Properties

Label 65.2.a.a.1.1
Level $65$
Weight $2$
Character 65.1
Self dual yes
Analytic conductor $0.519$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [65,2,Mod(1,65)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("65.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(65, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 65.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.519027613138\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 65.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -4.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.00000 q^{11} +2.00000 q^{12} -1.00000 q^{13} +4.00000 q^{14} +2.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -6.00000 q^{19} +1.00000 q^{20} +8.00000 q^{21} -2.00000 q^{22} -6.00000 q^{23} -6.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} +4.00000 q^{27} +4.00000 q^{28} +2.00000 q^{29} -2.00000 q^{30} -10.0000 q^{31} -5.00000 q^{32} -4.00000 q^{33} -2.00000 q^{34} +4.00000 q^{35} -1.00000 q^{36} -2.00000 q^{37} +6.00000 q^{38} +2.00000 q^{39} -3.00000 q^{40} -6.00000 q^{41} -8.00000 q^{42} +10.0000 q^{43} -2.00000 q^{44} -1.00000 q^{45} +6.00000 q^{46} +4.00000 q^{47} +2.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} -4.00000 q^{51} +1.00000 q^{52} +2.00000 q^{53} -4.00000 q^{54} -2.00000 q^{55} -12.0000 q^{56} +12.0000 q^{57} -2.00000 q^{58} +6.00000 q^{59} -2.00000 q^{60} +2.00000 q^{61} +10.0000 q^{62} -4.00000 q^{63} +7.00000 q^{64} +1.00000 q^{65} +4.00000 q^{66} -4.00000 q^{67} -2.00000 q^{68} +12.0000 q^{69} -4.00000 q^{70} +6.00000 q^{71} +3.00000 q^{72} -6.00000 q^{73} +2.00000 q^{74} -2.00000 q^{75} +6.00000 q^{76} -8.00000 q^{77} -2.00000 q^{78} -12.0000 q^{79} +1.00000 q^{80} -11.0000 q^{81} +6.00000 q^{82} -16.0000 q^{83} -8.00000 q^{84} -2.00000 q^{85} -10.0000 q^{86} -4.00000 q^{87} +6.00000 q^{88} +2.00000 q^{89} +1.00000 q^{90} +4.00000 q^{91} +6.00000 q^{92} +20.0000 q^{93} -4.00000 q^{94} +6.00000 q^{95} +10.0000 q^{96} -2.00000 q^{97} -9.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) −2.00000 −1.15470 −0.577350 − 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) −1.00000 −0.500000
\(5\) −1.00000 −0.447214
\(6\) 2.00000 0.816497
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 2.00000 0.577350
\(13\) −1.00000 −0.277350
\(14\) 4.00000 1.06904
\(15\) 2.00000 0.516398
\(16\) −1.00000 −0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.00000 −0.235702
\(19\) −6.00000 −1.37649 −0.688247 − 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 1.00000 0.223607
\(21\) 8.00000 1.74574
\(22\) −2.00000 −0.426401
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) −6.00000 −1.22474
\(25\) 1.00000 0.200000
\(26\) 1.00000 0.196116
\(27\) 4.00000 0.769800
\(28\) 4.00000 0.755929
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −2.00000 −0.365148
\(31\) −10.0000 −1.79605 −0.898027 − 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) −5.00000 −0.883883
\(33\) −4.00000 −0.696311
\(34\) −2.00000 −0.342997
\(35\) 4.00000 0.676123
\(36\) −1.00000 −0.166667
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 6.00000 0.973329
\(39\) 2.00000 0.320256
\(40\) −3.00000 −0.474342
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −8.00000 −1.23443
\(43\) 10.0000 1.52499 0.762493 − 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) −2.00000 −0.301511
\(45\) −1.00000 −0.149071
\(46\) 6.00000 0.884652
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 2.00000 0.288675
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) −4.00000 −0.560112
\(52\) 1.00000 0.138675
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −4.00000 −0.544331
\(55\) −2.00000 −0.269680
\(56\) −12.0000 −1.60357
\(57\) 12.0000 1.58944
\(58\) −2.00000 −0.262613
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) −2.00000 −0.258199
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 10.0000 1.27000
\(63\) −4.00000 −0.503953
\(64\) 7.00000 0.875000
\(65\) 1.00000 0.124035
\(66\) 4.00000 0.492366
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −2.00000 −0.242536
\(69\) 12.0000 1.44463
\(70\) −4.00000 −0.478091
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 3.00000 0.353553
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 2.00000 0.232495
\(75\) −2.00000 −0.230940
\(76\) 6.00000 0.688247
\(77\) −8.00000 −0.911685
\(78\) −2.00000 −0.226455
\(79\) −12.0000 −1.35011 −0.675053 − 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 1.00000 0.111803
\(81\) −11.0000 −1.22222
\(82\) 6.00000 0.662589
\(83\) −16.0000 −1.75623 −0.878114 − 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) −8.00000 −0.872872
\(85\) −2.00000 −0.216930
\(86\) −10.0000 −1.07833
\(87\) −4.00000 −0.428845
\(88\) 6.00000 0.639602
\(89\) 2.00000 0.212000 0.106000 − 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 1.00000 0.105409
\(91\) 4.00000 0.419314
\(92\) 6.00000 0.625543
\(93\) 20.0000 2.07390
\(94\) −4.00000 −0.412568
\(95\) 6.00000 0.615587
\(96\) 10.0000 1.02062
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −9.00000 −0.909137
\(99\) 2.00000 0.201008
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 65.2.a.a.1.1 ✓ 1
3.2 odd 2 585.2.a.h.1.1 1
4.3 odd 2 1040.2.a.f.1.1 1
5.2 odd 4 325.2.b.b.274.1 2
5.3 odd 4 325.2.b.b.274.2 2
5.4 even 2 325.2.a.d.1.1 1
7.6 odd 2 3185.2.a.e.1.1 1
8.3 odd 2 4160.2.a.f.1.1 1
8.5 even 2 4160.2.a.q.1.1 1
11.10 odd 2 7865.2.a.c.1.1 1
12.11 even 2 9360.2.a.ca.1.1 1
13.2 odd 12 845.2.m.b.316.1 4
13.3 even 3 845.2.e.b.191.1 2
13.4 even 6 845.2.e.a.146.1 2
13.5 odd 4 845.2.c.a.506.2 2
13.6 odd 12 845.2.m.b.361.2 4
13.7 odd 12 845.2.m.b.361.1 4
13.8 odd 4 845.2.c.a.506.1 2
13.9 even 3 845.2.e.b.146.1 2
13.10 even 6 845.2.e.a.191.1 2
13.11 odd 12 845.2.m.b.316.2 4
13.12 even 2 845.2.a.a.1.1 1
15.2 even 4 2925.2.c.h.2224.2 2
15.8 even 4 2925.2.c.h.2224.1 2
15.14 odd 2 2925.2.a.f.1.1 1
20.19 odd 2 5200.2.a.d.1.1 1
39.38 odd 2 7605.2.a.f.1.1 1
65.64 even 2 4225.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
325.2.a.d.1.1 1 5.4 even 2
325.2.b.b.274.1 2 5.2 odd 4
325.2.b.b.274.2 2 5.3 odd 4
585.2.a.h.1.1 1 3.2 odd 2
845.2.a.a.1.1 1 13.12 even 2
845.2.c.a.506.1 2 13.8 odd 4
845.2.c.a.506.2 2 13.5 odd 4
845.2.e.a.146.1 2 13.4 even 6
845.2.e.a.191.1 2 13.10 even 6
845.2.e.b.146.1 2 13.9 even 3
845.2.e.b.191.1 2 13.3 even 3
845.2.m.b.316.1 4 13.2 odd 12
845.2.m.b.316.2 4 13.11 odd 12
845.2.m.b.361.1 4 13.7 odd 12
845.2.m.b.361.2 4 13.6 odd 12
1040.2.a.f.1.1 1 4.3 odd 2
2925.2.a.f.1.1 1 15.14 odd 2
2925.2.c.h.2224.1 2 15.8 even 4
2925.2.c.h.2224.2 2 15.2 even 4
3185.2.a.e.1.1 1 7.6 odd 2
4160.2.a.f.1.1 1 8.3 odd 2
4160.2.a.q.1.1 1 8.5 even 2
4225.2.a.g.1.1 1 65.64 even 2
5200.2.a.d.1.1 1 20.19 odd 2
7605.2.a.f.1.1 1 39.38 odd 2
7865.2.a.c.1.1 1 11.10 odd 2
9360.2.a.ca.1.1 1 12.11 even 2