LMFDB - Embedded newform 6038.2.a.b.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6038,2,Mod(1,6038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6038, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6038 = 2 \cdot 3019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6038.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:	$48.2136727404$
Analytic rank:	$1$
Dimension:	$54$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	6038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.77973 q^{3} +1.00000 q^{4} +3.72096 q^{5} -2.77973 q^{6} -2.52302 q^{7} +1.00000 q^{8} +4.72690 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.77973 q^{3} +1.00000 q^{4} +3.72096 q^{5} -2.77973 q^{6} -2.52302 q^{7} +1.00000 q^{8} +4.72690 q^{9} +3.72096 q^{10} -4.59453 q^{11} -2.77973 q^{12} -1.95619 q^{13} -2.52302 q^{14} -10.3433 q^{15} +1.00000 q^{16} -5.40848 q^{17} +4.72690 q^{18} +3.48856 q^{19} +3.72096 q^{20} +7.01331 q^{21} -4.59453 q^{22} +5.79011 q^{23} -2.77973 q^{24} +8.84558 q^{25} -1.95619 q^{26} -4.80033 q^{27} -2.52302 q^{28} +3.69889 q^{29} -10.3433 q^{30} +1.33615 q^{31} +1.00000 q^{32} +12.7716 q^{33} -5.40848 q^{34} -9.38806 q^{35} +4.72690 q^{36} +7.29094 q^{37} +3.48856 q^{38} +5.43769 q^{39} +3.72096 q^{40} -2.92263 q^{41} +7.01331 q^{42} -4.04220 q^{43} -4.59453 q^{44} +17.5886 q^{45} +5.79011 q^{46} -8.19014 q^{47} -2.77973 q^{48} -0.634376 q^{49} +8.84558 q^{50} +15.0341 q^{51} -1.95619 q^{52} +0.430740 q^{53} -4.80033 q^{54} -17.0961 q^{55} -2.52302 q^{56} -9.69725 q^{57} +3.69889 q^{58} -0.545913 q^{59} -10.3433 q^{60} +1.27859 q^{61} +1.33615 q^{62} -11.9261 q^{63} +1.00000 q^{64} -7.27893 q^{65} +12.7716 q^{66} -10.9527 q^{67} -5.40848 q^{68} -16.0949 q^{69} -9.38806 q^{70} +7.93737 q^{71} +4.72690 q^{72} +11.7016 q^{73} +7.29094 q^{74} -24.5883 q^{75} +3.48856 q^{76} +11.5921 q^{77} +5.43769 q^{78} -8.70193 q^{79} +3.72096 q^{80} -0.837092 q^{81} -2.92263 q^{82} -16.4197 q^{83} +7.01331 q^{84} -20.1248 q^{85} -4.04220 q^{86} -10.2819 q^{87} -4.59453 q^{88} -11.1970 q^{89} +17.5886 q^{90} +4.93551 q^{91} +5.79011 q^{92} -3.71414 q^{93} -8.19014 q^{94} +12.9808 q^{95} -2.77973 q^{96} -9.99859 q^{97} -0.634376 q^{98} -21.7179 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 54 q + 54 q^{2} - 21 q^{3} + 54 q^{4} - 14 q^{5} - 21 q^{6} - 44 q^{7} + 54 q^{8} + 39 q^{9}+O(q^{10}) $ 54 * q + 54 * q^2 - 21 * q^3 + 54 * q^4 - 14 * q^5 - 21 * q^6 - 44 * q^7 + 54 * q^8 + 39 * q^9	\( 54 q + 54 q^{2} - 21 q^{3} + 54 q^{4} - 14 q^{5} - 21 q^{6} - 44 q^{7} + 54 q^{8} + 39 q^{9} - 14 q^{10} - 31 q^{11} - 21 q^{12} - 34 q^{13} - 44 q^{14} - 22 q^{15} + 54 q^{16} - 40 q^{17} + 39 q^{18} - 44 q^{19} - 14 q^{20} - 3 q^{21} - 31 q^{22} - 33 q^{23} - 21 q^{24} + 14 q^{25} - 34 q^{26} - 66 q^{27} - 44 q^{28} - 22 q^{30} - 65 q^{31} + 54 q^{32} - 43 q^{33} - 40 q^{34} - 46 q^{35} + 39 q^{36} - 58 q^{37} - 44 q^{38} - 36 q^{39} - 14 q^{40} - 49 q^{41} - 3 q^{42} - 47 q^{43} - 31 q^{44} - 45 q^{45} - 33 q^{46} - 66 q^{47} - 21 q^{48} + 16 q^{49} + 14 q^{50} - 33 q^{51} - 34 q^{52} - 16 q^{53} - 66 q^{54} - 50 q^{55} - 44 q^{56} - 33 q^{57} - 70 q^{59} - 22 q^{60} - 40 q^{61} - 65 q^{62} - 117 q^{63} + 54 q^{64} - 33 q^{65} - 43 q^{66} - 82 q^{67} - 40 q^{68} - q^{69} - 46 q^{70} - 60 q^{71} + 39 q^{72} - 92 q^{73} - 58 q^{74} - 68 q^{75} - 44 q^{76} + 13 q^{77} - 36 q^{78} - 57 q^{79} - 14 q^{80} + 26 q^{81} - 49 q^{82} - 77 q^{83} - 3 q^{84} - 24 q^{85} - 47 q^{86} - 61 q^{87} - 31 q^{88} - 54 q^{89} - 45 q^{90} - 46 q^{91} - 33 q^{92} - 24 q^{93} - 66 q^{94} - 66 q^{95} - 21 q^{96} - 137 q^{97} + 16 q^{98} - 71 q^{99}+O(q^{100}) \) 54 * q + 54 * q^2 - 21 * q^3 + 54 * q^4 - 14 * q^5 - 21 * q^6 - 44 * q^7 + 54 * q^8 + 39 * q^9 - 14 * q^10 - 31 * q^11 - 21 * q^12 - 34 * q^13 - 44 * q^14 - 22 * q^15 + 54 * q^16 - 40 * q^17 + 39 * q^18 - 44 * q^19 - 14 * q^20 - 3 * q^21 - 31 * q^22 - 33 * q^23 - 21 * q^24 + 14 * q^25 - 34 * q^26 - 66 * q^27 - 44 * q^28 - 22 * q^30 - 65 * q^31 + 54 * q^32 - 43 * q^33 - 40 * q^34 - 46 * q^35 + 39 * q^36 - 58 * q^37 - 44 * q^38 - 36 * q^39 - 14 * q^40 - 49 * q^41 - 3 * q^42 - 47 * q^43 - 31 * q^44 - 45 * q^45 - 33 * q^46 - 66 * q^47 - 21 * q^48 + 16 * q^49 + 14 * q^50 - 33 * q^51 - 34 * q^52 - 16 * q^53 - 66 * q^54 - 50 * q^55 - 44 * q^56 - 33 * q^57 - 70 * q^59 - 22 * q^60 - 40 * q^61 - 65 * q^62 - 117 * q^63 + 54 * q^64 - 33 * q^65 - 43 * q^66 - 82 * q^67 - 40 * q^68 - q^69 - 46 * q^70 - 60 * q^71 + 39 * q^72 - 92 * q^73 - 58 * q^74 - 68 * q^75 - 44 * q^76 + 13 * q^77 - 36 * q^78 - 57 * q^79 - 14 * q^80 + 26 * q^81 - 49 * q^82 - 77 * q^83 - 3 * q^84 - 24 * q^85 - 47 * q^86 - 61 * q^87 - 31 * q^88 - 54 * q^89 - 45 * q^90 - 46 * q^91 - 33 * q^92 - 24 * q^93 - 66 * q^94 - 66 * q^95 - 21 * q^96 - 137 * q^97 + 16 * q^98 - 71 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.77973	−1.60488	−0.802439	−	0.596734i	$-0.796465\pi$
$3$	−2.77973	−1.60488	−0.802439	+	0.596734i	$0.796465\pi$
$4$	1.00000	0.500000
$5$	3.72096	1.66407	0.832033	−	0.554726i	$-0.187177\pi$
$5$	3.72096	1.66407	0.832033	+	0.554726i	$0.187177\pi$
$6$	−2.77973	−1.13482
$7$	−2.52302	−0.953611	−0.476806	−	0.879009i	$-0.658205\pi$
$7$	−2.52302	−0.953611	−0.476806	+	0.879009i	$0.658205\pi$
$8$	1.00000	0.353553
$9$	4.72690	1.57563
$10$	3.72096	1.17667
$11$	−4.59453	−1.38530	−0.692652	−	0.721272i	$-0.743558\pi$
$11$	−4.59453	−1.38530	−0.692652	+	0.721272i	$0.743558\pi$
$12$	−2.77973	−0.802439
$13$	−1.95619	−0.542551	−0.271275	−	0.962502i	$-0.587445\pi$
$13$	−1.95619	−0.542551	−0.271275	+	0.962502i	$0.587445\pi$
$14$	−2.52302	−0.674305
$15$	−10.3433	−2.67062
$16$	1.00000	0.250000
$17$	−5.40848	−1.31175	−0.655874	−	0.754870i	$-0.727700\pi$
$17$	−5.40848	−1.31175	−0.655874	+	0.754870i	$0.727700\pi$
$18$	4.72690	1.11414
$19$	3.48856	0.800330	0.400165	−	0.916443i	$-0.368953\pi$
$19$	3.48856	0.800330	0.400165	+	0.916443i	$0.368953\pi$
$20$	3.72096	0.832033
$21$	7.01331	1.53043
$22$	−4.59453	−0.979558
$23$	5.79011	1.20732	0.603660	−	0.797242i	$-0.293708\pi$
$23$	5.79011	1.20732	0.603660	+	0.797242i	$0.293708\pi$
$24$	−2.77973	−0.567410
$25$	8.84558	1.76912
$26$	−1.95619	−0.383641
$27$	−4.80033	−0.923823
$28$	−2.52302	−0.476806
$29$	3.69889	0.686866	0.343433	−	0.939177i	$-0.388410\pi$
$29$	3.69889	0.686866	0.343433	+	0.939177i	$0.388410\pi$
$30$	−10.3433	−1.88842
$31$	1.33615	0.239980	0.119990	−	0.992775i	$-0.461714\pi$
$31$	1.33615	0.239980	0.119990	+	0.992775i	$0.461714\pi$
$32$	1.00000	0.176777
$33$	12.7716	2.22325
$34$	−5.40848	−0.927546
$35$	−9.38806	−1.58687
$36$	4.72690	0.787817
$37$	7.29094	1.19862	0.599312	−	0.800516i	$-0.295441\pi$
$37$	7.29094	1.19862	0.599312	+	0.800516i	$0.295441\pi$
$38$	3.48856	0.565919
$39$	5.43769	0.870728
$40$	3.72096	0.588336
$41$	−2.92263	−0.456439	−0.228219	−	0.973610i	$-0.573290\pi$
$41$	−2.92263	−0.456439	−0.228219	+	0.973610i	$0.573290\pi$
$42$	7.01331	1.08218
$43$	−4.04220	−0.616430	−0.308215	−	0.951317i	$-0.599732\pi$
$43$	−4.04220	−0.616430	−0.308215	+	0.951317i	$0.599732\pi$
$44$	−4.59453	−0.692652
$45$	17.5886	2.62196
$46$	5.79011	0.853704
$47$	−8.19014	−1.19465	−0.597327	−	0.801998i	$-0.703771\pi$
$47$	−8.19014	−1.19465	−0.597327	+	0.801998i	$0.703771\pi$
$48$	−2.77973	−0.401220
$49$	−0.634376	−0.0906252
$50$	8.84558	1.25095
$51$	15.0341	2.10520
$52$	−1.95619	−0.271275
$53$	0.430740	0.0591667	0.0295834	−	0.999562i	$-0.490582\pi$
$53$	0.430740	0.0591667	0.0295834	+	0.999562i	$0.490582\pi$
$54$	−4.80033	−0.653242
$55$	−17.0961	−2.30524
$56$	−2.52302	−0.337153
$57$	−9.69725	−1.28443
$58$	3.69889	0.485688
$59$	−0.545913	−0.0710718	−0.0355359	−	0.999368i	$-0.511314\pi$
$59$	−0.545913	−0.0710718	−0.0355359	+	0.999368i	$0.511314\pi$
$60$	−10.3433	−1.33531
$61$	1.27859	0.163707	0.0818534	−	0.996644i	$-0.473916\pi$
$61$	1.27859	0.163707	0.0818534	+	0.996644i	$0.473916\pi$
$62$	1.33615	0.169691
$63$	−11.9261	−1.50254
$64$	1.00000	0.125000
$65$	−7.27893	−0.902840
$66$	12.7716	1.57207
$67$	−10.9527	−1.33809	−0.669045	−	0.743222i	$-0.733297\pi$
$67$	−10.9527	−1.33809	−0.669045	+	0.743222i	$0.733297\pi$
$68$	−5.40848	−0.655874
$69$	−16.0949	−1.93760
$70$	−9.38806	−1.12209
$71$	7.93737	0.941992	0.470996	−	0.882135i	$-0.343895\pi$
$71$	7.93737	0.941992	0.470996	+	0.882135i	$0.343895\pi$
$72$	4.72690	0.557071
$73$	11.7016	1.36957	0.684786	−	0.728744i	$-0.259896\pi$
$73$	11.7016	1.36957	0.684786	+	0.728744i	$0.259896\pi$
$74$	7.29094	0.847555
$75$	−24.5883	−2.83922
$76$	3.48856	0.400165
$77$	11.5921	1.32104
$78$	5.43769	0.615698
$79$	−8.70193	−0.979043	−0.489522	−	0.871991i	$-0.662829\pi$
$79$	−8.70193	−0.979043	−0.489522	+	0.871991i	$0.662829\pi$
$80$	3.72096	0.416016
$81$	−0.837092	−0.0930103
$82$	−2.92263	−0.322751
$83$	−16.4197	−1.80229	−0.901147	−	0.433515i	$-0.857273\pi$
$83$	−16.4197	−1.80229	−0.901147	+	0.433515i	$0.857273\pi$
$84$	7.01331	0.765215
$85$	−20.1248	−2.18284
$86$	−4.04220	−0.435882
$87$	−10.2819	−1.10234
$88$	−4.59453	−0.489779
$89$	−11.1970	−1.18688	−0.593439	−	0.804879i	$-0.702230\pi$
$89$	−11.1970	−1.18688	−0.593439	+	0.804879i	$0.702230\pi$
$90$	17.5886	1.85401
$91$	4.93551	0.517383
$92$	5.79011	0.603660
$93$	−3.71414	−0.385138
$94$	−8.19014	−0.844748
$95$	12.9808	1.33180
$96$	−2.77973	−0.283705
$97$	−9.99859	−1.01520	−0.507602	−	0.861592i	$-0.669468\pi$
$97$	−9.99859	−1.01520	−0.507602	+	0.861592i	$0.669468\pi$
$98$	−0.634376	−0.0640817
$99$	−21.7179	−2.18273
$100$	8.84558	0.884558
$101$	1.58807	0.158019	0.0790095	−	0.996874i	$-0.474824\pi$
$101$	1.58807	0.158019	0.0790095	+	0.996874i	$0.474824\pi$
$102$	15.0341	1.48860
$103$	−17.8688	−1.76066	−0.880332	−	0.474358i	$-0.842680\pi$
$103$	−17.8688	−1.76066	−0.880332	+	0.474358i	$0.842680\pi$
$104$	−1.95619	−0.191821
$105$	26.0963	2.54674
$106$	0.430740	0.0418372
$107$	−9.80647	−0.948027	−0.474014	−	0.880518i	$-0.657195\pi$
$107$	−9.80647	−0.948027	−0.474014	+	0.880518i	$0.657195\pi$
$108$	−4.80033	−0.461912
$109$	−2.38119	−0.228077	−0.114039	−	0.993476i	$-0.536379\pi$
$109$	−2.38119	−0.228077	−0.114039	+	0.993476i	$0.536379\pi$
$110$	−17.0961	−1.63005
$111$	−20.2669	−1.92364
$112$	−2.52302	−0.238403
$113$	15.8072	1.48701	0.743507	−	0.668728i	$-0.233161\pi$
$113$	15.8072	1.48701	0.743507	+	0.668728i	$0.233161\pi$
$114$	−9.69725	−0.908231
$115$	21.5448	2.00906
$116$	3.69889	0.343433
$117$	−9.24674	−0.854862
$118$	−0.545913	−0.0502554
$119$	13.6457	1.25090
$120$	−10.3433	−0.944208
$121$	10.1098	0.919068
$122$	1.27859	0.115758
$123$	8.12414	0.732529
$124$	1.33615	0.119990
$125$	14.3093	1.27986
$126$	−11.9261	−1.06246
$127$	−5.42430	−0.481329	−0.240665	−	0.970608i	$-0.577365\pi$
$127$	−5.42430	−0.481329	−0.240665	+	0.970608i	$0.577365\pi$
$128$	1.00000	0.0883883
$129$	11.2362	0.989295
$130$	−7.27893	−0.638404
$131$	0.816465	0.0713349	0.0356675	−	0.999364i	$-0.488644\pi$
$131$	0.816465	0.0713349	0.0356675	+	0.999364i	$0.488644\pi$
$132$	12.7716	1.11162
$133$	−8.80170	−0.763204
$134$	−10.9527	−0.946172
$135$	−17.8618	−1.53730
$136$	−5.40848	−0.463773
$137$	6.06321	0.518015	0.259008	−	0.965875i	$-0.416605\pi$
$137$	6.06321	0.518015	0.259008	+	0.965875i	$0.416605\pi$
$138$	−16.0949	−1.37009
$139$	−7.29227	−0.618522	−0.309261	−	0.950977i	$-0.600082\pi$
$139$	−7.29227	−0.618522	−0.309261	+	0.950977i	$0.600082\pi$
$140$	−9.38806	−0.793436
$141$	22.7664	1.91727
$142$	7.93737	0.666089
$143$	8.98780	0.751598
$144$	4.72690	0.393909
$145$	13.7634	1.14299
$146$	11.7016	0.968433
$147$	1.76340	0.145442
$148$	7.29094	0.599312
$149$	7.82319	0.640901	0.320450	−	0.947265i	$-0.396166\pi$
$149$	7.82319	0.640901	0.320450	+	0.947265i	$0.396166\pi$
$150$	−24.5883	−2.00763
$151$	23.9677	1.95046	0.975230	−	0.221192i	$-0.0709949\pi$
$151$	23.9677	1.95046	0.975230	+	0.221192i	$0.0709949\pi$
$152$	3.48856	0.282959
$153$	−25.5654	−2.06684
$154$	11.5921	0.934118
$155$	4.97177	0.399342
$156$	5.43769	0.435364
$157$	−21.6663	−1.72916	−0.864579	−	0.502497i	$-0.832415\pi$
$157$	−21.6663	−1.72916	−0.864579	+	0.502497i	$0.832415\pi$
$158$	−8.70193	−0.692288
$159$	−1.19734	−0.0949554
$160$	3.72096	0.294168
$161$	−14.6085	−1.15131
$162$	−0.837092	−0.0657682
$163$	−15.1387	−1.18576	−0.592879	−	0.805292i	$-0.702009\pi$
$163$	−15.1387	−1.18576	−0.592879	+	0.805292i	$0.702009\pi$
$164$	−2.92263	−0.228219
$165$	47.5226	3.69963
$166$	−16.4197	−1.27441
$167$	−0.00600935	−0.000465018 0	−0.000232509	−	1.00000i	$-0.500074\pi$
$167$	−0.00600935	−0.000465018 0	−0.000232509		1.00000i	$0.500074\pi$
$168$	7.01331	0.541089
$169$	−9.17330	−0.705639
$170$	−20.1248	−1.54350
$171$	16.4901	1.26103
$172$	−4.04220	−0.308215
$173$	−8.76056	−0.666053	−0.333027	−	0.942917i	$-0.608070\pi$
$173$	−8.76056	−0.666053	−0.333027	+	0.942917i	$0.608070\pi$
$174$	−10.2819	−0.779470
$175$	−22.3176	−1.68705
$176$	−4.59453	−0.346326
$177$	1.51749	0.114062
$178$	−11.1970	−0.839249
$179$	−1.61508	−0.120717	−0.0603584	−	0.998177i	$-0.519224\pi$
$179$	−1.61508	−0.120717	−0.0603584	+	0.998177i	$0.519224\pi$
$180$	17.5886	1.31098
$181$	−10.3637	−0.770330	−0.385165	−	0.922848i	$-0.625855\pi$
$181$	−10.3637	−0.770330	−0.385165	+	0.922848i	$0.625855\pi$
$182$	4.93551	0.365845
$183$	−3.55414	−0.262729
$184$	5.79011	0.426852
$185$	27.1293	1.99459
$186$	−3.71414	−0.272334
$187$	24.8494	1.81717
$188$	−8.19014	−0.597327
$189$	12.1113	0.880969
$190$	12.9808	0.941726
$191$	−9.93616	−0.718956	−0.359478	−	0.933154i	$-0.617045\pi$
$191$	−9.93616	−0.718956	−0.359478	+	0.933154i	$0.617045\pi$
$192$	−2.77973	−0.200610
$193$	−1.81515	−0.130658	−0.0653288	−	0.997864i	$-0.520810\pi$
$193$	−1.81515	−0.130658	−0.0653288	+	0.997864i	$0.520810\pi$
$194$	−9.99859	−0.717857
$195$	20.2335	1.44895
$196$	−0.634376	−0.0453126
$197$	9.05102	0.644858	0.322429	−	0.946594i	$-0.395501\pi$
$197$	9.05102	0.644858	0.322429	+	0.946594i	$0.395501\pi$
$198$	−21.7179	−1.54343
$199$	8.67157	0.614711	0.307356	−	0.951595i	$-0.400556\pi$
$199$	8.67157	0.614711	0.307356	+	0.951595i	$0.400556\pi$
$200$	8.84558	0.625477
$201$	30.4457	2.14747
$202$	1.58807	0.111736
$203$	−9.33236	−0.655003
$204$	15.0341	1.05260
$205$	−10.8750	−0.759544
$206$	−17.8688	−1.24498
$207$	27.3693	1.90230
$208$	−1.95619	−0.135638
$209$	−16.0283	−1.10870
$210$	26.0963	1.80082
$211$	−25.3084	−1.74230	−0.871151	−	0.491015i	$-0.836626\pi$
$211$	−25.3084	−1.74230	−0.871151	+	0.491015i	$0.836626\pi$
$212$	0.430740	0.0295834
$213$	−22.0637	−1.51178
$214$	−9.80647	−0.670356
$215$	−15.0409	−1.02578
$216$	−4.80033	−0.326621
$217$	−3.37113	−0.228847
$218$	−2.38119	−0.161275
$219$	−32.5274	−2.19800
$220$	−17.0961	−1.15262
$221$	10.5800	0.711690
$222$	−20.2669	−1.36022
$223$	−15.6434	−1.04756	−0.523779	−	0.851854i	$-0.675478\pi$
$223$	−15.6434	−1.04756	−0.523779	+	0.851854i	$0.675478\pi$
$224$	−2.52302	−0.168576
$225$	41.8122	2.78748
$226$	15.8072	1.05148
$227$	−4.57766	−0.303830	−0.151915	−	0.988394i	$-0.548544\pi$
$227$	−4.57766	−0.303830	−0.151915	+	0.988394i	$0.548544\pi$
$228$	−9.69725	−0.642216
$229$	11.0915	0.732950	0.366475	−	0.930428i	$-0.380565\pi$
$229$	11.0915	0.732950	0.366475	+	0.930428i	$0.380565\pi$
$230$	21.5448	1.42062
$231$	−32.2229	−2.12011
$232$	3.69889	0.242844
$233$	−22.8703	−1.49828	−0.749141	−	0.662411i	$-0.769533\pi$
$233$	−22.8703	−1.49828	−0.749141	+	0.662411i	$0.769533\pi$
$234$	−9.24674	−0.604478
$235$	−30.4752	−1.98798
$236$	−0.545913	−0.0355359
$237$	24.1890	1.57125
$238$	13.6457	0.884519
$239$	−17.8804	−1.15659	−0.578294	−	0.815828i	$-0.696281\pi$
$239$	−17.8804	−1.15659	−0.578294	+	0.815828i	$0.696281\pi$
$240$	−10.3433	−0.667656
$241$	17.3245	1.11597	0.557984	−	0.829851i	$-0.311575\pi$
$241$	17.3245	1.11597	0.557984	+	0.829851i	$0.311575\pi$
$242$	10.1098	0.649879
$243$	16.7279	1.07309
$244$	1.27859	0.0818534
$245$	−2.36049	−0.150806
$246$	8.12414	0.517976
$247$	−6.82430	−0.434220
$248$	1.33615	0.0848456
$249$	45.6423	2.89246
$250$	14.3093	0.904997
$251$	21.7570	1.37329	0.686646	−	0.726992i	$-0.259082\pi$
$251$	21.7570	1.37329	0.686646	+	0.726992i	$0.259082\pi$
$252$	−11.9261	−0.751272
$253$	−26.6028	−1.67251
$254$	−5.42430	−0.340351
$255$	55.9414	3.50319
$256$	1.00000	0.0625000
$257$	−12.7479	−0.795193	−0.397596	−	0.917560i	$-0.630156\pi$
$257$	−12.7479	−0.795193	−0.397596	+	0.917560i	$0.630156\pi$
$258$	11.2362	0.699537
$259$	−18.3952	−1.14302
$260$	−7.27893	−0.451420
$261$	17.4843	1.08225
$262$	0.816465	0.0504414
$263$	20.5321	1.26606	0.633031	−	0.774127i	$-0.281811\pi$
$263$	20.5321	1.26606	0.633031	+	0.774127i	$0.281811\pi$
$264$	12.7716	0.786036
$265$	1.60277	0.0984573
$266$	−8.80170	−0.539667
$267$	31.1246	1.90479
$268$	−10.9527	−0.669045
$269$	−23.0224	−1.40370	−0.701849	−	0.712326i	$-0.747642\pi$
$269$	−23.0224	−1.40370	−0.701849	+	0.712326i	$0.747642\pi$
$270$	−17.8618	−1.08704
$271$	−2.35089	−0.142807	−0.0714033	−	0.997448i	$-0.522748\pi$
$271$	−2.35089	−0.142807	−0.0714033	+	0.997448i	$0.522748\pi$
$272$	−5.40848	−0.327937
$273$	−13.7194	−0.830336
$274$	6.06321	0.366292
$275$	−40.6413	−2.45076
$276$	−16.0949	−0.968801
$277$	−0.975853	−0.0586333	−0.0293166	−	0.999570i	$-0.509333\pi$
$277$	−0.975853	−0.0586333	−0.0293166	+	0.999570i	$0.509333\pi$
$278$	−7.29227	−0.437361
$279$	6.31585	0.378120
$280$	−9.38806	−0.561044
$281$	−10.3703	−0.618643	−0.309321	−	0.950958i	$-0.600102\pi$
$281$	−10.3703	−0.618643	−0.309321	+	0.950958i	$0.600102\pi$
$282$	22.7664	1.35572
$283$	21.8684	1.29994	0.649969	−	0.759960i	$-0.274782\pi$
$283$	21.8684	1.29994	0.649969	+	0.759960i	$0.274782\pi$
$284$	7.93737	0.470996
$285$	−36.0831	−2.13738
$286$	8.98780	0.531460
$287$	7.37386	0.435265
$288$	4.72690	0.278535
$289$	12.2516	0.720685
$290$	13.7634	0.808216
$291$	27.7934	1.62928
$292$	11.7016	0.684786
$293$	28.5904	1.67027	0.835134	−	0.550047i	$-0.185390\pi$
$293$	28.5904	1.67027	0.835134	+	0.550047i	$0.185390\pi$
$294$	1.76340	0.102843
$295$	−2.03132	−0.118268
$296$	7.29094	0.423777
$297$	22.0553	1.27978
$298$	7.82319	0.453185
$299$	−11.3266	−0.655033
$300$	−24.5883	−1.41961
$301$	10.1985	0.587834
$302$	23.9677	1.37918
$303$	−4.41441	−0.253601
$304$	3.48856	0.200083
$305$	4.75759	0.272419
$306$	−25.5654	−1.46147
$307$	26.9812	1.53990	0.769949	−	0.638106i	$-0.220282\pi$
$307$	26.9812	1.53990	0.769949	+	0.638106i	$0.220282\pi$
$308$	11.5921	0.660521
$309$	49.6704	2.82565
$310$	4.97177	0.282377
$311$	−19.4471	−1.10274	−0.551371	−	0.834260i	$-0.685895\pi$
$311$	−19.4471	−1.10274	−0.551371	+	0.834260i	$0.685895\pi$
$312$	5.43769	0.307849
$313$	2.50205	0.141424	0.0707120	−	0.997497i	$-0.477473\pi$
$313$	2.50205	0.141424	0.0707120	+	0.997497i	$0.477473\pi$
$314$	−21.6663	−1.22270
$315$	−44.3765	−2.50033
$316$	−8.70193	−0.489522
$317$	−13.2624	−0.744888	−0.372444	−	0.928055i	$-0.621480\pi$
$317$	−13.2624	−0.744888	−0.372444	+	0.928055i	$0.621480\pi$
$318$	−1.19734	−0.0671436
$319$	−16.9947	−0.951519
$320$	3.72096	0.208008
$321$	27.2593	1.52147
$322$	−14.6085	−0.814102
$323$	−18.8678	−1.04983
$324$	−0.837092	−0.0465051
$325$	−17.3037	−0.959835
$326$	−15.1387	−0.838457
$327$	6.61908	0.366036
$328$	−2.92263	−0.161375
$329$	20.6639	1.13924
$330$	47.5226	2.61603
$331$	18.6037	1.02255	0.511275	−	0.859417i	$-0.329173\pi$
$331$	18.6037	1.02255	0.511275	+	0.859417i	$0.329173\pi$
$332$	−16.4197	−0.901147
$333$	34.4636	1.88859
$334$	−0.00600935	−0.000328817 0
$335$	−40.7547	−2.22667
$336$	7.01331	0.382608
$337$	−26.5160	−1.44442	−0.722208	−	0.691676i	$-0.756873\pi$
$337$	−26.5160	−1.44442	−0.722208	+	0.691676i	$0.756873\pi$
$338$	−9.17330	−0.498962
$339$	−43.9397	−2.38648
$340$	−20.1248	−1.09142
$341$	−6.13899	−0.332445
$342$	16.4901	0.891681
$343$	19.2617	1.04003
$344$	−4.04220	−0.217941
$345$	−59.8887	−3.22430
$346$	−8.76056	−0.470971
$347$	−7.56571	−0.406148	−0.203074	−	0.979163i	$-0.565093\pi$
$347$	−7.56571	−0.406148	−0.203074	+	0.979163i	$0.565093\pi$
$348$	−10.2819	−0.551168
$349$	−17.1462	−0.917817	−0.458909	−	0.888484i	$-0.651759\pi$
$349$	−17.1462	−0.917817	−0.458909	+	0.888484i	$0.651759\pi$
$350$	−22.3176	−1.19292
$351$	9.39037	0.501221
$352$	−4.59453	−0.244890
$353$	−9.06116	−0.482277	−0.241138	−	0.970491i	$-0.577521\pi$
$353$	−9.06116	−0.482277	−0.241138	+	0.970491i	$0.577521\pi$
$354$	1.51749	0.0806537
$355$	29.5347	1.56754
$356$	−11.1970	−0.593439
$357$	−37.9314	−2.00754
$358$	−1.61508	−0.0853597
$359$	30.5993	1.61497	0.807485	−	0.589889i	$-0.200828\pi$
$359$	30.5993	1.61497	0.807485	+	0.589889i	$0.200828\pi$
$360$	17.5886	0.927003
$361$	−6.82996	−0.359472
$362$	−10.3637	−0.544705
$363$	−28.1024	−1.47499
$364$	4.93551	0.258691
$365$	43.5413	2.27906
$366$	−3.55414	−0.185778
$367$	−30.2276	−1.57787	−0.788933	−	0.614479i	$-0.789366\pi$
$367$	−30.2276	−1.57787	−0.788933	+	0.614479i	$0.789366\pi$
$368$	5.79011	0.301830
$369$	−13.8150	−0.719181
$370$	27.1293	1.41039
$371$	−1.08677	−0.0564221
$372$	−3.71414	−0.192569
$373$	−26.0530	−1.34898	−0.674488	−	0.738286i	$-0.735635\pi$
$373$	−26.0530	−1.34898	−0.674488	+	0.738286i	$0.735635\pi$
$374$	24.8494	1.28493
$375$	−39.7759	−2.05402
$376$	−8.19014	−0.422374
$377$	−7.23574	−0.372660
$378$	12.1113	0.622939
$379$	21.5786	1.10842	0.554209	−	0.832378i	$-0.313021\pi$
$379$	21.5786	1.10842	0.554209	+	0.832378i	$0.313021\pi$
$380$	12.9808	0.665901
$381$	15.0781	0.772475
$382$	−9.93616	−0.508378
$383$	−25.3460	−1.29512	−0.647560	−	0.762014i	$-0.724211\pi$
$383$	−25.3460	−1.29512	−0.647560	+	0.762014i	$0.724211\pi$
$384$	−2.77973	−0.141853
$385$	43.1338	2.19830
$386$	−1.81515	−0.0923889
$387$	−19.1071	−0.971268
$388$	−9.99859	−0.507602
$389$	−20.3548	−1.03203	−0.516015	−	0.856579i	$-0.672585\pi$
$389$	−20.3548	−1.03203	−0.516015	+	0.856579i	$0.672585\pi$
$390$	20.2335	1.02456
$391$	−31.3157	−1.58370
$392$	−0.634376	−0.0320408
$393$	−2.26955	−0.114484
$394$	9.05102	0.455984
$395$	−32.3796	−1.62919
$396$	−21.7179	−1.09137
$397$	6.36952	0.319677	0.159839	−	0.987143i	$-0.448903\pi$
$397$	6.36952	0.319677	0.159839	+	0.987143i	$0.448903\pi$
$398$	8.67157	0.434667
$399$	24.4663	1.22485
$400$	8.84558	0.442279
$401$	−27.7910	−1.38781	−0.693907	−	0.720065i	$-0.744112\pi$
$401$	−27.7910	−1.38781	−0.693907	+	0.720065i	$0.744112\pi$
$402$	30.4457	1.51849
$403$	−2.61377	−0.130201
$404$	1.58807	0.0790095
$405$	−3.11479	−0.154775
$406$	−9.33236	−0.463157
$407$	−33.4985	−1.66046
$408$	15.0341	0.744300
$409$	11.9357	0.590181	0.295091	−	0.955469i	$-0.404650\pi$
$409$	11.9357	0.590181	0.295091	+	0.955469i	$0.404650\pi$
$410$	−10.8750	−0.537079
$411$	−16.8541	−0.831351
$412$	−17.8688	−0.880332
$413$	1.37735	0.0677749
$414$	27.3693	1.34513
$415$	−61.0970	−2.99913
$416$	−1.95619	−0.0959103
$417$	20.2706	0.992653
$418$	−16.0283	−0.783970
$419$	−31.4267	−1.53530	−0.767648	−	0.640872i	$-0.778573\pi$
$419$	−31.4267	−1.53530	−0.767648	+	0.640872i	$0.778573\pi$
$420$	26.0963	1.27337
$421$	−27.1218	−1.32184	−0.660918	−	0.750458i	$-0.729833\pi$
$421$	−27.1218	−1.32184	−0.660918	+	0.750458i	$0.729833\pi$
$422$	−25.3084	−1.23199
$423$	−38.7140	−1.88234
$424$	0.430740	0.0209186
$425$	−47.8411	−2.32064
$426$	−22.0637	−1.06899
$427$	−3.22591	−0.156113
$428$	−9.80647	−0.474014
$429$	−24.9837	−1.20622
$430$	−15.0409	−0.725336
$431$	9.60015	0.462423	0.231211	−	0.972904i	$-0.425731\pi$
$431$	9.60015	0.462423	0.231211	+	0.972904i	$0.425731\pi$
$432$	−4.80033	−0.230956
$433$	−1.67026	−0.0802675	−0.0401337	−	0.999194i	$-0.512778\pi$
$433$	−1.67026	−0.0802675	−0.0401337	+	0.999194i	$0.512778\pi$
$434$	−3.37113	−0.161819
$435$	−38.2586	−1.83436
$436$	−2.38119	−0.114039
$437$	20.1991	0.966255
$438$	−32.5274	−1.55422
$439$	9.12837	0.435673	0.217837	−	0.975985i	$-0.430100\pi$
$439$	9.12837	0.435673	0.217837	+	0.975985i	$0.430100\pi$
$440$	−17.0961	−0.815025
$441$	−2.99864	−0.142792
$442$	10.5800	0.503241
$443$	−1.13267	−0.0538148	−0.0269074	−	0.999638i	$-0.508566\pi$
$443$	−1.13267	−0.0538148	−0.0269074	+	0.999638i	$0.508566\pi$
$444$	−20.2669	−0.961822
$445$	−41.6636	−1.97504
$446$	−15.6434	−0.740736
$447$	−21.7464	−1.02857
$448$	−2.52302	−0.119201
$449$	0.578765	0.0273136	0.0136568	−	0.999907i	$-0.495653\pi$
$449$	0.578765	0.0273136	0.0136568	+	0.999907i	$0.495653\pi$
$450$	41.8122	1.97105
$451$	13.4281	0.632307
$452$	15.8072	0.743507
$453$	−66.6236	−3.13025
$454$	−4.57766	−0.214840
$455$	18.3649	0.860959
$456$	−9.69725	−0.454115
$457$	32.3426	1.51292	0.756461	−	0.654039i	$-0.226927\pi$
$457$	32.3426	1.51292	0.756461	+	0.654039i	$0.226927\pi$
$458$	11.0915	0.518274
$459$	25.9625	1.21182
$460$	21.5448	1.00453
$461$	−3.97335	−0.185058	−0.0925288	−	0.995710i	$-0.529495\pi$
$461$	−3.97335	−0.185058	−0.0925288	+	0.995710i	$0.529495\pi$
$462$	−32.2229	−1.49915
$463$	3.69345	0.171649	0.0858247	−	0.996310i	$-0.472648\pi$
$463$	3.69345	0.171649	0.0858247	+	0.996310i	$0.472648\pi$
$464$	3.69889	0.171717
$465$	−13.8202	−0.640895
$466$	−22.8703	−1.05944
$467$	13.9692	0.646418	0.323209	−	0.946328i	$-0.395238\pi$
$467$	13.9692	0.646418	0.323209	+	0.946328i	$0.395238\pi$
$468$	−9.24674	−0.427431
$469$	27.6340	1.27602
$470$	−30.4752	−1.40572
$471$	60.2264	2.77509
$472$	−0.545913	−0.0251277
$473$	18.5720	0.853943
$474$	24.1890	1.11104
$475$	30.8583	1.41588
$476$	13.6457	0.625449
$477$	2.03607	0.0932251
$478$	−17.8804	−0.817831
$479$	2.54601	0.116330	0.0581650	−	0.998307i	$-0.481475\pi$
$479$	2.54601	0.116330	0.0581650	+	0.998307i	$0.481475\pi$
$480$	−10.3433	−0.472104
$481$	−14.2625	−0.650314
$482$	17.3245	0.789109
$483$	40.6078	1.84772
$484$	10.1098	0.459534
$485$	−37.2044	−1.68936
$486$	16.7279	0.758792
$487$	15.9351	0.722090	0.361045	−	0.932548i	$-0.382420\pi$
$487$	15.9351	0.722090	0.361045	+	0.932548i	$0.382420\pi$
$488$	1.27859	0.0578791
$489$	42.0816	1.90300
$490$	−2.36049	−0.106636
$491$	25.7912	1.16394	0.581969	−	0.813211i	$-0.302282\pi$
$491$	25.7912	1.16394	0.581969	+	0.813211i	$0.302282\pi$
$492$	8.12414	0.366264
$493$	−20.0054	−0.900996
$494$	−6.82430	−0.307040
$495$	−80.8116	−3.63221
$496$	1.33615	0.0599949
$497$	−20.0261	−0.898294
$498$	45.6423	2.04528
$499$	29.9163	1.33924	0.669618	−	0.742705i	$-0.266458\pi$
$499$	29.9163	1.33924	0.669618	+	0.742705i	$0.266458\pi$
$500$	14.3093	0.639929
$501$	0.0167044	0.000746297 0
$502$	21.7570	0.971064
$503$	14.7726	0.658676	0.329338	−	0.944212i	$-0.393174\pi$
$503$	14.7726	0.658676	0.329338	+	0.944212i	$0.393174\pi$
$504$	−11.9261	−0.531229
$505$	5.90915	0.262954
$506$	−26.6028	−1.18264
$507$	25.4993	1.13246
$508$	−5.42430	−0.240665
$509$	6.84080	0.303213	0.151606	−	0.988441i	$-0.451555\pi$
$509$	6.84080	0.303213	0.151606	+	0.988441i	$0.451555\pi$
$510$	55.9414	2.47713
$511$	−29.5234	−1.30604
$512$	1.00000	0.0441942
$513$	−16.7462	−0.739364
$514$	−12.7479	−0.562286
$515$	−66.4891	−2.92986
$516$	11.2362	0.494647
$517$	37.6299	1.65496
$518$	−18.3952	−0.808238
$519$	24.3520	1.06893
$520$	−7.27893	−0.319202
$521$	28.7982	1.26167	0.630836	−	0.775916i	$-0.282712\pi$
$521$	28.7982	1.26167	0.630836	+	0.775916i	$0.282712\pi$
$522$	17.4843	0.765266
$523$	−19.7158	−0.862112	−0.431056	−	0.902325i	$-0.641859\pi$
$523$	−19.7158	−0.862112	−0.431056	+	0.902325i	$0.641859\pi$
$524$	0.816465	0.0356675
$525$	62.0368	2.70751
$526$	20.5321	0.895241
$527$	−7.22654	−0.314793
$528$	12.7716	0.555811
$529$	10.5253	0.457623
$530$	1.60277	0.0696198
$531$	−2.58048	−0.111983
$532$	−8.80170	−0.381602
$533$	5.71724	0.247641
$534$	31.1246	1.34689
$535$	−36.4895	−1.57758
$536$	−10.9527	−0.473086
$537$	4.48949	0.193736
$538$	−23.0224	−0.992564
$539$	2.91466	0.125543
$540$	−17.8618	−0.768652
$541$	−18.1160	−0.778869	−0.389434	−	0.921054i	$-0.627329\pi$
$541$	−18.1160	−0.778869	−0.389434	+	0.921054i	$0.627329\pi$
$542$	−2.35089	−0.100980
$543$	28.8084	1.23629
$544$	−5.40848	−0.231887
$545$	−8.86034	−0.379535
$546$	−13.7194	−0.587136
$547$	−12.1271	−0.518516	−0.259258	−	0.965808i	$-0.583478\pi$
$547$	−12.1271	−0.518516	−0.259258	+	0.965808i	$0.583478\pi$
$548$	6.06321	0.259008
$549$	6.04378	0.257942
$550$	−40.6413	−1.73295
$551$	12.9038	0.549720
$552$	−16.0949	−0.685046
$553$	21.9551	0.933627
$554$	−0.975853	−0.0414600
$555$	−75.4123	−3.20107
$556$	−7.29227	−0.309261
$557$	20.8938	0.885300	0.442650	−	0.896694i	$-0.354038\pi$
$557$	20.8938	0.885300	0.442650	+	0.896694i	$0.354038\pi$
$558$	6.31585	0.267371
$559$	7.90733	0.334444
$560$	−9.38806	−0.396718
$561$	−69.0748	−2.91634
$562$	−10.3703	−0.437447
$563$	−25.0180	−1.05438	−0.527190	−	0.849747i	$-0.676755\pi$
$563$	−25.0180	−1.05438	−0.527190	+	0.849747i	$0.676755\pi$
$564$	22.7664	0.958637
$565$	58.8179	2.47449
$566$	21.8684	0.919196
$567$	2.11200	0.0886957
$568$	7.93737	0.333044
$569$	16.3624	0.685949	0.342974	−	0.939345i	$-0.388566\pi$
$569$	16.3624	0.685949	0.342974	+	0.939345i	$0.388566\pi$
$570$	−36.0831	−1.51136
$571$	−17.9762	−0.752281	−0.376141	−	0.926563i	$-0.622749\pi$
$571$	−17.9762	−0.752281	−0.376141	+	0.926563i	$0.622749\pi$
$572$	8.98780	0.375799
$573$	27.6199	1.15384
$574$	7.37386	0.307779
$575$	51.2168	2.13589
$576$	4.72690	0.196954
$577$	−5.32018	−0.221482	−0.110741	−	0.993849i	$-0.535322\pi$
$577$	−5.32018	−0.221482	−0.110741	+	0.993849i	$0.535322\pi$
$578$	12.2516	0.509601
$579$	5.04564	0.209690
$580$	13.7634	0.571495
$581$	41.4271	1.71869
$582$	27.7934	1.15207
$583$	−1.97905	−0.0819639
$584$	11.7016	0.484217
$585$	−34.4068	−1.42255
$586$	28.5904	1.18106
$587$	7.47774	0.308639	0.154320	−	0.988021i	$-0.450681\pi$
$587$	7.47774	0.308639	0.154320	+	0.988021i	$0.450681\pi$
$588$	1.76340	0.0727212
$589$	4.66124	0.192063
$590$	−2.03132	−0.0836282
$591$	−25.1594	−1.03492
$592$	7.29094	0.299656
$593$	−0.321463	−0.0132009	−0.00660044	−	0.999978i	$-0.502101\pi$
$593$	−0.321463	−0.0132009	−0.00660044	+	0.999978i	$0.502101\pi$
$594$	22.0553	0.904939
$595$	50.7751	2.08158
$596$	7.82319	0.320450
$597$	−24.1046	−0.986537
$598$	−11.3266	−0.463178
$599$	−36.1793	−1.47824	−0.739122	−	0.673572i	$-0.764759\pi$
$599$	−36.1793	−1.47824	−0.739122	+	0.673572i	$0.764759\pi$
$600$	−24.5883	−1.00381
$601$	38.8933	1.58649	0.793245	−	0.608902i	$-0.208390\pi$
$601$	38.8933	1.58649	0.793245	+	0.608902i	$0.208390\pi$
$602$	10.1985	0.415662
$603$	−51.7725	−2.10834
$604$	23.9677	0.975230
$605$	37.6180	1.52939
$606$	−4.41441	−0.179323
$607$	5.14847	0.208970	0.104485	−	0.994526i	$-0.466681\pi$
$607$	5.14847	0.208970	0.104485	+	0.994526i	$0.466681\pi$
$608$	3.48856	0.141480
$609$	25.9415	1.05120
$610$	4.75759	0.192629
$611$	16.0215	0.648160
$612$	−25.5654	−1.03342
$613$	−11.0421	−0.445987	−0.222994	−	0.974820i	$-0.571583\pi$
$613$	−11.0421	−0.445987	−0.222994	+	0.974820i	$0.571583\pi$
$614$	26.9812	1.08887
$615$	30.2296	1.21898
$616$	11.5921	0.467059
$617$	27.7563	1.11743	0.558713	−	0.829361i	$-0.311296\pi$
$617$	27.7563	1.11743	0.558713	+	0.829361i	$0.311296\pi$
$618$	49.6704	1.99804
$619$	9.27809	0.372918	0.186459	−	0.982463i	$-0.440299\pi$
$619$	9.27809	0.372918	0.186459	+	0.982463i	$0.440299\pi$
$620$	4.97177	0.199671
$621$	−27.7944	−1.11535
$622$	−19.4471	−0.779757
$623$	28.2502	1.13182
$624$	5.43769	0.217682
$625$	9.01635	0.360654
$626$	2.50205	0.100002
$627$	44.5544	1.77933
$628$	−21.6663	−0.864579
$629$	−39.4329	−1.57229
$630$	−44.3765	−1.76800
$631$	19.5346	0.777661	0.388831	−	0.921309i	$-0.372879\pi$
$631$	19.5346	0.777661	0.388831	+	0.921309i	$0.372879\pi$
$632$	−8.70193	−0.346144
$633$	70.3505	2.79618
$634$	−13.2624	−0.526715
$635$	−20.1836	−0.800964
$636$	−1.19734	−0.0474777
$637$	1.24096	0.0491688
$638$	−16.9947	−0.672825
$639$	37.5192	1.48424
$640$	3.72096	0.147084
$641$	22.6643	0.895187	0.447593	−	0.894237i	$-0.352281\pi$
$641$	22.6643	0.895187	0.447593	+	0.894237i	$0.352281\pi$
$642$	27.2593	1.07584
$643$	−34.9789	−1.37943	−0.689717	−	0.724079i	$-0.742265\pi$
$643$	−34.9789	−1.37943	−0.689717	+	0.724079i	$0.742265\pi$
$644$	−14.6085	−0.575657
$645$	41.8096	1.64625
$646$	−18.8678	−0.742343
$647$	12.2297	0.480800	0.240400	−	0.970674i	$-0.422721\pi$
$647$	12.2297	0.480800	0.240400	+	0.970674i	$0.422721\pi$
$648$	−0.837092	−0.0328841
$649$	2.50822	0.0984561
$650$	−17.3037	−0.678706
$651$	9.37084	0.367272
$652$	−15.1387	−0.592879
$653$	−9.97337	−0.390288	−0.195144	−	0.980775i	$-0.562517\pi$
$653$	−9.97337	−0.390288	−0.195144	+	0.980775i	$0.562517\pi$
$654$	6.61908	0.258827
$655$	3.03804	0.118706
$656$	−2.92263	−0.114110
$657$	55.3124	2.15794
$658$	20.6639	0.805561
$659$	−10.0692	−0.392238	−0.196119	−	0.980580i	$-0.562834\pi$
$659$	−10.0692	−0.392238	−0.196119	+	0.980580i	$0.562834\pi$
$660$	47.5226	1.84981
$661$	9.98803	0.388489	0.194245	−	0.980953i	$-0.437774\pi$
$661$	9.98803	0.388489	0.194245	+	0.980953i	$0.437774\pi$
$662$	18.6037	0.723052
$663$	−29.4096	−1.14218
$664$	−16.4197	−0.637207
$665$	−32.7508	−1.27002
$666$	34.4636	1.33544
$667$	21.4169	0.829267
$668$	−0.00600935	−0.000232509 0
$669$	43.4844	1.68120
$670$	−40.7547	−1.57449
$671$	−5.87453	−0.226784
$672$	7.01331	0.270544
$673$	−48.3469	−1.86363	−0.931817	−	0.362928i	$-0.881777\pi$
$673$	−48.3469	−1.86363	−0.931817	+	0.362928i	$0.881777\pi$
$674$	−26.5160	−1.02136
$675$	−42.4617	−1.63435
$676$	−9.17330	−0.352819
$677$	−10.7323	−0.412476	−0.206238	−	0.978502i	$-0.566122\pi$
$677$	−10.7323	−0.412476	−0.206238	+	0.978502i	$0.566122\pi$
$678$	−43.9397	−1.68749
$679$	25.2266	0.968109
$680$	−20.1248	−0.771749
$681$	12.7247	0.487610
$682$	−6.13899	−0.235074
$683$	−33.3896	−1.27762	−0.638809	−	0.769365i	$-0.720573\pi$
$683$	−33.3896	−1.27762	−0.638809	+	0.769365i	$0.720573\pi$
$684$	16.4901	0.630514
$685$	22.5610	0.862011
$686$	19.2617	0.735414
$687$	−30.8315	−1.17630
$688$	−4.04220	−0.154107
$689$	−0.842611	−0.0321009
$690$	−59.8887	−2.27992
$691$	−42.1061	−1.60179	−0.800896	−	0.598804i	$-0.795643\pi$
$691$	−42.1061	−1.60179	−0.800896	+	0.598804i	$0.795643\pi$
$692$	−8.76056	−0.333027
$693$	54.7947	2.08148
$694$	−7.56571	−0.287190
$695$	−27.1343	−1.02926
$696$	−10.2819	−0.389735
$697$	15.8070	0.598733
$698$	−17.1462	−0.648995
$699$	63.5732	2.40456
$700$	−22.3176	−0.843524
$701$	8.08867	0.305505	0.152752	−	0.988264i	$-0.451186\pi$
$701$	8.08867	0.305505	0.152752	+	0.988264i	$0.451186\pi$
$702$	9.39037	0.354417
$703$	25.4349	0.959294
$704$	−4.59453	−0.173163
$705$	84.7129	3.19047
$706$	−9.06116	−0.341021
$707$	−4.00673	−0.150689
$708$	1.51749	0.0570308
$709$	31.4648	1.18169	0.590843	−	0.806786i	$-0.298795\pi$
$709$	31.4648	1.18169	0.590843	+	0.806786i	$0.298795\pi$
$710$	29.5347	1.10842
$711$	−41.1332	−1.54261
$712$	−11.1970	−0.419624
$713$	7.73645	0.289732
$714$	−37.9314	−1.41955
$715$	33.4433	1.25071
$716$	−1.61508	−0.0603584
$717$	49.7028	1.85618
$718$	30.5993	1.14196
$719$	−43.4561	−1.62064	−0.810320	−	0.585988i	$-0.800707\pi$
$719$	−43.4561	−1.62064	−0.810320	+	0.585988i	$0.800707\pi$
$720$	17.5886	0.655490
$721$	45.0833	1.67899
$722$	−6.82996	−0.254185
$723$	−48.1574	−1.79099
$724$	−10.3637	−0.385165
$725$	32.7188	1.21515
$726$	−28.1024	−1.04298
$727$	−12.5844	−0.466729	−0.233365	−	0.972389i	$-0.574974\pi$
$727$	−12.5844	−0.466729	−0.233365	+	0.972389i	$0.574974\pi$
$728$	4.93551	0.182922
$729$	−43.9877	−1.62917
$730$	43.5413	1.61154
$731$	21.8622	0.808601
$732$	−3.55414	−0.131365
$733$	17.1656	0.634024	0.317012	−	0.948422i	$-0.397320\pi$
$733$	17.1656	0.634024	0.317012	+	0.948422i	$0.397320\pi$
$734$	−30.2276	−1.11572
$735$	6.56153	0.242026
$736$	5.79011	0.213426
$737$	50.3227	1.85366
$738$	−13.8150	−0.508537
$739$	14.7172	0.541382	0.270691	−	0.962666i	$-0.412748\pi$
$739$	14.7172	0.541382	0.270691	+	0.962666i	$0.412748\pi$
$740$	27.1293	0.997294
$741$	18.9697	0.696870
$742$	−1.08677	−0.0398964
$743$	7.78072	0.285447	0.142724	−	0.989763i	$-0.454414\pi$
$743$	7.78072	0.285447	0.142724	+	0.989763i	$0.454414\pi$
$744$	−3.71414	−0.136167
$745$	29.1098	1.06650
$746$	−26.0530	−0.953869
$747$	−77.6142	−2.83976
$748$	24.8494	0.908586
$749$	24.7419	0.904050
$750$	−39.7759	−1.45241
$751$	31.3970	1.14569	0.572846	−	0.819663i	$-0.305839\pi$
$751$	31.3970	1.14569	0.572846	+	0.819663i	$0.305839\pi$
$752$	−8.19014	−0.298664
$753$	−60.4787	−2.20397
$754$	−7.23574	−0.263510
$755$	89.1828	3.24569
$756$	12.1113	0.440484
$757$	13.1000	0.476128	0.238064	−	0.971249i	$-0.423487\pi$
$757$	13.1000	0.476128	0.238064	+	0.971249i	$0.423487\pi$
$758$	21.5786	0.783770
$759$	73.9487	2.68417
$760$	12.9808	0.470863
$761$	5.19854	0.188447	0.0942235	−	0.995551i	$-0.469963\pi$
$761$	5.19854	0.188447	0.0942235	+	0.995551i	$0.469963\pi$
$762$	15.0781	0.546222
$763$	6.00780	0.217497
$764$	−9.93616	−0.359478
$765$	−95.1278	−3.43935
$766$	−25.3460	−0.915788
$767$	1.06791	0.0385601
$768$	−2.77973	−0.100305
$769$	−12.6538	−0.456308	−0.228154	−	0.973625i	$-0.573269\pi$
$769$	−12.6538	−0.456308	−0.228154	+	0.973625i	$0.573269\pi$
$770$	43.1338	1.55443
$771$	35.4358	1.27619
$772$	−1.81515	−0.0653288
$773$	28.3918	1.02118	0.510591	−	0.859824i	$-0.329427\pi$
$773$	28.3918	1.02118	0.510591	+	0.859824i	$0.329427\pi$
$774$	−19.1071	−0.686790
$775$	11.8190	0.424552
$776$	−9.99859	−0.358929
$777$	51.1337	1.83441
$778$	−20.3548	−0.729756
$779$	−10.1958	−0.365302
$780$	20.2335	0.724474
$781$	−36.4685	−1.30495
$782$	−31.3157	−1.11985
$783$	−17.7559	−0.634543
$784$	−0.634376	−0.0226563
$785$	−80.6195	−2.87743
$786$	−2.26955	−0.0809523
$787$	−7.57706	−0.270093	−0.135046	−	0.990839i	$-0.543118\pi$
$787$	−7.57706	−0.270093	−0.135046	+	0.990839i	$0.543118\pi$
$788$	9.05102	0.322429
$789$	−57.0736	−2.03187
$790$	−32.3796	−1.15201
$791$	−39.8818	−1.41803
$792$	−21.7179	−0.771713
$793$	−2.50117	−0.0888192
$794$	6.36952	0.226046
$795$	−4.45527	−0.158012
$796$	8.67157	0.307356
$797$	46.1916	1.63619	0.818095	−	0.575083i	$-0.195030\pi$
$797$	46.1916	1.63619	0.818095	+	0.575083i	$0.195030\pi$
$798$	24.4663	0.866099
$799$	44.2962	1.56709
$800$	8.84558	0.312738
$801$	−52.9270	−1.87008
$802$	−27.7910	−0.981333
$803$	−53.7635	−1.89727
$804$	30.4457	1.07374
$805$	−54.3579	−1.91586
$806$	−2.61377	−0.0920661
$807$	63.9960	2.25276
$808$	1.58807	0.0558681
$809$	5.99755	0.210863	0.105431	−	0.994427i	$-0.466378\pi$
$809$	5.99755	0.210863	0.105431	+	0.994427i	$0.466378\pi$
$810$	−3.11479	−0.109443
$811$	33.1014	1.16235	0.581174	−	0.813780i	$-0.302594\pi$
$811$	33.1014	1.16235	0.581174	+	0.813780i	$0.302594\pi$
$812$	−9.33236	−0.327502
$813$	6.53486	0.229187
$814$	−33.4985	−1.17412
$815$	−56.3307	−1.97318
$816$	15.0341	0.526299
$817$	−14.1014	−0.493347
$818$	11.9357	0.417321
$819$	23.3297	0.815206
$820$	−10.8750	−0.379772
$821$	18.4904	0.645319	0.322660	−	0.946515i	$-0.395423\pi$
$821$	18.4904	0.645319	0.322660	+	0.946515i	$0.395423\pi$
$822$	−16.8541	−0.587854
$823$	−8.34308	−0.290822	−0.145411	−	0.989371i	$-0.546450\pi$
$823$	−8.34308	−0.290822	−0.145411	+	0.989371i	$0.546450\pi$
$824$	−17.8688	−0.622489
$825$	112.972	3.93318
$826$	1.37735	0.0479241
$827$	31.3176	1.08902	0.544509	−	0.838755i	$-0.316716\pi$
$827$	31.3176	1.08902	0.544509	+	0.838755i	$0.316716\pi$
$828$	27.3693	0.951148
$829$	−16.5178	−0.573687	−0.286843	−	0.957977i	$-0.592606\pi$
$829$	−16.5178	−0.573687	−0.286843	+	0.957977i	$0.592606\pi$
$830$	−61.0970	−2.12071
$831$	2.71261	0.0940993
$832$	−1.95619	−0.0678188
$833$	3.43101	0.118878
$834$	20.2706	0.701912
$835$	−0.0223606	−0.000773820 0
$836$	−16.0283	−0.554350
$837$	−6.41396	−0.221699
$838$	−31.4267	−1.08562
$839$	52.0628	1.79741	0.898705	−	0.438555i	$-0.144509\pi$
$839$	52.0628	1.79741	0.898705	+	0.438555i	$0.144509\pi$
$840$	26.0963	0.900408
$841$	−15.3182	−0.528215
$842$	−27.1218	−0.934679
$843$	28.8268	0.992847
$844$	−25.3084	−0.871151
$845$	−34.1335	−1.17423
$846$	−38.7140	−1.33101
$847$	−25.5071	−0.876434
$848$	0.430740	0.0147917
$849$	−60.7881	−2.08624
$850$	−47.8411	−1.64094
$851$	42.2153	1.44712
$852$	−22.0637	−0.755891
$853$	22.7997	0.780647	0.390324	−	0.920678i	$-0.372363\pi$
$853$	22.7997	0.780647	0.390324	+	0.920678i	$0.372363\pi$
$854$	−3.22591	−0.110388
$855$	61.3590	2.09843
$856$	−9.80647	−0.335178
$857$	−0.0646309	−0.00220775	−0.00110387	−	0.999999i	$-0.500351\pi$
$857$	−0.0646309	−0.00220775	−0.00110387	+	0.999999i	$0.500351\pi$
$858$	−24.9837	−0.852929
$859$	17.1318	0.584530	0.292265	−	0.956337i	$-0.405591\pi$
$859$	17.1318	0.584530	0.292265	+	0.956337i	$0.405591\pi$
$860$	−15.0409	−0.512890
$861$	−20.4973	−0.698548
$862$	9.60015	0.326982
$863$	28.5078	0.970416	0.485208	−	0.874399i	$-0.338744\pi$
$863$	28.5078	0.970416	0.485208	+	0.874399i	$0.338744\pi$
$864$	−4.80033	−0.163310
$865$	−32.5977	−1.10836
$866$	−1.67026	−0.0567577
$867$	−34.0563	−1.15661
$868$	−3.37113	−0.114424
$869$	39.9813	1.35627
$870$	−38.2586	−1.29709
$871$	21.4257	0.725981
$872$	−2.38119	−0.0806374
$873$	−47.2624	−1.59959
$874$	20.1991	0.683245
$875$	−36.1025	−1.22049
$876$	−32.5274	−1.09900
$877$	46.9937	1.58687	0.793433	−	0.608658i	$-0.208292\pi$
$877$	46.9937	1.58687	0.793433	+	0.608658i	$0.208292\pi$
$878$	9.12837	0.308067
$879$	−79.4736	−2.68058
$880$	−17.0961	−0.576309
$881$	−6.26562	−0.211094	−0.105547	−	0.994414i	$-0.533659\pi$
$881$	−6.26562	−0.211094	−0.105547	+	0.994414i	$0.533659\pi$
$882$	−2.99864	−0.100969
$883$	31.5800	1.06275	0.531376	−	0.847136i	$-0.321675\pi$
$883$	31.5800	1.06275	0.531376	+	0.847136i	$0.321675\pi$
$884$	10.5800	0.355845
$885$	5.64653	0.189806
$886$	−1.13267	−0.0380528
$887$	−41.5535	−1.39523	−0.697615	−	0.716473i	$-0.745755\pi$
$887$	−41.5535	−1.39523	−0.697615	+	0.716473i	$0.745755\pi$
$888$	−20.2669	−0.680111
$889$	13.6856	0.459001
$890$	−41.6636	−1.39657
$891$	3.84605	0.128848
$892$	−15.6434	−0.523779
$893$	−28.5718	−0.956118
$894$	−21.7464	−0.727307
$895$	−6.00966	−0.200881
$896$	−2.52302	−0.0842881
$897$	31.4848	1.05125
$898$	0.578765	0.0193136
$899$	4.94227	0.164834
$900$	41.8122	1.39374
$901$	−2.32965	−0.0776119
$902$	13.4281	0.447108
$903$	−28.3492	−0.943403
$904$	15.8072	0.525739
$905$	−38.5631	−1.28188
$906$	−66.6236	−2.21342
$907$	−44.5302	−1.47860	−0.739300	−	0.673377i	$-0.764843\pi$
$907$	−44.5302	−1.47860	−0.739300	+	0.673377i	$0.764843\pi$
$908$	−4.57766	−0.151915
$909$	7.50666	0.248980
$910$	18.3649	0.608790
$911$	−32.2905	−1.06983	−0.534916	−	0.844905i	$-0.679657\pi$
$911$	−32.2905	−1.06983	−0.534916	+	0.844905i	$0.679657\pi$
$912$	−9.69725	−0.321108
$913$	75.4408	2.49672
$914$	32.3426	1.06980
$915$	−13.2248	−0.437199
$916$	11.0915	0.366475
$917$	−2.05996	−0.0680258
$918$	25.9625	0.856889
$919$	−33.1119	−1.09226	−0.546131	−	0.837700i	$-0.683900\pi$
$919$	−33.1119	−1.09226	−0.546131	+	0.837700i	$0.683900\pi$
$920$	21.5448	0.710310
$921$	−75.0004	−2.47135
$922$	−3.97335	−0.130855
$923$	−15.5270	−0.511078
$924$	−32.2229	−1.06006
$925$	64.4926	2.12050
$926$	3.69345	0.121374
$927$	−84.4640	−2.77416
$928$	3.69889	0.121422
$929$	−22.6686	−0.743734	−0.371867	−	0.928286i	$-0.621282\pi$
$929$	−22.6686	−0.743734	−0.371867	+	0.928286i	$0.621282\pi$
$930$	−13.8202	−0.453181
$931$	−2.21306	−0.0725301
$932$	−22.8703	−0.749141
$933$	54.0576	1.76977
$934$	13.9692	0.457087
$935$	92.4639	3.02389
$936$	−9.24674	−0.302239
$937$	−29.6822	−0.969675	−0.484838	−	0.874604i	$-0.661121\pi$
$937$	−29.6822	−0.969675	−0.484838	+	0.874604i	$0.661121\pi$
$938$	27.6340	0.902281
$939$	−6.95502	−0.226968
$940$	−30.4752	−0.993992
$941$	−27.4029	−0.893308	−0.446654	−	0.894707i	$-0.647385\pi$
$941$	−27.4029	−0.893308	−0.446654	+	0.894707i	$0.647385\pi$
$942$	60.2264	1.96228
$943$	−16.9224	−0.551068
$944$	−0.545913	−0.0177680
$945$	45.0658	1.46599
$946$	18.5720	0.603829
$947$	−36.5209	−1.18677	−0.593385	−	0.804919i	$-0.702209\pi$
$947$	−36.5209	−1.18677	−0.593385	+	0.804919i	$0.702209\pi$
$948$	24.1890	0.785623
$949$	−22.8906	−0.743062
$950$	30.8583	1.00118
$951$	36.8658	1.19545
$952$	13.6457	0.442259
$953$	−5.00177	−0.162023	−0.0810117	−	0.996713i	$-0.525815\pi$
$953$	−5.00177	−0.162023	−0.0810117	+	0.996713i	$0.525815\pi$
$954$	2.03607	0.0659201
$955$	−36.9721	−1.19639
$956$	−17.8804	−0.578294
$957$	47.2406	1.52707
$958$	2.54601	0.0822578
$959$	−15.2976	−0.493985
$960$	−10.3433	−0.333828
$961$	−29.2147	−0.942410
$962$	−14.2625	−0.459841
$963$	−46.3542	−1.49374
$964$	17.3245	0.557984
$965$	−6.75412	−0.217423
$966$	40.6078	1.30654
$967$	−39.3526	−1.26549	−0.632747	−	0.774359i	$-0.718073\pi$
$967$	−39.3526	−1.26549	−0.632747	+	0.774359i	$0.718073\pi$
$968$	10.1098	0.324940
$969$	52.4474	1.68485
$970$	−37.2044	−1.19456
$971$	32.7792	1.05194	0.525968	−	0.850505i	$-0.323703\pi$
$971$	32.7792	1.05194	0.525968	+	0.850505i	$0.323703\pi$
$972$	16.7279	0.536547
$973$	18.3985	0.589830
$974$	15.9351	0.510595
$975$	48.0995	1.54042
$976$	1.27859	0.0409267
$977$	−30.8363	−0.986540	−0.493270	−	0.869876i	$-0.664199\pi$
$977$	−30.8363	−0.986540	−0.493270	+	0.869876i	$0.664199\pi$
$978$	42.0816	1.34562
$979$	51.4449	1.64419
$980$	−2.36049	−0.0754032
$981$	−11.2557	−0.359366
$982$	25.7912	0.823029
$983$	2.20718	0.0703981	0.0351990	−	0.999380i	$-0.488793\pi$
$983$	2.20718	0.0703981	0.0351990	+	0.999380i	$0.488793\pi$
$984$	8.12414	0.258988
$985$	33.6785	1.07309
$986$	−20.0054	−0.637100
$987$	−57.4400	−1.82834
$988$	−6.82430	−0.217110
$989$	−23.4048	−0.744228
$990$	−80.8116	−2.56836
$991$	−3.76298	−0.119535	−0.0597676	−	0.998212i	$-0.519036\pi$
$991$	−3.76298	−0.119535	−0.0597676	+	0.998212i	$0.519036\pi$
$992$	1.33615	0.0424228
$993$	−51.7132	−1.64107
$994$	−20.0261	−0.635190
$995$	32.2666	1.02292
$996$	45.6423	1.44623
$997$	49.5879	1.57046	0.785232	−	0.619201i	$-0.212543\pi$
$997$	49.5879	1.57046	0.785232	+	0.619201i	$0.212543\pi$
$998$	29.9163	0.946984
$999$	−34.9989	−1.10732

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6038.2.a.b.1.7	✓	54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6038.2.a.b.1.7	✓	54	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:	\( N \)	\(=\)	\( 6038 = 2 \cdot 3019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6038.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.77973 q^{3} +1.00000 q^{4} +3.72096 q^{5} -2.77973 q^{6} -2.52302 q^{7} +1.00000 q^{8} +4.72690 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.77973 q^{3} +1.00000 q^{4} +3.72096 q^{5} -2.77973 q^{6} -2.52302 q^{7} +1.00000 q^{8} +4.72690 q^{9} +3.72096 q^{10} -4.59453 q^{11} -2.77973 q^{12} -1.95619 q^{13} -2.52302 q^{14} -10.3433 q^{15} +1.00000 q^{16} -5.40848 q^{17} +4.72690 q^{18} +3.48856 q^{19} +3.72096 q^{20} +7.01331 q^{21} -4.59453 q^{22} +5.79011 q^{23} -2.77973 q^{24} +8.84558 q^{25} -1.95619 q^{26} -4.80033 q^{27} -2.52302 q^{28} +3.69889 q^{29} -10.3433 q^{30} +1.33615 q^{31} +1.00000 q^{32} +12.7716 q^{33} -5.40848 q^{34} -9.38806 q^{35} +4.72690 q^{36} +7.29094 q^{37} +3.48856 q^{38} +5.43769 q^{39} +3.72096 q^{40} -2.92263 q^{41} +7.01331 q^{42} -4.04220 q^{43} -4.59453 q^{44} +17.5886 q^{45} +5.79011 q^{46} -8.19014 q^{47} -2.77973 q^{48} -0.634376 q^{49} +8.84558 q^{50} +15.0341 q^{51} -1.95619 q^{52} +0.430740 q^{53} -4.80033 q^{54} -17.0961 q^{55} -2.52302 q^{56} -9.69725 q^{57} +3.69889 q^{58} -0.545913 q^{59} -10.3433 q^{60} +1.27859 q^{61} +1.33615 q^{62} -11.9261 q^{63} +1.00000 q^{64} -7.27893 q^{65} +12.7716 q^{66} -10.9527 q^{67} -5.40848 q^{68} -16.0949 q^{69} -9.38806 q^{70} +7.93737 q^{71} +4.72690 q^{72} +11.7016 q^{73} +7.29094 q^{74} -24.5883 q^{75} +3.48856 q^{76} +11.5921 q^{77} +5.43769 q^{78} -8.70193 q^{79} +3.72096 q^{80} -0.837092 q^{81} -2.92263 q^{82} -16.4197 q^{83} +7.01331 q^{84} -20.1248 q^{85} -4.04220 q^{86} -10.2819 q^{87} -4.59453 q^{88} -11.1970 q^{89} +17.5886 q^{90} +4.93551 q^{91} +5.79011 q^{92} -3.71414 q^{93} -8.19014 q^{94} +12.9808 q^{95} -2.77973 q^{96} -9.99859 q^{97} -0.634376 q^{98} -21.7179 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 54 q^{2} - 21 q^{3} + 54 q^{4} - 14 q^{5} - 21 q^{6} - 44 q^{7} + 54 q^{8} + 39 q^{9}+O(q^{10}) \) 54 * q + 54 * q^2 - 21 * q^3 + 54 * q^4 - 14 * q^5 - 21 * q^6 - 44 * q^7 + 54 * q^8 + 39 * q^9	\( 54 q + 54 q^{2} - 21 q^{3} + 54 q^{4} - 14 q^{5} - 21 q^{6} - 44 q^{7} + 54 q^{8} + 39 q^{9} - 14 q^{10} - 31 q^{11} - 21 q^{12} - 34 q^{13} - 44 q^{14} - 22 q^{15} + 54 q^{16} - 40 q^{17} + 39 q^{18} - 44 q^{19} - 14 q^{20} - 3 q^{21} - 31 q^{22} - 33 q^{23} - 21 q^{24} + 14 q^{25} - 34 q^{26} - 66 q^{27} - 44 q^{28} - 22 q^{30} - 65 q^{31} + 54 q^{32} - 43 q^{33} - 40 q^{34} - 46 q^{35} + 39 q^{36} - 58 q^{37} - 44 q^{38} - 36 q^{39} - 14 q^{40} - 49 q^{41} - 3 q^{42} - 47 q^{43} - 31 q^{44} - 45 q^{45} - 33 q^{46} - 66 q^{47} - 21 q^{48} + 16 q^{49} + 14 q^{50} - 33 q^{51} - 34 q^{52} - 16 q^{53} - 66 q^{54} - 50 q^{55} - 44 q^{56} - 33 q^{57} - 70 q^{59} - 22 q^{60} - 40 q^{61} - 65 q^{62} - 117 q^{63} + 54 q^{64} - 33 q^{65} - 43 q^{66} - 82 q^{67} - 40 q^{68} - q^{69} - 46 q^{70} - 60 q^{71} + 39 q^{72} - 92 q^{73} - 58 q^{74} - 68 q^{75} - 44 q^{76} + 13 q^{77} - 36 q^{78} - 57 q^{79} - 14 q^{80} + 26 q^{81} - 49 q^{82} - 77 q^{83} - 3 q^{84} - 24 q^{85} - 47 q^{86} - 61 q^{87} - 31 q^{88} - 54 q^{89} - 45 q^{90} - 46 q^{91} - 33 q^{92} - 24 q^{93} - 66 q^{94} - 66 q^{95} - 21 q^{96} - 137 q^{97} + 16 q^{98} - 71 q^{99}+O(q^{100}) \) 54 * q + 54 * q^2 - 21 * q^3 + 54 * q^4 - 14 * q^5 - 21 * q^6 - 44 * q^7 + 54 * q^8 + 39 * q^9 - 14 * q^10 - 31 * q^11 - 21 * q^12 - 34 * q^13 - 44 * q^14 - 22 * q^15 + 54 * q^16 - 40 * q^17 + 39 * q^18 - 44 * q^19 - 14 * q^20 - 3 * q^21 - 31 * q^22 - 33 * q^23 - 21 * q^24 + 14 * q^25 - 34 * q^26 - 66 * q^27 - 44 * q^28 - 22 * q^30 - 65 * q^31 + 54 * q^32 - 43 * q^33 - 40 * q^34 - 46 * q^35 + 39 * q^36 - 58 * q^37 - 44 * q^38 - 36 * q^39 - 14 * q^40 - 49 * q^41 - 3 * q^42 - 47 * q^43 - 31 * q^44 - 45 * q^45 - 33 * q^46 - 66 * q^47 - 21 * q^48 + 16 * q^49 + 14 * q^50 - 33 * q^51 - 34 * q^52 - 16 * q^53 - 66 * q^54 - 50 * q^55 - 44 * q^56 - 33 * q^57 - 70 * q^59 - 22 * q^60 - 40 * q^61 - 65 * q^62 - 117 * q^63 + 54 * q^64 - 33 * q^65 - 43 * q^66 - 82 * q^67 - 40 * q^68 - q^69 - 46 * q^70 - 60 * q^71 + 39 * q^72 - 92 * q^73 - 58 * q^74 - 68 * q^75 - 44 * q^76 + 13 * q^77 - 36 * q^78 - 57 * q^79 - 14 * q^80 + 26 * q^81 - 49 * q^82 - 77 * q^83 - 3 * q^84 - 24 * q^85 - 47 * q^86 - 61 * q^87 - 31 * q^88 - 54 * q^89 - 45 * q^90 - 46 * q^91 - 33 * q^92 - 24 * q^93 - 66 * q^94 - 66 * q^95 - 21 * q^96 - 137 * q^97 + 16 * q^98 - 71 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more