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Properties

Label 64.20.a.l.1.3
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 1.3
Root \(37.1696\) 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)
 
\(f(q)\) \(=\) \(q+34307.5 q^{3} -2.59881e6 q^{5} -1.93114e8 q^{7} +1.47441e7 q^{9} -8.32029e8 q^{11} -4.50621e10 q^{13} -8.91588e10 q^{15} +2.73062e11 q^{17} -9.22463e11 q^{19} -6.62527e12 q^{21} +1.26167e13 q^{23} -1.23197e13 q^{25} -3.93685e13 q^{27} -1.27884e14 q^{29} -1.96283e14 q^{31} -2.85448e13 q^{33} +5.01868e14 q^{35} +7.60758e14 q^{37} -1.54597e15 q^{39} +2.82783e15 q^{41} -9.55031e13 q^{43} -3.83173e13 q^{45} -4.05928e15 q^{47} +2.58942e16 q^{49} +9.36809e15 q^{51} -3.19955e16 q^{53} +2.16229e15 q^{55} -3.16474e16 q^{57} +5.99553e16 q^{59} +1.56065e17 q^{61} -2.84731e15 q^{63} +1.17108e17 q^{65} -1.35957e17 q^{67} +4.32847e17 q^{69} +1.18135e17 q^{71} +5.66518e17 q^{73} -4.22657e17 q^{75} +1.60677e17 q^{77} +1.38949e18 q^{79} -1.36777e18 q^{81} -1.99954e18 q^{83} -7.09638e17 q^{85} -4.38737e18 q^{87} -3.16978e18 q^{89} +8.70213e18 q^{91} -6.73399e18 q^{93} +2.39731e18 q^{95} +1.60346e18 q^{97} -1.22676e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 23732 q^{3} - 2140218 q^{5} + 55851720 q^{7} + 646753951 q^{9} + 297392964 q^{11} + 14862401022 q^{13} + 292635653528 q^{15} + 803332464534 q^{17} - 3212269666884 q^{19} - 11192319829728 q^{21} + 24948509305560 q^{23}+ \cdots + 11\!\cdots\!60 q^{99}+O(q^{100}) \) 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
\(3\) 34307.5 1.00632 0.503161 − 0.864192i \(-0.332170\pi\)
0.503161 + 0.864192i \(0.332170\pi\)
\(4\) 0 0
\(5\) −2.59881e6 −0.595059 −0.297529 − 0.954713i \(-0.596163\pi\)
−0.297529 + 0.954713i \(0.596163\pi\)
\(6\) 0 0
\(7\) −1.93114e8 −1.80877 −0.904384 − 0.426719i \(-0.859669\pi\)
−0.904384 + 0.426719i \(0.859669\pi\)
\(8\) 0 0
\(9\) 1.47441e7 0.0126857
\(10\) 0 0
\(11\) −8.32029e8 −0.106392 −0.0531958 − 0.998584i \(-0.516941\pi\)
−0.0531958 + 0.998584i \(0.516941\pi\)
\(12\) 0 0
\(13\) −4.50621e10 −1.17855 −0.589277 − 0.807931i \(-0.700587\pi\)
−0.589277 + 0.807931i \(0.700587\pi\)
\(14\) 0 0
\(15\) −8.91588e10 −0.598821
\(16\) 0 0
\(17\) 2.73062e11 0.558467 0.279233 − 0.960223i \(-0.409920\pi\)
0.279233 + 0.960223i \(0.409920\pi\)
\(18\) 0 0
\(19\) −9.22463e11 −0.655828 −0.327914 − 0.944708i \(-0.606346\pi\)
−0.327914 + 0.944708i \(0.606346\pi\)
\(20\) 0 0
\(21\) −6.62527e12 −1.82020
\(22\) 0 0
\(23\) 1.26167e13 1.46060 0.730299 − 0.683128i \(-0.239381\pi\)
0.730299 + 0.683128i \(0.239381\pi\)
\(24\) 0 0
\(25\) −1.23197e13 −0.645905
\(26\) 0 0
\(27\) −3.93685e13 −0.993557
\(28\) 0 0
\(29\) −1.27884e14 −1.63695 −0.818473 − 0.574545i \(-0.805179\pi\)
−0.818473 + 0.574545i \(0.805179\pi\)
\(30\) 0 0
\(31\) −1.96283e14 −1.33336 −0.666679 − 0.745345i \(-0.732285\pi\)
−0.666679 + 0.745345i \(0.732285\pi\)
\(32\) 0 0
\(33\) −2.85448e13 −0.107064
\(34\) 0 0
\(35\) 5.01868e14 1.07632
\(36\) 0 0
\(37\) 7.60758e14 0.962343 0.481172 − 0.876626i \(-0.340211\pi\)
0.481172 + 0.876626i \(0.340211\pi\)
\(38\) 0 0
\(39\) −1.54597e15 −1.18601
\(40\) 0 0
\(41\) 2.82783e15 1.34898 0.674492 − 0.738282i \(-0.264363\pi\)
0.674492 + 0.738282i \(0.264363\pi\)
\(42\) 0 0
\(43\) −9.55031e13 −0.0289779 −0.0144890 − 0.999895i \(-0.504612\pi\)
−0.0144890 + 0.999895i \(0.504612\pi\)
\(44\) 0 0
\(45\) −3.83173e13 −0.00754876
\(46\) 0 0
\(47\) −4.05928e15 −0.529078 −0.264539 − 0.964375i \(-0.585220\pi\)
−0.264539 + 0.964375i \(0.585220\pi\)
\(48\) 0 0
\(49\) 2.58942e16 2.27164
\(50\) 0 0
\(51\) 9.36809e15 0.561998
\(52\) 0 0
\(53\) −3.19955e16 −1.33189 −0.665944 − 0.746002i \(-0.731971\pi\)
−0.665944 + 0.746002i \(0.731971\pi\)
\(54\) 0 0
\(55\) 2.16229e15 0.0633093
\(56\) 0 0
\(57\) −3.16474e16 −0.659975
\(58\) 0 0
\(59\) 5.99553e16 0.901019 0.450509 − 0.892772i \(-0.351242\pi\)
0.450509 + 0.892772i \(0.351242\pi\)
\(60\) 0 0
\(61\) 1.56065e17 1.70872 0.854361 − 0.519679i \(-0.173949\pi\)
0.854361 + 0.519679i \(0.173949\pi\)
\(62\) 0 0
\(63\) −2.84731e15 −0.0229456
\(64\) 0 0
\(65\) 1.17108e17 0.701309
\(66\) 0 0
\(67\) −1.35957e17 −0.610506 −0.305253 − 0.952271i \(-0.598741\pi\)
−0.305253 + 0.952271i \(0.598741\pi\)
\(68\) 0 0
\(69\) 4.32847e17 1.46983
\(70\) 0 0
\(71\) 1.18135e17 0.305790 0.152895 − 0.988242i \(-0.451140\pi\)
0.152895 + 0.988242i \(0.451140\pi\)
\(72\) 0 0
\(73\) 5.66518e17 1.12628 0.563140 − 0.826361i \(-0.309593\pi\)
0.563140 + 0.826361i \(0.309593\pi\)
\(74\) 0 0
\(75\) −4.22657e17 −0.649989
\(76\) 0 0
\(77\) 1.60677e17 0.192438
\(78\) 0 0
\(79\) 1.38949e18 1.30437 0.652183 − 0.758061i \(-0.273853\pi\)
0.652183 + 0.758061i \(0.273853\pi\)
\(80\) 0 0
\(81\) −1.36777e18 −1.01252
\(82\) 0 0
\(83\) −1.99954e18 −1.17405 −0.587027 − 0.809567i \(-0.699702\pi\)
−0.587027 + 0.809567i \(0.699702\pi\)
\(84\) 0 0
\(85\) −7.09638e17 −0.332320
\(86\) 0 0
\(87\) −4.38737e18 −1.64730
\(88\) 0 0
\(89\) −3.16978e18 −0.959012 −0.479506 − 0.877539i \(-0.659184\pi\)
−0.479506 + 0.877539i \(0.659184\pi\)
\(90\) 0 0
\(91\) 8.70213e18 2.13173
\(92\) 0 0
\(93\) −6.73399e18 −1.34179
\(94\) 0 0
\(95\) 2.39731e18 0.390256
\(96\) 0 0
\(97\) 1.60346e18 0.214154 0.107077 − 0.994251i \(-0.465851\pi\)
0.107077 + 0.994251i \(0.465851\pi\)
\(98\) 0 0
\(99\) −1.22676e16 −0.00134966
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.1.3 3
4.3 odd 2 64.20.a.m.1.1 3
8.3 odd 2 16.20.a.f.1.3 3
8.5 even 2 8.20.a.b.1.1 ✓ 3
24.5 odd 2 72.20.a.f.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.20.a.b.1.1 ✓ 3 8.5 even 2
16.20.a.f.1.3 3 8.3 odd 2
64.20.a.l.1.3 3 1.1 even 1 trivial
64.20.a.m.1.1 3 4.3 odd 2
72.20.a.f.1.2 3 24.5 odd 2