LMFDB - Embedded newform 8046.2.a.g.1.7

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:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 9x^{7} + 25x^{6} + 29x^{5} - 58x^{4} - 43x^{3} + 34x^{2} + 25x + 2 $ x^9 - 3x^8 - 9x^7 + 25x^6 + 29x^5 - 58x^4 - 43x^3 + 34x^2 + 25x + 2
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			1.7
Root			$-2.13687$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.65274 q^{5} +0.0358535 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +1.65274 q^{5} +0.0358535 q^{7} -1.00000 q^{8} -1.65274 q^{10} -1.10979 q^{11} +1.20305 q^{13} -0.0358535 q^{14} +1.00000 q^{16} +3.50418 q^{17} +4.17906 q^{19} +1.65274 q^{20} +1.10979 q^{22} +5.77941 q^{23} -2.26846 q^{25} -1.20305 q^{26} +0.0358535 q^{28} -8.50659 q^{29} -7.39492 q^{31} -1.00000 q^{32} -3.50418 q^{34} +0.0592564 q^{35} -8.83693 q^{37} -4.17906 q^{38} -1.65274 q^{40} +5.77287 q^{41} -7.59277 q^{43} -1.10979 q^{44} -5.77941 q^{46} -11.7891 q^{47} -6.99871 q^{49} +2.26846 q^{50} +1.20305 q^{52} -8.49664 q^{53} -1.83419 q^{55} -0.0358535 q^{56} +8.50659 q^{58} -9.24179 q^{59} -6.48240 q^{61} +7.39492 q^{62} +1.00000 q^{64} +1.98832 q^{65} -3.00093 q^{67} +3.50418 q^{68} -0.0592564 q^{70} +10.9241 q^{71} +3.64076 q^{73} +8.83693 q^{74} +4.17906 q^{76} -0.0397898 q^{77} +4.74320 q^{79} +1.65274 q^{80} -5.77287 q^{82} -2.78704 q^{83} +5.79149 q^{85} +7.59277 q^{86} +1.10979 q^{88} +6.07566 q^{89} +0.0431334 q^{91} +5.77941 q^{92} +11.7891 q^{94} +6.90688 q^{95} +0.383046 q^{97} +6.99871 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{2} + 9 q^{4} + 4 q^{5} - 4 q^{7} - 9 q^{8}+O(q^{10}) $ 9 * q - 9 * q^2 + 9 * q^4 + 4 * q^5 - 4 * q^7 - 9 * q^8	$ 9 q - 9 q^{2} + 9 q^{4} + 4 q^{5} - 4 q^{7} - 9 q^{8} - 4 q^{10} + 4 q^{11} - 8 q^{13} + 4 q^{14} + 9 q^{16} + q^{17} - 10 q^{19} + 4 q^{20} - 4 q^{22} + 8 q^{23} - 3 q^{25} + 8 q^{26} - 4 q^{28} + 4 q^{29} - 17 q^{31} - 9 q^{32} - q^{34} + 10 q^{35} - 11 q^{37} + 10 q^{38} - 4 q^{40} - 16 q^{43} + 4 q^{44} - 8 q^{46} + 7 q^{47} - 5 q^{49} + 3 q^{50} - 8 q^{52} + 12 q^{53} - 23 q^{55} + 4 q^{56} - 4 q^{58} + 6 q^{59} - 13 q^{61} + 17 q^{62} + 9 q^{64} - 24 q^{65} - 14 q^{67} + q^{68} - 10 q^{70} + 30 q^{71} - 12 q^{73} + 11 q^{74} - 10 q^{76} + 12 q^{77} - 35 q^{79} + 4 q^{80} + 5 q^{83} - 27 q^{85} + 16 q^{86} - 4 q^{88} + 23 q^{89} - 28 q^{91} + 8 q^{92} - 7 q^{94} + 32 q^{95} - 21 q^{97} + 5 q^{98}+O(q^{100}) $ 9 * q - 9 * q^2 + 9 * q^4 + 4 * q^5 - 4 * q^7 - 9 * q^8 - 4 * q^10 + 4 * q^11 - 8 * q^13 + 4 * q^14 + 9 * q^16 + q^17 - 10 * q^19 + 4 * q^20 - 4 * q^22 + 8 * q^23 - 3 * q^25 + 8 * q^26 - 4 * q^28 + 4 * q^29 - 17 * q^31 - 9 * q^32 - q^34 + 10 * q^35 - 11 * q^37 + 10 * q^38 - 4 * q^40 - 16 * q^43 + 4 * q^44 - 8 * q^46 + 7 * q^47 - 5 * q^49 + 3 * q^50 - 8 * q^52 + 12 * q^53 - 23 * q^55 + 4 * q^56 - 4 * q^58 + 6 * q^59 - 13 * q^61 + 17 * q^62 + 9 * q^64 - 24 * q^65 - 14 * q^67 + q^68 - 10 * q^70 + 30 * q^71 - 12 * q^73 + 11 * q^74 - 10 * q^76 + 12 * q^77 - 35 * q^79 + 4 * q^80 + 5 * q^83 - 27 * q^85 + 16 * q^86 - 4 * q^88 + 23 * q^89 - 28 * q^91 + 8 * q^92 - 7 * q^94 + 32 * q^95 - 21 * q^97 + 5 * 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
$5$	1.65274	0.739126	0.369563	−	0.929206i	$-0.379507\pi$
$5$	1.65274	0.739126	0.369563	+	0.929206i	$0.379507\pi$
$6$	0	0
$7$	0.0358535	0.0135514	0.00677568	−	0.999977i	$-0.497843\pi$
$7$	0.0358535	0.0135514	0.00677568	+	0.999977i	$0.497843\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.65274	−0.522641
$11$	−1.10979	−0.334614	−0.167307	−	0.985905i	$-0.553507\pi$
$11$	−1.10979	−0.334614	−0.167307	+	0.985905i	$0.553507\pi$
$12$	0	0
$13$	1.20305	0.333665	0.166832	−	0.985985i	$-0.446646\pi$
$13$	1.20305	0.333665	0.166832	+	0.985985i	$0.446646\pi$
$14$	−0.0358535	−0.00958226
$15$	0	0
$16$	1.00000	0.250000
$17$	3.50418	0.849889	0.424944	−	0.905219i	$-0.360294\pi$
$17$	3.50418	0.849889	0.424944	+	0.905219i	$0.360294\pi$
$18$	0	0
$19$	4.17906	0.958741	0.479371	−	0.877613i	$-0.340865\pi$
$19$	4.17906	0.958741	0.479371	+	0.877613i	$0.340865\pi$
$20$	1.65274	0.369563
$21$	0	0
$22$	1.10979	0.236608
$23$	5.77941	1.20509	0.602545	−	0.798085i	$-0.294153\pi$
$23$	5.77941	1.20509	0.602545	+	0.798085i	$0.294153\pi$
$24$	0	0
$25$	−2.26846	−0.453692
$26$	−1.20305	−0.235937
$27$	0	0
$28$	0.0358535	0.00677568
$29$	−8.50659	−1.57963	−0.789817	−	0.613342i	$-0.789825\pi$
$29$	−8.50659	−1.57963	−0.789817	+	0.613342i	$0.789825\pi$
$30$	0	0
$31$	−7.39492	−1.32817	−0.664083	−	0.747659i	$-0.731178\pi$
$31$	−7.39492	−1.32817	−0.664083	+	0.747659i	$0.731178\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.50418	−0.600962
$35$	0.0592564	0.0100162
$36$	0	0
$37$	−8.83693	−1.45278	−0.726391	−	0.687282i	$-0.758804\pi$
$37$	−8.83693	−1.45278	−0.726391	+	0.687282i	$0.758804\pi$
$38$	−4.17906	−0.677933
$39$	0	0
$40$	−1.65274	−0.261321
$41$	5.77287	0.901571	0.450786	−	0.892632i	$-0.351144\pi$
$41$	5.77287	0.901571	0.450786	+	0.892632i	$0.351144\pi$
$42$	0	0
$43$	−7.59277	−1.15789	−0.578943	−	0.815368i	$-0.696535\pi$
$43$	−7.59277	−1.15789	−0.578943	+	0.815368i	$0.696535\pi$
$44$	−1.10979	−0.167307
$45$	0	0
$46$	−5.77941	−0.852127
$47$	−11.7891	−1.71962	−0.859810	−	0.510615i	$-0.829418\pi$
$47$	−11.7891	−1.71962	−0.859810	+	0.510615i	$0.829418\pi$
$48$	0	0
$49$	−6.99871	−0.999816
$50$	2.26846	0.320809
$51$	0	0
$52$	1.20305	0.166832
$53$	−8.49664	−1.16710	−0.583552	−	0.812076i	$-0.698338\pi$
$53$	−8.49664	−1.16710	−0.583552	+	0.812076i	$0.698338\pi$
$54$	0	0
$55$	−1.83419	−0.247322
$56$	−0.0358535	−0.00479113
$57$	0	0
$58$	8.50659	1.11697
$59$	−9.24179	−1.20318	−0.601590	−	0.798805i	$-0.705466\pi$
$59$	−9.24179	−1.20318	−0.601590	+	0.798805i	$0.705466\pi$
$60$	0	0
$61$	−6.48240	−0.829986	−0.414993	−	0.909825i	$-0.636216\pi$
$61$	−6.48240	−0.829986	−0.414993	+	0.909825i	$0.636216\pi$
$62$	7.39492	0.939156
$63$	0	0
$64$	1.00000	0.125000
$65$	1.98832	0.246620
$66$	0	0
$67$	−3.00093	−0.366622	−0.183311	−	0.983055i	$-0.558682\pi$
$67$	−3.00093	−0.366622	−0.183311	+	0.983055i	$0.558682\pi$
$68$	3.50418	0.424944
$69$	0	0
$70$	−0.0592564	−0.00708250
$71$	10.9241	1.29645	0.648226	−	0.761448i	$-0.275511\pi$
$71$	10.9241	1.29645	0.648226	+	0.761448i	$0.275511\pi$
$72$	0	0
$73$	3.64076	0.426118	0.213059	−	0.977039i	$-0.431657\pi$
$73$	3.64076	0.426118	0.213059	+	0.977039i	$0.431657\pi$
$74$	8.83693	1.02727
$75$	0	0
$76$	4.17906	0.479371
$77$	−0.0397898	−0.00453447
$78$	0	0
$79$	4.74320	0.533651	0.266826	−	0.963745i	$-0.414025\pi$
$79$	4.74320	0.533651	0.266826	+	0.963745i	$0.414025\pi$
$80$	1.65274	0.184782
$81$	0	0
$82$	−5.77287	−0.637507
$83$	−2.78704	−0.305918	−0.152959	−	0.988233i	$-0.548880\pi$
$83$	−2.78704	−0.305918	−0.152959	+	0.988233i	$0.548880\pi$
$84$	0	0
$85$	5.79149	0.628175
$86$	7.59277	0.818749
$87$	0	0
$88$	1.10979	0.118304
$89$	6.07566	0.644019	0.322010	−	0.946736i	$-0.395642\pi$
$89$	6.07566	0.644019	0.322010	+	0.946736i	$0.395642\pi$
$90$	0	0
$91$	0.0431334	0.00452161
$92$	5.77941	0.602545
$93$	0	0
$94$	11.7891	1.21595
$95$	6.90688	0.708631
$96$	0	0
$97$	0.383046	0.0388924	0.0194462	−	0.999811i	$-0.493810\pi$
$97$	0.383046	0.0388924	0.0194462	+	0.999811i	$0.493810\pi$
$98$	6.99871	0.706977
$99$	0	0
$100$	−2.26846	−0.226846
$101$	6.26977	0.623865	0.311933	−	0.950104i	$-0.399024\pi$
$101$	6.26977	0.623865	0.311933	+	0.950104i	$0.399024\pi$
$102$	0	0
$103$	−11.5950	−1.14249	−0.571244	−	0.820780i	$-0.693539\pi$
$103$	−11.5950	−1.14249	−0.571244	+	0.820780i	$0.693539\pi$
$104$	−1.20305	−0.117968
$105$	0	0
$106$	8.49664	0.825267
$107$	9.58849	0.926954	0.463477	−	0.886109i	$-0.346602\pi$
$107$	9.58849	0.926954	0.463477	+	0.886109i	$0.346602\pi$
$108$	0	0
$109$	7.07628	0.677784	0.338892	−	0.940825i	$-0.389948\pi$
$109$	7.07628	0.677784	0.338892	+	0.940825i	$0.389948\pi$
$110$	1.83419	0.174883
$111$	0	0
$112$	0.0358535	0.00338784
$113$	−1.56605	−0.147322	−0.0736609	−	0.997283i	$-0.523468\pi$
$113$	−1.56605	−0.147322	−0.0736609	+	0.997283i	$0.523468\pi$
$114$	0	0
$115$	9.55184	0.890713
$116$	−8.50659	−0.789817
$117$	0	0
$118$	9.24179	0.850776
$119$	0.125637	0.0115172
$120$	0	0
$121$	−9.76837	−0.888034
$122$	6.48240	0.586889
$123$	0	0
$124$	−7.39492	−0.664083
$125$	−12.0129	−1.07446
$126$	0	0
$127$	−4.38796	−0.389368	−0.194684	−	0.980866i	$-0.562368\pi$
$127$	−4.38796	−0.389368	−0.194684	+	0.980866i	$0.562368\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−1.98832	−0.174387
$131$	−11.9230	−1.04172	−0.520859	−	0.853643i	$-0.674388\pi$
$131$	−11.9230	−1.04172	−0.520859	+	0.853643i	$0.674388\pi$
$132$	0	0
$133$	0.149834	0.0129923
$134$	3.00093	0.259241
$135$	0	0
$136$	−3.50418	−0.300481
$137$	−7.57559	−0.647226	−0.323613	−	0.946190i	$-0.604898\pi$
$137$	−7.57559	−0.647226	−0.323613	+	0.946190i	$0.604898\pi$
$138$	0	0
$139$	4.27059	0.362227	0.181113	−	0.983462i	$-0.442030\pi$
$139$	4.27059	0.362227	0.181113	+	0.983462i	$0.442030\pi$
$140$	0.0592564	0.00500808
$141$	0	0
$142$	−10.9241	−0.916730
$143$	−1.33513	−0.111649
$144$	0	0
$145$	−14.0592	−1.16755
$146$	−3.64076	−0.301311
$147$	0	0
$148$	−8.83693	−0.726391
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−12.1677	−0.990194	−0.495097	−	0.868838i	$-0.664868\pi$
$151$	−12.1677	−0.990194	−0.495097	+	0.868838i	$0.664868\pi$
$152$	−4.17906	−0.338966
$153$	0	0
$154$	0.0397898	0.00320636
$155$	−12.2219	−0.981683
$156$	0	0
$157$	5.15135	0.411122	0.205561	−	0.978644i	$-0.434098\pi$
$157$	5.15135	0.411122	0.205561	+	0.978644i	$0.434098\pi$
$158$	−4.74320	−0.377348
$159$	0	0
$160$	−1.65274	−0.130660
$161$	0.207212	0.0163306
$162$	0	0
$163$	20.4207	1.59947	0.799735	−	0.600353i	$-0.204973\pi$
$163$	20.4207	1.59947	0.799735	+	0.600353i	$0.204973\pi$
$164$	5.77287	0.450786
$165$	0	0
$166$	2.78704	0.216316
$167$	16.9356	1.31052	0.655259	−	0.755404i	$-0.272559\pi$
$167$	16.9356	1.31052	0.655259	+	0.755404i	$0.272559\pi$
$168$	0	0
$169$	−11.5527	−0.888668
$170$	−5.79149	−0.444187
$171$	0	0
$172$	−7.59277	−0.578943
$173$	22.2660	1.69285	0.846427	−	0.532505i	$-0.178749\pi$
$173$	22.2660	1.69285	0.846427	+	0.532505i	$0.178749\pi$
$174$	0	0
$175$	−0.0813324	−0.00614815
$176$	−1.10979	−0.0836534
$177$	0	0
$178$	−6.07566	−0.455390
$179$	−4.06027	−0.303478	−0.151739	−	0.988421i	$-0.548487\pi$
$179$	−4.06027	−0.303478	−0.151739	+	0.988421i	$0.548487\pi$
$180$	0	0
$181$	−13.2135	−0.982154	−0.491077	−	0.871116i	$-0.663397\pi$
$181$	−13.2135	−0.982154	−0.491077	+	0.871116i	$0.663397\pi$
$182$	−0.0431334	−0.00319726
$183$	0	0
$184$	−5.77941	−0.426064
$185$	−14.6051	−1.07379
$186$	0	0
$187$	−3.88890	−0.284384
$188$	−11.7891	−0.859810
$189$	0	0
$190$	−6.90688	−0.501078
$191$	10.4427	0.755605	0.377803	−	0.925886i	$-0.376680\pi$
$191$	10.4427	0.755605	0.377803	+	0.925886i	$0.376680\pi$
$192$	0	0
$193$	−10.2254	−0.736042	−0.368021	−	0.929817i	$-0.619965\pi$
$193$	−10.2254	−0.736042	−0.368021	+	0.929817i	$0.619965\pi$
$194$	−0.383046	−0.0275011
$195$	0	0
$196$	−6.99871	−0.499908
$197$	22.4981	1.60292	0.801460	−	0.598048i	$-0.204057\pi$
$197$	22.4981	1.60292	0.801460	+	0.598048i	$0.204057\pi$
$198$	0	0
$199$	−1.33281	−0.0944802	−0.0472401	−	0.998884i	$-0.515043\pi$
$199$	−1.33281	−0.0944802	−0.0472401	+	0.998884i	$0.515043\pi$
$200$	2.26846	0.160404
$201$	0	0
$202$	−6.26977	−0.441140
$203$	−0.304991	−0.0214062
$204$	0	0
$205$	9.54104	0.666375
$206$	11.5950	0.807861
$207$	0	0
$208$	1.20305	0.0834162
$209$	−4.63787	−0.320808
$210$	0	0
$211$	−14.2623	−0.981857	−0.490928	−	0.871200i	$-0.663342\pi$
$211$	−14.2623	−0.981857	−0.490928	+	0.871200i	$0.663342\pi$
$212$	−8.49664	−0.583552
$213$	0	0
$214$	−9.58849	−0.655455
$215$	−12.5488	−0.855824
$216$	0	0
$217$	−0.265134	−0.0179985
$218$	−7.07628	−0.479266
$219$	0	0
$220$	−1.83419	−0.123661
$221$	4.21569	0.283578
$222$	0	0
$223$	−28.6033	−1.91542	−0.957710	−	0.287734i	$-0.907098\pi$
$223$	−28.6033	−1.91542	−0.957710	+	0.287734i	$0.907098\pi$
$224$	−0.0358535	−0.00239557
$225$	0	0
$226$	1.56605	0.104172
$227$	−2.37525	−0.157651	−0.0788253	−	0.996888i	$-0.525117\pi$
$227$	−2.37525	−0.157651	−0.0788253	+	0.996888i	$0.525117\pi$
$228$	0	0
$229$	9.02486	0.596380	0.298190	−	0.954507i	$-0.403617\pi$
$229$	9.02486	0.596380	0.298190	+	0.954507i	$0.403617\pi$
$230$	−9.55184	−0.629829
$231$	0	0
$232$	8.50659	0.558485
$233$	−14.5151	−0.950918	−0.475459	−	0.879738i	$-0.657718\pi$
$233$	−14.5151	−0.950918	−0.475459	+	0.879738i	$0.657718\pi$
$234$	0	0
$235$	−19.4843	−1.27102
$236$	−9.24179	−0.601590
$237$	0	0
$238$	−0.125637	−0.00814386
$239$	3.69277	0.238865	0.119433	−	0.992842i	$-0.461892\pi$
$239$	3.69277	0.238865	0.119433	+	0.992842i	$0.461892\pi$
$240$	0	0
$241$	−30.9002	−1.99045	−0.995227	−	0.0975856i	$-0.968888\pi$
$241$	−30.9002	−1.99045	−0.995227	+	0.0975856i	$0.968888\pi$
$242$	9.76837	0.627935
$243$	0	0
$244$	−6.48240	−0.414993
$245$	−11.5670	−0.738991
$246$	0	0
$247$	5.02760	0.319898
$248$	7.39492	0.469578
$249$	0	0
$250$	12.0129	0.759760
$251$	27.6898	1.74777	0.873884	−	0.486135i	$-0.161594\pi$
$251$	27.6898	1.74777	0.873884	+	0.486135i	$0.161594\pi$
$252$	0	0
$253$	−6.41392	−0.403239
$254$	4.38796	0.275325
$255$	0	0
$256$	1.00000	0.0625000
$257$	20.6618	1.28884	0.644422	−	0.764670i	$-0.277098\pi$
$257$	20.6618	1.28884	0.644422	+	0.764670i	$0.277098\pi$
$258$	0	0
$259$	−0.316835	−0.0196872
$260$	1.98832	0.123310
$261$	0	0
$262$	11.9230	0.736606
$263$	−30.9793	−1.91027	−0.955134	−	0.296174i	$-0.904289\pi$
$263$	−30.9793	−1.91027	−0.955134	+	0.296174i	$0.904289\pi$
$264$	0	0
$265$	−14.0427	−0.862637
$266$	−0.149834	−0.00918691
$267$	0	0
$268$	−3.00093	−0.183311
$269$	10.4072	0.634537	0.317268	−	0.948336i	$-0.397234\pi$
$269$	10.4072	0.634537	0.317268	+	0.948336i	$0.397234\pi$
$270$	0	0
$271$	4.03824	0.245306	0.122653	−	0.992450i	$-0.460860\pi$
$271$	4.03824	0.245306	0.122653	+	0.992450i	$0.460860\pi$
$272$	3.50418	0.212472
$273$	0	0
$274$	7.57559	0.457658
$275$	2.51751	0.151812
$276$	0	0
$277$	22.8113	1.37060	0.685300	−	0.728261i	$-0.259671\pi$
$277$	22.8113	1.37060	0.685300	+	0.728261i	$0.259671\pi$
$278$	−4.27059	−0.256133
$279$	0	0
$280$	−0.0592564	−0.00354125
$281$	−13.2889	−0.792750	−0.396375	−	0.918089i	$-0.629732\pi$
$281$	−13.2889	−0.792750	−0.396375	+	0.918089i	$0.629732\pi$
$282$	0	0
$283$	2.74147	0.162963	0.0814817	−	0.996675i	$-0.474035\pi$
$283$	2.74147	0.162963	0.0814817	+	0.996675i	$0.474035\pi$
$284$	10.9241	0.648226
$285$	0	0
$286$	1.33513	0.0789476
$287$	0.206978	0.0122175
$288$	0	0
$289$	−4.72071	−0.277689
$290$	14.0592	0.825582
$291$	0	0
$292$	3.64076	0.213059
$293$	1.14676	0.0669942	0.0334971	−	0.999439i	$-0.489336\pi$
$293$	1.14676	0.0669942	0.0334971	+	0.999439i	$0.489336\pi$
$294$	0	0
$295$	−15.2742	−0.889301
$296$	8.83693	0.513636
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	6.95289	0.402096
$300$	0	0
$301$	−0.272227	−0.0156909
$302$	12.1677	0.700173
$303$	0	0
$304$	4.17906	0.239685
$305$	−10.7137	−0.613465
$306$	0	0
$307$	−9.96224	−0.568575	−0.284288	−	0.958739i	$-0.591757\pi$
$307$	−9.96224	−0.568575	−0.284288	+	0.958739i	$0.591757\pi$
$308$	−0.0397898	−0.00226724
$309$	0	0
$310$	12.2219	0.694155
$311$	3.58000	0.203003	0.101502	−	0.994835i	$-0.467635\pi$
$311$	3.58000	0.203003	0.101502	+	0.994835i	$0.467635\pi$
$312$	0	0
$313$	−1.86731	−0.105547	−0.0527733	−	0.998607i	$-0.516806\pi$
$313$	−1.86731	−0.105547	−0.0527733	+	0.998607i	$0.516806\pi$
$314$	−5.15135	−0.290707
$315$	0	0
$316$	4.74320	0.266826
$317$	−18.7083	−1.05076	−0.525382	−	0.850867i	$-0.676077\pi$
$317$	−18.7083	−1.05076	−0.525382	+	0.850867i	$0.676077\pi$
$318$	0	0
$319$	9.44051	0.528567
$320$	1.65274	0.0923908
$321$	0	0
$322$	−0.207212	−0.0115475
$323$	14.6442	0.814824
$324$	0	0
$325$	−2.72906	−0.151381
$326$	−20.4207	−1.13100
$327$	0	0
$328$	−5.77287	−0.318754
$329$	−0.422681	−0.0233032
$330$	0	0
$331$	26.2800	1.44448	0.722239	−	0.691644i	$-0.243113\pi$
$331$	26.2800	1.44448	0.722239	+	0.691644i	$0.243113\pi$
$332$	−2.78704	−0.152959
$333$	0	0
$334$	−16.9356	−0.926676
$335$	−4.95975	−0.270980
$336$	0	0
$337$	−24.4996	−1.33458	−0.667288	−	0.744799i	$-0.732545\pi$
$337$	−24.4996	−1.33458	−0.667288	+	0.744799i	$0.732545\pi$
$338$	11.5527	0.628383
$339$	0	0
$340$	5.79149	0.314088
$341$	8.20679	0.444423
$342$	0	0
$343$	−0.501903	−0.0271002
$344$	7.59277	0.409374
$345$	0	0
$346$	−22.2660	−1.19703
$347$	−8.34721	−0.448102	−0.224051	−	0.974577i	$-0.571928\pi$
$347$	−8.34721	−0.448102	−0.224051	+	0.974577i	$0.571928\pi$
$348$	0	0
$349$	10.1250	0.541977	0.270988	−	0.962583i	$-0.412650\pi$
$349$	10.1250	0.541977	0.270988	+	0.962583i	$0.412650\pi$
$350$	0.0813324	0.00434740
$351$	0	0
$352$	1.10979	0.0591519
$353$	12.9436	0.688916	0.344458	−	0.938802i	$-0.388063\pi$
$353$	12.9436	0.688916	0.344458	+	0.938802i	$0.388063\pi$
$354$	0	0
$355$	18.0547	0.958241
$356$	6.07566	0.322010
$357$	0	0
$358$	4.06027	0.214592
$359$	12.7585	0.673369	0.336685	−	0.941617i	$-0.390694\pi$
$359$	12.7585	0.673369	0.336685	+	0.941617i	$0.390694\pi$
$360$	0	0
$361$	−1.53548	−0.0808148
$362$	13.2135	0.694488
$363$	0	0
$364$	0.0431334	0.00226081
$365$	6.01721	0.314955
$366$	0	0
$367$	−5.74079	−0.299667	−0.149833	−	0.988711i	$-0.547874\pi$
$367$	−5.74079	−0.299667	−0.149833	+	0.988711i	$0.547874\pi$
$368$	5.77941	0.301272
$369$	0	0
$370$	14.6051	0.759284
$371$	−0.304635	−0.0158158
$372$	0	0
$373$	20.9995	1.08731	0.543655	−	0.839309i	$-0.317040\pi$
$373$	20.9995	1.08731	0.543655	+	0.839309i	$0.317040\pi$
$374$	3.88890	0.201090
$375$	0	0
$376$	11.7891	0.607977
$377$	−10.2338	−0.527068
$378$	0	0
$379$	−19.9938	−1.02701	−0.513506	−	0.858086i	$-0.671653\pi$
$379$	−19.9938	−1.02701	−0.513506	+	0.858086i	$0.671653\pi$
$380$	6.90688	0.354315
$381$	0	0
$382$	−10.4427	−0.534294
$383$	4.52949	0.231446	0.115723	−	0.993282i	$-0.463082\pi$
$383$	4.52949	0.231446	0.115723	+	0.993282i	$0.463082\pi$
$384$	0	0
$385$	−0.0657621	−0.00335155
$386$	10.2254	0.520460
$387$	0	0
$388$	0.383046	0.0194462
$389$	−20.0034	−1.01421	−0.507106	−	0.861884i	$-0.669285\pi$
$389$	−20.0034	−1.01421	−0.507106	+	0.861884i	$0.669285\pi$
$390$	0	0
$391$	20.2521	1.02419
$392$	6.99871	0.353488
$393$	0	0
$394$	−22.4981	−1.13344
$395$	7.83925	0.394436
$396$	0	0
$397$	−14.0741	−0.706357	−0.353178	−	0.935556i	$-0.614899\pi$
$397$	−14.0741	−0.706357	−0.353178	+	0.935556i	$0.614899\pi$
$398$	1.33281	0.0668076
$399$	0	0
$400$	−2.26846	−0.113423
$401$	−7.17872	−0.358488	−0.179244	−	0.983805i	$-0.557365\pi$
$401$	−7.17872	−0.358488	−0.179244	+	0.983805i	$0.557365\pi$
$402$	0	0
$403$	−8.89642	−0.443162
$404$	6.26977	0.311933
$405$	0	0
$406$	0.304991	0.0151365
$407$	9.80711	0.486121
$408$	0	0
$409$	−12.2911	−0.607755	−0.303878	−	0.952711i	$-0.598281\pi$
$409$	−12.2911	−0.607755	−0.303878	+	0.952711i	$0.598281\pi$
$410$	−9.54104	−0.471198
$411$	0	0
$412$	−11.5950	−0.571244
$413$	−0.331351	−0.0163047
$414$	0	0
$415$	−4.60625	−0.226112
$416$	−1.20305	−0.0589842
$417$	0	0
$418$	4.63787	0.226845
$419$	11.9663	0.584594	0.292297	−	0.956328i	$-0.405580\pi$
$419$	11.9663	0.584594	0.292297	+	0.956328i	$0.405580\pi$
$420$	0	0
$421$	0.00355896	0.000173453 0	8.67265e−5	−	1.00000i	$-0.499972\pi$
$421$	0.00355896	0.000173453 0	8.67265e−5		1.00000i	$0.499972\pi$
$422$	14.2623	0.694278
$423$	0	0
$424$	8.49664	0.412634
$425$	−7.94910	−0.385588
$426$	0	0
$427$	−0.232417	−0.0112474
$428$	9.58849	0.463477
$429$	0	0
$430$	12.5488	0.605159
$431$	31.6067	1.52244	0.761220	−	0.648494i	$-0.224601\pi$
$431$	31.6067	1.52244	0.761220	+	0.648494i	$0.224601\pi$
$432$	0	0
$433$	2.95878	0.142190	0.0710950	−	0.997470i	$-0.477351\pi$
$433$	2.95878	0.142190	0.0710950	+	0.997470i	$0.477351\pi$
$434$	0.265134	0.0127268
$435$	0	0
$436$	7.07628	0.338892
$437$	24.1525	1.15537
$438$	0	0
$439$	34.0207	1.62372	0.811860	−	0.583852i	$-0.198455\pi$
$439$	34.0207	1.62372	0.811860	+	0.583852i	$0.198455\pi$
$440$	1.83419	0.0874414
$441$	0	0
$442$	−4.21569	−0.200520
$443$	5.70534	0.271069	0.135534	−	0.990773i	$-0.456725\pi$
$443$	5.70534	0.271069	0.135534	+	0.990773i	$0.456725\pi$
$444$	0	0
$445$	10.0415	0.476011
$446$	28.6033	1.35441
$447$	0	0
$448$	0.0358535	0.00169392
$449$	−10.8071	−0.510018	−0.255009	−	0.966939i	$-0.582078\pi$
$449$	−10.8071	−0.510018	−0.255009	+	0.966939i	$0.582078\pi$
$450$	0	0
$451$	−6.40667	−0.301678
$452$	−1.56605	−0.0736609
$453$	0	0
$454$	2.37525	0.111476
$455$	0.0712882	0.00334204
$456$	0	0
$457$	−26.9583	−1.26106	−0.630528	−	0.776167i	$-0.717162\pi$
$457$	−26.9583	−1.26106	−0.630528	+	0.776167i	$0.717162\pi$
$458$	−9.02486	−0.421704
$459$	0	0
$460$	9.55184	0.445357
$461$	−30.0843	−1.40117	−0.700583	−	0.713571i	$-0.747077\pi$
$461$	−30.0843	−1.40117	−0.700583	+	0.713571i	$0.747077\pi$
$462$	0	0
$463$	40.5192	1.88309	0.941543	−	0.336893i	$-0.109376\pi$
$463$	40.5192	1.88309	0.941543	+	0.336893i	$0.109376\pi$
$464$	−8.50659	−0.394909
$465$	0	0
$466$	14.5151	0.672401
$467$	−15.9889	−0.739876	−0.369938	−	0.929056i	$-0.620621\pi$
$467$	−15.9889	−0.739876	−0.369938	+	0.929056i	$0.620621\pi$
$468$	0	0
$469$	−0.107594	−0.00496823
$470$	19.4843	0.898744
$471$	0	0
$472$	9.24179	0.425388
$473$	8.42636	0.387444
$474$	0	0
$475$	−9.48003	−0.434974
$476$	0.125637	0.00575858
$477$	0	0
$478$	−3.69277	−0.168903
$479$	26.4486	1.20847	0.604235	−	0.796807i	$-0.293479\pi$
$479$	26.4486	1.20847	0.604235	+	0.796807i	$0.293479\pi$
$480$	0	0
$481$	−10.6312	−0.484742
$482$	30.9002	1.40746
$483$	0	0
$484$	−9.76837	−0.444017
$485$	0.633073	0.0287464
$486$	0	0
$487$	−43.0978	−1.95295	−0.976473	−	0.215638i	$-0.930817\pi$
$487$	−43.0978	−1.95295	−0.976473	+	0.215638i	$0.930817\pi$
$488$	6.48240	0.293444
$489$	0	0
$490$	11.5670	0.522545
$491$	20.3362	0.917758	0.458879	−	0.888499i	$-0.348251\pi$
$491$	20.3362	0.917758	0.458879	+	0.888499i	$0.348251\pi$
$492$	0	0
$493$	−29.8086	−1.34251
$494$	−5.02760	−0.226202
$495$	0	0
$496$	−7.39492	−0.332042
$497$	0.391667	0.0175687
$498$	0	0
$499$	−28.4668	−1.27435	−0.637174	−	0.770720i	$-0.719897\pi$
$499$	−28.4668	−1.27435	−0.637174	+	0.770720i	$0.719897\pi$
$500$	−12.0129	−0.537231
$501$	0	0
$502$	−27.6898	−1.23586
$503$	−12.0413	−0.536895	−0.268448	−	0.963294i	$-0.586511\pi$
$503$	−12.0413	−0.536895	−0.268448	+	0.963294i	$0.586511\pi$
$504$	0	0
$505$	10.3623	0.461115
$506$	6.41392	0.285133
$507$	0	0
$508$	−4.38796	−0.194684
$509$	−16.8133	−0.745236	−0.372618	−	0.927985i	$-0.621540\pi$
$509$	−16.8133	−0.745236	−0.372618	+	0.927985i	$0.621540\pi$
$510$	0	0
$511$	0.130534	0.00577449
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−20.6618	−0.911351
$515$	−19.1635	−0.844443
$516$	0	0
$517$	13.0834	0.575408
$518$	0.316835	0.0139209
$519$	0	0
$520$	−1.98832	−0.0871935
$521$	−19.6095	−0.859109	−0.429554	−	0.903041i	$-0.641329\pi$
$521$	−19.6095	−0.859109	−0.429554	+	0.903041i	$0.641329\pi$
$522$	0	0
$523$	13.7743	0.602306	0.301153	−	0.953576i	$-0.402628\pi$
$523$	13.7743	0.602306	0.301153	+	0.953576i	$0.402628\pi$
$524$	−11.9230	−0.520859
$525$	0	0
$526$	30.9793	1.35076
$527$	−25.9131	−1.12879
$528$	0	0
$529$	10.4015	0.452241
$530$	14.0427	0.609977
$531$	0	0
$532$	0.149834	0.00649613
$533$	6.94503	0.300823
$534$	0	0
$535$	15.8472	0.685136
$536$	3.00093	0.129621
$537$	0	0
$538$	−10.4072	−0.448685
$539$	7.76709	0.334552
$540$	0	0
$541$	−34.1862	−1.46978	−0.734890	−	0.678186i	$-0.762766\pi$
$541$	−34.1862	−1.46978	−0.734890	+	0.678186i	$0.762766\pi$
$542$	−4.03824	−0.173457
$543$	0	0
$544$	−3.50418	−0.150241
$545$	11.6952	0.500968
$546$	0	0
$547$	38.2519	1.63553	0.817766	−	0.575551i	$-0.195212\pi$
$547$	38.2519	1.63553	0.817766	+	0.575551i	$0.195212\pi$
$548$	−7.57559	−0.323613
$549$	0	0
$550$	−2.51751	−0.107347
$551$	−35.5495	−1.51446
$552$	0	0
$553$	0.170060	0.00723170
$554$	−22.8113	−0.969161
$555$	0	0
$556$	4.27059	0.181113
$557$	−8.45230	−0.358136	−0.179068	−	0.983837i	$-0.557308\pi$
$557$	−8.45230	−0.358136	−0.179068	+	0.983837i	$0.557308\pi$
$558$	0	0
$559$	−9.13444	−0.386346
$560$	0.0592564	0.00250404
$561$	0	0
$562$	13.2889	0.560559
$563$	−41.2408	−1.73809	−0.869047	−	0.494729i	$-0.835267\pi$
$563$	−41.2408	−1.73809	−0.869047	+	0.494729i	$0.835267\pi$
$564$	0	0
$565$	−2.58827	−0.108889
$566$	−2.74147	−0.115233
$567$	0	0
$568$	−10.9241	−0.458365
$569$	31.1589	1.30625	0.653124	−	0.757251i	$-0.273458\pi$
$569$	31.1589	1.30625	0.653124	+	0.757251i	$0.273458\pi$
$570$	0	0
$571$	5.09780	0.213336	0.106668	−	0.994295i	$-0.465982\pi$
$571$	5.09780	0.213336	0.106668	+	0.994295i	$0.465982\pi$
$572$	−1.33513	−0.0558244
$573$	0	0
$574$	−0.206978	−0.00863909
$575$	−13.1104	−0.546740
$576$	0	0
$577$	−26.7328	−1.11290	−0.556451	−	0.830880i	$-0.687837\pi$
$577$	−26.7328	−1.11290	−0.556451	+	0.830880i	$0.687837\pi$
$578$	4.72071	0.196356
$579$	0	0
$580$	−14.0592	−0.583775
$581$	−0.0999253	−0.00414560
$582$	0	0
$583$	9.42947	0.390529
$584$	−3.64076	−0.150656
$585$	0	0
$586$	−1.14676	−0.0473721
$587$	−42.7943	−1.76631	−0.883155	−	0.469081i	$-0.844585\pi$
$587$	−42.7943	−1.76631	−0.883155	+	0.469081i	$0.844585\pi$
$588$	0	0
$589$	−30.9038	−1.27337
$590$	15.2742	0.628831
$591$	0	0
$592$	−8.83693	−0.363195
$593$	−47.7664	−1.96153	−0.980765	−	0.195194i	$-0.937466\pi$
$593$	−47.7664	−1.96153	−0.980765	+	0.195194i	$0.937466\pi$
$594$	0	0
$595$	0.207645	0.00851263
$596$	1.00000	0.0409616
$597$	0	0
$598$	−6.95289	−0.284325
$599$	4.56549	0.186541	0.0932703	−	0.995641i	$-0.470268\pi$
$599$	4.56549	0.186541	0.0932703	+	0.995641i	$0.470268\pi$
$600$	0	0
$601$	24.5480	1.00133	0.500667	−	0.865640i	$-0.333088\pi$
$601$	24.5480	1.00133	0.500667	+	0.865640i	$0.333088\pi$
$602$	0.272227	0.0110952
$603$	0	0
$604$	−12.1677	−0.495097
$605$	−16.1445	−0.656369
$606$	0	0
$607$	44.1390	1.79154	0.895772	−	0.444513i	$-0.146623\pi$
$607$	44.1390	1.79154	0.895772	+	0.444513i	$0.146623\pi$
$608$	−4.17906	−0.169483
$609$	0	0
$610$	10.7137	0.433785
$611$	−14.1828	−0.573776
$612$	0	0
$613$	−8.56263	−0.345841	−0.172921	−	0.984936i	$-0.555320\pi$
$613$	−8.56263	−0.345841	−0.172921	+	0.984936i	$0.555320\pi$
$614$	9.96224	0.402043
$615$	0	0
$616$	0.0397898	0.00160318
$617$	−34.3672	−1.38357	−0.691785	−	0.722103i	$-0.743175\pi$
$617$	−34.3672	−1.38357	−0.691785	+	0.722103i	$0.743175\pi$
$618$	0	0
$619$	35.5991	1.43085	0.715424	−	0.698690i	$-0.246234\pi$
$619$	35.5991	1.43085	0.715424	+	0.698690i	$0.246234\pi$
$620$	−12.2219	−0.490841
$621$	0	0
$622$	−3.58000	−0.143545
$623$	0.217834	0.00872734
$624$	0	0
$625$	−8.51177	−0.340471
$626$	1.86731	0.0746328
$627$	0	0
$628$	5.15135	0.205561
$629$	−30.9662	−1.23470
$630$	0	0
$631$	−7.47160	−0.297440	−0.148720	−	0.988879i	$-0.547515\pi$
$631$	−7.47160	−0.297440	−0.148720	+	0.988879i	$0.547515\pi$
$632$	−4.74320	−0.188674
$633$	0	0
$634$	18.7083	0.743002
$635$	−7.25214	−0.287792
$636$	0	0
$637$	−8.41977	−0.333604
$638$	−9.44051	−0.373753
$639$	0	0
$640$	−1.65274	−0.0653301
$641$	28.8283	1.13865	0.569324	−	0.822113i	$-0.307205\pi$
$641$	28.8283	1.13865	0.569324	+	0.822113i	$0.307205\pi$
$642$	0	0
$643$	26.5086	1.04540	0.522700	−	0.852517i	$-0.324925\pi$
$643$	26.5086	1.04540	0.522700	+	0.852517i	$0.324925\pi$
$644$	0.207212	0.00816530
$645$	0	0
$646$	−14.6442	−0.576167
$647$	−12.6198	−0.496134	−0.248067	−	0.968743i	$-0.579795\pi$
$647$	−12.6198	−0.496134	−0.248067	+	0.968743i	$0.579795\pi$
$648$	0	0
$649$	10.2564	0.402600
$650$	2.72906	0.107043
$651$	0	0
$652$	20.4207	0.799735
$653$	−5.31321	−0.207922	−0.103961	−	0.994581i	$-0.533152\pi$
$653$	−5.31321	−0.207922	−0.103961	+	0.994581i	$0.533152\pi$
$654$	0	0
$655$	−19.7056	−0.769961
$656$	5.77287	0.225393
$657$	0	0
$658$	0.422681	0.0164778
$659$	22.4705	0.875327	0.437663	−	0.899139i	$-0.355806\pi$
$659$	22.4705	0.875327	0.437663	+	0.899139i	$0.355806\pi$
$660$	0	0
$661$	−37.9144	−1.47470	−0.737349	−	0.675511i	$-0.763923\pi$
$661$	−37.9144	−1.47470	−0.737349	+	0.675511i	$0.763923\pi$
$662$	−26.2800	−1.02140
$663$	0	0
$664$	2.78704	0.108158
$665$	0.247636	0.00960292
$666$	0	0
$667$	−49.1631	−1.90360
$668$	16.9356	0.655259
$669$	0	0
$670$	4.95975	0.191612
$671$	7.19409	0.277725
$672$	0	0
$673$	−34.3673	−1.32476	−0.662381	−	0.749167i	$-0.730454\pi$
$673$	−34.3673	−1.32476	−0.662381	+	0.749167i	$0.730454\pi$
$674$	24.4996	0.943688
$675$	0	0
$676$	−11.5527	−0.444334
$677$	−29.1518	−1.12039	−0.560197	−	0.828360i	$-0.689274\pi$
$677$	−29.1518	−1.12039	−0.560197	+	0.828360i	$0.689274\pi$
$678$	0	0
$679$	0.0137335	0.000527045 0
$680$	−5.79149	−0.222093
$681$	0	0
$682$	−8.20679	−0.314254
$683$	15.1256	0.578766	0.289383	−	0.957213i	$-0.406550\pi$
$683$	15.1256	0.578766	0.289383	+	0.957213i	$0.406550\pi$
$684$	0	0
$685$	−12.5205	−0.478382
$686$	0.501903	0.0191628
$687$	0	0
$688$	−7.59277	−0.289471
$689$	−10.2219	−0.389421
$690$	0	0
$691$	−42.7381	−1.62584	−0.812918	−	0.582378i	$-0.802122\pi$
$691$	−42.7381	−1.62584	−0.812918	+	0.582378i	$0.802122\pi$
$692$	22.2660	0.846427
$693$	0	0
$694$	8.34721	0.316856
$695$	7.05816	0.267731
$696$	0	0
$697$	20.2292	0.766235
$698$	−10.1250	−0.383235
$699$	0	0
$700$	−0.0813324	−0.00307408
$701$	15.0197	0.567285	0.283642	−	0.958930i	$-0.408457\pi$
$701$	15.0197	0.567285	0.283642	+	0.958930i	$0.408457\pi$
$702$	0	0
$703$	−36.9300	−1.39284
$704$	−1.10979	−0.0418267
$705$	0	0
$706$	−12.9436	−0.487137
$707$	0.224793	0.00845423
$708$	0	0
$709$	35.9651	1.35070	0.675349	−	0.737498i	$-0.263993\pi$
$709$	35.9651	1.35070	0.675349	+	0.737498i	$0.263993\pi$
$710$	−18.0547	−0.677579
$711$	0	0
$712$	−6.07566	−0.227695
$713$	−42.7382	−1.60056
$714$	0	0
$715$	−2.20661	−0.0825226
$716$	−4.06027	−0.151739
$717$	0	0
$718$	−12.7585	−0.476144
$719$	−0.962903	−0.0359102	−0.0179551	−	0.999839i	$-0.505716\pi$
$719$	−0.962903	−0.0359102	−0.0179551	+	0.999839i	$0.505716\pi$
$720$	0	0
$721$	−0.415722	−0.0154823
$722$	1.53548	0.0571447
$723$	0	0
$724$	−13.2135	−0.491077
$725$	19.2969	0.716668
$726$	0	0
$727$	52.3053	1.93990	0.969948	−	0.243312i	$-0.0782340\pi$
$727$	52.3053	1.93990	0.969948	+	0.243312i	$0.0782340\pi$
$728$	−0.0431334	−0.00159863
$729$	0	0
$730$	−6.01721	−0.222707
$731$	−26.6064	−0.984074
$732$	0	0
$733$	−5.25248	−0.194005	−0.0970024	−	0.995284i	$-0.530925\pi$
$733$	−5.25248	−0.194005	−0.0970024	+	0.995284i	$0.530925\pi$
$734$	5.74079	0.211896
$735$	0	0
$736$	−5.77941	−0.213032
$737$	3.33040	0.122677
$738$	0	0
$739$	−5.09563	−0.187446	−0.0937229	−	0.995598i	$-0.529877\pi$
$739$	−5.09563	−0.187446	−0.0937229	+	0.995598i	$0.529877\pi$
$740$	−14.6051	−0.536895
$741$	0	0
$742$	0.304635	0.0111835
$743$	38.6427	1.41766	0.708832	−	0.705378i	$-0.249223\pi$
$743$	38.6427	1.41766	0.708832	+	0.705378i	$0.249223\pi$
$744$	0	0
$745$	1.65274	0.0605516
$746$	−20.9995	−0.768845
$747$	0	0
$748$	−3.88890	−0.142192
$749$	0.343781	0.0125615
$750$	0	0
$751$	−30.7607	−1.12247	−0.561237	−	0.827655i	$-0.689674\pi$
$751$	−30.7607	−1.12247	−0.561237	+	0.827655i	$0.689674\pi$
$752$	−11.7891	−0.429905
$753$	0	0
$754$	10.2338	0.372694
$755$	−20.1100	−0.731878
$756$	0	0
$757$	48.3050	1.75568	0.877838	−	0.478957i	$-0.158985\pi$
$757$	48.3050	1.75568	0.877838	+	0.478957i	$0.158985\pi$
$758$	19.9938	0.726207
$759$	0	0
$760$	−6.90688	−0.250539
$761$	22.0555	0.799512	0.399756	−	0.916622i	$-0.369095\pi$
$761$	22.0555	0.799512	0.399756	+	0.916622i	$0.369095\pi$
$762$	0	0
$763$	0.253710	0.00918490
$764$	10.4427	0.377803
$765$	0	0
$766$	−4.52949	−0.163657
$767$	−11.1183	−0.401458
$768$	0	0
$769$	9.38769	0.338529	0.169264	−	0.985571i	$-0.445861\pi$
$769$	9.38769	0.338529	0.169264	+	0.985571i	$0.445861\pi$
$770$	0.0657621	0.00236990
$771$	0	0
$772$	−10.2254	−0.368021
$773$	28.4620	1.02371	0.511854	−	0.859073i	$-0.328959\pi$
$773$	28.4620	1.02371	0.511854	+	0.859073i	$0.328959\pi$
$774$	0	0
$775$	16.7751	0.602579
$776$	−0.383046	−0.0137505
$777$	0	0
$778$	20.0034	0.717156
$779$	24.1252	0.864374
$780$	0	0
$781$	−12.1234	−0.433810
$782$	−20.2521	−0.724213
$783$	0	0
$784$	−6.99871	−0.249954
$785$	8.51382	0.303871
$786$	0	0
$787$	51.5947	1.83915	0.919577	−	0.392911i	$-0.128532\pi$
$787$	51.5947	1.83915	0.919577	+	0.392911i	$0.128532\pi$
$788$	22.4981	0.801460
$789$	0	0
$790$	−7.83925	−0.278908
$791$	−0.0561485	−0.00199641
$792$	0	0
$793$	−7.79862	−0.276937
$794$	14.0741	0.499470
$795$	0	0
$796$	−1.33281	−0.0472401
$797$	−11.3644	−0.402546	−0.201273	−	0.979535i	$-0.564508\pi$
$797$	−11.3644	−0.402546	−0.201273	+	0.979535i	$0.564508\pi$
$798$	0	0
$799$	−41.3112	−1.46148
$800$	2.26846	0.0802022
$801$	0	0
$802$	7.17872	0.253489
$803$	−4.04047	−0.142585
$804$	0	0
$805$	0.342467	0.0120704
$806$	8.89642	0.313363
$807$	0	0
$808$	−6.26977	−0.220570
$809$	−9.11122	−0.320334	−0.160167	−	0.987090i	$-0.551203\pi$
$809$	−9.11122	−0.320334	−0.160167	+	0.987090i	$0.551203\pi$
$810$	0	0
$811$	39.0028	1.36957	0.684787	−	0.728744i	$-0.259895\pi$
$811$	39.0028	1.36957	0.684787	+	0.728744i	$0.259895\pi$
$812$	−0.304991	−0.0107031
$813$	0	0
$814$	−9.80711	−0.343739
$815$	33.7500	1.18221
$816$	0	0
$817$	−31.7306	−1.11011
$818$	12.2911	0.429748
$819$	0	0
$820$	9.54104	0.333188
$821$	1.67739	0.0585412	0.0292706	−	0.999572i	$-0.490682\pi$
$821$	1.67739	0.0585412	0.0292706	+	0.999572i	$0.490682\pi$
$822$	0	0
$823$	−3.48182	−0.121368	−0.0606842	−	0.998157i	$-0.519328\pi$
$823$	−3.48182	−0.121368	−0.0606842	+	0.998157i	$0.519328\pi$
$824$	11.5950	0.403931
$825$	0	0
$826$	0.331351	0.0115292
$827$	−14.9045	−0.518282	−0.259141	−	0.965840i	$-0.583439\pi$
$827$	−14.9045	−0.518282	−0.259141	+	0.965840i	$0.583439\pi$
$828$	0	0
$829$	−46.2140	−1.60508	−0.802538	−	0.596600i	$-0.796518\pi$
$829$	−46.2140	−1.60508	−0.802538	+	0.596600i	$0.796518\pi$
$830$	4.60625	0.159885
$831$	0	0
$832$	1.20305	0.0417081
$833$	−24.5248	−0.849733
$834$	0	0
$835$	27.9901	0.968639
$836$	−4.63787	−0.160404
$837$	0	0
$838$	−11.9663	−0.413370
$839$	3.27807	0.113171	0.0565857	−	0.998398i	$-0.481979\pi$
$839$	3.27807	0.113171	0.0565857	+	0.998398i	$0.481979\pi$
$840$	0	0
$841$	43.3621	1.49524
$842$	−0.00355896	−0.000122650 0
$843$	0	0
$844$	−14.2623	−0.490928
$845$	−19.0935	−0.656838
$846$	0	0
$847$	−0.350231	−0.0120341
$848$	−8.49664	−0.291776
$849$	0	0
$850$	7.94910	0.272652
$851$	−51.0722	−1.75073
$852$	0	0
$853$	46.6604	1.59762	0.798810	−	0.601583i	$-0.205463\pi$
$853$	46.6604	1.59762	0.798810	+	0.601583i	$0.205463\pi$
$854$	0.232417	0.00795315
$855$	0	0
$856$	−9.58849	−0.327728
$857$	−41.9297	−1.43229	−0.716145	−	0.697952i	$-0.754095\pi$
$857$	−41.9297	−1.43229	−0.716145	+	0.697952i	$0.754095\pi$
$858$	0	0
$859$	2.38586	0.0814044	0.0407022	−	0.999171i	$-0.487041\pi$
$859$	2.38586	0.0814044	0.0407022	+	0.999171i	$0.487041\pi$
$860$	−12.5488	−0.427912
$861$	0	0
$862$	−31.6067	−1.07653
$863$	−32.0932	−1.09247	−0.546233	−	0.837633i	$-0.683939\pi$
$863$	−32.0932	−1.09247	−0.546233	+	0.837633i	$0.683939\pi$
$864$	0	0
$865$	36.7999	1.25123
$866$	−2.95878	−0.100544
$867$	0	0
$868$	−0.265134	−0.00899923
$869$	−5.26394	−0.178567
$870$	0	0
$871$	−3.61026	−0.122329
$872$	−7.07628	−0.239633
$873$	0	0
$874$	−24.1525	−0.816970
$875$	−0.430703	−0.0145604
$876$	0	0
$877$	−42.7400	−1.44323	−0.721614	−	0.692296i	$-0.756599\pi$
$877$	−42.7400	−1.44323	−0.721614	+	0.692296i	$0.756599\pi$
$878$	−34.0207	−1.14814
$879$	0	0
$880$	−1.83419	−0.0618304
$881$	−34.7863	−1.17198	−0.585990	−	0.810318i	$-0.699294\pi$
$881$	−34.7863	−1.17198	−0.585990	+	0.810318i	$0.699294\pi$
$882$	0	0
$883$	50.0834	1.68544	0.842720	−	0.538352i	$-0.180953\pi$
$883$	50.0834	1.68544	0.842720	+	0.538352i	$0.180953\pi$
$884$	4.21569	0.141789
$885$	0	0
$886$	−5.70534	−0.191675
$887$	37.3808	1.25512	0.627562	−	0.778566i	$-0.284053\pi$
$887$	37.3808	1.25512	0.627562	+	0.778566i	$0.284053\pi$
$888$	0	0
$889$	−0.157324	−0.00527647
$890$	−10.0415	−0.336591
$891$	0	0
$892$	−28.6033	−0.957710
$893$	−49.2674	−1.64867
$894$	0	0
$895$	−6.71055	−0.224309
$896$	−0.0358535	−0.00119778
$897$	0	0
$898$	10.8071	0.360637
$899$	62.9056	2.09802
$900$	0	0
$901$	−29.7738	−0.991909
$902$	6.40667	0.213319
$903$	0	0
$904$	1.56605	0.0520861
$905$	−21.8385	−0.725936
$906$	0	0
$907$	18.0117	0.598069	0.299035	−	0.954242i	$-0.403335\pi$
$907$	18.0117	0.598069	0.299035	+	0.954242i	$0.403335\pi$
$908$	−2.37525	−0.0788253
$909$	0	0
$910$	−0.0712882	−0.00236318
$911$	29.2095	0.967753	0.483877	−	0.875136i	$-0.339228\pi$
$911$	29.2095	0.967753	0.483877	+	0.875136i	$0.339228\pi$
$912$	0	0
$913$	3.09303	0.102364
$914$	26.9583	0.891701
$915$	0	0
$916$	9.02486	0.298190
$917$	−0.427482	−0.0141167
$918$	0	0
$919$	−9.48599	−0.312914	−0.156457	−	0.987685i	$-0.550007\pi$
$919$	−9.48599	−0.312914	−0.156457	+	0.987685i	$0.550007\pi$
$920$	−9.55184	−0.314915
$921$	0	0
$922$	30.0843	0.990774
$923$	13.1422	0.432580
$924$	0	0
$925$	20.0462	0.659116
$926$	−40.5192	−1.33154
$927$	0	0
$928$	8.50659	0.279243
$929$	9.36643	0.307302	0.153651	−	0.988125i	$-0.450897\pi$
$929$	9.36643	0.307302	0.153651	+	0.988125i	$0.450897\pi$
$930$	0	0
$931$	−29.2480	−0.958565
$932$	−14.5151	−0.475459
$933$	0	0
$934$	15.9889	0.523171
$935$	−6.42732	−0.210196
$936$	0	0
$937$	34.3928	1.12356	0.561782	−	0.827285i	$-0.310116\pi$
$937$	34.3928	1.12356	0.561782	+	0.827285i	$0.310116\pi$
$938$	0.107594	0.00351307
$939$	0	0
$940$	−19.4843	−0.635508
$941$	−22.1703	−0.722730	−0.361365	−	0.932424i	$-0.617689\pi$
$941$	−22.1703	−0.722730	−0.361365	+	0.932424i	$0.617689\pi$
$942$	0	0
$943$	33.3638	1.08647
$944$	−9.24179	−0.300795
$945$	0	0
$946$	−8.42636	−0.273965
$947$	10.6321	0.345495	0.172748	−	0.984966i	$-0.444735\pi$
$947$	10.6321	0.345495	0.172748	+	0.984966i	$0.444735\pi$
$948$	0	0
$949$	4.38000	0.142181
$950$	9.48003	0.307573
$951$	0	0
$952$	−0.125637	−0.00407193
$953$	32.0340	1.03768	0.518841	−	0.854871i	$-0.326364\pi$
$953$	32.0340	1.03768	0.518841	+	0.854871i	$0.326364\pi$
$954$	0	0
$955$	17.2590	0.558488
$956$	3.69277	0.119433
$957$	0	0
$958$	−26.4486	−0.854517
$959$	−0.271612	−0.00877080
$960$	0	0
$961$	23.6848	0.764027
$962$	10.6312	0.342764
$963$	0	0
$964$	−30.9002	−0.995227
$965$	−16.8999	−0.544028
$966$	0	0
$967$	9.52983	0.306459	0.153229	−	0.988191i	$-0.451033\pi$
$967$	9.52983	0.306459	0.153229	+	0.988191i	$0.451033\pi$
$968$	9.76837	0.313967
$969$	0	0
$970$	−0.633073	−0.0203268
$971$	−40.8006	−1.30935	−0.654677	−	0.755909i	$-0.727195\pi$
$971$	−40.8006	−1.30935	−0.654677	+	0.755909i	$0.727195\pi$
$972$	0	0
$973$	0.153116	0.00490867
$974$	43.0978	1.38094
$975$	0	0
$976$	−6.48240	−0.207497
$977$	16.4171	0.525228	0.262614	−	0.964901i	$-0.415415\pi$
$977$	16.4171	0.525228	0.262614	+	0.964901i	$0.415415\pi$
$978$	0	0
$979$	−6.74270	−0.215498
$980$	−11.5670	−0.369495
$981$	0	0
$982$	−20.3362	−0.648953
$983$	35.1472	1.12102	0.560510	−	0.828148i	$-0.310605\pi$
$983$	35.1472	1.12102	0.560510	+	0.828148i	$0.310605\pi$
$984$	0	0
$985$	37.1834	1.18476
$986$	29.8086	0.949300
$987$	0	0
$988$	5.02760	0.159949
$989$	−43.8817	−1.39536
$990$	0	0
$991$	−49.7613	−1.58072	−0.790361	−	0.612642i	$-0.790107\pi$
$991$	−49.7613	−1.58072	−0.790361	+	0.612642i	$0.790107\pi$
$992$	7.39492	0.234789
$993$	0	0
$994$	−0.391667	−0.0124229
$995$	−2.20278	−0.0698328
$996$	0	0
$997$	−41.4428	−1.31251	−0.656254	−	0.754540i	$-0.727860\pi$
$997$	−41.4428	−1.31251	−0.656254	+	0.754540i	$0.727860\pi$
$998$	28.4668	0.901101
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.g.1.7	✓	9
3.2	odd	2		8046.2.a.h.1.3	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.g.1.7	✓	9	1.1	even	1	trivial
8046.2.a.h.1.3	yes	9	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.65274 q^{5} +0.0358535 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.65274 q^{5} +0.0358535 q^{7} -1.00000 q^{8} -1.65274 q^{10} -1.10979 q^{11} +1.20305 q^{13} -0.0358535 q^{14} +1.00000 q^{16} +3.50418 q^{17} +4.17906 q^{19} +1.65274 q^{20} +1.10979 q^{22} +5.77941 q^{23} -2.26846 q^{25} -1.20305 q^{26} +0.0358535 q^{28} -8.50659 q^{29} -7.39492 q^{31} -1.00000 q^{32} -3.50418 q^{34} +0.0592564 q^{35} -8.83693 q^{37} -4.17906 q^{38} -1.65274 q^{40} +5.77287 q^{41} -7.59277 q^{43} -1.10979 q^{44} -5.77941 q^{46} -11.7891 q^{47} -6.99871 q^{49} +2.26846 q^{50} +1.20305 q^{52} -8.49664 q^{53} -1.83419 q^{55} -0.0358535 q^{56} +8.50659 q^{58} -9.24179 q^{59} -6.48240 q^{61} +7.39492 q^{62} +1.00000 q^{64} +1.98832 q^{65} -3.00093 q^{67} +3.50418 q^{68} -0.0592564 q^{70} +10.9241 q^{71} +3.64076 q^{73} +8.83693 q^{74} +4.17906 q^{76} -0.0397898 q^{77} +4.74320 q^{79} +1.65274 q^{80} -5.77287 q^{82} -2.78704 q^{83} +5.79149 q^{85} +7.59277 q^{86} +1.10979 q^{88} +6.07566 q^{89} +0.0431334 q^{91} +5.77941 q^{92} +11.7891 q^{94} +6.90688 q^{95} +0.383046 q^{97} +6.99871 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{2} + 9 q^{4} + 4 q^{5} - 4 q^{7} - 9 q^{8}+O(q^{10}) \) 9 * q - 9 * q^2 + 9 * q^4 + 4 * q^5 - 4 * q^7 - 9 * q^8	\( 9 q - 9 q^{2} + 9 q^{4} + 4 q^{5} - 4 q^{7} - 9 q^{8} - 4 q^{10} + 4 q^{11} - 8 q^{13} + 4 q^{14} + 9 q^{16} + q^{17} - 10 q^{19} + 4 q^{20} - 4 q^{22} + 8 q^{23} - 3 q^{25} + 8 q^{26} - 4 q^{28} + 4 q^{29} - 17 q^{31} - 9 q^{32} - q^{34} + 10 q^{35} - 11 q^{37} + 10 q^{38} - 4 q^{40} - 16 q^{43} + 4 q^{44} - 8 q^{46} + 7 q^{47} - 5 q^{49} + 3 q^{50} - 8 q^{52} + 12 q^{53} - 23 q^{55} + 4 q^{56} - 4 q^{58} + 6 q^{59} - 13 q^{61} + 17 q^{62} + 9 q^{64} - 24 q^{65} - 14 q^{67} + q^{68} - 10 q^{70} + 30 q^{71} - 12 q^{73} + 11 q^{74} - 10 q^{76} + 12 q^{77} - 35 q^{79} + 4 q^{80} + 5 q^{83} - 27 q^{85} + 16 q^{86} - 4 q^{88} + 23 q^{89} - 28 q^{91} + 8 q^{92} - 7 q^{94} + 32 q^{95} - 21 q^{97} + 5 q^{98}+O(q^{100}) \) 9 * q - 9 * q^2 + 9 * q^4 + 4 * q^5 - 4 * q^7 - 9 * q^8 - 4 * q^10 + 4 * q^11 - 8 * q^13 + 4 * q^14 + 9 * q^16 + q^17 - 10 * q^19 + 4 * q^20 - 4 * q^22 + 8 * q^23 - 3 * q^25 + 8 * q^26 - 4 * q^28 + 4 * q^29 - 17 * q^31 - 9 * q^32 - q^34 + 10 * q^35 - 11 * q^37 + 10 * q^38 - 4 * q^40 - 16 * q^43 + 4 * q^44 - 8 * q^46 + 7 * q^47 - 5 * q^49 + 3 * q^50 - 8 * q^52 + 12 * q^53 - 23 * q^55 + 4 * q^56 - 4 * q^58 + 6 * q^59 - 13 * q^61 + 17 * q^62 + 9 * q^64 - 24 * q^65 - 14 * q^67 + q^68 - 10 * q^70 + 30 * q^71 - 12 * q^73 + 11 * q^74 - 10 * q^76 + 12 * q^77 - 35 * q^79 + 4 * q^80 + 5 * q^83 - 27 * q^85 + 16 * q^86 - 4 * q^88 + 23 * q^89 - 28 * q^91 + 8 * q^92 - 7 * q^94 + 32 * q^95 - 21 * q^97 + 5 * q^98

Introduction

L-functions

Modular forms

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Database

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