Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9072,2,Mod(1,9072)] 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("9072.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9072, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9072.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,-3,0,-3,0,0,0,6,0,3,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(72.4402847137\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 567)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.523976\) of defining polynomial
Character \(\chi\) \(=\) 9072.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.20147 q^{5} -1.00000 q^{7} +5.20147 q^{11} +3.15352 q^{13} +3.24943 q^{17} -7.45090 q^{19} +4.40294 q^{23} -0.153520 q^{25} +1.15352 q^{29} -2.00000 q^{31} -2.20147 q^{35} +5.00000 q^{37} +11.4509 q^{41} -9.29738 q^{43} -1.04795 q^{47} +1.00000 q^{49} +0.249425 q^{53} +11.4509 q^{55} -8.09591 q^{59} +8.60442 q^{61} +6.94239 q^{65} +7.60442 q^{67} +9.60442 q^{71} +0.846480 q^{73} -5.20147 q^{77} +7.60442 q^{79} +11.4509 q^{83} +7.15352 q^{85} -9.24943 q^{89} -3.15352 q^{91} -16.4029 q^{95} -3.45090 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} - 3 q^{7} + 6 q^{11} + 3 q^{13} - 3 q^{17} - 6 q^{23} + 6 q^{25} - 3 q^{29} - 6 q^{31} + 3 q^{35} + 15 q^{37} + 12 q^{41} - 12 q^{43} + 3 q^{49} - 12 q^{53} + 12 q^{55} - 18 q^{59} - 3 q^{61}+ \cdots + 12 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) 2.20147 0.984528 0.492264 − 0.870446i \(-0.336169\pi\)
0.492264 + 0.870446i \(0.336169\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.20147 1.56830 0.784151 − 0.620569i \(-0.213099\pi\)
0.784151 + 0.620569i \(0.213099\pi\)
\(12\) 0 0
\(13\) 3.15352 0.874629 0.437314 − 0.899309i \(-0.355930\pi\)
0.437314 + 0.899309i \(0.355930\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.24943 0.788101 0.394051 − 0.919089i \(-0.371073\pi\)
0.394051 + 0.919089i \(0.371073\pi\)
\(18\) 0 0
\(19\) −7.45090 −1.70935 −0.854677 − 0.519161i \(-0.826245\pi\)
−0.854677 + 0.519161i \(0.826245\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.40294 0.918077 0.459039 − 0.888416i \(-0.348194\pi\)
0.459039 + 0.888416i \(0.348194\pi\)
\(24\) 0 0
\(25\) −0.153520 −0.0307039
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.15352 0.214203 0.107102 − 0.994248i \(-0.465843\pi\)
0.107102 + 0.994248i \(0.465843\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.20147 −0.372117
\(36\) 0 0
\(37\) 5.00000 0.821995 0.410997 − 0.911636i \(-0.365181\pi\)
0.410997 + 0.911636i \(0.365181\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.4509 1.78833 0.894165 − 0.447738i \(-0.147770\pi\)
0.894165 + 0.447738i \(0.147770\pi\)
\(42\) 0 0
\(43\) −9.29738 −1.41784 −0.708918 − 0.705290i \(-0.750817\pi\)
−0.708918 + 0.705290i \(0.750817\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.04795 −0.152860 −0.0764298 − 0.997075i \(-0.524352\pi\)
−0.0764298 + 0.997075i \(0.524352\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.249425 0.0342612 0.0171306 − 0.999853i \(-0.494547\pi\)
0.0171306 + 0.999853i \(0.494547\pi\)
\(54\) 0 0
\(55\) 11.4509 1.54404
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.09591 −1.05400 −0.526999 − 0.849866i \(-0.676683\pi\)
−0.526999 + 0.849866i \(0.676683\pi\)
\(60\) 0 0
\(61\) 8.60442 1.10168 0.550841 − 0.834610i \(-0.314307\pi\)
0.550841 + 0.834610i \(0.314307\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.94239 0.861097
\(66\) 0 0
\(67\) 7.60442 0.929027 0.464514 − 0.885566i \(-0.346229\pi\)
0.464514 + 0.885566i \(0.346229\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.60442 1.13983 0.569917 − 0.821702i \(-0.306975\pi\)
0.569917 + 0.821702i \(0.306975\pi\)
\(72\) 0 0
\(73\) 0.846480 0.0990730 0.0495365 − 0.998772i \(-0.484226\pi\)
0.0495365 + 0.998772i \(0.484226\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.20147 −0.592763
\(78\) 0 0
\(79\) 7.60442 0.855564 0.427782 − 0.903882i \(-0.359295\pi\)
0.427782 + 0.903882i \(0.359295\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 11.4509 1.25690 0.628450 − 0.777850i \(-0.283690\pi\)
0.628450 + 0.777850i \(0.283690\pi\)
\(84\) 0 0
\(85\) 7.15352 0.775908
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.24943 −0.980437 −0.490219 − 0.871600i \(-0.663083\pi\)
−0.490219 + 0.871600i \(0.663083\pi\)
\(90\) 0 0
\(91\) −3.15352 −0.330579
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −16.4029 −1.68291
\(96\) 0 0
\(97\) −3.45090 −0.350386 −0.175193 − 0.984534i \(-0.556055\pi\)
−0.175193 + 0.984534i \(0.556055\pi\)
\(98\) 0 0
\(99\) 0 0
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 9072.2.a.bu.1.3 3
3.2 odd 2 9072.2.a.cb.1.1 3
4.3 odd 2 567.2.a.e.1.2 ✓ 3
12.11 even 2 567.2.a.f.1.2 yes 3
28.27 even 2 3969.2.a.o.1.2 3
36.7 odd 6 567.2.f.m.190.2 6
36.11 even 6 567.2.f.l.190.2 6
36.23 even 6 567.2.f.l.379.2 6
36.31 odd 6 567.2.f.m.379.2 6
84.83 odd 2 3969.2.a.n.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.a.e.1.2 ✓ 3 4.3 odd 2
567.2.a.f.1.2 yes 3 12.11 even 2
567.2.f.l.190.2 6 36.11 even 6
567.2.f.l.379.2 6 36.23 even 6
567.2.f.m.190.2 6 36.7 odd 6
567.2.f.m.379.2 6 36.31 odd 6
3969.2.a.n.1.2 3 84.83 odd 2
3969.2.a.o.1.2 3 28.27 even 2
9072.2.a.bu.1.3 3 1.1 even 1 trivial
9072.2.a.cb.1.1 3 3.2 odd 2