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Properties

Label 64.24.a.j.1.3
Level $64$
Weight $24$
Character 64.1
Self dual yes
Analytic conductor $214.531$
Analytic rank $1$
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 = [3,0,32708] 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: \(1\)
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: \( x^{3} - 1841764x - 103489260 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{21}\cdot 3\cdot 5 \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-56.2871\) 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)
 
\(f(q)\) \(=\) \(q+442003. q^{3} +2.22890e7 q^{5} -8.79699e8 q^{7} +1.01223e11 q^{9} +1.55363e12 q^{11} -1.97761e12 q^{13} +9.85181e12 q^{15} -2.09169e14 q^{17} +4.91986e14 q^{19} -3.88830e14 q^{21} -8.38926e15 q^{23} -1.14241e16 q^{25} +3.12947e15 q^{27} -9.20414e16 q^{29} -1.51495e17 q^{31} +6.86711e17 q^{33} -1.96076e16 q^{35} -1.71122e17 q^{37} -8.74110e17 q^{39} -4.58022e17 q^{41} +5.33564e18 q^{43} +2.25617e18 q^{45} +2.08351e19 q^{47} -2.65949e19 q^{49} -9.24534e19 q^{51} -5.59140e19 q^{53} +3.46290e19 q^{55} +2.17459e20 q^{57} -9.00041e19 q^{59} -1.75379e20 q^{61} -8.90461e19 q^{63} -4.40790e19 q^{65} +3.44632e20 q^{67} -3.70808e21 q^{69} +1.58288e21 q^{71} -2.05963e20 q^{73} -5.04950e21 q^{75} -1.36673e21 q^{77} +1.20889e22 q^{79} -8.14626e21 q^{81} +2.39773e21 q^{83} -4.66218e21 q^{85} -4.06826e22 q^{87} -4.69027e22 q^{89} +1.73970e21 q^{91} -6.69610e22 q^{93} +1.09659e22 q^{95} -6.79984e22 q^{97} +1.57264e23 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 32708 q^{3} - 31480650 q^{5} - 993025320 q^{7} + 1389317071 q^{9} - 23441525844 q^{11} - 2019379246962 q^{13} + 4994553094600 q^{15} - 2160517821354 q^{17} + 312191787410964 q^{19} - 825707464014048 q^{21}+ \cdots + 25\!\cdots\!40 q^{99}+O(q^{100}) \) 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
\(3\) 442003. 1.44056 0.720279 − 0.693685i \(-0.244014\pi\)
0.720279 + 0.693685i \(0.244014\pi\)
\(4\) 0 0
\(5\) 2.22890e7 0.204144 0.102072 − 0.994777i \(-0.467453\pi\)
0.102072 + 0.994777i \(0.467453\pi\)
\(6\) 0 0
\(7\) −8.79699e8 −0.168154 −0.0840769 − 0.996459i \(-0.526794\pi\)
−0.0840769 + 0.996459i \(0.526794\pi\)
\(8\) 0 0
\(9\) 1.01223e11 1.07521
\(10\) 0 0
\(11\) 1.55363e12 1.64185 0.820924 − 0.571037i \(-0.193459\pi\)
0.820924 + 0.571037i \(0.193459\pi\)
\(12\) 0 0
\(13\) −1.97761e12 −0.306050 −0.153025 − 0.988222i \(-0.548901\pi\)
−0.153025 + 0.988222i \(0.548901\pi\)
\(14\) 0 0
\(15\) 9.85181e12 0.294081
\(16\) 0 0
\(17\) −2.09169e14 −1.48025 −0.740125 − 0.672470i \(-0.765234\pi\)
−0.740125 + 0.672470i \(0.765234\pi\)
\(18\) 0 0
\(19\) 4.91986e14 0.968916 0.484458 − 0.874814i \(-0.339017\pi\)
0.484458 + 0.874814i \(0.339017\pi\)
\(20\) 0 0
\(21\) −3.88830e14 −0.242235
\(22\) 0 0
\(23\) −8.38926e15 −1.83592 −0.917959 − 0.396675i \(-0.870164\pi\)
−0.917959 + 0.396675i \(0.870164\pi\)
\(24\) 0 0
\(25\) −1.14241e16 −0.958325
\(26\) 0 0
\(27\) 3.12947e15 0.108340
\(28\) 0 0
\(29\) −9.20414e16 −1.40090 −0.700449 − 0.713703i \(-0.747017\pi\)
−0.700449 + 0.713703i \(0.747017\pi\)
\(30\) 0 0
\(31\) −1.51495e17 −1.07087 −0.535436 − 0.844576i \(-0.679853\pi\)
−0.535436 + 0.844576i \(0.679853\pi\)
\(32\) 0 0
\(33\) 6.86711e17 2.36518
\(34\) 0 0
\(35\) −1.96076e16 −0.0343275
\(36\) 0 0
\(37\) −1.71122e17 −0.158120 −0.0790598 − 0.996870i \(-0.525192\pi\)
−0.0790598 + 0.996870i \(0.525192\pi\)
\(38\) 0 0
\(39\) −8.74110e17 −0.440883
\(40\) 0 0
\(41\) −4.58022e17 −0.129979 −0.0649893 − 0.997886i \(-0.520701\pi\)
−0.0649893 + 0.997886i \(0.520701\pi\)
\(42\) 0 0
\(43\) 5.33564e18 0.875587 0.437794 − 0.899075i \(-0.355760\pi\)
0.437794 + 0.899075i \(0.355760\pi\)
\(44\) 0 0
\(45\) 2.25617e18 0.219497
\(46\) 0 0
\(47\) 2.08351e19 1.22933 0.614666 − 0.788787i \(-0.289291\pi\)
0.614666 + 0.788787i \(0.289291\pi\)
\(48\) 0 0
\(49\) −2.65949e19 −0.971724
\(50\) 0 0
\(51\) −9.24534e19 −2.13239
\(52\) 0 0
\(53\) −5.59140e19 −0.828607 −0.414303 − 0.910139i \(-0.635975\pi\)
−0.414303 + 0.910139i \(0.635975\pi\)
\(54\) 0 0
\(55\) 3.46290e19 0.335173
\(56\) 0 0
\(57\) 2.17459e20 1.39578
\(58\) 0 0
\(59\) −9.00041e19 −0.388565 −0.194283 − 0.980946i \(-0.562238\pi\)
−0.194283 + 0.980946i \(0.562238\pi\)
\(60\) 0 0
\(61\) −1.75379e20 −0.516041 −0.258021 − 0.966139i \(-0.583070\pi\)
−0.258021 + 0.966139i \(0.583070\pi\)
\(62\) 0 0
\(63\) −8.90461e19 −0.180800
\(64\) 0 0
\(65\) −4.40790e19 −0.0624782
\(66\) 0 0
\(67\) 3.44632e20 0.344743 0.172372 − 0.985032i \(-0.444857\pi\)
0.172372 + 0.985032i \(0.444857\pi\)
\(68\) 0 0
\(69\) −3.70808e21 −2.64475
\(70\) 0 0
\(71\) 1.58288e21 0.812787 0.406394 − 0.913698i \(-0.366786\pi\)
0.406394 + 0.913698i \(0.366786\pi\)
\(72\) 0 0
\(73\) −2.05963e20 −0.0768380 −0.0384190 − 0.999262i \(-0.512232\pi\)
−0.0384190 + 0.999262i \(0.512232\pi\)
\(74\) 0 0
\(75\) −5.04950e21 −1.38052
\(76\) 0 0
\(77\) −1.36673e21 −0.276083
\(78\) 0 0
\(79\) 1.20889e22 1.81834 0.909172 − 0.416420i \(-0.136716\pi\)
0.909172 + 0.416420i \(0.136716\pi\)
\(80\) 0 0
\(81\) −8.14626e21 −0.919137
\(82\) 0 0
\(83\) 2.39773e21 0.204363 0.102181 − 0.994766i \(-0.467418\pi\)
0.102181 + 0.994766i \(0.467418\pi\)
\(84\) 0 0
\(85\) −4.66218e21 −0.302184
\(86\) 0 0
\(87\) −4.06826e22 −2.01807
\(88\) 0 0
\(89\) −4.69027e22 −1.79148 −0.895742 − 0.444574i \(-0.853355\pi\)
−0.895742 + 0.444574i \(0.853355\pi\)
\(90\) 0 0
\(91\) 1.73970e21 0.0514635
\(92\) 0 0
\(93\) −6.69610e22 −1.54265
\(94\) 0 0
\(95\) 1.09659e22 0.197798
\(96\) 0 0
\(97\) −6.79984e22 −0.965214 −0.482607 − 0.875837i \(-0.660310\pi\)
−0.482607 + 0.875837i \(0.660310\pi\)
\(98\) 0 0
\(99\) 1.57264e23 1.76533
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.j.1.3 3
4.3 odd 2 64.24.a.i.1.1 3
8.3 odd 2 8.24.a.b.1.3 ✓ 3
8.5 even 2 16.24.a.d.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.24.a.b.1.3 ✓ 3 8.3 odd 2
16.24.a.d.1.1 3 8.5 even 2
64.24.a.i.1.1 3 4.3 odd 2
64.24.a.j.1.3 3 1.1 even 1 trivial