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 \(-0.167449\) 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)
 
\(f(q)\) \(=\) \(q-3.80451 q^{5} -1.00000 q^{7} -5.13941 q^{11} -5.80451 q^{13} +6.13941 q^{17} -0.334898 q^{19} +5.94392 q^{23} +9.47431 q^{25} +5.46961 q^{29} -0.334898 q^{31} +3.80451 q^{35} -5.33490 q^{37} -2.00000 q^{41} +11.0833 q^{43} +10.6137 q^{47} +1.00000 q^{49} +6.13471 q^{53} +19.5529 q^{55} -8.33490 q^{59} -6.47431 q^{61} +22.0833 q^{65} -0.469613 q^{67} -3.19549 q^{71} +0.139410 q^{73} +5.13941 q^{77} -9.13941 q^{79} +3.66510 q^{83} -23.3575 q^{85} +11.7484 q^{89} +5.80451 q^{91} +1.27412 q^{95} -5.88784 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{7} - 3 q^{11} - 6 q^{13} + 6 q^{17} - 6 q^{23} + 15 q^{25} + 6 q^{29} - 15 q^{37} - 6 q^{41} - 3 q^{43} + 6 q^{47} + 3 q^{49} + 9 q^{53} + 12 q^{55} - 24 q^{59} - 6 q^{61} + 30 q^{65} + 9 q^{67}+ \cdots + 30 q^{97}+O(q^{100}) \) 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
\(5\) −3.80451 −1.70143 −0.850715 − 0.525628i \(-0.823830\pi\)
−0.850715 + 0.525628i \(0.823830\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.13941 −1.54959 −0.774795 − 0.632212i \(-0.782147\pi\)
−0.774795 + 0.632212i \(0.782147\pi\)
\(12\) 0 0
\(13\) −5.80451 −1.60988 −0.804941 − 0.593355i \(-0.797803\pi\)
−0.804941 + 0.593355i \(0.797803\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.13941 1.48903 0.744513 − 0.667608i \(-0.232682\pi\)
0.744513 + 0.667608i \(0.232682\pi\)
\(18\) 0 0
\(19\) −0.334898 −0.0768310 −0.0384155 − 0.999262i \(-0.512231\pi\)
−0.0384155 + 0.999262i \(0.512231\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.94392 1.23939 0.619697 − 0.784841i \(-0.287256\pi\)
0.619697 + 0.784841i \(0.287256\pi\)
\(24\) 0 0
\(25\) 9.47431 1.89486
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.46961 1.01568 0.507841 − 0.861451i \(-0.330444\pi\)
0.507841 + 0.861451i \(0.330444\pi\)
\(30\) 0 0
\(31\) −0.334898 −0.0601495 −0.0300748 − 0.999548i \(-0.509575\pi\)
−0.0300748 + 0.999548i \(0.509575\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.80451 0.643080
\(36\) 0 0
\(37\) −5.33490 −0.877052 −0.438526 − 0.898719i \(-0.644499\pi\)
−0.438526 + 0.898719i \(0.644499\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 11.0833 1.69019 0.845096 − 0.534614i \(-0.179543\pi\)
0.845096 + 0.534614i \(0.179543\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.6137 1.54817 0.774085 − 0.633082i \(-0.218210\pi\)
0.774085 + 0.633082i \(0.218210\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.13471 0.842668 0.421334 − 0.906906i \(-0.361562\pi\)
0.421334 + 0.906906i \(0.361562\pi\)
\(54\) 0 0
\(55\) 19.5529 2.63652
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.33490 −1.08511 −0.542556 − 0.840020i \(-0.682543\pi\)
−0.542556 + 0.840020i \(0.682543\pi\)
\(60\) 0 0
\(61\) −6.47431 −0.828950 −0.414475 − 0.910061i \(-0.636035\pi\)
−0.414475 + 0.910061i \(0.636035\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 22.0833 2.73910
\(66\) 0 0
\(67\) −0.469613 −0.0573724 −0.0286862 − 0.999588i \(-0.509132\pi\)
−0.0286862 + 0.999588i \(0.509132\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.19549 −0.379235 −0.189617 − 0.981858i \(-0.560725\pi\)
−0.189617 + 0.981858i \(0.560725\pi\)
\(72\) 0 0
\(73\) 0.139410 0.0163167 0.00815835 − 0.999967i \(-0.497403\pi\)
0.00815835 + 0.999967i \(0.497403\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.13941 0.585690
\(78\) 0 0
\(79\) −9.13941 −1.02826 −0.514132 − 0.857711i \(-0.671886\pi\)
−0.514132 + 0.857711i \(0.671886\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.66510 0.402297 0.201149 − 0.979561i \(-0.435533\pi\)
0.201149 + 0.979561i \(0.435533\pi\)
\(84\) 0 0
\(85\) −23.3575 −2.53347
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 11.7484 1.24533 0.622666 − 0.782488i \(-0.286050\pi\)
0.622666 + 0.782488i \(0.286050\pi\)
\(90\) 0 0
\(91\) 5.80451 0.608478
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.27412 0.130722
\(96\) 0 0
\(97\) −5.88784 −0.597820 −0.298910 − 0.954281i \(-0.596623\pi\)
−0.298910 + 0.954281i \(0.596623\pi\)
\(98\) 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.bw.1.1 3
3.2 odd 2 9072.2.a.bx.1.3 3
4.3 odd 2 4536.2.a.u.1.1 yes 3
12.11 even 2 4536.2.a.t.1.3 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4536.2.a.t.1.3 ✓ 3 12.11 even 2
4536.2.a.u.1.1 yes 3 4.3 odd 2
9072.2.a.bw.1.1 3 1.1 even 1 trivial
9072.2.a.bx.1.3 3 3.2 odd 2