LMFDB - Embedded newform 5610.2.a.cd.1.3

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:	$3$
Coefficient field:	3.3.2089.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 13x - 4 $ x^3 - 13*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			1.3
Root			$-3.44055$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +3.63897 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +3.63897 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -1.63897 q^{13} +3.63897 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +7.63897 q^{19} -1.00000 q^{20} +3.63897 q^{21} -1.00000 q^{22} -0.361026 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.63897 q^{26} +1.00000 q^{27} +3.63897 q^{28} +2.00000 q^{29} -1.00000 q^{30} -8.52008 q^{31} +1.00000 q^{32} -1.00000 q^{33} +1.00000 q^{34} -3.63897 q^{35} +1.00000 q^{36} -3.24213 q^{37} +7.63897 q^{38} -1.63897 q^{39} -1.00000 q^{40} -2.00000 q^{41} +3.63897 q^{42} +8.88110 q^{43} -1.00000 q^{44} -1.00000 q^{45} -0.361026 q^{46} +10.4843 q^{47} +1.00000 q^{48} +6.24213 q^{49} +1.00000 q^{50} +1.00000 q^{51} -1.63897 q^{52} -2.88110 q^{53} +1.00000 q^{54} +1.00000 q^{55} +3.63897 q^{56} +7.63897 q^{57} +2.00000 q^{58} +8.88110 q^{59} -1.00000 q^{60} +7.24213 q^{61} -8.52008 q^{62} +3.63897 q^{63} +1.00000 q^{64} +1.63897 q^{65} -1.00000 q^{66} +13.2421 q^{67} +1.00000 q^{68} -0.361026 q^{69} -3.63897 q^{70} +1.60316 q^{71} +1.00000 q^{72} -10.1591 q^{73} -3.24213 q^{74} +1.00000 q^{75} +7.63897 q^{76} -3.63897 q^{77} -1.63897 q^{78} +2.39684 q^{79} -1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -16.5201 q^{83} +3.63897 q^{84} -1.00000 q^{85} +8.88110 q^{86} +2.00000 q^{87} -1.00000 q^{88} -1.20631 q^{89} -1.00000 q^{90} -5.96418 q^{91} -0.361026 q^{92} -8.52008 q^{93} +10.4843 q^{94} -7.63897 q^{95} +1.00000 q^{96} -13.7980 q^{97} +6.24213 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} + q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^5 + 3 * q^6 + q^7 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} + q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} - 3 q^{11} + 3 q^{12} + 5 q^{13} + q^{14} - 3 q^{15} + 3 q^{16} + 3 q^{17} + 3 q^{18} + 13 q^{19} - 3 q^{20} + q^{21} - 3 q^{22} - 11 q^{23} + 3 q^{24} + 3 q^{25} + 5 q^{26} + 3 q^{27} + q^{28} + 6 q^{29} - 3 q^{30} + 5 q^{31} + 3 q^{32} - 3 q^{33} + 3 q^{34} - q^{35} + 3 q^{36} + q^{37} + 13 q^{38} + 5 q^{39} - 3 q^{40} - 6 q^{41} + q^{42} + 6 q^{43} - 3 q^{44} - 3 q^{45} - 11 q^{46} + 10 q^{47} + 3 q^{48} + 8 q^{49} + 3 q^{50} + 3 q^{51} + 5 q^{52} + 12 q^{53} + 3 q^{54} + 3 q^{55} + q^{56} + 13 q^{57} + 6 q^{58} + 6 q^{59} - 3 q^{60} + 11 q^{61} + 5 q^{62} + q^{63} + 3 q^{64} - 5 q^{65} - 3 q^{66} + 29 q^{67} + 3 q^{68} - 11 q^{69} - q^{70} + 4 q^{71} + 3 q^{72} + 10 q^{73} + q^{74} + 3 q^{75} + 13 q^{76} - q^{77} + 5 q^{78} + 8 q^{79} - 3 q^{80} + 3 q^{81} - 6 q^{82} - 19 q^{83} + q^{84} - 3 q^{85} + 6 q^{86} + 6 q^{87} - 3 q^{88} - 2 q^{89} - 3 q^{90} - 27 q^{91} - 11 q^{92} + 5 q^{93} + 10 q^{94} - 13 q^{95} + 3 q^{96} + 9 q^{97} + 8 q^{98} - 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^5 + 3 * q^6 + q^7 + 3 * q^8 + 3 * q^9 - 3 * q^10 - 3 * q^11 + 3 * q^12 + 5 * q^13 + q^14 - 3 * q^15 + 3 * q^16 + 3 * q^17 + 3 * q^18 + 13 * q^19 - 3 * q^20 + q^21 - 3 * q^22 - 11 * q^23 + 3 * q^24 + 3 * q^25 + 5 * q^26 + 3 * q^27 + q^28 + 6 * q^29 - 3 * q^30 + 5 * q^31 + 3 * q^32 - 3 * q^33 + 3 * q^34 - q^35 + 3 * q^36 + q^37 + 13 * q^38 + 5 * q^39 - 3 * q^40 - 6 * q^41 + q^42 + 6 * q^43 - 3 * q^44 - 3 * q^45 - 11 * q^46 + 10 * q^47 + 3 * q^48 + 8 * q^49 + 3 * q^50 + 3 * q^51 + 5 * q^52 + 12 * q^53 + 3 * q^54 + 3 * q^55 + q^56 + 13 * q^57 + 6 * q^58 + 6 * q^59 - 3 * q^60 + 11 * q^61 + 5 * q^62 + q^63 + 3 * q^64 - 5 * q^65 - 3 * q^66 + 29 * q^67 + 3 * q^68 - 11 * q^69 - q^70 + 4 * q^71 + 3 * q^72 + 10 * q^73 + q^74 + 3 * q^75 + 13 * q^76 - q^77 + 5 * q^78 + 8 * q^79 - 3 * q^80 + 3 * q^81 - 6 * q^82 - 19 * q^83 + q^84 - 3 * q^85 + 6 * q^86 + 6 * q^87 - 3 * q^88 - 2 * q^89 - 3 * q^90 - 27 * q^91 - 11 * q^92 + 5 * q^93 + 10 * q^94 - 13 * q^95 + 3 * q^96 + 9 * q^97 + 8 * q^98 - 3 * 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
$7$	3.63897	1.37540	0.687701	−	0.725994i	$-0.258620\pi$
$7$	3.63897	1.37540	0.687701	+	0.725994i	$0.258620\pi$
$8$	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
$13$	−1.63897	−0.454570	−0.227285	−	0.973828i	$-0.572985\pi$
$13$	−1.63897	−0.454570	−0.227285	+	0.973828i	$0.572985\pi$
$14$	3.63897	0.972557
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	1.00000	0.235702
$19$	7.63897	1.75250	0.876250	−	0.481856i	$-0.160037\pi$
$19$	7.63897	1.75250	0.876250	+	0.481856i	$0.160037\pi$
$20$	−1.00000	−0.223607
$21$	3.63897	0.794089
$22$	−1.00000	−0.213201
$23$	−0.361026	−0.0752792	−0.0376396	−	0.999291i	$-0.511984\pi$
$23$	−0.361026	−0.0752792	−0.0376396	+	0.999291i	$0.511984\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−1.63897	−0.321429
$27$	1.00000	0.192450
$28$	3.63897	0.687701
$29$	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
$31$	−8.52008	−1.53025	−0.765126	−	0.643881i	$-0.777323\pi$
$31$	−8.52008	−1.53025	−0.765126	+	0.643881i	$0.777323\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	1.00000	0.171499
$35$	−3.63897	−0.615099
$36$	1.00000	0.166667
$37$	−3.24213	−0.533003	−0.266501	−	0.963835i	$-0.585868\pi$
$37$	−3.24213	−0.533003	−0.266501	+	0.963835i	$0.585868\pi$
$38$	7.63897	1.23921
$39$	−1.63897	−0.262446
$40$	−1.00000	−0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	3.63897	0.561506
$43$	8.88110	1.35436	0.677178	−	0.735819i	$-0.263203\pi$
$43$	8.88110	1.35436	0.677178	+	0.735819i	$0.263203\pi$
$44$	−1.00000	−0.150756
$45$	−1.00000	−0.149071
$46$	−0.361026	−0.0532304
$47$	10.4843	1.52929	0.764643	−	0.644454i	$-0.222915\pi$
$47$	10.4843	1.52929	0.764643	+	0.644454i	$0.222915\pi$
$48$	1.00000	0.144338
$49$	6.24213	0.891733
$50$	1.00000	0.141421
$51$	1.00000	0.140028
$52$	−1.63897	−0.227285
$53$	−2.88110	−0.395750	−0.197875	−	0.980227i	$-0.563404\pi$
$53$	−2.88110	−0.395750	−0.197875	+	0.980227i	$0.563404\pi$
$54$	1.00000	0.136083
$55$	1.00000	0.134840
$56$	3.63897	0.486278
$57$	7.63897	1.01181
$58$	2.00000	0.262613
$59$	8.88110	1.15622	0.578111	−	0.815958i	$-0.303790\pi$
$59$	8.88110	1.15622	0.578111	+	0.815958i	$0.303790\pi$
$60$	−1.00000	−0.129099
$61$	7.24213	0.927260	0.463630	−	0.886029i	$-0.346547\pi$
$61$	7.24213	0.927260	0.463630	+	0.886029i	$0.346547\pi$
$62$	−8.52008	−1.08205
$63$	3.63897	0.458468
$64$	1.00000	0.125000
$65$	1.63897	0.203290
$66$	−1.00000	−0.123091
$67$	13.2421	1.61778	0.808892	−	0.587958i	$-0.200068\pi$
$67$	13.2421	1.61778	0.808892	+	0.587958i	$0.200068\pi$
$68$	1.00000	0.121268
$69$	−0.361026	−0.0434625
$70$	−3.63897	−0.434941
$71$	1.60316	0.190260	0.0951298	−	0.995465i	$-0.469673\pi$
$71$	1.60316	0.190260	0.0951298	+	0.995465i	$0.469673\pi$
$72$	1.00000	0.117851
$73$	−10.1591	−1.18903	−0.594513	−	0.804086i	$-0.702655\pi$
$73$	−10.1591	−1.18903	−0.594513	+	0.804086i	$0.702655\pi$
$74$	−3.24213	−0.376890
$75$	1.00000	0.115470
$76$	7.63897	0.876250
$77$	−3.63897	−0.414700
$78$	−1.63897	−0.185577
$79$	2.39684	0.269666	0.134833	−	0.990868i	$-0.456950\pi$
$79$	2.39684	0.269666	0.134833	+	0.990868i	$0.456950\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	−16.5201	−1.81331	−0.906657	−	0.421869i	$-0.861374\pi$
$83$	−16.5201	−1.81331	−0.906657	+	0.421869i	$0.861374\pi$
$84$	3.63897	0.397045
$85$	−1.00000	−0.108465
$86$	8.88110	0.957674
$87$	2.00000	0.214423
$88$	−1.00000	−0.106600
$89$	−1.20631	−0.127869	−0.0639344	−	0.997954i	$-0.520365\pi$
$89$	−1.20631	−0.127869	−0.0639344	+	0.997954i	$0.520365\pi$
$90$	−1.00000	−0.105409
$91$	−5.96418	−0.625216
$92$	−0.361026	−0.0376396
$93$	−8.52008	−0.883491
$94$	10.4843	1.08137
$95$	−7.63897	−0.783742
$96$	1.00000	0.102062
$97$	−13.7980	−1.40098	−0.700489	−	0.713664i	$-0.747035\pi$
$97$	−13.7980	−1.40098	−0.700489	+	0.713664i	$0.747035\pi$
$98$	6.24213	0.630550
$99$	−1.00000	−0.100504
$100$	1.00000	0.100000
$101$	−12.5559	−1.24936	−0.624679	−	0.780882i	$-0.714770\pi$
$101$	−12.5559	−1.24936	−0.624679	+	0.780882i	$0.714770\pi$
$102$	1.00000	0.0990148
$103$	10.0358	0.988859	0.494429	−	0.869218i	$-0.335377\pi$
$103$	10.0358	0.988859	0.494429	+	0.869218i	$0.335377\pi$
$104$	−1.63897	−0.160715
$105$	−3.63897	−0.355127
$106$	−2.88110	−0.279837
$107$	−14.4843	−1.40025	−0.700123	−	0.714022i	$-0.746872\pi$
$107$	−14.4843	−1.40025	−0.700123	+	0.714022i	$0.746872\pi$
$108$	1.00000	0.0962250
$109$	6.52008	0.624510	0.312255	−	0.949998i	$-0.398916\pi$
$109$	6.52008	0.624510	0.312255	+	0.949998i	$0.398916\pi$
$110$	1.00000	0.0953463
$111$	−3.24213	−0.307729
$112$	3.63897	0.343851
$113$	10.8811	1.02361	0.511804	−	0.859102i	$-0.328977\pi$
$113$	10.8811	1.02361	0.511804	+	0.859102i	$0.328977\pi$
$114$	7.63897	0.715455
$115$	0.361026	0.0336659
$116$	2.00000	0.185695
$117$	−1.63897	−0.151523
$118$	8.88110	0.817572
$119$	3.63897	0.333584
$120$	−1.00000	−0.0912871
$121$	1.00000	0.0909091
$122$	7.24213	0.655672
$123$	−2.00000	−0.180334
$124$	−8.52008	−0.765126
$125$	−1.00000	−0.0894427
$126$	3.63897	0.324186
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	1.00000	0.0883883
$129$	8.88110	0.781938
$130$	1.63897	0.143748
$131$	−0.520077	−0.0454393	−0.0227197	−	0.999742i	$-0.507233\pi$
$131$	−0.520077	−0.0454393	−0.0227197	+	0.999742i	$0.507233\pi$
$132$	−1.00000	−0.0870388
$133$	27.7980	2.41039
$134$	13.2421	1.14395
$135$	−1.00000	−0.0860663
$136$	1.00000	0.0857493
$137$	4.03582	0.344803	0.172402	−	0.985027i	$-0.444847\pi$
$137$	4.03582	0.344803	0.172402	+	0.985027i	$0.444847\pi$
$138$	−0.361026	−0.0307326
$139$	−11.2779	−0.956583	−0.478292	−	0.878201i	$-0.658744\pi$
$139$	−11.2779	−0.956583	−0.478292	+	0.878201i	$0.658744\pi$
$140$	−3.63897	−0.307549
$141$	10.4843	0.882934
$142$	1.60316	0.134534
$143$	1.63897	0.137058
$144$	1.00000	0.0833333
$145$	−2.00000	−0.166091
$146$	−10.1591	−0.840769
$147$	6.24213	0.514842
$148$	−3.24213	−0.266501
$149$	11.3138	0.926860	0.463430	−	0.886134i	$-0.346619\pi$
$149$	11.3138	0.926860	0.463430	+	0.886134i	$0.346619\pi$
$150$	1.00000	0.0816497
$151$	−2.75787	−0.224432	−0.112216	−	0.993684i	$-0.535795\pi$
$151$	−2.75787	−0.224432	−0.112216	+	0.993684i	$0.535795\pi$
$152$	7.63897	0.619603
$153$	1.00000	0.0808452
$154$	−3.63897	−0.293237
$155$	8.52008	0.684349
$156$	−1.63897	−0.131223
$157$	12.4843	0.996352	0.498176	−	0.867076i	$-0.334003\pi$
$157$	12.4843	0.996352	0.498176	+	0.867076i	$0.334003\pi$
$158$	2.39684	0.190683
$159$	−2.88110	−0.228486
$160$	−1.00000	−0.0790569
$161$	−1.31377	−0.103539
$162$	1.00000	0.0785674
$163$	10.5559	0.826801	0.413401	−	0.910549i	$-0.364341\pi$
$163$	10.5559	0.826801	0.413401	+	0.910549i	$0.364341\pi$
$164$	−2.00000	−0.156174
$165$	1.00000	0.0778499
$166$	−16.5201	−1.28221
$167$	−4.00000	−0.309529	−0.154765	−	0.987951i	$-0.549462\pi$
$167$	−4.00000	−0.309529	−0.154765	+	0.987951i	$0.549462\pi$
$168$	3.63897	0.280753
$169$	−10.3138	−0.793367
$170$	−1.00000	−0.0766965
$171$	7.63897	0.584167
$172$	8.88110	0.677178
$173$	−0.757870	−0.0576198	−0.0288099	−	0.999585i	$-0.509172\pi$
$173$	−0.757870	−0.0576198	−0.0288099	+	0.999585i	$0.509172\pi$
$174$	2.00000	0.151620
$175$	3.63897	0.275081
$176$	−1.00000	−0.0753778
$177$	8.88110	0.667545
$178$	−1.20631	−0.0904169
$179$	−21.4012	−1.59960	−0.799800	−	0.600267i	$-0.795061\pi$
$179$	−21.4012	−1.59960	−0.799800	+	0.600267i	$0.795061\pi$
$180$	−1.00000	−0.0745356
$181$	−5.27795	−0.392307	−0.196153	−	0.980573i	$-0.562845\pi$
$181$	−5.27795	−0.392307	−0.196153	+	0.980573i	$0.562845\pi$
$182$	−5.96418	−0.442095
$183$	7.24213	0.535354
$184$	−0.361026	−0.0266152
$185$	3.24213	0.238366
$186$	−8.52008	−0.624722
$187$	−1.00000	−0.0731272
$188$	10.4843	0.764643
$189$	3.63897	0.264696
$190$	−7.63897	−0.554189
$191$	17.7622	1.28523	0.642614	−	0.766190i	$-0.277850\pi$
$191$	17.7622	1.28523	0.642614	+	0.766190i	$0.277850\pi$
$192$	1.00000	0.0721688
$193$	14.1591	1.01919	0.509595	−	0.860414i	$-0.329795\pi$
$193$	14.1591	1.01919	0.509595	+	0.860414i	$0.329795\pi$
$194$	−13.7980	−0.990640
$195$	1.63897	0.117369
$196$	6.24213	0.445866
$197$	−17.0760	−1.21661	−0.608306	−	0.793702i	$-0.708151\pi$
$197$	−17.0760	−1.21661	−0.608306	+	0.793702i	$0.708151\pi$
$198$	−1.00000	−0.0710669
$199$	20.5917	1.45971	0.729854	−	0.683603i	$-0.239588\pi$
$199$	20.5917	1.45971	0.729854	+	0.683603i	$0.239588\pi$
$200$	1.00000	0.0707107
$201$	13.2421	0.934028
$202$	−12.5559	−0.883430
$203$	7.27795	0.510812
$204$	1.00000	0.0700140
$205$	2.00000	0.139686
$206$	10.0358	0.699229
$207$	−0.361026	−0.0250931
$208$	−1.63897	−0.113642
$209$	−7.63897	−0.528399
$210$	−3.63897	−0.251113
$211$	3.27795	0.225663	0.112832	−	0.993614i	$-0.464008\pi$
$211$	3.27795	0.225663	0.112832	+	0.993614i	$0.464008\pi$
$212$	−2.88110	−0.197875
$213$	1.60316	0.109846
$214$	−14.4843	−0.990124
$215$	−8.88110	−0.605686
$216$	1.00000	0.0680414
$217$	−31.0043	−2.10471
$218$	6.52008	0.441595
$219$	−10.1591	−0.686485
$220$	1.00000	0.0674200
$221$	−1.63897	−0.110249
$222$	−3.24213	−0.217597
$223$	−2.03582	−0.136328	−0.0681642	−	0.997674i	$-0.521714\pi$
$223$	−2.03582	−0.136328	−0.0681642	+	0.997674i	$0.521714\pi$
$224$	3.63897	0.243139
$225$	1.00000	0.0666667
$226$	10.8811	0.723800
$227$	6.48426	0.430375	0.215188	−	0.976573i	$-0.430964\pi$
$227$	6.48426	0.430375	0.215188	+	0.976573i	$0.430964\pi$
$228$	7.63897	0.505903
$229$	19.2421	1.27156	0.635778	−	0.771872i	$-0.280679\pi$
$229$	19.2421	1.27156	0.635778	+	0.771872i	$0.280679\pi$
$230$	0.361026	0.0238054
$231$	−3.63897	−0.239427
$232$	2.00000	0.131306
$233$	5.20631	0.341077	0.170538	−	0.985351i	$-0.445449\pi$
$233$	5.20631	0.341077	0.170538	+	0.985351i	$0.445449\pi$
$234$	−1.63897	−0.107143
$235$	−10.4843	−0.683918
$236$	8.88110	0.578111
$237$	2.39684	0.155692
$238$	3.63897	0.235880
$239$	2.48426	0.160693	0.0803467	−	0.996767i	$-0.474397\pi$
$239$	2.48426	0.160693	0.0803467	+	0.996767i	$0.474397\pi$
$240$	−1.00000	−0.0645497
$241$	−12.1232	−0.780926	−0.390463	−	0.920619i	$-0.627685\pi$
$241$	−12.1232	−0.780926	−0.390463	+	0.920619i	$0.627685\pi$
$242$	1.00000	0.0642824
$243$	1.00000	0.0641500
$244$	7.24213	0.463630
$245$	−6.24213	−0.398795
$246$	−2.00000	−0.127515
$247$	−12.5201	−0.796633
$248$	−8.52008	−0.541025
$249$	−16.5201	−1.04692
$250$	−1.00000	−0.0632456
$251$	−19.9571	−1.25968	−0.629840	−	0.776725i	$-0.716879\pi$
$251$	−19.9571	−1.25968	−0.629840	+	0.776725i	$0.716879\pi$
$252$	3.63897	0.229234
$253$	0.361026	0.0226975
$254$	16.0000	1.00393
$255$	−1.00000	−0.0626224
$256$	1.00000	0.0625000
$257$	−28.5559	−1.78127	−0.890634	−	0.454721i	$-0.849739\pi$
$257$	−28.5559	−1.78127	−0.890634	+	0.454721i	$0.849739\pi$
$258$	8.88110	0.552913
$259$	−11.7980	−0.733094
$260$	1.63897	0.101645
$261$	2.00000	0.123797
$262$	−0.520077	−0.0321305
$263$	−5.96418	−0.367767	−0.183884	−	0.982948i	$-0.558867\pi$
$263$	−5.96418	−0.367767	−0.183884	+	0.982948i	$0.558867\pi$
$264$	−1.00000	−0.0615457
$265$	2.88110	0.176985
$266$	27.7980	1.70441
$267$	−1.20631	−0.0738251
$268$	13.2421	0.808892
$269$	2.44844	0.149284	0.0746421	−	0.997210i	$-0.476219\pi$
$269$	2.44844	0.149284	0.0746421	+	0.997210i	$0.476219\pi$
$270$	−1.00000	−0.0608581
$271$	22.5559	1.37017	0.685087	−	0.728462i	$-0.259764\pi$
$271$	22.5559	1.37017	0.685087	+	0.728462i	$0.259764\pi$
$272$	1.00000	0.0606339
$273$	−5.96418	−0.360969
$274$	4.03582	0.243813
$275$	−1.00000	−0.0603023
$276$	−0.361026	−0.0217312
$277$	−31.0402	−1.86502	−0.932511	−	0.361141i	$-0.882387\pi$
$277$	−31.0402	−1.86502	−0.932511	+	0.361141i	$0.882387\pi$
$278$	−11.2779	−0.676406
$279$	−8.52008	−0.510084
$280$	−3.63897	−0.217470
$281$	−30.3181	−1.80863	−0.904313	−	0.426870i	$-0.859616\pi$
$281$	−30.3181	−1.80863	−0.904313	+	0.426870i	$0.859616\pi$
$282$	10.4843	0.624329
$283$	5.76221	0.342528	0.171264	−	0.985225i	$-0.445215\pi$
$283$	5.76221	0.342528	0.171264	+	0.985225i	$0.445215\pi$
$284$	1.60316	0.0951298
$285$	−7.63897	−0.452494
$286$	1.63897	0.0969145
$287$	−7.27795	−0.429604
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	−2.00000	−0.117444
$291$	−13.7980	−0.808855
$292$	−10.1591	−0.594513
$293$	−3.04015	−0.177608	−0.0888038	−	0.996049i	$-0.528304\pi$
$293$	−3.04015	−0.177608	−0.0888038	+	0.996049i	$0.528304\pi$
$294$	6.24213	0.364048
$295$	−8.88110	−0.517078
$296$	−3.24213	−0.188445
$297$	−1.00000	−0.0580259
$298$	11.3138	0.655389
$299$	0.591713	0.0342196
$300$	1.00000	0.0577350
$301$	32.3181	1.86278
$302$	−2.75787	−0.158698
$303$	−12.5559	−0.721317
$304$	7.63897	0.438125
$305$	−7.24213	−0.414683
$306$	1.00000	0.0571662
$307$	33.1992	1.89478	0.947389	−	0.320083i	$-0.103711\pi$
$307$	33.1992	1.89478	0.947389	+	0.320083i	$0.103711\pi$
$308$	−3.63897	−0.207350
$309$	10.0358	0.570918
$310$	8.52008	0.483908
$311$	30.7149	1.74168	0.870842	−	0.491562i	$-0.163574\pi$
$311$	30.7149	1.74168	0.870842	+	0.491562i	$0.163574\pi$
$312$	−1.63897	−0.0927886
$313$	−25.8697	−1.46224	−0.731120	−	0.682249i	$-0.761002\pi$
$313$	−25.8697	−1.46224	−0.731120	+	0.682249i	$0.761002\pi$
$314$	12.4843	0.704528
$315$	−3.63897	−0.205033
$316$	2.39684	0.134833
$317$	−2.79369	−0.156909	−0.0784546	−	0.996918i	$-0.524999\pi$
$317$	−2.79369	−0.156909	−0.0784546	+	0.996918i	$0.524999\pi$
$318$	−2.88110	−0.161564
$319$	−2.00000	−0.111979
$320$	−1.00000	−0.0559017
$321$	−14.4843	−0.808433
$322$	−1.31377	−0.0732133
$323$	7.63897	0.425044
$324$	1.00000	0.0555556
$325$	−1.63897	−0.0909139
$326$	10.5559	0.584637
$327$	6.52008	0.360561
$328$	−2.00000	−0.110432
$329$	38.1519	2.10338
$330$	1.00000	0.0550482
$331$	18.5559	1.01992	0.509962	−	0.860197i	$-0.329659\pi$
$331$	18.5559	1.01992	0.509962	+	0.860197i	$0.329659\pi$
$332$	−16.5201	−0.906657
$333$	−3.24213	−0.177668
$334$	−4.00000	−0.218870
$335$	−13.2421	−0.723495
$336$	3.63897	0.198522
$337$	−16.7149	−0.910521	−0.455261	−	0.890358i	$-0.650454\pi$
$337$	−16.7149	−0.910521	−0.455261	+	0.890358i	$0.650454\pi$
$338$	−10.3138	−0.560995
$339$	10.8811	0.590980
$340$	−1.00000	−0.0542326
$341$	8.52008	0.461388
$342$	7.63897	0.413068
$343$	−2.75787	−0.148911
$344$	8.88110	0.478837
$345$	0.361026	0.0194370
$346$	−0.757870	−0.0407434
$347$	13.0402	0.700032	0.350016	−	0.936744i	$-0.386176\pi$
$347$	13.0402	0.700032	0.350016	+	0.936744i	$0.386176\pi$
$348$	2.00000	0.107211
$349$	−10.8811	−0.582452	−0.291226	−	0.956654i	$-0.594063\pi$
$349$	−10.8811	−0.582452	−0.291226	+	0.956654i	$0.594063\pi$
$350$	3.63897	0.194511
$351$	−1.63897	−0.0874819
$352$	−1.00000	−0.0533002
$353$	−29.0043	−1.54375	−0.771873	−	0.635777i	$-0.780680\pi$
$353$	−29.0043	−1.54375	−0.771873	+	0.635777i	$0.780680\pi$
$354$	8.88110	0.472025
$355$	−1.60316	−0.0850867
$356$	−1.20631	−0.0639344
$357$	3.63897	0.192595
$358$	−21.4012	−1.13109
$359$	−4.07164	−0.214893	−0.107446	−	0.994211i	$-0.534267\pi$
$359$	−4.07164	−0.214893	−0.107446	+	0.994211i	$0.534267\pi$
$360$	−1.00000	−0.0527046
$361$	39.3539	2.07126
$362$	−5.27795	−0.277403
$363$	1.00000	0.0524864
$364$	−5.96418	−0.312608
$365$	10.1591	0.531749
$366$	7.24213	0.378552
$367$	16.3181	0.851798	0.425899	−	0.904771i	$-0.359958\pi$
$367$	16.3181	0.851798	0.425899	+	0.904771i	$0.359958\pi$
$368$	−0.361026	−0.0188198
$369$	−2.00000	−0.104116
$370$	3.24213	0.168550
$371$	−10.4843	−0.544316
$372$	−8.52008	−0.441745
$373$	−18.0874	−0.936531	−0.468265	−	0.883588i	$-0.655121\pi$
$373$	−18.0874	−0.936531	−0.468265	+	0.883588i	$0.655121\pi$
$374$	−1.00000	−0.0517088
$375$	−1.00000	−0.0516398
$376$	10.4843	0.540684
$377$	−3.27795	−0.168823
$378$	3.63897	0.187169
$379$	4.44844	0.228501	0.114251	−	0.993452i	$-0.463553\pi$
$379$	4.44844	0.228501	0.114251	+	0.993452i	$0.463553\pi$
$380$	−7.63897	−0.391871
$381$	16.0000	0.819705
$382$	17.7622	0.908794
$383$	13.8338	0.706876	0.353438	−	0.935458i	$-0.385012\pi$
$383$	13.8338	0.706876	0.353438	+	0.935458i	$0.385012\pi$
$384$	1.00000	0.0510310
$385$	3.63897	0.185459
$386$	14.1591	0.720677
$387$	8.88110	0.451452
$388$	−13.7980	−0.700489
$389$	−25.3654	−1.28607	−0.643037	−	0.765835i	$-0.722326\pi$
$389$	−25.3654	−1.28607	−0.643037	+	0.765835i	$0.722326\pi$
$390$	1.63897	0.0829927
$391$	−0.361026	−0.0182579
$392$	6.24213	0.315275
$393$	−0.520077	−0.0262344
$394$	−17.0760	−0.860275
$395$	−2.39684	−0.120598
$396$	−1.00000	−0.0502519
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	20.5917	1.03217
$399$	27.7980	1.39164
$400$	1.00000	0.0500000
$401$	−22.3610	−1.11666	−0.558328	−	0.829620i	$-0.688557\pi$
$401$	−22.3610	−1.11666	−0.558328	+	0.829620i	$0.688557\pi$
$402$	13.2421	0.660457
$403$	13.9642	0.695605
$404$	−12.5559	−0.624679
$405$	−1.00000	−0.0496904
$406$	7.27795	0.361198
$407$	3.24213	0.160706
$408$	1.00000	0.0495074
$409$	−4.23779	−0.209545	−0.104773	−	0.994496i	$-0.533412\pi$
$409$	−4.23779	−0.209545	−0.104773	+	0.994496i	$0.533412\pi$
$410$	2.00000	0.0987730
$411$	4.03582	0.199072
$412$	10.0358	0.494429
$413$	32.3181	1.59027
$414$	−0.361026	−0.0177435
$415$	16.5201	0.810939
$416$	−1.63897	−0.0803573
$417$	−11.2779	−0.552283
$418$	−7.63897	−0.373634
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	−3.63897	−0.177564
$421$	−6.52008	−0.317769	−0.158885	−	0.987297i	$-0.550790\pi$
$421$	−6.52008	−0.317769	−0.158885	+	0.987297i	$0.550790\pi$
$422$	3.27795	0.159568
$423$	10.4843	0.509762
$424$	−2.88110	−0.139919
$425$	1.00000	0.0485071
$426$	1.60316	0.0776732
$427$	26.3539	1.27536
$428$	−14.4843	−0.700123
$429$	1.63897	0.0791304
$430$	−8.88110	−0.428285
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	1.00000	0.0481125
$433$	6.79369	0.326484	0.163242	−	0.986586i	$-0.447805\pi$
$433$	6.79369	0.326484	0.163242	+	0.986586i	$0.447805\pi$
$434$	−31.0043	−1.48826
$435$	−2.00000	−0.0958927
$436$	6.52008	0.312255
$437$	−2.75787	−0.131927
$438$	−10.1591	−0.485418
$439$	−28.1591	−1.34396	−0.671979	−	0.740570i	$-0.734556\pi$
$439$	−28.1591	−1.34396	−0.671979	+	0.740570i	$0.734556\pi$
$440$	1.00000	0.0476731
$441$	6.24213	0.297244
$442$	−1.63897	−0.0779580
$443$	−31.3654	−1.49021	−0.745107	−	0.666945i	$-0.767601\pi$
$443$	−31.3654	−1.49021	−0.745107	+	0.666945i	$0.767601\pi$
$444$	−3.24213	−0.153865
$445$	1.20631	0.0571847
$446$	−2.03582	−0.0963988
$447$	11.3138	0.535123
$448$	3.63897	0.171925
$449$	15.4728	0.730207	0.365104	−	0.930967i	$-0.381034\pi$
$449$	15.4728	0.730207	0.365104	+	0.930967i	$0.381034\pi$
$450$	1.00000	0.0471405
$451$	2.00000	0.0941763
$452$	10.8811	0.511804
$453$	−2.75787	−0.129576
$454$	6.48426	0.304321
$455$	5.96418	0.279605
$456$	7.63897	0.357728
$457$	3.31377	0.155011	0.0775057	−	0.996992i	$-0.475304\pi$
$457$	3.31377	0.155011	0.0775057	+	0.996992i	$0.475304\pi$
$458$	19.2421	0.899126
$459$	1.00000	0.0466760
$460$	0.361026	0.0168329
$461$	−14.3181	−0.666860	−0.333430	−	0.942775i	$-0.608206\pi$
$461$	−14.3181	−0.666860	−0.333430	+	0.942775i	$0.608206\pi$
$462$	−3.63897	−0.169300
$463$	−7.00434	−0.325519	−0.162760	−	0.986666i	$-0.552039\pi$
$463$	−7.00434	−0.325519	−0.162760	+	0.986666i	$0.552039\pi$
$464$	2.00000	0.0928477
$465$	8.52008	0.395109
$466$	5.20631	0.241178
$467$	−25.8496	−1.19618	−0.598089	−	0.801430i	$-0.704073\pi$
$467$	−25.8496	−1.19618	−0.598089	+	0.801430i	$0.704073\pi$
$468$	−1.63897	−0.0757616
$469$	48.1878	2.22510
$470$	−10.4843	−0.483603
$471$	12.4843	0.575244
$472$	8.88110	0.408786
$473$	−8.88110	−0.408354
$474$	2.39684	0.110091
$475$	7.63897	0.350500
$476$	3.63897	0.166792
$477$	−2.88110	−0.131917
$478$	2.48426	0.113627
$479$	28.8382	1.31765	0.658825	−	0.752296i	$-0.271054\pi$
$479$	28.8382	1.31765	0.658825	+	0.752296i	$0.271054\pi$
$480$	−1.00000	−0.0456435
$481$	5.31377	0.242287
$482$	−12.1232	−0.552198
$483$	−1.31377	−0.0597784
$484$	1.00000	0.0454545
$485$	13.7980	0.626536
$486$	1.00000	0.0453609
$487$	−16.3181	−0.739444	−0.369722	−	0.929142i	$-0.620547\pi$
$487$	−16.3181	−0.739444	−0.369722	+	0.929142i	$0.620547\pi$
$488$	7.24213	0.327836
$489$	10.5559	0.477354
$490$	−6.24213	−0.281991
$491$	4.07164	0.183750	0.0918752	−	0.995771i	$-0.470714\pi$
$491$	4.07164	0.183750	0.0918752	+	0.995771i	$0.470714\pi$
$492$	−2.00000	−0.0901670
$493$	2.00000	0.0900755
$494$	−12.5201	−0.563305
$495$	1.00000	0.0449467
$496$	−8.52008	−0.382563
$497$	5.83384	0.261684
$498$	−16.5201	−0.740282
$499$	−23.5244	−1.05310	−0.526549	−	0.850145i	$-0.676514\pi$
$499$	−23.5244	−1.05310	−0.526549	+	0.850145i	$0.676514\pi$
$500$	−1.00000	−0.0447214
$501$	−4.00000	−0.178707
$502$	−19.9571	−0.890728
$503$	−7.34958	−0.327702	−0.163851	−	0.986485i	$-0.552392\pi$
$503$	−7.34958	−0.327702	−0.163851	+	0.986485i	$0.552392\pi$
$504$	3.63897	0.162093
$505$	12.5559	0.558730
$506$	0.361026	0.0160496
$507$	−10.3138	−0.458050
$508$	16.0000	0.709885
$509$	−6.88110	−0.305000	−0.152500	−	0.988304i	$-0.548732\pi$
$509$	−6.88110	−0.305000	−0.152500	+	0.988304i	$0.548732\pi$
$510$	−1.00000	−0.0442807
$511$	−36.9685	−1.63539
$512$	1.00000	0.0441942
$513$	7.63897	0.337269
$514$	−28.5559	−1.25955
$515$	−10.0358	−0.442231
$516$	8.88110	0.390969
$517$	−10.4843	−0.461097
$518$	−11.7980	−0.518375
$519$	−0.757870	−0.0332668
$520$	1.63897	0.0718738
$521$	30.7508	1.34722	0.673608	−	0.739089i	$-0.264744\pi$
$521$	30.7508	1.34722	0.673608	+	0.739089i	$0.264744\pi$
$522$	2.00000	0.0875376
$523$	28.0874	1.22818	0.614088	−	0.789237i	$-0.289524\pi$
$523$	28.0874	1.22818	0.614088	+	0.789237i	$0.289524\pi$
$524$	−0.520077	−0.0227197
$525$	3.63897	0.158818
$526$	−5.96418	−0.260051
$527$	−8.52008	−0.371140
$528$	−1.00000	−0.0435194
$529$	−22.8697	−0.994333
$530$	2.88110	0.125147
$531$	8.88110	0.385407
$532$	27.7980	1.20520
$533$	3.27795	0.141984
$534$	−1.20631	−0.0522022
$535$	14.4843	0.626209
$536$	13.2421	0.571973
$537$	−21.4012	−0.923529
$538$	2.44844	0.105560
$539$	−6.24213	−0.268868
$540$	−1.00000	−0.0430331
$541$	−17.3496	−0.745917	−0.372958	−	0.927848i	$-0.621657\pi$
$541$	−17.3496	−0.745917	−0.372958	+	0.927848i	$0.621657\pi$
$542$	22.5559	0.968859
$543$	−5.27795	−0.226498
$544$	1.00000	0.0428746
$545$	−6.52008	−0.279289
$546$	−5.96418	−0.255243
$547$	−24.5201	−1.04840	−0.524201	−	0.851594i	$-0.675636\pi$
$547$	−24.5201	−1.04840	−0.524201	+	0.851594i	$0.675636\pi$
$548$	4.03582	0.172402
$549$	7.24213	0.309087
$550$	−1.00000	−0.0426401
$551$	15.2779	0.650862
$552$	−0.361026	−0.0153663
$553$	8.72205	0.370899
$554$	−31.0402	−1.31877
$555$	3.24213	0.137621
$556$	−11.2779	−0.478292
$557$	8.48426	0.359490	0.179745	−	0.983713i	$-0.442473\pi$
$557$	8.48426	0.359490	0.179745	+	0.983713i	$0.442473\pi$
$558$	−8.52008	−0.360684
$559$	−14.5559	−0.615649
$560$	−3.63897	−0.153775
$561$	−1.00000	−0.0422200
$562$	−30.3181	−1.27889
$563$	−9.56023	−0.402916	−0.201458	−	0.979497i	$-0.564568\pi$
$563$	−9.56023	−0.402916	−0.201458	+	0.979497i	$0.564568\pi$
$564$	10.4843	0.441467
$565$	−10.8811	−0.457771
$566$	5.76221	0.242204
$567$	3.63897	0.152823
$568$	1.60316	0.0672669
$569$	23.5602	0.987696	0.493848	−	0.869548i	$-0.335590\pi$
$569$	23.5602	0.987696	0.493848	+	0.869548i	$0.335590\pi$
$570$	−7.63897	−0.319961
$571$	30.8024	1.28904	0.644519	−	0.764588i	$-0.277058\pi$
$571$	30.8024	1.28904	0.644519	+	0.764588i	$0.277058\pi$
$572$	1.63897	0.0685289
$573$	17.7622	0.742027
$574$	−7.27795	−0.303776
$575$	−0.361026	−0.0150558
$576$	1.00000	0.0416667
$577$	−33.3496	−1.38836	−0.694181	−	0.719801i	$-0.744233\pi$
$577$	−33.3496	−1.38836	−0.694181	+	0.719801i	$0.744233\pi$
$578$	1.00000	0.0415945
$579$	14.1591	0.588430
$580$	−2.00000	−0.0830455
$581$	−60.1161	−2.49404
$582$	−13.7980	−0.571947
$583$	2.88110	0.119323
$584$	−10.1591	−0.420385
$585$	1.63897	0.0677632
$586$	−3.04015	−0.125588
$587$	−26.3968	−1.08951	−0.544757	−	0.838594i	$-0.683378\pi$
$587$	−26.3968	−1.08951	−0.544757	+	0.838594i	$0.683378\pi$
$588$	6.24213	0.257421
$589$	−65.0846	−2.68177
$590$	−8.88110	−0.365629
$591$	−17.0760	−0.702412
$592$	−3.24213	−0.133251
$593$	5.20631	0.213798	0.106899	−	0.994270i	$-0.465908\pi$
$593$	5.20631	0.213798	0.106899	+	0.994270i	$0.465908\pi$
$594$	−1.00000	−0.0410305
$595$	−3.63897	−0.149183
$596$	11.3138	0.463430
$597$	20.5917	0.842763
$598$	0.591713	0.0241969
$599$	1.31377	0.0536790	0.0268395	−	0.999640i	$-0.491456\pi$
$599$	1.31377	0.0536790	0.0268395	+	0.999640i	$0.491456\pi$
$600$	1.00000	0.0408248
$601$	−18.3610	−0.748962	−0.374481	−	0.927235i	$-0.622179\pi$
$601$	−18.3610	−0.748962	−0.374481	+	0.927235i	$0.622179\pi$
$602$	32.3181	1.31719
$603$	13.2421	0.539261
$604$	−2.75787	−0.112216
$605$	−1.00000	−0.0406558
$606$	−12.5559	−0.510048
$607$	39.1634	1.58959	0.794796	−	0.606876i	$-0.207578\pi$
$607$	39.1634	1.58959	0.794796	+	0.606876i	$0.207578\pi$
$608$	7.63897	0.309801
$609$	7.27795	0.294917
$610$	−7.24213	−0.293225
$611$	−17.1834	−0.695167
$612$	1.00000	0.0404226
$613$	26.8811	1.08572	0.542859	−	0.839824i	$-0.317342\pi$
$613$	26.8811	1.08572	0.542859	+	0.839824i	$0.317342\pi$
$614$	33.1992	1.33981
$615$	2.00000	0.0806478
$616$	−3.63897	−0.146618
$617$	−3.35669	−0.135135	−0.0675676	−	0.997715i	$-0.521524\pi$
$617$	−3.35669	−0.135135	−0.0675676	+	0.997715i	$0.521524\pi$
$618$	10.0358	0.403700
$619$	−14.0358	−0.564147	−0.282074	−	0.959393i	$-0.591022\pi$
$619$	−14.0358	−0.564147	−0.282074	+	0.959393i	$0.591022\pi$
$620$	8.52008	0.342175
$621$	−0.361026	−0.0144875
$622$	30.7149	1.23156
$623$	−4.38974	−0.175871
$624$	−1.63897	−0.0656115
$625$	1.00000	0.0400000
$626$	−25.8697	−1.03396
$627$	−7.63897	−0.305071
$628$	12.4843	0.498176
$629$	−3.24213	−0.129272
$630$	−3.63897	−0.144980
$631$	−5.51574	−0.219578	−0.109789	−	0.993955i	$-0.535018\pi$
$631$	−5.51574	−0.219578	−0.109789	+	0.993955i	$0.535018\pi$
$632$	2.39684	0.0953413
$633$	3.27795	0.130287
$634$	−2.79369	−0.110952
$635$	−16.0000	−0.634941
$636$	−2.88110	−0.114243
$637$	−10.2307	−0.405355
$638$	−2.00000	−0.0791808
$639$	1.60316	0.0634199
$640$	−1.00000	−0.0395285
$641$	39.1992	1.54828	0.774138	−	0.633017i	$-0.218184\pi$
$641$	39.1992	1.54828	0.774138	+	0.633017i	$0.218184\pi$
$642$	−14.4843	−0.571648
$643$	−18.5559	−0.731773	−0.365887	−	0.930659i	$-0.619234\pi$
$643$	−18.5559	−0.731773	−0.365887	+	0.930659i	$0.619234\pi$
$644$	−1.31377	−0.0517696
$645$	−8.88110	−0.349693
$646$	7.63897	0.300551
$647$	−11.9284	−0.468952	−0.234476	−	0.972122i	$-0.575337\pi$
$647$	−11.9284	−0.468952	−0.234476	+	0.972122i	$0.575337\pi$
$648$	1.00000	0.0392837
$649$	−8.88110	−0.348614
$650$	−1.63897	−0.0642858
$651$	−31.0043	−1.21516
$652$	10.5559	0.413401
$653$	11.0402	0.432035	0.216017	−	0.976390i	$-0.430693\pi$
$653$	11.0402	0.432035	0.216017	+	0.976390i	$0.430693\pi$
$654$	6.52008	0.254955
$655$	0.520077	0.0203211
$656$	−2.00000	−0.0780869
$657$	−10.1591	−0.396342
$658$	38.1519	1.48732
$659$	−6.95985	−0.271117	−0.135559	−	0.990769i	$-0.543283\pi$
$659$	−6.95985	−0.271117	−0.135559	+	0.990769i	$0.543283\pi$
$660$	1.00000	0.0389249
$661$	13.7264	0.533895	0.266947	−	0.963711i	$-0.413985\pi$
$661$	13.7264	0.533895	0.266947	+	0.963711i	$0.413985\pi$
$662$	18.5559	0.721195
$663$	−1.63897	−0.0636525
$664$	−16.5201	−0.641103
$665$	−27.7980	−1.07796
$666$	−3.24213	−0.125630
$667$	−0.722053	−0.0279580
$668$	−4.00000	−0.154765
$669$	−2.03582	−0.0787093
$670$	−13.2421	−0.511588
$671$	−7.24213	−0.279579
$672$	3.63897	0.140376
$673$	−19.9213	−0.767908	−0.383954	−	0.923352i	$-0.625438\pi$
$673$	−19.9213	−0.767908	−0.383954	+	0.923352i	$0.625438\pi$
$674$	−16.7149	−0.643836
$675$	1.00000	0.0384900
$676$	−10.3138	−0.396683
$677$	34.6362	1.33118	0.665589	−	0.746319i	$-0.268181\pi$
$677$	34.6362	1.33118	0.665589	+	0.746319i	$0.268181\pi$
$678$	10.8811	0.417886
$679$	−50.2106	−1.92691
$680$	−1.00000	−0.0383482
$681$	6.48426	0.248477
$682$	8.52008	0.326251
$683$	−0.924028	−0.0353570	−0.0176785	−	0.999844i	$-0.505628\pi$
$683$	−0.924028	−0.0353570	−0.0176785	+	0.999844i	$0.505628\pi$
$684$	7.63897	0.292083
$685$	−4.03582	−0.154201
$686$	−2.75787	−0.105296
$687$	19.2421	0.734133
$688$	8.88110	0.338589
$689$	4.72205	0.179896
$690$	0.361026	0.0137440
$691$	45.6319	1.73592	0.867959	−	0.496636i	$-0.165432\pi$
$691$	45.6319	1.73592	0.867959	+	0.496636i	$0.165432\pi$
$692$	−0.757870	−0.0288099
$693$	−3.63897	−0.138233
$694$	13.0402	0.494997
$695$	11.2779	0.427797
$696$	2.00000	0.0758098
$697$	−2.00000	−0.0757554
$698$	−10.8811	−0.411856
$699$	5.20631	0.196921
$700$	3.63897	0.137540
$701$	34.3181	1.29618	0.648088	−	0.761565i	$-0.275569\pi$
$701$	34.3181	1.29618	0.648088	+	0.761565i	$0.275569\pi$
$702$	−1.63897	−0.0618591
$703$	−24.7665	−0.934088
$704$	−1.00000	−0.0376889
$705$	−10.4843	−0.394860
$706$	−29.0043	−1.09159
$707$	−45.6906	−1.71837
$708$	8.88110	0.333772
$709$	−42.9685	−1.61372	−0.806858	−	0.590745i	$-0.798834\pi$
$709$	−42.9685	−1.61372	−0.806858	+	0.590745i	$0.798834\pi$
$710$	−1.60316	−0.0601654
$711$	2.39684	0.0898887
$712$	−1.20631	−0.0452084
$713$	3.07597	0.115196
$714$	3.63897	0.136185
$715$	−1.63897	−0.0612941
$716$	−21.4012	−0.799800
$717$	2.48426	0.0927764
$718$	−4.07164	−0.151952
$719$	−22.7149	−0.847125	−0.423562	−	0.905867i	$-0.639221\pi$
$719$	−22.7149	−0.847125	−0.423562	+	0.905867i	$0.639221\pi$
$720$	−1.00000	−0.0372678
$721$	36.5201	1.36008
$722$	39.3539	1.46460
$723$	−12.1232	−0.450868
$724$	−5.27795	−0.196153
$725$	2.00000	0.0742781
$726$	1.00000	0.0371135
$727$	−11.5244	−0.427417	−0.213708	−	0.976897i	$-0.568554\pi$
$727$	−11.5244	−0.427417	−0.213708	+	0.976897i	$0.568554\pi$
$728$	−5.96418	−0.221047
$729$	1.00000	0.0370370
$730$	10.1591	0.376003
$731$	8.88110	0.328479
$732$	7.24213	0.267677
$733$	−52.7594	−1.94871	−0.974357	−	0.225007i	$-0.927759\pi$
$733$	−52.7594	−1.94871	−0.974357	+	0.225007i	$0.927759\pi$
$734$	16.3181	0.602312
$735$	−6.24213	−0.230244
$736$	−0.361026	−0.0133076
$737$	−13.2421	−0.487780
$738$	−2.00000	−0.0736210
$739$	−25.4012	−0.934398	−0.467199	−	0.884152i	$-0.654737\pi$
$739$	−25.4012	−0.934398	−0.467199	+	0.884152i	$0.654737\pi$
$740$	3.24213	0.119183
$741$	−12.5201	−0.459937
$742$	−10.4843	−0.384889
$743$	8.79369	0.322609	0.161305	−	0.986905i	$-0.448430\pi$
$743$	8.79369	0.322609	0.161305	+	0.986905i	$0.448430\pi$
$744$	−8.52008	−0.312361
$745$	−11.3138	−0.414504
$746$	−18.0874	−0.662227
$747$	−16.5201	−0.604438
$748$	−1.00000	−0.0365636
$749$	−52.7078	−1.92590
$750$	−1.00000	−0.0365148
$751$	−20.3181	−0.741418	−0.370709	−	0.928749i	$-0.620885\pi$
$751$	−20.3181	−0.741418	−0.370709	+	0.928749i	$0.620885\pi$
$752$	10.4843	0.382322
$753$	−19.9571	−0.727276
$754$	−3.27795	−0.119376
$755$	2.75787	0.100369
$756$	3.63897	0.132348
$757$	−33.3496	−1.21211	−0.606056	−	0.795422i	$-0.707249\pi$
$757$	−33.3496	−1.21211	−0.606056	+	0.795422i	$0.707249\pi$
$758$	4.44844	0.161575
$759$	0.361026	0.0131044
$760$	−7.63897	−0.277095
$761$	−13.7264	−0.497581	−0.248791	−	0.968557i	$-0.580033\pi$
$761$	−13.7264	−0.497581	−0.248791	+	0.968557i	$0.580033\pi$
$762$	16.0000	0.579619
$763$	23.7264	0.858953
$764$	17.7622	0.642614
$765$	−1.00000	−0.0361551
$766$	13.8338	0.499837
$767$	−14.5559	−0.525583
$768$	1.00000	0.0360844
$769$	28.4843	1.02717	0.513584	−	0.858039i	$-0.328317\pi$
$769$	28.4843	1.02717	0.513584	+	0.858039i	$0.328317\pi$
$770$	3.63897	0.131140
$771$	−28.5559	−1.02842
$772$	14.1591	0.509595
$773$	27.4012	0.985552	0.492776	−	0.870156i	$-0.335982\pi$
$773$	27.4012	0.985552	0.492776	+	0.870156i	$0.335982\pi$
$774$	8.88110	0.319225
$775$	−8.52008	−0.306050
$776$	−13.7980	−0.495320
$777$	−11.7980	−0.423252
$778$	−25.3654	−0.909392
$779$	−15.2779	−0.547389
$780$	1.63897	0.0586847
$781$	−1.60316	−0.0573654
$782$	−0.361026	−0.0129103
$783$	2.00000	0.0714742
$784$	6.24213	0.222933
$785$	−12.4843	−0.445582
$786$	−0.520077	−0.0185505
$787$	9.96418	0.355185	0.177592	−	0.984104i	$-0.443169\pi$
$787$	9.96418	0.355185	0.177592	+	0.984104i	$0.443169\pi$
$788$	−17.0760	−0.608306
$789$	−5.96418	−0.212331
$790$	−2.39684	−0.0852759
$791$	39.5960	1.40787
$792$	−1.00000	−0.0355335
$793$	−11.8697	−0.421504
$794$	2.00000	0.0709773
$795$	2.88110	0.102182
$796$	20.5917	0.729854
$797$	12.1232	0.429427	0.214713	−	0.976677i	$-0.431118\pi$
$797$	12.1232	0.429427	0.214713	+	0.976677i	$0.431118\pi$
$798$	27.7980	0.984039
$799$	10.4843	0.370906
$800$	1.00000	0.0353553
$801$	−1.20631	−0.0426229
$802$	−22.3610	−0.789595
$803$	10.1591	0.358505
$804$	13.2421	0.467014
$805$	1.31377	0.0463041
$806$	13.9642	0.491867
$807$	2.44844	0.0861892
$808$	−12.5559	−0.441715
$809$	−19.7622	−0.694802	−0.347401	−	0.937717i	$-0.612936\pi$
$809$	−19.7622	−0.694802	−0.347401	+	0.937717i	$0.612936\pi$
$810$	−1.00000	−0.0351364
$811$	−35.5960	−1.24995	−0.624973	−	0.780646i	$-0.714890\pi$
$811$	−35.5960	−1.24995	−0.624973	+	0.780646i	$0.714890\pi$
$812$	7.27795	0.255406
$813$	22.5559	0.791070
$814$	3.24213	0.113637
$815$	−10.5559	−0.369757
$816$	1.00000	0.0350070
$817$	67.8425	2.37351
$818$	−4.23779	−0.148171
$819$	−5.96418	−0.208405
$820$	2.00000	0.0698430
$821$	−36.8740	−1.28691	−0.643456	−	0.765483i	$-0.722500\pi$
$821$	−36.8740	−1.28691	−0.643456	+	0.765483i	$0.722500\pi$
$822$	4.03582	0.140765
$823$	−19.9284	−0.694659	−0.347330	−	0.937743i	$-0.612911\pi$
$823$	−19.9284	−0.694659	−0.347330	+	0.937743i	$0.612911\pi$
$824$	10.0358	0.349614
$825$	−1.00000	−0.0348155
$826$	32.3181	1.12449
$827$	25.5157	0.887269	0.443635	−	0.896208i	$-0.353689\pi$
$827$	25.5157	0.887269	0.443635	+	0.896208i	$0.353689\pi$
$828$	−0.361026	−0.0125465
$829$	−53.2192	−1.84838	−0.924190	−	0.381932i	$-0.875259\pi$
$829$	−53.2192	−1.84838	−0.924190	+	0.381932i	$0.875259\pi$
$830$	16.5201	0.573420
$831$	−31.0402	−1.07677
$832$	−1.63897	−0.0568212
$833$	6.24213	0.216277
$834$	−11.2779	−0.390523
$835$	4.00000	0.138426
$836$	−7.63897	−0.264199
$837$	−8.52008	−0.294497
$838$	−4.00000	−0.138178
$839$	−29.9929	−1.03547	−0.517735	−	0.855541i	$-0.673225\pi$
$839$	−29.9929	−1.03547	−0.517735	+	0.855541i	$0.673225\pi$
$840$	−3.63897	−0.125557
$841$	−25.0000	−0.862069
$842$	−6.52008	−0.224697
$843$	−30.3181	−1.04421
$844$	3.27795	0.112832
$845$	10.3138	0.354804
$846$	10.4843	0.360456
$847$	3.63897	0.125037
$848$	−2.88110	−0.0989375
$849$	5.76221	0.197758
$850$	1.00000	0.0342997
$851$	1.17049	0.0401240
$852$	1.60316	0.0549232
$853$	25.4528	0.871487	0.435743	−	0.900071i	$-0.356486\pi$
$853$	25.4528	0.871487	0.435743	+	0.900071i	$0.356486\pi$
$854$	26.3539	0.901813
$855$	−7.63897	−0.261247
$856$	−14.4843	−0.495062
$857$	−31.4886	−1.07563	−0.537815	−	0.843063i	$-0.680750\pi$
$857$	−31.4886	−1.07563	−0.537815	+	0.843063i	$0.680750\pi$
$858$	1.63897	0.0559536
$859$	22.0803	0.753370	0.376685	−	0.926341i	$-0.377064\pi$
$859$	22.0803	0.753370	0.376685	+	0.926341i	$0.377064\pi$
$860$	−8.88110	−0.302843
$861$	−7.27795	−0.248032
$862$	−8.00000	−0.272481
$863$	22.8740	0.778640	0.389320	−	0.921103i	$-0.372710\pi$
$863$	22.8740	0.778640	0.389320	+	0.921103i	$0.372710\pi$
$864$	1.00000	0.0340207
$865$	0.757870	0.0257684
$866$	6.79369	0.230859
$867$	1.00000	0.0339618
$868$	−31.0043	−1.05236
$869$	−2.39684	−0.0813074
$870$	−2.00000	−0.0678064
$871$	−21.7035	−0.735395
$872$	6.52008	0.220798
$873$	−13.7980	−0.466992
$874$	−2.75787	−0.0932864
$875$	−3.63897	−0.123020
$876$	−10.1591	−0.343243
$877$	6.07164	0.205025	0.102512	−	0.994732i	$-0.467312\pi$
$877$	6.07164	0.205025	0.102512	+	0.994732i	$0.467312\pi$
$878$	−28.1591	−0.950322
$879$	−3.04015	−0.102542
$880$	1.00000	0.0337100
$881$	−52.6433	−1.77360	−0.886799	−	0.462155i	$-0.847076\pi$
$881$	−52.6433	−1.77360	−0.886799	+	0.462155i	$0.847076\pi$
$882$	6.24213	0.210183
$883$	−28.9685	−0.974868	−0.487434	−	0.873160i	$-0.662067\pi$
$883$	−28.9685	−0.974868	−0.487434	+	0.873160i	$0.662067\pi$
$884$	−1.63897	−0.0551247
$885$	−8.88110	−0.298535
$886$	−31.3654	−1.05374
$887$	−40.9685	−1.37559	−0.687794	−	0.725906i	$-0.741421\pi$
$887$	−40.9685	−1.37559	−0.687794	+	0.725906i	$0.741421\pi$
$888$	−3.24213	−0.108799
$889$	58.2236	1.95276
$890$	1.20631	0.0404357
$891$	−1.00000	−0.0335013
$892$	−2.03582	−0.0681642
$893$	80.0890	2.68008
$894$	11.3138	0.378389
$895$	21.4012	0.715363
$896$	3.63897	0.121570
$897$	0.591713	0.0197567
$898$	15.4728	0.516335
$899$	−17.0402	−0.568321
$900$	1.00000	0.0333333
$901$	−2.88110	−0.0959835
$902$	2.00000	0.0665927
$903$	32.3181	1.07548
$904$	10.8811	0.361900
$905$	5.27795	0.175445
$906$	−2.75787	−0.0916241
$907$	0.968518	0.0321591	0.0160796	−	0.999871i	$-0.494881\pi$
$907$	0.968518	0.0321591	0.0160796	+	0.999871i	$0.494881\pi$
$908$	6.48426	0.215188
$909$	−12.5559	−0.416453
$910$	5.96418	0.197711
$911$	4.08742	0.135422	0.0677111	−	0.997705i	$-0.478430\pi$
$911$	4.08742	0.135422	0.0677111	+	0.997705i	$0.478430\pi$
$912$	7.63897	0.252952
$913$	16.5201	0.546735
$914$	3.31377	0.109610
$915$	−7.24213	−0.239417
$916$	19.2421	0.635778
$917$	−1.89255	−0.0624974
$918$	1.00000	0.0330049
$919$	−33.3138	−1.09892	−0.549460	−	0.835520i	$-0.685166\pi$
$919$	−33.3138	−1.09892	−0.549460	+	0.835520i	$0.685166\pi$
$920$	0.361026	0.0119027
$921$	33.1992	1.09395
$922$	−14.3181	−0.471542
$923$	−2.62753	−0.0864862
$924$	−3.63897	−0.119713
$925$	−3.24213	−0.106601
$926$	−7.00434	−0.230177
$927$	10.0358	0.329620
$928$	2.00000	0.0656532
$929$	31.3295	1.02789	0.513945	−	0.857823i	$-0.328184\pi$
$929$	31.3295	1.02789	0.513945	+	0.857823i	$0.328184\pi$
$930$	8.52008	0.279384
$931$	47.6835	1.56276
$932$	5.20631	0.170538
$933$	30.7149	1.00556
$934$	−25.8496	−0.845825
$935$	1.00000	0.0327035
$936$	−1.63897	−0.0535715
$937$	9.72639	0.317747	0.158874	−	0.987299i	$-0.449214\pi$
$937$	9.72639	0.317747	0.158874	+	0.987299i	$0.449214\pi$
$938$	48.1878	1.57339
$939$	−25.8697	−0.844224
$940$	−10.4843	−0.341959
$941$	38.3897	1.25147	0.625735	−	0.780036i	$-0.284799\pi$
$941$	38.3897	1.25147	0.625735	+	0.780036i	$0.284799\pi$
$942$	12.4843	0.406759
$943$	0.722053	0.0235133
$944$	8.88110	0.289055
$945$	−3.63897	−0.118376
$946$	−8.88110	−0.288750
$947$	−23.3496	−0.758759	−0.379380	−	0.925241i	$-0.623863\pi$
$947$	−23.3496	−0.758759	−0.379380	+	0.925241i	$0.623863\pi$
$948$	2.39684	0.0778459
$949$	16.6504	0.540495
$950$	7.63897	0.247841
$951$	−2.79369	−0.0905915
$952$	3.63897	0.117940
$953$	31.8338	1.03120	0.515600	−	0.856830i	$-0.327569\pi$
$953$	31.8338	1.03120	0.515600	+	0.856830i	$0.327569\pi$
$954$	−2.88110	−0.0932792
$955$	−17.7622	−0.574772
$956$	2.48426	0.0803467
$957$	−2.00000	−0.0646508
$958$	28.8382	0.931719
$959$	14.6862	0.474243
$960$	−1.00000	−0.0322749
$961$	41.5917	1.34167
$962$	5.31377	0.171323
$963$	−14.4843	−0.466749
$964$	−12.1232	−0.390463
$965$	−14.1591	−0.455796
$966$	−1.31377	−0.0422697
$967$	28.0716	0.902723	0.451361	−	0.892341i	$-0.350939\pi$
$967$	28.0716	0.902723	0.451361	+	0.892341i	$0.350939\pi$
$968$	1.00000	0.0321412
$969$	7.63897	0.245399
$970$	13.7980	0.443028
$971$	−2.74209	−0.0879979	−0.0439989	−	0.999032i	$-0.514010\pi$
$971$	−2.74209	−0.0879979	−0.0439989	+	0.999032i	$0.514010\pi$
$972$	1.00000	0.0320750
$973$	−41.0402	−1.31569
$974$	−16.3181	−0.522866
$975$	−1.63897	−0.0524892
$976$	7.24213	0.231815
$977$	−27.5602	−0.881730	−0.440865	−	0.897573i	$-0.645328\pi$
$977$	−27.5602	−0.881730	−0.440865	+	0.897573i	$0.645328\pi$
$978$	10.5559	0.337540
$979$	1.20631	0.0385539
$980$	−6.24213	−0.199398
$981$	6.52008	0.208170
$982$	4.07164	0.129931
$983$	−25.8496	−0.824475	−0.412237	−	0.911076i	$-0.635253\pi$
$983$	−25.8496	−0.824475	−0.412237	+	0.911076i	$0.635253\pi$
$984$	−2.00000	−0.0637577
$985$	17.0760	0.544086
$986$	2.00000	0.0636930
$987$	38.1519	1.21439
$988$	−12.5201	−0.398317
$989$	−3.20631	−0.101955
$990$	1.00000	0.0317821
$991$	21.3138	0.677054	0.338527	−	0.940957i	$-0.390071\pi$
$991$	21.3138	0.677054	0.338527	+	0.940957i	$0.390071\pi$
$992$	−8.52008	−0.270513
$993$	18.5559	0.588854
$994$	5.83384	0.185038
$995$	−20.5917	−0.652801
$996$	−16.5201	−0.523459
$997$	8.23779	0.260894	0.130447	−	0.991455i	$-0.458359\pi$
$997$	8.23779	0.260894	0.130447	+	0.991455i	$0.458359\pi$
$998$	−23.5244	−0.744652
$999$	−3.24213	−0.102576

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5610.2.a.cd.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5610.2.a.cd.1.3	✓	3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +3.63897 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +3.63897 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -1.63897 q^{13} +3.63897 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +7.63897 q^{19} -1.00000 q^{20} +3.63897 q^{21} -1.00000 q^{22} -0.361026 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.63897 q^{26} +1.00000 q^{27} +3.63897 q^{28} +2.00000 q^{29} -1.00000 q^{30} -8.52008 q^{31} +1.00000 q^{32} -1.00000 q^{33} +1.00000 q^{34} -3.63897 q^{35} +1.00000 q^{36} -3.24213 q^{37} +7.63897 q^{38} -1.63897 q^{39} -1.00000 q^{40} -2.00000 q^{41} +3.63897 q^{42} +8.88110 q^{43} -1.00000 q^{44} -1.00000 q^{45} -0.361026 q^{46} +10.4843 q^{47} +1.00000 q^{48} +6.24213 q^{49} +1.00000 q^{50} +1.00000 q^{51} -1.63897 q^{52} -2.88110 q^{53} +1.00000 q^{54} +1.00000 q^{55} +3.63897 q^{56} +7.63897 q^{57} +2.00000 q^{58} +8.88110 q^{59} -1.00000 q^{60} +7.24213 q^{61} -8.52008 q^{62} +3.63897 q^{63} +1.00000 q^{64} +1.63897 q^{65} -1.00000 q^{66} +13.2421 q^{67} +1.00000 q^{68} -0.361026 q^{69} -3.63897 q^{70} +1.60316 q^{71} +1.00000 q^{72} -10.1591 q^{73} -3.24213 q^{74} +1.00000 q^{75} +7.63897 q^{76} -3.63897 q^{77} -1.63897 q^{78} +2.39684 q^{79} -1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -16.5201 q^{83} +3.63897 q^{84} -1.00000 q^{85} +8.88110 q^{86} +2.00000 q^{87} -1.00000 q^{88} -1.20631 q^{89} -1.00000 q^{90} -5.96418 q^{91} -0.361026 q^{92} -8.52008 q^{93} +10.4843 q^{94} -7.63897 q^{95} +1.00000 q^{96} -13.7980 q^{97} +6.24213 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} + q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^5 + 3 * q^6 + q^7 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} + q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} - 3 q^{11} + 3 q^{12} + 5 q^{13} + q^{14} - 3 q^{15} + 3 q^{16} + 3 q^{17} + 3 q^{18} + 13 q^{19} - 3 q^{20} + q^{21} - 3 q^{22} - 11 q^{23} + 3 q^{24} + 3 q^{25} + 5 q^{26} + 3 q^{27} + q^{28} + 6 q^{29} - 3 q^{30} + 5 q^{31} + 3 q^{32} - 3 q^{33} + 3 q^{34} - q^{35} + 3 q^{36} + q^{37} + 13 q^{38} + 5 q^{39} - 3 q^{40} - 6 q^{41} + q^{42} + 6 q^{43} - 3 q^{44} - 3 q^{45} - 11 q^{46} + 10 q^{47} + 3 q^{48} + 8 q^{49} + 3 q^{50} + 3 q^{51} + 5 q^{52} + 12 q^{53} + 3 q^{54} + 3 q^{55} + q^{56} + 13 q^{57} + 6 q^{58} + 6 q^{59} - 3 q^{60} + 11 q^{61} + 5 q^{62} + q^{63} + 3 q^{64} - 5 q^{65} - 3 q^{66} + 29 q^{67} + 3 q^{68} - 11 q^{69} - q^{70} + 4 q^{71} + 3 q^{72} + 10 q^{73} + q^{74} + 3 q^{75} + 13 q^{76} - q^{77} + 5 q^{78} + 8 q^{79} - 3 q^{80} + 3 q^{81} - 6 q^{82} - 19 q^{83} + q^{84} - 3 q^{85} + 6 q^{86} + 6 q^{87} - 3 q^{88} - 2 q^{89} - 3 q^{90} - 27 q^{91} - 11 q^{92} + 5 q^{93} + 10 q^{94} - 13 q^{95} + 3 q^{96} + 9 q^{97} + 8 q^{98} - 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^5 + 3 * q^6 + q^7 + 3 * q^8 + 3 * q^9 - 3 * q^10 - 3 * q^11 + 3 * q^12 + 5 * q^13 + q^14 - 3 * q^15 + 3 * q^16 + 3 * q^17 + 3 * q^18 + 13 * q^19 - 3 * q^20 + q^21 - 3 * q^22 - 11 * q^23 + 3 * q^24 + 3 * q^25 + 5 * q^26 + 3 * q^27 + q^28 + 6 * q^29 - 3 * q^30 + 5 * q^31 + 3 * q^32 - 3 * q^33 + 3 * q^34 - q^35 + 3 * q^36 + q^37 + 13 * q^38 + 5 * q^39 - 3 * q^40 - 6 * q^41 + q^42 + 6 * q^43 - 3 * q^44 - 3 * q^45 - 11 * q^46 + 10 * q^47 + 3 * q^48 + 8 * q^49 + 3 * q^50 + 3 * q^51 + 5 * q^52 + 12 * q^53 + 3 * q^54 + 3 * q^55 + q^56 + 13 * q^57 + 6 * q^58 + 6 * q^59 - 3 * q^60 + 11 * q^61 + 5 * q^62 + q^63 + 3 * q^64 - 5 * q^65 - 3 * q^66 + 29 * q^67 + 3 * q^68 - 11 * q^69 - q^70 + 4 * q^71 + 3 * q^72 + 10 * q^73 + q^74 + 3 * q^75 + 13 * q^76 - q^77 + 5 * q^78 + 8 * q^79 - 3 * q^80 + 3 * q^81 - 6 * q^82 - 19 * q^83 + q^84 - 3 * q^85 + 6 * q^86 + 6 * q^87 - 3 * q^88 - 2 * q^89 - 3 * q^90 - 27 * q^91 - 11 * q^92 + 5 * q^93 + 10 * q^94 - 13 * q^95 + 3 * q^96 + 9 * q^97 + 8 * q^98 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more