LMFDB - Embedded newform 1690.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1690,2,Mod(1,1690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1690, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1690.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1690 = 2 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1690.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:	$13.4947179416$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	1690.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.73205 q^{6} -4.46410 q^{7} -1.00000 q^{8} +4.46410 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.73205 q^{6} -4.46410 q^{7} -1.00000 q^{8} +4.46410 q^{9} -1.00000 q^{10} +1.73205 q^{11} +2.73205 q^{12} +4.46410 q^{14} +2.73205 q^{15} +1.00000 q^{16} +4.73205 q^{17} -4.46410 q^{18} +7.19615 q^{19} +1.00000 q^{20} -12.1962 q^{21} -1.73205 q^{22} -3.46410 q^{23} -2.73205 q^{24} +1.00000 q^{25} +4.00000 q^{27} -4.46410 q^{28} +2.53590 q^{29} -2.73205 q^{30} +6.73205 q^{31} -1.00000 q^{32} +4.73205 q^{33} -4.73205 q^{34} -4.46410 q^{35} +4.46410 q^{36} +1.19615 q^{37} -7.19615 q^{38} -1.00000 q^{40} -6.92820 q^{41} +12.1962 q^{42} -4.92820 q^{43} +1.73205 q^{44} +4.46410 q^{45} +3.46410 q^{46} +0.464102 q^{47} +2.73205 q^{48} +12.9282 q^{49} -1.00000 q^{50} +12.9282 q^{51} +1.73205 q^{53} -4.00000 q^{54} +1.73205 q^{55} +4.46410 q^{56} +19.6603 q^{57} -2.53590 q^{58} +9.46410 q^{59} +2.73205 q^{60} +3.26795 q^{61} -6.73205 q^{62} -19.9282 q^{63} +1.00000 q^{64} -4.73205 q^{66} +12.3923 q^{67} +4.73205 q^{68} -9.46410 q^{69} +4.46410 q^{70} +8.53590 q^{71} -4.46410 q^{72} +0.732051 q^{73} -1.19615 q^{74} +2.73205 q^{75} +7.19615 q^{76} -7.73205 q^{77} +6.73205 q^{79} +1.00000 q^{80} -2.46410 q^{81} +6.92820 q^{82} -5.66025 q^{83} -12.1962 q^{84} +4.73205 q^{85} +4.92820 q^{86} +6.92820 q^{87} -1.73205 q^{88} -9.00000 q^{89} -4.46410 q^{90} -3.46410 q^{92} +18.3923 q^{93} -0.464102 q^{94} +7.19615 q^{95} -2.73205 q^{96} -8.73205 q^{97} -12.9282 q^{98} +7.73205 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 - 2 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{12} + 2 q^{14} + 2 q^{15} + 2 q^{16} + 6 q^{17} - 2 q^{18} + 4 q^{19} + 2 q^{20} - 14 q^{21} - 2 q^{24} + 2 q^{25} + 8 q^{27} - 2 q^{28} + 12 q^{29} - 2 q^{30} + 10 q^{31} - 2 q^{32} + 6 q^{33} - 6 q^{34} - 2 q^{35} + 2 q^{36} - 8 q^{37} - 4 q^{38} - 2 q^{40} + 14 q^{42} + 4 q^{43} + 2 q^{45} - 6 q^{47} + 2 q^{48} + 12 q^{49} - 2 q^{50} + 12 q^{51} - 8 q^{54} + 2 q^{56} + 22 q^{57} - 12 q^{58} + 12 q^{59} + 2 q^{60} + 10 q^{61} - 10 q^{62} - 26 q^{63} + 2 q^{64} - 6 q^{66} + 4 q^{67} + 6 q^{68} - 12 q^{69} + 2 q^{70} + 24 q^{71} - 2 q^{72} - 2 q^{73} + 8 q^{74} + 2 q^{75} + 4 q^{76} - 12 q^{77} + 10 q^{79} + 2 q^{80} + 2 q^{81} + 6 q^{83} - 14 q^{84} + 6 q^{85} - 4 q^{86} - 18 q^{89} - 2 q^{90} + 16 q^{93} + 6 q^{94} + 4 q^{95} - 2 q^{96} - 14 q^{97} - 12 q^{98} + 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 - 2 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^10 + 2 * q^12 + 2 * q^14 + 2 * q^15 + 2 * q^16 + 6 * q^17 - 2 * q^18 + 4 * q^19 + 2 * q^20 - 14 * q^21 - 2 * q^24 + 2 * q^25 + 8 * q^27 - 2 * q^28 + 12 * q^29 - 2 * q^30 + 10 * q^31 - 2 * q^32 + 6 * q^33 - 6 * q^34 - 2 * q^35 + 2 * q^36 - 8 * q^37 - 4 * q^38 - 2 * q^40 + 14 * q^42 + 4 * q^43 + 2 * q^45 - 6 * q^47 + 2 * q^48 + 12 * q^49 - 2 * q^50 + 12 * q^51 - 8 * q^54 + 2 * q^56 + 22 * q^57 - 12 * q^58 + 12 * q^59 + 2 * q^60 + 10 * q^61 - 10 * q^62 - 26 * q^63 + 2 * q^64 - 6 * q^66 + 4 * q^67 + 6 * q^68 - 12 * q^69 + 2 * q^70 + 24 * q^71 - 2 * q^72 - 2 * q^73 + 8 * q^74 + 2 * q^75 + 4 * q^76 - 12 * q^77 + 10 * q^79 + 2 * q^80 + 2 * q^81 + 6 * q^83 - 14 * q^84 + 6 * q^85 - 4 * q^86 - 18 * q^89 - 2 * q^90 + 16 * q^93 + 6 * q^94 + 4 * q^95 - 2 * q^96 - 14 * q^97 - 12 * q^98 + 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	2.73205	1.57735	0.788675	−	0.614810i	$-0.210767\pi$
$3$	2.73205	1.57735	0.788675	+	0.614810i	$0.210767\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.73205	−1.11536
$7$	−4.46410	−1.68727	−0.843636	−	0.536916i	$-0.819589\pi$
$7$	−4.46410	−1.68727	−0.843636	+	0.536916i	$0.819589\pi$
$8$	−1.00000	−0.353553
$9$	4.46410	1.48803
$10$	−1.00000	−0.316228
$11$	1.73205	0.522233	0.261116	−	0.965307i	$-0.415909\pi$
$11$	1.73205	0.522233	0.261116	+	0.965307i	$0.415909\pi$
$12$	2.73205	0.788675
$13$	0	0
$13$	0	0
$14$	4.46410	1.19308
$15$	2.73205	0.705412
$16$	1.00000	0.250000
$17$	4.73205	1.14769	0.573845	−	0.818964i	$-0.305451\pi$
$17$	4.73205	1.14769	0.573845	+	0.818964i	$0.305451\pi$
$18$	−4.46410	−1.05220
$19$	7.19615	1.65091	0.825455	−	0.564467i	$-0.190918\pi$
$19$	7.19615	1.65091	0.825455	+	0.564467i	$0.190918\pi$
$20$	1.00000	0.223607
$21$	−12.1962	−2.66142
$22$	−1.73205	−0.369274
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	−2.73205	−0.557678
$25$	1.00000	0.200000
$26$	0	0
$27$	4.00000	0.769800
$28$	−4.46410	−0.843636
$29$	2.53590	0.470905	0.235452	−	0.971886i	$-0.424343\pi$
$29$	2.53590	0.470905	0.235452	+	0.971886i	$0.424343\pi$
$30$	−2.73205	−0.498802
$31$	6.73205	1.20911	0.604556	−	0.796563i	$-0.293351\pi$
$31$	6.73205	1.20911	0.604556	+	0.796563i	$0.293351\pi$
$32$	−1.00000	−0.176777
$33$	4.73205	0.823744
$34$	−4.73205	−0.811540
$35$	−4.46410	−0.754571
$36$	4.46410	0.744017
$37$	1.19615	0.196646	0.0983231	−	0.995155i	$-0.468652\pi$
$37$	1.19615	0.196646	0.0983231	+	0.995155i	$0.468652\pi$
$38$	−7.19615	−1.16737
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−6.92820	−1.08200	−0.541002	−	0.841021i	$-0.681955\pi$
$41$	−6.92820	−1.08200	−0.541002	+	0.841021i	$0.681955\pi$
$42$	12.1962	1.88191
$43$	−4.92820	−0.751544	−0.375772	−	0.926712i	$-0.622622\pi$
$43$	−4.92820	−0.751544	−0.375772	+	0.926712i	$0.622622\pi$
$44$	1.73205	0.261116
$45$	4.46410	0.665469
$46$	3.46410	0.510754
$47$	0.464102	0.0676962	0.0338481	−	0.999427i	$-0.489224\pi$
$47$	0.464102	0.0676962	0.0338481	+	0.999427i	$0.489224\pi$
$48$	2.73205	0.394338
$49$	12.9282	1.84689
$50$	−1.00000	−0.141421
$51$	12.9282	1.81031
$52$	0	0
$53$	1.73205	0.237915	0.118958	−	0.992899i	$-0.462045\pi$
$53$	1.73205	0.237915	0.118958	+	0.992899i	$0.462045\pi$
$54$	−4.00000	−0.544331
$55$	1.73205	0.233550
$56$	4.46410	0.596541
$57$	19.6603	2.60406
$58$	−2.53590	−0.332980
$59$	9.46410	1.23212	0.616061	−	0.787699i	$-0.288728\pi$
$59$	9.46410	1.23212	0.616061	+	0.787699i	$0.288728\pi$
$60$	2.73205	0.352706
$61$	3.26795	0.418418	0.209209	−	0.977871i	$-0.432911\pi$
$61$	3.26795	0.418418	0.209209	+	0.977871i	$0.432911\pi$
$62$	−6.73205	−0.854971
$63$	−19.9282	−2.51072
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.73205	−0.582475
$67$	12.3923	1.51396	0.756980	−	0.653437i	$-0.226674\pi$
$67$	12.3923	1.51396	0.756980	+	0.653437i	$0.226674\pi$
$68$	4.73205	0.573845
$69$	−9.46410	−1.13934
$70$	4.46410	0.533562
$71$	8.53590	1.01302	0.506512	−	0.862233i	$-0.330934\pi$
$71$	8.53590	1.01302	0.506512	+	0.862233i	$0.330934\pi$
$72$	−4.46410	−0.526099
$73$	0.732051	0.0856801	0.0428400	−	0.999082i	$-0.486359\pi$
$73$	0.732051	0.0856801	0.0428400	+	0.999082i	$0.486359\pi$
$74$	−1.19615	−0.139050
$75$	2.73205	0.315470
$76$	7.19615	0.825455
$77$	−7.73205	−0.881149
$78$	0	0
$79$	6.73205	0.757415	0.378707	−	0.925516i	$-0.376369\pi$
$79$	6.73205	0.757415	0.378707	+	0.925516i	$0.376369\pi$
$80$	1.00000	0.111803
$81$	−2.46410	−0.273789
$82$	6.92820	0.765092
$83$	−5.66025	−0.621294	−0.310647	−	0.950525i	$-0.600546\pi$
$83$	−5.66025	−0.621294	−0.310647	+	0.950525i	$0.600546\pi$
$84$	−12.1962	−1.33071
$85$	4.73205	0.513263
$86$	4.92820	0.531422
$87$	6.92820	0.742781
$88$	−1.73205	−0.184637
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	−4.46410	−0.470558
$91$	0	0
$92$	−3.46410	−0.361158
$93$	18.3923	1.90719
$94$	−0.464102	−0.0478684
$95$	7.19615	0.738310
$96$	−2.73205	−0.278839
$97$	−8.73205	−0.886605	−0.443303	−	0.896372i	$-0.646193\pi$
$97$	−8.73205	−0.886605	−0.443303	+	0.896372i	$0.646193\pi$
$98$	−12.9282	−1.30595
$99$	7.73205	0.777100
$100$	1.00000	0.100000
$101$	3.80385	0.378497	0.189248	−	0.981929i	$-0.439395\pi$
$101$	3.80385	0.378497	0.189248	+	0.981929i	$0.439395\pi$
$102$	−12.9282	−1.28008
$103$	−9.53590	−0.939600	−0.469800	−	0.882773i	$-0.655674\pi$
$103$	−9.53590	−0.939600	−0.469800	+	0.882773i	$0.655674\pi$
$104$	0	0
$105$	−12.1962	−1.19022
$106$	−1.73205	−0.168232
$107$	16.7321	1.61755	0.808774	−	0.588119i	$-0.200131\pi$
$107$	16.7321	1.61755	0.808774	+	0.588119i	$0.200131\pi$
$108$	4.00000	0.384900
$109$	−11.8564	−1.13564	−0.567819	−	0.823154i	$-0.692213\pi$
$109$	−11.8564	−1.13564	−0.567819	+	0.823154i	$0.692213\pi$
$110$	−1.73205	−0.165145
$111$	3.26795	0.310180
$112$	−4.46410	−0.421818
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	−19.6603	−1.84135
$115$	−3.46410	−0.323029
$116$	2.53590	0.235452
$117$	0	0
$118$	−9.46410	−0.871241
$119$	−21.1244	−1.93647
$120$	−2.73205	−0.249401
$121$	−8.00000	−0.727273
$122$	−3.26795	−0.295866
$123$	−18.9282	−1.70670
$124$	6.73205	0.604556
$125$	1.00000	0.0894427
$126$	19.9282	1.77535
$127$	−5.39230	−0.478490	−0.239245	−	0.970959i	$-0.576900\pi$
$127$	−5.39230	−0.478490	−0.239245	+	0.970959i	$0.576900\pi$
$128$	−1.00000	−0.0883883
$129$	−13.4641	−1.18545
$130$	0	0
$131$	0.803848	0.0702325	0.0351162	−	0.999383i	$-0.488820\pi$
$131$	0.803848	0.0702325	0.0351162	+	0.999383i	$0.488820\pi$
$132$	4.73205	0.411872
$133$	−32.1244	−2.78553
$134$	−12.3923	−1.07053
$135$	4.00000	0.344265
$136$	−4.73205	−0.405770
$137$	11.6603	0.996203	0.498101	−	0.867119i	$-0.334031\pi$
$137$	11.6603	0.996203	0.498101	+	0.867119i	$0.334031\pi$
$138$	9.46410	0.805638
$139$	−2.26795	−0.192365	−0.0961825	−	0.995364i	$-0.530663\pi$
$139$	−2.26795	−0.192365	−0.0961825	+	0.995364i	$0.530663\pi$
$140$	−4.46410	−0.377285
$141$	1.26795	0.106781
$142$	−8.53590	−0.716317
$143$	0	0
$144$	4.46410	0.372008
$145$	2.53590	0.210595
$146$	−0.732051	−0.0605850
$147$	35.3205	2.91319
$148$	1.19615	0.0983231
$149$	−17.3205	−1.41895	−0.709476	−	0.704730i	$-0.751068\pi$
$149$	−17.3205	−1.41895	−0.709476	+	0.704730i	$0.751068\pi$
$150$	−2.73205	−0.223071
$151$	−12.1962	−0.992509	−0.496254	−	0.868177i	$-0.665292\pi$
$151$	−12.1962	−0.992509	−0.496254	+	0.868177i	$0.665292\pi$
$152$	−7.19615	−0.583685
$153$	21.1244	1.70780
$154$	7.73205	0.623066
$155$	6.73205	0.540731
$156$	0	0
$157$	−10.1244	−0.808012	−0.404006	−	0.914756i	$-0.632382\pi$
$157$	−10.1244	−0.808012	−0.404006	+	0.914756i	$0.632382\pi$
$158$	−6.73205	−0.535573
$159$	4.73205	0.375276
$160$	−1.00000	−0.0790569
$161$	15.4641	1.21874
$162$	2.46410	0.193598
$163$	16.1962	1.26858	0.634290	−	0.773095i	$-0.281292\pi$
$163$	16.1962	1.26858	0.634290	+	0.773095i	$0.281292\pi$
$164$	−6.92820	−0.541002
$165$	4.73205	0.368390
$166$	5.66025	0.439321
$167$	11.5359	0.892675	0.446337	−	0.894865i	$-0.352728\pi$
$167$	11.5359	0.892675	0.446337	+	0.894865i	$0.352728\pi$
$168$	12.1962	0.940954
$169$	0	0
$170$	−4.73205	−0.362932
$171$	32.1244	2.45661
$172$	−4.92820	−0.375772
$173$	−11.1962	−0.851228	−0.425614	−	0.904905i	$-0.639942\pi$
$173$	−11.1962	−0.851228	−0.425614	+	0.904905i	$0.639942\pi$
$174$	−6.92820	−0.525226
$175$	−4.46410	−0.337454
$176$	1.73205	0.130558
$177$	25.8564	1.94349
$178$	9.00000	0.674579
$179$	16.3923	1.22522	0.612609	−	0.790386i	$-0.290120\pi$
$179$	16.3923	1.22522	0.612609	+	0.790386i	$0.290120\pi$
$180$	4.46410	0.332734
$181$	−10.5885	−0.787034	−0.393517	−	0.919317i	$-0.628742\pi$
$181$	−10.5885	−0.787034	−0.393517	+	0.919317i	$0.628742\pi$
$182$	0	0
$183$	8.92820	0.659992
$184$	3.46410	0.255377
$185$	1.19615	0.0879429
$186$	−18.3923	−1.34859
$187$	8.19615	0.599362
$188$	0.464102	0.0338481
$189$	−17.8564	−1.29886
$190$	−7.19615	−0.522064
$191$	−12.5885	−0.910869	−0.455434	−	0.890269i	$-0.650516\pi$
$191$	−12.5885	−0.910869	−0.455434	+	0.890269i	$0.650516\pi$
$192$	2.73205	0.197169
$193$	2.92820	0.210777	0.105388	−	0.994431i	$-0.466391\pi$
$193$	2.92820	0.210777	0.105388	+	0.994431i	$0.466391\pi$
$194$	8.73205	0.626925
$195$	0	0
$196$	12.9282	0.923443
$197$	−25.7321	−1.83333	−0.916666	−	0.399653i	$-0.869131\pi$
$197$	−25.7321	−1.83333	−0.916666	+	0.399653i	$0.869131\pi$
$198$	−7.73205	−0.549493
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	−1.00000	−0.0707107
$201$	33.8564	2.38805
$202$	−3.80385	−0.267638
$203$	−11.3205	−0.794544
$204$	12.9282	0.905155
$205$	−6.92820	−0.483887
$206$	9.53590	0.664398
$207$	−15.4641	−1.07483
$208$	0	0
$209$	12.4641	0.862160
$210$	12.1962	0.841614
$211$	−16.8038	−1.15682	−0.578412	−	0.815745i	$-0.696327\pi$
$211$	−16.8038	−1.15682	−0.578412	+	0.815745i	$0.696327\pi$
$212$	1.73205	0.118958
$213$	23.3205	1.59789
$214$	−16.7321	−1.14378
$215$	−4.92820	−0.336101
$216$	−4.00000	−0.272166
$217$	−30.0526	−2.04010
$218$	11.8564	0.803017
$219$	2.00000	0.135147
$220$	1.73205	0.116775
$221$	0	0
$222$	−3.26795	−0.219330
$223$	−21.7846	−1.45881	−0.729403	−	0.684085i	$-0.760202\pi$
$223$	−21.7846	−1.45881	−0.729403	+	0.684085i	$0.760202\pi$
$224$	4.46410	0.298270
$225$	4.46410	0.297607
$226$	0	0
$227$	23.3205	1.54784	0.773918	−	0.633286i	$-0.218294\pi$
$227$	23.3205	1.54784	0.773918	+	0.633286i	$0.218294\pi$
$228$	19.6603	1.30203
$229$	26.5885	1.75701	0.878507	−	0.477729i	$-0.158540\pi$
$229$	26.5885	1.75701	0.878507	+	0.477729i	$0.158540\pi$
$230$	3.46410	0.228416
$231$	−21.1244	−1.38988
$232$	−2.53590	−0.166490
$233$	−8.19615	−0.536948	−0.268474	−	0.963287i	$-0.586519\pi$
$233$	−8.19615	−0.536948	−0.268474	+	0.963287i	$0.586519\pi$
$234$	0	0
$235$	0.464102	0.0302747
$236$	9.46410	0.616061
$237$	18.3923	1.19471
$238$	21.1244	1.36929
$239$	−13.2679	−0.858232	−0.429116	−	0.903249i	$-0.641175\pi$
$239$	−13.2679	−0.858232	−0.429116	+	0.903249i	$0.641175\pi$
$240$	2.73205	0.176353
$241$	−7.00000	−0.450910	−0.225455	−	0.974254i	$-0.572387\pi$
$241$	−7.00000	−0.450910	−0.225455	+	0.974254i	$0.572387\pi$
$242$	8.00000	0.514259
$243$	−18.7321	−1.20166
$244$	3.26795	0.209209
$245$	12.9282	0.825953
$246$	18.9282	1.20682
$247$	0	0
$248$	−6.73205	−0.427486
$249$	−15.4641	−0.979998
$250$	−1.00000	−0.0632456
$251$	−17.1962	−1.08541	−0.542706	−	0.839923i	$-0.682600\pi$
$251$	−17.1962	−1.08541	−0.542706	+	0.839923i	$0.682600\pi$
$252$	−19.9282	−1.25536
$253$	−6.00000	−0.377217
$254$	5.39230	0.338343
$255$	12.9282	0.809595
$256$	1.00000	0.0625000
$257$	25.8564	1.61288	0.806439	−	0.591317i	$-0.201392\pi$
$257$	25.8564	1.61288	0.806439	+	0.591317i	$0.201392\pi$
$258$	13.4641	0.838238
$259$	−5.33975	−0.331796
$260$	0	0
$261$	11.3205	0.700722
$262$	−0.803848	−0.0496619
$263$	12.4641	0.768569	0.384285	−	0.923215i	$-0.374448\pi$
$263$	12.4641	0.768569	0.384285	+	0.923215i	$0.374448\pi$
$264$	−4.73205	−0.291238
$265$	1.73205	0.106399
$266$	32.1244	1.96967
$267$	−24.5885	−1.50479
$268$	12.3923	0.756980
$269$	22.0526	1.34457	0.672284	−	0.740293i	$-0.265313\pi$
$269$	22.0526	1.34457	0.672284	+	0.740293i	$0.265313\pi$
$270$	−4.00000	−0.243432
$271$	−3.32051	−0.201707	−0.100853	−	0.994901i	$-0.532157\pi$
$271$	−3.32051	−0.201707	−0.100853	+	0.994901i	$0.532157\pi$
$272$	4.73205	0.286923
$273$	0	0
$274$	−11.6603	−0.704422
$275$	1.73205	0.104447
$276$	−9.46410	−0.569672
$277$	−23.9808	−1.44086	−0.720432	−	0.693525i	$-0.756057\pi$
$277$	−23.9808	−1.44086	−0.720432	+	0.693525i	$0.756057\pi$
$278$	2.26795	0.136023
$279$	30.0526	1.79920
$280$	4.46410	0.266781
$281$	−5.07180	−0.302558	−0.151279	−	0.988491i	$-0.548339\pi$
$281$	−5.07180	−0.302558	−0.151279	+	0.988491i	$0.548339\pi$
$282$	−1.26795	−0.0755053
$283$	−23.8564	−1.41812	−0.709058	−	0.705150i	$-0.750880\pi$
$283$	−23.8564	−1.41812	−0.709058	+	0.705150i	$0.750880\pi$
$284$	8.53590	0.506512
$285$	19.6603	1.16457
$286$	0	0
$287$	30.9282	1.82563
$288$	−4.46410	−0.263050
$289$	5.39230	0.317194
$290$	−2.53590	−0.148913
$291$	−23.8564	−1.39849
$292$	0.732051	0.0428400
$293$	−6.80385	−0.397485	−0.198743	−	0.980052i	$-0.563686\pi$
$293$	−6.80385	−0.397485	−0.198743	+	0.980052i	$0.563686\pi$
$294$	−35.3205	−2.05993
$295$	9.46410	0.551021
$296$	−1.19615	−0.0695249
$297$	6.92820	0.402015
$298$	17.3205	1.00335
$299$	0	0
$300$	2.73205	0.157735
$301$	22.0000	1.26806
$302$	12.1962	0.701810
$303$	10.3923	0.597022
$304$	7.19615	0.412728
$305$	3.26795	0.187122
$306$	−21.1244	−1.20760
$307$	−28.2487	−1.61224	−0.806120	−	0.591753i	$-0.798436\pi$
$307$	−28.2487	−1.61224	−0.806120	+	0.591753i	$0.798436\pi$
$308$	−7.73205	−0.440574
$309$	−26.0526	−1.48208
$310$	−6.73205	−0.382355
$311$	21.8038	1.23638	0.618191	−	0.786028i	$-0.287866\pi$
$311$	21.8038	1.23638	0.618191	+	0.786028i	$0.287866\pi$
$312$	0	0
$313$	−5.60770	−0.316966	−0.158483	−	0.987362i	$-0.550660\pi$
$313$	−5.60770	−0.316966	−0.158483	+	0.987362i	$0.550660\pi$
$314$	10.1244	0.571350
$315$	−19.9282	−1.12283
$316$	6.73205	0.378707
$317$	−1.73205	−0.0972817	−0.0486408	−	0.998816i	$-0.515489\pi$
$317$	−1.73205	−0.0972817	−0.0486408	+	0.998816i	$0.515489\pi$
$318$	−4.73205	−0.265360
$319$	4.39230	0.245922
$320$	1.00000	0.0559017
$321$	45.7128	2.55144
$322$	−15.4641	−0.861781
$323$	34.0526	1.89474
$324$	−2.46410	−0.136895
$325$	0	0
$326$	−16.1962	−0.897022
$327$	−32.3923	−1.79130
$328$	6.92820	0.382546
$329$	−2.07180	−0.114222
$330$	−4.73205	−0.260491
$331$	6.14359	0.337682	0.168841	−	0.985643i	$-0.445997\pi$
$331$	6.14359	0.337682	0.168841	+	0.985643i	$0.445997\pi$
$332$	−5.66025	−0.310647
$333$	5.33975	0.292616
$334$	−11.5359	−0.631216
$335$	12.3923	0.677064
$336$	−12.1962	−0.665355
$337$	−0.196152	−0.0106851	−0.00534255	−	0.999986i	$-0.501701\pi$
$337$	−0.196152	−0.0106851	−0.00534255	+	0.999986i	$0.501701\pi$
$338$	0	0
$339$	0	0
$340$	4.73205	0.256631
$341$	11.6603	0.631438
$342$	−32.1244	−1.73709
$343$	−26.4641	−1.42893
$344$	4.92820	0.265711
$345$	−9.46410	−0.509530
$346$	11.1962	0.601909
$347$	−25.2679	−1.35645	−0.678227	−	0.734852i	$-0.737252\pi$
$347$	−25.2679	−1.35645	−0.678227	+	0.734852i	$0.737252\pi$
$348$	6.92820	0.371391
$349$	−17.2679	−0.924332	−0.462166	−	0.886793i	$-0.652928\pi$
$349$	−17.2679	−0.924332	−0.462166	+	0.886793i	$0.652928\pi$
$350$	4.46410	0.238616
$351$	0	0
$352$	−1.73205	−0.0923186
$353$	2.19615	0.116889	0.0584447	−	0.998291i	$-0.481386\pi$
$353$	2.19615	0.116889	0.0584447	+	0.998291i	$0.481386\pi$
$354$	−25.8564	−1.37425
$355$	8.53590	0.453038
$356$	−9.00000	−0.476999
$357$	−57.7128	−3.05449
$358$	−16.3923	−0.866360
$359$	−19.8564	−1.04798	−0.523991	−	0.851724i	$-0.675557\pi$
$359$	−19.8564	−1.04798	−0.523991	+	0.851724i	$0.675557\pi$
$360$	−4.46410	−0.235279
$361$	32.7846	1.72551
$362$	10.5885	0.556517
$363$	−21.8564	−1.14716
$364$	0	0
$365$	0.732051	0.0383173
$366$	−8.92820	−0.466685
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	−3.46410	−0.180579
$369$	−30.9282	−1.61006
$370$	−1.19615	−0.0621850
$371$	−7.73205	−0.401428
$372$	18.3923	0.953597
$373$	−3.07180	−0.159052	−0.0795258	−	0.996833i	$-0.525341\pi$
$373$	−3.07180	−0.159052	−0.0795258	+	0.996833i	$0.525341\pi$
$374$	−8.19615	−0.423813
$375$	2.73205	0.141082
$376$	−0.464102	−0.0239342
$377$	0	0
$378$	17.8564	0.918434
$379$	−7.58846	−0.389793	−0.194896	−	0.980824i	$-0.562437\pi$
$379$	−7.58846	−0.389793	−0.194896	+	0.980824i	$0.562437\pi$
$380$	7.19615	0.369155
$381$	−14.7321	−0.754746
$382$	12.5885	0.644082
$383$	1.60770	0.0821494	0.0410747	−	0.999156i	$-0.486922\pi$
$383$	1.60770	0.0821494	0.0410747	+	0.999156i	$0.486922\pi$
$384$	−2.73205	−0.139419
$385$	−7.73205	−0.394062
$386$	−2.92820	−0.149042
$387$	−22.0000	−1.11832
$388$	−8.73205	−0.443303
$389$	5.66025	0.286986	0.143493	−	0.989651i	$-0.454167\pi$
$389$	5.66025	0.286986	0.143493	+	0.989651i	$0.454167\pi$
$390$	0	0
$391$	−16.3923	−0.828994
$392$	−12.9282	−0.652973
$393$	2.19615	0.110781
$394$	25.7321	1.29636
$395$	6.73205	0.338726
$396$	7.73205	0.388550
$397$	−7.33975	−0.368371	−0.184186	−	0.982891i	$-0.558965\pi$
$397$	−7.33975	−0.368371	−0.184186	+	0.982891i	$0.558965\pi$
$398$	−2.00000	−0.100251
$399$	−87.7654	−4.39376
$400$	1.00000	0.0500000
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	−33.8564	−1.68860
$403$	0	0
$404$	3.80385	0.189248
$405$	−2.46410	−0.122442
$406$	11.3205	0.561827
$407$	2.07180	0.102695
$408$	−12.9282	−0.640041
$409$	15.3923	0.761100	0.380550	−	0.924760i	$-0.375735\pi$
$409$	15.3923	0.761100	0.380550	+	0.924760i	$0.375735\pi$
$410$	6.92820	0.342160
$411$	31.8564	1.57136
$412$	−9.53590	−0.469800
$413$	−42.2487	−2.07892
$414$	15.4641	0.760019
$415$	−5.66025	−0.277851
$416$	0	0
$417$	−6.19615	−0.303427
$418$	−12.4641	−0.609639
$419$	−5.32051	−0.259924	−0.129962	−	0.991519i	$-0.541486\pi$
$419$	−5.32051	−0.259924	−0.129962	+	0.991519i	$0.541486\pi$
$420$	−12.1962	−0.595111
$421$	−19.1244	−0.932064	−0.466032	−	0.884768i	$-0.654317\pi$
$421$	−19.1244	−0.932064	−0.466032	+	0.884768i	$0.654317\pi$
$422$	16.8038	0.817999
$423$	2.07180	0.100734
$424$	−1.73205	−0.0841158
$425$	4.73205	0.229538
$426$	−23.3205	−1.12988
$427$	−14.5885	−0.705985
$428$	16.7321	0.808774
$429$	0	0
$430$	4.92820	0.237659
$431$	21.1244	1.01752	0.508762	−	0.860907i	$-0.330103\pi$
$431$	21.1244	1.01752	0.508762	+	0.860907i	$0.330103\pi$
$432$	4.00000	0.192450
$433$	7.32051	0.351801	0.175901	−	0.984408i	$-0.443716\pi$
$433$	7.32051	0.351801	0.175901	+	0.984408i	$0.443716\pi$
$434$	30.0526	1.44257
$435$	6.92820	0.332182
$436$	−11.8564	−0.567819
$437$	−24.9282	−1.19248
$438$	−2.00000	−0.0955637
$439$	22.4449	1.07123	0.535617	−	0.844461i	$-0.320079\pi$
$439$	22.4449	1.07123	0.535617	+	0.844461i	$0.320079\pi$
$440$	−1.73205	−0.0825723
$441$	57.7128	2.74823
$442$	0	0
$443$	−21.7128	−1.03161	−0.515803	−	0.856707i	$-0.672507\pi$
$443$	−21.7128	−1.03161	−0.515803	+	0.856707i	$0.672507\pi$
$444$	3.26795	0.155090
$445$	−9.00000	−0.426641
$446$	21.7846	1.03153
$447$	−47.3205	−2.23818
$448$	−4.46410	−0.210909
$449$	−21.0000	−0.991051	−0.495526	−	0.868593i	$-0.665025\pi$
$449$	−21.0000	−0.991051	−0.495526	+	0.868593i	$0.665025\pi$
$450$	−4.46410	−0.210440
$451$	−12.0000	−0.565058
$452$	0	0
$453$	−33.3205	−1.56553
$454$	−23.3205	−1.09449
$455$	0	0
$456$	−19.6603	−0.920676
$457$	−7.80385	−0.365049	−0.182524	−	0.983201i	$-0.558427\pi$
$457$	−7.80385	−0.365049	−0.182524	+	0.983201i	$0.558427\pi$
$458$	−26.5885	−1.24240
$459$	18.9282	0.883493
$460$	−3.46410	−0.161515
$461$	22.7321	1.05874	0.529369	−	0.848392i	$-0.322429\pi$
$461$	22.7321	1.05874	0.529369	+	0.848392i	$0.322429\pi$
$462$	21.1244	0.982794
$463$	−3.07180	−0.142759	−0.0713793	−	0.997449i	$-0.522740\pi$
$463$	−3.07180	−0.142759	−0.0713793	+	0.997449i	$0.522740\pi$
$464$	2.53590	0.117726
$465$	18.3923	0.852923
$466$	8.19615	0.379679
$467$	20.7846	0.961797	0.480899	−	0.876776i	$-0.340311\pi$
$467$	20.7846	0.961797	0.480899	+	0.876776i	$0.340311\pi$
$468$	0	0
$469$	−55.3205	−2.55446
$470$	−0.464102	−0.0214074
$471$	−27.6603	−1.27452
$472$	−9.46410	−0.435621
$473$	−8.53590	−0.392481
$474$	−18.3923	−0.844787
$475$	7.19615	0.330182
$476$	−21.1244	−0.968233
$477$	7.73205	0.354026
$478$	13.2679	0.606862
$479$	4.73205	0.216213	0.108106	−	0.994139i	$-0.465521\pi$
$479$	4.73205	0.216213	0.108106	+	0.994139i	$0.465521\pi$
$480$	−2.73205	−0.124700
$481$	0	0
$482$	7.00000	0.318841
$483$	42.2487	1.92238
$484$	−8.00000	−0.363636
$485$	−8.73205	−0.396502
$486$	18.7321	0.849703
$487$	−31.0000	−1.40474	−0.702372	−	0.711810i	$-0.747876\pi$
$487$	−31.0000	−1.40474	−0.702372	+	0.711810i	$0.747876\pi$
$488$	−3.26795	−0.147933
$489$	44.2487	2.00100
$490$	−12.9282	−0.584037
$491$	−15.3397	−0.692273	−0.346137	−	0.938184i	$-0.612507\pi$
$491$	−15.3397	−0.692273	−0.346137	+	0.938184i	$0.612507\pi$
$492$	−18.9282	−0.853349
$493$	12.0000	0.540453
$494$	0	0
$495$	7.73205	0.347530
$496$	6.73205	0.302278
$497$	−38.1051	−1.70925
$498$	15.4641	0.692963
$499$	39.1769	1.75380	0.876900	−	0.480673i	$-0.159608\pi$
$499$	39.1769	1.75380	0.876900	+	0.480673i	$0.159608\pi$
$500$	1.00000	0.0447214
$501$	31.5167	1.40806
$502$	17.1962	0.767502
$503$	15.9282	0.710203	0.355102	−	0.934828i	$-0.384446\pi$
$503$	15.9282	0.710203	0.355102	+	0.934828i	$0.384446\pi$
$504$	19.9282	0.887673
$505$	3.80385	0.169269
$506$	6.00000	0.266733
$507$	0	0
$508$	−5.39230	−0.239245
$509$	24.2487	1.07481	0.537403	−	0.843326i	$-0.319406\pi$
$509$	24.2487	1.07481	0.537403	+	0.843326i	$0.319406\pi$
$510$	−12.9282	−0.572470
$511$	−3.26795	−0.144566
$512$	−1.00000	−0.0441942
$513$	28.7846	1.27087
$514$	−25.8564	−1.14048
$515$	−9.53590	−0.420202
$516$	−13.4641	−0.592724
$517$	0.803848	0.0353532
$518$	5.33975	0.234615
$519$	−30.5885	−1.34268
$520$	0	0
$521$	8.32051	0.364528	0.182264	−	0.983250i	$-0.441657\pi$
$521$	8.32051	0.364528	0.182264	+	0.983250i	$0.441657\pi$
$522$	−11.3205	−0.495485
$523$	2.00000	0.0874539	0.0437269	−	0.999044i	$-0.486077\pi$
$523$	2.00000	0.0874539	0.0437269	+	0.999044i	$0.486077\pi$
$524$	0.803848	0.0351162
$525$	−12.1962	−0.532284
$526$	−12.4641	−0.543461
$527$	31.8564	1.38769
$528$	4.73205	0.205936
$529$	−11.0000	−0.478261
$530$	−1.73205	−0.0752355
$531$	42.2487	1.83344
$532$	−32.1244	−1.39277
$533$	0	0
$534$	24.5885	1.06405
$535$	16.7321	0.723390
$536$	−12.3923	−0.535266
$537$	44.7846	1.93260
$538$	−22.0526	−0.950753
$539$	22.3923	0.964505
$540$	4.00000	0.172133
$541$	−25.1244	−1.08018	−0.540090	−	0.841607i	$-0.681610\pi$
$541$	−25.1244	−1.08018	−0.540090	+	0.841607i	$0.681610\pi$
$542$	3.32051	0.142628
$543$	−28.9282	−1.24143
$544$	−4.73205	−0.202885
$545$	−11.8564	−0.507873
$546$	0	0
$547$	29.7128	1.27043	0.635214	−	0.772336i	$-0.280912\pi$
$547$	29.7128	1.27043	0.635214	+	0.772336i	$0.280912\pi$
$548$	11.6603	0.498101
$549$	14.5885	0.622620
$550$	−1.73205	−0.0738549
$551$	18.2487	0.777421
$552$	9.46410	0.402819
$553$	−30.0526	−1.27796
$554$	23.9808	1.01884
$555$	3.26795	0.138717
$556$	−2.26795	−0.0961825
$557$	−19.9808	−0.846612	−0.423306	−	0.905987i	$-0.639130\pi$
$557$	−19.9808	−0.846612	−0.423306	+	0.905987i	$0.639130\pi$
$558$	−30.0526	−1.27223
$559$	0	0
$560$	−4.46410	−0.188643
$561$	22.3923	0.945404
$562$	5.07180	0.213941
$563$	−34.6410	−1.45994	−0.729972	−	0.683477i	$-0.760467\pi$
$563$	−34.6410	−1.45994	−0.729972	+	0.683477i	$0.760467\pi$
$564$	1.26795	0.0533903
$565$	0	0
$566$	23.8564	1.00276
$567$	11.0000	0.461957
$568$	−8.53590	−0.358158
$569$	−7.39230	−0.309902	−0.154951	−	0.987922i	$-0.549522\pi$
$569$	−7.39230	−0.309902	−0.154951	+	0.987922i	$0.549522\pi$
$570$	−19.6603	−0.823477
$571$	−3.19615	−0.133755	−0.0668774	−	0.997761i	$-0.521304\pi$
$571$	−3.19615	−0.133755	−0.0668774	+	0.997761i	$0.521304\pi$
$572$	0	0
$573$	−34.3923	−1.43676
$574$	−30.9282	−1.29092
$575$	−3.46410	−0.144463
$576$	4.46410	0.186004
$577$	−16.5885	−0.690587	−0.345293	−	0.938495i	$-0.612221\pi$
$577$	−16.5885	−0.690587	−0.345293	+	0.938495i	$0.612221\pi$
$578$	−5.39230	−0.224290
$579$	8.00000	0.332469
$580$	2.53590	0.105297
$581$	25.2679	1.04829
$582$	23.8564	0.988880
$583$	3.00000	0.124247
$584$	−0.732051	−0.0302925
$585$	0	0
$586$	6.80385	0.281064
$587$	−41.3205	−1.70548	−0.852740	−	0.522336i	$-0.825061\pi$
$587$	−41.3205	−1.70548	−0.852740	+	0.522336i	$0.825061\pi$
$588$	35.3205	1.45659
$589$	48.4449	1.99614
$590$	−9.46410	−0.389631
$591$	−70.3013	−2.89181
$592$	1.19615	0.0491616
$593$	39.7128	1.63081	0.815405	−	0.578891i	$-0.196514\pi$
$593$	39.7128	1.63081	0.815405	+	0.578891i	$0.196514\pi$
$594$	−6.92820	−0.284268
$595$	−21.1244	−0.866014
$596$	−17.3205	−0.709476
$597$	5.46410	0.223631
$598$	0	0
$599$	42.0000	1.71607	0.858037	−	0.513588i	$-0.171684\pi$
$599$	42.0000	1.71607	0.858037	+	0.513588i	$0.171684\pi$
$600$	−2.73205	−0.111536
$601$	−39.7846	−1.62285	−0.811424	−	0.584458i	$-0.801307\pi$
$601$	−39.7846	−1.62285	−0.811424	+	0.584458i	$0.801307\pi$
$602$	−22.0000	−0.896653
$603$	55.3205	2.25283
$604$	−12.1962	−0.496254
$605$	−8.00000	−0.325246
$606$	−10.3923	−0.422159
$607$	43.7846	1.77716	0.888581	−	0.458719i	$-0.151692\pi$
$607$	43.7846	1.77716	0.888581	+	0.458719i	$0.151692\pi$
$608$	−7.19615	−0.291843
$609$	−30.9282	−1.25327
$610$	−3.26795	−0.132315
$611$	0	0
$612$	21.1244	0.853901
$613$	−3.87564	−0.156536	−0.0782679	−	0.996932i	$-0.524939\pi$
$613$	−3.87564	−0.156536	−0.0782679	+	0.996932i	$0.524939\pi$
$614$	28.2487	1.14003
$615$	−18.9282	−0.763259
$616$	7.73205	0.311533
$617$	−4.14359	−0.166815	−0.0834074	−	0.996516i	$-0.526580\pi$
$617$	−4.14359	−0.166815	−0.0834074	+	0.996516i	$0.526580\pi$
$618$	26.0526	1.04799
$619$	26.1244	1.05003	0.525013	−	0.851094i	$-0.324060\pi$
$619$	26.1244	1.05003	0.525013	+	0.851094i	$0.324060\pi$
$620$	6.73205	0.270366
$621$	−13.8564	−0.556038
$622$	−21.8038	−0.874255
$623$	40.1769	1.60965
$624$	0	0
$625$	1.00000	0.0400000
$626$	5.60770	0.224129
$627$	34.0526	1.35993
$628$	−10.1244	−0.404006
$629$	5.66025	0.225689
$630$	19.9282	0.793959
$631$	36.3923	1.44875	0.724377	−	0.689404i	$-0.242127\pi$
$631$	36.3923	1.44875	0.724377	+	0.689404i	$0.242127\pi$
$632$	−6.73205	−0.267787
$633$	−45.9090	−1.82472
$634$	1.73205	0.0687885
$635$	−5.39230	−0.213987
$636$	4.73205	0.187638
$637$	0	0
$638$	−4.39230	−0.173893
$639$	38.1051	1.50742
$640$	−1.00000	−0.0395285
$641$	−28.8564	−1.13976	−0.569880	−	0.821728i	$-0.693010\pi$
$641$	−28.8564	−1.13976	−0.569880	+	0.821728i	$0.693010\pi$
$642$	−45.7128	−1.80414
$643$	−12.1962	−0.480969	−0.240485	−	0.970653i	$-0.577306\pi$
$643$	−12.1962	−0.480969	−0.240485	+	0.970653i	$0.577306\pi$
$644$	15.4641	0.609371
$645$	−13.4641	−0.530148
$646$	−34.0526	−1.33978
$647$	15.9282	0.626202	0.313101	−	0.949720i	$-0.398632\pi$
$647$	15.9282	0.626202	0.313101	+	0.949720i	$0.398632\pi$
$648$	2.46410	0.0967991
$649$	16.3923	0.643454
$650$	0	0
$651$	−82.1051	−3.21795
$652$	16.1962	0.634290
$653$	48.1244	1.88325	0.941626	−	0.336661i	$-0.109298\pi$
$653$	48.1244	1.88325	0.941626	+	0.336661i	$0.109298\pi$
$654$	32.3923	1.26664
$655$	0.803848	0.0314089
$656$	−6.92820	−0.270501
$657$	3.26795	0.127495
$658$	2.07180	0.0807670
$659$	34.3923	1.33973	0.669867	−	0.742481i	$-0.266351\pi$
$659$	34.3923	1.33973	0.669867	+	0.742481i	$0.266351\pi$
$660$	4.73205	0.184195
$661$	32.8372	1.27722	0.638609	−	0.769531i	$-0.279510\pi$
$661$	32.8372	1.27722	0.638609	+	0.769531i	$0.279510\pi$
$662$	−6.14359	−0.238778
$663$	0	0
$664$	5.66025	0.219660
$665$	−32.1244	−1.24573
$666$	−5.33975	−0.206911
$667$	−8.78461	−0.340141
$668$	11.5359	0.446337
$669$	−59.5167	−2.30105
$670$	−12.3923	−0.478757
$671$	5.66025	0.218512
$672$	12.1962	0.470477
$673$	31.3205	1.20732	0.603658	−	0.797243i	$-0.293709\pi$
$673$	31.3205	1.20732	0.603658	+	0.797243i	$0.293709\pi$
$674$	0.196152	0.00755551
$675$	4.00000	0.153960
$676$	0	0
$677$	−22.3923	−0.860606	−0.430303	−	0.902684i	$-0.641593\pi$
$677$	−22.3923	−0.860606	−0.430303	+	0.902684i	$0.641593\pi$
$678$	0	0
$679$	38.9808	1.49594
$680$	−4.73205	−0.181466
$681$	63.7128	2.44148
$682$	−11.6603	−0.446494
$683$	−3.46410	−0.132550	−0.0662751	−	0.997801i	$-0.521111\pi$
$683$	−3.46410	−0.132550	−0.0662751	+	0.997801i	$0.521111\pi$
$684$	32.1244	1.22831
$685$	11.6603	0.445515
$686$	26.4641	1.01040
$687$	72.6410	2.77143
$688$	−4.92820	−0.187886
$689$	0	0
$690$	9.46410	0.360292
$691$	−28.3731	−1.07936	−0.539681	−	0.841869i	$-0.681455\pi$
$691$	−28.3731	−1.07936	−0.539681	+	0.841869i	$0.681455\pi$
$692$	−11.1962	−0.425614
$693$	−34.5167	−1.31118
$694$	25.2679	0.959158
$695$	−2.26795	−0.0860282
$696$	−6.92820	−0.262613
$697$	−32.7846	−1.24181
$698$	17.2679	0.653602
$699$	−22.3923	−0.846955
$700$	−4.46410	−0.168727
$701$	3.80385	0.143669	0.0718347	−	0.997417i	$-0.477115\pi$
$701$	3.80385	0.143669	0.0718347	+	0.997417i	$0.477115\pi$
$702$	0	0
$703$	8.60770	0.324645
$704$	1.73205	0.0652791
$705$	1.26795	0.0477537
$706$	−2.19615	−0.0826533
$707$	−16.9808	−0.638627
$708$	25.8564	0.971743
$709$	−38.0526	−1.42909	−0.714547	−	0.699588i	$-0.753367\pi$
$709$	−38.0526	−1.42909	−0.714547	+	0.699588i	$0.753367\pi$
$710$	−8.53590	−0.320347
$711$	30.0526	1.12706
$712$	9.00000	0.337289
$713$	−23.3205	−0.873360
$714$	57.7128	2.15985
$715$	0	0
$716$	16.3923	0.612609
$717$	−36.2487	−1.35373
$718$	19.8564	0.741035
$719$	−10.1436	−0.378292	−0.189146	−	0.981949i	$-0.560572\pi$
$719$	−10.1436	−0.378292	−0.189146	+	0.981949i	$0.560572\pi$
$720$	4.46410	0.166367
$721$	42.5692	1.58536
$722$	−32.7846	−1.22012
$723$	−19.1244	−0.711242
$724$	−10.5885	−0.393517
$725$	2.53590	0.0941809
$726$	21.8564	0.811167
$727$	24.8564	0.921873	0.460937	−	0.887433i	$-0.347514\pi$
$727$	24.8564	0.921873	0.460937	+	0.887433i	$0.347514\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	−0.732051	−0.0270944
$731$	−23.3205	−0.862540
$732$	8.92820	0.329996
$733$	−43.5885	−1.60998	−0.804988	−	0.593291i	$-0.797828\pi$
$733$	−43.5885	−1.60998	−0.804988	+	0.593291i	$0.797828\pi$
$734$	16.0000	0.590571
$735$	35.3205	1.30282
$736$	3.46410	0.127688
$737$	21.4641	0.790640
$738$	30.9282	1.13848
$739$	−9.19615	−0.338286	−0.169143	−	0.985592i	$-0.554100\pi$
$739$	−9.19615	−0.338286	−0.169143	+	0.985592i	$0.554100\pi$
$740$	1.19615	0.0439714
$741$	0	0
$742$	7.73205	0.283853
$743$	−22.3923	−0.821494	−0.410747	−	0.911749i	$-0.634732\pi$
$743$	−22.3923	−0.821494	−0.410747	+	0.911749i	$0.634732\pi$
$744$	−18.3923	−0.674295
$745$	−17.3205	−0.634574
$746$	3.07180	0.112466
$747$	−25.2679	−0.924506
$748$	8.19615	0.299681
$749$	−74.6936	−2.72924
$750$	−2.73205	−0.0997604
$751$	−8.39230	−0.306240	−0.153120	−	0.988208i	$-0.548932\pi$
$751$	−8.39230	−0.306240	−0.153120	+	0.988208i	$0.548932\pi$
$752$	0.464102	0.0169240
$753$	−46.9808	−1.71207
$754$	0	0
$755$	−12.1962	−0.443863
$756$	−17.8564	−0.649431
$757$	−35.9808	−1.30774	−0.653872	−	0.756606i	$-0.726856\pi$
$757$	−35.9808	−1.30774	−0.653872	+	0.756606i	$0.726856\pi$
$758$	7.58846	0.275625
$759$	−16.3923	−0.595003
$760$	−7.19615	−0.261032
$761$	−37.6410	−1.36449	−0.682243	−	0.731126i	$-0.738995\pi$
$761$	−37.6410	−1.36449	−0.682243	+	0.731126i	$0.738995\pi$
$762$	14.7321	0.533686
$763$	52.9282	1.91613
$764$	−12.5885	−0.455434
$765$	21.1244	0.763753
$766$	−1.60770	−0.0580884
$767$	0	0
$768$	2.73205	0.0985844
$769$	32.0000	1.15395	0.576975	−	0.816762i	$-0.304233\pi$
$769$	32.0000	1.15395	0.576975	+	0.816762i	$0.304233\pi$
$770$	7.73205	0.278644
$771$	70.6410	2.54407
$772$	2.92820	0.105388
$773$	32.6603	1.17471	0.587354	−	0.809330i	$-0.300170\pi$
$773$	32.6603	1.17471	0.587354	+	0.809330i	$0.300170\pi$
$774$	22.0000	0.790774
$775$	6.73205	0.241822
$776$	8.73205	0.313462
$777$	−14.5885	−0.523358
$778$	−5.66025	−0.202930
$779$	−49.8564	−1.78629
$780$	0	0
$781$	14.7846	0.529035
$782$	16.3923	0.586188
$783$	10.1436	0.362502
$784$	12.9282	0.461722
$785$	−10.1244	−0.361354
$786$	−2.19615	−0.0783342
$787$	13.6603	0.486935	0.243468	−	0.969909i	$-0.421715\pi$
$787$	13.6603	0.486935	0.243468	+	0.969909i	$0.421715\pi$
$788$	−25.7321	−0.916666
$789$	34.0526	1.21230
$790$	−6.73205	−0.239516
$791$	0	0
$792$	−7.73205	−0.274746
$793$	0	0
$794$	7.33975	0.260478
$795$	4.73205	0.167829
$796$	2.00000	0.0708881
$797$	−5.07180	−0.179652	−0.0898261	−	0.995957i	$-0.528631\pi$
$797$	−5.07180	−0.179652	−0.0898261	+	0.995957i	$0.528631\pi$
$798$	87.7654	3.10686
$799$	2.19615	0.0776943
$800$	−1.00000	−0.0353553
$801$	−40.1769	−1.41958
$802$	−3.00000	−0.105934
$803$	1.26795	0.0447450
$804$	33.8564	1.19402
$805$	15.4641	0.545038
$806$	0	0
$807$	60.2487	2.12086
$808$	−3.80385	−0.133819
$809$	36.0000	1.26569	0.632846	−	0.774277i	$-0.281886\pi$
$809$	36.0000	1.26569	0.632846	+	0.774277i	$0.281886\pi$
$810$	2.46410	0.0865797
$811$	−1.33975	−0.0470448	−0.0235224	−	0.999723i	$-0.507488\pi$
$811$	−1.33975	−0.0470448	−0.0235224	+	0.999723i	$0.507488\pi$
$812$	−11.3205	−0.397272
$813$	−9.07180	−0.318162
$814$	−2.07180	−0.0726164
$815$	16.1962	0.567326
$816$	12.9282	0.452578
$817$	−35.4641	−1.24073
$818$	−15.3923	−0.538179
$819$	0	0
$820$	−6.92820	−0.241943
$821$	−21.1244	−0.737245	−0.368623	−	0.929579i	$-0.620171\pi$
$821$	−21.1244	−0.737245	−0.368623	+	0.929579i	$0.620171\pi$
$822$	−31.8564	−1.11112
$823$	1.78461	0.0622076	0.0311038	−	0.999516i	$-0.490098\pi$
$823$	1.78461	0.0622076	0.0311038	+	0.999516i	$0.490098\pi$
$824$	9.53590	0.332199
$825$	4.73205	0.164749
$826$	42.2487	1.47002
$827$	20.4449	0.710938	0.355469	−	0.934688i	$-0.384321\pi$
$827$	20.4449	0.710938	0.355469	+	0.934688i	$0.384321\pi$
$828$	−15.4641	−0.537415
$829$	−17.8564	−0.620179	−0.310089	−	0.950707i	$-0.600359\pi$
$829$	−17.8564	−0.620179	−0.310089	+	0.950707i	$0.600359\pi$
$830$	5.66025	0.196470
$831$	−65.5167	−2.27275
$832$	0	0
$833$	61.1769	2.11965
$834$	6.19615	0.214555
$835$	11.5359	0.399216
$836$	12.4641	0.431080
$837$	26.9282	0.930775
$838$	5.32051	0.183794
$839$	−31.5167	−1.08808	−0.544038	−	0.839061i	$-0.683105\pi$
$839$	−31.5167	−1.08808	−0.544038	+	0.839061i	$0.683105\pi$
$840$	12.1962	0.420807
$841$	−22.5692	−0.778249
$842$	19.1244	0.659069
$843$	−13.8564	−0.477240
$844$	−16.8038	−0.578412
$845$	0	0
$846$	−2.07180	−0.0712298
$847$	35.7128	1.22711
$848$	1.73205	0.0594789
$849$	−65.1769	−2.23687
$850$	−4.73205	−0.162308
$851$	−4.14359	−0.142041
$852$	23.3205	0.798947
$853$	−16.0000	−0.547830	−0.273915	−	0.961754i	$-0.588319\pi$
$853$	−16.0000	−0.547830	−0.273915	+	0.961754i	$0.588319\pi$
$854$	14.5885	0.499207
$855$	32.1244	1.09863
$856$	−16.7321	−0.571890
$857$	−22.0526	−0.753301	−0.376651	−	0.926355i	$-0.622924\pi$
$857$	−22.0526	−0.753301	−0.376651	+	0.926355i	$0.622924\pi$
$858$	0	0
$859$	−8.94744	−0.305283	−0.152641	−	0.988282i	$-0.548778\pi$
$859$	−8.94744	−0.305283	−0.152641	+	0.988282i	$0.548778\pi$
$860$	−4.92820	−0.168050
$861$	84.4974	2.87966
$862$	−21.1244	−0.719498
$863$	−17.3205	−0.589597	−0.294798	−	0.955559i	$-0.595253\pi$
$863$	−17.3205	−0.589597	−0.294798	+	0.955559i	$0.595253\pi$
$864$	−4.00000	−0.136083
$865$	−11.1962	−0.380681
$866$	−7.32051	−0.248761
$867$	14.7321	0.500327
$868$	−30.0526	−1.02005
$869$	11.6603	0.395547
$870$	−6.92820	−0.234888
$871$	0	0
$872$	11.8564	0.401509
$873$	−38.9808	−1.31930
$874$	24.9282	0.843209
$875$	−4.46410	−0.150914
$876$	2.00000	0.0675737
$877$	−26.3923	−0.891205	−0.445602	−	0.895231i	$-0.647010\pi$
$877$	−26.3923	−0.891205	−0.445602	+	0.895231i	$0.647010\pi$
$878$	−22.4449	−0.757477
$879$	−18.5885	−0.626973
$880$	1.73205	0.0583874
$881$	53.1051	1.78916	0.894578	−	0.446911i	$-0.147476\pi$
$881$	53.1051	1.78916	0.894578	+	0.446911i	$0.147476\pi$
$882$	−57.7128	−1.94329
$883$	3.26795	0.109975	0.0549876	−	0.998487i	$-0.482488\pi$
$883$	3.26795	0.109975	0.0549876	+	0.998487i	$0.482488\pi$
$884$	0	0
$885$	25.8564	0.869154
$886$	21.7128	0.729456
$887$	49.6410	1.66678	0.833391	−	0.552684i	$-0.186396\pi$
$887$	49.6410	1.66678	0.833391	+	0.552684i	$0.186396\pi$
$888$	−3.26795	−0.109665
$889$	24.0718	0.807342
$890$	9.00000	0.301681
$891$	−4.26795	−0.142982
$892$	−21.7846	−0.729403
$893$	3.33975	0.111760
$894$	47.3205	1.58263
$895$	16.3923	0.547934
$896$	4.46410	0.149135
$897$	0	0
$898$	21.0000	0.700779
$899$	17.0718	0.569376
$900$	4.46410	0.148803
$901$	8.19615	0.273053
$902$	12.0000	0.399556
$903$	60.1051	2.00017
$904$	0	0
$905$	−10.5885	−0.351972
$906$	33.3205	1.10700
$907$	2.58846	0.0859483	0.0429742	−	0.999076i	$-0.486317\pi$
$907$	2.58846	0.0859483	0.0429742	+	0.999076i	$0.486317\pi$
$908$	23.3205	0.773918
$909$	16.9808	0.563216
$910$	0	0
$911$	30.9282	1.02470	0.512349	−	0.858778i	$-0.328776\pi$
$911$	30.9282	1.02470	0.512349	+	0.858778i	$0.328776\pi$
$912$	19.6603	0.651016
$913$	−9.80385	−0.324460
$914$	7.80385	0.258128
$915$	8.92820	0.295157
$916$	26.5885	0.878507
$917$	−3.58846	−0.118501
$918$	−18.9282	−0.624724
$919$	39.1769	1.29233	0.646164	−	0.763199i	$-0.276372\pi$
$919$	39.1769	1.29233	0.646164	+	0.763199i	$0.276372\pi$
$920$	3.46410	0.114208
$921$	−77.1769	−2.54307
$922$	−22.7321	−0.748640
$923$	0	0
$924$	−21.1244	−0.694940
$925$	1.19615	0.0393292
$926$	3.07180	0.100946
$927$	−42.5692	−1.39816
$928$	−2.53590	−0.0832449
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	−18.3923	−0.603107
$931$	93.0333	3.04904
$932$	−8.19615	−0.268474
$933$	59.5692	1.95021
$934$	−20.7846	−0.680093
$935$	8.19615	0.268043
$936$	0	0
$937$	27.1769	0.887831	0.443916	−	0.896069i	$-0.353589\pi$
$937$	27.1769	0.887831	0.443916	+	0.896069i	$0.353589\pi$
$938$	55.3205	1.80628
$939$	−15.3205	−0.499966
$940$	0.464102	0.0151373
$941$	19.6077	0.639193	0.319596	−	0.947554i	$-0.396453\pi$
$941$	19.6077	0.639193	0.319596	+	0.947554i	$0.396453\pi$
$942$	27.6603	0.901220
$943$	24.0000	0.781548
$944$	9.46410	0.308030
$945$	−17.8564	−0.580869
$946$	8.53590	0.277526
$947$	33.1244	1.07640	0.538198	−	0.842818i	$-0.319105\pi$
$947$	33.1244	1.07640	0.538198	+	0.842818i	$0.319105\pi$
$948$	18.3923	0.597354
$949$	0	0
$950$	−7.19615	−0.233474
$951$	−4.73205	−0.153447
$952$	21.1244	0.684644
$953$	57.3731	1.85850	0.929248	−	0.369457i	$-0.120456\pi$
$953$	57.3731	1.85850	0.929248	+	0.369457i	$0.120456\pi$
$954$	−7.73205	−0.250334
$955$	−12.5885	−0.407353
$956$	−13.2679	−0.429116
$957$	12.0000	0.387905
$958$	−4.73205	−0.152886
$959$	−52.0526	−1.68086
$960$	2.73205	0.0881766
$961$	14.3205	0.461952
$962$	0	0
$963$	74.6936	2.40697
$964$	−7.00000	−0.225455
$965$	2.92820	0.0942622
$966$	−42.2487	−1.35933
$967$	42.1769	1.35632	0.678159	−	0.734915i	$-0.262778\pi$
$967$	42.1769	1.35632	0.678159	+	0.734915i	$0.262778\pi$
$968$	8.00000	0.257130
$969$	93.0333	2.98866
$970$	8.73205	0.280369
$971$	−30.8038	−0.988543	−0.494271	−	0.869308i	$-0.664565\pi$
$971$	−30.8038	−0.988543	−0.494271	+	0.869308i	$0.664565\pi$
$972$	−18.7321	−0.600831
$973$	10.1244	0.324572
$974$	31.0000	0.993304
$975$	0	0
$976$	3.26795	0.104605
$977$	18.2487	0.583828	0.291914	−	0.956445i	$-0.405708\pi$
$977$	18.2487	0.583828	0.291914	+	0.956445i	$0.405708\pi$
$978$	−44.2487	−1.41492
$979$	−15.5885	−0.498209
$980$	12.9282	0.412976
$981$	−52.9282	−1.68987
$982$	15.3397	0.489511
$983$	3.24871	0.103618	0.0518089	−	0.998657i	$-0.483501\pi$
$983$	3.24871	0.103618	0.0518089	+	0.998657i	$0.483501\pi$
$984$	18.9282	0.603409
$985$	−25.7321	−0.819891
$986$	−12.0000	−0.382158
$987$	−5.66025	−0.180168
$988$	0	0
$989$	17.0718	0.542852
$990$	−7.73205	−0.245741
$991$	−42.7846	−1.35910	−0.679549	−	0.733630i	$-0.737824\pi$
$991$	−42.7846	−1.35910	−0.679549	+	0.733630i	$0.737824\pi$
$992$	−6.73205	−0.213743
$993$	16.7846	0.532643
$994$	38.1051	1.20862
$995$	2.00000	0.0634043
$996$	−15.4641	−0.489999
$997$	32.8038	1.03891	0.519454	−	0.854498i	$-0.326135\pi$
$997$	32.8038	1.03891	0.519454	+	0.854498i	$0.326135\pi$
$998$	−39.1769	−1.24012
$999$	4.78461	0.151378

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1690.2.a.l.1.2		2
5.4	even	2		8450.2.a.bg.1.1		2
13.2	odd	12		1690.2.l.d.1161.1		4
13.3	even	3		130.2.e.d.61.1	✓	4
13.4	even	6		1690.2.e.k.991.1		4
13.5	odd	4		1690.2.d.h.1351.4		4
13.6	odd	12		1690.2.l.c.361.2		4
13.7	odd	12		1690.2.l.d.361.1		4
13.8	odd	4		1690.2.d.h.1351.2		4
13.9	even	3		130.2.e.d.81.1	yes	4
13.10	even	6		1690.2.e.k.191.1		4
13.11	odd	12		1690.2.l.c.1161.2		4
13.12	even	2		1690.2.a.o.1.2		2
39.29	odd	6		1170.2.i.n.451.2		4
39.35	odd	6		1170.2.i.n.991.2		4
52.3	odd	6		1040.2.q.p.321.2		4
52.35	odd	6		1040.2.q.p.81.2		4
65.3	odd	12		650.2.o.a.399.1		4
65.9	even	6		650.2.e.f.601.2		4
65.22	odd	12		650.2.o.a.549.1		4
65.29	even	6		650.2.e.f.451.2		4
65.42	odd	12		650.2.o.e.399.2		4
65.48	odd	12		650.2.o.e.549.2		4
65.64	even	2		8450.2.a.z.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.e.d.61.1	✓	4	13.3	even	3
130.2.e.d.81.1	yes	4	13.9	even	3
650.2.e.f.451.2		4	65.29	even	6
650.2.e.f.601.2		4	65.9	even	6
650.2.o.a.399.1		4	65.3	odd	12
650.2.o.a.549.1		4	65.22	odd	12
650.2.o.e.399.2		4	65.42	odd	12
650.2.o.e.549.2		4	65.48	odd	12
1040.2.q.p.81.2		4	52.35	odd	6
1040.2.q.p.321.2		4	52.3	odd	6
1170.2.i.n.451.2		4	39.29	odd	6
1170.2.i.n.991.2		4	39.35	odd	6
1690.2.a.l.1.2		2	1.1	even	1	trivial
1690.2.a.o.1.2		2	13.12	even	2
1690.2.d.h.1351.2		4	13.8	odd	4
1690.2.d.h.1351.4		4	13.5	odd	4
1690.2.e.k.191.1		4	13.10	even	6
1690.2.e.k.991.1		4	13.4	even	6
1690.2.l.c.361.2		4	13.6	odd	12
1690.2.l.c.1161.2		4	13.11	odd	12
1690.2.l.d.361.1		4	13.7	odd	12
1690.2.l.d.1161.1		4	13.2	odd	12
8450.2.a.z.1.1		2	65.64	even	2
8450.2.a.bg.1.1		2	5.4	even	2

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:	\( N \)	\(=\)	\( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1690.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.73205 q^{6} -4.46410 q^{7} -1.00000 q^{8} +4.46410 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.73205 q^{6} -4.46410 q^{7} -1.00000 q^{8} +4.46410 q^{9} -1.00000 q^{10} +1.73205 q^{11} +2.73205 q^{12} +4.46410 q^{14} +2.73205 q^{15} +1.00000 q^{16} +4.73205 q^{17} -4.46410 q^{18} +7.19615 q^{19} +1.00000 q^{20} -12.1962 q^{21} -1.73205 q^{22} -3.46410 q^{23} -2.73205 q^{24} +1.00000 q^{25} +4.00000 q^{27} -4.46410 q^{28} +2.53590 q^{29} -2.73205 q^{30} +6.73205 q^{31} -1.00000 q^{32} +4.73205 q^{33} -4.73205 q^{34} -4.46410 q^{35} +4.46410 q^{36} +1.19615 q^{37} -7.19615 q^{38} -1.00000 q^{40} -6.92820 q^{41} +12.1962 q^{42} -4.92820 q^{43} +1.73205 q^{44} +4.46410 q^{45} +3.46410 q^{46} +0.464102 q^{47} +2.73205 q^{48} +12.9282 q^{49} -1.00000 q^{50} +12.9282 q^{51} +1.73205 q^{53} -4.00000 q^{54} +1.73205 q^{55} +4.46410 q^{56} +19.6603 q^{57} -2.53590 q^{58} +9.46410 q^{59} +2.73205 q^{60} +3.26795 q^{61} -6.73205 q^{62} -19.9282 q^{63} +1.00000 q^{64} -4.73205 q^{66} +12.3923 q^{67} +4.73205 q^{68} -9.46410 q^{69} +4.46410 q^{70} +8.53590 q^{71} -4.46410 q^{72} +0.732051 q^{73} -1.19615 q^{74} +2.73205 q^{75} +7.19615 q^{76} -7.73205 q^{77} +6.73205 q^{79} +1.00000 q^{80} -2.46410 q^{81} +6.92820 q^{82} -5.66025 q^{83} -12.1962 q^{84} +4.73205 q^{85} +4.92820 q^{86} +6.92820 q^{87} -1.73205 q^{88} -9.00000 q^{89} -4.46410 q^{90} -3.46410 q^{92} +18.3923 q^{93} -0.464102 q^{94} +7.19615 q^{95} -2.73205 q^{96} -8.73205 q^{97} -12.9282 q^{98} +7.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 - 2 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{12} + 2 q^{14} + 2 q^{15} + 2 q^{16} + 6 q^{17} - 2 q^{18} + 4 q^{19} + 2 q^{20} - 14 q^{21} - 2 q^{24} + 2 q^{25} + 8 q^{27} - 2 q^{28} + 12 q^{29} - 2 q^{30} + 10 q^{31} - 2 q^{32} + 6 q^{33} - 6 q^{34} - 2 q^{35} + 2 q^{36} - 8 q^{37} - 4 q^{38} - 2 q^{40} + 14 q^{42} + 4 q^{43} + 2 q^{45} - 6 q^{47} + 2 q^{48} + 12 q^{49} - 2 q^{50} + 12 q^{51} - 8 q^{54} + 2 q^{56} + 22 q^{57} - 12 q^{58} + 12 q^{59} + 2 q^{60} + 10 q^{61} - 10 q^{62} - 26 q^{63} + 2 q^{64} - 6 q^{66} + 4 q^{67} + 6 q^{68} - 12 q^{69} + 2 q^{70} + 24 q^{71} - 2 q^{72} - 2 q^{73} + 8 q^{74} + 2 q^{75} + 4 q^{76} - 12 q^{77} + 10 q^{79} + 2 q^{80} + 2 q^{81} + 6 q^{83} - 14 q^{84} + 6 q^{85} - 4 q^{86} - 18 q^{89} - 2 q^{90} + 16 q^{93} + 6 q^{94} + 4 q^{95} - 2 q^{96} - 14 q^{97} - 12 q^{98} + 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 - 2 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^10 + 2 * q^12 + 2 * q^14 + 2 * q^15 + 2 * q^16 + 6 * q^17 - 2 * q^18 + 4 * q^19 + 2 * q^20 - 14 * q^21 - 2 * q^24 + 2 * q^25 + 8 * q^27 - 2 * q^28 + 12 * q^29 - 2 * q^30 + 10 * q^31 - 2 * q^32 + 6 * q^33 - 6 * q^34 - 2 * q^35 + 2 * q^36 - 8 * q^37 - 4 * q^38 - 2 * q^40 + 14 * q^42 + 4 * q^43 + 2 * q^45 - 6 * q^47 + 2 * q^48 + 12 * q^49 - 2 * q^50 + 12 * q^51 - 8 * q^54 + 2 * q^56 + 22 * q^57 - 12 * q^58 + 12 * q^59 + 2 * q^60 + 10 * q^61 - 10 * q^62 - 26 * q^63 + 2 * q^64 - 6 * q^66 + 4 * q^67 + 6 * q^68 - 12 * q^69 + 2 * q^70 + 24 * q^71 - 2 * q^72 - 2 * q^73 + 8 * q^74 + 2 * q^75 + 4 * q^76 - 12 * q^77 + 10 * q^79 + 2 * q^80 + 2 * q^81 + 6 * q^83 - 14 * q^84 + 6 * q^85 - 4 * q^86 - 18 * q^89 - 2 * q^90 + 16 * q^93 + 6 * q^94 + 4 * q^95 - 2 * q^96 - 14 * q^97 - 12 * q^98 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more