LMFDB - Embedded newform 6042.2.a.bd.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6042,2,Mod(1,6042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6042, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6042 = 2 \cdot 3 \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6042.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:	$48.2456129013$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 35 x^{9} + 77 x^{8} + 394 x^{7} - 994 x^{6} - 1477 x^{5} + 4683 x^{4} + 563 x^{3} + \cdots + 1534 $ x^11 - 2x^10 - 35x^9 + 77x^8 + 394x^7 - 994x^6 - 1477x^5 + 4683x^4 + 563x^3 - 6194x^2 + 1258x + 1534
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$4.06414$ of defining polynomial
Character	$\chi$	$=$	6042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.06414 q^{5} +1.00000 q^{6} +3.98638 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.06414 q^{5} +1.00000 q^{6} +3.98638 q^{7} -1.00000 q^{8} +1.00000 q^{9} -4.06414 q^{10} -4.83665 q^{11} -1.00000 q^{12} -4.31849 q^{13} -3.98638 q^{14} -4.06414 q^{15} +1.00000 q^{16} -5.40155 q^{17} -1.00000 q^{18} +1.00000 q^{19} +4.06414 q^{20} -3.98638 q^{21} +4.83665 q^{22} -5.82002 q^{23} +1.00000 q^{24} +11.5172 q^{25} +4.31849 q^{26} -1.00000 q^{27} +3.98638 q^{28} -10.2047 q^{29} +4.06414 q^{30} +5.35251 q^{31} -1.00000 q^{32} +4.83665 q^{33} +5.40155 q^{34} +16.2012 q^{35} +1.00000 q^{36} -5.45024 q^{37} -1.00000 q^{38} +4.31849 q^{39} -4.06414 q^{40} +4.22566 q^{41} +3.98638 q^{42} +11.8821 q^{43} -4.83665 q^{44} +4.06414 q^{45} +5.82002 q^{46} -2.62661 q^{47} -1.00000 q^{48} +8.89126 q^{49} -11.5172 q^{50} +5.40155 q^{51} -4.31849 q^{52} +1.00000 q^{53} +1.00000 q^{54} -19.6568 q^{55} -3.98638 q^{56} -1.00000 q^{57} +10.2047 q^{58} -2.75182 q^{59} -4.06414 q^{60} +5.87161 q^{61} -5.35251 q^{62} +3.98638 q^{63} +1.00000 q^{64} -17.5509 q^{65} -4.83665 q^{66} -7.78222 q^{67} -5.40155 q^{68} +5.82002 q^{69} -16.2012 q^{70} -6.39344 q^{71} -1.00000 q^{72} -16.1520 q^{73} +5.45024 q^{74} -11.5172 q^{75} +1.00000 q^{76} -19.2807 q^{77} -4.31849 q^{78} -15.6618 q^{79} +4.06414 q^{80} +1.00000 q^{81} -4.22566 q^{82} +0.441737 q^{83} -3.98638 q^{84} -21.9526 q^{85} -11.8821 q^{86} +10.2047 q^{87} +4.83665 q^{88} +17.5920 q^{89} -4.06414 q^{90} -17.2152 q^{91} -5.82002 q^{92} -5.35251 q^{93} +2.62661 q^{94} +4.06414 q^{95} +1.00000 q^{96} -13.7075 q^{97} -8.89126 q^{98} -4.83665 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 11 q^{2} - 11 q^{3} + 11 q^{4} + 2 q^{5} + 11 q^{6} - 2 q^{7} - 11 q^{8} + 11 q^{9}+O(q^{10}) $ 11 * q - 11 * q^2 - 11 * q^3 + 11 * q^4 + 2 * q^5 + 11 * q^6 - 2 * q^7 - 11 * q^8 + 11 * q^9	\( 11 q - 11 q^{2} - 11 q^{3} + 11 q^{4} + 2 q^{5} + 11 q^{6} - 2 q^{7} - 11 q^{8} + 11 q^{9} - 2 q^{10} - 6 q^{11} - 11 q^{12} - 7 q^{13} + 2 q^{14} - 2 q^{15} + 11 q^{16} - 16 q^{17} - 11 q^{18} + 11 q^{19} + 2 q^{20} + 2 q^{21} + 6 q^{22} - 11 q^{23} + 11 q^{24} + 19 q^{25} + 7 q^{26} - 11 q^{27} - 2 q^{28} - 7 q^{29} + 2 q^{30} + 12 q^{31} - 11 q^{32} + 6 q^{33} + 16 q^{34} - 5 q^{35} + 11 q^{36} + 3 q^{37} - 11 q^{38} + 7 q^{39} - 2 q^{40} + 11 q^{41} - 2 q^{42} + 7 q^{43} - 6 q^{44} + 2 q^{45} + 11 q^{46} - 37 q^{47} - 11 q^{48} + 23 q^{49} - 19 q^{50} + 16 q^{51} - 7 q^{52} + 11 q^{53} + 11 q^{54} - 11 q^{55} + 2 q^{56} - 11 q^{57} + 7 q^{58} - 16 q^{59} - 2 q^{60} + 24 q^{61} - 12 q^{62} - 2 q^{63} + 11 q^{64} - 7 q^{65} - 6 q^{66} - 28 q^{67} - 16 q^{68} + 11 q^{69} + 5 q^{70} + 4 q^{71} - 11 q^{72} - 7 q^{73} - 3 q^{74} - 19 q^{75} + 11 q^{76} - 13 q^{77} - 7 q^{78} + 5 q^{79} + 2 q^{80} + 11 q^{81} - 11 q^{82} - 11 q^{83} + 2 q^{84} - 11 q^{85} - 7 q^{86} + 7 q^{87} + 6 q^{88} - 10 q^{89} - 2 q^{90} - 24 q^{91} - 11 q^{92} - 12 q^{93} + 37 q^{94} + 2 q^{95} + 11 q^{96} - 24 q^{97} - 23 q^{98} - 6 q^{99}+O(q^{100}) \) 11 * q - 11 * q^2 - 11 * q^3 + 11 * q^4 + 2 * q^5 + 11 * q^6 - 2 * q^7 - 11 * q^8 + 11 * q^9 - 2 * q^10 - 6 * q^11 - 11 * q^12 - 7 * q^13 + 2 * q^14 - 2 * q^15 + 11 * q^16 - 16 * q^17 - 11 * q^18 + 11 * q^19 + 2 * q^20 + 2 * q^21 + 6 * q^22 - 11 * q^23 + 11 * q^24 + 19 * q^25 + 7 * q^26 - 11 * q^27 - 2 * q^28 - 7 * q^29 + 2 * q^30 + 12 * q^31 - 11 * q^32 + 6 * q^33 + 16 * q^34 - 5 * q^35 + 11 * q^36 + 3 * q^37 - 11 * q^38 + 7 * q^39 - 2 * q^40 + 11 * q^41 - 2 * q^42 + 7 * q^43 - 6 * q^44 + 2 * q^45 + 11 * q^46 - 37 * q^47 - 11 * q^48 + 23 * q^49 - 19 * q^50 + 16 * q^51 - 7 * q^52 + 11 * q^53 + 11 * q^54 - 11 * q^55 + 2 * q^56 - 11 * q^57 + 7 * q^58 - 16 * q^59 - 2 * q^60 + 24 * q^61 - 12 * q^62 - 2 * q^63 + 11 * q^64 - 7 * q^65 - 6 * q^66 - 28 * q^67 - 16 * q^68 + 11 * q^69 + 5 * q^70 + 4 * q^71 - 11 * q^72 - 7 * q^73 - 3 * q^74 - 19 * q^75 + 11 * q^76 - 13 * q^77 - 7 * q^78 + 5 * q^79 + 2 * q^80 + 11 * q^81 - 11 * q^82 - 11 * q^83 + 2 * q^84 - 11 * q^85 - 7 * q^86 + 7 * q^87 + 6 * q^88 - 10 * q^89 - 2 * q^90 - 24 * q^91 - 11 * q^92 - 12 * q^93 + 37 * q^94 + 2 * q^95 + 11 * q^96 - 24 * q^97 - 23 * q^98 - 6 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	4.06414	1.81754	0.908769	−	0.417301i	$-0.137024\pi$
$5$	4.06414	1.81754	0.908769	+	0.417301i	$0.137024\pi$
$6$	1.00000	0.408248
$7$	3.98638	1.50671	0.753356	−	0.657613i	$-0.228434\pi$
$7$	3.98638	1.50671	0.753356	+	0.657613i	$0.228434\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−4.06414	−1.28519
$11$	−4.83665	−1.45830	−0.729152	−	0.684352i	$-0.760085\pi$
$11$	−4.83665	−1.45830	−0.729152	+	0.684352i	$0.760085\pi$
$12$	−1.00000	−0.288675
$13$	−4.31849	−1.19773	−0.598866	−	0.800849i	$-0.704382\pi$
$13$	−4.31849	−1.19773	−0.598866	+	0.800849i	$0.704382\pi$
$14$	−3.98638	−1.06541
$15$	−4.06414	−1.04936
$16$	1.00000	0.250000
$17$	−5.40155	−1.31007	−0.655034	−	0.755599i	$-0.727346\pi$
$17$	−5.40155	−1.31007	−0.655034	+	0.755599i	$0.727346\pi$
$18$	−1.00000	−0.235702
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	4.06414	0.908769
$21$	−3.98638	−0.869900
$22$	4.83665	1.03118
$23$	−5.82002	−1.21356	−0.606779	−	0.794871i	$-0.707539\pi$
$23$	−5.82002	−1.21356	−0.606779	+	0.794871i	$0.707539\pi$
$24$	1.00000	0.204124
$25$	11.5172	2.30344
$26$	4.31849	0.846925
$27$	−1.00000	−0.192450
$28$	3.98638	0.753356
$29$	−10.2047	−1.89496	−0.947481	−	0.319811i	$-0.896380\pi$
$29$	−10.2047	−1.89496	−0.947481	+	0.319811i	$0.896380\pi$
$30$	4.06414	0.742006
$31$	5.35251	0.961340	0.480670	−	0.876902i	$-0.340394\pi$
$31$	5.35251	0.961340	0.480670	+	0.876902i	$0.340394\pi$
$32$	−1.00000	−0.176777
$33$	4.83665	0.841952
$34$	5.40155	0.926359
$35$	16.2012	2.73850
$36$	1.00000	0.166667
$37$	−5.45024	−0.896013	−0.448007	−	0.894030i	$-0.647866\pi$
$37$	−5.45024	−0.896013	−0.448007	+	0.894030i	$0.647866\pi$
$38$	−1.00000	−0.162221
$39$	4.31849	0.691511
$40$	−4.06414	−0.642596
$41$	4.22566	0.659938	0.329969	−	0.943992i	$-0.392962\pi$
$41$	4.22566	0.659938	0.329969	+	0.943992i	$0.392962\pi$
$42$	3.98638	0.615112
$43$	11.8821	1.81200	0.906002	−	0.423274i	$-0.139119\pi$
$43$	11.8821	1.81200	0.906002	+	0.423274i	$0.139119\pi$
$44$	−4.83665	−0.729152
$45$	4.06414	0.605846
$46$	5.82002	0.858115
$47$	−2.62661	−0.383131	−0.191566	−	0.981480i	$-0.561356\pi$
$47$	−2.62661	−0.383131	−0.191566	+	0.981480i	$0.561356\pi$
$48$	−1.00000	−0.144338
$49$	8.89126	1.27018
$50$	−11.5172	−1.62878
$51$	5.40155	0.756369
$52$	−4.31849	−0.598866
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	1.00000	0.136083
$55$	−19.6568	−2.65052
$56$	−3.98638	−0.532703
$57$	−1.00000	−0.132453
$58$	10.2047	1.33994
$59$	−2.75182	−0.358256	−0.179128	−	0.983826i	$-0.557328\pi$
$59$	−2.75182	−0.358256	−0.179128	+	0.983826i	$0.557328\pi$
$60$	−4.06414	−0.524678
$61$	5.87161	0.751783	0.375892	−	0.926664i	$-0.377337\pi$
$61$	5.87161	0.751783	0.375892	+	0.926664i	$0.377337\pi$
$62$	−5.35251	−0.679770
$63$	3.98638	0.502237
$64$	1.00000	0.125000
$65$	−17.5509	−2.17692
$66$	−4.83665	−0.595350
$67$	−7.78222	−0.950749	−0.475375	−	0.879783i	$-0.657687\pi$
$67$	−7.78222	−0.950749	−0.475375	+	0.879783i	$0.657687\pi$
$68$	−5.40155	−0.655034
$69$	5.82002	0.700648
$70$	−16.2012	−1.93641
$71$	−6.39344	−0.758761	−0.379381	−	0.925241i	$-0.623863\pi$
$71$	−6.39344	−0.758761	−0.379381	+	0.925241i	$0.623863\pi$
$72$	−1.00000	−0.117851
$73$	−16.1520	−1.89045	−0.945224	−	0.326422i	$-0.894157\pi$
$73$	−16.1520	−1.89045	−0.945224	+	0.326422i	$0.894157\pi$
$74$	5.45024	0.633577
$75$	−11.5172	−1.32989
$76$	1.00000	0.114708
$77$	−19.2807	−2.19724
$78$	−4.31849	−0.488972
$79$	−15.6618	−1.76209	−0.881047	−	0.473028i	$-0.843161\pi$
$79$	−15.6618	−1.76209	−0.881047	+	0.473028i	$0.843161\pi$
$80$	4.06414	0.454384
$81$	1.00000	0.111111
$82$	−4.22566	−0.466646
$83$	0.441737	0.0484869	0.0242434	−	0.999706i	$-0.492282\pi$
$83$	0.441737	0.0484869	0.0242434	+	0.999706i	$0.492282\pi$
$84$	−3.98638	−0.434950
$85$	−21.9526	−2.38110
$86$	−11.8821	−1.28128
$87$	10.2047	1.09406
$88$	4.83665	0.515588
$89$	17.5920	1.86475	0.932374	−	0.361496i	$-0.117734\pi$
$89$	17.5920	1.86475	0.932374	+	0.361496i	$0.117734\pi$
$90$	−4.06414	−0.428398
$91$	−17.2152	−1.80464
$92$	−5.82002	−0.606779
$93$	−5.35251	−0.555030
$94$	2.62661	0.270915
$95$	4.06414	0.416972
$96$	1.00000	0.102062
$97$	−13.7075	−1.39179	−0.695894	−	0.718145i	$-0.744991\pi$
$97$	−13.7075	−1.39179	−0.695894	+	0.718145i	$0.744991\pi$
$98$	−8.89126	−0.898153
$99$	−4.83665	−0.486101
$100$	11.5172	1.15172
$101$	2.70800	0.269456	0.134728	−	0.990883i	$-0.456984\pi$
$101$	2.70800	0.269456	0.134728	+	0.990883i	$0.456984\pi$
$102$	−5.40155	−0.534833
$103$	8.35848	0.823586	0.411793	−	0.911277i	$-0.364903\pi$
$103$	8.35848	0.823586	0.411793	+	0.911277i	$0.364903\pi$
$104$	4.31849	0.423463
$105$	−16.2012	−1.58108
$106$	−1.00000	−0.0971286
$107$	−11.9475	−1.15501	−0.577504	−	0.816388i	$-0.695973\pi$
$107$	−11.9475	−1.15501	−0.577504	+	0.816388i	$0.695973\pi$
$108$	−1.00000	−0.0962250
$109$	3.45593	0.331018	0.165509	−	0.986208i	$-0.447073\pi$
$109$	3.45593	0.331018	0.165509	+	0.986208i	$0.447073\pi$
$110$	19.6568	1.87420
$111$	5.45024	0.517313
$112$	3.98638	0.376678
$113$	16.7904	1.57951	0.789755	−	0.613423i	$-0.210208\pi$
$113$	16.7904	1.57951	0.789755	+	0.613423i	$0.210208\pi$
$114$	1.00000	0.0936586
$115$	−23.6533	−2.20569
$116$	−10.2047	−0.947481
$117$	−4.31849	−0.399244
$118$	2.75182	0.253326
$119$	−21.5327	−1.97390
$120$	4.06414	0.371003
$121$	12.3932	1.12665
$122$	−5.87161	−0.531591
$123$	−4.22566	−0.381015
$124$	5.35251	0.480670
$125$	26.4868	2.36905
$126$	−3.98638	−0.355135
$127$	−4.73489	−0.420153	−0.210077	−	0.977685i	$-0.567371\pi$
$127$	−4.73489	−0.420153	−0.210077	+	0.977685i	$0.567371\pi$
$128$	−1.00000	−0.0883883
$129$	−11.8821	−1.04616
$130$	17.5509	1.53932
$131$	−18.9772	−1.65804	−0.829021	−	0.559217i	$-0.811102\pi$
$131$	−18.9772	−1.65804	−0.829021	+	0.559217i	$0.811102\pi$
$132$	4.83665	0.420976
$133$	3.98638	0.345663
$134$	7.78222	0.672281
$135$	−4.06414	−0.349785
$136$	5.40155	0.463179
$137$	−6.22073	−0.531473	−0.265737	−	0.964046i	$-0.585615\pi$
$137$	−6.22073	−0.531473	−0.265737	+	0.964046i	$0.585615\pi$
$138$	−5.82002	−0.495433
$139$	19.0796	1.61831	0.809154	−	0.587597i	$-0.199926\pi$
$139$	19.0796	1.61831	0.809154	+	0.587597i	$0.199926\pi$
$140$	16.2012	1.36925
$141$	2.62661	0.221201
$142$	6.39344	0.536525
$143$	20.8870	1.74666
$144$	1.00000	0.0833333
$145$	−41.4732	−3.44416
$146$	16.1520	1.33675
$147$	−8.89126	−0.733339
$148$	−5.45024	−0.448007
$149$	−11.8280	−0.968991	−0.484496	−	0.874794i	$-0.660997\pi$
$149$	−11.8280	−0.968991	−0.484496	+	0.874794i	$0.660997\pi$
$150$	11.5172	0.940376
$151$	−7.96637	−0.648294	−0.324147	−	0.946007i	$-0.605077\pi$
$151$	−7.96637	−0.648294	−0.324147	+	0.946007i	$0.605077\pi$
$152$	−1.00000	−0.0811107
$153$	−5.40155	−0.436690
$154$	19.2807	1.55369
$155$	21.7533	1.74727
$156$	4.31849	0.345756
$157$	−2.81312	−0.224512	−0.112256	−	0.993679i	$-0.535808\pi$
$157$	−2.81312	−0.224512	−0.112256	+	0.993679i	$0.535808\pi$
$158$	15.6618	1.24599
$159$	−1.00000	−0.0793052
$160$	−4.06414	−0.321298
$161$	−23.2008	−1.82848
$162$	−1.00000	−0.0785674
$163$	−0.650159	−0.0509244	−0.0254622	−	0.999676i	$-0.508106\pi$
$163$	−0.650159	−0.0509244	−0.0254622	+	0.999676i	$0.508106\pi$
$164$	4.22566	0.329969
$165$	19.6568	1.53028
$166$	−0.441737	−0.0342854
$167$	−11.7727	−0.911000	−0.455500	−	0.890236i	$-0.650540\pi$
$167$	−11.7727	−0.911000	−0.455500	+	0.890236i	$0.650540\pi$
$168$	3.98638	0.307556
$169$	5.64934	0.434564
$170$	21.9526	1.68369
$171$	1.00000	0.0764719
$172$	11.8821	0.906002
$173$	2.01884	0.153490	0.0767448	−	0.997051i	$-0.475547\pi$
$173$	2.01884	0.153490	0.0767448	+	0.997051i	$0.475547\pi$
$174$	−10.2047	−0.773615
$175$	45.9120	3.47062
$176$	−4.83665	−0.364576
$177$	2.75182	0.206839
$178$	−17.5920	−1.31858
$179$	−19.3756	−1.44820	−0.724099	−	0.689696i	$-0.757744\pi$
$179$	−19.3756	−1.44820	−0.724099	+	0.689696i	$0.757744\pi$
$180$	4.06414	0.302923
$181$	−2.33353	−0.173450	−0.0867249	−	0.996232i	$-0.527640\pi$
$181$	−2.33353	−0.173450	−0.0867249	+	0.996232i	$0.527640\pi$
$182$	17.2152	1.27607
$183$	−5.87161	−0.434042
$184$	5.82002	0.429057
$185$	−22.1505	−1.62854
$186$	5.35251	0.392465
$187$	26.1254	1.91048
$188$	−2.62661	−0.191566
$189$	−3.98638	−0.289967
$190$	−4.06414	−0.294843
$191$	−6.24094	−0.451578	−0.225789	−	0.974176i	$-0.572496\pi$
$191$	−6.24094	−0.451578	−0.225789	+	0.974176i	$0.572496\pi$
$192$	−1.00000	−0.0721688
$193$	−18.8880	−1.35959	−0.679794	−	0.733403i	$-0.737931\pi$
$193$	−18.8880	−1.35959	−0.679794	+	0.733403i	$0.737931\pi$
$194$	13.7075	0.984142
$195$	17.5509	1.25685
$196$	8.89126	0.635090
$197$	9.81626	0.699379	0.349690	−	0.936866i	$-0.386287\pi$
$197$	9.81626	0.699379	0.349690	+	0.936866i	$0.386287\pi$
$198$	4.83665	0.343726
$199$	−16.9932	−1.20462	−0.602309	−	0.798263i	$-0.705752\pi$
$199$	−16.9932	−1.20462	−0.602309	+	0.798263i	$0.705752\pi$
$200$	−11.5172	−0.814389
$201$	7.78222	0.548915
$202$	−2.70800	−0.190534
$203$	−40.6798	−2.85516
$204$	5.40155	0.378184
$205$	17.1737	1.19946
$206$	−8.35848	−0.582363
$207$	−5.82002	−0.404519
$208$	−4.31849	−0.299433
$209$	−4.83665	−0.334558
$210$	16.2012	1.11799
$211$	−20.1443	−1.38679	−0.693396	−	0.720557i	$-0.743886\pi$
$211$	−20.1443	−1.38679	−0.693396	+	0.720557i	$0.743886\pi$
$212$	1.00000	0.0686803
$213$	6.39344	0.438071
$214$	11.9475	0.816713
$215$	48.2905	3.29338
$216$	1.00000	0.0680414
$217$	21.3372	1.44846
$218$	−3.45593	−0.234065
$219$	16.1520	1.09145
$220$	−19.6568	−1.32526
$221$	23.3265	1.56911
$222$	−5.45024	−0.365796
$223$	−1.54909	−0.103735	−0.0518675	−	0.998654i	$-0.516517\pi$
$223$	−1.54909	−0.103735	−0.0518675	+	0.998654i	$0.516517\pi$
$224$	−3.98638	−0.266352
$225$	11.5172	0.767814
$226$	−16.7904	−1.11688
$227$	−3.56380	−0.236538	−0.118269	−	0.992982i	$-0.537735\pi$
$227$	−3.56380	−0.236538	−0.118269	+	0.992982i	$0.537735\pi$
$228$	−1.00000	−0.0662266
$229$	28.4053	1.87708	0.938538	−	0.345175i	$-0.112181\pi$
$229$	28.4053	1.87708	0.938538	+	0.345175i	$0.112181\pi$
$230$	23.6533	1.55966
$231$	19.2807	1.26858
$232$	10.2047	0.669970
$233$	9.31482	0.610234	0.305117	−	0.952315i	$-0.401304\pi$
$233$	9.31482	0.610234	0.305117	+	0.952315i	$0.401304\pi$
$234$	4.31849	0.282308
$235$	−10.6749	−0.696355
$236$	−2.75182	−0.179128
$237$	15.6618	1.01735
$238$	21.5327	1.39576
$239$	12.9830	0.839803	0.419902	−	0.907570i	$-0.362065\pi$
$239$	12.9830	0.839803	0.419902	+	0.907570i	$0.362065\pi$
$240$	−4.06414	−0.262339
$241$	9.71605	0.625866	0.312933	−	0.949775i	$-0.398688\pi$
$241$	9.71605	0.625866	0.312933	+	0.949775i	$0.398688\pi$
$242$	−12.3932	−0.796663
$243$	−1.00000	−0.0641500
$244$	5.87161	0.375892
$245$	36.1353	2.30860
$246$	4.22566	0.269418
$247$	−4.31849	−0.274779
$248$	−5.35251	−0.339885
$249$	−0.441737	−0.0279939
$250$	−26.4868	−1.67517
$251$	−15.1387	−0.955545	−0.477773	−	0.878484i	$-0.658556\pi$
$251$	−15.1387	−0.955545	−0.477773	+	0.878484i	$0.658556\pi$
$252$	3.98638	0.251119
$253$	28.1494	1.76974
$254$	4.73489	0.297093
$255$	21.9526	1.37473
$256$	1.00000	0.0625000
$257$	8.64346	0.539164	0.269582	−	0.962977i	$-0.413114\pi$
$257$	8.64346	0.539164	0.269582	+	0.962977i	$0.413114\pi$
$258$	11.8821	0.739747
$259$	−21.7267	−1.35003
$260$	−17.5509	−1.08846
$261$	−10.2047	−0.631654
$262$	18.9772	1.17241
$263$	−28.9284	−1.78380	−0.891902	−	0.452228i	$-0.850629\pi$
$263$	−28.9284	−1.78380	−0.891902	+	0.452228i	$0.850629\pi$
$264$	−4.83665	−0.297675
$265$	4.06414	0.249658
$266$	−3.98638	−0.244421
$267$	−17.5920	−1.07661
$268$	−7.78222	−0.475375
$269$	0.513136	0.0312864	0.0156432	−	0.999878i	$-0.495020\pi$
$269$	0.513136	0.0312864	0.0156432	+	0.999878i	$0.495020\pi$
$270$	4.06414	0.247335
$271$	23.0655	1.40113	0.700565	−	0.713588i	$-0.252931\pi$
$271$	23.0655	1.40113	0.700565	+	0.713588i	$0.252931\pi$
$272$	−5.40155	−0.327517
$273$	17.2152	1.04191
$274$	6.22073	0.375808
$275$	−55.7047	−3.35912
$276$	5.82002	0.350324
$277$	3.36234	0.202024	0.101012	−	0.994885i	$-0.467792\pi$
$277$	3.36234	0.202024	0.101012	+	0.994885i	$0.467792\pi$
$278$	−19.0796	−1.14432
$279$	5.35251	0.320447
$280$	−16.2012	−0.968207
$281$	−5.06448	−0.302122	−0.151061	−	0.988524i	$-0.548269\pi$
$281$	−5.06448	−0.302122	−0.151061	+	0.988524i	$0.548269\pi$
$282$	−2.62661	−0.156413
$283$	−27.4233	−1.63015	−0.815073	−	0.579359i	$-0.803303\pi$
$283$	−27.4233	−1.63015	−0.815073	+	0.579359i	$0.803303\pi$
$284$	−6.39344	−0.379381
$285$	−4.06414	−0.240739
$286$	−20.8870	−1.23507
$287$	16.8451	0.994336
$288$	−1.00000	−0.0589256
$289$	12.1768	0.716281
$290$	41.4732	2.43539
$291$	13.7075	0.803549
$292$	−16.1520	−0.945224
$293$	−3.62708	−0.211896	−0.105948	−	0.994372i	$-0.533788\pi$
$293$	−3.62708	−0.211896	−0.105948	+	0.994372i	$0.533788\pi$
$294$	8.89126	0.518549
$295$	−11.1838	−0.651144
$296$	5.45024	0.316788
$297$	4.83665	0.280651
$298$	11.8280	0.685180
$299$	25.1337	1.45352
$300$	−11.5172	−0.664946
$301$	47.3666	2.73017
$302$	7.96637	0.458413
$303$	−2.70800	−0.155570
$304$	1.00000	0.0573539
$305$	23.8630	1.36639
$306$	5.40155	0.308786
$307$	−11.2165	−0.640157	−0.320078	−	0.947391i	$-0.603709\pi$
$307$	−11.2165	−0.640157	−0.320078	+	0.947391i	$0.603709\pi$
$308$	−19.2807	−1.09862
$309$	−8.35848	−0.475497
$310$	−21.7533	−1.23551
$311$	21.0056	1.19112	0.595560	−	0.803311i	$-0.296930\pi$
$311$	21.0056	1.19112	0.595560	+	0.803311i	$0.296930\pi$
$312$	−4.31849	−0.244486
$313$	−4.68076	−0.264572	−0.132286	−	0.991212i	$-0.542232\pi$
$313$	−4.68076	−0.264572	−0.132286	+	0.991212i	$0.542232\pi$
$314$	2.81312	0.158754
$315$	16.2012	0.912835
$316$	−15.6618	−0.881047
$317$	−5.51208	−0.309589	−0.154795	−	0.987947i	$-0.549472\pi$
$317$	−5.51208	−0.309589	−0.154795	+	0.987947i	$0.549472\pi$
$318$	1.00000	0.0560772
$319$	49.3565	2.76343
$320$	4.06414	0.227192
$321$	11.9475	0.666844
$322$	23.2008	1.29293
$323$	−5.40155	−0.300550
$324$	1.00000	0.0555556
$325$	−49.7369	−2.75891
$326$	0.650159	0.0360090
$327$	−3.45593	−0.191113
$328$	−4.22566	−0.233323
$329$	−10.4707	−0.577268
$330$	−19.6568	−1.08207
$331$	−13.3675	−0.734746	−0.367373	−	0.930074i	$-0.619743\pi$
$331$	−13.3675	−0.734746	−0.367373	+	0.930074i	$0.619743\pi$
$332$	0.441737	0.0242434
$333$	−5.45024	−0.298671
$334$	11.7727	0.644175
$335$	−31.6280	−1.72802
$336$	−3.98638	−0.217475
$337$	13.7890	0.751133	0.375567	−	0.926795i	$-0.377448\pi$
$337$	13.7890	0.751133	0.375567	+	0.926795i	$0.377448\pi$
$338$	−5.64934	−0.307283
$339$	−16.7904	−0.911930
$340$	−21.9526	−1.19055
$341$	−25.8882	−1.40193
$342$	−1.00000	−0.0540738
$343$	7.53930	0.407084
$344$	−11.8821	−0.640640
$345$	23.6533	1.27345
$346$	−2.01884	−0.108533
$347$	18.7091	1.00436	0.502178	−	0.864764i	$-0.332532\pi$
$347$	18.7091	1.00436	0.502178	+	0.864764i	$0.332532\pi$
$348$	10.2047	0.547029
$349$	−6.15438	−0.329437	−0.164718	−	0.986341i	$-0.552671\pi$
$349$	−6.15438	−0.329437	−0.164718	+	0.986341i	$0.552671\pi$
$350$	−45.9120	−2.45410
$351$	4.31849	0.230504
$352$	4.83665	0.257794
$353$	18.6337	0.991773	0.495886	−	0.868387i	$-0.334843\pi$
$353$	18.6337	0.991773	0.495886	+	0.868387i	$0.334843\pi$
$354$	−2.75182	−0.146258
$355$	−25.9838	−1.37908
$356$	17.5920	0.932374
$357$	21.5327	1.13963
$358$	19.3756	1.02403
$359$	−9.75552	−0.514877	−0.257438	−	0.966295i	$-0.582878\pi$
$359$	−9.75552	−0.514877	−0.257438	+	0.966295i	$0.582878\pi$
$360$	−4.06414	−0.214199
$361$	1.00000	0.0526316
$362$	2.33353	0.122648
$363$	−12.3932	−0.650473
$364$	−17.2152	−0.902319
$365$	−65.6439	−3.43596
$366$	5.87161	0.306914
$367$	−29.1451	−1.52136	−0.760682	−	0.649125i	$-0.775135\pi$
$367$	−29.1451	−1.52136	−0.760682	+	0.649125i	$0.775135\pi$
$368$	−5.82002	−0.303389
$369$	4.22566	0.219979
$370$	22.1505	1.15155
$371$	3.98638	0.206963
$372$	−5.35251	−0.277515
$373$	10.7578	0.557018	0.278509	−	0.960434i	$-0.410160\pi$
$373$	10.7578	0.557018	0.278509	+	0.960434i	$0.410160\pi$
$374$	−26.1254	−1.35091
$375$	−26.4868	−1.36777
$376$	2.62661	0.135457
$377$	44.0688	2.26966
$378$	3.98638	0.205037
$379$	−6.50927	−0.334359	−0.167179	−	0.985926i	$-0.553466\pi$
$379$	−6.50927	−0.334359	−0.167179	+	0.985926i	$0.553466\pi$
$380$	4.06414	0.208486
$381$	4.73489	0.242576
$382$	6.24094	0.319314
$383$	7.68981	0.392931	0.196465	−	0.980511i	$-0.437054\pi$
$383$	7.68981	0.392931	0.196465	+	0.980511i	$0.437054\pi$
$384$	1.00000	0.0510310
$385$	−78.3596	−3.99357
$386$	18.8880	0.961374
$387$	11.8821	0.604001
$388$	−13.7075	−0.695894
$389$	−16.0872	−0.815655	−0.407827	−	0.913059i	$-0.633713\pi$
$389$	−16.0872	−0.815655	−0.407827	+	0.913059i	$0.633713\pi$
$390$	−17.5509	−0.888726
$391$	31.4371	1.58984
$392$	−8.89126	−0.449077
$393$	18.9772	0.957271
$394$	−9.81626	−0.494536
$395$	−63.6519	−3.20267
$396$	−4.83665	−0.243051
$397$	20.3431	1.02099	0.510495	−	0.859881i	$-0.329462\pi$
$397$	20.3431	1.02099	0.510495	+	0.859881i	$0.329462\pi$
$398$	16.9932	0.851793
$399$	−3.98638	−0.199569
$400$	11.5172	0.575860
$401$	18.7938	0.938517	0.469258	−	0.883061i	$-0.344521\pi$
$401$	18.7938	0.938517	0.469258	+	0.883061i	$0.344521\pi$
$402$	−7.78222	−0.388142
$403$	−23.1148	−1.15143
$404$	2.70800	0.134728
$405$	4.06414	0.201949
$406$	40.6798	2.01890
$407$	26.3609	1.30666
$408$	−5.40155	−0.267417
$409$	35.9036	1.77532	0.887661	−	0.460498i	$-0.152329\pi$
$409$	35.9036	1.77532	0.887661	+	0.460498i	$0.152329\pi$
$410$	−17.1737	−0.848147
$411$	6.22073	0.306846
$412$	8.35848	0.411793
$413$	−10.9698	−0.539789
$414$	5.82002	0.286038
$415$	1.79528	0.0881267
$416$	4.31849	0.211731
$417$	−19.0796	−0.934330
$418$	4.83665	0.236568
$419$	−1.88387	−0.0920329	−0.0460165	−	0.998941i	$-0.514653\pi$
$419$	−1.88387	−0.0920329	−0.0460165	+	0.998941i	$0.514653\pi$
$420$	−16.2012	−0.790538
$421$	−28.4344	−1.38581	−0.692904	−	0.721030i	$-0.743669\pi$
$421$	−28.4344	−1.38581	−0.692904	+	0.721030i	$0.743669\pi$
$422$	20.1443	0.980610
$423$	−2.62661	−0.127710
$424$	−1.00000	−0.0485643
$425$	−62.2108	−3.01767
$426$	−6.39344	−0.309763
$427$	23.4065	1.13272
$428$	−11.9475	−0.577504
$429$	−20.8870	−1.00843
$430$	−48.2905	−2.32877
$431$	−6.90149	−0.332433	−0.166216	−	0.986089i	$-0.553155\pi$
$431$	−6.90149	−0.332433	−0.166216	+	0.986089i	$0.553155\pi$
$432$	−1.00000	−0.0481125
$433$	−11.3429	−0.545105	−0.272552	−	0.962141i	$-0.587868\pi$
$433$	−11.3429	−0.545105	−0.272552	+	0.962141i	$0.587868\pi$
$434$	−21.3372	−1.02422
$435$	41.4732	1.98849
$436$	3.45593	0.165509
$437$	−5.82002	−0.278409
$438$	−16.1520	−0.771772
$439$	−21.1503	−1.00945	−0.504724	−	0.863281i	$-0.668406\pi$
$439$	−21.1503	−1.00945	−0.504724	+	0.863281i	$0.668406\pi$
$440$	19.6568	0.937101
$441$	8.89126	0.423393
$442$	−23.3265	−1.10953
$443$	−19.5875	−0.930630	−0.465315	−	0.885145i	$-0.654059\pi$
$443$	−19.5875	−0.930630	−0.465315	+	0.885145i	$0.654059\pi$
$444$	5.45024	0.258657
$445$	71.4962	3.38925
$446$	1.54909	0.0733517
$447$	11.8280	0.559447
$448$	3.98638	0.188339
$449$	−23.6299	−1.11516	−0.557581	−	0.830122i	$-0.688270\pi$
$449$	−23.6299	−1.11516	−0.557581	+	0.830122i	$0.688270\pi$
$450$	−11.5172	−0.542926
$451$	−20.4380	−0.962390
$452$	16.7904	0.789755
$453$	7.96637	0.374293
$454$	3.56380	0.167258
$455$	−69.9647	−3.28000
$456$	1.00000	0.0468293
$457$	8.46055	0.395768	0.197884	−	0.980225i	$-0.436593\pi$
$457$	8.46055	0.395768	0.197884	+	0.980225i	$0.436593\pi$
$458$	−28.4053	−1.32729
$459$	5.40155	0.252123
$460$	−23.6533	−1.10284
$461$	−26.8590	−1.25095	−0.625475	−	0.780244i	$-0.715095\pi$
$461$	−26.8590	−1.25095	−0.625475	+	0.780244i	$0.715095\pi$
$462$	−19.2807	−0.897021
$463$	0.416670	0.0193643	0.00968215	−	0.999953i	$-0.496918\pi$
$463$	0.416670	0.0193643	0.00968215	+	0.999953i	$0.496918\pi$
$464$	−10.2047	−0.473741
$465$	−21.7533	−1.00879
$466$	−9.31482	−0.431501
$467$	−6.43132	−0.297606	−0.148803	−	0.988867i	$-0.547542\pi$
$467$	−6.43132	−0.297606	−0.148803	+	0.988867i	$0.547542\pi$
$468$	−4.31849	−0.199622
$469$	−31.0229	−1.43251
$470$	10.6749	0.492397
$471$	2.81312	0.129622
$472$	2.75182	0.126663
$473$	−57.4695	−2.64245
$474$	−15.6618	−0.719372
$475$	11.5172	0.528446
$476$	−21.5327	−0.986948
$477$	1.00000	0.0457869
$478$	−12.9830	−0.593831
$479$	11.1911	0.511336	0.255668	−	0.966765i	$-0.417705\pi$
$479$	11.1911	0.511336	0.255668	+	0.966765i	$0.417705\pi$
$480$	4.06414	0.185502
$481$	23.5368	1.07318
$482$	−9.71605	−0.442554
$483$	23.2008	1.05567
$484$	12.3932	0.563326
$485$	−55.7092	−2.52962
$486$	1.00000	0.0453609
$487$	41.3110	1.87198	0.935990	−	0.352027i	$-0.114507\pi$
$487$	41.3110	1.87198	0.935990	+	0.352027i	$0.114507\pi$
$488$	−5.87161	−0.265795
$489$	0.650159	0.0294012
$490$	−36.1353	−1.63243
$491$	−6.38926	−0.288343	−0.144172	−	0.989553i	$-0.546052\pi$
$491$	−6.38926	−0.288343	−0.144172	+	0.989553i	$0.546052\pi$
$492$	−4.22566	−0.190508
$493$	55.1211	2.48253
$494$	4.31849	0.194298
$495$	−19.6568	−0.883507
$496$	5.35251	0.240335
$497$	−25.4867	−1.14323
$498$	0.441737	0.0197947
$499$	−12.9201	−0.578383	−0.289191	−	0.957271i	$-0.593386\pi$
$499$	−12.9201	−0.578383	−0.289191	+	0.957271i	$0.593386\pi$
$500$	26.4868	1.18453
$501$	11.7727	0.525966
$502$	15.1387	0.675672
$503$	13.4389	0.599211	0.299605	−	0.954063i	$-0.403145\pi$
$503$	13.4389	0.599211	0.299605	+	0.954063i	$0.403145\pi$
$504$	−3.98638	−0.177568
$505$	11.0057	0.489746
$506$	−28.1494	−1.25139
$507$	−5.64934	−0.250896
$508$	−4.73489	−0.210077
$509$	31.4105	1.39225	0.696124	−	0.717922i	$-0.254907\pi$
$509$	31.4105	1.39225	0.696124	+	0.717922i	$0.254907\pi$
$510$	−21.9526	−0.972079
$511$	−64.3881	−2.84836
$512$	−1.00000	−0.0441942
$513$	−1.00000	−0.0441511
$514$	−8.64346	−0.381247
$515$	33.9700	1.49690
$516$	−11.8821	−0.523080
$517$	12.7040	0.558722
$518$	21.7267	0.954618
$519$	−2.01884	−0.0886172
$520$	17.5509	0.769659
$521$	−11.5151	−0.504484	−0.252242	−	0.967664i	$-0.581168\pi$
$521$	−11.5151	−0.504484	−0.252242	+	0.967664i	$0.581168\pi$
$522$	10.2047	0.446647
$523$	5.34433	0.233691	0.116846	−	0.993150i	$-0.462722\pi$
$523$	5.34433	0.233691	0.116846	+	0.993150i	$0.462722\pi$
$524$	−18.9772	−0.829021
$525$	−45.9120	−2.00376
$526$	28.9284	1.26134
$527$	−28.9119	−1.25942
$528$	4.83665	0.210488
$529$	10.8726	0.472722
$530$	−4.06414	−0.176535
$531$	−2.75182	−0.119419
$532$	3.98638	0.172832
$533$	−18.2485	−0.790429
$534$	17.5920	0.761280
$535$	−48.5562	−2.09927
$536$	7.78222	0.336141
$537$	19.3756	0.836118
$538$	−0.513136	−0.0221229
$539$	−43.0039	−1.85231
$540$	−4.06414	−0.174893
$541$	1.15553	0.0496800	0.0248400	−	0.999691i	$-0.492092\pi$
$541$	1.15553	0.0496800	0.0248400	+	0.999691i	$0.492092\pi$
$542$	−23.0655	−0.990749
$543$	2.33353	0.100141
$544$	5.40155	0.231590
$545$	14.0454	0.601638
$546$	−17.2152	−0.736741
$547$	30.5199	1.30494	0.652468	−	0.757816i	$-0.273733\pi$
$547$	30.5199	1.30494	0.652468	+	0.757816i	$0.273733\pi$
$548$	−6.22073	−0.265737
$549$	5.87161	0.250594
$550$	55.7047	2.37525
$551$	−10.2047	−0.434734
$552$	−5.82002	−0.247716
$553$	−62.4341	−2.65497
$554$	−3.36234	−0.142852
$555$	22.1505	0.940236
$556$	19.0796	0.809154
$557$	−16.8295	−0.713089	−0.356545	−	0.934278i	$-0.616045\pi$
$557$	−16.8295	−0.713089	−0.356545	+	0.934278i	$0.616045\pi$
$558$	−5.35251	−0.226590
$559$	−51.3127	−2.17030
$560$	16.2012	0.684626
$561$	−26.1254	−1.10302
$562$	5.06448	0.213632
$563$	18.7495	0.790196	0.395098	−	0.918639i	$-0.370711\pi$
$563$	18.7495	0.790196	0.395098	+	0.918639i	$0.370711\pi$
$564$	2.62661	0.110600
$565$	68.2385	2.87082
$566$	27.4233	1.15269
$567$	3.98638	0.167412
$568$	6.39344	0.268263
$569$	11.6178	0.487044	0.243522	−	0.969895i	$-0.421697\pi$
$569$	11.6178	0.487044	0.243522	+	0.969895i	$0.421697\pi$
$570$	4.06414	0.170228
$571$	−29.6634	−1.24138	−0.620688	−	0.784058i	$-0.713147\pi$
$571$	−29.6634	−1.24138	−0.620688	+	0.784058i	$0.713147\pi$
$572$	20.8870	0.873330
$573$	6.24094	0.260719
$574$	−16.8451	−0.703101
$575$	−67.0303	−2.79536
$576$	1.00000	0.0416667
$577$	31.4499	1.30928	0.654639	−	0.755942i	$-0.272821\pi$
$577$	31.4499	1.30928	0.654639	+	0.755942i	$0.272821\pi$
$578$	−12.1768	−0.506487
$579$	18.8880	0.784959
$580$	−41.4732	−1.72208
$581$	1.76093	0.0730558
$582$	−13.7075	−0.568195
$583$	−4.83665	−0.200314
$584$	16.1520	0.668375
$585$	−17.5509	−0.725641
$586$	3.62708	0.149833
$587$	25.0920	1.03566	0.517828	−	0.855485i	$-0.326741\pi$
$587$	25.0920	1.03566	0.517828	+	0.855485i	$0.326741\pi$
$588$	−8.89126	−0.366669
$589$	5.35251	0.220546
$590$	11.1838	0.460429
$591$	−9.81626	−0.403787
$592$	−5.45024	−0.224003
$593$	15.9767	0.656086	0.328043	−	0.944663i	$-0.393611\pi$
$593$	15.9767	0.656086	0.328043	+	0.944663i	$0.393611\pi$
$594$	−4.83665	−0.198450
$595$	−87.5117	−3.58763
$596$	−11.8280	−0.484496
$597$	16.9932	0.695486
$598$	−25.1337	−1.02779
$599$	36.2020	1.47918	0.739588	−	0.673060i	$-0.235020\pi$
$599$	36.2020	1.47918	0.739588	+	0.673060i	$0.235020\pi$
$600$	11.5172	0.470188
$601$	36.4930	1.48858	0.744290	−	0.667856i	$-0.232788\pi$
$601$	36.4930	1.48858	0.744290	+	0.667856i	$0.232788\pi$
$602$	−47.3666	−1.93052
$603$	−7.78222	−0.316916
$604$	−7.96637	−0.324147
$605$	50.3675	2.04773
$606$	2.70800	0.110005
$607$	−12.1491	−0.493117	−0.246558	−	0.969128i	$-0.579300\pi$
$607$	−12.1491	−0.493117	−0.246558	+	0.969128i	$0.579300\pi$
$608$	−1.00000	−0.0405554
$609$	40.6798	1.64843
$610$	−23.8630	−0.966186
$611$	11.3430	0.458889
$612$	−5.40155	−0.218345
$613$	−10.5801	−0.427325	−0.213663	−	0.976908i	$-0.568539\pi$
$613$	−10.5801	−0.427325	−0.213663	+	0.976908i	$0.568539\pi$
$614$	11.2165	0.452659
$615$	−17.1737	−0.692509
$616$	19.2807	0.776843
$617$	−2.45694	−0.0989127	−0.0494564	−	0.998776i	$-0.515749\pi$
$617$	−2.45694	−0.0989127	−0.0494564	+	0.998776i	$0.515749\pi$
$618$	8.35848	0.336227
$619$	−3.91703	−0.157439	−0.0787195	−	0.996897i	$-0.525083\pi$
$619$	−3.91703	−0.157439	−0.0787195	+	0.996897i	$0.525083\pi$
$620$	21.7533	0.873635
$621$	5.82002	0.233549
$622$	−21.0056	−0.842249
$623$	70.1284	2.80964
$624$	4.31849	0.172878
$625$	50.0600	2.00240
$626$	4.68076	0.187081
$627$	4.83665	0.193157
$628$	−2.81312	−0.112256
$629$	29.4397	1.17384
$630$	−16.2012	−0.645472
$631$	−18.8545	−0.750587	−0.375294	−	0.926906i	$-0.622458\pi$
$631$	−18.8545	−0.750587	−0.375294	+	0.926906i	$0.622458\pi$
$632$	15.6618	0.622995
$633$	20.1443	0.800665
$634$	5.51208	0.218913
$635$	−19.2432	−0.763644
$636$	−1.00000	−0.0396526
$637$	−38.3968	−1.52134
$638$	−49.3565	−1.95404
$639$	−6.39344	−0.252920
$640$	−4.06414	−0.160649
$641$	−31.0943	−1.22815	−0.614075	−	0.789247i	$-0.710471\pi$
$641$	−31.0943	−1.22815	−0.614075	+	0.789247i	$0.710471\pi$
$642$	−11.9475	−0.471530
$643$	−34.7950	−1.37218	−0.686090	−	0.727516i	$-0.740675\pi$
$643$	−34.7950	−1.37218	−0.686090	+	0.727516i	$0.740675\pi$
$644$	−23.2008	−0.914241
$645$	−48.2905	−1.90144
$646$	5.40155	0.212521
$647$	26.7292	1.05084	0.525418	−	0.850845i	$-0.323909\pi$
$647$	26.7292	1.05084	0.525418	+	0.850845i	$0.323909\pi$
$648$	−1.00000	−0.0392837
$649$	13.3096	0.522447
$650$	49.7369	1.95084
$651$	−21.3372	−0.836270
$652$	−0.650159	−0.0254622
$653$	−15.9912	−0.625784	−0.312892	−	0.949789i	$-0.601298\pi$
$653$	−15.9912	−0.625784	−0.312892	+	0.949789i	$0.601298\pi$
$654$	3.45593	0.135138
$655$	−77.1258	−3.01355
$656$	4.22566	0.164984
$657$	−16.1520	−0.630150
$658$	10.4707	0.408190
$659$	−4.16095	−0.162088	−0.0810439	−	0.996711i	$-0.525825\pi$
$659$	−4.16095	−0.162088	−0.0810439	+	0.996711i	$0.525825\pi$
$660$	19.6568	0.765140
$661$	31.0689	1.20844	0.604220	−	0.796818i	$-0.293485\pi$
$661$	31.0689	1.20844	0.604220	+	0.796818i	$0.293485\pi$
$662$	13.3675	0.519544
$663$	−23.3265	−0.905928
$664$	−0.441737	−0.0171427
$665$	16.2012	0.628256
$666$	5.45024	0.211192
$667$	59.3914	2.29965
$668$	−11.7727	−0.455500
$669$	1.54909	0.0598914
$670$	31.6280	1.22190
$671$	−28.3989	−1.09633
$672$	3.98638	0.153778
$673$	2.80381	0.108079	0.0540394	−	0.998539i	$-0.482790\pi$
$673$	2.80381	0.108079	0.0540394	+	0.998539i	$0.482790\pi$
$674$	−13.7890	−0.531131
$675$	−11.5172	−0.443297
$676$	5.64934	0.217282
$677$	−22.9053	−0.880321	−0.440161	−	0.897919i	$-0.645079\pi$
$677$	−22.9053	−0.880321	−0.440161	+	0.897919i	$0.645079\pi$
$678$	16.7904	0.644832
$679$	−54.6434	−2.09702
$680$	21.9526	0.841846
$681$	3.56380	0.136565
$682$	25.8882	0.991311
$683$	25.8715	0.989946	0.494973	−	0.868908i	$-0.335178\pi$
$683$	25.8715	0.989946	0.494973	+	0.868908i	$0.335178\pi$
$684$	1.00000	0.0382360
$685$	−25.2819	−0.965972
$686$	−7.53930	−0.287852
$687$	−28.4053	−1.08373
$688$	11.8821	0.453001
$689$	−4.31849	−0.164521
$690$	−23.6533	−0.900467
$691$	31.7010	1.20596	0.602982	−	0.797755i	$-0.293979\pi$
$691$	31.7010	1.20596	0.602982	+	0.797755i	$0.293979\pi$
$692$	2.01884	0.0767448
$693$	−19.2807	−0.732415
$694$	−18.7091	−0.710187
$695$	77.5420	2.94133
$696$	−10.2047	−0.386808
$697$	−22.8251	−0.864564
$698$	6.15438	0.232947
$699$	−9.31482	−0.352319
$700$	45.9120	1.73531
$701$	39.3240	1.48525	0.742623	−	0.669710i	$-0.233582\pi$
$701$	39.3240	1.48525	0.742623	+	0.669710i	$0.233582\pi$
$702$	−4.31849	−0.162991
$703$	−5.45024	−0.205560
$704$	−4.83665	−0.182288
$705$	10.6749	0.402041
$706$	−18.6337	−0.701289
$707$	10.7951	0.405993
$708$	2.75182	0.103420
$709$	46.9623	1.76371	0.881853	−	0.471525i	$-0.156296\pi$
$709$	46.9623	1.76371	0.881853	+	0.471525i	$0.156296\pi$
$710$	25.9838	0.975155
$711$	−15.6618	−0.587365
$712$	−17.5920	−0.659288
$713$	−31.1517	−1.16664
$714$	−21.5327	−0.805840
$715$	84.8876	3.17462
$716$	−19.3756	−0.724099
$717$	−12.9830	−0.484861
$718$	9.75552	0.364073
$719$	8.41253	0.313735	0.156867	−	0.987620i	$-0.449861\pi$
$719$	8.41253	0.313735	0.156867	+	0.987620i	$0.449861\pi$
$720$	4.06414	0.151461
$721$	33.3201	1.24091
$722$	−1.00000	−0.0372161
$723$	−9.71605	−0.361344
$724$	−2.33353	−0.0867249
$725$	−117.529	−4.36493
$726$	12.3932	0.459954
$727$	31.2472	1.15889	0.579447	−	0.815010i	$-0.303268\pi$
$727$	31.2472	1.15889	0.579447	+	0.815010i	$0.303268\pi$
$728$	17.2152	0.638036
$729$	1.00000	0.0370370
$730$	65.6439	2.42959
$731$	−64.1818	−2.37385
$732$	−5.87161	−0.217021
$733$	−18.3913	−0.679297	−0.339648	−	0.940552i	$-0.610308\pi$
$733$	−18.3913	−0.679297	−0.339648	+	0.940552i	$0.610308\pi$
$734$	29.1451	1.07577
$735$	−36.1353	−1.33287
$736$	5.82002	0.214529
$737$	37.6399	1.38648
$738$	−4.22566	−0.155549
$739$	5.37066	0.197563	0.0987815	−	0.995109i	$-0.468506\pi$
$739$	5.37066	0.197563	0.0987815	+	0.995109i	$0.468506\pi$
$740$	−22.1505	−0.814269
$741$	4.31849	0.158644
$742$	−3.98638	−0.146345
$743$	−5.81487	−0.213327	−0.106663	−	0.994295i	$-0.534017\pi$
$743$	−5.81487	−0.213327	−0.106663	+	0.994295i	$0.534017\pi$
$744$	5.35251	0.196233
$745$	−48.0708	−1.76118
$746$	−10.7578	−0.393871
$747$	0.441737	0.0161623
$748$	26.1254	0.955240
$749$	−47.6273	−1.74026
$750$	26.4868	0.967161
$751$	35.4339	1.29300	0.646501	−	0.762913i	$-0.276231\pi$
$751$	35.4339	1.29300	0.646501	+	0.762913i	$0.276231\pi$
$752$	−2.62661	−0.0957828
$753$	15.1387	0.551684
$754$	−44.0688	−1.60489
$755$	−32.3764	−1.17830
$756$	−3.98638	−0.144983
$757$	−41.7397	−1.51705	−0.758527	−	0.651641i	$-0.774081\pi$
$757$	−41.7397	−1.51705	−0.758527	+	0.651641i	$0.774081\pi$
$758$	6.50927	0.236427
$759$	−28.1494	−1.02176
$760$	−4.06414	−0.147422
$761$	25.9848	0.941948	0.470974	−	0.882147i	$-0.343902\pi$
$761$	25.9848	0.941948	0.470974	+	0.882147i	$0.343902\pi$
$762$	−4.73489	−0.171527
$763$	13.7767	0.498749
$764$	−6.24094	−0.225789
$765$	−21.9526	−0.793700
$766$	−7.68981	−0.277844
$767$	11.8837	0.429096
$768$	−1.00000	−0.0360844
$769$	−34.3832	−1.23989	−0.619945	−	0.784645i	$-0.712845\pi$
$769$	−34.3832	−1.23989	−0.619945	+	0.784645i	$0.712845\pi$
$770$	78.3596	2.82388
$771$	−8.64346	−0.311287
$772$	−18.8880	−0.679794
$773$	−33.1823	−1.19349	−0.596743	−	0.802432i	$-0.703539\pi$
$773$	−33.1823	−1.19349	−0.596743	+	0.802432i	$0.703539\pi$
$774$	−11.8821	−0.427093
$775$	61.6460	2.21439
$776$	13.7075	0.492071
$777$	21.7267	0.779442
$778$	16.0872	0.576755
$779$	4.22566	0.151400
$780$	17.5509	0.628424
$781$	30.9228	1.10650
$782$	−31.4371	−1.12419
$783$	10.2047	0.364686
$784$	8.89126	0.317545
$785$	−11.4329	−0.408058
$786$	−18.9772	−0.676893
$787$	−41.1409	−1.46651	−0.733257	−	0.679952i	$-0.762001\pi$
$787$	−41.1409	−1.46651	−0.733257	+	0.679952i	$0.762001\pi$
$788$	9.81626	0.349690
$789$	28.9284	1.02988
$790$	63.6519	2.26463
$791$	66.9330	2.37987
$792$	4.83665	0.171863
$793$	−25.3565	−0.900436
$794$	−20.3431	−0.721949
$795$	−4.06414	−0.144140
$796$	−16.9932	−0.602309
$797$	11.1089	0.393496	0.196748	−	0.980454i	$-0.436962\pi$
$797$	11.1089	0.393496	0.196748	+	0.980454i	$0.436962\pi$
$798$	3.98638	0.141116
$799$	14.1878	0.501928
$800$	−11.5172	−0.407195
$801$	17.5920	0.621582
$802$	−18.7938	−0.663632
$803$	78.1215	2.75685
$804$	7.78222	0.274458
$805$	−94.2913	−3.32333
$806$	23.1148	0.814183
$807$	−0.513136	−0.0180632
$808$	−2.70800	−0.0952671
$809$	−13.1286	−0.461579	−0.230789	−	0.973004i	$-0.574131\pi$
$809$	−13.1286	−0.461579	−0.230789	+	0.973004i	$0.574131\pi$
$810$	−4.06414	−0.142799
$811$	−37.5771	−1.31951	−0.659755	−	0.751481i	$-0.729340\pi$
$811$	−37.5771	−1.31951	−0.659755	+	0.751481i	$0.729340\pi$
$812$	−40.6798	−1.42758
$813$	−23.0655	−0.808943
$814$	−26.3609	−0.923948
$815$	−2.64234	−0.0925570
$816$	5.40155	0.189092
$817$	11.8821	0.415702
$818$	−35.9036	−1.25534
$819$	−17.2152	−0.601546
$820$	17.1737	0.599730
$821$	39.9993	1.39598	0.697992	−	0.716106i	$-0.254077\pi$
$821$	39.9993	1.39598	0.697992	+	0.716106i	$0.254077\pi$
$822$	−6.22073	−0.216973
$823$	33.4415	1.16570	0.582849	−	0.812581i	$-0.301938\pi$
$823$	33.4415	1.16570	0.582849	+	0.812581i	$0.301938\pi$
$824$	−8.35848	−0.291182
$825$	55.7047	1.93939
$826$	10.9698	0.381689
$827$	−43.8958	−1.52641	−0.763203	−	0.646158i	$-0.776375\pi$
$827$	−43.8958	−1.52641	−0.763203	+	0.646158i	$0.776375\pi$
$828$	−5.82002	−0.202260
$829$	8.59435	0.298494	0.149247	−	0.988800i	$-0.452315\pi$
$829$	8.59435	0.298494	0.149247	+	0.988800i	$0.452315\pi$
$830$	−1.79528	−0.0623150
$831$	−3.36234	−0.116638
$832$	−4.31849	−0.149717
$833$	−48.0266	−1.66402
$834$	19.0796	0.660671
$835$	−47.8459	−1.65578
$836$	−4.83665	−0.167279
$837$	−5.35251	−0.185010
$838$	1.88387	0.0650771
$839$	48.4348	1.67216	0.836078	−	0.548611i	$-0.184843\pi$
$839$	48.4348	1.67216	0.836078	+	0.548611i	$0.184843\pi$
$840$	16.2012	0.558995
$841$	75.1356	2.59088
$842$	28.4344	0.979914
$843$	5.06448	0.174430
$844$	−20.1443	−0.693396
$845$	22.9597	0.789837
$846$	2.62661	0.0903048
$847$	49.4039	1.69754
$848$	1.00000	0.0343401
$849$	27.4233	0.941165
$850$	62.2108	2.13381
$851$	31.7205	1.08736
$852$	6.39344	0.219036
$853$	−3.27808	−0.112239	−0.0561196	−	0.998424i	$-0.517873\pi$
$853$	−3.27808	−0.112239	−0.0561196	+	0.998424i	$0.517873\pi$
$854$	−23.4065	−0.800954
$855$	4.06414	0.138991
$856$	11.9475	0.408357
$857$	16.4745	0.562759	0.281379	−	0.959597i	$-0.409208\pi$
$857$	16.4745	0.562759	0.281379	+	0.959597i	$0.409208\pi$
$858$	20.8870	0.713071
$859$	4.76142	0.162457	0.0812287	−	0.996695i	$-0.474116\pi$
$859$	4.76142	0.162457	0.0812287	+	0.996695i	$0.474116\pi$
$860$	48.2905	1.64669
$861$	−16.8451	−0.574080
$862$	6.90149	0.235066
$863$	−32.7300	−1.11414	−0.557070	−	0.830465i	$-0.688075\pi$
$863$	−32.7300	−1.11414	−0.557070	+	0.830465i	$0.688075\pi$
$864$	1.00000	0.0340207
$865$	8.20484	0.278973
$866$	11.3429	0.385447
$867$	−12.1768	−0.413545
$868$	21.3372	0.724231
$869$	75.7508	2.56967
$870$	−41.4732	−1.40607
$871$	33.6074	1.13874
$872$	−3.45593	−0.117033
$873$	−13.7075	−0.463929
$874$	5.82002	0.196865
$875$	105.587	3.56948
$876$	16.1520	0.545726
$877$	−44.6355	−1.50723	−0.753617	−	0.657314i	$-0.771693\pi$
$877$	−44.6355	−1.50723	−0.753617	+	0.657314i	$0.771693\pi$
$878$	21.1503	0.713788
$879$	3.62708	0.122338
$880$	−19.6568	−0.662631
$881$	−26.2420	−0.884116	−0.442058	−	0.896987i	$-0.645751\pi$
$881$	−26.2420	−0.884116	−0.442058	+	0.896987i	$0.645751\pi$
$882$	−8.89126	−0.299384
$883$	−3.47060	−0.116795	−0.0583975	−	0.998293i	$-0.518599\pi$
$883$	−3.47060	−0.116795	−0.0583975	+	0.998293i	$0.518599\pi$
$884$	23.3265	0.784556
$885$	11.1838	0.375938
$886$	19.5875	0.658055
$887$	−17.6409	−0.592322	−0.296161	−	0.955138i	$-0.595707\pi$
$887$	−17.6409	−0.592322	−0.296161	+	0.955138i	$0.595707\pi$
$888$	−5.45024	−0.182898
$889$	−18.8751	−0.633050
$890$	−71.4962	−2.39656
$891$	−4.83665	−0.162034
$892$	−1.54909	−0.0518675
$893$	−2.62661	−0.0878963
$894$	−11.8280	−0.395589
$895$	−78.7450	−2.63216
$896$	−3.98638	−0.133176
$897$	−25.1337	−0.839189
$898$	23.6299	0.788539
$899$	−54.6207	−1.82170
$900$	11.5172	0.383907
$901$	−5.40155	−0.179952
$902$	20.4380	0.680512
$903$	−47.3666	−1.57626
$904$	−16.7904	−0.558441
$905$	−9.48378	−0.315252
$906$	−7.96637	−0.264665
$907$	47.2421	1.56865	0.784325	−	0.620350i	$-0.213010\pi$
$907$	47.2421	1.56865	0.784325	+	0.620350i	$0.213010\pi$
$908$	−3.56380	−0.118269
$909$	2.70800	0.0898187
$910$	69.9647	2.31931
$911$	−22.8852	−0.758220	−0.379110	−	0.925352i	$-0.623770\pi$
$911$	−22.8852	−0.758220	−0.379110	+	0.925352i	$0.623770\pi$
$912$	−1.00000	−0.0331133
$913$	−2.13652	−0.0707086
$914$	−8.46055	−0.279850
$915$	−23.8630	−0.788888
$916$	28.4053	0.938538
$917$	−75.6503	−2.49819
$918$	−5.40155	−0.178278
$919$	27.5182	0.907742	0.453871	−	0.891067i	$-0.350043\pi$
$919$	27.5182	0.907742	0.453871	+	0.891067i	$0.350043\pi$
$920$	23.6533	0.779828
$921$	11.2165	0.369595
$922$	26.8590	0.884555
$923$	27.6100	0.908793
$924$	19.2807	0.634290
$925$	−62.7715	−2.06391
$926$	−0.416670	−0.0136926
$927$	8.35848	0.274529
$928$	10.2047	0.334985
$929$	55.3364	1.81553	0.907764	−	0.419480i	$-0.137788\pi$
$929$	55.3364	1.81553	0.907764	+	0.419480i	$0.137788\pi$
$930$	21.7533	0.713320
$931$	8.89126	0.291399
$932$	9.31482	0.305117
$933$	−21.0056	−0.687694
$934$	6.43132	0.210439
$935$	106.177	3.47237
$936$	4.31849	0.141154
$937$	9.69374	0.316681	0.158340	−	0.987385i	$-0.449386\pi$
$937$	9.69374	0.316681	0.158340	+	0.987385i	$0.449386\pi$
$938$	31.0229	1.01293
$939$	4.68076	0.152751
$940$	−10.6749	−0.348177
$941$	−18.0604	−0.588752	−0.294376	−	0.955690i	$-0.595112\pi$
$941$	−18.0604	−0.588752	−0.294376	+	0.955690i	$0.595112\pi$
$942$	−2.81312	−0.0916565
$943$	−24.5934	−0.800872
$944$	−2.75182	−0.0895641
$945$	−16.2012	−0.527025
$946$	57.4695	1.86850
$947$	−46.1270	−1.49893	−0.749463	−	0.662046i	$-0.769688\pi$
$947$	−46.1270	−1.49893	−0.749463	+	0.662046i	$0.769688\pi$
$948$	15.6618	0.508673
$949$	69.7522	2.26425
$950$	−11.5172	−0.373667
$951$	5.51208	0.178741
$952$	21.5327	0.697878
$953$	−47.2022	−1.52903	−0.764515	−	0.644606i	$-0.777022\pi$
$953$	−47.2022	−1.52903	−0.764515	+	0.644606i	$0.777022\pi$
$954$	−1.00000	−0.0323762
$955$	−25.3640	−0.820760
$956$	12.9830	0.419902
$957$	−49.3565	−1.59547
$958$	−11.1911	−0.361569
$959$	−24.7982	−0.800777
$960$	−4.06414	−0.131169
$961$	−2.35061	−0.0758260
$962$	−23.5368	−0.758856
$963$	−11.9475	−0.385002
$964$	9.71605	0.312933
$965$	−76.7635	−2.47110
$966$	−23.2008	−0.746474
$967$	20.2890	0.652451	0.326226	−	0.945292i	$-0.394223\pi$
$967$	20.2890	0.652451	0.326226	+	0.945292i	$0.394223\pi$
$968$	−12.3932	−0.398331
$969$	5.40155	0.173523
$970$	55.7092	1.78871
$971$	14.8979	0.478096	0.239048	−	0.971008i	$-0.423165\pi$
$971$	14.8979	0.478096	0.239048	+	0.971008i	$0.423165\pi$
$972$	−1.00000	−0.0320750
$973$	76.0585	2.43832
$974$	−41.3110	−1.32369
$975$	49.7369	1.59286
$976$	5.87161	0.187946
$977$	50.2517	1.60769	0.803847	−	0.594835i	$-0.202783\pi$
$977$	50.2517	1.60769	0.803847	+	0.594835i	$0.202783\pi$
$978$	−0.650159	−0.0207898
$979$	−85.0863	−2.71937
$980$	36.1353	1.15430
$981$	3.45593	0.110339
$982$	6.38926	0.203889
$983$	33.1687	1.05792	0.528959	−	0.848647i	$-0.322583\pi$
$983$	33.1687	1.05792	0.528959	+	0.848647i	$0.322583\pi$
$984$	4.22566	0.134709
$985$	39.8946	1.27115
$986$	−55.1211	−1.75541
$987$	10.4707	0.333286
$988$	−4.31849	−0.137389
$989$	−69.1540	−2.19897
$990$	19.6568	0.624734
$991$	−5.71310	−0.181483	−0.0907413	−	0.995874i	$-0.528924\pi$
$991$	−5.71310	−0.181483	−0.0907413	+	0.995874i	$0.528924\pi$
$992$	−5.35251	−0.169942
$993$	13.3675	0.424206
$994$	25.4867	0.808389
$995$	−69.0628	−2.18944
$996$	−0.441737	−0.0139970
$997$	48.1535	1.52504	0.762518	−	0.646967i	$-0.223963\pi$
$997$	48.1535	1.52504	0.762518	+	0.646967i	$0.223963\pi$
$998$	12.9201	0.408978
$999$	5.45024	0.172438

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6042.2.a.bd.1.11	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.bd.1.11	✓	11	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:	\( N \)	\(=\)	\( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6042.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.06414 q^{5} +1.00000 q^{6} +3.98638 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.06414 q^{5} +1.00000 q^{6} +3.98638 q^{7} -1.00000 q^{8} +1.00000 q^{9} -4.06414 q^{10} -4.83665 q^{11} -1.00000 q^{12} -4.31849 q^{13} -3.98638 q^{14} -4.06414 q^{15} +1.00000 q^{16} -5.40155 q^{17} -1.00000 q^{18} +1.00000 q^{19} +4.06414 q^{20} -3.98638 q^{21} +4.83665 q^{22} -5.82002 q^{23} +1.00000 q^{24} +11.5172 q^{25} +4.31849 q^{26} -1.00000 q^{27} +3.98638 q^{28} -10.2047 q^{29} +4.06414 q^{30} +5.35251 q^{31} -1.00000 q^{32} +4.83665 q^{33} +5.40155 q^{34} +16.2012 q^{35} +1.00000 q^{36} -5.45024 q^{37} -1.00000 q^{38} +4.31849 q^{39} -4.06414 q^{40} +4.22566 q^{41} +3.98638 q^{42} +11.8821 q^{43} -4.83665 q^{44} +4.06414 q^{45} +5.82002 q^{46} -2.62661 q^{47} -1.00000 q^{48} +8.89126 q^{49} -11.5172 q^{50} +5.40155 q^{51} -4.31849 q^{52} +1.00000 q^{53} +1.00000 q^{54} -19.6568 q^{55} -3.98638 q^{56} -1.00000 q^{57} +10.2047 q^{58} -2.75182 q^{59} -4.06414 q^{60} +5.87161 q^{61} -5.35251 q^{62} +3.98638 q^{63} +1.00000 q^{64} -17.5509 q^{65} -4.83665 q^{66} -7.78222 q^{67} -5.40155 q^{68} +5.82002 q^{69} -16.2012 q^{70} -6.39344 q^{71} -1.00000 q^{72} -16.1520 q^{73} +5.45024 q^{74} -11.5172 q^{75} +1.00000 q^{76} -19.2807 q^{77} -4.31849 q^{78} -15.6618 q^{79} +4.06414 q^{80} +1.00000 q^{81} -4.22566 q^{82} +0.441737 q^{83} -3.98638 q^{84} -21.9526 q^{85} -11.8821 q^{86} +10.2047 q^{87} +4.83665 q^{88} +17.5920 q^{89} -4.06414 q^{90} -17.2152 q^{91} -5.82002 q^{92} -5.35251 q^{93} +2.62661 q^{94} +4.06414 q^{95} +1.00000 q^{96} -13.7075 q^{97} -8.89126 q^{98} -4.83665 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 11 q^{2} - 11 q^{3} + 11 q^{4} + 2 q^{5} + 11 q^{6} - 2 q^{7} - 11 q^{8} + 11 q^{9}+O(q^{10}) \) 11 * q - 11 * q^2 - 11 * q^3 + 11 * q^4 + 2 * q^5 + 11 * q^6 - 2 * q^7 - 11 * q^8 + 11 * q^9	\( 11 q - 11 q^{2} - 11 q^{3} + 11 q^{4} + 2 q^{5} + 11 q^{6} - 2 q^{7} - 11 q^{8} + 11 q^{9} - 2 q^{10} - 6 q^{11} - 11 q^{12} - 7 q^{13} + 2 q^{14} - 2 q^{15} + 11 q^{16} - 16 q^{17} - 11 q^{18} + 11 q^{19} + 2 q^{20} + 2 q^{21} + 6 q^{22} - 11 q^{23} + 11 q^{24} + 19 q^{25} + 7 q^{26} - 11 q^{27} - 2 q^{28} - 7 q^{29} + 2 q^{30} + 12 q^{31} - 11 q^{32} + 6 q^{33} + 16 q^{34} - 5 q^{35} + 11 q^{36} + 3 q^{37} - 11 q^{38} + 7 q^{39} - 2 q^{40} + 11 q^{41} - 2 q^{42} + 7 q^{43} - 6 q^{44} + 2 q^{45} + 11 q^{46} - 37 q^{47} - 11 q^{48} + 23 q^{49} - 19 q^{50} + 16 q^{51} - 7 q^{52} + 11 q^{53} + 11 q^{54} - 11 q^{55} + 2 q^{56} - 11 q^{57} + 7 q^{58} - 16 q^{59} - 2 q^{60} + 24 q^{61} - 12 q^{62} - 2 q^{63} + 11 q^{64} - 7 q^{65} - 6 q^{66} - 28 q^{67} - 16 q^{68} + 11 q^{69} + 5 q^{70} + 4 q^{71} - 11 q^{72} - 7 q^{73} - 3 q^{74} - 19 q^{75} + 11 q^{76} - 13 q^{77} - 7 q^{78} + 5 q^{79} + 2 q^{80} + 11 q^{81} - 11 q^{82} - 11 q^{83} + 2 q^{84} - 11 q^{85} - 7 q^{86} + 7 q^{87} + 6 q^{88} - 10 q^{89} - 2 q^{90} - 24 q^{91} - 11 q^{92} - 12 q^{93} + 37 q^{94} + 2 q^{95} + 11 q^{96} - 24 q^{97} - 23 q^{98} - 6 q^{99}+O(q^{100}) \) 11 * q - 11 * q^2 - 11 * q^3 + 11 * q^4 + 2 * q^5 + 11 * q^6 - 2 * q^7 - 11 * q^8 + 11 * q^9 - 2 * q^10 - 6 * q^11 - 11 * q^12 - 7 * q^13 + 2 * q^14 - 2 * q^15 + 11 * q^16 - 16 * q^17 - 11 * q^18 + 11 * q^19 + 2 * q^20 + 2 * q^21 + 6 * q^22 - 11 * q^23 + 11 * q^24 + 19 * q^25 + 7 * q^26 - 11 * q^27 - 2 * q^28 - 7 * q^29 + 2 * q^30 + 12 * q^31 - 11 * q^32 + 6 * q^33 + 16 * q^34 - 5 * q^35 + 11 * q^36 + 3 * q^37 - 11 * q^38 + 7 * q^39 - 2 * q^40 + 11 * q^41 - 2 * q^42 + 7 * q^43 - 6 * q^44 + 2 * q^45 + 11 * q^46 - 37 * q^47 - 11 * q^48 + 23 * q^49 - 19 * q^50 + 16 * q^51 - 7 * q^52 + 11 * q^53 + 11 * q^54 - 11 * q^55 + 2 * q^56 - 11 * q^57 + 7 * q^58 - 16 * q^59 - 2 * q^60 + 24 * q^61 - 12 * q^62 - 2 * q^63 + 11 * q^64 - 7 * q^65 - 6 * q^66 - 28 * q^67 - 16 * q^68 + 11 * q^69 + 5 * q^70 + 4 * q^71 - 11 * q^72 - 7 * q^73 - 3 * q^74 - 19 * q^75 + 11 * q^76 - 13 * q^77 - 7 * q^78 + 5 * q^79 + 2 * q^80 + 11 * q^81 - 11 * q^82 - 11 * q^83 + 2 * q^84 - 11 * q^85 - 7 * q^86 + 7 * q^87 + 6 * q^88 - 10 * q^89 - 2 * q^90 - 24 * q^91 - 11 * q^92 - 12 * q^93 + 37 * q^94 + 2 * q^95 + 11 * q^96 - 24 * q^97 - 23 * q^98 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more