Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [64,22,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, 22); 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, 22, names="a")
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 64.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,96764] 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: \(178.865500344\)
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: \( x^{3} - x^{2} - 4963x + 96223 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{21}\cdot 3\cdot 5\cdot 7 \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(21.2235\) 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)
 
\(f(q)\) \(=\) \(q+201833. q^{3} +2.11555e7 q^{5} -7.32981e8 q^{7} +3.02761e10 q^{9} -5.59634e9 q^{11} +6.30207e10 q^{13} +4.26988e12 q^{15} +1.35490e13 q^{17} +1.39028e13 q^{19} -1.47940e14 q^{21} +2.80711e14 q^{23} -2.92803e13 q^{25} +3.99948e15 q^{27} -1.19637e15 q^{29} -3.88487e15 q^{31} -1.12953e15 q^{33} -1.55066e16 q^{35} -2.72004e16 q^{37} +1.27197e16 q^{39} -6.89865e16 q^{41} +3.24557e16 q^{43} +6.40508e17 q^{45} +2.07417e17 q^{47} -2.12844e16 q^{49} +2.73464e18 q^{51} +6.38971e17 q^{53} -1.18394e17 q^{55} +2.80603e18 q^{57} -3.04610e18 q^{59} +5.64497e18 q^{61} -2.21918e19 q^{63} +1.33324e18 q^{65} -3.96718e18 q^{67} +5.66566e19 q^{69} +2.61878e19 q^{71} +1.37659e19 q^{73} -5.90972e18 q^{75} +4.10201e18 q^{77} +1.18258e20 q^{79} +4.90526e20 q^{81} -1.60950e19 q^{83} +2.86637e20 q^{85} -2.41466e20 q^{87} -2.30103e20 q^{89} -4.61930e19 q^{91} -7.84094e20 q^{93} +2.94120e20 q^{95} +2.72050e20 q^{97} -1.69436e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 96764 q^{3} + 24111774 q^{5} - 295988280 q^{7} + 18844697239 q^{9} - 40335108684 q^{11} - 133734425946 q^{13} + 1223136458200 q^{15} + 7797732274422 q^{17} + 35788199781996 q^{19} - 198539224853088 q^{21}+ \cdots - 94\!\cdots\!60 q^{99}+O(q^{100}) \) 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
\(3\) 201833. 1.97342 0.986708 − 0.162503i \(-0.0519569\pi\)
0.986708 + 0.162503i \(0.0519569\pi\)
\(4\) 0 0
\(5\) 2.11555e7 0.968811 0.484406 − 0.874844i \(-0.339036\pi\)
0.484406 + 0.874844i \(0.339036\pi\)
\(6\) 0 0
\(7\) −7.32981e8 −0.980761 −0.490381 − 0.871508i \(-0.663142\pi\)
−0.490381 + 0.871508i \(0.663142\pi\)
\(8\) 0 0
\(9\) 3.02761e10 2.89437
\(10\) 0 0
\(11\) −5.59634e9 −0.0650551 −0.0325275 − 0.999471i \(-0.510356\pi\)
−0.0325275 + 0.999471i \(0.510356\pi\)
\(12\) 0 0
\(13\) 6.30207e10 0.126788 0.0633940 − 0.997989i \(-0.479808\pi\)
0.0633940 + 0.997989i \(0.479808\pi\)
\(14\) 0 0
\(15\) 4.26988e12 1.91187
\(16\) 0 0
\(17\) 1.35490e13 1.63003 0.815013 − 0.579443i \(-0.196730\pi\)
0.815013 + 0.579443i \(0.196730\pi\)
\(18\) 0 0
\(19\) 1.39028e13 0.520221 0.260111 − 0.965579i \(-0.416241\pi\)
0.260111 + 0.965579i \(0.416241\pi\)
\(20\) 0 0
\(21\) −1.47940e14 −1.93545
\(22\) 0 0
\(23\) 2.80711e14 1.41292 0.706458 − 0.707754i \(-0.250292\pi\)
0.706458 + 0.707754i \(0.250292\pi\)
\(24\) 0 0
\(25\) −2.92803e13 −0.0614052
\(26\) 0 0
\(27\) 3.99948e15 3.73838
\(28\) 0 0
\(29\) −1.19637e15 −0.528063 −0.264031 − 0.964514i \(-0.585052\pi\)
−0.264031 + 0.964514i \(0.585052\pi\)
\(30\) 0 0
\(31\) −3.88487e15 −0.851292 −0.425646 − 0.904890i \(-0.639953\pi\)
−0.425646 + 0.904890i \(0.639953\pi\)
\(32\) 0 0
\(33\) −1.12953e15 −0.128381
\(34\) 0 0
\(35\) −1.55066e16 −0.950173
\(36\) 0 0
\(37\) −2.72004e16 −0.929944 −0.464972 − 0.885325i \(-0.653936\pi\)
−0.464972 + 0.885325i \(0.653936\pi\)
\(38\) 0 0
\(39\) 1.27197e16 0.250205
\(40\) 0 0
\(41\) −6.89865e16 −0.802663 −0.401332 − 0.915933i \(-0.631453\pi\)
−0.401332 + 0.915933i \(0.631453\pi\)
\(42\) 0 0
\(43\) 3.24557e16 0.229020 0.114510 − 0.993422i \(-0.463470\pi\)
0.114510 + 0.993422i \(0.463470\pi\)
\(44\) 0 0
\(45\) 6.40508e17 2.80410
\(46\) 0 0
\(47\) 2.07417e17 0.575196 0.287598 − 0.957751i \(-0.407143\pi\)
0.287598 + 0.957751i \(0.407143\pi\)
\(48\) 0 0
\(49\) −2.12844e16 −0.0381069
\(50\) 0 0
\(51\) 2.73464e18 3.21672
\(52\) 0 0
\(53\) 6.38971e17 0.501862 0.250931 − 0.968005i \(-0.419263\pi\)
0.250931 + 0.968005i \(0.419263\pi\)
\(54\) 0 0
\(55\) −1.18394e17 −0.0630261
\(56\) 0 0
\(57\) 2.80603e18 1.02661
\(58\) 0 0
\(59\) −3.04610e18 −0.775887 −0.387944 − 0.921683i \(-0.626814\pi\)
−0.387944 + 0.921683i \(0.626814\pi\)
\(60\) 0 0
\(61\) 5.64497e18 1.01321 0.506604 − 0.862179i \(-0.330901\pi\)
0.506604 + 0.862179i \(0.330901\pi\)
\(62\) 0 0
\(63\) −2.21918e19 −2.83869
\(64\) 0 0
\(65\) 1.33324e18 0.122834
\(66\) 0 0
\(67\) −3.96718e18 −0.265886 −0.132943 − 0.991124i \(-0.542443\pi\)
−0.132943 + 0.991124i \(0.542443\pi\)
\(68\) 0 0
\(69\) 5.66566e19 2.78827
\(70\) 0 0
\(71\) 2.61878e19 0.954743 0.477371 − 0.878702i \(-0.341590\pi\)
0.477371 + 0.878702i \(0.341590\pi\)
\(72\) 0 0
\(73\) 1.37659e19 0.374899 0.187449 − 0.982274i \(-0.439978\pi\)
0.187449 + 0.982274i \(0.439978\pi\)
\(74\) 0 0
\(75\) −5.90972e18 −0.121178
\(76\) 0 0
\(77\) 4.10201e18 0.0638035
\(78\) 0 0
\(79\) 1.18258e20 1.40523 0.702615 − 0.711570i \(-0.252016\pi\)
0.702615 + 0.711570i \(0.252016\pi\)
\(80\) 0 0
\(81\) 4.90526e20 4.48301
\(82\) 0 0
\(83\) −1.60950e19 −0.113860 −0.0569301 − 0.998378i \(-0.518131\pi\)
−0.0569301 + 0.998378i \(0.518131\pi\)
\(84\) 0 0
\(85\) 2.86637e20 1.57919
\(86\) 0 0
\(87\) −2.41466e20 −1.04209
\(88\) 0 0
\(89\) −2.30103e20 −0.782217 −0.391109 − 0.920345i \(-0.627908\pi\)
−0.391109 + 0.920345i \(0.627908\pi\)
\(90\) 0 0
\(91\) −4.61930e19 −0.124349
\(92\) 0 0
\(93\) −7.84094e20 −1.67995
\(94\) 0 0
\(95\) 2.94120e20 0.503996
\(96\) 0 0
\(97\) 2.72050e20 0.374581 0.187290 − 0.982305i \(-0.440029\pi\)
0.187290 + 0.982305i \(0.440029\pi\)
\(98\) 0 0
\(99\) −1.69436e20 −0.188293
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 64.22.a.m.1.3 3
4.3 odd 2 64.22.a.l.1.1 3
8.3 odd 2 8.22.a.b.1.3 ✓ 3
8.5 even 2 16.22.a.f.1.1 3
24.11 even 2 72.22.a.f.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.22.a.b.1.3 ✓ 3 8.3 odd 2
16.22.a.f.1.1 3 8.5 even 2
64.22.a.l.1.1 3 4.3 odd 2
64.22.a.m.1.3 3 1.1 even 1 trivial
72.22.a.f.1.2 3 24.11 even 2