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 1.2
Root \(2.66908\) 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)
 
\(f(q)\) \(=\) \(q-1.45490 q^{5} -1.00000 q^{7} +1.54510 q^{11} +5.88325 q^{13} -6.79306 q^{17} +6.24797 q^{19} -2.90981 q^{23} -2.88325 q^{25} +3.88325 q^{29} -2.00000 q^{31} +1.45490 q^{35} +5.00000 q^{37} -2.24797 q^{41} +7.13122 q^{43} +5.33816 q^{47} +1.00000 q^{49} -9.79306 q^{53} -2.24797 q^{55} +4.67632 q^{59} -2.36471 q^{61} -8.55957 q^{65} -3.36471 q^{67} -1.36471 q^{71} -1.88325 q^{73} -1.54510 q^{77} -3.36471 q^{79} -2.24797 q^{83} +9.88325 q^{85} +0.793062 q^{89} -5.88325 q^{91} -9.09019 q^{95} +10.2480 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} - 3 q^{7} + 6 q^{11} + 3 q^{13} - 3 q^{17} - 6 q^{23} + 6 q^{25} - 3 q^{29} - 6 q^{31} + 3 q^{35} + 15 q^{37} + 12 q^{41} - 12 q^{43} + 3 q^{49} - 12 q^{53} + 12 q^{55} - 18 q^{59} - 3 q^{61}+ \cdots + 12 q^{97}+O(q^{100}) \) 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
\(5\) −1.45490 −0.650653 −0.325326 − 0.945602i \(-0.605474\pi\)
−0.325326 + 0.945602i \(0.605474\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.54510 0.465864 0.232932 − 0.972493i \(-0.425168\pi\)
0.232932 + 0.972493i \(0.425168\pi\)
\(12\) 0 0
\(13\) 5.88325 1.63172 0.815861 − 0.578249i \(-0.196264\pi\)
0.815861 + 0.578249i \(0.196264\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.79306 −1.64756 −0.823780 − 0.566910i \(-0.808139\pi\)
−0.823780 + 0.566910i \(0.808139\pi\)
\(18\) 0 0
\(19\) 6.24797 1.43338 0.716691 − 0.697391i \(-0.245656\pi\)
0.716691 + 0.697391i \(0.245656\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.90981 −0.606737 −0.303368 − 0.952873i \(-0.598111\pi\)
−0.303368 + 0.952873i \(0.598111\pi\)
\(24\) 0 0
\(25\) −2.88325 −0.576651
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.88325 0.721102 0.360551 − 0.932739i \(-0.382589\pi\)
0.360551 + 0.932739i \(0.382589\pi\)
\(30\) 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
\(35\) 1.45490 0.245924
\(36\) 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
\(41\) −2.24797 −0.351073 −0.175537 − 0.984473i \(-0.556166\pi\)
−0.175537 + 0.984473i \(0.556166\pi\)
\(42\) 0 0
\(43\) 7.13122 1.08750 0.543750 − 0.839247i \(-0.317004\pi\)
0.543750 + 0.839247i \(0.317004\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.33816 0.778650 0.389325 − 0.921100i \(-0.372708\pi\)
0.389325 + 0.921100i \(0.372708\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.79306 −1.34518 −0.672590 − 0.740015i \(-0.734818\pi\)
−0.672590 + 0.740015i \(0.734818\pi\)
\(54\) 0 0
\(55\) −2.24797 −0.303116
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.67632 0.608805 0.304402 − 0.952544i \(-0.401543\pi\)
0.304402 + 0.952544i \(0.401543\pi\)
\(60\) 0 0
\(61\) −2.36471 −0.302770 −0.151385 − 0.988475i \(-0.548373\pi\)
−0.151385 + 0.988475i \(0.548373\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −8.55957 −1.06168
\(66\) 0 0
\(67\) −3.36471 −0.411065 −0.205533 − 0.978650i \(-0.565893\pi\)
−0.205533 + 0.978650i \(0.565893\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.36471 −0.161962 −0.0809808 − 0.996716i \(-0.525805\pi\)
−0.0809808 + 0.996716i \(0.525805\pi\)
\(72\) 0 0
\(73\) −1.88325 −0.220418 −0.110209 − 0.993908i \(-0.535152\pi\)
−0.110209 + 0.993908i \(0.535152\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.54510 −0.176080
\(78\) 0 0
\(79\) −3.36471 −0.378560 −0.189280 − 0.981923i \(-0.560615\pi\)
−0.189280 + 0.981923i \(0.560615\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.24797 −0.246746 −0.123373 − 0.992360i \(-0.539371\pi\)
−0.123373 + 0.992360i \(0.539371\pi\)
\(84\) 0 0
\(85\) 9.88325 1.07199
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.793062 0.0840644 0.0420322 − 0.999116i \(-0.486617\pi\)
0.0420322 + 0.999116i \(0.486617\pi\)
\(90\) 0 0
\(91\) −5.88325 −0.616733
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −9.09019 −0.932634
\(96\) 0 0
\(97\) 10.2480 1.04052 0.520262 − 0.854007i \(-0.325834\pi\)
0.520262 + 0.854007i \(0.325834\pi\)
\(98\) 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.1.2 3
3.2 odd 2 9072.2.a.cb.1.2 3
4.3 odd 2 567.2.a.e.1.3 ✓ 3
12.11 even 2 567.2.a.f.1.1 yes 3
28.27 even 2 3969.2.a.o.1.3 3
36.7 odd 6 567.2.f.m.190.1 6
36.11 even 6 567.2.f.l.190.3 6
36.23 even 6 567.2.f.l.379.3 6
36.31 odd 6 567.2.f.m.379.1 6
84.83 odd 2 3969.2.a.n.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.a.e.1.3 ✓ 3 4.3 odd 2
567.2.a.f.1.1 yes 3 12.11 even 2
567.2.f.l.190.3 6 36.11 even 6
567.2.f.l.379.3 6 36.23 even 6
567.2.f.m.190.1 6 36.7 odd 6
567.2.f.m.379.1 6 36.31 odd 6
3969.2.a.n.1.1 3 84.83 odd 2
3969.2.a.o.1.3 3 28.27 even 2
9072.2.a.bu.1.2 3 1.1 even 1 trivial
9072.2.a.cb.1.2 3 3.2 odd 2