Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [64,24,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, 24); 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, 24, names="a")
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 64.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,213948] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(214.530583901\)
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: \( x^{3} - 166408x - 10560732 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(436.575\) 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)
 
\(f(q)\) \(=\) \(q+574251. q^{3} -8.41392e7 q^{5} -5.95738e9 q^{7} +2.35621e11 q^{9} -1.24326e12 q^{11} -7.21479e12 q^{13} -4.83170e13 q^{15} -6.50936e13 q^{17} +5.56139e14 q^{19} -3.42103e15 q^{21} -3.08126e15 q^{23} -4.84153e15 q^{25} +8.12435e16 q^{27} +4.69935e16 q^{29} +2.40563e17 q^{31} -7.13941e17 q^{33} +5.01249e17 q^{35} +5.28193e17 q^{37} -4.14310e18 q^{39} +1.59837e18 q^{41} -7.35994e18 q^{43} -1.98249e19 q^{45} -1.63997e19 q^{47} +8.12160e18 q^{49} -3.73800e19 q^{51} +2.64089e19 q^{53} +1.04607e20 q^{55} +3.19363e20 q^{57} +3.24947e20 q^{59} +3.22520e20 q^{61} -1.40368e21 q^{63} +6.07047e20 q^{65} +1.09896e21 q^{67} -1.76941e21 q^{69} -2.30177e20 q^{71} +8.33706e20 q^{73} -2.78025e21 q^{75} +7.40655e21 q^{77} +2.59992e21 q^{79} +2.44721e22 q^{81} +1.05807e22 q^{83} +5.47692e21 q^{85} +2.69861e22 q^{87} +4.07434e22 q^{89} +4.29812e22 q^{91} +1.38143e23 q^{93} -4.67930e22 q^{95} -9.40029e22 q^{97} -2.92937e23 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 213948 q^{3} - 95628618 q^{5} - 8647912920 q^{7} + 174509823951 q^{9} - 35420906796 q^{11} - 3164858452338 q^{13} - 19825526344392 q^{15} - 30233487828906 q^{17} + 382754784400236 q^{19} - 27788918984928 q^{21}+ \cdots - 19\!\cdots\!00 q^{99}+O(q^{100}) \) 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
\(3\) 574251. 1.87157 0.935787 − 0.352565i \(-0.114691\pi\)
0.935787 + 0.352565i \(0.114691\pi\)
\(4\) 0 0
\(5\) −8.41392e7 −0.770625 −0.385312 − 0.922786i \(-0.625906\pi\)
−0.385312 + 0.922786i \(0.625906\pi\)
\(6\) 0 0
\(7\) −5.95738e9 −1.13875 −0.569374 − 0.822079i \(-0.692814\pi\)
−0.569374 + 0.822079i \(0.692814\pi\)
\(8\) 0 0
\(9\) 2.35621e11 2.50279
\(10\) 0 0
\(11\) −1.24326e12 −1.31385 −0.656924 − 0.753957i \(-0.728143\pi\)
−0.656924 + 0.753957i \(0.728143\pi\)
\(12\) 0 0
\(13\) −7.21479e12 −1.11654 −0.558271 − 0.829658i \(-0.688535\pi\)
−0.558271 + 0.829658i \(0.688535\pi\)
\(14\) 0 0
\(15\) −4.83170e13 −1.44228
\(16\) 0 0
\(17\) −6.50936e13 −0.460655 −0.230327 − 0.973113i \(-0.573980\pi\)
−0.230327 + 0.973113i \(0.573980\pi\)
\(18\) 0 0
\(19\) 5.56139e14 1.09526 0.547629 − 0.836721i \(-0.315530\pi\)
0.547629 + 0.836721i \(0.315530\pi\)
\(20\) 0 0
\(21\) −3.42103e15 −2.13125
\(22\) 0 0
\(23\) −3.08126e15 −0.674307 −0.337154 − 0.941450i \(-0.609464\pi\)
−0.337154 + 0.941450i \(0.609464\pi\)
\(24\) 0 0
\(25\) −4.84153e15 −0.406137
\(26\) 0 0
\(27\) 8.12435e16 2.81258
\(28\) 0 0
\(29\) 4.69935e16 0.715255 0.357628 − 0.933864i \(-0.383586\pi\)
0.357628 + 0.933864i \(0.383586\pi\)
\(30\) 0 0
\(31\) 2.40563e17 1.70047 0.850234 − 0.526405i \(-0.176460\pi\)
0.850234 + 0.526405i \(0.176460\pi\)
\(32\) 0 0
\(33\) −7.13941e17 −2.45896
\(34\) 0 0
\(35\) 5.01249e17 0.877548
\(36\) 0 0
\(37\) 5.28193e17 0.488059 0.244030 − 0.969768i \(-0.421531\pi\)
0.244030 + 0.969768i \(0.421531\pi\)
\(38\) 0 0
\(39\) −4.14310e18 −2.08969
\(40\) 0 0
\(41\) 1.59837e18 0.453589 0.226795 − 0.973943i \(-0.427175\pi\)
0.226795 + 0.973943i \(0.427175\pi\)
\(42\) 0 0
\(43\) −7.35994e18 −1.20778 −0.603889 − 0.797068i \(-0.706383\pi\)
−0.603889 + 0.797068i \(0.706383\pi\)
\(44\) 0 0
\(45\) −1.98249e19 −1.92871
\(46\) 0 0
\(47\) −1.63997e19 −0.967631 −0.483815 − 0.875170i \(-0.660749\pi\)
−0.483815 + 0.875170i \(0.660749\pi\)
\(48\) 0 0
\(49\) 8.12160e18 0.296747
\(50\) 0 0
\(51\) −3.73800e19 −0.862149
\(52\) 0 0
\(53\) 2.64089e19 0.391361 0.195681 − 0.980668i \(-0.437308\pi\)
0.195681 + 0.980668i \(0.437308\pi\)
\(54\) 0 0
\(55\) 1.04607e20 1.01248
\(56\) 0 0
\(57\) 3.19363e20 2.04986
\(58\) 0 0
\(59\) 3.24947e20 1.40286 0.701429 − 0.712739i \(-0.252546\pi\)
0.701429 + 0.712739i \(0.252546\pi\)
\(60\) 0 0
\(61\) 3.22520e20 0.948995 0.474498 − 0.880257i \(-0.342630\pi\)
0.474498 + 0.880257i \(0.342630\pi\)
\(62\) 0 0
\(63\) −1.40368e21 −2.85005
\(64\) 0 0
\(65\) 6.07047e20 0.860436
\(66\) 0 0
\(67\) 1.09896e21 1.09932 0.549658 − 0.835390i \(-0.314758\pi\)
0.549658 + 0.835390i \(0.314758\pi\)
\(68\) 0 0
\(69\) −1.76941e21 −1.26202
\(70\) 0 0
\(71\) −2.30177e20 −0.118193 −0.0590964 − 0.998252i \(-0.518822\pi\)
−0.0590964 + 0.998252i \(0.518822\pi\)
\(72\) 0 0
\(73\) 8.33706e20 0.311028 0.155514 − 0.987834i \(-0.450297\pi\)
0.155514 + 0.987834i \(0.450297\pi\)
\(74\) 0 0
\(75\) −2.78025e21 −0.760116
\(76\) 0 0
\(77\) 7.40655e21 1.49614
\(78\) 0 0
\(79\) 2.59992e21 0.391064 0.195532 − 0.980697i \(-0.437357\pi\)
0.195532 + 0.980697i \(0.437357\pi\)
\(80\) 0 0
\(81\) 2.44721e22 2.76117
\(82\) 0 0
\(83\) 1.05807e22 0.901817 0.450908 − 0.892570i \(-0.351100\pi\)
0.450908 + 0.892570i \(0.351100\pi\)
\(84\) 0 0
\(85\) 5.47692e21 0.354992
\(86\) 0 0
\(87\) 2.69861e22 1.33865
\(88\) 0 0
\(89\) 4.07434e22 1.55622 0.778112 − 0.628126i \(-0.216178\pi\)
0.778112 + 0.628126i \(0.216178\pi\)
\(90\) 0 0
\(91\) 4.29812e22 1.27146
\(92\) 0 0
\(93\) 1.38143e23 3.18255
\(94\) 0 0
\(95\) −4.67930e22 −0.844034
\(96\) 0 0
\(97\) −9.40029e22 −1.33434 −0.667169 − 0.744906i \(-0.732494\pi\)
−0.667169 + 0.744906i \(0.732494\pi\)
\(98\) 0 0
\(99\) −2.92937e23 −3.28828
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 64.24.a.k.1.3 3
4.3 odd 2 64.24.a.h.1.1 3
8.3 odd 2 16.24.a.e.1.3 3
8.5 even 2 8.24.a.a.1.1 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.24.a.a.1.1 ✓ 3 8.5 even 2
16.24.a.e.1.3 3 8.3 odd 2
64.24.a.h.1.1 3 4.3 odd 2
64.24.a.k.1.3 3 1.1 even 1 trivial