LMFDB - Embedded newform 966.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,4,Mod(1,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("966.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 966 = 2 \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	966.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:	$56.9958450655$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	966.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -9.30278 q^{5} +6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -9.30278 q^{5} +6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -18.6056 q^{10} -29.4222 q^{11} +12.0000 q^{12} -35.2750 q^{13} +14.0000 q^{14} -27.9083 q^{15} +16.0000 q^{16} -29.7889 q^{17} +18.0000 q^{18} +43.7166 q^{19} -37.2111 q^{20} +21.0000 q^{21} -58.8444 q^{22} -23.0000 q^{23} +24.0000 q^{24} -38.4584 q^{25} -70.5500 q^{26} +27.0000 q^{27} +28.0000 q^{28} +100.066 q^{29} -55.8167 q^{30} -190.194 q^{31} +32.0000 q^{32} -88.2666 q^{33} -59.5778 q^{34} -65.1194 q^{35} +36.0000 q^{36} -311.994 q^{37} +87.4332 q^{38} -105.825 q^{39} -74.4222 q^{40} -336.966 q^{41} +42.0000 q^{42} -123.553 q^{43} -117.689 q^{44} -83.7250 q^{45} -46.0000 q^{46} +120.094 q^{47} +48.0000 q^{48} +49.0000 q^{49} -76.9167 q^{50} -89.3667 q^{51} -141.100 q^{52} +330.480 q^{53} +54.0000 q^{54} +273.708 q^{55} +56.0000 q^{56} +131.150 q^{57} +200.133 q^{58} -170.497 q^{59} -111.633 q^{60} -275.586 q^{61} -380.389 q^{62} +63.0000 q^{63} +64.0000 q^{64} +328.156 q^{65} -176.533 q^{66} -672.063 q^{67} -119.156 q^{68} -69.0000 q^{69} -130.239 q^{70} -660.113 q^{71} +72.0000 q^{72} -617.367 q^{73} -623.988 q^{74} -115.375 q^{75} +174.866 q^{76} -205.955 q^{77} -211.650 q^{78} +1169.77 q^{79} -148.844 q^{80} +81.0000 q^{81} -673.933 q^{82} +1238.34 q^{83} +84.0000 q^{84} +277.119 q^{85} -247.105 q^{86} +300.199 q^{87} -235.378 q^{88} +851.896 q^{89} -167.450 q^{90} -246.925 q^{91} -92.0000 q^{92} -570.583 q^{93} +240.188 q^{94} -406.686 q^{95} +96.0000 q^{96} -1261.78 q^{97} +98.0000 q^{98} -264.800 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 15 q^{5} + 12 q^{6} + 14 q^{7} + 16 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 - 15 * q^5 + 12 * q^6 + 14 * q^7 + 16 * q^8 + 18 * q^9	\( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 15 q^{5} + 12 q^{6} + 14 q^{7} + 16 q^{8} + 18 q^{9} - 30 q^{10} - 30 q^{11} + 24 q^{12} - 103 q^{13} + 28 q^{14} - 45 q^{15} + 32 q^{16} - 74 q^{17} + 36 q^{18} - 64 q^{19} - 60 q^{20} + 42 q^{21} - 60 q^{22} - 46 q^{23} + 48 q^{24} - 131 q^{25} - 206 q^{26} + 54 q^{27} + 56 q^{28} - 146 q^{29} - 90 q^{30} - 128 q^{31} + 64 q^{32} - 90 q^{33} - 148 q^{34} - 105 q^{35} + 72 q^{36} - 112 q^{37} - 128 q^{38} - 309 q^{39} - 120 q^{40} - 198 q^{41} + 84 q^{42} + 81 q^{43} - 120 q^{44} - 135 q^{45} - 92 q^{46} - 142 q^{47} + 96 q^{48} + 98 q^{49} - 262 q^{50} - 222 q^{51} - 412 q^{52} + 167 q^{53} + 108 q^{54} + 277 q^{55} + 112 q^{56} - 192 q^{57} - 292 q^{58} - 85 q^{59} - 180 q^{60} - 699 q^{61} - 256 q^{62} + 126 q^{63} + 128 q^{64} + 714 q^{65} - 180 q^{66} - 93 q^{67} - 296 q^{68} - 138 q^{69} - 210 q^{70} + 515 q^{71} + 144 q^{72} - 1278 q^{73} - 224 q^{74} - 393 q^{75} - 256 q^{76} - 210 q^{77} - 618 q^{78} + 306 q^{79} - 240 q^{80} + 162 q^{81} - 396 q^{82} - 40 q^{83} + 168 q^{84} + 529 q^{85} + 162 q^{86} - 438 q^{87} - 240 q^{88} - 629 q^{89} - 270 q^{90} - 721 q^{91} - 184 q^{92} - 384 q^{93} - 284 q^{94} + 207 q^{95} + 192 q^{96} - 1514 q^{97} + 196 q^{98} - 270 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 - 15 * q^5 + 12 * q^6 + 14 * q^7 + 16 * q^8 + 18 * q^9 - 30 * q^10 - 30 * q^11 + 24 * q^12 - 103 * q^13 + 28 * q^14 - 45 * q^15 + 32 * q^16 - 74 * q^17 + 36 * q^18 - 64 * q^19 - 60 * q^20 + 42 * q^21 - 60 * q^22 - 46 * q^23 + 48 * q^24 - 131 * q^25 - 206 * q^26 + 54 * q^27 + 56 * q^28 - 146 * q^29 - 90 * q^30 - 128 * q^31 + 64 * q^32 - 90 * q^33 - 148 * q^34 - 105 * q^35 + 72 * q^36 - 112 * q^37 - 128 * q^38 - 309 * q^39 - 120 * q^40 - 198 * q^41 + 84 * q^42 + 81 * q^43 - 120 * q^44 - 135 * q^45 - 92 * q^46 - 142 * q^47 + 96 * q^48 + 98 * q^49 - 262 * q^50 - 222 * q^51 - 412 * q^52 + 167 * q^53 + 108 * q^54 + 277 * q^55 + 112 * q^56 - 192 * q^57 - 292 * q^58 - 85 * q^59 - 180 * q^60 - 699 * q^61 - 256 * q^62 + 126 * q^63 + 128 * q^64 + 714 * q^65 - 180 * q^66 - 93 * q^67 - 296 * q^68 - 138 * q^69 - 210 * q^70 + 515 * q^71 + 144 * q^72 - 1278 * q^73 - 224 * q^74 - 393 * q^75 - 256 * q^76 - 210 * q^77 - 618 * q^78 + 306 * q^79 - 240 * q^80 + 162 * q^81 - 396 * q^82 - 40 * q^83 + 168 * q^84 + 529 * q^85 + 162 * q^86 - 438 * q^87 - 240 * q^88 - 629 * q^89 - 270 * q^90 - 721 * q^91 - 184 * q^92 - 384 * q^93 - 284 * q^94 + 207 * q^95 + 192 * q^96 - 1514 * q^97 + 196 * q^98 - 270 * 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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	−9.30278	−0.832066	−0.416033	−	0.909350i	$-0.636580\pi$
$5$	−9.30278	−0.832066	−0.416033	+	0.909350i	$0.636580\pi$
$6$	6.00000	0.408248
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−18.6056	−0.588359
$11$	−29.4222	−0.806466	−0.403233	−	0.915097i	$-0.632114\pi$
$11$	−29.4222	−0.806466	−0.403233	+	0.915097i	$0.632114\pi$
$12$	12.0000	0.288675
$13$	−35.2750	−0.752579	−0.376290	−	0.926502i	$-0.622800\pi$
$13$	−35.2750	−0.752579	−0.376290	+	0.926502i	$0.622800\pi$
$14$	14.0000	0.267261
$15$	−27.9083	−0.480393
$16$	16.0000	0.250000
$17$	−29.7889	−0.424992	−0.212496	−	0.977162i	$-0.568159\pi$
$17$	−29.7889	−0.424992	−0.212496	+	0.977162i	$0.568159\pi$
$18$	18.0000	0.235702
$19$	43.7166	0.527856	0.263928	−	0.964542i	$-0.414982\pi$
$19$	43.7166	0.527856	0.263928	+	0.964542i	$0.414982\pi$
$20$	−37.2111	−0.416033
$21$	21.0000	0.218218
$22$	−58.8444	−0.570258
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	24.0000	0.204124
$25$	−38.4584	−0.307667
$26$	−70.5500	−0.532154
$27$	27.0000	0.192450
$28$	28.0000	0.188982
$29$	100.066	0.640754	0.320377	−	0.947290i	$-0.396190\pi$
$29$	100.066	0.640754	0.320377	+	0.947290i	$0.396190\pi$
$30$	−55.8167	−0.339689
$31$	−190.194	−1.10193	−0.550966	−	0.834528i	$-0.685741\pi$
$31$	−190.194	−1.10193	−0.550966	+	0.834528i	$0.685741\pi$
$32$	32.0000	0.176777
$33$	−88.2666	−0.465613
$34$	−59.5778	−0.300515
$35$	−65.1194	−0.314491
$36$	36.0000	0.166667
$37$	−311.994	−1.38626	−0.693129	−	0.720814i	$-0.743768\pi$
$37$	−311.994	−1.38626	−0.693129	+	0.720814i	$0.743768\pi$
$38$	87.4332	0.373251
$39$	−105.825	−0.434502
$40$	−74.4222	−0.294180
$41$	−336.966	−1.28354	−0.641772	−	0.766895i	$-0.721800\pi$
$41$	−336.966	−1.28354	−0.641772	+	0.766895i	$0.721800\pi$
$42$	42.0000	0.154303
$43$	−123.553	−0.438177	−0.219088	−	0.975705i	$-0.570308\pi$
$43$	−123.553	−0.438177	−0.219088	+	0.975705i	$0.570308\pi$
$44$	−117.689	−0.403233
$45$	−83.7250	−0.277355
$46$	−46.0000	−0.147442
$47$	120.094	0.372714	0.186357	−	0.982482i	$-0.440332\pi$
$47$	120.094	0.372714	0.186357	+	0.982482i	$0.440332\pi$
$48$	48.0000	0.144338
$49$	49.0000	0.142857
$50$	−76.9167	−0.217553
$51$	−89.3667	−0.245369
$52$	−141.100	−0.376290
$53$	330.480	0.856509	0.428254	−	0.903658i	$-0.359129\pi$
$53$	330.480	0.856509	0.428254	+	0.903658i	$0.359129\pi$
$54$	54.0000	0.136083
$55$	273.708	0.671033
$56$	56.0000	0.133631
$57$	131.150	0.304758
$58$	200.133	0.453082
$59$	−170.497	−0.376217	−0.188109	−	0.982148i	$-0.560236\pi$
$59$	−170.497	−0.376217	−0.188109	+	0.982148i	$0.560236\pi$
$60$	−111.633	−0.240197
$61$	−275.586	−0.578446	−0.289223	−	0.957262i	$-0.593397\pi$
$61$	−275.586	−0.578446	−0.289223	+	0.957262i	$0.593397\pi$
$62$	−380.389	−0.779184
$63$	63.0000	0.125988
$64$	64.0000	0.125000
$65$	328.156	0.626195
$66$	−176.533	−0.329238
$67$	−672.063	−1.22546	−0.612728	−	0.790294i	$-0.709928\pi$
$67$	−672.063	−1.22546	−0.612728	+	0.790294i	$0.709928\pi$
$68$	−119.156	−0.212496
$69$	−69.0000	−0.120386
$70$	−130.239	−0.222379
$71$	−660.113	−1.10339	−0.551697	−	0.834045i	$-0.686020\pi$
$71$	−660.113	−1.10339	−0.551697	+	0.834045i	$0.686020\pi$
$72$	72.0000	0.117851
$73$	−617.367	−0.989826	−0.494913	−	0.868943i	$-0.664800\pi$
$73$	−617.367	−0.989826	−0.494913	+	0.868943i	$0.664800\pi$
$74$	−623.988	−0.980232
$75$	−115.375	−0.177632
$76$	174.866	0.263928
$77$	−205.955	−0.304816
$78$	−211.650	−0.307239
$79$	1169.77	1.66594	0.832968	−	0.553322i	$-0.186640\pi$
$79$	1169.77	1.66594	0.832968	+	0.553322i	$0.186640\pi$
$80$	−148.844	−0.208016
$81$	81.0000	0.111111
$82$	−673.933	−0.907603
$83$	1238.34	1.63765	0.818826	−	0.574041i	$-0.194625\pi$
$83$	1238.34	1.63765	0.818826	+	0.574041i	$0.194625\pi$
$84$	84.0000	0.109109
$85$	277.119	0.353621
$86$	−247.105	−0.309838
$87$	300.199	0.369940
$88$	−235.378	−0.285129
$89$	851.896	1.01462	0.507308	−	0.861765i	$-0.330641\pi$
$89$	851.896	1.01462	0.507308	+	0.861765i	$0.330641\pi$
$90$	−167.450	−0.196120
$91$	−246.925	−0.284448
$92$	−92.0000	−0.104257
$93$	−570.583	−0.636201
$94$	240.188	0.263548
$95$	−406.686	−0.439211
$96$	96.0000	0.102062
$97$	−1261.78	−1.32076	−0.660382	−	0.750930i	$-0.729606\pi$
$97$	−1261.78	−1.32076	−0.660382	+	0.750930i	$0.729606\pi$
$98$	98.0000	0.101015
$99$	−264.800	−0.268822
$100$	−153.833	−0.153833
$101$	−1457.54	−1.43594	−0.717971	−	0.696073i	$-0.754929\pi$
$101$	−1457.54	−1.43594	−0.717971	+	0.696073i	$0.754929\pi$
$102$	−178.733	−0.173502
$103$	−1934.11	−1.85023	−0.925114	−	0.379690i	$-0.876031\pi$
$103$	−1934.11	−1.85023	−0.925114	+	0.379690i	$0.876031\pi$
$104$	−282.200	−0.266077
$105$	−195.358	−0.181572
$106$	660.961	0.605643
$107$	320.264	0.289356	0.144678	−	0.989479i	$-0.453785\pi$
$107$	320.264	0.289356	0.144678	+	0.989479i	$0.453785\pi$
$108$	108.000	0.0962250
$109$	−273.865	−0.240656	−0.120328	−	0.992734i	$-0.538395\pi$
$109$	−273.865	−0.240656	−0.120328	+	0.992734i	$0.538395\pi$
$110$	547.416	0.474492
$111$	−935.982	−0.800356
$112$	112.000	0.0944911
$113$	28.5862	0.0237979	0.0118990	−	0.999929i	$-0.496212\pi$
$113$	28.5862	0.0237979	0.0118990	+	0.999929i	$0.496212\pi$
$114$	262.299	0.215496
$115$	213.964	0.173498
$116$	400.266	0.320377
$117$	−317.475	−0.250860
$118$	−340.994	−0.266026
$119$	−208.522	−0.160632
$120$	−223.267	−0.169845
$121$	−465.334	−0.349612
$122$	−551.172	−0.409023
$123$	−1010.90	−0.741055
$124$	−760.777	−0.550966
$125$	1520.62	1.08806
$126$	126.000	0.0890871
$127$	−897.154	−0.626847	−0.313423	−	0.949614i	$-0.601476\pi$
$127$	−897.154	−0.626847	−0.313423	+	0.949614i	$0.601476\pi$
$128$	128.000	0.0883883
$129$	−370.658	−0.252981
$130$	656.311	0.442787
$131$	36.7342	0.0244998	0.0122499	−	0.999925i	$-0.496101\pi$
$131$	36.7342	0.0244998	0.0122499	+	0.999925i	$0.496101\pi$
$132$	−353.066	−0.232807
$133$	306.016	0.199511
$134$	−1344.13	−0.866529
$135$	−251.175	−0.160131
$136$	−238.311	−0.150257
$137$	1465.86	0.914141	0.457070	−	0.889431i	$-0.348899\pi$
$137$	1465.86	0.914141	0.457070	+	0.889431i	$0.348899\pi$
$138$	−138.000	−0.0851257
$139$	−1.93685	−0.00118188	−0.000590942	−	1.00000i	$-0.500188\pi$
$139$	−1.93685	−0.00118188	−0.000590942		1.00000i	$0.500188\pi$
$140$	−260.478	−0.157246
$141$	360.283	0.215186
$142$	−1320.23	−0.780217
$143$	1037.87	0.606930
$144$	144.000	0.0833333
$145$	−930.896	−0.533150
$146$	−1234.73	−0.699913
$147$	147.000	0.0824786
$148$	−1247.98	−0.693129
$149$	10.2068	0.00561189	0.00280595	−	0.999996i	$-0.499107\pi$
$149$	10.2068	0.00561189	0.00280595	+	0.999996i	$0.499107\pi$
$150$	−230.750	−0.125604
$151$	1827.57	0.984939	0.492469	−	0.870330i	$-0.336094\pi$
$151$	1827.57	0.984939	0.492469	+	0.870330i	$0.336094\pi$
$152$	349.733	0.186625
$153$	−268.100	−0.141664
$154$	−411.911	−0.215537
$155$	1769.33	0.916880
$156$	−423.300	−0.217251
$157$	653.139	0.332014	0.166007	−	0.986125i	$-0.446913\pi$
$157$	653.139	0.332014	0.166007	+	0.986125i	$0.446913\pi$
$158$	2339.53	1.17799
$159$	991.441	0.494505
$160$	−297.689	−0.147090
$161$	−161.000	−0.0788110
$162$	162.000	0.0785674
$163$	2088.11	1.00339	0.501697	−	0.865044i	$-0.332709\pi$
$163$	2088.11	1.00339	0.501697	+	0.865044i	$0.332709\pi$
$164$	−1347.87	−0.641772
$165$	821.125	0.387421
$166$	2476.67	1.15800
$167$	551.212	0.255414	0.127707	−	0.991812i	$-0.459238\pi$
$167$	551.212	0.255414	0.127707	+	0.991812i	$0.459238\pi$
$168$	168.000	0.0771517
$169$	−952.673	−0.433624
$170$	554.239	0.250048
$171$	393.449	0.175952
$172$	−494.210	−0.219088
$173$	−3092.49	−1.35906	−0.679530	−	0.733648i	$-0.737816\pi$
$173$	−3092.49	−1.35906	−0.679530	+	0.733648i	$0.737816\pi$
$174$	600.399	0.261587
$175$	−269.209	−0.116287
$176$	−470.755	−0.201617
$177$	−511.491	−0.217209
$178$	1703.79	0.717441
$179$	−2181.13	−0.910757	−0.455379	−	0.890298i	$-0.650496\pi$
$179$	−2181.13	−0.910757	−0.455379	+	0.890298i	$0.650496\pi$
$180$	−334.900	−0.138678
$181$	−1837.14	−0.754440	−0.377220	−	0.926124i	$-0.623120\pi$
$181$	−1837.14	−0.754440	−0.377220	+	0.926124i	$0.623120\pi$
$182$	−493.850	−0.201135
$183$	−826.759	−0.333966
$184$	−184.000	−0.0737210
$185$	2902.41	1.15346
$186$	−1141.17	−0.449862
$187$	876.455	0.342742
$188$	480.377	0.186357
$189$	189.000	0.0727393
$190$	−813.371	−0.310569
$191$	3932.23	1.48966	0.744832	−	0.667252i	$-0.232530\pi$
$191$	3932.23	1.48966	0.744832	+	0.667252i	$0.232530\pi$
$192$	192.000	0.0721688
$193$	4631.68	1.72744	0.863719	−	0.503974i	$-0.168129\pi$
$193$	4631.68	1.72744	0.863719	+	0.503974i	$0.168129\pi$
$194$	−2523.55	−0.933921
$195$	984.467	0.361534
$196$	196.000	0.0714286
$197$	−813.797	−0.294318	−0.147159	−	0.989113i	$-0.547013\pi$
$197$	−813.797	−0.294318	−0.147159	+	0.989113i	$0.547013\pi$
$198$	−529.600	−0.190086
$199$	5438.68	1.93738	0.968688	−	0.248281i	$-0.0798657\pi$
$199$	5438.68	1.93738	0.968688	+	0.248281i	$0.0798657\pi$
$200$	−307.667	−0.108777
$201$	−2016.19	−0.707518
$202$	−2915.07	−1.01536
$203$	700.465	0.242182
$204$	−357.467	−0.122685
$205$	3134.72	1.06799
$206$	−3868.22	−1.30831
$207$	−207.000	−0.0695048
$208$	−564.400	−0.188145
$209$	−1286.24	−0.425698
$210$	−390.717	−0.128391
$211$	−2565.68	−0.837104	−0.418552	−	0.908193i	$-0.637462\pi$
$211$	−2565.68	−0.837104	−0.418552	+	0.908193i	$0.637462\pi$
$212$	1321.92	0.428254
$213$	−1980.34	−0.637045
$214$	640.528	0.204606
$215$	1149.38	0.364592
$216$	216.000	0.0680414
$217$	−1331.36	−0.416491
$218$	−547.729	−0.170169
$219$	−1852.10	−0.571476
$220$	1094.83	0.335516
$221$	1050.80	0.319840
$222$	−1871.96	−0.565937
$223$	2606.93	0.782838	0.391419	−	0.920213i	$-0.371984\pi$
$223$	2606.93	0.782838	0.391419	+	0.920213i	$0.371984\pi$
$224$	224.000	0.0668153
$225$	−346.125	−0.102556
$226$	57.1724	0.0168277
$227$	−1976.52	−0.577913	−0.288957	−	0.957342i	$-0.593308\pi$
$227$	−1976.52	−0.577913	−0.288957	+	0.957342i	$0.593308\pi$
$228$	524.599	0.152379
$229$	−4597.58	−1.32671	−0.663354	−	0.748305i	$-0.730868\pi$
$229$	−4597.58	−1.32671	−0.663354	+	0.748305i	$0.730868\pi$
$230$	427.928	0.122681
$231$	−617.866	−0.175985
$232$	800.532	0.226541
$233$	3973.65	1.11726	0.558632	−	0.829416i	$-0.311326\pi$
$233$	3973.65	1.11726	0.558632	+	0.829416i	$0.311326\pi$
$234$	−634.950	−0.177385
$235$	−1117.21	−0.310122
$236$	−681.988	−0.188109
$237$	3509.30	0.961828
$238$	−417.045	−0.113584
$239$	421.811	0.114162	0.0570809	−	0.998370i	$-0.481821\pi$
$239$	421.811	0.114162	0.0570809	+	0.998370i	$0.481821\pi$
$240$	−446.533	−0.120098
$241$	1526.62	0.408043	0.204021	−	0.978966i	$-0.434599\pi$
$241$	1526.62	0.408043	0.204021	+	0.978966i	$0.434599\pi$
$242$	−930.668	−0.247213
$243$	243.000	0.0641500
$244$	−1102.34	−0.289223
$245$	−455.836	−0.118867
$246$	−2021.80	−0.524005
$247$	−1542.10	−0.397254
$248$	−1521.55	−0.389592
$249$	3715.01	0.945499
$250$	3041.23	0.769378
$251$	−6246.42	−1.57080	−0.785400	−	0.618989i	$-0.787543\pi$
$251$	−6246.42	−1.57080	−0.785400	+	0.618989i	$0.787543\pi$
$252$	252.000	0.0629941
$253$	676.711	0.168160
$254$	−1794.31	−0.443248
$255$	831.358	0.204163
$256$	256.000	0.0625000
$257$	−3299.31	−0.800798	−0.400399	−	0.916341i	$-0.631128\pi$
$257$	−3299.31	−0.800798	−0.400399	+	0.916341i	$0.631128\pi$
$258$	−741.315	−0.178885
$259$	−2183.96	−0.523956
$260$	1312.62	0.313098
$261$	900.598	0.213585
$262$	73.4683	0.0173240
$263$	3846.77	0.901909	0.450955	−	0.892547i	$-0.351084\pi$
$263$	3846.77	0.901909	0.450955	+	0.892547i	$0.351084\pi$
$264$	−706.133	−0.164619
$265$	−3074.38	−0.712671
$266$	612.032	0.141076
$267$	2555.69	0.585789
$268$	−2688.25	−0.612728
$269$	4039.21	0.915520	0.457760	−	0.889076i	$-0.348652\pi$
$269$	4039.21	0.915520	0.457760	+	0.889076i	$0.348652\pi$
$270$	−502.350	−0.113230
$271$	−1955.68	−0.438373	−0.219187	−	0.975683i	$-0.570340\pi$
$271$	−1955.68	−0.438373	−0.219187	+	0.975683i	$0.570340\pi$
$272$	−476.622	−0.106248
$273$	−740.775	−0.164226
$274$	2931.73	0.646395
$275$	1131.53	0.248123
$276$	−276.000	−0.0601929
$277$	540.407	0.117220	0.0586100	−	0.998281i	$-0.481333\pi$
$277$	540.407	0.117220	0.0586100	+	0.998281i	$0.481333\pi$
$278$	−3.87371	−0.000835718 0
$279$	−1711.75	−0.367311
$280$	−520.955	−0.111189
$281$	−4017.14	−0.852819	−0.426410	−	0.904530i	$-0.640222\pi$
$281$	−4017.14	−0.852819	−0.426410	+	0.904530i	$0.640222\pi$
$282$	720.565	0.152160
$283$	2974.63	0.624817	0.312409	−	0.949948i	$-0.398864\pi$
$283$	2974.63	0.624817	0.312409	+	0.949948i	$0.398864\pi$
$284$	−2640.45	−0.551697
$285$	−1220.06	−0.253579
$286$	2075.74	0.429164
$287$	−2358.76	−0.485134
$288$	288.000	0.0589256
$289$	−4025.62	−0.819382
$290$	−1861.79	−0.376994
$291$	−3785.33	−0.762543
$292$	−2469.47	−0.494913
$293$	6164.95	1.22922	0.614608	−	0.788833i	$-0.289314\pi$
$293$	6164.95	1.22922	0.614608	+	0.788833i	$0.289314\pi$
$294$	294.000	0.0583212
$295$	1586.10	0.313038
$296$	−2495.95	−0.490116
$297$	−794.400	−0.155204
$298$	20.4136	0.00396821
$299$	811.325	0.156924
$300$	−461.500	−0.0888158
$301$	−864.868	−0.165615
$302$	3655.15	0.696457
$303$	−4372.61	−0.829042
$304$	699.465	0.131964
$305$	2563.72	0.481305
$306$	−536.200	−0.100172
$307$	6167.03	1.14649	0.573243	−	0.819385i	$-0.305685\pi$
$307$	6167.03	1.14649	0.573243	+	0.819385i	$0.305685\pi$
$308$	−823.822	−0.152408
$309$	−5802.33	−1.06823
$310$	3538.67	0.648332
$311$	1359.88	0.247948	0.123974	−	0.992285i	$-0.460436\pi$
$311$	1359.88	0.247948	0.123974	+	0.992285i	$0.460436\pi$
$312$	−846.600	−0.153620
$313$	−7580.32	−1.36890	−0.684449	−	0.729061i	$-0.739957\pi$
$313$	−7580.32	−1.36890	−0.684449	+	0.729061i	$0.739957\pi$
$314$	1306.28	0.234769
$315$	−586.075	−0.104830
$316$	4679.06	0.832968
$317$	−6074.83	−1.07633	−0.538165	−	0.842840i	$-0.680882\pi$
$317$	−6074.83	−1.07633	−0.538165	+	0.842840i	$0.680882\pi$
$318$	1982.88	0.349668
$319$	−2944.18	−0.516747
$320$	−595.378	−0.104008
$321$	960.792	0.167060
$322$	−322.000	−0.0557278
$323$	−1302.27	−0.224335
$324$	324.000	0.0555556
$325$	1356.62	0.231544
$326$	4176.21	0.709506
$327$	−821.594	−0.138943
$328$	−2695.73	−0.453801
$329$	840.660	0.140873
$330$	1642.25	0.273948
$331$	6375.77	1.05874	0.529371	−	0.848390i	$-0.322428\pi$
$331$	6375.77	1.05874	0.529371	+	0.848390i	$0.322428\pi$
$332$	4953.35	0.818826
$333$	−2807.95	−0.462086
$334$	1102.42	0.180605
$335$	6252.05	1.01966
$336$	336.000	0.0545545
$337$	320.349	0.0517819	0.0258909	−	0.999665i	$-0.491758\pi$
$337$	320.349	0.0517819	0.0258909	+	0.999665i	$0.491758\pi$
$338$	−1905.35	−0.306619
$339$	85.7586	0.0137397
$340$	1108.48	0.176811
$341$	5595.94	0.888671
$342$	786.898	0.124417
$343$	343.000	0.0539949
$344$	−988.421	−0.154919
$345$	641.892	0.100169
$346$	−6184.97	−0.961001
$347$	2424.27	0.375048	0.187524	−	0.982260i	$-0.439954\pi$
$347$	2424.27	0.375048	0.187524	+	0.982260i	$0.439954\pi$
$348$	1200.80	0.184970
$349$	2474.68	0.379561	0.189780	−	0.981827i	$-0.439222\pi$
$349$	2474.68	0.379561	0.189780	+	0.981827i	$0.439222\pi$
$350$	−538.417	−0.0822274
$351$	−952.426	−0.144834
$352$	−941.511	−0.142564
$353$	−100.445	−0.0151449	−0.00757244	−	0.999971i	$-0.502410\pi$
$353$	−100.445	−0.0151449	−0.00757244	+	0.999971i	$0.502410\pi$
$354$	−1022.98	−0.153590
$355$	6140.88	0.918096
$356$	3407.58	0.507308
$357$	−625.567	−0.0927409
$358$	−4362.27	−0.644003
$359$	10195.1	1.49882	0.749410	−	0.662106i	$-0.230337\pi$
$359$	10195.1	1.49882	0.749410	+	0.662106i	$0.230337\pi$
$360$	−669.800	−0.0980599
$361$	−4947.86	−0.721368
$362$	−3674.28	−0.533470
$363$	−1396.00	−0.201849
$364$	−987.701	−0.142224
$365$	5743.22	0.823600
$366$	−1653.52	−0.236150
$367$	−1112.14	−0.158184	−0.0790919	−	0.996867i	$-0.525202\pi$
$367$	−1112.14	−0.158184	−0.0790919	+	0.996867i	$0.525202\pi$
$368$	−368.000	−0.0521286
$369$	−3032.70	−0.427848
$370$	5804.82	0.815617
$371$	2313.36	0.323730
$372$	−2282.33	−0.318100
$373$	8609.80	1.19517	0.597585	−	0.801805i	$-0.296127\pi$
$373$	8609.80	1.19517	0.597585	+	0.801805i	$0.296127\pi$
$374$	1752.91	0.242355
$375$	4561.85	0.628194
$376$	960.754	0.131774
$377$	−3529.85	−0.482218
$378$	378.000	0.0514344
$379$	9108.77	1.23453	0.617264	−	0.786756i	$-0.288241\pi$
$379$	9108.77	1.23453	0.617264	+	0.786756i	$0.288241\pi$
$380$	−1626.74	−0.219606
$381$	−2691.46	−0.361910
$382$	7864.45	1.05335
$383$	−4605.77	−0.614475	−0.307237	−	0.951633i	$-0.599405\pi$
$383$	−4605.77	−0.614475	−0.307237	+	0.951633i	$0.599405\pi$
$384$	384.000	0.0510310
$385$	1915.96	0.253627
$386$	9263.36	1.22148
$387$	−1111.97	−0.146059
$388$	−5047.11	−0.660382
$389$	8289.45	1.08044	0.540221	−	0.841523i	$-0.318341\pi$
$389$	8289.45	1.08044	0.540221	+	0.841523i	$0.318341\pi$
$390$	1968.93	0.255643
$391$	685.145	0.0886170
$392$	392.000	0.0505076
$393$	110.202	0.0141450
$394$	−1627.59	−0.208114
$395$	−10882.1	−1.38617
$396$	−1059.20	−0.134411
$397$	7154.99	0.904530	0.452265	−	0.891884i	$-0.350616\pi$
$397$	7154.99	0.904530	0.452265	+	0.891884i	$0.350616\pi$
$398$	10877.4	1.36993
$399$	918.048	0.115188
$400$	−615.334	−0.0769167
$401$	480.167	0.0597965	0.0298982	−	0.999553i	$-0.490482\pi$
$401$	480.167	0.0597965	0.0298982	+	0.999553i	$0.490482\pi$
$402$	−4032.38	−0.500290
$403$	6709.11	0.829291
$404$	−5830.14	−0.717971
$405$	−753.525	−0.0924517
$406$	1400.93	0.171249
$407$	9179.56	1.11797
$408$	−714.934	−0.0867512
$409$	−2609.26	−0.315451	−0.157725	−	0.987483i	$-0.550416\pi$
$409$	−2609.26	−0.315451	−0.157725	+	0.987483i	$0.550416\pi$
$410$	6269.45	0.755185
$411$	4397.59	0.527779
$412$	−7736.44	−0.925114
$413$	−1193.48	−0.142197
$414$	−414.000	−0.0491473
$415$	−11520.0	−1.36263
$416$	−1128.80	−0.133038
$417$	−5.81056	−0.000682361 0
$418$	−2572.48	−0.301014
$419$	9511.29	1.10897	0.554483	−	0.832195i	$-0.312916\pi$
$419$	9511.29	1.10897	0.554483	+	0.832195i	$0.312916\pi$
$420$	−781.433	−0.0907858
$421$	−6725.67	−0.778597	−0.389298	−	0.921112i	$-0.627283\pi$
$421$	−6725.67	−0.778597	−0.389298	+	0.921112i	$0.627283\pi$
$422$	−5131.37	−0.591922
$423$	1080.85	0.124238
$424$	2643.84	0.302822
$425$	1145.63	0.130756
$426$	−3960.68	−0.450459
$427$	−1929.10	−0.218632
$428$	1281.06	0.144678
$429$	3113.61	0.350411
$430$	2298.76	0.257805
$431$	10503.9	1.17391	0.586954	−	0.809620i	$-0.300327\pi$
$431$	10503.9	1.17391	0.586954	+	0.809620i	$0.300327\pi$
$432$	432.000	0.0481125
$433$	−6500.59	−0.721475	−0.360737	−	0.932667i	$-0.617475\pi$
$433$	−6500.59	−0.721475	−0.360737	+	0.932667i	$0.617475\pi$
$434$	−2662.72	−0.294504
$435$	−2792.69	−0.307814
$436$	−1095.46	−0.120328
$437$	−1005.48	−0.110066
$438$	−3704.20	−0.404095
$439$	−12356.2	−1.34335	−0.671675	−	0.740846i	$-0.734425\pi$
$439$	−12356.2	−1.34335	−0.671675	+	0.740846i	$0.734425\pi$
$440$	2189.67	0.237246
$441$	441.000	0.0476190
$442$	2101.61	0.226161
$443$	9065.77	0.972297	0.486149	−	0.873876i	$-0.338401\pi$
$443$	9065.77	0.972297	0.486149	+	0.873876i	$0.338401\pi$
$444$	−3743.93	−0.400178
$445$	−7925.00	−0.844227
$446$	5213.86	0.553550
$447$	30.6204	0.00324003
$448$	448.000	0.0472456
$449$	−5776.73	−0.607173	−0.303586	−	0.952804i	$-0.598184\pi$
$449$	−5776.73	−0.607173	−0.303586	+	0.952804i	$0.598184\pi$
$450$	−692.251	−0.0725178
$451$	9914.29	1.03513
$452$	114.345	0.0118990
$453$	5482.72	0.568655
$454$	−3953.04	−0.408646
$455$	2297.09	0.236680
$456$	1049.20	0.107748
$457$	14415.8	1.47558	0.737792	−	0.675028i	$-0.235868\pi$
$457$	14415.8	1.47558	0.737792	+	0.675028i	$0.235868\pi$
$458$	−9195.15	−0.938125
$459$	−804.300	−0.0817898
$460$	855.855	0.0867488
$461$	3524.83	0.356113	0.178056	−	0.984020i	$-0.443019\pi$
$461$	3524.83	0.356113	0.178056	+	0.984020i	$0.443019\pi$
$462$	−1235.73	−0.124440
$463$	−5241.94	−0.526164	−0.263082	−	0.964774i	$-0.584739\pi$
$463$	−5241.94	−0.526164	−0.263082	+	0.964774i	$0.584739\pi$
$464$	1601.06	0.160189
$465$	5308.00	0.529361
$466$	7947.30	0.790025
$467$	8723.97	0.864448	0.432224	−	0.901766i	$-0.357729\pi$
$467$	8723.97	0.864448	0.432224	+	0.901766i	$0.357729\pi$
$468$	−1269.90	−0.125430
$469$	−4704.44	−0.463179
$470$	−2234.42	−0.219290
$471$	1959.42	0.191688
$472$	−1363.98	−0.133013
$473$	3635.19	0.353375
$474$	7018.59	0.680115
$475$	−1681.27	−0.162404
$476$	−834.089	−0.0803160
$477$	2974.32	0.285503
$478$	843.622	0.0807246
$479$	5074.19	0.484020	0.242010	−	0.970274i	$-0.422193\pi$
$479$	5074.19	0.484020	0.242010	+	0.970274i	$0.422193\pi$
$480$	−893.066	−0.0849223
$481$	11005.6	1.04327
$482$	3053.24	0.288530
$483$	−483.000	−0.0455016
$484$	−1861.34	−0.174806
$485$	11738.0	1.09896
$486$	486.000	0.0453609
$487$	−18062.4	−1.68067	−0.840334	−	0.542068i	$-0.817641\pi$
$487$	−18062.4	−1.68067	−0.840334	+	0.542068i	$0.817641\pi$
$488$	−2204.69	−0.204511
$489$	6264.32	0.579310
$490$	−911.672	−0.0840513
$491$	−11236.1	−1.03275	−0.516374	−	0.856363i	$-0.672719\pi$
$491$	−11236.1	−1.03275	−0.516374	+	0.856363i	$0.672719\pi$
$492$	−4043.60	−0.370527
$493$	−2980.87	−0.272316
$494$	−3084.21	−0.280901
$495$	2463.37	0.223678
$496$	−3043.11	−0.275483
$497$	−4620.79	−0.417044
$498$	7430.02	0.668569
$499$	964.459	0.0865233	0.0432616	−	0.999064i	$-0.486225\pi$
$499$	964.459	0.0865233	0.0432616	+	0.999064i	$0.486225\pi$
$500$	6082.47	0.544032
$501$	1653.64	0.147463
$502$	−12492.8	−1.11072
$503$	−17035.4	−1.51009	−0.755043	−	0.655676i	$-0.772384\pi$
$503$	−17035.4	−1.51009	−0.755043	+	0.655676i	$0.772384\pi$
$504$	504.000	0.0445435
$505$	13559.1	1.19480
$506$	1353.42	0.118907
$507$	−2858.02	−0.250353
$508$	−3588.62	−0.313423
$509$	13635.6	1.18740	0.593701	−	0.804686i	$-0.297666\pi$
$509$	13635.6	1.18740	0.593701	+	0.804686i	$0.297666\pi$
$510$	1662.72	0.144365
$511$	−4321.57	−0.374119
$512$	512.000	0.0441942
$513$	1180.35	0.101586
$514$	−6598.61	−0.566250
$515$	17992.6	1.53951
$516$	−1482.63	−0.126491
$517$	−3533.44	−0.300581
$518$	−4367.92	−0.370493
$519$	−9277.46	−0.784654
$520$	2625.24	0.221393
$521$	−19628.3	−1.65054	−0.825269	−	0.564740i	$-0.808977\pi$
$521$	−19628.3	−1.65054	−0.825269	+	0.564740i	$0.808977\pi$
$522$	1801.20	0.151027
$523$	1140.50	0.0953546	0.0476773	−	0.998863i	$-0.484818\pi$
$523$	1140.50	0.0953546	0.0476773	+	0.998863i	$0.484818\pi$
$524$	146.937	0.0122499
$525$	−807.626	−0.0671384
$526$	7693.55	0.637746
$527$	5665.68	0.468313
$528$	−1412.27	−0.116403
$529$	529.000	0.0434783
$530$	−6148.77	−0.503935
$531$	−1534.47	−0.125406
$532$	1224.06	0.0997555
$533$	11886.5	0.965969
$534$	5111.38	0.414215
$535$	−2979.34	−0.240763
$536$	−5376.51	−0.433264
$537$	−6543.40	−0.525826
$538$	8078.41	0.647370
$539$	−1441.69	−0.115209
$540$	−1004.70	−0.0800655
$541$	3223.60	0.256180	0.128090	−	0.991763i	$-0.459115\pi$
$541$	3223.60	0.256180	0.128090	+	0.991763i	$0.459115\pi$
$542$	−3911.36	−0.309977
$543$	−5511.42	−0.435576
$544$	−953.245	−0.0751287
$545$	2547.70	0.200241
$546$	−1481.55	−0.116125
$547$	−21080.5	−1.64779	−0.823893	−	0.566745i	$-0.808202\pi$
$547$	−21080.5	−1.64779	−0.823893	+	0.566745i	$0.808202\pi$
$548$	5863.46	0.457070
$549$	−2480.28	−0.192815
$550$	2263.06	0.175449
$551$	4374.56	0.338226
$552$	−552.000	−0.0425628
$553$	8188.36	0.629664
$554$	1080.81	0.0828870
$555$	8707.23	0.665949
$556$	−7.74741	−0.000590942 0
$557$	19355.4	1.47238	0.736189	−	0.676776i	$-0.236624\pi$
$557$	19355.4	1.47238	0.736189	+	0.676776i	$0.236624\pi$
$558$	−3423.50	−0.259728
$559$	4358.32	0.329763
$560$	−1041.91	−0.0786228
$561$	2629.37	0.197882
$562$	−8034.27	−0.603034
$563$	1709.99	0.128006	0.0640029	−	0.997950i	$-0.479613\pi$
$563$	1709.99	0.128006	0.0640029	+	0.997950i	$0.479613\pi$
$564$	1441.13	0.107593
$565$	−265.931	−0.0198014
$566$	5949.26	0.441813
$567$	567.000	0.0419961
$568$	−5280.90	−0.390109
$569$	−2703.61	−0.199194	−0.0995970	−	0.995028i	$-0.531755\pi$
$569$	−2703.61	−0.199194	−0.0995970	+	0.995028i	$0.531755\pi$
$570$	−2440.11	−0.179307
$571$	−7800.45	−0.571696	−0.285848	−	0.958275i	$-0.592275\pi$
$571$	−7800.45	−0.571696	−0.285848	+	0.958275i	$0.592275\pi$
$572$	4151.48	0.303465
$573$	11796.7	0.860058
$574$	−4717.53	−0.343042
$575$	884.542	0.0641530
$576$	576.000	0.0416667
$577$	−11137.8	−0.803593	−0.401797	−	0.915729i	$-0.631614\pi$
$577$	−11137.8	−0.803593	−0.401797	+	0.915729i	$0.631614\pi$
$578$	−8051.24	−0.579390
$579$	13895.0	0.997337
$580$	−3723.58	−0.266575
$581$	8668.36	0.618975
$582$	−7570.66	−0.539199
$583$	−9723.46	−0.690745
$584$	−4938.93	−0.349956
$585$	2953.40	0.208732
$586$	12329.9	0.869187
$587$	20373.5	1.43254	0.716272	−	0.697821i	$-0.245847\pi$
$587$	20373.5	1.43254	0.716272	+	0.697821i	$0.245847\pi$
$588$	588.000	0.0412393
$589$	−8314.64	−0.581662
$590$	3172.19	0.221351
$591$	−2441.39	−0.169924
$592$	−4991.91	−0.346564
$593$	7856.89	0.544087	0.272044	−	0.962285i	$-0.412300\pi$
$593$	7856.89	0.544087	0.272044	+	0.962285i	$0.412300\pi$
$594$	−1588.80	−0.109746
$595$	1939.84	0.133656
$596$	40.8271	0.00280595
$597$	16316.0	1.11854
$598$	1622.65	0.110962
$599$	3383.00	0.230760	0.115380	−	0.993321i	$-0.463191\pi$
$599$	3383.00	0.230760	0.115380	+	0.993321i	$0.463191\pi$
$600$	−923.001	−0.0628022
$601$	−21987.8	−1.49235	−0.746175	−	0.665750i	$-0.768112\pi$
$601$	−21987.8	−1.49235	−0.746175	+	0.665750i	$0.768112\pi$
$602$	−1729.74	−0.117108
$603$	−6048.57	−0.408485
$604$	7310.29	0.492469
$605$	4328.90	0.290900
$606$	−8745.21	−0.586221
$607$	−19448.1	−1.30045	−0.650226	−	0.759740i	$-0.725326\pi$
$607$	−19448.1	−1.30045	−0.650226	+	0.759740i	$0.725326\pi$
$608$	1398.93	0.0933127
$609$	2101.40	0.139824
$610$	5127.43	0.340334
$611$	−4236.33	−0.280497
$612$	−1072.40	−0.0708320
$613$	3577.48	0.235715	0.117857	−	0.993031i	$-0.462397\pi$
$613$	3577.48	0.235715	0.117857	+	0.993031i	$0.462397\pi$
$614$	12334.1	0.810688
$615$	9404.17	0.616606
$616$	−1647.64	−0.107769
$617$	−12308.9	−0.803141	−0.401571	−	0.915828i	$-0.631536\pi$
$617$	−12308.9	−0.803141	−0.401571	+	0.915828i	$0.631536\pi$
$618$	−11604.7	−0.755352
$619$	−26471.6	−1.71887	−0.859437	−	0.511242i	$-0.829186\pi$
$619$	−26471.6	−1.71887	−0.859437	+	0.511242i	$0.829186\pi$
$620$	7077.34	0.458440
$621$	−621.000	−0.0401286
$622$	2719.76	0.175326
$623$	5963.27	0.383489
$624$	−1693.20	−0.108625
$625$	−9338.66	−0.597674
$626$	−15160.6	−0.967957
$627$	−3858.71	−0.245777
$628$	2612.56	0.166007
$629$	9293.96	0.589149
$630$	−1172.15	−0.0741263
$631$	12674.6	0.799635	0.399817	−	0.916595i	$-0.369074\pi$
$631$	12674.6	0.799635	0.399817	+	0.916595i	$0.369074\pi$
$632$	9358.12	0.588997
$633$	−7697.05	−0.483302
$634$	−12149.7	−0.761080
$635$	8346.02	0.521578
$636$	3965.76	0.247253
$637$	−1728.48	−0.107511
$638$	−5888.35	−0.365395
$639$	−5941.02	−0.367798
$640$	−1190.76	−0.0735449
$641$	29693.8	1.82970	0.914848	−	0.403797i	$-0.132310\pi$
$641$	29693.8	1.82970	0.914848	+	0.403797i	$0.132310\pi$
$642$	1921.58	0.118129
$643$	−13286.3	−0.814867	−0.407434	−	0.913235i	$-0.633576\pi$
$643$	−13286.3	−0.814867	−0.407434	+	0.913235i	$0.633576\pi$
$644$	−644.000	−0.0394055
$645$	3448.15	0.210497
$646$	−2604.54	−0.158629
$647$	−8555.27	−0.519849	−0.259925	−	0.965629i	$-0.583698\pi$
$647$	−8555.27	−0.519849	−0.259925	+	0.965629i	$0.583698\pi$
$648$	648.000	0.0392837
$649$	5016.40	0.303407
$650$	2713.24	0.163726
$651$	−3994.08	−0.240461
$652$	8352.43	0.501697
$653$	−1307.50	−0.0783557	−0.0391779	−	0.999232i	$-0.512474\pi$
$653$	−1307.50	−0.0783557	−0.0391779	+	0.999232i	$0.512474\pi$
$654$	−1643.19	−0.0982473
$655$	−341.730	−0.0203855
$656$	−5391.46	−0.320886
$657$	−5556.30	−0.329942
$658$	1681.32	0.0996119
$659$	−23070.6	−1.36374	−0.681870	−	0.731473i	$-0.738833\pi$
$659$	−23070.6	−1.36374	−0.681870	+	0.731473i	$0.738833\pi$
$660$	3284.50	0.193710
$661$	−731.582	−0.0430488	−0.0215244	−	0.999768i	$-0.506852\pi$
$661$	−731.582	−0.0430488	−0.0215244	+	0.999768i	$0.506852\pi$
$662$	12751.5	0.748644
$663$	3152.41	0.184660
$664$	9906.70	0.578998
$665$	−2846.80	−0.166006
$666$	−5615.89	−0.326744
$667$	−2301.53	−0.133607
$668$	2204.85	0.127707
$669$	7820.78	0.451972
$670$	12504.1	0.721009
$671$	8108.35	0.466497
$672$	672.000	0.0385758
$673$	22527.1	1.29027	0.645137	−	0.764067i	$-0.276800\pi$
$673$	22527.1	1.29027	0.645137	+	0.764067i	$0.276800\pi$
$674$	640.697	0.0366153
$675$	−1038.38	−0.0592105
$676$	−3810.69	−0.216812
$677$	11572.6	0.656975	0.328487	−	0.944508i	$-0.393461\pi$
$677$	11572.6	0.656975	0.328487	+	0.944508i	$0.393461\pi$
$678$	171.517	0.00971545
$679$	−8832.44	−0.499202
$680$	2216.96	0.125024
$681$	−5929.56	−0.333658
$682$	11191.9	0.628385
$683$	−25949.8	−1.45379	−0.726896	−	0.686748i	$-0.759038\pi$
$683$	−25949.8	−1.45379	−0.726896	+	0.686748i	$0.759038\pi$
$684$	1573.80	0.0879761
$685$	−13636.6	−0.760625
$686$	686.000	0.0381802
$687$	−13792.7	−0.765976
$688$	−1976.84	−0.109544
$689$	−11657.7	−0.644591
$690$	1283.78	0.0708301
$691$	−2429.41	−0.133747	−0.0668735	−	0.997761i	$-0.521302\pi$
$691$	−2429.41	−0.133747	−0.0668735	+	0.997761i	$0.521302\pi$
$692$	−12369.9	−0.679530
$693$	−1853.60	−0.101605
$694$	4848.55	0.265199
$695$	18.0181	0.000983404 0
$696$	2401.60	0.130793
$697$	10037.9	0.545496
$698$	4949.36	0.268390
$699$	11921.0	0.645053
$700$	−1076.83	−0.0581436
$701$	3696.96	0.199190	0.0995950	−	0.995028i	$-0.468245\pi$
$701$	3696.96	0.199190	0.0995950	+	0.995028i	$0.468245\pi$
$702$	−1904.85	−0.102413
$703$	−13639.3	−0.731745
$704$	−1883.02	−0.100808
$705$	−3351.63	−0.179049
$706$	−200.890	−0.0107090
$707$	−10202.7	−0.542735
$708$	−2045.96	−0.108605
$709$	20136.2	1.06662	0.533308	−	0.845921i	$-0.320949\pi$
$709$	20136.2	1.06662	0.533308	+	0.845921i	$0.320949\pi$
$710$	12281.8	0.649192
$711$	10527.9	0.555312
$712$	6815.17	0.358721
$713$	4374.47	0.229769
$714$	−1251.13	−0.0655777
$715$	−9655.06	−0.505005
$716$	−8724.53	−0.455379
$717$	1265.43	0.0659114
$718$	20390.2	1.05983
$719$	9463.30	0.490850	0.245425	−	0.969416i	$-0.421072\pi$
$719$	9463.30	0.490850	0.245425	+	0.969416i	$0.421072\pi$
$720$	−1339.60	−0.0693388
$721$	−13538.8	−0.699320
$722$	−9895.72	−0.510084
$723$	4579.86	0.235584
$724$	−7348.57	−0.377220
$725$	−3848.39	−0.197139
$726$	−2792.00	−0.142729
$727$	−31788.0	−1.62167	−0.810833	−	0.585278i	$-0.800986\pi$
$727$	−31788.0	−1.62167	−0.810833	+	0.585278i	$0.800986\pi$
$728$	−1975.40	−0.100568
$729$	729.000	0.0370370
$730$	11486.4	0.582373
$731$	3680.50	0.186222
$732$	−3307.03	−0.166983
$733$	−22024.8	−1.10983	−0.554916	−	0.831907i	$-0.687249\pi$
$733$	−22024.8	−1.10983	−0.554916	+	0.831907i	$0.687249\pi$
$734$	−2224.29	−0.111853
$735$	−1367.51	−0.0686276
$736$	−736.000	−0.0368605
$737$	19773.6	0.988289
$738$	−6065.39	−0.302534
$739$	27500.1	1.36889	0.684444	−	0.729066i	$-0.260045\pi$
$739$	27500.1	1.36889	0.684444	+	0.729066i	$0.260045\pi$
$740$	11609.6	0.576728
$741$	−4626.31	−0.229355
$742$	4626.72	0.228912
$743$	−13131.6	−0.648387	−0.324194	−	0.945991i	$-0.605093\pi$
$743$	−13131.6	−0.648387	−0.324194	+	0.945991i	$0.605093\pi$
$744$	−4564.66	−0.224931
$745$	−94.9514	−0.00466946
$746$	17219.6	0.845113
$747$	11145.0	0.545884
$748$	3505.82	0.171371
$749$	2241.85	0.109366
$750$	9123.70	0.444201
$751$	−4557.21	−0.221432	−0.110716	−	0.993852i	$-0.535314\pi$
$751$	−4557.21	−0.221432	−0.110716	+	0.993852i	$0.535314\pi$
$752$	1921.51	0.0931784
$753$	−18739.3	−0.906901
$754$	−7059.69	−0.340980
$755$	−17001.5	−0.819534
$756$	756.000	0.0363696
$757$	4636.75	0.222623	0.111311	−	0.993786i	$-0.464495\pi$
$757$	4636.75	0.222623	0.111311	+	0.993786i	$0.464495\pi$
$758$	18217.5	0.872943
$759$	2030.13	0.0970871
$760$	−3253.48	−0.155285
$761$	−24997.0	−1.19072	−0.595361	−	0.803458i	$-0.702991\pi$
$761$	−24997.0	−1.19072	−0.595361	+	0.803458i	$0.702991\pi$
$762$	−5382.92	−0.255909
$763$	−1917.05	−0.0909593
$764$	15728.9	0.744832
$765$	2494.07	0.117874
$766$	−9211.54	−0.434499
$767$	6014.29	0.283133
$768$	768.000	0.0360844
$769$	19248.0	0.902602	0.451301	−	0.892372i	$-0.350960\pi$
$769$	19248.0	0.902602	0.451301	+	0.892372i	$0.350960\pi$
$770$	3831.91	0.179341
$771$	−9897.92	−0.462341
$772$	18526.7	0.863719
$773$	23413.9	1.08944	0.544721	−	0.838617i	$-0.316635\pi$
$773$	23413.9	1.08944	0.544721	+	0.838617i	$0.316635\pi$
$774$	−2223.95	−0.103279
$775$	7314.56	0.339028
$776$	−10094.2	−0.466960
$777$	−6551.88	−0.302506
$778$	16578.9	0.763988
$779$	−14731.0	−0.677527
$780$	3937.87	0.180767
$781$	19422.0	0.889850
$782$	1370.29	0.0626617
$783$	2701.79	0.123313
$784$	784.000	0.0357143
$785$	−6076.00	−0.276257
$786$	220.405	0.0100020
$787$	−18841.6	−0.853406	−0.426703	−	0.904392i	$-0.640325\pi$
$787$	−18841.6	−0.853406	−0.426703	+	0.904392i	$0.640325\pi$
$788$	−3255.19	−0.147159
$789$	11540.3	0.520718
$790$	−21764.1	−0.980168
$791$	200.103	0.00899476
$792$	−2118.40	−0.0950430
$793$	9721.31	0.435326
$794$	14310.0	0.639600
$795$	−9223.15	−0.411461
$796$	21754.7	0.968688
$797$	−29181.1	−1.29692	−0.648460	−	0.761248i	$-0.724587\pi$
$797$	−29181.1	−1.29692	−0.648460	+	0.761248i	$0.724587\pi$
$798$	1836.10	0.0814500
$799$	−3577.47	−0.158400
$800$	−1230.67	−0.0543883
$801$	7667.06	0.338205
$802$	960.334	0.0422825
$803$	18164.3	0.798261
$804$	−8064.76	−0.353759
$805$	1497.75	0.0655760
$806$	13418.2	0.586398
$807$	12117.6	0.528576
$808$	−11660.3	−0.507682
$809$	−35530.2	−1.54410	−0.772049	−	0.635562i	$-0.780768\pi$
$809$	−35530.2	−1.54410	−0.772049	+	0.635562i	$0.780768\pi$
$810$	−1507.05	−0.0653732
$811$	−28096.3	−1.21652	−0.608259	−	0.793739i	$-0.708132\pi$
$811$	−28096.3	−1.21652	−0.608259	+	0.793739i	$0.708132\pi$
$812$	2801.86	0.121091
$813$	−5867.04	−0.253095
$814$	18359.1	0.790524
$815$	−19425.2	−0.834889
$816$	−1429.87	−0.0613424
$817$	−5401.30	−0.231294
$818$	−5218.51	−0.223058
$819$	−2222.33	−0.0948161
$820$	12538.9	0.533996
$821$	14687.0	0.624337	0.312169	−	0.950027i	$-0.398945\pi$
$821$	14687.0	0.624337	0.312169	+	0.950027i	$0.398945\pi$
$822$	8795.19	0.373196
$823$	−9663.95	−0.409312	−0.204656	−	0.978834i	$-0.565608\pi$
$823$	−9663.95	−0.409312	−0.204656	+	0.978834i	$0.565608\pi$
$824$	−15472.9	−0.654154
$825$	3394.59	0.143254
$826$	−2386.96	−0.100548
$827$	27797.4	1.16882	0.584409	−	0.811460i	$-0.301327\pi$
$827$	27797.4	1.16882	0.584409	+	0.811460i	$0.301327\pi$
$828$	−828.000	−0.0347524
$829$	−22737.1	−0.952585	−0.476292	−	0.879287i	$-0.658020\pi$
$829$	−22737.1	−0.952585	−0.476292	+	0.879287i	$0.658020\pi$
$830$	−23039.9	−0.963528
$831$	1621.22	0.0676770
$832$	−2257.60	−0.0940724
$833$	−1459.66	−0.0607132
$834$	−11.6211	−0.000482502 0
$835$	−5127.80	−0.212521
$836$	−5144.95	−0.212849
$837$	−5135.25	−0.212067
$838$	19022.6	0.784158
$839$	−26680.3	−1.09786	−0.548931	−	0.835868i	$-0.684965\pi$
$839$	−26680.3	−1.09786	−0.548931	+	0.835868i	$0.684965\pi$
$840$	−1562.87	−0.0641953
$841$	−14375.7	−0.589434
$842$	−13451.3	−0.550551
$843$	−12051.4	−0.492375
$844$	−10262.7	−0.418552
$845$	8862.50	0.360804
$846$	2161.70	0.0878495
$847$	−3257.34	−0.132141
$848$	5287.68	0.214127
$849$	8923.88	0.360738
$850$	2291.26	0.0924585
$851$	7175.87	0.289055
$852$	−7921.35	−0.318522
$853$	−8937.47	−0.358749	−0.179375	−	0.983781i	$-0.557407\pi$
$853$	−8937.47	−0.358749	−0.179375	+	0.983781i	$0.557407\pi$
$854$	−3858.21	−0.154596
$855$	−3660.17	−0.146404
$856$	2562.11	0.102303
$857$	−9792.48	−0.390321	−0.195160	−	0.980771i	$-0.562523\pi$
$857$	−9792.48	−0.390321	−0.195160	+	0.980771i	$0.562523\pi$
$858$	6227.21	0.247778
$859$	16007.7	0.635827	0.317914	−	0.948120i	$-0.397018\pi$
$859$	16007.7	0.635827	0.317914	+	0.948120i	$0.397018\pi$
$860$	4597.53	0.182296
$861$	−7076.29	−0.280092
$862$	21007.8	0.830078
$863$	−28793.6	−1.13574	−0.567872	−	0.823117i	$-0.692233\pi$
$863$	−28793.6	−1.13574	−0.567872	+	0.823117i	$0.692233\pi$
$864$	864.000	0.0340207
$865$	28768.7	1.13083
$866$	−13001.2	−0.510160
$867$	−12076.9	−0.473070
$868$	−5325.44	−0.208246
$869$	−34417.1	−1.34352
$870$	−5585.38	−0.217657
$871$	23707.0	0.922253
$872$	−2190.92	−0.0850846
$873$	−11356.0	−0.440255
$874$	−2010.96	−0.0778282
$875$	10644.3	0.411250
$876$	−7408.40	−0.285738
$877$	34746.7	1.33787	0.668936	−	0.743320i	$-0.266750\pi$
$877$	34746.7	1.33787	0.668936	+	0.743320i	$0.266750\pi$
$878$	−24712.5	−0.949893
$879$	18494.9	0.709688
$880$	4379.33	0.167758
$881$	−12951.3	−0.495277	−0.247638	−	0.968853i	$-0.579654\pi$
$881$	−12951.3	−0.495277	−0.247638	+	0.968853i	$0.579654\pi$
$882$	882.000	0.0336718
$883$	−40084.7	−1.52770	−0.763848	−	0.645396i	$-0.776692\pi$
$883$	−40084.7	−1.52770	−0.763848	+	0.645396i	$0.776692\pi$
$884$	4203.22	0.159920
$885$	4758.29	0.180732
$886$	18131.5	0.687518
$887$	−7138.15	−0.270209	−0.135105	−	0.990831i	$-0.543137\pi$
$887$	−7138.15	−0.270209	−0.135105	+	0.990831i	$0.543137\pi$
$888$	−7487.86	−0.282969
$889$	−6280.08	−0.236926
$890$	−15850.0	−0.596958
$891$	−2383.20	−0.0896074
$892$	10427.7	0.391419
$893$	5250.11	0.196739
$894$	61.2407	0.00229105
$895$	20290.6	0.757810
$896$	896.000	0.0334077
$897$	2433.98	0.0905999
$898$	−11553.5	−0.429336
$899$	−19032.1	−0.706068
$900$	−1384.50	−0.0512778
$901$	−9844.64	−0.364010
$902$	19828.6	0.731951
$903$	−2594.60	−0.0956180
$904$	228.690	0.00841383
$905$	17090.5	0.627744
$906$	10965.4	0.402100
$907$	−41184.7	−1.50773	−0.753867	−	0.657027i	$-0.771814\pi$
$907$	−41184.7	−1.50773	−0.753867	+	0.657027i	$0.771814\pi$
$908$	−7906.08	−0.288957
$909$	−13117.8	−0.478647
$910$	4594.18	0.167358
$911$	−13537.8	−0.492346	−0.246173	−	0.969226i	$-0.579173\pi$
$911$	−13537.8	−0.492346	−0.246173	+	0.969226i	$0.579173\pi$
$912$	2098.40	0.0761895
$913$	−36434.6	−1.32071
$914$	28831.6	1.04340
$915$	7691.15	0.277881
$916$	−18390.3	−0.663354
$917$	257.139	0.00926007
$918$	−1608.60	−0.0578341
$919$	−48511.8	−1.74130	−0.870651	−	0.491902i	$-0.836302\pi$
$919$	−48511.8	−1.74130	−0.870651	+	0.491902i	$0.836302\pi$
$920$	1711.71	0.0613407
$921$	18501.1	0.661924
$922$	7049.67	0.251810
$923$	23285.5	0.830391
$924$	−2471.47	−0.0879927
$925$	11998.8	0.426506
$926$	−10483.9	−0.372054
$927$	−17407.0	−0.616743
$928$	3202.13	0.113270
$929$	26726.8	0.943893	0.471946	−	0.881627i	$-0.343552\pi$
$929$	26726.8	0.943893	0.471946	+	0.881627i	$0.343552\pi$
$930$	10616.0	0.374315
$931$	2142.11	0.0754080
$932$	15894.6	0.558632
$933$	4079.64	0.143153
$934$	17447.9	0.611257
$935$	−8153.46	−0.285184
$936$	−2539.80	−0.0886923
$937$	−38539.8	−1.34369	−0.671846	−	0.740690i	$-0.734499\pi$
$937$	−38539.8	−1.34369	−0.671846	+	0.740690i	$0.734499\pi$
$938$	−9408.88	−0.327517
$939$	−22741.0	−0.790334
$940$	−4468.84	−0.155061
$941$	−1704.24	−0.0590401	−0.0295200	−	0.999564i	$-0.509398\pi$
$941$	−1704.24	−0.0590401	−0.0295200	+	0.999564i	$0.509398\pi$
$942$	3918.83	0.135544
$943$	7750.23	0.267637
$944$	−2727.95	−0.0940544
$945$	−1758.22	−0.0605239
$946$	7270.38	0.249874
$947$	4439.87	0.152351	0.0761755	−	0.997094i	$-0.475729\pi$
$947$	4439.87	0.152351	0.0761755	+	0.997094i	$0.475729\pi$
$948$	14037.2	0.480914
$949$	21777.6	0.744923
$950$	−3362.54	−0.114837
$951$	−18224.5	−0.621419
$952$	−1668.18	−0.0567920
$953$	1116.24	0.0379417	0.0189709	−	0.999820i	$-0.493961\pi$
$953$	1116.24	0.0379417	0.0189709	+	0.999820i	$0.493961\pi$
$954$	5948.64	0.201881
$955$	−36580.6	−1.23950
$956$	1687.24	0.0570809
$957$	−8832.53	−0.298344
$958$	10148.4	0.342254
$959$	10261.1	0.345513
$960$	−1786.13	−0.0600492
$961$	6382.87	0.214255
$962$	22011.2	0.737702
$963$	2882.38	0.0964520
$964$	6106.49	0.204021
$965$	−43087.5	−1.43734
$966$	−966.000	−0.0321745
$967$	−17622.5	−0.586041	−0.293021	−	0.956106i	$-0.594660\pi$
$967$	−17622.5	−0.586041	−0.293021	+	0.956106i	$0.594660\pi$
$968$	−3722.67	−0.123607
$969$	−3906.81	−0.129520
$970$	23476.1	0.777083
$971$	−43042.5	−1.42255	−0.711277	−	0.702911i	$-0.751883\pi$
$971$	−43042.5	−1.42255	−0.711277	+	0.702911i	$0.751883\pi$
$972$	972.000	0.0320750
$973$	−13.5580	−0.000446710 0
$974$	−36124.8	−1.18841
$975$	4069.86	0.133682
$976$	−4409.38	−0.144611
$977$	−2582.63	−0.0845708	−0.0422854	−	0.999106i	$-0.513464\pi$
$977$	−2582.63	−0.0845708	−0.0422854	+	0.999106i	$0.513464\pi$
$978$	12528.6	0.409634
$979$	−25064.7	−0.818253
$980$	−1823.34	−0.0594333
$981$	−2464.78	−0.0802185
$982$	−22472.3	−0.730263
$983$	−53604.7	−1.73929	−0.869647	−	0.493674i	$-0.835653\pi$
$983$	−53604.7	−1.73929	−0.869647	+	0.493674i	$0.835653\pi$
$984$	−8087.19	−0.262002
$985$	7570.57	0.244892
$986$	−5961.74	−0.192556
$987$	2521.98	0.0813328
$988$	−6168.41	−0.198627
$989$	2841.71	0.0913661
$990$	4926.75	0.158164
$991$	−46140.4	−1.47901	−0.739504	−	0.673152i	$-0.764940\pi$
$991$	−46140.4	−1.47901	−0.739504	+	0.673152i	$0.764940\pi$
$992$	−6086.22	−0.194796
$993$	19127.3	0.611265
$994$	−9241.58	−0.294894
$995$	−50594.8	−1.61202
$996$	14860.0	0.472750
$997$	−14104.8	−0.448046	−0.224023	−	0.974584i	$-0.571919\pi$
$997$	−14104.8	−0.448046	−0.224023	+	0.974584i	$0.571919\pi$
$998$	1928.92	0.0611812
$999$	−8423.84	−0.266785

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	966.4.a.c.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.4.a.c.1.1	✓	2	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:	\( N \)	\(=\)	\( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	966.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -9.30278 q^{5} +6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -9.30278 q^{5} +6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -18.6056 q^{10} -29.4222 q^{11} +12.0000 q^{12} -35.2750 q^{13} +14.0000 q^{14} -27.9083 q^{15} +16.0000 q^{16} -29.7889 q^{17} +18.0000 q^{18} +43.7166 q^{19} -37.2111 q^{20} +21.0000 q^{21} -58.8444 q^{22} -23.0000 q^{23} +24.0000 q^{24} -38.4584 q^{25} -70.5500 q^{26} +27.0000 q^{27} +28.0000 q^{28} +100.066 q^{29} -55.8167 q^{30} -190.194 q^{31} +32.0000 q^{32} -88.2666 q^{33} -59.5778 q^{34} -65.1194 q^{35} +36.0000 q^{36} -311.994 q^{37} +87.4332 q^{38} -105.825 q^{39} -74.4222 q^{40} -336.966 q^{41} +42.0000 q^{42} -123.553 q^{43} -117.689 q^{44} -83.7250 q^{45} -46.0000 q^{46} +120.094 q^{47} +48.0000 q^{48} +49.0000 q^{49} -76.9167 q^{50} -89.3667 q^{51} -141.100 q^{52} +330.480 q^{53} +54.0000 q^{54} +273.708 q^{55} +56.0000 q^{56} +131.150 q^{57} +200.133 q^{58} -170.497 q^{59} -111.633 q^{60} -275.586 q^{61} -380.389 q^{62} +63.0000 q^{63} +64.0000 q^{64} +328.156 q^{65} -176.533 q^{66} -672.063 q^{67} -119.156 q^{68} -69.0000 q^{69} -130.239 q^{70} -660.113 q^{71} +72.0000 q^{72} -617.367 q^{73} -623.988 q^{74} -115.375 q^{75} +174.866 q^{76} -205.955 q^{77} -211.650 q^{78} +1169.77 q^{79} -148.844 q^{80} +81.0000 q^{81} -673.933 q^{82} +1238.34 q^{83} +84.0000 q^{84} +277.119 q^{85} -247.105 q^{86} +300.199 q^{87} -235.378 q^{88} +851.896 q^{89} -167.450 q^{90} -246.925 q^{91} -92.0000 q^{92} -570.583 q^{93} +240.188 q^{94} -406.686 q^{95} +96.0000 q^{96} -1261.78 q^{97} +98.0000 q^{98} -264.800 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 15 q^{5} + 12 q^{6} + 14 q^{7} + 16 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 - 15 * q^5 + 12 * q^6 + 14 * q^7 + 16 * q^8 + 18 * q^9	\( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 15 q^{5} + 12 q^{6} + 14 q^{7} + 16 q^{8} + 18 q^{9} - 30 q^{10} - 30 q^{11} + 24 q^{12} - 103 q^{13} + 28 q^{14} - 45 q^{15} + 32 q^{16} - 74 q^{17} + 36 q^{18} - 64 q^{19} - 60 q^{20} + 42 q^{21} - 60 q^{22} - 46 q^{23} + 48 q^{24} - 131 q^{25} - 206 q^{26} + 54 q^{27} + 56 q^{28} - 146 q^{29} - 90 q^{30} - 128 q^{31} + 64 q^{32} - 90 q^{33} - 148 q^{34} - 105 q^{35} + 72 q^{36} - 112 q^{37} - 128 q^{38} - 309 q^{39} - 120 q^{40} - 198 q^{41} + 84 q^{42} + 81 q^{43} - 120 q^{44} - 135 q^{45} - 92 q^{46} - 142 q^{47} + 96 q^{48} + 98 q^{49} - 262 q^{50} - 222 q^{51} - 412 q^{52} + 167 q^{53} + 108 q^{54} + 277 q^{55} + 112 q^{56} - 192 q^{57} - 292 q^{58} - 85 q^{59} - 180 q^{60} - 699 q^{61} - 256 q^{62} + 126 q^{63} + 128 q^{64} + 714 q^{65} - 180 q^{66} - 93 q^{67} - 296 q^{68} - 138 q^{69} - 210 q^{70} + 515 q^{71} + 144 q^{72} - 1278 q^{73} - 224 q^{74} - 393 q^{75} - 256 q^{76} - 210 q^{77} - 618 q^{78} + 306 q^{79} - 240 q^{80} + 162 q^{81} - 396 q^{82} - 40 q^{83} + 168 q^{84} + 529 q^{85} + 162 q^{86} - 438 q^{87} - 240 q^{88} - 629 q^{89} - 270 q^{90} - 721 q^{91} - 184 q^{92} - 384 q^{93} - 284 q^{94} + 207 q^{95} + 192 q^{96} - 1514 q^{97} + 196 q^{98} - 270 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 - 15 * q^5 + 12 * q^6 + 14 * q^7 + 16 * q^8 + 18 * q^9 - 30 * q^10 - 30 * q^11 + 24 * q^12 - 103 * q^13 + 28 * q^14 - 45 * q^15 + 32 * q^16 - 74 * q^17 + 36 * q^18 - 64 * q^19 - 60 * q^20 + 42 * q^21 - 60 * q^22 - 46 * q^23 + 48 * q^24 - 131 * q^25 - 206 * q^26 + 54 * q^27 + 56 * q^28 - 146 * q^29 - 90 * q^30 - 128 * q^31 + 64 * q^32 - 90 * q^33 - 148 * q^34 - 105 * q^35 + 72 * q^36 - 112 * q^37 - 128 * q^38 - 309 * q^39 - 120 * q^40 - 198 * q^41 + 84 * q^42 + 81 * q^43 - 120 * q^44 - 135 * q^45 - 92 * q^46 - 142 * q^47 + 96 * q^48 + 98 * q^49 - 262 * q^50 - 222 * q^51 - 412 * q^52 + 167 * q^53 + 108 * q^54 + 277 * q^55 + 112 * q^56 - 192 * q^57 - 292 * q^58 - 85 * q^59 - 180 * q^60 - 699 * q^61 - 256 * q^62 + 126 * q^63 + 128 * q^64 + 714 * q^65 - 180 * q^66 - 93 * q^67 - 296 * q^68 - 138 * q^69 - 210 * q^70 + 515 * q^71 + 144 * q^72 - 1278 * q^73 - 224 * q^74 - 393 * q^75 - 256 * q^76 - 210 * q^77 - 618 * q^78 + 306 * q^79 - 240 * q^80 + 162 * q^81 - 396 * q^82 - 40 * q^83 + 168 * q^84 + 529 * q^85 + 162 * q^86 - 438 * q^87 - 240 * q^88 - 629 * q^89 - 270 * q^90 - 721 * q^91 - 184 * q^92 - 384 * q^93 - 284 * q^94 + 207 * q^95 + 192 * q^96 - 1514 * q^97 + 196 * q^98 - 270 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more