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,-1,0,3,0,0,0,2,0,3,0,0,0,-2] 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: 3.3.321.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 252)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.239123\) 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)
 
\(f(q)\) \(=\) \(q-0.239123 q^{5} +1.00000 q^{7} -5.12476 q^{11} -4.88564 q^{13} -3.70370 q^{17} -3.66019 q^{19} +7.42107 q^{23} -4.94282 q^{25} -3.46457 q^{29} +0.717370 q^{31} -0.239123 q^{35} +4.60301 q^{37} -5.60301 q^{41} -12.4887 q^{43} +4.33981 q^{47} +1.00000 q^{49} +0.942820 q^{53} +1.22545 q^{55} +7.57893 q^{59} +5.50808 q^{61} +1.16827 q^{65} +0.660190 q^{67} +13.7414 q^{71} -3.66019 q^{73} -5.12476 q^{77} +6.22545 q^{79} +9.70370 q^{83} +0.885640 q^{85} +7.48865 q^{89} -4.88564 q^{91} +0.875237 q^{95} -17.1488 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{5} + 3 q^{7} + 2 q^{11} + 3 q^{13} - 2 q^{17} - 3 q^{19} + 14 q^{23} - 6 q^{25} - q^{29} + 3 q^{31} - q^{35} - 3 q^{37} - 3 q^{43} + 21 q^{47} + 3 q^{49} - 6 q^{53} - 6 q^{55} + 31 q^{59} + 6 q^{61}+ \cdots - 9 q^{97}+O(q^{100}) \) 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
\(5\) −0.239123 −0.106939 −0.0534696 − 0.998569i \(-0.517028\pi\)
−0.0534696 + 0.998569i \(0.517028\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.12476 −1.54517 −0.772587 − 0.634909i \(-0.781038\pi\)
−0.772587 + 0.634909i \(0.781038\pi\)
\(12\) 0 0
\(13\) −4.88564 −1.35503 −0.677516 − 0.735508i \(-0.736944\pi\)
−0.677516 + 0.735508i \(0.736944\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.70370 −0.898278 −0.449139 − 0.893462i \(-0.648269\pi\)
−0.449139 + 0.893462i \(0.648269\pi\)
\(18\) 0 0
\(19\) −3.66019 −0.839705 −0.419853 − 0.907592i \(-0.637918\pi\)
−0.419853 + 0.907592i \(0.637918\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.42107 1.54740 0.773700 − 0.633553i \(-0.218404\pi\)
0.773700 + 0.633553i \(0.218404\pi\)
\(24\) 0 0
\(25\) −4.94282 −0.988564
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.46457 −0.643355 −0.321678 − 0.946849i \(-0.604247\pi\)
−0.321678 + 0.946849i \(0.604247\pi\)
\(30\) 0 0
\(31\) 0.717370 0.128843 0.0644217 − 0.997923i \(-0.479480\pi\)
0.0644217 + 0.997923i \(0.479480\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.239123 −0.0404192
\(36\) 0 0
\(37\) 4.60301 0.756730 0.378365 − 0.925656i \(-0.376486\pi\)
0.378365 + 0.925656i \(0.376486\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.60301 −0.875043 −0.437522 − 0.899208i \(-0.644144\pi\)
−0.437522 + 0.899208i \(0.644144\pi\)
\(42\) 0 0
\(43\) −12.4887 −1.90450 −0.952251 − 0.305317i \(-0.901237\pi\)
−0.952251 + 0.305317i \(0.901237\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.33981 0.633026 0.316513 − 0.948588i \(-0.397488\pi\)
0.316513 + 0.948588i \(0.397488\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.942820 0.129506 0.0647531 − 0.997901i \(-0.479374\pi\)
0.0647531 + 0.997901i \(0.479374\pi\)
\(54\) 0 0
\(55\) 1.22545 0.165240
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.57893 0.986693 0.493347 − 0.869833i \(-0.335773\pi\)
0.493347 + 0.869833i \(0.335773\pi\)
\(60\) 0 0
\(61\) 5.50808 0.705237 0.352619 − 0.935767i \(-0.385291\pi\)
0.352619 + 0.935767i \(0.385291\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.16827 0.144906
\(66\) 0 0
\(67\) 0.660190 0.0806550 0.0403275 − 0.999187i \(-0.487160\pi\)
0.0403275 + 0.999187i \(0.487160\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.7414 1.63081 0.815405 − 0.578891i \(-0.196514\pi\)
0.815405 + 0.578891i \(0.196514\pi\)
\(72\) 0 0
\(73\) −3.66019 −0.428393 −0.214196 − 0.976791i \(-0.568713\pi\)
−0.214196 + 0.976791i \(0.568713\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.12476 −0.584021
\(78\) 0 0
\(79\) 6.22545 0.700418 0.350209 − 0.936672i \(-0.386111\pi\)
0.350209 + 0.936672i \(0.386111\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.70370 1.06512 0.532560 − 0.846393i \(-0.321230\pi\)
0.532560 + 0.846393i \(0.321230\pi\)
\(84\) 0 0
\(85\) 0.885640 0.0960612
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.48865 0.793795 0.396898 − 0.917863i \(-0.370087\pi\)
0.396898 + 0.917863i \(0.370087\pi\)
\(90\) 0 0
\(91\) −4.88564 −0.512154
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.875237 0.0897974
\(96\) 0 0
\(97\) −17.1488 −1.74120 −0.870600 − 0.491991i \(-0.836269\pi\)
−0.870600 + 0.491991i \(0.836269\pi\)
\(98\) 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.bv.1.2 3
3.2 odd 2 9072.2.a.by.1.2 3
4.3 odd 2 2268.2.a.h.1.2 3
9.2 odd 6 1008.2.r.j.337.3 6
9.4 even 3 3024.2.r.j.2017.2 6
9.5 odd 6 1008.2.r.j.673.3 6
9.7 even 3 3024.2.r.j.1009.2 6
12.11 even 2 2268.2.a.i.1.2 3
36.7 odd 6 756.2.j.b.253.2 6
36.11 even 6 252.2.j.a.85.1 ✓ 6
36.23 even 6 252.2.j.a.169.1 yes 6
36.31 odd 6 756.2.j.b.505.2 6
252.11 even 6 1764.2.i.g.373.2 6
252.23 even 6 1764.2.i.g.1537.2 6
252.31 even 6 5292.2.l.f.3313.2 6
252.47 odd 6 1764.2.l.f.949.1 6
252.59 odd 6 1764.2.l.f.961.1 6
252.67 odd 6 5292.2.l.e.3313.2 6
252.79 odd 6 5292.2.l.e.361.2 6
252.83 odd 6 1764.2.j.e.589.3 6
252.95 even 6 1764.2.l.e.961.3 6
252.103 even 6 5292.2.i.e.2125.2 6
252.115 even 6 5292.2.i.e.1549.2 6
252.131 odd 6 1764.2.i.d.1537.2 6
252.139 even 6 5292.2.j.d.3529.2 6
252.151 odd 6 5292.2.i.f.1549.2 6
252.167 odd 6 1764.2.j.e.1177.3 6
252.187 even 6 5292.2.l.f.361.2 6
252.191 even 6 1764.2.l.e.949.3 6
252.223 even 6 5292.2.j.d.1765.2 6
252.227 odd 6 1764.2.i.d.373.2 6
252.247 odd 6 5292.2.i.f.2125.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.j.a.85.1 ✓ 6 36.11 even 6
252.2.j.a.169.1 yes 6 36.23 even 6
756.2.j.b.253.2 6 36.7 odd 6
756.2.j.b.505.2 6 36.31 odd 6
1008.2.r.j.337.3 6 9.2 odd 6
1008.2.r.j.673.3 6 9.5 odd 6
1764.2.i.d.373.2 6 252.227 odd 6
1764.2.i.d.1537.2 6 252.131 odd 6
1764.2.i.g.373.2 6 252.11 even 6
1764.2.i.g.1537.2 6 252.23 even 6
1764.2.j.e.589.3 6 252.83 odd 6
1764.2.j.e.1177.3 6 252.167 odd 6
1764.2.l.e.949.3 6 252.191 even 6
1764.2.l.e.961.3 6 252.95 even 6
1764.2.l.f.949.1 6 252.47 odd 6
1764.2.l.f.961.1 6 252.59 odd 6
2268.2.a.h.1.2 3 4.3 odd 2
2268.2.a.i.1.2 3 12.11 even 2
3024.2.r.j.1009.2 6 9.7 even 3
3024.2.r.j.2017.2 6 9.4 even 3
5292.2.i.e.1549.2 6 252.115 even 6
5292.2.i.e.2125.2 6 252.103 even 6
5292.2.i.f.1549.2 6 252.151 odd 6
5292.2.i.f.2125.2 6 252.247 odd 6
5292.2.j.d.1765.2 6 252.223 even 6
5292.2.j.d.3529.2 6 252.139 even 6
5292.2.l.e.361.2 6 252.79 odd 6
5292.2.l.e.3313.2 6 252.67 odd 6
5292.2.l.f.361.2 6 252.187 even 6
5292.2.l.f.3313.2 6 252.31 even 6
9072.2.a.bv.1.2 3 1.1 even 1 trivial
9072.2.a.by.1.2 3 3.2 odd 2