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Properties

Label 64.24.a.k.1.2
Level $64$
Weight $24$
Character 64.1
Self dual yes
Analytic conductor $214.531$
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] = [64,24,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 1.2
Root \(-65.1225\) 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)
 
\(f(q)\) \(=\) \(q-3705.18 q^{3} +6.91268e7 q^{5} +6.89672e9 q^{7} -9.41295e10 q^{9} -4.43283e11 q^{11} +7.05085e11 q^{13} -2.56127e11 q^{15} -2.06386e13 q^{17} +2.50525e13 q^{19} -2.55536e13 q^{21} -3.34652e15 q^{23} -7.14241e15 q^{25} +6.97583e14 q^{27} +8.58309e16 q^{29} -2.39339e17 q^{31} +1.64244e15 q^{33} +4.76749e17 q^{35} +2.03171e18 q^{37} -2.61247e15 q^{39} +2.96442e18 q^{41} +4.96274e18 q^{43} -6.50687e18 q^{45} +1.94347e19 q^{47} +2.01961e19 q^{49} +7.64695e16 q^{51} -9.94325e19 q^{53} -3.06427e19 q^{55} -9.28240e16 q^{57} +2.50005e20 q^{59} -1.62386e20 q^{61} -6.49185e20 q^{63} +4.87403e19 q^{65} +1.42919e21 q^{67} +1.23995e19 q^{69} -6.49160e20 q^{71} +3.52414e21 q^{73} +2.64639e19 q^{75} -3.05720e21 q^{77} -2.14128e21 q^{79} +8.85906e21 q^{81} -1.39600e22 q^{83} -1.42668e21 q^{85} -3.18019e20 q^{87} -3.87945e22 q^{89} +4.86278e21 q^{91} +8.86793e20 q^{93} +1.73180e21 q^{95} -7.34335e22 q^{97} +4.17259e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 213948 q^{3} - 95628618 q^{5} - 8647912920 q^{7} + 174509823951 q^{9} - 35420906796 q^{11} - 3164858452338 q^{13} - 19825526344392 q^{15} - 30233487828906 q^{17} + 382754784400236 q^{19} - 27788918984928 q^{21}+ \cdots - 19\!\cdots\!00 q^{99}+O(q^{100}) \) 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
\(3\) −3705.18 −0.0120758 −0.00603788 − 0.999982i \(-0.501922\pi\)
−0.00603788 + 0.999982i \(0.501922\pi\)
\(4\) 0 0
\(5\) 6.91268e7 0.633128 0.316564 − 0.948571i \(-0.397471\pi\)
0.316564 + 0.948571i \(0.397471\pi\)
\(6\) 0 0
\(7\) 6.89672e9 1.31830 0.659152 − 0.752010i \(-0.270915\pi\)
0.659152 + 0.752010i \(0.270915\pi\)
\(8\) 0 0
\(9\) −9.41295e10 −0.999854
\(10\) 0 0
\(11\) −4.43283e11 −0.468452 −0.234226 − 0.972182i \(-0.575256\pi\)
−0.234226 + 0.972182i \(0.575256\pi\)
\(12\) 0 0
\(13\) 7.05085e11 0.109117 0.0545586 − 0.998511i \(-0.482625\pi\)
0.0545586 + 0.998511i \(0.482625\pi\)
\(14\) 0 0
\(15\) −2.56127e11 −0.00764550
\(16\) 0 0
\(17\) −2.06386e13 −0.146055 −0.0730276 − 0.997330i \(-0.523266\pi\)
−0.0730276 + 0.997330i \(0.523266\pi\)
\(18\) 0 0
\(19\) 2.50525e13 0.0493384 0.0246692 − 0.999696i \(-0.492147\pi\)
0.0246692 + 0.999696i \(0.492147\pi\)
\(20\) 0 0
\(21\) −2.55536e13 −0.0159195
\(22\) 0 0
\(23\) −3.34652e15 −0.732358 −0.366179 − 0.930544i \(-0.619334\pi\)
−0.366179 + 0.930544i \(0.619334\pi\)
\(24\) 0 0
\(25\) −7.14241e15 −0.599149
\(26\) 0 0
\(27\) 6.97583e14 0.0241498
\(28\) 0 0
\(29\) 8.58309e16 1.30637 0.653186 − 0.757198i \(-0.273432\pi\)
0.653186 + 0.757198i \(0.273432\pi\)
\(30\) 0 0
\(31\) −2.39339e17 −1.69182 −0.845910 − 0.533326i \(-0.820942\pi\)
−0.845910 + 0.533326i \(0.820942\pi\)
\(32\) 0 0
\(33\) 1.64244e15 0.00565691
\(34\) 0 0
\(35\) 4.76749e17 0.834655
\(36\) 0 0
\(37\) 2.03171e18 1.87734 0.938670 − 0.344817i \(-0.112059\pi\)
0.938670 + 0.344817i \(0.112059\pi\)
\(38\) 0 0
\(39\) −2.61247e15 −0.00131767
\(40\) 0 0
\(41\) 2.96442e18 0.841251 0.420626 − 0.907234i \(-0.361811\pi\)
0.420626 + 0.907234i \(0.361811\pi\)
\(42\) 0 0
\(43\) 4.96274e18 0.814393 0.407197 − 0.913341i \(-0.366506\pi\)
0.407197 + 0.913341i \(0.366506\pi\)
\(44\) 0 0
\(45\) −6.50687e18 −0.633036
\(46\) 0 0
\(47\) 1.94347e19 1.14671 0.573355 − 0.819307i \(-0.305642\pi\)
0.573355 + 0.819307i \(0.305642\pi\)
\(48\) 0 0
\(49\) 2.01961e19 0.737924
\(50\) 0 0
\(51\) 7.64695e16 0.00176373
\(52\) 0 0
\(53\) −9.94325e19 −1.47352 −0.736759 − 0.676155i \(-0.763645\pi\)
−0.736759 + 0.676155i \(0.763645\pi\)
\(54\) 0 0
\(55\) −3.06427e19 −0.296590
\(56\) 0 0
\(57\) −9.28240e16 −0.000595799 0
\(58\) 0 0
\(59\) 2.50005e20 1.07932 0.539660 − 0.841883i \(-0.318553\pi\)
0.539660 + 0.841883i \(0.318553\pi\)
\(60\) 0 0
\(61\) −1.62386e20 −0.477810 −0.238905 − 0.971043i \(-0.576788\pi\)
−0.238905 + 0.971043i \(0.576788\pi\)
\(62\) 0 0
\(63\) −6.49185e20 −1.31811
\(64\) 0 0
\(65\) 4.87403e19 0.0690852
\(66\) 0 0
\(67\) 1.42919e21 1.42965 0.714825 − 0.699304i \(-0.246507\pi\)
0.714825 + 0.699304i \(0.246507\pi\)
\(68\) 0 0
\(69\) 1.23995e19 0.00884378
\(70\) 0 0
\(71\) −6.49160e20 −0.333335 −0.166667 − 0.986013i \(-0.553301\pi\)
−0.166667 + 0.986013i \(0.553301\pi\)
\(72\) 0 0
\(73\) 3.52414e21 1.31474 0.657370 − 0.753568i \(-0.271669\pi\)
0.657370 + 0.753568i \(0.271669\pi\)
\(74\) 0 0
\(75\) 2.64639e19 0.00723518
\(76\) 0 0
\(77\) −3.05720e21 −0.617562
\(78\) 0 0
\(79\) −2.14128e21 −0.322079 −0.161040 − 0.986948i \(-0.551485\pi\)
−0.161040 + 0.986948i \(0.551485\pi\)
\(80\) 0 0
\(81\) 8.85906e21 0.999563
\(82\) 0 0
\(83\) −1.39600e22 −1.18984 −0.594920 − 0.803785i \(-0.702816\pi\)
−0.594920 + 0.803785i \(0.702816\pi\)
\(84\) 0 0
\(85\) −1.42668e21 −0.0924716
\(86\) 0 0
\(87\) −3.18019e20 −0.0157754
\(88\) 0 0
\(89\) −3.87945e22 −1.48179 −0.740893 − 0.671624i \(-0.765597\pi\)
−0.740893 + 0.671624i \(0.765597\pi\)
\(90\) 0 0
\(91\) 4.86278e21 0.143850
\(92\) 0 0
\(93\) 8.86793e20 0.0204300
\(94\) 0 0
\(95\) 1.73180e21 0.0312375
\(96\) 0 0
\(97\) −7.34335e22 −1.04236 −0.521182 − 0.853446i \(-0.674509\pi\)
−0.521182 + 0.853446i \(0.674509\pi\)
\(98\) 0 0
\(99\) 4.17259e22 0.468383
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.1.2 3
4.3 odd 2 64.24.a.h.1.2 3
8.3 odd 2 16.24.a.e.1.2 3
8.5 even 2 8.24.a.a.1.2 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.24.a.a.1.2 ✓ 3 8.5 even 2
16.24.a.e.1.2 3 8.3 odd 2
64.24.a.h.1.2 3 4.3 odd 2
64.24.a.k.1.2 3 1.1 even 1 trivial