LMFDB - Embedded newform 6045.2.a.bh.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6045,2,Mod(1,6045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6045, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.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:	$48.2695680219$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 2 x^{16} - 25 x^{15} + 47 x^{14} + 252 x^{13} - 437 x^{12} - 1319 x^{11} + 2056 x^{10} + \cdots + 8 $ x^17 - 2x^16 - 25x^15 + 47x^14 + 252x^13 - 437x^12 - 1319x^11 + 2056x^10 + 3854x^9 - 5201x^8 - 6304x^7 + 6915x^6 + 5469x^5 - 4238x^4 - 2198x^3 + 760x^2 + 306x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$0.551239$ of defining polynomial
Character	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.551239 q^{2} -1.00000 q^{3} -1.69614 q^{4} +1.00000 q^{5} -0.551239 q^{6} +4.16860 q^{7} -2.03745 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.551239 q^{2} -1.00000 q^{3} -1.69614 q^{4} +1.00000 q^{5} -0.551239 q^{6} +4.16860 q^{7} -2.03745 q^{8} +1.00000 q^{9} +0.551239 q^{10} -3.03033 q^{11} +1.69614 q^{12} +1.00000 q^{13} +2.29790 q^{14} -1.00000 q^{15} +2.26915 q^{16} -7.33654 q^{17} +0.551239 q^{18} -2.32871 q^{19} -1.69614 q^{20} -4.16860 q^{21} -1.67043 q^{22} -4.61769 q^{23} +2.03745 q^{24} +1.00000 q^{25} +0.551239 q^{26} -1.00000 q^{27} -7.07052 q^{28} +1.99980 q^{29} -0.551239 q^{30} +1.00000 q^{31} +5.32575 q^{32} +3.03033 q^{33} -4.04418 q^{34} +4.16860 q^{35} -1.69614 q^{36} +4.21074 q^{37} -1.28368 q^{38} -1.00000 q^{39} -2.03745 q^{40} +2.79017 q^{41} -2.29790 q^{42} +8.03700 q^{43} +5.13985 q^{44} +1.00000 q^{45} -2.54545 q^{46} +12.3549 q^{47} -2.26915 q^{48} +10.3773 q^{49} +0.551239 q^{50} +7.33654 q^{51} -1.69614 q^{52} -5.15186 q^{53} -0.551239 q^{54} -3.03033 q^{55} -8.49334 q^{56} +2.32871 q^{57} +1.10237 q^{58} -5.70964 q^{59} +1.69614 q^{60} +6.28967 q^{61} +0.551239 q^{62} +4.16860 q^{63} -1.60253 q^{64} +1.00000 q^{65} +1.67043 q^{66} +8.65649 q^{67} +12.4438 q^{68} +4.61769 q^{69} +2.29790 q^{70} -12.3911 q^{71} -2.03745 q^{72} +0.108776 q^{73} +2.32112 q^{74} -1.00000 q^{75} +3.94981 q^{76} -12.6322 q^{77} -0.551239 q^{78} +4.32017 q^{79} +2.26915 q^{80} +1.00000 q^{81} +1.53805 q^{82} -12.9036 q^{83} +7.07052 q^{84} -7.33654 q^{85} +4.43031 q^{86} -1.99980 q^{87} +6.17415 q^{88} +12.2569 q^{89} +0.551239 q^{90} +4.16860 q^{91} +7.83222 q^{92} -1.00000 q^{93} +6.81053 q^{94} -2.32871 q^{95} -5.32575 q^{96} -0.0930997 q^{97} +5.72035 q^{98} -3.03033 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 2 q^{2} - 17 q^{3} + 20 q^{4} + 17 q^{5} - 2 q^{6} + 18 q^{7} + 9 q^{8} + 17 q^{9}+O(q^{10}) $ 17 * q + 2 * q^2 - 17 * q^3 + 20 * q^4 + 17 * q^5 - 2 * q^6 + 18 * q^7 + 9 * q^8 + 17 * q^9	\( 17 q + 2 q^{2} - 17 q^{3} + 20 q^{4} + 17 q^{5} - 2 q^{6} + 18 q^{7} + 9 q^{8} + 17 q^{9} + 2 q^{10} + 3 q^{11} - 20 q^{12} + 17 q^{13} + q^{14} - 17 q^{15} + 26 q^{16} + 2 q^{18} + 10 q^{19} + 20 q^{20} - 18 q^{21} + 5 q^{22} + 16 q^{23} - 9 q^{24} + 17 q^{25} + 2 q^{26} - 17 q^{27} + 36 q^{28} - 3 q^{29} - 2 q^{30} + 17 q^{31} + 20 q^{32} - 3 q^{33} + q^{34} + 18 q^{35} + 20 q^{36} + 14 q^{37} + 22 q^{38} - 17 q^{39} + 9 q^{40} - 6 q^{41} - q^{42} + 24 q^{43} - 15 q^{44} + 17 q^{45} + 6 q^{46} + 25 q^{47} - 26 q^{48} + 31 q^{49} + 2 q^{50} + 20 q^{52} - 15 q^{53} - 2 q^{54} + 3 q^{55} + 31 q^{56} - 10 q^{57} + 44 q^{58} + 16 q^{59} - 20 q^{60} - 5 q^{61} + 2 q^{62} + 18 q^{63} + 35 q^{64} + 17 q^{65} - 5 q^{66} + 50 q^{67} + 13 q^{68} - 16 q^{69} + q^{70} + 16 q^{71} + 9 q^{72} + 33 q^{73} + 2 q^{74} - 17 q^{75} + 9 q^{77} - 2 q^{78} - 10 q^{79} + 26 q^{80} + 17 q^{81} + 61 q^{82} + 27 q^{83} - 36 q^{84} - 12 q^{86} + 3 q^{87} + 23 q^{88} - 24 q^{89} + 2 q^{90} + 18 q^{91} - 21 q^{92} - 17 q^{93} + 6 q^{94} + 10 q^{95} - 20 q^{96} + 48 q^{97} + 14 q^{98} + 3 q^{99}+O(q^{100}) \) 17 * q + 2 * q^2 - 17 * q^3 + 20 * q^4 + 17 * q^5 - 2 * q^6 + 18 * q^7 + 9 * q^8 + 17 * q^9 + 2 * q^10 + 3 * q^11 - 20 * q^12 + 17 * q^13 + q^14 - 17 * q^15 + 26 * q^16 + 2 * q^18 + 10 * q^19 + 20 * q^20 - 18 * q^21 + 5 * q^22 + 16 * q^23 - 9 * q^24 + 17 * q^25 + 2 * q^26 - 17 * q^27 + 36 * q^28 - 3 * q^29 - 2 * q^30 + 17 * q^31 + 20 * q^32 - 3 * q^33 + q^34 + 18 * q^35 + 20 * q^36 + 14 * q^37 + 22 * q^38 - 17 * q^39 + 9 * q^40 - 6 * q^41 - q^42 + 24 * q^43 - 15 * q^44 + 17 * q^45 + 6 * q^46 + 25 * q^47 - 26 * q^48 + 31 * q^49 + 2 * q^50 + 20 * q^52 - 15 * q^53 - 2 * q^54 + 3 * q^55 + 31 * q^56 - 10 * q^57 + 44 * q^58 + 16 * q^59 - 20 * q^60 - 5 * q^61 + 2 * q^62 + 18 * q^63 + 35 * q^64 + 17 * q^65 - 5 * q^66 + 50 * q^67 + 13 * q^68 - 16 * q^69 + q^70 + 16 * q^71 + 9 * q^72 + 33 * q^73 + 2 * q^74 - 17 * q^75 + 9 * q^77 - 2 * q^78 - 10 * q^79 + 26 * q^80 + 17 * q^81 + 61 * q^82 + 27 * q^83 - 36 * q^84 - 12 * q^86 + 3 * q^87 + 23 * q^88 - 24 * q^89 + 2 * q^90 + 18 * q^91 - 21 * q^92 - 17 * q^93 + 6 * q^94 + 10 * q^95 - 20 * q^96 + 48 * q^97 + 14 * q^98 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.551239	0.389785	0.194892	−	0.980825i	$-0.437564\pi$
$2$	0.551239	0.389785	0.194892	+	0.980825i	$0.437564\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.69614	−0.848068
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.551239	−0.225042
$7$	4.16860	1.57558	0.787792	−	0.615941i	$-0.211224\pi$
$7$	4.16860	1.57558	0.787792	+	0.615941i	$0.211224\pi$
$8$	−2.03745	−0.720349
$9$	1.00000	0.333333
$10$	0.551239	0.174317
$11$	−3.03033	−0.913678	−0.456839	−	0.889549i	$-0.651018\pi$
$11$	−3.03033	−0.913678	−0.456839	+	0.889549i	$0.651018\pi$
$12$	1.69614	0.489632
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	2.29790	0.614139
$15$	−1.00000	−0.258199
$16$	2.26915	0.567287
$17$	−7.33654	−1.77937	−0.889686	−	0.456574i	$-0.849076\pi$
$17$	−7.33654	−1.77937	−0.889686	+	0.456574i	$0.849076\pi$
$18$	0.551239	0.129928
$19$	−2.32871	−0.534243	−0.267122	−	0.963663i	$-0.586073\pi$
$19$	−2.32871	−0.534243	−0.267122	+	0.963663i	$0.586073\pi$
$20$	−1.69614	−0.379267
$21$	−4.16860	−0.909664
$22$	−1.67043	−0.356138
$23$	−4.61769	−0.962854	−0.481427	−	0.876486i	$-0.659881\pi$
$23$	−4.61769	−0.962854	−0.481427	+	0.876486i	$0.659881\pi$
$24$	2.03745	0.415894
$25$	1.00000	0.200000
$26$	0.551239	0.108107
$27$	−1.00000	−0.192450
$28$	−7.07052	−1.33620
$29$	1.99980	0.371353	0.185677	−	0.982611i	$-0.440552\pi$
$29$	1.99980	0.371353	0.185677	+	0.982611i	$0.440552\pi$
$30$	−0.551239	−0.100642
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	5.32575	0.941469
$33$	3.03033	0.527512
$34$	−4.04418	−0.693572
$35$	4.16860	0.704623
$36$	−1.69614	−0.282689
$37$	4.21074	0.692242	0.346121	−	0.938190i	$-0.387499\pi$
$37$	4.21074	0.692242	0.346121	+	0.938190i	$0.387499\pi$
$38$	−1.28368	−0.208240
$39$	−1.00000	−0.160128
$40$	−2.03745	−0.322150
$41$	2.79017	0.435751	0.217875	−	0.975977i	$-0.430087\pi$
$41$	2.79017	0.435751	0.217875	+	0.975977i	$0.430087\pi$
$42$	−2.29790	−0.354573
$43$	8.03700	1.22563	0.612816	−	0.790226i	$-0.290037\pi$
$43$	8.03700	1.22563	0.612816	+	0.790226i	$0.290037\pi$
$44$	5.13985	0.774861
$45$	1.00000	0.149071
$46$	−2.54545	−0.375306
$47$	12.3549	1.80215	0.901077	−	0.433660i	$-0.142778\pi$
$47$	12.3549	1.80215	0.901077	+	0.433660i	$0.142778\pi$
$48$	−2.26915	−0.327523
$49$	10.3773	1.48247
$50$	0.551239	0.0779570
$51$	7.33654	1.02732
$52$	−1.69614	−0.235212
$53$	−5.15186	−0.707662	−0.353831	−	0.935309i	$-0.615121\pi$
$53$	−5.15186	−0.707662	−0.353831	+	0.935309i	$0.615121\pi$
$54$	−0.551239	−0.0750141
$55$	−3.03033	−0.408609
$56$	−8.49334	−1.13497
$57$	2.32871	0.308445
$58$	1.10237	0.144748
$59$	−5.70964	−0.743332	−0.371666	−	0.928367i	$-0.621213\pi$
$59$	−5.70964	−0.743332	−0.371666	+	0.928367i	$0.621213\pi$
$60$	1.69614	0.218970
$61$	6.28967	0.805310	0.402655	−	0.915352i	$-0.368087\pi$
$61$	6.28967	0.805310	0.402655	+	0.915352i	$0.368087\pi$
$62$	0.551239	0.0700074
$63$	4.16860	0.525195
$64$	−1.60253	−0.200317
$65$	1.00000	0.124035
$66$	1.67043	0.205616
$67$	8.65649	1.05756	0.528779	−	0.848759i	$-0.322650\pi$
$67$	8.65649	1.05756	0.528779	+	0.848759i	$0.322650\pi$
$68$	12.4438	1.50903
$69$	4.61769	0.555904
$70$	2.29790	0.274651
$71$	−12.3911	−1.47056	−0.735279	−	0.677765i	$-0.762949\pi$
$71$	−12.3911	−1.47056	−0.735279	+	0.677765i	$0.762949\pi$
$72$	−2.03745	−0.240116
$73$	0.108776	0.0127312	0.00636561	−	0.999980i	$-0.497974\pi$
$73$	0.108776	0.0127312	0.00636561	+	0.999980i	$0.497974\pi$
$74$	2.32112	0.269825
$75$	−1.00000	−0.115470
$76$	3.94981	0.453074
$77$	−12.6322	−1.43958
$78$	−0.551239	−0.0624155
$79$	4.32017	0.486057	0.243028	−	0.970019i	$-0.421859\pi$
$79$	4.32017	0.486057	0.243028	+	0.970019i	$0.421859\pi$
$80$	2.26915	0.253698
$81$	1.00000	0.111111
$82$	1.53805	0.169849
$83$	−12.9036	−1.41636	−0.708179	−	0.706033i	$-0.750483\pi$
$83$	−12.9036	−1.41636	−0.708179	+	0.706033i	$0.750483\pi$
$84$	7.07052	0.771457
$85$	−7.33654	−0.795759
$86$	4.43031	0.477733
$87$	−1.99980	−0.214401
$88$	6.17415	0.658167
$89$	12.2569	1.29923	0.649614	−	0.760264i	$-0.274930\pi$
$89$	12.2569	1.29923	0.649614	+	0.760264i	$0.274930\pi$
$90$	0.551239	0.0581057
$91$	4.16860	0.436988
$92$	7.83222	0.816566
$93$	−1.00000	−0.103695
$94$	6.81053	0.702452
$95$	−2.32871	−0.238921
$96$	−5.32575	−0.543557
$97$	−0.0930997	−0.00945284	−0.00472642	−	0.999989i	$-0.501504\pi$
$97$	−0.0930997	−0.00945284	−0.00472642	+	0.999989i	$0.501504\pi$
$98$	5.72035	0.577843
$99$	−3.03033	−0.304559
$100$	−1.69614	−0.169614
$101$	−15.8232	−1.57447	−0.787236	−	0.616652i	$-0.788489\pi$
$101$	−15.8232	−1.57447	−0.787236	+	0.616652i	$0.788489\pi$
$102$	4.04418	0.400434
$103$	11.8702	1.16960	0.584801	−	0.811177i	$-0.301173\pi$
$103$	11.8702	1.16960	0.584801	+	0.811177i	$0.301173\pi$
$104$	−2.03745	−0.199789
$105$	−4.16860	−0.406814
$106$	−2.83991	−0.275836
$107$	2.86304	0.276781	0.138390	−	0.990378i	$-0.455807\pi$
$107$	2.86304	0.276781	0.138390	+	0.990378i	$0.455807\pi$
$108$	1.69614	0.163211
$109$	10.2431	0.981115	0.490558	−	0.871409i	$-0.336793\pi$
$109$	10.2431	0.981115	0.490558	+	0.871409i	$0.336793\pi$
$110$	−1.67043	−0.159270
$111$	−4.21074	−0.399666
$112$	9.45918	0.893808
$113$	−0.427099	−0.0401781	−0.0200890	−	0.999798i	$-0.506395\pi$
$113$	−0.427099	−0.0401781	−0.0200890	+	0.999798i	$0.506395\pi$
$114$	1.28368	0.120227
$115$	−4.61769	−0.430601
$116$	−3.39193	−0.314933
$117$	1.00000	0.0924500
$118$	−3.14738	−0.289739
$119$	−30.5831	−2.80355
$120$	2.03745	0.185993
$121$	−1.81712	−0.165192
$122$	3.46711	0.313898
$123$	−2.79017	−0.251581
$124$	−1.69614	−0.152317
$125$	1.00000	0.0894427
$126$	2.29790	0.204713
$127$	−2.19087	−0.194408	−0.0972042	−	0.995264i	$-0.530990\pi$
$127$	−2.19087	−0.194408	−0.0972042	+	0.995264i	$0.530990\pi$
$128$	−11.5349	−1.01955
$129$	−8.03700	−0.707619
$130$	0.551239	0.0483469
$131$	8.24782	0.720615	0.360308	−	0.932834i	$-0.382672\pi$
$131$	8.24782	0.720615	0.360308	+	0.932834i	$0.382672\pi$
$132$	−5.13985	−0.447366
$133$	−9.70748	−0.841745
$134$	4.77180	0.412220
$135$	−1.00000	−0.0860663
$136$	14.9479	1.28177
$137$	−12.2773	−1.04892	−0.524462	−	0.851434i	$-0.675734\pi$
$137$	−12.2773	−1.04892	−0.524462	+	0.851434i	$0.675734\pi$
$138$	2.54545	0.216683
$139$	0.291863	0.0247555	0.0123778	−	0.999923i	$-0.496060\pi$
$139$	0.291863	0.0247555	0.0123778	+	0.999923i	$0.496060\pi$
$140$	−7.07052	−0.597568
$141$	−12.3549	−1.04047
$142$	−6.83048	−0.573201
$143$	−3.03033	−0.253409
$144$	2.26915	0.189096
$145$	1.99980	0.166074
$146$	0.0599613	0.00496244
$147$	−10.3773	−0.855902
$148$	−7.14199	−0.587068
$149$	16.0576	1.31549	0.657746	−	0.753240i	$-0.271510\pi$
$149$	16.0576	1.31549	0.657746	+	0.753240i	$0.271510\pi$
$150$	−0.551239	−0.0450085
$151$	22.8441	1.85902	0.929512	−	0.368793i	$-0.120229\pi$
$151$	22.8441	1.85902	0.929512	+	0.368793i	$0.120229\pi$
$152$	4.74464	0.384841
$153$	−7.33654	−0.593124
$154$	−6.96338	−0.561125
$155$	1.00000	0.0803219
$156$	1.69614	0.135800
$157$	−17.5950	−1.40423	−0.702115	−	0.712063i	$-0.747761\pi$
$157$	−17.5950	−1.40423	−0.702115	+	0.712063i	$0.747761\pi$
$158$	2.38144	0.189457
$159$	5.15186	0.408569
$160$	5.32575	0.421038
$161$	−19.2493	−1.51706
$162$	0.551239	0.0433094
$163$	−12.5960	−0.986597	−0.493299	−	0.869860i	$-0.664209\pi$
$163$	−12.5960	−0.986597	−0.493299	+	0.869860i	$0.664209\pi$
$164$	−4.73250	−0.369546
$165$	3.03033	0.235911
$166$	−7.11298	−0.552075
$167$	20.0758	1.55352	0.776758	−	0.629799i	$-0.216863\pi$
$167$	20.0758	1.55352	0.776758	+	0.629799i	$0.216863\pi$
$168$	8.49334	0.655275
$169$	1.00000	0.0769231
$170$	−4.04418	−0.310175
$171$	−2.32871	−0.178081
$172$	−13.6318	−1.03942
$173$	21.5668	1.63970	0.819848	−	0.572581i	$-0.194058\pi$
$173$	21.5668	1.63970	0.819848	+	0.572581i	$0.194058\pi$
$174$	−1.10237	−0.0835702
$175$	4.16860	0.315117
$176$	−6.87626	−0.518317
$177$	5.70964	0.429163
$178$	6.75648	0.506420
$179$	22.9385	1.71450	0.857251	−	0.514899i	$-0.172170\pi$
$179$	22.9385	1.71450	0.857251	+	0.514899i	$0.172170\pi$
$180$	−1.69614	−0.126422
$181$	−1.74061	−0.129379	−0.0646893	−	0.997905i	$-0.520606\pi$
$181$	−1.74061	−0.129379	−0.0646893	+	0.997905i	$0.520606\pi$
$182$	2.29790	0.170331
$183$	−6.28967	−0.464946
$184$	9.40832	0.693591
$185$	4.21074	0.309580
$186$	−0.551239	−0.0404188
$187$	22.2321	1.62577
$188$	−20.9557	−1.52835
$189$	−4.16860	−0.303221
$190$	−1.28368	−0.0931277
$191$	3.61464	0.261546	0.130773	−	0.991412i	$-0.458254\pi$
$191$	3.61464	0.261546	0.130773	+	0.991412i	$0.458254\pi$
$192$	1.60253	0.115653
$193$	22.2720	1.60317	0.801586	−	0.597879i	$-0.203990\pi$
$193$	22.2720	1.60317	0.801586	+	0.597879i	$0.203990\pi$
$194$	−0.0513202	−0.00368457
$195$	−1.00000	−0.0716115
$196$	−17.6012	−1.25723
$197$	1.29932	0.0925725	0.0462862	−	0.998928i	$-0.485261\pi$
$197$	1.29932	0.0925725	0.0462862	+	0.998928i	$0.485261\pi$
$198$	−1.67043	−0.118713
$199$	−2.91024	−0.206301	−0.103151	−	0.994666i	$-0.532892\pi$
$199$	−2.91024	−0.206301	−0.103151	+	0.994666i	$0.532892\pi$
$200$	−2.03745	−0.144070
$201$	−8.65649	−0.610582
$202$	−8.72239	−0.613705
$203$	8.33637	0.585099
$204$	−12.4438	−0.871237
$205$	2.79017	0.194874
$206$	6.54330	0.455893
$207$	−4.61769	−0.320951
$208$	2.26915	0.157337
$209$	7.05676	0.488126
$210$	−2.29790	−0.158570
$211$	13.5530	0.933028	0.466514	−	0.884514i	$-0.345510\pi$
$211$	13.5530	0.933028	0.466514	+	0.884514i	$0.345510\pi$
$212$	8.73825	0.600146
$213$	12.3911	0.849027
$214$	1.57822	0.107885
$215$	8.03700	0.548119
$216$	2.03745	0.138631
$217$	4.16860	0.282983
$218$	5.64642	0.382424
$219$	−0.108776	−0.00735037
$220$	5.13985	0.346528
$221$	−7.33654	−0.493509
$222$	−2.32112	−0.155784
$223$	−21.9261	−1.46828	−0.734140	−	0.678998i	$-0.762414\pi$
$223$	−21.9261	−1.46828	−0.734140	+	0.678998i	$0.762414\pi$
$224$	22.2009	1.48336
$225$	1.00000	0.0666667
$226$	−0.235433	−0.0156608
$227$	16.4044	1.08880	0.544398	−	0.838827i	$-0.316758\pi$
$227$	16.4044	1.08880	0.544398	+	0.838827i	$0.316758\pi$
$228$	−3.94981	−0.261583
$229$	14.4318	0.953683	0.476842	−	0.878989i	$-0.341782\pi$
$229$	14.4318	0.953683	0.476842	+	0.878989i	$0.341782\pi$
$230$	−2.54545	−0.167842
$231$	12.6322	0.831140
$232$	−4.07450	−0.267504
$233$	−20.2749	−1.32825	−0.664127	−	0.747619i	$-0.731197\pi$
$233$	−20.2749	−1.32825	−0.664127	+	0.747619i	$0.731197\pi$
$234$	0.551239	0.0360356
$235$	12.3549	0.805948
$236$	9.68432	0.630396
$237$	−4.32017	−0.280625
$238$	−16.8586	−1.09278
$239$	−16.1566	−1.04508	−0.522541	−	0.852614i	$-0.675016\pi$
$239$	−16.1566	−1.04508	−0.522541	+	0.852614i	$0.675016\pi$
$240$	−2.26915	−0.146473
$241$	9.20295	0.592814	0.296407	−	0.955062i	$-0.404211\pi$
$241$	9.20295	0.592814	0.296407	+	0.955062i	$0.404211\pi$
$242$	−1.00167	−0.0643895
$243$	−1.00000	−0.0641500
$244$	−10.6681	−0.682958
$245$	10.3773	0.662979
$246$	−1.53805	−0.0980624
$247$	−2.32871	−0.148172
$248$	−2.03745	−0.129378
$249$	12.9036	0.817734
$250$	0.551239	0.0348634
$251$	20.4326	1.28969	0.644846	−	0.764313i	$-0.276922\pi$
$251$	20.4326	1.28969	0.644846	+	0.764313i	$0.276922\pi$
$252$	−7.07052	−0.445401
$253$	13.9931	0.879739
$254$	−1.20769	−0.0757775
$255$	7.33654	0.459432
$256$	−3.15341	−0.197088
$257$	25.1338	1.56780	0.783901	−	0.620885i	$-0.213227\pi$
$257$	25.1338	1.56780	0.783901	+	0.620885i	$0.213227\pi$
$258$	−4.43031	−0.275819
$259$	17.5529	1.09069
$260$	−1.69614	−0.105190
$261$	1.99980	0.123784
$262$	4.54652	0.280885
$263$	16.2088	0.999477	0.499739	−	0.866176i	$-0.333429\pi$
$263$	16.2088	0.999477	0.499739	+	0.866176i	$0.333429\pi$
$264$	−6.17415	−0.379993
$265$	−5.15186	−0.316476
$266$	−5.35114	−0.328099
$267$	−12.2569	−0.750110
$268$	−14.6826	−0.896882
$269$	30.3100	1.84804	0.924018	−	0.382349i	$-0.124885\pi$
$269$	30.3100	1.84804	0.924018	+	0.382349i	$0.124885\pi$
$270$	−0.551239	−0.0335473
$271$	−0.667252	−0.0405327	−0.0202663	−	0.999795i	$-0.506451\pi$
$271$	−0.667252	−0.0405327	−0.0202663	+	0.999795i	$0.506451\pi$
$272$	−16.6477	−1.00941
$273$	−4.16860	−0.252295
$274$	−6.76775	−0.408855
$275$	−3.03033	−0.182736
$276$	−7.83222	−0.471444
$277$	−15.8762	−0.953910	−0.476955	−	0.878928i	$-0.658259\pi$
$277$	−15.8762	−0.953910	−0.476955	+	0.878928i	$0.658259\pi$
$278$	0.160886	0.00964932
$279$	1.00000	0.0598684
$280$	−8.49334	−0.507574
$281$	19.5615	1.16694	0.583472	−	0.812133i	$-0.301694\pi$
$281$	19.5615	1.16694	0.583472	+	0.812133i	$0.301694\pi$
$282$	−6.81053	−0.405561
$283$	16.6232	0.988148	0.494074	−	0.869420i	$-0.335507\pi$
$283$	16.6232	0.988148	0.494074	+	0.869420i	$0.335507\pi$
$284$	21.0171	1.24713
$285$	2.32871	0.137941
$286$	−1.67043	−0.0987749
$287$	11.6311	0.686562
$288$	5.32575	0.313823
$289$	36.8247	2.16616
$290$	1.10237	0.0647332
$291$	0.0930997	0.00545760
$292$	−0.184498	−0.0107969
$293$	8.27321	0.483326	0.241663	−	0.970360i	$-0.422307\pi$
$293$	8.27321	0.483326	0.241663	+	0.970360i	$0.422307\pi$
$294$	−5.72035	−0.333618
$295$	−5.70964	−0.332428
$296$	−8.57919	−0.498655
$297$	3.03033	0.175837
$298$	8.85158	0.512758
$299$	−4.61769	−0.267048
$300$	1.69614	0.0979264
$301$	33.5031	1.93109
$302$	12.5925	0.724619
$303$	15.8232	0.909022
$304$	−5.28419	−0.303069
$305$	6.28967	0.360146
$306$	−4.04418	−0.231191
$307$	6.25116	0.356773	0.178386	−	0.983961i	$-0.442912\pi$
$307$	6.25116	0.356773	0.178386	+	0.983961i	$0.442912\pi$
$308$	21.4260	1.22086
$309$	−11.8702	−0.675270
$310$	0.551239	0.0313083
$311$	−29.1576	−1.65338	−0.826689	−	0.562660i	$-0.809778\pi$
$311$	−29.1576	−1.65338	−0.826689	+	0.562660i	$0.809778\pi$
$312$	2.03745	0.115348
$313$	−32.4430	−1.83378	−0.916892	−	0.399135i	$-0.869311\pi$
$313$	−32.4430	−1.83378	−0.916892	+	0.399135i	$0.869311\pi$
$314$	−9.69902	−0.547348
$315$	4.16860	0.234874
$316$	−7.32759	−0.412209
$317$	4.23471	0.237845	0.118922	−	0.992904i	$-0.462056\pi$
$317$	4.23471	0.237845	0.118922	+	0.992904i	$0.462056\pi$
$318$	2.83991	0.159254
$319$	−6.06005	−0.339297
$320$	−1.60253	−0.0895843
$321$	−2.86304	−0.159799
$322$	−10.6110	−0.591326
$323$	17.0847	0.950617
$324$	−1.69614	−0.0942298
$325$	1.00000	0.0554700
$326$	−6.94342	−0.384561
$327$	−10.2431	−0.566447
$328$	−5.68484	−0.313893
$329$	51.5029	2.83945
$330$	1.67043	0.0919544
$331$	−33.3875	−1.83515	−0.917573	−	0.397568i	$-0.869854\pi$
$331$	−33.3875	−1.83515	−0.917573	+	0.397568i	$0.869854\pi$
$332$	21.8863	1.20117
$333$	4.21074	0.230747
$334$	11.0666	0.605537
$335$	8.65649	0.472955
$336$	−9.45918	−0.516040
$337$	−20.2920	−1.10538	−0.552688	−	0.833388i	$-0.686398\pi$
$337$	−20.2920	−1.10538	−0.552688	+	0.833388i	$0.686398\pi$
$338$	0.551239	0.0299834
$339$	0.427099	0.0231968
$340$	12.4438	0.674858
$341$	−3.03033	−0.164101
$342$	−1.28368	−0.0694133
$343$	14.0785	0.760166
$344$	−16.3750	−0.882882
$345$	4.61769	0.248608
$346$	11.8885	0.639129
$347$	−6.54748	−0.351487	−0.175744	−	0.984436i	$-0.556233\pi$
$347$	−6.54748	−0.351487	−0.175744	+	0.984436i	$0.556233\pi$
$348$	3.39193	0.181827
$349$	19.3420	1.03535	0.517677	−	0.855576i	$-0.326797\pi$
$349$	19.3420	1.03535	0.517677	+	0.855576i	$0.326797\pi$
$350$	2.29790	0.122828
$351$	−1.00000	−0.0533761
$352$	−16.1388	−0.860199
$353$	−5.76305	−0.306736	−0.153368	−	0.988169i	$-0.549012\pi$
$353$	−5.76305	−0.306736	−0.153368	+	0.988169i	$0.549012\pi$
$354$	3.14738	0.167281
$355$	−12.3911	−0.657653
$356$	−20.7894	−1.10183
$357$	30.5831	1.61863
$358$	12.6446	0.668287
$359$	36.3632	1.91918	0.959588	−	0.281410i	$-0.0908020\pi$
$359$	36.3632	1.91918	0.959588	+	0.281410i	$0.0908020\pi$
$360$	−2.03745	−0.107383
$361$	−13.5771	−0.714584
$362$	−0.959493	−0.0504298
$363$	1.81712	0.0953739
$364$	−7.07052	−0.370596
$365$	0.108776	0.00569357
$366$	−3.46711	−0.181229
$367$	2.29521	0.119809	0.0599046	−	0.998204i	$-0.480920\pi$
$367$	2.29521	0.119809	0.0599046	+	0.998204i	$0.480920\pi$
$368$	−10.4782	−0.546214
$369$	2.79017	0.145250
$370$	2.32112	0.120670
$371$	−21.4761	−1.11498
$372$	1.69614	0.0879405
$373$	31.0940	1.60999	0.804993	−	0.593284i	$-0.202169\pi$
$373$	31.0940	1.60999	0.804993	+	0.593284i	$0.202169\pi$
$374$	12.2552	0.633701
$375$	−1.00000	−0.0516398
$376$	−25.1726	−1.29818
$377$	1.99980	0.102995
$378$	−2.29790	−0.118191
$379$	3.35559	0.172365	0.0861827	−	0.996279i	$-0.472533\pi$
$379$	3.35559	0.172365	0.0861827	+	0.996279i	$0.472533\pi$
$380$	3.94981	0.202621
$381$	2.19087	0.112242
$382$	1.99253	0.101947
$383$	3.28242	0.167724	0.0838619	−	0.996477i	$-0.473275\pi$
$383$	3.28242	0.167724	0.0838619	+	0.996477i	$0.473275\pi$
$384$	11.5349	0.588637
$385$	−12.6322	−0.643798
$386$	12.2772	0.624892
$387$	8.03700	0.408544
$388$	0.157910	0.00801665
$389$	−19.2841	−0.977745	−0.488872	−	0.872355i	$-0.662592\pi$
$389$	−19.2841	−0.977745	−0.488872	+	0.872355i	$0.662592\pi$
$390$	−0.551239	−0.0279131
$391$	33.8778	1.71327
$392$	−21.1432	−1.06789
$393$	−8.24782	−0.416047
$394$	0.716234	0.0360833
$395$	4.32017	0.217371
$396$	5.13985	0.258287
$397$	−33.6549	−1.68909	−0.844545	−	0.535485i	$-0.820129\pi$
$397$	−33.6549	−1.68909	−0.844545	+	0.535485i	$0.820129\pi$
$398$	−1.60424	−0.0804132
$399$	9.70748	0.485982
$400$	2.26915	0.113457
$401$	−13.0248	−0.650426	−0.325213	−	0.945641i	$-0.605436\pi$
$401$	−13.0248	−0.650426	−0.325213	+	0.945641i	$0.605436\pi$
$402$	−4.77180	−0.237996
$403$	1.00000	0.0498135
$404$	26.8384	1.33526
$405$	1.00000	0.0496904
$406$	4.59533	0.228063
$407$	−12.7599	−0.632486
$408$	−14.9479	−0.740029
$409$	11.0758	0.547663	0.273832	−	0.961778i	$-0.411709\pi$
$409$	11.0758	0.547663	0.273832	+	0.961778i	$0.411709\pi$
$410$	1.53805	0.0759588
$411$	12.2773	0.605597
$412$	−20.1334	−0.991902
$413$	−23.8012	−1.17118
$414$	−2.54545	−0.125102
$415$	−12.9036	−0.633414
$416$	5.32575	0.261116
$417$	−0.291863	−0.0142926
$418$	3.88996	0.190264
$419$	32.4490	1.58524	0.792620	−	0.609717i	$-0.208717\pi$
$419$	32.4490	1.58524	0.792620	+	0.609717i	$0.208717\pi$
$420$	7.07052	0.345006
$421$	12.4988	0.609153	0.304576	−	0.952488i	$-0.401485\pi$
$421$	12.4988	0.609153	0.304576	+	0.952488i	$0.401485\pi$
$422$	7.47095	0.363680
$423$	12.3549	0.600718
$424$	10.4967	0.509764
$425$	−7.33654	−0.355874
$426$	6.83048	0.330938
$427$	26.2192	1.26883
$428$	−4.85611	−0.234729
$429$	3.03033	0.146306
$430$	4.43031	0.213649
$431$	−20.9236	−1.00785	−0.503926	−	0.863747i	$-0.668112\pi$
$431$	−20.9236	−1.00785	−0.503926	+	0.863747i	$0.668112\pi$
$432$	−2.26915	−0.109174
$433$	20.3506	0.977985	0.488993	−	0.872288i	$-0.337365\pi$
$433$	20.3506	0.977985	0.488993	+	0.872288i	$0.337365\pi$
$434$	2.29790	0.110303
$435$	−1.99980	−0.0958830
$436$	−17.3738	−0.832052
$437$	10.7533	0.514398
$438$	−0.0599613	−0.00286506
$439$	−9.52677	−0.454688	−0.227344	−	0.973815i	$-0.573004\pi$
$439$	−9.52677	−0.454688	−0.227344	+	0.973815i	$0.573004\pi$
$440$	6.17415	0.294341
$441$	10.3773	0.494155
$442$	−4.04418	−0.192362
$443$	−12.9460	−0.615084	−0.307542	−	0.951535i	$-0.599506\pi$
$443$	−12.9460	−0.615084	−0.307542	+	0.951535i	$0.599506\pi$
$444$	7.14199	0.338944
$445$	12.2569	0.581033
$446$	−12.0865	−0.572313
$447$	−16.0576	−0.759499
$448$	−6.68033	−0.315616
$449$	−34.3240	−1.61985	−0.809924	−	0.586534i	$-0.800492\pi$
$449$	−34.3240	−1.61985	−0.809924	+	0.586534i	$0.800492\pi$
$450$	0.551239	0.0259857
$451$	−8.45512	−0.398136
$452$	0.724417	0.0340737
$453$	−22.8441	−1.07331
$454$	9.04273	0.424396
$455$	4.16860	0.195427
$456$	−4.74464	−0.222188
$457$	28.2812	1.32294	0.661470	−	0.749972i	$-0.269933\pi$
$457$	28.2812	1.32294	0.661470	+	0.749972i	$0.269933\pi$
$458$	7.95539	0.371731
$459$	7.33654	0.342440
$460$	7.83222	0.365179
$461$	−22.7585	−1.05997	−0.529984	−	0.848007i	$-0.677802\pi$
$461$	−22.7585	−1.05997	−0.529984	+	0.848007i	$0.677802\pi$
$462$	6.96338	0.323966
$463$	15.4611	0.718537	0.359269	−	0.933234i	$-0.383026\pi$
$463$	15.4611	0.718537	0.359269	+	0.933234i	$0.383026\pi$
$464$	4.53784	0.210664
$465$	−1.00000	−0.0463739
$466$	−11.1763	−0.517734
$467$	21.6167	1.00030	0.500152	−	0.865938i	$-0.333277\pi$
$467$	21.6167	1.00030	0.500152	+	0.865938i	$0.333277\pi$
$468$	−1.69614	−0.0784039
$469$	36.0855	1.66627
$470$	6.81053	0.314146
$471$	17.5950	0.810733
$472$	11.6331	0.535458
$473$	−24.3548	−1.11983
$474$	−2.38144	−0.109383
$475$	−2.32871	−0.106849
$476$	51.8731	2.37760
$477$	−5.15186	−0.235887
$478$	−8.90614	−0.407357
$479$	−5.67475	−0.259286	−0.129643	−	0.991561i	$-0.541383\pi$
$479$	−5.67475	−0.259286	−0.129643	+	0.991561i	$0.541383\pi$
$480$	−5.32575	−0.243086
$481$	4.21074	0.191993
$482$	5.07303	0.231070
$483$	19.2493	0.875874
$484$	3.08208	0.140094
$485$	−0.0930997	−0.00422744
$486$	−0.551239	−0.0250047
$487$	−7.14258	−0.323661	−0.161831	−	0.986819i	$-0.551740\pi$
$487$	−7.14258	−0.323661	−0.161831	+	0.986819i	$0.551740\pi$
$488$	−12.8149	−0.580104
$489$	12.5960	0.569612
$490$	5.72035	0.258419
$491$	−0.333652	−0.0150575	−0.00752875	−	0.999972i	$-0.502396\pi$
$491$	−0.333652	−0.0150575	−0.00752875	+	0.999972i	$0.502396\pi$
$492$	4.73250	0.213358
$493$	−14.6716	−0.660775
$494$	−1.28368	−0.0577553
$495$	−3.03033	−0.136203
$496$	2.26915	0.101888
$497$	−51.6538	−2.31699
$498$	7.11298	0.318740
$499$	9.26640	0.414821	0.207411	−	0.978254i	$-0.433496\pi$
$499$	9.26640	0.414821	0.207411	+	0.978254i	$0.433496\pi$
$500$	−1.69614	−0.0758535
$501$	−20.0758	−0.896923
$502$	11.2632	0.502702
$503$	−25.0790	−1.11822	−0.559108	−	0.829095i	$-0.688856\pi$
$503$	−25.0790	−1.11822	−0.559108	+	0.829095i	$0.688856\pi$
$504$	−8.49334	−0.378323
$505$	−15.8232	−0.704125
$506$	7.71354	0.342909
$507$	−1.00000	−0.0444116
$508$	3.71602	0.164872
$509$	−9.08271	−0.402584	−0.201292	−	0.979531i	$-0.564514\pi$
$509$	−9.08271	−0.402584	−0.201292	+	0.979531i	$0.564514\pi$
$510$	4.04418	0.179079
$511$	0.453442	0.0200591
$512$	21.3315	0.942727
$513$	2.32871	0.102815
$514$	13.8547	0.611106
$515$	11.8702	0.523062
$516$	13.6318	0.600109
$517$	−37.4395	−1.64659
$518$	9.67585	0.425132
$519$	−21.5668	−0.946679
$520$	−2.03745	−0.0893483
$521$	−3.55980	−0.155958	−0.0779788	−	0.996955i	$-0.524847\pi$
$521$	−3.55980	−0.155958	−0.0779788	+	0.996955i	$0.524847\pi$
$522$	1.10237	0.0482493
$523$	−17.9023	−0.782812	−0.391406	−	0.920218i	$-0.628011\pi$
$523$	−17.9023	−0.782812	−0.391406	+	0.920218i	$0.628011\pi$
$524$	−13.9894	−0.611131
$525$	−4.16860	−0.181933
$526$	8.93492	0.389581
$527$	−7.33654	−0.319584
$528$	6.87626	0.299251
$529$	−1.67697	−0.0729119
$530$	−2.83991	−0.123358
$531$	−5.70964	−0.247777
$532$	16.4652	0.713857
$533$	2.79017	0.120856
$534$	−6.75648	−0.292382
$535$	2.86304	0.123780
$536$	−17.6372	−0.761811
$537$	−22.9385	−0.989868
$538$	16.7081	0.720336
$539$	−31.4465	−1.35450
$540$	1.69614	0.0729901
$541$	−11.9671	−0.514507	−0.257254	−	0.966344i	$-0.582818\pi$
$541$	−11.9671	−0.514507	−0.257254	+	0.966344i	$0.582818\pi$
$542$	−0.367815	−0.0157990
$543$	1.74061	0.0746968
$544$	−39.0726	−1.67522
$545$	10.2431	0.438768
$546$	−2.29790	−0.0983409
$547$	6.30960	0.269779	0.134890	−	0.990861i	$-0.456932\pi$
$547$	6.30960	0.269779	0.134890	+	0.990861i	$0.456932\pi$
$548$	20.8240	0.889559
$549$	6.28967	0.268437
$550$	−1.67043	−0.0712276
$551$	−4.65696	−0.198393
$552$	−9.40832	−0.400445
$553$	18.0091	0.765823
$554$	−8.75159	−0.371820
$555$	−4.21074	−0.178736
$556$	−0.495039	−0.0209943
$557$	34.9243	1.47979	0.739895	−	0.672723i	$-0.234875\pi$
$557$	34.9243	1.47979	0.739895	+	0.672723i	$0.234875\pi$
$558$	0.551239	0.0233358
$559$	8.03700	0.339929
$560$	9.45918	0.399723
$561$	−22.2321	−0.938640
$562$	10.7831	0.454857
$563$	15.8604	0.668436	0.334218	−	0.942496i	$-0.391528\pi$
$563$	15.8604	0.668436	0.334218	+	0.942496i	$0.391528\pi$
$564$	20.9557	0.882392
$565$	−0.427099	−0.0179682
$566$	9.16337	0.385165
$567$	4.16860	0.175065
$568$	25.2464	1.05931
$569$	12.0035	0.503214	0.251607	−	0.967829i	$-0.419041\pi$
$569$	12.0035	0.503214	0.251607	+	0.967829i	$0.419041\pi$
$570$	1.28368	0.0537673
$571$	−1.36415	−0.0570881	−0.0285440	−	0.999593i	$-0.509087\pi$
$571$	−1.36415	−0.0570881	−0.0285440	+	0.999593i	$0.509087\pi$
$572$	5.13985	0.214908
$573$	−3.61464	−0.151004
$574$	6.41152	0.267612
$575$	−4.61769	−0.192571
$576$	−1.60253	−0.0667722
$577$	−45.1461	−1.87946	−0.939729	−	0.341920i	$-0.888923\pi$
$577$	−45.1461	−1.87946	−0.939729	+	0.341920i	$0.888923\pi$
$578$	20.2992	0.844337
$579$	−22.2720	−0.925592
$580$	−3.39193	−0.140842
$581$	−53.7901	−2.23159
$582$	0.0513202	0.00212729
$583$	15.6118	0.646576
$584$	−0.221625	−0.00917092
$585$	1.00000	0.0413449
$586$	4.56052	0.188393
$587$	18.7346	0.773260	0.386630	−	0.922235i	$-0.373639\pi$
$587$	18.7346	0.773260	0.386630	+	0.922235i	$0.373639\pi$
$588$	17.6012	0.725863
$589$	−2.32871	−0.0959529
$590$	−3.14738	−0.129575
$591$	−1.29932	−0.0534467
$592$	9.55479	0.392699
$593$	24.0955	0.989483	0.494742	−	0.869040i	$-0.335263\pi$
$593$	24.0955	0.989483	0.494742	+	0.869040i	$0.335263\pi$
$594$	1.67043	0.0685388
$595$	−30.5831	−1.25379
$596$	−27.2359	−1.11563
$597$	2.91024	0.119108
$598$	−2.54545	−0.104091
$599$	−20.2181	−0.826088	−0.413044	−	0.910711i	$-0.635535\pi$
$599$	−20.2181	−0.826088	−0.413044	+	0.910711i	$0.635535\pi$
$600$	2.03745	0.0831787
$601$	13.8574	0.565254	0.282627	−	0.959230i	$-0.408794\pi$
$601$	13.8574	0.565254	0.282627	+	0.959230i	$0.408794\pi$
$602$	18.4682	0.752708
$603$	8.65649	0.352520
$604$	−38.7466	−1.57658
$605$	−1.81712	−0.0738763
$606$	8.72239	0.354323
$607$	28.2486	1.14657	0.573287	−	0.819355i	$-0.305668\pi$
$607$	28.2486	1.14657	0.573287	+	0.819355i	$0.305668\pi$
$608$	−12.4021	−0.502973
$609$	−8.33637	−0.337807
$610$	3.46711	0.140379
$611$	12.3549	0.499827
$612$	12.4438	0.503009
$613$	7.13534	0.288194	0.144097	−	0.989564i	$-0.453972\pi$
$613$	7.13534	0.288194	0.144097	+	0.989564i	$0.453972\pi$
$614$	3.44588	0.139065
$615$	−2.79017	−0.112510
$616$	25.7376	1.03700
$617$	23.7639	0.956699	0.478349	−	0.878170i	$-0.341235\pi$
$617$	23.7639	0.956699	0.478349	+	0.878170i	$0.341235\pi$
$618$	−6.54330	−0.263210
$619$	11.0077	0.442436	0.221218	−	0.975224i	$-0.428997\pi$
$619$	11.0077	0.442436	0.221218	+	0.975224i	$0.428997\pi$
$620$	−1.69614	−0.0681184
$621$	4.61769	0.185301
$622$	−16.0728	−0.644461
$623$	51.0942	2.04704
$624$	−2.26915	−0.0908386
$625$	1.00000	0.0400000
$626$	−17.8838	−0.714781
$627$	−7.05676	−0.281820
$628$	29.8434	1.19088
$629$	−30.8923	−1.23175
$630$	2.29790	0.0915504
$631$	−43.6896	−1.73926	−0.869628	−	0.493708i	$-0.835641\pi$
$631$	−43.6896	−1.73926	−0.869628	+	0.493708i	$0.835641\pi$
$632$	−8.80214	−0.350130
$633$	−13.5530	−0.538684
$634$	2.33434	0.0927083
$635$	−2.19087	−0.0869421
$636$	−8.73825	−0.346494
$637$	10.3773	0.411162
$638$	−3.34053	−0.132253
$639$	−12.3911	−0.490186
$640$	−11.5349	−0.455956
$641$	−7.98349	−0.315329	−0.157664	−	0.987493i	$-0.550396\pi$
$641$	−7.98349	−0.315329	−0.157664	+	0.987493i	$0.550396\pi$
$642$	−1.57822	−0.0622874
$643$	42.3075	1.66844	0.834222	−	0.551428i	$-0.185917\pi$
$643$	42.3075	1.66844	0.834222	+	0.551428i	$0.185917\pi$
$644$	32.6494	1.28657
$645$	−8.03700	−0.316457
$646$	9.41774	0.370536
$647$	8.25955	0.324716	0.162358	−	0.986732i	$-0.448090\pi$
$647$	8.25955	0.324716	0.162358	+	0.986732i	$0.448090\pi$
$648$	−2.03745	−0.0800388
$649$	17.3021	0.679166
$650$	0.551239	0.0216214
$651$	−4.16860	−0.163380
$652$	21.3646	0.836701
$653$	46.9778	1.83838	0.919191	−	0.393811i	$-0.128844\pi$
$653$	46.9778	1.83838	0.919191	+	0.393811i	$0.128844\pi$
$654$	−5.64642	−0.220792
$655$	8.24782	0.322269
$656$	6.33130	0.247196
$657$	0.108776	0.00424374
$658$	28.3904	1.10677
$659$	0.970199	0.0377936	0.0188968	−	0.999821i	$-0.493985\pi$
$659$	0.970199	0.0377936	0.0188968	+	0.999821i	$0.493985\pi$
$660$	−5.13985	−0.200068
$661$	−1.35721	−0.0527895	−0.0263948	−	0.999652i	$-0.508403\pi$
$661$	−1.35721	−0.0527895	−0.0263948	+	0.999652i	$0.508403\pi$
$662$	−18.4045	−0.715312
$663$	7.33654	0.284927
$664$	26.2906	1.02027
$665$	−9.70748	−0.376440
$666$	2.32112	0.0899418
$667$	−9.23445	−0.357559
$668$	−34.0514	−1.31749
$669$	21.9261	0.847712
$670$	4.77180	0.184351
$671$	−19.0598	−0.735794
$672$	−22.2009	−0.856420
$673$	9.81256	0.378246	0.189123	−	0.981953i	$-0.439435\pi$
$673$	9.81256	0.378246	0.189123	+	0.981953i	$0.439435\pi$
$674$	−11.1857	−0.430859
$675$	−1.00000	−0.0384900
$676$	−1.69614	−0.0652360
$677$	38.7854	1.49064	0.745322	−	0.666705i	$-0.232296\pi$
$677$	38.7854	1.49064	0.745322	+	0.666705i	$0.232296\pi$
$678$	0.235433	0.00904176
$679$	−0.388096	−0.0148937
$680$	14.9479	0.573224
$681$	−16.4044	−0.628617
$682$	−1.67043	−0.0639642
$683$	12.9338	0.494898	0.247449	−	0.968901i	$-0.420408\pi$
$683$	12.9338	0.494898	0.247449	+	0.968901i	$0.420408\pi$
$684$	3.94981	0.151025
$685$	−12.2773	−0.469093
$686$	7.76060	0.296301
$687$	−14.4318	−0.550609
$688$	18.2371	0.695285
$689$	−5.15186	−0.196270
$690$	2.54545	0.0969036
$691$	−1.75372	−0.0667146	−0.0333573	−	0.999443i	$-0.510620\pi$
$691$	−1.75372	−0.0667146	−0.0333573	+	0.999443i	$0.510620\pi$
$692$	−36.5803	−1.39057
$693$	−12.6322	−0.479859
$694$	−3.60923	−0.137004
$695$	0.291863	0.0110710
$696$	4.07450	0.154443
$697$	−20.4702	−0.775363
$698$	10.6621	0.403566
$699$	20.2749	0.766868
$700$	−7.07052	−0.267240
$701$	34.8813	1.31745	0.658725	−	0.752384i	$-0.271096\pi$
$701$	34.8813	1.31745	0.658725	+	0.752384i	$0.271096\pi$
$702$	−0.551239	−0.0208052
$703$	−9.80560	−0.369825
$704$	4.85620	0.183025
$705$	−12.3549	−0.465314
$706$	−3.17682	−0.119561
$707$	−65.9609	−2.48071
$708$	−9.68432	−0.363959
$709$	−15.0101	−0.563715	−0.281858	−	0.959456i	$-0.590951\pi$
$709$	−15.0101	−0.563715	−0.281858	+	0.959456i	$0.590951\pi$
$710$	−6.83048	−0.256343
$711$	4.32017	0.162019
$712$	−24.9729	−0.935898
$713$	−4.61769	−0.172934
$714$	16.8586	0.630917
$715$	−3.03033	−0.113328
$716$	−38.9068	−1.45401
$717$	16.1566	0.603379
$718$	20.0448	0.748065
$719$	15.9718	0.595648	0.297824	−	0.954621i	$-0.403739\pi$
$719$	15.9718	0.595648	0.297824	+	0.954621i	$0.403739\pi$
$720$	2.26915	0.0845661
$721$	49.4820	1.84281
$722$	−7.48423	−0.278534
$723$	−9.20295	−0.342262
$724$	2.95231	0.109722
$725$	1.99980	0.0742707
$726$	1.00167	0.0371753
$727$	42.6883	1.58322	0.791611	−	0.611026i	$-0.209243\pi$
$727$	42.6883	1.58322	0.791611	+	0.611026i	$0.209243\pi$
$728$	−8.49334	−0.314784
$729$	1.00000	0.0370370
$730$	0.0599613	0.00221927
$731$	−58.9638	−2.18085
$732$	10.6681	0.394306
$733$	−14.5590	−0.537750	−0.268875	−	0.963175i	$-0.586652\pi$
$733$	−14.5590	−0.537750	−0.268875	+	0.963175i	$0.586652\pi$
$734$	1.26521	0.0466998
$735$	−10.3773	−0.382771
$736$	−24.5926	−0.906497
$737$	−26.2320	−0.966268
$738$	1.53805	0.0566164
$739$	−38.4802	−1.41552	−0.707758	−	0.706455i	$-0.750293\pi$
$739$	−38.4802	−1.41552	−0.707758	+	0.706455i	$0.750293\pi$
$740$	−7.14199	−0.262545
$741$	2.32871	0.0855474
$742$	−11.8384	−0.434603
$743$	48.2624	1.77057	0.885287	−	0.465045i	$-0.153962\pi$
$743$	48.2624	1.77057	0.885287	+	0.465045i	$0.153962\pi$
$744$	2.03745	0.0746967
$745$	16.0576	0.588306
$746$	17.1402	0.627548
$747$	−12.9036	−0.472119
$748$	−37.7087	−1.37877
$749$	11.9349	0.436091
$750$	−0.551239	−0.0201284
$751$	30.6759	1.11938	0.559690	−	0.828702i	$-0.310920\pi$
$751$	30.6759	1.11938	0.559690	+	0.828702i	$0.310920\pi$
$752$	28.0352	1.02234
$753$	−20.4326	−0.744604
$754$	1.10237	0.0401458
$755$	22.8441	0.831381
$756$	7.07052	0.257152
$757$	−5.72384	−0.208036	−0.104018	−	0.994575i	$-0.533170\pi$
$757$	−5.72384	−0.208036	−0.104018	+	0.994575i	$0.533170\pi$
$758$	1.84973	0.0671854
$759$	−13.9931	−0.507917
$760$	4.74464	0.172106
$761$	−14.9672	−0.542561	−0.271281	−	0.962500i	$-0.587447\pi$
$761$	−14.9672	−0.542561	−0.271281	+	0.962500i	$0.587447\pi$
$762$	1.20769	0.0437501
$763$	42.6996	1.54583
$764$	−6.13092	−0.221809
$765$	−7.33654	−0.265253
$766$	1.80940	0.0653762
$767$	−5.70964	−0.206163
$768$	3.15341	0.113789
$769$	−19.5263	−0.704137	−0.352068	−	0.935974i	$-0.614522\pi$
$769$	−19.5263	−0.704137	−0.352068	+	0.935974i	$0.614522\pi$
$770$	−6.96338	−0.250943
$771$	−25.1338	−0.905171
$772$	−37.7763	−1.35960
$773$	−29.2483	−1.05199	−0.525993	−	0.850489i	$-0.676306\pi$
$773$	−29.2483	−1.05199	−0.525993	+	0.850489i	$0.676306\pi$
$774$	4.43031	0.159244
$775$	1.00000	0.0359211
$776$	0.189686	0.00680934
$777$	−17.5529	−0.629707
$778$	−10.6302	−0.381110
$779$	−6.49750	−0.232797
$780$	1.69614	0.0607314
$781$	37.5492	1.34362
$782$	18.6748	0.667809
$783$	−1.99980	−0.0714670
$784$	23.5475	0.840983
$785$	−17.5950	−0.627991
$786$	−4.54652	−0.162169
$787$	4.49375	0.160185	0.0800925	−	0.996787i	$-0.474478\pi$
$787$	4.49375	0.160185	0.0800925	+	0.996787i	$0.474478\pi$
$788$	−2.20382	−0.0785077
$789$	−16.2088	−0.577049
$790$	2.38144	0.0847280
$791$	−1.78040	−0.0633039
$792$	6.17415	0.219389
$793$	6.28967	0.223353
$794$	−18.5519	−0.658382
$795$	5.15186	0.182718
$796$	4.93616	0.174958
$797$	−33.9403	−1.20223	−0.601114	−	0.799164i	$-0.705276\pi$
$797$	−33.9403	−1.20223	−0.601114	+	0.799164i	$0.705276\pi$
$798$	5.35114	0.189428
$799$	−90.6425	−3.20670
$800$	5.32575	0.188294
$801$	12.2569	0.433076
$802$	−7.17977	−0.253526
$803$	−0.329626	−0.0116322
$804$	14.6826	0.517815
$805$	−19.2493	−0.678449
$806$	0.551239	0.0194166
$807$	−30.3100	−1.06696
$808$	32.2391	1.13417
$809$	−30.3477	−1.06697	−0.533485	−	0.845810i	$-0.679118\pi$
$809$	−30.3477	−1.06697	−0.533485	+	0.845810i	$0.679118\pi$
$810$	0.551239	0.0193686
$811$	−33.4504	−1.17460	−0.587302	−	0.809368i	$-0.699810\pi$
$811$	−33.4504	−1.17460	−0.587302	+	0.809368i	$0.699810\pi$
$812$	−14.1396	−0.496203
$813$	0.667252	0.0234016
$814$	−7.03377	−0.246533
$815$	−12.5960	−0.441220
$816$	16.6477	0.582785
$817$	−18.7159	−0.654785
$818$	6.10541	0.213471
$819$	4.16860	0.145663
$820$	−4.73250	−0.165266
$821$	−21.0286	−0.733903	−0.366952	−	0.930240i	$-0.619599\pi$
$821$	−21.0286	−0.733903	−0.366952	+	0.930240i	$0.619599\pi$
$822$	6.76775	0.236052
$823$	−17.2214	−0.600300	−0.300150	−	0.953892i	$-0.597037\pi$
$823$	−17.2214	−0.600300	−0.300150	+	0.953892i	$0.597037\pi$
$824$	−24.1849	−0.842521
$825$	3.03033	0.105502
$826$	−13.1202	−0.456509
$827$	−40.6397	−1.41318	−0.706591	−	0.707622i	$-0.749768\pi$
$827$	−40.6397	−1.41318	−0.706591	+	0.707622i	$0.749768\pi$
$828$	7.83222	0.272189
$829$	−23.9635	−0.832288	−0.416144	−	0.909299i	$-0.636619\pi$
$829$	−23.9635	−0.832288	−0.416144	+	0.909299i	$0.636619\pi$
$830$	−7.11298	−0.246895
$831$	15.8762	0.550740
$832$	−1.60253	−0.0555578
$833$	−76.1332	−2.63786
$834$	−0.160886	−0.00557104
$835$	20.0758	0.694753
$836$	−11.9692	−0.413964
$837$	−1.00000	−0.0345651
$838$	17.8872	0.617902
$839$	38.8815	1.34234	0.671169	−	0.741305i	$-0.265793\pi$
$839$	38.8815	1.34234	0.671169	+	0.741305i	$0.265793\pi$
$840$	8.49334	0.293048
$841$	−25.0008	−0.862097
$842$	6.88981	0.237439
$843$	−19.5615	−0.673735
$844$	−22.9877	−0.791271
$845$	1.00000	0.0344010
$846$	6.81053	0.234151
$847$	−7.57484	−0.260275
$848$	−11.6903	−0.401447
$849$	−16.6232	−0.570508
$850$	−4.04418	−0.138714
$851$	−19.4439	−0.666528
$852$	−21.0171	−0.720032
$853$	−14.2354	−0.487411	−0.243706	−	0.969849i	$-0.578363\pi$
$853$	−14.2354	−0.487411	−0.243706	+	0.969849i	$0.578363\pi$
$854$	14.4530	0.494572
$855$	−2.32871	−0.0796403
$856$	−5.83332	−0.199379
$857$	6.41637	0.219179	0.109590	−	0.993977i	$-0.465046\pi$
$857$	6.41637	0.219179	0.109590	+	0.993977i	$0.465046\pi$
$858$	1.67043	0.0570277
$859$	−43.1623	−1.47268	−0.736339	−	0.676613i	$-0.763447\pi$
$859$	−43.1623	−1.47268	−0.736339	+	0.676613i	$0.763447\pi$
$860$	−13.6318	−0.464842
$861$	−11.6311	−0.396387
$862$	−11.5339	−0.392845
$863$	6.09449	0.207459	0.103730	−	0.994606i	$-0.466922\pi$
$863$	6.09449	0.207459	0.103730	+	0.994606i	$0.466922\pi$
$864$	−5.32575	−0.181186
$865$	21.5668	0.733294
$866$	11.2180	0.381204
$867$	−36.8247	−1.25063
$868$	−7.07052	−0.239989
$869$	−13.0915	−0.444099
$870$	−1.10237	−0.0373738
$871$	8.65649	0.293314
$872$	−20.8699	−0.706745
$873$	−0.0930997	−0.00315095
$874$	5.92762	0.200505
$875$	4.16860	0.140925
$876$	0.184498	0.00623362
$877$	22.8111	0.770277	0.385138	−	0.922859i	$-0.374154\pi$
$877$	22.8111	0.770277	0.385138	+	0.922859i	$0.374154\pi$
$878$	−5.25153	−0.177230
$879$	−8.27321	−0.279049
$880$	−6.87626	−0.231799
$881$	42.5114	1.43225	0.716123	−	0.697974i	$-0.245915\pi$
$881$	42.5114	1.43225	0.716123	+	0.697974i	$0.245915\pi$
$882$	5.72035	0.192614
$883$	3.83027	0.128899	0.0644495	−	0.997921i	$-0.479471\pi$
$883$	3.83027	0.128899	0.0644495	+	0.997921i	$0.479471\pi$
$884$	12.4438	0.418529
$885$	5.70964	0.191927
$886$	−7.13635	−0.239750
$887$	−15.6435	−0.525258	−0.262629	−	0.964897i	$-0.584590\pi$
$887$	−15.6435	−0.525258	−0.262629	+	0.964897i	$0.584590\pi$
$888$	8.57919	0.287899
$889$	−9.13288	−0.306307
$890$	6.75648	0.226478
$891$	−3.03033	−0.101520
$892$	37.1896	1.24520
$893$	−28.7711	−0.962788
$894$	−8.85158	−0.296041
$895$	22.9385	0.766749
$896$	−48.0843	−1.60639
$897$	4.61769	0.154180
$898$	−18.9207	−0.631392
$899$	1.99980	0.0666970
$900$	−1.69614	−0.0565379
$901$	37.7968	1.25919
$902$	−4.66079	−0.155187
$903$	−33.5031	−1.11491
$904$	0.870194	0.0289422
$905$	−1.74061	−0.0578599
$906$	−12.5925	−0.418359
$907$	47.7877	1.58676	0.793382	−	0.608724i	$-0.208318\pi$
$907$	47.7877	1.58676	0.793382	+	0.608724i	$0.208318\pi$
$908$	−27.8240	−0.923373
$909$	−15.8232	−0.524824
$910$	2.29790	0.0761746
$911$	18.3571	0.608199	0.304100	−	0.952640i	$-0.401644\pi$
$911$	18.3571	0.608199	0.304100	+	0.952640i	$0.401644\pi$
$912$	5.28419	0.174977
$913$	39.1022	1.29409
$914$	15.5897	0.515662
$915$	−6.28967	−0.207930
$916$	−24.4784	−0.808788
$917$	34.3819	1.13539
$918$	4.04418	0.133478
$919$	54.3372	1.79242	0.896210	−	0.443630i	$-0.146309\pi$
$919$	54.3372	1.79242	0.896210	+	0.443630i	$0.146309\pi$
$920$	9.40832	0.310183
$921$	−6.25116	−0.205983
$922$	−12.5454	−0.413160
$923$	−12.3911	−0.407859
$924$	−21.4260	−0.704863
$925$	4.21074	0.138448
$926$	8.52275	0.280075
$927$	11.8702	0.389867
$928$	10.6504	0.349618
$929$	−5.77638	−0.189517	−0.0947584	−	0.995500i	$-0.530208\pi$
$929$	−5.77638	−0.189517	−0.0947584	+	0.995500i	$0.530208\pi$
$930$	−0.551239	−0.0180758
$931$	−24.1657	−0.791997
$932$	34.3890	1.12645
$933$	29.1576	0.954578
$934$	11.9160	0.389903
$935$	22.2321	0.727067
$936$	−2.03745	−0.0665963
$937$	−23.0175	−0.751948	−0.375974	−	0.926630i	$-0.622692\pi$
$937$	−23.0175	−0.751948	−0.375974	+	0.926630i	$0.622692\pi$
$938$	19.8917	0.649488
$939$	32.4430	1.05874
$940$	−20.9557	−0.683498
$941$	25.9548	0.846101	0.423051	−	0.906106i	$-0.360959\pi$
$941$	25.9548	0.846101	0.423051	+	0.906106i	$0.360959\pi$
$942$	9.69902	0.316011
$943$	−12.8841	−0.419565
$944$	−12.9560	−0.421682
$945$	−4.16860	−0.135605
$946$	−13.4253	−0.436494
$947$	3.78083	0.122860	0.0614302	−	0.998111i	$-0.480434\pi$
$947$	3.78083	0.122860	0.0614302	+	0.998111i	$0.480434\pi$
$948$	7.32759	0.237989
$949$	0.108776	0.00353101
$950$	−1.28368	−0.0416480
$951$	−4.23471	−0.137320
$952$	62.3117	2.01953
$953$	−1.11739	−0.0361959	−0.0180980	−	0.999836i	$-0.505761\pi$
$953$	−1.11739	−0.0361959	−0.0180980	+	0.999836i	$0.505761\pi$
$954$	−2.83991	−0.0919454
$955$	3.61464	0.116967
$956$	27.4038	0.886301
$957$	6.06005	0.195893
$958$	−3.12814	−0.101066
$959$	−51.1794	−1.65267
$960$	1.60253	0.0517215
$961$	1.00000	0.0322581
$962$	2.32112	0.0748361
$963$	2.86304	0.0922603
$964$	−15.6095	−0.502747
$965$	22.2720	0.716961
$966$	10.6110	0.341402
$967$	13.3349	0.428820	0.214410	−	0.976744i	$-0.431217\pi$
$967$	13.3349	0.428820	0.214410	+	0.976744i	$0.431217\pi$
$968$	3.70229	0.118996
$969$	−17.0847	−0.548839
$970$	−0.0513202	−0.00164779
$971$	−11.0319	−0.354029	−0.177014	−	0.984208i	$-0.556644\pi$
$971$	−11.0319	−0.354029	−0.177014	+	0.984208i	$0.556644\pi$
$972$	1.69614	0.0544036
$973$	1.21666	0.0390044
$974$	−3.93727	−0.126158
$975$	−1.00000	−0.0320256
$976$	14.2722	0.456842
$977$	−41.7950	−1.33714	−0.668569	−	0.743650i	$-0.733093\pi$
$977$	−41.7950	−1.33714	−0.668569	+	0.743650i	$0.733093\pi$
$978$	6.94342	0.222026
$979$	−37.1424	−1.18708
$980$	−17.6012	−0.562251
$981$	10.2431	0.327038
$982$	−0.183922	−0.00586919
$983$	33.7650	1.07694	0.538468	−	0.842646i	$-0.319003\pi$
$983$	33.7650	1.07694	0.538468	+	0.842646i	$0.319003\pi$
$984$	5.68484	0.181226
$985$	1.29932	0.0413997
$986$	−8.08756	−0.257560
$987$	−51.5029	−1.63935
$988$	3.94981	0.125660
$989$	−37.1124	−1.18010
$990$	−1.67043	−0.0530899
$991$	−16.1947	−0.514443	−0.257222	−	0.966352i	$-0.582807\pi$
$991$	−16.1947	−0.514443	−0.257222	+	0.966352i	$0.582807\pi$
$992$	5.32575	0.169093
$993$	33.3875	1.05952
$994$	−28.4736	−0.903127
$995$	−2.91024	−0.0922608
$996$	−21.8863	−0.693494
$997$	−12.1342	−0.384295	−0.192147	−	0.981366i	$-0.561545\pi$
$997$	−12.1342	−0.384295	−0.192147	+	0.981366i	$0.561545\pi$
$998$	5.10800	0.161691
$999$	−4.21074	−0.133222

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.bh.1.10	✓	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.bh.1.10	✓	17	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:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

\(f(q)\)	\(=\)	\(q+0.551239 q^{2} -1.00000 q^{3} -1.69614 q^{4} +1.00000 q^{5} -0.551239 q^{6} +4.16860 q^{7} -2.03745 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.551239 q^{2} -1.00000 q^{3} -1.69614 q^{4} +1.00000 q^{5} -0.551239 q^{6} +4.16860 q^{7} -2.03745 q^{8} +1.00000 q^{9} +0.551239 q^{10} -3.03033 q^{11} +1.69614 q^{12} +1.00000 q^{13} +2.29790 q^{14} -1.00000 q^{15} +2.26915 q^{16} -7.33654 q^{17} +0.551239 q^{18} -2.32871 q^{19} -1.69614 q^{20} -4.16860 q^{21} -1.67043 q^{22} -4.61769 q^{23} +2.03745 q^{24} +1.00000 q^{25} +0.551239 q^{26} -1.00000 q^{27} -7.07052 q^{28} +1.99980 q^{29} -0.551239 q^{30} +1.00000 q^{31} +5.32575 q^{32} +3.03033 q^{33} -4.04418 q^{34} +4.16860 q^{35} -1.69614 q^{36} +4.21074 q^{37} -1.28368 q^{38} -1.00000 q^{39} -2.03745 q^{40} +2.79017 q^{41} -2.29790 q^{42} +8.03700 q^{43} +5.13985 q^{44} +1.00000 q^{45} -2.54545 q^{46} +12.3549 q^{47} -2.26915 q^{48} +10.3773 q^{49} +0.551239 q^{50} +7.33654 q^{51} -1.69614 q^{52} -5.15186 q^{53} -0.551239 q^{54} -3.03033 q^{55} -8.49334 q^{56} +2.32871 q^{57} +1.10237 q^{58} -5.70964 q^{59} +1.69614 q^{60} +6.28967 q^{61} +0.551239 q^{62} +4.16860 q^{63} -1.60253 q^{64} +1.00000 q^{65} +1.67043 q^{66} +8.65649 q^{67} +12.4438 q^{68} +4.61769 q^{69} +2.29790 q^{70} -12.3911 q^{71} -2.03745 q^{72} +0.108776 q^{73} +2.32112 q^{74} -1.00000 q^{75} +3.94981 q^{76} -12.6322 q^{77} -0.551239 q^{78} +4.32017 q^{79} +2.26915 q^{80} +1.00000 q^{81} +1.53805 q^{82} -12.9036 q^{83} +7.07052 q^{84} -7.33654 q^{85} +4.43031 q^{86} -1.99980 q^{87} +6.17415 q^{88} +12.2569 q^{89} +0.551239 q^{90} +4.16860 q^{91} +7.83222 q^{92} -1.00000 q^{93} +6.81053 q^{94} -2.32871 q^{95} -5.32575 q^{96} -0.0930997 q^{97} +5.72035 q^{98} -3.03033 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 2 q^{2} - 17 q^{3} + 20 q^{4} + 17 q^{5} - 2 q^{6} + 18 q^{7} + 9 q^{8} + 17 q^{9}+O(q^{10}) \) 17 * q + 2 * q^2 - 17 * q^3 + 20 * q^4 + 17 * q^5 - 2 * q^6 + 18 * q^7 + 9 * q^8 + 17 * q^9	\( 17 q + 2 q^{2} - 17 q^{3} + 20 q^{4} + 17 q^{5} - 2 q^{6} + 18 q^{7} + 9 q^{8} + 17 q^{9} + 2 q^{10} + 3 q^{11} - 20 q^{12} + 17 q^{13} + q^{14} - 17 q^{15} + 26 q^{16} + 2 q^{18} + 10 q^{19} + 20 q^{20} - 18 q^{21} + 5 q^{22} + 16 q^{23} - 9 q^{24} + 17 q^{25} + 2 q^{26} - 17 q^{27} + 36 q^{28} - 3 q^{29} - 2 q^{30} + 17 q^{31} + 20 q^{32} - 3 q^{33} + q^{34} + 18 q^{35} + 20 q^{36} + 14 q^{37} + 22 q^{38} - 17 q^{39} + 9 q^{40} - 6 q^{41} - q^{42} + 24 q^{43} - 15 q^{44} + 17 q^{45} + 6 q^{46} + 25 q^{47} - 26 q^{48} + 31 q^{49} + 2 q^{50} + 20 q^{52} - 15 q^{53} - 2 q^{54} + 3 q^{55} + 31 q^{56} - 10 q^{57} + 44 q^{58} + 16 q^{59} - 20 q^{60} - 5 q^{61} + 2 q^{62} + 18 q^{63} + 35 q^{64} + 17 q^{65} - 5 q^{66} + 50 q^{67} + 13 q^{68} - 16 q^{69} + q^{70} + 16 q^{71} + 9 q^{72} + 33 q^{73} + 2 q^{74} - 17 q^{75} + 9 q^{77} - 2 q^{78} - 10 q^{79} + 26 q^{80} + 17 q^{81} + 61 q^{82} + 27 q^{83} - 36 q^{84} - 12 q^{86} + 3 q^{87} + 23 q^{88} - 24 q^{89} + 2 q^{90} + 18 q^{91} - 21 q^{92} - 17 q^{93} + 6 q^{94} + 10 q^{95} - 20 q^{96} + 48 q^{97} + 14 q^{98} + 3 q^{99}+O(q^{100}) \) 17 * q + 2 * q^2 - 17 * q^3 + 20 * q^4 + 17 * q^5 - 2 * q^6 + 18 * q^7 + 9 * q^8 + 17 * q^9 + 2 * q^10 + 3 * q^11 - 20 * q^12 + 17 * q^13 + q^14 - 17 * q^15 + 26 * q^16 + 2 * q^18 + 10 * q^19 + 20 * q^20 - 18 * q^21 + 5 * q^22 + 16 * q^23 - 9 * q^24 + 17 * q^25 + 2 * q^26 - 17 * q^27 + 36 * q^28 - 3 * q^29 - 2 * q^30 + 17 * q^31 + 20 * q^32 - 3 * q^33 + q^34 + 18 * q^35 + 20 * q^36 + 14 * q^37 + 22 * q^38 - 17 * q^39 + 9 * q^40 - 6 * q^41 - q^42 + 24 * q^43 - 15 * q^44 + 17 * q^45 + 6 * q^46 + 25 * q^47 - 26 * q^48 + 31 * q^49 + 2 * q^50 + 20 * q^52 - 15 * q^53 - 2 * q^54 + 3 * q^55 + 31 * q^56 - 10 * q^57 + 44 * q^58 + 16 * q^59 - 20 * q^60 - 5 * q^61 + 2 * q^62 + 18 * q^63 + 35 * q^64 + 17 * q^65 - 5 * q^66 + 50 * q^67 + 13 * q^68 - 16 * q^69 + q^70 + 16 * q^71 + 9 * q^72 + 33 * q^73 + 2 * q^74 - 17 * q^75 + 9 * q^77 - 2 * q^78 - 10 * q^79 + 26 * q^80 + 17 * q^81 + 61 * q^82 + 27 * q^83 - 36 * q^84 - 12 * q^86 + 3 * q^87 + 23 * q^88 - 24 * q^89 + 2 * q^90 + 18 * q^91 - 21 * q^92 - 17 * q^93 + 6 * q^94 + 10 * q^95 - 20 * q^96 + 48 * q^97 + 14 * q^98 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more