LMFDB - Embedded newform 690.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [690,2,Mod(1,690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(690, 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("690.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 690 = 2 \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	690.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:	$5.50967773947$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	690.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -5.12311 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -5.12311 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +5.12311 q^{11} -1.00000 q^{12} +2.00000 q^{13} -5.12311 q^{14} -1.00000 q^{15} +1.00000 q^{16} +7.12311 q^{17} +1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} +5.12311 q^{21} +5.12311 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -5.12311 q^{28} +2.00000 q^{29} -1.00000 q^{30} +1.00000 q^{32} -5.12311 q^{33} +7.12311 q^{34} -5.12311 q^{35} +1.00000 q^{36} -7.12311 q^{37} +4.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} +2.00000 q^{41} +5.12311 q^{42} +5.12311 q^{44} +1.00000 q^{45} +1.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} +19.2462 q^{49} +1.00000 q^{50} -7.12311 q^{51} +2.00000 q^{52} -4.24621 q^{53} -1.00000 q^{54} +5.12311 q^{55} -5.12311 q^{56} -4.00000 q^{57} +2.00000 q^{58} -14.2462 q^{59} -1.00000 q^{60} +0.876894 q^{61} -5.12311 q^{63} +1.00000 q^{64} +2.00000 q^{65} -5.12311 q^{66} -8.00000 q^{67} +7.12311 q^{68} -1.00000 q^{69} -5.12311 q^{70} +6.24621 q^{71} +1.00000 q^{72} +12.2462 q^{73} -7.12311 q^{74} -1.00000 q^{75} +4.00000 q^{76} -26.2462 q^{77} -2.00000 q^{78} -5.12311 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +11.3693 q^{83} +5.12311 q^{84} +7.12311 q^{85} -2.00000 q^{87} +5.12311 q^{88} +3.12311 q^{89} +1.00000 q^{90} -10.2462 q^{91} +1.00000 q^{92} -8.00000 q^{94} +4.00000 q^{95} -1.00000 q^{96} +0.246211 q^{97} +19.2462 q^{98} +5.12311 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{11} - 2 q^{12} + 4 q^{13} - 2 q^{14} - 2 q^{15} + 2 q^{16} + 6 q^{17} + 2 q^{18} + 8 q^{19} + 2 q^{20} + 2 q^{21} + 2 q^{22} + 2 q^{23} - 2 q^{24} + 2 q^{25} + 4 q^{26} - 2 q^{27} - 2 q^{28} + 4 q^{29} - 2 q^{30} + 2 q^{32} - 2 q^{33} + 6 q^{34} - 2 q^{35} + 2 q^{36} - 6 q^{37} + 8 q^{38} - 4 q^{39} + 2 q^{40} + 4 q^{41} + 2 q^{42} + 2 q^{44} + 2 q^{45} + 2 q^{46} - 16 q^{47} - 2 q^{48} + 22 q^{49} + 2 q^{50} - 6 q^{51} + 4 q^{52} + 8 q^{53} - 2 q^{54} + 2 q^{55} - 2 q^{56} - 8 q^{57} + 4 q^{58} - 12 q^{59} - 2 q^{60} + 10 q^{61} - 2 q^{63} + 2 q^{64} + 4 q^{65} - 2 q^{66} - 16 q^{67} + 6 q^{68} - 2 q^{69} - 2 q^{70} - 4 q^{71} + 2 q^{72} + 8 q^{73} - 6 q^{74} - 2 q^{75} + 8 q^{76} - 36 q^{77} - 4 q^{78} - 2 q^{79} + 2 q^{80} + 2 q^{81} + 4 q^{82} - 2 q^{83} + 2 q^{84} + 6 q^{85} - 4 q^{87} + 2 q^{88} - 2 q^{89} + 2 q^{90} - 4 q^{91} + 2 q^{92} - 16 q^{94} + 8 q^{95} - 2 q^{96} - 16 q^{97} + 22 q^{98} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^11 - 2 * q^12 + 4 * q^13 - 2 * q^14 - 2 * q^15 + 2 * q^16 + 6 * q^17 + 2 * q^18 + 8 * q^19 + 2 * q^20 + 2 * q^21 + 2 * q^22 + 2 * q^23 - 2 * q^24 + 2 * q^25 + 4 * q^26 - 2 * q^27 - 2 * q^28 + 4 * q^29 - 2 * q^30 + 2 * q^32 - 2 * q^33 + 6 * q^34 - 2 * q^35 + 2 * q^36 - 6 * q^37 + 8 * q^38 - 4 * q^39 + 2 * q^40 + 4 * q^41 + 2 * q^42 + 2 * q^44 + 2 * q^45 + 2 * q^46 - 16 * q^47 - 2 * q^48 + 22 * q^49 + 2 * q^50 - 6 * q^51 + 4 * q^52 + 8 * q^53 - 2 * q^54 + 2 * q^55 - 2 * q^56 - 8 * q^57 + 4 * q^58 - 12 * q^59 - 2 * q^60 + 10 * q^61 - 2 * q^63 + 2 * q^64 + 4 * q^65 - 2 * q^66 - 16 * q^67 + 6 * q^68 - 2 * q^69 - 2 * q^70 - 4 * q^71 + 2 * q^72 + 8 * q^73 - 6 * q^74 - 2 * q^75 + 8 * q^76 - 36 * q^77 - 4 * q^78 - 2 * q^79 + 2 * q^80 + 2 * q^81 + 4 * q^82 - 2 * q^83 + 2 * q^84 + 6 * q^85 - 4 * q^87 + 2 * q^88 - 2 * q^89 + 2 * q^90 - 4 * q^91 + 2 * q^92 - 16 * q^94 + 8 * q^95 - 2 * q^96 - 16 * q^97 + 22 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	−5.12311	−1.93635	−0.968176	−	0.250270i	$-0.919480\pi$
$7$	−5.12311	−1.93635	−0.968176	+	0.250270i	$0.919480\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	5.12311	1.54467	0.772337	−	0.635213i	$-0.219088\pi$
$11$	5.12311	1.54467	0.772337	+	0.635213i	$0.219088\pi$
$12$	−1.00000	−0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−5.12311	−1.36921
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	7.12311	1.72761	0.863803	−	0.503829i	$-0.168076\pi$
$17$	7.12311	1.72761	0.863803	+	0.503829i	$0.168076\pi$
$18$	1.00000	0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	1.00000	0.223607
$21$	5.12311	1.11795
$22$	5.12311	1.09225
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	−5.12311	−0.968176
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	−1.00000	−0.182574
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	−5.12311	−0.891818
$34$	7.12311	1.22160
$35$	−5.12311	−0.865963
$36$	1.00000	0.166667
$37$	−7.12311	−1.17103	−0.585516	−	0.810661i	$-0.699108\pi$
$37$	−7.12311	−1.17103	−0.585516	+	0.810661i	$0.699108\pi$
$38$	4.00000	0.648886
$39$	−2.00000	−0.320256
$40$	1.00000	0.158114
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	5.12311	0.790512
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	5.12311	0.772337
$45$	1.00000	0.149071
$46$	1.00000	0.147442
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	−1.00000	−0.144338
$49$	19.2462	2.74946
$50$	1.00000	0.141421
$51$	−7.12311	−0.997434
$52$	2.00000	0.277350
$53$	−4.24621	−0.583262	−0.291631	−	0.956531i	$-0.594198\pi$
$53$	−4.24621	−0.583262	−0.291631	+	0.956531i	$0.594198\pi$
$54$	−1.00000	−0.136083
$55$	5.12311	0.690799
$56$	−5.12311	−0.684604
$57$	−4.00000	−0.529813
$58$	2.00000	0.262613
$59$	−14.2462	−1.85470	−0.927349	−	0.374197i	$-0.877918\pi$
$59$	−14.2462	−1.85470	−0.927349	+	0.374197i	$0.877918\pi$
$60$	−1.00000	−0.129099
$61$	0.876894	0.112275	0.0561374	−	0.998423i	$-0.482122\pi$
$61$	0.876894	0.112275	0.0561374	+	0.998423i	$0.482122\pi$
$62$	0	0
$63$	−5.12311	−0.645451
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	−5.12311	−0.630611
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	7.12311	0.863803
$69$	−1.00000	−0.120386
$70$	−5.12311	−0.612328
$71$	6.24621	0.741289	0.370644	−	0.928775i	$-0.379137\pi$
$71$	6.24621	0.741289	0.370644	+	0.928775i	$0.379137\pi$
$72$	1.00000	0.117851
$73$	12.2462	1.43331	0.716655	−	0.697428i	$-0.245672\pi$
$73$	12.2462	1.43331	0.716655	+	0.697428i	$0.245672\pi$
$74$	−7.12311	−0.828044
$75$	−1.00000	−0.115470
$76$	4.00000	0.458831
$77$	−26.2462	−2.99103
$78$	−2.00000	−0.226455
$79$	−5.12311	−0.576394	−0.288197	−	0.957571i	$-0.593056\pi$
$79$	−5.12311	−0.576394	−0.288197	+	0.957571i	$0.593056\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	11.3693	1.24794	0.623972	−	0.781446i	$-0.285518\pi$
$83$	11.3693	1.24794	0.623972	+	0.781446i	$0.285518\pi$
$84$	5.12311	0.558977
$85$	7.12311	0.772609
$86$	0	0
$87$	−2.00000	−0.214423
$88$	5.12311	0.546125
$89$	3.12311	0.331049	0.165524	−	0.986206i	$-0.447068\pi$
$89$	3.12311	0.331049	0.165524	+	0.986206i	$0.447068\pi$
$90$	1.00000	0.105409
$91$	−10.2462	−1.07409
$92$	1.00000	0.104257
$93$	0	0
$94$	−8.00000	−0.825137
$95$	4.00000	0.410391
$96$	−1.00000	−0.102062
$97$	0.246211	0.0249990	0.0124995	−	0.999922i	$-0.496021\pi$
$97$	0.246211	0.0249990	0.0124995	+	0.999922i	$0.496021\pi$
$98$	19.2462	1.94416
$99$	5.12311	0.514891
$100$	1.00000	0.100000
$101$	−16.2462	−1.61656	−0.808279	−	0.588799i	$-0.799601\pi$
$101$	−16.2462	−1.61656	−0.808279	+	0.588799i	$0.799601\pi$
$102$	−7.12311	−0.705293
$103$	15.3693	1.51438	0.757192	−	0.653192i	$-0.226571\pi$
$103$	15.3693	1.51438	0.757192	+	0.653192i	$0.226571\pi$
$104$	2.00000	0.196116
$105$	5.12311	0.499964
$106$	−4.24621	−0.412428
$107$	11.3693	1.09911	0.549557	−	0.835456i	$-0.314797\pi$
$107$	11.3693	1.09911	0.549557	+	0.835456i	$0.314797\pi$
$108$	−1.00000	−0.0962250
$109$	11.1231	1.06540	0.532700	−	0.846304i	$-0.321177\pi$
$109$	11.1231	1.06540	0.532700	+	0.846304i	$0.321177\pi$
$110$	5.12311	0.488469
$111$	7.12311	0.676095
$112$	−5.12311	−0.484088
$113$	−0.876894	−0.0824913	−0.0412456	−	0.999149i	$-0.513133\pi$
$113$	−0.876894	−0.0824913	−0.0412456	+	0.999149i	$0.513133\pi$
$114$	−4.00000	−0.374634
$115$	1.00000	0.0932505
$116$	2.00000	0.185695
$117$	2.00000	0.184900
$118$	−14.2462	−1.31147
$119$	−36.4924	−3.34525
$120$	−1.00000	−0.0912871
$121$	15.2462	1.38602
$122$	0.876894	0.0793903
$123$	−2.00000	−0.180334
$124$	0	0
$125$	1.00000	0.0894427
$126$	−5.12311	−0.456403
$127$	−22.2462	−1.97403	−0.987016	−	0.160622i	$-0.948650\pi$
$127$	−22.2462	−1.97403	−0.987016	+	0.160622i	$0.948650\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	2.00000	0.175412
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	−5.12311	−0.445909
$133$	−20.4924	−1.77692
$134$	−8.00000	−0.691095
$135$	−1.00000	−0.0860663
$136$	7.12311	0.610801
$137$	−19.1231	−1.63380	−0.816899	−	0.576781i	$-0.804308\pi$
$137$	−19.1231	−1.63380	−0.816899	+	0.576781i	$0.804308\pi$
$138$	−1.00000	−0.0851257
$139$	−16.4924	−1.39887	−0.699435	−	0.714697i	$-0.746565\pi$
$139$	−16.4924	−1.39887	−0.699435	+	0.714697i	$0.746565\pi$
$140$	−5.12311	−0.432981
$141$	8.00000	0.673722
$142$	6.24621	0.524170
$143$	10.2462	0.856831
$144$	1.00000	0.0833333
$145$	2.00000	0.166091
$146$	12.2462	1.01350
$147$	−19.2462	−1.58740
$148$	−7.12311	−0.585516
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	−1.00000	−0.0816497
$151$	−10.2462	−0.833825	−0.416912	−	0.908947i	$-0.636888\pi$
$151$	−10.2462	−0.833825	−0.416912	+	0.908947i	$0.636888\pi$
$152$	4.00000	0.324443
$153$	7.12311	0.575869
$154$	−26.2462	−2.11498
$155$	0	0
$156$	−2.00000	−0.160128
$157$	0.876894	0.0699838	0.0349919	−	0.999388i	$-0.488859\pi$
$157$	0.876894	0.0699838	0.0349919	+	0.999388i	$0.488859\pi$
$158$	−5.12311	−0.407572
$159$	4.24621	0.336746
$160$	1.00000	0.0790569
$161$	−5.12311	−0.403757
$162$	1.00000	0.0785674
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	2.00000	0.156174
$165$	−5.12311	−0.398833
$166$	11.3693	0.882430
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	5.12311	0.395256
$169$	−9.00000	−0.692308
$170$	7.12311	0.546317
$171$	4.00000	0.305888
$172$	0	0
$173$	3.75379	0.285395	0.142698	−	0.989766i	$-0.454422\pi$
$173$	3.75379	0.285395	0.142698	+	0.989766i	$0.454422\pi$
$174$	−2.00000	−0.151620
$175$	−5.12311	−0.387270
$176$	5.12311	0.386169
$177$	14.2462	1.07081
$178$	3.12311	0.234087
$179$	24.4924	1.83065	0.915325	−	0.402716i	$-0.131934\pi$
$179$	24.4924	1.83065	0.915325	+	0.402716i	$0.131934\pi$
$180$	1.00000	0.0745356
$181$	19.1231	1.42141	0.710705	−	0.703491i	$-0.248376\pi$
$181$	19.1231	1.42141	0.710705	+	0.703491i	$0.248376\pi$
$182$	−10.2462	−0.759500
$183$	−0.876894	−0.0648219
$184$	1.00000	0.0737210
$185$	−7.12311	−0.523701
$186$	0	0
$187$	36.4924	2.66859
$188$	−8.00000	−0.583460
$189$	5.12311	0.372651
$190$	4.00000	0.290191
$191$	−20.4924	−1.48278	−0.741390	−	0.671075i	$-0.765833\pi$
$191$	−20.4924	−1.48278	−0.741390	+	0.671075i	$0.765833\pi$
$192$	−1.00000	−0.0721688
$193$	−8.24621	−0.593575	−0.296788	−	0.954944i	$-0.595915\pi$
$193$	−8.24621	−0.593575	−0.296788	+	0.954944i	$0.595915\pi$
$194$	0.246211	0.0176769
$195$	−2.00000	−0.143223
$196$	19.2462	1.37473
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	5.12311	0.364083
$199$	10.8769	0.771043	0.385521	−	0.922699i	$-0.374022\pi$
$199$	10.8769	0.771043	0.385521	+	0.922699i	$0.374022\pi$
$200$	1.00000	0.0707107
$201$	8.00000	0.564276
$202$	−16.2462	−1.14308
$203$	−10.2462	−0.719143
$204$	−7.12311	−0.498717
$205$	2.00000	0.139686
$206$	15.3693	1.07083
$207$	1.00000	0.0695048
$208$	2.00000	0.138675
$209$	20.4924	1.41749
$210$	5.12311	0.353528
$211$	16.4924	1.13539	0.567693	−	0.823241i	$-0.307836\pi$
$211$	16.4924	1.13539	0.567693	+	0.823241i	$0.307836\pi$
$212$	−4.24621	−0.291631
$213$	−6.24621	−0.427983
$214$	11.3693	0.777191
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	11.1231	0.753352
$219$	−12.2462	−0.827522
$220$	5.12311	0.345400
$221$	14.2462	0.958304
$222$	7.12311	0.478072
$223$	6.24621	0.418277	0.209139	−	0.977886i	$-0.432934\pi$
$223$	6.24621	0.418277	0.209139	+	0.977886i	$0.432934\pi$
$224$	−5.12311	−0.342302
$225$	1.00000	0.0666667
$226$	−0.876894	−0.0583301
$227$	19.3693	1.28559	0.642793	−	0.766040i	$-0.277775\pi$
$227$	19.3693	1.28559	0.642793	+	0.766040i	$0.277775\pi$
$228$	−4.00000	−0.264906
$229$	−1.36932	−0.0904870	−0.0452435	−	0.998976i	$-0.514406\pi$
$229$	−1.36932	−0.0904870	−0.0452435	+	0.998976i	$0.514406\pi$
$230$	1.00000	0.0659380
$231$	26.2462	1.72687
$232$	2.00000	0.131306
$233$	−20.7386	−1.35863	−0.679317	−	0.733845i	$-0.737724\pi$
$233$	−20.7386	−1.35863	−0.679317	+	0.733845i	$0.737724\pi$
$234$	2.00000	0.130744
$235$	−8.00000	−0.521862
$236$	−14.2462	−0.927349
$237$	5.12311	0.332781
$238$	−36.4924	−2.36545
$239$	−1.75379	−0.113443	−0.0567216	−	0.998390i	$-0.518065\pi$
$239$	−1.75379	−0.113443	−0.0567216	+	0.998390i	$0.518065\pi$
$240$	−1.00000	−0.0645497
$241$	−2.49242	−0.160551	−0.0802755	−	0.996773i	$-0.525580\pi$
$241$	−2.49242	−0.160551	−0.0802755	+	0.996773i	$0.525580\pi$
$242$	15.2462	0.980064
$243$	−1.00000	−0.0641500
$244$	0.876894	0.0561374
$245$	19.2462	1.22960
$246$	−2.00000	−0.127515
$247$	8.00000	0.509028
$248$	0	0
$249$	−11.3693	−0.720501
$250$	1.00000	0.0632456
$251$	−7.36932	−0.465147	−0.232574	−	0.972579i	$-0.574715\pi$
$251$	−7.36932	−0.465147	−0.232574	+	0.972579i	$0.574715\pi$
$252$	−5.12311	−0.322725
$253$	5.12311	0.322087
$254$	−22.2462	−1.39585
$255$	−7.12311	−0.446066
$256$	1.00000	0.0625000
$257$	−8.24621	−0.514385	−0.257192	−	0.966360i	$-0.582797\pi$
$257$	−8.24621	−0.514385	−0.257192	+	0.966360i	$0.582797\pi$
$258$	0	0
$259$	36.4924	2.26753
$260$	2.00000	0.124035
$261$	2.00000	0.123797
$262$	−4.00000	−0.247121
$263$	10.2462	0.631808	0.315904	−	0.948791i	$-0.397692\pi$
$263$	10.2462	0.631808	0.315904	+	0.948791i	$0.397692\pi$
$264$	−5.12311	−0.315305
$265$	−4.24621	−0.260843
$266$	−20.4924	−1.25647
$267$	−3.12311	−0.191131
$268$	−8.00000	−0.488678
$269$	−16.2462	−0.990549	−0.495274	−	0.868737i	$-0.664933\pi$
$269$	−16.2462	−0.990549	−0.495274	+	0.868737i	$0.664933\pi$
$270$	−1.00000	−0.0608581
$271$	10.2462	0.622413	0.311207	−	0.950342i	$-0.399267\pi$
$271$	10.2462	0.622413	0.311207	+	0.950342i	$0.399267\pi$
$272$	7.12311	0.431902
$273$	10.2462	0.620129
$274$	−19.1231	−1.15527
$275$	5.12311	0.308935
$276$	−1.00000	−0.0601929
$277$	20.2462	1.21648	0.608238	−	0.793754i	$-0.291876\pi$
$277$	20.2462	1.21648	0.608238	+	0.793754i	$0.291876\pi$
$278$	−16.4924	−0.989150
$279$	0	0
$280$	−5.12311	−0.306164
$281$	−12.8769	−0.768171	−0.384086	−	0.923298i	$-0.625483\pi$
$281$	−12.8769	−0.768171	−0.384086	+	0.923298i	$0.625483\pi$
$282$	8.00000	0.476393
$283$	−2.24621	−0.133523	−0.0667617	−	0.997769i	$-0.521267\pi$
$283$	−2.24621	−0.133523	−0.0667617	+	0.997769i	$0.521267\pi$
$284$	6.24621	0.370644
$285$	−4.00000	−0.236940
$286$	10.2462	0.605871
$287$	−10.2462	−0.604815
$288$	1.00000	0.0589256
$289$	33.7386	1.98463
$290$	2.00000	0.117444
$291$	−0.246211	−0.0144332
$292$	12.2462	0.716655
$293$	11.7538	0.686664	0.343332	−	0.939214i	$-0.388444\pi$
$293$	11.7538	0.686664	0.343332	+	0.939214i	$0.388444\pi$
$294$	−19.2462	−1.12246
$295$	−14.2462	−0.829446
$296$	−7.12311	−0.414022
$297$	−5.12311	−0.297273
$298$	−10.0000	−0.579284
$299$	2.00000	0.115663
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	−10.2462	−0.589603
$303$	16.2462	0.933320
$304$	4.00000	0.229416
$305$	0.876894	0.0502108
$306$	7.12311	0.407201
$307$	0.492423	0.0281040	0.0140520	−	0.999901i	$-0.495527\pi$
$307$	0.492423	0.0281040	0.0140520	+	0.999901i	$0.495527\pi$
$308$	−26.2462	−1.49552
$309$	−15.3693	−0.874330
$310$	0	0
$311$	26.7386	1.51621	0.758104	−	0.652133i	$-0.226126\pi$
$311$	26.7386	1.51621	0.758104	+	0.652133i	$0.226126\pi$
$312$	−2.00000	−0.113228
$313$	10.4924	0.593067	0.296533	−	0.955022i	$-0.404169\pi$
$313$	10.4924	0.593067	0.296533	+	0.955022i	$0.404169\pi$
$314$	0.876894	0.0494860
$315$	−5.12311	−0.288654
$316$	−5.12311	−0.288197
$317$	20.7386	1.16480	0.582399	−	0.812903i	$-0.302114\pi$
$317$	20.7386	1.16480	0.582399	+	0.812903i	$0.302114\pi$
$318$	4.24621	0.238116
$319$	10.2462	0.573678
$320$	1.00000	0.0559017
$321$	−11.3693	−0.634573
$322$	−5.12311	−0.285500
$323$	28.4924	1.58536
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	−12.0000	−0.664619
$327$	−11.1231	−0.615109
$328$	2.00000	0.110432
$329$	40.9848	2.25957
$330$	−5.12311	−0.282018
$331$	−32.4924	−1.78595	−0.892973	−	0.450111i	$-0.851384\pi$
$331$	−32.4924	−1.78595	−0.892973	+	0.450111i	$0.851384\pi$
$332$	11.3693	0.623972
$333$	−7.12311	−0.390344
$334$	−8.00000	−0.437741
$335$	−8.00000	−0.437087
$336$	5.12311	0.279488
$337$	−6.49242	−0.353665	−0.176832	−	0.984241i	$-0.556585\pi$
$337$	−6.49242	−0.353665	−0.176832	+	0.984241i	$0.556585\pi$
$338$	−9.00000	−0.489535
$339$	0.876894	0.0476264
$340$	7.12311	0.386305
$341$	0	0
$342$	4.00000	0.216295
$343$	−62.7386	−3.38757
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	3.75379	0.201805
$347$	−34.7386	−1.86487	−0.932434	−	0.361341i	$-0.882319\pi$
$347$	−34.7386	−1.86487	−0.932434	+	0.361341i	$0.882319\pi$
$348$	−2.00000	−0.107211
$349$	−24.7386	−1.32423	−0.662114	−	0.749403i	$-0.730341\pi$
$349$	−24.7386	−1.32423	−0.662114	+	0.749403i	$0.730341\pi$
$350$	−5.12311	−0.273842
$351$	−2.00000	−0.106752
$352$	5.12311	0.273062
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	14.2462	0.757178
$355$	6.24621	0.331514
$356$	3.12311	0.165524
$357$	36.4924	1.93138
$358$	24.4924	1.29446
$359$	−12.4924	−0.659325	−0.329662	−	0.944099i	$-0.606935\pi$
$359$	−12.4924	−0.659325	−0.329662	+	0.944099i	$0.606935\pi$
$360$	1.00000	0.0527046
$361$	−3.00000	−0.157895
$362$	19.1231	1.00509
$363$	−15.2462	−0.800219
$364$	−10.2462	−0.537047
$365$	12.2462	0.640996
$366$	−0.876894	−0.0458360
$367$	13.1231	0.685021	0.342510	−	0.939514i	$-0.388723\pi$
$367$	13.1231	0.685021	0.342510	+	0.939514i	$0.388723\pi$
$368$	1.00000	0.0521286
$369$	2.00000	0.104116
$370$	−7.12311	−0.370313
$371$	21.7538	1.12940
$372$	0	0
$373$	13.3693	0.692237	0.346118	−	0.938191i	$-0.387500\pi$
$373$	13.3693	0.692237	0.346118	+	0.938191i	$0.387500\pi$
$374$	36.4924	1.88698
$375$	−1.00000	−0.0516398
$376$	−8.00000	−0.412568
$377$	4.00000	0.206010
$378$	5.12311	0.263504
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	4.00000	0.205196
$381$	22.2462	1.13971
$382$	−20.4924	−1.04848
$383$	13.7538	0.702786	0.351393	−	0.936228i	$-0.385708\pi$
$383$	13.7538	0.702786	0.351393	+	0.936228i	$0.385708\pi$
$384$	−1.00000	−0.0510310
$385$	−26.2462	−1.33763
$386$	−8.24621	−0.419721
$387$	0	0
$388$	0.246211	0.0124995
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	−2.00000	−0.101274
$391$	7.12311	0.360231
$392$	19.2462	0.972080
$393$	4.00000	0.201773
$394$	−10.0000	−0.503793
$395$	−5.12311	−0.257771
$396$	5.12311	0.257446
$397$	−34.4924	−1.73113	−0.865563	−	0.500801i	$-0.833039\pi$
$397$	−34.4924	−1.73113	−0.865563	+	0.500801i	$0.833039\pi$
$398$	10.8769	0.545209
$399$	20.4924	1.02590
$400$	1.00000	0.0500000
$401$	−17.3693	−0.867382	−0.433691	−	0.901062i	$-0.642789\pi$
$401$	−17.3693	−0.867382	−0.433691	+	0.901062i	$0.642789\pi$
$402$	8.00000	0.399004
$403$	0	0
$404$	−16.2462	−0.808279
$405$	1.00000	0.0496904
$406$	−10.2462	−0.508511
$407$	−36.4924	−1.80886
$408$	−7.12311	−0.352646
$409$	−16.2462	−0.803323	−0.401662	−	0.915788i	$-0.631567\pi$
$409$	−16.2462	−0.803323	−0.401662	+	0.915788i	$0.631567\pi$
$410$	2.00000	0.0987730
$411$	19.1231	0.943273
$412$	15.3693	0.757192
$413$	72.9848	3.59135
$414$	1.00000	0.0491473
$415$	11.3693	0.558098
$416$	2.00000	0.0980581
$417$	16.4924	0.807637
$418$	20.4924	1.00232
$419$	−1.61553	−0.0789237	−0.0394619	−	0.999221i	$-0.512564\pi$
$419$	−1.61553	−0.0789237	−0.0394619	+	0.999221i	$0.512564\pi$
$420$	5.12311	0.249982
$421$	13.3693	0.651581	0.325790	−	0.945442i	$-0.394370\pi$
$421$	13.3693	0.651581	0.325790	+	0.945442i	$0.394370\pi$
$422$	16.4924	0.802839
$423$	−8.00000	−0.388973
$424$	−4.24621	−0.206214
$425$	7.12311	0.345521
$426$	−6.24621	−0.302630
$427$	−4.49242	−0.217404
$428$	11.3693	0.549557
$429$	−10.2462	−0.494692
$430$	0	0
$431$	26.2462	1.26424	0.632118	−	0.774872i	$-0.282186\pi$
$431$	26.2462	1.26424	0.632118	+	0.774872i	$0.282186\pi$
$432$	−1.00000	−0.0481125
$433$	10.4924	0.504234	0.252117	−	0.967697i	$-0.418873\pi$
$433$	10.4924	0.504234	0.252117	+	0.967697i	$0.418873\pi$
$434$	0	0
$435$	−2.00000	−0.0958927
$436$	11.1231	0.532700
$437$	4.00000	0.191346
$438$	−12.2462	−0.585147
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	5.12311	0.244234
$441$	19.2462	0.916486
$442$	14.2462	0.677623
$443$	−24.4924	−1.16367	−0.581835	−	0.813307i	$-0.697665\pi$
$443$	−24.4924	−1.16367	−0.581835	+	0.813307i	$0.697665\pi$
$444$	7.12311	0.338048
$445$	3.12311	0.148049
$446$	6.24621	0.295767
$447$	10.0000	0.472984
$448$	−5.12311	−0.242044
$449$	−10.4924	−0.495168	−0.247584	−	0.968866i	$-0.579637\pi$
$449$	−10.4924	−0.495168	−0.247584	+	0.968866i	$0.579637\pi$
$450$	1.00000	0.0471405
$451$	10.2462	0.482475
$452$	−0.876894	−0.0412456
$453$	10.2462	0.481409
$454$	19.3693	0.909047
$455$	−10.2462	−0.480350
$456$	−4.00000	−0.187317
$457$	−32.7386	−1.53145	−0.765724	−	0.643169i	$-0.777619\pi$
$457$	−32.7386	−1.53145	−0.765724	+	0.643169i	$0.777619\pi$
$458$	−1.36932	−0.0639840
$459$	−7.12311	−0.332478
$460$	1.00000	0.0466252
$461$	32.7386	1.52479	0.762395	−	0.647112i	$-0.224023\pi$
$461$	32.7386	1.52479	0.762395	+	0.647112i	$0.224023\pi$
$462$	26.2462	1.22108
$463$	−36.0000	−1.67306	−0.836531	−	0.547920i	$-0.815420\pi$
$463$	−36.0000	−1.67306	−0.836531	+	0.547920i	$0.815420\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−20.7386	−0.960699
$467$	−3.36932	−0.155913	−0.0779567	−	0.996957i	$-0.524840\pi$
$467$	−3.36932	−0.155913	−0.0779567	+	0.996957i	$0.524840\pi$
$468$	2.00000	0.0924500
$469$	40.9848	1.89250
$470$	−8.00000	−0.369012
$471$	−0.876894	−0.0404052
$472$	−14.2462	−0.655735
$473$	0	0
$474$	5.12311	0.235312
$475$	4.00000	0.183533
$476$	−36.4924	−1.67263
$477$	−4.24621	−0.194421
$478$	−1.75379	−0.0802164
$479$	−20.4924	−0.936323	−0.468161	−	0.883643i	$-0.655083\pi$
$479$	−20.4924	−0.936323	−0.468161	+	0.883643i	$0.655083\pi$
$480$	−1.00000	−0.0456435
$481$	−14.2462	−0.649571
$482$	−2.49242	−0.113527
$483$	5.12311	0.233109
$484$	15.2462	0.693010
$485$	0.246211	0.0111799
$486$	−1.00000	−0.0453609
$487$	40.4924	1.83489	0.917443	−	0.397866i	$-0.130249\pi$
$487$	40.4924	1.83489	0.917443	+	0.397866i	$0.130249\pi$
$488$	0.876894	0.0396951
$489$	12.0000	0.542659
$490$	19.2462	0.869455
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	−2.00000	−0.0901670
$493$	14.2462	0.641617
$494$	8.00000	0.359937
$495$	5.12311	0.230266
$496$	0	0
$497$	−32.0000	−1.43540
$498$	−11.3693	−0.509471
$499$	−28.9848	−1.29754	−0.648770	−	0.760985i	$-0.724716\pi$
$499$	−28.9848	−1.29754	−0.648770	+	0.760985i	$0.724716\pi$
$500$	1.00000	0.0447214
$501$	8.00000	0.357414
$502$	−7.36932	−0.328909
$503$	−6.73863	−0.300461	−0.150230	−	0.988651i	$-0.548002\pi$
$503$	−6.73863	−0.300461	−0.150230	+	0.988651i	$0.548002\pi$
$504$	−5.12311	−0.228201
$505$	−16.2462	−0.722947
$506$	5.12311	0.227750
$507$	9.00000	0.399704
$508$	−22.2462	−0.987016
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	−7.12311	−0.315416
$511$	−62.7386	−2.77539
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	−8.24621	−0.363725
$515$	15.3693	0.677253
$516$	0	0
$517$	−40.9848	−1.80251
$518$	36.4924	1.60338
$519$	−3.75379	−0.164773
$520$	2.00000	0.0877058
$521$	8.87689	0.388904	0.194452	−	0.980912i	$-0.437707\pi$
$521$	8.87689	0.388904	0.194452	+	0.980912i	$0.437707\pi$
$522$	2.00000	0.0875376
$523$	−28.4924	−1.24589	−0.622943	−	0.782267i	$-0.714063\pi$
$523$	−28.4924	−1.24589	−0.622943	+	0.782267i	$0.714063\pi$
$524$	−4.00000	−0.174741
$525$	5.12311	0.223591
$526$	10.2462	0.446756
$527$	0	0
$528$	−5.12311	−0.222955
$529$	1.00000	0.0434783
$530$	−4.24621	−0.184444
$531$	−14.2462	−0.618233
$532$	−20.4924	−0.888459
$533$	4.00000	0.173259
$534$	−3.12311	−0.135150
$535$	11.3693	0.491538
$536$	−8.00000	−0.345547
$537$	−24.4924	−1.05693
$538$	−16.2462	−0.700424
$539$	98.6004	4.24702
$540$	−1.00000	−0.0430331
$541$	−15.7538	−0.677308	−0.338654	−	0.940911i	$-0.609972\pi$
$541$	−15.7538	−0.677308	−0.338654	+	0.940911i	$0.609972\pi$
$542$	10.2462	0.440112
$543$	−19.1231	−0.820651
$544$	7.12311	0.305401
$545$	11.1231	0.476461
$546$	10.2462	0.438497
$547$	0.492423	0.0210545	0.0105272	−	0.999945i	$-0.496649\pi$
$547$	0.492423	0.0210545	0.0105272	+	0.999945i	$0.496649\pi$
$548$	−19.1231	−0.816899
$549$	0.876894	0.0374249
$550$	5.12311	0.218450
$551$	8.00000	0.340811
$552$	−1.00000	−0.0425628
$553$	26.2462	1.11610
$554$	20.2462	0.860179
$555$	7.12311	0.302359
$556$	−16.4924	−0.699435
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	0	0
$559$	0	0
$560$	−5.12311	−0.216491
$561$	−36.4924	−1.54071
$562$	−12.8769	−0.543179
$563$	11.3693	0.479160	0.239580	−	0.970877i	$-0.422990\pi$
$563$	11.3693	0.479160	0.239580	+	0.970877i	$0.422990\pi$
$564$	8.00000	0.336861
$565$	−0.876894	−0.0368912
$566$	−2.24621	−0.0944153
$567$	−5.12311	−0.215150
$568$	6.24621	0.262085
$569$	27.1231	1.13706	0.568530	−	0.822663i	$-0.307512\pi$
$569$	27.1231	1.13706	0.568530	+	0.822663i	$0.307512\pi$
$570$	−4.00000	−0.167542
$571$	34.7386	1.45377	0.726883	−	0.686761i	$-0.240968\pi$
$571$	34.7386	1.45377	0.726883	+	0.686761i	$0.240968\pi$
$572$	10.2462	0.428416
$573$	20.4924	0.856083
$574$	−10.2462	−0.427669
$575$	1.00000	0.0417029
$576$	1.00000	0.0416667
$577$	14.4924	0.603327	0.301664	−	0.953414i	$-0.402458\pi$
$577$	14.4924	0.603327	0.301664	+	0.953414i	$0.402458\pi$
$578$	33.7386	1.40334
$579$	8.24621	0.342701
$580$	2.00000	0.0830455
$581$	−58.2462	−2.41646
$582$	−0.246211	−0.0102058
$583$	−21.7538	−0.900950
$584$	12.2462	0.506752
$585$	2.00000	0.0826898
$586$	11.7538	0.485545
$587$	−9.75379	−0.402582	−0.201291	−	0.979531i	$-0.564514\pi$
$587$	−9.75379	−0.402582	−0.201291	+	0.979531i	$0.564514\pi$
$588$	−19.2462	−0.793700
$589$	0	0
$590$	−14.2462	−0.586507
$591$	10.0000	0.411345
$592$	−7.12311	−0.292758
$593$	7.75379	0.318410	0.159205	−	0.987246i	$-0.449107\pi$
$593$	7.75379	0.318410	0.159205	+	0.987246i	$0.449107\pi$
$594$	−5.12311	−0.210204
$595$	−36.4924	−1.49604
$596$	−10.0000	−0.409616
$597$	−10.8769	−0.445162
$598$	2.00000	0.0817861
$599$	−6.24621	−0.255213	−0.127607	−	0.991825i	$-0.540730\pi$
$599$	−6.24621	−0.255213	−0.127607	+	0.991825i	$0.540730\pi$
$600$	−1.00000	−0.0408248
$601$	−6.00000	−0.244745	−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000	−0.244745	−0.122373	+	0.992484i	$0.539050\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	−10.2462	−0.416912
$605$	15.2462	0.619847
$606$	16.2462	0.659957
$607$	26.7386	1.08529	0.542644	−	0.839963i	$-0.317423\pi$
$607$	26.7386	1.08529	0.542644	+	0.839963i	$0.317423\pi$
$608$	4.00000	0.162221
$609$	10.2462	0.415197
$610$	0.876894	0.0355044
$611$	−16.0000	−0.647291
$612$	7.12311	0.287934
$613$	43.1231	1.74173	0.870863	−	0.491526i	$-0.163561\pi$
$613$	43.1231	1.74173	0.870863	+	0.491526i	$0.163561\pi$
$614$	0.492423	0.0198726
$615$	−2.00000	−0.0806478
$616$	−26.2462	−1.05749
$617$	40.1080	1.61469	0.807343	−	0.590083i	$-0.200905\pi$
$617$	40.1080	1.61469	0.807343	+	0.590083i	$0.200905\pi$
$618$	−15.3693	−0.618245
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	26.7386	1.07212
$623$	−16.0000	−0.641026
$624$	−2.00000	−0.0800641
$625$	1.00000	0.0400000
$626$	10.4924	0.419362
$627$	−20.4924	−0.818389
$628$	0.876894	0.0349919
$629$	−50.7386	−2.02308
$630$	−5.12311	−0.204109
$631$	−39.3693	−1.56727	−0.783634	−	0.621223i	$-0.786636\pi$
$631$	−39.3693	−1.56727	−0.783634	+	0.621223i	$0.786636\pi$
$632$	−5.12311	−0.203786
$633$	−16.4924	−0.655515
$634$	20.7386	0.823636
$635$	−22.2462	−0.882814
$636$	4.24621	0.168373
$637$	38.4924	1.52513
$638$	10.2462	0.405651
$639$	6.24621	0.247096
$640$	1.00000	0.0395285
$641$	−9.36932	−0.370066	−0.185033	−	0.982732i	$-0.559239\pi$
$641$	−9.36932	−0.370066	−0.185033	+	0.982732i	$0.559239\pi$
$642$	−11.3693	−0.448711
$643$	36.4924	1.43912	0.719560	−	0.694430i	$-0.244343\pi$
$643$	36.4924	1.43912	0.719560	+	0.694430i	$0.244343\pi$
$644$	−5.12311	−0.201879
$645$	0	0
$646$	28.4924	1.12102
$647$	11.5076	0.452410	0.226205	−	0.974080i	$-0.427368\pi$
$647$	11.5076	0.452410	0.226205	+	0.974080i	$0.427368\pi$
$648$	1.00000	0.0392837
$649$	−72.9848	−2.86491
$650$	2.00000	0.0784465
$651$	0	0
$652$	−12.0000	−0.469956
$653$	10.4924	0.410600	0.205300	−	0.978699i	$-0.434183\pi$
$653$	10.4924	0.410600	0.205300	+	0.978699i	$0.434183\pi$
$654$	−11.1231	−0.434948
$655$	−4.00000	−0.156293
$656$	2.00000	0.0780869
$657$	12.2462	0.477770
$658$	40.9848	1.59776
$659$	−7.36932	−0.287068	−0.143534	−	0.989645i	$-0.545847\pi$
$659$	−7.36932	−0.287068	−0.143534	+	0.989645i	$0.545847\pi$
$660$	−5.12311	−0.199417
$661$	33.8617	1.31707	0.658535	−	0.752551i	$-0.271177\pi$
$661$	33.8617	1.31707	0.658535	+	0.752551i	$0.271177\pi$
$662$	−32.4924	−1.26285
$663$	−14.2462	−0.553277
$664$	11.3693	0.441215
$665$	−20.4924	−0.794662
$666$	−7.12311	−0.276015
$667$	2.00000	0.0774403
$668$	−8.00000	−0.309529
$669$	−6.24621	−0.241492
$670$	−8.00000	−0.309067
$671$	4.49242	0.173428
$672$	5.12311	0.197628
$673$	−46.9848	−1.81113	−0.905566	−	0.424205i	$-0.860554\pi$
$673$	−46.9848	−1.81113	−0.905566	+	0.424205i	$0.860554\pi$
$674$	−6.49242	−0.250079
$675$	−1.00000	−0.0384900
$676$	−9.00000	−0.346154
$677$	−46.4924	−1.78685	−0.893424	−	0.449213i	$-0.851704\pi$
$677$	−46.4924	−1.78685	−0.893424	+	0.449213i	$0.851704\pi$
$678$	0.876894	0.0336769
$679$	−1.26137	−0.0484068
$680$	7.12311	0.273159
$681$	−19.3693	−0.742234
$682$	0	0
$683$	17.7538	0.679330	0.339665	−	0.940547i	$-0.389686\pi$
$683$	17.7538	0.679330	0.339665	+	0.940547i	$0.389686\pi$
$684$	4.00000	0.152944
$685$	−19.1231	−0.730656
$686$	−62.7386	−2.39537
$687$	1.36932	0.0522427
$688$	0	0
$689$	−8.49242	−0.323536
$690$	−1.00000	−0.0380693
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	3.75379	0.142698
$693$	−26.2462	−0.997011
$694$	−34.7386	−1.31866
$695$	−16.4924	−0.625593
$696$	−2.00000	−0.0758098
$697$	14.2462	0.539614
$698$	−24.7386	−0.936371
$699$	20.7386	0.784407
$700$	−5.12311	−0.193635
$701$	−36.2462	−1.36900	−0.684500	−	0.729013i	$-0.739980\pi$
$701$	−36.2462	−1.36900	−0.684500	+	0.729013i	$0.739980\pi$
$702$	−2.00000	−0.0754851
$703$	−28.4924	−1.07461
$704$	5.12311	0.193084
$705$	8.00000	0.301297
$706$	−14.0000	−0.526897
$707$	83.2311	3.13023
$708$	14.2462	0.535405
$709$	13.3693	0.502095	0.251048	−	0.967975i	$-0.419225\pi$
$709$	13.3693	0.502095	0.251048	+	0.967975i	$0.419225\pi$
$710$	6.24621	0.234416
$711$	−5.12311	−0.192131
$712$	3.12311	0.117043
$713$	0	0
$714$	36.4924	1.36569
$715$	10.2462	0.383187
$716$	24.4924	0.915325
$717$	1.75379	0.0654964
$718$	−12.4924	−0.466213
$719$	−9.75379	−0.363755	−0.181877	−	0.983321i	$-0.558217\pi$
$719$	−9.75379	−0.363755	−0.181877	+	0.983321i	$0.558217\pi$
$720$	1.00000	0.0372678
$721$	−78.7386	−2.93238
$722$	−3.00000	−0.111648
$723$	2.49242	0.0926942
$724$	19.1231	0.710705
$725$	2.00000	0.0742781
$726$	−15.2462	−0.565840
$727$	27.8617	1.03333	0.516667	−	0.856186i	$-0.327172\pi$
$727$	27.8617	1.03333	0.516667	+	0.856186i	$0.327172\pi$
$728$	−10.2462	−0.379750
$729$	1.00000	0.0370370
$730$	12.2462	0.453253
$731$	0	0
$732$	−0.876894	−0.0324109
$733$	−13.8617	−0.511995	−0.255998	−	0.966677i	$-0.582404\pi$
$733$	−13.8617	−0.511995	−0.255998	+	0.966677i	$0.582404\pi$
$734$	13.1231	0.484383
$735$	−19.2462	−0.709907
$736$	1.00000	0.0368605
$737$	−40.9848	−1.50970
$738$	2.00000	0.0736210
$739$	40.4924	1.48954	0.744769	−	0.667322i	$-0.232560\pi$
$739$	40.4924	1.48954	0.744769	+	0.667322i	$0.232560\pi$
$740$	−7.12311	−0.261851
$741$	−8.00000	−0.293887
$742$	21.7538	0.798607
$743$	−14.7386	−0.540708	−0.270354	−	0.962761i	$-0.587141\pi$
$743$	−14.7386	−0.540708	−0.270354	+	0.962761i	$0.587141\pi$
$744$	0	0
$745$	−10.0000	−0.366372
$746$	13.3693	0.489485
$747$	11.3693	0.415982
$748$	36.4924	1.33430
$749$	−58.2462	−2.12827
$750$	−1.00000	−0.0365148
$751$	43.8617	1.60054	0.800269	−	0.599641i	$-0.204690\pi$
$751$	43.8617	1.60054	0.800269	+	0.599641i	$0.204690\pi$
$752$	−8.00000	−0.291730
$753$	7.36932	0.268553
$754$	4.00000	0.145671
$755$	−10.2462	−0.372898
$756$	5.12311	0.186326
$757$	9.86174	0.358431	0.179216	−	0.983810i	$-0.442644\pi$
$757$	9.86174	0.358431	0.179216	+	0.983810i	$0.442644\pi$
$758$	4.00000	0.145287
$759$	−5.12311	−0.185957
$760$	4.00000	0.145095
$761$	−48.2462	−1.74892	−0.874462	−	0.485094i	$-0.838785\pi$
$761$	−48.2462	−1.74892	−0.874462	+	0.485094i	$0.838785\pi$
$762$	22.2462	0.805895
$763$	−56.9848	−2.06299
$764$	−20.4924	−0.741390
$765$	7.12311	0.257536
$766$	13.7538	0.496945
$767$	−28.4924	−1.02880
$768$	−1.00000	−0.0360844
$769$	−20.7386	−0.747854	−0.373927	−	0.927458i	$-0.621989\pi$
$769$	−20.7386	−0.747854	−0.373927	+	0.927458i	$0.621989\pi$
$770$	−26.2462	−0.945848
$771$	8.24621	0.296980
$772$	−8.24621	−0.296788
$773$	−36.2462	−1.30369	−0.651843	−	0.758354i	$-0.726004\pi$
$773$	−36.2462	−1.30369	−0.651843	+	0.758354i	$0.726004\pi$
$774$	0	0
$775$	0	0
$776$	0.246211	0.00883847
$777$	−36.4924	−1.30916
$778$	−18.0000	−0.645331
$779$	8.00000	0.286630
$780$	−2.00000	−0.0716115
$781$	32.0000	1.14505
$782$	7.12311	0.254722
$783$	−2.00000	−0.0714742
$784$	19.2462	0.687365
$785$	0.876894	0.0312977
$786$	4.00000	0.142675
$787$	48.0000	1.71102	0.855508	−	0.517790i	$-0.173245\pi$
$787$	48.0000	1.71102	0.855508	+	0.517790i	$0.173245\pi$
$788$	−10.0000	−0.356235
$789$	−10.2462	−0.364775
$790$	−5.12311	−0.182272
$791$	4.49242	0.159732
$792$	5.12311	0.182042
$793$	1.75379	0.0622789
$794$	−34.4924	−1.22409
$795$	4.24621	0.150598
$796$	10.8769	0.385521
$797$	−22.4924	−0.796722	−0.398361	−	0.917229i	$-0.630421\pi$
$797$	−22.4924	−0.796722	−0.398361	+	0.917229i	$0.630421\pi$
$798$	20.4924	0.725424
$799$	−56.9848	−2.01598
$800$	1.00000	0.0353553
$801$	3.12311	0.110350
$802$	−17.3693	−0.613332
$803$	62.7386	2.21400
$804$	8.00000	0.282138
$805$	−5.12311	−0.180566
$806$	0	0
$807$	16.2462	0.571894
$808$	−16.2462	−0.571540
$809$	4.24621	0.149289	0.0746444	−	0.997210i	$-0.476218\pi$
$809$	4.24621	0.149289	0.0746444	+	0.997210i	$0.476218\pi$
$810$	1.00000	0.0351364
$811$	−36.0000	−1.26413	−0.632065	−	0.774915i	$-0.717793\pi$
$811$	−36.0000	−1.26413	−0.632065	+	0.774915i	$0.717793\pi$
$812$	−10.2462	−0.359572
$813$	−10.2462	−0.359350
$814$	−36.4924	−1.27906
$815$	−12.0000	−0.420342
$816$	−7.12311	−0.249359
$817$	0	0
$818$	−16.2462	−0.568035
$819$	−10.2462	−0.358032
$820$	2.00000	0.0698430
$821$	−19.7538	−0.689412	−0.344706	−	0.938711i	$-0.612021\pi$
$821$	−19.7538	−0.689412	−0.344706	+	0.938711i	$0.612021\pi$
$822$	19.1231	0.666995
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	15.3693	0.535416
$825$	−5.12311	−0.178364
$826$	72.9848	2.53947
$827$	17.1231	0.595429	0.297714	−	0.954655i	$-0.403776\pi$
$827$	17.1231	0.595429	0.297714	+	0.954655i	$0.403776\pi$
$828$	1.00000	0.0347524
$829$	32.2462	1.11996	0.559979	−	0.828507i	$-0.310809\pi$
$829$	32.2462	1.11996	0.559979	+	0.828507i	$0.310809\pi$
$830$	11.3693	0.394635
$831$	−20.2462	−0.702333
$832$	2.00000	0.0693375
$833$	137.093	4.74998
$834$	16.4924	0.571086
$835$	−8.00000	−0.276851
$836$	20.4924	0.708745
$837$	0	0
$838$	−1.61553	−0.0558075
$839$	−26.2462	−0.906120	−0.453060	−	0.891480i	$-0.649668\pi$
$839$	−26.2462	−0.906120	−0.453060	+	0.891480i	$0.649668\pi$
$840$	5.12311	0.176764
$841$	−25.0000	−0.862069
$842$	13.3693	0.460737
$843$	12.8769	0.443504
$844$	16.4924	0.567693
$845$	−9.00000	−0.309609
$846$	−8.00000	−0.275046
$847$	−78.1080	−2.68382
$848$	−4.24621	−0.145815
$849$	2.24621	0.0770898
$850$	7.12311	0.244321
$851$	−7.12311	−0.244177
$852$	−6.24621	−0.213992
$853$	−6.00000	−0.205436	−0.102718	−	0.994711i	$-0.532754\pi$
$853$	−6.00000	−0.205436	−0.102718	+	0.994711i	$0.532754\pi$
$854$	−4.49242	−0.153728
$855$	4.00000	0.136797
$856$	11.3693	0.388595
$857$	24.7386	0.845056	0.422528	−	0.906350i	$-0.361143\pi$
$857$	24.7386	0.845056	0.422528	+	0.906350i	$0.361143\pi$
$858$	−10.2462	−0.349800
$859$	48.4924	1.65454	0.827270	−	0.561804i	$-0.189893\pi$
$859$	48.4924	1.65454	0.827270	+	0.561804i	$0.189893\pi$
$860$	0	0
$861$	10.2462	0.349190
$862$	26.2462	0.893950
$863$	48.9848	1.66746	0.833732	−	0.552170i	$-0.186200\pi$
$863$	48.9848	1.66746	0.833732	+	0.552170i	$0.186200\pi$
$864$	−1.00000	−0.0340207
$865$	3.75379	0.127633
$866$	10.4924	0.356547
$867$	−33.7386	−1.14582
$868$	0	0
$869$	−26.2462	−0.890342
$870$	−2.00000	−0.0678064
$871$	−16.0000	−0.542139
$872$	11.1231	0.376676
$873$	0.246211	0.00833299
$874$	4.00000	0.135302
$875$	−5.12311	−0.173193
$876$	−12.2462	−0.413761
$877$	3.26137	0.110129	0.0550643	−	0.998483i	$-0.482464\pi$
$877$	3.26137	0.110129	0.0550643	+	0.998483i	$0.482464\pi$
$878$	0	0
$879$	−11.7538	−0.396445
$880$	5.12311	0.172700
$881$	31.6155	1.06515	0.532577	−	0.846381i	$-0.321224\pi$
$881$	31.6155	1.06515	0.532577	+	0.846381i	$0.321224\pi$
$882$	19.2462	0.648054
$883$	−20.9848	−0.706196	−0.353098	−	0.935586i	$-0.614872\pi$
$883$	−20.9848	−0.706196	−0.353098	+	0.935586i	$0.614872\pi$
$884$	14.2462	0.479152
$885$	14.2462	0.478881
$886$	−24.4924	−0.822839
$887$	52.4924	1.76252	0.881262	−	0.472629i	$-0.156695\pi$
$887$	52.4924	1.76252	0.881262	+	0.472629i	$0.156695\pi$
$888$	7.12311	0.239036
$889$	113.970	3.82242
$890$	3.12311	0.104687
$891$	5.12311	0.171630
$892$	6.24621	0.209139
$893$	−32.0000	−1.07084
$894$	10.0000	0.334450
$895$	24.4924	0.818691
$896$	−5.12311	−0.171151
$897$	−2.00000	−0.0667781
$898$	−10.4924	−0.350137
$899$	0	0
$900$	1.00000	0.0333333
$901$	−30.2462	−1.00765
$902$	10.2462	0.341162
$903$	0	0
$904$	−0.876894	−0.0291651
$905$	19.1231	0.635674
$906$	10.2462	0.340408
$907$	37.7538	1.25359	0.626797	−	0.779183i	$-0.284366\pi$
$907$	37.7538	1.25359	0.626797	+	0.779183i	$0.284366\pi$
$908$	19.3693	0.642793
$909$	−16.2462	−0.538853
$910$	−10.2462	−0.339659
$911$	−13.7538	−0.455683	−0.227842	−	0.973698i	$-0.573167\pi$
$911$	−13.7538	−0.455683	−0.227842	+	0.973698i	$0.573167\pi$
$912$	−4.00000	−0.132453
$913$	58.2462	1.92767
$914$	−32.7386	−1.08290
$915$	−0.876894	−0.0289892
$916$	−1.36932	−0.0452435
$917$	20.4924	0.676719
$918$	−7.12311	−0.235098
$919$	10.8769	0.358796	0.179398	−	0.983777i	$-0.442585\pi$
$919$	10.8769	0.358796	0.179398	+	0.983777i	$0.442585\pi$
$920$	1.00000	0.0329690
$921$	−0.492423	−0.0162259
$922$	32.7386	1.07819
$923$	12.4924	0.411193
$924$	26.2462	0.863437
$925$	−7.12311	−0.234206
$926$	−36.0000	−1.18303
$927$	15.3693	0.504795
$928$	2.00000	0.0656532
$929$	−30.9848	−1.01658	−0.508290	−	0.861186i	$-0.669722\pi$
$929$	−30.9848	−1.01658	−0.508290	+	0.861186i	$0.669722\pi$
$930$	0	0
$931$	76.9848	2.52308
$932$	−20.7386	−0.679317
$933$	−26.7386	−0.875384
$934$	−3.36932	−0.110247
$935$	36.4924	1.19343
$936$	2.00000	0.0653720
$937$	−16.7386	−0.546827	−0.273414	−	0.961897i	$-0.588153\pi$
$937$	−16.7386	−0.546827	−0.273414	+	0.961897i	$0.588153\pi$
$938$	40.9848	1.33820
$939$	−10.4924	−0.342407
$940$	−8.00000	−0.260931
$941$	−6.49242	−0.211647	−0.105823	−	0.994385i	$-0.533748\pi$
$941$	−6.49242	−0.211647	−0.105823	+	0.994385i	$0.533748\pi$
$942$	−0.876894	−0.0285708
$943$	2.00000	0.0651290
$944$	−14.2462	−0.463675
$945$	5.12311	0.166655
$946$	0	0
$947$	22.2462	0.722905	0.361452	−	0.932391i	$-0.382281\pi$
$947$	22.2462	0.722905	0.361452	+	0.932391i	$0.382281\pi$
$948$	5.12311	0.166391
$949$	24.4924	0.795058
$950$	4.00000	0.129777
$951$	−20.7386	−0.672496
$952$	−36.4924	−1.18273
$953$	−33.8617	−1.09689	−0.548445	−	0.836187i	$-0.684780\pi$
$953$	−33.8617	−1.09689	−0.548445	+	0.836187i	$0.684780\pi$
$954$	−4.24621	−0.137476
$955$	−20.4924	−0.663119
$956$	−1.75379	−0.0567216
$957$	−10.2462	−0.331213
$958$	−20.4924	−0.662080
$959$	97.9697	3.16361
$960$	−1.00000	−0.0322749
$961$	−31.0000	−1.00000
$962$	−14.2462	−0.459316
$963$	11.3693	0.366371
$964$	−2.49242	−0.0802755
$965$	−8.24621	−0.265455
$966$	5.12311	0.164833
$967$	−4.98485	−0.160302	−0.0801509	−	0.996783i	$-0.525540\pi$
$967$	−4.98485	−0.160302	−0.0801509	+	0.996783i	$0.525540\pi$
$968$	15.2462	0.490032
$969$	−28.4924	−0.915308
$970$	0.246211	0.00790537
$971$	34.8769	1.11925	0.559626	−	0.828745i	$-0.310945\pi$
$971$	34.8769	1.11925	0.559626	+	0.828745i	$0.310945\pi$
$972$	−1.00000	−0.0320750
$973$	84.4924	2.70870
$974$	40.4924	1.29746
$975$	−2.00000	−0.0640513
$976$	0.876894	0.0280687
$977$	21.8617	0.699419	0.349710	−	0.936858i	$-0.386280\pi$
$977$	21.8617	0.699419	0.349710	+	0.936858i	$0.386280\pi$
$978$	12.0000	0.383718
$979$	16.0000	0.511362
$980$	19.2462	0.614798
$981$	11.1231	0.355133
$982$	−12.0000	−0.382935
$983$	−26.2462	−0.837124	−0.418562	−	0.908188i	$-0.637466\pi$
$983$	−26.2462	−0.837124	−0.418562	+	0.908188i	$0.637466\pi$
$984$	−2.00000	−0.0637577
$985$	−10.0000	−0.318626
$986$	14.2462	0.453692
$987$	−40.9848	−1.30456
$988$	8.00000	0.254514
$989$	0	0
$990$	5.12311	0.162823
$991$	−37.4773	−1.19050	−0.595252	−	0.803539i	$-0.702948\pi$
$991$	−37.4773	−1.19050	−0.595252	+	0.803539i	$0.702948\pi$
$992$	0	0
$993$	32.4924	1.03112
$994$	−32.0000	−1.01498
$995$	10.8769	0.344821
$996$	−11.3693	−0.360251
$997$	20.2462	0.641204	0.320602	−	0.947214i	$-0.396115\pi$
$997$	20.2462	0.641204	0.320602	+	0.947214i	$0.396115\pi$
$998$	−28.9848	−0.917499
$999$	7.12311	0.225365

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.2.a.l.1.1	✓	2
3.2	odd	2		2070.2.a.t.1.1		2
4.3	odd	2		5520.2.a.bs.1.2		2
5.2	odd	4		3450.2.d.v.2899.3		4
5.3	odd	4		3450.2.d.v.2899.2		4
5.4	even	2		3450.2.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.a.l.1.1	✓	2	1.1	even	1	trivial
2070.2.a.t.1.1		2	3.2	odd	2
3450.2.a.bi.1.2		2	5.4	even	2
3450.2.d.v.2899.2		4	5.3	odd	4
3450.2.d.v.2899.3		4	5.2	odd	4
5520.2.a.bs.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:	\( N \)	\(=\)	\( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	690.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -5.12311 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -5.12311 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +5.12311 q^{11} -1.00000 q^{12} +2.00000 q^{13} -5.12311 q^{14} -1.00000 q^{15} +1.00000 q^{16} +7.12311 q^{17} +1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} +5.12311 q^{21} +5.12311 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -5.12311 q^{28} +2.00000 q^{29} -1.00000 q^{30} +1.00000 q^{32} -5.12311 q^{33} +7.12311 q^{34} -5.12311 q^{35} +1.00000 q^{36} -7.12311 q^{37} +4.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} +2.00000 q^{41} +5.12311 q^{42} +5.12311 q^{44} +1.00000 q^{45} +1.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} +19.2462 q^{49} +1.00000 q^{50} -7.12311 q^{51} +2.00000 q^{52} -4.24621 q^{53} -1.00000 q^{54} +5.12311 q^{55} -5.12311 q^{56} -4.00000 q^{57} +2.00000 q^{58} -14.2462 q^{59} -1.00000 q^{60} +0.876894 q^{61} -5.12311 q^{63} +1.00000 q^{64} +2.00000 q^{65} -5.12311 q^{66} -8.00000 q^{67} +7.12311 q^{68} -1.00000 q^{69} -5.12311 q^{70} +6.24621 q^{71} +1.00000 q^{72} +12.2462 q^{73} -7.12311 q^{74} -1.00000 q^{75} +4.00000 q^{76} -26.2462 q^{77} -2.00000 q^{78} -5.12311 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +11.3693 q^{83} +5.12311 q^{84} +7.12311 q^{85} -2.00000 q^{87} +5.12311 q^{88} +3.12311 q^{89} +1.00000 q^{90} -10.2462 q^{91} +1.00000 q^{92} -8.00000 q^{94} +4.00000 q^{95} -1.00000 q^{96} +0.246211 q^{97} +19.2462 q^{98} +5.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{11} - 2 q^{12} + 4 q^{13} - 2 q^{14} - 2 q^{15} + 2 q^{16} + 6 q^{17} + 2 q^{18} + 8 q^{19} + 2 q^{20} + 2 q^{21} + 2 q^{22} + 2 q^{23} - 2 q^{24} + 2 q^{25} + 4 q^{26} - 2 q^{27} - 2 q^{28} + 4 q^{29} - 2 q^{30} + 2 q^{32} - 2 q^{33} + 6 q^{34} - 2 q^{35} + 2 q^{36} - 6 q^{37} + 8 q^{38} - 4 q^{39} + 2 q^{40} + 4 q^{41} + 2 q^{42} + 2 q^{44} + 2 q^{45} + 2 q^{46} - 16 q^{47} - 2 q^{48} + 22 q^{49} + 2 q^{50} - 6 q^{51} + 4 q^{52} + 8 q^{53} - 2 q^{54} + 2 q^{55} - 2 q^{56} - 8 q^{57} + 4 q^{58} - 12 q^{59} - 2 q^{60} + 10 q^{61} - 2 q^{63} + 2 q^{64} + 4 q^{65} - 2 q^{66} - 16 q^{67} + 6 q^{68} - 2 q^{69} - 2 q^{70} - 4 q^{71} + 2 q^{72} + 8 q^{73} - 6 q^{74} - 2 q^{75} + 8 q^{76} - 36 q^{77} - 4 q^{78} - 2 q^{79} + 2 q^{80} + 2 q^{81} + 4 q^{82} - 2 q^{83} + 2 q^{84} + 6 q^{85} - 4 q^{87} + 2 q^{88} - 2 q^{89} + 2 q^{90} - 4 q^{91} + 2 q^{92} - 16 q^{94} + 8 q^{95} - 2 q^{96} - 16 q^{97} + 22 q^{98} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^11 - 2 * q^12 + 4 * q^13 - 2 * q^14 - 2 * q^15 + 2 * q^16 + 6 * q^17 + 2 * q^18 + 8 * q^19 + 2 * q^20 + 2 * q^21 + 2 * q^22 + 2 * q^23 - 2 * q^24 + 2 * q^25 + 4 * q^26 - 2 * q^27 - 2 * q^28 + 4 * q^29 - 2 * q^30 + 2 * q^32 - 2 * q^33 + 6 * q^34 - 2 * q^35 + 2 * q^36 - 6 * q^37 + 8 * q^38 - 4 * q^39 + 2 * q^40 + 4 * q^41 + 2 * q^42 + 2 * q^44 + 2 * q^45 + 2 * q^46 - 16 * q^47 - 2 * q^48 + 22 * q^49 + 2 * q^50 - 6 * q^51 + 4 * q^52 + 8 * q^53 - 2 * q^54 + 2 * q^55 - 2 * q^56 - 8 * q^57 + 4 * q^58 - 12 * q^59 - 2 * q^60 + 10 * q^61 - 2 * q^63 + 2 * q^64 + 4 * q^65 - 2 * q^66 - 16 * q^67 + 6 * q^68 - 2 * q^69 - 2 * q^70 - 4 * q^71 + 2 * q^72 + 8 * q^73 - 6 * q^74 - 2 * q^75 + 8 * q^76 - 36 * q^77 - 4 * q^78 - 2 * q^79 + 2 * q^80 + 2 * q^81 + 4 * q^82 - 2 * q^83 + 2 * q^84 + 6 * q^85 - 4 * q^87 + 2 * q^88 - 2 * q^89 + 2 * q^90 - 4 * q^91 + 2 * q^92 - 16 * q^94 + 8 * q^95 - 2 * q^96 - 16 * q^97 + 22 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more