LMFDB - Embedded newform 8043.2.a.t.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, 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("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.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:	$64.2236783457$
Analytic rank:	$0$
Dimension:	$52$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.837626 q^{2} -1.00000 q^{3} -1.29838 q^{4} +2.34855 q^{5} +0.837626 q^{6} +1.00000 q^{7} +2.76281 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.837626 q^{2} -1.00000 q^{3} -1.29838 q^{4} +2.34855 q^{5} +0.837626 q^{6} +1.00000 q^{7} +2.76281 q^{8} +1.00000 q^{9} -1.96721 q^{10} +2.64878 q^{11} +1.29838 q^{12} -4.81100 q^{13} -0.837626 q^{14} -2.34855 q^{15} +0.282564 q^{16} -7.16278 q^{17} -0.837626 q^{18} +4.98734 q^{19} -3.04932 q^{20} -1.00000 q^{21} -2.21869 q^{22} +3.89405 q^{23} -2.76281 q^{24} +0.515710 q^{25} +4.02981 q^{26} -1.00000 q^{27} -1.29838 q^{28} -9.78351 q^{29} +1.96721 q^{30} -7.90929 q^{31} -5.76230 q^{32} -2.64878 q^{33} +5.99973 q^{34} +2.34855 q^{35} -1.29838 q^{36} +3.14302 q^{37} -4.17752 q^{38} +4.81100 q^{39} +6.48861 q^{40} -4.49750 q^{41} +0.837626 q^{42} +3.34830 q^{43} -3.43914 q^{44} +2.34855 q^{45} -3.26176 q^{46} +4.41618 q^{47} -0.282564 q^{48} +1.00000 q^{49} -0.431972 q^{50} +7.16278 q^{51} +6.24652 q^{52} +7.48954 q^{53} +0.837626 q^{54} +6.22082 q^{55} +2.76281 q^{56} -4.98734 q^{57} +8.19492 q^{58} +4.10404 q^{59} +3.04932 q^{60} -0.167344 q^{61} +6.62503 q^{62} +1.00000 q^{63} +4.26153 q^{64} -11.2989 q^{65} +2.21869 q^{66} -13.3627 q^{67} +9.30003 q^{68} -3.89405 q^{69} -1.96721 q^{70} +15.3017 q^{71} +2.76281 q^{72} -8.69102 q^{73} -2.63268 q^{74} -0.515710 q^{75} -6.47548 q^{76} +2.64878 q^{77} -4.02981 q^{78} +12.2587 q^{79} +0.663618 q^{80} +1.00000 q^{81} +3.76723 q^{82} +15.2779 q^{83} +1.29838 q^{84} -16.8222 q^{85} -2.80462 q^{86} +9.78351 q^{87} +7.31809 q^{88} +8.16472 q^{89} -1.96721 q^{90} -4.81100 q^{91} -5.05597 q^{92} +7.90929 q^{93} -3.69910 q^{94} +11.7130 q^{95} +5.76230 q^{96} +8.35604 q^{97} -0.837626 q^{98} +2.64878 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) $ 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9} - 2 q^{10} + 9 q^{11} - 61 q^{12} + 44 q^{13} + 3 q^{14} + 7 q^{15} + 95 q^{16} - 6 q^{17} + 3 q^{18} + 7 q^{19} - 21 q^{20} - 52 q^{21} + 19 q^{22} - 4 q^{23} - 24 q^{24} + 83 q^{25} - 5 q^{26} - 52 q^{27} + 61 q^{28} + 31 q^{29} + 2 q^{30} + 11 q^{31} + 71 q^{32} - 9 q^{33} + 17 q^{34} - 7 q^{35} + 61 q^{36} + 71 q^{37} - 8 q^{38} - 44 q^{39} + 20 q^{40} - 25 q^{41} - 3 q^{42} + 75 q^{43} + 14 q^{44} - 7 q^{45} + 36 q^{46} - 20 q^{47} - 95 q^{48} + 52 q^{49} + 26 q^{50} + 6 q^{51} + 88 q^{52} + 70 q^{53} - 3 q^{54} + 12 q^{55} + 24 q^{56} - 7 q^{57} + 48 q^{58} - 27 q^{59} + 21 q^{60} + 59 q^{61} - 23 q^{62} + 52 q^{63} + 138 q^{64} + 44 q^{65} - 19 q^{66} + 65 q^{67} - 8 q^{68} + 4 q^{69} - 2 q^{70} - 11 q^{71} + 24 q^{72} + 34 q^{73} + 38 q^{74} - 83 q^{75} + 31 q^{76} + 9 q^{77} + 5 q^{78} + 74 q^{79} - 5 q^{80} + 52 q^{81} + 51 q^{82} - 30 q^{83} - 61 q^{84} + 70 q^{85} + 29 q^{86} - 31 q^{87} + 90 q^{88} - q^{89} - 2 q^{90} + 44 q^{91} + 34 q^{92} - 11 q^{93} + 27 q^{94} + 9 q^{95} - 71 q^{96} + 73 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9 - 2 * q^10 + 9 * q^11 - 61 * q^12 + 44 * q^13 + 3 * q^14 + 7 * q^15 + 95 * q^16 - 6 * q^17 + 3 * q^18 + 7 * q^19 - 21 * q^20 - 52 * q^21 + 19 * q^22 - 4 * q^23 - 24 * q^24 + 83 * q^25 - 5 * q^26 - 52 * q^27 + 61 * q^28 + 31 * q^29 + 2 * q^30 + 11 * q^31 + 71 * q^32 - 9 * q^33 + 17 * q^34 - 7 * q^35 + 61 * q^36 + 71 * q^37 - 8 * q^38 - 44 * q^39 + 20 * q^40 - 25 * q^41 - 3 * q^42 + 75 * q^43 + 14 * q^44 - 7 * q^45 + 36 * q^46 - 20 * q^47 - 95 * q^48 + 52 * q^49 + 26 * q^50 + 6 * q^51 + 88 * q^52 + 70 * q^53 - 3 * q^54 + 12 * q^55 + 24 * q^56 - 7 * q^57 + 48 * q^58 - 27 * q^59 + 21 * q^60 + 59 * q^61 - 23 * q^62 + 52 * q^63 + 138 * q^64 + 44 * q^65 - 19 * q^66 + 65 * q^67 - 8 * q^68 + 4 * q^69 - 2 * q^70 - 11 * q^71 + 24 * q^72 + 34 * q^73 + 38 * q^74 - 83 * q^75 + 31 * q^76 + 9 * q^77 + 5 * q^78 + 74 * q^79 - 5 * q^80 + 52 * q^81 + 51 * q^82 - 30 * q^83 - 61 * q^84 + 70 * q^85 + 29 * q^86 - 31 * q^87 + 90 * q^88 - q^89 - 2 * q^90 + 44 * q^91 + 34 * q^92 - 11 * q^93 + 27 * q^94 + 9 * q^95 - 71 * q^96 + 73 * q^97 + 3 * q^98 + 9 * 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 $

$2$	−0.837626	−0.592291	−0.296145	−	0.955143i	$-0.595701\pi$
$2$	−0.837626	−0.592291	−0.296145	+	0.955143i	$0.595701\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.29838	−0.649192
$5$	2.34855	1.05031	0.525153	−	0.851008i	$-0.324008\pi$
$5$	2.34855	1.05031	0.525153	+	0.851008i	$0.324008\pi$
$6$	0.837626	0.341959
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.76281	0.976801
$9$	1.00000	0.333333
$10$	−1.96721	−0.622086
$11$	2.64878	0.798639	0.399319	−	0.916812i	$-0.369246\pi$
$11$	2.64878	0.798639	0.399319	+	0.916812i	$0.369246\pi$
$12$	1.29838	0.374811
$13$	−4.81100	−1.33433	−0.667165	−	0.744910i	$-0.732492\pi$
$13$	−4.81100	−1.33433	−0.667165	+	0.744910i	$0.732492\pi$
$14$	−0.837626	−0.223865
$15$	−2.34855	−0.606394
$16$	0.282564	0.0706411
$17$	−7.16278	−1.73723	−0.868614	−	0.495489i	$-0.834989\pi$
$17$	−7.16278	−1.73723	−0.868614	+	0.495489i	$0.834989\pi$
$18$	−0.837626	−0.197430
$19$	4.98734	1.14417	0.572087	−	0.820193i	$-0.306134\pi$
$19$	4.98734	1.14417	0.572087	+	0.820193i	$0.306134\pi$
$20$	−3.04932	−0.681849
$21$	−1.00000	−0.218218
$22$	−2.21869	−0.473026
$23$	3.89405	0.811966	0.405983	−	0.913881i	$-0.366929\pi$
$23$	3.89405	0.811966	0.405983	+	0.913881i	$0.366929\pi$
$24$	−2.76281	−0.563956
$25$	0.515710	0.103142
$26$	4.02981	0.790312
$27$	−1.00000	−0.192450
$28$	−1.29838	−0.245371
$29$	−9.78351	−1.81675	−0.908376	−	0.418153i	$-0.862677\pi$
$29$	−9.78351	−1.81675	−0.908376	+	0.418153i	$0.862677\pi$
$30$	1.96721	0.359162
$31$	−7.90929	−1.42055	−0.710276	−	0.703924i	$-0.751430\pi$
$31$	−7.90929	−1.42055	−0.710276	+	0.703924i	$0.751430\pi$
$32$	−5.76230	−1.01864
$33$	−2.64878	−0.461094
$34$	5.99973	1.02894
$35$	2.34855	0.396978
$36$	−1.29838	−0.216397
$37$	3.14302	0.516709	0.258355	−	0.966050i	$-0.416820\pi$
$37$	3.14302	0.516709	0.258355	+	0.966050i	$0.416820\pi$
$38$	−4.17752	−0.677684
$39$	4.81100	0.770376
$40$	6.48861	1.02594
$41$	−4.49750	−0.702392	−0.351196	−	0.936302i	$-0.614225\pi$
$41$	−4.49750	−0.702392	−0.351196	+	0.936302i	$0.614225\pi$
$42$	0.837626	0.129248
$43$	3.34830	0.510611	0.255305	−	0.966860i	$-0.417824\pi$
$43$	3.34830	0.510611	0.255305	+	0.966860i	$0.417824\pi$
$44$	−3.43914	−0.518469
$45$	2.34855	0.350102
$46$	−3.26176	−0.480920
$47$	4.41618	0.644166	0.322083	−	0.946711i	$-0.395617\pi$
$47$	4.41618	0.644166	0.322083	+	0.946711i	$0.395617\pi$
$48$	−0.282564	−0.0407847
$49$	1.00000	0.142857
$50$	−0.431972	−0.0610900
$51$	7.16278	1.00299
$52$	6.24652	0.866236
$53$	7.48954	1.02877	0.514384	−	0.857560i	$-0.328021\pi$
$53$	7.48954	1.02877	0.514384	+	0.857560i	$0.328021\pi$
$54$	0.837626	0.113986
$55$	6.22082	0.838815
$56$	2.76281	0.369196
$57$	−4.98734	−0.660589
$58$	8.19492	1.07605
$59$	4.10404	0.534300	0.267150	−	0.963655i	$-0.413918\pi$
$59$	4.10404	0.534300	0.267150	+	0.963655i	$0.413918\pi$
$60$	3.04932	0.393666
$61$	−0.167344	−0.0214262	−0.0107131	−	0.999943i	$-0.503410\pi$
$61$	−0.167344	−0.0214262	−0.0107131	+	0.999943i	$0.503410\pi$
$62$	6.62503	0.841379
$63$	1.00000	0.125988
$64$	4.26153	0.532691
$65$	−11.2989	−1.40145
$66$	2.21869	0.273102
$67$	−13.3627	−1.63252	−0.816259	−	0.577686i	$-0.803956\pi$
$67$	−13.3627	−1.63252	−0.816259	+	0.577686i	$0.803956\pi$
$68$	9.30003	1.12779
$69$	−3.89405	−0.468789
$70$	−1.96721	−0.235127
$71$	15.3017	1.81597	0.907986	−	0.419000i	$-0.137619\pi$
$71$	15.3017	1.81597	0.907986	+	0.419000i	$0.137619\pi$
$72$	2.76281	0.325600
$73$	−8.69102	−1.01721	−0.508604	−	0.861001i	$-0.669838\pi$
$73$	−8.69102	−1.01721	−0.508604	+	0.861001i	$0.669838\pi$
$74$	−2.63268	−0.306042
$75$	−0.515710	−0.0595490
$76$	−6.47548	−0.742788
$77$	2.64878	0.301857
$78$	−4.02981	−0.456287
$79$	12.2587	1.37921	0.689606	−	0.724185i	$-0.257784\pi$
$79$	12.2587	1.37921	0.689606	+	0.724185i	$0.257784\pi$
$80$	0.663618	0.0741948
$81$	1.00000	0.111111
$82$	3.76723	0.416021
$83$	15.2779	1.67697	0.838485	−	0.544925i	$-0.183442\pi$
$83$	15.2779	1.67697	0.838485	+	0.544925i	$0.183442\pi$
$84$	1.29838	0.141665
$85$	−16.8222	−1.82462
$86$	−2.80462	−0.302430
$87$	9.78351	1.04890
$88$	7.31809	0.780111
$89$	8.16472	0.865459	0.432729	−	0.901524i	$-0.357551\pi$
$89$	8.16472	0.865459	0.432729	+	0.901524i	$0.357551\pi$
$90$	−1.96721	−0.207362
$91$	−4.81100	−0.504329
$92$	−5.05597	−0.527121
$93$	7.90929	0.820156
$94$	−3.69910	−0.381534
$95$	11.7130	1.20173
$96$	5.76230	0.588113
$97$	8.35604	0.848427	0.424213	−	0.905562i	$-0.360551\pi$
$97$	8.35604	0.848427	0.424213	+	0.905562i	$0.360551\pi$
$98$	−0.837626	−0.0846130
$99$	2.64878	0.266213
$100$	−0.669589	−0.0669589
$101$	−8.56358	−0.852108	−0.426054	−	0.904698i	$-0.640097\pi$
$101$	−8.56358	−0.852108	−0.426054	+	0.904698i	$0.640097\pi$
$102$	−5.99973	−0.594061
$103$	8.28638	0.816481	0.408241	−	0.912874i	$-0.366142\pi$
$103$	8.28638	0.816481	0.408241	+	0.912874i	$0.366142\pi$
$104$	−13.2919	−1.30338
$105$	−2.34855	−0.229195
$106$	−6.27343	−0.609330
$107$	13.1463	1.27090	0.635449	−	0.772143i	$-0.280815\pi$
$107$	13.1463	1.27090	0.635449	+	0.772143i	$0.280815\pi$
$108$	1.29838	0.124937
$109$	−4.71131	−0.451262	−0.225631	−	0.974213i	$-0.572444\pi$
$109$	−4.71131	−0.451262	−0.225631	+	0.974213i	$0.572444\pi$
$110$	−5.21072	−0.496822
$111$	−3.14302	−0.298322
$112$	0.282564	0.0266998
$113$	−17.2264	−1.62052	−0.810261	−	0.586070i	$-0.800674\pi$
$113$	−17.2264	−1.62052	−0.810261	+	0.586070i	$0.800674\pi$
$114$	4.17752	0.391261
$115$	9.14539	0.852812
$116$	12.7027	1.17942
$117$	−4.81100	−0.444777
$118$	−3.43765	−0.316461
$119$	−7.16278	−0.656611
$120$	−6.48861	−0.592327
$121$	−3.98394	−0.362176
$122$	0.140172	0.0126905
$123$	4.49750	0.405526
$124$	10.2693	0.922210
$125$	−10.5316	−0.941975
$126$	−0.837626	−0.0746216
$127$	4.63437	0.411233	0.205617	−	0.978633i	$-0.434080\pi$
$127$	4.63437	0.411233	0.205617	+	0.978633i	$0.434080\pi$
$128$	7.95505	0.703133
$129$	−3.34830	−0.294801
$130$	9.46424	0.830069
$131$	9.28952	0.811629	0.405815	−	0.913955i	$-0.366988\pi$
$131$	9.28952	0.811629	0.405815	+	0.913955i	$0.366988\pi$
$132$	3.43914	0.299338
$133$	4.98734	0.432457
$134$	11.1930	0.966926
$135$	−2.34855	−0.202131
$136$	−19.7894	−1.69693
$137$	9.31778	0.796072	0.398036	−	0.917370i	$-0.369692\pi$
$137$	9.31778	0.796072	0.398036	+	0.917370i	$0.369692\pi$
$138$	3.26176	0.277659
$139$	−14.0412	−1.19095	−0.595477	−	0.803372i	$-0.703037\pi$
$139$	−14.0412	−1.19095	−0.595477	+	0.803372i	$0.703037\pi$
$140$	−3.04932	−0.257715
$141$	−4.41618	−0.371909
$142$	−12.8171	−1.07558
$143$	−12.7433	−1.06565
$144$	0.282564	0.0235470
$145$	−22.9771	−1.90815
$146$	7.27982	0.602482
$147$	−1.00000	−0.0824786
$148$	−4.08084	−0.335443
$149$	10.7903	0.883979	0.441989	−	0.897020i	$-0.354273\pi$
$149$	10.7903	0.883979	0.441989	+	0.897020i	$0.354273\pi$
$150$	0.431972	0.0352703
$151$	16.9627	1.38041	0.690203	−	0.723616i	$-0.257521\pi$
$151$	16.9627	1.38041	0.690203	+	0.723616i	$0.257521\pi$
$152$	13.7791	1.11763
$153$	−7.16278	−0.579076
$154$	−2.21869	−0.178787
$155$	−18.5754	−1.49201
$156$	−6.24652	−0.500122
$157$	−12.1546	−0.970044	−0.485022	−	0.874502i	$-0.661188\pi$
$157$	−12.1546	−0.970044	−0.485022	+	0.874502i	$0.661188\pi$
$158$	−10.2682	−0.816895
$159$	−7.48954	−0.593959
$160$	−13.5331	−1.06988
$161$	3.89405	0.306894
$162$	−0.837626	−0.0658101
$163$	2.60894	0.204348	0.102174	−	0.994767i	$-0.467420\pi$
$163$	2.60894	0.204348	0.102174	+	0.994767i	$0.467420\pi$
$164$	5.83948	0.455987
$165$	−6.22082	−0.484290
$166$	−12.7972	−0.993254
$167$	4.36606	0.337856	0.168928	−	0.985628i	$-0.445969\pi$
$167$	4.36606	0.337856	0.168928	+	0.985628i	$0.445969\pi$
$168$	−2.76281	−0.213155
$169$	10.1457	0.780437
$170$	14.0907	1.08071
$171$	4.98734	0.381391
$172$	−4.34737	−0.331484
$173$	−10.7265	−0.815520	−0.407760	−	0.913089i	$-0.633690\pi$
$173$	−10.7265	−0.815520	−0.407760	+	0.913089i	$0.633690\pi$
$174$	−8.19492	−0.621256
$175$	0.515710	0.0389840
$176$	0.748452	0.0564167
$177$	−4.10404	−0.308478
$178$	−6.83898	−0.512603
$179$	15.2824	1.14226	0.571129	−	0.820860i	$-0.306506\pi$
$179$	15.2824	1.14226	0.571129	+	0.820860i	$0.306506\pi$
$180$	−3.04932	−0.227283
$181$	6.31187	0.469157	0.234579	−	0.972097i	$-0.424629\pi$
$181$	6.31187	0.469157	0.234579	+	0.972097i	$0.424629\pi$
$182$	4.02981	0.298710
$183$	0.167344	0.0123704
$184$	10.7585	0.793129
$185$	7.38156	0.542703
$186$	−6.62503	−0.485771
$187$	−18.9727	−1.38742
$188$	−5.73389	−0.418187
$189$	−1.00000	−0.0727393
$190$	−9.81114	−0.711775
$191$	−1.99011	−0.144000	−0.0719998	−	0.997405i	$-0.522938\pi$
$191$	−1.99011	−0.144000	−0.0719998	+	0.997405i	$0.522938\pi$
$192$	−4.26153	−0.307549
$193$	−1.32056	−0.0950559	−0.0475280	−	0.998870i	$-0.515134\pi$
$193$	−1.32056	−0.0950559	−0.0475280	+	0.998870i	$0.515134\pi$
$194$	−6.99923	−0.502516
$195$	11.2989	0.809130
$196$	−1.29838	−0.0927416
$197$	22.2380	1.58439	0.792196	−	0.610267i	$-0.208938\pi$
$197$	22.2380	1.58439	0.792196	+	0.610267i	$0.208938\pi$
$198$	−2.21869	−0.157675
$199$	−7.77181	−0.550929	−0.275464	−	0.961311i	$-0.588832\pi$
$199$	−7.77181	−0.550929	−0.275464	+	0.961311i	$0.588832\pi$
$200$	1.42481	0.100749
$201$	13.3627	0.942535
$202$	7.17308	0.504696
$203$	−9.78351	−0.686668
$204$	−9.30003	−0.651132
$205$	−10.5626	−0.737727
$206$	−6.94089	−0.483594
$207$	3.89405	0.270655
$208$	−1.35942	−0.0942586
$209$	13.2104	0.913782
$210$	1.96721	0.135750
$211$	17.7994	1.22536	0.612680	−	0.790331i	$-0.290092\pi$
$211$	17.7994	1.22536	0.612680	+	0.790331i	$0.290092\pi$
$212$	−9.72429	−0.667867
$213$	−15.3017	−1.04845
$214$	−11.0116	−0.752741
$215$	7.86366	0.536297
$216$	−2.76281	−0.187985
$217$	−7.90929	−0.536918
$218$	3.94632	0.267278
$219$	8.69102	0.587285
$220$	−8.07700	−0.544551
$221$	34.4601	2.31804
$222$	2.63268	0.176694
$223$	27.3971	1.83464	0.917322	−	0.398146i	$-0.130346\pi$
$223$	27.3971	1.83464	0.917322	+	0.398146i	$0.130346\pi$
$224$	−5.76230	−0.385010
$225$	0.515710	0.0343806
$226$	14.4293	0.959820
$227$	−17.0761	−1.13338	−0.566692	−	0.823930i	$-0.691777\pi$
$227$	−17.0761	−1.13338	−0.566692	+	0.823930i	$0.691777\pi$
$228$	6.47548	0.428849
$229$	11.6393	0.769146	0.384573	−	0.923095i	$-0.374349\pi$
$229$	11.6393	0.769146	0.384573	+	0.923095i	$0.374349\pi$
$230$	−7.66042	−0.505113
$231$	−2.64878	−0.174277
$232$	−27.0300	−1.77461
$233$	22.6650	1.48484	0.742418	−	0.669937i	$-0.233679\pi$
$233$	22.6650	1.48484	0.742418	+	0.669937i	$0.233679\pi$
$234$	4.02981	0.263437
$235$	10.3716	0.676571
$236$	−5.32861	−0.346863
$237$	−12.2587	−0.796288
$238$	5.99973	0.388904
$239$	−19.9399	−1.28981	−0.644903	−	0.764264i	$-0.723102\pi$
$239$	−19.9399	−1.28981	−0.644903	+	0.764264i	$0.723102\pi$
$240$	−0.663618	−0.0428364
$241$	−15.5798	−1.00358	−0.501791	−	0.864989i	$-0.667325\pi$
$241$	−15.5798	−1.00358	−0.501791	+	0.864989i	$0.667325\pi$
$242$	3.33705	0.214514
$243$	−1.00000	−0.0641500
$244$	0.217277	0.0139097
$245$	2.34855	0.150044
$246$	−3.76723	−0.240190
$247$	−23.9941	−1.52671
$248$	−21.8519	−1.38760
$249$	−15.2779	−0.968199
$250$	8.82154	0.557923
$251$	−18.9220	−1.19435	−0.597173	−	0.802112i	$-0.703709\pi$
$251$	−18.9220	−1.19435	−0.597173	+	0.802112i	$0.703709\pi$
$252$	−1.29838	−0.0817904
$253$	10.3145	0.648467
$254$	−3.88186	−0.243570
$255$	16.8222	1.05345
$256$	−15.1864	−0.949150
$257$	17.2789	1.07783	0.538915	−	0.842360i	$-0.318834\pi$
$257$	17.2789	1.07783	0.538915	+	0.842360i	$0.318834\pi$
$258$	2.80462	0.174608
$259$	3.14302	0.195298
$260$	14.6703	0.909812
$261$	−9.78351	−0.605584
$262$	−7.78114	−0.480721
$263$	−3.26425	−0.201282	−0.100641	−	0.994923i	$-0.532089\pi$
$263$	−3.26425	−0.201282	−0.100641	+	0.994923i	$0.532089\pi$
$264$	−7.31809	−0.450397
$265$	17.5896	1.08052
$266$	−4.17752	−0.256140
$267$	−8.16472	−0.499673
$268$	17.3500	1.05982
$269$	−2.18341	−0.133125	−0.0665623	−	0.997782i	$-0.521203\pi$
$269$	−2.18341	−0.133125	−0.0665623	+	0.997782i	$0.521203\pi$
$270$	1.96721	0.119721
$271$	−19.2778	−1.17104	−0.585522	−	0.810656i	$-0.699111\pi$
$271$	−19.2778	−1.17104	−0.585522	+	0.810656i	$0.699111\pi$
$272$	−2.02395	−0.122720
$273$	4.81100	0.291175
$274$	−7.80481	−0.471506
$275$	1.36600	0.0823731
$276$	5.05597	0.304334
$277$	6.06900	0.364651	0.182325	−	0.983238i	$-0.441638\pi$
$277$	6.06900	0.364651	0.182325	+	0.983238i	$0.441638\pi$
$278$	11.7612	0.705392
$279$	−7.90929	−0.473517
$280$	6.48861	0.387769
$281$	5.35034	0.319174	0.159587	−	0.987184i	$-0.448984\pi$
$281$	5.35034	0.319174	0.159587	+	0.987184i	$0.448984\pi$
$282$	3.69910	0.220278
$283$	30.4642	1.81091	0.905456	−	0.424441i	$-0.139529\pi$
$283$	30.4642	1.81091	0.905456	+	0.424441i	$0.139529\pi$
$284$	−19.8674	−1.17891
$285$	−11.7130	−0.693821
$286$	10.6741	0.631173
$287$	−4.49750	−0.265479
$288$	−5.76230	−0.339547
$289$	34.3054	2.01796
$290$	19.2462	1.13018
$291$	−8.35604	−0.489840
$292$	11.2843	0.660362
$293$	15.3100	0.894422	0.447211	−	0.894429i	$-0.352417\pi$
$293$	15.3100	0.894422	0.447211	+	0.894429i	$0.352417\pi$
$294$	0.837626	0.0488513
$295$	9.63855	0.561178
$296$	8.68357	0.504722
$297$	−2.64878	−0.153698
$298$	−9.03826	−0.523573
$299$	−18.7343	−1.08343
$300$	0.669589	0.0386587
$301$	3.34830	0.192993
$302$	−14.2084	−0.817602
$303$	8.56358	0.491965
$304$	1.40924	0.0808257
$305$	−0.393017	−0.0225041
$306$	5.99973	0.342982
$307$	11.6405	0.664360	0.332180	−	0.943216i	$-0.392216\pi$
$307$	11.6405	0.664360	0.332180	+	0.943216i	$0.392216\pi$
$308$	−3.43914	−0.195963
$309$	−8.28638	−0.471396
$310$	15.5592	0.883706
$311$	−34.2209	−1.94049	−0.970244	−	0.242129i	$-0.922154\pi$
$311$	−34.2209	−1.94049	−0.970244	+	0.242129i	$0.922154\pi$
$312$	13.2919	0.752504
$313$	−15.1728	−0.857616	−0.428808	−	0.903396i	$-0.641066\pi$
$313$	−15.1728	−0.857616	−0.428808	+	0.903396i	$0.641066\pi$
$314$	10.1810	0.574548
$315$	2.34855	0.132326
$316$	−15.9165	−0.895373
$317$	−2.64579	−0.148602	−0.0743011	−	0.997236i	$-0.523673\pi$
$317$	−2.64579	−0.148602	−0.0743011	+	0.997236i	$0.523673\pi$
$318$	6.27343	0.351797
$319$	−25.9144	−1.45093
$320$	10.0084	0.559488
$321$	−13.1463	−0.733753
$322$	−3.26176	−0.181771
$323$	−35.7232	−1.98769
$324$	−1.29838	−0.0721324
$325$	−2.48108	−0.137625
$326$	−2.18532	−0.121033
$327$	4.71131	0.260536
$328$	−12.4258	−0.686098
$329$	4.41618	0.243472
$330$	5.21072	0.286840
$331$	11.4480	0.629237	0.314618	−	0.949218i	$-0.398123\pi$
$331$	11.4480	0.629237	0.314618	+	0.949218i	$0.398123\pi$
$332$	−19.8366	−1.08867
$333$	3.14302	0.172236
$334$	−3.65712	−0.200109
$335$	−31.3831	−1.71464
$336$	−0.282564	−0.0154152
$337$	3.70485	0.201816	0.100908	−	0.994896i	$-0.467825\pi$
$337$	3.70485	0.201816	0.100908	+	0.994896i	$0.467825\pi$
$338$	−8.49829	−0.462246
$339$	17.2264	0.935608
$340$	21.8416	1.18453
$341$	−20.9500	−1.13451
$342$	−4.17752	−0.225895
$343$	1.00000	0.0539949
$344$	9.25071	0.498765
$345$	−9.14539	−0.492371
$346$	8.98479	0.483025
$347$	−6.95866	−0.373560	−0.186780	−	0.982402i	$-0.559805\pi$
$347$	−6.95866	−0.373560	−0.186780	+	0.982402i	$0.559805\pi$
$348$	−12.7027	−0.680939
$349$	−4.96087	−0.265549	−0.132775	−	0.991146i	$-0.542389\pi$
$349$	−4.96087	−0.265549	−0.132775	+	0.991146i	$0.542389\pi$
$350$	−0.431972	−0.0230899
$351$	4.81100	0.256792
$352$	−15.2631	−0.813526
$353$	2.33459	0.124258	0.0621289	−	0.998068i	$-0.480211\pi$
$353$	2.33459	0.124258	0.0621289	+	0.998068i	$0.480211\pi$
$354$	3.43765	0.182709
$355$	35.9368	1.90733
$356$	−10.6009	−0.561848
$357$	7.16278	0.379094
$358$	−12.8009	−0.676549
$359$	−23.8904	−1.26089	−0.630444	−	0.776235i	$-0.717127\pi$
$359$	−23.8904	−1.26089	−0.630444	+	0.776235i	$0.717127\pi$
$360$	6.48861	0.341980
$361$	5.87355	0.309134
$362$	−5.28698	−0.277878
$363$	3.98394	0.209103
$364$	6.24652	0.327406
$365$	−20.4113	−1.06838
$366$	−0.140172	−0.00732689
$367$	7.68336	0.401068	0.200534	−	0.979687i	$-0.435732\pi$
$367$	7.68336	0.401068	0.200534	+	0.979687i	$0.435732\pi$
$368$	1.10032	0.0573582
$369$	−4.49750	−0.234131
$370$	−6.18298	−0.321438
$371$	7.48954	0.388838
$372$	−10.2693	−0.532438
$373$	−4.55697	−0.235951	−0.117975	−	0.993017i	$-0.537640\pi$
$373$	−4.55697	−0.235951	−0.117975	+	0.993017i	$0.537640\pi$
$374$	15.8920	0.821755
$375$	10.5316	0.543850
$376$	12.2011	0.629222
$377$	47.0684	2.42415
$378$	0.837626	0.0430828
$379$	15.2378	0.782714	0.391357	−	0.920239i	$-0.372006\pi$
$379$	15.2378	0.782714	0.391357	+	0.920239i	$0.372006\pi$
$380$	−15.2080	−0.780154
$381$	−4.63437	−0.237426
$382$	1.66697	0.0852897
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	−7.95505	−0.405954
$385$	6.22082	0.317042
$386$	1.10613	0.0563008
$387$	3.34830	0.170204
$388$	−10.8493	−0.550792
$389$	−0.845771	−0.0428823	−0.0214412	−	0.999770i	$-0.506825\pi$
$389$	−0.845771	−0.0428823	−0.0214412	+	0.999770i	$0.506825\pi$
$390$	−9.46424	−0.479240
$391$	−27.8922	−1.41057
$392$	2.76281	0.139543
$393$	−9.28952	−0.468594
$394$	−18.6271	−0.938421
$395$	28.7902	1.44859
$396$	−3.43914	−0.172823
$397$	−5.44659	−0.273357	−0.136678	−	0.990615i	$-0.543643\pi$
$397$	−5.44659	−0.273357	−0.136678	+	0.990615i	$0.543643\pi$
$398$	6.50987	0.326310
$399$	−4.98734	−0.249679
$400$	0.145721	0.00728606
$401$	−37.3410	−1.86472	−0.932360	−	0.361532i	$-0.882254\pi$
$401$	−37.3410	−1.86472	−0.932360	+	0.361532i	$0.882254\pi$
$402$	−11.1930	−0.558255
$403$	38.0516	1.89548
$404$	11.1188	0.553182
$405$	2.34855	0.116701
$406$	8.19492	0.406707
$407$	8.32519	0.412664
$408$	19.7894	0.979721
$409$	17.0992	0.845500	0.422750	−	0.906246i	$-0.361065\pi$
$409$	17.0992	0.845500	0.422750	+	0.906246i	$0.361065\pi$
$410$	8.84754	0.436949
$411$	−9.31778	−0.459612
$412$	−10.7589	−0.530053
$413$	4.10404	0.201946
$414$	−3.26176	−0.160307
$415$	35.8810	1.76133
$416$	27.7224	1.35920
$417$	14.0412	0.687598
$418$	−11.0654	−0.541225
$419$	6.48552	0.316838	0.158419	−	0.987372i	$-0.449360\pi$
$419$	6.48552	0.316838	0.158419	+	0.987372i	$0.449360\pi$
$420$	3.04932	0.148792
$421$	2.53483	0.123540	0.0617700	−	0.998090i	$-0.480325\pi$
$421$	2.53483	0.123540	0.0617700	+	0.998090i	$0.480325\pi$
$422$	−14.9092	−0.725769
$423$	4.41618	0.214722
$424$	20.6922	1.00490
$425$	−3.69391	−0.179181
$426$	12.8171	0.620989
$427$	−0.167344	−0.00809834
$428$	−17.0689	−0.825056
$429$	12.7433	0.615252
$430$	−6.58681	−0.317644
$431$	−26.6193	−1.28220	−0.641102	−	0.767455i	$-0.721523\pi$
$431$	−26.6193	−1.28220	−0.641102	+	0.767455i	$0.721523\pi$
$432$	−0.282564	−0.0135949
$433$	21.2305	1.02027	0.510135	−	0.860094i	$-0.329595\pi$
$433$	21.2305	1.02027	0.510135	+	0.860094i	$0.329595\pi$
$434$	6.62503	0.318012
$435$	22.9771	1.10167
$436$	6.11709	0.292955
$437$	19.4209	0.929030
$438$	−7.27982	−0.347843
$439$	−4.56215	−0.217740	−0.108870	−	0.994056i	$-0.534723\pi$
$439$	−4.56215	−0.217740	−0.108870	+	0.994056i	$0.534723\pi$
$440$	17.1869	0.819355
$441$	1.00000	0.0476190
$442$	−28.8647	−1.37295
$443$	11.6793	0.554899	0.277450	−	0.960740i	$-0.410511\pi$
$443$	11.6793	0.554899	0.277450	+	0.960740i	$0.410511\pi$
$444$	4.08084	0.193668
$445$	19.1753	0.908996
$446$	−22.9485	−1.08664
$447$	−10.7903	−0.510365
$448$	4.26153	0.201338
$449$	7.88702	0.372212	0.186106	−	0.982530i	$-0.440413\pi$
$449$	7.88702	0.372212	0.186106	+	0.982530i	$0.440413\pi$
$450$	−0.431972	−0.0203633
$451$	−11.9129	−0.560958
$452$	22.3664	1.05203
$453$	−16.9627	−0.796978
$454$	14.3034	0.671293
$455$	−11.2989	−0.529700
$456$	−13.7791	−0.645264
$457$	27.9515	1.30752	0.653759	−	0.756703i	$-0.273191\pi$
$457$	27.9515	1.30752	0.653759	+	0.756703i	$0.273191\pi$
$458$	−9.74937	−0.455558
$459$	7.16278	0.334330
$460$	−11.8742	−0.553638
$461$	38.3961	1.78828	0.894142	−	0.447784i	$-0.147787\pi$
$461$	38.3961	1.78828	0.894142	+	0.447784i	$0.147787\pi$
$462$	2.21869	0.103223
$463$	−1.00827	−0.0468585	−0.0234292	−	0.999725i	$-0.507458\pi$
$463$	−1.00827	−0.0468585	−0.0234292	+	0.999725i	$0.507458\pi$
$464$	−2.76447	−0.128337
$465$	18.5754	0.861414
$466$	−18.9848	−0.879455
$467$	−6.73587	−0.311699	−0.155849	−	0.987781i	$-0.549811\pi$
$467$	−6.73587	−0.311699	−0.155849	+	0.987781i	$0.549811\pi$
$468$	6.24652	0.288745
$469$	−13.3627	−0.617034
$470$	−8.68755	−0.400727
$471$	12.1546	0.560055
$472$	11.3387	0.521905
$473$	8.86892	0.407793
$474$	10.2682	0.471634
$475$	2.57202	0.118012
$476$	9.30003	0.426266
$477$	7.48954	0.342923
$478$	16.7022	0.763940
$479$	29.9094	1.36660	0.683298	−	0.730139i	$-0.260545\pi$
$479$	29.9094	1.36660	0.683298	+	0.730139i	$0.260545\pi$
$480$	13.5331	0.617698
$481$	−15.1211	−0.689461
$482$	13.0500	0.594412
$483$	−3.89405	−0.177185
$484$	5.17268	0.235122
$485$	19.6246	0.891108
$486$	0.837626	0.0379955
$487$	33.0486	1.49758	0.748788	−	0.662809i	$-0.230636\pi$
$487$	33.0486	1.49758	0.748788	+	0.662809i	$0.230636\pi$
$488$	−0.462340	−0.0209291
$489$	−2.60894	−0.117980
$490$	−1.96721	−0.0888695
$491$	6.79917	0.306842	0.153421	−	0.988161i	$-0.450971\pi$
$491$	6.79917	0.306842	0.153421	+	0.988161i	$0.450971\pi$
$492$	−5.83948	−0.263264
$493$	70.0771	3.15611
$494$	20.0981	0.904254
$495$	6.22082	0.279605
$496$	−2.23489	−0.100349
$497$	15.3017	0.686373
$498$	12.7972	0.573455
$499$	4.92460	0.220455	0.110228	−	0.993906i	$-0.464842\pi$
$499$	4.92460	0.220455	0.110228	+	0.993906i	$0.464842\pi$
$500$	13.6741	0.611522
$501$	−4.36606	−0.195061
$502$	15.8496	0.707400
$503$	16.5025	0.735812	0.367906	−	0.929863i	$-0.380075\pi$
$503$	16.5025	0.735812	0.367906	+	0.929863i	$0.380075\pi$
$504$	2.76281	0.123065
$505$	−20.1120	−0.894974
$506$	−8.63969	−0.384081
$507$	−10.1457	−0.450586
$508$	−6.01718	−0.266969
$509$	−24.4469	−1.08359	−0.541794	−	0.840511i	$-0.682255\pi$
$509$	−24.4469	−1.08359	−0.541794	+	0.840511i	$0.682255\pi$
$510$	−14.0907	−0.623946
$511$	−8.69102	−0.384468
$512$	−3.18957	−0.140960
$513$	−4.98734	−0.220196
$514$	−14.4733	−0.638389
$515$	19.4610	0.857555
$516$	4.34737	0.191382
$517$	11.6975	0.514456
$518$	−2.63268	−0.115673
$519$	10.7265	0.470841
$520$	−31.2167	−1.36894
$521$	34.1206	1.49485	0.747426	−	0.664345i	$-0.231289\pi$
$521$	34.1206	1.49485	0.747426	+	0.664345i	$0.231289\pi$
$522$	8.19492	0.358682
$523$	3.62918	0.158693	0.0793464	−	0.996847i	$-0.474717\pi$
$523$	3.62918	0.158693	0.0793464	+	0.996847i	$0.474717\pi$
$524$	−12.0614	−0.526903
$525$	−0.515710	−0.0225074
$526$	2.73422	0.119218
$527$	56.6525	2.46782
$528$	−0.748452	−0.0325722
$529$	−7.83637	−0.340712
$530$	−14.7335	−0.639982
$531$	4.10404	0.178100
$532$	−6.47548	−0.280747
$533$	21.6375	0.937223
$534$	6.83898	0.295952
$535$	30.8747	1.33483
$536$	−36.9187	−1.59465
$537$	−15.2824	−0.659483
$538$	1.82888	0.0788485
$539$	2.64878	0.114091
$540$	3.04932	0.131222
$541$	−8.60482	−0.369950	−0.184975	−	0.982743i	$-0.559220\pi$
$541$	−8.60482	−0.369950	−0.184975	+	0.982743i	$0.559220\pi$
$542$	16.1476	0.693599
$543$	−6.31187	−0.270868
$544$	41.2741	1.76961
$545$	−11.0648	−0.473963
$546$	−4.02981	−0.172460
$547$	−26.6304	−1.13864	−0.569318	−	0.822118i	$-0.692793\pi$
$547$	−26.6304	−1.13864	−0.569318	+	0.822118i	$0.692793\pi$
$548$	−12.0980	−0.516803
$549$	−0.167344	−0.00714207
$550$	−1.14420	−0.0487889
$551$	−48.7937	−2.07868
$552$	−10.7585	−0.457913
$553$	12.2587	0.521293
$554$	−5.08355	−0.215979
$555$	−7.38156	−0.313330
$556$	18.2308	0.773158
$557$	−37.8231	−1.60262	−0.801308	−	0.598252i	$-0.795862\pi$
$557$	−37.8231	−1.60262	−0.801308	+	0.598252i	$0.795862\pi$
$558$	6.62503	0.280460
$559$	−16.1087	−0.681323
$560$	0.663618	0.0280430
$561$	18.9727	0.801026
$562$	−4.48158	−0.189044
$563$	−3.49391	−0.147251	−0.0736254	−	0.997286i	$-0.523457\pi$
$563$	−3.49391	−0.147251	−0.0736254	+	0.997286i	$0.523457\pi$
$564$	5.73389	0.241440
$565$	−40.4571	−1.70204
$566$	−25.5176	−1.07259
$567$	1.00000	0.0419961
$568$	42.2756	1.77384
$569$	9.86383	0.413513	0.206757	−	0.978392i	$-0.433709\pi$
$569$	9.86383	0.413513	0.206757	+	0.978392i	$0.433709\pi$
$570$	9.81114	0.410944
$571$	45.2735	1.89464	0.947318	−	0.320293i	$-0.103781\pi$
$571$	45.2735	1.89464	0.947318	+	0.320293i	$0.103781\pi$
$572$	16.5457	0.691809
$573$	1.99011	0.0831382
$574$	3.76723	0.157241
$575$	2.00820	0.0837477
$576$	4.26153	0.177564
$577$	6.72576	0.279997	0.139998	−	0.990152i	$-0.455290\pi$
$577$	6.72576	0.279997	0.139998	+	0.990152i	$0.455290\pi$
$578$	−28.7351	−1.19522
$579$	1.32056	0.0548806
$580$	29.8331	1.23875
$581$	15.2779	0.633835
$582$	6.99923	0.290128
$583$	19.8382	0.821614
$584$	−24.0116	−0.993609
$585$	−11.2989	−0.467152
$586$	−12.8241	−0.529758
$587$	−11.9195	−0.491972	−0.245986	−	0.969273i	$-0.579112\pi$
$587$	−11.9195	−0.491972	−0.245986	+	0.969273i	$0.579112\pi$
$588$	1.29838	0.0535444
$589$	−39.4463	−1.62536
$590$	−8.07350	−0.332381
$591$	−22.2380	−0.914749
$592$	0.888106	0.0365009
$593$	−8.73816	−0.358833	−0.179417	−	0.983773i	$-0.557421\pi$
$593$	−8.73816	−0.358833	−0.179417	+	0.983773i	$0.557421\pi$
$594$	2.21869	0.0910340
$595$	−16.8222	−0.689642
$596$	−14.0100	−0.573872
$597$	7.77181	0.318079
$598$	15.6923	0.641706
$599$	30.6671	1.25302	0.626512	−	0.779412i	$-0.284482\pi$
$599$	30.6671	1.25302	0.626512	+	0.779412i	$0.284482\pi$
$600$	−1.42481	−0.0581676
$601$	21.9578	0.895677	0.447838	−	0.894115i	$-0.352194\pi$
$601$	21.9578	0.895677	0.447838	+	0.894115i	$0.352194\pi$
$602$	−2.80462	−0.114308
$603$	−13.3627	−0.544173
$604$	−22.0241	−0.896148
$605$	−9.35650	−0.380396
$606$	−7.17308	−0.291386
$607$	45.8303	1.86019	0.930097	−	0.367313i	$-0.119722\pi$
$607$	45.8303	1.86019	0.930097	+	0.367313i	$0.119722\pi$
$608$	−28.7386	−1.16550
$609$	9.78351	0.396448
$610$	0.329201	0.0133290
$611$	−21.2462	−0.859530
$612$	9.30003	0.375931
$613$	38.0357	1.53625	0.768124	−	0.640302i	$-0.221191\pi$
$613$	38.0357	1.53625	0.768124	+	0.640302i	$0.221191\pi$
$614$	−9.75040	−0.393494
$615$	10.5626	0.425927
$616$	7.31809	0.294854
$617$	37.9265	1.52686	0.763431	−	0.645889i	$-0.223513\pi$
$617$	37.9265	1.52686	0.763431	+	0.645889i	$0.223513\pi$
$618$	6.94089	0.279203
$619$	15.4194	0.619759	0.309880	−	0.950776i	$-0.399711\pi$
$619$	15.4194	0.619759	0.309880	+	0.950776i	$0.399711\pi$
$620$	24.1180	0.968602
$621$	−3.89405	−0.156263
$622$	28.6643	1.14933
$623$	8.16472	0.327113
$624$	1.35942	0.0544202
$625$	−27.3126	−1.09250
$626$	12.7091	0.507958
$627$	−13.2104	−0.527572
$628$	15.7814	0.629745
$629$	−22.5128	−0.897642
$630$	−1.96721	−0.0783755
$631$	37.8063	1.50504	0.752522	−	0.658567i	$-0.228837\pi$
$631$	37.8063	1.50504	0.752522	+	0.658567i	$0.228837\pi$
$632$	33.8685	1.34722
$633$	−17.7994	−0.707462
$634$	2.21618	0.0880157
$635$	10.8841	0.431921
$636$	9.72429	0.385593
$637$	−4.81100	−0.190619
$638$	21.7066	0.859372
$639$	15.3017	0.605324
$640$	18.6829	0.738505
$641$	19.6878	0.777622	0.388811	−	0.921317i	$-0.372886\pi$
$641$	19.6878	0.777622	0.388811	+	0.921317i	$0.372886\pi$
$642$	11.0116	0.434595
$643$	35.8761	1.41482	0.707408	−	0.706806i	$-0.249864\pi$
$643$	35.8761	1.41482	0.707408	+	0.706806i	$0.249864\pi$
$644$	−5.05597	−0.199233
$645$	−7.86366	−0.309631
$646$	29.9227	1.17729
$647$	34.8738	1.37103	0.685515	−	0.728058i	$-0.259577\pi$
$647$	34.8738	1.37103	0.685515	+	0.728058i	$0.259577\pi$
$648$	2.76281	0.108533
$649$	10.8707	0.426713
$650$	2.07821	0.0815143
$651$	7.90929	0.309990
$652$	−3.38740	−0.132661
$653$	18.8728	0.738549	0.369275	−	0.929320i	$-0.379606\pi$
$653$	18.8728	0.738549	0.369275	+	0.929320i	$0.379606\pi$
$654$	−3.94632	−0.154313
$655$	21.8170	0.852459
$656$	−1.27083	−0.0496178
$657$	−8.69102	−0.339069
$658$	−3.69910	−0.144206
$659$	7.13241	0.277839	0.138920	−	0.990304i	$-0.455637\pi$
$659$	7.13241	0.277839	0.138920	+	0.990304i	$0.455637\pi$
$660$	8.07700	0.314397
$661$	−20.9940	−0.816571	−0.408285	−	0.912854i	$-0.633873\pi$
$661$	−20.9940	−0.816571	−0.408285	+	0.912854i	$0.633873\pi$
$662$	−9.58910	−0.372691
$663$	−34.4601	−1.33832
$664$	42.2100	1.63807
$665$	11.7130	0.454212
$666$	−2.63268	−0.102014
$667$	−38.0975	−1.47514
$668$	−5.66882	−0.219333
$669$	−27.3971	−1.05923
$670$	26.2873	1.01557
$671$	−0.443258	−0.0171118
$672$	5.76230	0.222286
$673$	−17.0045	−0.655476	−0.327738	−	0.944769i	$-0.606286\pi$
$673$	−17.0045	−0.655476	−0.327738	+	0.944769i	$0.606286\pi$
$674$	−3.10328	−0.119534
$675$	−0.515710	−0.0198497
$676$	−13.1730	−0.506653
$677$	35.1770	1.35196	0.675982	−	0.736918i	$-0.263720\pi$
$677$	35.1770	1.35196	0.675982	+	0.736918i	$0.263720\pi$
$678$	−14.4293	−0.554152
$679$	8.35604	0.320675
$680$	−46.4765	−1.78229
$681$	17.0761	0.654359
$682$	17.5483	0.671958
$683$	−15.8819	−0.607703	−0.303852	−	0.952719i	$-0.598273\pi$
$683$	−15.8819	−0.607703	−0.303852	+	0.952719i	$0.598273\pi$
$684$	−6.47548	−0.247596
$685$	21.8833	0.836119
$686$	−0.837626	−0.0319807
$687$	−11.6393	−0.444067
$688$	0.946110	0.0360701
$689$	−36.0322	−1.37272
$690$	7.66042	0.291627
$691$	−27.3079	−1.03884	−0.519421	−	0.854519i	$-0.673852\pi$
$691$	−27.3079	−1.03884	−0.519421	+	0.854519i	$0.673852\pi$
$692$	13.9271	0.529429
$693$	2.64878	0.100619
$694$	5.82875	0.221256
$695$	−32.9764	−1.25087
$696$	27.0300	1.02457
$697$	32.2146	1.22022
$698$	4.15535	0.157282
$699$	−22.6650	−0.857271
$700$	−0.669589	−0.0253081
$701$	−9.45633	−0.357161	−0.178580	−	0.983925i	$-0.557150\pi$
$701$	−9.45633	−0.357161	−0.178580	+	0.983925i	$0.557150\pi$
$702$	−4.02981	−0.152096
$703$	15.6753	0.591206
$704$	11.2879	0.425427
$705$	−10.3716	−0.390618
$706$	−1.95552	−0.0735968
$707$	−8.56358	−0.322067
$708$	5.32861	0.200261
$709$	12.6737	0.475970	0.237985	−	0.971269i	$-0.423513\pi$
$709$	12.6737	0.475970	0.237985	+	0.971269i	$0.423513\pi$
$710$	−30.1016	−1.12969
$711$	12.2587	0.459737
$712$	22.5576	0.845381
$713$	−30.7992	−1.15344
$714$	−5.99973	−0.224534
$715$	−29.9283	−1.11926
$716$	−19.8424	−0.741544
$717$	19.9399	0.744670
$718$	20.0112	0.746812
$719$	5.48988	0.204738	0.102369	−	0.994746i	$-0.467358\pi$
$719$	5.48988	0.204738	0.102369	+	0.994746i	$0.467358\pi$
$720$	0.663618	0.0247316
$721$	8.28638	0.308601
$722$	−4.91984	−0.183097
$723$	15.5798	0.579418
$724$	−8.19522	−0.304573
$725$	−5.04545	−0.187383
$726$	−3.33705	−0.123850
$727$	−1.41600	−0.0525167	−0.0262583	−	0.999655i	$-0.508359\pi$
$727$	−1.41600	−0.0525167	−0.0262583	+	0.999655i	$0.508359\pi$
$728$	−13.2919	−0.492630
$729$	1.00000	0.0370370
$730$	17.0971	0.632791
$731$	−23.9831	−0.887047
$732$	−0.217277	−0.00803077
$733$	−13.2592	−0.489739	−0.244869	−	0.969556i	$-0.578745\pi$
$733$	−13.2592	−0.489739	−0.244869	+	0.969556i	$0.578745\pi$
$734$	−6.43578	−0.237549
$735$	−2.34855	−0.0866277
$736$	−22.4387	−0.827102
$737$	−35.3950	−1.30379
$738$	3.76723	0.138674
$739$	−8.96306	−0.329711	−0.164856	−	0.986318i	$-0.552716\pi$
$739$	−8.96306	−0.329711	−0.164856	+	0.986318i	$0.552716\pi$
$740$	−9.58409	−0.352318
$741$	23.9941	0.881444
$742$	−6.27343	−0.230305
$743$	−7.29561	−0.267650	−0.133825	−	0.991005i	$-0.542726\pi$
$743$	−7.29561	−0.267650	−0.133825	+	0.991005i	$0.542726\pi$
$744$	21.8519	0.801129
$745$	25.3417	0.928448
$746$	3.81703	0.139751
$747$	15.2779	0.558990
$748$	24.6338	0.900700
$749$	13.1463	0.480354
$750$	−8.82154	−0.322117
$751$	8.92028	0.325506	0.162753	−	0.986667i	$-0.447963\pi$
$751$	8.92028	0.325506	0.162753	+	0.986667i	$0.447963\pi$
$752$	1.24785	0.0455046
$753$	18.9220	0.689556
$754$	−39.4257	−1.43580
$755$	39.8379	1.44985
$756$	1.29838	0.0472217
$757$	8.89494	0.323292	0.161646	−	0.986849i	$-0.448320\pi$
$757$	8.89494	0.323292	0.161646	+	0.986849i	$0.448320\pi$
$758$	−12.7636	−0.463594
$759$	−10.3145	−0.374393
$760$	32.3609	1.17385
$761$	−37.4227	−1.35657	−0.678286	−	0.734798i	$-0.737277\pi$
$761$	−37.4227	−1.35657	−0.678286	+	0.734798i	$0.737277\pi$
$762$	3.88186	0.140625
$763$	−4.71131	−0.170561
$764$	2.58393	0.0934834
$765$	−16.8222	−0.608207
$766$	0.837626	0.0302647
$767$	−19.7445	−0.712933
$768$	15.1864	0.547992
$769$	−11.5675	−0.417133	−0.208566	−	0.978008i	$-0.566880\pi$
$769$	−11.5675	−0.417133	−0.208566	+	0.978008i	$0.566880\pi$
$770$	−5.21072	−0.187781
$771$	−17.2789	−0.622286
$772$	1.71459	0.0617095
$773$	−30.3031	−1.08993	−0.544963	−	0.838460i	$-0.683456\pi$
$773$	−30.3031	−1.08993	−0.544963	+	0.838460i	$0.683456\pi$
$774$	−2.80462	−0.100810
$775$	−4.07890	−0.146518
$776$	23.0861	0.828744
$777$	−3.14302	−0.112755
$778$	0.708440	0.0253988
$779$	−22.4306	−0.803659
$780$	−14.6703	−0.525280
$781$	40.5308	1.45031
$782$	23.3632	0.835468
$783$	9.78351	0.349634
$784$	0.282564	0.0100916
$785$	−28.5458	−1.01884
$786$	7.78114	0.277544
$787$	−33.5523	−1.19601	−0.598006	−	0.801492i	$-0.704040\pi$
$787$	−33.5523	−1.19601	−0.598006	+	0.801492i	$0.704040\pi$
$788$	−28.8734	−1.02857
$789$	3.26425	0.116210
$790$	−24.1154	−0.857989
$791$	−17.2264	−0.612499
$792$	7.31809	0.260037
$793$	0.805091	0.0285896
$794$	4.56220	0.161907
$795$	−17.5896	−0.623839
$796$	10.0908	0.357658
$797$	34.7862	1.23219	0.616096	−	0.787671i	$-0.288713\pi$
$797$	34.7862	1.23219	0.616096	+	0.787671i	$0.288713\pi$
$798$	4.17752	0.147883
$799$	−31.6321	−1.11906
$800$	−2.97168	−0.105065
$801$	8.16472	0.288486
$802$	31.2778	1.10446
$803$	−23.0206	−0.812381
$804$	−17.3500	−0.611886
$805$	9.14539	0.322333
$806$	−31.8730	−1.12268
$807$	2.18341	0.0768595
$808$	−23.6596	−0.832340
$809$	15.7501	0.553746	0.276873	−	0.960907i	$-0.410702\pi$
$809$	15.7501	0.553746	0.276873	+	0.960907i	$0.410702\pi$
$810$	−1.96721	−0.0691207
$811$	−37.8786	−1.33010	−0.665049	−	0.746800i	$-0.731589\pi$
$811$	−37.8786	−1.33010	−0.665049	+	0.746800i	$0.731589\pi$
$812$	12.7027	0.445779
$813$	19.2778	0.676103
$814$	−6.97339	−0.244417
$815$	6.12724	0.214628
$816$	2.02395	0.0708523
$817$	16.6991	0.584227
$818$	−14.3227	−0.500782
$819$	−4.81100	−0.168110
$820$	13.7143	0.478926
$821$	−43.1350	−1.50542	−0.752711	−	0.658351i	$-0.771254\pi$
$821$	−43.1350	−1.50542	−0.752711	+	0.658351i	$0.771254\pi$
$822$	7.80481	0.272224
$823$	−45.1329	−1.57324	−0.786618	−	0.617441i	$-0.788170\pi$
$823$	−45.1329	−1.57324	−0.786618	+	0.617441i	$0.788170\pi$
$824$	22.8937	0.797540
$825$	−1.36600	−0.0475582
$826$	−3.43765	−0.119611
$827$	15.3556	0.533967	0.266984	−	0.963701i	$-0.413973\pi$
$827$	15.3556	0.533967	0.266984	+	0.963701i	$0.413973\pi$
$828$	−5.05597	−0.175707
$829$	−12.0825	−0.419643	−0.209822	−	0.977740i	$-0.567288\pi$
$829$	−12.0825	−0.419643	−0.209822	+	0.977740i	$0.567288\pi$
$830$	−30.0549	−1.04322
$831$	−6.06900	−0.210531
$832$	−20.5022	−0.710785
$833$	−7.16278	−0.248175
$834$	−11.7612	−0.407258
$835$	10.2539	0.354852
$836$	−17.1521	−0.593219
$837$	7.90929	0.273385
$838$	−5.43244	−0.187660
$839$	26.0001	0.897623	0.448812	−	0.893626i	$-0.351847\pi$
$839$	26.0001	0.897623	0.448812	+	0.893626i	$0.351847\pi$
$840$	−6.48861	−0.223878
$841$	66.7171	2.30059
$842$	−2.12324	−0.0731717
$843$	−5.35034	−0.184275
$844$	−23.1104	−0.795493
$845$	23.8277	0.819698
$846$	−3.69910	−0.127178
$847$	−3.98394	−0.136890
$848$	2.11628	0.0726733
$849$	−30.4642	−1.04553
$850$	3.09412	0.106127
$851$	12.2391	0.419550
$852$	19.8674	0.680646
$853$	−18.8302	−0.644735	−0.322367	−	0.946615i	$-0.604479\pi$
$853$	−18.8302	−0.644735	−0.322367	+	0.946615i	$0.604479\pi$
$854$	0.140172	0.00479658
$855$	11.7130	0.400577
$856$	36.3206	1.24141
$857$	9.65738	0.329890	0.164945	−	0.986303i	$-0.447255\pi$
$857$	9.65738	0.329890	0.164945	+	0.986303i	$0.447255\pi$
$858$	−10.6741	−0.364408
$859$	42.9535	1.46556	0.732778	−	0.680468i	$-0.238223\pi$
$859$	42.9535	1.46556	0.732778	+	0.680468i	$0.238223\pi$
$860$	−10.2100	−0.348160
$861$	4.49750	0.153275
$862$	22.2970	0.759438
$863$	27.3304	0.930338	0.465169	−	0.885222i	$-0.345994\pi$
$863$	27.3304	0.930338	0.465169	+	0.885222i	$0.345994\pi$
$864$	5.76230	0.196038
$865$	−25.1918	−0.856545
$866$	−17.7832	−0.604297
$867$	−34.3054	−1.16507
$868$	10.2693	0.348563
$869$	32.4707	1.10149
$870$	−19.2462	−0.652508
$871$	64.2881	2.17832
$872$	−13.0165	−0.440793
$873$	8.35604	0.282809
$874$	−16.2675	−0.550256
$875$	−10.5316	−0.356033
$876$	−11.2843	−0.381260
$877$	−37.9119	−1.28019	−0.640096	−	0.768295i	$-0.721106\pi$
$877$	−37.9119	−1.28019	−0.640096	+	0.768295i	$0.721106\pi$
$878$	3.82138	0.128965
$879$	−15.3100	−0.516395
$880$	1.75778	0.0592548
$881$	−25.3289	−0.853351	−0.426675	−	0.904405i	$-0.640315\pi$
$881$	−25.3289	−0.853351	−0.426675	+	0.904405i	$0.640315\pi$
$882$	−0.837626	−0.0282043
$883$	−35.1092	−1.18152	−0.590760	−	0.806847i	$-0.701172\pi$
$883$	−35.1092	−1.18152	−0.590760	+	0.806847i	$0.701172\pi$
$884$	−44.7424	−1.50485
$885$	−9.63855	−0.323996
$886$	−9.78287	−0.328662
$887$	40.3424	1.35456	0.677282	−	0.735724i	$-0.263158\pi$
$887$	40.3424	1.35456	0.677282	+	0.735724i	$0.263158\pi$
$888$	−8.68357	−0.291402
$889$	4.63437	0.155432
$890$	−16.0617	−0.538390
$891$	2.64878	0.0887376
$892$	−35.5719	−1.19104
$893$	22.0250	0.737038
$894$	9.03826	0.302285
$895$	35.8915	1.19972
$896$	7.95505	0.265759
$897$	18.7343	0.625519
$898$	−6.60637	−0.220458
$899$	77.3807	2.58079
$900$	−0.669589	−0.0223196
$901$	−53.6459	−1.78720
$902$	9.97857	0.332250
$903$	−3.34830	−0.111424
$904$	−47.5932	−1.58293
$905$	14.8238	0.492759
$906$	14.2084	0.472043
$907$	−36.1674	−1.20092	−0.600459	−	0.799656i	$-0.705015\pi$
$907$	−36.1674	−1.20092	−0.600459	+	0.799656i	$0.705015\pi$
$908$	22.1714	0.735783
$909$	−8.56358	−0.284036
$910$	9.46424	0.313737
$911$	−48.1370	−1.59485	−0.797425	−	0.603418i	$-0.793805\pi$
$911$	−48.1370	−1.59485	−0.797425	+	0.603418i	$0.793805\pi$
$912$	−1.40924	−0.0466648
$913$	40.4679	1.33929
$914$	−23.4129	−0.774431
$915$	0.393017	0.0129927
$916$	−15.1123	−0.499323
$917$	9.28952	0.306767
$918$	−5.99973	−0.198020
$919$	22.3951	0.738745	0.369372	−	0.929282i	$-0.379573\pi$
$919$	22.3951	0.738745	0.369372	+	0.929282i	$0.379573\pi$
$920$	25.2670	0.833028
$921$	−11.6405	−0.383568
$922$	−32.1615	−1.05918
$923$	−73.6162	−2.42311
$924$	3.43914	0.113139
$925$	1.62089	0.0532944
$926$	0.844557	0.0277539
$927$	8.28638	0.272160
$928$	56.3756	1.85062
$929$	−47.3719	−1.55422	−0.777111	−	0.629364i	$-0.783315\pi$
$929$	−47.3719	−1.55422	−0.777111	+	0.629364i	$0.783315\pi$
$930$	−15.5592	−0.510208
$931$	4.98734	0.163453
$932$	−29.4279	−0.963943
$933$	34.2209	1.12034
$934$	5.64214	0.184616
$935$	−44.5583	−1.45721
$936$	−13.2919	−0.434458
$937$	19.8073	0.647078	0.323539	−	0.946215i	$-0.395127\pi$
$937$	19.8073	0.647078	0.323539	+	0.946215i	$0.395127\pi$
$938$	11.1930	0.365464
$939$	15.1728	0.495145
$940$	−13.4664	−0.439224
$941$	−22.6040	−0.736871	−0.368435	−	0.929653i	$-0.620106\pi$
$941$	−22.6040	−0.736871	−0.368435	+	0.929653i	$0.620106\pi$
$942$	−10.1810	−0.331716
$943$	−17.5135	−0.570318
$944$	1.15965	0.0377435
$945$	−2.34855	−0.0763985
$946$	−7.42884	−0.241532
$947$	−33.4604	−1.08732	−0.543658	−	0.839307i	$-0.682961\pi$
$947$	−33.4604	−1.08732	−0.543658	+	0.839307i	$0.682961\pi$
$948$	15.9165	0.516944
$949$	41.8125	1.35729
$950$	−2.15439	−0.0698976
$951$	2.64579	0.0857955
$952$	−19.7894	−0.641378
$953$	4.65073	0.150652	0.0753260	−	0.997159i	$-0.476000\pi$
$953$	4.65073	0.150652	0.0753260	+	0.997159i	$0.476000\pi$
$954$	−6.27343	−0.203110
$955$	−4.67389	−0.151244
$956$	25.8896	0.837331
$957$	25.9144	0.837694
$958$	−25.0529	−0.809423
$959$	9.31778	0.300887
$960$	−10.0084	−0.323021
$961$	31.5569	1.01797
$962$	12.6658	0.408362
$963$	13.1463	0.423632
$964$	20.2285	0.651516
$965$	−3.10141	−0.0998378
$966$	3.26176	0.104945
$967$	56.5439	1.81833	0.909164	−	0.416438i	$-0.136722\pi$
$967$	56.5439	1.81833	0.909164	+	0.416438i	$0.136722\pi$
$968$	−11.0069	−0.353774
$969$	35.7232	1.14759
$970$	−16.4381	−0.527795
$971$	50.9483	1.63501	0.817504	−	0.575922i	$-0.195357\pi$
$971$	50.9483	1.63501	0.817504	+	0.575922i	$0.195357\pi$
$972$	1.29838	0.0416457
$973$	−14.0412	−0.450139
$974$	−27.6824	−0.887001
$975$	2.48108	0.0794581
$976$	−0.0472855	−0.00151357
$977$	−10.1428	−0.324497	−0.162248	−	0.986750i	$-0.551875\pi$
$977$	−10.1428	−0.324497	−0.162248	+	0.986750i	$0.551875\pi$
$978$	2.18532	0.0698787
$979$	21.6266	0.691189
$980$	−3.04932	−0.0974071
$981$	−4.71131	−0.150421
$982$	−5.69516	−0.181740
$983$	−62.2972	−1.98697	−0.993486	−	0.113957i	$-0.963647\pi$
$983$	−62.2972	−1.98697	−0.993486	+	0.113957i	$0.963647\pi$
$984$	12.4258	0.396119
$985$	52.2272	1.66410
$986$	−58.6984	−1.86934
$987$	−4.41618	−0.140568
$988$	31.1535	0.991125
$989$	13.0384	0.414598
$990$	−5.21072	−0.165607
$991$	33.3558	1.05958	0.529791	−	0.848128i	$-0.322270\pi$
$991$	33.3558	1.05958	0.529791	+	0.848128i	$0.322270\pi$
$992$	45.5758	1.44703
$993$	−11.4480	−0.363290
$994$	−12.8171	−0.406533
$995$	−18.2525	−0.578644
$996$	19.8366	0.628547
$997$	−7.15965	−0.226748	−0.113374	−	0.993552i	$-0.536166\pi$
$997$	−7.15965	−0.226748	−0.113374	+	0.993552i	$0.536166\pi$
$998$	−4.12497	−0.130574
$999$	−3.14302	−0.0994408

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.t.1.20	✓	52

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.t.1.20	✓	52	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:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q-0.837626 q^{2} -1.00000 q^{3} -1.29838 q^{4} +2.34855 q^{5} +0.837626 q^{6} +1.00000 q^{7} +2.76281 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.837626 q^{2} -1.00000 q^{3} -1.29838 q^{4} +2.34855 q^{5} +0.837626 q^{6} +1.00000 q^{7} +2.76281 q^{8} +1.00000 q^{9} -1.96721 q^{10} +2.64878 q^{11} +1.29838 q^{12} -4.81100 q^{13} -0.837626 q^{14} -2.34855 q^{15} +0.282564 q^{16} -7.16278 q^{17} -0.837626 q^{18} +4.98734 q^{19} -3.04932 q^{20} -1.00000 q^{21} -2.21869 q^{22} +3.89405 q^{23} -2.76281 q^{24} +0.515710 q^{25} +4.02981 q^{26} -1.00000 q^{27} -1.29838 q^{28} -9.78351 q^{29} +1.96721 q^{30} -7.90929 q^{31} -5.76230 q^{32} -2.64878 q^{33} +5.99973 q^{34} +2.34855 q^{35} -1.29838 q^{36} +3.14302 q^{37} -4.17752 q^{38} +4.81100 q^{39} +6.48861 q^{40} -4.49750 q^{41} +0.837626 q^{42} +3.34830 q^{43} -3.43914 q^{44} +2.34855 q^{45} -3.26176 q^{46} +4.41618 q^{47} -0.282564 q^{48} +1.00000 q^{49} -0.431972 q^{50} +7.16278 q^{51} +6.24652 q^{52} +7.48954 q^{53} +0.837626 q^{54} +6.22082 q^{55} +2.76281 q^{56} -4.98734 q^{57} +8.19492 q^{58} +4.10404 q^{59} +3.04932 q^{60} -0.167344 q^{61} +6.62503 q^{62} +1.00000 q^{63} +4.26153 q^{64} -11.2989 q^{65} +2.21869 q^{66} -13.3627 q^{67} +9.30003 q^{68} -3.89405 q^{69} -1.96721 q^{70} +15.3017 q^{71} +2.76281 q^{72} -8.69102 q^{73} -2.63268 q^{74} -0.515710 q^{75} -6.47548 q^{76} +2.64878 q^{77} -4.02981 q^{78} +12.2587 q^{79} +0.663618 q^{80} +1.00000 q^{81} +3.76723 q^{82} +15.2779 q^{83} +1.29838 q^{84} -16.8222 q^{85} -2.80462 q^{86} +9.78351 q^{87} +7.31809 q^{88} +8.16472 q^{89} -1.96721 q^{90} -4.81100 q^{91} -5.05597 q^{92} +7.90929 q^{93} -3.69910 q^{94} +11.7130 q^{95} +5.76230 q^{96} +8.35604 q^{97} -0.837626 q^{98} +2.64878 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9} - 2 q^{10} + 9 q^{11} - 61 q^{12} + 44 q^{13} + 3 q^{14} + 7 q^{15} + 95 q^{16} - 6 q^{17} + 3 q^{18} + 7 q^{19} - 21 q^{20} - 52 q^{21} + 19 q^{22} - 4 q^{23} - 24 q^{24} + 83 q^{25} - 5 q^{26} - 52 q^{27} + 61 q^{28} + 31 q^{29} + 2 q^{30} + 11 q^{31} + 71 q^{32} - 9 q^{33} + 17 q^{34} - 7 q^{35} + 61 q^{36} + 71 q^{37} - 8 q^{38} - 44 q^{39} + 20 q^{40} - 25 q^{41} - 3 q^{42} + 75 q^{43} + 14 q^{44} - 7 q^{45} + 36 q^{46} - 20 q^{47} - 95 q^{48} + 52 q^{49} + 26 q^{50} + 6 q^{51} + 88 q^{52} + 70 q^{53} - 3 q^{54} + 12 q^{55} + 24 q^{56} - 7 q^{57} + 48 q^{58} - 27 q^{59} + 21 q^{60} + 59 q^{61} - 23 q^{62} + 52 q^{63} + 138 q^{64} + 44 q^{65} - 19 q^{66} + 65 q^{67} - 8 q^{68} + 4 q^{69} - 2 q^{70} - 11 q^{71} + 24 q^{72} + 34 q^{73} + 38 q^{74} - 83 q^{75} + 31 q^{76} + 9 q^{77} + 5 q^{78} + 74 q^{79} - 5 q^{80} + 52 q^{81} + 51 q^{82} - 30 q^{83} - 61 q^{84} + 70 q^{85} + 29 q^{86} - 31 q^{87} + 90 q^{88} - q^{89} - 2 q^{90} + 44 q^{91} + 34 q^{92} - 11 q^{93} + 27 q^{94} + 9 q^{95} - 71 q^{96} + 73 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9 - 2 * q^10 + 9 * q^11 - 61 * q^12 + 44 * q^13 + 3 * q^14 + 7 * q^15 + 95 * q^16 - 6 * q^17 + 3 * q^18 + 7 * q^19 - 21 * q^20 - 52 * q^21 + 19 * q^22 - 4 * q^23 - 24 * q^24 + 83 * q^25 - 5 * q^26 - 52 * q^27 + 61 * q^28 + 31 * q^29 + 2 * q^30 + 11 * q^31 + 71 * q^32 - 9 * q^33 + 17 * q^34 - 7 * q^35 + 61 * q^36 + 71 * q^37 - 8 * q^38 - 44 * q^39 + 20 * q^40 - 25 * q^41 - 3 * q^42 + 75 * q^43 + 14 * q^44 - 7 * q^45 + 36 * q^46 - 20 * q^47 - 95 * q^48 + 52 * q^49 + 26 * q^50 + 6 * q^51 + 88 * q^52 + 70 * q^53 - 3 * q^54 + 12 * q^55 + 24 * q^56 - 7 * q^57 + 48 * q^58 - 27 * q^59 + 21 * q^60 + 59 * q^61 - 23 * q^62 + 52 * q^63 + 138 * q^64 + 44 * q^65 - 19 * q^66 + 65 * q^67 - 8 * q^68 + 4 * q^69 - 2 * q^70 - 11 * q^71 + 24 * q^72 + 34 * q^73 + 38 * q^74 - 83 * q^75 + 31 * q^76 + 9 * q^77 + 5 * q^78 + 74 * q^79 - 5 * q^80 + 52 * q^81 + 51 * q^82 - 30 * q^83 - 61 * q^84 + 70 * q^85 + 29 * q^86 - 31 * q^87 + 90 * q^88 - q^89 - 2 * q^90 + 44 * q^91 + 34 * q^92 - 11 * q^93 + 27 * q^94 + 9 * q^95 - 71 * q^96 + 73 * q^97 + 3 * q^98 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more