LMFDB - Embedded newform 8046.2.a.e.1.2

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 17x^{6} - 2x^{5} + 71x^{4} - 18x^{3} - 81x^{2} + 36x + 9 $ x^8 - 17x^6 - 2x^5 + 71x^4 - 18x^3 - 81x^2 + 36x + 9
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			1.2
Root			$-2.60917$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -2.60917 q^{5} -0.922875 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -2.60917 q^{5} -0.922875 q^{7} -1.00000 q^{8} +2.60917 q^{10} -0.544660 q^{11} -5.57121 q^{13} +0.922875 q^{14} +1.00000 q^{16} +3.06744 q^{17} +3.41958 q^{19} -2.60917 q^{20} +0.544660 q^{22} +1.81880 q^{23} +1.80779 q^{25} +5.57121 q^{26} -0.922875 q^{28} +8.11961 q^{29} -6.23285 q^{31} -1.00000 q^{32} -3.06744 q^{34} +2.40794 q^{35} -0.227618 q^{37} -3.41958 q^{38} +2.60917 q^{40} +5.22409 q^{41} -7.81375 q^{43} -0.544660 q^{44} -1.81880 q^{46} +0.855959 q^{47} -6.14830 q^{49} -1.80779 q^{50} -5.57121 q^{52} +5.74979 q^{53} +1.42111 q^{55} +0.922875 q^{56} -8.11961 q^{58} -1.43107 q^{59} -8.26935 q^{61} +6.23285 q^{62} +1.00000 q^{64} +14.5363 q^{65} +8.90466 q^{67} +3.06744 q^{68} -2.40794 q^{70} +9.28047 q^{71} +10.9621 q^{73} +0.227618 q^{74} +3.41958 q^{76} +0.502653 q^{77} +9.53232 q^{79} -2.60917 q^{80} -5.22409 q^{82} -2.48928 q^{83} -8.00349 q^{85} +7.81375 q^{86} +0.544660 q^{88} -17.2399 q^{89} +5.14153 q^{91} +1.81880 q^{92} -0.855959 q^{94} -8.92228 q^{95} +2.17657 q^{97} +6.14830 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{4} - 5 q^{7} - 8 q^{8}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^4 - 5 * q^7 - 8 * q^8	$ 8 q - 8 q^{2} + 8 q^{4} - 5 q^{7} - 8 q^{8} + 6 q^{11} - 8 q^{13} + 5 q^{14} + 8 q^{16} - 5 q^{17} - 14 q^{19} - 6 q^{22} + 21 q^{23} - 6 q^{25} + 8 q^{26} - 5 q^{28} + 3 q^{29} - 4 q^{31} - 8 q^{32} + 5 q^{34} - 2 q^{35} - 3 q^{37} + 14 q^{38} + 7 q^{41} - 12 q^{43} + 6 q^{44} - 21 q^{46} + 25 q^{47} - 7 q^{49} + 6 q^{50} - 8 q^{52} + 3 q^{53} - 9 q^{55} + 5 q^{56} - 3 q^{58} + 2 q^{59} - 17 q^{61} + 4 q^{62} + 8 q^{64} + 32 q^{65} - 14 q^{67} - 5 q^{68} + 2 q^{70} + 7 q^{71} - 10 q^{73} + 3 q^{74} - 14 q^{76} + 12 q^{77} - 33 q^{79} - 7 q^{82} + 13 q^{83} - 33 q^{85} + 12 q^{86} - 6 q^{88} - 22 q^{89} - 22 q^{91} + 21 q^{92} - 25 q^{94} - 14 q^{95} - 11 q^{97} + 7 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^4 - 5 * q^7 - 8 * q^8 + 6 * q^11 - 8 * q^13 + 5 * q^14 + 8 * q^16 - 5 * q^17 - 14 * q^19 - 6 * q^22 + 21 * q^23 - 6 * q^25 + 8 * q^26 - 5 * q^28 + 3 * q^29 - 4 * q^31 - 8 * q^32 + 5 * q^34 - 2 * q^35 - 3 * q^37 + 14 * q^38 + 7 * q^41 - 12 * q^43 + 6 * q^44 - 21 * q^46 + 25 * q^47 - 7 * q^49 + 6 * q^50 - 8 * q^52 + 3 * q^53 - 9 * q^55 + 5 * q^56 - 3 * q^58 + 2 * q^59 - 17 * q^61 + 4 * q^62 + 8 * q^64 + 32 * q^65 - 14 * q^67 - 5 * q^68 + 2 * q^70 + 7 * q^71 - 10 * q^73 + 3 * q^74 - 14 * q^76 + 12 * q^77 - 33 * q^79 - 7 * q^82 + 13 * q^83 - 33 * q^85 + 12 * q^86 - 6 * q^88 - 22 * q^89 - 22 * q^91 + 21 * q^92 - 25 * q^94 - 14 * q^95 - 11 * q^97 + 7 * 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
$5$	−2.60917	−1.16686	−0.583429	−	0.812164i	$-0.698289\pi$
$5$	−2.60917	−1.16686	−0.583429	+	0.812164i	$0.698289\pi$
$6$	0	0
$7$	−0.922875	−0.348814	−0.174407	−	0.984674i	$-0.555801\pi$
$7$	−0.922875	−0.348814	−0.174407	+	0.984674i	$0.555801\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	2.60917	0.825093
$11$	−0.544660	−0.164221	−0.0821106	−	0.996623i	$-0.526166\pi$
$11$	−0.544660	−0.164221	−0.0821106	+	0.996623i	$0.526166\pi$
$12$	0	0
$13$	−5.57121	−1.54518	−0.772588	−	0.634908i	$-0.781038\pi$
$13$	−5.57121	−1.54518	−0.772588	+	0.634908i	$0.781038\pi$
$14$	0.922875	0.246649
$15$	0	0
$16$	1.00000	0.250000
$17$	3.06744	0.743964	0.371982	−	0.928240i	$-0.378678\pi$
$17$	3.06744	0.743964	0.371982	+	0.928240i	$0.378678\pi$
$18$	0	0
$19$	3.41958	0.784505	0.392253	−	0.919857i	$-0.371696\pi$
$19$	3.41958	0.784505	0.392253	+	0.919857i	$0.371696\pi$
$20$	−2.60917	−0.583429
$21$	0	0
$22$	0.544660	0.116122
$23$	1.81880	0.379247	0.189623	−	0.981857i	$-0.439273\pi$
$23$	1.81880	0.379247	0.189623	+	0.981857i	$0.439273\pi$
$24$	0	0
$25$	1.80779	0.361558
$26$	5.57121	1.09260
$27$	0	0
$28$	−0.922875	−0.174407
$29$	8.11961	1.50777	0.753887	−	0.657005i	$-0.228177\pi$
$29$	8.11961	1.50777	0.753887	+	0.657005i	$0.228177\pi$
$30$	0	0
$31$	−6.23285	−1.11945	−0.559727	−	0.828677i	$-0.689094\pi$
$31$	−6.23285	−1.11945	−0.559727	+	0.828677i	$0.689094\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.06744	−0.526062
$35$	2.40794	0.407016
$36$	0	0
$37$	−0.227618	−0.0374202	−0.0187101	−	0.999825i	$-0.505956\pi$
$37$	−0.227618	−0.0374202	−0.0187101	+	0.999825i	$0.505956\pi$
$38$	−3.41958	−0.554729
$39$	0	0
$40$	2.60917	0.412547
$41$	5.22409	0.815866	0.407933	−	0.913012i	$-0.366250\pi$
$41$	5.22409	0.815866	0.407933	+	0.913012i	$0.366250\pi$
$42$	0	0
$43$	−7.81375	−1.19159	−0.595793	−	0.803138i	$-0.703162\pi$
$43$	−7.81375	−1.19159	−0.595793	+	0.803138i	$0.703162\pi$
$44$	−0.544660	−0.0821106
$45$	0	0
$46$	−1.81880	−0.268168
$47$	0.855959	0.124854	0.0624272	−	0.998050i	$-0.480116\pi$
$47$	0.855959	0.124854	0.0624272	+	0.998050i	$0.480116\pi$
$48$	0	0
$49$	−6.14830	−0.878329
$50$	−1.80779	−0.255660
$51$	0	0
$52$	−5.57121	−0.772588
$53$	5.74979	0.789794	0.394897	−	0.918725i	$-0.370780\pi$
$53$	5.74979	0.789794	0.394897	+	0.918725i	$0.370780\pi$
$54$	0	0
$55$	1.42111	0.191623
$56$	0.922875	0.123324
$57$	0	0
$58$	−8.11961	−1.06616
$59$	−1.43107	−0.186310	−0.0931550	−	0.995652i	$-0.529695\pi$
$59$	−1.43107	−0.186310	−0.0931550	+	0.995652i	$0.529695\pi$
$60$	0	0
$61$	−8.26935	−1.05878	−0.529391	−	0.848378i	$-0.677579\pi$
$61$	−8.26935	−1.05878	−0.529391	+	0.848378i	$0.677579\pi$
$62$	6.23285	0.791573
$63$	0	0
$64$	1.00000	0.125000
$65$	14.5363	1.80300
$66$	0	0
$67$	8.90466	1.08788	0.543939	−	0.839125i	$-0.316932\pi$
$67$	8.90466	1.08788	0.543939	+	0.839125i	$0.316932\pi$
$68$	3.06744	0.371982
$69$	0	0
$70$	−2.40794	−0.287804
$71$	9.28047	1.10139	0.550695	−	0.834707i	$-0.314363\pi$
$71$	9.28047	1.10139	0.550695	+	0.834707i	$0.314363\pi$
$72$	0	0
$73$	10.9621	1.28302	0.641508	−	0.767116i	$-0.278309\pi$
$73$	10.9621	1.28302	0.641508	+	0.767116i	$0.278309\pi$
$74$	0.227618	0.0264601
$75$	0	0
$76$	3.41958	0.392253
$77$	0.502653	0.0572826
$78$	0	0
$79$	9.53232	1.07247	0.536235	−	0.844069i	$-0.319846\pi$
$79$	9.53232	1.07247	0.536235	+	0.844069i	$0.319846\pi$
$80$	−2.60917	−0.291714
$81$	0	0
$82$	−5.22409	−0.576905
$83$	−2.48928	−0.273234	−0.136617	−	0.990624i	$-0.543623\pi$
$83$	−2.48928	−0.273234	−0.136617	+	0.990624i	$0.543623\pi$
$84$	0	0
$85$	−8.00349	−0.868100
$86$	7.81375	0.842578
$87$	0	0
$88$	0.544660	0.0580610
$89$	−17.2399	−1.82743	−0.913715	−	0.406355i	$-0.866800\pi$
$89$	−17.2399	−1.82743	−0.913715	+	0.406355i	$0.866800\pi$
$90$	0	0
$91$	5.14153	0.538979
$92$	1.81880	0.189623
$93$	0	0
$94$	−0.855959	−0.0882854
$95$	−8.92228	−0.915406
$96$	0	0
$97$	2.17657	0.220997	0.110498	−	0.993876i	$-0.464755\pi$
$97$	2.17657	0.220997	0.110498	+	0.993876i	$0.464755\pi$
$98$	6.14830	0.621072
$99$	0	0
$100$	1.80779	0.180779
$101$	14.9673	1.48931	0.744653	−	0.667452i	$-0.232615\pi$
$101$	14.9673	1.48931	0.744653	+	0.667452i	$0.232615\pi$
$102$	0	0
$103$	17.9956	1.77316	0.886580	−	0.462575i	$-0.153075\pi$
$103$	17.9956	1.77316	0.886580	+	0.462575i	$0.153075\pi$
$104$	5.57121	0.546302
$105$	0	0
$106$	−5.74979	−0.558469
$107$	16.9821	1.64172	0.820862	−	0.571126i	$-0.193493\pi$
$107$	16.9821	1.64172	0.820862	+	0.571126i	$0.193493\pi$
$108$	0	0
$109$	3.86769	0.370457	0.185229	−	0.982695i	$-0.440697\pi$
$109$	3.86769	0.370457	0.185229	+	0.982695i	$0.440697\pi$
$110$	−1.42111	−0.135498
$111$	0	0
$112$	−0.922875	−0.0872035
$113$	−8.33306	−0.783908	−0.391954	−	0.919985i	$-0.628201\pi$
$113$	−8.33306	−0.783908	−0.391954	+	0.919985i	$0.628201\pi$
$114$	0	0
$115$	−4.74558	−0.442527
$116$	8.11961	0.753887
$117$	0	0
$118$	1.43107	0.131741
$119$	−2.83087	−0.259505
$120$	0	0
$121$	−10.7033	−0.973031
$122$	8.26935	0.748672
$123$	0	0
$124$	−6.23285	−0.559727
$125$	8.32904	0.744972
$126$	0	0
$127$	−6.37495	−0.565686	−0.282843	−	0.959166i	$-0.591277\pi$
$127$	−6.37495	−0.565686	−0.282843	+	0.959166i	$0.591277\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−14.5363	−1.27491
$131$	0.138349	0.0120876	0.00604380	−	0.999982i	$-0.498076\pi$
$131$	0.138349	0.0120876	0.00604380	+	0.999982i	$0.498076\pi$
$132$	0	0
$133$	−3.15584	−0.273646
$134$	−8.90466	−0.769246
$135$	0	0
$136$	−3.06744	−0.263031
$137$	−15.5431	−1.32794	−0.663968	−	0.747761i	$-0.731129\pi$
$137$	−15.5431	−1.32794	−0.663968	+	0.747761i	$0.731129\pi$
$138$	0	0
$139$	−16.2027	−1.37429	−0.687146	−	0.726520i	$-0.741137\pi$
$139$	−16.2027	−1.37429	−0.687146	+	0.726520i	$0.741137\pi$
$140$	2.40794	0.203508
$141$	0	0
$142$	−9.28047	−0.778800
$143$	3.03442	0.253751
$144$	0	0
$145$	−21.1855	−1.75936
$146$	−10.9621	−0.907230
$147$	0	0
$148$	−0.227618	−0.0187101
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−10.0876	−0.820917	−0.410458	−	0.911879i	$-0.634631\pi$
$151$	−10.0876	−0.820917	−0.410458	+	0.911879i	$0.634631\pi$
$152$	−3.41958	−0.277365
$153$	0	0
$154$	−0.502653	−0.0405049
$155$	16.2626	1.30624
$156$	0	0
$157$	−9.88620	−0.789004	−0.394502	−	0.918895i	$-0.629083\pi$
$157$	−9.88620	−0.789004	−0.394502	+	0.918895i	$0.629083\pi$
$158$	−9.53232	−0.758351
$159$	0	0
$160$	2.60917	0.206273
$161$	−1.67853	−0.132287
$162$	0	0
$163$	6.27758	0.491698	0.245849	−	0.969308i	$-0.420933\pi$
$163$	6.27758	0.491698	0.245849	+	0.969308i	$0.420933\pi$
$164$	5.22409	0.407933
$165$	0	0
$166$	2.48928	0.193206
$167$	23.6519	1.83024	0.915118	−	0.403186i	$-0.132097\pi$
$167$	23.6519	1.83024	0.915118	+	0.403186i	$0.132097\pi$
$168$	0	0
$169$	18.0384	1.38757
$170$	8.00349	0.613840
$171$	0	0
$172$	−7.81375	−0.595793
$173$	−4.21154	−0.320197	−0.160099	−	0.987101i	$-0.551181\pi$
$173$	−4.21154	−0.320197	−0.160099	+	0.987101i	$0.551181\pi$
$174$	0	0
$175$	−1.66836	−0.126116
$176$	−0.544660	−0.0410553
$177$	0	0
$178$	17.2399	1.29219
$179$	−21.6329	−1.61692	−0.808458	−	0.588554i	$-0.799698\pi$
$179$	−21.6329	−1.61692	−0.808458	+	0.588554i	$0.799698\pi$
$180$	0	0
$181$	22.2885	1.65669	0.828344	−	0.560219i	$-0.189283\pi$
$181$	22.2885	1.65669	0.828344	+	0.560219i	$0.189283\pi$
$182$	−5.14153	−0.381116
$183$	0	0
$184$	−1.81880	−0.134084
$185$	0.593896	0.0436641
$186$	0	0
$187$	−1.67071	−0.122175
$188$	0.855959	0.0624272
$189$	0	0
$190$	8.92228	0.647290
$191$	−17.5751	−1.27169	−0.635844	−	0.771818i	$-0.719348\pi$
$191$	−17.5751	−1.27169	−0.635844	+	0.771818i	$0.719348\pi$
$192$	0	0
$193$	−7.86631	−0.566229	−0.283115	−	0.959086i	$-0.591368\pi$
$193$	−7.86631	−0.566229	−0.283115	+	0.959086i	$0.591368\pi$
$194$	−2.17657	−0.156268
$195$	0	0
$196$	−6.14830	−0.439164
$197$	−4.42276	−0.315108	−0.157554	−	0.987510i	$-0.550361\pi$
$197$	−4.42276	−0.315108	−0.157554	+	0.987510i	$0.550361\pi$
$198$	0	0
$199$	−25.5079	−1.80821	−0.904105	−	0.427311i	$-0.859461\pi$
$199$	−25.5079	−1.80821	−0.904105	+	0.427311i	$0.859461\pi$
$200$	−1.80779	−0.127830
$201$	0	0
$202$	−14.9673	−1.05310
$203$	−7.49338	−0.525932
$204$	0	0
$205$	−13.6306	−0.952000
$206$	−17.9956	−1.25381
$207$	0	0
$208$	−5.57121	−0.386294
$209$	−1.86251	−0.128832
$210$	0	0
$211$	−6.52937	−0.449500	−0.224750	−	0.974416i	$-0.572157\pi$
$211$	−6.52937	−0.449500	−0.224750	+	0.974416i	$0.572157\pi$
$212$	5.74979	0.394897
$213$	0	0
$214$	−16.9821	−1.16087
$215$	20.3874	1.39041
$216$	0	0
$217$	5.75214	0.390481
$218$	−3.86769	−0.261953
$219$	0	0
$220$	1.42111	0.0958114
$221$	−17.0894	−1.14956
$222$	0	0
$223$	3.38817	0.226889	0.113444	−	0.993544i	$-0.463812\pi$
$223$	3.38817	0.226889	0.113444	+	0.993544i	$0.463812\pi$
$224$	0.922875	0.0616622
$225$	0	0
$226$	8.33306	0.554307
$227$	16.2110	1.07596	0.537981	−	0.842957i	$-0.319187\pi$
$227$	16.2110	1.07596	0.537981	+	0.842957i	$0.319187\pi$
$228$	0	0
$229$	−1.12778	−0.0745256	−0.0372628	−	0.999306i	$-0.511864\pi$
$229$	−1.12778	−0.0745256	−0.0372628	+	0.999306i	$0.511864\pi$
$230$	4.74558	0.312914
$231$	0	0
$232$	−8.11961	−0.533078
$233$	12.7193	0.833270	0.416635	−	0.909074i	$-0.363209\pi$
$233$	12.7193	0.833270	0.416635	+	0.909074i	$0.363209\pi$
$234$	0	0
$235$	−2.23335	−0.145687
$236$	−1.43107	−0.0931550
$237$	0	0
$238$	2.83087	0.183498
$239$	−14.9893	−0.969578	−0.484789	−	0.874631i	$-0.661104\pi$
$239$	−14.9893	−0.969578	−0.484789	+	0.874631i	$0.661104\pi$
$240$	0	0
$241$	19.7096	1.26961	0.634804	−	0.772673i	$-0.281081\pi$
$241$	19.7096	1.26961	0.634804	+	0.772673i	$0.281081\pi$
$242$	10.7033	0.688037
$243$	0	0
$244$	−8.26935	−0.529391
$245$	16.0420	1.02488
$246$	0	0
$247$	−19.0512	−1.21220
$248$	6.23285	0.395786
$249$	0	0
$250$	−8.32904	−0.526775
$251$	−22.1408	−1.39752	−0.698758	−	0.715358i	$-0.746264\pi$
$251$	−22.1408	−1.39752	−0.698758	+	0.715358i	$0.746264\pi$
$252$	0	0
$253$	−0.990630	−0.0622804
$254$	6.37495	0.400000
$255$	0	0
$256$	1.00000	0.0625000
$257$	−7.89930	−0.492745	−0.246372	−	0.969175i	$-0.579239\pi$
$257$	−7.89930	−0.492745	−0.246372	+	0.969175i	$0.579239\pi$
$258$	0	0
$259$	0.210063	0.0130527
$260$	14.5363	0.901501
$261$	0	0
$262$	−0.138349	−0.00854723
$263$	−12.9967	−0.801413	−0.400707	−	0.916206i	$-0.631235\pi$
$263$	−12.9967	−0.801413	−0.400707	+	0.916206i	$0.631235\pi$
$264$	0	0
$265$	−15.0022	−0.921578
$266$	3.15584	0.193497
$267$	0	0
$268$	8.90466	0.543939
$269$	−9.64407	−0.588009	−0.294005	−	0.955804i	$-0.594988\pi$
$269$	−9.64407	−0.588009	−0.294005	+	0.955804i	$0.594988\pi$
$270$	0	0
$271$	−7.66876	−0.465844	−0.232922	−	0.972495i	$-0.574829\pi$
$271$	−7.66876	−0.465844	−0.232922	+	0.972495i	$0.574829\pi$
$272$	3.06744	0.185991
$273$	0	0
$274$	15.5431	0.938992
$275$	−0.984630	−0.0593754
$276$	0	0
$277$	32.3375	1.94297	0.971487	−	0.237091i	$-0.0761939\pi$
$277$	32.3375	1.94297	0.971487	+	0.237091i	$0.0761939\pi$
$278$	16.2027	0.971771
$279$	0	0
$280$	−2.40794	−0.143902
$281$	−14.4255	−0.860551	−0.430275	−	0.902698i	$-0.641584\pi$
$281$	−14.4255	−0.860551	−0.430275	+	0.902698i	$0.641584\pi$
$282$	0	0
$283$	−10.2717	−0.610587	−0.305294	−	0.952258i	$-0.598755\pi$
$283$	−10.2717	−0.610587	−0.305294	+	0.952258i	$0.598755\pi$
$284$	9.28047	0.550695
$285$	0	0
$286$	−3.03442	−0.179429
$287$	−4.82119	−0.284586
$288$	0	0
$289$	−7.59080	−0.446518
$290$	21.1855	1.24405
$291$	0	0
$292$	10.9621	0.641508
$293$	16.1337	0.942539	0.471269	−	0.881989i	$-0.343796\pi$
$293$	16.1337	0.942539	0.471269	+	0.881989i	$0.343796\pi$
$294$	0	0
$295$	3.73392	0.217397
$296$	0.227618	0.0132300
$297$	0	0
$298$	1.00000	0.0579284
$299$	−10.1329	−0.586003
$300$	0	0
$301$	7.21111	0.415642
$302$	10.0876	0.580476
$303$	0	0
$304$	3.41958	0.196126
$305$	21.5762	1.23545
$306$	0	0
$307$	14.2360	0.812494	0.406247	−	0.913763i	$-0.366837\pi$
$307$	14.2360	0.812494	0.406247	+	0.913763i	$0.366837\pi$
$308$	0.502653	0.0286413
$309$	0	0
$310$	−16.2626	−0.923653
$311$	6.65202	0.377201	0.188601	−	0.982054i	$-0.439605\pi$
$311$	6.65202	0.377201	0.188601	+	0.982054i	$0.439605\pi$
$312$	0	0
$313$	23.9290	1.35255	0.676275	−	0.736649i	$-0.263593\pi$
$313$	23.9290	1.35255	0.676275	+	0.736649i	$0.263593\pi$
$314$	9.88620	0.557910
$315$	0	0
$316$	9.53232	0.536235
$317$	13.7721	0.773521	0.386760	−	0.922180i	$-0.373594\pi$
$317$	13.7721	0.773521	0.386760	+	0.922180i	$0.373594\pi$
$318$	0	0
$319$	−4.42243	−0.247608
$320$	−2.60917	−0.145857
$321$	0	0
$322$	1.67853	0.0935408
$323$	10.4894	0.583644
$324$	0	0
$325$	−10.0716	−0.558670
$326$	−6.27758	−0.347683
$327$	0	0
$328$	−5.22409	−0.288452
$329$	−0.789943	−0.0435510
$330$	0	0
$331$	−16.1954	−0.890178	−0.445089	−	0.895486i	$-0.646828\pi$
$331$	−16.1954	−0.890178	−0.445089	+	0.895486i	$0.646828\pi$
$332$	−2.48928	−0.136617
$333$	0	0
$334$	−23.6519	−1.29417
$335$	−23.2338	−1.26940
$336$	0	0
$337$	−22.3172	−1.21570	−0.607848	−	0.794054i	$-0.707967\pi$
$337$	−22.3172	−1.21570	−0.607848	+	0.794054i	$0.707967\pi$
$338$	−18.0384	−0.981159
$339$	0	0
$340$	−8.00349	−0.434050
$341$	3.39479	0.183838
$342$	0	0
$343$	12.1342	0.655187
$344$	7.81375	0.421289
$345$	0	0
$346$	4.21154	0.226414
$347$	−34.3276	−1.84280	−0.921400	−	0.388615i	$-0.872954\pi$
$347$	−34.3276	−1.84280	−0.921400	+	0.388615i	$0.872954\pi$
$348$	0	0
$349$	−21.6118	−1.15685	−0.578425	−	0.815735i	$-0.696333\pi$
$349$	−21.6118	−1.15685	−0.578425	+	0.815735i	$0.696333\pi$
$350$	1.66836	0.0891777
$351$	0	0
$352$	0.544660	0.0290305
$353$	−34.6860	−1.84615	−0.923075	−	0.384620i	$-0.874333\pi$
$353$	−34.6860	−1.84615	−0.923075	+	0.384620i	$0.874333\pi$
$354$	0	0
$355$	−24.2144	−1.28517
$356$	−17.2399	−0.913715
$357$	0	0
$358$	21.6329	1.14333
$359$	−28.4637	−1.50226	−0.751129	−	0.660156i	$-0.770490\pi$
$359$	−28.4637	−1.50226	−0.751129	+	0.660156i	$0.770490\pi$
$360$	0	0
$361$	−7.30647	−0.384551
$362$	−22.2885	−1.17146
$363$	0	0
$364$	5.14153	0.269490
$365$	−28.6020	−1.49710
$366$	0	0
$367$	−15.7763	−0.823516	−0.411758	−	0.911293i	$-0.635085\pi$
$367$	−15.7763	−0.823516	−0.411758	+	0.911293i	$0.635085\pi$
$368$	1.81880	0.0948117
$369$	0	0
$370$	−0.593896	−0.0308752
$371$	−5.30634	−0.275491
$372$	0	0
$373$	−26.4504	−1.36955	−0.684774	−	0.728755i	$-0.740099\pi$
$373$	−26.4504	−1.36955	−0.684774	+	0.728755i	$0.740099\pi$
$374$	1.67071	0.0863905
$375$	0	0
$376$	−0.855959	−0.0441427
$377$	−45.2360	−2.32977
$378$	0	0
$379$	−21.0878	−1.08321	−0.541603	−	0.840635i	$-0.682182\pi$
$379$	−21.0878	−1.08321	−0.541603	+	0.840635i	$0.682182\pi$
$380$	−8.92228	−0.457703
$381$	0	0
$382$	17.5751	0.899219
$383$	18.5574	0.948237	0.474119	−	0.880461i	$-0.342767\pi$
$383$	18.5574	0.948237	0.474119	+	0.880461i	$0.342767\pi$
$384$	0	0
$385$	−1.31151	−0.0668407
$386$	7.86631	0.400385
$387$	0	0
$388$	2.17657	0.110498
$389$	26.2971	1.33332	0.666658	−	0.745364i	$-0.267724\pi$
$389$	26.2971	1.33332	0.666658	+	0.745364i	$0.267724\pi$
$390$	0	0
$391$	5.57907	0.282146
$392$	6.14830	0.310536
$393$	0	0
$394$	4.42276	0.222815
$395$	−24.8715	−1.25142
$396$	0	0
$397$	2.42889	0.121902	0.0609512	−	0.998141i	$-0.480587\pi$
$397$	2.42889	0.121902	0.0609512	+	0.998141i	$0.480587\pi$
$398$	25.5079	1.27860
$399$	0	0
$400$	1.80779	0.0903894
$401$	−5.83729	−0.291501	−0.145750	−	0.989321i	$-0.546560\pi$
$401$	−5.83729	−0.291501	−0.145750	+	0.989321i	$0.546560\pi$
$402$	0	0
$403$	34.7245	1.72975
$404$	14.9673	0.744653
$405$	0	0
$406$	7.49338	0.371890
$407$	0.123975	0.00614519
$408$	0	0
$409$	38.0056	1.87925	0.939627	−	0.342199i	$-0.111172\pi$
$409$	38.0056	1.87925	0.939627	+	0.342199i	$0.111172\pi$
$410$	13.6306	0.673166
$411$	0	0
$412$	17.9956	0.886580
$413$	1.32070	0.0649875
$414$	0	0
$415$	6.49496	0.318825
$416$	5.57121	0.273151
$417$	0	0
$418$	1.86251	0.0910983
$419$	15.3523	0.750008	0.375004	−	0.927023i	$-0.377641\pi$
$419$	15.3523	0.750008	0.375004	+	0.927023i	$0.377641\pi$
$420$	0	0
$421$	36.8285	1.79491	0.897456	−	0.441104i	$-0.145413\pi$
$421$	36.8285	1.79491	0.897456	+	0.441104i	$0.145413\pi$
$422$	6.52937	0.317845
$423$	0	0
$424$	−5.74979	−0.279234
$425$	5.54528	0.268986
$426$	0	0
$427$	7.63158	0.369318
$428$	16.9821	0.820862
$429$	0	0
$430$	−20.3874	−0.983169
$431$	−1.98851	−0.0957829	−0.0478915	−	0.998853i	$-0.515250\pi$
$431$	−1.98851	−0.0957829	−0.0478915	+	0.998853i	$0.515250\pi$
$432$	0	0
$433$	−14.1948	−0.682159	−0.341079	−	0.940034i	$-0.610793\pi$
$433$	−14.1948	−0.682159	−0.341079	+	0.940034i	$0.610793\pi$
$434$	−5.75214	−0.276112
$435$	0	0
$436$	3.86769	0.185229
$437$	6.21955	0.297521
$438$	0	0
$439$	−35.4726	−1.69301	−0.846507	−	0.532377i	$-0.821299\pi$
$439$	−35.4726	−1.69301	−0.846507	+	0.532377i	$0.821299\pi$
$440$	−1.42111	−0.0677489
$441$	0	0
$442$	17.0894	0.812858
$443$	−2.95745	−0.140513	−0.0702563	−	0.997529i	$-0.522382\pi$
$443$	−2.95745	−0.140513	−0.0702563	+	0.997529i	$0.522382\pi$
$444$	0	0
$445$	44.9820	2.13235
$446$	−3.38817	−0.160434
$447$	0	0
$448$	−0.922875	−0.0436017
$449$	−9.25264	−0.436659	−0.218329	−	0.975875i	$-0.570061\pi$
$449$	−9.25264	−0.436659	−0.218329	+	0.975875i	$0.570061\pi$
$450$	0	0
$451$	−2.84536	−0.133983
$452$	−8.33306	−0.391954
$453$	0	0
$454$	−16.2110	−0.760819
$455$	−13.4152	−0.628912
$456$	0	0
$457$	−42.0109	−1.96519	−0.982594	−	0.185764i	$-0.940524\pi$
$457$	−42.0109	−1.96519	−0.982594	+	0.185764i	$0.940524\pi$
$458$	1.12778	0.0526975
$459$	0	0
$460$	−4.74558	−0.221264
$461$	12.0399	0.560752	0.280376	−	0.959890i	$-0.409541\pi$
$461$	12.0399	0.560752	0.280376	+	0.959890i	$0.409541\pi$
$462$	0	0
$463$	−4.77301	−0.221821	−0.110910	−	0.993830i	$-0.535377\pi$
$463$	−4.77301	−0.221821	−0.110910	+	0.993830i	$0.535377\pi$
$464$	8.11961	0.376943
$465$	0	0
$466$	−12.7193	−0.589211
$467$	−14.3296	−0.663093	−0.331546	−	0.943439i	$-0.607570\pi$
$467$	−14.3296	−0.663093	−0.331546	+	0.943439i	$0.607570\pi$
$468$	0	0
$469$	−8.21789	−0.379467
$470$	2.23335	0.103017
$471$	0	0
$472$	1.43107	0.0658705
$473$	4.25584	0.195684
$474$	0	0
$475$	6.18187	0.283644
$476$	−2.83087	−0.129752
$477$	0	0
$478$	14.9893	0.685595
$479$	14.1676	0.647334	0.323667	−	0.946171i	$-0.395084\pi$
$479$	14.1676	0.647334	0.323667	+	0.946171i	$0.395084\pi$
$480$	0	0
$481$	1.26811	0.0578208
$482$	−19.7096	−0.897748
$483$	0	0
$484$	−10.7033	−0.486516
$485$	−5.67904	−0.257872
$486$	0	0
$487$	−3.25466	−0.147483	−0.0737414	−	0.997277i	$-0.523494\pi$
$487$	−3.25466	−0.147483	−0.0737414	+	0.997277i	$0.523494\pi$
$488$	8.26935	0.374336
$489$	0	0
$490$	−16.0420	−0.724703
$491$	8.44806	0.381256	0.190628	−	0.981662i	$-0.438948\pi$
$491$	8.44806	0.381256	0.190628	+	0.981662i	$0.438948\pi$
$492$	0	0
$493$	24.9064	1.12173
$494$	19.0512	0.857154
$495$	0	0
$496$	−6.23285	−0.279863
$497$	−8.56472	−0.384180
$498$	0	0
$499$	−31.1209	−1.39316	−0.696581	−	0.717478i	$-0.745296\pi$
$499$	−31.1209	−1.39316	−0.696581	+	0.717478i	$0.745296\pi$
$500$	8.32904	0.372486
$501$	0	0
$502$	22.1408	0.988193
$503$	39.4877	1.76067	0.880334	−	0.474354i	$-0.157318\pi$
$503$	39.4877	1.76067	0.880334	+	0.474354i	$0.157318\pi$
$504$	0	0
$505$	−39.0524	−1.73781
$506$	0.990630	0.0440389
$507$	0	0
$508$	−6.37495	−0.282843
$509$	−9.92301	−0.439830	−0.219915	−	0.975519i	$-0.570578\pi$
$509$	−9.92301	−0.439830	−0.219915	+	0.975519i	$0.570578\pi$
$510$	0	0
$511$	−10.1166	−0.447534
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	7.89930	0.348423
$515$	−46.9537	−2.06903
$516$	0	0
$517$	−0.466207	−0.0205037
$518$	−0.210063	−0.00922965
$519$	0	0
$520$	−14.5363	−0.637457
$521$	9.62338	0.421608	0.210804	−	0.977528i	$-0.432392\pi$
$521$	9.62338	0.421608	0.210804	+	0.977528i	$0.432392\pi$
$522$	0	0
$523$	15.3859	0.672780	0.336390	−	0.941723i	$-0.390794\pi$
$523$	15.3859	0.672780	0.336390	+	0.941723i	$0.390794\pi$
$524$	0.138349	0.00604380
$525$	0	0
$526$	12.9967	0.566685
$527$	−19.1189	−0.832833
$528$	0	0
$529$	−19.6920	−0.856172
$530$	15.0022	0.651654
$531$	0	0
$532$	−3.15584	−0.136823
$533$	−29.1045	−1.26066
$534$	0	0
$535$	−44.3093	−1.91566
$536$	−8.90466	−0.384623
$537$	0	0
$538$	9.64407	0.415785
$539$	3.34873	0.144240
$540$	0	0
$541$	−24.6359	−1.05918	−0.529589	−	0.848254i	$-0.677654\pi$
$541$	−24.6359	−1.05918	−0.529589	+	0.848254i	$0.677654\pi$
$542$	7.66876	0.329402
$543$	0	0
$544$	−3.06744	−0.131515
$545$	−10.0915	−0.432271
$546$	0	0
$547$	−28.7329	−1.22853	−0.614264	−	0.789100i	$-0.710547\pi$
$547$	−28.7329	−1.22853	−0.614264	+	0.789100i	$0.710547\pi$
$548$	−15.5431	−0.663968
$549$	0	0
$550$	0.984630	0.0419848
$551$	27.7656	1.18286
$552$	0	0
$553$	−8.79714	−0.374093
$554$	−32.3375	−1.37389
$555$	0	0
$556$	−16.2027	−0.687146
$557$	16.4780	0.698193	0.349097	−	0.937087i	$-0.386488\pi$
$557$	16.4780	0.698193	0.349097	+	0.937087i	$0.386488\pi$
$558$	0	0
$559$	43.5320	1.84121
$560$	2.40794	0.101754
$561$	0	0
$562$	14.4255	0.608501
$563$	−21.2433	−0.895299	−0.447649	−	0.894209i	$-0.647739\pi$
$563$	−21.2433	−0.895299	−0.447649	+	0.894209i	$0.647739\pi$
$564$	0	0
$565$	21.7424	0.914709
$566$	10.2717	0.431750
$567$	0	0
$568$	−9.28047	−0.389400
$569$	−4.02345	−0.168672	−0.0843360	−	0.996437i	$-0.526877\pi$
$569$	−4.02345	−0.168672	−0.0843360	+	0.996437i	$0.526877\pi$
$570$	0	0
$571$	−26.0458	−1.08998	−0.544992	−	0.838441i	$-0.683467\pi$
$571$	−26.0458	−1.08998	−0.544992	+	0.838441i	$0.683467\pi$
$572$	3.03442	0.126875
$573$	0	0
$574$	4.82119	0.201232
$575$	3.28801	0.137120
$576$	0	0
$577$	32.1220	1.33726	0.668629	−	0.743596i	$-0.266881\pi$
$577$	32.1220	1.33726	0.668629	+	0.743596i	$0.266881\pi$
$578$	7.59080	0.315736
$579$	0	0
$580$	−21.1855	−0.879679
$581$	2.29729	0.0953078
$582$	0	0
$583$	−3.13168	−0.129701
$584$	−10.9621	−0.453615
$585$	0	0
$586$	−16.1337	−0.666476
$587$	27.8724	1.15042	0.575209	−	0.818007i	$-0.304921\pi$
$587$	27.8724	1.15042	0.575209	+	0.818007i	$0.304921\pi$
$588$	0	0
$589$	−21.3137	−0.878217
$590$	−3.73392	−0.153723
$591$	0	0
$592$	−0.227618	−0.00935506
$593$	27.5269	1.13040	0.565198	−	0.824955i	$-0.308800\pi$
$593$	27.5269	1.13040	0.565198	+	0.824955i	$0.308800\pi$
$594$	0	0
$595$	7.38622	0.302805
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	10.1329	0.414367
$599$	27.4843	1.12298	0.561489	−	0.827484i	$-0.310229\pi$
$599$	27.4843	1.12298	0.561489	+	0.827484i	$0.310229\pi$
$600$	0	0
$601$	0.362618	0.0147915	0.00739575	−	0.999973i	$-0.497646\pi$
$601$	0.362618	0.0147915	0.00739575	+	0.999973i	$0.497646\pi$
$602$	−7.21111	−0.293903
$603$	0	0
$604$	−10.0876	−0.410458
$605$	27.9269	1.13539
$606$	0	0
$607$	−3.91221	−0.158792	−0.0793958	−	0.996843i	$-0.525299\pi$
$607$	−3.91221	−0.158792	−0.0793958	+	0.996843i	$0.525299\pi$
$608$	−3.41958	−0.138682
$609$	0	0
$610$	−21.5762	−0.873594
$611$	−4.76873	−0.192922
$612$	0	0
$613$	−14.8015	−0.597829	−0.298914	−	0.954280i	$-0.596625\pi$
$613$	−14.8015	−0.597829	−0.298914	+	0.954280i	$0.596625\pi$
$614$	−14.2360	−0.574520
$615$	0	0
$616$	−0.502653	−0.0202525
$617$	−31.5714	−1.27102	−0.635509	−	0.772094i	$-0.719210\pi$
$617$	−31.5714	−1.27102	−0.635509	+	0.772094i	$0.719210\pi$
$618$	0	0
$619$	−16.6962	−0.671076	−0.335538	−	0.942027i	$-0.608918\pi$
$619$	−16.6962	−0.671076	−0.335538	+	0.942027i	$0.608918\pi$
$620$	16.2626	0.653121
$621$	0	0
$622$	−6.65202	−0.266722
$623$	15.9103	0.637433
$624$	0	0
$625$	−30.7708	−1.23083
$626$	−23.9290	−0.956397
$627$	0	0
$628$	−9.88620	−0.394502
$629$	−0.698206	−0.0278393
$630$	0	0
$631$	−28.9397	−1.15207	−0.576035	−	0.817425i	$-0.695401\pi$
$631$	−28.9397	−1.15207	−0.576035	+	0.817425i	$0.695401\pi$
$632$	−9.53232	−0.379175
$633$	0	0
$634$	−13.7721	−0.546962
$635$	16.6334	0.660075
$636$	0	0
$637$	34.2535	1.35717
$638$	4.42243	0.175086
$639$	0	0
$640$	2.60917	0.103137
$641$	−16.2745	−0.642804	−0.321402	−	0.946943i	$-0.604154\pi$
$641$	−16.2745	−0.642804	−0.321402	+	0.946943i	$0.604154\pi$
$642$	0	0
$643$	−0.663338	−0.0261595	−0.0130798	−	0.999914i	$-0.504164\pi$
$643$	−0.663338	−0.0261595	−0.0130798	+	0.999914i	$0.504164\pi$
$644$	−1.67853	−0.0661433
$645$	0	0
$646$	−10.4894	−0.412698
$647$	−16.1211	−0.633786	−0.316893	−	0.948461i	$-0.602640\pi$
$647$	−16.1211	−0.633786	−0.316893	+	0.948461i	$0.602640\pi$
$648$	0	0
$649$	0.779449	0.0305961
$650$	10.0716	0.395039
$651$	0	0
$652$	6.27758	0.245849
$653$	−19.8926	−0.778458	−0.389229	−	0.921141i	$-0.627259\pi$
$653$	−19.8926	−0.778458	−0.389229	+	0.921141i	$0.627259\pi$
$654$	0	0
$655$	−0.360976	−0.0141045
$656$	5.22409	0.203967
$657$	0	0
$658$	0.789943	0.0307952
$659$	−10.1162	−0.394071	−0.197035	−	0.980396i	$-0.563131\pi$
$659$	−10.1162	−0.394071	−0.197035	+	0.980396i	$0.563131\pi$
$660$	0	0
$661$	11.1467	0.433558	0.216779	−	0.976221i	$-0.430445\pi$
$661$	11.1467	0.433558	0.216779	+	0.976221i	$0.430445\pi$
$662$	16.1954	0.629451
$663$	0	0
$664$	2.48928	0.0966028
$665$	8.23415	0.319307
$666$	0	0
$667$	14.7680	0.571818
$668$	23.6519	0.915118
$669$	0	0
$670$	23.2338	0.897600
$671$	4.50398	0.173874
$672$	0	0
$673$	−35.9187	−1.38456	−0.692282	−	0.721627i	$-0.743395\pi$
$673$	−35.9187	−1.38456	−0.692282	+	0.721627i	$0.743395\pi$
$674$	22.3172	0.859626
$675$	0	0
$676$	18.0384	0.693785
$677$	23.3124	0.895968	0.447984	−	0.894042i	$-0.352142\pi$
$677$	23.3124	0.895968	0.447984	+	0.894042i	$0.352142\pi$
$678$	0	0
$679$	−2.00870	−0.0770868
$680$	8.00349	0.306920
$681$	0	0
$682$	−3.39479	−0.129993
$683$	10.3232	0.395006	0.197503	−	0.980302i	$-0.436717\pi$
$683$	10.3232	0.395006	0.197503	+	0.980302i	$0.436717\pi$
$684$	0	0
$685$	40.5546	1.54951
$686$	−12.1342	−0.463287
$687$	0	0
$688$	−7.81375	−0.297896
$689$	−32.0333	−1.22037
$690$	0	0
$691$	23.3001	0.886377	0.443189	−	0.896428i	$-0.353847\pi$
$691$	23.3001	0.886377	0.443189	+	0.896428i	$0.353847\pi$
$692$	−4.21154	−0.160099
$693$	0	0
$694$	34.3276	1.30306
$695$	42.2755	1.60360
$696$	0	0
$697$	16.0246	0.606975
$698$	21.6118	0.818017
$699$	0	0
$700$	−1.66836	−0.0630582
$701$	−45.6250	−1.72323	−0.861616	−	0.507561i	$-0.830547\pi$
$701$	−45.6250	−1.72323	−0.861616	+	0.507561i	$0.830547\pi$
$702$	0	0
$703$	−0.778359	−0.0293564
$704$	−0.544660	−0.0205276
$705$	0	0
$706$	34.6860	1.30543
$707$	−13.8130	−0.519491
$708$	0	0
$709$	−4.87235	−0.182985	−0.0914925	−	0.995806i	$-0.529164\pi$
$709$	−4.87235	−0.182985	−0.0914925	+	0.995806i	$0.529164\pi$
$710$	24.2144	0.908749
$711$	0	0
$712$	17.2399	0.646094
$713$	−11.3363	−0.424549
$714$	0	0
$715$	−7.91732	−0.296091
$716$	−21.6329	−0.808458
$717$	0	0
$718$	28.4637	1.06226
$719$	−7.96672	−0.297108	−0.148554	−	0.988904i	$-0.547462\pi$
$719$	−7.96672	−0.297108	−0.148554	+	0.988904i	$0.547462\pi$
$720$	0	0
$721$	−16.6077	−0.618503
$722$	7.30647	0.271919
$723$	0	0
$724$	22.2885	0.828344
$725$	14.6785	0.545147
$726$	0	0
$727$	−27.7202	−1.02809	−0.514043	−	0.857764i	$-0.671853\pi$
$727$	−27.7202	−1.02809	−0.514043	+	0.857764i	$0.671853\pi$
$728$	−5.14153	−0.190558
$729$	0	0
$730$	28.6020	1.05861
$731$	−23.9682	−0.886496
$732$	0	0
$733$	−6.86671	−0.253628	−0.126814	−	0.991927i	$-0.540475\pi$
$733$	−6.86671	−0.253628	−0.126814	+	0.991927i	$0.540475\pi$
$734$	15.7763	0.582313
$735$	0	0
$736$	−1.81880	−0.0670420
$737$	−4.85001	−0.178653
$738$	0	0
$739$	22.7806	0.837996	0.418998	−	0.907987i	$-0.362381\pi$
$739$	22.7806	0.837996	0.418998	+	0.907987i	$0.362381\pi$
$740$	0.593896	0.0218320
$741$	0	0
$742$	5.30634	0.194802
$743$	25.2485	0.926278	0.463139	−	0.886286i	$-0.346723\pi$
$743$	25.2485	0.926278	0.463139	+	0.886286i	$0.346723\pi$
$744$	0	0
$745$	2.60917	0.0955927
$746$	26.4504	0.968417
$747$	0	0
$748$	−1.67071	−0.0610873
$749$	−15.6724	−0.572656
$750$	0	0
$751$	−39.2865	−1.43358	−0.716792	−	0.697287i	$-0.754390\pi$
$751$	−39.2865	−1.43358	−0.716792	+	0.697287i	$0.754390\pi$
$752$	0.855959	0.0312136
$753$	0	0
$754$	45.2360	1.64740
$755$	26.3203	0.957893
$756$	0	0
$757$	−3.17944	−0.115559	−0.0577793	−	0.998329i	$-0.518402\pi$
$757$	−3.17944	−0.115559	−0.0577793	+	0.998329i	$0.518402\pi$
$758$	21.0878	0.765942
$759$	0	0
$760$	8.92228	0.323645
$761$	−11.2489	−0.407771	−0.203886	−	0.978995i	$-0.565357\pi$
$761$	−11.2489	−0.407771	−0.203886	+	0.978995i	$0.565357\pi$
$762$	0	0
$763$	−3.56939	−0.129221
$764$	−17.5751	−0.635844
$765$	0	0
$766$	−18.5574	−0.670505
$767$	7.97282	0.287882
$768$	0	0
$769$	−42.8806	−1.54631	−0.773157	−	0.634215i	$-0.781324\pi$
$769$	−42.8806	−1.54631	−0.773157	+	0.634215i	$0.781324\pi$
$770$	1.31151	0.0472635
$771$	0	0
$772$	−7.86631	−0.283115
$773$	−35.4721	−1.27584	−0.637921	−	0.770102i	$-0.720205\pi$
$773$	−35.4721	−1.27584	−0.637921	+	0.770102i	$0.720205\pi$
$774$	0	0
$775$	−11.2677	−0.404747
$776$	−2.17657	−0.0781342
$777$	0	0
$778$	−26.2971	−0.942797
$779$	17.8642	0.640052
$780$	0	0
$781$	−5.05470	−0.180872
$782$	−5.57907	−0.199507
$783$	0	0
$784$	−6.14830	−0.219582
$785$	25.7948	0.920656
$786$	0	0
$787$	19.4675	0.693943	0.346971	−	0.937876i	$-0.387210\pi$
$787$	19.4675	0.693943	0.346971	+	0.937876i	$0.387210\pi$
$788$	−4.42276	−0.157554
$789$	0	0
$790$	24.8715	0.884888
$791$	7.69037	0.273438
$792$	0	0
$793$	46.0703	1.63600
$794$	−2.42889	−0.0861981
$795$	0	0
$796$	−25.5079	−0.904105
$797$	−19.2389	−0.681478	−0.340739	−	0.940158i	$-0.610677\pi$
$797$	−19.2389	−0.681478	−0.340739	+	0.940158i	$0.610677\pi$
$798$	0	0
$799$	2.62560	0.0928872
$800$	−1.80779	−0.0639149
$801$	0	0
$802$	5.83729	0.206122
$803$	−5.97062	−0.210698
$804$	0	0
$805$	4.37957	0.154360
$806$	−34.7245	−1.22312
$807$	0	0
$808$	−14.9673	−0.526549
$809$	−38.3793	−1.34935	−0.674673	−	0.738117i	$-0.735715\pi$
$809$	−38.3793	−1.34935	−0.674673	+	0.738117i	$0.735715\pi$
$810$	0	0
$811$	32.8630	1.15397	0.576987	−	0.816753i	$-0.304228\pi$
$811$	32.8630	1.15397	0.576987	+	0.816753i	$0.304228\pi$
$812$	−7.49338	−0.262966
$813$	0	0
$814$	−0.123975	−0.00434531
$815$	−16.3793	−0.573742
$816$	0	0
$817$	−26.7197	−0.934805
$818$	−38.0056	−1.32883
$819$	0	0
$820$	−13.6306	−0.476000
$821$	9.81222	0.342449	0.171224	−	0.985232i	$-0.445228\pi$
$821$	9.81222	0.342449	0.171224	+	0.985232i	$0.445228\pi$
$822$	0	0
$823$	41.2373	1.43744	0.718721	−	0.695298i	$-0.244728\pi$
$823$	41.2373	1.43744	0.718721	+	0.695298i	$0.244728\pi$
$824$	−17.9956	−0.626907
$825$	0	0
$826$	−1.32070	−0.0459531
$827$	44.8213	1.55859	0.779295	−	0.626658i	$-0.215578\pi$
$827$	44.8213	1.55859	0.779295	+	0.626658i	$0.215578\pi$
$828$	0	0
$829$	−0.0227422	−0.000789868 0	−0.000394934	−	1.00000i	$-0.500126\pi$
$829$	−0.0227422	−0.000789868 0	−0.000394934		1.00000i	$0.500126\pi$
$830$	−6.49496	−0.225443
$831$	0	0
$832$	−5.57121	−0.193147
$833$	−18.8596	−0.653445
$834$	0	0
$835$	−61.7118	−2.13563
$836$	−1.86251	−0.0644162
$837$	0	0
$838$	−15.3523	−0.530336
$839$	36.7304	1.26807	0.634037	−	0.773303i	$-0.281397\pi$
$839$	36.7304	1.26807	0.634037	+	0.773303i	$0.281397\pi$
$840$	0	0
$841$	36.9280	1.27338
$842$	−36.8285	−1.26919
$843$	0	0
$844$	−6.52937	−0.224750
$845$	−47.0653	−1.61910
$846$	0	0
$847$	9.87785	0.339407
$848$	5.74979	0.197449
$849$	0	0
$850$	−5.54528	−0.190202
$851$	−0.413993	−0.0141915
$852$	0	0
$853$	38.3155	1.31190	0.655949	−	0.754805i	$-0.272268\pi$
$853$	38.3155	1.31190	0.655949	+	0.754805i	$0.272268\pi$
$854$	−7.63158	−0.261147
$855$	0	0
$856$	−16.9821	−0.580437
$857$	42.0149	1.43520	0.717601	−	0.696454i	$-0.245240\pi$
$857$	42.0149	1.43520	0.717601	+	0.696454i	$0.245240\pi$
$858$	0	0
$859$	−44.5252	−1.51918	−0.759590	−	0.650402i	$-0.774600\pi$
$859$	−44.5252	−1.51918	−0.759590	+	0.650402i	$0.774600\pi$
$860$	20.3874	0.695205
$861$	0	0
$862$	1.98851	0.0677288
$863$	−19.7897	−0.673651	−0.336825	−	0.941567i	$-0.609353\pi$
$863$	−19.7897	−0.673651	−0.336825	+	0.941567i	$0.609353\pi$
$864$	0	0
$865$	10.9886	0.373625
$866$	14.1948	0.482359
$867$	0	0
$868$	5.75214	0.195240
$869$	−5.19188	−0.176122
$870$	0	0
$871$	−49.6098	−1.68096
$872$	−3.86769	−0.130976
$873$	0	0
$874$	−6.21955	−0.210379
$875$	−7.68666	−0.259857
$876$	0	0
$877$	36.2479	1.22401	0.612003	−	0.790856i	$-0.290364\pi$
$877$	36.2479	1.22401	0.612003	+	0.790856i	$0.290364\pi$
$878$	35.4726	1.19714
$879$	0	0
$880$	1.42111	0.0479057
$881$	−26.1698	−0.881685	−0.440842	−	0.897585i	$-0.645320\pi$
$881$	−26.1698	−0.881685	−0.440842	+	0.897585i	$0.645320\pi$
$882$	0	0
$883$	49.6246	1.67000	0.835001	−	0.550249i	$-0.185467\pi$
$883$	49.6246	1.67000	0.835001	+	0.550249i	$0.185467\pi$
$884$	−17.0894	−0.574778
$885$	0	0
$886$	2.95745	0.0993575
$887$	−31.9737	−1.07357	−0.536785	−	0.843719i	$-0.680361\pi$
$887$	−31.9737	−1.07357	−0.536785	+	0.843719i	$0.680361\pi$
$888$	0	0
$889$	5.88329	0.197319
$890$	−44.9820	−1.50780
$891$	0	0
$892$	3.38817	0.113444
$893$	2.92702	0.0979490
$894$	0	0
$895$	56.4439	1.88671
$896$	0.922875	0.0308311
$897$	0	0
$898$	9.25264	0.308765
$899$	−50.6083	−1.68788
$900$	0	0
$901$	17.6371	0.587578
$902$	2.84536	0.0947400
$903$	0	0
$904$	8.33306	0.277153
$905$	−58.1545	−1.93312
$906$	0	0
$907$	−6.27851	−0.208474	−0.104237	−	0.994552i	$-0.533240\pi$
$907$	−6.27851	−0.208474	−0.104237	+	0.994552i	$0.533240\pi$
$908$	16.2110	0.537981
$909$	0	0
$910$	13.4152	0.444708
$911$	47.4189	1.57106	0.785530	−	0.618824i	$-0.212391\pi$
$911$	47.4189	1.57106	0.785530	+	0.618824i	$0.212391\pi$
$912$	0	0
$913$	1.35581	0.0448708
$914$	42.0109	1.38960
$915$	0	0
$916$	−1.12778	−0.0372628
$917$	−0.127679	−0.00421632
$918$	0	0
$919$	9.13197	0.301236	0.150618	−	0.988592i	$-0.451874\pi$
$919$	9.13197	0.301236	0.150618	+	0.988592i	$0.451874\pi$
$920$	4.74558	0.156457
$921$	0	0
$922$	−12.0399	−0.396512
$923$	−51.7035	−1.70184
$924$	0	0
$925$	−0.411486	−0.0135296
$926$	4.77301	0.156851
$927$	0	0
$928$	−8.11961	−0.266539
$929$	10.4328	0.342288	0.171144	−	0.985246i	$-0.445254\pi$
$929$	10.4328	0.342288	0.171144	+	0.985246i	$0.445254\pi$
$930$	0	0
$931$	−21.0246	−0.689054
$932$	12.7193	0.416635
$933$	0	0
$934$	14.3296	0.468877
$935$	4.35918	0.142560
$936$	0	0
$937$	35.1160	1.14719	0.573596	−	0.819139i	$-0.305548\pi$
$937$	35.1160	1.14719	0.573596	+	0.819139i	$0.305548\pi$
$938$	8.21789	0.268324
$939$	0	0
$940$	−2.23335	−0.0728437
$941$	12.3960	0.404098	0.202049	−	0.979375i	$-0.435240\pi$
$941$	12.3960	0.404098	0.202049	+	0.979375i	$0.435240\pi$
$942$	0	0
$943$	9.50160	0.309415
$944$	−1.43107	−0.0465775
$945$	0	0
$946$	−4.25584	−0.138369
$947$	24.9520	0.810830	0.405415	−	0.914133i	$-0.367127\pi$
$947$	24.9520	0.810830	0.405415	+	0.914133i	$0.367127\pi$
$948$	0	0
$949$	−61.0722	−1.98249
$950$	−6.18187	−0.200567
$951$	0	0
$952$	2.83087	0.0917489
$953$	−10.2568	−0.332251	−0.166125	−	0.986105i	$-0.553126\pi$
$953$	−10.2568	−0.332251	−0.166125	+	0.986105i	$0.553126\pi$
$954$	0	0
$955$	45.8564	1.48388
$956$	−14.9893	−0.484789
$957$	0	0
$958$	−14.1676	−0.457734
$959$	14.3443	0.463202
$960$	0	0
$961$	7.84844	0.253176
$962$	−1.26811	−0.0408855
$963$	0	0
$964$	19.7096	0.634804
$965$	20.5246	0.660709
$966$	0	0
$967$	−14.9579	−0.481012	−0.240506	−	0.970648i	$-0.577313\pi$
$967$	−14.9579	−0.481012	−0.240506	+	0.970648i	$0.577313\pi$
$968$	10.7033	0.344019
$969$	0	0
$970$	5.67904	0.182343
$971$	15.3686	0.493201	0.246600	−	0.969117i	$-0.420686\pi$
$971$	15.3686	0.493201	0.246600	+	0.969117i	$0.420686\pi$
$972$	0	0
$973$	14.9530	0.479372
$974$	3.25466	0.104286
$975$	0	0
$976$	−8.26935	−0.264695
$977$	−28.9854	−0.927326	−0.463663	−	0.886012i	$-0.653465\pi$
$977$	−28.9854	−0.927326	−0.463663	+	0.886012i	$0.653465\pi$
$978$	0	0
$979$	9.38991	0.300103
$980$	16.0420	0.512442
$981$	0	0
$982$	−8.44806	−0.269589
$983$	−39.3208	−1.25414	−0.627070	−	0.778963i	$-0.715746\pi$
$983$	−39.3208	−1.25414	−0.627070	+	0.778963i	$0.715746\pi$
$984$	0	0
$985$	11.5397	0.367687
$986$	−24.9064	−0.793182
$987$	0	0
$988$	−19.0512	−0.606100
$989$	−14.2117	−0.451905
$990$	0	0
$991$	46.0270	1.46210	0.731048	−	0.682326i	$-0.239032\pi$
$991$	46.0270	1.46210	0.731048	+	0.682326i	$0.239032\pi$
$992$	6.23285	0.197893
$993$	0	0
$994$	8.56472	0.271656
$995$	66.5546	2.10992
$996$	0	0
$997$	−43.1324	−1.36602	−0.683008	−	0.730411i	$-0.739329\pi$
$997$	−43.1324	−1.36602	−0.683008	+	0.730411i	$0.739329\pi$
$998$	31.1209	0.985115
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.e.1.2	✓	8
3.2	odd	2		8046.2.a.f.1.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.e.1.2	✓	8	1.1	even	1	trivial
8046.2.a.f.1.7	yes	8	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.60917 q^{5} -0.922875 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.60917 q^{5} -0.922875 q^{7} -1.00000 q^{8} +2.60917 q^{10} -0.544660 q^{11} -5.57121 q^{13} +0.922875 q^{14} +1.00000 q^{16} +3.06744 q^{17} +3.41958 q^{19} -2.60917 q^{20} +0.544660 q^{22} +1.81880 q^{23} +1.80779 q^{25} +5.57121 q^{26} -0.922875 q^{28} +8.11961 q^{29} -6.23285 q^{31} -1.00000 q^{32} -3.06744 q^{34} +2.40794 q^{35} -0.227618 q^{37} -3.41958 q^{38} +2.60917 q^{40} +5.22409 q^{41} -7.81375 q^{43} -0.544660 q^{44} -1.81880 q^{46} +0.855959 q^{47} -6.14830 q^{49} -1.80779 q^{50} -5.57121 q^{52} +5.74979 q^{53} +1.42111 q^{55} +0.922875 q^{56} -8.11961 q^{58} -1.43107 q^{59} -8.26935 q^{61} +6.23285 q^{62} +1.00000 q^{64} +14.5363 q^{65} +8.90466 q^{67} +3.06744 q^{68} -2.40794 q^{70} +9.28047 q^{71} +10.9621 q^{73} +0.227618 q^{74} +3.41958 q^{76} +0.502653 q^{77} +9.53232 q^{79} -2.60917 q^{80} -5.22409 q^{82} -2.48928 q^{83} -8.00349 q^{85} +7.81375 q^{86} +0.544660 q^{88} -17.2399 q^{89} +5.14153 q^{91} +1.81880 q^{92} -0.855959 q^{94} -8.92228 q^{95} +2.17657 q^{97} +6.14830 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} - 5 q^{7} - 8 q^{8}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^4 - 5 * q^7 - 8 * q^8	\( 8 q - 8 q^{2} + 8 q^{4} - 5 q^{7} - 8 q^{8} + 6 q^{11} - 8 q^{13} + 5 q^{14} + 8 q^{16} - 5 q^{17} - 14 q^{19} - 6 q^{22} + 21 q^{23} - 6 q^{25} + 8 q^{26} - 5 q^{28} + 3 q^{29} - 4 q^{31} - 8 q^{32} + 5 q^{34} - 2 q^{35} - 3 q^{37} + 14 q^{38} + 7 q^{41} - 12 q^{43} + 6 q^{44} - 21 q^{46} + 25 q^{47} - 7 q^{49} + 6 q^{50} - 8 q^{52} + 3 q^{53} - 9 q^{55} + 5 q^{56} - 3 q^{58} + 2 q^{59} - 17 q^{61} + 4 q^{62} + 8 q^{64} + 32 q^{65} - 14 q^{67} - 5 q^{68} + 2 q^{70} + 7 q^{71} - 10 q^{73} + 3 q^{74} - 14 q^{76} + 12 q^{77} - 33 q^{79} - 7 q^{82} + 13 q^{83} - 33 q^{85} + 12 q^{86} - 6 q^{88} - 22 q^{89} - 22 q^{91} + 21 q^{92} - 25 q^{94} - 14 q^{95} - 11 q^{97} + 7 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^4 - 5 * q^7 - 8 * q^8 + 6 * q^11 - 8 * q^13 + 5 * q^14 + 8 * q^16 - 5 * q^17 - 14 * q^19 - 6 * q^22 + 21 * q^23 - 6 * q^25 + 8 * q^26 - 5 * q^28 + 3 * q^29 - 4 * q^31 - 8 * q^32 + 5 * q^34 - 2 * q^35 - 3 * q^37 + 14 * q^38 + 7 * q^41 - 12 * q^43 + 6 * q^44 - 21 * q^46 + 25 * q^47 - 7 * q^49 + 6 * q^50 - 8 * q^52 + 3 * q^53 - 9 * q^55 + 5 * q^56 - 3 * q^58 + 2 * q^59 - 17 * q^61 + 4 * q^62 + 8 * q^64 + 32 * q^65 - 14 * q^67 - 5 * q^68 + 2 * q^70 + 7 * q^71 - 10 * q^73 + 3 * q^74 - 14 * q^76 + 12 * q^77 - 33 * q^79 - 7 * q^82 + 13 * q^83 - 33 * q^85 + 12 * q^86 - 6 * q^88 - 22 * q^89 - 22 * q^91 + 21 * q^92 - 25 * q^94 - 14 * q^95 - 11 * q^97 + 7 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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