LMFDB - Embedded newform 8021.2.a.d.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8021, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8021.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$64.0480074613$
Analytic rank:	$0$
Dimension:	$174$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	8021.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.63723 q^{2} -2.22050 q^{3} +4.95500 q^{4} +3.72301 q^{5} +5.85596 q^{6} -3.62454 q^{7} -7.79302 q^{8} +1.93060 q^{9} +O(q^{10})$	\(q-2.63723 q^{2} -2.22050 q^{3} +4.95500 q^{4} +3.72301 q^{5} +5.85596 q^{6} -3.62454 q^{7} -7.79302 q^{8} +1.93060 q^{9} -9.81845 q^{10} +1.52148 q^{11} -11.0026 q^{12} +1.00000 q^{13} +9.55876 q^{14} -8.26693 q^{15} +10.6420 q^{16} -5.82293 q^{17} -5.09145 q^{18} +0.602766 q^{19} +18.4475 q^{20} +8.04828 q^{21} -4.01251 q^{22} -4.63475 q^{23} +17.3044 q^{24} +8.86082 q^{25} -2.63723 q^{26} +2.37460 q^{27} -17.9596 q^{28} -7.60229 q^{29} +21.8018 q^{30} +9.63360 q^{31} -12.4794 q^{32} -3.37845 q^{33} +15.3564 q^{34} -13.4942 q^{35} +9.56613 q^{36} +4.57212 q^{37} -1.58963 q^{38} -2.22050 q^{39} -29.0135 q^{40} +6.19448 q^{41} -21.2252 q^{42} -2.05161 q^{43} +7.53895 q^{44} +7.18765 q^{45} +12.2229 q^{46} +1.96892 q^{47} -23.6305 q^{48} +6.13730 q^{49} -23.3681 q^{50} +12.9298 q^{51} +4.95500 q^{52} -9.66647 q^{53} -6.26236 q^{54} +5.66450 q^{55} +28.2461 q^{56} -1.33844 q^{57} +20.0490 q^{58} +5.88623 q^{59} -40.9626 q^{60} -14.5759 q^{61} -25.4061 q^{62} -6.99754 q^{63} +11.6271 q^{64} +3.72301 q^{65} +8.90976 q^{66} +7.42017 q^{67} -28.8526 q^{68} +10.2914 q^{69} +35.5874 q^{70} +0.0988246 q^{71} -15.0452 q^{72} -15.1210 q^{73} -12.0577 q^{74} -19.6754 q^{75} +2.98670 q^{76} -5.51468 q^{77} +5.85596 q^{78} -3.21738 q^{79} +39.6203 q^{80} -11.0646 q^{81} -16.3363 q^{82} +12.4635 q^{83} +39.8792 q^{84} -21.6789 q^{85} +5.41058 q^{86} +16.8808 q^{87} -11.8570 q^{88} +0.378790 q^{89} -18.9555 q^{90} -3.62454 q^{91} -22.9652 q^{92} -21.3914 q^{93} -5.19251 q^{94} +2.24410 q^{95} +27.7105 q^{96} -14.2878 q^{97} -16.1855 q^{98} +2.93738 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 174 q + 6 q^{2} + 37 q^{3} + 214 q^{4} + 10 q^{5} + 12 q^{6} + 28 q^{7} + 15 q^{8} + 211 q^{9}+O(q^{10}) $ 174 * q + 6 * q^2 + 37 * q^3 + 214 * q^4 + 10 * q^5 + 12 * q^6 + 28 * q^7 + 15 * q^8 + 211 * q^9	\( 174 q + 6 q^{2} + 37 q^{3} + 214 q^{4} + 10 q^{5} + 12 q^{6} + 28 q^{7} + 15 q^{8} + 211 q^{9} + 47 q^{10} + 47 q^{11} + 81 q^{12} + 174 q^{13} + 22 q^{14} + 26 q^{15} + 286 q^{16} + 27 q^{17} + 22 q^{18} + 91 q^{19} + 18 q^{20} + 8 q^{21} + 58 q^{22} + 62 q^{23} + 24 q^{24} + 244 q^{25} + 6 q^{26} + 139 q^{27} + 43 q^{28} + 42 q^{29} + 31 q^{30} + 82 q^{31} + 11 q^{32} + 12 q^{33} + 50 q^{34} + 74 q^{35} + 310 q^{36} + 47 q^{37} + 10 q^{38} + 37 q^{39} + 118 q^{40} + 16 q^{41} + 26 q^{42} + 136 q^{43} + 74 q^{44} + 18 q^{45} + 53 q^{46} + 15 q^{47} + 132 q^{48} + 254 q^{49} - 5 q^{50} + 121 q^{51} + 214 q^{52} + 39 q^{53} + 30 q^{54} + 188 q^{55} + 55 q^{56} + 11 q^{57} + 32 q^{58} + 58 q^{59} + 16 q^{60} + 128 q^{61} + 27 q^{62} + 42 q^{63} + 423 q^{64} + 10 q^{65} + 4 q^{66} + 132 q^{67} + 52 q^{68} + 63 q^{69} - 8 q^{70} + 78 q^{71} + 2 q^{72} + 21 q^{73} - 16 q^{74} + 188 q^{75} + 160 q^{76} + 20 q^{77} + 12 q^{78} + 232 q^{79} + 2 q^{80} + 302 q^{81} + 115 q^{82} + 18 q^{83} - 26 q^{84} + 47 q^{85} + 27 q^{86} + 127 q^{87} + 163 q^{88} + 14 q^{90} + 28 q^{91} + 68 q^{92} + 15 q^{93} + 91 q^{94} + 75 q^{95} - 26 q^{96} + 34 q^{97} - 60 q^{98} + 181 q^{99}+O(q^{100}) \) 174 * q + 6 * q^2 + 37 * q^3 + 214 * q^4 + 10 * q^5 + 12 * q^6 + 28 * q^7 + 15 * q^8 + 211 * q^9 + 47 * q^10 + 47 * q^11 + 81 * q^12 + 174 * q^13 + 22 * q^14 + 26 * q^15 + 286 * q^16 + 27 * q^17 + 22 * q^18 + 91 * q^19 + 18 * q^20 + 8 * q^21 + 58 * q^22 + 62 * q^23 + 24 * q^24 + 244 * q^25 + 6 * q^26 + 139 * q^27 + 43 * q^28 + 42 * q^29 + 31 * q^30 + 82 * q^31 + 11 * q^32 + 12 * q^33 + 50 * q^34 + 74 * q^35 + 310 * q^36 + 47 * q^37 + 10 * q^38 + 37 * q^39 + 118 * q^40 + 16 * q^41 + 26 * q^42 + 136 * q^43 + 74 * q^44 + 18 * q^45 + 53 * q^46 + 15 * q^47 + 132 * q^48 + 254 * q^49 - 5 * q^50 + 121 * q^51 + 214 * q^52 + 39 * q^53 + 30 * q^54 + 188 * q^55 + 55 * q^56 + 11 * q^57 + 32 * q^58 + 58 * q^59 + 16 * q^60 + 128 * q^61 + 27 * q^62 + 42 * q^63 + 423 * q^64 + 10 * q^65 + 4 * q^66 + 132 * q^67 + 52 * q^68 + 63 * q^69 - 8 * q^70 + 78 * q^71 + 2 * q^72 + 21 * q^73 - 16 * q^74 + 188 * q^75 + 160 * q^76 + 20 * q^77 + 12 * q^78 + 232 * q^79 + 2 * q^80 + 302 * q^81 + 115 * q^82 + 18 * q^83 - 26 * q^84 + 47 * q^85 + 27 * q^86 + 127 * q^87 + 163 * q^88 + 14 * q^90 + 28 * q^91 + 68 * q^92 + 15 * q^93 + 91 * q^94 + 75 * q^95 - 26 * q^96 + 34 * q^97 - 60 * q^98 + 181 * 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 $

$2$	−2.63723	−1.86481	−0.932403	−	0.361421i	$-0.882292\pi$
$2$	−2.63723	−1.86481	−0.932403	+	0.361421i	$0.882292\pi$
$3$	−2.22050	−1.28200	−0.641002	−	0.767539i	$-0.721481\pi$
$3$	−2.22050	−1.28200	−0.641002	+	0.767539i	$0.721481\pi$
$4$	4.95500	2.47750
$5$	3.72301	1.66498	0.832491	−	0.554039i	$-0.186914\pi$
$5$	3.72301	1.66498	0.832491	+	0.554039i	$0.186914\pi$
$6$	5.85596	2.39069
$7$	−3.62454	−1.36995	−0.684974	−	0.728568i	$-0.740186\pi$
$7$	−3.62454	−1.36995	−0.684974	+	0.728568i	$0.740186\pi$
$8$	−7.79302	−2.75525
$9$	1.93060	0.643534
$10$	−9.81845	−3.10487
$11$	1.52148	0.458745	0.229372	−	0.973339i	$-0.426333\pi$
$11$	1.52148	0.458745	0.229372	+	0.973339i	$0.426333\pi$
$12$	−11.0026	−3.17616
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	9.55876	2.55469
$15$	−8.26693	−2.13451
$16$	10.6420	2.66050
$17$	−5.82293	−1.41227	−0.706134	−	0.708078i	$-0.749563\pi$
$17$	−5.82293	−1.41227	−0.706134	+	0.708078i	$0.749563\pi$
$18$	−5.09145	−1.20007
$19$	0.602766	0.138284	0.0691420	−	0.997607i	$-0.477974\pi$
$19$	0.602766	0.138284	0.0691420	+	0.997607i	$0.477974\pi$
$20$	18.4475	4.12499
$21$	8.04828	1.75628
$22$	−4.01251	−0.855470
$23$	−4.63475	−0.966413	−0.483206	−	0.875506i	$-0.660528\pi$
$23$	−4.63475	−0.966413	−0.483206	+	0.875506i	$0.660528\pi$
$24$	17.3044	3.53224
$25$	8.86082	1.77216
$26$	−2.63723	−0.517204
$27$	2.37460	0.456991
$28$	−17.9596	−3.39404
$29$	−7.60229	−1.41171	−0.705854	−	0.708357i	$-0.749437\pi$
$29$	−7.60229	−1.41171	−0.705854	+	0.708357i	$0.749437\pi$
$30$	21.8018	3.98045
$31$	9.63360	1.73025	0.865123	−	0.501560i	$-0.167240\pi$
$31$	9.63360	1.73025	0.865123	+	0.501560i	$0.167240\pi$
$32$	−12.4794	−2.20607
$33$	−3.37845	−0.588112
$34$	15.3564	2.63361
$35$	−13.4942	−2.28094
$36$	9.56613	1.59435
$37$	4.57212	0.751652	0.375826	−	0.926690i	$-0.377359\pi$
$37$	4.57212	0.751652	0.375826	+	0.926690i	$0.377359\pi$
$38$	−1.58963	−0.257873
$39$	−2.22050	−0.355564
$40$	−29.0135	−4.58744
$41$	6.19448	0.967415	0.483707	−	0.875230i	$-0.339290\pi$
$41$	6.19448	0.967415	0.483707	+	0.875230i	$0.339290\pi$
$42$	−21.2252	−3.27512
$43$	−2.05161	−0.312868	−0.156434	−	0.987688i	$-0.550000\pi$
$43$	−2.05161	−0.312868	−0.156434	+	0.987688i	$0.550000\pi$
$44$	7.53895	1.13654
$45$	7.18765	1.07147
$46$	12.2229	1.80217
$47$	1.96892	0.287197	0.143599	−	0.989636i	$-0.454133\pi$
$47$	1.96892	0.287197	0.143599	+	0.989636i	$0.454133\pi$
$48$	−23.6305	−3.41077
$49$	6.13730	0.876757
$50$	−23.3681	−3.30474
$51$	12.9298	1.81053
$52$	4.95500	0.687135
$53$	−9.66647	−1.32779	−0.663896	−	0.747825i	$-0.731098\pi$
$53$	−9.66647	−1.32779	−0.663896	+	0.747825i	$0.731098\pi$
$54$	−6.26236	−0.852199
$55$	5.66450	0.763802
$56$	28.2461	3.77455
$57$	−1.33844	−0.177281
$58$	20.0490	2.63256
$59$	5.88623	0.766322	0.383161	−	0.923682i	$-0.374835\pi$
$59$	5.88623	0.766322	0.383161	+	0.923682i	$0.374835\pi$
$60$	−40.9626	−5.28825
$61$	−14.5759	−1.86625	−0.933124	−	0.359555i	$-0.882928\pi$
$61$	−14.5759	−1.86625	−0.933124	+	0.359555i	$0.882928\pi$
$62$	−25.4061	−3.22657
$63$	−6.99754	−0.881608
$64$	11.6271	1.45339
$65$	3.72301	0.461783
$66$	8.90976	1.09672
$67$	7.42017	0.906518	0.453259	−	0.891379i	$-0.350261\pi$
$67$	7.42017	0.906518	0.453259	+	0.891379i	$0.350261\pi$
$68$	−28.8526	−3.49889
$69$	10.2914	1.23894
$70$	35.5874	4.25351
$71$	0.0988246	0.0117283	0.00586416	−	0.999983i	$-0.498133\pi$
$71$	0.0988246	0.0117283	0.00586416	+	0.999983i	$0.498133\pi$
$72$	−15.0452	−1.77310
$73$	−15.1210	−1.76978	−0.884891	−	0.465798i	$-0.845767\pi$
$73$	−15.1210	−1.76978	−0.884891	+	0.465798i	$0.845767\pi$
$74$	−12.0577	−1.40168
$75$	−19.6754	−2.27192
$76$	2.98670	0.342598
$77$	−5.51468	−0.628456
$78$	5.85596	0.663057
$79$	−3.21738	−0.361984	−0.180992	−	0.983485i	$-0.557931\pi$
$79$	−3.21738	−0.361984	−0.180992	+	0.983485i	$0.557931\pi$
$80$	39.6203	4.42969
$81$	−11.0646	−1.22940
$82$	−16.3363	−1.80404
$83$	12.4635	1.36804	0.684022	−	0.729461i	$-0.260229\pi$
$83$	12.4635	1.36804	0.684022	+	0.729461i	$0.260229\pi$
$84$	39.8792	4.35118
$85$	−21.6789	−2.35140
$86$	5.41058	0.583438
$87$	16.8808	1.80982
$88$	−11.8570	−1.26396
$89$	0.378790	0.0401516	0.0200758	−	0.999798i	$-0.493609\pi$
$89$	0.378790	0.0401516	0.0200758	+	0.999798i	$0.493609\pi$
$90$	−18.9555	−1.99809
$91$	−3.62454	−0.379955
$92$	−22.9652	−2.39429
$93$	−21.3914	−2.21818
$94$	−5.19251	−0.535567
$95$	2.24410	0.230240
$96$	27.7105	2.82819
$97$	−14.2878	−1.45071	−0.725353	−	0.688377i	$-0.758323\pi$
$97$	−14.2878	−1.45071	−0.725353	+	0.688377i	$0.758323\pi$
$98$	−16.1855	−1.63498
$99$	2.93738	0.295218
$100$	43.9054	4.39054
$101$	7.30730	0.727103	0.363552	−	0.931574i	$-0.381564\pi$
$101$	7.30730	0.727103	0.363552	+	0.931574i	$0.381564\pi$
$102$	−34.0989	−3.37629
$103$	5.86714	0.578106	0.289053	−	0.957313i	$-0.406660\pi$
$103$	5.86714	0.578106	0.289053	+	0.957313i	$0.406660\pi$
$104$	−7.79302	−0.764168
$105$	29.9638	2.92417
$106$	25.4927	2.47607
$107$	−6.69445	−0.647177	−0.323588	−	0.946198i	$-0.604889\pi$
$107$	−6.69445	−0.647177	−0.323588	+	0.946198i	$0.604889\pi$
$108$	11.7661	1.13219
$109$	4.60425	0.441007	0.220503	−	0.975386i	$-0.429230\pi$
$109$	4.60425	0.441007	0.220503	+	0.975386i	$0.429230\pi$
$110$	−14.9386	−1.42434
$111$	−10.1524	−0.963620
$112$	−38.5724	−3.64475
$113$	−12.6007	−1.18537	−0.592686	−	0.805434i	$-0.701932\pi$
$113$	−12.6007	−1.18537	−0.592686	+	0.805434i	$0.701932\pi$
$114$	3.52978	0.330594
$115$	−17.2552	−1.60906
$116$	−37.6693	−3.49751
$117$	1.93060	0.178484
$118$	−15.5234	−1.42904
$119$	21.1055	1.93473
$120$	64.4244	5.88111
$121$	−8.68509	−0.789553
$122$	38.4399	3.48019
$123$	−13.7548	−1.24023
$124$	47.7345	4.28668
$125$	14.3739	1.28564
$126$	18.4542	1.64403
$127$	12.1478	1.07794	0.538971	−	0.842324i	$-0.318813\pi$
$127$	12.1478	1.07794	0.538971	+	0.842324i	$0.318813\pi$
$128$	−5.70460	−0.504220
$129$	4.55560	0.401098
$130$	−9.81845	−0.861135
$131$	−0.0751425	−0.00656523	−0.00328261	−	0.999995i	$-0.501045\pi$
$131$	−0.0751425	−0.00656523	−0.00328261	+	0.999995i	$0.501045\pi$
$132$	−16.7402	−1.45705
$133$	−2.18475	−0.189442
$134$	−19.5687	−1.69048
$135$	8.84065	0.760882
$136$	45.3782	3.89115
$137$	9.12464	0.779570	0.389785	−	0.920906i	$-0.372549\pi$
$137$	9.12464	0.779570	0.389785	+	0.920906i	$0.372549\pi$
$138$	−27.1410	−2.31039
$139$	−1.11222	−0.0943373	−0.0471687	−	0.998887i	$-0.515020\pi$
$139$	−1.11222	−0.0943373	−0.0471687	+	0.998887i	$0.515020\pi$
$140$	−66.8638	−5.65102
$141$	−4.37199	−0.368188
$142$	−0.260623	−0.0218710
$143$	1.52148	0.127233
$144$	20.5455	1.71212
$145$	−28.3034	−2.35047
$146$	39.8777	3.30030
$147$	−13.6278	−1.12401
$148$	22.6548	1.86222
$149$	13.8465	1.13435	0.567174	−	0.823598i	$-0.308037\pi$
$149$	13.8465	1.13435	0.567174	+	0.823598i	$0.308037\pi$
$150$	51.8887	4.23669
$151$	−12.6154	−1.02662	−0.513312	−	0.858202i	$-0.671582\pi$
$151$	−12.6154	−1.02662	−0.513312	+	0.858202i	$0.671582\pi$
$152$	−4.69737	−0.381007
$153$	−11.2418	−0.908843
$154$	14.5435	1.17195
$155$	35.8660	2.88083
$156$	−11.0026	−0.880909
$157$	9.22383	0.736142	0.368071	−	0.929798i	$-0.380018\pi$
$157$	9.22383	0.736142	0.368071	+	0.929798i	$0.380018\pi$
$158$	8.48499	0.675030
$159$	21.4644	1.70223
$160$	−46.4610	−3.67307
$161$	16.7989	1.32394
$162$	29.1799	2.29259
$163$	18.0762	1.41584	0.707919	−	0.706293i	$-0.249634\pi$
$163$	18.0762	1.41584	0.707919	+	0.706293i	$0.249634\pi$
$164$	30.6936	2.39677
$165$	−12.5780	−0.979197
$166$	−32.8691	−2.55114
$167$	−4.17674	−0.323205	−0.161603	−	0.986856i	$-0.551666\pi$
$167$	−4.17674	−0.323205	−0.161603	+	0.986856i	$0.551666\pi$
$168$	−62.7204	−4.83898
$169$	1.00000	0.0769231
$170$	57.1722	4.38491
$171$	1.16370	0.0889904
$172$	−10.1657	−0.775131
$173$	22.4320	1.70547	0.852736	−	0.522341i	$-0.174941\pi$
$173$	22.4320	1.70547	0.852736	+	0.522341i	$0.174941\pi$
$174$	−44.5187	−3.37495
$175$	−32.1164	−2.42777
$176$	16.1916	1.22049
$177$	−13.0704	−0.982428
$178$	−0.998957	−0.0748750
$179$	−16.0977	−1.20320	−0.601600	−	0.798798i	$-0.705470\pi$
$179$	−16.0977	−1.20320	−0.601600	+	0.798798i	$0.705470\pi$
$180$	35.6148	2.65457
$181$	15.1746	1.12792	0.563958	−	0.825803i	$-0.309278\pi$
$181$	15.1746	1.12792	0.563958	+	0.825803i	$0.309278\pi$
$182$	9.55876	0.708542
$183$	32.3656	2.39254
$184$	36.1187	2.66271
$185$	17.0221	1.25149
$186$	56.4140	4.13648
$187$	−8.85950	−0.647871
$188$	9.75601	0.711530
$189$	−8.60682	−0.626054
$190$	−5.91823	−0.429353
$191$	−7.85441	−0.568325	−0.284162	−	0.958776i	$-0.591715\pi$
$191$	−7.85441	−0.568325	−0.284162	+	0.958776i	$0.591715\pi$
$192$	−25.8180	−1.86325
$193$	13.0289	0.937838	0.468919	−	0.883241i	$-0.344644\pi$
$193$	13.0289	0.937838	0.468919	+	0.883241i	$0.344644\pi$
$194$	37.6802	2.70528
$195$	−8.26693	−0.592007
$196$	30.4103	2.17216
$197$	−18.7959	−1.33915	−0.669576	−	0.742744i	$-0.733524\pi$
$197$	−18.7959	−1.33915	−0.669576	+	0.742744i	$0.733524\pi$
$198$	−7.74655	−0.550524
$199$	18.0799	1.28165	0.640825	−	0.767687i	$-0.278592\pi$
$199$	18.0799	1.28165	0.640825	+	0.767687i	$0.278592\pi$
$200$	−69.0525	−4.88275
$201$	−16.4765	−1.16216
$202$	−19.2710	−1.35591
$203$	27.5548	1.93397
$204$	64.0671	4.48560
$205$	23.0621	1.61073
$206$	−15.4730	−1.07806
$207$	−8.94786	−0.621919
$208$	10.6420	0.737891
$209$	0.917099	0.0634370
$210$	−79.0216	−5.45301
$211$	26.9974	1.85858	0.929288	−	0.369355i	$-0.120422\pi$
$211$	26.9974	1.85858	0.929288	+	0.369355i	$0.120422\pi$
$212$	−47.8973	−3.28960
$213$	−0.219440	−0.0150357
$214$	17.6548	1.20686
$215$	−7.63818	−0.520920
$216$	−18.5053	−1.25912
$217$	−34.9174	−2.37035
$218$	−12.1425	−0.822392
$219$	33.5762	2.26887
$220$	28.0676	1.89232
$221$	−5.82293	−0.391693
$222$	26.7742	1.79696
$223$	−9.31518	−0.623791	−0.311895	−	0.950116i	$-0.600964\pi$
$223$	−9.31518	−0.623791	−0.311895	+	0.950116i	$0.600964\pi$
$224$	45.2322	3.02220
$225$	17.1067	1.14045
$226$	33.2309	2.21049
$227$	−2.49833	−0.165820	−0.0829101	−	0.996557i	$-0.526421\pi$
$227$	−2.49833	−0.165820	−0.0829101	+	0.996557i	$0.526421\pi$
$228$	−6.63196	−0.439212
$229$	12.3493	0.816066	0.408033	−	0.912967i	$-0.366215\pi$
$229$	12.3493	0.816066	0.408033	+	0.912967i	$0.366215\pi$
$230$	45.5061	3.00058
$231$	12.2453	0.805683
$232$	59.2447	3.88961
$233$	−9.92453	−0.650178	−0.325089	−	0.945684i	$-0.605394\pi$
$233$	−9.92453	−0.650178	−0.325089	+	0.945684i	$0.605394\pi$
$234$	−5.09145	−0.332838
$235$	7.33033	0.478178
$236$	29.1663	1.89856
$237$	7.14419	0.464065
$238$	−55.6600	−3.60790
$239$	−23.2175	−1.50182	−0.750908	−	0.660406i	$-0.770384\pi$
$239$	−23.2175	−1.50182	−0.750908	+	0.660406i	$0.770384\pi$
$240$	−87.9768	−5.67888
$241$	−15.4603	−0.995885	−0.497942	−	0.867210i	$-0.665911\pi$
$241$	−15.4603	−0.995885	−0.497942	+	0.867210i	$0.665911\pi$
$242$	22.9046	1.47236
$243$	17.4451	1.11910
$244$	−72.2234	−4.62363
$245$	22.8492	1.45978
$246$	36.2746	2.31279
$247$	0.602766	0.0383531
$248$	−75.0749	−4.76726
$249$	−27.6751	−1.75384
$250$	−37.9073	−2.39747
$251$	−27.8041	−1.75498	−0.877489	−	0.479598i	$-0.840783\pi$
$251$	−27.8041	−1.75498	−0.877489	+	0.479598i	$0.840783\pi$
$252$	−34.6728	−2.18418
$253$	−7.05170	−0.443337
$254$	−32.0366	−2.01015
$255$	48.1378	3.01451
$256$	−8.20990	−0.513119
$257$	−7.43601	−0.463845	−0.231923	−	0.972734i	$-0.574502\pi$
$257$	−7.43601	−0.463845	−0.231923	+	0.972734i	$0.574502\pi$
$258$	−12.0142	−0.747970
$259$	−16.5718	−1.02972
$260$	18.4475	1.14407
$261$	−14.6770	−0.908482
$262$	0.198168	0.0122429
$263$	27.8137	1.71506	0.857532	−	0.514430i	$-0.171997\pi$
$263$	27.8137	1.71506	0.857532	+	0.514430i	$0.171997\pi$
$264$	26.3283	1.62040
$265$	−35.9884	−2.21075
$266$	5.76169	0.353272
$267$	−0.841101	−0.0514745
$268$	36.7669	2.24590
$269$	−18.0317	−1.09941	−0.549706	−	0.835358i	$-0.685260\pi$
$269$	−18.0317	−1.09941	−0.549706	+	0.835358i	$0.685260\pi$
$270$	−23.3148	−1.41890
$271$	−12.6273	−0.767053	−0.383527	−	0.923530i	$-0.625291\pi$
$271$	−12.6273	−0.767053	−0.383527	+	0.923530i	$0.625291\pi$
$272$	−61.9677	−3.75734
$273$	8.04828	0.487104
$274$	−24.0638	−1.45375
$275$	13.4816	0.812971
$276$	50.9941	3.06948
$277$	14.7749	0.887738	0.443869	−	0.896092i	$-0.353606\pi$
$277$	14.7749	0.887738	0.443869	+	0.896092i	$0.353606\pi$
$278$	2.93318	0.175921
$279$	18.5986	1.11347
$280$	105.161	6.28455
$281$	17.6174	1.05097	0.525484	−	0.850804i	$-0.323884\pi$
$281$	17.6174	1.05097	0.525484	+	0.850804i	$0.323884\pi$
$282$	11.5299	0.686598
$283$	1.12002	0.0665783	0.0332892	−	0.999446i	$-0.489402\pi$
$283$	1.12002	0.0665783	0.0332892	+	0.999446i	$0.489402\pi$
$284$	0.489676	0.0290569
$285$	−4.98302	−0.295169
$286$	−4.01251	−0.237265
$287$	−22.4521	−1.32531
$288$	−24.0928	−1.41968
$289$	16.9066	0.994504
$290$	74.6427	4.38317
$291$	31.7260	1.85981
$292$	−74.9246	−4.38463
$293$	−17.7381	−1.03627	−0.518135	−	0.855299i	$-0.673374\pi$
$293$	−17.7381	−1.03627	−0.518135	+	0.855299i	$0.673374\pi$
$294$	35.9398	2.09605
$295$	21.9145	1.27591
$296$	−35.6306	−2.07099
$297$	3.61291	0.209642
$298$	−36.5164	−2.11534
$299$	−4.63475	−0.268035
$300$	−97.4916	−5.62868
$301$	7.43616	0.428613
$302$	33.2697	1.91445
$303$	−16.2258	−0.932149
$304$	6.41464	0.367905
$305$	−54.2661	−3.10727
$306$	29.6472	1.69481
$307$	−15.4086	−0.879413	−0.439706	−	0.898142i	$-0.644918\pi$
$307$	−15.4086	−0.879413	−0.439706	+	0.898142i	$0.644918\pi$
$308$	−27.3252	−1.55700
$309$	−13.0280	−0.741134
$310$	−94.5871	−5.37218
$311$	20.9448	1.18767	0.593835	−	0.804587i	$-0.297613\pi$
$311$	20.9448	1.18767	0.593835	+	0.804587i	$0.297613\pi$
$312$	17.3044	0.979667
$313$	24.7588	1.39945	0.699725	−	0.714412i	$-0.253306\pi$
$313$	24.7588	1.39945	0.699725	+	0.714412i	$0.253306\pi$
$314$	−24.3254	−1.37276
$315$	−26.0519	−1.46786
$316$	−15.9421	−0.896815
$317$	−16.2883	−0.914840	−0.457420	−	0.889251i	$-0.651226\pi$
$317$	−16.2883	−0.914840	−0.457420	+	0.889251i	$0.651226\pi$
$318$	−56.6065	−3.17434
$319$	−11.5668	−0.647614
$320$	43.2879	2.41987
$321$	14.8650	0.829683
$322$	−44.3025	−2.46888
$323$	−3.50987	−0.195294
$324$	−54.8250	−3.04583
$325$	8.86082	0.491510
$326$	−47.6712	−2.64026
$327$	−10.2237	−0.565373
$328$	−48.2737	−2.66547
$329$	−7.13645	−0.393445
$330$	33.1711	1.82601
$331$	−16.1242	−0.886266	−0.443133	−	0.896456i	$-0.646133\pi$
$331$	−16.1242	−0.886266	−0.443133	+	0.896456i	$0.646133\pi$
$332$	61.7565	3.38933
$333$	8.82694	0.483713
$334$	11.0150	0.602715
$335$	27.6254	1.50934
$336$	85.6499	4.67258
$337$	−20.1625	−1.09832	−0.549161	−	0.835716i	$-0.685053\pi$
$337$	−20.1625	−1.09832	−0.549161	+	0.835716i	$0.685053\pi$
$338$	−2.63723	−0.143447
$339$	27.9797	1.51965
$340$	−107.419	−5.82560
$341$	14.6574	0.793741
$342$	−3.06895	−0.165950
$343$	3.12690	0.168837
$344$	15.9883	0.862029
$345$	38.3152	2.06282
$346$	−59.1584	−3.18038
$347$	−3.40657	−0.182874	−0.0914371	−	0.995811i	$-0.529146\pi$
$347$	−3.40657	−0.182874	−0.0914371	+	0.995811i	$0.529146\pi$
$348$	83.6445	4.48382
$349$	3.23519	0.173176	0.0865879	−	0.996244i	$-0.472404\pi$
$349$	3.23519	0.173176	0.0865879	+	0.996244i	$0.472404\pi$
$350$	84.6985	4.52732
$351$	2.37460	0.126747
$352$	−18.9872	−1.01202
$353$	−22.0813	−1.17527	−0.587633	−	0.809127i	$-0.699940\pi$
$353$	−22.0813	−1.17527	−0.587633	+	0.809127i	$0.699940\pi$
$354$	34.4696	1.83204
$355$	0.367925	0.0195274
$356$	1.87690	0.0994756
$357$	−46.8646	−2.48034
$358$	42.4534	2.24373
$359$	32.9181	1.73735	0.868677	−	0.495379i	$-0.164971\pi$
$359$	32.9181	1.73735	0.868677	+	0.495379i	$0.164971\pi$
$360$	−56.0135	−2.95217
$361$	−18.6367	−0.980878
$362$	−40.0189	−2.10334
$363$	19.2852	1.01221
$364$	−17.9596	−0.941339
$365$	−56.2958	−2.94665
$366$	−85.3557	−4.46161
$367$	1.05341	0.0549878	0.0274939	−	0.999622i	$-0.491247\pi$
$367$	1.05341	0.0549878	0.0274939	+	0.999622i	$0.491247\pi$
$368$	−49.3231	−2.57114
$369$	11.9591	0.622564
$370$	−44.8911	−2.33378
$371$	35.0365	1.81901
$372$	−105.994	−5.49554
$373$	−28.5648	−1.47903	−0.739514	−	0.673141i	$-0.764945\pi$
$373$	−28.5648	−1.47903	−0.739514	+	0.673141i	$0.764945\pi$
$374$	23.3646	1.20815
$375$	−31.9172	−1.64819
$376$	−15.3439	−0.791299
$377$	−7.60229	−0.391538
$378$	22.6982	1.16747
$379$	4.61300	0.236954	0.118477	−	0.992957i	$-0.462199\pi$
$379$	4.61300	0.236954	0.118477	+	0.992957i	$0.462199\pi$
$380$	11.1195	0.570420
$381$	−26.9741	−1.38193
$382$	20.7139	1.05982
$383$	8.24768	0.421437	0.210718	−	0.977547i	$-0.432420\pi$
$383$	8.24768	0.421437	0.210718	+	0.977547i	$0.432420\pi$
$384$	12.6670	0.646412
$385$	−20.5312	−1.04637
$386$	−34.3601	−1.74889
$387$	−3.96085	−0.201341
$388$	−70.7960	−3.59412
$389$	24.2284	1.22843	0.614215	−	0.789139i	$-0.289473\pi$
$389$	24.2284	1.22843	0.614215	+	0.789139i	$0.289473\pi$
$390$	21.8018	1.10398
$391$	26.9879	1.36483
$392$	−47.8281	−2.41568
$393$	0.166853	0.00841665
$394$	49.5692	2.49726
$395$	−11.9784	−0.602697
$396$	14.5547	0.731402
$397$	6.68228	0.335374	0.167687	−	0.985840i	$-0.446370\pi$
$397$	6.68228	0.335374	0.167687	+	0.985840i	$0.446370\pi$
$398$	−47.6809	−2.39003
$399$	4.85123	0.242865
$400$	94.2969	4.71485
$401$	−13.5131	−0.674812	−0.337406	−	0.941359i	$-0.609549\pi$
$401$	−13.5131	−0.674812	−0.337406	+	0.941359i	$0.609549\pi$
$402$	43.4522	2.16720
$403$	9.63360	0.479884
$404$	36.2076	1.80140
$405$	−41.1936	−2.04693
$406$	−72.6684	−3.60647
$407$	6.95641	0.344816
$408$	−100.762	−4.98847
$409$	22.3735	1.10630	0.553148	−	0.833083i	$-0.313426\pi$
$409$	22.3735	1.10630	0.553148	+	0.833083i	$0.313426\pi$
$410$	−60.8202	−3.00369
$411$	−20.2612	−0.999412
$412$	29.0716	1.43226
$413$	−21.3349	−1.04982
$414$	23.5976	1.15976
$415$	46.4017	2.27777
$416$	−12.4794	−0.611854
$417$	2.46968	0.120941
$418$	−2.41860	−0.118298
$419$	38.3443	1.87324	0.936621	−	0.350345i	$-0.113936\pi$
$419$	38.3443	1.87324	0.936621	+	0.350345i	$0.113936\pi$
$420$	148.471	7.24463
$421$	−4.70572	−0.229343	−0.114671	−	0.993403i	$-0.536581\pi$
$421$	−4.70572	−0.229343	−0.114671	+	0.993403i	$0.536581\pi$
$422$	−71.1984	−3.46588
$423$	3.80121	0.184821
$424$	75.3310	3.65840
$425$	−51.5960	−2.50277
$426$	0.578713	0.0280387
$427$	52.8308	2.55666
$428$	−33.1710	−1.60338
$429$	−3.37845	−0.163113
$430$	20.1437	0.971414
$431$	−3.35162	−0.161442	−0.0807209	−	0.996737i	$-0.525722\pi$
$431$	−3.35162	−0.161442	−0.0807209	+	0.996737i	$0.525722\pi$
$432$	25.2705	1.21583
$433$	31.2045	1.49959	0.749796	−	0.661670i	$-0.230152\pi$
$433$	31.2045	1.49959	0.749796	+	0.661670i	$0.230152\pi$
$434$	92.0853	4.42024
$435$	62.8476	3.01331
$436$	22.8140	1.09259
$437$	−2.79367	−0.133639
$438$	−88.5482	−4.23100
$439$	−11.9494	−0.570316	−0.285158	−	0.958481i	$-0.592046\pi$
$439$	−11.9494	−0.570316	−0.285158	+	0.958481i	$0.592046\pi$
$440$	−44.1436	−2.10446
$441$	11.8487	0.564223
$442$	15.3564	0.730431
$443$	−10.5789	−0.502619	−0.251309	−	0.967907i	$-0.580861\pi$
$443$	−10.5789	−0.502619	−0.251309	+	0.967907i	$0.580861\pi$
$444$	−50.3050	−2.38737
$445$	1.41024	0.0668517
$446$	24.5663	1.16325
$447$	−30.7460	−1.45424
$448$	−42.1430	−1.99107
$449$	−19.4165	−0.916320	−0.458160	−	0.888870i	$-0.651491\pi$
$449$	−19.4165	−0.916320	−0.458160	+	0.888870i	$0.651491\pi$
$450$	−45.1144	−2.12671
$451$	9.42480	0.443796
$452$	−62.4363	−2.93676
$453$	28.0124	1.31614
$454$	6.58869	0.309222
$455$	−13.4942	−0.632618
$456$	10.4305	0.488452
$457$	38.6432	1.80765	0.903827	−	0.427898i	$-0.140746\pi$
$457$	38.6432	1.80765	0.903827	+	0.427898i	$0.140746\pi$
$458$	−32.5680	−1.52180
$459$	−13.8271	−0.645394
$460$	−85.4997	−3.98644
$461$	20.3257	0.946662	0.473331	−	0.880885i	$-0.343051\pi$
$461$	20.3257	0.946662	0.473331	+	0.880885i	$0.343051\pi$
$462$	−32.2938	−1.50244
$463$	34.3034	1.59422	0.797108	−	0.603837i	$-0.206362\pi$
$463$	34.3034	1.59422	0.797108	+	0.603837i	$0.206362\pi$
$464$	−80.9036	−3.75585
$465$	−79.6404	−3.69323
$466$	26.1733	1.21245
$467$	−16.1678	−0.748157	−0.374079	−	0.927397i	$-0.622041\pi$
$467$	−16.1678	−0.748157	−0.374079	+	0.927397i	$0.622041\pi$
$468$	9.56613	0.442194
$469$	−26.8947	−1.24188
$470$	−19.3318	−0.891709
$471$	−20.4815	−0.943737
$472$	−45.8715	−2.11141
$473$	−3.12150	−0.143527
$474$	−18.8409	−0.865391
$475$	5.34100	0.245062
$476$	104.578	4.79330
$477$	−18.6621	−0.854479
$478$	61.2300	2.80060
$479$	−4.10787	−0.187693	−0.0938467	−	0.995587i	$-0.529916\pi$
$479$	−4.10787	−0.187693	−0.0938467	+	0.995587i	$0.529916\pi$
$480$	103.167	4.70889
$481$	4.57212	0.208471
$482$	40.7724	1.85713
$483$	−37.3018	−1.69729
$484$	−43.0346	−1.95612
$485$	−53.1936	−2.41540
$486$	−46.0067	−2.08691
$487$	19.6123	0.888719	0.444359	−	0.895849i	$-0.353431\pi$
$487$	19.6123	0.888719	0.444359	+	0.895849i	$0.353431\pi$
$488$	113.590	5.14197
$489$	−40.1382	−1.81511
$490$	−60.2588	−2.72221
$491$	33.6265	1.51754	0.758771	−	0.651358i	$-0.225800\pi$
$491$	33.6265	1.51754	0.758771	+	0.651358i	$0.225800\pi$
$492$	−68.1550	−3.07267
$493$	44.2676	1.99371
$494$	−1.58963	−0.0715210
$495$	10.9359	0.491532
$496$	102.521	4.60332
$497$	−0.358194	−0.0160672
$498$	72.9857	3.27057
$499$	−16.9115	−0.757064	−0.378532	−	0.925588i	$-0.623571\pi$
$499$	−16.9115	−0.757064	−0.378532	+	0.925588i	$0.623571\pi$
$500$	71.2226	3.18517
$501$	9.27442	0.414351
$502$	73.3258	3.27269
$503$	−10.5370	−0.469822	−0.234911	−	0.972017i	$-0.575480\pi$
$503$	−10.5370	−0.469822	−0.234911	+	0.972017i	$0.575480\pi$
$504$	54.5320	2.42905
$505$	27.2052	1.21061
$506$	18.5970	0.826737
$507$	−2.22050	−0.0986157
$508$	60.1923	2.67060
$509$	−17.4612	−0.773952	−0.386976	−	0.922090i	$-0.626480\pi$
$509$	−17.4612	−0.773952	−0.386976	+	0.922090i	$0.626480\pi$
$510$	−126.951	−5.62147
$511$	54.8068	2.42451
$512$	33.0606	1.46109
$513$	1.43132	0.0631945
$514$	19.6105	0.864981
$515$	21.8434	0.962536
$516$	22.5730	0.993720
$517$	2.99569	0.131750
$518$	43.7038	1.92023
$519$	−49.8101	−2.18642
$520$	−29.0135	−1.27233
$521$	41.0735	1.79946	0.899731	−	0.436444i	$-0.143762\pi$
$521$	41.0735	1.79946	0.899731	+	0.436444i	$0.143762\pi$
$522$	38.7066	1.69414
$523$	−18.8918	−0.826081	−0.413041	−	0.910713i	$-0.635533\pi$
$523$	−18.8918	−0.826081	−0.413041	+	0.910713i	$0.635533\pi$
$524$	−0.372331	−0.0162653
$525$	71.3144	3.11241
$526$	−73.3511	−3.19826
$527$	−56.0958	−2.44357
$528$	−35.9535	−1.56467
$529$	−1.51906	−0.0660462
$530$	94.9098	4.12262
$531$	11.3640	0.493154
$532$	−10.8254	−0.469342
$533$	6.19448	0.268313
$534$	2.21818	0.0959900
$535$	−24.9235	−1.07754
$536$	−57.8255	−2.49768
$537$	35.7449	1.54251
$538$	47.5538	2.05019
$539$	9.33780	0.402208
$540$	43.8054	1.88508
$541$	6.99050	0.300545	0.150273	−	0.988645i	$-0.451985\pi$
$541$	6.99050	0.300545	0.150273	+	0.988645i	$0.451985\pi$
$542$	33.3011	1.43040
$543$	−33.6950	−1.44599
$544$	72.6669	3.11557
$545$	17.1417	0.734268
$546$	−21.2252	−0.908354
$547$	−5.72658	−0.244851	−0.122425	−	0.992478i	$-0.539067\pi$
$547$	−5.72658	−0.244851	−0.122425	+	0.992478i	$0.539067\pi$
$548$	45.2126	1.93138
$549$	−28.1402	−1.20099
$550$	−35.5541	−1.51603
$551$	−4.58240	−0.195217
$552$	−80.2015	−3.41360
$553$	11.6615	0.495899
$554$	−38.9648	−1.65546
$555$	−37.7974	−1.60441
$556$	−5.51105	−0.233721
$557$	−22.2344	−0.942103	−0.471051	−	0.882106i	$-0.656125\pi$
$557$	−22.2344	−0.942103	−0.471051	+	0.882106i	$0.656125\pi$
$558$	−49.0490	−2.07641
$559$	−2.05161	−0.0867740
$560$	−143.606	−6.06844
$561$	19.6725	0.830573
$562$	−46.4613	−1.95985
$563$	11.3097	0.476645	0.238323	−	0.971186i	$-0.423402\pi$
$563$	11.3097	0.476645	0.238323	+	0.971186i	$0.423402\pi$
$564$	−21.6632	−0.912185
$565$	−46.9125	−1.97362
$566$	−2.95376	−0.124156
$567$	40.1040	1.68421
$568$	−0.770142	−0.0323144
$569$	−35.7239	−1.49763	−0.748813	−	0.662782i	$-0.769376\pi$
$569$	−35.7239	−1.49763	−0.748813	+	0.662782i	$0.769376\pi$
$570$	13.1414	0.550433
$571$	−28.8739	−1.20833	−0.604167	−	0.796858i	$-0.706494\pi$
$571$	−28.8739	−1.20833	−0.604167	+	0.796858i	$0.706494\pi$
$572$	7.53895	0.315219
$573$	17.4407	0.728595
$574$	59.2115	2.47144
$575$	−41.0677	−1.71264
$576$	22.4473	0.935306
$577$	32.2720	1.34350	0.671751	−	0.740777i	$-0.265543\pi$
$577$	32.2720	1.34350	0.671751	+	0.740777i	$0.265543\pi$
$578$	−44.5865	−1.85456
$579$	−28.9305	−1.20231
$580$	−140.243	−5.82329
$581$	−45.1744	−1.87415
$582$	−83.6688	−3.46818
$583$	−14.7074	−0.609118
$584$	117.838	4.87619
$585$	7.18765	0.297173
$586$	46.7795	1.93244
$587$	21.1678	0.873687	0.436844	−	0.899537i	$-0.356096\pi$
$587$	21.1678	0.873687	0.436844	+	0.899537i	$0.356096\pi$
$588$	−67.5259	−2.78472
$589$	5.80681	0.239265
$590$	−57.7937	−2.37933
$591$	41.7362	1.71680
$592$	48.6565	1.99977
$593$	26.0547	1.06994	0.534968	−	0.844872i	$-0.320324\pi$
$593$	26.0547	1.06994	0.534968	+	0.844872i	$0.320324\pi$
$594$	−9.52808	−0.390942
$595$	78.5759	3.22130
$596$	68.6093	2.81034
$597$	−40.1464	−1.64308
$598$	12.2229	0.499833
$599$	−38.0968	−1.55659	−0.778296	−	0.627898i	$-0.783916\pi$
$599$	−38.0968	−1.55659	−0.778296	+	0.627898i	$0.783916\pi$
$600$	153.331	6.25971
$601$	−11.3515	−0.463036	−0.231518	−	0.972831i	$-0.574369\pi$
$601$	−11.3515	−0.463036	−0.231518	+	0.972831i	$0.574369\pi$
$602$	−19.6109	−0.799280
$603$	14.3254	0.583375
$604$	−62.5091	−2.54346
$605$	−32.3347	−1.31459
$606$	42.7913	1.73828
$607$	38.8453	1.57668	0.788341	−	0.615238i	$-0.210940\pi$
$607$	38.8453	1.57668	0.788341	+	0.615238i	$0.210940\pi$
$608$	−7.52217	−0.305064
$609$	−61.1853	−2.47935
$610$	143.112	5.79445
$611$	1.96892	0.0796541
$612$	−55.7029	−2.25166
$613$	−34.4527	−1.39153	−0.695765	−	0.718269i	$-0.744935\pi$
$613$	−34.4527	−1.39153	−0.695765	+	0.718269i	$0.744935\pi$
$614$	40.6360	1.63993
$615$	−51.2093	−2.06496
$616$	42.9760	1.73155
$617$	−1.00000	−0.0402585
$617$	−1.00000	−0.0402585
$618$	34.3577	1.38207
$619$	−4.20106	−0.168855	−0.0844275	−	0.996430i	$-0.526906\pi$
$619$	−4.20106	−0.168855	−0.0844275	+	0.996430i	$0.526906\pi$
$620$	177.716	7.13725
$621$	−11.0057	−0.441642
$622$	−55.2363	−2.21477
$623$	−1.37294	−0.0550056
$624$	−23.6305	−0.945979
$625$	9.21005	0.368402
$626$	−65.2947	−2.60970
$627$	−2.03641	−0.0813265
$628$	45.7041	1.82379
$629$	−26.6231	−1.06153
$630$	68.7050	2.73727
$631$	−16.7995	−0.668777	−0.334388	−	0.942435i	$-0.608530\pi$
$631$	−16.7995	−0.668777	−0.334388	+	0.942435i	$0.608530\pi$
$632$	25.0731	0.997356
$633$	−59.9476	−2.38270
$634$	42.9559	1.70600
$635$	45.2264	1.79476
$636$	106.356	4.21728
$637$	6.13730	0.243169
$638$	30.5042	1.20767
$639$	0.190791	0.00754757
$640$	−21.2383	−0.839517
$641$	30.8894	1.22006	0.610030	−	0.792379i	$-0.291158\pi$
$641$	30.8894	1.22006	0.610030	+	0.792379i	$0.291158\pi$
$642$	−39.2024	−1.54720
$643$	33.2266	1.31033	0.655165	−	0.755486i	$-0.272599\pi$
$643$	33.2266	1.31033	0.655165	+	0.755486i	$0.272599\pi$
$644$	83.2383	3.28005
$645$	16.9606	0.667821
$646$	9.25633	0.364186
$647$	−1.65566	−0.0650907	−0.0325453	−	0.999470i	$-0.510361\pi$
$647$	−1.65566	−0.0650907	−0.0325453	+	0.999470i	$0.510361\pi$
$648$	86.2265	3.38730
$649$	8.95581	0.351546
$650$	−23.3681	−0.916570
$651$	77.5339	3.03879
$652$	89.5676	3.50774
$653$	18.9708	0.742386	0.371193	−	0.928556i	$-0.378949\pi$
$653$	18.9708	0.742386	0.371193	+	0.928556i	$0.378949\pi$
$654$	26.9623	1.05431
$655$	−0.279756	−0.0109310
$656$	65.9217	2.57381
$657$	−29.1927	−1.13891
$658$	18.8205	0.733698
$659$	4.58871	0.178751	0.0893753	−	0.995998i	$-0.471513\pi$
$659$	4.58871	0.178751	0.0893753	+	0.995998i	$0.471513\pi$
$660$	−62.3240	−2.42596
$661$	27.2921	1.06154	0.530770	−	0.847516i	$-0.321903\pi$
$661$	27.2921	1.06154	0.530770	+	0.847516i	$0.321903\pi$
$662$	42.5233	1.65271
$663$	12.9298	0.502152
$664$	−97.1281	−3.76930
$665$	−8.13385	−0.315417
$666$	−23.2787	−0.902031
$667$	35.2347	1.36429
$668$	−20.6957	−0.800741
$669$	20.6843	0.799702
$670$	−72.8546	−2.81462
$671$	−22.1769	−0.856131
$672$	−100.438	−3.87447
$673$	38.1089	1.46899	0.734494	−	0.678615i	$-0.237420\pi$
$673$	38.1089	1.46899	0.734494	+	0.678615i	$0.237420\pi$
$674$	53.1733	2.04816
$675$	21.0409	0.809863
$676$	4.95500	0.190577
$677$	−24.2517	−0.932067	−0.466033	−	0.884767i	$-0.654317\pi$
$677$	−24.2517	−0.932067	−0.466033	+	0.884767i	$0.654317\pi$
$678$	−73.7891	−2.83385
$679$	51.7867	1.98739
$680$	168.944	6.47870
$681$	5.54754	0.212582
$682$	−38.6549	−1.48017
$683$	−44.6156	−1.70717	−0.853584	−	0.520956i	$-0.825576\pi$
$683$	−44.6156	−1.70717	−0.853584	+	0.520956i	$0.825576\pi$
$684$	5.76613	0.220474
$685$	33.9711	1.29797
$686$	−8.24635	−0.314847
$687$	−27.4216	−1.04620
$688$	−21.8333	−0.832386
$689$	−9.66647	−0.368263
$690$	−101.046	−3.84676
$691$	−9.46885	−0.360212	−0.180106	−	0.983647i	$-0.557644\pi$
$691$	−9.46885	−0.360212	−0.180106	+	0.983647i	$0.557644\pi$
$692$	111.150	4.22531
$693$	−10.6467	−0.404433
$694$	8.98392	0.341025
$695$	−4.14081	−0.157070
$696$	−131.553	−4.98649
$697$	−36.0700	−1.36625
$698$	−8.53196	−0.322939
$699$	22.0374	0.833530
$700$	−159.137	−6.01480
$701$	−26.0033	−0.982130	−0.491065	−	0.871123i	$-0.663392\pi$
$701$	−26.0033	−0.982130	−0.491065	+	0.871123i	$0.663392\pi$
$702$	−6.26236	−0.236358
$703$	2.75592	0.103941
$704$	17.6905	0.666735
$705$	−16.2770	−0.613026
$706$	58.2334	2.19164
$707$	−26.4856	−0.996093
$708$	−64.7636	−2.43396
$709$	20.6490	0.775490	0.387745	−	0.921767i	$-0.373254\pi$
$709$	20.6490	0.775490	0.387745	+	0.921767i	$0.373254\pi$
$710$	−0.970304	−0.0364149
$711$	−6.21149	−0.232949
$712$	−2.95191	−0.110628
$713$	−44.6494	−1.67213
$714$	123.593	4.62535
$715$	5.66450	0.211840
$716$	−79.7641	−2.98093
$717$	51.5544	1.92534
$718$	−86.8128	−3.23983
$719$	23.1127	0.861958	0.430979	−	0.902362i	$-0.358168\pi$
$719$	23.1127	0.861958	0.430979	+	0.902362i	$0.358168\pi$
$720$	76.4911	2.85065
$721$	−21.2657	−0.791975
$722$	49.1493	1.82915
$723$	34.3295	1.27673
$724$	75.1899	2.79441
$725$	−67.3625	−2.50178
$726$	−50.8596	−1.88758
$727$	42.0199	1.55843	0.779216	−	0.626756i	$-0.215618\pi$
$727$	42.0199	1.55843	0.779216	+	0.626756i	$0.215618\pi$
$728$	28.2461	1.04687
$729$	−5.54296	−0.205295
$730$	148.465	5.49494
$731$	11.9464	0.441854
$732$	160.372	5.92751
$733$	49.2895	1.82055	0.910274	−	0.414005i	$-0.135871\pi$
$733$	49.2895	1.82055	0.910274	+	0.414005i	$0.135871\pi$
$734$	−2.77810	−0.102541
$735$	−50.7366	−1.87145
$736$	57.8390	2.13198
$737$	11.2897	0.415860
$738$	−31.5388	−1.16096
$739$	−16.2358	−0.597244	−0.298622	−	0.954371i	$-0.596527\pi$
$739$	−16.2358	−0.597244	−0.298622	+	0.954371i	$0.596527\pi$
$740$	84.3442	3.10056
$741$	−1.33844	−0.0491688
$742$	−92.3995	−3.39209
$743$	6.06727	0.222587	0.111293	−	0.993788i	$-0.464501\pi$
$743$	6.06727	0.222587	0.111293	+	0.993788i	$0.464501\pi$
$744$	166.703	6.11164
$745$	51.5506	1.88867
$746$	75.3320	2.75810
$747$	24.0620	0.880382
$748$	−43.8988	−1.60510
$749$	24.2643	0.886598
$750$	84.1730	3.07356
$751$	−10.4461	−0.381182	−0.190591	−	0.981670i	$-0.561040\pi$
$751$	−10.4461	−0.381182	−0.190591	+	0.981670i	$0.561040\pi$
$752$	20.9533	0.764088
$753$	61.7388	2.24989
$754$	20.0490	0.730141
$755$	−46.9672	−1.70931
$756$	−42.6468	−1.55105
$757$	−22.4209	−0.814900	−0.407450	−	0.913227i	$-0.633582\pi$
$757$	−22.4209	−0.814900	−0.407450	+	0.913227i	$0.633582\pi$
$758$	−12.1656	−0.441873
$759$	15.6583	0.568359
$760$	−17.4883	−0.634369
$761$	3.97453	0.144077	0.0720384	−	0.997402i	$-0.477050\pi$
$761$	3.97453	0.144077	0.0720384	+	0.997402i	$0.477050\pi$
$762$	71.1371	2.57702
$763$	−16.6883	−0.604156
$764$	−38.9186	−1.40802
$765$	−41.8532	−1.51321
$766$	−21.7511	−0.785898
$767$	5.88623	0.212540
$768$	18.2301	0.657821
$769$	−5.71114	−0.205949	−0.102975	−	0.994684i	$-0.532836\pi$
$769$	−5.71114	−0.205949	−0.102975	+	0.994684i	$0.532836\pi$
$770$	54.1456	1.95127
$771$	16.5116	0.594651
$772$	64.5580	2.32349
$773$	42.4339	1.52624	0.763120	−	0.646256i	$-0.223666\pi$
$773$	42.4339	1.52624	0.763120	+	0.646256i	$0.223666\pi$
$774$	10.4457	0.375462
$775$	85.3616	3.06628
$776$	111.345	3.99705
$777$	36.7977	1.32011
$778$	−63.8960	−2.29078
$779$	3.73382	0.133778
$780$	−40.9626	−1.46670
$781$	0.150360	0.00538030
$782$	−71.1733	−2.54515
$783$	−18.0523	−0.645138
$784$	65.3132	2.33261
$785$	34.3404	1.22566
$786$	−0.440032	−0.0156954
$787$	−15.7265	−0.560589	−0.280295	−	0.959914i	$-0.590432\pi$
$787$	−15.7265	−0.560589	−0.280295	+	0.959914i	$0.590432\pi$
$788$	−93.1336	−3.31775
$789$	−61.7601	−2.19872
$790$	31.5897	1.12391
$791$	45.6717	1.62390
$792$	−22.8911	−0.813398
$793$	−14.5759	−0.517604
$794$	−17.6227	−0.625407
$795$	79.9121	2.83419
$796$	89.5859	3.17529
$797$	−23.4277	−0.829851	−0.414926	−	0.909855i	$-0.636192\pi$
$797$	−23.4277	−0.829851	−0.414926	+	0.909855i	$0.636192\pi$
$798$	−12.7938	−0.452896
$799$	−11.4649	−0.405599
$800$	−110.578	−3.90952
$801$	0.731292	0.0258389
$802$	35.6372	1.25839
$803$	−23.0064	−0.811878
$804$	−81.6408	−2.87925
$805$	62.5423	2.20433
$806$	−25.4061	−0.894890
$807$	40.0393	1.40945
$808$	−56.9459	−2.00335
$809$	−34.6254	−1.21736	−0.608681	−	0.793415i	$-0.708301\pi$
$809$	−34.6254	−1.21736	−0.608681	+	0.793415i	$0.708301\pi$
$810$	108.637	3.81712
$811$	19.8397	0.696666	0.348333	−	0.937371i	$-0.386748\pi$
$811$	19.8397	0.696666	0.348333	+	0.937371i	$0.386748\pi$
$812$	136.534	4.79140
$813$	28.0388	0.983365
$814$	−18.3457	−0.643015
$815$	67.2980	2.35734
$816$	137.599	4.81693
$817$	−1.23664	−0.0432647
$818$	−59.0040	−2.06303
$819$	−6.99754	−0.244514
$820$	114.273	3.99058
$821$	30.3226	1.05827	0.529133	−	0.848539i	$-0.322517\pi$
$821$	30.3226	1.05827	0.529133	+	0.848539i	$0.322517\pi$
$822$	53.4336	1.86371
$823$	−8.10459	−0.282508	−0.141254	−	0.989973i	$-0.545113\pi$
$823$	−8.10459	−0.282508	−0.141254	+	0.989973i	$0.545113\pi$
$824$	−45.7227	−1.59283
$825$	−29.9358	−1.04223
$826$	56.2651	1.95771
$827$	29.9835	1.04263	0.521314	−	0.853365i	$-0.325442\pi$
$827$	29.9835	1.04263	0.521314	+	0.853365i	$0.325442\pi$
$828$	−44.3366	−1.54080
$829$	13.6152	0.472876	0.236438	−	0.971647i	$-0.424020\pi$
$829$	13.6152	0.472876	0.236438	+	0.971647i	$0.424020\pi$
$830$	−122.372	−4.24759
$831$	−32.8076	−1.13808
$832$	11.6271	0.403098
$833$	−35.7371	−1.23822
$834$	−6.51312	−0.225531
$835$	−15.5500	−0.538131
$836$	4.54422	0.157165
$837$	22.8759	0.790707
$838$	−101.123	−3.49323
$839$	−10.2692	−0.354531	−0.177266	−	0.984163i	$-0.556725\pi$
$839$	−10.2692	−0.354531	−0.177266	+	0.984163i	$0.556725\pi$
$840$	−233.509	−8.05682
$841$	28.7947	0.992922
$842$	12.4101	0.427679
$843$	−39.1194	−1.34734
$844$	133.772	4.60462
$845$	3.72301	0.128076
$846$	−10.0247	−0.344655
$847$	31.4795	1.08165
$848$	−102.871	−3.53259
$849$	−2.48700	−0.0853537
$850$	136.071	4.66718
$851$	−21.1906	−0.726406
$852$	−1.08732	−0.0372511
$853$	26.4256	0.904796	0.452398	−	0.891816i	$-0.350569\pi$
$853$	26.4256	0.904796	0.452398	+	0.891816i	$0.350569\pi$
$854$	−139.327	−4.76768
$855$	4.33247	0.148167
$856$	52.1699	1.78313
$857$	−17.0167	−0.581278	−0.290639	−	0.956833i	$-0.593868\pi$
$857$	−17.0167	−0.581278	−0.290639	+	0.956833i	$0.593868\pi$
$858$	8.90976	0.304174
$859$	26.8278	0.915352	0.457676	−	0.889119i	$-0.348682\pi$
$859$	26.8278	0.915352	0.457676	+	0.889119i	$0.348682\pi$
$860$	−37.8472	−1.29058
$861$	49.8549	1.69905
$862$	8.83900	0.301057
$863$	18.5926	0.632900	0.316450	−	0.948609i	$-0.397509\pi$
$863$	18.5926	0.632900	0.316450	+	0.948609i	$0.397509\pi$
$864$	−29.6336	−1.00815
$865$	83.5146	2.83958
$866$	−82.2935	−2.79645
$867$	−37.5409	−1.27496
$868$	−173.016	−5.87253
$869$	−4.89520	−0.166058
$870$	−165.744	−5.61924
$871$	7.42017	0.251423
$872$	−35.8810	−1.21508
$873$	−27.5840	−0.933578
$874$	7.36756	0.249211
$875$	−52.0987	−1.76126
$876$	166.370	5.62112
$877$	−30.2256	−1.02065	−0.510324	−	0.859982i	$-0.670474\pi$
$877$	−30.2256	−1.02065	−0.510324	+	0.859982i	$0.670474\pi$
$878$	31.5135	1.06353
$879$	39.3874	1.32850
$880$	60.2817	2.03210
$881$	15.7053	0.529125	0.264562	−	0.964369i	$-0.414773\pi$
$881$	15.7053	0.529125	0.264562	+	0.964369i	$0.414773\pi$
$882$	−31.2477	−1.05217
$883$	0.666666	0.0224351	0.0112176	−	0.999937i	$-0.496429\pi$
$883$	0.666666	0.0224351	0.0112176	+	0.999937i	$0.496429\pi$
$884$	−28.8526	−0.970419
$885$	−48.6611	−1.63572
$886$	27.8990	0.937286
$887$	−20.3172	−0.682185	−0.341092	−	0.940030i	$-0.610797\pi$
$887$	−20.3172	−0.682185	−0.341092	+	0.940030i	$0.610797\pi$
$888$	79.1176	2.65501
$889$	−44.0302	−1.47673
$890$	−3.71913	−0.124665
$891$	−16.8346	−0.563980
$892$	−46.1567	−1.54544
$893$	1.18680	0.0397147
$894$	81.0845	2.71187
$895$	−59.9320	−2.00330
$896$	20.6765	0.690755
$897$	10.2914	0.343622
$898$	51.2058	1.70876
$899$	−73.2374	−2.44260
$900$	84.7637	2.82546
$901$	56.2872	1.87520
$902$	−24.8554	−0.827594
$903$	−16.5120	−0.549484
$904$	98.1973	3.26599
$905$	56.4951	1.87796
$906$	−73.8751	−2.45434
$907$	−31.7574	−1.05449	−0.527243	−	0.849715i	$-0.676774\pi$
$907$	−31.7574	−1.05449	−0.527243	+	0.849715i	$0.676774\pi$
$908$	−12.3792	−0.410819
$909$	14.1075	0.467915
$910$	35.5874	1.17971
$911$	−25.8884	−0.857721	−0.428860	−	0.903371i	$-0.641085\pi$
$911$	−25.8884	−0.857721	−0.428860	+	0.903371i	$0.641085\pi$
$912$	−14.2437	−0.471655
$913$	18.9630	0.627583
$914$	−101.911	−3.37092
$915$	120.498	3.98353
$916$	61.1908	2.02180
$917$	0.272357	0.00899402
$918$	36.4653	1.20353
$919$	43.7851	1.44434	0.722168	−	0.691717i	$-0.243146\pi$
$919$	43.7851	1.44434	0.722168	+	0.691717i	$0.243146\pi$
$920$	134.470	4.43336
$921$	34.2146	1.12741
$922$	−53.6036	−1.76534
$923$	0.0988246	0.00325285
$924$	60.6756	1.99608
$925$	40.5127	1.33205
$926$	−90.4661	−2.97290
$927$	11.3271	0.372031
$928$	94.8721	3.11433
$929$	−34.9911	−1.14802	−0.574010	−	0.818848i	$-0.694613\pi$
$929$	−34.9911	−1.14802	−0.574010	+	0.818848i	$0.694613\pi$
$930$	210.030	6.88716
$931$	3.69935	0.121241
$932$	−49.1760	−1.61081
$933$	−46.5078	−1.52260
$934$	42.6383	1.39517
$935$	−32.9840	−1.07869
$936$	−15.0452	−0.491768
$937$	51.8104	1.69257	0.846287	−	0.532728i	$-0.178833\pi$
$937$	51.8104	1.69257	0.846287	+	0.532728i	$0.178833\pi$
$938$	70.9276	2.31587
$939$	−54.9768	−1.79410
$940$	36.3218	1.18469
$941$	9.99351	0.325779	0.162889	−	0.986644i	$-0.447919\pi$
$941$	9.99351	0.325779	0.162889	+	0.986644i	$0.447919\pi$
$942$	54.0144	1.75989
$943$	−28.7099	−0.934922
$944$	62.6414	2.03880
$945$	−32.0433	−1.04237
$946$	8.23212	0.267649
$947$	33.6279	1.09276	0.546380	−	0.837537i	$-0.316005\pi$
$947$	33.6279	1.09276	0.546380	+	0.837537i	$0.316005\pi$
$948$	35.3994	1.14972
$949$	−15.1210	−0.490849
$950$	−14.0855	−0.456993
$951$	36.1680	1.17283
$952$	−164.475	−5.33067
$953$	31.3985	1.01710	0.508549	−	0.861033i	$-0.330182\pi$
$953$	31.3985	1.01710	0.508549	+	0.861033i	$0.330182\pi$
$954$	49.2163	1.59344
$955$	−29.2420	−0.946251
$956$	−115.043	−3.72075
$957$	25.6839	0.830244
$958$	10.8334	0.350012
$959$	−33.0726	−1.06797
$960$	−96.1207	−3.10228
$961$	61.8063	1.99375
$962$	−12.0577	−0.388757
$963$	−12.9243	−0.416480
$964$	−76.6057	−2.46730
$965$	48.5066	1.56148
$966$	98.3735	3.16512
$967$	−1.20571	−0.0387731	−0.0193865	−	0.999812i	$-0.506171\pi$
$967$	−1.20571	−0.0387731	−0.0193865	+	0.999812i	$0.506171\pi$
$968$	67.6830	2.17542
$969$	7.79364	0.250368
$970$	140.284	4.50425
$971$	36.2920	1.16466	0.582332	−	0.812951i	$-0.302140\pi$
$971$	36.2920	1.16466	0.582332	+	0.812951i	$0.302140\pi$
$972$	86.4403	2.77257
$973$	4.03129	0.129237
$974$	−51.7222	−1.65729
$975$	−19.6754	−0.630118
$976$	−155.116	−4.96516
$977$	17.2435	0.551667	0.275834	−	0.961205i	$-0.411046\pi$
$977$	17.2435	0.551667	0.275834	+	0.961205i	$0.411046\pi$
$978$	105.854	3.38483
$979$	0.576322	0.0184193
$980$	113.218	3.61661
$981$	8.88897	0.283803
$982$	−88.6809	−2.82992
$983$	13.6145	0.434235	0.217118	−	0.976145i	$-0.430334\pi$
$983$	13.6145	0.434235	0.217118	+	0.976145i	$0.430334\pi$
$984$	107.191	3.41714
$985$	−69.9773	−2.22966
$986$	−116.744	−3.71789
$987$	15.8464	0.504398
$988$	2.98670	0.0950197
$989$	9.50872	0.302360
$990$	−28.8405	−0.916612
$991$	11.9737	0.380358	0.190179	−	0.981749i	$-0.439093\pi$
$991$	11.9737	0.380358	0.190179	+	0.981749i	$0.439093\pi$
$992$	−120.222	−3.81705
$993$	35.8037	1.13620
$994$	0.944640	0.0299622
$995$	67.3117	2.13392
$996$	−137.130	−4.34513
$997$	46.7068	1.47922	0.739609	−	0.673037i	$-0.235010\pi$
$997$	46.7068	1.47922	0.739609	+	0.673037i	$0.235010\pi$
$998$	44.5997	1.41178
$999$	10.8569	0.343498

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8021.2.a.d.1.13	✓	174

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8021.2.a.d.1.13	✓	174	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.63723 q^{2} -2.22050 q^{3} +4.95500 q^{4} +3.72301 q^{5} +5.85596 q^{6} -3.62454 q^{7} -7.79302 q^{8} +1.93060 q^{9} +O(q^{10})\)	\(q-2.63723 q^{2} -2.22050 q^{3} +4.95500 q^{4} +3.72301 q^{5} +5.85596 q^{6} -3.62454 q^{7} -7.79302 q^{8} +1.93060 q^{9} -9.81845 q^{10} +1.52148 q^{11} -11.0026 q^{12} +1.00000 q^{13} +9.55876 q^{14} -8.26693 q^{15} +10.6420 q^{16} -5.82293 q^{17} -5.09145 q^{18} +0.602766 q^{19} +18.4475 q^{20} +8.04828 q^{21} -4.01251 q^{22} -4.63475 q^{23} +17.3044 q^{24} +8.86082 q^{25} -2.63723 q^{26} +2.37460 q^{27} -17.9596 q^{28} -7.60229 q^{29} +21.8018 q^{30} +9.63360 q^{31} -12.4794 q^{32} -3.37845 q^{33} +15.3564 q^{34} -13.4942 q^{35} +9.56613 q^{36} +4.57212 q^{37} -1.58963 q^{38} -2.22050 q^{39} -29.0135 q^{40} +6.19448 q^{41} -21.2252 q^{42} -2.05161 q^{43} +7.53895 q^{44} +7.18765 q^{45} +12.2229 q^{46} +1.96892 q^{47} -23.6305 q^{48} +6.13730 q^{49} -23.3681 q^{50} +12.9298 q^{51} +4.95500 q^{52} -9.66647 q^{53} -6.26236 q^{54} +5.66450 q^{55} +28.2461 q^{56} -1.33844 q^{57} +20.0490 q^{58} +5.88623 q^{59} -40.9626 q^{60} -14.5759 q^{61} -25.4061 q^{62} -6.99754 q^{63} +11.6271 q^{64} +3.72301 q^{65} +8.90976 q^{66} +7.42017 q^{67} -28.8526 q^{68} +10.2914 q^{69} +35.5874 q^{70} +0.0988246 q^{71} -15.0452 q^{72} -15.1210 q^{73} -12.0577 q^{74} -19.6754 q^{75} +2.98670 q^{76} -5.51468 q^{77} +5.85596 q^{78} -3.21738 q^{79} +39.6203 q^{80} -11.0646 q^{81} -16.3363 q^{82} +12.4635 q^{83} +39.8792 q^{84} -21.6789 q^{85} +5.41058 q^{86} +16.8808 q^{87} -11.8570 q^{88} +0.378790 q^{89} -18.9555 q^{90} -3.62454 q^{91} -22.9652 q^{92} -21.3914 q^{93} -5.19251 q^{94} +2.24410 q^{95} +27.7105 q^{96} -14.2878 q^{97} -16.1855 q^{98} +2.93738 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 174 q + 6 q^{2} + 37 q^{3} + 214 q^{4} + 10 q^{5} + 12 q^{6} + 28 q^{7} + 15 q^{8} + 211 q^{9}+O(q^{10}) \) 174 * q + 6 * q^2 + 37 * q^3 + 214 * q^4 + 10 * q^5 + 12 * q^6 + 28 * q^7 + 15 * q^8 + 211 * q^9	\( 174 q + 6 q^{2} + 37 q^{3} + 214 q^{4} + 10 q^{5} + 12 q^{6} + 28 q^{7} + 15 q^{8} + 211 q^{9} + 47 q^{10} + 47 q^{11} + 81 q^{12} + 174 q^{13} + 22 q^{14} + 26 q^{15} + 286 q^{16} + 27 q^{17} + 22 q^{18} + 91 q^{19} + 18 q^{20} + 8 q^{21} + 58 q^{22} + 62 q^{23} + 24 q^{24} + 244 q^{25} + 6 q^{26} + 139 q^{27} + 43 q^{28} + 42 q^{29} + 31 q^{30} + 82 q^{31} + 11 q^{32} + 12 q^{33} + 50 q^{34} + 74 q^{35} + 310 q^{36} + 47 q^{37} + 10 q^{38} + 37 q^{39} + 118 q^{40} + 16 q^{41} + 26 q^{42} + 136 q^{43} + 74 q^{44} + 18 q^{45} + 53 q^{46} + 15 q^{47} + 132 q^{48} + 254 q^{49} - 5 q^{50} + 121 q^{51} + 214 q^{52} + 39 q^{53} + 30 q^{54} + 188 q^{55} + 55 q^{56} + 11 q^{57} + 32 q^{58} + 58 q^{59} + 16 q^{60} + 128 q^{61} + 27 q^{62} + 42 q^{63} + 423 q^{64} + 10 q^{65} + 4 q^{66} + 132 q^{67} + 52 q^{68} + 63 q^{69} - 8 q^{70} + 78 q^{71} + 2 q^{72} + 21 q^{73} - 16 q^{74} + 188 q^{75} + 160 q^{76} + 20 q^{77} + 12 q^{78} + 232 q^{79} + 2 q^{80} + 302 q^{81} + 115 q^{82} + 18 q^{83} - 26 q^{84} + 47 q^{85} + 27 q^{86} + 127 q^{87} + 163 q^{88} + 14 q^{90} + 28 q^{91} + 68 q^{92} + 15 q^{93} + 91 q^{94} + 75 q^{95} - 26 q^{96} + 34 q^{97} - 60 q^{98} + 181 q^{99}+O(q^{100}) \) 174 * q + 6 * q^2 + 37 * q^3 + 214 * q^4 + 10 * q^5 + 12 * q^6 + 28 * q^7 + 15 * q^8 + 211 * q^9 + 47 * q^10 + 47 * q^11 + 81 * q^12 + 174 * q^13 + 22 * q^14 + 26 * q^15 + 286 * q^16 + 27 * q^17 + 22 * q^18 + 91 * q^19 + 18 * q^20 + 8 * q^21 + 58 * q^22 + 62 * q^23 + 24 * q^24 + 244 * q^25 + 6 * q^26 + 139 * q^27 + 43 * q^28 + 42 * q^29 + 31 * q^30 + 82 * q^31 + 11 * q^32 + 12 * q^33 + 50 * q^34 + 74 * q^35 + 310 * q^36 + 47 * q^37 + 10 * q^38 + 37 * q^39 + 118 * q^40 + 16 * q^41 + 26 * q^42 + 136 * q^43 + 74 * q^44 + 18 * q^45 + 53 * q^46 + 15 * q^47 + 132 * q^48 + 254 * q^49 - 5 * q^50 + 121 * q^51 + 214 * q^52 + 39 * q^53 + 30 * q^54 + 188 * q^55 + 55 * q^56 + 11 * q^57 + 32 * q^58 + 58 * q^59 + 16 * q^60 + 128 * q^61 + 27 * q^62 + 42 * q^63 + 423 * q^64 + 10 * q^65 + 4 * q^66 + 132 * q^67 + 52 * q^68 + 63 * q^69 - 8 * q^70 + 78 * q^71 + 2 * q^72 + 21 * q^73 - 16 * q^74 + 188 * q^75 + 160 * q^76 + 20 * q^77 + 12 * q^78 + 232 * q^79 + 2 * q^80 + 302 * q^81 + 115 * q^82 + 18 * q^83 - 26 * q^84 + 47 * q^85 + 27 * q^86 + 127 * q^87 + 163 * q^88 + 14 * q^90 + 28 * q^91 + 68 * q^92 + 15 * q^93 + 91 * q^94 + 75 * q^95 - 26 * q^96 + 34 * q^97 - 60 * q^98 + 181 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more