LMFDB - Embedded newform 435.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [435,2,Mod(1,435)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(435, 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("435.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 435 = 3 \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	435.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:	$3.47349248793$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	435.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -1.00000 q^{5} +0.618034 q^{6} -3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -1.00000 q^{5} +0.618034 q^{6} -3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} -0.618034 q^{10} -5.47214 q^{11} -1.61803 q^{12} -6.23607 q^{13} -1.85410 q^{14} -1.00000 q^{15} +1.85410 q^{16} +3.47214 q^{17} +0.618034 q^{18} +7.70820 q^{19} +1.61803 q^{20} -3.00000 q^{21} -3.38197 q^{22} -2.23607 q^{24} +1.00000 q^{25} -3.85410 q^{26} +1.00000 q^{27} +4.85410 q^{28} +1.00000 q^{29} -0.618034 q^{30} -8.00000 q^{31} +5.61803 q^{32} -5.47214 q^{33} +2.14590 q^{34} +3.00000 q^{35} -1.61803 q^{36} -8.00000 q^{37} +4.76393 q^{38} -6.23607 q^{39} +2.23607 q^{40} -4.47214 q^{41} -1.85410 q^{42} +3.23607 q^{43} +8.85410 q^{44} -1.00000 q^{45} +6.70820 q^{47} +1.85410 q^{48} +2.00000 q^{49} +0.618034 q^{50} +3.47214 q^{51} +10.0902 q^{52} -6.76393 q^{53} +0.618034 q^{54} +5.47214 q^{55} +6.70820 q^{56} +7.70820 q^{57} +0.618034 q^{58} +5.23607 q^{59} +1.61803 q^{60} -5.70820 q^{61} -4.94427 q^{62} -3.00000 q^{63} -0.236068 q^{64} +6.23607 q^{65} -3.38197 q^{66} -11.4721 q^{67} -5.61803 q^{68} +1.85410 q^{70} +7.23607 q^{71} -2.23607 q^{72} -8.00000 q^{73} -4.94427 q^{74} +1.00000 q^{75} -12.4721 q^{76} +16.4164 q^{77} -3.85410 q^{78} -6.18034 q^{79} -1.85410 q^{80} +1.00000 q^{81} -2.76393 q^{82} -3.70820 q^{83} +4.85410 q^{84} -3.47214 q^{85} +2.00000 q^{86} +1.00000 q^{87} +12.2361 q^{88} +11.1803 q^{89} -0.618034 q^{90} +18.7082 q^{91} -8.00000 q^{93} +4.14590 q^{94} -7.70820 q^{95} +5.61803 q^{96} +2.76393 q^{97} +1.23607 q^{98} -5.47214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 2 q^{3} - q^{4} - 2 q^{5} - q^{6} - 6 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - q^2 + 2 * q^3 - q^4 - 2 * q^5 - q^6 - 6 * q^7 + 2 * q^9	$ 2 q - q^{2} + 2 q^{3} - q^{4} - 2 q^{5} - q^{6} - 6 q^{7} + 2 q^{9} + q^{10} - 2 q^{11} - q^{12} - 8 q^{13} + 3 q^{14} - 2 q^{15} - 3 q^{16} - 2 q^{17} - q^{18} + 2 q^{19} + q^{20} - 6 q^{21} - 9 q^{22} + 2 q^{25} - q^{26} + 2 q^{27} + 3 q^{28} + 2 q^{29} + q^{30} - 16 q^{31} + 9 q^{32} - 2 q^{33} + 11 q^{34} + 6 q^{35} - q^{36} - 16 q^{37} + 14 q^{38} - 8 q^{39} + 3 q^{42} + 2 q^{43} + 11 q^{44} - 2 q^{45} - 3 q^{48} + 4 q^{49} - q^{50} - 2 q^{51} + 9 q^{52} - 18 q^{53} - q^{54} + 2 q^{55} + 2 q^{57} - q^{58} + 6 q^{59} + q^{60} + 2 q^{61} + 8 q^{62} - 6 q^{63} + 4 q^{64} + 8 q^{65} - 9 q^{66} - 14 q^{67} - 9 q^{68} - 3 q^{70} + 10 q^{71} - 16 q^{73} + 8 q^{74} + 2 q^{75} - 16 q^{76} + 6 q^{77} - q^{78} + 10 q^{79} + 3 q^{80} + 2 q^{81} - 10 q^{82} + 6 q^{83} + 3 q^{84} + 2 q^{85} + 4 q^{86} + 2 q^{87} + 20 q^{88} + q^{90} + 24 q^{91} - 16 q^{93} + 15 q^{94} - 2 q^{95} + 9 q^{96} + 10 q^{97} - 2 q^{98} - 2 q^{99}+O(q^{100}) $ 2 * q - q^2 + 2 * q^3 - q^4 - 2 * q^5 - q^6 - 6 * q^7 + 2 * q^9 + q^10 - 2 * q^11 - q^12 - 8 * q^13 + 3 * q^14 - 2 * q^15 - 3 * q^16 - 2 * q^17 - q^18 + 2 * q^19 + q^20 - 6 * q^21 - 9 * q^22 + 2 * q^25 - q^26 + 2 * q^27 + 3 * q^28 + 2 * q^29 + q^30 - 16 * q^31 + 9 * q^32 - 2 * q^33 + 11 * q^34 + 6 * q^35 - q^36 - 16 * q^37 + 14 * q^38 - 8 * q^39 + 3 * q^42 + 2 * q^43 + 11 * q^44 - 2 * q^45 - 3 * q^48 + 4 * q^49 - q^50 - 2 * q^51 + 9 * q^52 - 18 * q^53 - q^54 + 2 * q^55 + 2 * q^57 - q^58 + 6 * q^59 + q^60 + 2 * q^61 + 8 * q^62 - 6 * q^63 + 4 * q^64 + 8 * q^65 - 9 * q^66 - 14 * q^67 - 9 * q^68 - 3 * q^70 + 10 * q^71 - 16 * q^73 + 8 * q^74 + 2 * q^75 - 16 * q^76 + 6 * q^77 - q^78 + 10 * q^79 + 3 * q^80 + 2 * q^81 - 10 * q^82 + 6 * q^83 + 3 * q^84 + 2 * q^85 + 4 * q^86 + 2 * q^87 + 20 * q^88 + q^90 + 24 * q^91 - 16 * q^93 + 15 * q^94 - 2 * q^95 + 9 * q^96 + 10 * q^97 - 2 * q^98 - 2 * 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 $

$2$	0.618034	0.437016	0.218508	−	0.975835i	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.61803	−0.809017
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.618034	0.252311
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	−2.23607	−0.790569
$9$	1.00000	0.333333
$10$	−0.618034	−0.195440
$11$	−5.47214	−1.64991	−0.824956	−	0.565198i	$-0.808800\pi$
$11$	−5.47214	−1.64991	−0.824956	+	0.565198i	$0.808800\pi$
$12$	−1.61803	−0.467086
$13$	−6.23607	−1.72957	−0.864787	−	0.502139i	$-0.832547\pi$
$13$	−6.23607	−1.72957	−0.864787	+	0.502139i	$0.832547\pi$
$14$	−1.85410	−0.495530
$15$	−1.00000	−0.258199
$16$	1.85410	0.463525
$17$	3.47214	0.842117	0.421058	−	0.907034i	$-0.361659\pi$
$17$	3.47214	0.842117	0.421058	+	0.907034i	$0.361659\pi$
$18$	0.618034	0.145672
$19$	7.70820	1.76838	0.884192	−	0.467124i	$-0.154710\pi$
$19$	7.70820	1.76838	0.884192	+	0.467124i	$0.154710\pi$
$20$	1.61803	0.361803
$21$	−3.00000	−0.654654
$22$	−3.38197	−0.721038
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−2.23607	−0.456435
$25$	1.00000	0.200000
$26$	−3.85410	−0.755852
$27$	1.00000	0.192450
$28$	4.85410	0.917339
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−0.618034	−0.112837
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	5.61803	0.993137
$33$	−5.47214	−0.952577
$34$	2.14590	0.368018
$35$	3.00000	0.507093
$36$	−1.61803	−0.269672
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	4.76393	0.772812
$39$	−6.23607	−0.998570
$40$	2.23607	0.353553
$41$	−4.47214	−0.698430	−0.349215	−	0.937043i	$-0.613552\pi$
$41$	−4.47214	−0.698430	−0.349215	+	0.937043i	$0.613552\pi$
$42$	−1.85410	−0.286094
$43$	3.23607	0.493496	0.246748	−	0.969080i	$-0.420638\pi$
$43$	3.23607	0.493496	0.246748	+	0.969080i	$0.420638\pi$
$44$	8.85410	1.33481
$45$	−1.00000	−0.149071
$46$	0	0
$47$	6.70820	0.978492	0.489246	−	0.872146i	$-0.337272\pi$
$47$	6.70820	0.978492	0.489246	+	0.872146i	$0.337272\pi$
$48$	1.85410	0.267617
$49$	2.00000	0.285714
$50$	0.618034	0.0874032
$51$	3.47214	0.486196
$52$	10.0902	1.39925
$53$	−6.76393	−0.929098	−0.464549	−	0.885548i	$-0.653783\pi$
$53$	−6.76393	−0.929098	−0.464549	+	0.885548i	$0.653783\pi$
$54$	0.618034	0.0841038
$55$	5.47214	0.737863
$56$	6.70820	0.896421
$57$	7.70820	1.02098
$58$	0.618034	0.0811518
$59$	5.23607	0.681678	0.340839	−	0.940122i	$-0.389289\pi$
$59$	5.23607	0.681678	0.340839	+	0.940122i	$0.389289\pi$
$60$	1.61803	0.208887
$61$	−5.70820	−0.730861	−0.365430	−	0.930839i	$-0.619078\pi$
$61$	−5.70820	−0.730861	−0.365430	+	0.930839i	$0.619078\pi$
$62$	−4.94427	−0.627923
$63$	−3.00000	−0.377964
$64$	−0.236068	−0.0295085
$65$	6.23607	0.773489
$66$	−3.38197	−0.416291
$67$	−11.4721	−1.40154	−0.700772	−	0.713385i	$-0.747161\pi$
$67$	−11.4721	−1.40154	−0.700772	+	0.713385i	$0.747161\pi$
$68$	−5.61803	−0.681287
$69$	0	0
$70$	1.85410	0.221608
$71$	7.23607	0.858763	0.429382	−	0.903123i	$-0.358732\pi$
$71$	7.23607	0.858763	0.429382	+	0.903123i	$0.358732\pi$
$72$	−2.23607	−0.263523
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	−4.94427	−0.574760
$75$	1.00000	0.115470
$76$	−12.4721	−1.43065
$77$	16.4164	1.87082
$78$	−3.85410	−0.436391
$79$	−6.18034	−0.695343	−0.347671	−	0.937616i	$-0.613027\pi$
$79$	−6.18034	−0.695343	−0.347671	+	0.937616i	$0.613027\pi$
$80$	−1.85410	−0.207295
$81$	1.00000	0.111111
$82$	−2.76393	−0.305225
$83$	−3.70820	−0.407028	−0.203514	−	0.979072i	$-0.565236\pi$
$83$	−3.70820	−0.407028	−0.203514	+	0.979072i	$0.565236\pi$
$84$	4.85410	0.529626
$85$	−3.47214	−0.376606
$86$	2.00000	0.215666
$87$	1.00000	0.107211
$88$	12.2361	1.30437
$89$	11.1803	1.18511	0.592557	−	0.805529i	$-0.298119\pi$
$89$	11.1803	1.18511	0.592557	+	0.805529i	$0.298119\pi$
$90$	−0.618034	−0.0651465
$91$	18.7082	1.96115
$92$	0	0
$93$	−8.00000	−0.829561
$94$	4.14590	0.427617
$95$	−7.70820	−0.790845
$96$	5.61803	0.573388
$97$	2.76393	0.280635	0.140317	−	0.990107i	$-0.455188\pi$
$97$	2.76393	0.280635	0.140317	+	0.990107i	$0.455188\pi$
$98$	1.23607	0.124862
$99$	−5.47214	−0.549970
$100$	−1.61803	−0.161803
$101$	4.23607	0.421505	0.210752	−	0.977540i	$-0.432409\pi$
$101$	4.23607	0.421505	0.210752	+	0.977540i	$0.432409\pi$
$102$	2.14590	0.212476
$103$	7.41641	0.730760	0.365380	−	0.930858i	$-0.380939\pi$
$103$	7.41641	0.730760	0.365380	+	0.930858i	$0.380939\pi$
$104$	13.9443	1.36735
$105$	3.00000	0.292770
$106$	−4.18034	−0.406031
$107$	7.52786	0.727746	0.363873	−	0.931449i	$-0.381454\pi$
$107$	7.52786	0.727746	0.363873	+	0.931449i	$0.381454\pi$
$108$	−1.61803	−0.155695
$109$	−16.4164	−1.57241	−0.786203	−	0.617968i	$-0.787956\pi$
$109$	−16.4164	−1.57241	−0.786203	+	0.617968i	$0.787956\pi$
$110$	3.38197	0.322458
$111$	−8.00000	−0.759326
$112$	−5.56231	−0.525589
$113$	−7.94427	−0.747334	−0.373667	−	0.927563i	$-0.621900\pi$
$113$	−7.94427	−0.747334	−0.373667	+	0.927563i	$0.621900\pi$
$114$	4.76393	0.446183
$115$	0	0
$116$	−1.61803	−0.150231
$117$	−6.23607	−0.576525
$118$	3.23607	0.297904
$119$	−10.4164	−0.954871
$120$	2.23607	0.204124
$121$	18.9443	1.72221
$122$	−3.52786	−0.319398
$123$	−4.47214	−0.403239
$124$	12.9443	1.16243
$125$	−1.00000	−0.0894427
$126$	−1.85410	−0.165177
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	−11.3820	−1.00603
$129$	3.23607	0.284920
$130$	3.85410	0.338027
$131$	3.47214	0.303362	0.151681	−	0.988430i	$-0.451531\pi$
$131$	3.47214	0.303362	0.151681	+	0.988430i	$0.451531\pi$
$132$	8.85410	0.770651
$133$	−23.1246	−2.00516
$134$	−7.09017	−0.612497
$135$	−1.00000	−0.0860663
$136$	−7.76393	−0.665752
$137$	−10.9443	−0.935032	−0.467516	−	0.883985i	$-0.654851\pi$
$137$	−10.9443	−0.935032	−0.467516	+	0.883985i	$0.654851\pi$
$138$	0	0
$139$	−0.708204	−0.0600691	−0.0300345	−	0.999549i	$-0.509562\pi$
$139$	−0.708204	−0.0600691	−0.0300345	+	0.999549i	$0.509562\pi$
$140$	−4.85410	−0.410246
$141$	6.70820	0.564933
$142$	4.47214	0.375293
$143$	34.1246	2.85364
$144$	1.85410	0.154508
$145$	−1.00000	−0.0830455
$146$	−4.94427	−0.409191
$147$	2.00000	0.164957
$148$	12.9443	1.06401
$149$	20.1803	1.65324	0.826619	−	0.562762i	$-0.190261\pi$
$149$	20.1803	1.65324	0.826619	+	0.562762i	$0.190261\pi$
$150$	0.618034	0.0504623
$151$	2.47214	0.201180	0.100590	−	0.994928i	$-0.467927\pi$
$151$	2.47214	0.201180	0.100590	+	0.994928i	$0.467927\pi$
$152$	−17.2361	−1.39803
$153$	3.47214	0.280706
$154$	10.1459	0.817580
$155$	8.00000	0.642575
$156$	10.0902	0.807860
$157$	−11.7082	−0.934416	−0.467208	−	0.884147i	$-0.654740\pi$
$157$	−11.7082	−0.934416	−0.467208	+	0.884147i	$0.654740\pi$
$158$	−3.81966	−0.303876
$159$	−6.76393	−0.536415
$160$	−5.61803	−0.444145
$161$	0	0
$162$	0.618034	0.0485573
$163$	−15.1246	−1.18465	−0.592326	−	0.805699i	$-0.701790\pi$
$163$	−15.1246	−1.18465	−0.592326	+	0.805699i	$0.701790\pi$
$164$	7.23607	0.565042
$165$	5.47214	0.426005
$166$	−2.29180	−0.177878
$167$	−17.8885	−1.38426	−0.692129	−	0.721774i	$-0.743327\pi$
$167$	−17.8885	−1.38426	−0.692129	+	0.721774i	$0.743327\pi$
$168$	6.70820	0.517549
$169$	25.8885	1.99143
$170$	−2.14590	−0.164583
$171$	7.70820	0.589461
$172$	−5.23607	−0.399246
$173$	−19.4164	−1.47620	−0.738101	−	0.674690i	$-0.764277\pi$
$173$	−19.4164	−1.47620	−0.738101	+	0.674690i	$0.764277\pi$
$174$	0.618034	0.0468530
$175$	−3.00000	−0.226779
$176$	−10.1459	−0.764776
$177$	5.23607	0.393567
$178$	6.90983	0.517914
$179$	4.18034	0.312453	0.156227	−	0.987721i	$-0.450067\pi$
$179$	4.18034	0.312453	0.156227	+	0.987721i	$0.450067\pi$
$180$	1.61803	0.120601
$181$	8.41641	0.625587	0.312793	−	0.949821i	$-0.398735\pi$
$181$	8.41641	0.625587	0.312793	+	0.949821i	$0.398735\pi$
$182$	11.5623	0.857055
$183$	−5.70820	−0.421963
$184$	0	0
$185$	8.00000	0.588172
$186$	−4.94427	−0.362532
$187$	−19.0000	−1.38942
$188$	−10.8541	−0.791617
$189$	−3.00000	−0.218218
$190$	−4.76393	−0.345612
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−0.236068	−0.0170367
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	1.70820	0.122642
$195$	6.23607	0.446574
$196$	−3.23607	−0.231148
$197$	14.9443	1.06474	0.532368	−	0.846513i	$-0.321302\pi$
$197$	14.9443	1.06474	0.532368	+	0.846513i	$0.321302\pi$
$198$	−3.38197	−0.240346
$199$	20.7082	1.46797	0.733983	−	0.679168i	$-0.237659\pi$
$199$	20.7082	1.46797	0.733983	+	0.679168i	$0.237659\pi$
$200$	−2.23607	−0.158114
$201$	−11.4721	−0.809182
$202$	2.61803	0.184204
$203$	−3.00000	−0.210559
$204$	−5.61803	−0.393341
$205$	4.47214	0.312348
$206$	4.58359	0.319354
$207$	0	0
$208$	−11.5623	−0.801702
$209$	−42.1803	−2.91768
$210$	1.85410	0.127945
$211$	−27.8885	−1.91993	−0.959963	−	0.280126i	$-0.909624\pi$
$211$	−27.8885	−1.91993	−0.959963	+	0.280126i	$0.909624\pi$
$212$	10.9443	0.751656
$213$	7.23607	0.495807
$214$	4.65248	0.318037
$215$	−3.23607	−0.220698
$216$	−2.23607	−0.152145
$217$	24.0000	1.62923
$218$	−10.1459	−0.687167
$219$	−8.00000	−0.540590
$220$	−8.85410	−0.596943
$221$	−21.6525	−1.45650
$222$	−4.94427	−0.331838
$223$	−13.0000	−0.870544	−0.435272	−	0.900299i	$-0.643348\pi$
$223$	−13.0000	−0.870544	−0.435272	+	0.900299i	$0.643348\pi$
$224$	−16.8541	−1.12611
$225$	1.00000	0.0666667
$226$	−4.90983	−0.326597
$227$	−5.81966	−0.386264	−0.193132	−	0.981173i	$-0.561865\pi$
$227$	−5.81966	−0.386264	−0.193132	+	0.981173i	$0.561865\pi$
$228$	−12.4721	−0.825987
$229$	9.41641	0.622254	0.311127	−	0.950368i	$-0.399294\pi$
$229$	9.41641	0.622254	0.311127	+	0.950368i	$0.399294\pi$
$230$	0	0
$231$	16.4164	1.08012
$232$	−2.23607	−0.146805
$233$	1.41641	0.0927920	0.0463960	−	0.998923i	$-0.485226\pi$
$233$	1.41641	0.0927920	0.0463960	+	0.998923i	$0.485226\pi$
$234$	−3.85410	−0.251951
$235$	−6.70820	−0.437595
$236$	−8.47214	−0.551489
$237$	−6.18034	−0.401456
$238$	−6.43769	−0.417294
$239$	11.8885	0.769006	0.384503	−	0.923124i	$-0.374373\pi$
$239$	11.8885	0.769006	0.384503	+	0.923124i	$0.374373\pi$
$240$	−1.85410	−0.119682
$241$	7.00000	0.450910	0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000	0.450910	0.225455	+	0.974254i	$0.427613\pi$
$242$	11.7082	0.752632
$243$	1.00000	0.0641500
$244$	9.23607	0.591279
$245$	−2.00000	−0.127775
$246$	−2.76393	−0.176222
$247$	−48.0689	−3.05855
$248$	17.8885	1.13592
$249$	−3.70820	−0.234998
$250$	−0.618034	−0.0390879
$251$	−24.8885	−1.57095	−0.785475	−	0.618893i	$-0.787582\pi$
$251$	−24.8885	−1.57095	−0.785475	+	0.618893i	$0.787582\pi$
$252$	4.85410	0.305780
$253$	0	0
$254$	−3.70820	−0.232673
$255$	−3.47214	−0.217434
$256$	−6.56231	−0.410144
$257$	−9.70820	−0.605581	−0.302791	−	0.953057i	$-0.597918\pi$
$257$	−9.70820	−0.605581	−0.302791	+	0.953057i	$0.597918\pi$
$258$	2.00000	0.124515
$259$	24.0000	1.49129
$260$	−10.0902	−0.625766
$261$	1.00000	0.0618984
$262$	2.14590	0.132574
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	12.2361	0.753078
$265$	6.76393	0.415505
$266$	−14.2918	−0.876286
$267$	11.1803	0.684226
$268$	18.5623	1.13387
$269$	−30.2361	−1.84353	−0.921763	−	0.387754i	$-0.873251\pi$
$269$	−30.2361	−1.84353	−0.921763	+	0.387754i	$0.873251\pi$
$270$	−0.618034	−0.0376124
$271$	14.3607	0.872349	0.436175	−	0.899862i	$-0.356333\pi$
$271$	14.3607	0.872349	0.436175	+	0.899862i	$0.356333\pi$
$272$	6.43769	0.390343
$273$	18.7082	1.13227
$274$	−6.76393	−0.408624
$275$	−5.47214	−0.329982
$276$	0	0
$277$	8.70820	0.523225	0.261613	−	0.965173i	$-0.415746\pi$
$277$	8.70820	0.523225	0.261613	+	0.965173i	$0.415746\pi$
$278$	−0.437694	−0.0262511
$279$	−8.00000	−0.478947
$280$	−6.70820	−0.400892
$281$	−23.1246	−1.37950	−0.689749	−	0.724048i	$-0.742279\pi$
$281$	−23.1246	−1.37950	−0.689749	+	0.724048i	$0.742279\pi$
$282$	4.14590	0.246885
$283$	18.4721	1.09805	0.549027	−	0.835804i	$-0.314998\pi$
$283$	18.4721	1.09805	0.549027	+	0.835804i	$0.314998\pi$
$284$	−11.7082	−0.694754
$285$	−7.70820	−0.456595
$286$	21.0902	1.24709
$287$	13.4164	0.791946
$288$	5.61803	0.331046
$289$	−4.94427	−0.290840
$290$	−0.618034	−0.0362922
$291$	2.76393	0.162025
$292$	12.9443	0.757506
$293$	−17.9443	−1.04832	−0.524158	−	0.851621i	$-0.675620\pi$
$293$	−17.9443	−1.04832	−0.524158	+	0.851621i	$0.675620\pi$
$294$	1.23607	0.0720889
$295$	−5.23607	−0.304856
$296$	17.8885	1.03975
$297$	−5.47214	−0.317526
$298$	12.4721	0.722491
$299$	0	0
$300$	−1.61803	−0.0934172
$301$	−9.70820	−0.559572
$302$	1.52786	0.0879187
$303$	4.23607	0.243356
$304$	14.2918	0.819691
$305$	5.70820	0.326851
$306$	2.14590	0.122673
$307$	24.9443	1.42364	0.711822	−	0.702360i	$-0.247870\pi$
$307$	24.9443	1.42364	0.711822	+	0.702360i	$0.247870\pi$
$308$	−26.5623	−1.51353
$309$	7.41641	0.421905
$310$	4.94427	0.280816
$311$	23.4721	1.33098	0.665491	−	0.746406i	$-0.268222\pi$
$311$	23.4721	1.33098	0.665491	+	0.746406i	$0.268222\pi$
$312$	13.9443	0.789439
$313$	1.18034	0.0667168	0.0333584	−	0.999443i	$-0.489380\pi$
$313$	1.18034	0.0667168	0.0333584	+	0.999443i	$0.489380\pi$
$314$	−7.23607	−0.408355
$315$	3.00000	0.169031
$316$	10.0000	0.562544
$317$	−28.4164	−1.59602	−0.798012	−	0.602641i	$-0.794115\pi$
$317$	−28.4164	−1.59602	−0.798012	+	0.602641i	$0.794115\pi$
$318$	−4.18034	−0.234422
$319$	−5.47214	−0.306381
$320$	0.236068	0.0131966
$321$	7.52786	0.420164
$322$	0	0
$323$	26.7639	1.48919
$324$	−1.61803	−0.0898908
$325$	−6.23607	−0.345915
$326$	−9.34752	−0.517711
$327$	−16.4164	−0.907829
$328$	10.0000	0.552158
$329$	−20.1246	−1.10951
$330$	3.38197	0.186171
$331$	−15.8885	−0.873313	−0.436657	−	0.899628i	$-0.643838\pi$
$331$	−15.8885	−0.873313	−0.436657	+	0.899628i	$0.643838\pi$
$332$	6.00000	0.329293
$333$	−8.00000	−0.438397
$334$	−11.0557	−0.604943
$335$	11.4721	0.626790
$336$	−5.56231	−0.303449
$337$	−20.4721	−1.11519	−0.557594	−	0.830114i	$-0.688275\pi$
$337$	−20.4721	−1.11519	−0.557594	+	0.830114i	$0.688275\pi$
$338$	16.0000	0.870285
$339$	−7.94427	−0.431474
$340$	5.61803	0.304681
$341$	43.7771	2.37066
$342$	4.76393	0.257604
$343$	15.0000	0.809924
$344$	−7.23607	−0.390143
$345$	0	0
$346$	−12.0000	−0.645124
$347$	17.5279	0.940945	0.470473	−	0.882415i	$-0.344083\pi$
$347$	17.5279	0.940945	0.470473	+	0.882415i	$0.344083\pi$
$348$	−1.61803	−0.0867357
$349$	−18.9443	−1.01406	−0.507032	−	0.861927i	$-0.669257\pi$
$349$	−18.9443	−1.01406	−0.507032	+	0.861927i	$0.669257\pi$
$350$	−1.85410	−0.0991059
$351$	−6.23607	−0.332857
$352$	−30.7426	−1.63859
$353$	3.52786	0.187769	0.0938846	−	0.995583i	$-0.470072\pi$
$353$	3.52786	0.187769	0.0938846	+	0.995583i	$0.470072\pi$
$354$	3.23607	0.171995
$355$	−7.23607	−0.384051
$356$	−18.0902	−0.958777
$357$	−10.4164	−0.551295
$358$	2.58359	0.136547
$359$	−4.58359	−0.241913	−0.120956	−	0.992658i	$-0.538596\pi$
$359$	−4.58359	−0.241913	−0.120956	+	0.992658i	$0.538596\pi$
$360$	2.23607	0.117851
$361$	40.4164	2.12718
$362$	5.20163	0.273391
$363$	18.9443	0.994316
$364$	−30.2705	−1.58661
$365$	8.00000	0.418739
$366$	−3.52786	−0.184404
$367$	15.4164	0.804730	0.402365	−	0.915479i	$-0.368188\pi$
$367$	15.4164	0.804730	0.402365	+	0.915479i	$0.368188\pi$
$368$	0	0
$369$	−4.47214	−0.232810
$370$	4.94427	0.257040
$371$	20.2918	1.05350
$372$	12.9443	0.671129
$373$	15.8885	0.822678	0.411339	−	0.911483i	$-0.365061\pi$
$373$	15.8885	0.822678	0.411339	+	0.911483i	$0.365061\pi$
$374$	−11.7426	−0.607198
$375$	−1.00000	−0.0516398
$376$	−15.0000	−0.773566
$377$	−6.23607	−0.321174
$378$	−1.85410	−0.0953647
$379$	6.94427	0.356703	0.178352	−	0.983967i	$-0.442924\pi$
$379$	6.94427	0.356703	0.178352	+	0.983967i	$0.442924\pi$
$380$	12.4721	0.639807
$381$	−6.00000	−0.307389
$382$	7.41641	0.379456
$383$	−20.0689	−1.02547	−0.512736	−	0.858546i	$-0.671368\pi$
$383$	−20.0689	−1.02547	−0.512736	+	0.858546i	$0.671368\pi$
$384$	−11.3820	−0.580834
$385$	−16.4164	−0.836658
$386$	−3.70820	−0.188743
$387$	3.23607	0.164499
$388$	−4.47214	−0.227038
$389$	22.2361	1.12741	0.563707	−	0.825975i	$-0.309375\pi$
$389$	22.2361	1.12741	0.563707	+	0.825975i	$0.309375\pi$
$390$	3.85410	0.195160
$391$	0	0
$392$	−4.47214	−0.225877
$393$	3.47214	0.175146
$394$	9.23607	0.465306
$395$	6.18034	0.310967
$396$	8.85410	0.444935
$397$	−6.58359	−0.330421	−0.165211	−	0.986258i	$-0.552830\pi$
$397$	−6.58359	−0.330421	−0.165211	+	0.986258i	$0.552830\pi$
$398$	12.7984	0.641525
$399$	−23.1246	−1.15768
$400$	1.85410	0.0927051
$401$	−36.6525	−1.83034	−0.915169	−	0.403071i	$-0.867943\pi$
$401$	−36.6525	−1.83034	−0.915169	+	0.403071i	$0.867943\pi$
$402$	−7.09017	−0.353626
$403$	49.8885	2.48513
$404$	−6.85410	−0.341004
$405$	−1.00000	−0.0496904
$406$	−1.85410	−0.0920175
$407$	43.7771	2.16995
$408$	−7.76393	−0.384372
$409$	8.65248	0.427837	0.213919	−	0.976851i	$-0.431377\pi$
$409$	8.65248	0.427837	0.213919	+	0.976851i	$0.431377\pi$
$410$	2.76393	0.136501
$411$	−10.9443	−0.539841
$412$	−12.0000	−0.591198
$413$	−15.7082	−0.772950
$414$	0	0
$415$	3.70820	0.182029
$416$	−35.0344	−1.71770
$417$	−0.708204	−0.0346809
$418$	−26.0689	−1.27507
$419$	34.3607	1.67863	0.839315	−	0.543646i	$-0.182957\pi$
$419$	34.3607	1.67863	0.839315	+	0.543646i	$0.182957\pi$
$420$	−4.85410	−0.236856
$421$	−1.81966	−0.0886848	−0.0443424	−	0.999016i	$-0.514119\pi$
$421$	−1.81966	−0.0886848	−0.0443424	+	0.999016i	$0.514119\pi$
$422$	−17.2361	−0.839039
$423$	6.70820	0.326164
$424$	15.1246	0.734516
$425$	3.47214	0.168423
$426$	4.47214	0.216676
$427$	17.1246	0.828718
$428$	−12.1803	−0.588759
$429$	34.1246	1.64755
$430$	−2.00000	−0.0964486
$431$	34.4721	1.66046	0.830232	−	0.557418i	$-0.188208\pi$
$431$	34.4721	1.66046	0.830232	+	0.557418i	$0.188208\pi$
$432$	1.85410	0.0892055
$433$	−4.65248	−0.223584	−0.111792	−	0.993732i	$-0.535659\pi$
$433$	−4.65248	−0.223584	−0.111792	+	0.993732i	$0.535659\pi$
$434$	14.8328	0.711998
$435$	−1.00000	−0.0479463
$436$	26.5623	1.27210
$437$	0	0
$438$	−4.94427	−0.236246
$439$	−10.1246	−0.483221	−0.241611	−	0.970373i	$-0.577676\pi$
$439$	−10.1246	−0.483221	−0.241611	+	0.970373i	$0.577676\pi$
$440$	−12.2361	−0.583332
$441$	2.00000	0.0952381
$442$	−13.3820	−0.636515
$443$	23.7639	1.12906	0.564529	−	0.825413i	$-0.309058\pi$
$443$	23.7639	1.12906	0.564529	+	0.825413i	$0.309058\pi$
$444$	12.9443	0.614308
$445$	−11.1803	−0.529999
$446$	−8.03444	−0.380442
$447$	20.1803	0.954497
$448$	0.708204	0.0334595
$449$	1.76393	0.0832451	0.0416225	−	0.999133i	$-0.486747\pi$
$449$	1.76393	0.0832451	0.0416225	+	0.999133i	$0.486747\pi$
$450$	0.618034	0.0291344
$451$	24.4721	1.15235
$452$	12.8541	0.604606
$453$	2.47214	0.116151
$454$	−3.59675	−0.168804
$455$	−18.7082	−0.877054
$456$	−17.2361	−0.807153
$457$	6.12461	0.286497	0.143249	−	0.989687i	$-0.454245\pi$
$457$	6.12461	0.286497	0.143249	+	0.989687i	$0.454245\pi$
$458$	5.81966	0.271935
$459$	3.47214	0.162065
$460$	0	0
$461$	2.36068	0.109948	0.0549739	−	0.998488i	$-0.482492\pi$
$461$	2.36068	0.109948	0.0549739	+	0.998488i	$0.482492\pi$
$462$	10.1459	0.472030
$463$	−2.52786	−0.117480	−0.0587399	−	0.998273i	$-0.518708\pi$
$463$	−2.52786	−0.117480	−0.0587399	+	0.998273i	$0.518708\pi$
$464$	1.85410	0.0860745
$465$	8.00000	0.370991
$466$	0.875388	0.0405516
$467$	−12.9443	−0.598989	−0.299495	−	0.954098i	$-0.596818\pi$
$467$	−12.9443	−0.598989	−0.299495	+	0.954098i	$0.596818\pi$
$468$	10.0902	0.466418
$469$	34.4164	1.58920
$470$	−4.14590	−0.191236
$471$	−11.7082	−0.539486
$472$	−11.7082	−0.538914
$473$	−17.7082	−0.814224
$474$	−3.81966	−0.175443
$475$	7.70820	0.353677
$476$	16.8541	0.772506
$477$	−6.76393	−0.309699
$478$	7.34752	0.336068
$479$	−6.47214	−0.295719	−0.147860	−	0.989008i	$-0.547238\pi$
$479$	−6.47214	−0.295719	−0.147860	+	0.989008i	$0.547238\pi$
$480$	−5.61803	−0.256427
$481$	49.8885	2.27472
$482$	4.32624	0.197055
$483$	0	0
$484$	−30.6525	−1.39329
$485$	−2.76393	−0.125504
$486$	0.618034	0.0280346
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	12.7639	0.577796
$489$	−15.1246	−0.683959
$490$	−1.23607	−0.0558399
$491$	25.8885	1.16833	0.584167	−	0.811634i	$-0.301421\pi$
$491$	25.8885	1.16833	0.584167	+	0.811634i	$0.301421\pi$
$492$	7.23607	0.326227
$493$	3.47214	0.156377
$494$	−29.7082	−1.33664
$495$	5.47214	0.245954
$496$	−14.8328	−0.666013
$497$	−21.7082	−0.973746
$498$	−2.29180	−0.102698
$499$	−23.7639	−1.06382	−0.531910	−	0.846801i	$-0.678525\pi$
$499$	−23.7639	−1.06382	−0.531910	+	0.846801i	$0.678525\pi$
$500$	1.61803	0.0723607
$501$	−17.8885	−0.799201
$502$	−15.3820	−0.686531
$503$	−30.5967	−1.36424	−0.682121	−	0.731240i	$-0.738942\pi$
$503$	−30.5967	−1.36424	−0.682121	+	0.731240i	$0.738942\pi$
$504$	6.70820	0.298807
$505$	−4.23607	−0.188503
$506$	0	0
$507$	25.8885	1.14975
$508$	9.70820	0.430732
$509$	6.18034	0.273939	0.136969	−	0.990575i	$-0.456264\pi$
$509$	6.18034	0.273939	0.136969	+	0.990575i	$0.456264\pi$
$510$	−2.14590	−0.0950220
$511$	24.0000	1.06170
$512$	18.7082	0.826794
$513$	7.70820	0.340326
$514$	−6.00000	−0.264649
$515$	−7.41641	−0.326806
$516$	−5.23607	−0.230505
$517$	−36.7082	−1.61442
$518$	14.8328	0.651717
$519$	−19.4164	−0.852286
$520$	−13.9443	−0.611497
$521$	−20.7639	−0.909684	−0.454842	−	0.890572i	$-0.650304\pi$
$521$	−20.7639	−0.909684	−0.454842	+	0.890572i	$0.650304\pi$
$522$	0.618034	0.0270506
$523$	−29.8328	−1.30450	−0.652249	−	0.758005i	$-0.726174\pi$
$523$	−29.8328	−1.30450	−0.652249	+	0.758005i	$0.726174\pi$
$524$	−5.61803	−0.245425
$525$	−3.00000	−0.130931
$526$	14.8328	0.646741
$527$	−27.7771	−1.20999
$528$	−10.1459	−0.441544
$529$	−23.0000	−1.00000
$530$	4.18034	0.181582
$531$	5.23607	0.227226
$532$	37.4164	1.62221
$533$	27.8885	1.20799
$534$	6.90983	0.299018
$535$	−7.52786	−0.325458
$536$	25.6525	1.10802
$537$	4.18034	0.180395
$538$	−18.6869	−0.805650
$539$	−10.9443	−0.471403
$540$	1.61803	0.0696291
$541$	38.3607	1.64925	0.824627	−	0.565677i	$-0.191385\pi$
$541$	38.3607	1.64925	0.824627	+	0.565677i	$0.191385\pi$
$542$	8.87539	0.381231
$543$	8.41641	0.361183
$544$	19.5066	0.836338
$545$	16.4164	0.703202
$546$	11.5623	0.494821
$547$	27.4721	1.17462	0.587312	−	0.809361i	$-0.300186\pi$
$547$	27.4721	1.17462	0.587312	+	0.809361i	$0.300186\pi$
$548$	17.7082	0.756457
$549$	−5.70820	−0.243620
$550$	−3.38197	−0.144208
$551$	7.70820	0.328381
$552$	0	0
$553$	18.5410	0.788444
$554$	5.38197	0.228658
$555$	8.00000	0.339581
$556$	1.14590	0.0485969
$557$	2.76393	0.117112	0.0585558	−	0.998284i	$-0.481350\pi$
$557$	2.76393	0.117112	0.0585558	+	0.998284i	$0.481350\pi$
$558$	−4.94427	−0.209308
$559$	−20.1803	−0.853537
$560$	5.56231	0.235050
$561$	−19.0000	−0.802181
$562$	−14.2918	−0.602863
$563$	44.1246	1.85963	0.929815	−	0.368026i	$-0.119966\pi$
$563$	44.1246	1.85963	0.929815	+	0.368026i	$0.119966\pi$
$564$	−10.8541	−0.457040
$565$	7.94427	0.334218
$566$	11.4164	0.479867
$567$	−3.00000	−0.125988
$568$	−16.1803	−0.678912
$569$	−5.18034	−0.217171	−0.108586	−	0.994087i	$-0.534632\pi$
$569$	−5.18034	−0.217171	−0.108586	+	0.994087i	$0.534632\pi$
$570$	−4.76393	−0.199539
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	−55.2148	−2.30865
$573$	12.0000	0.501307
$574$	8.29180	0.346093
$575$	0	0
$576$	−0.236068	−0.00983617
$577$	−29.3050	−1.21998	−0.609991	−	0.792409i	$-0.708827\pi$
$577$	−29.3050	−1.21998	−0.609991	+	0.792409i	$0.708827\pi$
$578$	−3.05573	−0.127102
$579$	−6.00000	−0.249351
$580$	1.61803	0.0671852
$581$	11.1246	0.461527
$582$	1.70820	0.0708073
$583$	37.0132	1.53293
$584$	17.8885	0.740233
$585$	6.23607	0.257830
$586$	−11.0902	−0.458131
$587$	−1.81966	−0.0751054	−0.0375527	−	0.999295i	$-0.511956\pi$
$587$	−1.81966	−0.0751054	−0.0375527	+	0.999295i	$0.511956\pi$
$588$	−3.23607	−0.133453
$589$	−61.6656	−2.54089
$590$	−3.23607	−0.133227
$591$	14.9443	0.614725
$592$	−14.8328	−0.609625
$593$	−24.6525	−1.01236	−0.506178	−	0.862429i	$-0.668942\pi$
$593$	−24.6525	−1.01236	−0.506178	+	0.862429i	$0.668942\pi$
$594$	−3.38197	−0.138764
$595$	10.4164	0.427031
$596$	−32.6525	−1.33750
$597$	20.7082	0.847530
$598$	0	0
$599$	−8.88854	−0.363176	−0.181588	−	0.983375i	$-0.558124\pi$
$599$	−8.88854	−0.363176	−0.181588	+	0.983375i	$0.558124\pi$
$600$	−2.23607	−0.0912871
$601$	8.11146	0.330873	0.165437	−	0.986220i	$-0.447097\pi$
$601$	8.11146	0.330873	0.165437	+	0.986220i	$0.447097\pi$
$602$	−6.00000	−0.244542
$603$	−11.4721	−0.467181
$604$	−4.00000	−0.162758
$605$	−18.9443	−0.770194
$606$	2.61803	0.106350
$607$	40.6525	1.65003	0.825017	−	0.565109i	$-0.191166\pi$
$607$	40.6525	1.65003	0.825017	+	0.565109i	$0.191166\pi$
$608$	43.3050	1.75625
$609$	−3.00000	−0.121566
$610$	3.52786	0.142839
$611$	−41.8328	−1.69237
$612$	−5.61803	−0.227096
$613$	7.29180	0.294513	0.147256	−	0.989098i	$-0.452956\pi$
$613$	7.29180	0.294513	0.147256	+	0.989098i	$0.452956\pi$
$614$	15.4164	0.622156
$615$	4.47214	0.180334
$616$	−36.7082	−1.47902
$617$	43.3050	1.74339	0.871696	−	0.490047i	$-0.163020\pi$
$617$	43.3050	1.74339	0.871696	+	0.490047i	$0.163020\pi$
$618$	4.58359	0.184379
$619$	−3.52786	−0.141797	−0.0708984	−	0.997484i	$-0.522587\pi$
$619$	−3.52786	−0.141797	−0.0708984	+	0.997484i	$0.522587\pi$
$620$	−12.9443	−0.519854
$621$	0	0
$622$	14.5066	0.581661
$623$	−33.5410	−1.34379
$624$	−11.5623	−0.462863
$625$	1.00000	0.0400000
$626$	0.729490	0.0291563
$627$	−42.1803	−1.68452
$628$	18.9443	0.755959
$629$	−27.7771	−1.10755
$630$	1.85410	0.0738692
$631$	−38.4853	−1.53208	−0.766038	−	0.642796i	$-0.777774\pi$
$631$	−38.4853	−1.53208	−0.766038	+	0.642796i	$0.777774\pi$
$632$	13.8197	0.549717
$633$	−27.8885	−1.10847
$634$	−17.5623	−0.697488
$635$	6.00000	0.238103
$636$	10.9443	0.433969
$637$	−12.4721	−0.494164
$638$	−3.38197	−0.133893
$639$	7.23607	0.286254
$640$	11.3820	0.449912
$641$	46.2361	1.82621	0.913107	−	0.407719i	$-0.133676\pi$
$641$	46.2361	1.82621	0.913107	+	0.407719i	$0.133676\pi$
$642$	4.65248	0.183619
$643$	−35.8328	−1.41311	−0.706554	−	0.707659i	$-0.749751\pi$
$643$	−35.8328	−1.41311	−0.706554	+	0.707659i	$0.749751\pi$
$644$	0	0
$645$	−3.23607	−0.127420
$646$	16.5410	0.650798
$647$	11.2361	0.441735	0.220868	−	0.975304i	$-0.429111\pi$
$647$	11.2361	0.441735	0.220868	+	0.975304i	$0.429111\pi$
$648$	−2.23607	−0.0878410
$649$	−28.6525	−1.12471
$650$	−3.85410	−0.151170
$651$	24.0000	0.940634
$652$	24.4721	0.958403
$653$	8.88854	0.347836	0.173918	−	0.984760i	$-0.444357\pi$
$653$	8.88854	0.347836	0.173918	+	0.984760i	$0.444357\pi$
$654$	−10.1459	−0.396736
$655$	−3.47214	−0.135668
$656$	−8.29180	−0.323740
$657$	−8.00000	−0.312110
$658$	−12.4377	−0.484872
$659$	−21.0000	−0.818044	−0.409022	−	0.912525i	$-0.634130\pi$
$659$	−21.0000	−0.818044	−0.409022	+	0.912525i	$0.634130\pi$
$660$	−8.85410	−0.344645
$661$	−37.8328	−1.47153	−0.735763	−	0.677239i	$-0.763176\pi$
$661$	−37.8328	−1.47153	−0.735763	+	0.677239i	$0.763176\pi$
$662$	−9.81966	−0.381652
$663$	−21.6525	−0.840912
$664$	8.29180	0.321784
$665$	23.1246	0.896734
$666$	−4.94427	−0.191587
$667$	0	0
$668$	28.9443	1.11989
$669$	−13.0000	−0.502609
$670$	7.09017	0.273917
$671$	31.2361	1.20586
$672$	−16.8541	−0.650161
$673$	−33.2918	−1.28330	−0.641652	−	0.766996i	$-0.721751\pi$
$673$	−33.2918	−1.28330	−0.641652	+	0.766996i	$0.721751\pi$
$674$	−12.6525	−0.487355
$675$	1.00000	0.0384900
$676$	−41.8885	−1.61110
$677$	−20.8885	−0.802812	−0.401406	−	0.915900i	$-0.631478\pi$
$677$	−20.8885	−0.802812	−0.401406	+	0.915900i	$0.631478\pi$
$678$	−4.90983	−0.188561
$679$	−8.29180	−0.318210
$680$	7.76393	0.297733
$681$	−5.81966	−0.223010
$682$	27.0557	1.03602
$683$	7.41641	0.283781	0.141890	−	0.989882i	$-0.454682\pi$
$683$	7.41641	0.283781	0.141890	+	0.989882i	$0.454682\pi$
$684$	−12.4721	−0.476884
$685$	10.9443	0.418159
$686$	9.27051	0.353950
$687$	9.41641	0.359258
$688$	6.00000	0.228748
$689$	42.1803	1.60694
$690$	0	0
$691$	−34.7082	−1.32036	−0.660181	−	0.751106i	$-0.729520\pi$
$691$	−34.7082	−1.32036	−0.660181	+	0.751106i	$0.729520\pi$
$692$	31.4164	1.19427
$693$	16.4164	0.623608
$694$	10.8328	0.411208
$695$	0.708204	0.0268637
$696$	−2.23607	−0.0847579
$697$	−15.5279	−0.588160
$698$	−11.7082	−0.443162
$699$	1.41641	0.0535735
$700$	4.85410	0.183468
$701$	−5.12461	−0.193554	−0.0967770	−	0.995306i	$-0.530853\pi$
$701$	−5.12461	−0.193554	−0.0967770	+	0.995306i	$0.530853\pi$
$702$	−3.85410	−0.145464
$703$	−61.6656	−2.32576
$704$	1.29180	0.0486864
$705$	−6.70820	−0.252646
$706$	2.18034	0.0820582
$707$	−12.7082	−0.477941
$708$	−8.47214	−0.318402
$709$	19.8885	0.746930	0.373465	−	0.927644i	$-0.378170\pi$
$709$	19.8885	0.746930	0.373465	+	0.927644i	$0.378170\pi$
$710$	−4.47214	−0.167836
$711$	−6.18034	−0.231781
$712$	−25.0000	−0.936915
$713$	0	0
$714$	−6.43769	−0.240925
$715$	−34.1246	−1.27619
$716$	−6.76393	−0.252780
$717$	11.8885	0.443986
$718$	−2.83282	−0.105720
$719$	−11.2361	−0.419035	−0.209517	−	0.977805i	$-0.567189\pi$
$719$	−11.2361	−0.419035	−0.209517	+	0.977805i	$0.567189\pi$
$720$	−1.85410	−0.0690983
$721$	−22.2492	−0.828604
$722$	24.9787	0.929611
$723$	7.00000	0.260333
$724$	−13.6180	−0.506110
$725$	1.00000	0.0371391
$726$	11.7082	0.434532
$727$	2.11146	0.0783096	0.0391548	−	0.999233i	$-0.487533\pi$
$727$	2.11146	0.0783096	0.0391548	+	0.999233i	$0.487533\pi$
$728$	−41.8328	−1.55043
$729$	1.00000	0.0370370
$730$	4.94427	0.182996
$731$	11.2361	0.415581
$732$	9.23607	0.341375
$733$	−14.2918	−0.527880	−0.263940	−	0.964539i	$-0.585022\pi$
$733$	−14.2918	−0.527880	−0.263940	+	0.964539i	$0.585022\pi$
$734$	9.52786	0.351680
$735$	−2.00000	−0.0737711
$736$	0	0
$737$	62.7771	2.31242
$738$	−2.76393	−0.101742
$739$	41.4853	1.52606	0.763031	−	0.646362i	$-0.223711\pi$
$739$	41.4853	1.52606	0.763031	+	0.646362i	$0.223711\pi$
$740$	−12.9443	−0.475841
$741$	−48.0689	−1.76585
$742$	12.5410	0.460395
$743$	−10.8197	−0.396935	−0.198467	−	0.980107i	$-0.563596\pi$
$743$	−10.8197	−0.396935	−0.198467	+	0.980107i	$0.563596\pi$
$744$	17.8885	0.655826
$745$	−20.1803	−0.739350
$746$	9.81966	0.359523
$747$	−3.70820	−0.135676
$748$	30.7426	1.12406
$749$	−22.5836	−0.825186
$750$	−0.618034	−0.0225674
$751$	43.4164	1.58429	0.792144	−	0.610335i	$-0.208965\pi$
$751$	43.4164	1.58429	0.792144	+	0.610335i	$0.208965\pi$
$752$	12.4377	0.453556
$753$	−24.8885	−0.906989
$754$	−3.85410	−0.140358
$755$	−2.47214	−0.0899702
$756$	4.85410	0.176542
$757$	−34.8328	−1.26602	−0.633010	−	0.774144i	$-0.718181\pi$
$757$	−34.8328	−1.26602	−0.633010	+	0.774144i	$0.718181\pi$
$758$	4.29180	0.155885
$759$	0	0
$760$	17.2361	0.625218
$761$	33.0132	1.19673	0.598363	−	0.801225i	$-0.295818\pi$
$761$	33.0132	1.19673	0.598363	+	0.801225i	$0.295818\pi$
$762$	−3.70820	−0.134334
$763$	49.2492	1.78294
$764$	−19.4164	−0.702461
$765$	−3.47214	−0.125535
$766$	−12.4033	−0.448148
$767$	−32.6525	−1.17901
$768$	−6.56231	−0.236797
$769$	34.6525	1.24960	0.624800	−	0.780785i	$-0.285180\pi$
$769$	34.6525	1.24960	0.624800	+	0.780785i	$0.285180\pi$
$770$	−10.1459	−0.365633
$771$	−9.70820	−0.349632
$772$	9.70820	0.349406
$773$	−52.2492	−1.87927	−0.939637	−	0.342173i	$-0.888837\pi$
$773$	−52.2492	−1.87927	−0.939637	+	0.342173i	$0.888837\pi$
$774$	2.00000	0.0718885
$775$	−8.00000	−0.287368
$776$	−6.18034	−0.221861
$777$	24.0000	0.860995
$778$	13.7426	0.492698
$779$	−34.4721	−1.23509
$780$	−10.0902	−0.361286
$781$	−39.5967	−1.41688
$782$	0	0
$783$	1.00000	0.0357371
$784$	3.70820	0.132436
$785$	11.7082	0.417884
$786$	2.14590	0.0765416
$787$	1.88854	0.0673193	0.0336597	−	0.999433i	$-0.489284\pi$
$787$	1.88854	0.0673193	0.0336597	+	0.999433i	$0.489284\pi$
$788$	−24.1803	−0.861389
$789$	24.0000	0.854423
$790$	3.81966	0.135897
$791$	23.8328	0.847397
$792$	12.2361	0.434790
$793$	35.5967	1.26408
$794$	−4.06888	−0.144399
$795$	6.76393	0.239892
$796$	−33.5066	−1.18761
$797$	6.94427	0.245979	0.122989	−	0.992408i	$-0.460752\pi$
$797$	6.94427	0.245979	0.122989	+	0.992408i	$0.460752\pi$
$798$	−14.2918	−0.505924
$799$	23.2918	0.824005
$800$	5.61803	0.198627
$801$	11.1803	0.395038
$802$	−22.6525	−0.799887
$803$	43.7771	1.54486
$804$	18.5623	0.654642
$805$	0	0
$806$	30.8328	1.08604
$807$	−30.2361	−1.06436
$808$	−9.47214	−0.333229
$809$	44.2361	1.55526	0.777629	−	0.628724i	$-0.216422\pi$
$809$	44.2361	1.55526	0.777629	+	0.628724i	$0.216422\pi$
$810$	−0.618034	−0.0217155
$811$	−22.7082	−0.797393	−0.398696	−	0.917083i	$-0.630537\pi$
$811$	−22.7082	−0.797393	−0.398696	+	0.917083i	$0.630537\pi$
$812$	4.85410	0.170346
$813$	14.3607	0.503651
$814$	27.0557	0.948303
$815$	15.1246	0.529792
$816$	6.43769	0.225364
$817$	24.9443	0.872690
$818$	5.34752	0.186972
$819$	18.7082	0.653718
$820$	−7.23607	−0.252694
$821$	−10.5836	−0.369370	−0.184685	−	0.982798i	$-0.559126\pi$
$821$	−10.5836	−0.369370	−0.184685	+	0.982798i	$0.559126\pi$
$822$	−6.76393	−0.235919
$823$	−28.7639	−1.00265	−0.501324	−	0.865260i	$-0.667153\pi$
$823$	−28.7639	−1.00265	−0.501324	+	0.865260i	$0.667153\pi$
$824$	−16.5836	−0.577717
$825$	−5.47214	−0.190515
$826$	−9.70820	−0.337792
$827$	−7.41641	−0.257894	−0.128947	−	0.991652i	$-0.541160\pi$
$827$	−7.41641	−0.257894	−0.128947	+	0.991652i	$0.541160\pi$
$828$	0	0
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	2.29180	0.0795494
$831$	8.70820	0.302084
$832$	1.47214	0.0510371
$833$	6.94427	0.240605
$834$	−0.437694	−0.0151561
$835$	17.8885	0.619059
$836$	68.2492	2.36045
$837$	−8.00000	−0.276520
$838$	21.2361	0.733588
$839$	14.8885	0.514010	0.257005	−	0.966410i	$-0.417264\pi$
$839$	14.8885	0.514010	0.257005	+	0.966410i	$0.417264\pi$
$840$	−6.70820	−0.231455
$841$	1.00000	0.0344828
$842$	−1.12461	−0.0387567
$843$	−23.1246	−0.796454
$844$	45.1246	1.55325
$845$	−25.8885	−0.890593
$846$	4.14590	0.142539
$847$	−56.8328	−1.95280
$848$	−12.5410	−0.430660
$849$	18.4721	0.633962
$850$	2.14590	0.0736037
$851$	0	0
$852$	−11.7082	−0.401116
$853$	−28.7639	−0.984858	−0.492429	−	0.870353i	$-0.663891\pi$
$853$	−28.7639	−0.984858	−0.492429	+	0.870353i	$0.663891\pi$
$854$	10.5836	0.362163
$855$	−7.70820	−0.263615
$856$	−16.8328	−0.575334
$857$	−23.8885	−0.816017	−0.408009	−	0.912978i	$-0.633777\pi$
$857$	−23.8885	−0.816017	−0.408009	+	0.912978i	$0.633777\pi$
$858$	21.0902	0.720007
$859$	−15.7082	−0.535957	−0.267979	−	0.963425i	$-0.586356\pi$
$859$	−15.7082	−0.535957	−0.267979	+	0.963425i	$0.586356\pi$
$860$	5.23607	0.178548
$861$	13.4164	0.457230
$862$	21.3050	0.725650
$863$	15.5967	0.530919	0.265460	−	0.964122i	$-0.414476\pi$
$863$	15.5967	0.530919	0.265460	+	0.964122i	$0.414476\pi$
$864$	5.61803	0.191129
$865$	19.4164	0.660178
$866$	−2.87539	−0.0977097
$867$	−4.94427	−0.167916
$868$	−38.8328	−1.31807
$869$	33.8197	1.14725
$870$	−0.618034	−0.0209533
$871$	71.5410	2.42407
$872$	36.7082	1.24310
$873$	2.76393	0.0935449
$874$	0	0
$875$	3.00000	0.101419
$876$	12.9443	0.437346
$877$	48.8328	1.64897	0.824484	−	0.565886i	$-0.191466\pi$
$877$	48.8328	1.64897	0.824484	+	0.565886i	$0.191466\pi$
$878$	−6.25735	−0.211175
$879$	−17.9443	−0.605245
$880$	10.1459	0.342018
$881$	12.7082	0.428150	0.214075	−	0.976817i	$-0.431326\pi$
$881$	12.7082	0.428150	0.214075	+	0.976817i	$0.431326\pi$
$882$	1.23607	0.0416206
$883$	−32.0000	−1.07689	−0.538443	−	0.842662i	$-0.680987\pi$
$883$	−32.0000	−1.07689	−0.538443	+	0.842662i	$0.680987\pi$
$884$	35.0344	1.17834
$885$	−5.23607	−0.176008
$886$	14.6869	0.493417
$887$	−44.5967	−1.49741	−0.748706	−	0.662902i	$-0.769325\pi$
$887$	−44.5967	−1.49741	−0.748706	+	0.662902i	$0.769325\pi$
$888$	17.8885	0.600300
$889$	18.0000	0.603701
$890$	−6.90983	−0.231618
$891$	−5.47214	−0.183323
$892$	21.0344	0.704285
$893$	51.7082	1.73035
$894$	12.4721	0.417131
$895$	−4.18034	−0.139733
$896$	34.1459	1.14073
$897$	0	0
$898$	1.09017	0.0363794
$899$	−8.00000	−0.266815
$900$	−1.61803	−0.0539345
$901$	−23.4853	−0.782409
$902$	15.1246	0.503594
$903$	−9.70820	−0.323069
$904$	17.7639	0.590820
$905$	−8.41641	−0.279771
$906$	1.52786	0.0507599
$907$	26.7639	0.888682	0.444341	−	0.895858i	$-0.353438\pi$
$907$	26.7639	0.888682	0.444341	+	0.895858i	$0.353438\pi$
$908$	9.41641	0.312494
$909$	4.23607	0.140502
$910$	−11.5623	−0.383287
$911$	−18.0557	−0.598213	−0.299106	−	0.954220i	$-0.596689\pi$
$911$	−18.0557	−0.598213	−0.299106	+	0.954220i	$0.596689\pi$
$912$	14.2918	0.473249
$913$	20.2918	0.671560
$914$	3.78522	0.125204
$915$	5.70820	0.188707
$916$	−15.2361	−0.503414
$917$	−10.4164	−0.343980
$918$	2.14590	0.0708252
$919$	20.1246	0.663850	0.331925	−	0.943306i	$-0.392302\pi$
$919$	20.1246	0.663850	0.331925	+	0.943306i	$0.392302\pi$
$920$	0	0
$921$	24.9443	0.821942
$922$	1.45898	0.0480490
$923$	−45.1246	−1.48529
$924$	−26.5623	−0.873836
$925$	−8.00000	−0.263038
$926$	−1.56231	−0.0513406
$927$	7.41641	0.243587
$928$	5.61803	0.184421
$929$	−56.8328	−1.86462	−0.932312	−	0.361655i	$-0.882212\pi$
$929$	−56.8328	−1.86462	−0.932312	+	0.361655i	$0.882212\pi$
$930$	4.94427	0.162129
$931$	15.4164	0.505252
$932$	−2.29180	−0.0750703
$933$	23.4721	0.768443
$934$	−8.00000	−0.261768
$935$	19.0000	0.621366
$936$	13.9443	0.455783
$937$	19.7639	0.645660	0.322830	−	0.946457i	$-0.395366\pi$
$937$	19.7639	0.645660	0.322830	+	0.946457i	$0.395366\pi$
$938$	21.2705	0.694507
$939$	1.18034	0.0385189
$940$	10.8541	0.354022
$941$	−46.0689	−1.50180	−0.750901	−	0.660414i	$-0.770381\pi$
$941$	−46.0689	−1.50180	−0.750901	+	0.660414i	$0.770381\pi$
$942$	−7.23607	−0.235764
$943$	0	0
$944$	9.70820	0.315975
$945$	3.00000	0.0975900
$946$	−10.9443	−0.355829
$947$	−12.7082	−0.412961	−0.206481	−	0.978451i	$-0.566201\pi$
$947$	−12.7082	−0.412961	−0.206481	+	0.978451i	$0.566201\pi$
$948$	10.0000	0.324785
$949$	49.8885	1.61945
$950$	4.76393	0.154562
$951$	−28.4164	−0.921465
$952$	23.2918	0.754891
$953$	19.0132	0.615897	0.307948	−	0.951403i	$-0.400358\pi$
$953$	19.0132	0.615897	0.307948	+	0.951403i	$0.400358\pi$
$954$	−4.18034	−0.135344
$955$	−12.0000	−0.388311
$956$	−19.2361	−0.622139
$957$	−5.47214	−0.176889
$958$	−4.00000	−0.129234
$959$	32.8328	1.06023
$960$	0.236068	0.00761906
$961$	33.0000	1.06452
$962$	30.8328	0.994090
$963$	7.52786	0.242582
$964$	−11.3262	−0.364794
$965$	6.00000	0.193147
$966$	0	0
$967$	−35.1246	−1.12953	−0.564766	−	0.825251i	$-0.691033\pi$
$967$	−35.1246	−1.12953	−0.564766	+	0.825251i	$0.691033\pi$
$968$	−42.3607	−1.36152
$969$	26.7639	0.859781
$970$	−1.70820	−0.0548471
$971$	−23.0557	−0.739894	−0.369947	−	0.929053i	$-0.620624\pi$
$971$	−23.0557	−0.739894	−0.369947	+	0.929053i	$0.620624\pi$
$972$	−1.61803	−0.0518985
$973$	2.12461	0.0681119
$974$	−7.41641	−0.237637
$975$	−6.23607	−0.199714
$976$	−10.5836	−0.338773
$977$	−22.4721	−0.718947	−0.359474	−	0.933155i	$-0.617044\pi$
$977$	−22.4721	−0.718947	−0.359474	+	0.933155i	$0.617044\pi$
$978$	−9.34752	−0.298901
$979$	−61.1803	−1.95533
$980$	3.23607	0.103372
$981$	−16.4164	−0.524136
$982$	16.0000	0.510581
$983$	37.5279	1.19695	0.598476	−	0.801140i	$-0.295773\pi$
$983$	37.5279	1.19695	0.598476	+	0.801140i	$0.295773\pi$
$984$	10.0000	0.318788
$985$	−14.9443	−0.476164
$986$	2.14590	0.0683393
$987$	−20.1246	−0.640573
$988$	77.7771	2.47442
$989$	0	0
$990$	3.38197	0.107486
$991$	22.4853	0.714269	0.357134	−	0.934053i	$-0.383754\pi$
$991$	22.4853	0.714269	0.357134	+	0.934053i	$0.383754\pi$
$992$	−44.9443	−1.42698
$993$	−15.8885	−0.504208
$994$	−13.4164	−0.425543
$995$	−20.7082	−0.656494
$996$	6.00000	0.190117
$997$	7.41641	0.234880	0.117440	−	0.993080i	$-0.462531\pi$
$997$	7.41641	0.234880	0.117440	+	0.993080i	$0.462531\pi$
$998$	−14.6869	−0.464906
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	435.2.a.e.1.2	✓	2
3.2	odd	2		1305.2.a.k.1.1		2
4.3	odd	2		6960.2.a.bu.1.2		2
5.2	odd	4		2175.2.c.j.349.3		4
5.3	odd	4		2175.2.c.j.349.2		4
5.4	even	2		2175.2.a.q.1.1		2
15.14	odd	2		6525.2.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.a.e.1.2	✓	2	1.1	even	1	trivial
1305.2.a.k.1.1		2	3.2	odd	2
2175.2.a.q.1.1		2	5.4	even	2
2175.2.c.j.349.2		4	5.3	odd	4
2175.2.c.j.349.3		4	5.2	odd	4
6525.2.a.s.1.2		2	15.14	odd	2
6960.2.a.bu.1.2		2	4.3	odd	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:	\( N \)	\(=\)	\( 435 = 3 \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	435.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -1.00000 q^{5} +0.618034 q^{6} -3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -1.00000 q^{5} +0.618034 q^{6} -3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} -0.618034 q^{10} -5.47214 q^{11} -1.61803 q^{12} -6.23607 q^{13} -1.85410 q^{14} -1.00000 q^{15} +1.85410 q^{16} +3.47214 q^{17} +0.618034 q^{18} +7.70820 q^{19} +1.61803 q^{20} -3.00000 q^{21} -3.38197 q^{22} -2.23607 q^{24} +1.00000 q^{25} -3.85410 q^{26} +1.00000 q^{27} +4.85410 q^{28} +1.00000 q^{29} -0.618034 q^{30} -8.00000 q^{31} +5.61803 q^{32} -5.47214 q^{33} +2.14590 q^{34} +3.00000 q^{35} -1.61803 q^{36} -8.00000 q^{37} +4.76393 q^{38} -6.23607 q^{39} +2.23607 q^{40} -4.47214 q^{41} -1.85410 q^{42} +3.23607 q^{43} +8.85410 q^{44} -1.00000 q^{45} +6.70820 q^{47} +1.85410 q^{48} +2.00000 q^{49} +0.618034 q^{50} +3.47214 q^{51} +10.0902 q^{52} -6.76393 q^{53} +0.618034 q^{54} +5.47214 q^{55} +6.70820 q^{56} +7.70820 q^{57} +0.618034 q^{58} +5.23607 q^{59} +1.61803 q^{60} -5.70820 q^{61} -4.94427 q^{62} -3.00000 q^{63} -0.236068 q^{64} +6.23607 q^{65} -3.38197 q^{66} -11.4721 q^{67} -5.61803 q^{68} +1.85410 q^{70} +7.23607 q^{71} -2.23607 q^{72} -8.00000 q^{73} -4.94427 q^{74} +1.00000 q^{75} -12.4721 q^{76} +16.4164 q^{77} -3.85410 q^{78} -6.18034 q^{79} -1.85410 q^{80} +1.00000 q^{81} -2.76393 q^{82} -3.70820 q^{83} +4.85410 q^{84} -3.47214 q^{85} +2.00000 q^{86} +1.00000 q^{87} +12.2361 q^{88} +11.1803 q^{89} -0.618034 q^{90} +18.7082 q^{91} -8.00000 q^{93} +4.14590 q^{94} -7.70820 q^{95} +5.61803 q^{96} +2.76393 q^{97} +1.23607 q^{98} -5.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 2 q^{3} - q^{4} - 2 q^{5} - q^{6} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - q^2 + 2 * q^3 - q^4 - 2 * q^5 - q^6 - 6 * q^7 + 2 * q^9	\( 2 q - q^{2} + 2 q^{3} - q^{4} - 2 q^{5} - q^{6} - 6 q^{7} + 2 q^{9} + q^{10} - 2 q^{11} - q^{12} - 8 q^{13} + 3 q^{14} - 2 q^{15} - 3 q^{16} - 2 q^{17} - q^{18} + 2 q^{19} + q^{20} - 6 q^{21} - 9 q^{22} + 2 q^{25} - q^{26} + 2 q^{27} + 3 q^{28} + 2 q^{29} + q^{30} - 16 q^{31} + 9 q^{32} - 2 q^{33} + 11 q^{34} + 6 q^{35} - q^{36} - 16 q^{37} + 14 q^{38} - 8 q^{39} + 3 q^{42} + 2 q^{43} + 11 q^{44} - 2 q^{45} - 3 q^{48} + 4 q^{49} - q^{50} - 2 q^{51} + 9 q^{52} - 18 q^{53} - q^{54} + 2 q^{55} + 2 q^{57} - q^{58} + 6 q^{59} + q^{60} + 2 q^{61} + 8 q^{62} - 6 q^{63} + 4 q^{64} + 8 q^{65} - 9 q^{66} - 14 q^{67} - 9 q^{68} - 3 q^{70} + 10 q^{71} - 16 q^{73} + 8 q^{74} + 2 q^{75} - 16 q^{76} + 6 q^{77} - q^{78} + 10 q^{79} + 3 q^{80} + 2 q^{81} - 10 q^{82} + 6 q^{83} + 3 q^{84} + 2 q^{85} + 4 q^{86} + 2 q^{87} + 20 q^{88} + q^{90} + 24 q^{91} - 16 q^{93} + 15 q^{94} - 2 q^{95} + 9 q^{96} + 10 q^{97} - 2 q^{98} - 2 q^{99}+O(q^{100}) \) 2 * q - q^2 + 2 * q^3 - q^4 - 2 * q^5 - q^6 - 6 * q^7 + 2 * q^9 + q^10 - 2 * q^11 - q^12 - 8 * q^13 + 3 * q^14 - 2 * q^15 - 3 * q^16 - 2 * q^17 - q^18 + 2 * q^19 + q^20 - 6 * q^21 - 9 * q^22 + 2 * q^25 - q^26 + 2 * q^27 + 3 * q^28 + 2 * q^29 + q^30 - 16 * q^31 + 9 * q^32 - 2 * q^33 + 11 * q^34 + 6 * q^35 - q^36 - 16 * q^37 + 14 * q^38 - 8 * q^39 + 3 * q^42 + 2 * q^43 + 11 * q^44 - 2 * q^45 - 3 * q^48 + 4 * q^49 - q^50 - 2 * q^51 + 9 * q^52 - 18 * q^53 - q^54 + 2 * q^55 + 2 * q^57 - q^58 + 6 * q^59 + q^60 + 2 * q^61 + 8 * q^62 - 6 * q^63 + 4 * q^64 + 8 * q^65 - 9 * q^66 - 14 * q^67 - 9 * q^68 - 3 * q^70 + 10 * q^71 - 16 * q^73 + 8 * q^74 + 2 * q^75 - 16 * q^76 + 6 * q^77 - q^78 + 10 * q^79 + 3 * q^80 + 2 * q^81 - 10 * q^82 + 6 * q^83 + 3 * q^84 + 2 * q^85 + 4 * q^86 + 2 * q^87 + 20 * q^88 + q^90 + 24 * q^91 - 16 * q^93 + 15 * q^94 - 2 * q^95 + 9 * q^96 + 10 * q^97 - 2 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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