LMFDB - Embedded newform 5082.2.a.bt.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5082,2,Mod(1,5082)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5082, 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("5082.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5082 = 2 \cdot 3 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5082.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:	$40.5799743072$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 462)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	5082.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.61803 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.61803 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.61803 q^{10} +1.00000 q^{12} +2.47214 q^{13} -1.00000 q^{14} -1.61803 q^{15} +1.00000 q^{16} -7.85410 q^{17} +1.00000 q^{18} -1.38197 q^{19} -1.61803 q^{20} -1.00000 q^{21} +1.38197 q^{23} +1.00000 q^{24} -2.38197 q^{25} +2.47214 q^{26} +1.00000 q^{27} -1.00000 q^{28} -9.23607 q^{29} -1.61803 q^{30} +2.09017 q^{31} +1.00000 q^{32} -7.85410 q^{34} +1.61803 q^{35} +1.00000 q^{36} -1.14590 q^{37} -1.38197 q^{38} +2.47214 q^{39} -1.61803 q^{40} +5.32624 q^{41} -1.00000 q^{42} -1.52786 q^{43} -1.61803 q^{45} +1.38197 q^{46} -7.70820 q^{47} +1.00000 q^{48} +1.00000 q^{49} -2.38197 q^{50} -7.85410 q^{51} +2.47214 q^{52} -3.70820 q^{53} +1.00000 q^{54} -1.00000 q^{56} -1.38197 q^{57} -9.23607 q^{58} +3.52786 q^{59} -1.61803 q^{60} +3.52786 q^{61} +2.09017 q^{62} -1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -3.52786 q^{67} -7.85410 q^{68} +1.38197 q^{69} +1.61803 q^{70} -12.0000 q^{71} +1.00000 q^{72} -2.94427 q^{73} -1.14590 q^{74} -2.38197 q^{75} -1.38197 q^{76} +2.47214 q^{78} -5.23607 q^{79} -1.61803 q^{80} +1.00000 q^{81} +5.32624 q^{82} -11.4164 q^{83} -1.00000 q^{84} +12.7082 q^{85} -1.52786 q^{86} -9.23607 q^{87} +11.8541 q^{89} -1.61803 q^{90} -2.47214 q^{91} +1.38197 q^{92} +2.09017 q^{93} -7.70820 q^{94} +2.23607 q^{95} +1.00000 q^{96} -12.9443 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 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 - 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} - q^{5} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} - q^{10} + 2 q^{12} - 4 q^{13} - 2 q^{14} - q^{15} + 2 q^{16} - 9 q^{17} + 2 q^{18} - 5 q^{19} - q^{20} - 2 q^{21} + 5 q^{23} + 2 q^{24} - 7 q^{25} - 4 q^{26} + 2 q^{27} - 2 q^{28} - 14 q^{29} - q^{30} - 7 q^{31} + 2 q^{32} - 9 q^{34} + q^{35} + 2 q^{36} - 9 q^{37} - 5 q^{38} - 4 q^{39} - q^{40} - 5 q^{41} - 2 q^{42} - 12 q^{43} - q^{45} + 5 q^{46} - 2 q^{47} + 2 q^{48} + 2 q^{49} - 7 q^{50} - 9 q^{51} - 4 q^{52} + 6 q^{53} + 2 q^{54} - 2 q^{56} - 5 q^{57} - 14 q^{58} + 16 q^{59} - q^{60} + 16 q^{61} - 7 q^{62} - 2 q^{63} + 2 q^{64} - 8 q^{65} - 16 q^{67} - 9 q^{68} + 5 q^{69} + q^{70} - 24 q^{71} + 2 q^{72} + 12 q^{73} - 9 q^{74} - 7 q^{75} - 5 q^{76} - 4 q^{78} - 6 q^{79} - q^{80} + 2 q^{81} - 5 q^{82} + 4 q^{83} - 2 q^{84} + 12 q^{85} - 12 q^{86} - 14 q^{87} + 17 q^{89} - q^{90} + 4 q^{91} + 5 q^{92} - 7 q^{93} - 2 q^{94} + 2 q^{96} - 8 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - q^5 + 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 - q^10 + 2 * q^12 - 4 * q^13 - 2 * q^14 - q^15 + 2 * q^16 - 9 * q^17 + 2 * q^18 - 5 * q^19 - q^20 - 2 * q^21 + 5 * q^23 + 2 * q^24 - 7 * q^25 - 4 * q^26 + 2 * q^27 - 2 * q^28 - 14 * q^29 - q^30 - 7 * q^31 + 2 * q^32 - 9 * q^34 + q^35 + 2 * q^36 - 9 * q^37 - 5 * q^38 - 4 * q^39 - q^40 - 5 * q^41 - 2 * q^42 - 12 * q^43 - q^45 + 5 * q^46 - 2 * q^47 + 2 * q^48 + 2 * q^49 - 7 * q^50 - 9 * q^51 - 4 * q^52 + 6 * q^53 + 2 * q^54 - 2 * q^56 - 5 * q^57 - 14 * q^58 + 16 * q^59 - q^60 + 16 * q^61 - 7 * q^62 - 2 * q^63 + 2 * q^64 - 8 * q^65 - 16 * q^67 - 9 * q^68 + 5 * q^69 + q^70 - 24 * q^71 + 2 * q^72 + 12 * q^73 - 9 * q^74 - 7 * q^75 - 5 * q^76 - 4 * q^78 - 6 * q^79 - q^80 + 2 * q^81 - 5 * q^82 + 4 * q^83 - 2 * q^84 + 12 * q^85 - 12 * q^86 - 14 * q^87 + 17 * q^89 - q^90 + 4 * q^91 + 5 * q^92 - 7 * q^93 - 2 * q^94 + 2 * q^96 - 8 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−1.61803	−0.723607	−0.361803	−	0.932254i	$-0.617839\pi$
$5$	−1.61803	−0.723607	−0.361803	+	0.932254i	$0.617839\pi$
$6$	1.00000	0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.61803	−0.511667
$11$	0	0
$11$	0	0
$12$	1.00000	0.288675
$13$	2.47214	0.685647	0.342824	−	0.939400i	$-0.388617\pi$
$13$	2.47214	0.685647	0.342824	+	0.939400i	$0.388617\pi$
$14$	−1.00000	−0.267261
$15$	−1.61803	−0.417775
$16$	1.00000	0.250000
$17$	−7.85410	−1.90490	−0.952450	−	0.304696i	$-0.901445\pi$
$17$	−7.85410	−1.90490	−0.952450	+	0.304696i	$0.901445\pi$
$18$	1.00000	0.235702
$19$	−1.38197	−0.317045	−0.158522	−	0.987355i	$-0.550673\pi$
$19$	−1.38197	−0.317045	−0.158522	+	0.987355i	$0.550673\pi$
$20$	−1.61803	−0.361803
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.38197	0.288160	0.144080	−	0.989566i	$-0.453978\pi$
$23$	1.38197	0.288160	0.144080	+	0.989566i	$0.453978\pi$
$24$	1.00000	0.204124
$25$	−2.38197	−0.476393
$26$	2.47214	0.484826
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	−9.23607	−1.71509	−0.857547	−	0.514405i	$-0.828013\pi$
$29$	−9.23607	−1.71509	−0.857547	+	0.514405i	$0.828013\pi$
$30$	−1.61803	−0.295411
$31$	2.09017	0.375406	0.187703	−	0.982226i	$-0.439896\pi$
$31$	2.09017	0.375406	0.187703	+	0.982226i	$0.439896\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−7.85410	−1.34697
$35$	1.61803	0.273498
$36$	1.00000	0.166667
$37$	−1.14590	−0.188384	−0.0941922	−	0.995554i	$-0.530027\pi$
$37$	−1.14590	−0.188384	−0.0941922	+	0.995554i	$0.530027\pi$
$38$	−1.38197	−0.224184
$39$	2.47214	0.395859
$40$	−1.61803	−0.255834
$41$	5.32624	0.831819	0.415909	−	0.909406i	$-0.363463\pi$
$41$	5.32624	0.831819	0.415909	+	0.909406i	$0.363463\pi$
$42$	−1.00000	−0.154303
$43$	−1.52786	−0.232997	−0.116499	−	0.993191i	$-0.537167\pi$
$43$	−1.52786	−0.232997	−0.116499	+	0.993191i	$0.537167\pi$
$44$	0	0
$45$	−1.61803	−0.241202
$46$	1.38197	0.203760
$47$	−7.70820	−1.12436	−0.562179	−	0.827016i	$-0.690037\pi$
$47$	−7.70820	−1.12436	−0.562179	+	0.827016i	$0.690037\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	−2.38197	−0.336861
$51$	−7.85410	−1.09979
$52$	2.47214	0.342824
$53$	−3.70820	−0.509361	−0.254680	−	0.967025i	$-0.581970\pi$
$53$	−3.70820	−0.509361	−0.254680	+	0.967025i	$0.581970\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−1.38197	−0.183046
$58$	−9.23607	−1.21276
$59$	3.52786	0.459289	0.229644	−	0.973275i	$-0.426244\pi$
$59$	3.52786	0.459289	0.229644	+	0.973275i	$0.426244\pi$
$60$	−1.61803	−0.208887
$61$	3.52786	0.451697	0.225848	−	0.974162i	$-0.427485\pi$
$61$	3.52786	0.451697	0.225848	+	0.974162i	$0.427485\pi$
$62$	2.09017	0.265452
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	−3.52786	−0.430997	−0.215499	−	0.976504i	$-0.569138\pi$
$67$	−3.52786	−0.430997	−0.215499	+	0.976504i	$0.569138\pi$
$68$	−7.85410	−0.952450
$69$	1.38197	0.166369
$70$	1.61803	0.193392
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	1.00000	0.117851
$73$	−2.94427	−0.344601	−0.172300	−	0.985044i	$-0.555120\pi$
$73$	−2.94427	−0.344601	−0.172300	+	0.985044i	$0.555120\pi$
$74$	−1.14590	−0.133208
$75$	−2.38197	−0.275046
$76$	−1.38197	−0.158522
$77$	0	0
$78$	2.47214	0.279914
$79$	−5.23607	−0.589104	−0.294552	−	0.955636i	$-0.595170\pi$
$79$	−5.23607	−0.589104	−0.294552	+	0.955636i	$0.595170\pi$
$80$	−1.61803	−0.180902
$81$	1.00000	0.111111
$82$	5.32624	0.588185
$83$	−11.4164	−1.25311	−0.626557	−	0.779376i	$-0.715536\pi$
$83$	−11.4164	−1.25311	−0.626557	+	0.779376i	$0.715536\pi$
$84$	−1.00000	−0.109109
$85$	12.7082	1.37840
$86$	−1.52786	−0.164754
$87$	−9.23607	−0.990210
$88$	0	0
$89$	11.8541	1.25653	0.628266	−	0.777998i	$-0.283765\pi$
$89$	11.8541	1.25653	0.628266	+	0.777998i	$0.283765\pi$
$90$	−1.61803	−0.170556
$91$	−2.47214	−0.259150
$92$	1.38197	0.144080
$93$	2.09017	0.216741
$94$	−7.70820	−0.795041
$95$	2.23607	0.229416
$96$	1.00000	0.102062
$97$	−12.9443	−1.31429	−0.657146	−	0.753763i	$-0.728236\pi$
$97$	−12.9443	−1.31429	−0.657146	+	0.753763i	$0.728236\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	−2.38197	−0.238197
$101$	−0.854102	−0.0849863	−0.0424932	−	0.999097i	$-0.513530\pi$
$101$	−0.854102	−0.0849863	−0.0424932	+	0.999097i	$0.513530\pi$
$102$	−7.85410	−0.777672
$103$	9.85410	0.970954	0.485477	−	0.874250i	$-0.338646\pi$
$103$	9.85410	0.970954	0.485477	+	0.874250i	$0.338646\pi$
$104$	2.47214	0.242413
$105$	1.61803	0.157904
$106$	−3.70820	−0.360173
$107$	−17.8541	−1.72602	−0.863011	−	0.505186i	$-0.831424\pi$
$107$	−17.8541	−1.72602	−0.863011	+	0.505186i	$0.831424\pi$
$108$	1.00000	0.0962250
$109$	16.5623	1.58638	0.793191	−	0.608973i	$-0.208418\pi$
$109$	16.5623	1.58638	0.793191	+	0.608973i	$0.208418\pi$
$110$	0	0
$111$	−1.14590	−0.108764
$112$	−1.00000	−0.0944911
$113$	5.23607	0.492568	0.246284	−	0.969198i	$-0.420790\pi$
$113$	5.23607	0.492568	0.246284	+	0.969198i	$0.420790\pi$
$114$	−1.38197	−0.129433
$115$	−2.23607	−0.208514
$116$	−9.23607	−0.857547
$117$	2.47214	0.228549
$118$	3.52786	0.324766
$119$	7.85410	0.719984
$120$	−1.61803	−0.147706
$121$	0	0
$122$	3.52786	0.319398
$123$	5.32624	0.480251
$124$	2.09017	0.187703
$125$	11.9443	1.06833
$126$	−1.00000	−0.0890871
$127$	−16.4721	−1.46167	−0.730833	−	0.682556i	$-0.760868\pi$
$127$	−16.4721	−1.46167	−0.730833	+	0.682556i	$0.760868\pi$
$128$	1.00000	0.0883883
$129$	−1.52786	−0.134521
$130$	−4.00000	−0.350823
$131$	−18.9443	−1.65517	−0.827584	−	0.561341i	$-0.810285\pi$
$131$	−18.9443	−1.65517	−0.827584	+	0.561341i	$0.810285\pi$
$132$	0	0
$133$	1.38197	0.119832
$134$	−3.52786	−0.304761
$135$	−1.61803	−0.139258
$136$	−7.85410	−0.673484
$137$	4.47214	0.382080	0.191040	−	0.981582i	$-0.438814\pi$
$137$	4.47214	0.382080	0.191040	+	0.981582i	$0.438814\pi$
$138$	1.38197	0.117641
$139$	19.2705	1.63450	0.817252	−	0.576281i	$-0.195497\pi$
$139$	19.2705	1.63450	0.817252	+	0.576281i	$0.195497\pi$
$140$	1.61803	0.136749
$141$	−7.70820	−0.649148
$142$	−12.0000	−1.00702
$143$	0	0
$144$	1.00000	0.0833333
$145$	14.9443	1.24105
$146$	−2.94427	−0.243670
$147$	1.00000	0.0824786
$148$	−1.14590	−0.0941922
$149$	−14.4721	−1.18560	−0.592802	−	0.805348i	$-0.701978\pi$
$149$	−14.4721	−1.18560	−0.592802	+	0.805348i	$0.701978\pi$
$150$	−2.38197	−0.194487
$151$	−4.76393	−0.387683	−0.193842	−	0.981033i	$-0.562095\pi$
$151$	−4.76393	−0.387683	−0.193842	+	0.981033i	$0.562095\pi$
$152$	−1.38197	−0.112092
$153$	−7.85410	−0.634967
$154$	0	0
$155$	−3.38197	−0.271646
$156$	2.47214	0.197929
$157$	9.70820	0.774799	0.387400	−	0.921912i	$-0.373373\pi$
$157$	9.70820	0.774799	0.387400	+	0.921912i	$0.373373\pi$
$158$	−5.23607	−0.416559
$159$	−3.70820	−0.294080
$160$	−1.61803	−0.127917
$161$	−1.38197	−0.108914
$162$	1.00000	0.0785674
$163$	−14.9443	−1.17053	−0.585263	−	0.810844i	$-0.699009\pi$
$163$	−14.9443	−1.17053	−0.585263	+	0.810844i	$0.699009\pi$
$164$	5.32624	0.415909
$165$	0	0
$166$	−11.4164	−0.886085
$167$	23.4164	1.81202	0.906008	−	0.423261i	$-0.139114\pi$
$167$	23.4164	1.81202	0.906008	+	0.423261i	$0.139114\pi$
$168$	−1.00000	−0.0771517
$169$	−6.88854	−0.529888
$170$	12.7082	0.974675
$171$	−1.38197	−0.105682
$172$	−1.52786	−0.116499
$173$	−21.5623	−1.63935	−0.819676	−	0.572828i	$-0.805846\pi$
$173$	−21.5623	−1.63935	−0.819676	+	0.572828i	$0.805846\pi$
$174$	−9.23607	−0.700185
$175$	2.38197	0.180060
$176$	0	0
$177$	3.52786	0.265170
$178$	11.8541	0.888503
$179$	19.0344	1.42270	0.711350	−	0.702837i	$-0.248084\pi$
$179$	19.0344	1.42270	0.711350	+	0.702837i	$0.248084\pi$
$180$	−1.61803	−0.120601
$181$	8.18034	0.608040	0.304020	−	0.952666i	$-0.401671\pi$
$181$	8.18034	0.608040	0.304020	+	0.952666i	$0.401671\pi$
$182$	−2.47214	−0.183247
$183$	3.52786	0.260787
$184$	1.38197	0.101880
$185$	1.85410	0.136316
$186$	2.09017	0.153259
$187$	0	0
$188$	−7.70820	−0.562179
$189$	−1.00000	−0.0727393
$190$	2.23607	0.162221
$191$	3.90983	0.282905	0.141453	−	0.989945i	$-0.454823\pi$
$191$	3.90983	0.282905	0.141453	+	0.989945i	$0.454823\pi$
$192$	1.00000	0.0721688
$193$	−19.5066	−1.40411	−0.702057	−	0.712121i	$-0.747735\pi$
$193$	−19.5066	−1.40411	−0.702057	+	0.712121i	$0.747735\pi$
$194$	−12.9443	−0.929345
$195$	−4.00000	−0.286446
$196$	1.00000	0.0714286
$197$	−11.8885	−0.847024	−0.423512	−	0.905891i	$-0.639203\pi$
$197$	−11.8885	−0.847024	−0.423512	+	0.905891i	$0.639203\pi$
$198$	0	0
$199$	−0.326238	−0.0231264	−0.0115632	−	0.999933i	$-0.503681\pi$
$199$	−0.326238	−0.0231264	−0.0115632	+	0.999933i	$0.503681\pi$
$200$	−2.38197	−0.168430
$201$	−3.52786	−0.248836
$202$	−0.854102	−0.0600944
$203$	9.23607	0.648245
$204$	−7.85410	−0.549897
$205$	−8.61803	−0.601910
$206$	9.85410	0.686568
$207$	1.38197	0.0960533
$208$	2.47214	0.171412
$209$	0	0
$210$	1.61803	0.111655
$211$	−2.47214	−0.170189	−0.0850944	−	0.996373i	$-0.527119\pi$
$211$	−2.47214	−0.170189	−0.0850944	+	0.996373i	$0.527119\pi$
$212$	−3.70820	−0.254680
$213$	−12.0000	−0.822226
$214$	−17.8541	−1.22048
$215$	2.47214	0.168598
$216$	1.00000	0.0680414
$217$	−2.09017	−0.141890
$218$	16.5623	1.12174
$219$	−2.94427	−0.198955
$220$	0	0
$221$	−19.4164	−1.30609
$222$	−1.14590	−0.0769076
$223$	−2.43769	−0.163240	−0.0816200	−	0.996664i	$-0.526009\pi$
$223$	−2.43769	−0.163240	−0.0816200	+	0.996664i	$0.526009\pi$
$224$	−1.00000	−0.0668153
$225$	−2.38197	−0.158798
$226$	5.23607	0.348298
$227$	12.7639	0.847172	0.423586	−	0.905856i	$-0.360771\pi$
$227$	12.7639	0.847172	0.423586	+	0.905856i	$0.360771\pi$
$228$	−1.38197	−0.0915229
$229$	−5.70820	−0.377209	−0.188604	−	0.982053i	$-0.560396\pi$
$229$	−5.70820	−0.377209	−0.188604	+	0.982053i	$0.560396\pi$
$230$	−2.23607	−0.147442
$231$	0	0
$232$	−9.23607	−0.606378
$233$	−16.0000	−1.04819	−0.524097	−	0.851658i	$-0.675597\pi$
$233$	−16.0000	−1.04819	−0.524097	+	0.851658i	$0.675597\pi$
$234$	2.47214	0.161609
$235$	12.4721	0.813592
$236$	3.52786	0.229644
$237$	−5.23607	−0.340119
$238$	7.85410	0.509106
$239$	20.7984	1.34533	0.672667	−	0.739945i	$-0.265148\pi$
$239$	20.7984	1.34533	0.672667	+	0.739945i	$0.265148\pi$
$240$	−1.61803	−0.104444
$241$	−4.47214	−0.288076	−0.144038	−	0.989572i	$-0.546009\pi$
$241$	−4.47214	−0.288076	−0.144038	+	0.989572i	$0.546009\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	3.52786	0.225848
$245$	−1.61803	−0.103372
$246$	5.32624	0.339589
$247$	−3.41641	−0.217381
$248$	2.09017	0.132726
$249$	−11.4164	−0.723485
$250$	11.9443	0.755422
$251$	15.7082	0.991493	0.495747	−	0.868467i	$-0.334895\pi$
$251$	15.7082	0.991493	0.495747	+	0.868467i	$0.334895\pi$
$252$	−1.00000	−0.0629941
$253$	0	0
$254$	−16.4721	−1.03355
$255$	12.7082	0.795819
$256$	1.00000	0.0625000
$257$	14.3262	0.893646	0.446823	−	0.894622i	$-0.352555\pi$
$257$	14.3262	0.893646	0.446823	+	0.894622i	$0.352555\pi$
$258$	−1.52786	−0.0951207
$259$	1.14590	0.0712026
$260$	−4.00000	−0.248069
$261$	−9.23607	−0.571698
$262$	−18.9443	−1.17038
$263$	−26.6180	−1.64134	−0.820669	−	0.571404i	$-0.806399\pi$
$263$	−26.6180	−1.64134	−0.820669	+	0.571404i	$0.806399\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	1.38197	0.0847338
$267$	11.8541	0.725459
$268$	−3.52786	−0.215499
$269$	−1.05573	−0.0643689	−0.0321844	−	0.999482i	$-0.510246\pi$
$269$	−1.05573	−0.0643689	−0.0321844	+	0.999482i	$0.510246\pi$
$270$	−1.61803	−0.0984704
$271$	−31.2148	−1.89616	−0.948081	−	0.318028i	$-0.896980\pi$
$271$	−31.2148	−1.89616	−0.948081	+	0.318028i	$0.896980\pi$
$272$	−7.85410	−0.476225
$273$	−2.47214	−0.149620
$274$	4.47214	0.270172
$275$	0	0
$276$	1.38197	0.0831846
$277$	19.2705	1.15785	0.578926	−	0.815380i	$-0.303472\pi$
$277$	19.2705	1.15785	0.578926	+	0.815380i	$0.303472\pi$
$278$	19.2705	1.15577
$279$	2.09017	0.125135
$280$	1.61803	0.0966960
$281$	−5.52786	−0.329765	−0.164882	−	0.986313i	$-0.552724\pi$
$281$	−5.52786	−0.329765	−0.164882	+	0.986313i	$0.552724\pi$
$282$	−7.70820	−0.459017
$283$	−3.14590	−0.187004	−0.0935021	−	0.995619i	$-0.529806\pi$
$283$	−3.14590	−0.187004	−0.0935021	+	0.995619i	$0.529806\pi$
$284$	−12.0000	−0.712069
$285$	2.23607	0.132453
$286$	0	0
$287$	−5.32624	−0.314398
$288$	1.00000	0.0589256
$289$	44.6869	2.62864
$290$	14.9443	0.877558
$291$	−12.9443	−0.758807
$292$	−2.94427	−0.172300
$293$	−2.38197	−0.139156	−0.0695780	−	0.997577i	$-0.522165\pi$
$293$	−2.38197	−0.139156	−0.0695780	+	0.997577i	$0.522165\pi$
$294$	1.00000	0.0583212
$295$	−5.70820	−0.332344
$296$	−1.14590	−0.0666040
$297$	0	0
$298$	−14.4721	−0.838348
$299$	3.41641	0.197576
$300$	−2.38197	−0.137523
$301$	1.52786	0.0880646
$302$	−4.76393	−0.274133
$303$	−0.854102	−0.0490669
$304$	−1.38197	−0.0792612
$305$	−5.70820	−0.326851
$306$	−7.85410	−0.448989
$307$	−12.8541	−0.733622	−0.366811	−	0.930295i	$-0.619550\pi$
$307$	−12.8541	−0.733622	−0.366811	+	0.930295i	$0.619550\pi$
$308$	0	0
$309$	9.85410	0.560580
$310$	−3.38197	−0.192083
$311$	−29.5967	−1.67828	−0.839139	−	0.543917i	$-0.816941\pi$
$311$	−29.5967	−1.67828	−0.839139	+	0.543917i	$0.816941\pi$
$312$	2.47214	0.139957
$313$	11.7082	0.661787	0.330893	−	0.943668i	$-0.392650\pi$
$313$	11.7082	0.661787	0.330893	+	0.943668i	$0.392650\pi$
$314$	9.70820	0.547866
$315$	1.61803	0.0911659
$316$	−5.23607	−0.294552
$317$	−8.65248	−0.485971	−0.242986	−	0.970030i	$-0.578127\pi$
$317$	−8.65248	−0.485971	−0.242986	+	0.970030i	$0.578127\pi$
$318$	−3.70820	−0.207946
$319$	0	0
$320$	−1.61803	−0.0904508
$321$	−17.8541	−0.996519
$322$	−1.38197	−0.0770140
$323$	10.8541	0.603938
$324$	1.00000	0.0555556
$325$	−5.88854	−0.326638
$326$	−14.9443	−0.827687
$327$	16.5623	0.915898
$328$	5.32624	0.294092
$329$	7.70820	0.424967
$330$	0	0
$331$	−17.8885	−0.983243	−0.491622	−	0.870809i	$-0.663596\pi$
$331$	−17.8885	−0.983243	−0.491622	+	0.870809i	$0.663596\pi$
$332$	−11.4164	−0.626557
$333$	−1.14590	−0.0627948
$334$	23.4164	1.28129
$335$	5.70820	0.311872
$336$	−1.00000	−0.0545545
$337$	35.6869	1.94399	0.971995	−	0.235001i	$-0.0755093\pi$
$337$	35.6869	1.94399	0.971995	+	0.235001i	$0.0755093\pi$
$338$	−6.88854	−0.374687
$339$	5.23607	0.284384
$340$	12.7082	0.689199
$341$	0	0
$342$	−1.38197	−0.0747282
$343$	−1.00000	−0.0539949
$344$	−1.52786	−0.0823769
$345$	−2.23607	−0.120386
$346$	−21.5623	−1.15920
$347$	0.965558	0.0518339	0.0259169	−	0.999664i	$-0.491749\pi$
$347$	0.965558	0.0518339	0.0259169	+	0.999664i	$0.491749\pi$
$348$	−9.23607	−0.495105
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	2.38197	0.127321
$351$	2.47214	0.131953
$352$	0	0
$353$	−2.58359	−0.137511	−0.0687554	−	0.997634i	$-0.521903\pi$
$353$	−2.58359	−0.137511	−0.0687554	+	0.997634i	$0.521903\pi$
$354$	3.52786	0.187504
$355$	19.4164	1.03052
$356$	11.8541	0.628266
$357$	7.85410	0.415683
$358$	19.0344	1.00600
$359$	3.32624	0.175552	0.0877761	−	0.996140i	$-0.472024\pi$
$359$	3.32624	0.175552	0.0877761	+	0.996140i	$0.472024\pi$
$360$	−1.61803	−0.0852779
$361$	−17.0902	−0.899483
$362$	8.18034	0.429949
$363$	0	0
$364$	−2.47214	−0.129575
$365$	4.76393	0.249356
$366$	3.52786	0.184404
$367$	13.9098	0.726087	0.363044	−	0.931772i	$-0.381738\pi$
$367$	13.9098	0.726087	0.363044	+	0.931772i	$0.381738\pi$
$368$	1.38197	0.0720400
$369$	5.32624	0.277273
$370$	1.85410	0.0963902
$371$	3.70820	0.192520
$372$	2.09017	0.108370
$373$	19.0344	0.985566	0.492783	−	0.870152i	$-0.335980\pi$
$373$	19.0344	0.985566	0.492783	+	0.870152i	$0.335980\pi$
$374$	0	0
$375$	11.9443	0.616800
$376$	−7.70820	−0.397520
$377$	−22.8328	−1.17595
$378$	−1.00000	−0.0514344
$379$	30.6525	1.57451	0.787256	−	0.616626i	$-0.211501\pi$
$379$	30.6525	1.57451	0.787256	+	0.616626i	$0.211501\pi$
$380$	2.23607	0.114708
$381$	−16.4721	−0.843893
$382$	3.90983	0.200044
$383$	10.9443	0.559226	0.279613	−	0.960113i	$-0.409794\pi$
$383$	10.9443	0.559226	0.279613	+	0.960113i	$0.409794\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−19.5066	−0.992859
$387$	−1.52786	−0.0776657
$388$	−12.9443	−0.657146
$389$	−36.9443	−1.87315	−0.936574	−	0.350469i	$-0.886022\pi$
$389$	−36.9443	−1.87315	−0.936574	+	0.350469i	$0.886022\pi$
$390$	−4.00000	−0.202548
$391$	−10.8541	−0.548916
$392$	1.00000	0.0505076
$393$	−18.9443	−0.955612
$394$	−11.8885	−0.598936
$395$	8.47214	0.426279
$396$	0	0
$397$	−11.7082	−0.587618	−0.293809	−	0.955864i	$-0.594923\pi$
$397$	−11.7082	−0.587618	−0.293809	+	0.955864i	$0.594923\pi$
$398$	−0.326238	−0.0163528
$399$	1.38197	0.0691848
$400$	−2.38197	−0.119098
$401$	1.41641	0.0707320	0.0353660	−	0.999374i	$-0.488740\pi$
$401$	1.41641	0.0707320	0.0353660	+	0.999374i	$0.488740\pi$
$402$	−3.52786	−0.175954
$403$	5.16718	0.257396
$404$	−0.854102	−0.0424932
$405$	−1.61803	−0.0804008
$406$	9.23607	0.458378
$407$	0	0
$408$	−7.85410	−0.388836
$409$	−8.76393	−0.433349	−0.216674	−	0.976244i	$-0.569521\pi$
$409$	−8.76393	−0.433349	−0.216674	+	0.976244i	$0.569521\pi$
$410$	−8.61803	−0.425614
$411$	4.47214	0.220594
$412$	9.85410	0.485477
$413$	−3.52786	−0.173595
$414$	1.38197	0.0679199
$415$	18.4721	0.906761
$416$	2.47214	0.121206
$417$	19.2705	0.943681
$418$	0	0
$419$	38.6525	1.88830	0.944149	−	0.329520i	$-0.106887\pi$
$419$	38.6525	1.88830	0.944149	+	0.329520i	$0.106887\pi$
$420$	1.61803	0.0789520
$421$	38.6869	1.88548	0.942742	−	0.333521i	$-0.108237\pi$
$421$	38.6869	1.88548	0.942742	+	0.333521i	$0.108237\pi$
$422$	−2.47214	−0.120342
$423$	−7.70820	−0.374786
$424$	−3.70820	−0.180086
$425$	18.7082	0.907481
$426$	−12.0000	−0.581402
$427$	−3.52786	−0.170725
$428$	−17.8541	−0.863011
$429$	0	0
$430$	2.47214	0.119217
$431$	25.9787	1.25135	0.625675	−	0.780084i	$-0.284823\pi$
$431$	25.9787	1.25135	0.625675	+	0.780084i	$0.284823\pi$
$432$	1.00000	0.0481125
$433$	22.4721	1.07994	0.539971	−	0.841684i	$-0.318435\pi$
$433$	22.4721	1.07994	0.539971	+	0.841684i	$0.318435\pi$
$434$	−2.09017	−0.100331
$435$	14.9443	0.716523
$436$	16.5623	0.793191
$437$	−1.90983	−0.0913596
$438$	−2.94427	−0.140683
$439$	28.5623	1.36320	0.681602	−	0.731723i	$-0.261283\pi$
$439$	28.5623	1.36320	0.681602	+	0.731723i	$0.261283\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−19.4164	−0.923544
$443$	13.2016	0.627228	0.313614	−	0.949551i	$-0.398460\pi$
$443$	13.2016	0.627228	0.313614	+	0.949551i	$0.398460\pi$
$444$	−1.14590	−0.0543819
$445$	−19.1803	−0.909235
$446$	−2.43769	−0.115428
$447$	−14.4721	−0.684509
$448$	−1.00000	−0.0472456
$449$	16.1803	0.763597	0.381799	−	0.924245i	$-0.375305\pi$
$449$	16.1803	0.763597	0.381799	+	0.924245i	$0.375305\pi$
$450$	−2.38197	−0.112287
$451$	0	0
$452$	5.23607	0.246284
$453$	−4.76393	−0.223829
$454$	12.7639	0.599041
$455$	4.00000	0.187523
$456$	−1.38197	−0.0647165
$457$	5.41641	0.253369	0.126684	−	0.991943i	$-0.459566\pi$
$457$	5.41641	0.253369	0.126684	+	0.991943i	$0.459566\pi$
$458$	−5.70820	−0.266727
$459$	−7.85410	−0.366598
$460$	−2.23607	−0.104257
$461$	−30.3607	−1.41404	−0.707019	−	0.707195i	$-0.749960\pi$
$461$	−30.3607	−1.41404	−0.707019	+	0.707195i	$0.749960\pi$
$462$	0	0
$463$	2.18034	0.101329	0.0506645	−	0.998716i	$-0.483866\pi$
$463$	2.18034	0.101329	0.0506645	+	0.998716i	$0.483866\pi$
$464$	−9.23607	−0.428774
$465$	−3.38197	−0.156835
$466$	−16.0000	−0.741186
$467$	−27.7082	−1.28218	−0.641091	−	0.767465i	$-0.721518\pi$
$467$	−27.7082	−1.28218	−0.641091	+	0.767465i	$0.721518\pi$
$468$	2.47214	0.114275
$469$	3.52786	0.162902
$470$	12.4721	0.575297
$471$	9.70820	0.447330
$472$	3.52786	0.162383
$473$	0	0
$474$	−5.23607	−0.240501
$475$	3.29180	0.151038
$476$	7.85410	0.359992
$477$	−3.70820	−0.169787
$478$	20.7984	0.951295
$479$	−17.5279	−0.800869	−0.400434	−	0.916325i	$-0.631141\pi$
$479$	−17.5279	−0.800869	−0.400434	+	0.916325i	$0.631141\pi$
$480$	−1.61803	−0.0738528
$481$	−2.83282	−0.129165
$482$	−4.47214	−0.203700
$483$	−1.38197	−0.0628816
$484$	0	0
$485$	20.9443	0.951030
$486$	1.00000	0.0453609
$487$	3.12461	0.141590	0.0707948	−	0.997491i	$-0.477446\pi$
$487$	3.12461	0.141590	0.0707948	+	0.997491i	$0.477446\pi$
$488$	3.52786	0.159699
$489$	−14.9443	−0.675803
$490$	−1.61803	−0.0730953
$491$	24.3262	1.09783	0.548914	−	0.835879i	$-0.315041\pi$
$491$	24.3262	1.09783	0.548914	+	0.835879i	$0.315041\pi$
$492$	5.32624	0.240125
$493$	72.5410	3.26708
$494$	−3.41641	−0.153711
$495$	0	0
$496$	2.09017	0.0938514
$497$	12.0000	0.538274
$498$	−11.4164	−0.511581
$499$	−15.4164	−0.690133	−0.345067	−	0.938578i	$-0.612144\pi$
$499$	−15.4164	−0.690133	−0.345067	+	0.938578i	$0.612144\pi$
$500$	11.9443	0.534164
$501$	23.4164	1.04617
$502$	15.7082	0.701091
$503$	23.8885	1.06514	0.532569	−	0.846387i	$-0.321227\pi$
$503$	23.8885	1.06514	0.532569	+	0.846387i	$0.321227\pi$
$504$	−1.00000	−0.0445435
$505$	1.38197	0.0614967
$506$	0	0
$507$	−6.88854	−0.305931
$508$	−16.4721	−0.730833
$509$	15.3262	0.679324	0.339662	−	0.940548i	$-0.389687\pi$
$509$	15.3262	0.679324	0.339662	+	0.940548i	$0.389687\pi$
$510$	12.7082	0.562729
$511$	2.94427	0.130247
$512$	1.00000	0.0441942
$513$	−1.38197	−0.0610153
$514$	14.3262	0.631903
$515$	−15.9443	−0.702589
$516$	−1.52786	−0.0672605
$517$	0	0
$518$	1.14590	0.0503479
$519$	−21.5623	−0.946480
$520$	−4.00000	−0.175412
$521$	38.0902	1.66876	0.834380	−	0.551189i	$-0.185826\pi$
$521$	38.0902	1.66876	0.834380	+	0.551189i	$0.185826\pi$
$522$	−9.23607	−0.404252
$523$	19.8541	0.868159	0.434080	−	0.900875i	$-0.357074\pi$
$523$	19.8541	0.868159	0.434080	+	0.900875i	$0.357074\pi$
$524$	−18.9443	−0.827584
$525$	2.38197	0.103958
$526$	−26.6180	−1.16060
$527$	−16.4164	−0.715110
$528$	0	0
$529$	−21.0902	−0.916964
$530$	6.00000	0.260623
$531$	3.52786	0.153096
$532$	1.38197	0.0599158
$533$	13.1672	0.570334
$534$	11.8541	0.512977
$535$	28.8885	1.24896
$536$	−3.52786	−0.152381
$537$	19.0344	0.821397
$538$	−1.05573	−0.0455157
$539$	0	0
$540$	−1.61803	−0.0696291
$541$	32.1591	1.38263	0.691313	−	0.722556i	$-0.257033\pi$
$541$	32.1591	1.38263	0.691313	+	0.722556i	$0.257033\pi$
$542$	−31.2148	−1.34079
$543$	8.18034	0.351052
$544$	−7.85410	−0.336742
$545$	−26.7984	−1.14792
$546$	−2.47214	−0.105798
$547$	−3.59675	−0.153786	−0.0768929	−	0.997039i	$-0.524500\pi$
$547$	−3.59675	−0.153786	−0.0768929	+	0.997039i	$0.524500\pi$
$548$	4.47214	0.191040
$549$	3.52786	0.150566
$550$	0	0
$551$	12.7639	0.543762
$552$	1.38197	0.0588204
$553$	5.23607	0.222660
$554$	19.2705	0.818726
$555$	1.85410	0.0787022
$556$	19.2705	0.817252
$557$	−29.4164	−1.24641	−0.623207	−	0.782057i	$-0.714170\pi$
$557$	−29.4164	−1.24641	−0.623207	+	0.782057i	$0.714170\pi$
$558$	2.09017	0.0884839
$559$	−3.77709	−0.159754
$560$	1.61803	0.0683744
$561$	0	0
$562$	−5.52786	−0.233179
$563$	15.1246	0.637426	0.318713	−	0.947851i	$-0.396749\pi$
$563$	15.1246	0.637426	0.318713	+	0.947851i	$0.396749\pi$
$564$	−7.70820	−0.324574
$565$	−8.47214	−0.356425
$566$	−3.14590	−0.132232
$567$	−1.00000	−0.0419961
$568$	−12.0000	−0.503509
$569$	−35.4164	−1.48473	−0.742367	−	0.669994i	$-0.766297\pi$
$569$	−35.4164	−1.48473	−0.742367	+	0.669994i	$0.766297\pi$
$570$	2.23607	0.0936586
$571$	2.29180	0.0959087	0.0479543	−	0.998850i	$-0.484730\pi$
$571$	2.29180	0.0959087	0.0479543	+	0.998850i	$0.484730\pi$
$572$	0	0
$573$	3.90983	0.163335
$574$	−5.32624	−0.222313
$575$	−3.29180	−0.137277
$576$	1.00000	0.0416667
$577$	26.1803	1.08990	0.544951	−	0.838468i	$-0.316548\pi$
$577$	26.1803	1.08990	0.544951	+	0.838468i	$0.316548\pi$
$578$	44.6869	1.85873
$579$	−19.5066	−0.810666
$580$	14.9443	0.620527
$581$	11.4164	0.473632
$582$	−12.9443	−0.536557
$583$	0	0
$584$	−2.94427	−0.121835
$585$	−4.00000	−0.165380
$586$	−2.38197	−0.0983981
$587$	−21.4164	−0.883950	−0.441975	−	0.897027i	$-0.645722\pi$
$587$	−21.4164	−0.883950	−0.441975	+	0.897027i	$0.645722\pi$
$588$	1.00000	0.0412393
$589$	−2.88854	−0.119020
$590$	−5.70820	−0.235003
$591$	−11.8885	−0.489029
$592$	−1.14590	−0.0470961
$593$	31.9787	1.31321	0.656604	−	0.754235i	$-0.271992\pi$
$593$	31.9787	1.31321	0.656604	+	0.754235i	$0.271992\pi$
$594$	0	0
$595$	−12.7082	−0.520986
$596$	−14.4721	−0.592802
$597$	−0.326238	−0.0133520
$598$	3.41641	0.139707
$599$	−20.5066	−0.837876	−0.418938	−	0.908015i	$-0.637598\pi$
$599$	−20.5066	−0.837876	−0.418938	+	0.908015i	$0.637598\pi$
$600$	−2.38197	−0.0972434
$601$	−3.52786	−0.143905	−0.0719523	−	0.997408i	$-0.522923\pi$
$601$	−3.52786	−0.143905	−0.0719523	+	0.997408i	$0.522923\pi$
$602$	1.52786	0.0622711
$603$	−3.52786	−0.143666
$604$	−4.76393	−0.193842
$605$	0	0
$606$	−0.854102	−0.0346955
$607$	−11.0902	−0.450136	−0.225068	−	0.974343i	$-0.572260\pi$
$607$	−11.0902	−0.450136	−0.225068	+	0.974343i	$0.572260\pi$
$608$	−1.38197	−0.0560461
$609$	9.23607	0.374264
$610$	−5.70820	−0.231118
$611$	−19.0557	−0.770912
$612$	−7.85410	−0.317483
$613$	−28.5066	−1.15137	−0.575685	−	0.817672i	$-0.695265\pi$
$613$	−28.5066	−1.15137	−0.575685	+	0.817672i	$0.695265\pi$
$614$	−12.8541	−0.518749
$615$	−8.61803	−0.347513
$616$	0	0
$617$	−27.0132	−1.08751	−0.543754	−	0.839244i	$-0.682998\pi$
$617$	−27.0132	−1.08751	−0.543754	+	0.839244i	$0.682998\pi$
$618$	9.85410	0.396390
$619$	6.38197	0.256513	0.128256	−	0.991741i	$-0.459062\pi$
$619$	6.38197	0.256513	0.128256	+	0.991741i	$0.459062\pi$
$620$	−3.38197	−0.135823
$621$	1.38197	0.0554564
$622$	−29.5967	−1.18672
$623$	−11.8541	−0.474925
$624$	2.47214	0.0989646
$625$	−7.41641	−0.296656
$626$	11.7082	0.467954
$627$	0	0
$628$	9.70820	0.387400
$629$	9.00000	0.358854
$630$	1.61803	0.0644640
$631$	1.05573	0.0420279	0.0210139	−	0.999779i	$-0.493311\pi$
$631$	1.05573	0.0420279	0.0210139	+	0.999779i	$0.493311\pi$
$632$	−5.23607	−0.208280
$633$	−2.47214	−0.0982586
$634$	−8.65248	−0.343634
$635$	26.6525	1.05767
$636$	−3.70820	−0.147040
$637$	2.47214	0.0979496
$638$	0	0
$639$	−12.0000	−0.474713
$640$	−1.61803	−0.0639584
$641$	−30.9443	−1.22223	−0.611113	−	0.791544i	$-0.709278\pi$
$641$	−30.9443	−1.22223	−0.611113	+	0.791544i	$0.709278\pi$
$642$	−17.8541	−0.704645
$643$	8.14590	0.321243	0.160621	−	0.987016i	$-0.448650\pi$
$643$	8.14590	0.321243	0.160621	+	0.987016i	$0.448650\pi$
$644$	−1.38197	−0.0544571
$645$	2.47214	0.0973403
$646$	10.8541	0.427049
$647$	3.70820	0.145785	0.0728923	−	0.997340i	$-0.476777\pi$
$647$	3.70820	0.145785	0.0728923	+	0.997340i	$0.476777\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	−5.88854	−0.230968
$651$	−2.09017	−0.0819202
$652$	−14.9443	−0.585263
$653$	−13.8197	−0.540805	−0.270403	−	0.962747i	$-0.587157\pi$
$653$	−13.8197	−0.540805	−0.270403	+	0.962747i	$0.587157\pi$
$654$	16.5623	0.647637
$655$	30.6525	1.19769
$656$	5.32624	0.207955
$657$	−2.94427	−0.114867
$658$	7.70820	0.300497
$659$	5.72949	0.223189	0.111595	−	0.993754i	$-0.464404\pi$
$659$	5.72949	0.223189	0.111595	+	0.993754i	$0.464404\pi$
$660$	0	0
$661$	−37.1246	−1.44398	−0.721990	−	0.691903i	$-0.756772\pi$
$661$	−37.1246	−1.44398	−0.721990	+	0.691903i	$0.756772\pi$
$662$	−17.8885	−0.695258
$663$	−19.4164	−0.754071
$664$	−11.4164	−0.443043
$665$	−2.23607	−0.0867110
$666$	−1.14590	−0.0444026
$667$	−12.7639	−0.494221
$668$	23.4164	0.906008
$669$	−2.43769	−0.0942467
$670$	5.70820	0.220527
$671$	0	0
$672$	−1.00000	−0.0385758
$673$	23.8885	0.920836	0.460418	−	0.887702i	$-0.347700\pi$
$673$	23.8885	0.920836	0.460418	+	0.887702i	$0.347700\pi$
$674$	35.6869	1.37461
$675$	−2.38197	−0.0916819
$676$	−6.88854	−0.264944
$677$	17.4164	0.669367	0.334683	−	0.942331i	$-0.391371\pi$
$677$	17.4164	0.669367	0.334683	+	0.942331i	$0.391371\pi$
$678$	5.23607	0.201090
$679$	12.9443	0.496756
$680$	12.7082	0.487337
$681$	12.7639	0.489115
$682$	0	0
$683$	40.7426	1.55897	0.779487	−	0.626418i	$-0.215480\pi$
$683$	40.7426	1.55897	0.779487	+	0.626418i	$0.215480\pi$
$684$	−1.38197	−0.0528408
$685$	−7.23607	−0.276476
$686$	−1.00000	−0.0381802
$687$	−5.70820	−0.217782
$688$	−1.52786	−0.0582493
$689$	−9.16718	−0.349242
$690$	−2.23607	−0.0851257
$691$	12.1459	0.462052	0.231026	−	0.972948i	$-0.425792\pi$
$691$	12.1459	0.462052	0.231026	+	0.972948i	$0.425792\pi$
$692$	−21.5623	−0.819676
$693$	0	0
$694$	0.965558	0.0366521
$695$	−31.1803	−1.18274
$696$	−9.23607	−0.350092
$697$	−41.8328	−1.58453
$698$	−6.00000	−0.227103
$699$	−16.0000	−0.605176
$700$	2.38197	0.0900299
$701$	−7.23607	−0.273303	−0.136651	−	0.990619i	$-0.543634\pi$
$701$	−7.23607	−0.273303	−0.136651	+	0.990619i	$0.543634\pi$
$702$	2.47214	0.0933048
$703$	1.58359	0.0597263
$704$	0	0
$705$	12.4721	0.469728
$706$	−2.58359	−0.0972348
$707$	0.854102	0.0321218
$708$	3.52786	0.132585
$709$	11.6180	0.436324	0.218162	−	0.975913i	$-0.429994\pi$
$709$	11.6180	0.436324	0.218162	+	0.975913i	$0.429994\pi$
$710$	19.4164	0.728685
$711$	−5.23607	−0.196368
$712$	11.8541	0.444251
$713$	2.88854	0.108177
$714$	7.85410	0.293932
$715$	0	0
$716$	19.0344	0.711350
$717$	20.7984	0.776730
$718$	3.32624	0.124134
$719$	−10.2918	−0.383819	−0.191910	−	0.981413i	$-0.561468\pi$
$719$	−10.2918	−0.383819	−0.191910	+	0.981413i	$0.561468\pi$
$720$	−1.61803	−0.0603006
$721$	−9.85410	−0.366986
$722$	−17.0902	−0.636030
$723$	−4.47214	−0.166321
$724$	8.18034	0.304020
$725$	22.0000	0.817059
$726$	0	0
$727$	23.4508	0.869744	0.434872	−	0.900492i	$-0.356794\pi$
$727$	23.4508	0.869744	0.434872	+	0.900492i	$0.356794\pi$
$728$	−2.47214	−0.0916235
$729$	1.00000	0.0370370
$730$	4.76393	0.176321
$731$	12.0000	0.443836
$732$	3.52786	0.130394
$733$	−22.9443	−0.847466	−0.423733	−	0.905787i	$-0.639281\pi$
$733$	−22.9443	−0.847466	−0.423733	+	0.905787i	$0.639281\pi$
$734$	13.9098	0.513421
$735$	−1.61803	−0.0596821
$736$	1.38197	0.0509399
$737$	0	0
$738$	5.32624	0.196062
$739$	−20.6525	−0.759714	−0.379857	−	0.925045i	$-0.624027\pi$
$739$	−20.6525	−0.759714	−0.379857	+	0.925045i	$0.624027\pi$
$740$	1.85410	0.0681581
$741$	−3.41641	−0.125505
$742$	3.70820	0.136132
$743$	22.5623	0.827731	0.413865	−	0.910338i	$-0.364178\pi$
$743$	22.5623	0.827731	0.413865	+	0.910338i	$0.364178\pi$
$744$	2.09017	0.0766293
$745$	23.4164	0.857911
$746$	19.0344	0.696900
$747$	−11.4164	−0.417705
$748$	0	0
$749$	17.8541	0.652375
$750$	11.9443	0.436143
$751$	30.5410	1.11446	0.557229	−	0.830359i	$-0.311865\pi$
$751$	30.5410	1.11446	0.557229	+	0.830359i	$0.311865\pi$
$752$	−7.70820	−0.281089
$753$	15.7082	0.572439
$754$	−22.8328	−0.831522
$755$	7.70820	0.280530
$756$	−1.00000	−0.0363696
$757$	46.3951	1.68626	0.843130	−	0.537710i	$-0.180711\pi$
$757$	46.3951	1.68626	0.843130	+	0.537710i	$0.180711\pi$
$758$	30.6525	1.11335
$759$	0	0
$760$	2.23607	0.0811107
$761$	5.63932	0.204425	0.102213	−	0.994763i	$-0.467408\pi$
$761$	5.63932	0.204425	0.102213	+	0.994763i	$0.467408\pi$
$762$	−16.4721	−0.596723
$763$	−16.5623	−0.599596
$764$	3.90983	0.141453
$765$	12.7082	0.459466
$766$	10.9443	0.395433
$767$	8.72136	0.314910
$768$	1.00000	0.0360844
$769$	29.1246	1.05026	0.525130	−	0.851022i	$-0.324017\pi$
$769$	29.1246	1.05026	0.525130	+	0.851022i	$0.324017\pi$
$770$	0	0
$771$	14.3262	0.515947
$772$	−19.5066	−0.702057
$773$	−33.4164	−1.20190	−0.600952	−	0.799285i	$-0.705212\pi$
$773$	−33.4164	−1.20190	−0.600952	+	0.799285i	$0.705212\pi$
$774$	−1.52786	−0.0549179
$775$	−4.97871	−0.178841
$776$	−12.9443	−0.464672
$777$	1.14590	0.0411089
$778$	−36.9443	−1.32452
$779$	−7.36068	−0.263724
$780$	−4.00000	−0.143223
$781$	0	0
$782$	−10.8541	−0.388142
$783$	−9.23607	−0.330070
$784$	1.00000	0.0357143
$785$	−15.7082	−0.560650
$786$	−18.9443	−0.675720
$787$	−30.5623	−1.08943	−0.544714	−	0.838622i	$-0.683362\pi$
$787$	−30.5623	−1.08943	−0.544714	+	0.838622i	$0.683362\pi$
$788$	−11.8885	−0.423512
$789$	−26.6180	−0.947627
$790$	8.47214	0.301425
$791$	−5.23607	−0.186173
$792$	0	0
$793$	8.72136	0.309705
$794$	−11.7082	−0.415509
$795$	6.00000	0.212798
$796$	−0.326238	−0.0115632
$797$	27.8673	0.987109	0.493554	−	0.869715i	$-0.335697\pi$
$797$	27.8673	0.987109	0.493554	+	0.869715i	$0.335697\pi$
$798$	1.38197	0.0489211
$799$	60.5410	2.14179
$800$	−2.38197	−0.0842152
$801$	11.8541	0.418844
$802$	1.41641	0.0500151
$803$	0	0
$804$	−3.52786	−0.124418
$805$	2.23607	0.0788110
$806$	5.16718	0.182006
$807$	−1.05573	−0.0371634
$808$	−0.854102	−0.0300472
$809$	36.0000	1.26569	0.632846	−	0.774277i	$-0.281886\pi$
$809$	36.0000	1.26569	0.632846	+	0.774277i	$0.281886\pi$
$810$	−1.61803	−0.0568519
$811$	−43.4164	−1.52456	−0.762278	−	0.647250i	$-0.775919\pi$
$811$	−43.4164	−1.52456	−0.762278	+	0.647250i	$0.775919\pi$
$812$	9.23607	0.324122
$813$	−31.2148	−1.09475
$814$	0	0
$815$	24.1803	0.847000
$816$	−7.85410	−0.274949
$817$	2.11146	0.0738705
$818$	−8.76393	−0.306424
$819$	−2.47214	−0.0863834
$820$	−8.61803	−0.300955
$821$	−5.23607	−0.182740	−0.0913700	−	0.995817i	$-0.529125\pi$
$821$	−5.23607	−0.182740	−0.0913700	+	0.995817i	$0.529125\pi$
$822$	4.47214	0.155984
$823$	−53.1246	−1.85181	−0.925904	−	0.377759i	$-0.876695\pi$
$823$	−53.1246	−1.85181	−0.925904	+	0.377759i	$0.876695\pi$
$824$	9.85410	0.343284
$825$	0	0
$826$	−3.52786	−0.122750
$827$	−45.9230	−1.59690	−0.798449	−	0.602062i	$-0.794346\pi$
$827$	−45.9230	−1.59690	−0.798449	+	0.602062i	$0.794346\pi$
$828$	1.38197	0.0480266
$829$	−12.1803	−0.423041	−0.211520	−	0.977374i	$-0.567841\pi$
$829$	−12.1803	−0.423041	−0.211520	+	0.977374i	$0.567841\pi$
$830$	18.4721	0.641177
$831$	19.2705	0.668487
$832$	2.47214	0.0857059
$833$	−7.85410	−0.272129
$834$	19.2705	0.667283
$835$	−37.8885	−1.31119
$836$	0	0
$837$	2.09017	0.0722468
$838$	38.6525	1.33523
$839$	35.2361	1.21648	0.608242	−	0.793752i	$-0.291875\pi$
$839$	35.2361	1.21648	0.608242	+	0.793752i	$0.291875\pi$
$840$	1.61803	0.0558275
$841$	56.3050	1.94155
$842$	38.6869	1.33324
$843$	−5.52786	−0.190390
$844$	−2.47214	−0.0850944
$845$	11.1459	0.383431
$846$	−7.70820	−0.265014
$847$	0	0
$848$	−3.70820	−0.127340
$849$	−3.14590	−0.107967
$850$	18.7082	0.641686
$851$	−1.58359	−0.0542848
$852$	−12.0000	−0.411113
$853$	−6.76393	−0.231593	−0.115796	−	0.993273i	$-0.536942\pi$
$853$	−6.76393	−0.231593	−0.115796	+	0.993273i	$0.536942\pi$
$854$	−3.52786	−0.120721
$855$	2.23607	0.0764719
$856$	−17.8541	−0.610241
$857$	23.8885	0.816017	0.408009	−	0.912978i	$-0.366223\pi$
$857$	23.8885	0.816017	0.408009	+	0.912978i	$0.366223\pi$
$858$	0	0
$859$	32.9443	1.12404	0.562022	−	0.827122i	$-0.310024\pi$
$859$	32.9443	1.12404	0.562022	+	0.827122i	$0.310024\pi$
$860$	2.47214	0.0842991
$861$	−5.32624	−0.181518
$862$	25.9787	0.884839
$863$	−14.7426	−0.501845	−0.250923	−	0.968007i	$-0.580734\pi$
$863$	−14.7426	−0.501845	−0.250923	+	0.968007i	$0.580734\pi$
$864$	1.00000	0.0340207
$865$	34.8885	1.18625
$866$	22.4721	0.763634
$867$	44.6869	1.51765
$868$	−2.09017	−0.0709450
$869$	0	0
$870$	14.9443	0.506658
$871$	−8.72136	−0.295512
$872$	16.5623	0.560870
$873$	−12.9443	−0.438097
$874$	−1.90983	−0.0646010
$875$	−11.9443	−0.403790
$876$	−2.94427	−0.0994777
$877$	−43.3050	−1.46230	−0.731152	−	0.682214i	$-0.761017\pi$
$877$	−43.3050	−1.46230	−0.731152	+	0.682214i	$0.761017\pi$
$878$	28.5623	0.963931
$879$	−2.38197	−0.0803417
$880$	0	0
$881$	15.9230	0.536459	0.268230	−	0.963355i	$-0.413561\pi$
$881$	15.9230	0.536459	0.268230	+	0.963355i	$0.413561\pi$
$882$	1.00000	0.0336718
$883$	1.34752	0.0453478	0.0226739	−	0.999743i	$-0.492782\pi$
$883$	1.34752	0.0453478	0.0226739	+	0.999743i	$0.492782\pi$
$884$	−19.4164	−0.653044
$885$	−5.70820	−0.191879
$886$	13.2016	0.443517
$887$	−9.41641	−0.316172	−0.158086	−	0.987425i	$-0.550532\pi$
$887$	−9.41641	−0.316172	−0.158086	+	0.987425i	$0.550532\pi$
$888$	−1.14590	−0.0384538
$889$	16.4721	0.552458
$890$	−19.1803	−0.642926
$891$	0	0
$892$	−2.43769	−0.0816200
$893$	10.6525	0.356472
$894$	−14.4721	−0.484021
$895$	−30.7984	−1.02948
$896$	−1.00000	−0.0334077
$897$	3.41641	0.114071
$898$	16.1803	0.539945
$899$	−19.3050	−0.643856
$900$	−2.38197	−0.0793989
$901$	29.1246	0.970281
$902$	0	0
$903$	1.52786	0.0508441
$904$	5.23607	0.174149
$905$	−13.2361	−0.439982
$906$	−4.76393	−0.158271
$907$	4.18034	0.138806	0.0694030	−	0.997589i	$-0.477891\pi$
$907$	4.18034	0.138806	0.0694030	+	0.997589i	$0.477891\pi$
$908$	12.7639	0.423586
$909$	−0.854102	−0.0283288
$910$	4.00000	0.132599
$911$	5.52786	0.183146	0.0915732	−	0.995798i	$-0.470810\pi$
$911$	5.52786	0.183146	0.0915732	+	0.995798i	$0.470810\pi$
$912$	−1.38197	−0.0457615
$913$	0	0
$914$	5.41641	0.179159
$915$	−5.70820	−0.188707
$916$	−5.70820	−0.188604
$917$	18.9443	0.625595
$918$	−7.85410	−0.259224
$919$	33.4164	1.10231	0.551153	−	0.834404i	$-0.314188\pi$
$919$	33.4164	1.10231	0.551153	+	0.834404i	$0.314188\pi$
$920$	−2.23607	−0.0737210
$921$	−12.8541	−0.423557
$922$	−30.3607	−0.999876
$923$	−29.6656	−0.976456
$924$	0	0
$925$	2.72949	0.0897451
$926$	2.18034	0.0716504
$927$	9.85410	0.323651
$928$	−9.23607	−0.303189
$929$	−17.0902	−0.560710	−0.280355	−	0.959896i	$-0.590452\pi$
$929$	−17.0902	−0.560710	−0.280355	+	0.959896i	$0.590452\pi$
$930$	−3.38197	−0.110899
$931$	−1.38197	−0.0452921
$932$	−16.0000	−0.524097
$933$	−29.5967	−0.968954
$934$	−27.7082	−0.906640
$935$	0	0
$936$	2.47214	0.0808043
$937$	−51.6656	−1.68784	−0.843921	−	0.536467i	$-0.819759\pi$
$937$	−51.6656	−1.68784	−0.843921	+	0.536467i	$0.819759\pi$
$938$	3.52786	0.115189
$939$	11.7082	0.382083
$940$	12.4721	0.406796
$941$	−22.2148	−0.724181	−0.362091	−	0.932143i	$-0.617937\pi$
$941$	−22.2148	−0.724181	−0.362091	+	0.932143i	$0.617937\pi$
$942$	9.70820	0.316310
$943$	7.36068	0.239697
$944$	3.52786	0.114822
$945$	1.61803	0.0526346
$946$	0	0
$947$	50.6312	1.64529	0.822646	−	0.568553i	$-0.192497\pi$
$947$	50.6312	1.64529	0.822646	+	0.568553i	$0.192497\pi$
$948$	−5.23607	−0.170060
$949$	−7.27864	−0.236275
$950$	3.29180	0.106800
$951$	−8.65248	−0.280576
$952$	7.85410	0.254553
$953$	3.12461	0.101216	0.0506081	−	0.998719i	$-0.483884\pi$
$953$	3.12461	0.101216	0.0506081	+	0.998719i	$0.483884\pi$
$954$	−3.70820	−0.120058
$955$	−6.32624	−0.204712
$956$	20.7984	0.672667
$957$	0	0
$958$	−17.5279	−0.566300
$959$	−4.47214	−0.144413
$960$	−1.61803	−0.0522218
$961$	−26.6312	−0.859071
$962$	−2.83282	−0.0913336
$963$	−17.8541	−0.575340
$964$	−4.47214	−0.144038
$965$	31.5623	1.01603
$966$	−1.38197	−0.0444640
$967$	21.7082	0.698089	0.349044	−	0.937106i	$-0.386506\pi$
$967$	21.7082	0.698089	0.349044	+	0.937106i	$0.386506\pi$
$968$	0	0
$969$	10.8541	0.348684
$970$	20.9443	0.672480
$971$	−14.1115	−0.452858	−0.226429	−	0.974028i	$-0.572705\pi$
$971$	−14.1115	−0.452858	−0.226429	+	0.974028i	$0.572705\pi$
$972$	1.00000	0.0320750
$973$	−19.2705	−0.617784
$974$	3.12461	0.100119
$975$	−5.88854	−0.188584
$976$	3.52786	0.112924
$977$	−52.0689	−1.66583	−0.832916	−	0.553400i	$-0.813330\pi$
$977$	−52.0689	−1.66583	−0.832916	+	0.553400i	$0.813330\pi$
$978$	−14.9443	−0.477865
$979$	0	0
$980$	−1.61803	−0.0516862
$981$	16.5623	0.528794
$982$	24.3262	0.776281
$983$	−8.47214	−0.270219	−0.135110	−	0.990831i	$-0.543139\pi$
$983$	−8.47214	−0.270219	−0.135110	+	0.990831i	$0.543139\pi$
$984$	5.32624	0.169794
$985$	19.2361	0.612912
$986$	72.5410	2.31018
$987$	7.70820	0.245355
$988$	−3.41641	−0.108690
$989$	−2.11146	−0.0671404
$990$	0	0
$991$	−40.1803	−1.27637	−0.638185	−	0.769883i	$-0.720315\pi$
$991$	−40.1803	−1.27637	−0.638185	+	0.769883i	$0.720315\pi$
$992$	2.09017	0.0663630
$993$	−17.8885	−0.567676
$994$	12.0000	0.380617
$995$	0.527864	0.0167344
$996$	−11.4164	−0.361743
$997$	50.8328	1.60989	0.804946	−	0.593348i	$-0.202194\pi$
$997$	50.8328	1.60989	0.804946	+	0.593348i	$0.202194\pi$
$998$	−15.4164	−0.487998
$999$	−1.14590	−0.0362546

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5082.2.a.bt.1.1		2
11.5	even	5		462.2.j.a.421.1	yes	4
11.9	even	5		462.2.j.a.169.1	✓	4
11.10	odd	2		5082.2.a.bj.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.j.a.169.1	✓	4	11.9	even	5
462.2.j.a.421.1	yes	4	11.5	even	5
5082.2.a.bj.1.1		2	11.10	odd	2
5082.2.a.bt.1.1		2	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:	\( N \)	\(=\)	\( 5082 = 2 \cdot 3 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5082.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.61803 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.61803 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.61803 q^{10} +1.00000 q^{12} +2.47214 q^{13} -1.00000 q^{14} -1.61803 q^{15} +1.00000 q^{16} -7.85410 q^{17} +1.00000 q^{18} -1.38197 q^{19} -1.61803 q^{20} -1.00000 q^{21} +1.38197 q^{23} +1.00000 q^{24} -2.38197 q^{25} +2.47214 q^{26} +1.00000 q^{27} -1.00000 q^{28} -9.23607 q^{29} -1.61803 q^{30} +2.09017 q^{31} +1.00000 q^{32} -7.85410 q^{34} +1.61803 q^{35} +1.00000 q^{36} -1.14590 q^{37} -1.38197 q^{38} +2.47214 q^{39} -1.61803 q^{40} +5.32624 q^{41} -1.00000 q^{42} -1.52786 q^{43} -1.61803 q^{45} +1.38197 q^{46} -7.70820 q^{47} +1.00000 q^{48} +1.00000 q^{49} -2.38197 q^{50} -7.85410 q^{51} +2.47214 q^{52} -3.70820 q^{53} +1.00000 q^{54} -1.00000 q^{56} -1.38197 q^{57} -9.23607 q^{58} +3.52786 q^{59} -1.61803 q^{60} +3.52786 q^{61} +2.09017 q^{62} -1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -3.52786 q^{67} -7.85410 q^{68} +1.38197 q^{69} +1.61803 q^{70} -12.0000 q^{71} +1.00000 q^{72} -2.94427 q^{73} -1.14590 q^{74} -2.38197 q^{75} -1.38197 q^{76} +2.47214 q^{78} -5.23607 q^{79} -1.61803 q^{80} +1.00000 q^{81} +5.32624 q^{82} -11.4164 q^{83} -1.00000 q^{84} +12.7082 q^{85} -1.52786 q^{86} -9.23607 q^{87} +11.8541 q^{89} -1.61803 q^{90} -2.47214 q^{91} +1.38197 q^{92} +2.09017 q^{93} -7.70820 q^{94} +2.23607 q^{95} +1.00000 q^{96} -12.9443 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - q^{5} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - q^5 + 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - q^{5} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} - q^{10} + 2 q^{12} - 4 q^{13} - 2 q^{14} - q^{15} + 2 q^{16} - 9 q^{17} + 2 q^{18} - 5 q^{19} - q^{20} - 2 q^{21} + 5 q^{23} + 2 q^{24} - 7 q^{25} - 4 q^{26} + 2 q^{27} - 2 q^{28} - 14 q^{29} - q^{30} - 7 q^{31} + 2 q^{32} - 9 q^{34} + q^{35} + 2 q^{36} - 9 q^{37} - 5 q^{38} - 4 q^{39} - q^{40} - 5 q^{41} - 2 q^{42} - 12 q^{43} - q^{45} + 5 q^{46} - 2 q^{47} + 2 q^{48} + 2 q^{49} - 7 q^{50} - 9 q^{51} - 4 q^{52} + 6 q^{53} + 2 q^{54} - 2 q^{56} - 5 q^{57} - 14 q^{58} + 16 q^{59} - q^{60} + 16 q^{61} - 7 q^{62} - 2 q^{63} + 2 q^{64} - 8 q^{65} - 16 q^{67} - 9 q^{68} + 5 q^{69} + q^{70} - 24 q^{71} + 2 q^{72} + 12 q^{73} - 9 q^{74} - 7 q^{75} - 5 q^{76} - 4 q^{78} - 6 q^{79} - q^{80} + 2 q^{81} - 5 q^{82} + 4 q^{83} - 2 q^{84} + 12 q^{85} - 12 q^{86} - 14 q^{87} + 17 q^{89} - q^{90} + 4 q^{91} + 5 q^{92} - 7 q^{93} - 2 q^{94} + 2 q^{96} - 8 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - q^5 + 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 - q^10 + 2 * q^12 - 4 * q^13 - 2 * q^14 - q^15 + 2 * q^16 - 9 * q^17 + 2 * q^18 - 5 * q^19 - q^20 - 2 * q^21 + 5 * q^23 + 2 * q^24 - 7 * q^25 - 4 * q^26 + 2 * q^27 - 2 * q^28 - 14 * q^29 - q^30 - 7 * q^31 + 2 * q^32 - 9 * q^34 + q^35 + 2 * q^36 - 9 * q^37 - 5 * q^38 - 4 * q^39 - q^40 - 5 * q^41 - 2 * q^42 - 12 * q^43 - q^45 + 5 * q^46 - 2 * q^47 + 2 * q^48 + 2 * q^49 - 7 * q^50 - 9 * q^51 - 4 * q^52 + 6 * q^53 + 2 * q^54 - 2 * q^56 - 5 * q^57 - 14 * q^58 + 16 * q^59 - q^60 + 16 * q^61 - 7 * q^62 - 2 * q^63 + 2 * q^64 - 8 * q^65 - 16 * q^67 - 9 * q^68 + 5 * q^69 + q^70 - 24 * q^71 + 2 * q^72 + 12 * q^73 - 9 * q^74 - 7 * q^75 - 5 * q^76 - 4 * q^78 - 6 * q^79 - q^80 + 2 * q^81 - 5 * q^82 + 4 * q^83 - 2 * q^84 + 12 * q^85 - 12 * q^86 - 14 * q^87 + 17 * q^89 - q^90 + 4 * q^91 + 5 * q^92 - 7 * q^93 - 2 * q^94 + 2 * q^96 - 8 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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