LMFDB - Embedded newform 5610.2.a.bs.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.30278$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.60555 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.60555 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +0.605551 q^{13} +2.60555 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +0.605551 q^{19} -1.00000 q^{20} -2.60555 q^{21} +1.00000 q^{22} -4.60555 q^{23} -1.00000 q^{24} +1.00000 q^{25} -0.605551 q^{26} +1.00000 q^{27} -2.60555 q^{28} +6.00000 q^{29} +1.00000 q^{30} +5.21110 q^{31} -1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +2.60555 q^{35} +1.00000 q^{36} -7.21110 q^{37} -0.605551 q^{38} +0.605551 q^{39} +1.00000 q^{40} +9.21110 q^{41} +2.60555 q^{42} -5.39445 q^{43} -1.00000 q^{44} -1.00000 q^{45} +4.60555 q^{46} +12.0000 q^{47} +1.00000 q^{48} -0.211103 q^{49} -1.00000 q^{50} +1.00000 q^{51} +0.605551 q^{52} -1.39445 q^{53} -1.00000 q^{54} +1.00000 q^{55} +2.60555 q^{56} +0.605551 q^{57} -6.00000 q^{58} -10.6056 q^{59} -1.00000 q^{60} +2.00000 q^{61} -5.21110 q^{62} -2.60555 q^{63} +1.00000 q^{64} -0.605551 q^{65} +1.00000 q^{66} -10.0000 q^{67} +1.00000 q^{68} -4.60555 q^{69} -2.60555 q^{70} -1.39445 q^{71} -1.00000 q^{72} -8.60555 q^{73} +7.21110 q^{74} +1.00000 q^{75} +0.605551 q^{76} +2.60555 q^{77} -0.605551 q^{78} +0.605551 q^{79} -1.00000 q^{80} +1.00000 q^{81} -9.21110 q^{82} -2.60555 q^{84} -1.00000 q^{85} +5.39445 q^{86} +6.00000 q^{87} +1.00000 q^{88} +6.00000 q^{89} +1.00000 q^{90} -1.57779 q^{91} -4.60555 q^{92} +5.21110 q^{93} -12.0000 q^{94} -0.605551 q^{95} -1.00000 q^{96} -13.2111 q^{97} +0.211103 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{10} - 2 q^{11} + 2 q^{12} - 6 q^{13} - 2 q^{14} - 2 q^{15} + 2 q^{16} + 2 q^{17} - 2 q^{18} - 6 q^{19} - 2 q^{20} + 2 q^{21} + 2 q^{22} - 2 q^{23} - 2 q^{24} + 2 q^{25} + 6 q^{26} + 2 q^{27} + 2 q^{28} + 12 q^{29} + 2 q^{30} - 4 q^{31} - 2 q^{32} - 2 q^{33} - 2 q^{34} - 2 q^{35} + 2 q^{36} + 6 q^{38} - 6 q^{39} + 2 q^{40} + 4 q^{41} - 2 q^{42} - 18 q^{43} - 2 q^{44} - 2 q^{45} + 2 q^{46} + 24 q^{47} + 2 q^{48} + 14 q^{49} - 2 q^{50} + 2 q^{51} - 6 q^{52} - 10 q^{53} - 2 q^{54} + 2 q^{55} - 2 q^{56} - 6 q^{57} - 12 q^{58} - 14 q^{59} - 2 q^{60} + 4 q^{61} + 4 q^{62} + 2 q^{63} + 2 q^{64} + 6 q^{65} + 2 q^{66} - 20 q^{67} + 2 q^{68} - 2 q^{69} + 2 q^{70} - 10 q^{71} - 2 q^{72} - 10 q^{73} + 2 q^{75} - 6 q^{76} - 2 q^{77} + 6 q^{78} - 6 q^{79} - 2 q^{80} + 2 q^{81} - 4 q^{82} + 2 q^{84} - 2 q^{85} + 18 q^{86} + 12 q^{87} + 2 q^{88} + 12 q^{89} + 2 q^{90} - 32 q^{91} - 2 q^{92} - 4 q^{93} - 24 q^{94} + 6 q^{95} - 2 q^{96} - 12 q^{97} - 14 q^{98} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 + 2 * q^10 - 2 * q^11 + 2 * q^12 - 6 * q^13 - 2 * q^14 - 2 * q^15 + 2 * q^16 + 2 * q^17 - 2 * q^18 - 6 * q^19 - 2 * q^20 + 2 * q^21 + 2 * q^22 - 2 * q^23 - 2 * q^24 + 2 * q^25 + 6 * q^26 + 2 * q^27 + 2 * q^28 + 12 * q^29 + 2 * q^30 - 4 * q^31 - 2 * q^32 - 2 * q^33 - 2 * q^34 - 2 * q^35 + 2 * q^36 + 6 * q^38 - 6 * q^39 + 2 * q^40 + 4 * q^41 - 2 * q^42 - 18 * q^43 - 2 * q^44 - 2 * q^45 + 2 * q^46 + 24 * q^47 + 2 * q^48 + 14 * q^49 - 2 * q^50 + 2 * q^51 - 6 * q^52 - 10 * q^53 - 2 * q^54 + 2 * q^55 - 2 * q^56 - 6 * q^57 - 12 * q^58 - 14 * q^59 - 2 * q^60 + 4 * q^61 + 4 * q^62 + 2 * q^63 + 2 * q^64 + 6 * q^65 + 2 * q^66 - 20 * q^67 + 2 * q^68 - 2 * q^69 + 2 * q^70 - 10 * q^71 - 2 * q^72 - 10 * q^73 + 2 * q^75 - 6 * q^76 - 2 * q^77 + 6 * q^78 - 6 * q^79 - 2 * q^80 + 2 * q^81 - 4 * q^82 + 2 * q^84 - 2 * q^85 + 18 * q^86 + 12 * q^87 + 2 * q^88 + 12 * q^89 + 2 * q^90 - 32 * q^91 - 2 * q^92 - 4 * q^93 - 24 * q^94 + 6 * q^95 - 2 * q^96 - 12 * q^97 - 14 * 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
$7$	−2.60555	−0.984806	−0.492403	−	0.870367i	$-0.663881\pi$
$7$	−2.60555	−0.984806	−0.492403	+	0.870367i	$0.663881\pi$
$8$	−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
$13$	0.605551	0.167950	0.0839749	−	0.996468i	$-0.473238\pi$
$13$	0.605551	0.167950	0.0839749	+	0.996468i	$0.473238\pi$
$14$	2.60555	0.696363
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.00000	−0.235702
$19$	0.605551	0.138923	0.0694615	−	0.997585i	$-0.477872\pi$
$19$	0.605551	0.138923	0.0694615	+	0.997585i	$0.477872\pi$
$20$	−1.00000	−0.223607
$21$	−2.60555	−0.568578
$22$	1.00000	0.213201
$23$	−4.60555	−0.960324	−0.480162	−	0.877180i	$-0.659422\pi$
$23$	−4.60555	−0.960324	−0.480162	+	0.877180i	$0.659422\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	−0.605551	−0.118758
$27$	1.00000	0.192450
$28$	−2.60555	−0.492403
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	1.00000	0.182574
$31$	5.21110	0.935942	0.467971	−	0.883744i	$-0.344985\pi$
$31$	5.21110	0.935942	0.467971	+	0.883744i	$0.344985\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	−1.00000	−0.171499
$35$	2.60555	0.440419
$36$	1.00000	0.166667
$37$	−7.21110	−1.18550	−0.592749	−	0.805387i	$-0.701957\pi$
$37$	−7.21110	−1.18550	−0.592749	+	0.805387i	$0.701957\pi$
$38$	−0.605551	−0.0982334
$39$	0.605551	0.0969658
$40$	1.00000	0.158114
$41$	9.21110	1.43853	0.719266	−	0.694735i	$-0.244478\pi$
$41$	9.21110	1.43853	0.719266	+	0.694735i	$0.244478\pi$
$42$	2.60555	0.402045
$43$	−5.39445	−0.822646	−0.411323	−	0.911490i	$-0.634933\pi$
$43$	−5.39445	−0.822646	−0.411323	+	0.911490i	$0.634933\pi$
$44$	−1.00000	−0.150756
$45$	−1.00000	−0.149071
$46$	4.60555	0.679051
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	1.00000	0.144338
$49$	−0.211103	−0.0301575
$50$	−1.00000	−0.141421
$51$	1.00000	0.140028
$52$	0.605551	0.0839749
$53$	−1.39445	−0.191542	−0.0957711	−	0.995403i	$-0.530532\pi$
$53$	−1.39445	−0.191542	−0.0957711	+	0.995403i	$0.530532\pi$
$54$	−1.00000	−0.136083
$55$	1.00000	0.134840
$56$	2.60555	0.348181
$57$	0.605551	0.0802072
$58$	−6.00000	−0.787839
$59$	−10.6056	−1.38073	−0.690363	−	0.723464i	$-0.742549\pi$
$59$	−10.6056	−1.38073	−0.690363	+	0.723464i	$0.742549\pi$
$60$	−1.00000	−0.129099
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−5.21110	−0.661811
$63$	−2.60555	−0.328269
$64$	1.00000	0.125000
$65$	−0.605551	−0.0751094
$66$	1.00000	0.123091
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	1.00000	0.121268
$69$	−4.60555	−0.554443
$70$	−2.60555	−0.311423
$71$	−1.39445	−0.165491	−0.0827453	−	0.996571i	$-0.526369\pi$
$71$	−1.39445	−0.165491	−0.0827453	+	0.996571i	$0.526369\pi$
$72$	−1.00000	−0.117851
$73$	−8.60555	−1.00720	−0.503602	−	0.863936i	$-0.667992\pi$
$73$	−8.60555	−1.00720	−0.503602	+	0.863936i	$0.667992\pi$
$74$	7.21110	0.838274
$75$	1.00000	0.115470
$76$	0.605551	0.0694615
$77$	2.60555	0.296930
$78$	−0.605551	−0.0685652
$79$	0.605551	0.0681298	0.0340649	−	0.999420i	$-0.489155\pi$
$79$	0.605551	0.0681298	0.0340649	+	0.999420i	$0.489155\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−9.21110	−1.01720
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	−2.60555	−0.284289
$85$	−1.00000	−0.108465
$86$	5.39445	0.581698
$87$	6.00000	0.643268
$88$	1.00000	0.106600
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	1.00000	0.105409
$91$	−1.57779	−0.165398
$92$	−4.60555	−0.480162
$93$	5.21110	0.540366
$94$	−12.0000	−1.23771
$95$	−0.605551	−0.0621282
$96$	−1.00000	−0.102062
$97$	−13.2111	−1.34138	−0.670692	−	0.741736i	$-0.734003\pi$
$97$	−13.2111	−1.34138	−0.670692	+	0.741736i	$0.734003\pi$
$98$	0.211103	0.0213246
$99$	−1.00000	−0.100504
$100$	1.00000	0.100000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	−1.00000	−0.0990148
$103$	14.4222	1.42106	0.710531	−	0.703666i	$-0.248455\pi$
$103$	14.4222	1.42106	0.710531	+	0.703666i	$0.248455\pi$
$104$	−0.605551	−0.0593792
$105$	2.60555	0.254276
$106$	1.39445	0.135441
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	1.00000	0.0962250
$109$	8.42221	0.806701	0.403350	−	0.915046i	$-0.367846\pi$
$109$	8.42221	0.806701	0.403350	+	0.915046i	$0.367846\pi$
$110$	−1.00000	−0.0953463
$111$	−7.21110	−0.684448
$112$	−2.60555	−0.246201
$113$	19.8167	1.86419	0.932097	−	0.362209i	$-0.117977\pi$
$113$	19.8167	1.86419	0.932097	+	0.362209i	$0.117977\pi$
$114$	−0.605551	−0.0567151
$115$	4.60555	0.429470
$116$	6.00000	0.557086
$117$	0.605551	0.0559832
$118$	10.6056	0.976320
$119$	−2.60555	−0.238850
$120$	1.00000	0.0912871
$121$	1.00000	0.0909091
$122$	−2.00000	−0.181071
$123$	9.21110	0.830537
$124$	5.21110	0.467971
$125$	−1.00000	−0.0894427
$126$	2.60555	0.232121
$127$	−13.2111	−1.17230	−0.586148	−	0.810204i	$-0.699356\pi$
$127$	−13.2111	−1.17230	−0.586148	+	0.810204i	$0.699356\pi$
$128$	−1.00000	−0.0883883
$129$	−5.39445	−0.474955
$130$	0.605551	0.0531104
$131$	−18.4222	−1.60956	−0.804778	−	0.593576i	$-0.797716\pi$
$131$	−18.4222	−1.60956	−0.804778	+	0.593576i	$0.797716\pi$
$132$	−1.00000	−0.0870388
$133$	−1.57779	−0.136812
$134$	10.0000	0.863868
$135$	−1.00000	−0.0860663
$136$	−1.00000	−0.0857493
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	4.60555	0.392051
$139$	−10.4222	−0.884000	−0.442000	−	0.897015i	$-0.645731\pi$
$139$	−10.4222	−0.884000	−0.442000	+	0.897015i	$0.645731\pi$
$140$	2.60555	0.220209
$141$	12.0000	1.01058
$142$	1.39445	0.117020
$143$	−0.605551	−0.0506387
$144$	1.00000	0.0833333
$145$	−6.00000	−0.498273
$146$	8.60555	0.712200
$147$	−0.211103	−0.0174114
$148$	−7.21110	−0.592749
$149$	−21.2111	−1.73768	−0.868841	−	0.495092i	$-0.835134\pi$
$149$	−21.2111	−1.73768	−0.868841	+	0.495092i	$0.835134\pi$
$150$	−1.00000	−0.0816497
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−0.605551	−0.0491167
$153$	1.00000	0.0808452
$154$	−2.60555	−0.209961
$155$	−5.21110	−0.418566
$156$	0.605551	0.0484829
$157$	−13.2111	−1.05436	−0.527180	−	0.849753i	$-0.676751\pi$
$157$	−13.2111	−1.05436	−0.527180	+	0.849753i	$0.676751\pi$
$158$	−0.605551	−0.0481751
$159$	−1.39445	−0.110587
$160$	1.00000	0.0790569
$161$	12.0000	0.945732
$162$	−1.00000	−0.0785674
$163$	5.21110	0.408165	0.204083	−	0.978954i	$-0.434579\pi$
$163$	5.21110	0.408165	0.204083	+	0.978954i	$0.434579\pi$
$164$	9.21110	0.719266
$165$	1.00000	0.0778499
$166$	0	0
$167$	−21.2111	−1.64136	−0.820682	−	0.571385i	$-0.806406\pi$
$167$	−21.2111	−1.64136	−0.820682	+	0.571385i	$0.806406\pi$
$168$	2.60555	0.201023
$169$	−12.6333	−0.971793
$170$	1.00000	0.0766965
$171$	0.605551	0.0463077
$172$	−5.39445	−0.411323
$173$	−12.4222	−0.944443	−0.472221	−	0.881480i	$-0.656548\pi$
$173$	−12.4222	−0.944443	−0.472221	+	0.881480i	$0.656548\pi$
$174$	−6.00000	−0.454859
$175$	−2.60555	−0.196961
$176$	−1.00000	−0.0753778
$177$	−10.6056	−0.797162
$178$	−6.00000	−0.449719
$179$	−7.81665	−0.584244	−0.292122	−	0.956381i	$-0.594361\pi$
$179$	−7.81665	−0.584244	−0.292122	+	0.956381i	$0.594361\pi$
$180$	−1.00000	−0.0745356
$181$	20.4222	1.51797	0.758985	−	0.651108i	$-0.225695\pi$
$181$	20.4222	1.51797	0.758985	+	0.651108i	$0.225695\pi$
$182$	1.57779	0.116954
$183$	2.00000	0.147844
$184$	4.60555	0.339526
$185$	7.21110	0.530171
$186$	−5.21110	−0.382097
$187$	−1.00000	−0.0731272
$188$	12.0000	0.875190
$189$	−2.60555	−0.189526
$190$	0.605551	0.0439313
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	1.00000	0.0721688
$193$	12.6056	0.907367	0.453684	−	0.891163i	$-0.350110\pi$
$193$	12.6056	0.907367	0.453684	+	0.891163i	$0.350110\pi$
$194$	13.2111	0.948502
$195$	−0.605551	−0.0433644
$196$	−0.211103	−0.0150788
$197$	−15.2111	−1.08375	−0.541873	−	0.840460i	$-0.682285\pi$
$197$	−15.2111	−1.08375	−0.541873	+	0.840460i	$0.682285\pi$
$198$	1.00000	0.0710669
$199$	5.21110	0.369405	0.184703	−	0.982794i	$-0.440868\pi$
$199$	5.21110	0.369405	0.184703	+	0.982794i	$0.440868\pi$
$200$	−1.00000	−0.0707107
$201$	−10.0000	−0.705346
$202$	12.0000	0.844317
$203$	−15.6333	−1.09724
$204$	1.00000	0.0700140
$205$	−9.21110	−0.643331
$206$	−14.4222	−1.00484
$207$	−4.60555	−0.320108
$208$	0.605551	0.0419874
$209$	−0.605551	−0.0418869
$210$	−2.60555	−0.179800
$211$	−25.2111	−1.73560	−0.867802	−	0.496910i	$-0.834468\pi$
$211$	−25.2111	−1.73560	−0.867802	+	0.496910i	$0.834468\pi$
$212$	−1.39445	−0.0957711
$213$	−1.39445	−0.0955461
$214$	0	0
$215$	5.39445	0.367898
$216$	−1.00000	−0.0680414
$217$	−13.5778	−0.921721
$218$	−8.42221	−0.570424
$219$	−8.60555	−0.581509
$220$	1.00000	0.0674200
$221$	0.605551	0.0407338
$222$	7.21110	0.483978
$223$	−19.6333	−1.31474	−0.657372	−	0.753566i	$-0.728332\pi$
$223$	−19.6333	−1.31474	−0.657372	+	0.753566i	$0.728332\pi$
$224$	2.60555	0.174091
$225$	1.00000	0.0666667
$226$	−19.8167	−1.31818
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0.605551	0.0401036
$229$	−0.788897	−0.0521318	−0.0260659	−	0.999660i	$-0.508298\pi$
$229$	−0.788897	−0.0521318	−0.0260659	+	0.999660i	$0.508298\pi$
$230$	−4.60555	−0.303681
$231$	2.60555	0.171433
$232$	−6.00000	−0.393919
$233$	−15.2111	−0.996512	−0.498256	−	0.867030i	$-0.666026\pi$
$233$	−15.2111	−0.996512	−0.498256	+	0.867030i	$0.666026\pi$
$234$	−0.605551	−0.0395861
$235$	−12.0000	−0.782794
$236$	−10.6056	−0.690363
$237$	0.605551	0.0393348
$238$	2.60555	0.168893
$239$	15.6333	1.01123	0.505617	−	0.862758i	$-0.331265\pi$
$239$	15.6333	1.01123	0.505617	+	0.862758i	$0.331265\pi$
$240$	−1.00000	−0.0645497
$241$	−9.02776	−0.581529	−0.290764	−	0.956795i	$-0.593910\pi$
$241$	−9.02776	−0.581529	−0.290764	+	0.956795i	$0.593910\pi$
$242$	−1.00000	−0.0642824
$243$	1.00000	0.0641500
$244$	2.00000	0.128037
$245$	0.211103	0.0134868
$246$	−9.21110	−0.587278
$247$	0.366692	0.0233321
$248$	−5.21110	−0.330905
$249$	0	0
$250$	1.00000	0.0632456
$251$	22.6056	1.42685	0.713425	−	0.700732i	$-0.247143\pi$
$251$	22.6056	1.42685	0.713425	+	0.700732i	$0.247143\pi$
$252$	−2.60555	−0.164134
$253$	4.60555	0.289549
$254$	13.2111	0.828938
$255$	−1.00000	−0.0626224
$256$	1.00000	0.0625000
$257$	21.6333	1.34945	0.674724	−	0.738070i	$-0.264263\pi$
$257$	21.6333	1.34945	0.674724	+	0.738070i	$0.264263\pi$
$258$	5.39445	0.335844
$259$	18.7889	1.16749
$260$	−0.605551	−0.0375547
$261$	6.00000	0.371391
$262$	18.4222	1.13813
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	1.00000	0.0615457
$265$	1.39445	0.0856603
$266$	1.57779	0.0967408
$267$	6.00000	0.367194
$268$	−10.0000	−0.610847
$269$	21.6333	1.31901	0.659503	−	0.751702i	$-0.270767\pi$
$269$	21.6333	1.31901	0.659503	+	0.751702i	$0.270767\pi$
$270$	1.00000	0.0608581
$271$	−1.21110	−0.0735692	−0.0367846	−	0.999323i	$-0.511712\pi$
$271$	−1.21110	−0.0735692	−0.0367846	+	0.999323i	$0.511712\pi$
$272$	1.00000	0.0606339
$273$	−1.57779	−0.0954925
$274$	−6.00000	−0.362473
$275$	−1.00000	−0.0603023
$276$	−4.60555	−0.277222
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	10.4222	0.625082
$279$	5.21110	0.311981
$280$	−2.60555	−0.155711
$281$	−3.21110	−0.191558	−0.0957792	−	0.995403i	$-0.530534\pi$
$281$	−3.21110	−0.191558	−0.0957792	+	0.995403i	$0.530534\pi$
$282$	−12.0000	−0.714590
$283$	−13.2111	−0.785319	−0.392659	−	0.919684i	$-0.628445\pi$
$283$	−13.2111	−0.785319	−0.392659	+	0.919684i	$0.628445\pi$
$284$	−1.39445	−0.0827453
$285$	−0.605551	−0.0358698
$286$	0.605551	0.0358070
$287$	−24.0000	−1.41668
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	6.00000	0.352332
$291$	−13.2111	−0.774449
$292$	−8.60555	−0.503602
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	0.211103	0.0123118
$295$	10.6056	0.617479
$296$	7.21110	0.419137
$297$	−1.00000	−0.0580259
$298$	21.2111	1.22873
$299$	−2.78890	−0.161286
$300$	1.00000	0.0577350
$301$	14.0555	0.810146
$302$	16.0000	0.920697
$303$	−12.0000	−0.689382
$304$	0.605551	0.0347307
$305$	−2.00000	−0.114520
$306$	−1.00000	−0.0571662
$307$	−33.0278	−1.88499	−0.942497	−	0.334215i	$-0.891529\pi$
$307$	−33.0278	−1.88499	−0.942497	+	0.334215i	$0.891529\pi$
$308$	2.60555	0.148465
$309$	14.4222	0.820451
$310$	5.21110	0.295971
$311$	−26.2389	−1.48787	−0.743935	−	0.668252i	$-0.767043\pi$
$311$	−26.2389	−1.48787	−0.743935	+	0.668252i	$0.767043\pi$
$312$	−0.605551	−0.0342826
$313$	−18.7889	−1.06201	−0.531006	−	0.847368i	$-0.678186\pi$
$313$	−18.7889	−1.06201	−0.531006	+	0.847368i	$0.678186\pi$
$314$	13.2111	0.745546
$315$	2.60555	0.146806
$316$	0.605551	0.0340649
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	1.39445	0.0781968
$319$	−6.00000	−0.335936
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−12.0000	−0.668734
$323$	0.605551	0.0336938
$324$	1.00000	0.0555556
$325$	0.605551	0.0335899
$326$	−5.21110	−0.288616
$327$	8.42221	0.465749
$328$	−9.21110	−0.508598
$329$	−31.2666	−1.72378
$330$	−1.00000	−0.0550482
$331$	2.42221	0.133136	0.0665682	−	0.997782i	$-0.478795\pi$
$331$	2.42221	0.133136	0.0665682	+	0.997782i	$0.478795\pi$
$332$	0	0
$333$	−7.21110	−0.395166
$334$	21.2111	1.16062
$335$	10.0000	0.546358
$336$	−2.60555	−0.142144
$337$	9.81665	0.534747	0.267374	−	0.963593i	$-0.413844\pi$
$337$	9.81665	0.534747	0.267374	+	0.963593i	$0.413844\pi$
$338$	12.6333	0.687161
$339$	19.8167	1.07629
$340$	−1.00000	−0.0542326
$341$	−5.21110	−0.282197
$342$	−0.605551	−0.0327445
$343$	18.7889	1.01451
$344$	5.39445	0.290849
$345$	4.60555	0.247955
$346$	12.4222	0.667822
$347$	24.8444	1.33372	0.666859	−	0.745184i	$-0.267638\pi$
$347$	24.8444	1.33372	0.666859	+	0.745184i	$0.267638\pi$
$348$	6.00000	0.321634
$349$	−30.2389	−1.61865	−0.809325	−	0.587362i	$-0.800167\pi$
$349$	−30.2389	−1.61865	−0.809325	+	0.587362i	$0.800167\pi$
$350$	2.60555	0.139273
$351$	0.605551	0.0323219
$352$	1.00000	0.0533002
$353$	3.21110	0.170910	0.0854549	−	0.996342i	$-0.472766\pi$
$353$	3.21110	0.170910	0.0854549	+	0.996342i	$0.472766\pi$
$354$	10.6056	0.563679
$355$	1.39445	0.0740097
$356$	6.00000	0.317999
$357$	−2.60555	−0.137900
$358$	7.81665	0.413123
$359$	−14.7889	−0.780528	−0.390264	−	0.920703i	$-0.627616\pi$
$359$	−14.7889	−0.780528	−0.390264	+	0.920703i	$0.627616\pi$
$360$	1.00000	0.0527046
$361$	−18.6333	−0.980700
$362$	−20.4222	−1.07337
$363$	1.00000	0.0524864
$364$	−1.57779	−0.0826989
$365$	8.60555	0.450435
$366$	−2.00000	−0.104542
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	−4.60555	−0.240081
$369$	9.21110	0.479511
$370$	−7.21110	−0.374887
$371$	3.63331	0.188632
$372$	5.21110	0.270183
$373$	31.0278	1.60656	0.803278	−	0.595604i	$-0.203087\pi$
$373$	31.0278	1.60656	0.803278	+	0.595604i	$0.203087\pi$
$374$	1.00000	0.0517088
$375$	−1.00000	−0.0516398
$376$	−12.0000	−0.618853
$377$	3.63331	0.187125
$378$	2.60555	0.134015
$379$	−34.4222	−1.76815	−0.884075	−	0.467345i	$-0.845211\pi$
$379$	−34.4222	−1.76815	−0.884075	+	0.467345i	$0.845211\pi$
$380$	−0.605551	−0.0310641
$381$	−13.2111	−0.676825
$382$	−12.0000	−0.613973
$383$	30.4222	1.55450	0.777251	−	0.629191i	$-0.216614\pi$
$383$	30.4222	1.55450	0.777251	+	0.629191i	$0.216614\pi$
$384$	−1.00000	−0.0510310
$385$	−2.60555	−0.132791
$386$	−12.6056	−0.641606
$387$	−5.39445	−0.274215
$388$	−13.2111	−0.670692
$389$	23.0278	1.16755	0.583777	−	0.811914i	$-0.301574\pi$
$389$	23.0278	1.16755	0.583777	+	0.811914i	$0.301574\pi$
$390$	0.605551	0.0306633
$391$	−4.60555	−0.232913
$392$	0.211103	0.0106623
$393$	−18.4222	−0.929277
$394$	15.2111	0.766324
$395$	−0.605551	−0.0304686
$396$	−1.00000	−0.0502519
$397$	−19.2111	−0.964178	−0.482089	−	0.876122i	$-0.660122\pi$
$397$	−19.2111	−0.964178	−0.482089	+	0.876122i	$0.660122\pi$
$398$	−5.21110	−0.261209
$399$	−1.57779	−0.0789885
$400$	1.00000	0.0500000
$401$	25.8167	1.28922	0.644611	−	0.764511i	$-0.277019\pi$
$401$	25.8167	1.28922	0.644611	+	0.764511i	$0.277019\pi$
$402$	10.0000	0.498755
$403$	3.15559	0.157191
$404$	−12.0000	−0.597022
$405$	−1.00000	−0.0496904
$406$	15.6333	0.775868
$407$	7.21110	0.357441
$408$	−1.00000	−0.0495074
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	9.21110	0.454904
$411$	6.00000	0.295958
$412$	14.4222	0.710531
$413$	27.6333	1.35975
$414$	4.60555	0.226350
$415$	0	0
$416$	−0.605551	−0.0296896
$417$	−10.4222	−0.510378
$418$	0.605551	0.0296185
$419$	33.2111	1.62247	0.811234	−	0.584721i	$-0.198796\pi$
$419$	33.2111	1.62247	0.811234	+	0.584721i	$0.198796\pi$
$420$	2.60555	0.127138
$421$	−22.8444	−1.11337	−0.556684	−	0.830724i	$-0.687927\pi$
$421$	−22.8444	−1.11337	−0.556684	+	0.830724i	$0.687927\pi$
$422$	25.2111	1.22726
$423$	12.0000	0.583460
$424$	1.39445	0.0677204
$425$	1.00000	0.0485071
$426$	1.39445	0.0675613
$427$	−5.21110	−0.252183
$428$	0	0
$429$	−0.605551	−0.0292363
$430$	−5.39445	−0.260143
$431$	24.4222	1.17638	0.588188	−	0.808724i	$-0.299841\pi$
$431$	24.4222	1.17638	0.588188	+	0.808724i	$0.299841\pi$
$432$	1.00000	0.0481125
$433$	20.4222	0.981429	0.490714	−	0.871321i	$-0.336736\pi$
$433$	20.4222	0.981429	0.490714	+	0.871321i	$0.336736\pi$
$434$	13.5778	0.651755
$435$	−6.00000	−0.287678
$436$	8.42221	0.403350
$437$	−2.78890	−0.133411
$438$	8.60555	0.411189
$439$	0.605551	0.0289014	0.0144507	−	0.999896i	$-0.495400\pi$
$439$	0.605551	0.0289014	0.0144507	+	0.999896i	$0.495400\pi$
$440$	−1.00000	−0.0476731
$441$	−0.211103	−0.0100525
$442$	−0.605551	−0.0288031
$443$	−1.81665	−0.0863118	−0.0431559	−	0.999068i	$-0.513741\pi$
$443$	−1.81665	−0.0863118	−0.0431559	+	0.999068i	$0.513741\pi$
$444$	−7.21110	−0.342224
$445$	−6.00000	−0.284427
$446$	19.6333	0.929664
$447$	−21.2111	−1.00325
$448$	−2.60555	−0.123101
$449$	29.4500	1.38983	0.694915	−	0.719092i	$-0.255442\pi$
$449$	29.4500	1.38983	0.694915	+	0.719092i	$0.255442\pi$
$450$	−1.00000	−0.0471405
$451$	−9.21110	−0.433734
$452$	19.8167	0.932097
$453$	−16.0000	−0.751746
$454$	12.0000	0.563188
$455$	1.57779	0.0739682
$456$	−0.605551	−0.0283575
$457$	−22.8444	−1.06862	−0.534308	−	0.845290i	$-0.679428\pi$
$457$	−22.8444	−1.06862	−0.534308	+	0.845290i	$0.679428\pi$
$458$	0.788897	0.0368628
$459$	1.00000	0.0466760
$460$	4.60555	0.214735
$461$	33.2111	1.54680	0.773398	−	0.633921i	$-0.218556\pi$
$461$	33.2111	1.54680	0.773398	+	0.633921i	$0.218556\pi$
$462$	−2.60555	−0.121221
$463$	−25.2111	−1.17166	−0.585830	−	0.810434i	$-0.699231\pi$
$463$	−25.2111	−1.17166	−0.585830	+	0.810434i	$0.699231\pi$
$464$	6.00000	0.278543
$465$	−5.21110	−0.241659
$466$	15.2111	0.704641
$467$	−32.2389	−1.49184	−0.745918	−	0.666038i	$-0.767989\pi$
$467$	−32.2389	−1.49184	−0.745918	+	0.666038i	$0.767989\pi$
$468$	0.605551	0.0279916
$469$	26.0555	1.20313
$470$	12.0000	0.553519
$471$	−13.2111	−0.608735
$472$	10.6056	0.488160
$473$	5.39445	0.248037
$474$	−0.605551	−0.0278139
$475$	0.605551	0.0277846
$476$	−2.60555	−0.119425
$477$	−1.39445	−0.0638474
$478$	−15.6333	−0.715051
$479$	27.2111	1.24331	0.621654	−	0.783292i	$-0.286461\pi$
$479$	27.2111	1.24331	0.621654	+	0.783292i	$0.286461\pi$
$480$	1.00000	0.0456435
$481$	−4.36669	−0.199104
$482$	9.02776	0.411203
$483$	12.0000	0.546019
$484$	1.00000	0.0454545
$485$	13.2111	0.599885
$486$	−1.00000	−0.0453609
$487$	−28.4222	−1.28793	−0.643967	−	0.765054i	$-0.722712\pi$
$487$	−28.4222	−1.28793	−0.643967	+	0.765054i	$0.722712\pi$
$488$	−2.00000	−0.0905357
$489$	5.21110	0.235654
$490$	−0.211103	−0.00953664
$491$	−24.4222	−1.10216	−0.551079	−	0.834453i	$-0.685784\pi$
$491$	−24.4222	−1.10216	−0.551079	+	0.834453i	$0.685784\pi$
$492$	9.21110	0.415269
$493$	6.00000	0.270226
$494$	−0.366692	−0.0164983
$495$	1.00000	0.0449467
$496$	5.21110	0.233985
$497$	3.63331	0.162976
$498$	0	0
$499$	38.4222	1.72001	0.860007	−	0.510282i	$-0.170459\pi$
$499$	38.4222	1.72001	0.860007	+	0.510282i	$0.170459\pi$
$500$	−1.00000	−0.0447214
$501$	−21.2111	−0.947642
$502$	−22.6056	−1.00894
$503$	−18.4222	−0.821406	−0.410703	−	0.911769i	$-0.634717\pi$
$503$	−18.4222	−0.821406	−0.410703	+	0.911769i	$0.634717\pi$
$504$	2.60555	0.116060
$505$	12.0000	0.533993
$506$	−4.60555	−0.204742
$507$	−12.6333	−0.561065
$508$	−13.2111	−0.586148
$509$	−8.23886	−0.365181	−0.182591	−	0.983189i	$-0.558448\pi$
$509$	−8.23886	−0.365181	−0.182591	+	0.983189i	$0.558448\pi$
$510$	1.00000	0.0442807
$511$	22.4222	0.991900
$512$	−1.00000	−0.0441942
$513$	0.605551	0.0267357
$514$	−21.6333	−0.954204
$515$	−14.4222	−0.635518
$516$	−5.39445	−0.237477
$517$	−12.0000	−0.527759
$518$	−18.7889	−0.825537
$519$	−12.4222	−0.545274
$520$	0.605551	0.0265552
$521$	−35.0278	−1.53459	−0.767297	−	0.641292i	$-0.778399\pi$
$521$	−35.0278	−1.53459	−0.767297	+	0.641292i	$0.778399\pi$
$522$	−6.00000	−0.262613
$523$	−39.4500	−1.72503	−0.862513	−	0.506035i	$-0.831111\pi$
$523$	−39.4500	−1.72503	−0.862513	+	0.506035i	$0.831111\pi$
$524$	−18.4222	−0.804778
$525$	−2.60555	−0.113716
$526$	−12.0000	−0.523225
$527$	5.21110	0.226999
$528$	−1.00000	−0.0435194
$529$	−1.78890	−0.0777781
$530$	−1.39445	−0.0605710
$531$	−10.6056	−0.460242
$532$	−1.57779	−0.0684061
$533$	5.57779	0.241601
$534$	−6.00000	−0.259645
$535$	0	0
$536$	10.0000	0.431934
$537$	−7.81665	−0.337314
$538$	−21.6333	−0.932678
$539$	0.211103	0.00909283
$540$	−1.00000	−0.0430331
$541$	14.8444	0.638211	0.319106	−	0.947719i	$-0.396618\pi$
$541$	14.8444	0.638211	0.319106	+	0.947719i	$0.396618\pi$
$542$	1.21110	0.0520213
$543$	20.4222	0.876401
$544$	−1.00000	−0.0428746
$545$	−8.42221	−0.360768
$546$	1.57779	0.0675234
$547$	29.2111	1.24898	0.624488	−	0.781034i	$-0.285308\pi$
$547$	29.2111	1.24898	0.624488	+	0.781034i	$0.285308\pi$
$548$	6.00000	0.256307
$549$	2.00000	0.0853579
$550$	1.00000	0.0426401
$551$	3.63331	0.154784
$552$	4.60555	0.196025
$553$	−1.57779	−0.0670947
$554$	−2.00000	−0.0849719
$555$	7.21110	0.306094
$556$	−10.4222	−0.442000
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	−5.21110	−0.220604
$559$	−3.26662	−0.138163
$560$	2.60555	0.110105
$561$	−1.00000	−0.0422200
$562$	3.21110	0.135452
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	12.0000	0.505291
$565$	−19.8167	−0.833693
$566$	13.2111	0.555304
$567$	−2.60555	−0.109423
$568$	1.39445	0.0585098
$569$	−45.6333	−1.91305	−0.956524	−	0.291654i	$-0.905794\pi$
$569$	−45.6333	−1.91305	−0.956524	+	0.291654i	$0.905794\pi$
$570$	0.605551	0.0253638
$571$	2.42221	0.101366	0.0506831	−	0.998715i	$-0.483860\pi$
$571$	2.42221	0.101366	0.0506831	+	0.998715i	$0.483860\pi$
$572$	−0.605551	−0.0253194
$573$	12.0000	0.501307
$574$	24.0000	1.00174
$575$	−4.60555	−0.192065
$576$	1.00000	0.0416667
$577$	−3.57779	−0.148946	−0.0744728	−	0.997223i	$-0.523727\pi$
$577$	−3.57779	−0.148946	−0.0744728	+	0.997223i	$0.523727\pi$
$578$	−1.00000	−0.0415945
$579$	12.6056	0.523869
$580$	−6.00000	−0.249136
$581$	0	0
$582$	13.2111	0.547618
$583$	1.39445	0.0577522
$584$	8.60555	0.356100
$585$	−0.605551	−0.0250365
$586$	30.0000	1.23929
$587$	−31.3944	−1.29579	−0.647894	−	0.761731i	$-0.724350\pi$
$587$	−31.3944	−1.29579	−0.647894	+	0.761731i	$0.724350\pi$
$588$	−0.211103	−0.00870572
$589$	3.15559	0.130024
$590$	−10.6056	−0.436624
$591$	−15.2111	−0.625701
$592$	−7.21110	−0.296374
$593$	−39.2111	−1.61021	−0.805103	−	0.593134i	$-0.797890\pi$
$593$	−39.2111	−1.61021	−0.805103	+	0.593134i	$0.797890\pi$
$594$	1.00000	0.0410305
$595$	2.60555	0.106817
$596$	−21.2111	−0.868841
$597$	5.21110	0.213276
$598$	2.78890	0.114046
$599$	30.4222	1.24302	0.621509	−	0.783407i	$-0.286520\pi$
$599$	30.4222	1.24302	0.621509	+	0.783407i	$0.286520\pi$
$600$	−1.00000	−0.0408248
$601$	−35.8167	−1.46099	−0.730496	−	0.682917i	$-0.760711\pi$
$601$	−35.8167	−1.46099	−0.730496	+	0.682917i	$0.760711\pi$
$602$	−14.0555	−0.572860
$603$	−10.0000	−0.407231
$604$	−16.0000	−0.651031
$605$	−1.00000	−0.0406558
$606$	12.0000	0.487467
$607$	34.2389	1.38971	0.694856	−	0.719149i	$-0.255468\pi$
$607$	34.2389	1.38971	0.694856	+	0.719149i	$0.255468\pi$
$608$	−0.605551	−0.0245583
$609$	−15.6333	−0.633494
$610$	2.00000	0.0809776
$611$	7.26662	0.293976
$612$	1.00000	0.0404226
$613$	36.6056	1.47848	0.739242	−	0.673440i	$-0.235184\pi$
$613$	36.6056	1.47848	0.739242	+	0.673440i	$0.235184\pi$
$614$	33.0278	1.33289
$615$	−9.21110	−0.371428
$616$	−2.60555	−0.104981
$617$	−43.8167	−1.76399	−0.881996	−	0.471257i	$-0.843800\pi$
$617$	−43.8167	−1.76399	−0.881996	+	0.471257i	$0.843800\pi$
$618$	−14.4222	−0.580146
$619$	−40.8444	−1.64168	−0.820838	−	0.571161i	$-0.806493\pi$
$619$	−40.8444	−1.64168	−0.820838	+	0.571161i	$0.806493\pi$
$620$	−5.21110	−0.209283
$621$	−4.60555	−0.184814
$622$	26.2389	1.05208
$623$	−15.6333	−0.626335
$624$	0.605551	0.0242415
$625$	1.00000	0.0400000
$626$	18.7889	0.750955
$627$	−0.605551	−0.0241834
$628$	−13.2111	−0.527180
$629$	−7.21110	−0.287525
$630$	−2.60555	−0.103808
$631$	−4.00000	−0.159237	−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000	−0.159237	−0.0796187	+	0.996825i	$0.525370\pi$
$632$	−0.605551	−0.0240875
$633$	−25.2111	−1.00205
$634$	−6.00000	−0.238290
$635$	13.2111	0.524267
$636$	−1.39445	−0.0552935
$637$	−0.127833	−0.00506494
$638$	6.00000	0.237542
$639$	−1.39445	−0.0551635
$640$	1.00000	0.0395285
$641$	20.2389	0.799387	0.399693	−	0.916649i	$-0.369117\pi$
$641$	20.2389	0.799387	0.399693	+	0.916649i	$0.369117\pi$
$642$	0	0
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	12.0000	0.472866
$645$	5.39445	0.212406
$646$	−0.605551	−0.0238251
$647$	21.2111	0.833894	0.416947	−	0.908931i	$-0.363100\pi$
$647$	21.2111	0.833894	0.416947	+	0.908931i	$0.363100\pi$
$648$	−1.00000	−0.0392837
$649$	10.6056	0.416304
$650$	−0.605551	−0.0237517
$651$	−13.5778	−0.532156
$652$	5.21110	0.204083
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	−8.42221	−0.329334
$655$	18.4222	0.719815
$656$	9.21110	0.359633
$657$	−8.60555	−0.335735
$658$	31.2666	1.21890
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	1.00000	0.0389249
$661$	16.7889	0.653012	0.326506	−	0.945195i	$-0.394129\pi$
$661$	16.7889	0.653012	0.326506	+	0.945195i	$0.394129\pi$
$662$	−2.42221	−0.0941417
$663$	0.605551	0.0235177
$664$	0	0
$665$	1.57779	0.0611843
$666$	7.21110	0.279425
$667$	−27.6333	−1.06997
$668$	−21.2111	−0.820682
$669$	−19.6333	−0.759068
$670$	−10.0000	−0.386334
$671$	−2.00000	−0.0772091
$672$	2.60555	0.100511
$673$	−11.3944	−0.439224	−0.219612	−	0.975587i	$-0.570479\pi$
$673$	−11.3944	−0.439224	−0.219612	+	0.975587i	$0.570479\pi$
$674$	−9.81665	−0.378123
$675$	1.00000	0.0384900
$676$	−12.6333	−0.485896
$677$	−33.6333	−1.29263	−0.646317	−	0.763069i	$-0.723691\pi$
$677$	−33.6333	−1.29263	−0.646317	+	0.763069i	$0.723691\pi$
$678$	−19.8167	−0.761054
$679$	34.4222	1.32100
$680$	1.00000	0.0383482
$681$	−12.0000	−0.459841
$682$	5.21110	0.199543
$683$	6.42221	0.245739	0.122869	−	0.992423i	$-0.460790\pi$
$683$	6.42221	0.245739	0.122869	+	0.992423i	$0.460790\pi$
$684$	0.605551	0.0231538
$685$	−6.00000	−0.229248
$686$	−18.7889	−0.717363
$687$	−0.788897	−0.0300983
$688$	−5.39445	−0.205661
$689$	−0.844410	−0.0321695
$690$	−4.60555	−0.175330
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	−12.4222	−0.472221
$693$	2.60555	0.0989767
$694$	−24.8444	−0.943081
$695$	10.4222	0.395337
$696$	−6.00000	−0.227429
$697$	9.21110	0.348895
$698$	30.2389	1.14456
$699$	−15.2111	−0.575337
$700$	−2.60555	−0.0984806
$701$	3.63331	0.137228	0.0686141	−	0.997643i	$-0.478142\pi$
$701$	3.63331	0.137228	0.0686141	+	0.997643i	$0.478142\pi$
$702$	−0.605551	−0.0228551
$703$	−4.36669	−0.164693
$704$	−1.00000	−0.0376889
$705$	−12.0000	−0.451946
$706$	−3.21110	−0.120851
$707$	31.2666	1.17590
$708$	−10.6056	−0.398581
$709$	16.7889	0.630520	0.315260	−	0.949005i	$-0.397908\pi$
$709$	16.7889	0.630520	0.315260	+	0.949005i	$0.397908\pi$
$710$	−1.39445	−0.0523327
$711$	0.605551	0.0227099
$712$	−6.00000	−0.224860
$713$	−24.0000	−0.898807
$714$	2.60555	0.0975103
$715$	0.605551	0.0226463
$716$	−7.81665	−0.292122
$717$	15.6333	0.583837
$718$	14.7889	0.551917
$719$	−23.4500	−0.874536	−0.437268	−	0.899331i	$-0.644054\pi$
$719$	−23.4500	−0.874536	−0.437268	+	0.899331i	$0.644054\pi$
$720$	−1.00000	−0.0372678
$721$	−37.5778	−1.39947
$722$	18.6333	0.693460
$723$	−9.02776	−0.335746
$724$	20.4222	0.758985
$725$	6.00000	0.222834
$726$	−1.00000	−0.0371135
$727$	−16.8444	−0.624725	−0.312362	−	0.949963i	$-0.601120\pi$
$727$	−16.8444	−0.624725	−0.312362	+	0.949963i	$0.601120\pi$
$728$	1.57779	0.0584770
$729$	1.00000	0.0370370
$730$	−8.60555	−0.318506
$731$	−5.39445	−0.199521
$732$	2.00000	0.0739221
$733$	−11.3944	−0.420864	−0.210432	−	0.977609i	$-0.567487\pi$
$733$	−11.3944	−0.420864	−0.210432	+	0.977609i	$0.567487\pi$
$734$	22.0000	0.812035
$735$	0.211103	0.00778663
$736$	4.60555	0.169763
$737$	10.0000	0.368355
$738$	−9.21110	−0.339065
$739$	7.02776	0.258520	0.129260	−	0.991611i	$-0.458740\pi$
$739$	7.02776	0.258520	0.129260	+	0.991611i	$0.458740\pi$
$740$	7.21110	0.265085
$741$	0.366692	0.0134708
$742$	−3.63331	−0.133383
$743$	43.2666	1.58730	0.793649	−	0.608376i	$-0.208179\pi$
$743$	43.2666	1.58730	0.793649	+	0.608376i	$0.208179\pi$
$744$	−5.21110	−0.191048
$745$	21.2111	0.777115
$746$	−31.0278	−1.13601
$747$	0	0
$748$	−1.00000	−0.0365636
$749$	0	0
$750$	1.00000	0.0365148
$751$	13.5778	0.495461	0.247730	−	0.968829i	$-0.420315\pi$
$751$	13.5778	0.495461	0.247730	+	0.968829i	$0.420315\pi$
$752$	12.0000	0.437595
$753$	22.6056	0.823792
$754$	−3.63331	−0.132317
$755$	16.0000	0.582300
$756$	−2.60555	−0.0947630
$757$	−9.57779	−0.348111	−0.174055	−	0.984736i	$-0.555687\pi$
$757$	−9.57779	−0.348111	−0.174055	+	0.984736i	$0.555687\pi$
$758$	34.4222	1.25027
$759$	4.60555	0.167171
$760$	0.605551	0.0219657
$761$	−46.4777	−1.68482	−0.842408	−	0.538840i	$-0.818863\pi$
$761$	−46.4777	−1.68482	−0.842408	+	0.538840i	$0.818863\pi$
$762$	13.2111	0.478588
$763$	−21.9445	−0.794444
$764$	12.0000	0.434145
$765$	−1.00000	−0.0361551
$766$	−30.4222	−1.09920
$767$	−6.42221	−0.231892
$768$	1.00000	0.0360844
$769$	28.7889	1.03815	0.519077	−	0.854727i	$-0.326276\pi$
$769$	28.7889	1.03815	0.519077	+	0.854727i	$0.326276\pi$
$770$	2.60555	0.0938976
$771$	21.6333	0.779105
$772$	12.6056	0.453684
$773$	18.9722	0.682384	0.341192	−	0.939994i	$-0.389169\pi$
$773$	18.9722	0.682384	0.341192	+	0.939994i	$0.389169\pi$
$774$	5.39445	0.193899
$775$	5.21110	0.187188
$776$	13.2111	0.474251
$777$	18.7889	0.674048
$778$	−23.0278	−0.825585
$779$	5.57779	0.199845
$780$	−0.605551	−0.0216822
$781$	1.39445	0.0498973
$782$	4.60555	0.164694
$783$	6.00000	0.214423
$784$	−0.211103	−0.00753938
$785$	13.2111	0.471524
$786$	18.4222	0.657098
$787$	−21.5778	−0.769165	−0.384583	−	0.923091i	$-0.625655\pi$
$787$	−21.5778	−0.769165	−0.384583	+	0.923091i	$0.625655\pi$
$788$	−15.2111	−0.541873
$789$	12.0000	0.427211
$790$	0.605551	0.0215445
$791$	−51.6333	−1.83587
$792$	1.00000	0.0355335
$793$	1.21110	0.0430075
$794$	19.2111	0.681777
$795$	1.39445	0.0494560
$796$	5.21110	0.184703
$797$	−2.23886	−0.0793045	−0.0396522	−	0.999214i	$-0.512625\pi$
$797$	−2.23886	−0.0793045	−0.0396522	+	0.999214i	$0.512625\pi$
$798$	1.57779	0.0558533
$799$	12.0000	0.424529
$800$	−1.00000	−0.0353553
$801$	6.00000	0.212000
$802$	−25.8167	−0.911618
$803$	8.60555	0.303683
$804$	−10.0000	−0.352673
$805$	−12.0000	−0.422944
$806$	−3.15559	−0.111151
$807$	21.6333	0.761528
$808$	12.0000	0.422159
$809$	−12.8444	−0.451585	−0.225793	−	0.974175i	$-0.572497\pi$
$809$	−12.8444	−0.451585	−0.225793	+	0.974175i	$0.572497\pi$
$810$	1.00000	0.0351364
$811$	−1.21110	−0.0425276	−0.0212638	−	0.999774i	$-0.506769\pi$
$811$	−1.21110	−0.0425276	−0.0212638	+	0.999774i	$0.506769\pi$
$812$	−15.6333	−0.548622
$813$	−1.21110	−0.0424752
$814$	−7.21110	−0.252749
$815$	−5.21110	−0.182537
$816$	1.00000	0.0350070
$817$	−3.26662	−0.114284
$818$	−14.0000	−0.489499
$819$	−1.57779	−0.0551326
$820$	−9.21110	−0.321666
$821$	18.8444	0.657674	0.328837	−	0.944387i	$-0.393343\pi$
$821$	18.8444	0.657674	0.328837	+	0.944387i	$0.393343\pi$
$822$	−6.00000	−0.209274
$823$	5.63331	0.196365	0.0981824	−	0.995168i	$-0.468697\pi$
$823$	5.63331	0.196365	0.0981824	+	0.995168i	$0.468697\pi$
$824$	−14.4222	−0.502421
$825$	−1.00000	−0.0348155
$826$	−27.6333	−0.961486
$827$	12.8444	0.446644	0.223322	−	0.974745i	$-0.428310\pi$
$827$	12.8444	0.446644	0.223322	+	0.974745i	$0.428310\pi$
$828$	−4.60555	−0.160054
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	0.605551	0.0209937
$833$	−0.211103	−0.00731427
$834$	10.4222	0.360891
$835$	21.2111	0.734040
$836$	−0.605551	−0.0209434
$837$	5.21110	0.180122
$838$	−33.2111	−1.14726
$839$	−31.8167	−1.09843	−0.549216	−	0.835680i	$-0.685074\pi$
$839$	−31.8167	−1.09843	−0.549216	+	0.835680i	$0.685074\pi$
$840$	−2.60555	−0.0899001
$841$	7.00000	0.241379
$842$	22.8444	0.787270
$843$	−3.21110	−0.110596
$844$	−25.2111	−0.867802
$845$	12.6333	0.434599
$846$	−12.0000	−0.412568
$847$	−2.60555	−0.0895278
$848$	−1.39445	−0.0478856
$849$	−13.2111	−0.453404
$850$	−1.00000	−0.0342997
$851$	33.2111	1.13846
$852$	−1.39445	−0.0477730
$853$	−36.7889	−1.25963	−0.629814	−	0.776746i	$-0.716869\pi$
$853$	−36.7889	−1.25963	−0.629814	+	0.776746i	$0.716869\pi$
$854$	5.21110	0.178320
$855$	−0.605551	−0.0207094
$856$	0	0
$857$	−30.0000	−1.02478	−0.512390	−	0.858753i	$-0.671240\pi$
$857$	−30.0000	−1.02478	−0.512390	+	0.858753i	$0.671240\pi$
$858$	0.605551	0.0206732
$859$	−16.0000	−0.545913	−0.272956	−	0.962026i	$-0.588002\pi$
$859$	−16.0000	−0.545913	−0.272956	+	0.962026i	$0.588002\pi$
$860$	5.39445	0.183949
$861$	−24.0000	−0.817918
$862$	−24.4222	−0.831824
$863$	15.6333	0.532164	0.266082	−	0.963950i	$-0.414271\pi$
$863$	15.6333	0.532164	0.266082	+	0.963950i	$0.414271\pi$
$864$	−1.00000	−0.0340207
$865$	12.4222	0.422368
$866$	−20.4222	−0.693975
$867$	1.00000	0.0339618
$868$	−13.5778	−0.460860
$869$	−0.605551	−0.0205419
$870$	6.00000	0.203419
$871$	−6.05551	−0.205183
$872$	−8.42221	−0.285212
$873$	−13.2111	−0.447128
$874$	2.78890	0.0943359
$875$	2.60555	0.0880837
$876$	−8.60555	−0.290755
$877$	48.0555	1.62272	0.811360	−	0.584547i	$-0.198728\pi$
$877$	48.0555	1.62272	0.811360	+	0.584547i	$0.198728\pi$
$878$	−0.605551	−0.0204364
$879$	−30.0000	−1.01187
$880$	1.00000	0.0337100
$881$	−57.0833	−1.92318	−0.961592	−	0.274482i	$-0.911493\pi$
$881$	−57.0833	−1.92318	−0.961592	+	0.274482i	$0.911493\pi$
$882$	0.211103	0.00710819
$883$	−12.7889	−0.430381	−0.215190	−	0.976572i	$-0.569037\pi$
$883$	−12.7889	−0.430381	−0.215190	+	0.976572i	$0.569037\pi$
$884$	0.605551	0.0203669
$885$	10.6056	0.356502
$886$	1.81665	0.0610317
$887$	46.0555	1.54639	0.773196	−	0.634167i	$-0.218657\pi$
$887$	46.0555	1.54639	0.773196	+	0.634167i	$0.218657\pi$
$888$	7.21110	0.241989
$889$	34.4222	1.15448
$890$	6.00000	0.201120
$891$	−1.00000	−0.0335013
$892$	−19.6333	−0.657372
$893$	7.26662	0.243168
$894$	21.2111	0.709405
$895$	7.81665	0.261282
$896$	2.60555	0.0870454
$897$	−2.78890	−0.0931186
$898$	−29.4500	−0.982758
$899$	31.2666	1.04280
$900$	1.00000	0.0333333
$901$	−1.39445	−0.0464558
$902$	9.21110	0.306696
$903$	14.0555	0.467738
$904$	−19.8167	−0.659092
$905$	−20.4222	−0.678857
$906$	16.0000	0.531564
$907$	32.0000	1.06254	0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000	1.06254	0.531271	+	0.847202i	$0.321714\pi$
$908$	−12.0000	−0.398234
$909$	−12.0000	−0.398015
$910$	−1.57779	−0.0523034
$911$	−56.6611	−1.87726	−0.938632	−	0.344919i	$-0.887906\pi$
$911$	−56.6611	−1.87726	−0.938632	+	0.344919i	$0.887906\pi$
$912$	0.605551	0.0200518
$913$	0	0
$914$	22.8444	0.755626
$915$	−2.00000	−0.0661180
$916$	−0.788897	−0.0260659
$917$	48.0000	1.58510
$918$	−1.00000	−0.0330049
$919$	13.5778	0.447890	0.223945	−	0.974602i	$-0.428106\pi$
$919$	13.5778	0.447890	0.223945	+	0.974602i	$0.428106\pi$
$920$	−4.60555	−0.151841
$921$	−33.0278	−1.08830
$922$	−33.2111	−1.09375
$923$	−0.844410	−0.0277941
$924$	2.60555	0.0857163
$925$	−7.21110	−0.237100
$926$	25.2111	0.828488
$927$	14.4222	0.473687
$928$	−6.00000	−0.196960
$929$	−13.8167	−0.453310	−0.226655	−	0.973975i	$-0.572779\pi$
$929$	−13.8167	−0.453310	−0.226655	+	0.973975i	$0.572779\pi$
$930$	5.21110	0.170879
$931$	−0.127833	−0.00418957
$932$	−15.2111	−0.498256
$933$	−26.2389	−0.859022
$934$	32.2389	1.05489
$935$	1.00000	0.0327035
$936$	−0.605551	−0.0197931
$937$	10.3667	0.338665	0.169333	−	0.985559i	$-0.445839\pi$
$937$	10.3667	0.338665	0.169333	+	0.985559i	$0.445839\pi$
$938$	−26.0555	−0.850743
$939$	−18.7889	−0.613152
$940$	−12.0000	−0.391397
$941$	24.4222	0.796141	0.398071	−	0.917355i	$-0.369680\pi$
$941$	24.4222	0.796141	0.398071	+	0.917355i	$0.369680\pi$
$942$	13.2111	0.430441
$943$	−42.4222	−1.38146
$944$	−10.6056	−0.345181
$945$	2.60555	0.0847586
$946$	−5.39445	−0.175389
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0.605551	0.0196674
$949$	−5.21110	−0.169160
$950$	−0.605551	−0.0196467
$951$	6.00000	0.194563
$952$	2.60555	0.0844464
$953$	9.63331	0.312053	0.156027	−	0.987753i	$-0.450131\pi$
$953$	9.63331	0.312053	0.156027	+	0.987753i	$0.450131\pi$
$954$	1.39445	0.0451469
$955$	−12.0000	−0.388311
$956$	15.6333	0.505617
$957$	−6.00000	−0.193952
$958$	−27.2111	−0.879151
$959$	−15.6333	−0.504826
$960$	−1.00000	−0.0322749
$961$	−3.84441	−0.124013
$962$	4.36669	0.140788
$963$	0	0
$964$	−9.02776	−0.290764
$965$	−12.6056	−0.405787
$966$	−12.0000	−0.386094
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	−1.00000	−0.0321412
$969$	0.605551	0.0194531
$970$	−13.2111	−0.424183
$971$	38.2389	1.22714	0.613572	−	0.789639i	$-0.289732\pi$
$971$	38.2389	1.22714	0.613572	+	0.789639i	$0.289732\pi$
$972$	1.00000	0.0320750
$973$	27.1556	0.870568
$974$	28.4222	0.910706
$975$	0.605551	0.0193932
$976$	2.00000	0.0640184
$977$	−20.7889	−0.665096	−0.332548	−	0.943086i	$-0.607908\pi$
$977$	−20.7889	−0.665096	−0.332548	+	0.943086i	$0.607908\pi$
$978$	−5.21110	−0.166633
$979$	−6.00000	−0.191761
$980$	0.211103	0.00674342
$981$	8.42221	0.268900
$982$	24.4222	0.779344
$983$	31.3944	1.00133	0.500664	−	0.865642i	$-0.333089\pi$
$983$	31.3944	1.00133	0.500664	+	0.865642i	$0.333089\pi$
$984$	−9.21110	−0.293639
$985$	15.2111	0.484666
$986$	−6.00000	−0.191079
$987$	−31.2666	−0.995227
$988$	0.366692	0.0116660
$989$	24.8444	0.790006
$990$	−1.00000	−0.0317821
$991$	−35.2666	−1.12028	−0.560140	−	0.828398i	$-0.689253\pi$
$991$	−35.2666	−1.12028	−0.560140	+	0.828398i	$0.689253\pi$
$992$	−5.21110	−0.165453
$993$	2.42221	0.0768664
$994$	−3.63331	−0.115242
$995$	−5.21110	−0.165203
$996$	0	0
$997$	23.2111	0.735103	0.367551	−	0.930003i	$-0.380196\pi$
$997$	23.2111	0.735103	0.367551	+	0.930003i	$0.380196\pi$
$998$	−38.4222	−1.21623
$999$	−7.21110	−0.228149

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.60555 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.60555 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +0.605551 q^{13} +2.60555 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +0.605551 q^{19} -1.00000 q^{20} -2.60555 q^{21} +1.00000 q^{22} -4.60555 q^{23} -1.00000 q^{24} +1.00000 q^{25} -0.605551 q^{26} +1.00000 q^{27} -2.60555 q^{28} +6.00000 q^{29} +1.00000 q^{30} +5.21110 q^{31} -1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +2.60555 q^{35} +1.00000 q^{36} -7.21110 q^{37} -0.605551 q^{38} +0.605551 q^{39} +1.00000 q^{40} +9.21110 q^{41} +2.60555 q^{42} -5.39445 q^{43} -1.00000 q^{44} -1.00000 q^{45} +4.60555 q^{46} +12.0000 q^{47} +1.00000 q^{48} -0.211103 q^{49} -1.00000 q^{50} +1.00000 q^{51} +0.605551 q^{52} -1.39445 q^{53} -1.00000 q^{54} +1.00000 q^{55} +2.60555 q^{56} +0.605551 q^{57} -6.00000 q^{58} -10.6056 q^{59} -1.00000 q^{60} +2.00000 q^{61} -5.21110 q^{62} -2.60555 q^{63} +1.00000 q^{64} -0.605551 q^{65} +1.00000 q^{66} -10.0000 q^{67} +1.00000 q^{68} -4.60555 q^{69} -2.60555 q^{70} -1.39445 q^{71} -1.00000 q^{72} -8.60555 q^{73} +7.21110 q^{74} +1.00000 q^{75} +0.605551 q^{76} +2.60555 q^{77} -0.605551 q^{78} +0.605551 q^{79} -1.00000 q^{80} +1.00000 q^{81} -9.21110 q^{82} -2.60555 q^{84} -1.00000 q^{85} +5.39445 q^{86} +6.00000 q^{87} +1.00000 q^{88} +6.00000 q^{89} +1.00000 q^{90} -1.57779 q^{91} -4.60555 q^{92} +5.21110 q^{93} -12.0000 q^{94} -0.605551 q^{95} -1.00000 q^{96} -13.2111 q^{97} +0.211103 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{10} - 2 q^{11} + 2 q^{12} - 6 q^{13} - 2 q^{14} - 2 q^{15} + 2 q^{16} + 2 q^{17} - 2 q^{18} - 6 q^{19} - 2 q^{20} + 2 q^{21} + 2 q^{22} - 2 q^{23} - 2 q^{24} + 2 q^{25} + 6 q^{26} + 2 q^{27} + 2 q^{28} + 12 q^{29} + 2 q^{30} - 4 q^{31} - 2 q^{32} - 2 q^{33} - 2 q^{34} - 2 q^{35} + 2 q^{36} + 6 q^{38} - 6 q^{39} + 2 q^{40} + 4 q^{41} - 2 q^{42} - 18 q^{43} - 2 q^{44} - 2 q^{45} + 2 q^{46} + 24 q^{47} + 2 q^{48} + 14 q^{49} - 2 q^{50} + 2 q^{51} - 6 q^{52} - 10 q^{53} - 2 q^{54} + 2 q^{55} - 2 q^{56} - 6 q^{57} - 12 q^{58} - 14 q^{59} - 2 q^{60} + 4 q^{61} + 4 q^{62} + 2 q^{63} + 2 q^{64} + 6 q^{65} + 2 q^{66} - 20 q^{67} + 2 q^{68} - 2 q^{69} + 2 q^{70} - 10 q^{71} - 2 q^{72} - 10 q^{73} + 2 q^{75} - 6 q^{76} - 2 q^{77} + 6 q^{78} - 6 q^{79} - 2 q^{80} + 2 q^{81} - 4 q^{82} + 2 q^{84} - 2 q^{85} + 18 q^{86} + 12 q^{87} + 2 q^{88} + 12 q^{89} + 2 q^{90} - 32 q^{91} - 2 q^{92} - 4 q^{93} - 24 q^{94} + 6 q^{95} - 2 q^{96} - 12 q^{97} - 14 q^{98} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 + 2 * q^10 - 2 * q^11 + 2 * q^12 - 6 * q^13 - 2 * q^14 - 2 * q^15 + 2 * q^16 + 2 * q^17 - 2 * q^18 - 6 * q^19 - 2 * q^20 + 2 * q^21 + 2 * q^22 - 2 * q^23 - 2 * q^24 + 2 * q^25 + 6 * q^26 + 2 * q^27 + 2 * q^28 + 12 * q^29 + 2 * q^30 - 4 * q^31 - 2 * q^32 - 2 * q^33 - 2 * q^34 - 2 * q^35 + 2 * q^36 + 6 * q^38 - 6 * q^39 + 2 * q^40 + 4 * q^41 - 2 * q^42 - 18 * q^43 - 2 * q^44 - 2 * q^45 + 2 * q^46 + 24 * q^47 + 2 * q^48 + 14 * q^49 - 2 * q^50 + 2 * q^51 - 6 * q^52 - 10 * q^53 - 2 * q^54 + 2 * q^55 - 2 * q^56 - 6 * q^57 - 12 * q^58 - 14 * q^59 - 2 * q^60 + 4 * q^61 + 4 * q^62 + 2 * q^63 + 2 * q^64 + 6 * q^65 + 2 * q^66 - 20 * q^67 + 2 * q^68 - 2 * q^69 + 2 * q^70 - 10 * q^71 - 2 * q^72 - 10 * q^73 + 2 * q^75 - 6 * q^76 - 2 * q^77 + 6 * q^78 - 6 * q^79 - 2 * q^80 + 2 * q^81 - 4 * q^82 + 2 * q^84 - 2 * q^85 + 18 * q^86 + 12 * q^87 + 2 * q^88 + 12 * q^89 + 2 * q^90 - 32 * q^91 - 2 * q^92 - 4 * q^93 - 24 * q^94 + 6 * q^95 - 2 * q^96 - 12 * q^97 - 14 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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