LMFDB - Embedded newform 409.2.a.b.1.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [409,2,Mod(1,409)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(409, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("409.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 409 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	409.a (trivial)

Newform invariants

comment:select newform

sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$3.26588144267$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 5 x^{19} - 19 x^{18} + 126 x^{17} + 100 x^{16} - 1283 x^{15} + 247 x^{14} + 6767 x^{13} + \cdots - 4 $ x^20 - 5x^19 - 19x^18 + 126x^17 + 100x^16 - 1283x^15 + 247x^14 + 6767x^13 - 4554x^12 - 19689x^11 + 18771x^10 + 31011x^9 - 35515x^8 - 23548x^7 + 31466x^6 + 5354x^5 - 10552x^4 + 1129x^3 + 523x^2 - 54*x - 4
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			1.13
Root			$1.30315$ of defining polynomial
Character	$\chi$	$=$	409.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.30315 q^{2} +3.24667 q^{3} -0.301807 q^{4} -0.0792979 q^{5} +4.23089 q^{6} -0.257170 q^{7} -2.99959 q^{8} +7.54085 q^{9} -0.103337 q^{10} -1.41842 q^{11} -0.979866 q^{12} +2.01129 q^{13} -0.335130 q^{14} -0.257454 q^{15} -3.30530 q^{16} -0.527634 q^{17} +9.82683 q^{18} -2.79846 q^{19} +0.0239327 q^{20} -0.834944 q^{21} -1.84841 q^{22} +4.39762 q^{23} -9.73868 q^{24} -4.99371 q^{25} +2.62100 q^{26} +14.7426 q^{27} +0.0776155 q^{28} -5.67996 q^{29} -0.335501 q^{30} -5.41119 q^{31} +1.69189 q^{32} -4.60514 q^{33} -0.687585 q^{34} +0.0203930 q^{35} -2.27588 q^{36} +7.42947 q^{37} -3.64680 q^{38} +6.52998 q^{39} +0.237862 q^{40} -8.44997 q^{41} -1.08806 q^{42} -11.0307 q^{43} +0.428089 q^{44} -0.597974 q^{45} +5.73075 q^{46} +8.27772 q^{47} -10.7312 q^{48} -6.93386 q^{49} -6.50754 q^{50} -1.71305 q^{51} -0.607020 q^{52} +0.238203 q^{53} +19.2118 q^{54} +0.112478 q^{55} +0.771405 q^{56} -9.08566 q^{57} -7.40182 q^{58} +5.45179 q^{59} +0.0777013 q^{60} -0.709644 q^{61} -7.05157 q^{62} -1.93928 q^{63} +8.81539 q^{64} -0.159491 q^{65} -6.00118 q^{66} +10.4121 q^{67} +0.159244 q^{68} +14.2776 q^{69} +0.0265751 q^{70} +1.66186 q^{71} -22.6195 q^{72} +9.78378 q^{73} +9.68170 q^{74} -16.2129 q^{75} +0.844594 q^{76} +0.364775 q^{77} +8.50952 q^{78} +5.81360 q^{79} +0.262103 q^{80} +25.2418 q^{81} -11.0116 q^{82} -2.44766 q^{83} +0.251992 q^{84} +0.0418403 q^{85} -14.3747 q^{86} -18.4409 q^{87} +4.25469 q^{88} +0.484108 q^{89} -0.779248 q^{90} -0.517242 q^{91} -1.32723 q^{92} -17.5683 q^{93} +10.7871 q^{94} +0.221912 q^{95} +5.49302 q^{96} +11.4472 q^{97} -9.03585 q^{98} -10.6961 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 5 q^{2} + q^{3} + 23 q^{4} + 8 q^{5} - 4 q^{6} + 5 q^{7} + 12 q^{8} + 27 q^{9} - 5 q^{10} + 30 q^{11} - 5 q^{12} - 7 q^{13} + 17 q^{14} + 26 q^{15} + 29 q^{16} + 6 q^{17} - 4 q^{18} + q^{19} + 14 q^{20}+ \cdots + 43 q^{99}+O(q^{100}) $ 20 * q + 5 * q^2 + q^3 + 23 * q^4 + 8 * q^5 - 4 * q^6 + 5 * q^7 + 12 * q^8 + 27 * q^9 - 5 * q^10 + 30 * q^11 - 5 * q^12 - 7 * q^13 + 17 * q^14 + 26 * q^15 + 29 * q^16 + 6 * q^17 - 4 * q^18 + q^19 + 14 * q^20 - 3 * q^21 + q^22 + 37 * q^23 - 17 * q^24 + 14 * q^25 + 11 * q^26 - 5 * q^27 - 6 * q^28 + 12 * q^29 - 6 * q^30 + 10 * q^31 + 24 * q^32 - 22 * q^33 - 11 * q^34 + 12 * q^35 + 21 * q^36 - 4 * q^37 - 7 * q^38 + 31 * q^39 - 34 * q^40 - 7 * q^41 - 35 * q^42 - q^43 + 58 * q^44 - 10 * q^45 - 22 * q^46 + 51 * q^47 - 58 * q^48 + 5 * q^49 + q^50 + 6 * q^51 - 44 * q^52 + 10 * q^53 - 49 * q^54 - 5 * q^55 + 31 * q^56 - 30 * q^57 - 22 * q^58 + 43 * q^59 - 26 * q^60 - 21 * q^61 - 26 * q^62 + 8 * q^63 - 4 * q^64 - 10 * q^65 - 37 * q^66 + 4 * q^67 - 16 * q^68 - 19 * q^69 - 66 * q^70 + 97 * q^71 - 43 * q^72 - 36 * q^73 + 17 * q^74 - 19 * q^75 - 43 * q^76 - 20 * q^77 - 71 * q^78 + 16 * q^79 - 23 * q^80 + 16 * q^81 - 67 * q^82 + 28 * q^83 - 73 * q^84 - 30 * q^85 + 37 * q^86 - 14 * q^87 - 48 * q^88 + 2 * q^89 - 121 * q^90 - 25 * q^91 + 14 * q^92 - 27 * q^93 - 60 * q^94 + 61 * q^95 - 88 * q^96 - 36 * q^97 - 54 * q^98 + 43 * q^99

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 $

$2$	1.30315	0.921464	0.460732	−	0.887539i	$-0.347587\pi$
$2$	1.30315	0.921464	0.460732	+	0.887539i	$0.347587\pi$
$3$	3.24667	1.87446	0.937232	−	0.348706i	$-0.113379\pi$
$3$	3.24667	1.87446	0.937232	+	0.348706i	$0.113379\pi$
$4$	−0.301807	−0.150903
$5$	−0.0792979	−0.0354631	−0.0177316	−	0.999843i	$-0.505644\pi$
$5$	−0.0792979	−0.0354631	−0.0177316	+	0.999843i	$0.505644\pi$
$6$	4.23089	1.72725
$7$	−0.257170	−0.0972010	−0.0486005	−	0.998818i	$-0.515476\pi$
$7$	−0.257170	−0.0972010	−0.0486005	+	0.998818i	$0.515476\pi$
$8$	−2.99959	−1.06052
$9$	7.54085	2.51362
$10$	−0.103337	−0.0326780
$11$	−1.41842	−0.427670	−0.213835	−	0.976870i	$-0.568596\pi$
$11$	−1.41842	−0.427670	−0.213835	+	0.976870i	$0.568596\pi$
$12$	−0.979866	−0.282863
$13$	2.01129	0.557830	0.278915	−	0.960316i	$-0.410025\pi$
$13$	2.01129	0.557830	0.278915	+	0.960316i	$0.410025\pi$
$14$	−0.335130	−0.0895673
$15$	−0.257454	−0.0664743
$16$	−3.30530	−0.826325
$17$	−0.527634	−0.127970	−0.0639851	−	0.997951i	$-0.520381\pi$
$17$	−0.527634	−0.127970	−0.0639851	+	0.997951i	$0.520381\pi$
$18$	9.82683	2.31621
$19$	−2.79846	−0.642010	−0.321005	−	0.947077i	$-0.604021\pi$
$19$	−2.79846	−0.642010	−0.321005	+	0.947077i	$0.604021\pi$
$20$	0.0239327	0.00535150
$21$	−0.834944	−0.182200
$22$	−1.84841	−0.394083
$23$	4.39762	0.916967	0.458483	−	0.888703i	$-0.348393\pi$
$23$	4.39762	0.916967	0.458483	+	0.888703i	$0.348393\pi$
$24$	−9.73868	−1.98790
$25$	−4.99371	−0.998742
$26$	2.62100	0.514021
$27$	14.7426	2.83722
$28$	0.0776155	0.0146680
$29$	−5.67996	−1.05474	−0.527371	−	0.849635i	$-0.676822\pi$
$29$	−5.67996	−1.05474	−0.527371	+	0.849635i	$0.676822\pi$
$30$	−0.335501	−0.0612537
$31$	−5.41119	−0.971878	−0.485939	−	0.873993i	$-0.661522\pi$
$31$	−5.41119	−0.971878	−0.485939	+	0.873993i	$0.661522\pi$
$32$	1.69189	0.299088
$33$	−4.60514	−0.801653
$34$	−0.687585	−0.117920
$35$	0.0203930	0.00344705
$36$	−2.27588	−0.379313
$37$	7.42947	1.22140	0.610699	−	0.791863i	$-0.290889\pi$
$37$	7.42947	1.22140	0.610699	+	0.791863i	$0.290889\pi$
$38$	−3.64680	−0.591590
$39$	6.52998	1.04563
$40$	0.237862	0.0376092
$41$	−8.44997	−1.31966	−0.659832	−	0.751413i	$-0.729372\pi$
$41$	−8.44997	−1.31966	−0.659832	+	0.751413i	$0.729372\pi$
$42$	−1.08806	−0.167891
$43$	−11.0307	−1.68217	−0.841085	−	0.540903i	$-0.818083\pi$
$43$	−11.0307	−1.68217	−0.841085	+	0.540903i	$0.818083\pi$
$44$	0.428089	0.0645369
$45$	−0.597974	−0.0891406
$46$	5.73075	0.844952
$47$	8.27772	1.20743	0.603715	−	0.797200i	$-0.293686\pi$
$47$	8.27772	1.20743	0.603715	+	0.797200i	$0.293686\pi$
$48$	−10.7312	−1.54892
$49$	−6.93386	−0.990552
$50$	−6.50754	−0.920306
$51$	−1.71305	−0.239875
$52$	−0.607020	−0.0841785
$53$	0.238203	0.0327197	0.0163599	−	0.999866i	$-0.494792\pi$
$53$	0.238203	0.0327197	0.0163599	+	0.999866i	$0.494792\pi$
$54$	19.2118	2.61439
$55$	0.112478	0.0151665
$56$	0.771405	0.103083
$57$	−9.08566	−1.20343
$58$	−7.40182	−0.971907
$59$	5.45179	0.709762	0.354881	−	0.934911i	$-0.384521\pi$
$59$	5.45179	0.709762	0.354881	+	0.934911i	$0.384521\pi$
$60$	0.0777013	0.0100312
$61$	−0.709644	−0.0908605	−0.0454303	−	0.998968i	$-0.514466\pi$
$61$	−0.709644	−0.0908605	−0.0454303	+	0.998968i	$0.514466\pi$
$62$	−7.05157	−0.895551
$63$	−1.93928	−0.244326
$64$	8.81539	1.10192
$65$	−0.159491	−0.0197824
$66$	−6.00118	−0.738694
$67$	10.4121	1.27204	0.636021	−	0.771671i	$-0.280579\pi$
$67$	10.4121	1.27204	0.636021	+	0.771671i	$0.280579\pi$
$68$	0.159244	0.0193111
$69$	14.2776	1.71882
$70$	0.0265751	0.00317633
$71$	1.66186	0.197226	0.0986132	−	0.995126i	$-0.468559\pi$
$71$	1.66186	0.197226	0.0986132	+	0.995126i	$0.468559\pi$
$72$	−22.6195	−2.66573
$73$	9.78378	1.14510	0.572552	−	0.819868i	$-0.305953\pi$
$73$	9.78378	1.14510	0.572552	+	0.819868i	$0.305953\pi$
$74$	9.68170	1.12547
$75$	−16.2129	−1.87211
$76$	0.844594	0.0968815
$77$	0.364775	0.0415700
$78$	8.50952	0.963513
$79$	5.81360	0.654081	0.327041	−	0.945010i	$-0.393949\pi$
$79$	5.81360	0.654081	0.327041	+	0.945010i	$0.393949\pi$
$80$	0.262103	0.0293041
$81$	25.2418	2.80465
$82$	−11.0116	−1.21602
$83$	−2.44766	−0.268665	−0.134333	−	0.990936i	$-0.542889\pi$
$83$	−2.44766	−0.268665	−0.134333	+	0.990936i	$0.542889\pi$
$84$	0.251992	0.0274946
$85$	0.0418403	0.00453822
$86$	−14.3747	−1.55006
$87$	−18.4409	−1.97708
$88$	4.25469	0.453551
$89$	0.484108	0.0513154	0.0256577	−	0.999671i	$-0.491832\pi$
$89$	0.484108	0.0513154	0.0256577	+	0.999671i	$0.491832\pi$
$90$	−0.779248	−0.0821399
$91$	−0.517242	−0.0542217
$92$	−1.32723	−0.138373
$93$	−17.5683	−1.82175
$94$	10.7871	1.11260
$95$	0.221912	0.0227677
$96$	5.49302	0.560629
$97$	11.4472	1.16229	0.581143	−	0.813802i	$-0.302606\pi$
$97$	11.4472	1.16229	0.581143	+	0.813802i	$0.302606\pi$
$98$	−9.03585	−0.912758
$99$	−10.6961	−1.07500
$100$	1.50714	0.150714
$101$	6.57355	0.654093	0.327046	−	0.945008i	$-0.393947\pi$
$101$	6.57355	0.654093	0.327046	+	0.945008i	$0.393947\pi$
$102$	−2.23236	−0.221037
$103$	−0.246723	−0.0243104	−0.0121552	−	0.999926i	$-0.503869\pi$
$103$	−0.246723	−0.0243104	−0.0121552	+	0.999926i	$0.503869\pi$
$104$	−6.03304	−0.591588
$105$	0.0662094	0.00646137
$106$	0.310414	0.0301501
$107$	9.62130	0.930126	0.465063	−	0.885278i	$-0.346032\pi$
$107$	9.62130	0.930126	0.465063	+	0.885278i	$0.346032\pi$
$108$	−4.44942	−0.428146
$109$	0.981224	0.0939842	0.0469921	−	0.998895i	$-0.485036\pi$
$109$	0.981224	0.0939842	0.0469921	+	0.998895i	$0.485036\pi$
$110$	0.146575	0.0139754
$111$	24.1210	2.28947
$112$	0.850023	0.0803196
$113$	9.78799	0.920777	0.460388	−	0.887718i	$-0.347710\pi$
$113$	9.78799	0.920777	0.460388	+	0.887718i	$0.347710\pi$
$114$	−11.8400	−1.10891
$115$	−0.348722	−0.0325185
$116$	1.71425	0.159164
$117$	15.1668	1.40217
$118$	7.10448	0.654021
$119$	0.135692	0.0124388
$120$	0.772257	0.0704971
$121$	−8.98808	−0.817098
$122$	−0.924770	−0.0837248
$123$	−27.4342	−2.47366
$124$	1.63313	0.146660
$125$	0.792481	0.0708816
$126$	−2.52716	−0.225138
$127$	−15.5930	−1.38365	−0.691827	−	0.722063i	$-0.743194\pi$
$127$	−15.5930	−1.38365	−0.691827	+	0.722063i	$0.743194\pi$
$128$	8.10396	0.716296
$129$	−35.8131	−3.15317
$130$	−0.207840	−0.0182288
$131$	−1.17813	−0.102933	−0.0514667	−	0.998675i	$-0.516390\pi$
$131$	−1.17813	−0.102933	−0.0514667	+	0.998675i	$0.516390\pi$
$132$	1.38986	0.120972
$133$	0.719679	0.0624041
$134$	13.5685	1.17214
$135$	−1.16906	−0.100617
$136$	1.58269	0.135714
$137$	−14.4017	−1.23042	−0.615210	−	0.788364i	$-0.710929\pi$
$137$	−14.4017	−1.23042	−0.615210	+	0.788364i	$0.710929\pi$
$138$	18.6058	1.58383
$139$	−20.3316	−1.72450	−0.862251	−	0.506482i	$-0.830946\pi$
$139$	−20.3316	−1.72450	−0.862251	+	0.506482i	$0.830946\pi$
$140$	−0.00615475	−0.000520172 0
$141$	26.8750	2.26328
$142$	2.16565	0.181737
$143$	−2.85285	−0.238567
$144$	−24.9248	−2.07706
$145$	0.450409	0.0374044
$146$	12.7497	1.05517
$147$	−22.5119	−1.85675
$148$	−2.24226	−0.184313
$149$	0.247293	0.0202590	0.0101295	−	0.999949i	$-0.496776\pi$
$149$	0.247293	0.0202590	0.0101295	+	0.999949i	$0.496776\pi$
$150$	−21.1278	−1.72508
$151$	7.64628	0.622246	0.311123	−	0.950370i	$-0.399295\pi$
$151$	7.64628	0.622246	0.311123	+	0.950370i	$0.399295\pi$
$152$	8.39424	0.680863
$153$	−3.97881	−0.321668
$154$	0.475356	0.0383053
$155$	0.429096	0.0344658
$156$	−1.97079	−0.157789
$157$	15.6300	1.24741	0.623705	−	0.781660i	$-0.285627\pi$
$157$	15.6300	1.24741	0.623705	+	0.781660i	$0.285627\pi$
$158$	7.57598	0.602713
$159$	0.773366	0.0613319
$160$	−0.134164	−0.0106066
$161$	−1.13093	−0.0891301
$162$	32.8938	2.58438
$163$	−19.4203	−1.52111	−0.760556	−	0.649273i	$-0.775073\pi$
$163$	−19.4203	−1.52111	−0.760556	+	0.649273i	$0.775073\pi$
$164$	2.55026	0.199142
$165$	0.365178	0.0284291
$166$	−3.18966	−0.247566
$167$	17.7956	1.37707	0.688534	−	0.725204i	$-0.258255\pi$
$167$	17.7956	1.37707	0.688534	+	0.725204i	$0.258255\pi$
$168$	2.50449	0.193226
$169$	−8.95473	−0.688825
$170$	0.0545241	0.00418181
$171$	−21.1027	−1.61377
$172$	3.32915	0.253845
$173$	12.6809	0.964112	0.482056	−	0.876140i	$-0.339890\pi$
$173$	12.6809	0.964112	0.482056	+	0.876140i	$0.339890\pi$
$174$	−24.0313	−1.82180
$175$	1.28423	0.0970788
$176$	4.68831	0.353395
$177$	17.7001	1.33042
$178$	0.630865	0.0472853
$179$	−14.8252	−1.10809	−0.554045	−	0.832486i	$-0.686917\pi$
$179$	−14.8252	−1.10809	−0.554045	+	0.832486i	$0.686917\pi$
$180$	0.180472	0.0134516
$181$	−23.6180	−1.75551	−0.877757	−	0.479106i	$-0.840961\pi$
$181$	−23.6180	−1.75551	−0.877757	+	0.479106i	$0.840961\pi$
$182$	−0.674042	−0.0499633
$183$	−2.30398	−0.170315
$184$	−13.1911	−0.972459
$185$	−0.589142	−0.0433146
$186$	−22.8941	−1.67868
$187$	0.748408	0.0547290
$188$	−2.49827	−0.182205
$189$	−3.79135	−0.275780
$190$	0.289184	0.0209796
$191$	16.4708	1.19178	0.595892	−	0.803064i	$-0.296798\pi$
$191$	16.4708	1.19178	0.595892	+	0.803064i	$0.296798\pi$
$192$	28.6206	2.06552
$193$	17.8342	1.28373	0.641867	−	0.766816i	$-0.278160\pi$
$193$	17.8342	1.28373	0.641867	+	0.766816i	$0.278160\pi$
$194$	14.9174	1.07100
$195$	−0.517814	−0.0370814
$196$	2.09269	0.149478
$197$	5.98149	0.426163	0.213082	−	0.977034i	$-0.431650\pi$
$197$	5.98149	0.426163	0.213082	+	0.977034i	$0.431650\pi$
$198$	−13.9386	−0.990573
$199$	27.7176	1.96485	0.982423	−	0.186669i	$-0.0597693\pi$
$199$	27.7176	1.96485	0.982423	+	0.186669i	$0.0597693\pi$
$200$	14.9791	1.05918
$201$	33.8047	2.38440
$202$	8.56630	0.602723
$203$	1.46071	0.102522
$204$	0.517011	0.0361980
$205$	0.670065	0.0467994
$206$	−0.321517	−0.0224011
$207$	33.1618	2.30490
$208$	−6.64790	−0.460949
$209$	3.96940	0.274569
$210$	0.0862806	0.00595393
$211$	5.37734	0.370191	0.185096	−	0.982721i	$-0.440741\pi$
$211$	5.37734	0.370191	0.185096	+	0.982721i	$0.440741\pi$
$212$	−0.0718913	−0.00493751
$213$	5.39550	0.369694
$214$	12.5380	0.857078
$215$	0.874714	0.0596550
$216$	−44.2218	−3.00892
$217$	1.39159	0.0944675
$218$	1.27868	0.0866031
$219$	31.7647	2.14646
$220$	−0.0339466	−0.00228868
$221$	−1.06122	−0.0713856
$222$	31.4332	2.10966
$223$	1.56551	0.104834	0.0524170	−	0.998625i	$-0.483307\pi$
$223$	1.56551	0.104834	0.0524170	+	0.998625i	$0.483307\pi$
$224$	−0.435104	−0.0290716
$225$	−37.6568	−2.51045
$226$	12.7552	0.848463
$227$	15.3517	1.01893	0.509464	−	0.860492i	$-0.329844\pi$
$227$	15.3517	1.01893	0.509464	+	0.860492i	$0.329844\pi$
$228$	2.74211	0.181601
$229$	−2.82658	−0.186786	−0.0933930	−	0.995629i	$-0.529771\pi$
$229$	−2.82658	−0.186786	−0.0933930	+	0.995629i	$0.529771\pi$
$230$	−0.454436	−0.0299647
$231$	1.18430	0.0779214
$232$	17.0376	1.11857
$233$	−19.3158	−1.26542	−0.632710	−	0.774389i	$-0.718058\pi$
$233$	−19.3158	−1.26542	−0.632710	+	0.774389i	$0.718058\pi$
$234$	19.7646	1.29205
$235$	−0.656406	−0.0428192
$236$	−1.64539	−0.107105
$237$	18.8748	1.22605
$238$	0.176826	0.0114619
$239$	−1.08613	−0.0702560	−0.0351280	−	0.999383i	$-0.511184\pi$
$239$	−1.08613	−0.0702560	−0.0351280	+	0.999383i	$0.511184\pi$
$240$	0.850963	0.0549294
$241$	−18.0997	−1.16591	−0.582953	−	0.812506i	$-0.698103\pi$
$241$	−18.0997	−1.16591	−0.582953	+	0.812506i	$0.698103\pi$
$242$	−11.7128	−0.752927
$243$	37.7239	2.41999
$244$	0.214175	0.0137112
$245$	0.549841	0.0351281
$246$	−35.7508	−2.27939
$247$	−5.62850	−0.358133
$248$	16.2314	1.03069
$249$	−7.94673	−0.503604
$250$	1.03272	0.0653149
$251$	20.1908	1.27443	0.637215	−	0.770686i	$-0.280086\pi$
$251$	20.1908	1.27443	0.637215	+	0.770686i	$0.280086\pi$
$252$	0.585287	0.0368696
$253$	−6.23768	−0.392160
$254$	−20.3200	−1.27499
$255$	0.135842	0.00850673
$256$	−7.07012	−0.441882
$257$	18.7568	1.17002	0.585009	−	0.811027i	$-0.301091\pi$
$257$	18.7568	1.17002	0.585009	+	0.811027i	$0.301091\pi$
$258$	−46.6697	−2.90553
$259$	−1.91063	−0.118721
$260$	0.0481354	0.00298523
$261$	−42.8317	−2.65122
$262$	−1.53527	−0.0948495
$263$	8.47761	0.522752	0.261376	−	0.965237i	$-0.415824\pi$
$263$	8.47761	0.522752	0.261376	+	0.965237i	$0.415824\pi$
$264$	13.8136	0.850166
$265$	−0.0188890	−0.00116034
$266$	0.937848	0.0575031
$267$	1.57174	0.0961889
$268$	−3.14245	−0.191956
$269$	−23.6023	−1.43906	−0.719528	−	0.694463i	$-0.755642\pi$
$269$	−23.6023	−1.43906	−0.719528	+	0.694463i	$0.755642\pi$
$270$	−1.52346	−0.0927146
$271$	5.92870	0.360143	0.180071	−	0.983654i	$-0.442367\pi$
$271$	5.92870	0.360143	0.180071	+	0.983654i	$0.442367\pi$
$272$	1.74399	0.105745
$273$	−1.67931	−0.101637
$274$	−18.7675	−1.13379
$275$	7.08319	0.427132
$276$	−4.30908	−0.259376
$277$	−28.7263	−1.72600	−0.862998	−	0.505208i	$-0.831416\pi$
$277$	−28.7263	−1.72600	−0.862998	+	0.505208i	$0.831416\pi$
$278$	−26.4950	−1.58907
$279$	−40.8049	−2.44293
$280$	−0.0611708	−0.00365565
$281$	2.68307	0.160058	0.0800291	−	0.996793i	$-0.474499\pi$
$281$	2.68307	0.160058	0.0800291	+	0.996793i	$0.474499\pi$
$282$	35.0221	2.08554
$283$	−17.7178	−1.05321	−0.526606	−	0.850110i	$-0.676536\pi$
$283$	−17.7178	−1.05321	−0.526606	+	0.850110i	$0.676536\pi$
$284$	−0.501560	−0.0297621
$285$	0.720474	0.0426772
$286$	−3.71769	−0.219831
$287$	2.17308	0.128273
$288$	12.7583	0.751791
$289$	−16.7216	−0.983624
$290$	0.586949	0.0344669
$291$	37.1652	2.17866
$292$	−2.95281	−0.172800
$293$	21.5122	1.25676	0.628379	−	0.777907i	$-0.283719\pi$
$293$	21.5122	1.25676	0.628379	+	0.777907i	$0.283719\pi$
$294$	−29.3364	−1.71093
$295$	−0.432316	−0.0251704
$296$	−22.2854	−1.29531
$297$	−20.9112	−1.21339
$298$	0.322259	0.0186680
$299$	8.84487	0.511512
$300$	4.89317	0.282507
$301$	2.83677	0.163509
$302$	9.96423	0.573377
$303$	21.3421	1.22607
$304$	9.24974	0.530509
$305$	0.0562733	0.00322220
$306$	−5.18498	−0.296405
$307$	13.0749	0.746225	0.373113	−	0.927786i	$-0.378290\pi$
$307$	13.0749	0.746225	0.373113	+	0.927786i	$0.378290\pi$
$308$	−0.110092	−0.00627305
$309$	−0.801028	−0.0455689
$310$	0.559175	0.0317590
$311$	−22.3534	−1.26755	−0.633773	−	0.773519i	$-0.718495\pi$
$311$	−22.3534	−1.26755	−0.633773	+	0.773519i	$0.718495\pi$
$312$	−19.5873	−1.10891
$313$	−24.9904	−1.41254	−0.706271	−	0.707942i	$-0.749624\pi$
$313$	−24.9904	−1.41254	−0.706271	+	0.707942i	$0.749624\pi$
$314$	20.3682	1.14944
$315$	0.153781	0.00866456
$316$	−1.75458	−0.0987031
$317$	31.5740	1.77337	0.886685	−	0.462374i	$-0.153002\pi$
$317$	31.5740	1.77337	0.886685	+	0.462374i	$0.153002\pi$
$318$	1.00781	0.0565152
$319$	8.05658	0.451082
$320$	−0.699042	−0.0390776
$321$	31.2372	1.74349
$322$	−1.47377	−0.0821302
$323$	1.47656	0.0821582
$324$	−7.61815	−0.423231
$325$	−10.0438	−0.557129
$326$	−25.3075	−1.40165
$327$	3.18571	0.176170
$328$	25.3465	1.39952
$329$	−2.12878	−0.117363
$330$	0.475881	0.0261964
$331$	−7.61834	−0.418742	−0.209371	−	0.977836i	$-0.567142\pi$
$331$	−7.61834	−0.418742	−0.209371	+	0.977836i	$0.567142\pi$
$332$	0.738720	0.0405425
$333$	56.0245	3.07012
$334$	23.1903	1.26892
$335$	−0.825660	−0.0451106
$336$	2.75974	0.150556
$337$	7.76074	0.422754	0.211377	−	0.977405i	$-0.432205\pi$
$337$	7.76074	0.422754	0.211377	+	0.977405i	$0.432205\pi$
$338$	−11.6693	−0.634728
$339$	31.7783	1.72596
$340$	−0.0126277	−0.000684833 0
$341$	7.67535	0.415643
$342$	−27.5000	−1.48703
$343$	3.58337	0.193484
$344$	33.0877	1.78397
$345$	−1.13218	−0.0609548
$346$	16.5251	0.888395
$347$	19.8266	1.06435	0.532173	−	0.846636i	$-0.321376\pi$
$347$	19.8266	1.06435	0.532173	+	0.846636i	$0.321376\pi$
$348$	5.56560	0.298347
$349$	−4.56114	−0.244152	−0.122076	−	0.992521i	$-0.538955\pi$
$349$	−4.56114	−0.244152	−0.122076	+	0.992521i	$0.538955\pi$
$350$	1.67354	0.0894546
$351$	29.6516	1.58269
$352$	−2.39982	−0.127911
$353$	−22.9993	−1.22413	−0.612064	−	0.790808i	$-0.709660\pi$
$353$	−22.9993	−1.22413	−0.612064	+	0.790808i	$0.709660\pi$
$354$	23.0659	1.22594
$355$	−0.131782	−0.00699427
$356$	−0.146107	−0.00774367
$357$	0.440545	0.0233161
$358$	−19.3195	−1.02107
$359$	−17.0473	−0.899724	−0.449862	−	0.893098i	$-0.648527\pi$
$359$	−17.0473	−0.899724	−0.449862	+	0.893098i	$0.648527\pi$
$360$	1.79368	0.0945351
$361$	−11.1686	−0.587823
$362$	−30.7778	−1.61764
$363$	−29.1813	−1.53162
$364$	0.156107	0.00818223
$365$	−0.775834	−0.0406090
$366$	−3.00242	−0.156939
$367$	−14.3441	−0.748754	−0.374377	−	0.927277i	$-0.622143\pi$
$367$	−14.3441	−0.748754	−0.374377	+	0.927277i	$0.622143\pi$
$368$	−14.5354	−0.757713
$369$	−63.7199	−3.31713
$370$	−0.767739	−0.0399128
$371$	−0.0612586	−0.00318039
$372$	5.30224	0.274908
$373$	−28.8717	−1.49492	−0.747461	−	0.664306i	$-0.768727\pi$
$373$	−28.8717	−1.49492	−0.747461	+	0.664306i	$0.768727\pi$
$374$	0.975286	0.0504309
$375$	2.57292	0.132865
$376$	−24.8298	−1.28050
$377$	−11.4240	−0.588367
$378$	−4.94069	−0.254122
$379$	24.6795	1.26770	0.633850	−	0.773456i	$-0.281474\pi$
$379$	24.6795	1.26770	0.633850	+	0.773456i	$0.281474\pi$
$380$	−0.0669745	−0.00343572
$381$	−50.6253	−2.59361
$382$	21.4639	1.09819
$383$	−12.5038	−0.638913	−0.319457	−	0.947601i	$-0.603500\pi$
$383$	−12.5038	−0.638913	−0.319457	+	0.947601i	$0.603500\pi$
$384$	26.3109	1.34267
$385$	−0.0289259	−0.00147420
$386$	23.2406	1.18292
$387$	−83.1810	−4.22833
$388$	−3.45484	−0.175393
$389$	−28.6548	−1.45286	−0.726429	−	0.687242i	$-0.758821\pi$
$389$	−28.6548	−1.45286	−0.726429	+	0.687242i	$0.758821\pi$
$390$	−0.674788	−0.0341692
$391$	−2.32034	−0.117344
$392$	20.7988	1.05050
$393$	−3.82499	−0.192945
$394$	7.79476	0.392694
$395$	−0.461007	−0.0231958
$396$	3.22816	0.162221
$397$	29.0706	1.45901	0.729507	−	0.683974i	$-0.239750\pi$
$397$	29.0706	1.45901	0.729507	+	0.683974i	$0.239750\pi$
$398$	36.1201	1.81054
$399$	2.33656	0.116974
$400$	16.5057	0.825286
$401$	32.6462	1.63027	0.815136	−	0.579270i	$-0.196662\pi$
$401$	32.6462	1.63027	0.815136	+	0.579270i	$0.196662\pi$
$402$	44.0525	2.19714
$403$	−10.8834	−0.542143
$404$	−1.98394	−0.0987048
$405$	−2.00162	−0.0994615
$406$	1.90352	0.0944703
$407$	−10.5381	−0.522356
$408$	5.13846	0.254392
$409$	1.00000	0.0494468
$409$	1.00000	0.0494468
$410$	0.873194	0.0431240
$411$	−46.7575	−2.30638
$412$	0.0744627	0.00366851
$413$	−1.40203	−0.0689896
$414$	43.2147	2.12389
$415$	0.194094	0.00952772
$416$	3.40288	0.166840
$417$	−66.0098	−3.23252
$418$	5.17271	0.253005
$419$	−10.6982	−0.522642	−0.261321	−	0.965252i	$-0.584158\pi$
$419$	−10.6982	−0.522642	−0.261321	+	0.965252i	$0.584158\pi$
$420$	−0.0199824	−0.000975043 0
$421$	−25.9303	−1.26377	−0.631883	−	0.775064i	$-0.717718\pi$
$421$	−25.9303	−1.26377	−0.631883	+	0.775064i	$0.717718\pi$
$422$	7.00747	0.341118
$423$	62.4210	3.03501
$424$	−0.714513	−0.0346998
$425$	2.63485	0.127809
$426$	7.03114	0.340660
$427$	0.182499	0.00883174
$428$	−2.90377	−0.140359
$429$	−9.26226	−0.447186
$430$	1.13988	0.0549699
$431$	22.6333	1.09021	0.545104	−	0.838368i	$-0.316490\pi$
$431$	22.6333	1.09021	0.545104	+	0.838368i	$0.316490\pi$
$432$	−48.7288	−2.34446
$433$	−20.3248	−0.976746	−0.488373	−	0.872635i	$-0.662409\pi$
$433$	−20.3248	−0.976746	−0.488373	+	0.872635i	$0.662409\pi$
$434$	1.81345	0.0870484
$435$	1.46233	0.0701133
$436$	−0.296140	−0.0141825
$437$	−12.3066	−0.588702
$438$	41.3941	1.97788
$439$	−19.1914	−0.915957	−0.457978	−	0.888963i	$-0.651426\pi$
$439$	−19.1914	−0.915957	−0.457978	+	0.888963i	$0.651426\pi$
$440$	−0.337388	−0.0160843
$441$	−52.2872	−2.48987
$442$	−1.38293	−0.0657793
$443$	5.52024	0.262274	0.131137	−	0.991364i	$-0.458137\pi$
$443$	5.52024	0.262274	0.131137	+	0.991364i	$0.458137\pi$
$444$	−7.27988	−0.345488
$445$	−0.0383888	−0.00181980
$446$	2.04009	0.0966009
$447$	0.802878	0.0379748
$448$	−2.26705	−0.107108
$449$	−21.0318	−0.992554	−0.496277	−	0.868164i	$-0.665300\pi$
$449$	−21.0318	−0.992554	−0.496277	+	0.868164i	$0.665300\pi$
$450$	−49.0724	−2.31329
$451$	11.9856	0.564381
$452$	−2.95408	−0.138948
$453$	24.8249	1.16638
$454$	20.0055	0.938906
$455$	0.0410162	0.00192287
$456$	27.2533	1.27625
$457$	−5.27660	−0.246829	−0.123415	−	0.992355i	$-0.539384\pi$
$457$	−5.27660	−0.246829	−0.123415	+	0.992355i	$0.539384\pi$
$458$	−3.68346	−0.172117
$459$	−7.77871	−0.363079
$460$	0.105247	0.00490715
$461$	−12.0755	−0.562414	−0.281207	−	0.959647i	$-0.590735\pi$
$461$	−12.0755	−0.562414	−0.281207	+	0.959647i	$0.590735\pi$
$462$	1.54332	0.0718018
$463$	10.7139	0.497916	0.248958	−	0.968514i	$-0.419912\pi$
$463$	10.7139	0.497916	0.248958	+	0.968514i	$0.419912\pi$
$464$	18.7740	0.871559
$465$	1.39313	0.0646049
$466$	−25.1713	−1.16604
$467$	−20.7896	−0.962030	−0.481015	−	0.876712i	$-0.659732\pi$
$467$	−20.7896	−0.962030	−0.481015	+	0.876712i	$0.659732\pi$
$468$	−4.57744	−0.211592
$469$	−2.67768	−0.123644
$470$	−0.855394	−0.0394564
$471$	50.7454	2.33822
$472$	−16.3531	−0.752714
$473$	15.6462	0.719414
$474$	24.5967	1.12976
$475$	13.9747	0.641203
$476$	−0.0409526	−0.00187706
$477$	1.79625	0.0822448
$478$	−1.41539	−0.0647384
$479$	−39.1497	−1.78879	−0.894397	−	0.447273i	$-0.852395\pi$
$479$	−39.1497	−1.78879	−0.894397	+	0.447273i	$0.852395\pi$
$480$	−0.435585	−0.0198816
$481$	14.9428	0.681333
$482$	−23.5866	−1.07434
$483$	−3.67177	−0.167071
$484$	2.71266	0.123303
$485$	−0.907738	−0.0412183
$486$	49.1599	2.22994
$487$	12.1969	0.552692	0.276346	−	0.961058i	$-0.410876\pi$
$487$	12.1969	0.552692	0.276346	+	0.961058i	$0.410876\pi$
$488$	2.12864	0.0963591
$489$	−63.0511	−2.85127
$490$	0.716524	0.0323693
$491$	25.0876	1.13219	0.566094	−	0.824341i	$-0.308454\pi$
$491$	25.0876	1.13219	0.566094	+	0.824341i	$0.308454\pi$
$492$	8.27983	0.373284
$493$	2.99694	0.134975
$494$	−7.33477	−0.330007
$495$	0.848179	0.0381228
$496$	17.8856	0.803087
$497$	−0.427380	−0.0191706
$498$	−10.3558	−0.464053
$499$	−27.4769	−1.23004	−0.615018	−	0.788513i	$-0.710851\pi$
$499$	−27.4769	−1.23004	−0.615018	+	0.788513i	$0.710851\pi$
$500$	−0.239176	−0.0106963
$501$	57.7765	2.58126
$502$	26.3116	1.17434
$503$	28.9853	1.29239	0.646196	−	0.763172i	$-0.276359\pi$
$503$	28.9853	1.29239	0.646196	+	0.763172i	$0.276359\pi$
$504$	5.81704	0.259112
$505$	−0.521269	−0.0231962
$506$	−8.12862	−0.361361
$507$	−29.0730	−1.29118
$508$	4.70607	0.208798
$509$	29.7835	1.32013	0.660065	−	0.751208i	$-0.270529\pi$
$509$	29.7835	1.32013	0.660065	+	0.751208i	$0.270529\pi$
$510$	0.177022	0.00783865
$511$	−2.51609	−0.111305
$512$	−25.4213	−1.12347
$513$	−41.2566	−1.82152
$514$	24.4429	1.07813
$515$	0.0195646	0.000862121 0
$516$	10.8086	0.475823
$517$	−11.7413	−0.516382
$518$	−2.48984	−0.109397
$519$	41.1707	1.80719
$520$	0.478408	0.0209796
$521$	26.7743	1.17300	0.586501	−	0.809948i	$-0.300505\pi$
$521$	26.7743	1.17300	0.586501	+	0.809948i	$0.300505\pi$
$522$	−55.8160	−2.44300
$523$	−10.6602	−0.466140	−0.233070	−	0.972460i	$-0.574877\pi$
$523$	−10.6602	−0.466140	−0.233070	+	0.972460i	$0.574877\pi$
$524$	0.355567	0.0155330
$525$	4.16947	0.181971
$526$	11.0476	0.481697
$527$	2.85513	0.124371
$528$	15.2214	0.662426
$529$	−3.66095	−0.159172
$530$	−0.0246152	−0.00106921
$531$	41.1111	1.78407
$532$	−0.217204	−0.00941698
$533$	−16.9953	−0.736148
$534$	2.04821	0.0886346
$535$	−0.762949	−0.0329852
$536$	−31.2321	−1.34902
$537$	−48.1326	−2.07708
$538$	−30.7572	−1.32604
$539$	9.83515	0.423630
$540$	0.352830	0.0151834
$541$	−41.2307	−1.77265	−0.886323	−	0.463067i	$-0.846749\pi$
$541$	−41.2307	−1.77265	−0.886323	+	0.463067i	$0.846749\pi$
$542$	7.72597	0.331859
$543$	−76.6799	−3.29065
$544$	−0.892702	−0.0382743
$545$	−0.0778090	−0.00333297
$546$	−2.18839	−0.0936545
$547$	39.9688	1.70894	0.854471	−	0.519499i	$-0.173882\pi$
$547$	39.9688	1.70894	0.854471	+	0.519499i	$0.173882\pi$
$548$	4.34653	0.185674
$549$	−5.35131	−0.228388
$550$	9.23044	0.393587
$551$	15.8951	0.677155
$552$	−42.8270	−1.82284
$553$	−1.49508	−0.0635774
$554$	−37.4346	−1.59044
$555$	−1.91275	−0.0811916
$556$	6.13620	0.260233
$557$	−2.73835	−0.116027	−0.0580137	−	0.998316i	$-0.518477\pi$
$557$	−2.73835	−0.116027	−0.0580137	+	0.998316i	$0.518477\pi$
$558$	−53.1748	−2.25107
$559$	−22.1859	−0.938365
$560$	−0.0674051	−0.00284838
$561$	2.42983	0.102588
$562$	3.49643	0.147488
$563$	38.5408	1.62430	0.812150	−	0.583449i	$-0.198297\pi$
$563$	38.5408	1.62430	0.812150	+	0.583449i	$0.198297\pi$
$564$	−8.11106	−0.341537
$565$	−0.776168	−0.0326536
$566$	−23.0889	−0.970497
$567$	−6.49143	−0.272614
$568$	−4.98490	−0.209162
$569$	21.9552	0.920409	0.460204	−	0.887813i	$-0.347776\pi$
$569$	21.9552	0.920409	0.460204	+	0.887813i	$0.347776\pi$
$570$	0.938884	0.0393255
$571$	−27.2036	−1.13843	−0.569217	−	0.822187i	$-0.692754\pi$
$571$	−27.2036	−1.13843	−0.569217	+	0.822187i	$0.692754\pi$
$572$	0.861010	0.0360006
$573$	53.4752	2.23396
$574$	2.83184	0.118199
$575$	−21.9604	−0.915814
$576$	66.4755	2.76981
$577$	6.14567	0.255847	0.127924	−	0.991784i	$-0.459169\pi$
$577$	6.14567	0.255847	0.127924	+	0.991784i	$0.459169\pi$
$578$	−21.7907	−0.906374
$579$	57.9017	2.40631
$580$	−0.135936	−0.00564446
$581$	0.629464	0.0261146
$582$	48.4317	2.00756
$583$	−0.337873	−0.0139933
$584$	−29.3474	−1.21440
$585$	−1.20270	−0.0497254
$586$	28.0336	1.15806
$587$	3.76761	0.155506	0.0777529	−	0.996973i	$-0.475225\pi$
$587$	3.76761	0.155506	0.0777529	+	0.996973i	$0.475225\pi$
$588$	6.79426	0.280190
$589$	15.1430	0.623956
$590$	−0.563371	−0.0231936
$591$	19.4199	0.798828
$592$	−24.5566	−1.00927
$593$	−44.8123	−1.84022	−0.920110	−	0.391660i	$-0.871901\pi$
$593$	−44.8123	−1.84022	−0.920110	+	0.391660i	$0.871901\pi$
$594$	−27.2504	−1.11810
$595$	−0.0107601	−0.000441120 0
$596$	−0.0746347	−0.00305716
$597$	89.9897	3.68303
$598$	11.5262	0.471340
$599$	−4.44042	−0.181431	−0.0907154	−	0.995877i	$-0.528915\pi$
$599$	−4.44042	−0.181431	−0.0907154	+	0.995877i	$0.528915\pi$
$600$	48.6322	1.98540
$601$	−17.7180	−0.722731	−0.361366	−	0.932424i	$-0.617689\pi$
$601$	−17.7180	−0.722731	−0.361366	+	0.932424i	$0.617689\pi$
$602$	3.69673	0.150667
$603$	78.5162	3.19743
$604$	−2.30770	−0.0938990
$605$	0.712736	0.0289768
$606$	27.8119	1.12978
$607$	−39.9816	−1.62280	−0.811402	−	0.584489i	$-0.801295\pi$
$607$	−39.9816	−1.62280	−0.811402	+	0.584489i	$0.801295\pi$
$608$	−4.73470	−0.192017
$609$	4.74245	0.192174
$610$	0.0733324	0.00296914
$611$	16.6489	0.673541
$612$	1.20083	0.0485407
$613$	2.70621	0.109303	0.0546514	−	0.998505i	$-0.482595\pi$
$613$	2.70621	0.109303	0.0546514	+	0.998505i	$0.482595\pi$
$614$	17.0386	0.687620
$615$	2.17548	0.0877238
$616$	−1.09418	−0.0440857
$617$	−22.8230	−0.918819	−0.459409	−	0.888225i	$-0.651939\pi$
$617$	−22.8230	−0.918819	−0.459409	+	0.888225i	$0.651939\pi$
$618$	−1.04386	−0.0419901
$619$	8.04743	0.323454	0.161727	−	0.986836i	$-0.448294\pi$
$619$	8.04743	0.323454	0.161727	+	0.986836i	$0.448294\pi$
$620$	−0.129504	−0.00520101
$621$	64.8324	2.60163
$622$	−29.1298	−1.16800
$623$	−0.124498	−0.00498791
$624$	−21.5835	−0.864032
$625$	24.9057	0.996229
$626$	−32.5662	−1.30161
$627$	12.8873	0.514669
$628$	−4.71724	−0.188238
$629$	−3.92004	−0.156302
$630$	0.200399	0.00798408
$631$	44.8068	1.78373	0.891865	−	0.452302i	$-0.149397\pi$
$631$	44.8068	1.78373	0.891865	+	0.452302i	$0.149397\pi$
$632$	−17.4384	−0.693664
$633$	17.4584	0.693910
$634$	41.1455	1.63410
$635$	1.23649	0.0490687
$636$	−0.233407	−0.00925519
$637$	−13.9460	−0.552560
$638$	10.4989	0.415656
$639$	12.5318	0.495751
$640$	−0.642627	−0.0254021
$641$	15.4824	0.611517	0.305758	−	0.952109i	$-0.401090\pi$
$641$	15.4824	0.611517	0.305758	+	0.952109i	$0.401090\pi$
$642$	40.7066	1.60656
$643$	9.68629	0.381990	0.190995	−	0.981591i	$-0.438829\pi$
$643$	9.68629	0.381990	0.190995	+	0.981591i	$0.438829\pi$
$644$	0.341324	0.0134500
$645$	2.83990	0.111821
$646$	1.92418	0.0757058
$647$	17.1868	0.675683	0.337841	−	0.941203i	$-0.390303\pi$
$647$	17.1868	0.675683	0.337841	+	0.941203i	$0.390303\pi$
$648$	−75.7152	−2.97437
$649$	−7.73293	−0.303544
$650$	−13.0885	−0.513374
$651$	4.51804	0.177076
$652$	5.86116	0.229541
$653$	−36.6235	−1.43319	−0.716595	−	0.697490i	$-0.754300\pi$
$653$	−36.6235	−1.43319	−0.716595	+	0.697490i	$0.754300\pi$
$654$	4.15144	0.162334
$655$	0.0934231	0.00365034
$656$	27.9297	1.09047
$657$	73.7780	2.87835
$658$	−2.77411	−0.108146
$659$	5.68191	0.221336	0.110668	−	0.993857i	$-0.464701\pi$
$659$	5.68191	0.221336	0.110668	+	0.993857i	$0.464701\pi$
$660$	−0.110213	−0.00429005
$661$	−12.2718	−0.477317	−0.238659	−	0.971104i	$-0.576708\pi$
$661$	−12.2718	−0.477317	−0.238659	+	0.971104i	$0.576708\pi$
$662$	−9.92781	−0.385855
$663$	−3.44544	−0.133810
$664$	7.34198	0.284924
$665$	−0.0570690	−0.00221304
$666$	73.0082	2.82901
$667$	−24.9783	−0.967164
$668$	−5.37084	−0.207804
$669$	5.08268	0.196508
$670$	−1.07596	−0.0415678
$671$	1.00657	0.0388584
$672$	−1.41264	−0.0544937
$673$	2.49095	0.0960192	0.0480096	−	0.998847i	$-0.484712\pi$
$673$	2.49095	0.0960192	0.0480096	+	0.998847i	$0.484712\pi$
$674$	10.1134	0.389553
$675$	−73.6204	−2.83365
$676$	2.70260	0.103946
$677$	−36.6597	−1.40895	−0.704474	−	0.709730i	$-0.748817\pi$
$677$	−36.6597	−1.40895	−0.704474	+	0.709730i	$0.748817\pi$
$678$	41.4119	1.59041
$679$	−2.94387	−0.112975
$680$	−0.125504	−0.00481286
$681$	49.8419	1.90994
$682$	10.0021	0.383001
$683$	12.6766	0.485056	0.242528	−	0.970144i	$-0.422023\pi$
$683$	12.6766	0.485056	0.242528	+	0.970144i	$0.422023\pi$
$684$	6.36895	0.243523
$685$	1.14202	0.0436345
$686$	4.66966	0.178288
$687$	−9.17698	−0.350123
$688$	36.4598	1.39002
$689$	0.479095	0.0182520
$690$	−1.47540	−0.0561677
$691$	−19.2969	−0.734088	−0.367044	−	0.930204i	$-0.619630\pi$
$691$	−19.2969	−0.734088	−0.367044	+	0.930204i	$0.619630\pi$
$692$	−3.82718	−0.145488
$693$	2.75071	0.104491
$694$	25.8369	0.980757
$695$	1.61225	0.0611562
$696$	55.3153	2.09672
$697$	4.45849	0.168878
$698$	−5.94383	−0.224977
$699$	−62.7120	−2.37199
$700$	−0.387590	−0.0146495
$701$	13.5194	0.510619	0.255310	−	0.966859i	$-0.417823\pi$
$701$	13.5194	0.510619	0.255310	+	0.966859i	$0.417823\pi$
$702$	38.6404	1.45839
$703$	−20.7911	−0.784150
$704$	−12.5039	−0.471260
$705$	−2.13113	−0.0802631
$706$	−29.9714	−1.12799
$707$	−1.69052	−0.0635785
$708$	−5.34202	−0.200765
$709$	30.7803	1.15598	0.577990	−	0.816044i	$-0.303837\pi$
$709$	30.7803	1.15598	0.577990	+	0.816044i	$0.303837\pi$
$710$	−0.171731	−0.00644497
$711$	43.8395	1.64411
$712$	−1.45213	−0.0544208
$713$	−23.7963	−0.891180
$714$	0.574096	0.0214850
$715$	0.226225	0.00846035
$716$	4.47436	0.167215
$717$	−3.52631	−0.131692
$718$	−22.2152	−0.829063
$719$	5.83155	0.217480	0.108740	−	0.994070i	$-0.465318\pi$
$719$	5.83155	0.217480	0.108740	+	0.994070i	$0.465318\pi$
$720$	1.97648	0.0736591
$721$	0.0634497	0.00236299
$722$	−14.5544	−0.541658
$723$	−58.7638	−2.18545
$724$	7.12808	0.264913
$725$	28.3641	1.05342
$726$	−38.0275	−1.41133
$727$	−33.8612	−1.25584	−0.627921	−	0.778277i	$-0.716094\pi$
$727$	−33.8612	−1.25584	−0.627921	+	0.778277i	$0.716094\pi$
$728$	1.55151	0.0575030
$729$	46.7516	1.73154
$730$	−1.01103	−0.0374197
$731$	5.82019	0.215268
$732$	0.695355	0.0257011
$733$	−30.0453	−1.10975	−0.554875	−	0.831934i	$-0.687234\pi$
$733$	−30.0453	−1.10975	−0.554875	+	0.831934i	$0.687234\pi$
$734$	−18.6924	−0.689950
$735$	1.78515	0.0658463
$736$	7.44031	0.274253
$737$	−14.7688	−0.544015
$738$	−83.0364	−3.05661
$739$	−11.4350	−0.420644	−0.210322	−	0.977632i	$-0.567451\pi$
$739$	−11.4350	−0.420644	−0.210322	+	0.977632i	$0.567451\pi$
$740$	0.177807	0.00653631
$741$	−18.2739	−0.671307
$742$	−0.0798290	−0.00293062
$743$	33.0844	1.21375	0.606875	−	0.794797i	$-0.292423\pi$
$743$	33.0844	1.21375	0.606875	+	0.794797i	$0.292423\pi$
$744$	52.6978	1.93200
$745$	−0.0196098	−0.000718449 0
$746$	−37.6241	−1.37752
$747$	−18.4574	−0.675322
$748$	−0.225875	−0.00825879
$749$	−2.47431	−0.0904092
$750$	3.35290	0.122430
$751$	31.3123	1.14260	0.571302	−	0.820740i	$-0.306439\pi$
$751$	31.3123	1.14260	0.571302	+	0.820740i	$0.306439\pi$
$752$	−27.3604	−0.997729
$753$	65.5527	2.38887
$754$	−14.8872	−0.542159
$755$	−0.606335	−0.0220668
$756$	1.14426	0.0416162
$757$	53.0401	1.92778	0.963888	−	0.266308i	$-0.0858039\pi$
$757$	53.0401	1.92778	0.963888	+	0.266308i	$0.0858039\pi$
$758$	32.1610	1.16814
$759$	−20.2517	−0.735089
$760$	−0.665646	−0.0241455
$761$	−27.3441	−0.991222	−0.495611	−	0.868545i	$-0.665056\pi$
$761$	−27.3441	−0.991222	−0.495611	+	0.868545i	$0.665056\pi$
$762$	−65.9722	−2.38992
$763$	−0.252341	−0.00913536
$764$	−4.97100	−0.179844
$765$	0.315511	0.0114073
$766$	−16.2943	−0.588736
$767$	10.9651	0.395927
$768$	−22.9543	−0.828293
$769$	34.9373	1.25987	0.629935	−	0.776648i	$-0.283081\pi$
$769$	34.9373	1.25987	0.629935	+	0.776648i	$0.283081\pi$
$770$	−0.0376947	−0.00135842
$771$	60.8971	2.19316
$772$	−5.38248	−0.193720
$773$	−25.6077	−0.921043	−0.460522	−	0.887648i	$-0.652338\pi$
$773$	−25.6077	−0.921043	−0.460522	+	0.887648i	$0.652338\pi$
$774$	−108.397	−3.89625
$775$	27.0219	0.970656
$776$	−34.3369	−1.23262
$777$	−6.20319	−0.222538
$778$	−37.3415	−1.33876
$779$	23.6469	0.847238
$780$	0.156280	0.00559571
$781$	−2.35722	−0.0843479
$782$	−3.02374	−0.108129
$783$	−83.7374	−2.99253
$784$	22.9185	0.818518
$785$	−1.23943	−0.0442370
$786$	−4.98452	−0.177792
$787$	5.77938	0.206012	0.103006	−	0.994681i	$-0.467154\pi$
$787$	5.77938	0.206012	0.103006	+	0.994681i	$0.467154\pi$
$788$	−1.80525	−0.0643095
$789$	27.5240	0.979879
$790$	−0.600760	−0.0213741
$791$	−2.51717	−0.0895004
$792$	32.0840	1.14005
$793$	−1.42730	−0.0506848
$794$	37.8833	1.34443
$795$	−0.0613264	−0.00217502
$796$	−8.36534	−0.296502
$797$	36.0085	1.27549	0.637743	−	0.770250i	$-0.279868\pi$
$797$	36.0085	1.27549	0.637743	+	0.770250i	$0.279868\pi$
$798$	3.04488	0.107788
$799$	−4.36761	−0.154515
$800$	−8.44883	−0.298711
$801$	3.65059	0.128987
$802$	42.5428	1.50224
$803$	−13.8775	−0.489727
$804$	−10.2025	−0.359814
$805$	0.0896808	0.00316083
$806$	−14.1827	−0.499565
$807$	−76.6287	−2.69746
$808$	−19.7180	−0.693676
$809$	3.77871	0.132852	0.0664262	−	0.997791i	$-0.478840\pi$
$809$	3.77871	0.132852	0.0664262	+	0.997791i	$0.478840\pi$
$810$	−2.60841	−0.0916503
$811$	2.61255	0.0917391	0.0458695	−	0.998947i	$-0.485394\pi$
$811$	2.61255	0.0917391	0.0458695	+	0.998947i	$0.485394\pi$
$812$	−0.440853	−0.0154709
$813$	19.2485	0.675075
$814$	−13.7327	−0.481332
$815$	1.53999	0.0539434
$816$	5.66215	0.198215
$817$	30.8690	1.07997
$818$	1.30315	0.0455635
$819$	−3.90044	−0.136292
$820$	−0.202230	−0.00706218
$821$	−35.1769	−1.22768	−0.613841	−	0.789430i	$-0.710376\pi$
$821$	−35.1769	−1.22768	−0.613841	+	0.789430i	$0.710376\pi$
$822$	−60.9319	−2.12524
$823$	33.2639	1.15951	0.579754	−	0.814792i	$-0.303149\pi$
$823$	33.2639	1.15951	0.579754	+	0.814792i	$0.303149\pi$
$824$	0.740069	0.0257815
$825$	22.9968	0.800644
$826$	−1.82706	−0.0635715
$827$	−54.7760	−1.90475	−0.952375	−	0.304930i	$-0.901367\pi$
$827$	−54.7760	−1.90475	−0.952375	+	0.304930i	$0.901367\pi$
$828$	−10.0084	−0.347817
$829$	−2.60439	−0.0904543	−0.0452272	−	0.998977i	$-0.514401\pi$
$829$	−2.60439	−0.0904543	−0.0452272	+	0.998977i	$0.514401\pi$
$830$	0.252934	0.00877945
$831$	−93.2647	−3.23532
$832$	17.7303	0.614686
$833$	3.65855	0.126761
$834$	−86.0205	−2.97865
$835$	−1.41116	−0.0488351
$836$	−1.19799	−0.0414334
$837$	−79.7750	−2.75743
$838$	−13.9414	−0.481596
$839$	31.5303	1.08855	0.544273	−	0.838908i	$-0.316806\pi$
$839$	31.5303	1.08855	0.544273	+	0.838908i	$0.316806\pi$
$840$	−0.198601	−0.00685239
$841$	3.26193	0.112481
$842$	−33.7910	−1.16451
$843$	8.71102	0.300023
$844$	−1.62292	−0.0558631
$845$	0.710092	0.0244279
$846$	81.3438	2.79666
$847$	2.31146	0.0794228
$848$	−0.787333	−0.0270371
$849$	−57.5237	−1.97421
$850$	3.43360	0.117772
$851$	32.6720	1.11998
$852$	−1.62840	−0.0557880
$853$	−32.4941	−1.11258	−0.556288	−	0.830990i	$-0.687775\pi$
$853$	−32.4941	−1.11258	−0.556288	+	0.830990i	$0.687775\pi$
$854$	0.237823	0.00813813
$855$	1.67340	0.0572292
$856$	−28.8600	−0.986414
$857$	−23.7591	−0.811597	−0.405798	−	0.913963i	$-0.633006\pi$
$857$	−23.7591	−0.811597	−0.405798	+	0.913963i	$0.633006\pi$
$858$	−12.0701	−0.412066
$859$	−0.365751	−0.0124793	−0.00623963	−	0.999981i	$-0.501986\pi$
$859$	−0.365751	−0.0124793	−0.00623963	+	0.999981i	$0.501986\pi$
$860$	−0.263994	−0.00900214
$861$	7.05525	0.240442
$862$	29.4945	1.00459
$863$	−9.73662	−0.331438	−0.165719	−	0.986173i	$-0.552995\pi$
$863$	−9.73662	−0.331438	−0.165719	+	0.986173i	$0.552995\pi$
$864$	24.9429	0.848576
$865$	−1.00557	−0.0341904
$866$	−26.4862	−0.900037
$867$	−54.2895	−1.84377
$868$	−0.419992	−0.0142555
$869$	−8.24614	−0.279731
$870$	1.90563	0.0646069
$871$	20.9417	0.709584
$872$	−2.94327	−0.0996718
$873$	86.3215	2.92154
$874$	−16.0373	−0.542468
$875$	−0.203802	−0.00688977
$876$	−9.58679	−0.323908
$877$	18.3330	0.619060	0.309530	−	0.950890i	$-0.399828\pi$
$877$	18.3330	0.619060	0.309530	+	0.950890i	$0.399828\pi$
$878$	−25.0093	−0.844022
$879$	69.8431	2.35575
$880$	−0.371773	−0.0125325
$881$	−52.3916	−1.76512	−0.882559	−	0.470201i	$-0.844181\pi$
$881$	−52.3916	−1.76512	−0.882559	+	0.470201i	$0.844181\pi$
$882$	−68.1379	−2.29432
$883$	43.0331	1.44818	0.724090	−	0.689706i	$-0.242260\pi$
$883$	43.0331	1.44818	0.724090	+	0.689706i	$0.242260\pi$
$884$	0.320284	0.0107723
$885$	−1.40358	−0.0471810
$886$	7.19368	0.241677
$887$	−52.0606	−1.74802	−0.874011	−	0.485906i	$-0.838490\pi$
$887$	−52.0606	−1.74802	−0.874011	+	0.485906i	$0.838490\pi$
$888$	−72.3533	−2.42802
$889$	4.01005	0.134493
$890$	−0.0500263	−0.00167688
$891$	−35.8036	−1.19946
$892$	−0.472480	−0.0158198
$893$	−23.1649	−0.775183
$894$	1.04627	0.0349925
$895$	1.17561	0.0392964
$896$	−2.08409	−0.0696247
$897$	28.7163	0.958811
$898$	−27.4076	−0.914603
$899$	30.7353	1.02508
$900$	11.3651	0.378836
$901$	−0.125684	−0.00418715
$902$	15.6190	0.520057
$903$	9.21004	0.306491
$904$	−29.3600	−0.976499
$905$	1.87286	0.0622560
$906$	32.3506	1.07478
$907$	34.5288	1.14651	0.573255	−	0.819377i	$-0.305680\pi$
$907$	34.5288	1.14651	0.573255	+	0.819377i	$0.305680\pi$
$908$	−4.63325	−0.153760
$909$	49.5701	1.64414
$910$	0.0534502	0.00177186
$911$	−4.22715	−0.140052	−0.0700258	−	0.997545i	$-0.522308\pi$
$911$	−4.22715	−0.140052	−0.0700258	+	0.997545i	$0.522308\pi$
$912$	30.0308	0.994420
$913$	3.47181	0.114900
$914$	−6.87619	−0.227444
$915$	0.182701	0.00603990
$916$	0.853082	0.0281866
$917$	0.302979	0.0100052
$918$	−10.1368	−0.334565
$919$	28.9654	0.955479	0.477740	−	0.878501i	$-0.341456\pi$
$919$	28.9654	0.955479	0.477740	+	0.878501i	$0.341456\pi$
$920$	1.04602	0.0344864
$921$	42.4499	1.39877
$922$	−15.7362	−0.518245
$923$	3.34247	0.110019
$924$	−0.357431	−0.0117586
$925$	−37.1006	−1.21986
$926$	13.9618	0.458812
$927$	−1.86050	−0.0611069
$928$	−9.60989	−0.315460
$929$	50.4616	1.65559	0.827796	−	0.561029i	$-0.189594\pi$
$929$	50.4616	1.65559	0.827796	+	0.561029i	$0.189594\pi$
$930$	1.81546	0.0595312
$931$	19.4041	0.635945
$932$	5.82964	0.190956
$933$	−72.5741	−2.37597
$934$	−27.0920	−0.886477
$935$	−0.0593472	−0.00194086
$936$	−45.4942	−1.48703
$937$	29.4857	0.963255	0.481628	−	0.876376i	$-0.340046\pi$
$937$	29.4857	0.963255	0.481628	+	0.876376i	$0.340046\pi$
$938$	−3.48941	−0.113933
$939$	−81.1355	−2.64776
$940$	0.198108	0.00646157
$941$	−34.8624	−1.13648	−0.568241	−	0.822862i	$-0.692376\pi$
$941$	−34.8624	−1.13648	−0.568241	+	0.822862i	$0.692376\pi$
$942$	66.1287	2.15459
$943$	−37.1597	−1.21009
$944$	−18.0198	−0.586494
$945$	0.300647	0.00978003
$946$	20.3893	0.662914
$947$	20.2228	0.657153	0.328577	−	0.944477i	$-0.393431\pi$
$947$	20.2228	0.657153	0.328577	+	0.944477i	$0.393431\pi$
$948$	−5.69655	−0.185015
$949$	19.6780	0.638774
$950$	18.2111	0.590846
$951$	102.510	3.32412
$952$	−0.407020	−0.0131916
$953$	−0.996417	−0.0322771	−0.0161386	−	0.999870i	$-0.505137\pi$
$953$	−0.996417	−0.0322771	−0.0161386	+	0.999870i	$0.505137\pi$
$954$	2.34078	0.0757856
$955$	−1.30610	−0.0422644
$956$	0.327802	0.0106019
$957$	26.1570	0.845537
$958$	−51.0178	−1.64831
$959$	3.70368	0.119598
$960$	−2.26956	−0.0732496
$961$	−1.71905	−0.0554534
$962$	19.4727	0.627824
$963$	72.5527	2.33798
$964$	5.46262	0.175939
$965$	−1.41422	−0.0455252
$966$	−4.78485	−0.153950
$967$	−1.26858	−0.0407947	−0.0203973	−	0.999792i	$-0.506493\pi$
$967$	−1.26858	−0.0407947	−0.0203973	+	0.999792i	$0.506493\pi$
$968$	26.9606	0.866546
$969$	4.79391	0.154003
$970$	−1.18292	−0.0379812
$971$	−41.1238	−1.31973	−0.659863	−	0.751386i	$-0.729386\pi$
$971$	−41.1238	−1.31973	−0.659863	+	0.751386i	$0.729386\pi$
$972$	−11.3853	−0.365185
$973$	5.22866	0.167623
$974$	15.8943	0.509286
$975$	−32.6088	−1.04432
$976$	2.34558	0.0750803
$977$	15.6610	0.501039	0.250520	−	0.968112i	$-0.419399\pi$
$977$	15.6610	0.501039	0.250520	+	0.968112i	$0.419399\pi$
$978$	−82.1649	−2.62734
$979$	−0.686670	−0.0219461
$980$	−0.165946	−0.00530094
$981$	7.39926	0.236240
$982$	32.6928	1.04327
$983$	−17.6399	−0.562626	−0.281313	−	0.959616i	$-0.590770\pi$
$983$	−17.6399	−0.562626	−0.281313	+	0.959616i	$0.590770\pi$
$984$	82.2915	2.62336
$985$	−0.474320	−0.0151131
$986$	3.90546	0.124375
$987$	−6.91144	−0.219993
$988$	1.69872	0.0540435
$989$	−48.5089	−1.54249
$990$	1.10530	0.0351288
$991$	−16.1288	−0.512347	−0.256174	−	0.966631i	$-0.582462\pi$
$991$	−16.1288	−0.512347	−0.256174	+	0.966631i	$0.582462\pi$
$992$	−9.15516	−0.290677
$993$	−24.7342	−0.784916
$994$	−0.556939	−0.0176650
$995$	−2.19795	−0.0696796
$996$	2.39838	0.0759955
$997$	40.2154	1.27364	0.636818	−	0.771014i	$-0.280250\pi$
$997$	40.2154	1.27364	0.636818	+	0.771014i	$0.280250\pi$
$998$	−35.8065	−1.13343
$999$	109.530	3.46537

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	409.2.a.b.1.13	✓	20
3.2	odd	2		3681.2.a.i.1.8		20
4.3	odd	2		6544.2.a.i.1.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
409.2.a.b.1.13	✓	20	1.1	even	1	trivial
3681.2.a.i.1.8		20	3.2	odd	2
6544.2.a.i.1.1		20	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 409 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	409.a (trivial)

\(f(q)\)	\(=\)	\(q+1.30315 q^{2} +3.24667 q^{3} -0.301807 q^{4} -0.0792979 q^{5} +4.23089 q^{6} -0.257170 q^{7} -2.99959 q^{8} +7.54085 q^{9} -0.103337 q^{10} -1.41842 q^{11} -0.979866 q^{12} +2.01129 q^{13} -0.335130 q^{14} -0.257454 q^{15} -3.30530 q^{16} -0.527634 q^{17} +9.82683 q^{18} -2.79846 q^{19} +0.0239327 q^{20} -0.834944 q^{21} -1.84841 q^{22} +4.39762 q^{23} -9.73868 q^{24} -4.99371 q^{25} +2.62100 q^{26} +14.7426 q^{27} +0.0776155 q^{28} -5.67996 q^{29} -0.335501 q^{30} -5.41119 q^{31} +1.69189 q^{32} -4.60514 q^{33} -0.687585 q^{34} +0.0203930 q^{35} -2.27588 q^{36} +7.42947 q^{37} -3.64680 q^{38} +6.52998 q^{39} +0.237862 q^{40} -8.44997 q^{41} -1.08806 q^{42} -11.0307 q^{43} +0.428089 q^{44} -0.597974 q^{45} +5.73075 q^{46} +8.27772 q^{47} -10.7312 q^{48} -6.93386 q^{49} -6.50754 q^{50} -1.71305 q^{51} -0.607020 q^{52} +0.238203 q^{53} +19.2118 q^{54} +0.112478 q^{55} +0.771405 q^{56} -9.08566 q^{57} -7.40182 q^{58} +5.45179 q^{59} +0.0777013 q^{60} -0.709644 q^{61} -7.05157 q^{62} -1.93928 q^{63} +8.81539 q^{64} -0.159491 q^{65} -6.00118 q^{66} +10.4121 q^{67} +0.159244 q^{68} +14.2776 q^{69} +0.0265751 q^{70} +1.66186 q^{71} -22.6195 q^{72} +9.78378 q^{73} +9.68170 q^{74} -16.2129 q^{75} +0.844594 q^{76} +0.364775 q^{77} +8.50952 q^{78} +5.81360 q^{79} +0.262103 q^{80} +25.2418 q^{81} -11.0116 q^{82} -2.44766 q^{83} +0.251992 q^{84} +0.0418403 q^{85} -14.3747 q^{86} -18.4409 q^{87} +4.25469 q^{88} +0.484108 q^{89} -0.779248 q^{90} -0.517242 q^{91} -1.32723 q^{92} -17.5683 q^{93} +10.7871 q^{94} +0.221912 q^{95} +5.49302 q^{96} +11.4472 q^{97} -9.03585 q^{98} -10.6961 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 5 q^{2} + q^{3} + 23 q^{4} + 8 q^{5} - 4 q^{6} + 5 q^{7} + 12 q^{8} + 27 q^{9} - 5 q^{10} + 30 q^{11} - 5 q^{12} - 7 q^{13} + 17 q^{14} + 26 q^{15} + 29 q^{16} + 6 q^{17} - 4 q^{18} + q^{19} + 14 q^{20}+ \cdots + 43 q^{99}+O(q^{100}) \) 20 * q + 5 * q^2 + q^3 + 23 * q^4 + 8 * q^5 - 4 * q^6 + 5 * q^7 + 12 * q^8 + 27 * q^9 - 5 * q^10 + 30 * q^11 - 5 * q^12 - 7 * q^13 + 17 * q^14 + 26 * q^15 + 29 * q^16 + 6 * q^17 - 4 * q^18 + q^19 + 14 * q^20 - 3 * q^21 + q^22 + 37 * q^23 - 17 * q^24 + 14 * q^25 + 11 * q^26 - 5 * q^27 - 6 * q^28 + 12 * q^29 - 6 * q^30 + 10 * q^31 + 24 * q^32 - 22 * q^33 - 11 * q^34 + 12 * q^35 + 21 * q^36 - 4 * q^37 - 7 * q^38 + 31 * q^39 - 34 * q^40 - 7 * q^41 - 35 * q^42 - q^43 + 58 * q^44 - 10 * q^45 - 22 * q^46 + 51 * q^47 - 58 * q^48 + 5 * q^49 + q^50 + 6 * q^51 - 44 * q^52 + 10 * q^53 - 49 * q^54 - 5 * q^55 + 31 * q^56 - 30 * q^57 - 22 * q^58 + 43 * q^59 - 26 * q^60 - 21 * q^61 - 26 * q^62 + 8 * q^63 - 4 * q^64 - 10 * q^65 - 37 * q^66 + 4 * q^67 - 16 * q^68 - 19 * q^69 - 66 * q^70 + 97 * q^71 - 43 * q^72 - 36 * q^73 + 17 * q^74 - 19 * q^75 - 43 * q^76 - 20 * q^77 - 71 * q^78 + 16 * q^79 - 23 * q^80 + 16 * q^81 - 67 * q^82 + 28 * q^83 - 73 * q^84 - 30 * q^85 + 37 * q^86 - 14 * q^87 - 48 * q^88 + 2 * q^89 - 121 * q^90 - 25 * q^91 + 14 * q^92 - 27 * q^93 - 60 * q^94 + 61 * q^95 - 88 * q^96 - 36 * q^97 - 54 * q^98 + 43 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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