.283687i	$-0.908442\pi$
$641$	−48.5557	−1.91783	−0.958917	+	0.283687i	$0.908442\pi$
$642$	0	0
$643$	−8.17298	−0.322311	−0.161155	−	0.986929i	$-0.551522\pi$
$643$	−8.17298	−0.322311	−0.161155	+	0.986929i	$0.551522\pi$
$644$	7.63952	0.301039
$645$	0	0
$646$	0.671744	0.0264294
$647$	9.53758	0.374961	0.187480	−	0.982268i	$-0.439968\pi$
$647$	9.53758	0.374961	0.187480	+	0.982268i	$0.439968\pi$
$648$	0	0
$649$	17.6803	0.694014
$650$	−3.25248	−0.127573
$651$	0	0
$652$	−17.5841	−0.688645
$653$	−28.8269	−1.12808	−0.564041	−	0.825747i	$-0.690754\pi$
$653$	−28.8269	−1.12808	−0.564041	+	0.825747i	$0.690754\pi$
$654$	0	0
$655$	−17.1843	−0.671445
$656$	1.84886	0.0721857
$657$	0	0
$658$	42.3286	1.65014
$659$	31.8577	1.24100	0.620501	−	0.784206i	$-0.286929\pi$
$659$	31.8577	1.24100	0.620501	+	0.784206i	$0.286929\pi$
$660$	0	0
$661$	22.7257	0.883926	0.441963	−	0.897033i	$-0.354282\pi$
$661$	22.7257	0.883926	0.441963	+	0.897033i	$0.354282\pi$
$662$	25.3921	0.986893
$663$	0	0
$664$	5.97278	0.231789
$665$	1.88967	0.0732783
$666$	0	0
$667$	17.5677	0.680223
$668$	10.7254	0.414979
$669$	0	0
$670$	−28.3005	−1.09334
$671$	23.3766	0.902445
$672$	0	0
$673$	−25.8990	−0.998334	−0.499167	−	0.866506i	$-0.666361\pi$
$673$	−25.8990	−0.998334	−0.499167	+	0.866506i	$0.666361\pi$
$674$	6.97919	0.268828
$675$	0	0
$676$	9.78325	0.376279
$677$	−15.8297	−0.608384	−0.304192	−	0.952611i	$-0.598386\pi$
$677$	−15.8297	−0.608384	−0.304192	+	0.952611i	$0.598386\pi$
$678$	0	0
$679$	−31.7797	−1.21959
$680$	8.57089	0.328679
$681$	0	0
$682$	−8.32198	−0.318665
$683$	6.47713	0.247840	0.123920	−	0.992292i	$-0.460453\pi$
$683$	6.47713	0.247840	0.123920	+	0.992292i	$0.460453\pi$
$684$	0	0
$685$	−17.1957	−0.657014
$686$	17.0158	0.649668
$687$	0	0
$688$	8.66820	0.330472
$689$	6.16554	0.234888
$690$	0	0
$691$	−37.1230	−1.41223	−0.706113	−	0.708099i	$-0.749553\pi$
$691$	−37.1230	−1.41223	−0.706113	+	0.708099i	$0.749553\pi$
$692$	−1.33631	−0.0507988
$693$	0	0
$694$	−14.6682	−0.556796
$695$	43.2545	1.64074
$696$	0	0
$697$	−6.64815	−0.251817
$698$	−15.9194	−0.602557
$699$	0	0
$700$	2.89174	0.109297
$701$	4.49591	0.169808	0.0849041	−	0.996389i	$-0.472942\pi$
$701$	4.49591	0.169808	0.0849041	+	0.996389i	$0.472942\pi$
$702$	0	0
$703$	2.00997	0.0758076
$704$	3.02184	0.113890
$705$	0	0
$706$	29.0530	1.09342
$707$	−30.5588	−1.14928
$708$	0	0
$709$	28.5524	1.07231	0.536153	−	0.844121i	$-0.319877\pi$
$709$	28.5524	1.07231	0.536153	+	0.844121i	$0.319877\pi$
$710$	−22.4307	−0.841811
$711$	0	0
$712$	0.434476	0.0162827
$713$	−4.95757	−0.185662
$714$	0	0
$715$	34.3802	1.28575
$716$	−10.3669	−0.387429
$717$	0	0
$718$	−16.0434	−0.598733
$719$	−17.0459	−0.635706	−0.317853	−	0.948140i	$-0.602962\pi$
$719$	−17.0459	−0.635706	−0.317853	+	0.948140i	$0.602962\pi$
$720$	0	0
$721$	−64.3809	−2.39767
$722$	−18.9651	−0.705808
$723$	0	0
$724$	7.10722	0.264138
$725$	6.64978	0.246967
$726$	0	0
$727$	−38.8392	−1.44047	−0.720234	−	0.693731i	$-0.755965\pi$
$727$	−38.8392	−1.44047	−0.720234	+	0.693731i	$0.755965\pi$
$728$	−20.2563	−0.750748
$729$	0	0
$730$	−3.34439	−0.123781
$731$	−31.1693	−1.15284
$732$	0	0
$733$	−31.7494	−1.17269	−0.586345	−	0.810062i	$-0.699434\pi$
$733$	−31.7494	−1.17269	−0.586345	+	0.810062i	$0.699434\pi$
$734$	8.55330	0.315708
$735$	0	0
$736$	1.80017	0.0663552
$737$	35.8788	1.32161
$738$	0	0
$739$	6.75837	0.248611	0.124305	−	0.992244i	$-0.460330\pi$
$739$	6.75837	0.248611	0.124305	+	0.992244i	$0.460330\pi$
$740$	25.6456	0.942750
$741$	0	0
$742$	−5.48170	−0.201240
$743$	−33.2388	−1.21941	−0.609707	−	0.792627i	$-0.708713\pi$
$743$	−33.2388	−1.21941	−0.609707	+	0.792627i	$0.708713\pi$
$744$	0	0
$745$	2.38357	0.0873273
$746$	−19.1127	−0.699767
$747$	0	0
$748$	−10.8660	−0.397301
$749$	12.3161	0.450019
$750$	0	0
$751$	−19.3714	−0.706872	−0.353436	−	0.935459i	$-0.614987\pi$
$751$	−19.3714	−0.706872	−0.353436	+	0.935459i	$0.614987\pi$
$752$	9.97430	0.363725
$753$	0	0
$754$	−46.5809	−1.69638
$755$	0.644225	0.0234457
$756$	0	0
$757$	−12.4674	−0.453134	−0.226567	−	0.973996i	$-0.572750\pi$
$757$	−12.4674	−0.453134	−0.226567	+	0.973996i	$0.572750\pi$
$758$	32.7673	1.19016
$759$	0	0
$760$	0.445281	0.0161520
$761$	−14.2412	−0.516244	−0.258122	−	0.966112i	$-0.583104\pi$
$761$	−14.2412	−0.516244	−0.258122	+	0.966112i	$0.583104\pi$
$762$	0	0
$763$	68.0305	2.46287
$764$	15.8013	0.571669
$765$	0	0
$766$	−25.0882	−0.906472
$767$	−27.9271	−1.00839
$768$	0	0
$769$	−26.5644	−0.957937	−0.478969	−	0.877832i	$-0.658989\pi$
$769$	−26.5644	−0.957937	−0.478969	+	0.877832i	$0.658989\pi$
$770$	−30.5670	−1.10156
$771$	0	0
$772$	3.48510	0.125431
$773$	−30.4242	−1.09428	−0.547141	−	0.837040i	$-0.684284\pi$
$773$	−30.4242	−1.09428	−0.547141	+	0.837040i	$0.684284\pi$
$774$	0	0
$775$	−1.87656	−0.0674079
$776$	−7.48855	−0.268823
$777$	0	0
$778$	15.7312	0.563993
$779$	−0.345390	−0.0123749
$780$	0	0
$781$	28.4373	1.01757
$782$	−6.47310	−0.231477
$783$	0	0
$784$	11.0096	0.393200
$785$	−1.24662	−0.0444939
$786$	0	0
$787$	29.1672	1.03970	0.519850	−	0.854258i	$-0.325988\pi$
$787$	29.1672	1.03970	0.519850	+	0.854258i	$0.325988\pi$
$788$	19.3760	0.690241
$789$	0	0
$790$	21.5879	0.768063
$791$	13.4458	0.478079
$792$	0	0
$793$	−36.9247	−1.31124
$794$	26.3693	0.935810
$795$	0	0
$796$	12.1663	0.431224
$797$	32.0181	1.13414	0.567070	−	0.823669i	$-0.308077\pi$
$797$	32.0181	1.13414	0.567070	+	0.823669i	$0.308077\pi$
$798$	0	0
$799$	−35.8658	−1.26884
$800$	0.681408	0.0240914
$801$	0	0
$802$	14.1978	0.501342
$803$	4.23995	0.149625
$804$	0	0
$805$	−18.2093	−0.641795
$806$	13.1450	0.463015
$807$	0	0
$808$	−7.20086	−0.253325
$809$	22.0357	0.774733	0.387367	−	0.921926i	$-0.373385\pi$
$809$	22.0357	0.774733	0.387367	+	0.921926i	$0.373385\pi$
$810$	0	0
$811$	−15.0426	−0.528216	−0.264108	−	0.964493i	$-0.585078\pi$
$811$	−15.0426	−0.528216	−0.264108	+	0.964493i	$0.585078\pi$
$812$	41.4145	1.45336
$813$	0	0
$814$	−32.5130	−1.13958
$815$	41.9128	1.46814
$816$	0	0
$817$	−1.61933	−0.0566531
$818$	−34.4325	−1.20391
$819$	0	0
$820$	−4.40688	−0.153895
$821$	−48.5742	−1.69525	−0.847625	−	0.530596i	$-0.821968\pi$
$821$	−48.5742	−1.69525	−0.847625	+	0.530596i	$0.821968\pi$
$822$	0	0
$823$	17.9630	0.626151	0.313076	−	0.949728i	$-0.398641\pi$
$823$	17.9630	0.626151	0.313076	+	0.949728i	$0.398641\pi$
$824$	−15.1707	−0.528496
$825$	0	0
$826$	24.8297	0.863934
$827$	3.81442	0.132640	0.0663201	−	0.997798i	$-0.478874\pi$
$827$	3.81442	0.132640	0.0663201	+	0.997798i	$0.478874\pi$
$828$	0	0
$829$	35.9079	1.24713	0.623567	−	0.781770i	$-0.285683\pi$
$829$	35.9079	1.24713	0.623567	+	0.781770i	$0.285683\pi$
$830$	−14.2365	−0.494157
$831$	0	0
$832$	−4.77318	−0.165480
$833$	−39.5885	−1.37166
$834$	0	0
$835$	−25.5648	−0.884707
$836$	−0.564519	−0.0195243
$837$	0	0
$838$	−28.5257	−0.985403
$839$	5.79268	0.199985	0.0999927	−	0.994988i	$-0.468118\pi$
$839$	5.79268	0.199985	0.0999927	+	0.994988i	$0.468118\pi$
$840$	0	0
$841$	66.2358	2.28399
$842$	16.4084	0.565470
$843$	0	0
$844$	2.39138	0.0823146
$845$	−23.3191	−0.802200
$846$	0	0
$847$	−7.92929	−0.272454
$848$	−1.29170	−0.0443573
$849$	0	0
$850$	−2.45022	−0.0840418
$851$	−19.3686	−0.663948
$852$	0	0
$853$	13.9042	0.476070	0.238035	−	0.971257i	$-0.423497\pi$
$853$	13.9042	0.476070	0.238035	+	0.971257i	$0.423497\pi$
$854$	32.8293	1.12340
$855$	0	0
$856$	2.90215	0.0991935
$857$	8.26823	0.282437	0.141219	−	0.989978i	$-0.454898\pi$
$857$	8.26823	0.282437	0.141219	+	0.989978i	$0.454898\pi$
$858$	0	0
$859$	−12.4657	−0.425324	−0.212662	−	0.977126i	$-0.568213\pi$
$859$	−12.4657	−0.425324	−0.212662	+	0.977126i	$0.568213\pi$
$860$	−20.6613	−0.704543
$861$	0	0
$862$	11.7732	0.400998
$863$	−37.3748	−1.27225	−0.636127	−	0.771584i	$-0.719465\pi$
$863$	−37.3748	−1.27225	−0.636127	+	0.771584i	$0.719465\pi$
$864$	0	0
$865$	3.18518	0.108299
$866$	3.98250	0.135331
$867$	0	0
$868$	−11.6871	−0.396686
$869$	−27.3687	−0.928420
$870$	0	0
$871$	−56.6727	−1.92028
$872$	16.0307	0.542867
$873$	0	0
$874$	−0.336295	−0.0113753
$875$	43.6840	1.47679
$876$	0	0
$877$	−8.11044	−0.273870	−0.136935	−	0.990580i	$-0.543725\pi$
$877$	−8.11044	−0.273870	−0.136935	+	0.990580i	$0.543725\pi$
$878$	13.3979	0.452157
$879$	0	0
$880$	−7.20278	−0.242806
$881$	−45.5146	−1.53343	−0.766714	−	0.641989i	$-0.778109\pi$
$881$	−45.5146	−1.53343	−0.766714	+	0.641989i	$0.778109\pi$
$882$	0	0
$883$	39.9016	1.34280	0.671398	−	0.741097i	$-0.265694\pi$
$883$	39.9016	1.34280	0.671398	+	0.741097i	$0.265694\pi$
$884$	17.1635	0.577271
$885$	0	0
$886$	28.2871	0.950325
$887$	55.3591	1.85878	0.929388	−	0.369105i	$-0.120336\pi$
$887$	55.3591	1.85878	0.929388	+	0.369105i	$0.120336\pi$
$888$	0	0
$889$	−3.82642	−0.128334
$890$	−1.03560	−0.0347135
$891$	0	0
$892$	13.6534	0.457151
$893$	−1.86332	−0.0623538
$894$	0	0
$895$	24.7102	0.825971
$896$	4.24377	0.141774
$897$	0	0
$898$	15.9872	0.533501
$899$	−26.8754	−0.896344
$900$	0	0
$901$	4.64474	0.154739
$902$	5.58696	0.186025
$903$	0	0
$904$	3.16837	0.105378
$905$	−16.9406	−0.563123
$906$	0	0
$907$	−12.1697	−0.404090	−0.202045	−	0.979376i	$-0.564759\pi$
$907$	−12.1697	−0.404090	−0.202045	+	0.979376i	$0.564759\pi$
$908$	−3.02417	−0.100361
$909$	0	0
$910$	48.2823	1.60054
$911$	46.4607	1.53931	0.769656	−	0.638459i	$-0.220428\pi$
$911$	46.4607	1.53931	0.769656	+	0.638459i	$0.220428\pi$
$912$	0	0
$913$	18.0488	0.597328
$914$	−3.00001	−0.0992316
$915$	0	0
$916$	5.88592	0.194476
$917$	30.5953	1.01035
$918$	0	0
$919$	40.6558	1.34111	0.670556	−	0.741859i	$-0.266056\pi$
$919$	40.6558	1.34111	0.670556	+	0.741859i	$0.266056\pi$
$920$	−4.29084	−0.141465
$921$	0	0
$922$	14.2661	0.469828
$923$	−44.9183	−1.47850
$924$	0	0
$925$	−7.33148	−0.241058
$926$	−13.8671	−0.455703
$927$	0	0
$928$	9.75888	0.320351
$929$	36.2919	1.19070	0.595350	−	0.803467i	$-0.297014\pi$
$929$	36.2919	1.19070	0.595350	+	0.803467i	$0.297014\pi$
$930$	0	0
$931$	−2.05673	−0.0674067
$932$	4.86587	0.159387
$933$	0	0
$934$	−31.8022	−1.04060
$935$	25.8999	0.847017
$936$	0	0
$937$	−0.224517	−0.00733466	−0.00366733	−	0.999993i	$-0.501167\pi$
$937$	−0.224517	−0.00733466	−0.00366733	+	0.999993i	$0.501167\pi$
$938$	50.3869	1.64519
$939$	0	0
$940$	−23.7744	−0.775437
$941$	−61.0315	−1.98957	−0.994784	−	0.102002i	$-0.967475\pi$
$941$	−61.0315	−1.98957	−0.994784	+	0.102002i	$0.967475\pi$
$942$	0	0
$943$	3.32826	0.108383
$944$	5.85085	0.190429
$945$	0	0
$946$	26.1939	0.851639
$947$	−7.24306	−0.235368	−0.117684	−	0.993051i	$-0.537547\pi$
$947$	−7.24306	−0.235368	−0.117684	+	0.993051i	$0.537547\pi$
$948$	0	0
$949$	−6.69726	−0.217402
$950$	−0.127296	−0.00413001
$951$	0	0
$952$	−15.2598	−0.494574
$953$	27.9004	0.903783	0.451891	−	0.892073i	$-0.350750\pi$
$953$	27.9004	0.903783	0.451891	+	0.892073i	$0.350750\pi$
$954$	0	0
$955$	−37.6634	−1.21876
$956$	15.8490	0.512593
$957$	0	0
$958$	27.0793	0.874894
$959$	30.6157	0.988633
$960$	0	0
$961$	−23.4158	−0.755349
$962$	51.3561	1.65579
$963$	0	0
$964$	29.1733	0.939608
$965$	−8.30697	−0.267411
$966$	0	0
$967$	23.8209	0.766028	0.383014	−	0.923743i	$-0.374886\pi$
$967$	23.8209	0.766028	0.383014	+	0.923743i	$0.374886\pi$
$968$	−1.86845	−0.0600544
$969$	0	0
$970$	17.8495	0.573112
$971$	15.1655	0.486684	0.243342	−	0.969941i	$-0.421756\pi$
$971$	15.1655	0.486684	0.243342	+	0.969941i	$0.421756\pi$
$972$	0	0
$973$	−77.0115	−2.46888
$974$	−36.9878	−1.18516
$975$	0	0
$976$	7.73588	0.247619
$977$	−46.0645	−1.47373	−0.736867	−	0.676038i	$-0.763696\pi$
$977$	−46.0645	−1.47373	−0.736867	+	0.676038i	$0.763696\pi$
$978$	0	0
$979$	1.31292	0.0419611
$980$	−26.2422	−0.838275
$981$	0	0
$982$	−6.03716	−0.192654
$983$	−36.9451	−1.17836	−0.589182	−	0.808000i	$-0.700550\pi$
$983$	−36.9451	−1.17836	−0.589182	+	0.808000i	$0.700550\pi$
$984$	0	0
$985$	−46.1840	−1.47155
$986$	−35.0912	−1.11753
$987$	0	0
$988$	0.891690	0.0283684
$989$	15.6042	0.496186
$990$	0	0
$991$	22.1002	0.702037	0.351019	−	0.936369i	$-0.385835\pi$
$991$	22.1002	0.702037	0.351019	+	0.936369i	$0.385835\pi$
$992$	−2.75394	−0.0874377
$993$	0	0
$994$	39.9363	1.26670
$995$	−28.9993	−0.919340
$996$	0	0
$997$	−39.9001	−1.26365	−0.631824	−	0.775112i	$-0.717694\pi$
$997$	−39.9001	−1.26365	−0.631824	+	0.775112i	$0.717694\pi$
$998$	−22.7307	−0.719527
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.p.1.3	yes	12
3.2	odd	2		8046.2.a.i.1.10	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.i.1.10	✓	12	3.2	odd	2
8046.2.a.p.1.3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.38357 q^{5} +4.24377 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.38357 q^{5} +4.24377 q^{7} +1.00000 q^{8} -2.38357 q^{10} +3.02184 q^{11} -4.77318 q^{13} +4.24377 q^{14} +1.00000 q^{16} -3.59582 q^{17} -0.186813 q^{19} -2.38357 q^{20} +3.02184 q^{22} +1.80017 q^{23} +0.681408 q^{25} -4.77318 q^{26} +4.24377 q^{28} +9.75888 q^{29} -2.75394 q^{31} +1.00000 q^{32} -3.59582 q^{34} -10.1153 q^{35} -10.7593 q^{37} -0.186813 q^{38} -2.38357 q^{40} +1.84886 q^{41} +8.66820 q^{43} +3.02184 q^{44} +1.80017 q^{46} +9.97430 q^{47} +11.0096 q^{49} +0.681408 q^{50} -4.77318 q^{52} -1.29170 q^{53} -7.20278 q^{55} +4.24377 q^{56} +9.75888 q^{58} +5.85085 q^{59} +7.73588 q^{61} -2.75394 q^{62} +1.00000 q^{64} +11.3772 q^{65} +11.8732 q^{67} -3.59582 q^{68} -10.1153 q^{70} +9.41056 q^{71} +1.40310 q^{73} -10.7593 q^{74} -0.186813 q^{76} +12.8240 q^{77} -9.05695 q^{79} -2.38357 q^{80} +1.84886 q^{82} +5.97278 q^{83} +8.57089 q^{85} +8.66820 q^{86} +3.02184 q^{88} +0.434476 q^{89} -20.2563 q^{91} +1.80017 q^{92} +9.97430 q^{94} +0.445281 q^{95} -7.48855 q^{97} +11.0096 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8	\( 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8} + 5 q^{10} + 6 q^{11} + 3 q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} + 8 q^{19} + 5 q^{20} + 6 q^{22} + 11 q^{23} + 11 q^{25} + 3 q^{26} + 6 q^{28} + 29 q^{29} + 2 q^{31} + 12 q^{32} + 6 q^{34} + 4 q^{35} + 5 q^{37} + 8 q^{38} + 5 q^{40} + 22 q^{41} + 9 q^{43} + 6 q^{44} + 11 q^{46} + 15 q^{47} + 14 q^{49} + 11 q^{50} + 3 q^{52} + 12 q^{53} + 13 q^{55} + 6 q^{56} + 29 q^{58} + 34 q^{59} - 4 q^{61} + 2 q^{62} + 12 q^{64} + 12 q^{65} + q^{67} + 6 q^{68} + 4 q^{70} + 21 q^{71} - 2 q^{73} + 5 q^{74} + 8 q^{76} + 34 q^{77} + 9 q^{79} + 5 q^{80} + 22 q^{82} + 10 q^{83} + 5 q^{85} + 9 q^{86} + 6 q^{88} - 2 q^{89} + 17 q^{91} + 11 q^{92} + 15 q^{94} + 69 q^{95} - 13 q^{97} + 14 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8 + 5 * q^10 + 6 * q^11 + 3 * q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 + 8 * q^19 + 5 * q^20 + 6 * q^22 + 11 * q^23 + 11 * q^25 + 3 * q^26 + 6 * q^28 + 29 * q^29 + 2 * q^31 + 12 * q^32 + 6 * q^34 + 4 * q^35 + 5 * q^37 + 8 * q^38 + 5 * q^40 + 22 * q^41 + 9 * q^43 + 6 * q^44 + 11 * q^46 + 15 * q^47 + 14 * q^49 + 11 * q^50 + 3 * q^52 + 12 * q^53 + 13 * q^55 + 6 * q^56 + 29 * q^58 + 34 * q^59 - 4 * q^61 + 2 * q^62 + 12 * q^64 + 12 * q^65 + q^67 + 6 * q^68 + 4 * q^70 + 21 * q^71 - 2 * q^73 + 5 * q^74 + 8 * q^76 + 34 * q^77 + 9 * q^79 + 5 * q^80 + 22 * q^82 + 10 * q^83 + 5 * q^85 + 9 * q^86 + 6 * q^88 - 2 * q^89 + 17 * q^91 + 11 * q^92 + 15 * q^94 + 69 * q^95 - 13 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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