LMFDB - Embedded newform 6069.2.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6069,2,Mod(1,6069)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6069, 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("6069.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6069 = 3 \cdot 7 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6069.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:	$48.4612089867$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 12x^{7} - 3x^{6} + 45x^{5} + 21x^{4} - 53x^{3} - 39x^{2} + 3 $ x^9 - 12x^7 - 3x^6 + 45x^5 + 21x^4 - 53x^3 - 39x^2 + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.02180$ of defining polynomial
Character	$\chi$	$=$	6069.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.02180 q^{2} +1.00000 q^{3} +2.08766 q^{4} -0.796171 q^{5} -2.02180 q^{6} +1.00000 q^{7} -0.177223 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.02180 q^{2} +1.00000 q^{3} +2.08766 q^{4} -0.796171 q^{5} -2.02180 q^{6} +1.00000 q^{7} -0.177223 q^{8} +1.00000 q^{9} +1.60970 q^{10} -3.96044 q^{11} +2.08766 q^{12} -0.338889 q^{13} -2.02180 q^{14} -0.796171 q^{15} -3.81700 q^{16} -2.02180 q^{18} +5.37764 q^{19} -1.66213 q^{20} +1.00000 q^{21} +8.00719 q^{22} +9.14634 q^{23} -0.177223 q^{24} -4.36611 q^{25} +0.685164 q^{26} +1.00000 q^{27} +2.08766 q^{28} -2.16884 q^{29} +1.60970 q^{30} -10.6499 q^{31} +8.07165 q^{32} -3.96044 q^{33} -0.796171 q^{35} +2.08766 q^{36} -4.68236 q^{37} -10.8725 q^{38} -0.338889 q^{39} +0.141100 q^{40} +5.86902 q^{41} -2.02180 q^{42} -9.17251 q^{43} -8.26803 q^{44} -0.796171 q^{45} -18.4920 q^{46} +2.55127 q^{47} -3.81700 q^{48} +1.00000 q^{49} +8.82738 q^{50} -0.707484 q^{52} +4.13973 q^{53} -2.02180 q^{54} +3.15319 q^{55} -0.177223 q^{56} +5.37764 q^{57} +4.38495 q^{58} +4.55341 q^{59} -1.66213 q^{60} -9.58258 q^{61} +21.5320 q^{62} +1.00000 q^{63} -8.68521 q^{64} +0.269814 q^{65} +8.00719 q^{66} +2.80790 q^{67} +9.14634 q^{69} +1.60970 q^{70} -2.66873 q^{71} -0.177223 q^{72} +5.66157 q^{73} +9.46678 q^{74} -4.36611 q^{75} +11.2267 q^{76} -3.96044 q^{77} +0.685164 q^{78} +0.842146 q^{79} +3.03899 q^{80} +1.00000 q^{81} -11.8660 q^{82} +0.586268 q^{83} +2.08766 q^{84} +18.5449 q^{86} -2.16884 q^{87} +0.701881 q^{88} +17.6141 q^{89} +1.60970 q^{90} -0.338889 q^{91} +19.0944 q^{92} -10.6499 q^{93} -5.15815 q^{94} -4.28152 q^{95} +8.07165 q^{96} -14.5657 q^{97} -2.02180 q^{98} -3.96044 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{3} + 6 q^{4} - 3 q^{5} + 9 q^{7} + 9 q^{8} + 9 q^{9}+O(q^{10}) $ 9 * q + 9 * q^3 + 6 * q^4 - 3 * q^5 + 9 * q^7 + 9 * q^8 + 9 * q^9	$ 9 q + 9 q^{3} + 6 q^{4} - 3 q^{5} + 9 q^{7} + 9 q^{8} + 9 q^{9} - 12 q^{10} - 18 q^{11} + 6 q^{12} - 21 q^{13} - 3 q^{15} - 9 q^{19} - 15 q^{20} + 9 q^{21} - 6 q^{22} + 9 q^{24} + 3 q^{26} + 9 q^{27} + 6 q^{28} - 6 q^{29} - 12 q^{30} - 30 q^{31} + 3 q^{32} - 18 q^{33} - 3 q^{35} + 6 q^{36} - 12 q^{37} - 36 q^{38} - 21 q^{39} - 30 q^{40} - 9 q^{41} - 30 q^{44} - 3 q^{45} - 33 q^{46} - 9 q^{47} + 9 q^{49} - 12 q^{50} - 12 q^{52} + 30 q^{55} + 9 q^{56} - 9 q^{57} - 9 q^{58} - 3 q^{59} - 15 q^{60} - 33 q^{61} + 12 q^{62} + 9 q^{63} - 15 q^{64} + 15 q^{65} - 6 q^{66} - 15 q^{67} - 12 q^{70} + 9 q^{72} - 21 q^{73} - 6 q^{74} + 3 q^{76} - 18 q^{77} + 3 q^{78} - 30 q^{79} + 9 q^{81} - 9 q^{82} + 6 q^{84} + 72 q^{86} - 6 q^{87} - 48 q^{88} - 12 q^{89} - 12 q^{90} - 21 q^{91} - 48 q^{92} - 30 q^{93} + 48 q^{94} + 18 q^{95} + 3 q^{96} - 18 q^{99}+O(q^{100}) $ 9 * q + 9 * q^3 + 6 * q^4 - 3 * q^5 + 9 * q^7 + 9 * q^8 + 9 * q^9 - 12 * q^10 - 18 * q^11 + 6 * q^12 - 21 * q^13 - 3 * q^15 - 9 * q^19 - 15 * q^20 + 9 * q^21 - 6 * q^22 + 9 * q^24 + 3 * q^26 + 9 * q^27 + 6 * q^28 - 6 * q^29 - 12 * q^30 - 30 * q^31 + 3 * q^32 - 18 * q^33 - 3 * q^35 + 6 * q^36 - 12 * q^37 - 36 * q^38 - 21 * q^39 - 30 * q^40 - 9 * q^41 - 30 * q^44 - 3 * q^45 - 33 * q^46 - 9 * q^47 + 9 * q^49 - 12 * q^50 - 12 * q^52 + 30 * q^55 + 9 * q^56 - 9 * q^57 - 9 * q^58 - 3 * q^59 - 15 * q^60 - 33 * q^61 + 12 * q^62 + 9 * q^63 - 15 * q^64 + 15 * q^65 - 6 * q^66 - 15 * q^67 - 12 * q^70 + 9 * q^72 - 21 * q^73 - 6 * q^74 + 3 * q^76 - 18 * q^77 + 3 * q^78 - 30 * q^79 + 9 * q^81 - 9 * q^82 + 6 * q^84 + 72 * q^86 - 6 * q^87 - 48 * q^88 - 12 * q^89 - 12 * q^90 - 21 * q^91 - 48 * q^92 - 30 * q^93 + 48 * q^94 + 18 * q^95 + 3 * q^96 - 18 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.02180	−1.42963	−0.714813	−	0.699316i	$-0.753488\pi$
$2$	−2.02180	−1.42963	−0.714813	+	0.699316i	$0.753488\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.08766	1.04383
$5$	−0.796171	−0.356059	−0.178029	−	0.984025i	$-0.556972\pi$
$5$	−0.796171	−0.356059	−0.178029	+	0.984025i	$0.556972\pi$
$6$	−2.02180	−0.825394
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−0.177223	−0.0626578
$9$	1.00000	0.333333
$10$	1.60970	0.509030
$11$	−3.96044	−1.19412	−0.597058	−	0.802198i	$-0.703664\pi$
$11$	−3.96044	−1.19412	−0.597058	+	0.802198i	$0.703664\pi$
$12$	2.08766	0.602654
$13$	−0.338889	−0.0939909	−0.0469954	−	0.998895i	$-0.514965\pi$
$13$	−0.338889	−0.0939909	−0.0469954	+	0.998895i	$0.514965\pi$
$14$	−2.02180	−0.540348
$15$	−0.796171	−0.205571
$16$	−3.81700	−0.954251
$17$	0	0
$17$	0	0
$18$	−2.02180	−0.476542
$19$	5.37764	1.23371	0.616857	−	0.787075i	$-0.288406\pi$
$19$	5.37764	1.23371	0.616857	+	0.787075i	$0.288406\pi$
$20$	−1.66213	−0.371664
$21$	1.00000	0.218218
$22$	8.00719	1.70714
$23$	9.14634	1.90714	0.953572	−	0.301164i	$-0.0973751\pi$
$23$	9.14634	1.90714	0.953572	+	0.301164i	$0.0973751\pi$
$24$	−0.177223	−0.0361755
$25$	−4.36611	−0.873222
$26$	0.685164	0.134372
$27$	1.00000	0.192450
$28$	2.08766	0.394530
$29$	−2.16884	−0.402744	−0.201372	−	0.979515i	$-0.564540\pi$
$29$	−2.16884	−0.402744	−0.201372	+	0.979515i	$0.564540\pi$
$30$	1.60970	0.293889
$31$	−10.6499	−1.91279	−0.956394	−	0.292081i	$-0.905652\pi$
$31$	−10.6499	−1.91279	−0.956394	+	0.292081i	$0.905652\pi$
$32$	8.07165	1.42688
$33$	−3.96044	−0.689423
$34$	0	0
$35$	−0.796171	−0.134578
$36$	2.08766	0.347943
$37$	−4.68236	−0.769776	−0.384888	−	0.922963i	$-0.625760\pi$
$37$	−4.68236	−0.769776	−0.384888	+	0.922963i	$0.625760\pi$
$38$	−10.8725	−1.76375
$39$	−0.338889	−0.0542657
$40$	0.141100	0.0223099
$41$	5.86902	0.916587	0.458294	−	0.888801i	$-0.348461\pi$
$41$	5.86902	0.916587	0.458294	+	0.888801i	$0.348461\pi$
$42$	−2.02180	−0.311970
$43$	−9.17251	−1.39879	−0.699397	−	0.714733i	$-0.746548\pi$
$43$	−9.17251	−1.39879	−0.699397	+	0.714733i	$0.746548\pi$
$44$	−8.26803	−1.24645
$45$	−0.796171	−0.118686
$46$	−18.4920	−2.72650
$47$	2.55127	0.372141	0.186071	−	0.982536i	$-0.440425\pi$
$47$	2.55127	0.372141	0.186071	+	0.982536i	$0.440425\pi$
$48$	−3.81700	−0.550937
$49$	1.00000	0.142857
$50$	8.82738	1.24838
$51$	0	0
$52$	−0.707484	−0.0981103
$53$	4.13973	0.568636	0.284318	−	0.958730i	$-0.408233\pi$
$53$	4.13973	0.568636	0.284318	+	0.958730i	$0.408233\pi$
$54$	−2.02180	−0.275131
$55$	3.15319	0.425175
$56$	−0.177223	−0.0236824
$57$	5.37764	0.712286
$58$	4.38495	0.575772
$59$	4.55341	0.592803	0.296402	−	0.955063i	$-0.404213\pi$
$59$	4.55341	0.592803	0.296402	+	0.955063i	$0.404213\pi$
$60$	−1.66213	−0.214580
$61$	−9.58258	−1.22692	−0.613462	−	0.789724i	$-0.710224\pi$
$61$	−9.58258	−1.22692	−0.613462	+	0.789724i	$0.710224\pi$
$62$	21.5320	2.73457
$63$	1.00000	0.125988
$64$	−8.68521	−1.08565
$65$	0.269814	0.0334663
$66$	8.00719	0.985617
$67$	2.80790	0.343040	0.171520	−	0.985181i	$-0.445132\pi$
$67$	2.80790	0.343040	0.171520	+	0.985181i	$0.445132\pi$
$68$	0	0
$69$	9.14634	1.10109
$70$	1.60970	0.192395
$71$	−2.66873	−0.316720	−0.158360	−	0.987381i	$-0.550621\pi$
$71$	−2.66873	−0.316720	−0.158360	+	0.987381i	$0.550621\pi$
$72$	−0.177223	−0.0208859
$73$	5.66157	0.662636	0.331318	−	0.943519i	$-0.392507\pi$
$73$	5.66157	0.662636	0.331318	+	0.943519i	$0.392507\pi$
$74$	9.46678	1.10049
$75$	−4.36611	−0.504155
$76$	11.2267	1.28779
$77$	−3.96044	−0.451333
$78$	0.685164	0.0775795
$79$	0.842146	0.0947488	0.0473744	−	0.998877i	$-0.484915\pi$
$79$	0.842146	0.0947488	0.0473744	+	0.998877i	$0.484915\pi$
$80$	3.03899	0.339769
$81$	1.00000	0.111111
$82$	−11.8660	−1.31038
$83$	0.586268	0.0643513	0.0321756	−	0.999482i	$-0.489756\pi$
$83$	0.586268	0.0643513	0.0321756	+	0.999482i	$0.489756\pi$
$84$	2.08766	0.227782
$85$	0	0
$86$	18.5449	1.99975
$87$	−2.16884	−0.232524
$88$	0.701881	0.0748207
$89$	17.6141	1.86709	0.933546	−	0.358457i	$-0.116697\pi$
$89$	17.6141	1.86709	0.933546	+	0.358457i	$0.116697\pi$
$90$	1.60970	0.169677
$91$	−0.338889	−0.0355252
$92$	19.0944	1.99073
$93$	−10.6499	−1.10435
$94$	−5.15815	−0.532022
$95$	−4.28152	−0.439275
$96$	8.07165	0.823809
$97$	−14.5657	−1.47892	−0.739462	−	0.673199i	$-0.764920\pi$
$97$	−14.5657	−1.47892	−0.739462	+	0.673199i	$0.764920\pi$
$98$	−2.02180	−0.204232
$99$	−3.96044	−0.398039
$100$	−9.11494	−0.911494
$101$	−3.01541	−0.300045	−0.150022	−	0.988683i	$-0.547935\pi$
$101$	−3.01541	−0.300045	−0.150022	+	0.988683i	$0.547935\pi$
$102$	0	0
$103$	−12.9090	−1.27197	−0.635983	−	0.771703i	$-0.719405\pi$
$103$	−12.9090	−1.27197	−0.635983	+	0.771703i	$0.719405\pi$
$104$	0.0600589	0.00588926
$105$	−0.796171	−0.0776984
$106$	−8.36970	−0.812937
$107$	−5.98695	−0.578781	−0.289390	−	0.957211i	$-0.593453\pi$
$107$	−5.98695	−0.578781	−0.289390	+	0.957211i	$0.593453\pi$
$108$	2.08766	0.200885
$109$	17.0321	1.63138	0.815690	−	0.578490i	$-0.196358\pi$
$109$	17.0321	1.63138	0.815690	+	0.578490i	$0.196358\pi$
$110$	−6.37510	−0.607841
$111$	−4.68236	−0.444430
$112$	−3.81700	−0.360673
$113$	−15.5851	−1.46613	−0.733064	−	0.680160i	$-0.761910\pi$
$113$	−15.5851	−1.46613	−0.733064	+	0.680160i	$0.761910\pi$
$114$	−10.8725	−1.01830
$115$	−7.28206	−0.679055
$116$	−4.52779	−0.420395
$117$	−0.338889	−0.0313303
$118$	−9.20606	−0.847486
$119$	0	0
$120$	0.141100	0.0128806
$121$	4.68505	0.425913
$122$	19.3740	1.75404
$123$	5.86902	0.529192
$124$	−22.2334	−1.99662
$125$	7.45703	0.666977
$126$	−2.02180	−0.180116
$127$	−8.71907	−0.773692	−0.386846	−	0.922144i	$-0.626436\pi$
$127$	−8.71907	−0.773692	−0.386846	+	0.922144i	$0.626436\pi$
$128$	1.41642	0.125195
$129$	−9.17251	−0.807595
$130$	−0.545508	−0.0478442
$131$	−17.5095	−1.52981	−0.764904	−	0.644144i	$-0.777214\pi$
$131$	−17.5095	−1.52981	−0.764904	+	0.644144i	$0.777214\pi$
$132$	−8.26803	−0.719639
$133$	5.37764	0.466300
$134$	−5.67701	−0.490419
$135$	−0.796171	−0.0685235
$136$	0	0
$137$	−18.4988	−1.58046	−0.790229	−	0.612812i	$-0.790038\pi$
$137$	−18.4988	−1.58046	−0.790229	+	0.612812i	$0.790038\pi$
$138$	−18.4920	−1.57415
$139$	10.4827	0.889131	0.444566	−	0.895746i	$-0.353358\pi$
$139$	10.4827	0.889131	0.444566	+	0.895746i	$0.353358\pi$
$140$	−1.66213	−0.140476
$141$	2.55127	0.214856
$142$	5.39563	0.452791
$143$	1.34215	0.112236
$144$	−3.81700	−0.318084
$145$	1.72677	0.143400
$146$	−11.4465	−0.947322
$147$	1.00000	0.0824786
$148$	−9.77516	−0.803514
$149$	2.48701	0.203744	0.101872	−	0.994798i	$-0.467517\pi$
$149$	2.48701	0.203744	0.101872	+	0.994798i	$0.467517\pi$
$150$	8.82738	0.720753
$151$	5.59009	0.454915	0.227457	−	0.973788i	$-0.426959\pi$
$151$	5.59009	0.454915	0.227457	+	0.973788i	$0.426959\pi$
$152$	−0.953042	−0.0773019
$153$	0	0
$154$	8.00719	0.645238
$155$	8.47918	0.681064
$156$	−0.707484	−0.0566440
$157$	−8.13832	−0.649509	−0.324754	−	0.945798i	$-0.605282\pi$
$157$	−8.13832	−0.649509	−0.324754	+	0.945798i	$0.605282\pi$
$158$	−1.70265	−0.135455
$159$	4.13973	0.328302
$160$	−6.42641	−0.508053
$161$	9.14634	0.720833
$162$	−2.02180	−0.158847
$163$	20.7578	1.62588	0.812940	−	0.582347i	$-0.197866\pi$
$163$	20.7578	1.62588	0.812940	+	0.582347i	$0.197866\pi$
$164$	12.2525	0.956760
$165$	3.15319	0.245475
$166$	−1.18531	−0.0919982
$167$	−5.25596	−0.406719	−0.203359	−	0.979104i	$-0.565186\pi$
$167$	−5.25596	−0.406719	−0.203359	+	0.979104i	$0.565186\pi$
$168$	−0.177223	−0.0136731
$169$	−12.8852	−0.991166
$170$	0	0
$171$	5.37764	0.411238
$172$	−19.1491	−1.46010
$173$	−14.4999	−1.10241	−0.551205	−	0.834370i	$-0.685832\pi$
$173$	−14.4999	−1.10241	−0.551205	+	0.834370i	$0.685832\pi$
$174$	4.38495	0.332422
$175$	−4.36611	−0.330047
$176$	15.1170	1.13949
$177$	4.55341	0.342255
$178$	−35.6121	−2.66924
$179$	9.60523	0.717928	0.358964	−	0.933351i	$-0.383130\pi$
$179$	9.60523	0.717928	0.358964	+	0.933351i	$0.383130\pi$
$180$	−1.66213	−0.123888
$181$	8.10859	0.602707	0.301354	−	0.953512i	$-0.402562\pi$
$181$	8.10859	0.602707	0.301354	+	0.953512i	$0.402562\pi$
$182$	0.685164	0.0507877
$183$	−9.58258	−0.708365
$184$	−1.62094	−0.119498
$185$	3.72796	0.274085
$186$	21.5320	1.57880
$187$	0	0
$188$	5.32618	0.388451
$189$	1.00000	0.0727393
$190$	8.65636	0.627998
$191$	10.8846	0.787585	0.393793	−	0.919199i	$-0.371163\pi$
$191$	10.8846	0.787585	0.393793	+	0.919199i	$0.371163\pi$
$192$	−8.68521	−0.626801
$193$	−19.6918	−1.41744	−0.708722	−	0.705488i	$-0.750728\pi$
$193$	−19.6918	−1.41744	−0.708722	+	0.705488i	$0.750728\pi$
$194$	29.4489	2.11431
$195$	0.269814	0.0193218
$196$	2.08766	0.149118
$197$	−6.38860	−0.455169	−0.227585	−	0.973758i	$-0.573083\pi$
$197$	−6.38860	−0.455169	−0.227585	+	0.973758i	$0.573083\pi$
$198$	8.00719	0.569046
$199$	7.07076	0.501233	0.250616	−	0.968086i	$-0.419367\pi$
$199$	7.07076	0.501233	0.250616	+	0.968086i	$0.419367\pi$
$200$	0.773776	0.0547142
$201$	2.80790	0.198054
$202$	6.09655	0.428952
$203$	−2.16884	−0.152223
$204$	0	0
$205$	−4.67275	−0.326359
$206$	26.0995	1.81844
$207$	9.14634	0.635715
$208$	1.29354	0.0896909
$209$	−21.2978	−1.47320
$210$	1.60970	0.111080
$211$	−21.9882	−1.51373	−0.756864	−	0.653572i	$-0.773270\pi$
$211$	−21.9882	−1.51373	−0.756864	+	0.653572i	$0.773270\pi$
$212$	8.64234	0.593558
$213$	−2.66873	−0.182858
$214$	12.1044	0.827439
$215$	7.30289	0.498053
$216$	−0.177223	−0.0120585
$217$	−10.6499	−0.722966
$218$	−34.4354	−2.33226
$219$	5.66157	0.382573
$220$	6.58277	0.443810
$221$	0	0
$222$	9.46678	0.635369
$223$	14.4716	0.969091	0.484546	−	0.874766i	$-0.338985\pi$
$223$	14.4716	0.969091	0.484546	+	0.874766i	$0.338985\pi$
$224$	8.07165	0.539310
$225$	−4.36611	−0.291074
$226$	31.5100	2.09601
$227$	8.35861	0.554780	0.277390	−	0.960757i	$-0.410531\pi$
$227$	8.35861	0.554780	0.277390	+	0.960757i	$0.410531\pi$
$228$	11.2267	0.743504
$229$	0.113291	0.00748645	0.00374322	−	0.999993i	$-0.498808\pi$
$229$	0.113291	0.00748645	0.00374322	+	0.999993i	$0.498808\pi$
$230$	14.7228	0.970795
$231$	−3.96044	−0.260577
$232$	0.384369	0.0252350
$233$	−4.58747	−0.300535	−0.150268	−	0.988645i	$-0.548014\pi$
$233$	−4.58747	−0.300535	−0.150268	+	0.988645i	$0.548014\pi$
$234$	0.685164	0.0447906
$235$	−2.03125	−0.132504
$236$	9.50595	0.618785
$237$	0.842146	0.0547032
$238$	0	0
$239$	9.40874	0.608601	0.304300	−	0.952576i	$-0.401577\pi$
$239$	9.40874	0.608601	0.304300	+	0.952576i	$0.401577\pi$
$240$	3.03899	0.196166
$241$	6.74293	0.434350	0.217175	−	0.976133i	$-0.430316\pi$
$241$	6.74293	0.434350	0.217175	+	0.976133i	$0.430316\pi$
$242$	−9.47220	−0.608896
$243$	1.00000	0.0641500
$244$	−20.0051	−1.28070
$245$	−0.796171	−0.0508655
$246$	−11.8660	−0.756546
$247$	−1.82242	−0.115958
$248$	1.88742	0.119851
$249$	0.586268	0.0371532
$250$	−15.0766	−0.953527
$251$	−19.2601	−1.21569	−0.607843	−	0.794057i	$-0.707965\pi$
$251$	−19.2601	−1.21569	−0.607843	+	0.794057i	$0.707965\pi$
$252$	2.08766	0.131510
$253$	−36.2235	−2.27735
$254$	17.6282	1.10609
$255$	0	0
$256$	14.5067	0.906669
$257$	27.6076	1.72212	0.861058	−	0.508508i	$-0.169803\pi$
$257$	27.6076	1.72212	0.861058	+	0.508508i	$0.169803\pi$
$258$	18.5449	1.15456
$259$	−4.68236	−0.290948
$260$	0.563278	0.0349330
$261$	−2.16884	−0.134248
$262$	35.4005	2.18705
$263$	−14.1019	−0.869558	−0.434779	−	0.900537i	$-0.643173\pi$
$263$	−14.1019	−0.869558	−0.434779	+	0.900537i	$0.643173\pi$
$264$	0.701881	0.0431978
$265$	−3.29594	−0.202468
$266$	−10.8725	−0.666635
$267$	17.6141	1.07797
$268$	5.86194	0.358075
$269$	6.06716	0.369921	0.184961	−	0.982746i	$-0.440784\pi$
$269$	6.06716	0.369921	0.184961	+	0.982746i	$0.440784\pi$
$270$	1.60970	0.0979630
$271$	−10.5918	−0.643404	−0.321702	−	0.946841i	$-0.604255\pi$
$271$	−10.5918	−0.643404	−0.321702	+	0.946841i	$0.604255\pi$
$272$	0	0
$273$	−0.338889	−0.0205105
$274$	37.4008	2.25946
$275$	17.2917	1.04273
$276$	19.0944	1.14935
$277$	13.0667	0.785104	0.392552	−	0.919730i	$-0.371592\pi$
$277$	13.0667	0.785104	0.392552	+	0.919730i	$0.371592\pi$
$278$	−21.1939	−1.27112
$279$	−10.6499	−0.637596
$280$	0.141100	0.00843234
$281$	−15.2363	−0.908920	−0.454460	−	0.890767i	$-0.650168\pi$
$281$	−15.2363	−0.908920	−0.454460	+	0.890767i	$0.650168\pi$
$282$	−5.15815	−0.307163
$283$	−8.32069	−0.494614	−0.247307	−	0.968937i	$-0.579546\pi$
$283$	−8.32069	−0.494614	−0.247307	+	0.968937i	$0.579546\pi$
$284$	−5.57139	−0.330601
$285$	−4.28152	−0.253615
$286$	−2.71355	−0.160455
$287$	5.86902	0.346438
$288$	8.07165	0.475626
$289$	0	0
$290$	−3.49117	−0.205009
$291$	−14.5657	−0.853857
$292$	11.8194	0.691679
$293$	−14.6794	−0.857577	−0.428789	−	0.903405i	$-0.641060\pi$
$293$	−14.6794	−0.857577	−0.428789	+	0.903405i	$0.641060\pi$
$294$	−2.02180	−0.117913
$295$	−3.62529	−0.211073
$296$	0.829823	0.0482325
$297$	−3.96044	−0.229808
$298$	−5.02823	−0.291278
$299$	−3.09959	−0.179254
$300$	−9.11494	−0.526251
$301$	−9.17251	−0.528695
$302$	−11.3020	−0.650358
$303$	−3.01541	−0.173231
$304$	−20.5265	−1.17727
$305$	7.62938	0.436857
$306$	0	0
$307$	9.34393	0.533286	0.266643	−	0.963795i	$-0.414085\pi$
$307$	9.34393	0.533286	0.266643	+	0.963795i	$0.414085\pi$
$308$	−8.26803	−0.471115
$309$	−12.9090	−0.734370
$310$	−17.1432	−0.973667
$311$	−12.5466	−0.711455	−0.355727	−	0.934590i	$-0.615767\pi$
$311$	−12.5466	−0.711455	−0.355727	+	0.934590i	$0.615767\pi$
$312$	0.0600589	0.00340017
$313$	11.4295	0.646033	0.323017	−	0.946393i	$-0.395303\pi$
$313$	11.4295	0.646033	0.323017	+	0.946393i	$0.395303\pi$
$314$	16.4540	0.928554
$315$	−0.796171	−0.0448592
$316$	1.75811	0.0989015
$317$	−1.27266	−0.0714799	−0.0357400	−	0.999361i	$-0.511379\pi$
$317$	−1.27266	−0.0714799	−0.0357400	+	0.999361i	$0.511379\pi$
$318$	−8.36970	−0.469349
$319$	8.58955	0.480923
$320$	6.91492	0.386556
$321$	−5.98695	−0.334159
$322$	−18.4920	−1.03052
$323$	0	0
$324$	2.08766	0.115981
$325$	1.47963	0.0820749
$326$	−41.9681	−2.32440
$327$	17.0321	0.941877
$328$	−1.04013	−0.0574314
$329$	2.55127	0.140656
$330$	−6.37510	−0.350937
$331$	−32.8446	−1.80530	−0.902651	−	0.430373i	$-0.858382\pi$
$331$	−32.8446	−1.80530	−0.902651	+	0.430373i	$0.858382\pi$
$332$	1.22393	0.0671717
$333$	−4.68236	−0.256592
$334$	10.6265	0.581455
$335$	−2.23557	−0.122142
$336$	−3.81700	−0.208235
$337$	−11.3437	−0.617930	−0.308965	−	0.951073i	$-0.599983\pi$
$337$	−11.3437	−0.617930	−0.308965	+	0.951073i	$0.599983\pi$
$338$	26.0511	1.41700
$339$	−15.5851	−0.846469
$340$	0	0
$341$	42.1784	2.28409
$342$	−10.8725	−0.587917
$343$	1.00000	0.0539949
$344$	1.62558	0.0876455
$345$	−7.28206	−0.392053
$346$	29.3159	1.57603
$347$	8.67779	0.465848	0.232924	−	0.972495i	$-0.425171\pi$
$347$	8.67779	0.465848	0.232924	+	0.972495i	$0.425171\pi$
$348$	−4.52779	−0.242715
$349$	−11.6079	−0.621358	−0.310679	−	0.950515i	$-0.600556\pi$
$349$	−11.6079	−0.621358	−0.310679	+	0.950515i	$0.600556\pi$
$350$	8.82738	0.471843
$351$	−0.338889	−0.0180886
$352$	−31.9672	−1.70386
$353$	−21.5750	−1.14832	−0.574161	−	0.818742i	$-0.694672\pi$
$353$	−21.5750	−1.14832	−0.574161	+	0.818742i	$0.694672\pi$
$354$	−9.20606	−0.489297
$355$	2.12477	0.112771
$356$	36.7722	1.94892
$357$	0	0
$358$	−19.4198	−1.02637
$359$	14.7447	0.778195	0.389098	−	0.921196i	$-0.372787\pi$
$359$	14.7447	0.778195	0.389098	+	0.921196i	$0.372787\pi$
$360$	0.141100	0.00743662
$361$	9.91899	0.522052
$362$	−16.3939	−0.861645
$363$	4.68505	0.245901
$364$	−0.707484	−0.0370822
$365$	−4.50758	−0.235937
$366$	19.3740	1.01270
$367$	23.2530	1.21380	0.606898	−	0.794780i	$-0.292414\pi$
$367$	23.2530	1.21380	0.606898	+	0.794780i	$0.292414\pi$
$368$	−34.9116	−1.81989
$369$	5.86902	0.305529
$370$	−7.53718	−0.391839
$371$	4.13973	0.214924
$372$	−22.2334	−1.15275
$373$	8.22143	0.425689	0.212845	−	0.977086i	$-0.431727\pi$
$373$	8.22143	0.425689	0.212845	+	0.977086i	$0.431727\pi$
$374$	0	0
$375$	7.45703	0.385079
$376$	−0.452144	−0.0233176
$377$	0.734996	0.0378542
$378$	−2.02180	−0.103990
$379$	−36.7926	−1.88991	−0.944956	−	0.327199i	$-0.893895\pi$
$379$	−36.7926	−1.88991	−0.944956	+	0.327199i	$0.893895\pi$
$380$	−8.93835	−0.458528
$381$	−8.71907	−0.446691
$382$	−22.0065	−1.12595
$383$	20.2911	1.03683	0.518413	−	0.855130i	$-0.326523\pi$
$383$	20.2911	1.03683	0.518413	+	0.855130i	$0.326523\pi$
$384$	1.41642	0.0722815
$385$	3.15319	0.160701
$386$	39.8127	2.02641
$387$	−9.17251	−0.466265
$388$	−30.4082	−1.54374
$389$	0.186762	0.00946923	0.00473461	−	0.999989i	$-0.498493\pi$
$389$	0.186762	0.00946923	0.00473461	+	0.999989i	$0.498493\pi$
$390$	−0.545508	−0.0276229
$391$	0	0
$392$	−0.177223	−0.00895112
$393$	−17.5095	−0.883235
$394$	12.9165	0.650721
$395$	−0.670492	−0.0337361
$396$	−8.26803	−0.415484
$397$	−23.8728	−1.19814	−0.599071	−	0.800696i	$-0.704463\pi$
$397$	−23.8728	−1.19814	−0.599071	+	0.800696i	$0.704463\pi$
$398$	−14.2956	−0.716575
$399$	5.37764	0.269219
$400$	16.6655	0.833273
$401$	−33.3154	−1.66369	−0.831846	−	0.555006i	$-0.812716\pi$
$401$	−33.3154	−1.66369	−0.831846	+	0.555006i	$0.812716\pi$
$402$	−5.67701	−0.283143
$403$	3.60915	0.179785
$404$	−6.29515	−0.313195
$405$	−0.796171	−0.0395621
$406$	4.38495	0.217621
$407$	18.5442	0.919202
$408$	0	0
$409$	−19.1061	−0.944734	−0.472367	−	0.881402i	$-0.656600\pi$
$409$	−19.1061	−0.944734	−0.472367	+	0.881402i	$0.656600\pi$
$410$	9.44734	0.466571
$411$	−18.4988	−0.912478
$412$	−26.9497	−1.32771
$413$	4.55341	0.224059
$414$	−18.4920	−0.908834
$415$	−0.466770	−0.0229128
$416$	−2.73539	−0.134114
$417$	10.4827	0.513340
$418$	43.0598	2.10612
$419$	17.6152	0.860557	0.430278	−	0.902696i	$-0.358415\pi$
$419$	17.6152	0.860557	0.430278	+	0.902696i	$0.358415\pi$
$420$	−1.66213	−0.0811038
$421$	2.74504	0.133785	0.0668926	−	0.997760i	$-0.478692\pi$
$421$	2.74504	0.133785	0.0668926	+	0.997760i	$0.478692\pi$
$422$	44.4556	2.16406
$423$	2.55127	0.124047
$424$	−0.733657	−0.0356295
$425$	0	0
$426$	5.39563	0.261419
$427$	−9.58258	−0.463734
$428$	−12.4987	−0.604147
$429$	1.34215	0.0647995
$430$	−14.7650	−0.712029
$431$	−25.2262	−1.21511	−0.607553	−	0.794279i	$-0.707849\pi$
$431$	−25.2262	−1.21511	−0.607553	+	0.794279i	$0.707849\pi$
$432$	−3.81700	−0.183646
$433$	−12.3644	−0.594193	−0.297097	−	0.954847i	$-0.596018\pi$
$433$	−12.3644	−0.594193	−0.297097	+	0.954847i	$0.596018\pi$
$434$	21.5320	1.03357
$435$	1.72677	0.0827922
$436$	35.5572	1.70288
$437$	49.1857	2.35287
$438$	−11.4465	−0.546936
$439$	40.9891	1.95630	0.978152	−	0.207893i	$-0.0666605\pi$
$439$	40.9891	1.95630	0.978152	+	0.207893i	$0.0666605\pi$
$440$	−0.558817	−0.0266406
$441$	1.00000	0.0476190
$442$	0	0
$443$	35.2656	1.67552	0.837759	−	0.546039i	$-0.183865\pi$
$443$	35.2656	1.67552	0.837759	+	0.546039i	$0.183865\pi$
$444$	−9.77516	−0.463909
$445$	−14.0239	−0.664794
$446$	−29.2586	−1.38544
$447$	2.48701	0.117632
$448$	−8.68521	−0.410338
$449$	11.6904	0.551702	0.275851	−	0.961200i	$-0.411040\pi$
$449$	11.6904	0.551702	0.275851	+	0.961200i	$0.411040\pi$
$450$	8.82738	0.416127
$451$	−23.2439	−1.09451
$452$	−32.5364	−1.53039
$453$	5.59009	0.262645
$454$	−16.8994	−0.793128
$455$	0.269814	0.0126491
$456$	−0.953042	−0.0446303
$457$	6.91530	0.323484	0.161742	−	0.986833i	$-0.448289\pi$
$457$	6.91530	0.323484	0.161742	+	0.986833i	$0.448289\pi$
$458$	−0.229050	−0.0107028
$459$	0	0
$460$	−15.2024	−0.708817
$461$	−16.2536	−0.757004	−0.378502	−	0.925601i	$-0.623561\pi$
$461$	−16.2536	−0.757004	−0.378502	+	0.925601i	$0.623561\pi$
$462$	8.00719	0.372528
$463$	2.42960	0.112913	0.0564566	−	0.998405i	$-0.482020\pi$
$463$	2.42960	0.112913	0.0564566	+	0.998405i	$0.482020\pi$
$464$	8.27847	0.384318
$465$	8.47918	0.393213
$466$	9.27493	0.429653
$467$	−4.96726	−0.229857	−0.114929	−	0.993374i	$-0.536664\pi$
$467$	−4.96726	−0.229857	−0.114929	+	0.993374i	$0.536664\pi$
$468$	−0.707484	−0.0327034
$469$	2.80790	0.129657
$470$	4.10677	0.189431
$471$	−8.13832	−0.374994
$472$	−0.806969	−0.0371438
$473$	36.3271	1.67032
$474$	−1.70265	−0.0782051
$475$	−23.4794	−1.07731
$476$	0	0
$477$	4.13973	0.189545
$478$	−19.0225	−0.870071
$479$	−35.8535	−1.63819	−0.819095	−	0.573658i	$-0.805524\pi$
$479$	−35.8535	−1.63819	−0.819095	+	0.573658i	$0.805524\pi$
$480$	−6.42641	−0.293324
$481$	1.58680	0.0723519
$482$	−13.6328	−0.620958
$483$	9.14634	0.416173
$484$	9.78077	0.444580
$485$	11.5968	0.526584
$486$	−2.02180	−0.0917105
$487$	−35.4441	−1.60612	−0.803062	−	0.595895i	$-0.796797\pi$
$487$	−35.4441	−1.60612	−0.803062	+	0.595895i	$0.796797\pi$
$488$	1.69826	0.0768764
$489$	20.7578	0.938702
$490$	1.60970	0.0727186
$491$	−24.4787	−1.10471	−0.552354	−	0.833609i	$-0.686270\pi$
$491$	−24.4787	−1.10471	−0.552354	+	0.833609i	$0.686270\pi$
$492$	12.2525	0.552386
$493$	0	0
$494$	3.68456	0.165776
$495$	3.15319	0.141725
$496$	40.6509	1.82528
$497$	−2.66873	−0.119709
$498$	−1.18531	−0.0531152
$499$	−34.8150	−1.55853	−0.779267	−	0.626692i	$-0.784408\pi$
$499$	−34.8150	−1.55853	−0.779267	+	0.626692i	$0.784408\pi$
$500$	15.5677	0.696209
$501$	−5.25596	−0.234819
$502$	38.9400	1.73798
$503$	−38.7296	−1.72687	−0.863434	−	0.504462i	$-0.831691\pi$
$503$	−38.7296	−1.72687	−0.863434	+	0.504462i	$0.831691\pi$
$504$	−0.177223	−0.00789414
$505$	2.40079	0.106834
$506$	73.2365	3.25576
$507$	−12.8852	−0.572250
$508$	−18.2024	−0.807602
$509$	−42.7219	−1.89361	−0.946807	−	0.321802i	$-0.895712\pi$
$509$	−42.7219	−1.89361	−0.946807	+	0.321802i	$0.895712\pi$
$510$	0	0
$511$	5.66157	0.250453
$512$	−32.1624	−1.42139
$513$	5.37764	0.237429
$514$	−55.8169	−2.46198
$515$	10.2778	0.452895
$516$	−19.1491	−0.842990
$517$	−10.1041	−0.444380
$518$	9.46678	0.415946
$519$	−14.4999	−0.636477
$520$	−0.0478172	−0.00209692
$521$	−15.0271	−0.658351	−0.329175	−	0.944269i	$-0.606771\pi$
$521$	−15.0271	−0.658351	−0.329175	+	0.944269i	$0.606771\pi$
$522$	4.38495	0.191924
$523$	35.4305	1.54927	0.774634	−	0.632410i	$-0.217934\pi$
$523$	35.4305	1.54927	0.774634	+	0.632410i	$0.217934\pi$
$524$	−36.5537	−1.59686
$525$	−4.36611	−0.190553
$526$	28.5111	1.24314
$527$	0	0
$528$	15.1170	0.657883
$529$	60.6556	2.63720
$530$	6.66371	0.289453
$531$	4.55341	0.197601
$532$	11.2267	0.486737
$533$	−1.98895	−0.0861509
$534$	−35.6121	−1.54109
$535$	4.76664	0.206080
$536$	−0.497625	−0.0214941
$537$	9.60523	0.414496
$538$	−12.2666	−0.528849
$539$	−3.96044	−0.170588
$540$	−1.66213	−0.0715268
$541$	−28.4223	−1.22197	−0.610984	−	0.791643i	$-0.709226\pi$
$541$	−28.4223	−1.22197	−0.610984	+	0.791643i	$0.709226\pi$
$542$	21.4144	0.919826
$543$	8.10859	0.347973
$544$	0	0
$545$	−13.5605	−0.580867
$546$	0.685164	0.0293223
$547$	−38.3873	−1.64132	−0.820660	−	0.571416i	$-0.806394\pi$
$547$	−38.3873	−1.64132	−0.820660	+	0.571416i	$0.806394\pi$
$548$	−38.6191	−1.64973
$549$	−9.58258	−0.408975
$550$	−34.9603	−1.49071
$551$	−11.6632	−0.496871
$552$	−1.62094	−0.0689919
$553$	0.842146	0.0358117
$554$	−26.4183	−1.12240
$555$	3.72796	0.158243
$556$	21.8843	0.928100
$557$	29.2077	1.23757	0.618785	−	0.785561i	$-0.287625\pi$
$557$	29.2077	1.23757	0.618785	+	0.785561i	$0.287625\pi$
$558$	21.5320	0.911523
$559$	3.10846	0.131474
$560$	3.03899	0.128421
$561$	0	0
$562$	30.8046	1.29941
$563$	−7.13474	−0.300694	−0.150347	−	0.988633i	$-0.548039\pi$
$563$	−7.13474	−0.300694	−0.150347	+	0.988633i	$0.548039\pi$
$564$	5.32618	0.224273
$565$	12.4085	0.522027
$566$	16.8227	0.707113
$567$	1.00000	0.0419961
$568$	0.472961	0.0198450
$569$	32.9775	1.38249	0.691245	−	0.722620i	$-0.257062\pi$
$569$	32.9775	1.38249	0.691245	+	0.722620i	$0.257062\pi$
$570$	8.65636	0.362575
$571$	4.68900	0.196229	0.0981143	−	0.995175i	$-0.468719\pi$
$571$	4.68900	0.196229	0.0981143	+	0.995175i	$0.468719\pi$
$572$	2.80194	0.117155
$573$	10.8846	0.454713
$574$	−11.8660	−0.495276
$575$	−39.9340	−1.66536
$576$	−8.68521	−0.361884
$577$	−9.52490	−0.396527	−0.198263	−	0.980149i	$-0.563530\pi$
$577$	−9.52490	−0.396527	−0.198263	+	0.980149i	$0.563530\pi$
$578$	0	0
$579$	−19.6918	−0.818362
$580$	3.60490	0.149685
$581$	0.586268	0.0243225
$582$	29.4489	1.22070
$583$	−16.3951	−0.679018
$584$	−1.00336	−0.0415194
$585$	0.269814	0.0111554
$586$	29.6787	1.22601
$587$	−6.30065	−0.260056	−0.130028	−	0.991510i	$-0.541507\pi$
$587$	−6.30065	−0.260056	−0.130028	+	0.991510i	$0.541507\pi$
$588$	2.08766	0.0860935
$589$	−57.2716	−2.35983
$590$	7.32960	0.301755
$591$	−6.38860	−0.262792
$592$	17.8726	0.734559
$593$	44.1123	1.81148	0.905738	−	0.423838i	$-0.139317\pi$
$593$	44.1123	1.81148	0.905738	+	0.423838i	$0.139317\pi$
$594$	8.00719	0.328539
$595$	0	0
$596$	5.19203	0.212674
$597$	7.07076	0.289387
$598$	6.26675	0.256266
$599$	42.9971	1.75682	0.878408	−	0.477912i	$-0.158606\pi$
$599$	42.9971	1.75682	0.878408	+	0.477912i	$0.158606\pi$
$600$	0.773776	0.0315893
$601$	−0.455710	−0.0185888	−0.00929439	−	0.999957i	$-0.502959\pi$
$601$	−0.455710	−0.0185888	−0.00929439	+	0.999957i	$0.502959\pi$
$602$	18.5449	0.755835
$603$	2.80790	0.114347
$604$	11.6702	0.474853
$605$	−3.73010	−0.151650
$606$	6.09655	0.247655
$607$	−14.9373	−0.606285	−0.303143	−	0.952945i	$-0.598036\pi$
$607$	−14.9373	−0.606285	−0.303143	+	0.952945i	$0.598036\pi$
$608$	43.4064	1.76036
$609$	−2.16884	−0.0878859
$610$	−15.4250	−0.624542
$611$	−0.864598	−0.0349779
$612$	0	0
$613$	3.36159	0.135773	0.0678866	−	0.997693i	$-0.478374\pi$
$613$	3.36159	0.135773	0.0678866	+	0.997693i	$0.478374\pi$
$614$	−18.8915	−0.762400
$615$	−4.67275	−0.188423
$616$	0.701881	0.0282796
$617$	−2.51120	−0.101097	−0.0505485	−	0.998722i	$-0.516097\pi$
$617$	−2.51120	−0.101097	−0.0505485	+	0.998722i	$0.516097\pi$
$618$	26.0995	1.04987
$619$	−32.7230	−1.31525	−0.657625	−	0.753345i	$-0.728439\pi$
$619$	−32.7230	−1.31525	−0.657625	+	0.753345i	$0.728439\pi$
$620$	17.7016	0.710914
$621$	9.14634	0.367030
$622$	25.3667	1.01711
$623$	17.6141	0.705695
$624$	1.29354	0.0517831
$625$	15.8935	0.635739
$626$	−23.1081	−0.923585
$627$	−21.2978	−0.850552
$628$	−16.9900	−0.677976
$629$	0	0
$630$	1.60970	0.0641318
$631$	4.00974	0.159625	0.0798126	−	0.996810i	$-0.474568\pi$
$631$	4.00974	0.159625	0.0798126	+	0.996810i	$0.474568\pi$
$632$	−0.149248	−0.00593675
$633$	−21.9882	−0.873952
$634$	2.57307	0.102190
$635$	6.94187	0.275480
$636$	8.64234	0.342691
$637$	−0.338889	−0.0134273
$638$	−17.3663	−0.687539
$639$	−2.66873	−0.105573
$640$	−1.12772	−0.0445769
$641$	35.9660	1.42057	0.710286	−	0.703913i	$-0.248565\pi$
$641$	35.9660	1.42057	0.710286	+	0.703913i	$0.248565\pi$
$642$	12.1044	0.477722
$643$	29.0241	1.14460	0.572300	−	0.820044i	$-0.306051\pi$
$643$	29.0241	1.14460	0.572300	+	0.820044i	$0.306051\pi$
$644$	19.0944	0.752426
$645$	7.30289	0.287551
$646$	0	0
$647$	16.9266	0.665452	0.332726	−	0.943023i	$-0.392032\pi$
$647$	16.9266	0.665452	0.332726	+	0.943023i	$0.392032\pi$
$648$	−0.177223	−0.00696198
$649$	−18.0335	−0.707876
$650$	−2.99150	−0.117336
$651$	−10.6499	−0.417404
$652$	43.3352	1.69714
$653$	−27.6212	−1.08090	−0.540450	−	0.841376i	$-0.681746\pi$
$653$	−27.6212	−1.08090	−0.540450	+	0.841376i	$0.681746\pi$
$654$	−34.4354	−1.34653
$655$	13.9405	0.544702
$656$	−22.4021	−0.874654
$657$	5.66157	0.220879
$658$	−5.15815	−0.201086
$659$	35.3066	1.37535	0.687676	−	0.726018i	$-0.258631\pi$
$659$	35.3066	1.37535	0.687676	+	0.726018i	$0.258631\pi$
$660$	6.58277	0.256234
$661$	−22.7708	−0.885683	−0.442842	−	0.896600i	$-0.646030\pi$
$661$	−22.7708	−0.885683	−0.442842	+	0.896600i	$0.646030\pi$
$662$	66.4050	2.58091
$663$	0	0
$664$	−0.103900	−0.00403211
$665$	−4.28152	−0.166030
$666$	9.46678	0.366830
$667$	−19.8370	−0.768090
$668$	−10.9726	−0.424545
$669$	14.4716	0.559505
$670$	4.51987	0.174618
$671$	37.9512	1.46509
$672$	8.07165	0.311371
$673$	−29.2950	−1.12924	−0.564620	−	0.825351i	$-0.690977\pi$
$673$	−29.2950	−1.12924	−0.564620	+	0.825351i	$0.690977\pi$
$674$	22.9346	0.883408
$675$	−4.36611	−0.168052
$676$	−26.8998	−1.03461
$677$	−38.3426	−1.47363	−0.736814	−	0.676096i	$-0.763670\pi$
$677$	−38.3426	−1.47363	−0.736814	+	0.676096i	$0.763670\pi$
$678$	31.5100	1.21013
$679$	−14.5657	−0.558981
$680$	0	0
$681$	8.35861	0.320303
$682$	−85.2761	−3.26539
$683$	−7.80932	−0.298815	−0.149408	−	0.988776i	$-0.547737\pi$
$683$	−7.80932	−0.298815	−0.149408	+	0.988776i	$0.547737\pi$
$684$	11.2267	0.429262
$685$	14.7282	0.562736
$686$	−2.02180	−0.0771925
$687$	0.113291	0.00432230
$688$	35.0115	1.33480
$689$	−1.40291	−0.0534466
$690$	14.7228	0.560489
$691$	−14.1041	−0.536546	−0.268273	−	0.963343i	$-0.586453\pi$
$691$	−14.1041	−0.536546	−0.268273	+	0.963343i	$0.586453\pi$
$692$	−30.2709	−1.15073
$693$	−3.96044	−0.150444
$694$	−17.5447	−0.665988
$695$	−8.34603	−0.316583
$696$	0.384369	0.0145695
$697$	0	0
$698$	23.4689	0.888309
$699$	−4.58747	−0.173514
$700$	−9.11494	−0.344512
$701$	−24.3756	−0.920654	−0.460327	−	0.887750i	$-0.652268\pi$
$701$	−24.3756	−0.920654	−0.460327	+	0.887750i	$0.652268\pi$
$702$	0.685164	0.0258598
$703$	−25.1801	−0.949684
$704$	34.3972	1.29639
$705$	−2.03125	−0.0765013
$706$	43.6203	1.64167
$707$	−3.01541	−0.113406
$708$	9.50595	0.357256
$709$	−36.8892	−1.38540	−0.692701	−	0.721225i	$-0.743579\pi$
$709$	−36.8892	−1.38540	−0.692701	+	0.721225i	$0.743579\pi$
$710$	−4.29584	−0.161220
$711$	0.842146	0.0315829
$712$	−3.12163	−0.116988
$713$	−97.4081	−3.64796
$714$	0	0
$715$	−1.06858	−0.0399626
$716$	20.0524	0.749394
$717$	9.40874	0.351376
$718$	−29.8108	−1.11253
$719$	−43.1378	−1.60877	−0.804384	−	0.594110i	$-0.797504\pi$
$719$	−43.1378	−1.60877	−0.804384	+	0.594110i	$0.797504\pi$
$720$	3.03899	0.113256
$721$	−12.9090	−0.480758
$722$	−20.0542	−0.746339
$723$	6.74293	0.250772
$724$	16.9280	0.629123
$725$	9.46940	0.351685
$726$	−9.47220	−0.351546
$727$	−3.50565	−0.130017	−0.0650087	−	0.997885i	$-0.520708\pi$
$727$	−3.50565	−0.130017	−0.0650087	+	0.997885i	$0.520708\pi$
$728$	0.0600589	0.00222593
$729$	1.00000	0.0370370
$730$	9.11340	0.337302
$731$	0	0
$732$	−20.0051	−0.739411
$733$	−14.4182	−0.532547	−0.266274	−	0.963898i	$-0.585792\pi$
$733$	−14.4182	−0.532547	−0.266274	+	0.963898i	$0.585792\pi$
$734$	−47.0128	−1.73527
$735$	−0.796171	−0.0293672
$736$	73.8261	2.72126
$737$	−11.1205	−0.409630
$738$	−11.8660	−0.436792
$739$	−45.0374	−1.65673	−0.828364	−	0.560191i	$-0.810728\pi$
$739$	−45.0374	−1.65673	−0.828364	+	0.560191i	$0.810728\pi$
$740$	7.78271	0.286098
$741$	−1.82242	−0.0669483
$742$	−8.36970	−0.307261
$743$	−0.171900	−0.00630638	−0.00315319	−	0.999995i	$-0.501004\pi$
$743$	−0.171900	−0.00630638	−0.00315319	+	0.999995i	$0.501004\pi$
$744$	1.88742	0.0691961
$745$	−1.98009	−0.0725448
$746$	−16.6220	−0.608576
$747$	0.586268	0.0214504
$748$	0	0
$749$	−5.98695	−0.218758
$750$	−15.0766	−0.550519
$751$	6.14404	0.224199	0.112100	−	0.993697i	$-0.464242\pi$
$751$	6.14404	0.224199	0.112100	+	0.993697i	$0.464242\pi$
$752$	−9.73821	−0.355116
$753$	−19.2601	−0.701877
$754$	−1.48601	−0.0541173
$755$	−4.45067	−0.161976
$756$	2.08766	0.0759273
$757$	−28.9832	−1.05341	−0.526707	−	0.850047i	$-0.676573\pi$
$757$	−28.9832	−1.05341	−0.526707	+	0.850047i	$0.676573\pi$
$758$	74.3872	2.70186
$759$	−36.2235	−1.31483
$760$	0.758785	0.0275240
$761$	−36.8527	−1.33591	−0.667954	−	0.744202i	$-0.732830\pi$
$761$	−36.8527	−1.33591	−0.667954	+	0.744202i	$0.732830\pi$
$762$	17.6282	0.638601
$763$	17.0321	0.616603
$764$	22.7234	0.822104
$765$	0	0
$766$	−41.0244	−1.48227
$767$	−1.54310	−0.0557181
$768$	14.5067	0.523466
$769$	6.86931	0.247714	0.123857	−	0.992300i	$-0.460474\pi$
$769$	6.86931	0.247714	0.123857	+	0.992300i	$0.460474\pi$
$770$	−6.37510	−0.229742
$771$	27.6076	0.994264
$772$	−41.1096	−1.47957
$773$	−22.1401	−0.796325	−0.398162	−	0.917315i	$-0.630352\pi$
$773$	−22.1401	−0.796325	−0.398162	+	0.917315i	$0.630352\pi$
$774$	18.5449	0.666584
$775$	46.4989	1.67029
$776$	2.58138	0.0926661
$777$	−4.68236	−0.167979
$778$	−0.377595	−0.0135374
$779$	31.5615	1.13081
$780$	0.563278	0.0201686
$781$	10.5693	0.378201
$782$	0	0
$783$	−2.16884	−0.0775080
$784$	−3.81700	−0.136322
$785$	6.47950	0.231263
$786$	35.4005	1.26270
$787$	−16.0847	−0.573358	−0.286679	−	0.958027i	$-0.592551\pi$
$787$	−16.0847	−0.573358	−0.286679	+	0.958027i	$0.592551\pi$
$788$	−13.3372	−0.475118
$789$	−14.1019	−0.502039
$790$	1.35560	0.0482300
$791$	−15.5851	−0.554144
$792$	0.701881	0.0249402
$793$	3.24743	0.115320
$794$	48.2659	1.71289
$795$	−3.29594	−0.116895
$796$	14.7613	0.523201
$797$	−2.05076	−0.0726418	−0.0363209	−	0.999340i	$-0.511564\pi$
$797$	−2.05076	−0.0726418	−0.0363209	+	0.999340i	$0.511564\pi$
$798$	−10.8725	−0.384882
$799$	0	0
$800$	−35.2417	−1.24598
$801$	17.6141	0.622364
$802$	67.3569	2.37846
$803$	−22.4223	−0.791265
$804$	5.86194	0.206735
$805$	−7.28206	−0.256659
$806$	−7.29696	−0.257024
$807$	6.06716	0.213574
$808$	0.534401	0.0188002
$809$	24.9418	0.876906	0.438453	−	0.898754i	$-0.355527\pi$
$809$	24.9418	0.876906	0.438453	+	0.898754i	$0.355527\pi$
$810$	1.60970	0.0565589
$811$	−38.1085	−1.33817	−0.669085	−	0.743186i	$-0.733314\pi$
$811$	−38.1085	−1.33817	−0.669085	+	0.743186i	$0.733314\pi$
$812$	−4.52779	−0.158894
$813$	−10.5918	−0.371469
$814$	−37.4926	−1.31411
$815$	−16.5268	−0.578909
$816$	0	0
$817$	−49.3265	−1.72571
$818$	38.6285	1.35061
$819$	−0.338889	−0.0118417
$820$	−9.75509	−0.340663
$821$	39.9313	1.39361	0.696806	−	0.717259i	$-0.254604\pi$
$821$	39.9313	1.39361	0.696806	+	0.717259i	$0.254604\pi$
$822$	37.4008	1.30450
$823$	−25.3478	−0.883568	−0.441784	−	0.897122i	$-0.645654\pi$
$823$	−25.3478	−0.883568	−0.441784	+	0.897122i	$0.645654\pi$
$824$	2.28778	0.0796987
$825$	17.2917	0.602020
$826$	−9.20606	−0.320320
$827$	−25.0190	−0.869998	−0.434999	−	0.900431i	$-0.643251\pi$
$827$	−25.0190	−0.869998	−0.434999	+	0.900431i	$0.643251\pi$
$828$	19.0944	0.663577
$829$	36.3621	1.26291	0.631455	−	0.775413i	$-0.282458\pi$
$829$	36.3621	1.26291	0.631455	+	0.775413i	$0.282458\pi$
$830$	0.943713	0.0327568
$831$	13.0667	0.453280
$832$	2.94332	0.102041
$833$	0	0
$834$	−21.1939	−0.733884
$835$	4.18465	0.144816
$836$	−44.4625	−1.53777
$837$	−10.6499	−0.368116
$838$	−35.6143	−1.23027
$839$	−47.5867	−1.64287	−0.821437	−	0.570299i	$-0.806827\pi$
$839$	−47.5867	−1.64287	−0.821437	+	0.570299i	$0.806827\pi$
$840$	0.141100	0.00486841
$841$	−24.2961	−0.837798
$842$	−5.54992	−0.191263
$843$	−15.2363	−0.524765
$844$	−45.9038	−1.58007
$845$	10.2588	0.352913
$846$	−5.15815	−0.177341
$847$	4.68505	0.160980
$848$	−15.8014	−0.542622
$849$	−8.32069	−0.285565
$850$	0	0
$851$	−42.8265	−1.46807
$852$	−5.57139	−0.190873
$853$	−40.8389	−1.39830	−0.699148	−	0.714977i	$-0.746437\pi$
$853$	−40.8389	−1.39830	−0.699148	+	0.714977i	$0.746437\pi$
$854$	19.3740	0.662965
$855$	−4.28152	−0.146425
$856$	1.06103	0.0362651
$857$	−29.8891	−1.02099	−0.510496	−	0.859880i	$-0.670538\pi$
$857$	−29.8891	−1.02099	−0.510496	+	0.859880i	$0.670538\pi$
$858$	−2.71355	−0.0926390
$859$	22.8761	0.780524	0.390262	−	0.920704i	$-0.372384\pi$
$859$	22.8761	0.780524	0.390262	+	0.920704i	$0.372384\pi$
$860$	15.2459	0.519882
$861$	5.86902	0.200016
$862$	51.0023	1.73715
$863$	−21.8280	−0.743035	−0.371518	−	0.928426i	$-0.621162\pi$
$863$	−21.8280	−0.743035	−0.371518	+	0.928426i	$0.621162\pi$
$864$	8.07165	0.274603
$865$	11.5444	0.392523
$866$	24.9982	0.849474
$867$	0	0
$868$	−22.2334	−0.754652
$869$	−3.33526	−0.113141
$870$	−3.49117	−0.118362
$871$	−0.951567	−0.0322426
$872$	−3.01848	−0.102219
$873$	−14.5657	−0.492975
$874$	−99.4435	−3.36373
$875$	7.45703	0.252094
$876$	11.8194	0.399341
$877$	15.0344	0.507674	0.253837	−	0.967247i	$-0.418307\pi$
$877$	15.0344	0.507674	0.253837	+	0.967247i	$0.418307\pi$
$878$	−82.8716	−2.79678
$879$	−14.6794	−0.495123
$880$	−12.0357	−0.405724
$881$	−43.6520	−1.47067	−0.735336	−	0.677703i	$-0.762976\pi$
$881$	−43.6520	−1.47067	−0.735336	+	0.677703i	$0.762976\pi$
$882$	−2.02180	−0.0680774
$883$	−5.48802	−0.184687	−0.0923434	−	0.995727i	$-0.529436\pi$
$883$	−5.48802	−0.184687	−0.0923434	+	0.995727i	$0.529436\pi$
$884$	0	0
$885$	−3.62529	−0.121863
$886$	−71.2998	−2.39536
$887$	−10.5926	−0.355664	−0.177832	−	0.984061i	$-0.556908\pi$
$887$	−10.5926	−0.355664	−0.177832	+	0.984061i	$0.556908\pi$
$888$	0.829823	0.0278470
$889$	−8.71907	−0.292428
$890$	28.3534	0.950407
$891$	−3.96044	−0.132680
$892$	30.2118	1.01156
$893$	13.7198	0.459116
$894$	−5.02823	−0.168169
$895$	−7.64741	−0.255625
$896$	1.41642	0.0473194
$897$	−3.09959	−0.103492
$898$	−23.6355	−0.788728
$899$	23.0980	0.770363
$900$	−9.11494	−0.303831
$901$	0	0
$902$	46.9944	1.56474
$903$	−9.17251	−0.305242
$904$	2.76205	0.0918644
$905$	−6.45583	−0.214599
$906$	−11.3020	−0.375484
$907$	4.56022	0.151419	0.0757097	−	0.997130i	$-0.475878\pi$
$907$	4.56022	0.151419	0.0757097	+	0.997130i	$0.475878\pi$
$908$	17.4499	0.579095
$909$	−3.01541	−0.100015
$910$	−0.545508	−0.0180834
$911$	−5.03634	−0.166862	−0.0834308	−	0.996514i	$-0.526588\pi$
$911$	−5.03634	−0.166862	−0.0834308	+	0.996514i	$0.526588\pi$
$912$	−20.5265	−0.679699
$913$	−2.32188	−0.0768429
$914$	−13.9813	−0.462461
$915$	7.62938	0.252219
$916$	0.236512	0.00781457
$917$	−17.5095	−0.578213
$918$	0	0
$919$	41.2557	1.36090	0.680450	−	0.732794i	$-0.261784\pi$
$919$	41.2557	1.36090	0.680450	+	0.732794i	$0.261784\pi$
$920$	1.29055	0.0425481
$921$	9.34393	0.307893
$922$	32.8614	1.08223
$923$	0.904403	0.0297688
$924$	−8.26803	−0.271998
$925$	20.4437	0.672185
$926$	−4.91216	−0.161424
$927$	−12.9090	−0.423989
$928$	−17.5061	−0.574666
$929$	53.2696	1.74772	0.873859	−	0.486180i	$-0.161610\pi$
$929$	53.2696	1.74772	0.873859	+	0.486180i	$0.161610\pi$
$930$	−17.1432	−0.562147
$931$	5.37764	0.176245
$932$	−9.57706	−0.313707
$933$	−12.5466	−0.410759
$934$	10.0428	0.328610
$935$	0	0
$936$	0.0600589	0.00196309
$937$	12.1965	0.398442	0.199221	−	0.979955i	$-0.436159\pi$
$937$	12.1965	0.398442	0.199221	+	0.979955i	$0.436159\pi$
$938$	−5.67701	−0.185361
$939$	11.4295	0.372987
$940$	−4.24055	−0.138312
$941$	26.8763	0.876143	0.438071	−	0.898940i	$-0.355662\pi$
$941$	26.8763	0.876143	0.438071	+	0.898940i	$0.355662\pi$
$942$	16.4540	0.536101
$943$	53.6801	1.74807
$944$	−17.3804	−0.565683
$945$	−0.796171	−0.0258995
$946$	−73.4460	−2.38794
$947$	−28.8441	−0.937308	−0.468654	−	0.883382i	$-0.655261\pi$
$947$	−28.8441	−0.937308	−0.468654	+	0.883382i	$0.655261\pi$
$948$	1.75811	0.0571008
$949$	−1.91864	−0.0622818
$950$	47.4705	1.54015
$951$	−1.27266	−0.0412690
$952$	0	0
$953$	−7.36090	−0.238443	−0.119221	−	0.992868i	$-0.538040\pi$
$953$	−7.36090	−0.238443	−0.119221	+	0.992868i	$0.538040\pi$
$954$	−8.36970	−0.270979
$955$	−8.66604	−0.280427
$956$	19.6422	0.635275
$957$	8.58955	0.277661
$958$	72.4885	2.34200
$959$	−18.4988	−0.597357
$960$	6.91492	0.223178
$961$	82.4214	2.65875
$962$	−3.20819	−0.103436
$963$	−5.98695	−0.192927
$964$	14.0769	0.453387
$965$	15.6780	0.504693
$966$	−18.4920	−0.594972
$967$	−42.3882	−1.36311	−0.681556	−	0.731766i	$-0.738696\pi$
$967$	−42.3882	−1.36311	−0.681556	+	0.731766i	$0.738696\pi$
$968$	−0.830298	−0.0266868
$969$	0	0
$970$	−23.4464	−0.752817
$971$	−28.2002	−0.904988	−0.452494	−	0.891767i	$-0.649466\pi$
$971$	−28.2002	−0.904988	−0.452494	+	0.891767i	$0.649466\pi$
$972$	2.08766	0.0669616
$973$	10.4827	0.336060
$974$	71.6607	2.29616
$975$	1.47963	0.0473860
$976$	36.5768	1.17079
$977$	32.1878	1.02978	0.514890	−	0.857256i	$-0.327833\pi$
$977$	32.1878	1.02978	0.514890	+	0.857256i	$0.327833\pi$
$978$	−41.9681	−1.34199
$979$	−69.7596	−2.22953
$980$	−1.66213	−0.0530949
$981$	17.0321	0.543793
$982$	49.4909	1.57932
$983$	25.5348	0.814433	0.407216	−	0.913332i	$-0.366500\pi$
$983$	25.5348	0.814433	0.407216	+	0.913332i	$0.366500\pi$
$984$	−1.04013	−0.0331580
$985$	5.08642	0.162067
$986$	0	0
$987$	2.55127	0.0812079
$988$	−3.80459	−0.121040
$989$	−83.8950	−2.66770
$990$	−6.37510	−0.202614
$991$	−3.61661	−0.114885	−0.0574426	−	0.998349i	$-0.518295\pi$
$991$	−3.61661	−0.114885	−0.0574426	+	0.998349i	$0.518295\pi$
$992$	−85.9626	−2.72932
$993$	−32.8446	−1.04229
$994$	5.39563	0.171139
$995$	−5.62954	−0.178468
$996$	1.22393	0.0387816
$997$	49.0890	1.55467	0.777333	−	0.629090i	$-0.216572\pi$
$997$	49.0890	1.55467	0.777333	+	0.629090i	$0.216572\pi$
$998$	70.3888	2.22812
$999$	−4.68236	−0.148143

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6069.2.a.bc.1.2	yes	9
17.16	even	2		6069.2.a.z.1.2	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6069.2.a.z.1.2	✓	9	17.16	even	2
6069.2.a.bc.1.2	yes	9	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:	\( N \)	\(=\)	\( 6069 = 3 \cdot 7 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6069.a (trivial)

\(f(q)\)	\(=\)	\(q-2.02180 q^{2} +1.00000 q^{3} +2.08766 q^{4} -0.796171 q^{5} -2.02180 q^{6} +1.00000 q^{7} -0.177223 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.02180 q^{2} +1.00000 q^{3} +2.08766 q^{4} -0.796171 q^{5} -2.02180 q^{6} +1.00000 q^{7} -0.177223 q^{8} +1.00000 q^{9} +1.60970 q^{10} -3.96044 q^{11} +2.08766 q^{12} -0.338889 q^{13} -2.02180 q^{14} -0.796171 q^{15} -3.81700 q^{16} -2.02180 q^{18} +5.37764 q^{19} -1.66213 q^{20} +1.00000 q^{21} +8.00719 q^{22} +9.14634 q^{23} -0.177223 q^{24} -4.36611 q^{25} +0.685164 q^{26} +1.00000 q^{27} +2.08766 q^{28} -2.16884 q^{29} +1.60970 q^{30} -10.6499 q^{31} +8.07165 q^{32} -3.96044 q^{33} -0.796171 q^{35} +2.08766 q^{36} -4.68236 q^{37} -10.8725 q^{38} -0.338889 q^{39} +0.141100 q^{40} +5.86902 q^{41} -2.02180 q^{42} -9.17251 q^{43} -8.26803 q^{44} -0.796171 q^{45} -18.4920 q^{46} +2.55127 q^{47} -3.81700 q^{48} +1.00000 q^{49} +8.82738 q^{50} -0.707484 q^{52} +4.13973 q^{53} -2.02180 q^{54} +3.15319 q^{55} -0.177223 q^{56} +5.37764 q^{57} +4.38495 q^{58} +4.55341 q^{59} -1.66213 q^{60} -9.58258 q^{61} +21.5320 q^{62} +1.00000 q^{63} -8.68521 q^{64} +0.269814 q^{65} +8.00719 q^{66} +2.80790 q^{67} +9.14634 q^{69} +1.60970 q^{70} -2.66873 q^{71} -0.177223 q^{72} +5.66157 q^{73} +9.46678 q^{74} -4.36611 q^{75} +11.2267 q^{76} -3.96044 q^{77} +0.685164 q^{78} +0.842146 q^{79} +3.03899 q^{80} +1.00000 q^{81} -11.8660 q^{82} +0.586268 q^{83} +2.08766 q^{84} +18.5449 q^{86} -2.16884 q^{87} +0.701881 q^{88} +17.6141 q^{89} +1.60970 q^{90} -0.338889 q^{91} +19.0944 q^{92} -10.6499 q^{93} -5.15815 q^{94} -4.28152 q^{95} +8.07165 q^{96} -14.5657 q^{97} -2.02180 q^{98} -3.96044 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} + 6 q^{4} - 3 q^{5} + 9 q^{7} + 9 q^{8} + 9 q^{9}+O(q^{10}) \) 9 * q + 9 * q^3 + 6 * q^4 - 3 * q^5 + 9 * q^7 + 9 * q^8 + 9 * q^9	\( 9 q + 9 q^{3} + 6 q^{4} - 3 q^{5} + 9 q^{7} + 9 q^{8} + 9 q^{9} - 12 q^{10} - 18 q^{11} + 6 q^{12} - 21 q^{13} - 3 q^{15} - 9 q^{19} - 15 q^{20} + 9 q^{21} - 6 q^{22} + 9 q^{24} + 3 q^{26} + 9 q^{27} + 6 q^{28} - 6 q^{29} - 12 q^{30} - 30 q^{31} + 3 q^{32} - 18 q^{33} - 3 q^{35} + 6 q^{36} - 12 q^{37} - 36 q^{38} - 21 q^{39} - 30 q^{40} - 9 q^{41} - 30 q^{44} - 3 q^{45} - 33 q^{46} - 9 q^{47} + 9 q^{49} - 12 q^{50} - 12 q^{52} + 30 q^{55} + 9 q^{56} - 9 q^{57} - 9 q^{58} - 3 q^{59} - 15 q^{60} - 33 q^{61} + 12 q^{62} + 9 q^{63} - 15 q^{64} + 15 q^{65} - 6 q^{66} - 15 q^{67} - 12 q^{70} + 9 q^{72} - 21 q^{73} - 6 q^{74} + 3 q^{76} - 18 q^{77} + 3 q^{78} - 30 q^{79} + 9 q^{81} - 9 q^{82} + 6 q^{84} + 72 q^{86} - 6 q^{87} - 48 q^{88} - 12 q^{89} - 12 q^{90} - 21 q^{91} - 48 q^{92} - 30 q^{93} + 48 q^{94} + 18 q^{95} + 3 q^{96} - 18 q^{99}+O(q^{100}) \) 9 * q + 9 * q^3 + 6 * q^4 - 3 * q^5 + 9 * q^7 + 9 * q^8 + 9 * q^9 - 12 * q^10 - 18 * q^11 + 6 * q^12 - 21 * q^13 - 3 * q^15 - 9 * q^19 - 15 * q^20 + 9 * q^21 - 6 * q^22 + 9 * q^24 + 3 * q^26 + 9 * q^27 + 6 * q^28 - 6 * q^29 - 12 * q^30 - 30 * q^31 + 3 * q^32 - 18 * q^33 - 3 * q^35 + 6 * q^36 - 12 * q^37 - 36 * q^38 - 21 * q^39 - 30 * q^40 - 9 * q^41 - 30 * q^44 - 3 * q^45 - 33 * q^46 - 9 * q^47 + 9 * q^49 - 12 * q^50 - 12 * q^52 + 30 * q^55 + 9 * q^56 - 9 * q^57 - 9 * q^58 - 3 * q^59 - 15 * q^60 - 33 * q^61 + 12 * q^62 + 9 * q^63 - 15 * q^64 + 15 * q^65 - 6 * q^66 - 15 * q^67 - 12 * q^70 + 9 * q^72 - 21 * q^73 - 6 * q^74 + 3 * q^76 - 18 * q^77 + 3 * q^78 - 30 * q^79 + 9 * q^81 - 9 * q^82 + 6 * q^84 + 72 * q^86 - 6 * q^87 - 48 * q^88 - 12 * q^89 - 12 * q^90 - 21 * q^91 - 48 * q^92 - 30 * q^93 + 48 * q^94 + 18 * q^95 + 3 * q^96 - 18 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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