LMFDB - Embedded newform 6031.2.a.e.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6031,2,Mod(1,6031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6031, 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("6031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6031 = 37 \cdot 163 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6031.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:	$48.1577774590$
Analytic rank:	$0$
Dimension:	$134$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	6031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.39066 q^{2} -3.44347 q^{3} +3.71525 q^{4} -1.21628 q^{5} +8.23217 q^{6} -4.84430 q^{7} -4.10059 q^{8} +8.85751 q^{9} +O(q^{10})$	\(q-2.39066 q^{2} -3.44347 q^{3} +3.71525 q^{4} -1.21628 q^{5} +8.23217 q^{6} -4.84430 q^{7} -4.10059 q^{8} +8.85751 q^{9} +2.90771 q^{10} -0.135468 q^{11} -12.7934 q^{12} -2.82151 q^{13} +11.5811 q^{14} +4.18823 q^{15} +2.37260 q^{16} -1.47149 q^{17} -21.1753 q^{18} +2.49646 q^{19} -4.51879 q^{20} +16.6812 q^{21} +0.323859 q^{22} +2.68073 q^{23} +14.1203 q^{24} -3.52066 q^{25} +6.74527 q^{26} -20.1702 q^{27} -17.9978 q^{28} -9.84315 q^{29} -10.0126 q^{30} +2.01105 q^{31} +2.52910 q^{32} +0.466482 q^{33} +3.51784 q^{34} +5.89203 q^{35} +32.9079 q^{36} +1.00000 q^{37} -5.96818 q^{38} +9.71580 q^{39} +4.98746 q^{40} +3.23832 q^{41} -39.8791 q^{42} +9.12887 q^{43} -0.503300 q^{44} -10.7732 q^{45} -6.40871 q^{46} -0.500209 q^{47} -8.16998 q^{48} +16.4672 q^{49} +8.41670 q^{50} +5.06704 q^{51} -10.4826 q^{52} -0.367546 q^{53} +48.2200 q^{54} +0.164768 q^{55} +19.8645 q^{56} -8.59648 q^{57} +23.5316 q^{58} -3.17992 q^{59} +15.5603 q^{60} +4.45697 q^{61} -4.80773 q^{62} -42.9084 q^{63} -10.7914 q^{64} +3.43175 q^{65} -1.11520 q^{66} -5.70935 q^{67} -5.46696 q^{68} -9.23102 q^{69} -14.0858 q^{70} -10.2470 q^{71} -36.3210 q^{72} +11.3144 q^{73} -2.39066 q^{74} +12.1233 q^{75} +9.27497 q^{76} +0.656250 q^{77} -23.2272 q^{78} -14.1824 q^{79} -2.88574 q^{80} +42.8829 q^{81} -7.74172 q^{82} -1.22978 q^{83} +61.9749 q^{84} +1.78975 q^{85} -21.8240 q^{86} +33.8946 q^{87} +0.555500 q^{88} +14.6789 q^{89} +25.7551 q^{90} +13.6682 q^{91} +9.95959 q^{92} -6.92498 q^{93} +1.19583 q^{94} -3.03639 q^{95} -8.70888 q^{96} -18.6249 q^{97} -39.3676 q^{98} -1.19991 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 134 q + 9 q^{2} + 7 q^{3} + 149 q^{4} + 22 q^{5} + 20 q^{6} + 11 q^{7} + 27 q^{8} + 181 q^{9}+O(q^{10}) $ 134 * q + 9 * q^2 + 7 * q^3 + 149 * q^4 + 22 * q^5 + 20 * q^6 + 11 * q^7 + 27 * q^8 + 181 * q^9	\( 134 q + 9 q^{2} + 7 q^{3} + 149 q^{4} + 22 q^{5} + 20 q^{6} + 11 q^{7} + 27 q^{8} + 181 q^{9} + 15 q^{10} + 20 q^{11} + 28 q^{12} + 11 q^{13} + 17 q^{14} - 13 q^{15} + 143 q^{16} + 76 q^{17} + 23 q^{18} + 15 q^{19} + 67 q^{20} + 63 q^{21} + 2 q^{22} + 22 q^{23} + 33 q^{24} + 160 q^{25} + 65 q^{26} + 31 q^{27} + 10 q^{28} + 73 q^{29} + 20 q^{30} + 10 q^{31} + 53 q^{32} + 72 q^{33} - 7 q^{34} + 52 q^{35} + 201 q^{36} + 134 q^{37} + 70 q^{38} + 6 q^{39} + 11 q^{40} + 182 q^{41} - 15 q^{42} + 12 q^{43} + 33 q^{44} + 29 q^{45} + 24 q^{46} + 80 q^{47} + 21 q^{48} + 229 q^{49} + 37 q^{50} + 57 q^{51} - 15 q^{52} + 75 q^{53} + 95 q^{54} - 9 q^{55} + 39 q^{56} + 19 q^{57} - 21 q^{58} + 91 q^{59} + 62 q^{60} + 58 q^{61} + 108 q^{62} + 9 q^{63} + 167 q^{64} + 76 q^{65} + 105 q^{66} - 17 q^{67} + 109 q^{68} + 48 q^{69} - 55 q^{70} + 56 q^{71} + 48 q^{72} + 54 q^{73} + 9 q^{74} + 28 q^{75} + 82 q^{76} + 156 q^{77} + 16 q^{78} - 2 q^{79} + 98 q^{80} + 270 q^{81} - 42 q^{82} + 130 q^{83} + 229 q^{84} + 22 q^{85} + 72 q^{86} + 22 q^{87} + 61 q^{88} + 157 q^{89} + 176 q^{90} + 31 q^{91} - 18 q^{92} + 36 q^{93} + 83 q^{94} + 98 q^{95} + 111 q^{96} + 35 q^{97} + 53 q^{98} + 55 q^{99}+O(q^{100}) \) 134 * q + 9 * q^2 + 7 * q^3 + 149 * q^4 + 22 * q^5 + 20 * q^6 + 11 * q^7 + 27 * q^8 + 181 * q^9 + 15 * q^10 + 20 * q^11 + 28 * q^12 + 11 * q^13 + 17 * q^14 - 13 * q^15 + 143 * q^16 + 76 * q^17 + 23 * q^18 + 15 * q^19 + 67 * q^20 + 63 * q^21 + 2 * q^22 + 22 * q^23 + 33 * q^24 + 160 * q^25 + 65 * q^26 + 31 * q^27 + 10 * q^28 + 73 * q^29 + 20 * q^30 + 10 * q^31 + 53 * q^32 + 72 * q^33 - 7 * q^34 + 52 * q^35 + 201 * q^36 + 134 * q^37 + 70 * q^38 + 6 * q^39 + 11 * q^40 + 182 * q^41 - 15 * q^42 + 12 * q^43 + 33 * q^44 + 29 * q^45 + 24 * q^46 + 80 * q^47 + 21 * q^48 + 229 * q^49 + 37 * q^50 + 57 * q^51 - 15 * q^52 + 75 * q^53 + 95 * q^54 - 9 * q^55 + 39 * q^56 + 19 * q^57 - 21 * q^58 + 91 * q^59 + 62 * q^60 + 58 * q^61 + 108 * q^62 + 9 * q^63 + 167 * q^64 + 76 * q^65 + 105 * q^66 - 17 * q^67 + 109 * q^68 + 48 * q^69 - 55 * q^70 + 56 * q^71 + 48 * q^72 + 54 * q^73 + 9 * q^74 + 28 * q^75 + 82 * q^76 + 156 * q^77 + 16 * q^78 - 2 * q^79 + 98 * q^80 + 270 * q^81 - 42 * q^82 + 130 * q^83 + 229 * q^84 + 22 * q^85 + 72 * q^86 + 22 * q^87 + 61 * q^88 + 157 * q^89 + 176 * q^90 + 31 * q^91 - 18 * q^92 + 36 * q^93 + 83 * q^94 + 98 * q^95 + 111 * q^96 + 35 * q^97 + 53 * q^98 + 55 * 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 $

$2$	−2.39066	−1.69045	−0.845226	−	0.534409i	$-0.820534\pi$
$2$	−2.39066	−1.69045	−0.845226	+	0.534409i	$0.820534\pi$
$3$	−3.44347	−1.98809	−0.994045	−	0.108970i	$-0.965245\pi$
$3$	−3.44347	−1.98809	−0.994045	+	0.108970i	$0.965245\pi$
$4$	3.71525	1.85763
$5$	−1.21628	−0.543937	−0.271969	−	0.962306i	$-0.587675\pi$
$5$	−1.21628	−0.543937	−0.271969	+	0.962306i	$0.587675\pi$
$6$	8.23217	3.36077
$7$	−4.84430	−1.83097	−0.915487	−	0.402349i	$-0.868194\pi$
$7$	−4.84430	−1.83097	−0.915487	+	0.402349i	$0.868194\pi$
$8$	−4.10059	−1.44978
$9$	8.85751	2.95250
$10$	2.90771	0.919499
$11$	−0.135468	−0.0408453	−0.0204226	−	0.999791i	$-0.506501\pi$
$11$	−0.135468	−0.0408453	−0.0204226	+	0.999791i	$0.506501\pi$
$12$	−12.7934	−3.69313
$13$	−2.82151	−0.782546	−0.391273	−	0.920275i	$-0.627965\pi$
$13$	−2.82151	−0.782546	−0.391273	+	0.920275i	$0.627965\pi$
$14$	11.5811	3.09517
$15$	4.18823	1.08140
$16$	2.37260	0.593149
$17$	−1.47149	−0.356889	−0.178445	−	0.983950i	$-0.557107\pi$
$17$	−1.47149	−0.356889	−0.178445	+	0.983950i	$0.557107\pi$
$18$	−21.1753	−4.99106
$19$	2.49646	0.572726	0.286363	−	0.958121i	$-0.407554\pi$
$19$	2.49646	0.572726	0.286363	+	0.958121i	$0.407554\pi$
$20$	−4.51879	−1.01043
$21$	16.6812	3.64014
$22$	0.323859	0.0690470
$23$	2.68073	0.558971	0.279485	−	0.960150i	$-0.409836\pi$
$23$	2.68073	0.558971	0.279485	+	0.960150i	$0.409836\pi$
$24$	14.1203	2.88228
$25$	−3.52066	−0.704132
$26$	6.74527	1.32286
$27$	−20.1702	−3.88175
$28$	−17.9978	−3.40126
$29$	−9.84315	−1.82783	−0.913913	−	0.405909i	$-0.866955\pi$
$29$	−9.84315	−1.82783	−0.913913	+	0.405909i	$0.866955\pi$
$30$	−10.0126	−1.82805
$31$	2.01105	0.361195	0.180597	−	0.983557i	$-0.442197\pi$
$31$	2.01105	0.361195	0.180597	+	0.983557i	$0.442197\pi$
$32$	2.52910	0.447085
$33$	0.466482	0.0812041
$34$	3.51784	0.603304
$35$	5.89203	0.995934
$36$	32.9079	5.48465
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−5.96818	−0.968166
$39$	9.71580	1.55577
$40$	4.98746	0.788587
$41$	3.23832	0.505741	0.252870	−	0.967500i	$-0.418625\pi$
$41$	3.23832	0.505741	0.252870	+	0.967500i	$0.418625\pi$
$42$	−39.8791	−6.15348
$43$	9.12887	1.39214	0.696069	−	0.717974i	$-0.254931\pi$
$43$	9.12887	1.39214	0.696069	+	0.717974i	$0.254931\pi$
$44$	−0.503300	−0.0758753
$45$	−10.7732	−1.60598
$46$	−6.40871	−0.944913
$47$	−0.500209	−0.0729629	−0.0364815	−	0.999334i	$-0.511615\pi$
$47$	−0.500209	−0.0729629	−0.0364815	+	0.999334i	$0.511615\pi$
$48$	−8.16998	−1.17923
$49$	16.4672	2.35246
$50$	8.41670	1.19030
$51$	5.06704	0.709528
$52$	−10.4826	−1.45368
$53$	−0.367546	−0.0504863	−0.0252431	−	0.999681i	$-0.508036\pi$
$53$	−0.367546	−0.0504863	−0.0252431	+	0.999681i	$0.508036\pi$
$54$	48.2200	6.56191
$55$	0.164768	0.0222173
$56$	19.8645	2.65450
$57$	−8.59648	−1.13863
$58$	23.5316	3.08985
$59$	−3.17992	−0.413990	−0.206995	−	0.978342i	$-0.566368\pi$
$59$	−3.17992	−0.413990	−0.206995	+	0.978342i	$0.566368\pi$
$60$	15.5603	2.00883
$61$	4.45697	0.570657	0.285328	−	0.958430i	$-0.407897\pi$
$61$	4.45697	0.570657	0.285328	+	0.958430i	$0.407897\pi$
$62$	−4.80773	−0.610582
$63$	−42.9084	−5.40595
$64$	−10.7914	−1.34893
$65$	3.43175	0.425656
$66$	−1.11520	−0.137272
$67$	−5.70935	−0.697508	−0.348754	−	0.937214i	$-0.613395\pi$
$67$	−5.70935	−0.697508	−0.348754	+	0.937214i	$0.613395\pi$
$68$	−5.46696	−0.662967
$69$	−9.23102	−1.11128
$70$	−14.0858	−1.68358
$71$	−10.2470	−1.21610	−0.608050	−	0.793899i	$-0.708048\pi$
$71$	−10.2470	−1.21610	−0.608050	+	0.793899i	$0.708048\pi$
$72$	−36.3210	−4.28047
$73$	11.3144	1.32425	0.662126	−	0.749393i	$-0.269654\pi$
$73$	11.3144	1.32425	0.662126	+	0.749393i	$0.269654\pi$
$74$	−2.39066	−0.277909
$75$	12.1233	1.39988
$76$	9.27497	1.06391
$77$	0.656250	0.0747866
$78$	−23.2272	−2.62996
$79$	−14.1824	−1.59564	−0.797821	−	0.602894i	$-0.794014\pi$
$79$	−14.1824	−1.59564	−0.797821	+	0.602894i	$0.794014\pi$
$80$	−2.88574	−0.322636
$81$	42.8829	4.76476
$82$	−7.74172	−0.854930
$83$	−1.22978	−0.134986	−0.0674931	−	0.997720i	$-0.521500\pi$
$83$	−1.22978	−0.134986	−0.0674931	+	0.997720i	$0.521500\pi$
$84$	61.9749	6.76202
$85$	1.78975	0.194125
$86$	−21.8240	−2.35334
$87$	33.8946	3.63388
$88$	0.555500	0.0592165
$89$	14.6789	1.55596	0.777979	−	0.628290i	$-0.216245\pi$
$89$	14.6789	1.55596	0.777979	+	0.628290i	$0.216245\pi$
$90$	25.7551	2.71482
$91$	13.6682	1.43282
$92$	9.95959	1.03836
$93$	−6.92498	−0.718087
$94$	1.19583	0.123340
$95$	−3.03639	−0.311527
$96$	−8.70888	−0.888846
$97$	−18.6249	−1.89107	−0.945535	−	0.325521i	$-0.894460\pi$
$97$	−18.6249	−1.89107	−0.945535	+	0.325521i	$0.894460\pi$
$98$	−39.3676	−3.97672
$99$	−1.19991	−0.120596
$100$	−13.0801	−1.30801
$101$	8.84752	0.880361	0.440180	−	0.897909i	$-0.354914\pi$
$101$	8.84752	0.880361	0.440180	+	0.897909i	$0.354914\pi$
$102$	−12.1136	−1.19942
$103$	−14.8523	−1.46344	−0.731722	−	0.681603i	$-0.761283\pi$
$103$	−14.8523	−1.46344	−0.731722	+	0.681603i	$0.761283\pi$
$104$	11.5698	1.13452
$105$	−20.2890	−1.98001
$106$	0.878676	0.0853446
$107$	−16.7233	−1.61670	−0.808349	−	0.588704i	$-0.799638\pi$
$107$	−16.7233	−1.61670	−0.808349	+	0.588704i	$0.799638\pi$
$108$	−74.9372	−7.21084
$109$	2.43512	0.233242	0.116621	−	0.993176i	$-0.462794\pi$
$109$	2.43512	0.233242	0.116621	+	0.993176i	$0.462794\pi$
$110$	−0.393903	−0.0375572
$111$	−3.44347	−0.326840
$112$	−11.4936	−1.08604
$113$	2.09540	0.197119	0.0985593	−	0.995131i	$-0.468577\pi$
$113$	2.09540	0.197119	0.0985593	+	0.995131i	$0.468577\pi$
$114$	20.5513	1.92480
$115$	−3.26052	−0.304045
$116$	−36.5698	−3.39542
$117$	−24.9916	−2.31047
$118$	7.60210	0.699830
$119$	7.12835	0.653454
$120$	−17.1742	−1.56778
$121$	−10.9816	−0.998332
$122$	−10.6551	−0.964667
$123$	−11.1511	−1.00546
$124$	7.47154	0.670964
$125$	10.3635	0.926941
$126$	102.579	9.13850
$127$	8.66784	0.769147	0.384573	−	0.923094i	$-0.374349\pi$
$127$	8.66784	0.769147	0.384573	+	0.923094i	$0.374349\pi$
$128$	20.7404	1.83321
$129$	−31.4350	−2.76770
$130$	−8.20414	−0.719551
$131$	14.4173	1.25964	0.629821	−	0.776740i	$-0.283128\pi$
$131$	14.4173	1.25964	0.629821	+	0.776740i	$0.283128\pi$
$132$	1.73310	0.150847
$133$	−12.0936	−1.04865
$134$	13.6491	1.17910
$135$	24.5326	2.11143
$136$	6.03398	0.517409
$137$	16.4247	1.40326	0.701630	−	0.712542i	$-0.252456\pi$
$137$	16.4247	1.40326	0.701630	+	0.712542i	$0.252456\pi$
$138$	22.0682	1.87857
$139$	−8.16156	−0.692255	−0.346127	−	0.938188i	$-0.612503\pi$
$139$	−8.16156	−0.692255	−0.346127	+	0.938188i	$0.612503\pi$
$140$	21.8904	1.85007
$141$	1.72245	0.145057
$142$	24.4972	2.05576
$143$	0.382226	0.0319633
$144$	21.0153	1.75127
$145$	11.9720	0.994223
$146$	−27.0489	−2.23858
$147$	−56.7045	−4.67691
$148$	3.71525	0.305392
$149$	−4.82510	−0.395288	−0.197644	−	0.980274i	$-0.563329\pi$
$149$	−4.82510	−0.395288	−0.197644	+	0.980274i	$0.563329\pi$
$150$	−28.9827	−2.36643
$151$	−1.38176	−0.112446	−0.0562232	−	0.998418i	$-0.517906\pi$
$151$	−1.38176	−0.112446	−0.0562232	+	0.998418i	$0.517906\pi$
$152$	−10.2369	−0.830325
$153$	−13.0337	−1.05372
$154$	−1.56887	−0.126423
$155$	−2.44600	−0.196467
$156$	36.0966	2.89004
$157$	3.19329	0.254852	0.127426	−	0.991848i	$-0.459328\pi$
$157$	3.19329	0.254852	0.127426	+	0.991848i	$0.459328\pi$
$158$	33.9053	2.69736
$159$	1.26563	0.100371
$160$	−3.07609	−0.243186
$161$	−12.9863	−1.02346
$162$	−102.518	−8.05460
$163$	−1.00000	−0.0783260
$163$	−1.00000	−0.0783260
$164$	12.0312	0.939478
$165$	−0.567373	−0.0441699
$166$	2.93999	0.228188
$167$	−3.16645	−0.245028	−0.122514	−	0.992467i	$-0.539096\pi$
$167$	−3.16645	−0.245028	−0.122514	+	0.992467i	$0.539096\pi$
$168$	−68.4027	−5.27739
$169$	−5.03907	−0.387621
$170$	−4.27867	−0.328159
$171$	22.1124	1.69098
$172$	33.9160	2.58607
$173$	−18.5326	−1.40901	−0.704503	−	0.709701i	$-0.748830\pi$
$173$	−18.5326	−1.40901	−0.704503	+	0.709701i	$0.748830\pi$
$174$	−81.0305	−6.14290
$175$	17.0551	1.28925
$176$	−0.321412	−0.0242274
$177$	10.9500	0.823050
$178$	−35.0922	−2.63027
$179$	−21.6275	−1.61652	−0.808258	−	0.588829i	$-0.799589\pi$
$179$	−21.6275	−1.61652	−0.808258	+	0.588829i	$0.799589\pi$
$180$	−40.0252	−2.98330
$181$	−8.83530	−0.656722	−0.328361	−	0.944552i	$-0.606496\pi$
$181$	−8.83530	−0.656722	−0.328361	+	0.944552i	$0.606496\pi$
$182$	−32.6761	−2.42212
$183$	−15.3475	−1.13452
$184$	−10.9926	−0.810382
$185$	−1.21628	−0.0894227
$186$	16.5553	1.21389
$187$	0.199341	0.0145772
$188$	−1.85840	−0.135538
$189$	97.7103	7.10738
$190$	7.25898	0.526622
$191$	−9.98659	−0.722604	−0.361302	−	0.932449i	$-0.617668\pi$
$191$	−9.98659	−0.722604	−0.361302	+	0.932449i	$0.617668\pi$
$192$	37.1599	2.68179
$193$	−10.4260	−0.750479	−0.375239	−	0.926928i	$-0.622439\pi$
$193$	−10.4260	−0.750479	−0.375239	+	0.926928i	$0.622439\pi$
$194$	44.5257	3.19676
$195$	−11.8171	−0.846243
$196$	61.1800	4.37000
$197$	−5.41021	−0.385462	−0.192731	−	0.981252i	$-0.561734\pi$
$197$	−5.41021	−0.385462	−0.192731	+	0.981252i	$0.561734\pi$
$198$	2.86858	0.203861
$199$	−25.5069	−1.80813	−0.904067	−	0.427390i	$-0.859433\pi$
$199$	−25.5069	−1.80813	−0.904067	+	0.427390i	$0.859433\pi$
$200$	14.4368	1.02083
$201$	19.6600	1.38671
$202$	−21.1514	−1.48821
$203$	47.6832	3.34670
$204$	18.8253	1.31804
$205$	−3.93871	−0.275091
$206$	35.5069	2.47388
$207$	23.7446	1.65036
$208$	−6.69431	−0.464167
$209$	−0.338191	−0.0233932
$210$	48.5042	3.34711
$211$	−15.3996	−1.06015	−0.530075	−	0.847951i	$-0.677836\pi$
$211$	−15.3996	−1.06015	−0.530075	+	0.847951i	$0.677836\pi$
$212$	−1.36552	−0.0937846
$213$	35.2854	2.41772
$214$	39.9796	2.73295
$215$	−11.1033	−0.757236
$216$	82.7095	5.62767
$217$	−9.74211	−0.661337
$218$	−5.82154	−0.394284
$219$	−38.9609	−2.63273
$220$	0.612154	0.0412714
$221$	4.15183	0.279282
$222$	8.23217	0.552507
$223$	−15.6357	−1.04705	−0.523523	−	0.852012i	$-0.675382\pi$
$223$	−15.6357	−1.04705	−0.523523	+	0.852012i	$0.675382\pi$
$224$	−12.2517	−0.818601
$225$	−31.1843	−2.07895
$226$	−5.00939	−0.333219
$227$	−16.7705	−1.11310	−0.556550	−	0.830814i	$-0.687875\pi$
$227$	−16.7705	−1.11310	−0.556550	+	0.830814i	$0.687875\pi$
$228$	−31.9381	−2.11515
$229$	−21.1078	−1.39484	−0.697420	−	0.716662i	$-0.745669\pi$
$229$	−21.1078	−1.39484	−0.697420	+	0.716662i	$0.745669\pi$
$230$	7.79479	0.513973
$231$	−2.25978	−0.148683
$232$	40.3627	2.64994
$233$	−3.81406	−0.249868	−0.124934	−	0.992165i	$-0.539872\pi$
$233$	−3.81406	−0.249868	−0.124934	+	0.992165i	$0.539872\pi$
$234$	59.7463	3.90574
$235$	0.608394	0.0396873
$236$	−11.8142	−0.769039
$237$	48.8367	3.17228
$238$	−17.0414	−1.10463
$239$	−15.0787	−0.975357	−0.487679	−	0.873023i	$-0.662156\pi$
$239$	−15.0787	−0.975357	−0.487679	+	0.873023i	$0.662156\pi$
$240$	9.93698	0.641430
$241$	29.5507	1.90353	0.951763	−	0.306833i	$-0.0992694\pi$
$241$	29.5507	1.90353	0.951763	+	0.306833i	$0.0992694\pi$
$242$	26.2534	1.68763
$243$	−87.1556	−5.59103
$244$	16.5588	1.06007
$245$	−20.0288	−1.27959
$246$	26.6584	1.69968
$247$	−7.04378	−0.448185
$248$	−8.24647	−0.523651
$249$	4.23472	0.268365
$250$	−24.7756	−1.56695
$251$	6.38933	0.403291	0.201645	−	0.979459i	$-0.435371\pi$
$251$	6.38933	0.403291	0.201645	+	0.979459i	$0.435371\pi$
$252$	−159.416	−10.0422
$253$	−0.363154	−0.0228313
$254$	−20.7219	−1.30021
$255$	−6.16294	−0.385939
$256$	−28.0004	−1.75002
$257$	8.97001	0.559534	0.279767	−	0.960068i	$-0.409743\pi$
$257$	8.97001	0.559534	0.279767	+	0.960068i	$0.409743\pi$
$258$	75.1504	4.67866
$259$	−4.84430	−0.301010
$260$	12.7498	0.790710
$261$	−87.1857	−5.39666
$262$	−34.4668	−2.12936
$263$	−9.09313	−0.560706	−0.280353	−	0.959897i	$-0.590452\pi$
$263$	−9.09313	−0.560706	−0.280353	+	0.959897i	$0.590452\pi$
$264$	−1.91285	−0.117728
$265$	0.447038	0.0274614
$266$	28.9116	1.77269
$267$	−50.5463	−3.09338
$268$	−21.2117	−1.29571
$269$	−21.9185	−1.33639	−0.668197	−	0.743984i	$-0.732934\pi$
$269$	−21.9185	−1.33639	−0.668197	+	0.743984i	$0.732934\pi$
$270$	−58.6490	−3.56927
$271$	−4.44341	−0.269918	−0.134959	−	0.990851i	$-0.543090\pi$
$271$	−4.44341	−0.269918	−0.134959	+	0.990851i	$0.543090\pi$
$272$	−3.49126	−0.211689
$273$	−47.0662	−2.84858
$274$	−39.2659	−2.37214
$275$	0.476939	0.0287605
$276$	−34.2956	−2.06435
$277$	−28.1067	−1.68877	−0.844384	−	0.535738i	$-0.820034\pi$
$277$	−28.1067	−1.68877	−0.844384	+	0.535738i	$0.820034\pi$
$278$	19.5115	1.17022
$279$	17.8129	1.06643
$280$	−24.1608	−1.44388
$281$	18.0490	1.07671	0.538356	−	0.842718i	$-0.319046\pi$
$281$	18.0490	1.07671	0.538356	+	0.842718i	$0.319046\pi$
$282$	−4.11780	−0.245212
$283$	−14.5187	−0.863047	−0.431524	−	0.902102i	$-0.642024\pi$
$283$	−14.5187	−0.863047	−0.431524	+	0.902102i	$0.642024\pi$
$284$	−38.0703	−2.25906
$285$	10.4557	0.619344
$286$	−0.913772	−0.0540325
$287$	−15.6874	−0.925998
$288$	22.4015	1.32002
$289$	−14.8347	−0.872630
$290$	−28.6210	−1.68069
$291$	64.1343	3.75962
$292$	42.0359	2.45996
$293$	−5.66495	−0.330950	−0.165475	−	0.986214i	$-0.552916\pi$
$293$	−5.66495	−0.330950	−0.165475	+	0.986214i	$0.552916\pi$
$294$	135.561	7.90609
$295$	3.86767	0.225185
$296$	−4.10059	−0.238342
$297$	2.73242	0.158551
$298$	11.5352	0.668214
$299$	−7.56371	−0.437421
$300$	45.0411	2.60045
$301$	−44.2230	−2.54897
$302$	3.30333	0.190085
$303$	−30.4662	−1.75024
$304$	5.92309	0.339712
$305$	−5.42093	−0.310401
$306$	31.1592	1.78126
$307$	2.50243	0.142821	0.0714105	−	0.997447i	$-0.477250\pi$
$307$	2.50243	0.142821	0.0714105	+	0.997447i	$0.477250\pi$
$308$	2.43813	0.138926
$309$	51.1436	2.90946
$310$	5.84754	0.332118
$311$	−0.198251	−0.0112418	−0.00562090	−	0.999984i	$-0.501789\pi$
$311$	−0.198251	−0.0112418	−0.00562090	+	0.999984i	$0.501789\pi$
$312$	−39.8405	−2.25552
$313$	−3.47697	−0.196530	−0.0982650	−	0.995160i	$-0.531329\pi$
$313$	−3.47697	−0.196530	−0.0982650	+	0.995160i	$0.531329\pi$
$314$	−7.63407	−0.430815
$315$	52.1887	2.94050
$316$	−52.6911	−2.96411
$317$	1.46768	0.0824330	0.0412165	−	0.999150i	$-0.486877\pi$
$317$	1.46768	0.0824330	0.0412165	+	0.999150i	$0.486877\pi$
$318$	−3.02570	−0.169673
$319$	1.33344	0.0746581
$320$	13.1254	0.733731
$321$	57.5861	3.21414
$322$	31.0457	1.73011
$323$	−3.67352	−0.204400
$324$	159.321	8.85115
$325$	9.93359	0.551016
$326$	2.39066	0.132406
$327$	−8.38526	−0.463706
$328$	−13.2790	−0.733211
$329$	2.42316	0.133593
$330$	1.35640	0.0746671
$331$	−17.5039	−0.962099	−0.481049	−	0.876693i	$-0.659744\pi$
$331$	−17.5039	−0.962099	−0.481049	+	0.876693i	$0.659744\pi$
$332$	−4.56895	−0.250754
$333$	8.85751	0.485388
$334$	7.56991	0.414207
$335$	6.94417	0.379400
$336$	39.5778	2.15915
$337$	7.41249	0.403784	0.201892	−	0.979408i	$-0.435291\pi$
$337$	7.41249	0.403784	0.201892	+	0.979408i	$0.435291\pi$
$338$	12.0467	0.655255
$339$	−7.21545	−0.391890
$340$	6.64936	0.360612
$341$	−0.272433	−0.0147531
$342$	−52.8632	−2.85851
$343$	−45.8621	−2.47632
$344$	−37.4337	−2.01829
$345$	11.2275	0.604469
$346$	44.3051	2.38186
$347$	6.47808	0.347762	0.173881	−	0.984767i	$-0.444369\pi$
$347$	6.47808	0.347762	0.173881	+	0.984767i	$0.444369\pi$
$348$	125.927	6.75040
$349$	−11.1506	−0.596880	−0.298440	−	0.954428i	$-0.596466\pi$
$349$	−11.1506	−0.596880	−0.298440	+	0.954428i	$0.596466\pi$
$350$	−40.7730	−2.17941
$351$	56.9103	3.03765
$352$	−0.342613	−0.0182613
$353$	0.676731	0.0360188	0.0180094	−	0.999838i	$-0.494267\pi$
$353$	0.676731	0.0360188	0.0180094	+	0.999838i	$0.494267\pi$
$354$	−26.1776	−1.39133
$355$	12.4633	0.661482
$356$	54.5357	2.89039
$357$	−24.5463	−1.29913
$358$	51.7040	2.73264
$359$	−19.1354	−1.00993	−0.504965	−	0.863140i	$-0.668494\pi$
$359$	−19.1354	−1.00993	−0.504965	+	0.863140i	$0.668494\pi$
$360$	44.1765	2.32830
$361$	−12.7677	−0.671984
$362$	21.1222	1.11016
$363$	37.8150	1.98477
$364$	50.7810	2.66165
$365$	−13.7615	−0.720310
$366$	36.6905	1.91785
$367$	7.68946	0.401386	0.200693	−	0.979654i	$-0.435681\pi$
$367$	7.68946	0.401386	0.200693	+	0.979654i	$0.435681\pi$
$368$	6.36029	0.331553
$369$	28.6835	1.49320
$370$	2.90771	0.151165
$371$	1.78050	0.0924390
$372$	−25.7281	−1.33394
$373$	21.3883	1.10744	0.553722	−	0.832702i	$-0.313207\pi$
$373$	21.3883	1.10744	0.553722	+	0.832702i	$0.313207\pi$
$374$	−0.476556	−0.0246421
$375$	−35.6865	−1.84284
$376$	2.05115	0.105780
$377$	27.7726	1.43036
$378$	−233.592	−12.0147
$379$	10.4266	0.535579	0.267789	−	0.963477i	$-0.413707\pi$
$379$	10.4266	0.535579	0.267789	+	0.963477i	$0.413707\pi$
$380$	−11.2810	−0.578701
$381$	−29.8475	−1.52913
$382$	23.8745	1.22153
$383$	24.8371	1.26912	0.634559	−	0.772874i	$-0.281182\pi$
$383$	24.8371	1.26912	0.634559	+	0.772874i	$0.281182\pi$
$384$	−71.4189	−3.64458
$385$	−0.798184	−0.0406792
$386$	24.9250	1.26865
$387$	80.8590	4.11029
$388$	−69.1961	−3.51290
$389$	19.6415	0.995862	0.497931	−	0.867217i	$-0.334093\pi$
$389$	19.6415	0.995862	0.497931	+	0.867217i	$0.334093\pi$
$390$	28.2507	1.43053
$391$	−3.94467	−0.199491
$392$	−67.5253	−3.41054
$393$	−49.6454	−2.50428
$394$	12.9340	0.651604
$395$	17.2498	0.867929
$396$	−4.45798	−0.224022
$397$	−23.2987	−1.16933	−0.584665	−	0.811275i	$-0.698774\pi$
$397$	−23.2987	−1.16933	−0.584665	+	0.811275i	$0.698774\pi$
$398$	60.9783	3.05656
$399$	41.6439	2.08480
$400$	−8.35311	−0.417656
$401$	29.8234	1.48931	0.744654	−	0.667451i	$-0.232615\pi$
$401$	29.8234	1.48931	0.744654	+	0.667451i	$0.232615\pi$
$402$	−47.0003	−2.34416
$403$	−5.67419	−0.282651
$404$	32.8708	1.63538
$405$	−52.1576	−2.59173
$406$	−113.994	−5.65744
$407$	−0.135468	−0.00671492
$408$	−20.7778	−1.02866
$409$	21.0010	1.03843	0.519217	−	0.854642i	$-0.326224\pi$
$409$	21.0010	1.03843	0.519217	+	0.854642i	$0.326224\pi$
$410$	9.41611	0.465028
$411$	−56.5581	−2.78981
$412$	−55.1802	−2.71853
$413$	15.4045	0.758005
$414$	−56.7652	−2.78986
$415$	1.49576	0.0734240
$416$	−7.13588	−0.349865
$417$	28.1041	1.37626
$418$	0.808500	0.0395450
$419$	36.8214	1.79884	0.899421	−	0.437084i	$-0.143989\pi$
$419$	36.8214	1.79884	0.899421	+	0.437084i	$0.143989\pi$
$420$	−75.3789	−3.67811
$421$	−1.82707	−0.0890461	−0.0445230	−	0.999008i	$-0.514177\pi$
$421$	−1.82707	−0.0890461	−0.0445230	+	0.999008i	$0.514177\pi$
$422$	36.8151	1.79213
$423$	−4.43060	−0.215423
$424$	1.50715	0.0731938
$425$	5.18062	0.251297
$426$	−84.3554	−4.08703
$427$	−21.5909	−1.04486
$428$	−62.1311	−3.00322
$429$	−1.31618	−0.0635460
$430$	26.5441	1.28007
$431$	6.28909	0.302935	0.151467	−	0.988462i	$-0.451600\pi$
$431$	6.28909	0.302935	0.151467	+	0.988462i	$0.451600\pi$
$432$	−47.8557	−2.30246
$433$	18.3178	0.880300	0.440150	−	0.897924i	$-0.354925\pi$
$433$	18.3178	0.880300	0.440150	+	0.897924i	$0.354925\pi$
$434$	23.2901	1.11796
$435$	−41.2254	−1.97660
$436$	9.04708	0.433276
$437$	6.69233	0.320137
$438$	93.1422	4.45051
$439$	−33.6213	−1.60466	−0.802330	−	0.596881i	$-0.796406\pi$
$439$	−33.6213	−1.60466	−0.802330	+	0.596881i	$0.796406\pi$
$440$	−0.675644	−0.0322101
$441$	145.859	6.94565
$442$	−9.92561	−0.472113
$443$	33.0363	1.56960	0.784801	−	0.619748i	$-0.212765\pi$
$443$	33.0363	1.56960	0.784801	+	0.619748i	$0.212765\pi$
$444$	−12.7934	−0.607147
$445$	−17.8536	−0.846343
$446$	37.3797	1.76998
$447$	16.6151	0.785867
$448$	52.2768	2.46985
$449$	−7.93248	−0.374357	−0.187178	−	0.982326i	$-0.559934\pi$
$449$	−7.93248	−0.374357	−0.187178	+	0.982326i	$0.559934\pi$
$450$	74.5510	3.51437
$451$	−0.438691	−0.0206571
$452$	7.78494	0.366173
$453$	4.75807	0.223553
$454$	40.0927	1.88164
$455$	−16.6244	−0.779365
$456$	35.2506	1.65076
$457$	11.2727	0.527313	0.263657	−	0.964617i	$-0.415071\pi$
$457$	11.2727	0.527313	0.263657	+	0.964617i	$0.415071\pi$
$458$	50.4615	2.35791
$459$	29.6802	1.38535
$460$	−12.1137	−0.564802
$461$	−26.7251	−1.24471	−0.622357	−	0.782733i	$-0.713825\pi$
$461$	−26.7251	−1.24471	−0.622357	+	0.782733i	$0.713825\pi$
$462$	5.40236	0.251341
$463$	−12.4635	−0.579230	−0.289615	−	0.957143i	$-0.593527\pi$
$463$	−12.4635	−0.579230	−0.289615	+	0.957143i	$0.593527\pi$
$464$	−23.3538	−1.08417
$465$	8.42272	0.390594
$466$	9.11813	0.422389
$467$	−17.3791	−0.804208	−0.402104	−	0.915594i	$-0.631721\pi$
$467$	−17.3791	−0.804208	−0.402104	+	0.915594i	$0.631721\pi$
$468$	−92.8499	−4.29199
$469$	27.6578	1.27712
$470$	−1.45446	−0.0670894
$471$	−10.9960	−0.506669
$472$	13.0395	0.600193
$473$	−1.23667	−0.0568623
$474$	−116.752	−5.36259
$475$	−8.78918	−0.403275
$476$	26.4836	1.21387
$477$	−3.25554	−0.149061
$478$	36.0479	1.64879
$479$	25.8020	1.17892	0.589462	−	0.807796i	$-0.299340\pi$
$479$	25.8020	1.17892	0.589462	+	0.807796i	$0.299340\pi$
$480$	10.5924	0.483476
$481$	−2.82151	−0.128650
$482$	−70.6456	−3.21782
$483$	44.7178	2.03473
$484$	−40.7996	−1.85453
$485$	22.6531	1.02862
$486$	208.359	9.45137
$487$	−15.4927	−0.702043	−0.351021	−	0.936367i	$-0.614166\pi$
$487$	−15.4927	−0.702043	−0.351021	+	0.936367i	$0.614166\pi$
$488$	−18.2762	−0.827324
$489$	3.44347	0.155719
$490$	47.8820	2.16309
$491$	16.1284	0.727863	0.363931	−	0.931426i	$-0.381434\pi$
$491$	16.1284	0.727863	0.363931	+	0.931426i	$0.381434\pi$
$492$	−41.4291	−1.86777
$493$	14.4841	0.652331
$494$	16.8393	0.757635
$495$	1.45943	0.0655965
$496$	4.77140	0.214242
$497$	49.6397	2.22665
$498$	−10.1238	−0.453657
$499$	20.3846	0.912541	0.456271	−	0.889841i	$-0.349185\pi$
$499$	20.3846	0.912541	0.456271	+	0.889841i	$0.349185\pi$
$500$	38.5031	1.72191
$501$	10.9036	0.487137
$502$	−15.2747	−0.681743
$503$	24.6042	1.09705	0.548523	−	0.836135i	$-0.315190\pi$
$503$	24.6042	1.09705	0.548523	+	0.836135i	$0.315190\pi$
$504$	175.950	7.83742
$505$	−10.7611	−0.478861
$506$	0.868178	0.0385952
$507$	17.3519	0.770625
$508$	32.2032	1.42879
$509$	−26.0142	−1.15306	−0.576529	−	0.817077i	$-0.695593\pi$
$509$	−26.0142	−1.15306	−0.576529	+	0.817077i	$0.695593\pi$
$510$	14.7335	0.652410
$511$	−54.8104	−2.42467
$512$	25.4586	1.12512
$513$	−50.3539	−2.22318
$514$	−21.4442	−0.945865
$515$	18.0646	0.796022
$516$	−116.789	−5.14135
$517$	0.0677625	0.00298019
$518$	11.5811	0.508843
$519$	63.8165	2.80123
$520$	−14.0722	−0.617106
$521$	25.9645	1.13752	0.568762	−	0.822502i	$-0.307423\pi$
$521$	25.9645	1.13752	0.568762	+	0.822502i	$0.307423\pi$
$522$	208.431	9.12279
$523$	−8.46869	−0.370310	−0.185155	−	0.982709i	$-0.559279\pi$
$523$	−8.46869	−0.370310	−0.185155	+	0.982709i	$0.559279\pi$
$524$	53.5638	2.33994
$525$	−58.7289	−2.56314
$526$	21.7386	0.947847
$527$	−2.95924	−0.128906
$528$	1.10677	0.0481662
$529$	−15.8137	−0.687552
$530$	−1.06872	−0.0464221
$531$	−28.1662	−1.22231
$532$	−44.9307	−1.94799
$533$	−9.13696	−0.395766
$534$	120.839	5.22922
$535$	20.3402	0.879382
$536$	23.4117	1.01123
$537$	74.4737	3.21378
$538$	52.3997	2.25911
$539$	−2.23079	−0.0960870
$540$	91.1447	3.92224
$541$	−37.9044	−1.62964	−0.814820	−	0.579714i	$-0.803164\pi$
$541$	−37.9044	−1.62964	−0.814820	+	0.579714i	$0.803164\pi$
$542$	10.6227	0.456283
$543$	30.4241	1.30562
$544$	−3.72155	−0.159560
$545$	−2.96179	−0.126869
$546$	112.519	4.81538
$547$	−9.37223	−0.400727	−0.200364	−	0.979722i	$-0.564212\pi$
$547$	−9.37223	−0.400727	−0.200364	+	0.979722i	$0.564212\pi$
$548$	61.0220	2.60673
$549$	39.4776	1.68486
$550$	−1.14020	−0.0486182
$551$	−24.5730	−1.04684
$552$	37.8526	1.61111
$553$	68.7037	2.92158
$554$	67.1936	2.85478
$555$	4.18823	0.177780
$556$	−30.3223	−1.28595
$557$	−24.1449	−1.02305	−0.511526	−	0.859268i	$-0.670920\pi$
$557$	−24.1449	−1.02305	−0.511526	+	0.859268i	$0.670920\pi$
$558$	−42.5845	−1.80274
$559$	−25.7572	−1.08941
$560$	13.9794	0.590738
$561$	−0.686424	−0.0289809
$562$	−43.1489	−1.82013
$563$	−41.0577	−1.73038	−0.865188	−	0.501448i	$-0.832801\pi$
$563$	−41.0577	−1.73038	−0.865188	+	0.501448i	$0.832801\pi$
$564$	6.39936	0.269461
$565$	−2.54859	−0.107220
$566$	34.7093	1.45894
$567$	−207.738	−8.72416
$568$	42.0189	1.76307
$569$	−10.7259	−0.449654	−0.224827	−	0.974399i	$-0.572182\pi$
$569$	−10.7259	−0.449654	−0.224827	+	0.974399i	$0.572182\pi$
$570$	−24.9961	−1.04697
$571$	−4.91688	−0.205765	−0.102882	−	0.994694i	$-0.532807\pi$
$571$	−4.91688	−0.205765	−0.102882	+	0.994694i	$0.532807\pi$
$572$	1.42007	0.0593759
$573$	34.3886	1.43660
$574$	37.5032	1.56535
$575$	−9.43794	−0.393589
$576$	−95.5849	−3.98271
$577$	−33.8928	−1.41098	−0.705488	−	0.708722i	$-0.749272\pi$
$577$	−33.8928	−1.41098	−0.705488	+	0.708722i	$0.749272\pi$
$578$	35.4647	1.47514
$579$	35.9016	1.49202
$580$	44.4791	1.84689
$581$	5.95743	0.247156
$582$	−153.323	−6.35545
$583$	0.0497908	0.00206213
$584$	−46.3957	−1.91987
$585$	30.3967	1.25675
$586$	13.5430	0.559455
$587$	3.47997	0.143634	0.0718169	−	0.997418i	$-0.477120\pi$
$587$	3.47997	0.143634	0.0718169	+	0.997418i	$0.477120\pi$
$588$	−210.672	−8.68795
$589$	5.02049	0.206866
$590$	−9.24629	−0.380664
$591$	18.6299	0.766333
$592$	2.37260	0.0975132
$593$	−42.5094	−1.74565	−0.872825	−	0.488033i	$-0.837715\pi$
$593$	−42.5094	−1.74565	−0.872825	+	0.488033i	$0.837715\pi$
$594$	−6.53229	−0.268023
$595$	−8.67007	−0.355438
$596$	−17.9265	−0.734297
$597$	87.8322	3.59473
$598$	18.0823	0.739438
$599$	29.1479	1.19095	0.595476	−	0.803373i	$-0.296963\pi$
$599$	29.1479	1.19095	0.595476	+	0.803373i	$0.296963\pi$
$600$	−49.7126	−2.02951
$601$	−4.18290	−0.170624	−0.0853120	−	0.996354i	$-0.527189\pi$
$601$	−4.18290	−0.170624	−0.0853120	+	0.996354i	$0.527189\pi$
$602$	105.722	4.30891
$603$	−50.5706	−2.05939
$604$	−5.13360	−0.208883
$605$	13.3568	0.543030
$606$	72.8343	2.95869
$607$	6.05556	0.245788	0.122894	−	0.992420i	$-0.460783\pi$
$607$	6.05556	0.245788	0.122894	+	0.992420i	$0.460783\pi$
$608$	6.31378	0.256058
$609$	−164.196	−6.65354
$610$	12.9596	0.524718
$611$	1.41134	0.0570969
$612$	−48.4237	−1.95741
$613$	25.6424	1.03569	0.517844	−	0.855475i	$-0.326735\pi$
$613$	25.6424	1.03569	0.517844	+	0.855475i	$0.326735\pi$
$614$	−5.98245	−0.241432
$615$	13.5628	0.546906
$616$	−2.69101	−0.108424
$617$	30.6352	1.23333	0.616663	−	0.787227i	$-0.288484\pi$
$617$	30.6352	1.23333	0.616663	+	0.787227i	$0.288484\pi$
$618$	−122.267	−4.91830
$619$	−24.5398	−0.986336	−0.493168	−	0.869934i	$-0.664161\pi$
$619$	−24.5398	−0.986336	−0.493168	+	0.869934i	$0.664161\pi$
$620$	−9.08749	−0.364963
$621$	−54.0707	−2.16978
$622$	0.473951	0.0190037
$623$	−71.1089	−2.84892
$624$	23.0517	0.922806
$625$	4.99837	0.199935
$626$	8.31226	0.332225
$627$	1.16455	0.0465077
$628$	11.8639	0.473420
$629$	−1.47149	−0.0586722
$630$	−124.765	−4.97077
$631$	−38.2059	−1.52095	−0.760476	−	0.649366i	$-0.775034\pi$
$631$	−38.2059	−1.52095	−0.760476	+	0.649366i	$0.775034\pi$
$632$	58.1561	2.31332
$633$	53.0280	2.10767
$634$	−3.50872	−0.139349
$635$	−10.5425	−0.418367
$636$	4.70215	0.186452
$637$	−46.4625	−1.84091
$638$	−3.18779	−0.126206
$639$	−90.7632	−3.59054
$640$	−25.2261	−0.997150
$641$	0.879418	0.0347349	0.0173675	−	0.999849i	$-0.494471\pi$
$641$	0.879418	0.0347349	0.0173675	+	0.999849i	$0.494471\pi$
$642$	−137.669	−5.43335
$643$	−14.1166	−0.556702	−0.278351	−	0.960479i	$-0.589788\pi$
$643$	−14.1166	−0.556702	−0.278351	+	0.960479i	$0.589788\pi$
$644$	−48.2472	−1.90121
$645$	38.2338	1.50545
$646$	8.78212	0.345528
$647$	39.5192	1.55366	0.776831	−	0.629710i	$-0.216826\pi$
$647$	39.5192	1.55366	0.776831	+	0.629710i	$0.216826\pi$
$648$	−175.845	−6.90784
$649$	0.430779	0.0169096
$650$	−23.7478	−0.931466
$651$	33.5467	1.31480
$652$	−3.71525	−0.145501
$653$	5.16929	0.202290	0.101145	−	0.994872i	$-0.467749\pi$
$653$	5.16929	0.202290	0.101145	+	0.994872i	$0.467749\pi$
$654$	20.0463	0.783872
$655$	−17.5354	−0.685166
$656$	7.68323	0.299980
$657$	100.217	3.90986
$658$	−5.79295	−0.225833
$659$	−16.1383	−0.628659	−0.314329	−	0.949314i	$-0.601780\pi$
$659$	−16.1383	−0.628659	−0.314329	+	0.949314i	$0.601780\pi$
$660$	−2.10793	−0.0820512
$661$	36.1994	1.40799	0.703996	−	0.710204i	$-0.251397\pi$
$661$	36.1994	1.40799	0.703996	+	0.710204i	$0.251397\pi$
$662$	41.8458	1.62638
$663$	−14.2967	−0.555238
$664$	5.04283	0.195700
$665$	14.7092	0.570398
$666$	−21.1753	−0.820525
$667$	−26.3868	−1.02170
$668$	−11.7642	−0.455170
$669$	53.8412	2.08162
$670$	−16.6011	−0.641358
$671$	−0.603779	−0.0233086
$672$	42.1884	1.62745
$673$	−1.69631	−0.0653879	−0.0326940	−	0.999465i	$-0.510409\pi$
$673$	−1.69631	−0.0653879	−0.0326940	+	0.999465i	$0.510409\pi$
$674$	−17.7207	−0.682577
$675$	71.0123	2.73327
$676$	−18.7214	−0.720055
$677$	38.6560	1.48567	0.742835	−	0.669474i	$-0.233481\pi$
$677$	38.6560	1.48567	0.742835	+	0.669474i	$0.233481\pi$
$678$	17.2497	0.662470
$679$	90.2245	3.46250
$680$	−7.33901	−0.281438
$681$	57.7489	2.21294
$682$	0.651295	0.0249394
$683$	−48.6319	−1.86085	−0.930425	−	0.366483i	$-0.880562\pi$
$683$	−48.6319	−1.86085	−0.930425	+	0.366483i	$0.880562\pi$
$684$	82.1531	3.14120
$685$	−19.9771	−0.763285
$686$	109.641	4.18610
$687$	72.6840	2.77307
$688$	21.6591	0.825746
$689$	1.03703	0.0395078
$690$	−26.8412	−1.02183
$691$	9.63903	0.366686	0.183343	−	0.983049i	$-0.441308\pi$
$691$	9.63903	0.366686	0.183343	+	0.983049i	$0.441308\pi$
$692$	−68.8533	−2.61741
$693$	5.81274	0.220808
$694$	−15.4869	−0.587874
$695$	9.92675	0.376543
$696$	−138.988	−5.26832
$697$	−4.76516	−0.180493
$698$	26.6574	1.00900
$699$	13.1336	0.496759
$700$	63.3642	2.39494
$701$	26.9865	1.01927	0.509633	−	0.860392i	$-0.329781\pi$
$701$	26.9865	1.01927	0.509633	+	0.860392i	$0.329781\pi$
$702$	−136.053	−5.13500
$703$	2.49646	0.0941557
$704$	1.46190	0.0550973
$705$	−2.09499	−0.0789018
$706$	−1.61783	−0.0608880
$707$	−42.8600	−1.61192
$708$	40.6819	1.52892
$709$	4.55408	0.171032	0.0855160	−	0.996337i	$-0.472746\pi$
$709$	4.55408	0.171032	0.0855160	+	0.996337i	$0.472746\pi$
$710$	−29.7954	−1.11820
$711$	−125.621	−4.71114
$712$	−60.1920	−2.25579
$713$	5.39107	0.201897
$714$	58.6818	2.19611
$715$	−0.464894	−0.0173860
$716$	−80.3516	−3.00288
$717$	51.9229	1.93910
$718$	45.7463	1.70724
$719$	16.1375	0.601828	0.300914	−	0.953651i	$-0.402708\pi$
$719$	16.1375	0.601828	0.300914	+	0.953651i	$0.402708\pi$
$720$	−25.5605	−0.952584
$721$	71.9492	2.67953
$722$	30.5232	1.13596
$723$	−101.757	−3.78438
$724$	−32.8254	−1.21994
$725$	34.6544	1.28703
$726$	−90.4028	−3.35516
$727$	49.2264	1.82570	0.912852	−	0.408290i	$-0.133875\pi$
$727$	49.2264	1.82570	0.912852	+	0.408290i	$0.133875\pi$
$728$	−56.0478	−2.07727
$729$	171.469	6.35071
$730$	32.8991	1.21765
$731$	−13.4330	−0.496839
$732$	−57.0197	−2.10751
$733$	11.6928	0.431885	0.215943	−	0.976406i	$-0.430718\pi$
$733$	11.6928	0.431885	0.215943	+	0.976406i	$0.430718\pi$
$734$	−18.3829	−0.678524
$735$	68.9686	2.54394
$736$	6.77983	0.249908
$737$	0.773437	0.0284899
$738$	−68.5724	−2.52418
$739$	−35.9020	−1.32068	−0.660338	−	0.750969i	$-0.729587\pi$
$739$	−35.9020	−1.32068	−0.660338	+	0.750969i	$0.729587\pi$
$740$	−4.51879	−0.166114
$741$	24.2551	0.891032
$742$	−4.25657	−0.156264
$743$	−10.8646	−0.398582	−0.199291	−	0.979940i	$-0.563864\pi$
$743$	−10.8646	−0.398582	−0.199291	+	0.979940i	$0.563864\pi$
$744$	28.3965	1.04107
$745$	5.86867	0.215012
$746$	−51.1321	−1.87208
$747$	−10.8928	−0.398547
$748$	0.740601	0.0270791
$749$	81.0124	2.96013
$750$	85.3142	3.11524
$751$	10.4183	0.380171	0.190085	−	0.981768i	$-0.439124\pi$
$751$	10.4183	0.380171	0.190085	+	0.981768i	$0.439124\pi$
$752$	−1.18679	−0.0432779
$753$	−22.0015	−0.801778
$754$	−66.3947	−2.41795
$755$	1.68061	0.0611638
$756$	363.018	13.2029
$757$	−3.71037	−0.134856	−0.0674278	−	0.997724i	$-0.521479\pi$
$757$	−3.71037	−0.134856	−0.0674278	+	0.997724i	$0.521479\pi$
$758$	−24.9265	−0.905370
$759$	1.25051	0.0453907
$760$	12.4510	0.451645
$761$	14.2602	0.516932	0.258466	−	0.966020i	$-0.416783\pi$
$761$	14.2602	0.516932	0.258466	+	0.966020i	$0.416783\pi$
$762$	71.3552	2.58492
$763$	−11.7964	−0.427060
$764$	−37.1027	−1.34233
$765$	15.8527	0.573155
$766$	−59.3772	−2.14538
$767$	8.97218	0.323967
$768$	96.4186	3.47920
$769$	−11.1981	−0.403814	−0.201907	−	0.979405i	$-0.564714\pi$
$769$	−11.1981	−0.403814	−0.201907	+	0.979405i	$0.564714\pi$
$770$	1.90819	0.0687663
$771$	−30.8880	−1.11240
$772$	−38.7352	−1.39411
$773$	11.7467	0.422500	0.211250	−	0.977432i	$-0.432247\pi$
$773$	11.7467	0.422500	0.211250	+	0.977432i	$0.432247\pi$
$774$	−193.306	−6.94825
$775$	−7.08021	−0.254329
$776$	76.3729	2.74163
$777$	16.6812	0.598435
$778$	−46.9561	−1.68346
$779$	8.08433	0.289651
$780$	−43.9036	−1.57200
$781$	1.38815	0.0496719
$782$	9.43037	0.337229
$783$	198.538	7.09516
$784$	39.0701	1.39536
$785$	−3.88394	−0.138624
$786$	118.685	4.23337
$787$	32.6730	1.16467	0.582333	−	0.812950i	$-0.302140\pi$
$787$	32.6730	1.16467	0.582333	+	0.812950i	$0.302140\pi$
$788$	−20.1003	−0.716044
$789$	31.3119	1.11473
$790$	−41.2383	−1.46719
$791$	−10.1507	−0.360919
$792$	4.92035	0.174837
$793$	−12.5754	−0.446565
$794$	55.6993	1.97670
$795$	−1.53936	−0.0545956
$796$	−94.7645	−3.35884
$797$	22.2309	0.787457	0.393729	−	0.919227i	$-0.371185\pi$
$797$	22.2309	0.787457	0.393729	+	0.919227i	$0.371185\pi$
$798$	−99.5565	−3.52426
$799$	0.736053	0.0260397
$800$	−8.90410	−0.314807
$801$	130.018	4.59397
$802$	−71.2975	−2.51760
$803$	−1.53275	−0.0540894
$804$	73.0418	2.57599
$805$	15.7949	0.556698
$806$	13.5651	0.477809
$807$	75.4758	2.65687
$808$	−36.2800	−1.27633
$809$	−20.5211	−0.721485	−0.360742	−	0.932665i	$-0.617477\pi$
$809$	−20.5211	−0.721485	−0.360742	+	0.932665i	$0.617477\pi$
$810$	124.691	4.38120
$811$	−47.1246	−1.65477	−0.827385	−	0.561635i	$-0.810172\pi$
$811$	−47.1246	−1.65477	−0.827385	+	0.561635i	$0.810172\pi$
$812$	177.155	6.21692
$813$	15.3008	0.536621
$814$	0.323859	0.0113513
$815$	1.21628	0.0426044
$816$	12.0221	0.420856
$817$	22.7898	0.797315
$818$	−50.2063	−1.75542
$819$	121.067	4.23041
$820$	−14.6333	−0.511017
$821$	−0.239658	−0.00836413	−0.00418206	−	0.999991i	$-0.501331\pi$
$821$	−0.239658	−0.00836413	−0.00418206	+	0.999991i	$0.501331\pi$
$822$	135.211	4.71603
$823$	−47.4038	−1.65239	−0.826197	−	0.563381i	$-0.809500\pi$
$823$	−47.4038	−1.65239	−0.826197	+	0.563381i	$0.809500\pi$
$824$	60.9033	2.12167
$825$	−1.64233	−0.0571784
$826$	−36.8269	−1.28137
$827$	10.0816	0.350571	0.175286	−	0.984518i	$-0.443915\pi$
$827$	10.0816	0.350571	0.175286	+	0.984518i	$0.443915\pi$
$828$	88.2171	3.06576
$829$	−29.4496	−1.02283	−0.511413	−	0.859335i	$-0.670878\pi$
$829$	−29.4496	−1.02283	−0.511413	+	0.859335i	$0.670878\pi$
$830$	−3.57585	−0.124120
$831$	96.7847	3.35742
$832$	30.4481	1.05560
$833$	−24.2314	−0.839568
$834$	−67.1874	−2.32651
$835$	3.85130	0.133280
$836$	−1.25647	−0.0434558
$837$	−40.5631	−1.40207
$838$	−88.0273	−3.04085
$839$	−3.79823	−0.131129	−0.0655647	−	0.997848i	$-0.520885\pi$
$839$	−3.79823	−0.131129	−0.0655647	+	0.997848i	$0.520885\pi$
$840$	83.1969	2.87057
$841$	67.8875	2.34095
$842$	4.36791	0.150528
$843$	−62.1511	−2.14060
$844$	−57.2133	−1.96936
$845$	6.12893	0.210841
$846$	10.5921	0.364162
$847$	53.1984	1.82792
$848$	−0.872038	−0.0299459
$849$	49.9948	1.71582
$850$	−12.3851	−0.424806
$851$	2.68073	0.0918942
$852$	131.094	4.49121
$853$	−9.19968	−0.314991	−0.157496	−	0.987520i	$-0.550342\pi$
$853$	−9.19968	−0.314991	−0.157496	+	0.987520i	$0.550342\pi$
$854$	51.6165	1.76628
$855$	−26.8949	−0.919785
$856$	68.5751	2.34385
$857$	31.8111	1.08665	0.543324	−	0.839523i	$-0.317166\pi$
$857$	31.8111	1.08665	0.543324	+	0.839523i	$0.317166\pi$
$858$	3.14655	0.107421
$859$	−21.1043	−0.720068	−0.360034	−	0.932939i	$-0.617235\pi$
$859$	−21.1043	−0.720068	−0.360034	+	0.932939i	$0.617235\pi$
$860$	−41.2514	−1.40666
$861$	54.0191	1.84097
$862$	−15.0351	−0.512096
$863$	−3.17301	−0.108010	−0.0540052	−	0.998541i	$-0.517199\pi$
$863$	−3.17301	−0.108010	−0.0540052	+	0.998541i	$0.517199\pi$
$864$	−51.0123	−1.73547
$865$	22.5408	0.766411
$866$	−43.7917	−1.48810
$867$	51.0829	1.73487
$868$	−36.1944	−1.22852
$869$	1.92127	0.0651745
$870$	98.5558	3.34135
$871$	16.1090	0.545832
$872$	−9.98541	−0.338149
$873$	−164.970	−5.58339
$874$	−15.9991	−0.541177
$875$	−50.2040	−1.69720
$876$	−144.749	−4.89063
$877$	15.6874	0.529726	0.264863	−	0.964286i	$-0.414673\pi$
$877$	15.6874	0.529726	0.264863	+	0.964286i	$0.414673\pi$
$878$	80.3772	2.71260
$879$	19.5071	0.657959
$880$	0.390927	0.0131782
$881$	19.3735	0.652709	0.326355	−	0.945247i	$-0.394180\pi$
$881$	19.3735	0.652709	0.326355	+	0.945247i	$0.394180\pi$
$882$	−348.698	−11.7413
$883$	55.2554	1.85949	0.929747	−	0.368200i	$-0.120026\pi$
$883$	55.2554	1.85949	0.929747	+	0.368200i	$0.120026\pi$
$884$	15.4251	0.518802
$885$	−13.3182	−0.447687
$886$	−78.9786	−2.65334
$887$	30.0646	1.00947	0.504735	−	0.863274i	$-0.331590\pi$
$887$	30.0646	1.00947	0.504735	+	0.863274i	$0.331590\pi$
$888$	14.1203	0.473845
$889$	−41.9896	−1.40829
$890$	42.6820	1.43070
$891$	−5.80928	−0.194618
$892$	−58.0907	−1.94502
$893$	−1.24875	−0.0417878
$894$	−39.7210	−1.32847
$895$	26.3051	0.879283
$896$	−100.473	−3.35655
$897$	26.0454	0.869632
$898$	18.9639	0.632832
$899$	−19.7950	−0.660201
$900$	−115.857	−3.86192
$901$	0.540840	0.0180180
$902$	1.04876	0.0349199
$903$	152.281	5.06758
$904$	−8.59236	−0.285778
$905$	10.7462	0.357216
$906$	−11.3749	−0.377906
$907$	−0.0685022	−0.00227458	−0.00113729	−	0.999999i	$-0.500362\pi$
$907$	−0.0685022	−0.00227458	−0.00113729	+	0.999999i	$0.500362\pi$
$908$	−62.3068	−2.06772
$909$	78.3669	2.59927
$910$	39.7433	1.31748
$911$	−10.2965	−0.341139	−0.170570	−	0.985346i	$-0.554561\pi$
$911$	−10.2965	−0.341139	−0.170570	+	0.985346i	$0.554561\pi$
$912$	−20.3960	−0.675379
$913$	0.166597	0.00551355
$914$	−26.9491	−0.891398
$915$	18.6668	0.617106
$916$	−78.4207	−2.59109
$917$	−69.8415	−2.30637
$918$	−70.9553	−2.34187
$919$	33.7802	1.11430	0.557152	−	0.830410i	$-0.311894\pi$
$919$	33.7802	1.11430	0.557152	+	0.830410i	$0.311894\pi$
$920$	13.3700	0.440797
$921$	−8.61704	−0.283941
$922$	63.8907	2.10413
$923$	28.9121	0.951655
$924$	−8.39565	−0.276197
$925$	−3.52066	−0.115759
$926$	29.7961	0.979160
$927$	−131.555	−4.32082
$928$	−24.8943	−0.817195
$929$	−48.7448	−1.59927	−0.799633	−	0.600489i	$-0.794973\pi$
$929$	−48.7448	−1.59927	−0.799633	+	0.600489i	$0.794973\pi$
$930$	−20.1359	−0.660281
$931$	41.1097	1.34732
$932$	−14.1702	−0.464161
$933$	0.682673	0.0223497
$934$	41.5475	1.35947
$935$	−0.242454	−0.00792910
$936$	102.480	3.34966
$937$	32.7285	1.06919	0.534597	−	0.845107i	$-0.320463\pi$
$937$	32.7285	1.06919	0.534597	+	0.845107i	$0.320463\pi$
$938$	−66.1204	−2.15891
$939$	11.9729	0.390720
$940$	2.26034	0.0737241
$941$	59.1937	1.92966	0.964829	−	0.262877i	$-0.0846713\pi$
$941$	59.1937	1.92966	0.964829	+	0.262877i	$0.0846713\pi$
$942$	26.2877	0.856500
$943$	8.68106	0.282694
$944$	−7.54467	−0.245558
$945$	−118.843	−3.86597
$946$	2.95647	0.0961230
$947$	−0.381041	−0.0123822	−0.00619109	−	0.999981i	$-0.501971\pi$
$947$	−0.381041	−0.0123822	−0.00619109	+	0.999981i	$0.501971\pi$
$948$	181.441	5.89291
$949$	−31.9237	−1.03629
$950$	21.0119	0.681717
$951$	−5.05391	−0.163884
$952$	−29.2304	−0.947363
$953$	−21.4792	−0.695779	−0.347889	−	0.937536i	$-0.613101\pi$
$953$	−21.4792	−0.695779	−0.347889	+	0.937536i	$0.613101\pi$
$954$	7.78288	0.251980
$955$	12.1465	0.393051
$956$	−56.0210	−1.81185
$957$	−4.59165	−0.148427
$958$	−61.6838	−1.99291
$959$	−79.5663	−2.56933
$960$	−45.1969	−1.45872
$961$	−26.9557	−0.869539
$962$	6.74527	0.217476
$963$	−148.126	−4.77330
$964$	109.788	3.53604
$965$	12.6809	0.408213
$966$	−106.905	−3.43961
$967$	−18.5038	−0.595041	−0.297521	−	0.954715i	$-0.596160\pi$
$967$	−18.5038	−0.595041	−0.297521	+	0.954715i	$0.596160\pi$
$968$	45.0312	1.44736
$969$	12.6497	0.406365
$970$	−54.1558	−1.73884
$971$	−22.8473	−0.733203	−0.366602	−	0.930378i	$-0.619479\pi$
$971$	−22.8473	−0.733203	−0.366602	+	0.930378i	$0.619479\pi$
$972$	−323.805	−10.3860
$973$	39.5371	1.26750
$974$	37.0378	1.18677
$975$	−34.2060	−1.09547
$976$	10.5746	0.338485
$977$	−50.9207	−1.62910	−0.814549	−	0.580095i	$-0.803016\pi$
$977$	−50.9207	−1.62910	−0.814549	+	0.580095i	$0.803016\pi$
$978$	−8.23217	−0.263236
$979$	−1.98853	−0.0635535
$980$	−74.4120	−2.37700
$981$	21.5691	0.688647
$982$	−38.5574	−1.23042
$983$	−33.2930	−1.06188	−0.530940	−	0.847409i	$-0.678161\pi$
$983$	−33.2930	−1.06188	−0.530940	+	0.847409i	$0.678161\pi$
$984$	45.7259	1.45769
$985$	6.58034	0.209667
$986$	−34.6266	−1.10273
$987$	−8.34409	−0.265595
$988$	−26.1694	−0.832560
$989$	24.4720	0.778165
$990$	−3.48900	−0.110888
$991$	14.1141	0.448350	0.224175	−	0.974549i	$-0.428031\pi$
$991$	14.1141	0.448350	0.224175	+	0.974549i	$0.428031\pi$
$992$	5.08613	0.161485
$993$	60.2741	1.91274
$994$	−118.672	−3.76404
$995$	31.0235	0.983512
$996$	15.7331	0.498521
$997$	36.7574	1.16412	0.582059	−	0.813147i	$-0.302247\pi$
$997$	36.7574	1.16412	0.582059	+	0.813147i	$0.302247\pi$
$998$	−48.7327	−1.54261
$999$	−20.1702	−0.638156

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6031.2.a.e.1.15	✓	134

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6031.2.a.e.1.15	✓	134	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:	\( N \)	\(=\)	\( 6031 = 37 \cdot 163 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.39066 q^{2} -3.44347 q^{3} +3.71525 q^{4} -1.21628 q^{5} +8.23217 q^{6} -4.84430 q^{7} -4.10059 q^{8} +8.85751 q^{9} +O(q^{10})\)	\(q-2.39066 q^{2} -3.44347 q^{3} +3.71525 q^{4} -1.21628 q^{5} +8.23217 q^{6} -4.84430 q^{7} -4.10059 q^{8} +8.85751 q^{9} +2.90771 q^{10} -0.135468 q^{11} -12.7934 q^{12} -2.82151 q^{13} +11.5811 q^{14} +4.18823 q^{15} +2.37260 q^{16} -1.47149 q^{17} -21.1753 q^{18} +2.49646 q^{19} -4.51879 q^{20} +16.6812 q^{21} +0.323859 q^{22} +2.68073 q^{23} +14.1203 q^{24} -3.52066 q^{25} +6.74527 q^{26} -20.1702 q^{27} -17.9978 q^{28} -9.84315 q^{29} -10.0126 q^{30} +2.01105 q^{31} +2.52910 q^{32} +0.466482 q^{33} +3.51784 q^{34} +5.89203 q^{35} +32.9079 q^{36} +1.00000 q^{37} -5.96818 q^{38} +9.71580 q^{39} +4.98746 q^{40} +3.23832 q^{41} -39.8791 q^{42} +9.12887 q^{43} -0.503300 q^{44} -10.7732 q^{45} -6.40871 q^{46} -0.500209 q^{47} -8.16998 q^{48} +16.4672 q^{49} +8.41670 q^{50} +5.06704 q^{51} -10.4826 q^{52} -0.367546 q^{53} +48.2200 q^{54} +0.164768 q^{55} +19.8645 q^{56} -8.59648 q^{57} +23.5316 q^{58} -3.17992 q^{59} +15.5603 q^{60} +4.45697 q^{61} -4.80773 q^{62} -42.9084 q^{63} -10.7914 q^{64} +3.43175 q^{65} -1.11520 q^{66} -5.70935 q^{67} -5.46696 q^{68} -9.23102 q^{69} -14.0858 q^{70} -10.2470 q^{71} -36.3210 q^{72} +11.3144 q^{73} -2.39066 q^{74} +12.1233 q^{75} +9.27497 q^{76} +0.656250 q^{77} -23.2272 q^{78} -14.1824 q^{79} -2.88574 q^{80} +42.8829 q^{81} -7.74172 q^{82} -1.22978 q^{83} +61.9749 q^{84} +1.78975 q^{85} -21.8240 q^{86} +33.8946 q^{87} +0.555500 q^{88} +14.6789 q^{89} +25.7551 q^{90} +13.6682 q^{91} +9.95959 q^{92} -6.92498 q^{93} +1.19583 q^{94} -3.03639 q^{95} -8.70888 q^{96} -18.6249 q^{97} -39.3676 q^{98} -1.19991 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 134 q + 9 q^{2} + 7 q^{3} + 149 q^{4} + 22 q^{5} + 20 q^{6} + 11 q^{7} + 27 q^{8} + 181 q^{9}+O(q^{10}) \) 134 * q + 9 * q^2 + 7 * q^3 + 149 * q^4 + 22 * q^5 + 20 * q^6 + 11 * q^7 + 27 * q^8 + 181 * q^9	\( 134 q + 9 q^{2} + 7 q^{3} + 149 q^{4} + 22 q^{5} + 20 q^{6} + 11 q^{7} + 27 q^{8} + 181 q^{9} + 15 q^{10} + 20 q^{11} + 28 q^{12} + 11 q^{13} + 17 q^{14} - 13 q^{15} + 143 q^{16} + 76 q^{17} + 23 q^{18} + 15 q^{19} + 67 q^{20} + 63 q^{21} + 2 q^{22} + 22 q^{23} + 33 q^{24} + 160 q^{25} + 65 q^{26} + 31 q^{27} + 10 q^{28} + 73 q^{29} + 20 q^{30} + 10 q^{31} + 53 q^{32} + 72 q^{33} - 7 q^{34} + 52 q^{35} + 201 q^{36} + 134 q^{37} + 70 q^{38} + 6 q^{39} + 11 q^{40} + 182 q^{41} - 15 q^{42} + 12 q^{43} + 33 q^{44} + 29 q^{45} + 24 q^{46} + 80 q^{47} + 21 q^{48} + 229 q^{49} + 37 q^{50} + 57 q^{51} - 15 q^{52} + 75 q^{53} + 95 q^{54} - 9 q^{55} + 39 q^{56} + 19 q^{57} - 21 q^{58} + 91 q^{59} + 62 q^{60} + 58 q^{61} + 108 q^{62} + 9 q^{63} + 167 q^{64} + 76 q^{65} + 105 q^{66} - 17 q^{67} + 109 q^{68} + 48 q^{69} - 55 q^{70} + 56 q^{71} + 48 q^{72} + 54 q^{73} + 9 q^{74} + 28 q^{75} + 82 q^{76} + 156 q^{77} + 16 q^{78} - 2 q^{79} + 98 q^{80} + 270 q^{81} - 42 q^{82} + 130 q^{83} + 229 q^{84} + 22 q^{85} + 72 q^{86} + 22 q^{87} + 61 q^{88} + 157 q^{89} + 176 q^{90} + 31 q^{91} - 18 q^{92} + 36 q^{93} + 83 q^{94} + 98 q^{95} + 111 q^{96} + 35 q^{97} + 53 q^{98} + 55 q^{99}+O(q^{100}) \) 134 * q + 9 * q^2 + 7 * q^3 + 149 * q^4 + 22 * q^5 + 20 * q^6 + 11 * q^7 + 27 * q^8 + 181 * q^9 + 15 * q^10 + 20 * q^11 + 28 * q^12 + 11 * q^13 + 17 * q^14 - 13 * q^15 + 143 * q^16 + 76 * q^17 + 23 * q^18 + 15 * q^19 + 67 * q^20 + 63 * q^21 + 2 * q^22 + 22 * q^23 + 33 * q^24 + 160 * q^25 + 65 * q^26 + 31 * q^27 + 10 * q^28 + 73 * q^29 + 20 * q^30 + 10 * q^31 + 53 * q^32 + 72 * q^33 - 7 * q^34 + 52 * q^35 + 201 * q^36 + 134 * q^37 + 70 * q^38 + 6 * q^39 + 11 * q^40 + 182 * q^41 - 15 * q^42 + 12 * q^43 + 33 * q^44 + 29 * q^45 + 24 * q^46 + 80 * q^47 + 21 * q^48 + 229 * q^49 + 37 * q^50 + 57 * q^51 - 15 * q^52 + 75 * q^53 + 95 * q^54 - 9 * q^55 + 39 * q^56 + 19 * q^57 - 21 * q^58 + 91 * q^59 + 62 * q^60 + 58 * q^61 + 108 * q^62 + 9 * q^63 + 167 * q^64 + 76 * q^65 + 105 * q^66 - 17 * q^67 + 109 * q^68 + 48 * q^69 - 55 * q^70 + 56 * q^71 + 48 * q^72 + 54 * q^73 + 9 * q^74 + 28 * q^75 + 82 * q^76 + 156 * q^77 + 16 * q^78 - 2 * q^79 + 98 * q^80 + 270 * q^81 - 42 * q^82 + 130 * q^83 + 229 * q^84 + 22 * q^85 + 72 * q^86 + 22 * q^87 + 61 * q^88 + 157 * q^89 + 176 * q^90 + 31 * q^91 - 18 * q^92 + 36 * q^93 + 83 * q^94 + 98 * q^95 + 111 * q^96 + 35 * q^97 + 53 * q^98 + 55 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more