LMFDB - Embedded newform 6013.2.a.f.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6013,2,Mod(1,6013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6013, 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("6013.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6013 = 7 \cdot 859 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6013.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:	$48.0140467354$
Analytic rank:	$0$
Dimension:	$110$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	6013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.59895 q^{2} -1.56127 q^{3} +4.75456 q^{4} +1.42920 q^{5} +4.05766 q^{6} -1.00000 q^{7} -7.15899 q^{8} -0.562451 q^{9} +O(q^{10})$	\(q-2.59895 q^{2} -1.56127 q^{3} +4.75456 q^{4} +1.42920 q^{5} +4.05766 q^{6} -1.00000 q^{7} -7.15899 q^{8} -0.562451 q^{9} -3.71443 q^{10} -4.88622 q^{11} -7.42314 q^{12} -4.16954 q^{13} +2.59895 q^{14} -2.23136 q^{15} +9.09675 q^{16} +2.07475 q^{17} +1.46179 q^{18} +5.99965 q^{19} +6.79522 q^{20} +1.56127 q^{21} +12.6991 q^{22} +5.45092 q^{23} +11.1771 q^{24} -2.95739 q^{25} +10.8364 q^{26} +5.56193 q^{27} -4.75456 q^{28} +3.49804 q^{29} +5.79920 q^{30} +3.09159 q^{31} -9.32407 q^{32} +7.62869 q^{33} -5.39219 q^{34} -1.42920 q^{35} -2.67421 q^{36} +9.67701 q^{37} -15.5928 q^{38} +6.50976 q^{39} -10.2316 q^{40} -3.21585 q^{41} -4.05766 q^{42} -12.1279 q^{43} -23.2319 q^{44} -0.803855 q^{45} -14.1667 q^{46} +1.49318 q^{47} -14.2024 q^{48} +1.00000 q^{49} +7.68612 q^{50} -3.23924 q^{51} -19.8244 q^{52} +2.88894 q^{53} -14.4552 q^{54} -6.98339 q^{55} +7.15899 q^{56} -9.36705 q^{57} -9.09125 q^{58} +2.91756 q^{59} -10.6091 q^{60} -10.9478 q^{61} -8.03490 q^{62} +0.562451 q^{63} +6.03934 q^{64} -5.95911 q^{65} -19.8266 q^{66} -8.28007 q^{67} +9.86454 q^{68} -8.51033 q^{69} +3.71443 q^{70} -13.8106 q^{71} +4.02658 q^{72} +13.2740 q^{73} -25.1501 q^{74} +4.61727 q^{75} +28.5257 q^{76} +4.88622 q^{77} -16.9186 q^{78} -14.1452 q^{79} +13.0011 q^{80} -6.99629 q^{81} +8.35785 q^{82} +10.8204 q^{83} +7.42314 q^{84} +2.96523 q^{85} +31.5198 q^{86} -5.46137 q^{87} +34.9804 q^{88} -5.84514 q^{89} +2.08918 q^{90} +4.16954 q^{91} +25.9168 q^{92} -4.82679 q^{93} -3.88071 q^{94} +8.57470 q^{95} +14.5574 q^{96} -3.40602 q^{97} -2.59895 q^{98} +2.74826 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 110 q + 16 q^{2} + 29 q^{3} + 118 q^{4} + 12 q^{6} - 110 q^{7} + 57 q^{8} + 127 q^{9}+O(q^{10}) $ 110 * q + 16 * q^2 + 29 * q^3 + 118 * q^4 + 12 * q^6 - 110 * q^7 + 57 * q^8 + 127 * q^9	\( 110 q + 16 q^{2} + 29 q^{3} + 118 q^{4} + 12 q^{6} - 110 q^{7} + 57 q^{8} + 127 q^{9} + 3 q^{10} + 52 q^{11} + 62 q^{12} - 9 q^{13} - 16 q^{14} + 39 q^{15} + 146 q^{16} + 11 q^{17} + 60 q^{18} + 14 q^{19} + 18 q^{20} - 29 q^{21} + 32 q^{22} + 73 q^{23} + 24 q^{24} + 132 q^{25} - 7 q^{26} + 116 q^{27} - 118 q^{28} + 35 q^{29} + 18 q^{30} + 36 q^{31} + 140 q^{32} + 42 q^{33} - 7 q^{34} + 180 q^{36} + 49 q^{37} + 45 q^{39} + 6 q^{40} - 14 q^{41} - 12 q^{42} + 58 q^{43} + 92 q^{44} + 17 q^{45} + 27 q^{46} + 87 q^{47} + 98 q^{48} + 110 q^{49} + 91 q^{50} + 42 q^{51} + 16 q^{52} + 95 q^{53} + 41 q^{54} + 8 q^{55} - 57 q^{56} + 61 q^{57} + 46 q^{58} + 114 q^{59} + 81 q^{60} - 47 q^{61} + 31 q^{62} - 127 q^{63} + 199 q^{64} + 62 q^{65} + 21 q^{66} + 95 q^{67} + 60 q^{68} - 39 q^{69} - 3 q^{70} + 131 q^{71} + 186 q^{72} + 31 q^{73} + 23 q^{74} + 121 q^{75} + 14 q^{76} - 52 q^{77} + 110 q^{78} + 9 q^{79} + 61 q^{80} + 194 q^{81} + 45 q^{82} + 73 q^{83} - 62 q^{84} + 59 q^{85} + 72 q^{86} + 64 q^{87} + 100 q^{88} - 17 q^{89} + 11 q^{90} + 9 q^{91} + 192 q^{92} + 85 q^{93} - 11 q^{94} + 108 q^{95} + 68 q^{96} + 32 q^{97} + 16 q^{98} + 160 q^{99}+O(q^{100}) \) 110 * q + 16 * q^2 + 29 * q^3 + 118 * q^4 + 12 * q^6 - 110 * q^7 + 57 * q^8 + 127 * q^9 + 3 * q^10 + 52 * q^11 + 62 * q^12 - 9 * q^13 - 16 * q^14 + 39 * q^15 + 146 * q^16 + 11 * q^17 + 60 * q^18 + 14 * q^19 + 18 * q^20 - 29 * q^21 + 32 * q^22 + 73 * q^23 + 24 * q^24 + 132 * q^25 - 7 * q^26 + 116 * q^27 - 118 * q^28 + 35 * q^29 + 18 * q^30 + 36 * q^31 + 140 * q^32 + 42 * q^33 - 7 * q^34 + 180 * q^36 + 49 * q^37 + 45 * q^39 + 6 * q^40 - 14 * q^41 - 12 * q^42 + 58 * q^43 + 92 * q^44 + 17 * q^45 + 27 * q^46 + 87 * q^47 + 98 * q^48 + 110 * q^49 + 91 * q^50 + 42 * q^51 + 16 * q^52 + 95 * q^53 + 41 * q^54 + 8 * q^55 - 57 * q^56 + 61 * q^57 + 46 * q^58 + 114 * q^59 + 81 * q^60 - 47 * q^61 + 31 * q^62 - 127 * q^63 + 199 * q^64 + 62 * q^65 + 21 * q^66 + 95 * q^67 + 60 * q^68 - 39 * q^69 - 3 * q^70 + 131 * q^71 + 186 * q^72 + 31 * q^73 + 23 * q^74 + 121 * q^75 + 14 * q^76 - 52 * q^77 + 110 * q^78 + 9 * q^79 + 61 * q^80 + 194 * q^81 + 45 * q^82 + 73 * q^83 - 62 * q^84 + 59 * q^85 + 72 * q^86 + 64 * q^87 + 100 * q^88 - 17 * q^89 + 11 * q^90 + 9 * q^91 + 192 * q^92 + 85 * q^93 - 11 * q^94 + 108 * q^95 + 68 * q^96 + 32 * q^97 + 16 * q^98 + 160 * 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 $

$2$	−2.59895	−1.83774	−0.918869	−	0.394562i	$-0.870896\pi$
$2$	−2.59895	−1.83774	−0.918869	+	0.394562i	$0.870896\pi$
$3$	−1.56127	−0.901397	−0.450698	−	0.892676i	$-0.648825\pi$
$3$	−1.56127	−0.901397	−0.450698	+	0.892676i	$0.648825\pi$
$4$	4.75456	2.37728
$5$	1.42920	0.639158	0.319579	−	0.947560i	$-0.396459\pi$
$5$	1.42920	0.639158	0.319579	+	0.947560i	$0.396459\pi$
$6$	4.05766	1.65653
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−7.15899	−2.53108
$9$	−0.562451	−0.187484
$10$	−3.71443	−1.17460
$11$	−4.88622	−1.47325	−0.736626	−	0.676300i	$-0.763582\pi$
$11$	−4.88622	−1.47325	−0.736626	+	0.676300i	$0.763582\pi$
$12$	−7.42314	−2.14287
$13$	−4.16954	−1.15642	−0.578211	−	0.815887i	$-0.696249\pi$
$13$	−4.16954	−1.15642	−0.578211	+	0.815887i	$0.696249\pi$
$14$	2.59895	0.694600
$15$	−2.23136	−0.576135
$16$	9.09675	2.27419
$17$	2.07475	0.503201	0.251601	−	0.967831i	$-0.419043\pi$
$17$	2.07475	0.503201	0.251601	+	0.967831i	$0.419043\pi$
$18$	1.46179	0.344546
$19$	5.99965	1.37641	0.688207	−	0.725514i	$-0.258398\pi$
$19$	5.99965	1.37641	0.688207	+	0.725514i	$0.258398\pi$
$20$	6.79522	1.51946
$21$	1.56127	0.340696
$22$	12.6991	2.70745
$23$	5.45092	1.13660	0.568298	−	0.822823i	$-0.307602\pi$
$23$	5.45092	1.13660	0.568298	+	0.822823i	$0.307602\pi$
$24$	11.1771	2.28151
$25$	−2.95739	−0.591478
$26$	10.8364	2.12520
$27$	5.56193	1.07039
$28$	−4.75456	−0.898528
$29$	3.49804	0.649570	0.324785	−	0.945788i	$-0.394708\pi$
$29$	3.49804	0.649570	0.324785	+	0.945788i	$0.394708\pi$
$30$	5.79920	1.05878
$31$	3.09159	0.555266	0.277633	−	0.960687i	$-0.410450\pi$
$31$	3.09159	0.555266	0.277633	+	0.960687i	$0.410450\pi$
$32$	−9.32407	−1.64828
$33$	7.62869	1.32798
$34$	−5.39219	−0.924752
$35$	−1.42920	−0.241579
$36$	−2.67421	−0.445702
$37$	9.67701	1.59089	0.795445	−	0.606025i	$-0.207237\pi$
$37$	9.67701	1.59089	0.795445	+	0.606025i	$0.207237\pi$
$38$	−15.5928	−2.52949
$39$	6.50976	1.04240
$40$	−10.2316	−1.61776
$41$	−3.21585	−0.502232	−0.251116	−	0.967957i	$-0.580798\pi$
$41$	−3.21585	−0.502232	−0.251116	+	0.967957i	$0.580798\pi$
$42$	−4.05766	−0.626110
$43$	−12.1279	−1.84948	−0.924741	−	0.380597i	$-0.875719\pi$
$43$	−12.1279	−1.84948	−0.924741	+	0.380597i	$0.875719\pi$
$44$	−23.2319	−3.50234
$45$	−0.803855	−0.119832
$46$	−14.1667	−2.08877
$47$	1.49318	0.217803	0.108901	−	0.994053i	$-0.465267\pi$
$47$	1.49318	0.217803	0.108901	+	0.994053i	$0.465267\pi$
$48$	−14.2024	−2.04995
$49$	1.00000	0.142857
$50$	7.68612	1.08698
$51$	−3.23924	−0.453584
$52$	−19.8244	−2.74914
$53$	2.88894	0.396827	0.198413	−	0.980118i	$-0.436421\pi$
$53$	2.88894	0.396827	0.198413	+	0.980118i	$0.436421\pi$
$54$	−14.4552	−1.96710
$55$	−6.98339	−0.941640
$56$	7.15899	0.956660
$57$	−9.36705	−1.24070
$58$	−9.09125	−1.19374
$59$	2.91756	0.379834	0.189917	−	0.981800i	$-0.439178\pi$
$59$	2.91756	0.379834	0.189917	+	0.981800i	$0.439178\pi$
$60$	−10.6091	−1.36963
$61$	−10.9478	−1.40172	−0.700861	−	0.713298i	$-0.747201\pi$
$61$	−10.9478	−1.40172	−0.700861	+	0.713298i	$0.747201\pi$
$62$	−8.03490	−1.02043
$63$	0.562451	0.0708622
$64$	6.03934	0.754917
$65$	−5.95911	−0.739136
$66$	−19.8266	−2.44049
$67$	−8.28007	−1.01157	−0.505786	−	0.862659i	$-0.668797\pi$
$67$	−8.28007	−1.01157	−0.505786	+	0.862659i	$0.668797\pi$
$68$	9.86454	1.19625
$69$	−8.51033	−1.02452
$70$	3.71443	0.443959
$71$	−13.8106	−1.63902	−0.819508	−	0.573068i	$-0.805753\pi$
$71$	−13.8106	−1.63902	−0.819508	+	0.573068i	$0.805753\pi$
$72$	4.02658	0.474537
$73$	13.2740	1.55360	0.776802	−	0.629745i	$-0.216841\pi$
$73$	13.2740	1.55360	0.776802	+	0.629745i	$0.216841\pi$
$74$	−25.1501	−2.92364
$75$	4.61727	0.533156
$76$	28.5257	3.27213
$77$	4.88622	0.556837
$78$	−16.9186	−1.91565
$79$	−14.1452	−1.59145	−0.795727	−	0.605655i	$-0.792911\pi$
$79$	−14.1452	−1.59145	−0.795727	+	0.605655i	$0.792911\pi$
$80$	13.0011	1.45356
$81$	−6.99629	−0.777366
$82$	8.35785	0.922970
$83$	10.8204	1.18769	0.593846	−	0.804578i	$-0.297609\pi$
$83$	10.8204	1.18769	0.593846	+	0.804578i	$0.297609\pi$
$84$	7.42314	0.809930
$85$	2.96523	0.321625
$86$	31.5198	3.39886
$87$	−5.46137	−0.585520
$88$	34.9804	3.72893
$89$	−5.84514	−0.619583	−0.309792	−	0.950804i	$-0.600259\pi$
$89$	−5.84514	−0.619583	−0.309792	+	0.950804i	$0.600259\pi$
$90$	2.08918	0.220219
$91$	4.16954	0.437087
$92$	25.9168	2.70201
$93$	−4.82679	−0.500515
$94$	−3.88071	−0.400265
$95$	8.57470	0.879746
$96$	14.5574	1.48575
$97$	−3.40602	−0.345829	−0.172915	−	0.984937i	$-0.555318\pi$
$97$	−3.40602	−0.345829	−0.172915	+	0.984937i	$0.555318\pi$
$98$	−2.59895	−0.262534
$99$	2.74826	0.276211
$100$	−14.0611	−1.40611
$101$	−11.4081	−1.13515	−0.567573	−	0.823323i	$-0.692118\pi$
$101$	−11.4081	−1.13515	−0.567573	+	0.823323i	$0.692118\pi$
$102$	8.41863	0.833569
$103$	−6.34532	−0.625223	−0.312611	−	0.949881i	$-0.601204\pi$
$103$	−6.34532	−0.625223	−0.312611	+	0.949881i	$0.601204\pi$
$104$	29.8497	2.92700
$105$	2.23136	0.217758
$106$	−7.50823	−0.729263
$107$	16.2383	1.56981	0.784907	−	0.619614i	$-0.212711\pi$
$107$	16.2383	1.56981	0.784907	+	0.619614i	$0.212711\pi$
$108$	26.4446	2.54463
$109$	−18.8885	−1.80919	−0.904593	−	0.426276i	$-0.859825\pi$
$109$	−18.8885	−1.80919	−0.904593	+	0.426276i	$0.859825\pi$
$110$	18.1495	1.73049
$111$	−15.1084	−1.43402
$112$	−9.09675	−0.859562
$113$	−11.5607	−1.08754	−0.543771	−	0.839233i	$-0.683004\pi$
$113$	−11.5607	−1.08754	−0.543771	+	0.839233i	$0.683004\pi$
$114$	24.3445	2.28007
$115$	7.79045	0.726464
$116$	16.6317	1.54421
$117$	2.34516	0.216811
$118$	−7.58261	−0.698036
$119$	−2.07475	−0.190192
$120$	15.9743	1.45825
$121$	12.8752	1.17047
$122$	28.4528	2.57600
$123$	5.02080	0.452710
$124$	14.6992	1.32002
$125$	−11.3727	−1.01720
$126$	−1.46179	−0.130226
$127$	1.10870	0.0983811	0.0491906	−	0.998789i	$-0.484336\pi$
$127$	1.10870	0.0983811	0.0491906	+	0.998789i	$0.484336\pi$
$128$	2.95219	0.260939
$129$	18.9348	1.66712
$130$	15.4874	1.35834
$131$	−9.37219	−0.818852	−0.409426	−	0.912343i	$-0.634271\pi$
$131$	−9.37219	−0.818852	−0.409426	+	0.912343i	$0.634271\pi$
$132$	36.2711	3.15699
$133$	−5.99965	−0.520236
$134$	21.5195	1.85900
$135$	7.94911	0.684151
$136$	−14.8531	−1.27364
$137$	13.2477	1.13183	0.565913	−	0.824465i	$-0.308524\pi$
$137$	13.2477	1.13183	0.565913	+	0.824465i	$0.308524\pi$
$138$	22.1180	1.88281
$139$	14.6892	1.24592	0.622959	−	0.782254i	$-0.285930\pi$
$139$	14.6892	1.24592	0.622959	+	0.782254i	$0.285930\pi$
$140$	−6.79522	−0.574301
$141$	−2.33125	−0.196327
$142$	35.8931	3.01208
$143$	20.3733	1.70370
$144$	−5.11648	−0.426374
$145$	4.99940	0.415177
$146$	−34.4985	−2.85512
$147$	−1.56127	−0.128771
$148$	46.0100	3.78200
$149$	−13.1498	−1.07727	−0.538636	−	0.842539i	$-0.681060\pi$
$149$	−13.1498	−1.07727	−0.538636	+	0.842539i	$0.681060\pi$
$150$	−12.0001	−0.979801
$151$	−15.7503	−1.28174	−0.640871	−	0.767648i	$-0.721427\pi$
$151$	−15.7503	−1.28174	−0.640871	+	0.767648i	$0.721427\pi$
$152$	−42.9514	−3.48382
$153$	−1.16695	−0.0943421
$154$	−12.6991	−1.02332
$155$	4.41850	0.354902
$156$	30.9511	2.47807
$157$	1.96741	0.157016	0.0785081	−	0.996913i	$-0.474984\pi$
$157$	1.96741	0.157016	0.0785081	+	0.996913i	$0.474984\pi$
$158$	36.7626	2.92468
$159$	−4.51040	−0.357698
$160$	−13.3260	−1.05351
$161$	−5.45092	−0.429593
$162$	18.1831	1.42860
$163$	−16.4078	−1.28515	−0.642577	−	0.766221i	$-0.722135\pi$
$163$	−16.4078	−1.28515	−0.642577	+	0.766221i	$0.722135\pi$
$164$	−15.2900	−1.19395
$165$	10.9029	0.848791
$166$	−28.1217	−2.18267
$167$	3.00021	0.232163	0.116082	−	0.993240i	$-0.462967\pi$
$167$	3.00021	0.232163	0.116082	+	0.993240i	$0.462967\pi$
$168$	−11.1771	−0.862330
$169$	4.38507	0.337313
$170$	−7.70651	−0.591062
$171$	−3.37451	−0.258055
$172$	−57.6627	−4.39674
$173$	−15.5099	−1.17920	−0.589598	−	0.807697i	$-0.700714\pi$
$173$	−15.5099	−1.17920	−0.589598	+	0.807697i	$0.700714\pi$
$174$	14.1938	1.07603
$175$	2.95739	0.223558
$176$	−44.4488	−3.35045
$177$	−4.55509	−0.342381
$178$	15.1912	1.13863
$179$	17.2606	1.29012	0.645059	−	0.764133i	$-0.276833\pi$
$179$	17.2606	1.29012	0.645059	+	0.764133i	$0.276833\pi$
$180$	−3.82198	−0.284874
$181$	11.3389	0.842813	0.421406	−	0.906872i	$-0.361537\pi$
$181$	11.3389	0.842813	0.421406	+	0.906872i	$0.361537\pi$
$182$	−10.8364	−0.803251
$183$	17.0924	1.26351
$184$	−39.0231	−2.87682
$185$	13.8304	1.01683
$186$	12.5446	0.919815
$187$	−10.1377	−0.741342
$188$	7.09943	0.517779
$189$	−5.56193	−0.404571
$190$	−22.2853	−1.61674
$191$	7.17463	0.519138	0.259569	−	0.965725i	$-0.416420\pi$
$191$	7.17463	0.519138	0.259569	+	0.965725i	$0.416420\pi$
$192$	−9.42901	−0.680480
$193$	−5.33698	−0.384164	−0.192082	−	0.981379i	$-0.561524\pi$
$193$	−5.33698	−0.384164	−0.192082	+	0.981379i	$0.561524\pi$
$194$	8.85209	0.635543
$195$	9.30375	0.666255
$196$	4.75456	0.339612
$197$	−4.02318	−0.286639	−0.143320	−	0.989676i	$-0.545778\pi$
$197$	−4.02318	−0.286639	−0.143320	+	0.989676i	$0.545778\pi$
$198$	−7.14261	−0.507603
$199$	18.1595	1.28729	0.643646	−	0.765323i	$-0.277421\pi$
$199$	18.1595	1.28729	0.643646	+	0.765323i	$0.277421\pi$
$200$	21.1719	1.49708
$201$	12.9274	0.911827
$202$	29.6491	2.08610
$203$	−3.49804	−0.245514
$204$	−15.4012	−1.07830
$205$	−4.59610	−0.321005
$206$	16.4912	1.14900
$207$	−3.06588	−0.213093
$208$	−37.9293	−2.62992
$209$	−29.3156	−2.02781
$210$	−5.79920	−0.400183
$211$	−4.26125	−0.293357	−0.146678	−	0.989184i	$-0.546858\pi$
$211$	−4.26125	−0.293357	−0.146678	+	0.989184i	$0.546858\pi$
$212$	13.7357	0.943369
$213$	21.5620	1.47740
$214$	−42.2025	−2.88491
$215$	−17.3331	−1.18211
$216$	−39.8178	−2.70926
$217$	−3.09159	−0.209871
$218$	49.0903	3.32481
$219$	−20.7242	−1.40041
$220$	−33.2030	−2.23854
$221$	−8.65076	−0.581913
$222$	39.2660	2.63536
$223$	−22.4937	−1.50629	−0.753143	−	0.657856i	$-0.771463\pi$
$223$	−22.4937	−1.50629	−0.753143	+	0.657856i	$0.771463\pi$
$224$	9.32407	0.622991
$225$	1.66339	0.110892
$226$	30.0458	1.99862
$227$	25.3634	1.68343	0.841714	−	0.539924i	$-0.181547\pi$
$227$	25.3634	1.68343	0.841714	+	0.539924i	$0.181547\pi$
$228$	−44.5362	−2.94948
$229$	4.24192	0.280314	0.140157	−	0.990129i	$-0.455239\pi$
$229$	4.24192	0.280314	0.140157	+	0.990129i	$0.455239\pi$
$230$	−20.2470	−1.33505
$231$	−7.62869	−0.501931
$232$	−25.0424	−1.64412
$233$	−4.58348	−0.300274	−0.150137	−	0.988665i	$-0.547971\pi$
$233$	−4.58348	−0.300274	−0.150137	+	0.988665i	$0.547971\pi$
$234$	−6.09498	−0.398441
$235$	2.13405	0.139210
$236$	13.8717	0.902973
$237$	22.0843	1.43453
$238$	5.39219	0.349523
$239$	26.8802	1.73873	0.869366	−	0.494168i	$-0.164527\pi$
$239$	26.8802	1.73873	0.869366	+	0.494168i	$0.164527\pi$
$240$	−20.2981	−1.31024
$241$	27.3838	1.76394	0.881972	−	0.471301i	$-0.156215\pi$
$241$	27.3838	1.76394	0.881972	+	0.471301i	$0.156215\pi$
$242$	−33.4620	−2.15102
$243$	−5.76272	−0.369679
$244$	−52.0520	−3.33229
$245$	1.42920	0.0913082
$246$	−13.0488	−0.831963
$247$	−25.0158	−1.59172
$248$	−22.1326	−1.40542
$249$	−16.8935	−1.07058
$250$	29.5571	1.86936
$251$	29.7424	1.87732	0.938662	−	0.344838i	$-0.112066\pi$
$251$	29.7424	1.87732	0.938662	+	0.344838i	$0.112066\pi$
$252$	2.67421	0.168459
$253$	−26.6344	−1.67449
$254$	−2.88146	−0.180799
$255$	−4.62952	−0.289912
$256$	−19.7513	−1.23445
$257$	18.4334	1.14984	0.574922	−	0.818208i	$-0.305032\pi$
$257$	18.4334	1.14984	0.574922	+	0.818208i	$0.305032\pi$
$258$	−49.2107	−3.06373
$259$	−9.67701	−0.601300
$260$	−28.3330	−1.75714
$261$	−1.96748	−0.121784
$262$	24.3579	1.50484
$263$	15.2750	0.941897	0.470948	−	0.882161i	$-0.343912\pi$
$263$	15.2750	0.941897	0.470948	+	0.882161i	$0.343912\pi$
$264$	−54.6137	−3.36124
$265$	4.12887	0.253635
$266$	15.5928	0.956057
$267$	9.12581	0.558490
$268$	−39.3681	−2.40479
$269$	−13.3552	−0.814282	−0.407141	−	0.913365i	$-0.633474\pi$
$269$	−13.3552	−0.814282	−0.407141	+	0.913365i	$0.633474\pi$
$270$	−20.6594	−1.25729
$271$	−8.01106	−0.486637	−0.243319	−	0.969946i	$-0.578236\pi$
$271$	−8.01106	−0.486637	−0.243319	+	0.969946i	$0.578236\pi$
$272$	18.8735	1.14437
$273$	−6.50976	−0.393989
$274$	−34.4301	−2.08000
$275$	14.4505	0.871396
$276$	−40.4629	−2.43558
$277$	17.7401	1.06590	0.532951	−	0.846146i	$-0.321083\pi$
$277$	17.7401	1.06590	0.532951	+	0.846146i	$0.321083\pi$
$278$	−38.1764	−2.28967
$279$	−1.73887	−0.104103
$280$	10.2316	0.611456
$281$	9.34502	0.557477	0.278739	−	0.960367i	$-0.410084\pi$
$281$	9.34502	0.557477	0.278739	+	0.960367i	$0.410084\pi$
$282$	6.05882	0.360797
$283$	−13.5380	−0.804752	−0.402376	−	0.915475i	$-0.631815\pi$
$283$	−13.5380	−0.804752	−0.402376	+	0.915475i	$0.631815\pi$
$284$	−65.6634	−3.89640
$285$	−13.3874	−0.793000
$286$	−52.9493	−3.13096
$287$	3.21585	0.189826
$288$	5.24434	0.309026
$289$	−12.6954	−0.746789
$290$	−12.9932	−0.762987
$291$	5.31770	0.311729
$292$	63.1120	3.69335
$293$	−9.76762	−0.570631	−0.285315	−	0.958434i	$-0.592098\pi$
$293$	−9.76762	−0.570631	−0.285315	+	0.958434i	$0.592098\pi$
$294$	4.05766	0.236647
$295$	4.16978	0.242774
$296$	−69.2776	−4.02668
$297$	−27.1768	−1.57696
$298$	34.1757	1.97974
$299$	−22.7278	−1.31438
$300$	21.9531	1.26746
$301$	12.1279	0.699039
$302$	40.9344	2.35551
$303$	17.8110	1.02322
$304$	54.5774	3.13023
$305$	−15.6466	−0.895921
$306$	3.03284	0.173376
$307$	1.23612	0.0705489	0.0352744	−	0.999378i	$-0.488769\pi$
$307$	1.23612	0.0705489	0.0352744	+	0.999378i	$0.488769\pi$
$308$	23.2319	1.32376
$309$	9.90672	0.563574
$310$	−11.4835	−0.652218
$311$	15.2358	0.863945	0.431972	−	0.901887i	$-0.357818\pi$
$311$	15.2358	0.863945	0.431972	+	0.901887i	$0.357818\pi$
$312$	−46.6033	−2.63839
$313$	−12.7061	−0.718190	−0.359095	−	0.933301i	$-0.616915\pi$
$313$	−12.7061	−0.718190	−0.359095	+	0.933301i	$0.616915\pi$
$314$	−5.11320	−0.288555
$315$	0.803855	0.0452921
$316$	−67.2541	−3.78334
$317$	12.3248	0.692227	0.346114	−	0.938193i	$-0.387501\pi$
$317$	12.3248	0.692227	0.346114	+	0.938193i	$0.387501\pi$
$318$	11.7223	0.657356
$319$	−17.0922	−0.956980
$320$	8.63142	0.482511
$321$	−25.3523	−1.41502
$322$	14.1667	0.789479
$323$	12.4478	0.692613
$324$	−33.2643	−1.84802
$325$	12.3310	0.683998
$326$	42.6430	2.36178
$327$	29.4899	1.63080
$328$	23.0222	1.27119
$329$	−1.49318	−0.0823218
$330$	−28.3362	−1.55986
$331$	−5.38664	−0.296077	−0.148038	−	0.988982i	$-0.547296\pi$
$331$	−5.38664	−0.296077	−0.148038	+	0.988982i	$0.547296\pi$
$332$	51.4463	2.82348
$333$	−5.44285	−0.298266
$334$	−7.79740	−0.426655
$335$	−11.8339	−0.646554
$336$	14.2024	0.774807
$337$	29.9227	1.62999	0.814997	−	0.579465i	$-0.196739\pi$
$337$	29.9227	1.62999	0.814997	+	0.579465i	$0.196739\pi$
$338$	−11.3966	−0.619894
$339$	18.0494	0.980307
$340$	14.0984	0.764593
$341$	−15.1062	−0.818046
$342$	8.77020	0.474238
$343$	−1.00000	−0.0539949
$344$	86.8232	4.68120
$345$	−12.1630	−0.654832
$346$	40.3096	2.16706
$347$	16.9585	0.910378	0.455189	−	0.890395i	$-0.349572\pi$
$347$	16.9585	0.910378	0.455189	+	0.890395i	$0.349572\pi$
$348$	−25.9664	−1.39195
$349$	−26.7061	−1.42954	−0.714771	−	0.699359i	$-0.753469\pi$
$349$	−26.7061	−1.42954	−0.714771	+	0.699359i	$0.753469\pi$
$350$	−7.68612	−0.410840
$351$	−23.1907	−1.23783
$352$	45.5595	2.42833
$353$	−11.5540	−0.614958	−0.307479	−	0.951555i	$-0.599485\pi$
$353$	−11.5540	−0.614958	−0.307479	+	0.951555i	$0.599485\pi$
$354$	11.8385	0.629207
$355$	−19.7381	−1.04759
$356$	−27.7911	−1.47292
$357$	3.23924	0.171439
$358$	−44.8595	−2.37090
$359$	−7.76978	−0.410073	−0.205037	−	0.978754i	$-0.565731\pi$
$359$	−7.76978	−0.410073	−0.205037	+	0.978754i	$0.565731\pi$
$360$	5.75479	0.303304
$361$	16.9958	0.894517
$362$	−29.4693	−1.54887
$363$	−20.1016	−1.05506
$364$	19.8244	1.03908
$365$	18.9712	0.992997
$366$	−44.4224	−2.32200
$367$	35.7841	1.86792	0.933958	−	0.357382i	$-0.116331\pi$
$367$	35.7841	1.86792	0.933958	+	0.357382i	$0.116331\pi$
$368$	49.5857	2.58483
$369$	1.80876	0.0941603
$370$	−35.9445	−1.86867
$371$	−2.88894	−0.149986
$372$	−22.9493	−1.18986
$373$	26.0936	1.35108	0.675539	−	0.737325i	$-0.263911\pi$
$373$	26.0936	1.35108	0.675539	+	0.737325i	$0.263911\pi$
$374$	26.3474	1.36239
$375$	17.7558	0.916905
$376$	−10.6897	−0.551278
$377$	−14.5852	−0.751177
$378$	14.4552	0.743496
$379$	15.4577	0.794006	0.397003	−	0.917817i	$-0.370050\pi$
$379$	15.4577	0.794006	0.397003	+	0.917817i	$0.370050\pi$
$380$	40.7690	2.09140
$381$	−1.73097	−0.0886804
$382$	−18.6465	−0.954039
$383$	9.82892	0.502234	0.251117	−	0.967957i	$-0.419202\pi$
$383$	9.82892	0.502234	0.251117	+	0.967957i	$0.419202\pi$
$384$	−4.60914	−0.235209
$385$	6.98339	0.355907
$386$	13.8706	0.705993
$387$	6.82133	0.346748
$388$	−16.1941	−0.822133
$389$	13.2016	0.669350	0.334675	−	0.942334i	$-0.391373\pi$
$389$	13.2016	0.669350	0.334675	+	0.942334i	$0.391373\pi$
$390$	−24.1800	−1.22440
$391$	11.3093	0.571936
$392$	−7.15899	−0.361583
$393$	14.6325	0.738110
$394$	10.4561	0.526768
$395$	−20.2163	−1.01719
$396$	13.0668	0.656631
$397$	13.0607	0.655497	0.327749	−	0.944765i	$-0.393710\pi$
$397$	13.0607	0.655497	0.327749	+	0.944765i	$0.393710\pi$
$398$	−47.1957	−2.36571
$399$	9.36705	0.468939
$400$	−26.9026	−1.34513
$401$	−17.4636	−0.872088	−0.436044	−	0.899925i	$-0.643621\pi$
$401$	−17.4636	−0.872088	−0.436044	+	0.899925i	$0.643621\pi$
$402$	−33.5977	−1.67570
$403$	−12.8905	−0.642122
$404$	−54.2404	−2.69856
$405$	−9.99910	−0.496859
$406$	9.09125	0.451191
$407$	−47.2840	−2.34378
$408$	23.1897	1.14806
$409$	29.5305	1.46019	0.730095	−	0.683345i	$-0.239476\pi$
$409$	29.5305	1.46019	0.730095	+	0.683345i	$0.239476\pi$
$410$	11.9450	0.589924
$411$	−20.6831	−1.02022
$412$	−30.1692	−1.48633
$413$	−2.91756	−0.143564
$414$	7.96808	0.391610
$415$	15.4645	0.759123
$416$	38.8771	1.90611
$417$	−22.9337	−1.12307
$418$	76.1900	3.72658
$419$	16.6890	0.815311	0.407655	−	0.913136i	$-0.366346\pi$
$419$	16.6890	0.815311	0.407655	+	0.913136i	$0.366346\pi$
$420$	10.6091	0.517673
$421$	12.0563	0.587590	0.293795	−	0.955868i	$-0.405082\pi$
$421$	12.0563	0.587590	0.293795	+	0.955868i	$0.405082\pi$
$422$	11.0748	0.539113
$423$	−0.839842	−0.0408345
$424$	−20.6819	−1.00440
$425$	−6.13585	−0.297632
$426$	−56.0386	−2.71508
$427$	10.9478	0.529801
$428$	77.2059	3.73189
$429$	−31.8081	−1.53571
$430$	45.0480	2.17241
$431$	−17.3474	−0.835592	−0.417796	−	0.908541i	$-0.637197\pi$
$431$	−17.3474	−0.835592	−0.417796	+	0.908541i	$0.637197\pi$
$432$	50.5955	2.43428
$433$	19.0816	0.917005	0.458503	−	0.888693i	$-0.348386\pi$
$433$	19.0816	0.917005	0.458503	+	0.888693i	$0.348386\pi$
$434$	8.03490	0.385688
$435$	−7.80539	−0.374240
$436$	−89.8064	−4.30095
$437$	32.7036	1.56443
$438$	53.8613	2.57359
$439$	−6.60724	−0.315346	−0.157673	−	0.987491i	$-0.550399\pi$
$439$	−6.60724	−0.315346	−0.157673	+	0.987491i	$0.550399\pi$
$440$	49.9940	2.38337
$441$	−0.562451	−0.0267834
$442$	22.4829	1.06940
$443$	−3.26022	−0.154898	−0.0774489	−	0.996996i	$-0.524677\pi$
$443$	−3.26022	−0.154898	−0.0774489	+	0.996996i	$0.524677\pi$
$444$	−71.8338	−3.40908
$445$	−8.35387	−0.396011
$446$	58.4600	2.76816
$447$	20.5303	0.971050
$448$	−6.03934	−0.285332
$449$	21.6364	1.02108	0.510542	−	0.859853i	$-0.329445\pi$
$449$	21.6364	1.02108	0.510542	+	0.859853i	$0.329445\pi$
$450$	−4.32307	−0.203791
$451$	15.7134	0.739914
$452$	−54.9663	−2.58540
$453$	24.5904	1.15536
$454$	−65.9183	−3.09370
$455$	5.95911	0.279367
$456$	67.0586	3.14031
$457$	4.20912	0.196894	0.0984472	−	0.995142i	$-0.468612\pi$
$457$	4.20912	0.196894	0.0984472	+	0.995142i	$0.468612\pi$
$458$	−11.0245	−0.515143
$459$	11.5396	0.538624
$460$	37.0402	1.72701
$461$	11.5876	0.539688	0.269844	−	0.962904i	$-0.413028\pi$
$461$	11.5876	0.539688	0.269844	+	0.962904i	$0.413028\pi$
$462$	19.8266	0.922418
$463$	−8.34527	−0.387837	−0.193919	−	0.981018i	$-0.562120\pi$
$463$	−8.34527	−0.387837	−0.193919	+	0.981018i	$0.562120\pi$
$464$	31.8208	1.47724
$465$	−6.89845	−0.319908
$466$	11.9123	0.551825
$467$	39.2711	1.81725	0.908624	−	0.417616i	$-0.137134\pi$
$467$	39.2711	1.81725	0.908624	+	0.417616i	$0.137134\pi$
$468$	11.1502	0.515420
$469$	8.28007	0.382338
$470$	−5.54631	−0.255832
$471$	−3.07164	−0.141534
$472$	−20.8868	−0.961393
$473$	59.2595	2.72475
$474$	−57.3962	−2.63629
$475$	−17.7433	−0.814118
$476$	−9.86454	−0.452140
$477$	−1.62489	−0.0743985
$478$	−69.8603	−3.19534
$479$	−0.322330	−0.0147276	−0.00736382	−	0.999973i	$-0.502344\pi$
$479$	−0.322330	−0.0147276	−0.00736382	+	0.999973i	$0.502344\pi$
$480$	20.8054	0.949631
$481$	−40.3487	−1.83974
$482$	−71.1692	−3.24167
$483$	8.51033	0.387234
$484$	61.2159	2.78254
$485$	−4.86788	−0.221039
$486$	14.9771	0.679373
$487$	31.0878	1.40872	0.704361	−	0.709842i	$-0.251234\pi$
$487$	31.0878	1.40872	0.704361	+	0.709842i	$0.251234\pi$
$488$	78.3752	3.54788
$489$	25.6169	1.15843
$490$	−3.71443	−0.167801
$491$	−17.4361	−0.786881	−0.393440	−	0.919350i	$-0.628715\pi$
$491$	−17.4361	−0.786881	−0.393440	+	0.919350i	$0.628715\pi$
$492$	23.8717	1.07622
$493$	7.25756	0.326864
$494$	65.0149	2.92516
$495$	3.92782	0.176542
$496$	28.1234	1.26278
$497$	13.8106	0.619490
$498$	43.9055	1.96745
$499$	−15.0777	−0.674969	−0.337484	−	0.941331i	$-0.609576\pi$
$499$	−15.0777	−0.674969	−0.337484	+	0.941331i	$0.609576\pi$
$500$	−54.0722	−2.41818
$501$	−4.68412	−0.209271
$502$	−77.2992	−3.45003
$503$	−10.8971	−0.485877	−0.242939	−	0.970042i	$-0.578111\pi$
$503$	−10.8971	−0.485877	−0.242939	+	0.970042i	$0.578111\pi$
$504$	−4.02658	−0.179358
$505$	−16.3044	−0.725537
$506$	69.2216	3.07728
$507$	−6.84626	−0.304053
$508$	5.27138	0.233880
$509$	23.3873	1.03662	0.518312	−	0.855192i	$-0.326561\pi$
$509$	23.3873	1.03662	0.518312	+	0.855192i	$0.326561\pi$
$510$	12.0319	0.532782
$511$	−13.2740	−0.587207
$512$	45.4283	2.00767
$513$	33.3696	1.47331
$514$	−47.9076	−2.11311
$515$	−9.06873	−0.399616
$516$	90.0268	3.96321
$517$	−7.29602	−0.320879
$518$	25.1501	1.10503
$519$	24.2151	1.06292
$520$	42.6612	1.87082
$521$	16.6219	0.728217	0.364108	−	0.931357i	$-0.381374\pi$
$521$	16.6219	0.728217	0.364108	+	0.931357i	$0.381374\pi$
$522$	5.11338	0.223807
$523$	14.8234	0.648182	0.324091	−	0.946026i	$-0.394942\pi$
$523$	14.8234	0.648182	0.324091	+	0.946026i	$0.394942\pi$
$524$	−44.5607	−1.94664
$525$	−4.61727	−0.201514
$526$	−39.6990	−1.73096
$527$	6.41428	0.279410
$528$	69.3963	3.02009
$529$	6.71254	0.291849
$530$	−10.7308	−0.466114
$531$	−1.64099	−0.0712128
$532$	−28.5257	−1.23675
$533$	13.4086	0.580792
$534$	−23.7176	−1.02636
$535$	23.2077	1.00336
$536$	59.2769	2.56037
$537$	−26.9484	−1.16291
$538$	34.7096	1.49644
$539$	−4.88622	−0.210465
$540$	37.7946	1.62642
$541$	−41.5913	−1.78815	−0.894075	−	0.447918i	$-0.852166\pi$
$541$	−41.5913	−1.78815	−0.894075	+	0.447918i	$0.852166\pi$
$542$	20.8204	0.894312
$543$	−17.7030	−0.759709
$544$	−19.3451	−0.829416
$545$	−26.9954	−1.15636
$546$	16.9186	0.724048
$547$	0.847172	0.0362225	0.0181112	−	0.999836i	$-0.494235\pi$
$547$	0.847172	0.0362225	0.0181112	+	0.999836i	$0.494235\pi$
$548$	62.9870	2.69067
$549$	6.15761	0.262800
$550$	−37.5561	−1.60140
$551$	20.9870	0.894077
$552$	60.9254	2.59316
$553$	14.1452	0.601513
$554$	−46.1058	−1.95885
$555$	−21.5929	−0.916567
$556$	69.8405	2.96190
$557$	8.17619	0.346436	0.173218	−	0.984883i	$-0.444583\pi$
$557$	8.17619	0.346436	0.173218	+	0.984883i	$0.444583\pi$
$558$	4.51924	0.191315
$559$	50.5676	2.13878
$560$	−13.0011	−0.549396
$561$	15.8276	0.668243
$562$	−24.2873	−1.02450
$563$	0.920167	0.0387804	0.0193902	−	0.999812i	$-0.493828\pi$
$563$	0.920167	0.0387804	0.0193902	+	0.999812i	$0.493828\pi$
$564$	−11.0841	−0.466724
$565$	−16.5226	−0.695111
$566$	35.1847	1.47892
$567$	6.99629	0.293817
$568$	98.8699	4.14849
$569$	−10.7714	−0.451561	−0.225781	−	0.974178i	$-0.572493\pi$
$569$	−10.7714	−0.451561	−0.225781	+	0.974178i	$0.572493\pi$
$570$	34.7932	1.45733
$571$	−32.2686	−1.35040	−0.675199	−	0.737636i	$-0.735942\pi$
$571$	−32.2686	−1.35040	−0.675199	+	0.737636i	$0.735942\pi$
$572$	96.8662	4.05018
$573$	−11.2015	−0.467949
$574$	−8.35785	−0.348850
$575$	−16.1205	−0.672271
$576$	−3.39683	−0.141535
$577$	−37.8826	−1.57707	−0.788537	−	0.614987i	$-0.789161\pi$
$577$	−37.8826	−1.57707	−0.788537	+	0.614987i	$0.789161\pi$
$578$	32.9948	1.37240
$579$	8.33244	0.346284
$580$	23.7700	0.986994
$581$	−10.8204	−0.448906
$582$	−13.8205	−0.572877
$583$	−14.1160	−0.584625
$584$	−95.0283	−3.93230
$585$	3.35171	0.138576
$586$	25.3856	1.04867
$587$	23.4220	0.966731	0.483366	−	0.875419i	$-0.339414\pi$
$587$	23.4220	0.966731	0.483366	+	0.875419i	$0.339414\pi$
$588$	−7.42314	−0.306125
$589$	18.5485	0.764276
$590$	−10.8371	−0.446155
$591$	6.28124	0.258376
$592$	88.0294	3.61799
$593$	−39.6804	−1.62948	−0.814739	−	0.579827i	$-0.803120\pi$
$593$	−39.6804	−1.62948	−0.814739	+	0.579827i	$0.803120\pi$
$594$	70.6314	2.89804
$595$	−2.96523	−0.121563
$596$	−62.5215	−2.56098
$597$	−28.3518	−1.16036
$598$	59.0686	2.41550
$599$	−33.0373	−1.34987	−0.674934	−	0.737878i	$-0.735828\pi$
$599$	−33.0373	−1.34987	−0.674934	+	0.737878i	$0.735828\pi$
$600$	−33.0550	−1.34946
$601$	−6.54902	−0.267140	−0.133570	−	0.991039i	$-0.542644\pi$
$601$	−6.54902	−0.267140	−0.133570	+	0.991039i	$0.542644\pi$
$602$	−31.5198	−1.28465
$603$	4.65714	0.189653
$604$	−74.8859	−3.04706
$605$	18.4012	0.748116
$606$	−46.2901	−1.88041
$607$	−30.9825	−1.25754	−0.628771	−	0.777591i	$-0.716442\pi$
$607$	−30.9825	−1.25754	−0.628771	+	0.777591i	$0.716442\pi$
$608$	−55.9412	−2.26872
$609$	5.46137	0.221306
$610$	40.6648	1.64647
$611$	−6.22588	−0.251872
$612$	−5.54832	−0.224278
$613$	−38.9482	−1.57310	−0.786551	−	0.617525i	$-0.788135\pi$
$613$	−38.9482	−1.57310	−0.786551	+	0.617525i	$0.788135\pi$
$614$	−3.21261	−0.129650
$615$	7.17572	0.289353
$616$	−34.9804	−1.40940
$617$	−5.62141	−0.226309	−0.113155	−	0.993577i	$-0.536096\pi$
$617$	−5.62141	−0.226309	−0.113155	+	0.993577i	$0.536096\pi$
$618$	−25.7471	−1.03570
$619$	45.1770	1.81582	0.907909	−	0.419166i	$-0.137678\pi$
$619$	45.1770	1.81582	0.907909	+	0.419166i	$0.137678\pi$
$620$	21.0080	0.843703
$621$	30.3176	1.21661
$622$	−39.5972	−1.58770
$623$	5.84514	0.234180
$624$	59.2177	2.37060
$625$	−1.46692	−0.0586767
$626$	33.0225	1.31985
$627$	45.7695	1.82786
$628$	9.35416	0.373272
$629$	20.0774	0.800538
$630$	−2.08918	−0.0832351
$631$	30.5901	1.21777	0.608887	−	0.793257i	$-0.291616\pi$
$631$	30.5901	1.21777	0.608887	+	0.793257i	$0.291616\pi$
$632$	101.265	4.02811
$633$	6.65294	0.264431
$634$	−32.0315	−1.27213
$635$	1.58455	0.0628810
$636$	−21.4450	−0.850349
$637$	−4.16954	−0.165203
$638$	44.4219	1.75868
$639$	7.76779	0.307289
$640$	4.21926	0.166781
$641$	−20.5202	−0.810500	−0.405250	−	0.914206i	$-0.632815\pi$
$641$	−20.5202	−0.810500	−0.405250	+	0.914206i	$0.632815\pi$
$642$	65.8894	2.60045
$643$	−40.1374	−1.58286	−0.791432	−	0.611257i	$-0.790664\pi$
$643$	−40.1374	−1.58286	−0.791432	+	0.611257i	$0.790664\pi$
$644$	−25.9168	−1.02126
$645$	27.0616	1.06555
$646$	−32.3512	−1.27284
$647$	36.3797	1.43023	0.715116	−	0.699006i	$-0.246374\pi$
$647$	36.3797	1.43023	0.715116	+	0.699006i	$0.246374\pi$
$648$	50.0864	1.96758
$649$	−14.2559	−0.559592
$650$	−32.0476	−1.25701
$651$	4.82679	0.189177
$652$	−78.0117	−3.05517
$653$	−13.6522	−0.534252	−0.267126	−	0.963662i	$-0.586074\pi$
$653$	−13.6522	−0.534252	−0.267126	+	0.963662i	$0.586074\pi$
$654$	−76.6429	−2.99697
$655$	−13.3947	−0.523375
$656$	−29.2538	−1.14217
$657$	−7.46598	−0.291275
$658$	3.88071	0.151286
$659$	40.9033	1.59337	0.796683	−	0.604397i	$-0.206586\pi$
$659$	40.9033	1.59337	0.796683	+	0.604397i	$0.206586\pi$
$660$	51.8387	2.01782
$661$	31.6368	1.23053	0.615264	−	0.788321i	$-0.289050\pi$
$661$	31.6368	1.23053	0.615264	+	0.788321i	$0.289050\pi$
$662$	13.9996	0.544112
$663$	13.5061	0.524535
$664$	−77.4631	−3.00615
$665$	−8.57470	−0.332513
$666$	14.1457	0.548135
$667$	19.0675	0.738298
$668$	14.2647	0.551917
$669$	35.1186	1.35776
$670$	30.7557	1.18820
$671$	53.4934	2.06509
$672$	−14.5574	−0.561562
$673$	3.18268	0.122683	0.0613417	−	0.998117i	$-0.480462\pi$
$673$	3.18268	0.122683	0.0613417	+	0.998117i	$0.480462\pi$
$674$	−77.7678	−2.99550
$675$	−16.4488	−0.633114
$676$	20.8491	0.801889
$677$	−10.1233	−0.389071	−0.194536	−	0.980895i	$-0.562320\pi$
$677$	−10.1233	−0.389071	−0.194536	+	0.980895i	$0.562320\pi$
$678$	−46.9095	−1.80155
$679$	3.40602	0.130711
$680$	−21.2281	−0.814060
$681$	−39.5990	−1.51744
$682$	39.2603	1.50336
$683$	21.5970	0.826388	0.413194	−	0.910643i	$-0.364413\pi$
$683$	21.5970	0.826388	0.413194	+	0.910643i	$0.364413\pi$
$684$	−16.0443	−0.613471
$685$	18.9336	0.723415
$686$	2.59895	0.0992285
$687$	−6.62276	−0.252674
$688$	−110.324	−4.20607
$689$	−12.0456	−0.458899
$690$	31.6110	1.20341
$691$	12.9997	0.494531	0.247266	−	0.968948i	$-0.420468\pi$
$691$	12.9997	0.494531	0.247266	+	0.968948i	$0.420468\pi$
$692$	−73.7429	−2.80328
$693$	−2.74826	−0.104398
$694$	−44.0743	−1.67304
$695$	20.9937	0.796338
$696$	39.0979	1.48200
$697$	−6.67209	−0.252724
$698$	69.4078	2.62712
$699$	7.15603	0.270666
$700$	14.0611	0.531459
$701$	32.4478	1.22554	0.612768	−	0.790263i	$-0.290056\pi$
$701$	32.4478	1.22554	0.612768	+	0.790263i	$0.290056\pi$
$702$	60.2716	2.27480
$703$	58.0587	2.18972
$704$	−29.5096	−1.11218
$705$	−3.33183	−0.125484
$706$	30.0284	1.13013
$707$	11.4081	0.429045
$708$	−21.6575	−0.813937
$709$	1.40078	0.0526073	0.0263036	−	0.999654i	$-0.491626\pi$
$709$	1.40078	0.0526073	0.0263036	+	0.999654i	$0.491626\pi$
$710$	51.2984	1.92520
$711$	7.95596	0.298372
$712$	41.8453	1.56822
$713$	16.8520	0.631113
$714$	−8.41863	−0.315059
$715$	29.1175	1.08893
$716$	82.0666	3.06697
$717$	−41.9671	−1.56729
$718$	20.1933	0.753608
$719$	39.7332	1.48180	0.740899	−	0.671616i	$-0.234400\pi$
$719$	39.7332	1.48180	0.740899	+	0.671616i	$0.234400\pi$
$720$	−7.31248	−0.272520
$721$	6.34532	0.236312
$722$	−44.1714	−1.64389
$723$	−42.7534	−1.59001
$724$	53.9115	2.00360
$725$	−10.3451	−0.384206
$726$	52.2431	1.93892
$727$	−1.42632	−0.0528994	−0.0264497	−	0.999650i	$-0.508420\pi$
$727$	−1.42632	−0.0528994	−0.0264497	+	0.999650i	$0.508420\pi$
$728$	−29.8497	−1.10630
$729$	29.9860	1.11059
$730$	−49.3052	−1.82487
$731$	−25.1623	−0.930662
$732$	81.2670	3.00372
$733$	−10.3521	−0.382365	−0.191183	−	0.981555i	$-0.561232\pi$
$733$	−10.3521	−0.382365	−0.191183	+	0.981555i	$0.561232\pi$
$734$	−93.0013	−3.43274
$735$	−2.23136	−0.0823049
$736$	−50.8248	−1.87343
$737$	40.4583	1.49030
$738$	−4.70089	−0.173042
$739$	−27.1365	−0.998231	−0.499115	−	0.866536i	$-0.666342\pi$
$739$	−27.1365	−0.998231	−0.499115	+	0.866536i	$0.666342\pi$
$740$	65.7574	2.41729
$741$	39.0563	1.43477
$742$	7.50823	0.275636
$743$	−19.5810	−0.718357	−0.359178	−	0.933269i	$-0.616943\pi$
$743$	−19.5810	−0.718357	−0.359178	+	0.933269i	$0.616943\pi$
$744$	34.5549	1.26685
$745$	−18.7937	−0.688547
$746$	−67.8162	−2.48293
$747$	−6.08595	−0.222673
$748$	−48.2004	−1.76238
$749$	−16.2383	−0.593334
$750$	−46.1465	−1.68503
$751$	46.4074	1.69343	0.846715	−	0.532047i	$-0.178577\pi$
$751$	46.4074	1.69343	0.846715	+	0.532047i	$0.178577\pi$
$752$	13.5831	0.495325
$753$	−46.4358	−1.69221
$754$	37.9063	1.38047
$755$	−22.5103	−0.819235
$756$	−26.4446	−0.961779
$757$	−18.7901	−0.682937	−0.341468	−	0.939893i	$-0.610924\pi$
$757$	−18.7901	−0.682937	−0.341468	+	0.939893i	$0.610924\pi$
$758$	−40.1737	−1.45918
$759$	41.5834	1.50938
$760$	−61.3862	−2.22671
$761$	−13.6495	−0.494794	−0.247397	−	0.968914i	$-0.579575\pi$
$761$	−13.6495	−0.494794	−0.247397	+	0.968914i	$0.579575\pi$
$762$	4.49872	0.162971
$763$	18.8885	0.683808
$764$	34.1122	1.23414
$765$	−1.66780	−0.0602995
$766$	−25.5449	−0.922975
$767$	−12.1649	−0.439249
$768$	30.8370	1.11273
$769$	−26.5689	−0.958101	−0.479050	−	0.877787i	$-0.659019\pi$
$769$	−26.5689	−0.958101	−0.479050	+	0.877787i	$0.659019\pi$
$770$	−18.1495	−0.654063
$771$	−28.7794	−1.03647
$772$	−25.3750	−0.913266
$773$	−16.0738	−0.578135	−0.289067	−	0.957309i	$-0.593345\pi$
$773$	−16.0738	−0.578135	−0.289067	+	0.957309i	$0.593345\pi$
$774$	−17.7283	−0.637232
$775$	−9.14303	−0.328427
$776$	24.3837	0.875323
$777$	15.1084	0.542010
$778$	−34.3105	−1.23009
$779$	−19.2940	−0.691279
$780$	44.2353	1.58388
$781$	67.4817	2.41468
$782$	−29.3924	−1.05107
$783$	19.4559	0.695296
$784$	9.09675	0.324884
$785$	2.81182	0.100358
$786$	−38.0291	−1.35645
$787$	−12.7825	−0.455646	−0.227823	−	0.973703i	$-0.573161\pi$
$787$	−12.7825	−0.455646	−0.227823	+	0.973703i	$0.573161\pi$
$788$	−19.1285	−0.681423
$789$	−23.8483	−0.849023
$790$	52.5411	1.86933
$791$	11.5607	0.411052
$792$	−19.6748	−0.699113
$793$	45.6473	1.62098
$794$	−33.9442	−1.20463
$795$	−6.44627	−0.228625
$796$	86.3405	3.06026
$797$	32.8223	1.16262	0.581312	−	0.813681i	$-0.302539\pi$
$797$	32.8223	1.16262	0.581312	+	0.813681i	$0.302539\pi$
$798$	−24.3445	−0.861787
$799$	3.09798	0.109599
$800$	27.5749	0.974920
$801$	3.28760	0.116162
$802$	45.3870	1.60267
$803$	−64.8597	−2.28885
$804$	61.4641	2.16767
$805$	−7.79045	−0.274577
$806$	33.5018	1.18005
$807$	20.8510	0.733991
$808$	81.6703	2.87315
$809$	−14.1588	−0.497798	−0.248899	−	0.968530i	$-0.580069\pi$
$809$	−14.1588	−0.497798	−0.248899	+	0.968530i	$0.580069\pi$
$810$	25.9872	0.913098
$811$	33.4528	1.17469	0.587343	−	0.809338i	$-0.300174\pi$
$811$	33.4528	1.17469	0.587343	+	0.809338i	$0.300174\pi$
$812$	−16.6317	−0.583657
$813$	12.5074	0.438653
$814$	122.889	4.30726
$815$	−23.4500	−0.821416
$816$	−29.4665	−1.03154
$817$	−72.7630	−2.54565
$818$	−76.7485	−2.68345
$819$	−2.34516	−0.0819467
$820$	−21.8524	−0.763120
$821$	−13.5731	−0.473703	−0.236851	−	0.971546i	$-0.576115\pi$
$821$	−13.5731	−0.473703	−0.236851	+	0.971546i	$0.576115\pi$
$822$	53.7546	1.87491
$823$	30.4040	1.05982	0.529908	−	0.848055i	$-0.322226\pi$
$823$	30.4040	1.05982	0.529908	+	0.848055i	$0.322226\pi$
$824$	45.4261	1.58249
$825$	−22.5610	−0.785473
$826$	7.58261	0.263833
$827$	4.23998	0.147439	0.0737193	−	0.997279i	$-0.476513\pi$
$827$	4.23998	0.147439	0.0737193	+	0.997279i	$0.476513\pi$
$828$	−14.5769	−0.506583
$829$	−22.0588	−0.766135	−0.383068	−	0.923720i	$-0.625132\pi$
$829$	−22.0588	−0.766135	−0.383068	+	0.923720i	$0.625132\pi$
$830$	−40.1916	−1.39507
$831$	−27.6971	−0.960800
$832$	−25.1813	−0.873003
$833$	2.07475	0.0718859
$834$	59.6036	2.06390
$835$	4.28790	0.148389
$836$	−139.383	−4.82067
$837$	17.1952	0.594353
$838$	−43.3739	−1.49833
$839$	23.6178	0.815378	0.407689	−	0.913121i	$-0.366335\pi$
$839$	23.6178	0.815378	0.407689	+	0.913121i	$0.366335\pi$
$840$	−15.9743	−0.551165
$841$	−16.7637	−0.578059
$842$	−31.3339	−1.07984
$843$	−14.5901	−0.502508
$844$	−20.2604	−0.697392
$845$	6.26715	0.215596
$846$	2.18271	0.0750432
$847$	−12.8752	−0.442397
$848$	26.2800	0.902458
$849$	21.1364	0.725400
$850$	15.9468	0.546970
$851$	52.7486	1.80820
$852$	102.518	3.51221
$853$	20.0762	0.687397	0.343698	−	0.939080i	$-0.388320\pi$
$853$	20.0762	0.687397	0.343698	+	0.939080i	$0.388320\pi$
$854$	−28.4528	−0.973636
$855$	−4.82285	−0.164938
$856$	−116.250	−3.97333
$857$	−28.9300	−0.988229	−0.494114	−	0.869397i	$-0.664508\pi$
$857$	−28.9300	−0.988229	−0.494114	+	0.869397i	$0.664508\pi$
$858$	82.6679	2.82224
$859$	1.00000	0.0341196
$859$	1.00000	0.0341196
$860$	−82.4115	−2.81021
$861$	−5.02080	−0.171108
$862$	45.0850	1.53560
$863$	15.2300	0.518436	0.259218	−	0.965819i	$-0.416535\pi$
$863$	15.2300	0.518436	0.259218	+	0.965819i	$0.416535\pi$
$864$	−51.8599	−1.76431
$865$	−22.1668	−0.753693
$866$	−49.5923	−1.68522
$867$	19.8209	0.673153
$868$	−14.6992	−0.498922
$869$	69.1164	2.34461
$870$	20.2858	0.687754
$871$	34.5241	1.16980
$872$	135.222	4.57920
$873$	1.91572	0.0648373
$874$	−84.9952	−2.87501
$875$	11.3727	0.384467
$876$	−98.5346	−3.32918
$877$	32.9175	1.11154	0.555772	−	0.831335i	$-0.312423\pi$
$877$	32.9175	1.11154	0.555772	+	0.831335i	$0.312423\pi$
$878$	17.1719	0.579524
$879$	15.2498	0.514365
$880$	−63.5262	−2.14147
$881$	−34.2002	−1.15224	−0.576118	−	0.817367i	$-0.695433\pi$
$881$	−34.2002	−1.15224	−0.576118	+	0.817367i	$0.695433\pi$
$882$	1.46179	0.0492209
$883$	−51.1106	−1.72001	−0.860005	−	0.510286i	$-0.829540\pi$
$883$	−51.1106	−1.72001	−0.860005	+	0.510286i	$0.829540\pi$
$884$	−41.1306	−1.38337
$885$	−6.51013	−0.218836
$886$	8.47317	0.284662
$887$	53.8436	1.80789	0.903946	−	0.427647i	$-0.140657\pi$
$887$	53.8436	1.80789	0.903946	+	0.427647i	$0.140657\pi$
$888$	108.161	3.62963
$889$	−1.10870	−0.0371846
$890$	21.7113	0.727765
$891$	34.1855	1.14526
$892$	−106.948	−3.58087
$893$	8.95857	0.299787
$894$	−53.3573	−1.78454
$895$	24.6688	0.824589
$896$	−2.95219	−0.0986256
$897$	35.4842	1.18478
$898$	−56.2320	−1.87649
$899$	10.8145	0.360684
$900$	7.90868	0.263623
$901$	5.99383	0.199684
$902$	−40.8383	−1.35977
$903$	−18.9348	−0.630111
$904$	82.7632	2.75266
$905$	16.2055	0.538690
$906$	−63.9094	−2.12325
$907$	40.2225	1.33557	0.667783	−	0.744356i	$-0.267244\pi$
$907$	40.2225	1.33557	0.667783	+	0.744356i	$0.267244\pi$
$908$	120.592	4.00198
$909$	6.41649	0.212822
$910$	−15.4874	−0.513404
$911$	−45.4597	−1.50615	−0.753073	−	0.657937i	$-0.771429\pi$
$911$	−45.4597	−1.50615	−0.753073	+	0.657937i	$0.771429\pi$
$912$	−85.2097	−2.82158
$913$	−52.8709	−1.74977
$914$	−10.9393	−0.361840
$915$	24.4285	0.807581
$916$	20.1685	0.666385
$917$	9.37219	0.309497
$918$	−29.9910	−0.989849
$919$	47.1014	1.55373	0.776866	−	0.629666i	$-0.216808\pi$
$919$	47.1014	1.55373	0.776866	+	0.629666i	$0.216808\pi$
$920$	−55.7718	−1.83874
$921$	−1.92990	−0.0635925
$922$	−30.1156	−0.991805
$923$	57.5838	1.89540
$924$	−36.2711	−1.19323
$925$	−28.6187	−0.940976
$926$	21.6890	0.712744
$927$	3.56893	0.117219
$928$	−32.6160	−1.07067
$929$	36.5595	1.19948	0.599739	−	0.800196i	$-0.295271\pi$
$929$	36.5595	1.19948	0.599739	+	0.800196i	$0.295271\pi$
$930$	17.9288	0.587907
$931$	5.99965	0.196631
$932$	−21.7925	−0.713836
$933$	−23.7872	−0.778757
$934$	−102.064	−3.33963
$935$	−14.4888	−0.473834
$936$	−16.7890	−0.548766
$937$	−44.1769	−1.44320	−0.721598	−	0.692312i	$-0.756592\pi$
$937$	−44.1769	−1.44320	−0.721598	+	0.692312i	$0.756592\pi$
$938$	−21.5195	−0.702637
$939$	19.8376	0.647375
$940$	10.1465	0.330942
$941$	28.5230	0.929824	0.464912	−	0.885357i	$-0.346086\pi$
$941$	28.5230	0.929824	0.464912	+	0.885357i	$0.346086\pi$
$942$	7.98306	0.260102
$943$	−17.5294	−0.570834
$944$	26.5404	0.863815
$945$	−7.94911	−0.258585
$946$	−154.013	−5.00738
$947$	34.4692	1.12010	0.560050	−	0.828459i	$-0.310782\pi$
$947$	34.4692	1.12010	0.560050	+	0.828459i	$0.310782\pi$
$948$	105.001	3.41029
$949$	−55.3465	−1.79662
$950$	46.1140	1.49614
$951$	−19.2422	−0.623971
$952$	14.8531	0.481392
$953$	30.2411	0.979605	0.489803	−	0.871833i	$-0.337069\pi$
$953$	30.2411	0.979605	0.489803	+	0.871833i	$0.337069\pi$
$954$	4.22301	0.136725
$955$	10.2540	0.331811
$956$	127.803	4.13346
$957$	26.6855	0.862619
$958$	0.837722	0.0270656
$959$	−13.2477	−0.427790
$960$	−13.4759	−0.434934
$961$	−21.4421	−0.691680
$962$	104.864	3.38096
$963$	−9.13324	−0.294315
$964$	130.198	4.19339
$965$	−7.62761	−0.245541
$966$	−22.1180	−0.711634
$967$	−32.4131	−1.04233	−0.521167	−	0.853455i	$-0.674503\pi$
$967$	−32.4131	−1.04233	−0.521167	+	0.853455i	$0.674503\pi$
$968$	−92.1733	−2.96256
$969$	−19.4343	−0.624320
$970$	12.6514	0.406212
$971$	−29.4443	−0.944913	−0.472457	−	0.881354i	$-0.656633\pi$
$971$	−29.4443	−0.944913	−0.472457	+	0.881354i	$0.656633\pi$
$972$	−27.3992	−0.878831
$973$	−14.6892	−0.470913
$974$	−80.7957	−2.58886
$975$	−19.2519	−0.616554
$976$	−99.5895	−3.18778
$977$	35.4280	1.13344	0.566721	−	0.823910i	$-0.308212\pi$
$977$	35.4280	1.13344	0.566721	+	0.823910i	$0.308212\pi$
$978$	−66.5770	−2.12890
$979$	28.5606	0.912802
$980$	6.79522	0.217065
$981$	10.6238	0.339193
$982$	45.3157	1.44608
$983$	−29.7608	−0.949222	−0.474611	−	0.880196i	$-0.657411\pi$
$983$	−29.7608	−0.949222	−0.474611	+	0.880196i	$0.657411\pi$
$984$	−35.9438	−1.14585
$985$	−5.74992	−0.183208
$986$	−18.8621	−0.600691
$987$	2.33125	0.0742046
$988$	−118.939	−3.78396
$989$	−66.1080	−2.10211
$990$	−10.2082	−0.324439
$991$	19.1984	0.609856	0.304928	−	0.952375i	$-0.401368\pi$
$991$	19.1984	0.609856	0.304928	+	0.952375i	$0.401368\pi$
$992$	−28.8262	−0.915233
$993$	8.40998	0.266883
$994$	−35.8931	−1.13846
$995$	25.9535	0.822783
$996$	−80.3213	−2.54508
$997$	48.2618	1.52847	0.764234	−	0.644939i	$-0.223117\pi$
$997$	48.2618	1.52847	0.764234	+	0.644939i	$0.223117\pi$
$998$	39.1862	1.24042
$999$	53.8229	1.70288

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6013.2.a.f.1.5	✓	110

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6013.2.a.f.1.5	✓	110	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:	\( N \)	\(=\)	\( 6013 = 7 \cdot 859 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6013.a (trivial)

\(f(q)\)	\(=\)	\(q-2.59895 q^{2} -1.56127 q^{3} +4.75456 q^{4} +1.42920 q^{5} +4.05766 q^{6} -1.00000 q^{7} -7.15899 q^{8} -0.562451 q^{9} +O(q^{10})\)	\(q-2.59895 q^{2} -1.56127 q^{3} +4.75456 q^{4} +1.42920 q^{5} +4.05766 q^{6} -1.00000 q^{7} -7.15899 q^{8} -0.562451 q^{9} -3.71443 q^{10} -4.88622 q^{11} -7.42314 q^{12} -4.16954 q^{13} +2.59895 q^{14} -2.23136 q^{15} +9.09675 q^{16} +2.07475 q^{17} +1.46179 q^{18} +5.99965 q^{19} +6.79522 q^{20} +1.56127 q^{21} +12.6991 q^{22} +5.45092 q^{23} +11.1771 q^{24} -2.95739 q^{25} +10.8364 q^{26} +5.56193 q^{27} -4.75456 q^{28} +3.49804 q^{29} +5.79920 q^{30} +3.09159 q^{31} -9.32407 q^{32} +7.62869 q^{33} -5.39219 q^{34} -1.42920 q^{35} -2.67421 q^{36} +9.67701 q^{37} -15.5928 q^{38} +6.50976 q^{39} -10.2316 q^{40} -3.21585 q^{41} -4.05766 q^{42} -12.1279 q^{43} -23.2319 q^{44} -0.803855 q^{45} -14.1667 q^{46} +1.49318 q^{47} -14.2024 q^{48} +1.00000 q^{49} +7.68612 q^{50} -3.23924 q^{51} -19.8244 q^{52} +2.88894 q^{53} -14.4552 q^{54} -6.98339 q^{55} +7.15899 q^{56} -9.36705 q^{57} -9.09125 q^{58} +2.91756 q^{59} -10.6091 q^{60} -10.9478 q^{61} -8.03490 q^{62} +0.562451 q^{63} +6.03934 q^{64} -5.95911 q^{65} -19.8266 q^{66} -8.28007 q^{67} +9.86454 q^{68} -8.51033 q^{69} +3.71443 q^{70} -13.8106 q^{71} +4.02658 q^{72} +13.2740 q^{73} -25.1501 q^{74} +4.61727 q^{75} +28.5257 q^{76} +4.88622 q^{77} -16.9186 q^{78} -14.1452 q^{79} +13.0011 q^{80} -6.99629 q^{81} +8.35785 q^{82} +10.8204 q^{83} +7.42314 q^{84} +2.96523 q^{85} +31.5198 q^{86} -5.46137 q^{87} +34.9804 q^{88} -5.84514 q^{89} +2.08918 q^{90} +4.16954 q^{91} +25.9168 q^{92} -4.82679 q^{93} -3.88071 q^{94} +8.57470 q^{95} +14.5574 q^{96} -3.40602 q^{97} -2.59895 q^{98} +2.74826 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 110 q + 16 q^{2} + 29 q^{3} + 118 q^{4} + 12 q^{6} - 110 q^{7} + 57 q^{8} + 127 q^{9}+O(q^{10}) \) 110 * q + 16 * q^2 + 29 * q^3 + 118 * q^4 + 12 * q^6 - 110 * q^7 + 57 * q^8 + 127 * q^9	\( 110 q + 16 q^{2} + 29 q^{3} + 118 q^{4} + 12 q^{6} - 110 q^{7} + 57 q^{8} + 127 q^{9} + 3 q^{10} + 52 q^{11} + 62 q^{12} - 9 q^{13} - 16 q^{14} + 39 q^{15} + 146 q^{16} + 11 q^{17} + 60 q^{18} + 14 q^{19} + 18 q^{20} - 29 q^{21} + 32 q^{22} + 73 q^{23} + 24 q^{24} + 132 q^{25} - 7 q^{26} + 116 q^{27} - 118 q^{28} + 35 q^{29} + 18 q^{30} + 36 q^{31} + 140 q^{32} + 42 q^{33} - 7 q^{34} + 180 q^{36} + 49 q^{37} + 45 q^{39} + 6 q^{40} - 14 q^{41} - 12 q^{42} + 58 q^{43} + 92 q^{44} + 17 q^{45} + 27 q^{46} + 87 q^{47} + 98 q^{48} + 110 q^{49} + 91 q^{50} + 42 q^{51} + 16 q^{52} + 95 q^{53} + 41 q^{54} + 8 q^{55} - 57 q^{56} + 61 q^{57} + 46 q^{58} + 114 q^{59} + 81 q^{60} - 47 q^{61} + 31 q^{62} - 127 q^{63} + 199 q^{64} + 62 q^{65} + 21 q^{66} + 95 q^{67} + 60 q^{68} - 39 q^{69} - 3 q^{70} + 131 q^{71} + 186 q^{72} + 31 q^{73} + 23 q^{74} + 121 q^{75} + 14 q^{76} - 52 q^{77} + 110 q^{78} + 9 q^{79} + 61 q^{80} + 194 q^{81} + 45 q^{82} + 73 q^{83} - 62 q^{84} + 59 q^{85} + 72 q^{86} + 64 q^{87} + 100 q^{88} - 17 q^{89} + 11 q^{90} + 9 q^{91} + 192 q^{92} + 85 q^{93} - 11 q^{94} + 108 q^{95} + 68 q^{96} + 32 q^{97} + 16 q^{98} + 160 q^{99}+O(q^{100}) \) 110 * q + 16 * q^2 + 29 * q^3 + 118 * q^4 + 12 * q^6 - 110 * q^7 + 57 * q^8 + 127 * q^9 + 3 * q^10 + 52 * q^11 + 62 * q^12 - 9 * q^13 - 16 * q^14 + 39 * q^15 + 146 * q^16 + 11 * q^17 + 60 * q^18 + 14 * q^19 + 18 * q^20 - 29 * q^21 + 32 * q^22 + 73 * q^23 + 24 * q^24 + 132 * q^25 - 7 * q^26 + 116 * q^27 - 118 * q^28 + 35 * q^29 + 18 * q^30 + 36 * q^31 + 140 * q^32 + 42 * q^33 - 7 * q^34 + 180 * q^36 + 49 * q^37 + 45 * q^39 + 6 * q^40 - 14 * q^41 - 12 * q^42 + 58 * q^43 + 92 * q^44 + 17 * q^45 + 27 * q^46 + 87 * q^47 + 98 * q^48 + 110 * q^49 + 91 * q^50 + 42 * q^51 + 16 * q^52 + 95 * q^53 + 41 * q^54 + 8 * q^55 - 57 * q^56 + 61 * q^57 + 46 * q^58 + 114 * q^59 + 81 * q^60 - 47 * q^61 + 31 * q^62 - 127 * q^63 + 199 * q^64 + 62 * q^65 + 21 * q^66 + 95 * q^67 + 60 * q^68 - 39 * q^69 - 3 * q^70 + 131 * q^71 + 186 * q^72 + 31 * q^73 + 23 * q^74 + 121 * q^75 + 14 * q^76 - 52 * q^77 + 110 * q^78 + 9 * q^79 + 61 * q^80 + 194 * q^81 + 45 * q^82 + 73 * q^83 - 62 * q^84 + 59 * q^85 + 72 * q^86 + 64 * q^87 + 100 * q^88 - 17 * q^89 + 11 * q^90 + 9 * q^91 + 192 * q^92 + 85 * q^93 - 11 * q^94 + 108 * q^95 + 68 * q^96 + 32 * q^97 + 16 * q^98 + 160 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more