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Properties

Label 64.20.a.l.1.2
Level $64$
Weight $20$
Character 64.1
Self dual yes
Analytic conductor $146.443$
Analytic rank $0$
Dimension $3$
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] = [64,20,Mod(1,64)] 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("64.1"); S:= CuspForms(chi, 20); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(64, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 20, names="a")
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 64.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-23732,0,-2140218] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(146.442685796\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2519x + 43659 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-56.8359\) of defining polynomial
Character \(\chi\) \(=\) 64.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3798.34 q^{3} +8.06198e6 q^{5} +1.77174e8 q^{7} -1.14783e9 q^{9} -3.21682e9 q^{11} -1.43866e10 q^{13} -3.06221e10 q^{15} +4.05993e11 q^{17} -1.96189e12 q^{19} -6.72966e11 q^{21} +7.72431e12 q^{23} +4.59220e13 q^{25} +8.77452e12 q^{27} +5.61183e13 q^{29} +1.62831e14 q^{31} +1.22186e13 q^{33} +1.42837e15 q^{35} -3.64918e14 q^{37} +5.46452e13 q^{39} -1.20992e15 q^{41} +3.11040e15 q^{43} -9.25382e15 q^{45} -1.96388e15 q^{47} +1.99917e16 q^{49} -1.54210e15 q^{51} -3.16153e15 q^{53} -2.59340e16 q^{55} +7.45193e15 q^{57} -1.75133e16 q^{59} +6.75521e16 q^{61} -2.03366e17 q^{63} -1.15985e17 q^{65} -1.62877e17 q^{67} -2.93395e16 q^{69} +6.94114e17 q^{71} -6.40681e17 q^{73} -1.74427e17 q^{75} -5.69937e17 q^{77} +1.59338e18 q^{79} +1.30075e18 q^{81} +2.39697e18 q^{83} +3.27310e18 q^{85} -2.13156e17 q^{87} -6.86597e17 q^{89} -2.54893e18 q^{91} -6.18486e17 q^{93} -1.58167e19 q^{95} +5.24866e18 q^{97} +3.69238e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 23732 q^{3} - 2140218 q^{5} + 55851720 q^{7} + 646753951 q^{9} + 297392964 q^{11} + 14862401022 q^{13} + 292635653528 q^{15} + 803332464534 q^{17} - 3212269666884 q^{19} - 11192319829728 q^{21} + 24948509305560 q^{23}+ \cdots + 11\!\cdots\!60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3798.34 −0.111414 −0.0557072 − 0.998447i \(-0.517741\pi\)
−0.0557072 + 0.998447i \(0.517741\pi\)
\(4\) 0 0
\(5\) 8.06198e6 1.84598 0.922989 − 0.384826i \(-0.125738\pi\)
0.922989 + 0.384826i \(0.125738\pi\)
\(6\) 0 0
\(7\) 1.77174e8 1.65947 0.829733 − 0.558160i \(-0.188493\pi\)
0.829733 + 0.558160i \(0.188493\pi\)
\(8\) 0 0
\(9\) −1.14783e9 −0.987587
\(10\) 0 0
\(11\) −3.21682e9 −0.411336 −0.205668 − 0.978622i \(-0.565937\pi\)
−0.205668 + 0.978622i \(0.565937\pi\)
\(12\) 0 0
\(13\) −1.43866e10 −0.376268 −0.188134 − 0.982143i \(-0.560244\pi\)
−0.188134 + 0.982143i \(0.560244\pi\)
\(14\) 0 0
\(15\) −3.06221e10 −0.205669
\(16\) 0 0
\(17\) 4.05993e11 0.830335 0.415168 − 0.909745i \(-0.363723\pi\)
0.415168 + 0.909745i \(0.363723\pi\)
\(18\) 0 0
\(19\) −1.96189e12 −1.39481 −0.697406 − 0.716676i \(-0.745663\pi\)
−0.697406 + 0.716676i \(0.745663\pi\)
\(20\) 0 0
\(21\) −6.72966e11 −0.184889
\(22\) 0 0
\(23\) 7.72431e12 0.894222 0.447111 − 0.894479i \(-0.352453\pi\)
0.447111 + 0.894479i \(0.352453\pi\)
\(24\) 0 0
\(25\) 4.59220e13 2.40764
\(26\) 0 0
\(27\) 8.77452e12 0.221446
\(28\) 0 0
\(29\) 5.61183e13 0.718330 0.359165 − 0.933274i \(-0.383062\pi\)
0.359165 + 0.933274i \(0.383062\pi\)
\(30\) 0 0
\(31\) 1.62831e14 1.10612 0.553058 − 0.833143i \(-0.313461\pi\)
0.553058 + 0.833143i \(0.313461\pi\)
\(32\) 0 0
\(33\) 1.22186e13 0.0458288
\(34\) 0 0
\(35\) 1.42837e15 3.06334
\(36\) 0 0
\(37\) −3.64918e14 −0.461615 −0.230807 − 0.972999i \(-0.574137\pi\)
−0.230807 + 0.972999i \(0.574137\pi\)
\(38\) 0 0
\(39\) 5.46452e13 0.0419217
\(40\) 0 0
\(41\) −1.20992e15 −0.577179 −0.288590 − 0.957453i \(-0.593186\pi\)
−0.288590 + 0.957453i \(0.593186\pi\)
\(42\) 0 0
\(43\) 3.11040e15 0.943768 0.471884 − 0.881661i \(-0.343574\pi\)
0.471884 + 0.881661i \(0.343574\pi\)
\(44\) 0 0
\(45\) −9.25382e15 −1.82306
\(46\) 0 0
\(47\) −1.96388e15 −0.255967 −0.127984 − 0.991776i \(-0.540851\pi\)
−0.127984 + 0.991776i \(0.540851\pi\)
\(48\) 0 0
\(49\) 1.99917e16 1.75383
\(50\) 0 0
\(51\) −1.54210e15 −0.0925114
\(52\) 0 0
\(53\) −3.16153e15 −0.131606 −0.0658032 − 0.997833i \(-0.520961\pi\)
−0.0658032 + 0.997833i \(0.520961\pi\)
\(54\) 0 0
\(55\) −2.59340e16 −0.759317
\(56\) 0 0
\(57\) 7.45193e15 0.155402
\(58\) 0 0
\(59\) −1.75133e16 −0.263193 −0.131596 − 0.991303i \(-0.542010\pi\)
−0.131596 + 0.991303i \(0.542010\pi\)
\(60\) 0 0
\(61\) 6.75521e16 0.739615 0.369807 − 0.929108i \(-0.379424\pi\)
0.369807 + 0.929108i \(0.379424\pi\)
\(62\) 0 0
\(63\) −2.03366e17 −1.63887
\(64\) 0 0
\(65\) −1.15985e17 −0.694582
\(66\) 0 0
\(67\) −1.62877e17 −0.731391 −0.365696 − 0.930734i \(-0.619169\pi\)
−0.365696 + 0.930734i \(0.619169\pi\)
\(68\) 0 0
\(69\) −2.93395e16 −0.0996293
\(70\) 0 0
\(71\) 6.94114e17 1.79670 0.898352 − 0.439275i \(-0.144765\pi\)
0.898352 + 0.439275i \(0.144765\pi\)
\(72\) 0 0
\(73\) −6.40681e17 −1.27372 −0.636861 − 0.770978i \(-0.719768\pi\)
−0.636861 + 0.770978i \(0.719768\pi\)
\(74\) 0 0
\(75\) −1.74427e17 −0.268246
\(76\) 0 0
\(77\) −5.69937e17 −0.682598
\(78\) 0 0
\(79\) 1.59338e18 1.49576 0.747882 − 0.663832i \(-0.231071\pi\)
0.747882 + 0.663832i \(0.231071\pi\)
\(80\) 0 0
\(81\) 1.30075e18 0.962915
\(82\) 0 0
\(83\) 2.39697e18 1.40741 0.703704 − 0.710493i \(-0.251528\pi\)
0.703704 + 0.710493i \(0.251528\pi\)
\(84\) 0 0
\(85\) 3.27310e18 1.53278
\(86\) 0 0
\(87\) −2.13156e17 −0.0800323
\(88\) 0 0
\(89\) −6.86597e17 −0.207729 −0.103864 − 0.994591i \(-0.533121\pi\)
−0.103864 + 0.994591i \(0.533121\pi\)
\(90\) 0 0
\(91\) −2.54893e18 −0.624404
\(92\) 0 0
\(93\) −6.18486e17 −0.123237
\(94\) 0 0
\(95\) −1.58167e19 −2.57479
\(96\) 0 0
\(97\) 5.24866e18 0.700999 0.350499 − 0.936563i \(-0.386012\pi\)
0.350499 + 0.936563i \(0.386012\pi\)
\(98\) 0 0
\(99\) 3.69238e18 0.406230
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 64.20.a.l.1.2 3
4.3 odd 2 64.20.a.m.1.2 3
8.3 odd 2 16.20.a.f.1.2 3
8.5 even 2 8.20.a.b.1.2 ✓ 3
24.5 odd 2 72.20.a.f.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.20.a.b.1.2 ✓ 3 8.5 even 2
16.20.a.f.1.2 3 8.3 odd 2
64.20.a.l.1.2 3 1.1 even 1 trivial
64.20.a.m.1.2 3 4.3 odd 2
72.20.a.f.1.3 3 24.5 odd 2