LMFDB - Embedded newform 8041.2.a.i.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8041,2,Mod(1,8041)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8041, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8041.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8041 = 11 \cdot 17 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8041.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:	$64.2077082653$
Analytic rank:	$0$
Dimension:	$78$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	8041.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.28473 q^{2} -0.648502 q^{3} +3.22000 q^{4} -1.19117 q^{5} +1.48165 q^{6} +3.55276 q^{7} -2.78738 q^{8} -2.57944 q^{9} +O(q^{10})$	\(q-2.28473 q^{2} -0.648502 q^{3} +3.22000 q^{4} -1.19117 q^{5} +1.48165 q^{6} +3.55276 q^{7} -2.78738 q^{8} -2.57944 q^{9} +2.72151 q^{10} +1.00000 q^{11} -2.08818 q^{12} +2.20951 q^{13} -8.11710 q^{14} +0.772478 q^{15} -0.0715963 q^{16} -1.00000 q^{17} +5.89334 q^{18} -2.00734 q^{19} -3.83558 q^{20} -2.30397 q^{21} -2.28473 q^{22} +3.40797 q^{23} +1.80762 q^{24} -3.58111 q^{25} -5.04814 q^{26} +3.61828 q^{27} +11.4399 q^{28} +7.34415 q^{29} -1.76491 q^{30} -2.81936 q^{31} +5.73833 q^{32} -0.648502 q^{33} +2.28473 q^{34} -4.23195 q^{35} -8.30581 q^{36} -6.89242 q^{37} +4.58623 q^{38} -1.43287 q^{39} +3.32025 q^{40} +2.63626 q^{41} +5.26396 q^{42} -1.00000 q^{43} +3.22000 q^{44} +3.07256 q^{45} -7.78629 q^{46} -0.309786 q^{47} +0.0464303 q^{48} +5.62208 q^{49} +8.18187 q^{50} +0.648502 q^{51} +7.11463 q^{52} +3.94179 q^{53} -8.26681 q^{54} -1.19117 q^{55} -9.90286 q^{56} +1.30176 q^{57} -16.7794 q^{58} -11.6969 q^{59} +2.48738 q^{60} -6.80995 q^{61} +6.44148 q^{62} -9.16414 q^{63} -12.9674 q^{64} -2.63191 q^{65} +1.48165 q^{66} +1.66914 q^{67} -3.22000 q^{68} -2.21007 q^{69} +9.66886 q^{70} +5.91650 q^{71} +7.18988 q^{72} +7.52804 q^{73} +15.7473 q^{74} +2.32236 q^{75} -6.46363 q^{76} +3.55276 q^{77} +3.27373 q^{78} +7.35099 q^{79} +0.0852835 q^{80} +5.39187 q^{81} -6.02316 q^{82} +8.75805 q^{83} -7.41879 q^{84} +1.19117 q^{85} +2.28473 q^{86} -4.76270 q^{87} -2.78738 q^{88} -13.9073 q^{89} -7.01999 q^{90} +7.84985 q^{91} +10.9737 q^{92} +1.82836 q^{93} +0.707779 q^{94} +2.39109 q^{95} -3.72132 q^{96} -12.7362 q^{97} -12.8449 q^{98} -2.57944 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 78 q + 7 q^{2} + 10 q^{3} + 91 q^{4} + 17 q^{5} + 12 q^{6} + 11 q^{7} + 33 q^{8} + 102 q^{9}+O(q^{10}) $ 78 * q + 7 * q^2 + 10 * q^3 + 91 * q^4 + 17 * q^5 + 12 * q^6 + 11 * q^7 + 33 * q^8 + 102 * q^9	\( 78 q + 7 q^{2} + 10 q^{3} + 91 q^{4} + 17 q^{5} + 12 q^{6} + 11 q^{7} + 33 q^{8} + 102 q^{9} + 3 q^{10} + 78 q^{11} + 31 q^{12} - 16 q^{13} + 31 q^{14} + 38 q^{15} + 121 q^{16} - 78 q^{17} + 11 q^{18} + 51 q^{20} + 6 q^{21} + 7 q^{22} + 48 q^{23} + 11 q^{24} + 101 q^{25} + 18 q^{26} + 46 q^{27} + 27 q^{28} + 22 q^{29} + 14 q^{30} + 56 q^{31} + 83 q^{32} + 10 q^{33} - 7 q^{34} + 24 q^{35} + 139 q^{36} + 53 q^{37} + 10 q^{38} + 79 q^{39} - q^{40} + 23 q^{41} + 17 q^{42} - 78 q^{43} + 91 q^{44} + 76 q^{45} + 21 q^{46} + 57 q^{47} + 78 q^{48} + 115 q^{49} + 58 q^{50} - 10 q^{51} - 63 q^{52} + 22 q^{53} - 18 q^{54} + 17 q^{55} + 111 q^{56} - 11 q^{57} + 36 q^{58} + 71 q^{59} + 36 q^{60} + 4 q^{61} - 5 q^{62} + 71 q^{63} + 183 q^{64} + 47 q^{65} + 12 q^{66} + 11 q^{67} - 91 q^{68} + 31 q^{69} + 33 q^{70} + 159 q^{71} + 59 q^{72} + 2 q^{73} - 4 q^{74} + 83 q^{75} - 44 q^{76} + 11 q^{77} + 101 q^{78} + 35 q^{79} + 85 q^{80} + 170 q^{81} + 98 q^{82} - 32 q^{83} + 44 q^{84} - 17 q^{85} - 7 q^{86} - 6 q^{87} + 33 q^{88} + 50 q^{89} - 5 q^{90} + 86 q^{91} + 106 q^{92} + 68 q^{93} - q^{94} + 109 q^{95} - 50 q^{96} + 40 q^{97} + 106 q^{98} + 102 q^{99}+O(q^{100}) \) 78 * q + 7 * q^2 + 10 * q^3 + 91 * q^4 + 17 * q^5 + 12 * q^6 + 11 * q^7 + 33 * q^8 + 102 * q^9 + 3 * q^10 + 78 * q^11 + 31 * q^12 - 16 * q^13 + 31 * q^14 + 38 * q^15 + 121 * q^16 - 78 * q^17 + 11 * q^18 + 51 * q^20 + 6 * q^21 + 7 * q^22 + 48 * q^23 + 11 * q^24 + 101 * q^25 + 18 * q^26 + 46 * q^27 + 27 * q^28 + 22 * q^29 + 14 * q^30 + 56 * q^31 + 83 * q^32 + 10 * q^33 - 7 * q^34 + 24 * q^35 + 139 * q^36 + 53 * q^37 + 10 * q^38 + 79 * q^39 - q^40 + 23 * q^41 + 17 * q^42 - 78 * q^43 + 91 * q^44 + 76 * q^45 + 21 * q^46 + 57 * q^47 + 78 * q^48 + 115 * q^49 + 58 * q^50 - 10 * q^51 - 63 * q^52 + 22 * q^53 - 18 * q^54 + 17 * q^55 + 111 * q^56 - 11 * q^57 + 36 * q^58 + 71 * q^59 + 36 * q^60 + 4 * q^61 - 5 * q^62 + 71 * q^63 + 183 * q^64 + 47 * q^65 + 12 * q^66 + 11 * q^67 - 91 * q^68 + 31 * q^69 + 33 * q^70 + 159 * q^71 + 59 * q^72 + 2 * q^73 - 4 * q^74 + 83 * q^75 - 44 * q^76 + 11 * q^77 + 101 * q^78 + 35 * q^79 + 85 * q^80 + 170 * q^81 + 98 * q^82 - 32 * q^83 + 44 * q^84 - 17 * q^85 - 7 * q^86 - 6 * q^87 + 33 * q^88 + 50 * q^89 - 5 * q^90 + 86 * q^91 + 106 * q^92 + 68 * q^93 - q^94 + 109 * q^95 - 50 * q^96 + 40 * q^97 + 106 * q^98 + 102 * 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 $

$2$	−2.28473	−1.61555	−0.807775	−	0.589491i	$-0.799328\pi$
$2$	−2.28473	−1.61555	−0.807775	+	0.589491i	$0.799328\pi$
$3$	−0.648502	−0.374413	−0.187207	−	0.982321i	$-0.559943\pi$
$3$	−0.648502	−0.374413	−0.187207	+	0.982321i	$0.559943\pi$
$4$	3.22000	1.61000
$5$	−1.19117	−0.532709	−0.266354	−	0.963875i	$-0.585819\pi$
$5$	−1.19117	−0.532709	−0.266354	+	0.963875i	$0.585819\pi$
$6$	1.48165	0.604883
$7$	3.55276	1.34282	0.671408	−	0.741088i	$-0.265690\pi$
$7$	3.55276	1.34282	0.671408	+	0.741088i	$0.265690\pi$
$8$	−2.78738	−0.985486
$9$	−2.57944	−0.859815
$10$	2.72151	0.860617
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−2.08818	−0.602805
$13$	2.20951	0.612808	0.306404	−	0.951902i	$-0.400874\pi$
$13$	2.20951	0.612808	0.306404	+	0.951902i	$0.400874\pi$
$14$	−8.11710	−2.16939
$15$	0.772478	0.199453
$16$	−0.0715963	−0.0178991
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	5.89334	1.38907
$19$	−2.00734	−0.460515	−0.230258	−	0.973130i	$-0.573957\pi$
$19$	−2.00734	−0.460515	−0.230258	+	0.973130i	$0.573957\pi$
$20$	−3.83558	−0.857661
$21$	−2.30397	−0.502768
$22$	−2.28473	−0.487107
$23$	3.40797	0.710610	0.355305	−	0.934750i	$-0.384377\pi$
$23$	3.40797	0.710610	0.355305	+	0.934750i	$0.384377\pi$
$24$	1.80762	0.368979
$25$	−3.58111	−0.716221
$26$	−5.04814	−0.990022
$27$	3.61828	0.696339
$28$	11.4399	2.16193
$29$	7.34415	1.36377	0.681887	−	0.731457i	$-0.261159\pi$
$29$	7.34415	1.36377	0.681887	+	0.731457i	$0.261159\pi$
$30$	−1.76491	−0.322226
$31$	−2.81936	−0.506371	−0.253186	−	0.967418i	$-0.581478\pi$
$31$	−2.81936	−0.506371	−0.253186	+	0.967418i	$0.581478\pi$
$32$	5.73833	1.01440
$33$	−0.648502	−0.112890
$34$	2.28473	0.391828
$35$	−4.23195	−0.715330
$36$	−8.30581	−1.38430
$37$	−6.89242	−1.13311	−0.566553	−	0.824025i	$-0.691723\pi$
$37$	−6.89242	−1.13311	−0.566553	+	0.824025i	$0.691723\pi$
$38$	4.58623	0.743985
$39$	−1.43287	−0.229443
$40$	3.32025	0.524977
$41$	2.63626	0.411715	0.205858	−	0.978582i	$-0.434002\pi$
$41$	2.63626	0.411715	0.205858	+	0.978582i	$0.434002\pi$
$42$	5.26396	0.812246
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	3.22000	0.485433
$45$	3.07256	0.458031
$46$	−7.78629	−1.14803
$47$	−0.309786	−0.0451870	−0.0225935	−	0.999745i	$-0.507192\pi$
$47$	−0.309786	−0.0451870	−0.0225935	+	0.999745i	$0.507192\pi$
$48$	0.0464303	0.00670164
$49$	5.62208	0.803154
$50$	8.18187	1.15709
$51$	0.648502	0.0908085
$52$	7.11463	0.986621
$53$	3.94179	0.541447	0.270723	−	0.962657i	$-0.412737\pi$
$53$	3.94179	0.541447	0.270723	+	0.962657i	$0.412737\pi$
$54$	−8.26681	−1.12497
$55$	−1.19117	−0.160618
$56$	−9.90286	−1.32333
$57$	1.30176	0.172423
$58$	−16.7794	−2.20325
$59$	−11.6969	−1.52280	−0.761402	−	0.648280i	$-0.775489\pi$
$59$	−11.6969	−1.52280	−0.761402	+	0.648280i	$0.775489\pi$
$60$	2.48738	0.321120
$61$	−6.80995	−0.871925	−0.435962	−	0.899965i	$-0.643592\pi$
$61$	−6.80995	−0.871925	−0.435962	+	0.899965i	$0.643592\pi$
$62$	6.44148	0.818068
$63$	−9.16414	−1.15457
$64$	−12.9674	−1.62092
$65$	−2.63191	−0.326448
$66$	1.48165	0.182379
$67$	1.66914	0.203918	0.101959	−	0.994789i	$-0.467489\pi$
$67$	1.66914	0.203918	0.101959	+	0.994789i	$0.467489\pi$
$68$	−3.22000	−0.390482
$69$	−2.21007	−0.266062
$70$	9.66886	1.15565
$71$	5.91650	0.702159	0.351080	−	0.936346i	$-0.385815\pi$
$71$	5.91650	0.702159	0.351080	+	0.936346i	$0.385815\pi$
$72$	7.18988	0.847335
$73$	7.52804	0.881091	0.440545	−	0.897730i	$-0.354785\pi$
$73$	7.52804	0.881091	0.440545	+	0.897730i	$0.354785\pi$
$74$	15.7473	1.83059
$75$	2.32236	0.268163
$76$	−6.46363	−0.741429
$77$	3.55276	0.404874
$78$	3.27373	0.370677
$79$	7.35099	0.827051	0.413526	−	0.910492i	$-0.364297\pi$
$79$	7.35099	0.827051	0.413526	+	0.910492i	$0.364297\pi$
$80$	0.0852835	0.00953499
$81$	5.39187	0.599097
$82$	−6.02316	−0.665146
$83$	8.75805	0.961320	0.480660	−	0.876907i	$-0.340397\pi$
$83$	8.75805	0.961320	0.480660	+	0.876907i	$0.340397\pi$
$84$	−7.41879	−0.809456
$85$	1.19117	0.129201
$86$	2.28473	0.246369
$87$	−4.76270	−0.510615
$88$	−2.78738	−0.297135
$89$	−13.9073	−1.47417	−0.737087	−	0.675798i	$-0.763799\pi$
$89$	−13.9073	−1.47417	−0.737087	+	0.675798i	$0.763799\pi$
$90$	−7.01999	−0.739972
$91$	7.84985	0.822888
$92$	10.9737	1.14408
$93$	1.82836	0.189592
$94$	0.707779	0.0730018
$95$	2.39109	0.245320
$96$	−3.72132	−0.379806
$97$	−12.7362	−1.29317	−0.646584	−	0.762843i	$-0.723803\pi$
$97$	−12.7362	−1.29317	−0.646584	+	0.762843i	$0.723803\pi$
$98$	−12.8449	−1.29753
$99$	−2.57944	−0.259244
$100$	−11.5312	−1.15312
$101$	9.09598	0.905084	0.452542	−	0.891743i	$-0.350517\pi$
$101$	9.09598	0.905084	0.452542	+	0.891743i	$0.350517\pi$
$102$	−1.48165	−0.146706
$103$	−2.34909	−0.231462	−0.115731	−	0.993281i	$-0.536921\pi$
$103$	−2.34909	−0.231462	−0.115731	+	0.993281i	$0.536921\pi$
$104$	−6.15873	−0.603914
$105$	2.74443	0.267829
$106$	−9.00594	−0.874734
$107$	11.3911	1.10122	0.550609	−	0.834763i	$-0.314395\pi$
$107$	11.3911	1.10122	0.550609	+	0.834763i	$0.314395\pi$
$108$	11.6509	1.12111
$109$	10.8657	1.04074	0.520371	−	0.853940i	$-0.325794\pi$
$109$	10.8657	1.04074	0.520371	+	0.853940i	$0.325794\pi$
$110$	2.72151	0.259486
$111$	4.46975	0.424250
$112$	−0.254364	−0.0240351
$113$	0.532901	0.0501311	0.0250656	−	0.999686i	$-0.492021\pi$
$113$	0.532901	0.0501311	0.0250656	+	0.999686i	$0.492021\pi$
$114$	−2.97418	−0.278558
$115$	−4.05948	−0.378548
$116$	23.6482	2.19568
$117$	−5.69931	−0.526901
$118$	26.7242	2.46017
$119$	−3.55276	−0.325681
$120$	−2.15319	−0.196558
$121$	1.00000	0.0909091
$122$	15.5589	1.40864
$123$	−1.70962	−0.154152
$124$	−9.07833	−0.815258
$125$	10.2216	0.914246
$126$	20.9376	1.86527
$127$	21.2140	1.88243	0.941217	−	0.337802i	$-0.109683\pi$
$127$	21.2140	1.88243	0.941217	+	0.337802i	$0.109683\pi$
$128$	18.1503	1.60427
$129$	0.648502	0.0570974
$130$	6.01321	0.527393
$131$	−22.6848	−1.98198	−0.990989	−	0.133945i	$-0.957235\pi$
$131$	−22.6848	−1.98198	−0.990989	+	0.133945i	$0.957235\pi$
$132$	−2.08818	−0.181753
$133$	−7.13159	−0.618387
$134$	−3.81354	−0.329440
$135$	−4.31000	−0.370946
$136$	2.78738	0.239015
$137$	17.7104	1.51310	0.756552	−	0.653934i	$-0.226883\pi$
$137$	17.7104	1.51310	0.756552	+	0.653934i	$0.226883\pi$
$138$	5.04943	0.429836
$139$	13.0865	1.10998	0.554992	−	0.831855i	$-0.312721\pi$
$139$	13.0865	1.10998	0.554992	+	0.831855i	$0.312721\pi$
$140$	−13.6269	−1.15168
$141$	0.200897	0.0169186
$142$	−13.5176	−1.13437
$143$	2.20951	0.184769
$144$	0.184679	0.0153899
$145$	−8.74815	−0.726495
$146$	−17.1996	−1.42345
$147$	−3.64593	−0.300711
$148$	−22.1936	−1.82430
$149$	5.89280	0.482757	0.241378	−	0.970431i	$-0.422401\pi$
$149$	5.89280	0.482757	0.241378	+	0.970431i	$0.422401\pi$
$150$	−5.30596	−0.433230
$151$	−3.39079	−0.275939	−0.137969	−	0.990436i	$-0.544058\pi$
$151$	−3.39079	−0.275939	−0.137969	+	0.990436i	$0.544058\pi$
$152$	5.59521	0.453831
$153$	2.57944	0.208536
$154$	−8.11710	−0.654094
$155$	3.35834	0.269748
$156$	−4.61385	−0.369404
$157$	−7.08938	−0.565794	−0.282897	−	0.959150i	$-0.591296\pi$
$157$	−7.08938	−0.565794	−0.282897	+	0.959150i	$0.591296\pi$
$158$	−16.7950	−1.33614
$159$	−2.55626	−0.202725
$160$	−6.83534	−0.540381
$161$	12.1077	0.954218
$162$	−12.3190	−0.967870
$163$	−0.823931	−0.0645353	−0.0322676	−	0.999479i	$-0.510273\pi$
$163$	−0.823931	−0.0645353	−0.0322676	+	0.999479i	$0.510273\pi$
$164$	8.48877	0.662862
$165$	0.772478	0.0601374
$166$	−20.0098	−1.55306
$167$	11.4728	0.887788	0.443894	−	0.896079i	$-0.353597\pi$
$167$	11.4728	0.887788	0.443894	+	0.896079i	$0.353597\pi$
$168$	6.42203	0.495470
$169$	−8.11806	−0.624466
$170$	−2.72151	−0.208730
$171$	5.17782	0.395958
$172$	−3.22000	−0.245523
$173$	−21.8738	−1.66303	−0.831517	−	0.555499i	$-0.812527\pi$
$173$	−21.8738	−1.66303	−0.831517	+	0.555499i	$0.812527\pi$
$174$	10.8815	0.824924
$175$	−12.7228	−0.961753
$176$	−0.0715963	−0.00539677
$177$	7.58545	0.570158
$178$	31.7745	2.38160
$179$	3.95534	0.295636	0.147818	−	0.989015i	$-0.452775\pi$
$179$	3.95534	0.295636	0.147818	+	0.989015i	$0.452775\pi$
$180$	9.89366	0.737430
$181$	−18.4422	−1.37080	−0.685398	−	0.728169i	$-0.740372\pi$
$181$	−18.4422	−1.37080	−0.685398	+	0.728169i	$0.740372\pi$
$182$	−17.9348	−1.32942
$183$	4.41627	0.326460
$184$	−9.49928	−0.700296
$185$	8.21006	0.603616
$186$	−4.17731	−0.306295
$187$	−1.00000	−0.0731272
$188$	−0.997512	−0.0727511
$189$	12.8549	0.935055
$190$	−5.46299	−0.396327
$191$	9.97761	0.721954	0.360977	−	0.932575i	$-0.382443\pi$
$191$	9.97761	0.721954	0.360977	+	0.932575i	$0.382443\pi$
$192$	8.40936	0.606893
$193$	14.4311	1.03877	0.519385	−	0.854540i	$-0.326161\pi$
$193$	14.4311	1.03877	0.519385	+	0.854540i	$0.326161\pi$
$194$	29.0989	2.08918
$195$	1.70680	0.122226
$196$	18.1031	1.29308
$197$	19.7895	1.40995	0.704973	−	0.709234i	$-0.250959\pi$
$197$	19.7895	1.40995	0.704973	+	0.709234i	$0.250959\pi$
$198$	5.89334	0.418821
$199$	−19.0828	−1.35274	−0.676371	−	0.736561i	$-0.736448\pi$
$199$	−19.0828	−1.35274	−0.676371	+	0.736561i	$0.736448\pi$
$200$	9.98189	0.705826
$201$	−1.08244	−0.0763496
$202$	−20.7819	−1.46221
$203$	26.0920	1.83130
$204$	2.08818	0.146202
$205$	−3.14025	−0.219324
$206$	5.36703	0.373939
$207$	−8.79066	−0.610993
$208$	−0.158193	−0.0109687
$209$	−2.00734	−0.138851
$210$	−6.27028	−0.432691
$211$	4.30128	0.296113	0.148056	−	0.988979i	$-0.452698\pi$
$211$	4.30128	0.296113	0.148056	+	0.988979i	$0.452698\pi$
$212$	12.6926	0.871730
$213$	−3.83686	−0.262898
$214$	−26.0256	−1.77907
$215$	1.19117	0.0812373
$216$	−10.0855	−0.686232
$217$	−10.0165	−0.679964
$218$	−24.8251	−1.68137
$219$	−4.88195	−0.329892
$220$	−3.83558	−0.258595
$221$	−2.20951	−0.148628
$222$	−10.2122	−0.685397
$223$	−13.9198	−0.932137	−0.466069	−	0.884749i	$-0.654330\pi$
$223$	−13.9198	−0.932137	−0.466069	+	0.884749i	$0.654330\pi$
$224$	20.3869	1.36216
$225$	9.23727	0.615818
$226$	−1.21754	−0.0809893
$227$	3.26121	0.216454	0.108227	−	0.994126i	$-0.465483\pi$
$227$	3.26121	0.216454	0.108227	+	0.994126i	$0.465483\pi$
$228$	4.19168	0.277601
$229$	−4.43344	−0.292970	−0.146485	−	0.989213i	$-0.546796\pi$
$229$	−4.43344	−0.292970	−0.146485	+	0.989213i	$0.546796\pi$
$230$	9.27481	0.611563
$231$	−2.30397	−0.151590
$232$	−20.4709	−1.34398
$233$	1.04788	0.0686488	0.0343244	−	0.999411i	$-0.489072\pi$
$233$	1.04788	0.0686488	0.0343244	+	0.999411i	$0.489072\pi$
$234$	13.0214	0.851235
$235$	0.369009	0.0240715
$236$	−37.6640	−2.45171
$237$	−4.76714	−0.309659
$238$	8.11710	0.526153
$239$	−8.56665	−0.554130	−0.277065	−	0.960851i	$-0.589362\pi$
$239$	−8.56665	−0.554130	−0.277065	+	0.960851i	$0.589362\pi$
$240$	−0.0553066	−0.00357002
$241$	−3.14689	−0.202709	−0.101354	−	0.994850i	$-0.532318\pi$
$241$	−3.14689	−0.202709	−0.101354	+	0.994850i	$0.532318\pi$
$242$	−2.28473	−0.146868
$243$	−14.3515	−0.920648
$244$	−21.9280	−1.40380
$245$	−6.69687	−0.427847
$246$	3.90603	0.249039
$247$	−4.43524	−0.282207
$248$	7.85861	0.499022
$249$	−5.67961	−0.359931
$250$	−23.3536	−1.47701
$251$	−21.3698	−1.34885	−0.674424	−	0.738344i	$-0.735608\pi$
$251$	−21.3698	−1.34885	−0.674424	+	0.738344i	$0.735608\pi$
$252$	−29.5085	−1.85886
$253$	3.40797	0.214257
$254$	−48.4682	−3.04117
$255$	−0.772478	−0.0483745
$256$	−15.5338	−0.970862
$257$	17.3554	1.08260	0.541301	−	0.840829i	$-0.317932\pi$
$257$	17.3554	1.08260	0.541301	+	0.840829i	$0.317932\pi$
$258$	−1.48165	−0.0922438
$259$	−24.4871	−1.52155
$260$	−8.47475	−0.525582
$261$	−18.9438	−1.17259
$262$	51.8286	3.20198
$263$	9.40210	0.579759	0.289879	−	0.957063i	$-0.406385\pi$
$263$	9.40210	0.579759	0.289879	+	0.957063i	$0.406385\pi$
$264$	1.80762	0.111251
$265$	−4.69536	−0.288433
$266$	16.2938	0.999035
$267$	9.01894	0.551950
$268$	5.37464	0.328308
$269$	−22.5602	−1.37552	−0.687760	−	0.725938i	$-0.741406\pi$
$269$	−22.5602	−1.37552	−0.687760	+	0.725938i	$0.741406\pi$
$270$	9.84720	0.599281
$271$	28.6420	1.73988	0.869938	−	0.493161i	$-0.164159\pi$
$271$	28.6420	1.73988	0.869938	+	0.493161i	$0.164159\pi$
$272$	0.0715963	0.00434116
$273$	−5.09065	−0.308100
$274$	−40.4636	−2.44449
$275$	−3.58111	−0.215949
$276$	−7.11644	−0.428359
$277$	7.54784	0.453506	0.226753	−	0.973952i	$-0.427189\pi$
$277$	7.54784	0.453506	0.226753	+	0.973952i	$0.427189\pi$
$278$	−29.8992	−1.79324
$279$	7.27238	0.435386
$280$	11.7960	0.704947
$281$	24.1984	1.44356	0.721778	−	0.692125i	$-0.243325\pi$
$281$	24.1984	1.44356	0.721778	+	0.692125i	$0.243325\pi$
$282$	−0.458996	−0.0273328
$283$	−5.84037	−0.347174	−0.173587	−	0.984819i	$-0.555536\pi$
$283$	−5.84037	−0.347174	−0.173587	+	0.984819i	$0.555536\pi$
$284$	19.0511	1.13048
$285$	−1.55063	−0.0918511
$286$	−5.04814	−0.298503
$287$	9.36600	0.552858
$288$	−14.8017	−0.872199
$289$	1.00000	0.0588235
$290$	19.9872	1.17369
$291$	8.25947	0.484179
$292$	24.2403	1.41856
$293$	21.5392	1.25833	0.629165	−	0.777272i	$-0.283397\pi$
$293$	21.5392	1.25833	0.629165	+	0.777272i	$0.283397\pi$
$294$	8.32997	0.485814
$295$	13.9330	0.811211
$296$	19.2118	1.11666
$297$	3.61828	0.209954
$298$	−13.4635	−0.779917
$299$	7.52993	0.435467
$300$	7.47799	0.431742
$301$	−3.55276	−0.204777
$302$	7.74705	0.445793
$303$	−5.89876	−0.338875
$304$	0.143718	0.00824279
$305$	8.11183	0.464482
$306$	−5.89334	−0.336900
$307$	−31.2816	−1.78533	−0.892666	−	0.450718i	$-0.851168\pi$
$307$	−31.2816	−1.78533	−0.892666	+	0.450718i	$0.851168\pi$
$308$	11.4399	0.651848
$309$	1.52339	0.0866625
$310$	−7.67291	−0.435792
$311$	8.11131	0.459950	0.229975	−	0.973197i	$-0.426136\pi$
$311$	8.11131	0.459950	0.229975	+	0.973197i	$0.426136\pi$
$312$	3.99395	0.226113
$313$	−9.34692	−0.528319	−0.264160	−	0.964479i	$-0.585095\pi$
$313$	−9.34692	−0.528319	−0.264160	+	0.964479i	$0.585095\pi$
$314$	16.1973	0.914069
$315$	10.9161	0.615051
$316$	23.6702	1.33155
$317$	23.1945	1.30273	0.651367	−	0.758763i	$-0.274196\pi$
$317$	23.1945	1.30273	0.651367	+	0.758763i	$0.274196\pi$
$318$	5.84037	0.327512
$319$	7.34415	0.411193
$320$	15.4464	0.863478
$321$	−7.38715	−0.412311
$322$	−27.6628	−1.54159
$323$	2.00734	0.111691
$324$	17.3618	0.964546
$325$	−7.91249	−0.438906
$326$	1.88246	0.104260
$327$	−7.04641	−0.389667
$328$	−7.34825	−0.405740
$329$	−1.10060	−0.0606778
$330$	−1.76491	−0.0971549
$331$	−7.80558	−0.429034	−0.214517	−	0.976720i	$-0.568818\pi$
$331$	−7.80558	−0.429034	−0.214517	+	0.976720i	$0.568818\pi$
$332$	28.2009	1.54773
$333$	17.7786	0.974262
$334$	−26.2122	−1.43427
$335$	−1.98824	−0.108629
$336$	0.164956	0.00899907
$337$	−1.36060	−0.0741166	−0.0370583	−	0.999313i	$-0.511799\pi$
$337$	−1.36060	−0.0741166	−0.0370583	+	0.999313i	$0.511799\pi$
$338$	18.5476	1.00886
$339$	−0.345588	−0.0187697
$340$	3.83558	0.208013
$341$	−2.81936	−0.152677
$342$	−11.8299	−0.639689
$343$	−4.89542	−0.264328
$344$	2.78738	0.150285
$345$	2.63258	0.141733
$346$	49.9758	2.68671
$347$	−15.7178	−0.843774	−0.421887	−	0.906648i	$-0.638632\pi$
$347$	−15.7178	−0.843774	−0.421887	+	0.906648i	$0.638632\pi$
$348$	−15.3359	−0.822090
$349$	−17.6120	−0.942747	−0.471373	−	0.881934i	$-0.656242\pi$
$349$	−17.6120	−0.942747	−0.471373	+	0.881934i	$0.656242\pi$
$350$	29.0682	1.55376
$351$	7.99463	0.426722
$352$	5.73833	0.305854
$353$	−9.92305	−0.528151	−0.264075	−	0.964502i	$-0.585067\pi$
$353$	−9.92305	−0.528151	−0.264075	+	0.964502i	$0.585067\pi$
$354$	−17.3307	−0.921118
$355$	−7.04757	−0.374046
$356$	−44.7816	−2.37342
$357$	2.30397	0.121939
$358$	−9.03689	−0.477615
$359$	−24.5876	−1.29768	−0.648842	−	0.760923i	$-0.724746\pi$
$359$	−24.5876	−1.29768	−0.648842	+	0.760923i	$0.724746\pi$
$360$	−8.56439	−0.451383
$361$	−14.9706	−0.787926
$362$	42.1354	2.21459
$363$	−0.648502	−0.0340375
$364$	25.2765	1.32485
$365$	−8.96720	−0.469365
$366$	−10.0900	−0.527412
$367$	−15.7572	−0.822517	−0.411258	−	0.911519i	$-0.634911\pi$
$367$	−15.7572	−0.822517	−0.411258	+	0.911519i	$0.634911\pi$
$368$	−0.243998	−0.0127193
$369$	−6.80010	−0.353999
$370$	−18.7578	−0.975171
$371$	14.0042	0.727063
$372$	5.88732	0.305243
$373$	22.1275	1.14572	0.572858	−	0.819654i	$-0.305835\pi$
$373$	22.1275	1.14572	0.572858	+	0.819654i	$0.305835\pi$
$374$	2.28473	0.118141
$375$	−6.62872	−0.342306
$376$	0.863491	0.0445311
$377$	16.2270	0.835732
$378$	−29.3700	−1.51063
$379$	13.2418	0.680184	0.340092	−	0.940392i	$-0.389542\pi$
$379$	13.2418	0.680184	0.340092	+	0.940392i	$0.389542\pi$
$380$	7.69930	0.394966
$381$	−13.7573	−0.704808
$382$	−22.7962	−1.16635
$383$	−15.4977	−0.791897	−0.395948	−	0.918273i	$-0.629584\pi$
$383$	−15.4977	−0.791897	−0.395948	+	0.918273i	$0.629584\pi$
$384$	−11.7705	−0.600660
$385$	−4.23195	−0.215680
$386$	−32.9711	−1.67819
$387$	2.57944	0.131121
$388$	−41.0107	−2.08200
$389$	28.7306	1.45670	0.728350	−	0.685205i	$-0.240287\pi$
$389$	28.7306	1.45670	0.728350	+	0.685205i	$0.240287\pi$
$390$	−3.89958	−0.197463
$391$	−3.40797	−0.172348
$392$	−15.6708	−0.791497
$393$	14.7111	0.742078
$394$	−45.2138	−2.27784
$395$	−8.75630	−0.440577
$396$	−8.30581	−0.417383
$397$	2.46298	0.123613	0.0618067	−	0.998088i	$-0.480314\pi$
$397$	2.46298	0.123613	0.0618067	+	0.998088i	$0.480314\pi$
$398$	43.5990	2.18542
$399$	4.62485	0.231532
$400$	0.256394	0.0128197
$401$	−1.07935	−0.0539001	−0.0269500	−	0.999637i	$-0.508580\pi$
$401$	−1.07935	−0.0539001	−0.0269500	+	0.999637i	$0.508580\pi$
$402$	2.47309	0.123347
$403$	−6.22940	−0.310308
$404$	29.2891	1.45719
$405$	−6.42265	−0.319144
$406$	−59.6132	−2.95855
$407$	−6.89242	−0.341644
$408$	−1.80762	−0.0894905
$409$	−5.14552	−0.254430	−0.127215	−	0.991875i	$-0.540604\pi$
$409$	−5.14552	−0.254430	−0.127215	+	0.991875i	$0.540604\pi$
$410$	7.17462	0.354329
$411$	−11.4853	−0.566526
$412$	−7.56406	−0.372654
$413$	−41.5562	−2.04484
$414$	20.0843	0.987089
$415$	−10.4323	−0.512104
$416$	12.6789	0.621634
$417$	−8.48664	−0.415593
$418$	4.58623	0.224320
$419$	21.3971	1.04532	0.522658	−	0.852543i	$-0.324941\pi$
$419$	21.3971	1.04532	0.522658	+	0.852543i	$0.324941\pi$
$420$	8.83706	0.431204
$421$	−0.216598	−0.0105563	−0.00527817	−	0.999986i	$-0.501680\pi$
$421$	−0.216598	−0.0105563	−0.00527817	+	0.999986i	$0.501680\pi$
$422$	−9.82728	−0.478384
$423$	0.799077	0.0388524
$424$	−10.9873	−0.533588
$425$	3.58111	0.173709
$426$	8.76620	0.424724
$427$	−24.1941	−1.17083
$428$	36.6793	1.77296
$429$	−1.43287	−0.0691797
$430$	−2.72151	−0.131243
$431$	−24.3837	−1.17452	−0.587262	−	0.809397i	$-0.699794\pi$
$431$	−24.3837	−1.17452	−0.587262	+	0.809397i	$0.699794\pi$
$432$	−0.259056	−0.0124638
$433$	19.2679	0.925955	0.462977	−	0.886370i	$-0.346781\pi$
$433$	19.2679	0.925955	0.462977	+	0.886370i	$0.346781\pi$
$434$	22.8850	1.09851
$435$	5.67320	0.272009
$436$	34.9874	1.67559
$437$	−6.84094	−0.327247
$438$	11.1540	0.532957
$439$	−27.0559	−1.29131	−0.645653	−	0.763631i	$-0.723415\pi$
$439$	−27.0559	−1.29131	−0.645653	+	0.763631i	$0.723415\pi$
$440$	3.32025	0.158286
$441$	−14.5018	−0.690564
$442$	5.04814	0.240116
$443$	16.8591	0.801000	0.400500	−	0.916297i	$-0.368836\pi$
$443$	16.8591	0.801000	0.400500	+	0.916297i	$0.368836\pi$
$444$	14.3926	0.683042
$445$	16.5660	0.785305
$446$	31.8030	1.50591
$447$	−3.82149	−0.180750
$448$	−46.0698	−2.17660
$449$	7.60495	0.358900	0.179450	−	0.983767i	$-0.442568\pi$
$449$	7.60495	0.358900	0.179450	+	0.983767i	$0.442568\pi$
$450$	−21.1047	−0.994884
$451$	2.63626	0.124137
$452$	1.71594	0.0807111
$453$	2.19894	0.103315
$454$	−7.45099	−0.349692
$455$	−9.35053	−0.438360
$456$	−3.62850	−0.169920
$457$	7.17726	0.335738	0.167869	−	0.985809i	$-0.446311\pi$
$457$	7.17726	0.335738	0.167869	+	0.985809i	$0.446311\pi$
$458$	10.1292	0.473307
$459$	−3.61828	−0.168887
$460$	−13.0715	−0.609463
$461$	34.7799	1.61986	0.809930	−	0.586526i	$-0.199505\pi$
$461$	34.7799	1.61986	0.809930	+	0.586526i	$0.199505\pi$
$462$	5.26396	0.244901
$463$	16.9796	0.789110	0.394555	−	0.918872i	$-0.370899\pi$
$463$	16.9796	0.789110	0.394555	+	0.918872i	$0.370899\pi$
$464$	−0.525814	−0.0244103
$465$	−2.17789	−0.100997
$466$	−2.39412	−0.110905
$467$	27.7214	1.28280	0.641398	−	0.767209i	$-0.278355\pi$
$467$	27.7214	1.28280	0.641398	+	0.767209i	$0.278355\pi$
$468$	−18.3518	−0.848312
$469$	5.93006	0.273825
$470$	−0.843087	−0.0388887
$471$	4.59748	0.211841
$472$	32.6036	1.50070
$473$	−1.00000	−0.0459800
$474$	10.8916	0.500269
$475$	7.18850	0.329831
$476$	−11.4399	−0.524346
$477$	−10.1676	−0.465544
$478$	19.5725	0.895225
$479$	−26.0165	−1.18873	−0.594363	−	0.804197i	$-0.702596\pi$
$479$	−26.0165	−1.18873	−0.594363	+	0.804197i	$0.702596\pi$
$480$	4.43273	0.202326
$481$	−15.2289	−0.694377
$482$	7.18980	0.327486
$483$	−7.85185	−0.357272
$484$	3.22000	0.146364
$485$	15.1711	0.688882
$486$	32.7893	1.48735
$487$	−19.4282	−0.880377	−0.440188	−	0.897905i	$-0.645088\pi$
$487$	−19.4282	−0.880377	−0.440188	+	0.897905i	$0.645088\pi$
$488$	18.9819	0.859270
$489$	0.534321	0.0241628
$490$	15.3005	0.691208
$491$	25.5428	1.15273	0.576365	−	0.817192i	$-0.304470\pi$
$491$	25.5428	1.15273	0.576365	+	0.817192i	$0.304470\pi$
$492$	−5.50499	−0.248184
$493$	−7.34415	−0.330764
$494$	10.1333	0.455920
$495$	3.07256	0.138102
$496$	0.201855	0.00906358
$497$	21.0199	0.942870
$498$	12.9764	0.581486
$499$	17.4977	0.783306	0.391653	−	0.920113i	$-0.371903\pi$
$499$	17.4977	0.783306	0.391653	+	0.920113i	$0.371903\pi$
$500$	32.9135	1.47194
$501$	−7.44011	−0.332400
$502$	48.8242	2.17913
$503$	−15.8960	−0.708769	−0.354385	−	0.935100i	$-0.615310\pi$
$503$	−15.8960	−0.708769	−0.354385	+	0.935100i	$0.615310\pi$
$504$	25.5439	1.13782
$505$	−10.8349	−0.482146
$506$	−7.78629	−0.346143
$507$	5.26458	0.233808
$508$	68.3090	3.03072
$509$	16.1262	0.714783	0.357392	−	0.933955i	$-0.383666\pi$
$509$	16.1262	0.714783	0.357392	+	0.933955i	$0.383666\pi$
$510$	1.76491	0.0781514
$511$	26.7453	1.18314
$512$	−0.809974	−0.0357961
$513$	−7.26312	−0.320675
$514$	−39.6525	−1.74900
$515$	2.79817	0.123302
$516$	2.08818	0.0919269
$517$	−0.309786	−0.0136244
$518$	55.9464	2.45814
$519$	14.1852	0.622662
$520$	7.33612	0.321710
$521$	33.4431	1.46517	0.732585	−	0.680676i	$-0.238314\pi$
$521$	33.4431	1.46517	0.732585	+	0.680676i	$0.238314\pi$
$522$	43.2816	1.89438
$523$	−14.7468	−0.644830	−0.322415	−	0.946598i	$-0.604495\pi$
$523$	−14.7468	−0.644830	−0.322415	+	0.946598i	$0.604495\pi$
$524$	−73.0450	−3.19098
$525$	8.25077	0.360093
$526$	−21.4813	−0.936629
$527$	2.81936	0.122813
$528$	0.0464303	0.00202062
$529$	−11.3858	−0.495034
$530$	10.7276	0.465979
$531$	30.1715	1.30933
$532$	−22.9637	−0.995603
$533$	5.82485	0.252302
$534$	−20.6059	−0.891702
$535$	−13.5688	−0.586629
$536$	−4.65253	−0.200958
$537$	−2.56505	−0.110690
$538$	51.5440	2.22222
$539$	5.62208	0.242160
$540$	−13.8782	−0.597223
$541$	24.8653	1.06904	0.534520	−	0.845156i	$-0.320492\pi$
$541$	24.8653	1.06904	0.534520	+	0.845156i	$0.320492\pi$
$542$	−65.4392	−2.81086
$543$	11.9598	0.513244
$544$	−5.73833	−0.246029
$545$	−12.9429	−0.554412
$546$	11.6308	0.497751
$547$	−6.56693	−0.280782	−0.140391	−	0.990096i	$-0.544836\pi$
$547$	−6.56693	−0.280782	−0.140391	+	0.990096i	$0.544836\pi$
$548$	57.0276	2.43610
$549$	17.5659	0.749694
$550$	8.18187	0.348876
$551$	−14.7422	−0.628039
$552$	6.16030	0.262200
$553$	26.1163	1.11058
$554$	−17.2448	−0.732661
$555$	−5.32424	−0.226002
$556$	42.1386	1.78708
$557$	−37.0106	−1.56819	−0.784096	−	0.620640i	$-0.786873\pi$
$557$	−37.0106	−1.56819	−0.784096	+	0.620640i	$0.786873\pi$
$558$	−16.6154	−0.703387
$559$	−2.20951	−0.0934523
$560$	0.302992	0.0128037
$561$	0.648502	0.0273798
$562$	−55.2869	−2.33214
$563$	−9.00574	−0.379547	−0.189773	−	0.981828i	$-0.560775\pi$
$563$	−9.00574	−0.379547	−0.189773	+	0.981828i	$0.560775\pi$
$564$	0.646889	0.0272389
$565$	−0.634777	−0.0267053
$566$	13.3437	0.560877
$567$	19.1560	0.804476
$568$	−16.4915	−0.691968
$569$	−3.64279	−0.152714	−0.0763569	−	0.997081i	$-0.524329\pi$
$569$	−3.64279	−0.152714	−0.0763569	+	0.997081i	$0.524329\pi$
$570$	3.54276	0.148390
$571$	39.3205	1.64551	0.822755	−	0.568395i	$-0.192436\pi$
$571$	39.3205	1.64551	0.822755	+	0.568395i	$0.192436\pi$
$572$	7.11463	0.297477
$573$	−6.47050	−0.270309
$574$	−21.3988	−0.893169
$575$	−12.2043	−0.508954
$576$	33.4486	1.39369
$577$	18.6552	0.776627	0.388314	−	0.921527i	$-0.373058\pi$
$577$	18.6552	0.776627	0.388314	+	0.921527i	$0.373058\pi$
$578$	−2.28473	−0.0950323
$579$	−9.35858	−0.388929
$580$	−28.1691	−1.16966
$581$	31.1152	1.29088
$582$	−18.8707	−0.782215
$583$	3.94179	0.163252
$584$	−20.9835	−0.868303
$585$	6.78886	0.280685
$586$	−49.2112	−2.03290
$587$	−12.3832	−0.511107	−0.255554	−	0.966795i	$-0.582258\pi$
$587$	−12.3832	−0.511107	−0.255554	+	0.966795i	$0.582258\pi$
$588$	−11.7399	−0.484145
$589$	5.65940	0.233192
$590$	−31.8332	−1.31055
$591$	−12.8336	−0.527902
$592$	0.493471	0.0202815
$593$	21.1433	0.868253	0.434126	−	0.900852i	$-0.357057\pi$
$593$	21.1433	0.868253	0.434126	+	0.900852i	$0.357057\pi$
$594$	−8.26681	−0.339191
$595$	4.23195	0.173493
$596$	18.9748	0.777239
$597$	12.3752	0.506484
$598$	−17.2039	−0.703519
$599$	−5.00603	−0.204541	−0.102270	−	0.994757i	$-0.532611\pi$
$599$	−5.00603	−0.204541	−0.102270	+	0.994757i	$0.532611\pi$
$600$	−6.47328	−0.264270
$601$	29.7526	1.21363	0.606817	−	0.794841i	$-0.292446\pi$
$601$	29.7526	1.21363	0.606817	+	0.794841i	$0.292446\pi$
$602$	8.11710	0.330828
$603$	−4.30546	−0.175332
$604$	−10.9184	−0.444262
$605$	−1.19117	−0.0484281
$606$	13.4771	0.547470
$607$	11.3253	0.459680	0.229840	−	0.973228i	$-0.426180\pi$
$607$	11.3253	0.459680	0.229840	+	0.973228i	$0.426180\pi$
$608$	−11.5188	−0.467148
$609$	−16.9207	−0.685662
$610$	−18.5334	−0.750394
$611$	−0.684476	−0.0276909
$612$	8.30581	0.335743
$613$	26.1633	1.05672	0.528362	−	0.849019i	$-0.322806\pi$
$613$	26.1633	1.05672	0.528362	+	0.849019i	$0.322806\pi$
$614$	71.4700	2.88429
$615$	2.03646	0.0821179
$616$	−9.90286	−0.398998
$617$	−4.15687	−0.167349	−0.0836747	−	0.996493i	$-0.526666\pi$
$617$	−4.15687	−0.167349	−0.0836747	+	0.996493i	$0.526666\pi$
$618$	−3.48053	−0.140008
$619$	14.6889	0.590398	0.295199	−	0.955436i	$-0.404614\pi$
$619$	14.6889	0.590398	0.295199	+	0.955436i	$0.404614\pi$
$620$	10.8139	0.434295
$621$	12.3310	0.494825
$622$	−18.5322	−0.743072
$623$	−49.4093	−1.97954
$624$	0.102588	0.00410682
$625$	5.72987	0.229195
$626$	21.3552	0.853526
$627$	1.30176	0.0519874
$628$	−22.8278	−0.910929
$629$	6.89242	0.274819
$630$	−24.9403	−0.993645
$631$	−21.1440	−0.841729	−0.420864	−	0.907124i	$-0.638273\pi$
$631$	−21.1440	−0.841729	−0.420864	+	0.907124i	$0.638273\pi$
$632$	−20.4900	−0.815047
$633$	−2.78939	−0.110868
$634$	−52.9933	−2.10463
$635$	−25.2695	−1.00279
$636$	−8.23117	−0.326387
$637$	12.4220	0.492179
$638$	−16.7794	−0.664303
$639$	−15.2613	−0.603727
$640$	−21.6201	−0.854610
$641$	10.2656	0.405468	0.202734	−	0.979234i	$-0.435017\pi$
$641$	10.2656	0.405468	0.202734	+	0.979234i	$0.435017\pi$
$642$	16.8777	0.666108
$643$	−24.3254	−0.959302	−0.479651	−	0.877459i	$-0.659237\pi$
$643$	−24.3254	−0.959302	−0.479651	+	0.877459i	$0.659237\pi$
$644$	38.9867	1.53629
$645$	−0.772478	−0.0304163
$646$	−4.58623	−0.180443
$647$	23.4820	0.923172	0.461586	−	0.887095i	$-0.347281\pi$
$647$	23.4820	0.923172	0.461586	+	0.887095i	$0.347281\pi$
$648$	−15.0292	−0.590401
$649$	−11.6969	−0.459143
$650$	18.0779	0.709075
$651$	6.49572	0.254587
$652$	−2.65306	−0.103902
$653$	−28.6780	−1.12226	−0.561128	−	0.827729i	$-0.689633\pi$
$653$	−28.6780	−1.12226	−0.561128	+	0.827729i	$0.689633\pi$
$654$	16.0991	0.629527
$655$	27.0215	1.05582
$656$	−0.188747	−0.00736932
$657$	−19.4182	−0.757575
$658$	2.51457	0.0980280
$659$	1.22280	0.0476334	0.0238167	−	0.999716i	$-0.492418\pi$
$659$	1.22280	0.0476334	0.0238167	+	0.999716i	$0.492418\pi$
$660$	2.48738	0.0968212
$661$	12.0950	0.470441	0.235221	−	0.971942i	$-0.424419\pi$
$661$	12.0950	0.470441	0.235221	+	0.971942i	$0.424419\pi$
$662$	17.8337	0.693125
$663$	1.43287	0.0556482
$664$	−24.4120	−0.947368
$665$	8.49495	0.329420
$666$	−40.6194	−1.57397
$667$	25.0286	0.969112
$668$	36.9423	1.42934
$669$	9.02701	0.349004
$670$	4.54259	0.175496
$671$	−6.80995	−0.262895
$672$	−13.2209	−0.510009
$673$	−4.39274	−0.169328	−0.0846639	−	0.996410i	$-0.526982\pi$
$673$	−4.39274	−0.169328	−0.0846639	+	0.996410i	$0.526982\pi$
$674$	3.10861	0.119739
$675$	−12.9575	−0.498733
$676$	−26.1402	−1.00539
$677$	−23.5799	−0.906248	−0.453124	−	0.891447i	$-0.649691\pi$
$677$	−23.5799	−0.906248	−0.453124	+	0.891447i	$0.649691\pi$
$678$	0.789575	0.0303234
$679$	−45.2487	−1.73649
$680$	−3.32025	−0.127326
$681$	−2.11490	−0.0810432
$682$	6.44148	0.246657
$683$	−10.9734	−0.419886	−0.209943	−	0.977714i	$-0.567328\pi$
$683$	−10.9734	−0.419886	−0.209943	+	0.977714i	$0.567328\pi$
$684$	16.6726	0.637492
$685$	−21.0962	−0.806043
$686$	11.1847	0.427035
$687$	2.87509	0.109692
$688$	0.0715963	0.00272958
$689$	8.70943	0.331803
$690$	−6.01474	−0.228977
$691$	−22.2785	−0.847515	−0.423757	−	0.905776i	$-0.639289\pi$
$691$	−22.2785	−0.847515	−0.423757	+	0.905776i	$0.639289\pi$
$692$	−70.4337	−2.67749
$693$	−9.16414	−0.348117
$694$	35.9109	1.36316
$695$	−15.5883	−0.591298
$696$	13.2754	0.503204
$697$	−2.63626	−0.0998556
$698$	40.2386	1.52305
$699$	−0.679551	−0.0257030
$700$	−40.9674	−1.54842
$701$	40.7156	1.53781	0.768904	−	0.639364i	$-0.220802\pi$
$701$	40.7156	1.53781	0.768904	+	0.639364i	$0.220802\pi$
$702$	−18.2656	−0.689391
$703$	13.8354	0.521813
$704$	−12.9674	−0.488725
$705$	−0.239303	−0.00901268
$706$	22.6715	0.853253
$707$	32.3158	1.21536
$708$	24.4252	0.917954
$709$	4.72375	0.177404	0.0887021	−	0.996058i	$-0.471728\pi$
$709$	4.72375	0.177404	0.0887021	+	0.996058i	$0.471728\pi$
$710$	16.1018	0.604290
$711$	−18.9615	−0.711111
$712$	38.7649	1.45278
$713$	−9.60827	−0.359833
$714$	−5.26396	−0.196999
$715$	−2.63191	−0.0984278
$716$	12.7362	0.475974
$717$	5.55549	0.207474
$718$	56.1761	2.09647
$719$	47.5144	1.77199	0.885995	−	0.463695i	$-0.153477\pi$
$719$	47.5144	1.77199	0.885995	+	0.463695i	$0.153477\pi$
$720$	−0.219984	−0.00819832
$721$	−8.34573	−0.310811
$722$	34.2038	1.27293
$723$	2.04077	0.0758969
$724$	−59.3838	−2.20698
$725$	−26.3002	−0.976765
$726$	1.48165	0.0549893
$727$	−5.62245	−0.208525	−0.104263	−	0.994550i	$-0.533248\pi$
$727$	−5.62245	−0.208525	−0.104263	+	0.994550i	$0.533248\pi$
$728$	−21.8805	−0.810945
$729$	−6.86863	−0.254394
$730$	20.4877	0.758282
$731$	1.00000	0.0369863
$732$	14.2204	0.525601
$733$	13.3269	0.492239	0.246119	−	0.969240i	$-0.420844\pi$
$733$	13.3269	0.492239	0.246119	+	0.969240i	$0.420844\pi$
$734$	36.0009	1.32882
$735$	4.34293	0.160191
$736$	19.5560	0.720845
$737$	1.66914	0.0614836
$738$	15.5364	0.571903
$739$	34.5361	1.27043	0.635216	−	0.772334i	$-0.280911\pi$
$739$	34.5361	1.27043	0.635216	+	0.772334i	$0.280911\pi$
$740$	26.4364	0.971821
$741$	2.87626	0.105662
$742$	−31.9959	−1.17461
$743$	2.33458	0.0856474	0.0428237	−	0.999083i	$-0.486365\pi$
$743$	2.33458	0.0856474	0.0428237	+	0.999083i	$0.486365\pi$
$744$	−5.09632	−0.186840
$745$	−7.01934	−0.257169
$746$	−50.5553	−1.85096
$747$	−22.5909	−0.826558
$748$	−3.22000	−0.117735
$749$	40.4698	1.47873
$750$	15.1448	0.553012
$751$	38.9039	1.41962	0.709811	−	0.704392i	$-0.248780\pi$
$751$	38.9039	1.41962	0.709811	+	0.704392i	$0.248780\pi$
$752$	0.0221795	0.000808805 0
$753$	13.8583	0.505026
$754$	−37.0743	−1.35017
$755$	4.03902	0.146995
$756$	41.3927	1.50544
$757$	−7.57459	−0.275303	−0.137652	−	0.990481i	$-0.543955\pi$
$757$	−7.57459	−0.275303	−0.137652	+	0.990481i	$0.543955\pi$
$758$	−30.2539	−1.09887
$759$	−2.21007	−0.0802206
$760$	−6.66486	−0.241760
$761$	19.0910	0.692048	0.346024	−	0.938226i	$-0.387532\pi$
$761$	19.0910	0.692048	0.346024	+	0.938226i	$0.387532\pi$
$762$	31.4318	1.13865
$763$	38.6030	1.39752
$764$	32.1279	1.16235
$765$	−3.07256	−0.111089
$766$	35.4082	1.27935
$767$	−25.8444	−0.933186
$768$	10.0737	0.363503
$769$	12.9331	0.466378	0.233189	−	0.972431i	$-0.425084\pi$
$769$	12.9331	0.466378	0.233189	+	0.972431i	$0.425084\pi$
$770$	9.66886	0.348442
$771$	−11.2550	−0.405340
$772$	46.4680	1.67242
$773$	−9.68816	−0.348459	−0.174230	−	0.984705i	$-0.555743\pi$
$773$	−9.68816	−0.348459	−0.174230	+	0.984705i	$0.555743\pi$
$774$	−5.89334	−0.211832
$775$	10.0964	0.362674
$776$	35.5006	1.27440
$777$	15.8799	0.569689
$778$	−65.6418	−2.35337
$779$	−5.29187	−0.189601
$780$	5.49589	0.196785
$781$	5.91650	0.211709
$782$	7.78629	0.278437
$783$	26.5732	0.949649
$784$	−0.402520	−0.0143757
$785$	8.44468	0.301404
$786$	−33.6110	−1.19886
$787$	−10.0288	−0.357487	−0.178744	−	0.983896i	$-0.557203\pi$
$787$	−10.0288	−0.357487	−0.178744	+	0.983896i	$0.557203\pi$
$788$	63.7224	2.27001
$789$	−6.09729	−0.217069
$790$	20.0058	0.711775
$791$	1.89327	0.0673168
$792$	7.18988	0.255481
$793$	−15.0467	−0.534322
$794$	−5.62725	−0.199704
$795$	3.04495	0.107993
$796$	−61.4465	−2.17791
$797$	20.4095	0.722941	0.361470	−	0.932384i	$-0.382275\pi$
$797$	20.4095	0.722941	0.361470	+	0.932384i	$0.382275\pi$
$798$	−10.5665	−0.374052
$799$	0.309786	0.0109595
$800$	−20.5496	−0.726537
$801$	35.8732	1.26752
$802$	2.46602	0.0870782
$803$	7.52804	0.265659
$804$	−3.48547	−0.122923
$805$	−14.4223	−0.508320
$806$	14.2325	0.501319
$807$	14.6304	0.515013
$808$	−25.3539	−0.891947
$809$	32.7107	1.15005	0.575024	−	0.818137i	$-0.304993\pi$
$809$	32.7107	1.15005	0.575024	+	0.818137i	$0.304993\pi$
$810$	14.6740	0.515593
$811$	45.7942	1.60805	0.804025	−	0.594595i	$-0.202688\pi$
$811$	45.7942	1.60805	0.804025	+	0.594595i	$0.202688\pi$
$812$	84.0162	2.94839
$813$	−18.5744	−0.651432
$814$	15.7473	0.551943
$815$	0.981444	0.0343785
$816$	−0.0464303	−0.00162539
$817$	2.00734	0.0702279
$818$	11.7561	0.411044
$819$	−20.2483	−0.707531
$820$	−10.1116	−0.353112
$821$	10.2577	0.357996	0.178998	−	0.983849i	$-0.442714\pi$
$821$	10.2577	0.357996	0.178998	+	0.983849i	$0.442714\pi$
$822$	26.2407	0.915250
$823$	27.4487	0.956802	0.478401	−	0.878141i	$-0.341216\pi$
$823$	27.4487	0.956802	0.478401	+	0.878141i	$0.341216\pi$
$824$	6.54778	0.228103
$825$	2.32236	0.0808541
$826$	94.9447	3.30355
$827$	−53.6382	−1.86518	−0.932592	−	0.360931i	$-0.882459\pi$
$827$	−53.6382	−1.86518	−0.932592	+	0.360931i	$0.882459\pi$
$828$	−28.3059	−0.983699
$829$	37.6464	1.30751	0.653756	−	0.756705i	$-0.273192\pi$
$829$	37.6464	1.30751	0.653756	+	0.756705i	$0.273192\pi$
$830$	23.8351	0.827329
$831$	−4.89480	−0.169799
$832$	−28.6515	−0.993312
$833$	−5.62208	−0.194793
$834$	19.3897	0.671411
$835$	−13.6660	−0.472933
$836$	−6.46363	−0.223549
$837$	−10.2012	−0.352606
$838$	−48.8866	−1.68876
$839$	25.8381	0.892031	0.446016	−	0.895025i	$-0.352843\pi$
$839$	25.8381	0.892031	0.446016	+	0.895025i	$0.352843\pi$
$840$	−7.64975	−0.263941
$841$	24.9365	0.859881
$842$	0.494868	0.0170543
$843$	−15.6927	−0.540486
$844$	13.8501	0.476741
$845$	9.67002	0.332659
$846$	−1.82568	−0.0627680
$847$	3.55276	0.122074
$848$	−0.282218	−0.00969139
$849$	3.78749	0.129986
$850$	−8.18187	−0.280636
$851$	−23.4891	−0.805197
$852$	−12.3547	−0.423265
$853$	30.6675	1.05003	0.525017	−	0.851092i	$-0.324059\pi$
$853$	30.6675	1.05003	0.525017	+	0.851092i	$0.324059\pi$
$854$	55.2770	1.89154
$855$	−6.16768	−0.210930
$856$	−31.7513	−1.08524
$857$	−4.19736	−0.143379	−0.0716896	−	0.997427i	$-0.522839\pi$
$857$	−4.19736	−0.143379	−0.0716896	+	0.997427i	$0.522839\pi$
$858$	3.27373	0.111763
$859$	37.1928	1.26900	0.634501	−	0.772922i	$-0.281205\pi$
$859$	37.1928	1.26900	0.634501	+	0.772922i	$0.281205\pi$
$860$	3.83558	0.130792
$861$	−6.07387	−0.206997
$862$	55.7103	1.89750
$863$	−9.66577	−0.329027	−0.164513	−	0.986375i	$-0.552605\pi$
$863$	−9.66577	−0.329027	−0.164513	+	0.986375i	$0.552605\pi$
$864$	20.7629	0.706368
$865$	26.0555	0.885913
$866$	−44.0219	−1.49593
$867$	−0.648502	−0.0220243
$868$	−32.2531	−1.09474
$869$	7.35099	0.249365
$870$	−12.9617	−0.439444
$871$	3.68799	0.124963
$872$	−30.2867	−1.02564
$873$	32.8524	1.11189
$874$	15.6297	0.528683
$875$	36.3148	1.22766
$876$	−15.7199	−0.531126
$877$	−42.0305	−1.41927	−0.709635	−	0.704569i	$-0.751140\pi$
$877$	−42.0305	−1.41927	−0.709635	+	0.704569i	$0.751140\pi$
$878$	61.8154	2.08617
$879$	−13.9682	−0.471135
$880$	0.0852835	0.00287491
$881$	20.1381	0.678468	0.339234	−	0.940702i	$-0.389832\pi$
$881$	20.1381	0.678468	0.339234	+	0.940702i	$0.389832\pi$
$882$	33.1328	1.11564
$883$	−16.5777	−0.557885	−0.278943	−	0.960308i	$-0.589984\pi$
$883$	−16.5777	−0.557885	−0.278943	+	0.960308i	$0.589984\pi$
$884$	−7.11463	−0.239291
$885$	−9.03559	−0.303728
$886$	−38.5185	−1.29406
$887$	15.3506	0.515424	0.257712	−	0.966222i	$-0.417031\pi$
$887$	15.3506	0.515424	0.257712	+	0.966222i	$0.417031\pi$
$888$	−12.4589	−0.418092
$889$	75.3680	2.52776
$890$	−37.8489	−1.26870
$891$	5.39187	0.180634
$892$	−44.8217	−1.50074
$893$	0.621846	0.0208093
$894$	8.73109	0.292011
$895$	−4.71149	−0.157488
$896$	64.4835	2.15424
$897$	−4.88318	−0.163045
$898$	−17.3753	−0.579820
$899$	−20.7058	−0.690577
$900$	29.7440	0.991467
$901$	−3.94179	−0.131320
$902$	−6.02316	−0.200549
$903$	2.30397	0.0766713
$904$	−1.48540	−0.0494035
$905$	21.9678	0.730235
$906$	−5.02398	−0.166911
$907$	22.9954	0.763551	0.381776	−	0.924255i	$-0.375313\pi$
$907$	22.9954	0.763551	0.381776	+	0.924255i	$0.375313\pi$
$908$	10.5011	0.348491
$909$	−23.4626	−0.778204
$910$	21.3635	0.708192
$911$	30.4863	1.01006	0.505029	−	0.863103i	$-0.331482\pi$
$911$	30.4863	1.01006	0.505029	+	0.863103i	$0.331482\pi$
$912$	−0.0932014	−0.00308621
$913$	8.75805	0.289849
$914$	−16.3981	−0.542401
$915$	−5.26054	−0.173908
$916$	−14.2757	−0.471681
$917$	−80.5935	−2.66143
$918$	8.26681	0.272845
$919$	−6.58041	−0.217068	−0.108534	−	0.994093i	$-0.534616\pi$
$919$	−6.58041	−0.217068	−0.108534	+	0.994093i	$0.534616\pi$
$920$	11.3153	0.373054
$921$	20.2862	0.668452
$922$	−79.4627	−2.61697
$923$	13.0726	0.430289
$924$	−7.41879	−0.244060
$925$	24.6825	0.811555
$926$	−38.7939	−1.27485
$927$	6.05933	0.199015
$928$	42.1431	1.38342
$929$	48.2706	1.58371	0.791853	−	0.610712i	$-0.209117\pi$
$929$	48.2706	1.58371	0.791853	+	0.610712i	$0.209117\pi$
$930$	4.97590	0.163166
$931$	−11.2854	−0.369864
$932$	3.37417	0.110525
$933$	−5.26020	−0.172211
$934$	−63.3361	−2.07242
$935$	1.19117	0.0389555
$936$	15.8861	0.519254
$937$	26.9123	0.879188	0.439594	−	0.898197i	$-0.355122\pi$
$937$	26.9123	0.879188	0.439594	+	0.898197i	$0.355122\pi$
$938$	−13.5486	−0.442377
$939$	6.06150	0.197810
$940$	1.18821	0.0387551
$941$	52.7134	1.71841	0.859204	−	0.511634i	$-0.170960\pi$
$941$	52.7134	1.71841	0.859204	+	0.511634i	$0.170960\pi$
$942$	−10.5040	−0.342239
$943$	8.98429	0.292569
$944$	0.837453	0.0272568
$945$	−15.3124	−0.498112
$946$	2.28473	0.0742830
$947$	−10.8298	−0.351922	−0.175961	−	0.984397i	$-0.556303\pi$
$947$	−10.8298	−0.351922	−0.175961	+	0.984397i	$0.556303\pi$
$948$	−15.3502	−0.498551
$949$	16.6333	0.539940
$950$	−16.4238	−0.532858
$951$	−15.0417	−0.487761
$952$	9.90286	0.320954
$953$	38.8834	1.25956	0.629778	−	0.776775i	$-0.283146\pi$
$953$	38.8834	1.25956	0.629778	+	0.776775i	$0.283146\pi$
$954$	23.2303	0.752110
$955$	−11.8851	−0.384591
$956$	−27.5846	−0.892150
$957$	−4.76270	−0.153956
$958$	59.4408	1.92045
$959$	62.9208	2.03182
$960$	−10.0170	−0.323297
$961$	−23.0512	−0.743588
$962$	34.7939	1.12180
$963$	−29.3827	−0.946844
$964$	−10.1330	−0.326362
$965$	−17.1899	−0.553362
$966$	17.9394	0.577190
$967$	51.8598	1.66770	0.833850	−	0.551991i	$-0.186132\pi$
$967$	51.8598	1.66770	0.833850	+	0.551991i	$0.186132\pi$
$968$	−2.78738	−0.0895896
$969$	−1.30176	−0.0418187
$970$	−34.6618	−1.11292
$971$	−4.51303	−0.144830	−0.0724150	−	0.997375i	$-0.523071\pi$
$971$	−4.51303	−0.144830	−0.0724150	+	0.997375i	$0.523071\pi$
$972$	−46.2118	−1.48224
$973$	46.4932	1.49050
$974$	44.3883	1.42229
$975$	5.13127	0.164332
$976$	0.487567	0.0156066
$977$	−52.9686	−1.69462	−0.847308	−	0.531102i	$-0.821778\pi$
$977$	−52.9686	−1.69462	−0.847308	+	0.531102i	$0.821778\pi$
$978$	−1.22078	−0.0390363
$979$	−13.9073	−0.444480
$980$	−21.5639	−0.688834
$981$	−28.0274	−0.894845
$982$	−58.3585	−1.86229
$983$	28.3015	0.902677	0.451339	−	0.892353i	$-0.350947\pi$
$983$	28.3015	0.902677	0.451339	+	0.892353i	$0.350947\pi$
$984$	4.76536	0.151914
$985$	−23.5728	−0.751091
$986$	16.7794	0.534365
$987$	0.713739	0.0227186
$988$	−14.2815	−0.454354
$989$	−3.40797	−0.108367
$990$	−7.01999	−0.223110
$991$	−18.8636	−0.599222	−0.299611	−	0.954061i	$-0.596857\pi$
$991$	−18.8636	−0.599222	−0.299611	+	0.954061i	$0.596857\pi$
$992$	−16.1784	−0.513665
$993$	5.06194	0.160636
$994$	−48.0248	−1.52325
$995$	22.7309	0.720617
$996$	−18.2884	−0.579489
$997$	−21.3459	−0.676030	−0.338015	−	0.941141i	$-0.609756\pi$
$997$	−21.3459	−0.676030	−0.338015	+	0.941141i	$0.609756\pi$
$998$	−39.9776	−1.26547
$999$	−24.9387	−0.789026

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8041.2.a.i.1.10	✓	78

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8041.2.a.i.1.10	✓	78	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:	\( N \)	\(=\)	\( 8041 = 11 \cdot 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8041.a (trivial)

\(f(q)\)	\(=\)	\(q-2.28473 q^{2} -0.648502 q^{3} +3.22000 q^{4} -1.19117 q^{5} +1.48165 q^{6} +3.55276 q^{7} -2.78738 q^{8} -2.57944 q^{9} +O(q^{10})\)	\(q-2.28473 q^{2} -0.648502 q^{3} +3.22000 q^{4} -1.19117 q^{5} +1.48165 q^{6} +3.55276 q^{7} -2.78738 q^{8} -2.57944 q^{9} +2.72151 q^{10} +1.00000 q^{11} -2.08818 q^{12} +2.20951 q^{13} -8.11710 q^{14} +0.772478 q^{15} -0.0715963 q^{16} -1.00000 q^{17} +5.89334 q^{18} -2.00734 q^{19} -3.83558 q^{20} -2.30397 q^{21} -2.28473 q^{22} +3.40797 q^{23} +1.80762 q^{24} -3.58111 q^{25} -5.04814 q^{26} +3.61828 q^{27} +11.4399 q^{28} +7.34415 q^{29} -1.76491 q^{30} -2.81936 q^{31} +5.73833 q^{32} -0.648502 q^{33} +2.28473 q^{34} -4.23195 q^{35} -8.30581 q^{36} -6.89242 q^{37} +4.58623 q^{38} -1.43287 q^{39} +3.32025 q^{40} +2.63626 q^{41} +5.26396 q^{42} -1.00000 q^{43} +3.22000 q^{44} +3.07256 q^{45} -7.78629 q^{46} -0.309786 q^{47} +0.0464303 q^{48} +5.62208 q^{49} +8.18187 q^{50} +0.648502 q^{51} +7.11463 q^{52} +3.94179 q^{53} -8.26681 q^{54} -1.19117 q^{55} -9.90286 q^{56} +1.30176 q^{57} -16.7794 q^{58} -11.6969 q^{59} +2.48738 q^{60} -6.80995 q^{61} +6.44148 q^{62} -9.16414 q^{63} -12.9674 q^{64} -2.63191 q^{65} +1.48165 q^{66} +1.66914 q^{67} -3.22000 q^{68} -2.21007 q^{69} +9.66886 q^{70} +5.91650 q^{71} +7.18988 q^{72} +7.52804 q^{73} +15.7473 q^{74} +2.32236 q^{75} -6.46363 q^{76} +3.55276 q^{77} +3.27373 q^{78} +7.35099 q^{79} +0.0852835 q^{80} +5.39187 q^{81} -6.02316 q^{82} +8.75805 q^{83} -7.41879 q^{84} +1.19117 q^{85} +2.28473 q^{86} -4.76270 q^{87} -2.78738 q^{88} -13.9073 q^{89} -7.01999 q^{90} +7.84985 q^{91} +10.9737 q^{92} +1.82836 q^{93} +0.707779 q^{94} +2.39109 q^{95} -3.72132 q^{96} -12.7362 q^{97} -12.8449 q^{98} -2.57944 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q + 7 q^{2} + 10 q^{3} + 91 q^{4} + 17 q^{5} + 12 q^{6} + 11 q^{7} + 33 q^{8} + 102 q^{9}+O(q^{10}) \) 78 * q + 7 * q^2 + 10 * q^3 + 91 * q^4 + 17 * q^5 + 12 * q^6 + 11 * q^7 + 33 * q^8 + 102 * q^9	\( 78 q + 7 q^{2} + 10 q^{3} + 91 q^{4} + 17 q^{5} + 12 q^{6} + 11 q^{7} + 33 q^{8} + 102 q^{9} + 3 q^{10} + 78 q^{11} + 31 q^{12} - 16 q^{13} + 31 q^{14} + 38 q^{15} + 121 q^{16} - 78 q^{17} + 11 q^{18} + 51 q^{20} + 6 q^{21} + 7 q^{22} + 48 q^{23} + 11 q^{24} + 101 q^{25} + 18 q^{26} + 46 q^{27} + 27 q^{28} + 22 q^{29} + 14 q^{30} + 56 q^{31} + 83 q^{32} + 10 q^{33} - 7 q^{34} + 24 q^{35} + 139 q^{36} + 53 q^{37} + 10 q^{38} + 79 q^{39} - q^{40} + 23 q^{41} + 17 q^{42} - 78 q^{43} + 91 q^{44} + 76 q^{45} + 21 q^{46} + 57 q^{47} + 78 q^{48} + 115 q^{49} + 58 q^{50} - 10 q^{51} - 63 q^{52} + 22 q^{53} - 18 q^{54} + 17 q^{55} + 111 q^{56} - 11 q^{57} + 36 q^{58} + 71 q^{59} + 36 q^{60} + 4 q^{61} - 5 q^{62} + 71 q^{63} + 183 q^{64} + 47 q^{65} + 12 q^{66} + 11 q^{67} - 91 q^{68} + 31 q^{69} + 33 q^{70} + 159 q^{71} + 59 q^{72} + 2 q^{73} - 4 q^{74} + 83 q^{75} - 44 q^{76} + 11 q^{77} + 101 q^{78} + 35 q^{79} + 85 q^{80} + 170 q^{81} + 98 q^{82} - 32 q^{83} + 44 q^{84} - 17 q^{85} - 7 q^{86} - 6 q^{87} + 33 q^{88} + 50 q^{89} - 5 q^{90} + 86 q^{91} + 106 q^{92} + 68 q^{93} - q^{94} + 109 q^{95} - 50 q^{96} + 40 q^{97} + 106 q^{98} + 102 q^{99}+O(q^{100}) \) 78 * q + 7 * q^2 + 10 * q^3 + 91 * q^4 + 17 * q^5 + 12 * q^6 + 11 * q^7 + 33 * q^8 + 102 * q^9 + 3 * q^10 + 78 * q^11 + 31 * q^12 - 16 * q^13 + 31 * q^14 + 38 * q^15 + 121 * q^16 - 78 * q^17 + 11 * q^18 + 51 * q^20 + 6 * q^21 + 7 * q^22 + 48 * q^23 + 11 * q^24 + 101 * q^25 + 18 * q^26 + 46 * q^27 + 27 * q^28 + 22 * q^29 + 14 * q^30 + 56 * q^31 + 83 * q^32 + 10 * q^33 - 7 * q^34 + 24 * q^35 + 139 * q^36 + 53 * q^37 + 10 * q^38 + 79 * q^39 - q^40 + 23 * q^41 + 17 * q^42 - 78 * q^43 + 91 * q^44 + 76 * q^45 + 21 * q^46 + 57 * q^47 + 78 * q^48 + 115 * q^49 + 58 * q^50 - 10 * q^51 - 63 * q^52 + 22 * q^53 - 18 * q^54 + 17 * q^55 + 111 * q^56 - 11 * q^57 + 36 * q^58 + 71 * q^59 + 36 * q^60 + 4 * q^61 - 5 * q^62 + 71 * q^63 + 183 * q^64 + 47 * q^65 + 12 * q^66 + 11 * q^67 - 91 * q^68 + 31 * q^69 + 33 * q^70 + 159 * q^71 + 59 * q^72 + 2 * q^73 - 4 * q^74 + 83 * q^75 - 44 * q^76 + 11 * q^77 + 101 * q^78 + 35 * q^79 + 85 * q^80 + 170 * q^81 + 98 * q^82 - 32 * q^83 + 44 * q^84 - 17 * q^85 - 7 * q^86 - 6 * q^87 + 33 * q^88 + 50 * q^89 - 5 * q^90 + 86 * q^91 + 106 * q^92 + 68 * q^93 - q^94 + 109 * q^95 - 50 * q^96 + 40 * q^97 + 106 * q^98 + 102 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more