LMFDB - Embedded newform 5610.2.a.bn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5610,2,Mod(1,5610)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5610, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5610.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5610.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:	$44.7960755339$
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:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	5610.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} -5.12311 q^{13} -1.56155 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} -1.56155 q^{21} -1.00000 q^{22} +2.43845 q^{23} +1.00000 q^{24} +1.00000 q^{25} +5.12311 q^{26} -1.00000 q^{27} +1.56155 q^{28} -0.438447 q^{29} -1.00000 q^{30} -8.68466 q^{31} -1.00000 q^{32} -1.00000 q^{33} +1.00000 q^{34} -1.56155 q^{35} +1.00000 q^{36} +6.00000 q^{37} +4.00000 q^{38} +5.12311 q^{39} +1.00000 q^{40} +1.12311 q^{41} +1.56155 q^{42} +7.80776 q^{43} +1.00000 q^{44} -1.00000 q^{45} -2.43845 q^{46} -10.2462 q^{47} -1.00000 q^{48} -4.56155 q^{49} -1.00000 q^{50} +1.00000 q^{51} -5.12311 q^{52} +2.87689 q^{53} +1.00000 q^{54} -1.00000 q^{55} -1.56155 q^{56} +4.00000 q^{57} +0.438447 q^{58} +3.12311 q^{59} +1.00000 q^{60} -6.00000 q^{61} +8.68466 q^{62} +1.56155 q^{63} +1.00000 q^{64} +5.12311 q^{65} +1.00000 q^{66} +12.0000 q^{67} -1.00000 q^{68} -2.43845 q^{69} +1.56155 q^{70} +4.87689 q^{71} -1.00000 q^{72} +4.24621 q^{73} -6.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} +1.56155 q^{77} -5.12311 q^{78} -12.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -1.12311 q^{82} +10.2462 q^{83} -1.56155 q^{84} +1.00000 q^{85} -7.80776 q^{86} +0.438447 q^{87} -1.00000 q^{88} -2.87689 q^{89} +1.00000 q^{90} -8.00000 q^{91} +2.43845 q^{92} +8.68466 q^{93} +10.2462 q^{94} +4.00000 q^{95} +1.00000 q^{96} +10.6847 q^{97} +4.56155 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 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 - q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{11} - 2 q^{12} - 2 q^{13} + q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{17} - 2 q^{18} - 8 q^{19} - 2 q^{20} + q^{21} - 2 q^{22} + 9 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{26} - 2 q^{27} - q^{28} - 5 q^{29} - 2 q^{30} - 5 q^{31} - 2 q^{32} - 2 q^{33} + 2 q^{34} + q^{35} + 2 q^{36} + 12 q^{37} + 8 q^{38} + 2 q^{39} + 2 q^{40} - 6 q^{41} - q^{42} - 5 q^{43} + 2 q^{44} - 2 q^{45} - 9 q^{46} - 4 q^{47} - 2 q^{48} - 5 q^{49} - 2 q^{50} + 2 q^{51} - 2 q^{52} + 14 q^{53} + 2 q^{54} - 2 q^{55} + q^{56} + 8 q^{57} + 5 q^{58} - 2 q^{59} + 2 q^{60} - 12 q^{61} + 5 q^{62} - q^{63} + 2 q^{64} + 2 q^{65} + 2 q^{66} + 24 q^{67} - 2 q^{68} - 9 q^{69} - q^{70} + 18 q^{71} - 2 q^{72} - 8 q^{73} - 12 q^{74} - 2 q^{75} - 8 q^{76} - q^{77} - 2 q^{78} - 24 q^{79} - 2 q^{80} + 2 q^{81} + 6 q^{82} + 4 q^{83} + q^{84} + 2 q^{85} + 5 q^{86} + 5 q^{87} - 2 q^{88} - 14 q^{89} + 2 q^{90} - 16 q^{91} + 9 q^{92} + 5 q^{93} + 4 q^{94} + 8 q^{95} + 2 q^{96} + 9 q^{97} + 5 q^{98} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^11 - 2 * q^12 - 2 * q^13 + q^14 + 2 * q^15 + 2 * q^16 - 2 * q^17 - 2 * q^18 - 8 * q^19 - 2 * q^20 + q^21 - 2 * q^22 + 9 * q^23 + 2 * q^24 + 2 * q^25 + 2 * q^26 - 2 * q^27 - q^28 - 5 * q^29 - 2 * q^30 - 5 * q^31 - 2 * q^32 - 2 * q^33 + 2 * q^34 + q^35 + 2 * q^36 + 12 * q^37 + 8 * q^38 + 2 * q^39 + 2 * q^40 - 6 * q^41 - q^42 - 5 * q^43 + 2 * q^44 - 2 * q^45 - 9 * q^46 - 4 * q^47 - 2 * q^48 - 5 * q^49 - 2 * q^50 + 2 * q^51 - 2 * q^52 + 14 * q^53 + 2 * q^54 - 2 * q^55 + q^56 + 8 * q^57 + 5 * q^58 - 2 * q^59 + 2 * q^60 - 12 * q^61 + 5 * q^62 - q^63 + 2 * q^64 + 2 * q^65 + 2 * q^66 + 24 * q^67 - 2 * q^68 - 9 * q^69 - q^70 + 18 * q^71 - 2 * q^72 - 8 * q^73 - 12 * q^74 - 2 * q^75 - 8 * q^76 - q^77 - 2 * q^78 - 24 * q^79 - 2 * q^80 + 2 * q^81 + 6 * q^82 + 4 * q^83 + q^84 + 2 * q^85 + 5 * q^86 + 5 * q^87 - 2 * q^88 - 14 * q^89 + 2 * q^90 - 16 * q^91 + 9 * q^92 + 5 * q^93 + 4 * q^94 + 8 * q^95 + 2 * q^96 + 9 * q^97 + 5 * 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
$7$	1.56155	0.590211	0.295106	−	0.955465i	$-0.404645\pi$
$7$	1.56155	0.590211	0.295106	+	0.955465i	$0.404645\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.00000	−0.288675
$13$	−5.12311	−1.42089	−0.710447	−	0.703751i	$-0.751507\pi$
$13$	−5.12311	−1.42089	−0.710447	+	0.703751i	$0.751507\pi$
$14$	−1.56155	−0.417343
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−1.00000	−0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−1.00000	−0.223607
$21$	−1.56155	−0.340759
$22$	−1.00000	−0.213201
$23$	2.43845	0.508451	0.254226	−	0.967145i	$-0.418179\pi$
$23$	2.43845	0.508451	0.254226	+	0.967145i	$0.418179\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	5.12311	1.00472
$27$	−1.00000	−0.192450
$28$	1.56155	0.295106
$29$	−0.438447	−0.0814176	−0.0407088	−	0.999171i	$-0.512962\pi$
$29$	−0.438447	−0.0814176	−0.0407088	+	0.999171i	$0.512962\pi$
$30$	−1.00000	−0.182574
$31$	−8.68466	−1.55981	−0.779905	−	0.625897i	$-0.784733\pi$
$31$	−8.68466	−1.55981	−0.779905	+	0.625897i	$0.784733\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	1.00000	0.171499
$35$	−1.56155	−0.263951
$36$	1.00000	0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	4.00000	0.648886
$39$	5.12311	0.820353
$40$	1.00000	0.158114
$41$	1.12311	0.175400	0.0876998	−	0.996147i	$-0.472048\pi$
$41$	1.12311	0.175400	0.0876998	+	0.996147i	$0.472048\pi$
$42$	1.56155	0.240953
$43$	7.80776	1.19067	0.595336	−	0.803477i	$-0.297019\pi$
$43$	7.80776	1.19067	0.595336	+	0.803477i	$0.297019\pi$
$44$	1.00000	0.150756
$45$	−1.00000	−0.149071
$46$	−2.43845	−0.359529
$47$	−10.2462	−1.49456	−0.747282	−	0.664507i	$-0.768641\pi$
$47$	−10.2462	−1.49456	−0.747282	+	0.664507i	$0.768641\pi$
$48$	−1.00000	−0.144338
$49$	−4.56155	−0.651650
$50$	−1.00000	−0.141421
$51$	1.00000	0.140028
$52$	−5.12311	−0.710447
$53$	2.87689	0.395172	0.197586	−	0.980286i	$-0.436690\pi$
$53$	2.87689	0.395172	0.197586	+	0.980286i	$0.436690\pi$
$54$	1.00000	0.136083
$55$	−1.00000	−0.134840
$56$	−1.56155	−0.208671
$57$	4.00000	0.529813
$58$	0.438447	0.0575709
$59$	3.12311	0.406594	0.203297	−	0.979117i	$-0.434834\pi$
$59$	3.12311	0.406594	0.203297	+	0.979117i	$0.434834\pi$
$60$	1.00000	0.129099
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	8.68466	1.10295
$63$	1.56155	0.196737
$64$	1.00000	0.125000
$65$	5.12311	0.635443
$66$	1.00000	0.123091
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	−1.00000	−0.121268
$69$	−2.43845	−0.293555
$70$	1.56155	0.186641
$71$	4.87689	0.578781	0.289390	−	0.957211i	$-0.406547\pi$
$71$	4.87689	0.578781	0.289390	+	0.957211i	$0.406547\pi$
$72$	−1.00000	−0.117851
$73$	4.24621	0.496981	0.248491	−	0.968634i	$-0.420065\pi$
$73$	4.24621	0.496981	0.248491	+	0.968634i	$0.420065\pi$
$74$	−6.00000	−0.697486
$75$	−1.00000	−0.115470
$76$	−4.00000	−0.458831
$77$	1.56155	0.177955
$78$	−5.12311	−0.580077
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−1.12311	−0.124026
$83$	10.2462	1.12467	0.562334	−	0.826910i	$-0.309904\pi$
$83$	10.2462	1.12467	0.562334	+	0.826910i	$0.309904\pi$
$84$	−1.56155	−0.170379
$85$	1.00000	0.108465
$86$	−7.80776	−0.841933
$87$	0.438447	0.0470065
$88$	−1.00000	−0.106600
$89$	−2.87689	−0.304950	−0.152475	−	0.988307i	$-0.548724\pi$
$89$	−2.87689	−0.304950	−0.152475	+	0.988307i	$0.548724\pi$
$90$	1.00000	0.105409
$91$	−8.00000	−0.838628
$92$	2.43845	0.254226
$93$	8.68466	0.900557
$94$	10.2462	1.05682
$95$	4.00000	0.410391
$96$	1.00000	0.102062
$97$	10.6847	1.08486	0.542431	−	0.840100i	$-0.317504\pi$
$97$	10.6847	1.08486	0.542431	+	0.840100i	$0.317504\pi$
$98$	4.56155	0.460786
$99$	1.00000	0.100504
$100$	1.00000	0.100000
$101$	8.24621	0.820529	0.410264	−	0.911967i	$-0.365436\pi$
$101$	8.24621	0.820529	0.410264	+	0.911967i	$0.365436\pi$
$102$	−1.00000	−0.0990148
$103$	8.68466	0.855725	0.427862	−	0.903844i	$-0.359267\pi$
$103$	8.68466	0.855725	0.427862	+	0.903844i	$0.359267\pi$
$104$	5.12311	0.502362
$105$	1.56155	0.152392
$106$	−2.87689	−0.279429
$107$	−7.80776	−0.754805	−0.377403	−	0.926049i	$-0.623183\pi$
$107$	−7.80776	−0.754805	−0.377403	+	0.926049i	$0.623183\pi$
$108$	−1.00000	−0.0962250
$109$	−15.3693	−1.47211	−0.736057	−	0.676920i	$-0.763314\pi$
$109$	−15.3693	−1.47211	−0.736057	+	0.676920i	$0.763314\pi$
$110$	1.00000	0.0953463
$111$	−6.00000	−0.569495
$112$	1.56155	0.147553
$113$	18.4924	1.73962	0.869810	−	0.493386i	$-0.164241\pi$
$113$	18.4924	1.73962	0.869810	+	0.493386i	$0.164241\pi$
$114$	−4.00000	−0.374634
$115$	−2.43845	−0.227386
$116$	−0.438447	−0.0407088
$117$	−5.12311	−0.473631
$118$	−3.12311	−0.287505
$119$	−1.56155	−0.143147
$120$	−1.00000	−0.0912871
$121$	1.00000	0.0909091
$122$	6.00000	0.543214
$123$	−1.12311	−0.101267
$124$	−8.68466	−0.779905
$125$	−1.00000	−0.0894427
$126$	−1.56155	−0.139114
$127$	−9.36932	−0.831392	−0.415696	−	0.909504i	$-0.636462\pi$
$127$	−9.36932	−0.831392	−0.415696	+	0.909504i	$0.636462\pi$
$128$	−1.00000	−0.0883883
$129$	−7.80776	−0.687435
$130$	−5.12311	−0.449326
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	−1.00000	−0.0870388
$133$	−6.24621	−0.541615
$134$	−12.0000	−1.03664
$135$	1.00000	0.0860663
$136$	1.00000	0.0857493
$137$	−9.80776	−0.837934	−0.418967	−	0.908001i	$-0.637608\pi$
$137$	−9.80776	−0.837934	−0.418967	+	0.908001i	$0.637608\pi$
$138$	2.43845	0.207574
$139$	−8.68466	−0.736623	−0.368312	−	0.929702i	$-0.620064\pi$
$139$	−8.68466	−0.736623	−0.368312	+	0.929702i	$0.620064\pi$
$140$	−1.56155	−0.131975
$141$	10.2462	0.862887
$142$	−4.87689	−0.409260
$143$	−5.12311	−0.428416
$144$	1.00000	0.0833333
$145$	0.438447	0.0364111
$146$	−4.24621	−0.351419
$147$	4.56155	0.376231
$148$	6.00000	0.493197
$149$	5.12311	0.419701	0.209851	−	0.977733i	$-0.432702\pi$
$149$	5.12311	0.419701	0.209851	+	0.977733i	$0.432702\pi$
$150$	1.00000	0.0816497
$151$	−1.75379	−0.142721	−0.0713607	−	0.997451i	$-0.522734\pi$
$151$	−1.75379	−0.142721	−0.0713607	+	0.997451i	$0.522734\pi$
$152$	4.00000	0.324443
$153$	−1.00000	−0.0808452
$154$	−1.56155	−0.125834
$155$	8.68466	0.697569
$156$	5.12311	0.410177
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	12.0000	0.954669
$159$	−2.87689	−0.228153
$160$	1.00000	0.0790569
$161$	3.80776	0.300094
$162$	−1.00000	−0.0785674
$163$	−7.80776	−0.611551	−0.305776	−	0.952104i	$-0.598916\pi$
$163$	−7.80776	−0.611551	−0.305776	+	0.952104i	$0.598916\pi$
$164$	1.12311	0.0876998
$165$	1.00000	0.0778499
$166$	−10.2462	−0.795260
$167$	−9.36932	−0.725020	−0.362510	−	0.931980i	$-0.618080\pi$
$167$	−9.36932	−0.725020	−0.362510	+	0.931980i	$0.618080\pi$
$168$	1.56155	0.120476
$169$	13.2462	1.01894
$170$	−1.00000	−0.0766965
$171$	−4.00000	−0.305888
$172$	7.80776	0.595336
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	−0.438447	−0.0332386
$175$	1.56155	0.118042
$176$	1.00000	0.0753778
$177$	−3.12311	−0.234747
$178$	2.87689	0.215632
$179$	23.6155	1.76511	0.882554	−	0.470212i	$-0.155822\pi$
$179$	23.6155	1.76511	0.882554	+	0.470212i	$0.155822\pi$
$180$	−1.00000	−0.0745356
$181$	0.438447	0.0325895	0.0162948	−	0.999867i	$-0.494813\pi$
$181$	0.438447	0.0325895	0.0162948	+	0.999867i	$0.494813\pi$
$182$	8.00000	0.592999
$183$	6.00000	0.443533
$184$	−2.43845	−0.179765
$185$	−6.00000	−0.441129
$186$	−8.68466	−0.636790
$187$	−1.00000	−0.0731272
$188$	−10.2462	−0.747282
$189$	−1.56155	−0.113586
$190$	−4.00000	−0.290191
$191$	0.192236	0.0139097	0.00695485	−	0.999976i	$-0.497786\pi$
$191$	0.192236	0.0139097	0.00695485	+	0.999976i	$0.497786\pi$
$192$	−1.00000	−0.0721688
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	−10.6847	−0.767114
$195$	−5.12311	−0.366873
$196$	−4.56155	−0.325825
$197$	−1.12311	−0.0800180	−0.0400090	−	0.999199i	$-0.512739\pi$
$197$	−1.12311	−0.0800180	−0.0400090	+	0.999199i	$0.512739\pi$
$198$	−1.00000	−0.0710669
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	−1.00000	−0.0707107
$201$	−12.0000	−0.846415
$202$	−8.24621	−0.580201
$203$	−0.684658	−0.0480536
$204$	1.00000	0.0700140
$205$	−1.12311	−0.0784411
$206$	−8.68466	−0.605089
$207$	2.43845	0.169484
$208$	−5.12311	−0.355223
$209$	−4.00000	−0.276686
$210$	−1.56155	−0.107757
$211$	−19.8078	−1.36362	−0.681811	−	0.731528i	$-0.738808\pi$
$211$	−19.8078	−1.36362	−0.681811	+	0.731528i	$0.738808\pi$
$212$	2.87689	0.197586
$213$	−4.87689	−0.334159
$214$	7.80776	0.533728
$215$	−7.80776	−0.532485
$216$	1.00000	0.0680414
$217$	−13.5616	−0.920618
$218$	15.3693	1.04094
$219$	−4.24621	−0.286932
$220$	−1.00000	−0.0674200
$221$	5.12311	0.344617
$222$	6.00000	0.402694
$223$	0.684658	0.0458481	0.0229241	−	0.999737i	$-0.492702\pi$
$223$	0.684658	0.0458481	0.0229241	+	0.999737i	$0.492702\pi$
$224$	−1.56155	−0.104336
$225$	1.00000	0.0666667
$226$	−18.4924	−1.23010
$227$	4.68466	0.310932	0.155466	−	0.987841i	$-0.450312\pi$
$227$	4.68466	0.310932	0.155466	+	0.987841i	$0.450312\pi$
$228$	4.00000	0.264906
$229$	−11.3693	−0.751306	−0.375653	−	0.926760i	$-0.622581\pi$
$229$	−11.3693	−0.751306	−0.375653	+	0.926760i	$0.622581\pi$
$230$	2.43845	0.160786
$231$	−1.56155	−0.102743
$232$	0.438447	0.0287855
$233$	8.43845	0.552821	0.276411	−	0.961040i	$-0.410855\pi$
$233$	8.43845	0.552821	0.276411	+	0.961040i	$0.410855\pi$
$234$	5.12311	0.334908
$235$	10.2462	0.668389
$236$	3.12311	0.203297
$237$	12.0000	0.779484
$238$	1.56155	0.101220
$239$	19.1231	1.23697	0.618485	−	0.785796i	$-0.287747\pi$
$239$	19.1231	1.23697	0.618485	+	0.785796i	$0.287747\pi$
$240$	1.00000	0.0645497
$241$	25.8078	1.66242	0.831212	−	0.555955i	$-0.187647\pi$
$241$	25.8078	1.66242	0.831212	+	0.555955i	$0.187647\pi$
$242$	−1.00000	−0.0642824
$243$	−1.00000	−0.0641500
$244$	−6.00000	−0.384111
$245$	4.56155	0.291427
$246$	1.12311	0.0716066
$247$	20.4924	1.30390
$248$	8.68466	0.551476
$249$	−10.2462	−0.649327
$250$	1.00000	0.0632456
$251$	14.2462	0.899213	0.449606	−	0.893227i	$-0.351564\pi$
$251$	14.2462	0.899213	0.449606	+	0.893227i	$0.351564\pi$
$252$	1.56155	0.0983686
$253$	2.43845	0.153304
$254$	9.36932	0.587883
$255$	−1.00000	−0.0626224
$256$	1.00000	0.0625000
$257$	15.5616	0.970703	0.485351	−	0.874319i	$-0.338692\pi$
$257$	15.5616	0.970703	0.485351	+	0.874319i	$0.338692\pi$
$258$	7.80776	0.486090
$259$	9.36932	0.582181
$260$	5.12311	0.317722
$261$	−0.438447	−0.0271392
$262$	−12.0000	−0.741362
$263$	7.31534	0.451083	0.225542	−	0.974234i	$-0.427585\pi$
$263$	7.31534	0.451083	0.225542	+	0.974234i	$0.427585\pi$
$264$	1.00000	0.0615457
$265$	−2.87689	−0.176726
$266$	6.24621	0.382980
$267$	2.87689	0.176063
$268$	12.0000	0.733017
$269$	−18.4924	−1.12750	−0.563751	−	0.825944i	$-0.690642\pi$
$269$	−18.4924	−1.12750	−0.563751	+	0.825944i	$0.690642\pi$
$270$	−1.00000	−0.0608581
$271$	−18.0540	−1.09670	−0.548350	−	0.836249i	$-0.684744\pi$
$271$	−18.0540	−1.09670	−0.548350	+	0.836249i	$0.684744\pi$
$272$	−1.00000	−0.0606339
$273$	8.00000	0.484182
$274$	9.80776	0.592509
$275$	1.00000	0.0603023
$276$	−2.43845	−0.146777
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	8.68466	0.520871
$279$	−8.68466	−0.519937
$280$	1.56155	0.0933206
$281$	−5.80776	−0.346462	−0.173231	−	0.984881i	$-0.555421\pi$
$281$	−5.80776	−0.346462	−0.173231	+	0.984881i	$0.555421\pi$
$282$	−10.2462	−0.610153
$283$	−17.3693	−1.03250	−0.516249	−	0.856438i	$-0.672672\pi$
$283$	−17.3693	−1.03250	−0.516249	+	0.856438i	$0.672672\pi$
$284$	4.87689	0.289390
$285$	−4.00000	−0.236940
$286$	5.12311	0.302936
$287$	1.75379	0.103523
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	−0.438447	−0.0257465
$291$	−10.6847	−0.626346
$292$	4.24621	0.248491
$293$	20.0540	1.17157	0.585783	−	0.810468i	$-0.300787\pi$
$293$	20.0540	1.17157	0.585783	+	0.810468i	$0.300787\pi$
$294$	−4.56155	−0.266035
$295$	−3.12311	−0.181834
$296$	−6.00000	−0.348743
$297$	−1.00000	−0.0580259
$298$	−5.12311	−0.296774
$299$	−12.4924	−0.722455
$300$	−1.00000	−0.0577350
$301$	12.1922	0.702749
$302$	1.75379	0.100919
$303$	−8.24621	−0.473732
$304$	−4.00000	−0.229416
$305$	6.00000	0.343559
$306$	1.00000	0.0571662
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	1.56155	0.0889777
$309$	−8.68466	−0.494053
$310$	−8.68466	−0.493255
$311$	14.2462	0.807829	0.403914	−	0.914797i	$-0.367649\pi$
$311$	14.2462	0.807829	0.403914	+	0.914797i	$0.367649\pi$
$312$	−5.12311	−0.290039
$313$	20.0540	1.13352	0.566759	−	0.823884i	$-0.308197\pi$
$313$	20.0540	1.13352	0.566759	+	0.823884i	$0.308197\pi$
$314$	18.0000	1.01580
$315$	−1.56155	−0.0879835
$316$	−12.0000	−0.675053
$317$	32.9309	1.84958	0.924791	−	0.380476i	$-0.124240\pi$
$317$	32.9309	1.84958	0.924791	+	0.380476i	$0.124240\pi$
$318$	2.87689	0.161328
$319$	−0.438447	−0.0245483
$320$	−1.00000	−0.0559017
$321$	7.80776	0.435787
$322$	−3.80776	−0.212198
$323$	4.00000	0.222566
$324$	1.00000	0.0555556
$325$	−5.12311	−0.284179
$326$	7.80776	0.432432
$327$	15.3693	0.849925
$328$	−1.12311	−0.0620131
$329$	−16.0000	−0.882109
$330$	−1.00000	−0.0550482
$331$	25.1771	1.38386	0.691929	−	0.721966i	$-0.256761\pi$
$331$	25.1771	1.38386	0.691929	+	0.721966i	$0.256761\pi$
$332$	10.2462	0.562334
$333$	6.00000	0.328798
$334$	9.36932	0.512666
$335$	−12.0000	−0.655630
$336$	−1.56155	−0.0851897
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	−13.2462	−0.720499
$339$	−18.4924	−1.00437
$340$	1.00000	0.0542326
$341$	−8.68466	−0.470301
$342$	4.00000	0.216295
$343$	−18.0540	−0.974823
$344$	−7.80776	−0.420966
$345$	2.43845	0.131282
$346$	−18.0000	−0.967686
$347$	−0.492423	−0.0264346	−0.0132173	−	0.999913i	$-0.504207\pi$
$347$	−0.492423	−0.0264346	−0.0132173	+	0.999913i	$0.504207\pi$
$348$	0.438447	0.0235032
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	−1.56155	−0.0834685
$351$	5.12311	0.273451
$352$	−1.00000	−0.0533002
$353$	−17.8078	−0.947812	−0.473906	−	0.880576i	$-0.657156\pi$
$353$	−17.8078	−0.947812	−0.473906	+	0.880576i	$0.657156\pi$
$354$	3.12311	0.165991
$355$	−4.87689	−0.258839
$356$	−2.87689	−0.152475
$357$	1.56155	0.0826461
$358$	−23.6155	−1.24812
$359$	−9.36932	−0.494494	−0.247247	−	0.968953i	$-0.579526\pi$
$359$	−9.36932	−0.494494	−0.247247	+	0.968953i	$0.579526\pi$
$360$	1.00000	0.0527046
$361$	−3.00000	−0.157895
$362$	−0.438447	−0.0230443
$363$	−1.00000	−0.0524864
$364$	−8.00000	−0.419314
$365$	−4.24621	−0.222257
$366$	−6.00000	−0.313625
$367$	−6.63068	−0.346119	−0.173059	−	0.984911i	$-0.555365\pi$
$367$	−6.63068	−0.346119	−0.173059	+	0.984911i	$0.555365\pi$
$368$	2.43845	0.127113
$369$	1.12311	0.0584665
$370$	6.00000	0.311925
$371$	4.49242	0.233235
$372$	8.68466	0.450279
$373$	24.7386	1.28092	0.640459	−	0.767992i	$-0.278744\pi$
$373$	24.7386	1.28092	0.640459	+	0.767992i	$0.278744\pi$
$374$	1.00000	0.0517088
$375$	1.00000	0.0516398
$376$	10.2462	0.528408
$377$	2.24621	0.115686
$378$	1.56155	0.0803176
$379$	24.4924	1.25809	0.629046	−	0.777368i	$-0.283446\pi$
$379$	24.4924	1.25809	0.629046	+	0.777368i	$0.283446\pi$
$380$	4.00000	0.205196
$381$	9.36932	0.480005
$382$	−0.192236	−0.00983565
$383$	16.4924	0.842723	0.421362	−	0.906893i	$-0.361552\pi$
$383$	16.4924	0.842723	0.421362	+	0.906893i	$0.361552\pi$
$384$	1.00000	0.0510310
$385$	−1.56155	−0.0795841
$386$	2.00000	0.101797
$387$	7.80776	0.396891
$388$	10.6847	0.542431
$389$	7.75379	0.393133	0.196566	−	0.980491i	$-0.437021\pi$
$389$	7.75379	0.393133	0.196566	+	0.980491i	$0.437021\pi$
$390$	5.12311	0.259419
$391$	−2.43845	−0.123318
$392$	4.56155	0.230393
$393$	−12.0000	−0.605320
$394$	1.12311	0.0565812
$395$	12.0000	0.603786
$396$	1.00000	0.0502519
$397$	12.2462	0.614620	0.307310	−	0.951610i	$-0.400571\pi$
$397$	12.2462	0.614620	0.307310	+	0.951610i	$0.400571\pi$
$398$	0	0
$399$	6.24621	0.312702
$400$	1.00000	0.0500000
$401$	8.05398	0.402196	0.201098	−	0.979571i	$-0.435549\pi$
$401$	8.05398	0.402196	0.201098	+	0.979571i	$0.435549\pi$
$402$	12.0000	0.598506
$403$	44.4924	2.21633
$404$	8.24621	0.410264
$405$	−1.00000	−0.0496904
$406$	0.684658	0.0339790
$407$	6.00000	0.297409
$408$	−1.00000	−0.0495074
$409$	25.6155	1.26661	0.633303	−	0.773904i	$-0.281699\pi$
$409$	25.6155	1.26661	0.633303	+	0.773904i	$0.281699\pi$
$410$	1.12311	0.0554662
$411$	9.80776	0.483781
$412$	8.68466	0.427862
$413$	4.87689	0.239976
$414$	−2.43845	−0.119843
$415$	−10.2462	−0.502967
$416$	5.12311	0.251181
$417$	8.68466	0.425290
$418$	4.00000	0.195646
$419$	5.06913	0.247643	0.123822	−	0.992304i	$-0.460485\pi$
$419$	5.06913	0.247643	0.123822	+	0.992304i	$0.460485\pi$
$420$	1.56155	0.0761960
$421$	33.1231	1.61432	0.807161	−	0.590332i	$-0.201003\pi$
$421$	33.1231	1.61432	0.807161	+	0.590332i	$0.201003\pi$
$422$	19.8078	0.964227
$423$	−10.2462	−0.498188
$424$	−2.87689	−0.139714
$425$	−1.00000	−0.0485071
$426$	4.87689	0.236286
$427$	−9.36932	−0.453413
$428$	−7.80776	−0.377403
$429$	5.12311	0.247346
$430$	7.80776	0.376524
$431$	−5.17708	−0.249371	−0.124686	−	0.992196i	$-0.539792\pi$
$431$	−5.17708	−0.249371	−0.124686	+	0.992196i	$0.539792\pi$
$432$	−1.00000	−0.0481125
$433$	−34.1080	−1.63912	−0.819562	−	0.572991i	$-0.805783\pi$
$433$	−34.1080	−1.63912	−0.819562	+	0.572991i	$0.805783\pi$
$434$	13.5616	0.650975
$435$	−0.438447	−0.0210219
$436$	−15.3693	−0.736057
$437$	−9.75379	−0.466587
$438$	4.24621	0.202892
$439$	37.3693	1.78354	0.891770	−	0.452489i	$-0.149464\pi$
$439$	37.3693	1.78354	0.891770	+	0.452489i	$0.149464\pi$
$440$	1.00000	0.0476731
$441$	−4.56155	−0.217217
$442$	−5.12311	−0.243681
$443$	5.56155	0.264237	0.132119	−	0.991234i	$-0.457822\pi$
$443$	5.56155	0.264237	0.132119	+	0.991234i	$0.457822\pi$
$444$	−6.00000	−0.284747
$445$	2.87689	0.136378
$446$	−0.684658	−0.0324195
$447$	−5.12311	−0.242315
$448$	1.56155	0.0737764
$449$	0.438447	0.0206916	0.0103458	−	0.999946i	$-0.496707\pi$
$449$	0.438447	0.0206916	0.0103458	+	0.999946i	$0.496707\pi$
$450$	−1.00000	−0.0471405
$451$	1.12311	0.0528850
$452$	18.4924	0.869810
$453$	1.75379	0.0824002
$454$	−4.68466	−0.219862
$455$	8.00000	0.375046
$456$	−4.00000	−0.187317
$457$	36.7386	1.71856	0.859280	−	0.511505i	$-0.170912\pi$
$457$	36.7386	1.71856	0.859280	+	0.511505i	$0.170912\pi$
$458$	11.3693	0.531253
$459$	1.00000	0.0466760
$460$	−2.43845	−0.113693
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	1.56155	0.0726500
$463$	−12.4924	−0.580572	−0.290286	−	0.956940i	$-0.593750\pi$
$463$	−12.4924	−0.580572	−0.290286	+	0.956940i	$0.593750\pi$
$464$	−0.438447	−0.0203544
$465$	−8.68466	−0.402741
$466$	−8.43845	−0.390904
$467$	17.7538	0.821547	0.410774	−	0.911737i	$-0.365259\pi$
$467$	17.7538	0.821547	0.410774	+	0.911737i	$0.365259\pi$
$468$	−5.12311	−0.236816
$469$	18.7386	0.865270
$470$	−10.2462	−0.472622
$471$	18.0000	0.829396
$472$	−3.12311	−0.143753
$473$	7.80776	0.359001
$474$	−12.0000	−0.551178
$475$	−4.00000	−0.183533
$476$	−1.56155	−0.0715737
$477$	2.87689	0.131724
$478$	−19.1231	−0.874670
$479$	13.1771	0.602076	0.301038	−	0.953612i	$-0.402667\pi$
$479$	13.1771	0.602076	0.301038	+	0.953612i	$0.402667\pi$
$480$	−1.00000	−0.0456435
$481$	−30.7386	−1.40156
$482$	−25.8078	−1.17551
$483$	−3.80776	−0.173259
$484$	1.00000	0.0454545
$485$	−10.6847	−0.485165
$486$	1.00000	0.0453609
$487$	33.3693	1.51211	0.756054	−	0.654509i	$-0.227125\pi$
$487$	33.3693	1.51211	0.756054	+	0.654509i	$0.227125\pi$
$488$	6.00000	0.271607
$489$	7.80776	0.353079
$490$	−4.56155	−0.206070
$491$	−10.2462	−0.462405	−0.231203	−	0.972906i	$-0.574266\pi$
$491$	−10.2462	−0.462405	−0.231203	+	0.972906i	$0.574266\pi$
$492$	−1.12311	−0.0506335
$493$	0.438447	0.0197467
$494$	−20.4924	−0.921998
$495$	−1.00000	−0.0449467
$496$	−8.68466	−0.389953
$497$	7.61553	0.341603
$498$	10.2462	0.459144
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	−1.00000	−0.0447214
$501$	9.36932	0.418590
$502$	−14.2462	−0.635840
$503$	−6.24621	−0.278505	−0.139252	−	0.990257i	$-0.544470\pi$
$503$	−6.24621	−0.278505	−0.139252	+	0.990257i	$0.544470\pi$
$504$	−1.56155	−0.0695571
$505$	−8.24621	−0.366952
$506$	−2.43845	−0.108402
$507$	−13.2462	−0.588285
$508$	−9.36932	−0.415696
$509$	−40.2462	−1.78388	−0.891941	−	0.452152i	$-0.850657\pi$
$509$	−40.2462	−1.78388	−0.891941	+	0.452152i	$0.850657\pi$
$510$	1.00000	0.0442807
$511$	6.63068	0.293324
$512$	−1.00000	−0.0441942
$513$	4.00000	0.176604
$514$	−15.5616	−0.686391
$515$	−8.68466	−0.382692
$516$	−7.80776	−0.343718
$517$	−10.2462	−0.450628
$518$	−9.36932	−0.411664
$519$	−18.0000	−0.790112
$520$	−5.12311	−0.224663
$521$	−12.7386	−0.558090	−0.279045	−	0.960278i	$-0.590018\pi$
$521$	−12.7386	−0.558090	−0.279045	+	0.960278i	$0.590018\pi$
$522$	0.438447	0.0191903
$523$	−12.6847	−0.554661	−0.277331	−	0.960775i	$-0.589450\pi$
$523$	−12.6847	−0.554661	−0.277331	+	0.960775i	$0.589450\pi$
$524$	12.0000	0.524222
$525$	−1.56155	−0.0681518
$526$	−7.31534	−0.318964
$527$	8.68466	0.378310
$528$	−1.00000	−0.0435194
$529$	−17.0540	−0.741477
$530$	2.87689	0.124964
$531$	3.12311	0.135531
$532$	−6.24621	−0.270808
$533$	−5.75379	−0.249224
$534$	−2.87689	−0.124495
$535$	7.80776	0.337559
$536$	−12.0000	−0.518321
$537$	−23.6155	−1.01909
$538$	18.4924	0.797265
$539$	−4.56155	−0.196480
$540$	1.00000	0.0430331
$541$	21.1231	0.908153	0.454077	−	0.890963i	$-0.349969\pi$
$541$	21.1231	0.908153	0.454077	+	0.890963i	$0.349969\pi$
$542$	18.0540	0.775485
$543$	−0.438447	−0.0188156
$544$	1.00000	0.0428746
$545$	15.3693	0.658349
$546$	−8.00000	−0.342368
$547$	−9.36932	−0.400603	−0.200302	−	0.979734i	$-0.564192\pi$
$547$	−9.36932	−0.400603	−0.200302	+	0.979734i	$0.564192\pi$
$548$	−9.80776	−0.418967
$549$	−6.00000	−0.256074
$550$	−1.00000	−0.0426401
$551$	1.75379	0.0747139
$552$	2.43845	0.103787
$553$	−18.7386	−0.796848
$554$	−2.00000	−0.0849719
$555$	6.00000	0.254686
$556$	−8.68466	−0.368312
$557$	10.6847	0.452724	0.226362	−	0.974043i	$-0.427317\pi$
$557$	10.6847	0.452724	0.226362	+	0.974043i	$0.427317\pi$
$558$	8.68466	0.367651
$559$	−40.0000	−1.69182
$560$	−1.56155	−0.0659877
$561$	1.00000	0.0422200
$562$	5.80776	0.244986
$563$	16.4924	0.695073	0.347536	−	0.937667i	$-0.387018\pi$
$563$	16.4924	0.695073	0.347536	+	0.937667i	$0.387018\pi$
$564$	10.2462	0.431443
$565$	−18.4924	−0.777982
$566$	17.3693	0.730087
$567$	1.56155	0.0655791
$568$	−4.87689	−0.204630
$569$	−32.2462	−1.35183	−0.675916	−	0.736979i	$-0.736252\pi$
$569$	−32.2462	−1.35183	−0.675916	+	0.736979i	$0.736252\pi$
$570$	4.00000	0.167542
$571$	30.2462	1.26576	0.632882	−	0.774248i	$-0.281872\pi$
$571$	30.2462	1.26576	0.632882	+	0.774248i	$0.281872\pi$
$572$	−5.12311	−0.214208
$573$	−0.192236	−0.00803077
$574$	−1.75379	−0.0732017
$575$	2.43845	0.101690
$576$	1.00000	0.0416667
$577$	−26.4924	−1.10289	−0.551447	−	0.834210i	$-0.685924\pi$
$577$	−26.4924	−1.10289	−0.551447	+	0.834210i	$0.685924\pi$
$578$	−1.00000	−0.0415945
$579$	2.00000	0.0831172
$580$	0.438447	0.0182055
$581$	16.0000	0.663792
$582$	10.6847	0.442893
$583$	2.87689	0.119149
$584$	−4.24621	−0.175709
$585$	5.12311	0.211814
$586$	−20.0540	−0.828422
$587$	−30.5464	−1.26078	−0.630392	−	0.776277i	$-0.717106\pi$
$587$	−30.5464	−1.26078	−0.630392	+	0.776277i	$0.717106\pi$
$588$	4.56155	0.188115
$589$	34.7386	1.43138
$590$	3.12311	0.128576
$591$	1.12311	0.0461984
$592$	6.00000	0.246598
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	1.00000	0.0410305
$595$	1.56155	0.0640174
$596$	5.12311	0.209851
$597$	0	0
$598$	12.4924	0.510853
$599$	19.3153	0.789203	0.394602	−	0.918852i	$-0.370883\pi$
$599$	19.3153	0.789203	0.394602	+	0.918852i	$0.370883\pi$
$600$	1.00000	0.0408248
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	−12.1922	−0.496918
$603$	12.0000	0.488678
$604$	−1.75379	−0.0713607
$605$	−1.00000	−0.0406558
$606$	8.24621	0.334979
$607$	−30.7386	−1.24764	−0.623821	−	0.781567i	$-0.714421\pi$
$607$	−30.7386	−1.24764	−0.623821	+	0.781567i	$0.714421\pi$
$608$	4.00000	0.162221
$609$	0.684658	0.0277438
$610$	−6.00000	−0.242933
$611$	52.4924	2.12362
$612$	−1.00000	−0.0404226
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	−20.0000	−0.807134
$615$	1.12311	0.0452880
$616$	−1.56155	−0.0629168
$617$	−3.75379	−0.151122	−0.0755609	−	0.997141i	$-0.524075\pi$
$617$	−3.75379	−0.151122	−0.0755609	+	0.997141i	$0.524075\pi$
$618$	8.68466	0.349348
$619$	−17.8617	−0.717924	−0.358962	−	0.933352i	$-0.616869\pi$
$619$	−17.8617	−0.717924	−0.358962	+	0.933352i	$0.616869\pi$
$620$	8.68466	0.348784
$621$	−2.43845	−0.0978515
$622$	−14.2462	−0.571221
$623$	−4.49242	−0.179985
$624$	5.12311	0.205088
$625$	1.00000	0.0400000
$626$	−20.0540	−0.801518
$627$	4.00000	0.159745
$628$	−18.0000	−0.718278
$629$	−6.00000	−0.239236
$630$	1.56155	0.0622138
$631$	41.3693	1.64689	0.823443	−	0.567399i	$-0.192050\pi$
$631$	41.3693	1.64689	0.823443	+	0.567399i	$0.192050\pi$
$632$	12.0000	0.477334
$633$	19.8078	0.787288
$634$	−32.9309	−1.30785
$635$	9.36932	0.371810
$636$	−2.87689	−0.114076
$637$	23.3693	0.925926
$638$	0.438447	0.0173583
$639$	4.87689	0.192927
$640$	1.00000	0.0395285
$641$	2.19224	0.0865881	0.0432941	−	0.999062i	$-0.486215\pi$
$641$	2.19224	0.0865881	0.0432941	+	0.999062i	$0.486215\pi$
$642$	−7.80776	−0.308148
$643$	31.8078	1.25438	0.627188	−	0.778868i	$-0.284206\pi$
$643$	31.8078	1.25438	0.627188	+	0.778868i	$0.284206\pi$
$644$	3.80776	0.150047
$645$	7.80776	0.307430
$646$	−4.00000	−0.157378
$647$	40.1080	1.57681	0.788403	−	0.615159i	$-0.210908\pi$
$647$	40.1080	1.57681	0.788403	+	0.615159i	$0.210908\pi$
$648$	−1.00000	−0.0392837
$649$	3.12311	0.122593
$650$	5.12311	0.200945
$651$	13.5616	0.531519
$652$	−7.80776	−0.305776
$653$	20.4384	0.799818	0.399909	−	0.916555i	$-0.369042\pi$
$653$	20.4384	0.799818	0.399909	+	0.916555i	$0.369042\pi$
$654$	−15.3693	−0.600988
$655$	−12.0000	−0.468879
$656$	1.12311	0.0438499
$657$	4.24621	0.165660
$658$	16.0000	0.623745
$659$	23.4233	0.912442	0.456221	−	0.889867i	$-0.349203\pi$
$659$	23.4233	0.912442	0.456221	+	0.889867i	$0.349203\pi$
$660$	1.00000	0.0389249
$661$	−27.3693	−1.06454	−0.532272	−	0.846574i	$-0.678661\pi$
$661$	−27.3693	−1.06454	−0.532272	+	0.846574i	$0.678661\pi$
$662$	−25.1771	−0.978535
$663$	−5.12311	−0.198965
$664$	−10.2462	−0.397630
$665$	6.24621	0.242218
$666$	−6.00000	−0.232495
$667$	−1.06913	−0.0413969
$668$	−9.36932	−0.362510
$669$	−0.684658	−0.0264704
$670$	12.0000	0.463600
$671$	−6.00000	−0.231627
$672$	1.56155	0.0602382
$673$	9.12311	0.351670	0.175835	−	0.984420i	$-0.443737\pi$
$673$	9.12311	0.351670	0.175835	+	0.984420i	$0.443737\pi$
$674$	−6.00000	−0.231111
$675$	−1.00000	−0.0384900
$676$	13.2462	0.509470
$677$	−2.49242	−0.0957916	−0.0478958	−	0.998852i	$-0.515252\pi$
$677$	−2.49242	−0.0957916	−0.0478958	+	0.998852i	$0.515252\pi$
$678$	18.4924	0.710197
$679$	16.6847	0.640298
$680$	−1.00000	−0.0383482
$681$	−4.68466	−0.179517
$682$	8.68466	0.332553
$683$	21.7538	0.832386	0.416193	−	0.909276i	$-0.363364\pi$
$683$	21.7538	0.832386	0.416193	+	0.909276i	$0.363364\pi$
$684$	−4.00000	−0.152944
$685$	9.80776	0.374735
$686$	18.0540	0.689304
$687$	11.3693	0.433766
$688$	7.80776	0.297668
$689$	−14.7386	−0.561497
$690$	−2.43845	−0.0928301
$691$	46.7386	1.77802	0.889011	−	0.457886i	$-0.151393\pi$
$691$	46.7386	1.77802	0.889011	+	0.457886i	$0.151393\pi$
$692$	18.0000	0.684257
$693$	1.56155	0.0593185
$694$	0.492423	0.0186921
$695$	8.68466	0.329428
$696$	−0.438447	−0.0166193
$697$	−1.12311	−0.0425407
$698$	2.00000	0.0757011
$699$	−8.43845	−0.319171
$700$	1.56155	0.0590211
$701$	−22.9848	−0.868126	−0.434063	−	0.900883i	$-0.642920\pi$
$701$	−22.9848	−0.868126	−0.434063	+	0.900883i	$0.642920\pi$
$702$	−5.12311	−0.193359
$703$	−24.0000	−0.905177
$704$	1.00000	0.0376889
$705$	−10.2462	−0.385895
$706$	17.8078	0.670204
$707$	12.8769	0.484285
$708$	−3.12311	−0.117373
$709$	−52.7386	−1.98064	−0.990320	−	0.138800i	$-0.955676\pi$
$709$	−52.7386	−1.98064	−0.990320	+	0.138800i	$0.955676\pi$
$710$	4.87689	0.183027
$711$	−12.0000	−0.450035
$712$	2.87689	0.107816
$713$	−21.1771	−0.793088
$714$	−1.56155	−0.0584396
$715$	5.12311	0.191593
$716$	23.6155	0.882554
$717$	−19.1231	−0.714165
$718$	9.36932	0.349660
$719$	14.6307	0.545632	0.272816	−	0.962066i	$-0.412045\pi$
$719$	14.6307	0.545632	0.272816	+	0.962066i	$0.412045\pi$
$720$	−1.00000	−0.0372678
$721$	13.5616	0.505059
$722$	3.00000	0.111648
$723$	−25.8078	−0.959801
$724$	0.438447	0.0162948
$725$	−0.438447	−0.0162835
$726$	1.00000	0.0371135
$727$	0.684658	0.0253926	0.0126963	−	0.999919i	$-0.495959\pi$
$727$	0.684658	0.0253926	0.0126963	+	0.999919i	$0.495959\pi$
$728$	8.00000	0.296500
$729$	1.00000	0.0370370
$730$	4.24621	0.157159
$731$	−7.80776	−0.288781
$732$	6.00000	0.221766
$733$	−1.61553	−0.0596709	−0.0298354	−	0.999555i	$-0.509498\pi$
$733$	−1.61553	−0.0596709	−0.0298354	+	0.999555i	$0.509498\pi$
$734$	6.63068	0.244743
$735$	−4.56155	−0.168255
$736$	−2.43845	−0.0898824
$737$	12.0000	0.442026
$738$	−1.12311	−0.0413421
$739$	48.1080	1.76968	0.884840	−	0.465895i	$-0.154268\pi$
$739$	48.1080	1.76968	0.884840	+	0.465895i	$0.154268\pi$
$740$	−6.00000	−0.220564
$741$	−20.4924	−0.752808
$742$	−4.49242	−0.164922
$743$	14.2462	0.522643	0.261321	−	0.965252i	$-0.415842\pi$
$743$	14.2462	0.522643	0.261321	+	0.965252i	$0.415842\pi$
$744$	−8.68466	−0.318395
$745$	−5.12311	−0.187696
$746$	−24.7386	−0.905746
$747$	10.2462	0.374889
$748$	−1.00000	−0.0365636
$749$	−12.1922	−0.445495
$750$	−1.00000	−0.0365148
$751$	16.6847	0.608832	0.304416	−	0.952539i	$-0.401539\pi$
$751$	16.6847	0.608832	0.304416	+	0.952539i	$0.401539\pi$
$752$	−10.2462	−0.373641
$753$	−14.2462	−0.519161
$754$	−2.24621	−0.0818022
$755$	1.75379	0.0638269
$756$	−1.56155	−0.0567931
$757$	28.5464	1.03754	0.518768	−	0.854915i	$-0.326391\pi$
$757$	28.5464	1.03754	0.518768	+	0.854915i	$0.326391\pi$
$758$	−24.4924	−0.889605
$759$	−2.43845	−0.0885100
$760$	−4.00000	−0.145095
$761$	19.5616	0.709106	0.354553	−	0.935036i	$-0.384633\pi$
$761$	19.5616	0.709106	0.354553	+	0.935036i	$0.384633\pi$
$762$	−9.36932	−0.339415
$763$	−24.0000	−0.868858
$764$	0.192236	0.00695485
$765$	1.00000	0.0361551
$766$	−16.4924	−0.595895
$767$	−16.0000	−0.577727
$768$	−1.00000	−0.0360844
$769$	−30.9848	−1.11734	−0.558671	−	0.829389i	$-0.688689\pi$
$769$	−30.9848	−1.11734	−0.558671	+	0.829389i	$0.688689\pi$
$770$	1.56155	0.0562745
$771$	−15.5616	−0.560436
$772$	−2.00000	−0.0719816
$773$	−30.8769	−1.11056	−0.555282	−	0.831662i	$-0.687390\pi$
$773$	−30.8769	−1.11056	−0.555282	+	0.831662i	$0.687390\pi$
$774$	−7.80776	−0.280644
$775$	−8.68466	−0.311962
$776$	−10.6847	−0.383557
$777$	−9.36932	−0.336122
$778$	−7.75379	−0.277987
$779$	−4.49242	−0.160958
$780$	−5.12311	−0.183437
$781$	4.87689	0.174509
$782$	2.43845	0.0871987
$783$	0.438447	0.0156688
$784$	−4.56155	−0.162913
$785$	18.0000	0.642448
$786$	12.0000	0.428026
$787$	44.4924	1.58598	0.792992	−	0.609232i	$-0.208522\pi$
$787$	44.4924	1.58598	0.792992	+	0.609232i	$0.208522\pi$
$788$	−1.12311	−0.0400090
$789$	−7.31534	−0.260433
$790$	−12.0000	−0.426941
$791$	28.8769	1.02674
$792$	−1.00000	−0.0355335
$793$	30.7386	1.09156
$794$	−12.2462	−0.434602
$795$	2.87689	0.102033
$796$	0	0
$797$	43.8617	1.55366	0.776831	−	0.629709i	$-0.216826\pi$
$797$	43.8617	1.55366	0.776831	+	0.629709i	$0.216826\pi$
$798$	−6.24621	−0.221113
$799$	10.2462	0.362485
$800$	−1.00000	−0.0353553
$801$	−2.87689	−0.101650
$802$	−8.05398	−0.284396
$803$	4.24621	0.149846
$804$	−12.0000	−0.423207
$805$	−3.80776	−0.134206
$806$	−44.4924	−1.56718
$807$	18.4924	0.650964
$808$	−8.24621	−0.290101
$809$	−50.9848	−1.79253	−0.896266	−	0.443517i	$-0.853730\pi$
$809$	−50.9848	−1.79253	−0.896266	+	0.443517i	$0.853730\pi$
$810$	1.00000	0.0351364
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	−0.684658	−0.0240268
$813$	18.0540	0.633181
$814$	−6.00000	−0.210300
$815$	7.80776	0.273494
$816$	1.00000	0.0350070
$817$	−31.2311	−1.09264
$818$	−25.6155	−0.895626
$819$	−8.00000	−0.279543
$820$	−1.12311	−0.0392205
$821$	−3.56155	−0.124299	−0.0621495	−	0.998067i	$-0.519796\pi$
$821$	−3.56155	−0.124299	−0.0621495	+	0.998067i	$0.519796\pi$
$822$	−9.80776	−0.342085
$823$	9.36932	0.326594	0.163297	−	0.986577i	$-0.447787\pi$
$823$	9.36932	0.326594	0.163297	+	0.986577i	$0.447787\pi$
$824$	−8.68466	−0.302544
$825$	−1.00000	−0.0348155
$826$	−4.87689	−0.169689
$827$	23.8078	0.827877	0.413939	−	0.910305i	$-0.364153\pi$
$827$	23.8078	0.827877	0.413939	+	0.910305i	$0.364153\pi$
$828$	2.43845	0.0847419
$829$	7.75379	0.269300	0.134650	−	0.990893i	$-0.457009\pi$
$829$	7.75379	0.269300	0.134650	+	0.990893i	$0.457009\pi$
$830$	10.2462	0.355651
$831$	−2.00000	−0.0693792
$832$	−5.12311	−0.177612
$833$	4.56155	0.158048
$834$	−8.68466	−0.300725
$835$	9.36932	0.324239
$836$	−4.00000	−0.138343
$837$	8.68466	0.300186
$838$	−5.06913	−0.175110
$839$	−33.3693	−1.15204	−0.576018	−	0.817437i	$-0.695394\pi$
$839$	−33.3693	−1.15204	−0.576018	+	0.817437i	$0.695394\pi$
$840$	−1.56155	−0.0538787
$841$	−28.8078	−0.993371
$842$	−33.1231	−1.14150
$843$	5.80776	0.200030
$844$	−19.8078	−0.681811
$845$	−13.2462	−0.455684
$846$	10.2462	0.352272
$847$	1.56155	0.0536556
$848$	2.87689	0.0987930
$849$	17.3693	0.596113
$850$	1.00000	0.0342997
$851$	14.6307	0.501533
$852$	−4.87689	−0.167080
$853$	−38.3002	−1.31137	−0.655687	−	0.755033i	$-0.727621\pi$
$853$	−38.3002	−1.31137	−0.655687	+	0.755033i	$0.727621\pi$
$854$	9.36932	0.320611
$855$	4.00000	0.136797
$856$	7.80776	0.266864
$857$	0.822919	0.0281104	0.0140552	−	0.999901i	$-0.495526\pi$
$857$	0.822919	0.0281104	0.0140552	+	0.999901i	$0.495526\pi$
$858$	−5.12311	−0.174900
$859$	20.3002	0.692633	0.346317	−	0.938118i	$-0.387432\pi$
$859$	20.3002	0.692633	0.346317	+	0.938118i	$0.387432\pi$
$860$	−7.80776	−0.266243
$861$	−1.75379	−0.0597690
$862$	5.17708	0.176332
$863$	8.49242	0.289085	0.144543	−	0.989499i	$-0.453829\pi$
$863$	8.49242	0.289085	0.144543	+	0.989499i	$0.453829\pi$
$864$	1.00000	0.0340207
$865$	−18.0000	−0.612018
$866$	34.1080	1.15904
$867$	−1.00000	−0.0339618
$868$	−13.5616	−0.460309
$869$	−12.0000	−0.407072
$870$	0.438447	0.0148648
$871$	−61.4773	−2.08308
$872$	15.3693	0.520471
$873$	10.6847	0.361621
$874$	9.75379	0.329927
$875$	−1.56155	−0.0527901
$876$	−4.24621	−0.143466
$877$	27.6695	0.934333	0.467166	−	0.884169i	$-0.345275\pi$
$877$	27.6695	0.934333	0.467166	+	0.884169i	$0.345275\pi$
$878$	−37.3693	−1.26115
$879$	−20.0540	−0.676404
$880$	−1.00000	−0.0337100
$881$	−48.9309	−1.64852	−0.824261	−	0.566209i	$-0.808409\pi$
$881$	−48.9309	−1.64852	−0.824261	+	0.566209i	$0.808409\pi$
$882$	4.56155	0.153595
$883$	38.7386	1.30366	0.651829	−	0.758366i	$-0.274002\pi$
$883$	38.7386	1.30366	0.651829	+	0.758366i	$0.274002\pi$
$884$	5.12311	0.172309
$885$	3.12311	0.104982
$886$	−5.56155	−0.186844
$887$	−28.4924	−0.956682	−0.478341	−	0.878174i	$-0.658762\pi$
$887$	−28.4924	−0.956682	−0.478341	+	0.878174i	$0.658762\pi$
$888$	6.00000	0.201347
$889$	−14.6307	−0.490697
$890$	−2.87689	−0.0964337
$891$	1.00000	0.0335013
$892$	0.684658	0.0229241
$893$	40.9848	1.37151
$894$	5.12311	0.171342
$895$	−23.6155	−0.789380
$896$	−1.56155	−0.0521678
$897$	12.4924	0.417110
$898$	−0.438447	−0.0146312
$899$	3.80776	0.126996
$900$	1.00000	0.0333333
$901$	−2.87689	−0.0958432
$902$	−1.12311	−0.0373953
$903$	−12.1922	−0.405732
$904$	−18.4924	−0.615049
$905$	−0.438447	−0.0145745
$906$	−1.75379	−0.0582657
$907$	−4.68466	−0.155552	−0.0777758	−	0.996971i	$-0.524782\pi$
$907$	−4.68466	−0.155552	−0.0777758	+	0.996971i	$0.524782\pi$
$908$	4.68466	0.155466
$909$	8.24621	0.273510
$910$	−8.00000	−0.265197
$911$	16.9848	0.562733	0.281367	−	0.959600i	$-0.409212\pi$
$911$	16.9848	0.562733	0.281367	+	0.959600i	$0.409212\pi$
$912$	4.00000	0.132453
$913$	10.2462	0.339100
$914$	−36.7386	−1.21521
$915$	−6.00000	−0.198354
$916$	−11.3693	−0.375653
$917$	18.7386	0.618804
$918$	−1.00000	−0.0330049
$919$	−47.9157	−1.58059	−0.790297	−	0.612724i	$-0.790074\pi$
$919$	−47.9157	−1.58059	−0.790297	+	0.612724i	$0.790074\pi$
$920$	2.43845	0.0803932
$921$	−20.0000	−0.659022
$922$	−18.0000	−0.592798
$923$	−24.9848	−0.822386
$924$	−1.56155	−0.0513713
$925$	6.00000	0.197279
$926$	12.4924	0.410526
$927$	8.68466	0.285242
$928$	0.438447	0.0143927
$929$	−23.5616	−0.773029	−0.386515	−	0.922283i	$-0.626321\pi$
$929$	−23.5616	−0.773029	−0.386515	+	0.922283i	$0.626321\pi$
$930$	8.68466	0.284781
$931$	18.2462	0.597995
$932$	8.43845	0.276411
$933$	−14.2462	−0.466400
$934$	−17.7538	−0.580922
$935$	1.00000	0.0327035
$936$	5.12311	0.167454
$937$	−32.7386	−1.06952	−0.534762	−	0.845003i	$-0.679599\pi$
$937$	−32.7386	−1.06952	−0.534762	+	0.845003i	$0.679599\pi$
$938$	−18.7386	−0.611838
$939$	−20.0540	−0.654437
$940$	10.2462	0.334195
$941$	11.7538	0.383163	0.191581	−	0.981477i	$-0.438638\pi$
$941$	11.7538	0.383163	0.191581	+	0.981477i	$0.438638\pi$
$942$	−18.0000	−0.586472
$943$	2.73863	0.0891822
$944$	3.12311	0.101648
$945$	1.56155	0.0507973
$946$	−7.80776	−0.253852
$947$	−16.8769	−0.548425	−0.274213	−	0.961669i	$-0.588417\pi$
$947$	−16.8769	−0.548425	−0.274213	+	0.961669i	$0.588417\pi$
$948$	12.0000	0.389742
$949$	−21.7538	−0.706158
$950$	4.00000	0.129777
$951$	−32.9309	−1.06786
$952$	1.56155	0.0506102
$953$	−13.5076	−0.437553	−0.218777	−	0.975775i	$-0.570207\pi$
$953$	−13.5076	−0.437553	−0.218777	+	0.975775i	$0.570207\pi$
$954$	−2.87689	−0.0931429
$955$	−0.192236	−0.00622061
$956$	19.1231	0.618485
$957$	0.438447	0.0141730
$958$	−13.1771	−0.425732
$959$	−15.3153	−0.494558
$960$	1.00000	0.0322749
$961$	44.4233	1.43301
$962$	30.7386	0.991053
$963$	−7.80776	−0.251602
$964$	25.8078	0.831212
$965$	2.00000	0.0643823
$966$	3.80776	0.122513
$967$	−14.6307	−0.470491	−0.235246	−	0.971936i	$-0.575589\pi$
$967$	−14.6307	−0.470491	−0.235246	+	0.971936i	$0.575589\pi$
$968$	−1.00000	−0.0321412
$969$	−4.00000	−0.128499
$970$	10.6847	0.343064
$971$	23.2311	0.745520	0.372760	−	0.927928i	$-0.378411\pi$
$971$	23.2311	0.745520	0.372760	+	0.927928i	$0.378411\pi$
$972$	−1.00000	−0.0320750
$973$	−13.5616	−0.434763
$974$	−33.3693	−1.06922
$975$	5.12311	0.164071
$976$	−6.00000	−0.192055
$977$	−57.7235	−1.84674	−0.923369	−	0.383914i	$-0.874576\pi$
$977$	−57.7235	−1.84674	−0.923369	+	0.383914i	$0.874576\pi$
$978$	−7.80776	−0.249665
$979$	−2.87689	−0.0919459
$980$	4.56155	0.145713
$981$	−15.3693	−0.490705
$982$	10.2462	0.326970
$983$	30.9309	0.986542	0.493271	−	0.869876i	$-0.335801\pi$
$983$	30.9309	0.986542	0.493271	+	0.869876i	$0.335801\pi$
$984$	1.12311	0.0358033
$985$	1.12311	0.0357851
$986$	−0.438447	−0.0139630
$987$	16.0000	0.509286
$988$	20.4924	0.651951
$989$	19.0388	0.605399
$990$	1.00000	0.0317821
$991$	35.4233	1.12526	0.562629	−	0.826710i	$-0.309790\pi$
$991$	35.4233	1.12526	0.562629	+	0.826710i	$0.309790\pi$
$992$	8.68466	0.275738
$993$	−25.1771	−0.798971
$994$	−7.61553	−0.241550
$995$	0	0
$996$	−10.2462	−0.324664
$997$	−2.19224	−0.0694288	−0.0347144	−	0.999397i	$-0.511052\pi$
$997$	−2.19224	−0.0694288	−0.0347144	+	0.999397i	$0.511052\pi$
$998$	−4.00000	−0.126618
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5610.2.a.bn.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5610.2.a.bn.1.2	✓	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5610.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} -5.12311 q^{13} -1.56155 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} -1.56155 q^{21} -1.00000 q^{22} +2.43845 q^{23} +1.00000 q^{24} +1.00000 q^{25} +5.12311 q^{26} -1.00000 q^{27} +1.56155 q^{28} -0.438447 q^{29} -1.00000 q^{30} -8.68466 q^{31} -1.00000 q^{32} -1.00000 q^{33} +1.00000 q^{34} -1.56155 q^{35} +1.00000 q^{36} +6.00000 q^{37} +4.00000 q^{38} +5.12311 q^{39} +1.00000 q^{40} +1.12311 q^{41} +1.56155 q^{42} +7.80776 q^{43} +1.00000 q^{44} -1.00000 q^{45} -2.43845 q^{46} -10.2462 q^{47} -1.00000 q^{48} -4.56155 q^{49} -1.00000 q^{50} +1.00000 q^{51} -5.12311 q^{52} +2.87689 q^{53} +1.00000 q^{54} -1.00000 q^{55} -1.56155 q^{56} +4.00000 q^{57} +0.438447 q^{58} +3.12311 q^{59} +1.00000 q^{60} -6.00000 q^{61} +8.68466 q^{62} +1.56155 q^{63} +1.00000 q^{64} +5.12311 q^{65} +1.00000 q^{66} +12.0000 q^{67} -1.00000 q^{68} -2.43845 q^{69} +1.56155 q^{70} +4.87689 q^{71} -1.00000 q^{72} +4.24621 q^{73} -6.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} +1.56155 q^{77} -5.12311 q^{78} -12.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -1.12311 q^{82} +10.2462 q^{83} -1.56155 q^{84} +1.00000 q^{85} -7.80776 q^{86} +0.438447 q^{87} -1.00000 q^{88} -2.87689 q^{89} +1.00000 q^{90} -8.00000 q^{91} +2.43845 q^{92} +8.68466 q^{93} +10.2462 q^{94} +4.00000 q^{95} +1.00000 q^{96} +10.6847 q^{97} +4.56155 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{11} - 2 q^{12} - 2 q^{13} + q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{17} - 2 q^{18} - 8 q^{19} - 2 q^{20} + q^{21} - 2 q^{22} + 9 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{26} - 2 q^{27} - q^{28} - 5 q^{29} - 2 q^{30} - 5 q^{31} - 2 q^{32} - 2 q^{33} + 2 q^{34} + q^{35} + 2 q^{36} + 12 q^{37} + 8 q^{38} + 2 q^{39} + 2 q^{40} - 6 q^{41} - q^{42} - 5 q^{43} + 2 q^{44} - 2 q^{45} - 9 q^{46} - 4 q^{47} - 2 q^{48} - 5 q^{49} - 2 q^{50} + 2 q^{51} - 2 q^{52} + 14 q^{53} + 2 q^{54} - 2 q^{55} + q^{56} + 8 q^{57} + 5 q^{58} - 2 q^{59} + 2 q^{60} - 12 q^{61} + 5 q^{62} - q^{63} + 2 q^{64} + 2 q^{65} + 2 q^{66} + 24 q^{67} - 2 q^{68} - 9 q^{69} - q^{70} + 18 q^{71} - 2 q^{72} - 8 q^{73} - 12 q^{74} - 2 q^{75} - 8 q^{76} - q^{77} - 2 q^{78} - 24 q^{79} - 2 q^{80} + 2 q^{81} + 6 q^{82} + 4 q^{83} + q^{84} + 2 q^{85} + 5 q^{86} + 5 q^{87} - 2 q^{88} - 14 q^{89} + 2 q^{90} - 16 q^{91} + 9 q^{92} + 5 q^{93} + 4 q^{94} + 8 q^{95} + 2 q^{96} + 9 q^{97} + 5 q^{98} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^11 - 2 * q^12 - 2 * q^13 + q^14 + 2 * q^15 + 2 * q^16 - 2 * q^17 - 2 * q^18 - 8 * q^19 - 2 * q^20 + q^21 - 2 * q^22 + 9 * q^23 + 2 * q^24 + 2 * q^25 + 2 * q^26 - 2 * q^27 - q^28 - 5 * q^29 - 2 * q^30 - 5 * q^31 - 2 * q^32 - 2 * q^33 + 2 * q^34 + q^35 + 2 * q^36 + 12 * q^37 + 8 * q^38 + 2 * q^39 + 2 * q^40 - 6 * q^41 - q^42 - 5 * q^43 + 2 * q^44 - 2 * q^45 - 9 * q^46 - 4 * q^47 - 2 * q^48 - 5 * q^49 - 2 * q^50 + 2 * q^51 - 2 * q^52 + 14 * q^53 + 2 * q^54 - 2 * q^55 + q^56 + 8 * q^57 + 5 * q^58 - 2 * q^59 + 2 * q^60 - 12 * q^61 + 5 * q^62 - q^63 + 2 * q^64 + 2 * q^65 + 2 * q^66 + 24 * q^67 - 2 * q^68 - 9 * q^69 - q^70 + 18 * q^71 - 2 * q^72 - 8 * q^73 - 12 * q^74 - 2 * q^75 - 8 * q^76 - q^77 - 2 * q^78 - 24 * q^79 - 2 * q^80 + 2 * q^81 + 6 * q^82 + 4 * q^83 + q^84 + 2 * q^85 + 5 * q^86 + 5 * q^87 - 2 * q^88 - 14 * q^89 + 2 * q^90 - 16 * q^91 + 9 * q^92 + 5 * q^93 + 4 * q^94 + 8 * q^95 + 2 * q^96 + 9 * q^97 + 5 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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