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 = [3,0,0,0,0,0,-3,0,0,0,3,0,-6,0,0,0,-6] 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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4536)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.36147\) 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)
 
\(f(q)\) \(=\) \(q-3.93800 q^{5} -1.00000 q^{7} +1.78493 q^{11} +1.93800 q^{13} -2.78493 q^{17} -4.72294 q^{19} +5.15307 q^{23} +10.5079 q^{25} +6.66094 q^{29} -4.72294 q^{31} +3.93800 q^{35} -9.72294 q^{37} +2.00000 q^{41} -3.36814 q^{43} -8.29281 q^{47} +1.00000 q^{49} +10.3839 q^{53} -7.02908 q^{55} +12.7229 q^{59} -7.50787 q^{61} -7.63186 q^{65} +11.6609 q^{67} +10.9380 q^{71} -3.21507 q^{73} -1.78493 q^{77} -5.78493 q^{79} +0.722938 q^{83} +10.9671 q^{85} +7.09107 q^{89} -1.93800 q^{91} +18.5989 q^{95} +16.3061 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{7} + 3 q^{11} - 6 q^{13} - 6 q^{17} + 6 q^{23} + 15 q^{25} - 6 q^{29} - 15 q^{37} + 6 q^{41} - 3 q^{43} - 6 q^{47} + 3 q^{49} - 9 q^{53} + 12 q^{55} + 24 q^{59} - 6 q^{61} - 30 q^{65} + 9 q^{67}+ \cdots + 30 q^{97}+O(q^{100}) \) 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
\(5\) −3.93800 −1.76113 −0.880564 − 0.473927i \(-0.842836\pi\)
−0.880564 + 0.473927i \(0.842836\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.78493 0.538178 0.269089 − 0.963115i \(-0.413277\pi\)
0.269089 + 0.963115i \(0.413277\pi\)
\(12\) 0 0
\(13\) 1.93800 0.537505 0.268753 − 0.963209i \(-0.413389\pi\)
0.268753 + 0.963209i \(0.413389\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.78493 −0.675446 −0.337723 − 0.941246i \(-0.609657\pi\)
−0.337723 + 0.941246i \(0.609657\pi\)
\(18\) 0 0
\(19\) −4.72294 −1.08352 −0.541758 − 0.840534i \(-0.682241\pi\)
−0.541758 + 0.840534i \(0.682241\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.15307 1.07449 0.537245 − 0.843426i \(-0.319465\pi\)
0.537245 + 0.843426i \(0.319465\pi\)
\(24\) 0 0
\(25\) 10.5079 2.10157
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.66094 1.23691 0.618453 − 0.785822i \(-0.287760\pi\)
0.618453 + 0.785822i \(0.287760\pi\)
\(30\) 0 0
\(31\) −4.72294 −0.848265 −0.424132 − 0.905600i \(-0.639421\pi\)
−0.424132 + 0.905600i \(0.639421\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.93800 0.665644
\(36\) 0 0
\(37\) −9.72294 −1.59844 −0.799221 − 0.601038i \(-0.794754\pi\)
−0.799221 + 0.601038i \(0.794754\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −3.36814 −0.513636 −0.256818 − 0.966460i \(-0.582674\pi\)
−0.256818 + 0.966460i \(0.582674\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.29281 −1.20963 −0.604815 − 0.796366i \(-0.706753\pi\)
−0.604815 + 0.796366i \(0.706753\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.3839 1.42634 0.713168 − 0.700993i \(-0.247260\pi\)
0.713168 + 0.700993i \(0.247260\pi\)
\(54\) 0 0
\(55\) −7.02908 −0.947800
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.7229 1.65639 0.828193 − 0.560443i \(-0.189369\pi\)
0.828193 + 0.560443i \(0.189369\pi\)
\(60\) 0 0
\(61\) −7.50787 −0.961284 −0.480642 − 0.876917i \(-0.659596\pi\)
−0.480642 + 0.876917i \(0.659596\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −7.63186 −0.946616
\(66\) 0 0
\(67\) 11.6609 1.42461 0.712305 − 0.701870i \(-0.247651\pi\)
0.712305 + 0.701870i \(0.247651\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.9380 1.29810 0.649051 − 0.760745i \(-0.275166\pi\)
0.649051 + 0.760745i \(0.275166\pi\)
\(72\) 0 0
\(73\) −3.21507 −0.376295 −0.188148 − 0.982141i \(-0.560248\pi\)
−0.188148 + 0.982141i \(0.560248\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.78493 −0.203412
\(78\) 0 0
\(79\) −5.78493 −0.650856 −0.325428 − 0.945567i \(-0.605508\pi\)
−0.325428 + 0.945567i \(0.605508\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.722938 0.0793527 0.0396764 − 0.999213i \(-0.487367\pi\)
0.0396764 + 0.999213i \(0.487367\pi\)
\(84\) 0 0
\(85\) 10.9671 1.18955
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.09107 0.751652 0.375826 − 0.926690i \(-0.377359\pi\)
0.375826 + 0.926690i \(0.377359\pi\)
\(90\) 0 0
\(91\) −1.93800 −0.203158
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 18.5989 1.90821
\(96\) 0 0
\(97\) 16.3061 1.65564 0.827819 − 0.560996i \(-0.189582\pi\)
0.827819 + 0.560996i \(0.189582\pi\)
\(98\) 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.bx.1.1 3
3.2 odd 2 9072.2.a.bw.1.3 3
4.3 odd 2 4536.2.a.t.1.1 ✓ 3
12.11 even 2 4536.2.a.u.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4536.2.a.t.1.1 ✓ 3 4.3 odd 2
4536.2.a.u.1.3 yes 3 12.11 even 2
9072.2.a.bw.1.3 3 3.2 odd 2
9072.2.a.bx.1.1 3 1.1 even 1 trivial