LMFDB - Embedded newform 8021.2.a.c.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8021,2,Mod(1,8021)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8021, 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("8021.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8021 = 13 \cdot 617 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8021.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:	$64.0480074613$
Analytic rank:	$0$
Dimension:	$169$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	8021.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44727 q^{2} -2.92033 q^{3} +3.98911 q^{4} -3.99296 q^{5} +7.14681 q^{6} +3.05388 q^{7} -4.86787 q^{8} +5.52831 q^{9} +O(q^{10})$	\(q-2.44727 q^{2} -2.92033 q^{3} +3.98911 q^{4} -3.99296 q^{5} +7.14681 q^{6} +3.05388 q^{7} -4.86787 q^{8} +5.52831 q^{9} +9.77182 q^{10} +4.22672 q^{11} -11.6495 q^{12} -1.00000 q^{13} -7.47366 q^{14} +11.6607 q^{15} +3.93476 q^{16} -4.58556 q^{17} -13.5292 q^{18} -1.38817 q^{19} -15.9283 q^{20} -8.91833 q^{21} -10.3439 q^{22} -3.48044 q^{23} +14.2158 q^{24} +10.9437 q^{25} +2.44727 q^{26} -7.38349 q^{27} +12.1823 q^{28} -0.287059 q^{29} -28.5369 q^{30} +7.40109 q^{31} +0.106336 q^{32} -12.3434 q^{33} +11.2221 q^{34} -12.1940 q^{35} +22.0530 q^{36} +0.916404 q^{37} +3.39722 q^{38} +2.92033 q^{39} +19.4372 q^{40} -10.5809 q^{41} +21.8255 q^{42} -0.118422 q^{43} +16.8608 q^{44} -22.0743 q^{45} +8.51757 q^{46} -9.08016 q^{47} -11.4908 q^{48} +2.32620 q^{49} -26.7821 q^{50} +13.3913 q^{51} -3.98911 q^{52} +11.8265 q^{53} +18.0694 q^{54} -16.8771 q^{55} -14.8659 q^{56} +4.05390 q^{57} +0.702509 q^{58} -3.54309 q^{59} +46.5159 q^{60} +8.33673 q^{61} -18.1124 q^{62} +16.8828 q^{63} -8.12976 q^{64} +3.99296 q^{65} +30.2076 q^{66} -1.42406 q^{67} -18.2923 q^{68} +10.1640 q^{69} +29.8420 q^{70} +11.9326 q^{71} -26.9111 q^{72} -10.0089 q^{73} -2.24268 q^{74} -31.9592 q^{75} -5.53755 q^{76} +12.9079 q^{77} -7.14681 q^{78} +7.74799 q^{79} -15.7113 q^{80} +4.97727 q^{81} +25.8942 q^{82} +6.00668 q^{83} -35.5762 q^{84} +18.3099 q^{85} +0.289810 q^{86} +0.838305 q^{87} -20.5751 q^{88} +6.40242 q^{89} +54.0216 q^{90} -3.05388 q^{91} -13.8839 q^{92} -21.6136 q^{93} +22.2216 q^{94} +5.54289 q^{95} -0.310535 q^{96} -1.38510 q^{97} -5.69282 q^{98} +23.3666 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 169 q + 9 q^{2} + 9 q^{3} + 199 q^{4} + 12 q^{5} + 22 q^{6} + 36 q^{7} + 30 q^{8} + 198 q^{9}+O(q^{10}) $ 169 * q + 9 * q^2 + 9 * q^3 + 199 * q^4 + 12 * q^5 + 22 * q^6 + 36 * q^7 + 30 * q^8 + 198 * q^9	\( 169 q + 9 q^{2} + 9 q^{3} + 199 q^{4} + 12 q^{5} + 22 q^{6} + 36 q^{7} + 30 q^{8} + 198 q^{9} - 3 q^{10} + 59 q^{11} + 11 q^{12} - 169 q^{13} + 30 q^{14} + 50 q^{15} + 267 q^{16} + q^{17} + 53 q^{18} + 107 q^{19} + 48 q^{20} + 36 q^{21} + 14 q^{22} + 12 q^{23} + 78 q^{24} + 217 q^{25} - 9 q^{26} + 39 q^{27} + 99 q^{28} + 30 q^{29} - q^{30} + 106 q^{31} + 74 q^{32} + 16 q^{33} + 56 q^{34} + 46 q^{35} + 271 q^{36} + 73 q^{37} + 2 q^{38} - 9 q^{39} - 16 q^{40} + 52 q^{41} - 2 q^{42} + 64 q^{43} + 124 q^{44} + 84 q^{45} + 105 q^{46} + 55 q^{47} + 26 q^{48} + 257 q^{49} + 60 q^{50} + 117 q^{51} - 199 q^{52} + 7 q^{53} + 78 q^{54} - 4 q^{55} + 63 q^{56} + 51 q^{57} + 84 q^{58} + 98 q^{59} + 94 q^{60} + 32 q^{61} - 25 q^{62} + 128 q^{63} + 380 q^{64} - 12 q^{65} + 16 q^{66} + 170 q^{67} - 10 q^{68} + 55 q^{69} + 70 q^{70} + 124 q^{71} + 173 q^{72} + 81 q^{73} + 54 q^{74} + 120 q^{75} + 212 q^{76} + 20 q^{77} - 22 q^{78} + 92 q^{79} + 66 q^{80} + 265 q^{81} + 21 q^{82} + 62 q^{83} + 98 q^{84} + 139 q^{85} + 51 q^{86} - 33 q^{87} + 31 q^{88} + 58 q^{89} + 16 q^{90} - 36 q^{91} + 40 q^{92} + 37 q^{93} + 55 q^{94} + 23 q^{95} + 164 q^{96} + 78 q^{97} + 69 q^{98} + 307 q^{99}+O(q^{100}) \) 169 * q + 9 * q^2 + 9 * q^3 + 199 * q^4 + 12 * q^5 + 22 * q^6 + 36 * q^7 + 30 * q^8 + 198 * q^9 - 3 * q^10 + 59 * q^11 + 11 * q^12 - 169 * q^13 + 30 * q^14 + 50 * q^15 + 267 * q^16 + q^17 + 53 * q^18 + 107 * q^19 + 48 * q^20 + 36 * q^21 + 14 * q^22 + 12 * q^23 + 78 * q^24 + 217 * q^25 - 9 * q^26 + 39 * q^27 + 99 * q^28 + 30 * q^29 - q^30 + 106 * q^31 + 74 * q^32 + 16 * q^33 + 56 * q^34 + 46 * q^35 + 271 * q^36 + 73 * q^37 + 2 * q^38 - 9 * q^39 - 16 * q^40 + 52 * q^41 - 2 * q^42 + 64 * q^43 + 124 * q^44 + 84 * q^45 + 105 * q^46 + 55 * q^47 + 26 * q^48 + 257 * q^49 + 60 * q^50 + 117 * q^51 - 199 * q^52 + 7 * q^53 + 78 * q^54 - 4 * q^55 + 63 * q^56 + 51 * q^57 + 84 * q^58 + 98 * q^59 + 94 * q^60 + 32 * q^61 - 25 * q^62 + 128 * q^63 + 380 * q^64 - 12 * q^65 + 16 * q^66 + 170 * q^67 - 10 * q^68 + 55 * q^69 + 70 * q^70 + 124 * q^71 + 173 * q^72 + 81 * q^73 + 54 * q^74 + 120 * q^75 + 212 * q^76 + 20 * q^77 - 22 * q^78 + 92 * q^79 + 66 * q^80 + 265 * q^81 + 21 * q^82 + 62 * q^83 + 98 * q^84 + 139 * q^85 + 51 * q^86 - 33 * q^87 + 31 * q^88 + 58 * q^89 + 16 * q^90 - 36 * q^91 + 40 * q^92 + 37 * q^93 + 55 * q^94 + 23 * q^95 + 164 * q^96 + 78 * q^97 + 69 * q^98 + 307 * q^99

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 $

$2$	−2.44727	−1.73048	−0.865239	−	0.501360i	$-0.832833\pi$
$2$	−2.44727	−1.73048	−0.865239	+	0.501360i	$0.832833\pi$
$3$	−2.92033	−1.68605	−0.843026	−	0.537873i	$-0.819228\pi$
$3$	−2.92033	−1.68605	−0.843026	+	0.537873i	$0.819228\pi$
$4$	3.98911	1.99455
$5$	−3.99296	−1.78570	−0.892852	−	0.450350i	$-0.851299\pi$
$5$	−3.99296	−1.78570	−0.892852	+	0.450350i	$0.851299\pi$
$6$	7.14681	2.91767
$7$	3.05388	1.15426	0.577129	−	0.816653i	$-0.304173\pi$
$7$	3.05388	1.15426	0.577129	+	0.816653i	$0.304173\pi$
$8$	−4.86787	−1.72105
$9$	5.52831	1.84277
$10$	9.77182	3.09012
$11$	4.22672	1.27440	0.637202	−	0.770697i	$-0.280092\pi$
$11$	4.22672	1.27440	0.637202	+	0.770697i	$0.280092\pi$
$12$	−11.6495	−3.36292
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−7.47366	−1.99742
$15$	11.6607	3.01079
$16$	3.93476	0.983691
$17$	−4.58556	−1.11216	−0.556081	−	0.831128i	$-0.687696\pi$
$17$	−4.58556	−1.11216	−0.556081	+	0.831128i	$0.687696\pi$
$18$	−13.5292	−3.18887
$19$	−1.38817	−0.318468	−0.159234	−	0.987241i	$-0.550902\pi$
$19$	−1.38817	−0.318468	−0.159234	+	0.987241i	$0.550902\pi$
$20$	−15.9283	−3.56168
$21$	−8.91833	−1.94614
$22$	−10.3439	−2.20533
$23$	−3.48044	−0.725723	−0.362861	−	0.931843i	$-0.618200\pi$
$23$	−3.48044	−0.725723	−0.362861	+	0.931843i	$0.618200\pi$
$24$	14.2158	2.90178
$25$	10.9437	2.18874
$26$	2.44727	0.479948
$27$	−7.38349	−1.42095
$28$	12.1823	2.30223
$29$	−0.287059	−0.0533055	−0.0266527	−	0.999645i	$-0.508485\pi$
$29$	−0.287059	−0.0533055	−0.0266527	+	0.999645i	$0.508485\pi$
$30$	−28.5369	−5.21010
$31$	7.40109	1.32927	0.664637	−	0.747166i	$-0.268586\pi$
$31$	7.40109	1.32927	0.664637	+	0.747166i	$0.268586\pi$
$32$	0.106336	0.0187976
$33$	−12.3434	−2.14871
$34$	11.2221	1.92457
$35$	−12.1940	−2.06116
$36$	22.0530	3.67550
$37$	0.916404	0.150656	0.0753280	−	0.997159i	$-0.476000\pi$
$37$	0.916404	0.150656	0.0753280	+	0.997159i	$0.476000\pi$
$38$	3.39722	0.551101
$39$	2.92033	0.467627
$40$	19.4372	3.07329
$41$	−10.5809	−1.65245	−0.826226	−	0.563338i	$-0.809517\pi$
$41$	−10.5809	−1.65245	−0.826226	+	0.563338i	$0.809517\pi$
$42$	21.8255	3.36775
$43$	−0.118422	−0.0180592	−0.00902960	−	0.999959i	$-0.502874\pi$
$43$	−0.118422	−0.0180592	−0.00902960	+	0.999959i	$0.502874\pi$
$44$	16.8608	2.54187
$45$	−22.0743	−3.29064
$46$	8.51757	1.25585
$47$	−9.08016	−1.32448	−0.662238	−	0.749293i	$-0.730393\pi$
$47$	−9.08016	−1.32448	−0.662238	+	0.749293i	$0.730393\pi$
$48$	−11.4908	−1.65855
$49$	2.32620	0.332314
$50$	−26.7821	−3.78756
$51$	13.3913	1.87516
$52$	−3.98911	−0.553190
$53$	11.8265	1.62449	0.812246	−	0.583315i	$-0.198245\pi$
$53$	11.8265	1.62449	0.812246	+	0.583315i	$0.198245\pi$
$54$	18.0694	2.45893
$55$	−16.8771	−2.27571
$56$	−14.8659	−1.98654
$57$	4.05390	0.536953
$58$	0.702509	0.0922439
$59$	−3.54309	−0.461271	−0.230635	−	0.973040i	$-0.574080\pi$
$59$	−3.54309	−0.461271	−0.230635	+	0.973040i	$0.574080\pi$
$60$	46.5159	6.00518
$61$	8.33673	1.06741	0.533704	−	0.845671i	$-0.320800\pi$
$61$	8.33673	1.06741	0.533704	+	0.845671i	$0.320800\pi$
$62$	−18.1124	−2.30028
$63$	16.8828	2.12703
$64$	−8.12976	−1.01622
$65$	3.99296	0.495265
$66$	30.2076	3.71829
$67$	−1.42406	−0.173977	−0.0869886	−	0.996209i	$-0.527724\pi$
$67$	−1.42406	−0.173977	−0.0869886	+	0.996209i	$0.527724\pi$
$68$	−18.2923	−2.21827
$69$	10.1640	1.22361
$70$	29.8420	3.56680
$71$	11.9326	1.41614	0.708068	−	0.706144i	$-0.249567\pi$
$71$	11.9326	1.41614	0.708068	+	0.706144i	$0.249567\pi$
$72$	−26.9111	−3.17150
$73$	−10.0089	−1.17146	−0.585729	−	0.810507i	$-0.699192\pi$
$73$	−10.0089	−1.17146	−0.585729	+	0.810507i	$0.699192\pi$
$74$	−2.24268	−0.260707
$75$	−31.9592	−3.69033
$76$	−5.53755	−0.635201
$77$	12.9079	1.47099
$78$	−7.14681	−0.809217
$79$	7.74799	0.871717	0.435859	−	0.900015i	$-0.356445\pi$
$79$	7.74799	0.871717	0.435859	+	0.900015i	$0.356445\pi$
$80$	−15.7113	−1.75658
$81$	4.97727	0.553030
$82$	25.8942	2.85953
$83$	6.00668	0.659319	0.329660	−	0.944100i	$-0.393066\pi$
$83$	6.00668	0.659319	0.329660	+	0.944100i	$0.393066\pi$
$84$	−35.5762	−3.88168
$85$	18.3099	1.98599
$86$	0.289810	0.0312511
$87$	0.838305	0.0898757
$88$	−20.5751	−2.19332
$89$	6.40242	0.678656	0.339328	−	0.940668i	$-0.389800\pi$
$89$	6.40242	0.678656	0.339328	+	0.940668i	$0.389800\pi$
$90$	54.0216	5.69438
$91$	−3.05388	−0.320134
$92$	−13.8839	−1.44749
$93$	−21.6136	−2.24123
$94$	22.2216	2.29198
$95$	5.54289	0.568689
$96$	−0.310535	−0.0316938
$97$	−1.38510	−0.140635	−0.0703176	−	0.997525i	$-0.522401\pi$
$97$	−1.38510	−0.140635	−0.0703176	+	0.997525i	$0.522401\pi$
$98$	−5.69282	−0.575061
$99$	23.3666	2.34843
$100$	43.6556	4.36556
$101$	18.8037	1.87103	0.935517	−	0.353281i	$-0.114934\pi$
$101$	18.8037	1.87103	0.935517	+	0.353281i	$0.114934\pi$
$102$	−32.7722	−3.24493
$103$	4.16594	0.410483	0.205241	−	0.978711i	$-0.434202\pi$
$103$	4.16594	0.410483	0.205241	+	0.978711i	$0.434202\pi$
$104$	4.86787	0.477334
$105$	35.6105	3.47523
$106$	−28.9425	−2.81115
$107$	−8.66509	−0.837686	−0.418843	−	0.908059i	$-0.637564\pi$
$107$	−8.66509	−0.837686	−0.418843	+	0.908059i	$0.637564\pi$
$108$	−29.4535	−2.83417
$109$	12.4995	1.19723	0.598616	−	0.801036i	$-0.295718\pi$
$109$	12.4995	1.19723	0.598616	+	0.801036i	$0.295718\pi$
$110$	41.3027	3.93806
$111$	−2.67620	−0.254014
$112$	12.0163	1.13543
$113$	−2.46406	−0.231800	−0.115900	−	0.993261i	$-0.536975\pi$
$113$	−2.46406	−0.231800	−0.115900	+	0.993261i	$0.536975\pi$
$114$	−9.92098	−0.929185
$115$	13.8973	1.29593
$116$	−1.14511	−0.106321
$117$	−5.52831	−0.511092
$118$	8.67087	0.798219
$119$	−14.0038	−1.28372
$120$	−56.7630	−5.18173
$121$	6.86514	0.624104
$122$	−20.4022	−1.84713
$123$	30.8996	2.78612
$124$	29.5237	2.65131
$125$	−23.7329	−2.12273
$126$	−41.3167	−3.68078
$127$	−9.44432	−0.838048	−0.419024	−	0.907975i	$-0.637628\pi$
$127$	−9.44432	−0.838048	−0.419024	+	0.907975i	$0.637628\pi$
$128$	19.6830	1.73975
$129$	0.345831	0.0304488
$130$	−9.77182	−0.857045
$131$	6.39332	0.558587	0.279293	−	0.960206i	$-0.409900\pi$
$131$	6.39332	0.558587	0.279293	+	0.960206i	$0.409900\pi$
$132$	−49.2391	−4.28572
$133$	−4.23930	−0.367594
$134$	3.48506	0.301064
$135$	29.4819	2.53740
$136$	22.3219	1.91409
$137$	5.70810	0.487676	0.243838	−	0.969816i	$-0.421593\pi$
$137$	5.70810	0.487676	0.243838	+	0.969816i	$0.421593\pi$
$138$	−24.8741	−2.11742
$139$	12.3613	1.04848	0.524238	−	0.851572i	$-0.324350\pi$
$139$	12.3613	1.04848	0.524238	+	0.851572i	$0.324350\pi$
$140$	−48.6432	−4.11110
$141$	26.5170	2.23314
$142$	−29.2022	−2.45059
$143$	−4.22672	−0.353456
$144$	21.7526	1.81272
$145$	1.14621	0.0951878
$146$	24.4945	2.02718
$147$	−6.79325	−0.560298
$148$	3.65564	0.300491
$149$	−23.2578	−1.90535	−0.952676	−	0.303987i	$-0.901682\pi$
$149$	−23.2578	−1.90535	−0.952676	+	0.303987i	$0.901682\pi$
$150$	78.2125	6.38603
$151$	9.04097	0.735743	0.367872	−	0.929877i	$-0.380087\pi$
$151$	9.04097	0.735743	0.367872	+	0.929877i	$0.380087\pi$
$152$	6.75743	0.548100
$153$	−25.3504	−2.04946
$154$	−31.5890	−2.54552
$155$	−29.5522	−2.37369
$156$	11.6495	0.932706
$157$	19.5112	1.55717	0.778583	−	0.627542i	$-0.215939\pi$
$157$	19.5112	1.55717	0.778583	+	0.627542i	$0.215939\pi$
$158$	−18.9614	−1.50849
$159$	−34.5372	−2.73898
$160$	−0.424593	−0.0335670
$161$	−10.6289	−0.837672
$162$	−12.1807	−0.957007
$163$	9.97451	0.781264	0.390632	−	0.920547i	$-0.372257\pi$
$163$	9.97451	0.781264	0.390632	+	0.920547i	$0.372257\pi$
$164$	−42.2082	−3.29591
$165$	49.2866	3.83696
$166$	−14.6999	−1.14094
$167$	−2.36580	−0.183071	−0.0915354	−	0.995802i	$-0.529177\pi$
$167$	−2.36580	−0.183071	−0.0915354	+	0.995802i	$0.529177\pi$
$168$	43.4133	3.34941
$169$	1.00000	0.0769231
$170$	−44.8093	−3.43671
$171$	−7.67422	−0.586862
$172$	−0.472399	−0.0360201
$173$	1.00259	0.0762253	0.0381127	−	0.999273i	$-0.487865\pi$
$173$	1.00259	0.0762253	0.0381127	+	0.999273i	$0.487865\pi$
$174$	−2.05156	−0.155528
$175$	33.4207	2.52637
$176$	16.6311	1.25362
$177$	10.3470	0.777726
$178$	−15.6684	−1.17440
$179$	−15.4134	−1.15205	−0.576027	−	0.817431i	$-0.695398\pi$
$179$	−15.4134	−1.15205	−0.576027	+	0.817431i	$0.695398\pi$
$180$	−88.0567	−6.56336
$181$	−26.1972	−1.94722	−0.973612	−	0.228209i	$-0.926713\pi$
$181$	−26.1972	−1.94722	−0.973612	+	0.228209i	$0.926713\pi$
$182$	7.47366	0.553985
$183$	−24.3460	−1.79971
$184$	16.9424	1.24901
$185$	−3.65916	−0.269027
$186$	52.8942	3.87839
$187$	−19.3819	−1.41734
$188$	−36.2217	−2.64174
$189$	−22.5483	−1.64015
$190$	−13.5649	−0.984103
$191$	−17.9787	−1.30090	−0.650448	−	0.759551i	$-0.725419\pi$
$191$	−17.9787	−1.30090	−0.650448	+	0.759551i	$0.725419\pi$
$192$	23.7416	1.71340
$193$	−4.29336	−0.309043	−0.154522	−	0.987989i	$-0.549384\pi$
$193$	−4.29336	−0.309043	−0.154522	+	0.987989i	$0.549384\pi$
$194$	3.38970	0.243366
$195$	−11.6607	−0.835043
$196$	9.27944	0.662817
$197$	16.1877	1.15332	0.576661	−	0.816983i	$-0.304355\pi$
$197$	16.1877	1.15332	0.576661	+	0.816983i	$0.304355\pi$
$198$	−57.1843	−4.06391
$199$	−7.33501	−0.519965	−0.259983	−	0.965613i	$-0.583717\pi$
$199$	−7.33501	−0.519965	−0.259983	+	0.965613i	$0.583717\pi$
$200$	−53.2725	−3.76694
$201$	4.15873	0.293334
$202$	−46.0175	−3.23778
$203$	−0.876643	−0.0615283
$204$	53.4195	3.74011
$205$	42.2489	2.95079
$206$	−10.1952	−0.710331
$207$	−19.2410	−1.33734
$208$	−3.93476	−0.272827
$209$	−5.86739	−0.405856
$210$	−87.1484	−6.01381
$211$	−14.1636	−0.975063	−0.487532	−	0.873105i	$-0.662103\pi$
$211$	−14.1636	−0.975063	−0.487532	+	0.873105i	$0.662103\pi$
$212$	47.1771	3.24014
$213$	−34.8470	−2.38768
$214$	21.2058	1.44960
$215$	0.472854	0.0322484
$216$	35.9419	2.44554
$217$	22.6020	1.53433
$218$	−30.5895	−2.07178
$219$	29.2294	1.97514
$220$	−67.3245	−4.53902
$221$	4.58556	0.308458
$222$	6.54937	0.439565
$223$	−20.4345	−1.36839	−0.684196	−	0.729298i	$-0.739847\pi$
$223$	−20.4345	−1.36839	−0.684196	+	0.729298i	$0.739847\pi$
$224$	0.324736	0.0216974
$225$	60.5001	4.03334
$226$	6.03022	0.401124
$227$	−29.1641	−1.93569	−0.967843	−	0.251555i	$-0.919058\pi$
$227$	−29.1641	−1.93569	−0.967843	+	0.251555i	$0.919058\pi$
$228$	16.1715	1.07098
$229$	−13.2411	−0.874997	−0.437498	−	0.899219i	$-0.644135\pi$
$229$	−13.2411	−0.874997	−0.437498	+	0.899219i	$0.644135\pi$
$230$	−34.0103	−2.24257
$231$	−37.6953	−2.48017
$232$	1.39737	0.0917415
$233$	2.18505	0.143147	0.0715736	−	0.997435i	$-0.477198\pi$
$233$	2.18505	0.143147	0.0715736	+	0.997435i	$0.477198\pi$
$234$	13.5292	0.884434
$235$	36.2567	2.36512
$236$	−14.1338	−0.920029
$237$	−22.6267	−1.46976
$238$	34.2709	2.22145
$239$	11.8881	0.768977	0.384489	−	0.923130i	$-0.374378\pi$
$239$	11.8881	0.768977	0.384489	+	0.923130i	$0.374378\pi$
$240$	45.8822	2.96169
$241$	3.75378	0.241802	0.120901	−	0.992665i	$-0.461422\pi$
$241$	3.75378	0.241802	0.120901	+	0.992665i	$0.461422\pi$
$242$	−16.8008	−1.08000
$243$	7.61520	0.488515
$244$	33.2561	2.12900
$245$	−9.28839	−0.593414
$246$	−75.6194	−4.82132
$247$	1.38817	0.0883270
$248$	−36.0276	−2.28775
$249$	−17.5415	−1.11165
$250$	58.0807	3.67335
$251$	5.02627	0.317255	0.158628	−	0.987338i	$-0.449293\pi$
$251$	5.02627	0.317255	0.158628	+	0.987338i	$0.449293\pi$
$252$	67.3473	4.24248
$253$	−14.7109	−0.924864
$254$	23.1128	1.45022
$255$	−53.4710	−3.34848
$256$	−31.9100	−1.99438
$257$	−0.521792	−0.0325485	−0.0162742	−	0.999868i	$-0.505180\pi$
$257$	−0.521792	−0.0325485	−0.0162742	+	0.999868i	$0.505180\pi$
$258$	−0.846341	−0.0526909
$259$	2.79859	0.173896
$260$	15.9283	0.987833
$261$	−1.58695	−0.0982297
$262$	−15.6461	−0.966622
$263$	2.27449	0.140251	0.0701255	−	0.997538i	$-0.477660\pi$
$263$	2.27449	0.140251	0.0701255	+	0.997538i	$0.477660\pi$
$264$	60.0861	3.69804
$265$	−47.2226	−2.90086
$266$	10.3747	0.636113
$267$	−18.6972	−1.14425
$268$	−5.68075	−0.347007
$269$	7.55732	0.460778	0.230389	−	0.973099i	$-0.426000\pi$
$269$	7.55732	0.460778	0.230389	+	0.973099i	$0.426000\pi$
$270$	−72.1501	−4.39092
$271$	−27.8479	−1.69164	−0.845820	−	0.533468i	$-0.820889\pi$
$271$	−27.8479	−1.69164	−0.845820	+	0.533468i	$0.820889\pi$
$272$	−18.0431	−1.09402
$273$	8.91833	0.539762
$274$	−13.9692	−0.843913
$275$	46.2559	2.78934
$276$	40.5454	2.44055
$277$	4.45275	0.267540	0.133770	−	0.991012i	$-0.457292\pi$
$277$	4.45275	0.267540	0.133770	+	0.991012i	$0.457292\pi$
$278$	−30.2515	−1.81436
$279$	40.9155	2.44955
$280$	59.3589	3.54737
$281$	−3.58555	−0.213896	−0.106948	−	0.994265i	$-0.534108\pi$
$281$	−3.58555	−0.213896	−0.106948	+	0.994265i	$0.534108\pi$
$282$	−64.8942	−3.86439
$283$	−6.81856	−0.405322	−0.202661	−	0.979249i	$-0.564959\pi$
$283$	−6.81856	−0.405322	−0.202661	+	0.979249i	$0.564959\pi$
$284$	47.6003	2.82456
$285$	−16.1871	−0.958839
$286$	10.3439	0.611648
$287$	−32.3127	−1.90736
$288$	0.587856	0.0346397
$289$	4.02737	0.236904
$290$	−2.80509	−0.164720
$291$	4.04493	0.237118
$292$	−39.9267	−2.33653
$293$	1.45952	0.0852661	0.0426331	−	0.999091i	$-0.486425\pi$
$293$	1.45952	0.0852661	0.0426331	+	0.999091i	$0.486425\pi$
$294$	16.6249	0.969583
$295$	14.1474	0.823693
$296$	−4.46094	−0.259287
$297$	−31.2079	−1.81087
$298$	56.9180	3.29717
$299$	3.48044	0.201279
$300$	−127.489	−7.36055
$301$	−0.361647	−0.0208450
$302$	−22.1256	−1.27319
$303$	−54.9128	−3.15466
$304$	−5.46211	−0.313274
$305$	−33.2882	−1.90608
$306$	62.0391	3.54654
$307$	−14.5627	−0.831138	−0.415569	−	0.909562i	$-0.636418\pi$
$307$	−14.5627	−0.831138	−0.415569	+	0.909562i	$0.636418\pi$
$308$	51.4910	2.93397
$309$	−12.1659	−0.692095
$310$	72.3221	4.10762
$311$	6.32496	0.358656	0.179328	−	0.983789i	$-0.442608\pi$
$311$	6.32496	0.358656	0.179328	+	0.983789i	$0.442608\pi$
$312$	−14.2158	−0.804810
$313$	24.6527	1.39345	0.696726	−	0.717338i	$-0.254640\pi$
$313$	24.6527	1.39345	0.696726	+	0.717338i	$0.254640\pi$
$314$	−47.7492	−2.69464
$315$	−67.4123	−3.79825
$316$	30.9076	1.73869
$317$	−20.5094	−1.15192	−0.575960	−	0.817478i	$-0.695372\pi$
$317$	−20.5094	−1.15192	−0.575960	+	0.817478i	$0.695372\pi$
$318$	84.5217	4.73974
$319$	−1.21332	−0.0679326
$320$	32.4618	1.81467
$321$	25.3049	1.41238
$322$	26.0117	1.44957
$323$	6.36553	0.354188
$324$	19.8549	1.10305
$325$	−10.9437	−0.607047
$326$	−24.4103	−1.35196
$327$	−36.5025	−2.01859
$328$	51.5063	2.84396
$329$	−27.7297	−1.52879
$330$	−120.617	−6.63977
$331$	22.9784	1.26301	0.631505	−	0.775372i	$-0.282438\pi$
$331$	22.9784	1.26301	0.631505	+	0.775372i	$0.282438\pi$
$332$	23.9613	1.31505
$333$	5.06617	0.277624
$334$	5.78973	0.316800
$335$	5.68623	0.310672
$336$	−35.0915	−1.91440
$337$	−33.3466	−1.81650	−0.908252	−	0.418424i	$-0.862583\pi$
$337$	−33.3466	−1.81650	−0.908252	+	0.418424i	$0.862583\pi$
$338$	−2.44727	−0.133114
$339$	7.19587	0.390826
$340$	73.0403	3.96117
$341$	31.2823	1.69403
$342$	18.7809	1.01555
$343$	−14.2732	−0.770683
$344$	0.576464	0.0310809
$345$	−40.5845	−2.18500
$346$	−2.45360	−0.131906
$347$	16.3794	0.879295	0.439647	−	0.898170i	$-0.355103\pi$
$347$	16.3794	0.879295	0.439647	+	0.898170i	$0.355103\pi$
$348$	3.34409	0.179262
$349$	36.8798	1.97413	0.987065	−	0.160321i	$-0.0512530\pi$
$349$	36.8798	1.97413	0.987065	+	0.160321i	$0.0512530\pi$
$350$	−81.7894	−4.37183
$351$	7.38349	0.394101
$352$	0.449450	0.0239558
$353$	3.70325	0.197104	0.0985520	−	0.995132i	$-0.468579\pi$
$353$	3.70325	0.197104	0.0985520	+	0.995132i	$0.468579\pi$
$354$	−25.3218	−1.34584
$355$	−47.6463	−2.52880
$356$	25.5400	1.35362
$357$	40.8956	2.16442
$358$	37.7208	1.99360
$359$	11.3399	0.598498	0.299249	−	0.954175i	$-0.403264\pi$
$359$	11.3399	0.598498	0.299249	+	0.954175i	$0.403264\pi$
$360$	107.455	5.66337
$361$	−17.0730	−0.898578
$362$	64.1116	3.36963
$363$	−20.0485	−1.05227
$364$	−12.1823	−0.638524
$365$	39.9652	2.09188
$366$	59.5810	3.11435
$367$	−30.7955	−1.60751	−0.803755	−	0.594961i	$-0.797168\pi$
$367$	−30.7955	−1.60751	−0.803755	+	0.594961i	$0.797168\pi$
$368$	−13.6947	−0.713887
$369$	−58.4943	−3.04509
$370$	8.95494	0.465545
$371$	36.1167	1.87509
$372$	−86.2190	−4.47024
$373$	17.0578	0.883221	0.441611	−	0.897207i	$-0.354407\pi$
$373$	17.0578	0.883221	0.441611	+	0.897207i	$0.354407\pi$
$374$	47.4326	2.45268
$375$	69.3078	3.57904
$376$	44.2011	2.27950
$377$	0.287059	0.0147843
$378$	55.1817	2.83824
$379$	22.9907	1.18095	0.590476	−	0.807055i	$-0.298940\pi$
$379$	22.9907	1.18095	0.590476	+	0.807055i	$0.298940\pi$
$380$	22.1112	1.13428
$381$	27.5805	1.41299
$382$	43.9987	2.25117
$383$	20.4029	1.04254	0.521269	−	0.853392i	$-0.325459\pi$
$383$	20.4029	1.04254	0.521269	+	0.853392i	$0.325459\pi$
$384$	−57.4808	−2.93331
$385$	−51.5407	−2.62675
$386$	10.5070	0.534792
$387$	−0.654674	−0.0332790
$388$	−5.52530	−0.280505
$389$	16.8782	0.855760	0.427880	−	0.903836i	$-0.359261\pi$
$389$	16.8782	0.855760	0.427880	+	0.903836i	$0.359261\pi$
$390$	28.5369	1.44502
$391$	15.9598	0.807121
$392$	−11.3236	−0.571929
$393$	−18.6706	−0.941806
$394$	−39.6155	−1.99580
$395$	−30.9374	−1.55663
$396$	93.2119	4.68407
$397$	34.4040	1.72668	0.863342	−	0.504618i	$-0.168367\pi$
$397$	34.4040	1.72668	0.863342	+	0.504618i	$0.168367\pi$
$398$	17.9507	0.899788
$399$	12.3801	0.619783
$400$	43.0609	2.15304
$401$	34.3640	1.71606	0.858028	−	0.513603i	$-0.171689\pi$
$401$	34.3640	1.71606	0.858028	+	0.513603i	$0.171689\pi$
$402$	−10.1775	−0.507609
$403$	−7.40109	−0.368674
$404$	75.0098	3.73188
$405$	−19.8740	−0.987548
$406$	2.14538	0.106473
$407$	3.87338	0.191996
$408$	−65.1873	−3.22725
$409$	29.2457	1.44611	0.723054	−	0.690791i	$-0.242738\pi$
$409$	29.2457	1.44611	0.723054	+	0.690791i	$0.242738\pi$
$410$	−103.394	−5.10628
$411$	−16.6695	−0.822247
$412$	16.6184	0.818730
$413$	−10.8202	−0.532426
$414$	47.0878	2.31424
$415$	−23.9844	−1.17735
$416$	−0.106336	−0.00521353
$417$	−36.0992	−1.76778
$418$	14.3591	0.702325
$419$	−15.6811	−0.766073	−0.383037	−	0.923733i	$-0.625122\pi$
$419$	−15.6811	−0.766073	−0.383037	+	0.923733i	$0.625122\pi$
$420$	142.054	6.93153
$421$	34.7108	1.69170	0.845850	−	0.533420i	$-0.179094\pi$
$421$	34.7108	1.69170	0.845850	+	0.533420i	$0.179094\pi$
$422$	34.6621	1.68733
$423$	−50.1979	−2.44071
$424$	−57.5698	−2.79584
$425$	−50.1830	−2.43423
$426$	85.2799	4.13183
$427$	25.4594	1.23207
$428$	−34.5660	−1.67081
$429$	12.3434	0.595945
$430$	−1.15720	−0.0558051
$431$	−20.7106	−0.997597	−0.498798	−	0.866718i	$-0.666225\pi$
$431$	−20.7106	−0.997597	−0.498798	+	0.866718i	$0.666225\pi$
$432$	−29.0523	−1.39778
$433$	−32.1788	−1.54641	−0.773207	−	0.634153i	$-0.781349\pi$
$433$	−32.1788	−1.54641	−0.773207	+	0.634153i	$0.781349\pi$
$434$	−55.3132	−2.65512
$435$	−3.34731	−0.160491
$436$	49.8617	2.38794
$437$	4.83144	0.231119
$438$	−71.5320	−3.41793
$439$	10.2475	0.489086	0.244543	−	0.969638i	$-0.421362\pi$
$439$	10.2475	0.489086	0.244543	+	0.969638i	$0.421362\pi$
$440$	82.1556	3.91661
$441$	12.8599	0.612377
$442$	−11.2221	−0.533780
$443$	6.39874	0.304013	0.152007	−	0.988379i	$-0.451427\pi$
$443$	6.39874	0.304013	0.152007	+	0.988379i	$0.451427\pi$
$444$	−10.6757	−0.506644
$445$	−25.5646	−1.21188
$446$	50.0086	2.36797
$447$	67.9203	3.21252
$448$	−24.8273	−1.17298
$449$	−18.5358	−0.874759	−0.437379	−	0.899277i	$-0.644093\pi$
$449$	−18.5358	−0.874759	−0.437379	+	0.899277i	$0.644093\pi$
$450$	−148.060	−6.97961
$451$	−44.7223	−2.10589
$452$	−9.82941	−0.462337
$453$	−26.4026	−1.24050
$454$	71.3722	3.34966
$455$	12.1940	0.571664
$456$	−19.7339	−0.924124
$457$	−32.2356	−1.50792	−0.753959	−	0.656921i	$-0.771858\pi$
$457$	−32.2356	−1.50792	−0.753959	+	0.656921i	$0.771858\pi$
$458$	32.4045	1.51416
$459$	33.8574	1.58033
$460$	55.4377	2.58479
$461$	28.5952	1.33181	0.665906	−	0.746035i	$-0.268045\pi$
$461$	28.5952	1.33181	0.665906	+	0.746035i	$0.268045\pi$
$462$	92.2503	4.29187
$463$	0.424350	0.0197212	0.00986061	−	0.999951i	$-0.496861\pi$
$463$	0.424350	0.0197212	0.00986061	+	0.999951i	$0.496861\pi$
$464$	−1.12951	−0.0524361
$465$	86.3021	4.00217
$466$	−5.34739	−0.247713
$467$	−15.1209	−0.699714	−0.349857	−	0.936803i	$-0.613770\pi$
$467$	−15.1209	−0.699714	−0.349857	+	0.936803i	$0.613770\pi$
$468$	−22.0530	−1.01940
$469$	−4.34893	−0.200815
$470$	−88.7297	−4.09279
$471$	−56.9792	−2.62546
$472$	17.2473	0.793871
$473$	−0.500537	−0.0230147
$474$	55.3735	2.54339
$475$	−15.1917	−0.697042
$476$	−55.8625	−2.56045
$477$	65.3805	2.99357
$478$	−29.0933	−1.33070
$479$	−27.1413	−1.24012	−0.620058	−	0.784556i	$-0.712891\pi$
$479$	−27.1413	−1.24012	−0.620058	+	0.784556i	$0.712891\pi$
$480$	1.23995	0.0565957
$481$	−0.916404	−0.0417844
$482$	−9.18649	−0.418433
$483$	31.0398	1.41236
$484$	27.3858	1.24481
$485$	5.53063	0.251133
$486$	−18.6364	−0.845365
$487$	−18.3009	−0.829292	−0.414646	−	0.909983i	$-0.636095\pi$
$487$	−18.3009	−0.829292	−0.414646	+	0.909983i	$0.636095\pi$
$488$	−40.5821	−1.83707
$489$	−29.1288	−1.31725
$490$	22.7312	1.02689
$491$	−26.8901	−1.21353	−0.606767	−	0.794880i	$-0.707534\pi$
$491$	−26.8901	−1.21353	−0.606767	+	0.794880i	$0.707534\pi$
$492$	123.262	5.55707
$493$	1.31632	0.0592843
$494$	−3.39722	−0.152848
$495$	−93.3018	−4.19360
$496$	29.1215	1.30760
$497$	36.4407	1.63459
$498$	42.9287	1.92368
$499$	−2.20530	−0.0987230	−0.0493615	−	0.998781i	$-0.515719\pi$
$499$	−2.20530	−0.0987230	−0.0493615	+	0.998781i	$0.515719\pi$
$500$	−94.6731	−4.23391
$501$	6.90890	0.308667
$502$	−12.3006	−0.549003
$503$	−3.65724	−0.163068	−0.0815341	−	0.996671i	$-0.525982\pi$
$503$	−3.65724	−0.163068	−0.0815341	+	0.996671i	$0.525982\pi$
$504$	−82.1834	−3.66074
$505$	−75.0822	−3.34111
$506$	36.0014	1.60046
$507$	−2.92033	−0.129696
$508$	−37.6744	−1.67153
$509$	−30.3282	−1.34427	−0.672137	−	0.740427i	$-0.734623\pi$
$509$	−30.3282	−1.34427	−0.672137	+	0.740427i	$0.734623\pi$
$510$	130.858	5.79448
$511$	−30.5661	−1.35216
$512$	38.7263	1.71148
$513$	10.2495	0.452528
$514$	1.27696	0.0563244
$515$	−16.6344	−0.733001
$516$	1.37956	0.0607317
$517$	−38.3793	−1.68792
$518$	−6.84889	−0.300923
$519$	−2.92788	−0.128520
$520$	−19.4372	−0.852378
$521$	5.47341	0.239794	0.119897	−	0.992786i	$-0.461744\pi$
$521$	5.47341	0.239794	0.119897	+	0.992786i	$0.461744\pi$
$522$	3.88369	0.169984
$523$	−25.7567	−1.12626	−0.563130	−	0.826368i	$-0.690403\pi$
$523$	−25.7567	−1.12626	−0.563130	+	0.826368i	$0.690403\pi$
$524$	25.5036	1.11413
$525$	−97.5995	−4.25959
$526$	−5.56628	−0.242701
$527$	−33.9381	−1.47837
$528$	−48.5684	−2.11367
$529$	−10.8865	−0.473326
$530$	115.566	5.01988
$531$	−19.5873	−0.850016
$532$	−16.9110	−0.733186
$533$	10.5809	0.458308
$534$	45.7569	1.98010
$535$	34.5993	1.49586
$536$	6.93217	0.299424
$537$	45.0123	1.94242
$538$	−18.4948	−0.797366
$539$	9.83217	0.423502
$540$	117.607	5.06098
$541$	24.5955	1.05744	0.528722	−	0.848795i	$-0.322671\pi$
$541$	24.5955	1.05744	0.528722	+	0.848795i	$0.322671\pi$
$542$	68.1513	2.92735
$543$	76.5045	3.28312
$544$	−0.487608	−0.0209060
$545$	−49.9098	−2.13790
$546$	−21.8255	−0.934046
$547$	−12.5900	−0.538310	−0.269155	−	0.963097i	$-0.586744\pi$
$547$	−12.5900	−0.538310	−0.269155	+	0.963097i	$0.586744\pi$
$548$	22.7702	0.972696
$549$	46.0880	1.96699
$550$	−113.200	−4.82688
$551$	0.398486	0.0169761
$552$	−49.4772	−2.10589
$553$	23.6614	1.00619
$554$	−10.8971	−0.462971
$555$	10.6859	0.453593
$556$	49.3107	2.09124
$557$	8.78772	0.372348	0.186174	−	0.982517i	$-0.440391\pi$
$557$	8.78772	0.372348	0.186174	+	0.982517i	$0.440391\pi$
$558$	−100.131	−4.23889
$559$	0.118422	0.00500872
$560$	−47.9806	−2.02755
$561$	56.6014	2.38971
$562$	8.77479	0.370142
$563$	−20.1794	−0.850459	−0.425229	−	0.905086i	$-0.639807\pi$
$563$	−20.1794	−0.850459	−0.425229	+	0.905086i	$0.639807\pi$
$564$	105.779	4.45411
$565$	9.83890	0.413925
$566$	16.6868	0.701400
$567$	15.2000	0.638340
$568$	−58.0863	−2.43725
$569$	−33.3361	−1.39752	−0.698762	−	0.715354i	$-0.746265\pi$
$569$	−33.3361	−1.39752	−0.698762	+	0.715354i	$0.746265\pi$
$570$	39.6140	1.65925
$571$	−6.00941	−0.251486	−0.125743	−	0.992063i	$-0.540131\pi$
$571$	−6.00941	−0.251486	−0.125743	+	0.992063i	$0.540131\pi$
$572$	−16.8608	−0.704987
$573$	52.5038	2.19338
$574$	79.0777	3.30064
$575$	−38.0889	−1.58842
$576$	−44.9438	−1.87266
$577$	23.1400	0.963329	0.481664	−	0.876356i	$-0.340032\pi$
$577$	23.1400	0.963329	0.481664	+	0.876356i	$0.340032\pi$
$578$	−9.85604	−0.409957
$579$	12.5380	0.521063
$580$	4.57236	0.189857
$581$	18.3437	0.761025
$582$	−9.89903	−0.410328
$583$	49.9872	2.07026
$584$	48.7222	2.01614
$585$	22.0743	0.912660
$586$	−3.57183	−0.147551
$587$	4.16401	0.171867	0.0859336	−	0.996301i	$-0.472613\pi$
$587$	4.16401	0.171867	0.0859336	+	0.996301i	$0.472613\pi$
$588$	−27.0990	−1.11754
$589$	−10.2740	−0.423331
$590$	−34.6224	−1.42538
$591$	−47.2733	−1.94456
$592$	3.60584	0.148199
$593$	14.7480	0.605627	0.302814	−	0.953050i	$-0.402074\pi$
$593$	14.7480	0.605627	0.302814	+	0.953050i	$0.402074\pi$
$594$	76.3741	3.13367
$595$	55.9164	2.29235
$596$	−92.7778	−3.80033
$597$	21.4206	0.876688
$598$	−8.51757	−0.348309
$599$	36.0562	1.47322	0.736608	−	0.676320i	$-0.236426\pi$
$599$	36.0562	1.47322	0.736608	+	0.676320i	$0.236426\pi$
$600$	155.573	6.35125
$601$	17.6525	0.720062	0.360031	−	0.932940i	$-0.382766\pi$
$601$	17.6525	0.720062	0.360031	+	0.932940i	$0.382766\pi$
$602$	0.885047	0.0360718
$603$	−7.87267	−0.320600
$604$	36.0654	1.46748
$605$	−27.4122	−1.11446
$606$	134.386	5.45907
$607$	26.4716	1.07445	0.537225	−	0.843439i	$-0.319473\pi$
$607$	26.4716	1.07445	0.537225	+	0.843439i	$0.319473\pi$
$608$	−0.147612	−0.00598644
$609$	2.56008	0.103740
$610$	81.4650	3.29842
$611$	9.08016	0.367344
$612$	−101.125	−4.08775
$613$	−19.3411	−0.781180	−0.390590	−	0.920565i	$-0.627729\pi$
$613$	−19.3411	−0.781180	−0.390590	+	0.920565i	$0.627729\pi$
$614$	35.6388	1.43827
$615$	−123.381	−4.97519
$616$	−62.8340	−2.53165
$617$	1.00000	0.0402585
$617$	1.00000	0.0402585
$618$	29.7732	1.19766
$619$	42.7369	1.71774	0.858871	−	0.512192i	$-0.171166\pi$
$619$	42.7369	1.71774	0.858871	+	0.512192i	$0.171166\pi$
$620$	−117.887	−4.73445
$621$	25.6978	1.03122
$622$	−15.4789	−0.620646
$623$	19.5522	0.783344
$624$	11.4908	0.460000
$625$	40.0459	1.60184
$626$	−60.3316	−2.41134
$627$	17.1347	0.684294
$628$	77.8324	3.10585
$629$	−4.20223	−0.167554
$630$	164.976	6.57279
$631$	−22.9693	−0.914394	−0.457197	−	0.889366i	$-0.651146\pi$
$631$	−22.9693	−0.914394	−0.457197	+	0.889366i	$0.651146\pi$
$632$	−37.7162	−1.50027
$633$	41.3624	1.64401
$634$	50.1918	1.99337
$635$	37.7107	1.49651
$636$	−137.773	−5.46304
$637$	−2.32620	−0.0921672
$638$	2.96931	0.117556
$639$	65.9670	2.60961
$640$	−78.5934	−3.10668
$641$	−33.5596	−1.32553	−0.662763	−	0.748829i	$-0.730616\pi$
$641$	−33.5596	−1.32553	−0.662763	+	0.748829i	$0.730616\pi$
$642$	−61.9278	−2.44410
$643$	−8.10712	−0.319714	−0.159857	−	0.987140i	$-0.551103\pi$
$643$	−8.10712	−0.319714	−0.159857	+	0.987140i	$0.551103\pi$
$644$	−42.3997	−1.67078
$645$	−1.38089	−0.0543725
$646$	−15.5781	−0.612914
$647$	7.30779	0.287299	0.143649	−	0.989629i	$-0.454116\pi$
$647$	7.30779	0.287299	0.143649	+	0.989629i	$0.454116\pi$
$648$	−24.2287	−0.951795
$649$	−14.9756	−0.587845
$650$	26.7821	1.05048
$651$	−66.0054	−2.58695
$652$	39.7894	1.55827
$653$	−2.15504	−0.0843332	−0.0421666	−	0.999111i	$-0.513426\pi$
$653$	−2.15504	−0.0843332	−0.0421666	+	0.999111i	$0.513426\pi$
$654$	89.3313	3.49313
$655$	−25.5282	−0.997470
$656$	−41.6332	−1.62550
$657$	−55.3325	−2.15873
$658$	67.8620	2.64554
$659$	20.1858	0.786327	0.393164	−	0.919469i	$-0.371381\pi$
$659$	20.1858	0.786327	0.393164	+	0.919469i	$0.371381\pi$
$660$	196.610	7.65302
$661$	−5.75251	−0.223747	−0.111873	−	0.993722i	$-0.535685\pi$
$661$	−5.75251	−0.223747	−0.111873	+	0.993722i	$0.535685\pi$
$662$	−56.2343	−2.18561
$663$	−13.3913	−0.520076
$664$	−29.2398	−1.13472
$665$	16.9273	0.656414
$666$	−12.3983	−0.480423
$667$	0.999092	0.0386850
$668$	−9.43742	−0.365145
$669$	59.6753	2.30718
$670$	−13.9157	−0.537611
$671$	35.2370	1.36031
$672$	−0.948336	−0.0365829
$673$	19.8393	0.764748	0.382374	−	0.924008i	$-0.375107\pi$
$673$	19.8393	0.764748	0.382374	+	0.924008i	$0.375107\pi$
$674$	81.6079	3.14342
$675$	−80.8026	−3.11009
$676$	3.98911	0.153427
$677$	−46.7115	−1.79527	−0.897634	−	0.440742i	$-0.854715\pi$
$677$	−46.7115	−1.79527	−0.897634	+	0.440742i	$0.854715\pi$
$678$	−17.6102	−0.676316
$679$	−4.22992	−0.162329
$680$	−89.1305	−3.41800
$681$	85.1686	3.26367
$682$	−76.5561	−2.93148
$683$	30.0249	1.14887	0.574435	−	0.818550i	$-0.305222\pi$
$683$	30.0249	1.14887	0.574435	+	0.818550i	$0.305222\pi$
$684$	−30.6133	−1.17053
$685$	−22.7922	−0.870845
$686$	34.9304	1.33365
$687$	38.6684	1.47529
$688$	−0.465963	−0.0177647
$689$	−11.8265	−0.450553
$690$	99.3211	3.78109
$691$	−34.4985	−1.31239	−0.656193	−	0.754593i	$-0.727834\pi$
$691$	−34.4985	−1.31239	−0.656193	+	0.754593i	$0.727834\pi$
$692$	3.99943	0.152036
$693$	71.3588	2.71070
$694$	−40.0849	−1.52160
$695$	−49.3583	−1.87227
$696$	−4.08076	−0.154681
$697$	48.5192	1.83779
$698$	−90.2546	−3.41619
$699$	−6.38105	−0.241353
$700$	133.319	5.03898
$701$	−11.9649	−0.451906	−0.225953	−	0.974138i	$-0.572550\pi$
$701$	−11.9649	−0.451906	−0.225953	+	0.974138i	$0.572550\pi$
$702$	−18.0694	−0.681984
$703$	−1.27212	−0.0479790
$704$	−34.3622	−1.29507
$705$	−105.881	−3.98772
$706$	−9.06283	−0.341084
$707$	57.4242	2.15966
$708$	41.2752	1.55122
$709$	−0.846445	−0.0317889	−0.0158945	−	0.999874i	$-0.505060\pi$
$709$	−0.846445	−0.0317889	−0.0158945	+	0.999874i	$0.505060\pi$
$710$	116.603	4.37603
$711$	42.8333	1.60637
$712$	−31.1662	−1.16800
$713$	−25.7591	−0.964685
$714$	−100.082	−3.74549
$715$	16.8771	0.631168
$716$	−61.4858	−2.29783
$717$	−34.7171	−1.29654
$718$	−27.7518	−1.03569
$719$	−5.17555	−0.193015	−0.0965076	−	0.995332i	$-0.530767\pi$
$719$	−5.17555	−0.193015	−0.0965076	+	0.995332i	$0.530767\pi$
$720$	−86.8571	−3.23697
$721$	12.7223	0.473803
$722$	41.7821	1.55497
$723$	−10.9623	−0.407691
$724$	−104.504	−3.88384
$725$	−3.14148	−0.116672
$726$	49.0639	1.82093
$727$	33.4540	1.24074	0.620370	−	0.784309i	$-0.286982\pi$
$727$	33.4540	1.24074	0.620370	+	0.784309i	$0.286982\pi$
$728$	14.8659	0.550967
$729$	−37.1707	−1.37669
$730$	−97.8055	−3.61994
$731$	0.543032	0.0200848
$732$	−97.1187	−3.58961
$733$	−2.09378	−0.0773356	−0.0386678	−	0.999252i	$-0.512311\pi$
$733$	−2.09378	−0.0773356	−0.0386678	+	0.999252i	$0.512311\pi$
$734$	75.3647	2.78176
$735$	27.1251	1.00053
$736$	−0.370095	−0.0136419
$737$	−6.01912	−0.221717
$738$	143.151	5.26946
$739$	−38.0346	−1.39913	−0.699563	−	0.714571i	$-0.746622\pi$
$739$	−38.0346	−1.39913	−0.699563	+	0.714571i	$0.746622\pi$
$740$	−14.5968	−0.536589
$741$	−4.05390	−0.148924
$742$	−88.3871	−3.24479
$743$	49.4749	1.81506	0.907529	−	0.419989i	$-0.137966\pi$
$743$	49.4749	1.81506	0.907529	+	0.419989i	$0.137966\pi$
$744$	105.212	3.85727
$745$	92.8673	3.40239
$746$	−41.7450	−1.52839
$747$	33.2068	1.21497
$748$	−77.3164	−2.82697
$749$	−26.4622	−0.966907
$750$	−169.615	−6.19345
$751$	−15.6507	−0.571103	−0.285552	−	0.958363i	$-0.592177\pi$
$751$	−15.6507	−0.571103	−0.285552	+	0.958363i	$0.592177\pi$
$752$	−35.7283	−1.30288
$753$	−14.6784	−0.534909
$754$	−0.702509	−0.0255839
$755$	−36.1002	−1.31382
$756$	−89.9476	−3.27136
$757$	3.70817	0.134776	0.0673879	−	0.997727i	$-0.478533\pi$
$757$	3.70817	0.134776	0.0673879	+	0.997727i	$0.478533\pi$
$758$	−56.2643	−2.04361
$759$	42.9605	1.55937
$760$	−26.9821	−0.978744
$761$	−1.85856	−0.0673728	−0.0336864	−	0.999432i	$-0.510725\pi$
$761$	−1.85856	−0.0673728	−0.0336864	+	0.999432i	$0.510725\pi$
$762$	−67.4968	−2.44515
$763$	38.1719	1.38191
$764$	−71.7191	−2.59471
$765$	101.223	3.65973
$766$	−49.9312	−1.80409
$767$	3.54309	0.127933
$768$	93.1877	3.36262
$769$	29.6114	1.06782	0.533908	−	0.845543i	$-0.320723\pi$
$769$	29.6114	1.06782	0.533908	+	0.845543i	$0.320723\pi$
$770$	126.134	4.54554
$771$	1.52380	0.0548784
$772$	−17.1267	−0.616403
$773$	42.7162	1.53639	0.768197	−	0.640213i	$-0.221154\pi$
$773$	42.7162	1.53639	0.768197	+	0.640213i	$0.221154\pi$
$774$	1.60216	0.0575885
$775$	80.9952	2.90943
$776$	6.74247	0.242041
$777$	−8.17280	−0.293198
$778$	−41.3055	−1.48087
$779$	14.6880	0.526253
$780$	−46.5159	−1.66554
$781$	50.4356	1.80473
$782$	−39.0578	−1.39671
$783$	2.11949	0.0757445
$784$	9.15303	0.326894
$785$	−77.9075	−2.78064
$786$	45.6918	1.62977
$787$	39.6256	1.41250	0.706249	−	0.707963i	$-0.250386\pi$
$787$	39.6256	1.41250	0.706249	+	0.707963i	$0.250386\pi$
$788$	64.5743	2.30036
$789$	−6.64226	−0.236471
$790$	75.7120	2.69371
$791$	−7.52496	−0.267557
$792$	−113.746	−4.04178
$793$	−8.33673	−0.296046
$794$	−84.1956	−2.98799
$795$	137.905	4.89100
$796$	−29.2601	−1.03710
$797$	−44.9644	−1.59272	−0.796360	−	0.604823i	$-0.793244\pi$
$797$	−44.9644	−1.59272	−0.796360	+	0.604823i	$0.793244\pi$
$798$	−30.2975	−1.07252
$799$	41.6376	1.47303
$800$	1.16370	0.0411431
$801$	35.3946	1.25061
$802$	−84.0978	−2.96960
$803$	−42.3049	−1.49291
$804$	16.5896	0.585071
$805$	42.4406	1.49583
$806$	18.1124	0.637983
$807$	−22.0699	−0.776896
$808$	−91.5339	−3.22015
$809$	15.4408	0.542871	0.271435	−	0.962457i	$-0.412502\pi$
$809$	15.4408	0.542871	0.271435	+	0.962457i	$0.412502\pi$
$810$	48.6370	1.70893
$811$	−11.7791	−0.413620	−0.206810	−	0.978381i	$-0.566308\pi$
$811$	−11.7791	−0.413620	−0.206810	+	0.978381i	$0.566308\pi$
$812$	−3.49702	−0.122722
$813$	81.3250	2.85219
$814$	−9.47919	−0.332246
$815$	−39.8278	−1.39511
$816$	52.6918	1.84458
$817$	0.164390	0.00575127
$818$	−71.5721	−2.50246
$819$	−16.8828	−0.589933
$820$	168.535	5.88551
$821$	42.3117	1.47669	0.738345	−	0.674423i	$-0.235608\pi$
$821$	42.3117	1.47669	0.738345	+	0.674423i	$0.235608\pi$
$822$	40.7947	1.42288
$823$	−15.4147	−0.537323	−0.268661	−	0.963235i	$-0.586581\pi$
$823$	−15.4147	−0.537323	−0.268661	+	0.963235i	$0.586581\pi$
$824$	−20.2793	−0.706463
$825$	−135.082	−4.70296
$826$	26.4798	0.921351
$827$	−7.69515	−0.267586	−0.133793	−	0.991009i	$-0.542716\pi$
$827$	−7.69515	−0.267586	−0.133793	+	0.991009i	$0.542716\pi$
$828$	−76.7543	−2.66740
$829$	30.9350	1.07442	0.537209	−	0.843449i	$-0.319479\pi$
$829$	30.9350	1.07442	0.537209	+	0.843449i	$0.319479\pi$
$830$	58.6962	2.03738
$831$	−13.0035	−0.451086
$832$	8.12976	0.281849
$833$	−10.6669	−0.369587
$834$	88.3443	3.05911
$835$	9.44652	0.326910
$836$	−23.4057	−0.809502
$837$	−54.6458	−1.88884
$838$	38.3759	1.32567
$839$	−9.04629	−0.312313	−0.156156	−	0.987732i	$-0.549910\pi$
$839$	−9.04629	−0.312313	−0.156156	+	0.987732i	$0.549910\pi$
$840$	−173.347	−5.98106
$841$	−28.9176	−0.997159
$842$	−84.9465	−2.92745
$843$	10.4710	0.360639
$844$	−56.5002	−1.94482
$845$	−3.99296	−0.137362
$846$	122.848	4.22359
$847$	20.9653	0.720377
$848$	46.5344	1.59800
$849$	19.9124	0.683393
$850$	122.811	4.21238
$851$	−3.18949	−0.109334
$852$	−139.009	−4.76235
$853$	−31.8131	−1.08926	−0.544629	−	0.838677i	$-0.683330\pi$
$853$	−31.8131	−1.08926	−0.544629	+	0.838677i	$0.683330\pi$
$854$	−62.3059	−2.13206
$855$	30.6428	1.04796
$856$	42.1806	1.44170
$857$	−40.3078	−1.37689	−0.688443	−	0.725290i	$-0.741706\pi$
$857$	−40.3078	−1.37689	−0.688443	+	0.725290i	$0.741706\pi$
$858$	−30.2076	−1.03127
$859$	26.2046	0.894090	0.447045	−	0.894511i	$-0.352476\pi$
$859$	26.2046	0.894090	0.447045	+	0.894511i	$0.352476\pi$
$860$	1.88627	0.0643212
$861$	94.3636	3.21590
$862$	50.6844	1.72632
$863$	17.3095	0.589222	0.294611	−	0.955617i	$-0.404810\pi$
$863$	17.3095	0.589222	0.294611	+	0.955617i	$0.404810\pi$
$864$	−0.785127	−0.0267106
$865$	−4.00329	−0.136116
$866$	78.7501	2.67604
$867$	−11.7612	−0.399432
$868$	90.1620	3.06030
$869$	32.7486	1.11092
$870$	8.19177	0.277727
$871$	1.42406	0.0482526
$872$	−60.8458	−2.06050
$873$	−7.65724	−0.259158
$874$	−11.8238	−0.399947
$875$	−72.4775	−2.45019
$876$	116.599	3.93952
$877$	−10.2702	−0.346799	−0.173399	−	0.984852i	$-0.555475\pi$
$877$	−10.2702	−0.346799	−0.173399	+	0.984852i	$0.555475\pi$
$878$	−25.0783	−0.846353
$879$	−4.26228	−0.143763
$880$	−66.4074	−2.23859
$881$	−30.3137	−1.02129	−0.510647	−	0.859790i	$-0.670594\pi$
$881$	−30.3137	−1.02129	−0.510647	+	0.859790i	$0.670594\pi$
$882$	−31.4717	−1.05971
$883$	28.4662	0.957965	0.478983	−	0.877824i	$-0.341006\pi$
$883$	28.4662	0.957965	0.478983	+	0.877824i	$0.341006\pi$
$884$	18.2923	0.615236
$885$	−41.3150	−1.38879
$886$	−15.6594	−0.526088
$887$	1.14152	0.0383287	0.0191643	−	0.999816i	$-0.493899\pi$
$887$	1.14152	0.0383287	0.0191643	+	0.999816i	$0.493899\pi$
$888$	13.0274	0.437171
$889$	−28.8418	−0.967324
$890$	62.5633	2.09713
$891$	21.0375	0.704784
$892$	−81.5153	−2.72933
$893$	12.6048	0.421803
$894$	−166.219	−5.55920
$895$	61.5451	2.05723
$896$	60.1096	2.00812
$897$	−10.1640	−0.339367
$898$	45.3620	1.51375
$899$	−2.12455	−0.0708576
$900$	241.341	8.04471
$901$	−54.2311	−1.80670
$902$	109.447	3.64420
$903$	1.05613	0.0351457
$904$	11.9948	0.398939
$905$	104.604	3.47717
$906$	64.6141	2.14666
$907$	−2.98177	−0.0990082	−0.0495041	−	0.998774i	$-0.515764\pi$
$907$	−2.98177	−0.0990082	−0.0495041	+	0.998774i	$0.515764\pi$
$908$	−116.339	−3.86083
$909$	103.952	3.44788
$910$	−29.8420	−0.989252
$911$	42.1441	1.39630	0.698148	−	0.715953i	$-0.254008\pi$
$911$	42.1441	1.39630	0.698148	+	0.715953i	$0.254008\pi$
$912$	15.9512	0.528196
$913$	25.3886	0.840238
$914$	78.8891	2.60942
$915$	97.2123	3.21374
$916$	−52.8202	−1.74523
$917$	19.5244	0.644754
$918$	−82.8581	−2.73473
$919$	−9.80473	−0.323428	−0.161714	−	0.986838i	$-0.551702\pi$
$919$	−9.80473	−0.323428	−0.161714	+	0.986838i	$0.551702\pi$
$920$	−67.6501	−2.23036
$921$	42.5279	1.40134
$922$	−69.9801	−2.30467
$923$	−11.9326	−0.392766
$924$	−150.371	−4.94683
$925$	10.0288	0.329746
$926$	−1.03850	−0.0341271
$927$	23.0306	0.756425
$928$	−0.0305245	−0.00100202
$929$	21.9562	0.720359	0.360179	−	0.932883i	$-0.382715\pi$
$929$	21.9562	0.720359	0.360179	+	0.932883i	$0.382715\pi$
$930$	−211.204	−6.92566
$931$	−3.22915	−0.105831
$932$	8.71638	0.285515
$933$	−18.4710	−0.604712
$934$	37.0050	1.21084
$935$	77.3909	2.53095
$936$	26.9111	0.879617
$937$	−35.8446	−1.17099	−0.585496	−	0.810676i	$-0.699100\pi$
$937$	−35.8446	−1.17099	−0.585496	+	0.810676i	$0.699100\pi$
$938$	10.6430	0.347505
$939$	−71.9938	−2.34943
$940$	144.632	4.71737
$941$	−54.5117	−1.77703	−0.888516	−	0.458846i	$-0.848263\pi$
$941$	−54.5117	−1.77703	−0.888516	+	0.458846i	$0.848263\pi$
$942$	139.443	4.54330
$943$	36.8261	1.19922
$944$	−13.9412	−0.453748
$945$	90.0344	2.92882
$946$	1.22495	0.0398265
$947$	38.2376	1.24255	0.621277	−	0.783591i	$-0.286614\pi$
$947$	38.2376	1.24255	0.621277	+	0.783591i	$0.286614\pi$
$948$	−90.2602	−2.93151
$949$	10.0089	0.324904
$950$	37.1781	1.20622
$951$	59.8940	1.94220
$952$	68.1686	2.20936
$953$	48.3668	1.56675	0.783377	−	0.621547i	$-0.213496\pi$
$953$	48.3668	1.56675	0.783377	+	0.621547i	$0.213496\pi$
$954$	−160.003	−5.18030
$955$	71.7883	2.32302
$956$	47.4229	1.53377
$957$	3.54328	0.114538
$958$	66.4219	2.14599
$959$	17.4319	0.562904
$960$	−94.7990	−3.05962
$961$	23.7761	0.766971
$962$	2.24268	0.0723071
$963$	−47.9033	−1.54366
$964$	14.9742	0.482288
$965$	17.1432	0.551860
$966$	−75.9625	−2.44405
$967$	50.3196	1.61817	0.809085	−	0.587692i	$-0.199963\pi$
$967$	50.3196	1.61817	0.809085	+	0.587692i	$0.199963\pi$
$968$	−33.4186	−1.07412
$969$	−18.5894	−0.597178
$970$	−13.5349	−0.434580
$971$	−17.3305	−0.556162	−0.278081	−	0.960558i	$-0.589698\pi$
$971$	−17.3305	−0.556162	−0.278081	+	0.960558i	$0.589698\pi$
$972$	30.3779	0.974370
$973$	37.7501	1.21021
$974$	44.7871	1.43507
$975$	31.9592	1.02351
$976$	32.8031	1.05000
$977$	38.8265	1.24217	0.621085	−	0.783743i	$-0.286692\pi$
$977$	38.8265	1.24217	0.621085	+	0.783743i	$0.286692\pi$
$978$	71.2859	2.27947
$979$	27.0612	0.864881
$980$	−37.0524	−1.18360
$981$	69.1009	2.20622
$982$	65.8072	2.09999
$983$	20.0229	0.638632	0.319316	−	0.947648i	$-0.396547\pi$
$983$	20.0229	0.638632	0.319316	+	0.947648i	$0.396547\pi$
$984$	−150.415	−4.79506
$985$	−64.6366	−2.05949
$986$	−3.22140	−0.102590
$987$	80.9799	2.57762
$988$	5.53755	0.176173
$989$	0.412162	0.0131060
$990$	228.334	7.25694
$991$	16.1646	0.513485	0.256742	−	0.966480i	$-0.417351\pi$
$991$	16.1646	0.513485	0.256742	+	0.966480i	$0.417351\pi$
$992$	0.786999	0.0249872
$993$	−67.1045	−2.12950
$994$	−89.1800	−2.82862
$995$	29.2884	0.928504
$996$	−69.9748	−2.21724
$997$	−23.8478	−0.755266	−0.377633	−	0.925955i	$-0.623262\pi$
$997$	−23.8478	−0.755266	−0.377633	+	0.925955i	$0.623262\pi$
$998$	5.39696	0.170838
$999$	−6.76626	−0.214075

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8021.2.a.c.1.16	✓	169

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8021.2.a.c.1.16	✓	169	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8021 = 13 \cdot 617 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8021.a (trivial)

\(f(q)\)	\(=\)	\(q-2.44727 q^{2} -2.92033 q^{3} +3.98911 q^{4} -3.99296 q^{5} +7.14681 q^{6} +3.05388 q^{7} -4.86787 q^{8} +5.52831 q^{9} +O(q^{10})\)	\(q-2.44727 q^{2} -2.92033 q^{3} +3.98911 q^{4} -3.99296 q^{5} +7.14681 q^{6} +3.05388 q^{7} -4.86787 q^{8} +5.52831 q^{9} +9.77182 q^{10} +4.22672 q^{11} -11.6495 q^{12} -1.00000 q^{13} -7.47366 q^{14} +11.6607 q^{15} +3.93476 q^{16} -4.58556 q^{17} -13.5292 q^{18} -1.38817 q^{19} -15.9283 q^{20} -8.91833 q^{21} -10.3439 q^{22} -3.48044 q^{23} +14.2158 q^{24} +10.9437 q^{25} +2.44727 q^{26} -7.38349 q^{27} +12.1823 q^{28} -0.287059 q^{29} -28.5369 q^{30} +7.40109 q^{31} +0.106336 q^{32} -12.3434 q^{33} +11.2221 q^{34} -12.1940 q^{35} +22.0530 q^{36} +0.916404 q^{37} +3.39722 q^{38} +2.92033 q^{39} +19.4372 q^{40} -10.5809 q^{41} +21.8255 q^{42} -0.118422 q^{43} +16.8608 q^{44} -22.0743 q^{45} +8.51757 q^{46} -9.08016 q^{47} -11.4908 q^{48} +2.32620 q^{49} -26.7821 q^{50} +13.3913 q^{51} -3.98911 q^{52} +11.8265 q^{53} +18.0694 q^{54} -16.8771 q^{55} -14.8659 q^{56} +4.05390 q^{57} +0.702509 q^{58} -3.54309 q^{59} +46.5159 q^{60} +8.33673 q^{61} -18.1124 q^{62} +16.8828 q^{63} -8.12976 q^{64} +3.99296 q^{65} +30.2076 q^{66} -1.42406 q^{67} -18.2923 q^{68} +10.1640 q^{69} +29.8420 q^{70} +11.9326 q^{71} -26.9111 q^{72} -10.0089 q^{73} -2.24268 q^{74} -31.9592 q^{75} -5.53755 q^{76} +12.9079 q^{77} -7.14681 q^{78} +7.74799 q^{79} -15.7113 q^{80} +4.97727 q^{81} +25.8942 q^{82} +6.00668 q^{83} -35.5762 q^{84} +18.3099 q^{85} +0.289810 q^{86} +0.838305 q^{87} -20.5751 q^{88} +6.40242 q^{89} +54.0216 q^{90} -3.05388 q^{91} -13.8839 q^{92} -21.6136 q^{93} +22.2216 q^{94} +5.54289 q^{95} -0.310535 q^{96} -1.38510 q^{97} -5.69282 q^{98} +23.3666 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 169 q + 9 q^{2} + 9 q^{3} + 199 q^{4} + 12 q^{5} + 22 q^{6} + 36 q^{7} + 30 q^{8} + 198 q^{9}+O(q^{10}) \) 169 * q + 9 * q^2 + 9 * q^3 + 199 * q^4 + 12 * q^5 + 22 * q^6 + 36 * q^7 + 30 * q^8 + 198 * q^9	\( 169 q + 9 q^{2} + 9 q^{3} + 199 q^{4} + 12 q^{5} + 22 q^{6} + 36 q^{7} + 30 q^{8} + 198 q^{9} - 3 q^{10} + 59 q^{11} + 11 q^{12} - 169 q^{13} + 30 q^{14} + 50 q^{15} + 267 q^{16} + q^{17} + 53 q^{18} + 107 q^{19} + 48 q^{20} + 36 q^{21} + 14 q^{22} + 12 q^{23} + 78 q^{24} + 217 q^{25} - 9 q^{26} + 39 q^{27} + 99 q^{28} + 30 q^{29} - q^{30} + 106 q^{31} + 74 q^{32} + 16 q^{33} + 56 q^{34} + 46 q^{35} + 271 q^{36} + 73 q^{37} + 2 q^{38} - 9 q^{39} - 16 q^{40} + 52 q^{41} - 2 q^{42} + 64 q^{43} + 124 q^{44} + 84 q^{45} + 105 q^{46} + 55 q^{47} + 26 q^{48} + 257 q^{49} + 60 q^{50} + 117 q^{51} - 199 q^{52} + 7 q^{53} + 78 q^{54} - 4 q^{55} + 63 q^{56} + 51 q^{57} + 84 q^{58} + 98 q^{59} + 94 q^{60} + 32 q^{61} - 25 q^{62} + 128 q^{63} + 380 q^{64} - 12 q^{65} + 16 q^{66} + 170 q^{67} - 10 q^{68} + 55 q^{69} + 70 q^{70} + 124 q^{71} + 173 q^{72} + 81 q^{73} + 54 q^{74} + 120 q^{75} + 212 q^{76} + 20 q^{77} - 22 q^{78} + 92 q^{79} + 66 q^{80} + 265 q^{81} + 21 q^{82} + 62 q^{83} + 98 q^{84} + 139 q^{85} + 51 q^{86} - 33 q^{87} + 31 q^{88} + 58 q^{89} + 16 q^{90} - 36 q^{91} + 40 q^{92} + 37 q^{93} + 55 q^{94} + 23 q^{95} + 164 q^{96} + 78 q^{97} + 69 q^{98} + 307 q^{99}+O(q^{100}) \) 169 * q + 9 * q^2 + 9 * q^3 + 199 * q^4 + 12 * q^5 + 22 * q^6 + 36 * q^7 + 30 * q^8 + 198 * q^9 - 3 * q^10 + 59 * q^11 + 11 * q^12 - 169 * q^13 + 30 * q^14 + 50 * q^15 + 267 * q^16 + q^17 + 53 * q^18 + 107 * q^19 + 48 * q^20 + 36 * q^21 + 14 * q^22 + 12 * q^23 + 78 * q^24 + 217 * q^25 - 9 * q^26 + 39 * q^27 + 99 * q^28 + 30 * q^29 - q^30 + 106 * q^31 + 74 * q^32 + 16 * q^33 + 56 * q^34 + 46 * q^35 + 271 * q^36 + 73 * q^37 + 2 * q^38 - 9 * q^39 - 16 * q^40 + 52 * q^41 - 2 * q^42 + 64 * q^43 + 124 * q^44 + 84 * q^45 + 105 * q^46 + 55 * q^47 + 26 * q^48 + 257 * q^49 + 60 * q^50 + 117 * q^51 - 199 * q^52 + 7 * q^53 + 78 * q^54 - 4 * q^55 + 63 * q^56 + 51 * q^57 + 84 * q^58 + 98 * q^59 + 94 * q^60 + 32 * q^61 - 25 * q^62 + 128 * q^63 + 380 * q^64 - 12 * q^65 + 16 * q^66 + 170 * q^67 - 10 * q^68 + 55 * q^69 + 70 * q^70 + 124 * q^71 + 173 * q^72 + 81 * q^73 + 54 * q^74 + 120 * q^75 + 212 * q^76 + 20 * q^77 - 22 * q^78 + 92 * q^79 + 66 * q^80 + 265 * q^81 + 21 * q^82 + 62 * q^83 + 98 * q^84 + 139 * q^85 + 51 * q^86 - 33 * q^87 + 31 * q^88 + 58 * q^89 + 16 * q^90 - 36 * q^91 + 40 * q^92 + 37 * q^93 + 55 * q^94 + 23 * q^95 + 164 * q^96 + 78 * q^97 + 69 * q^98 + 307 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more