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 = [2,0,0,0,-4,0,2,0,0,0,2,0,-6,0,0,0,-14] 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: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1134)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) 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)
 
\(f(q)\) \(=\) \(q-3.73205 q^{5} +1.00000 q^{7} -4.19615 q^{11} +0.464102 q^{13} -7.00000 q^{17} +2.73205 q^{19} +6.19615 q^{23} +8.92820 q^{25} -8.46410 q^{29} +2.19615 q^{31} -3.73205 q^{35} -6.66025 q^{37} -9.46410 q^{41} -5.46410 q^{43} +1.26795 q^{47} +1.00000 q^{49} +2.53590 q^{53} +15.6603 q^{55} -6.19615 q^{59} -9.92820 q^{61} -1.73205 q^{65} +3.26795 q^{67} +13.4641 q^{71} +11.7321 q^{73} -4.19615 q^{77} -15.1244 q^{79} -14.5885 q^{83} +26.1244 q^{85} +3.92820 q^{89} +0.464102 q^{91} -10.1962 q^{95} -2.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} + 2 q^{7} + 2 q^{11} - 6 q^{13} - 14 q^{17} + 2 q^{19} + 2 q^{23} + 4 q^{25} - 10 q^{29} - 6 q^{31} - 4 q^{35} + 4 q^{37} - 12 q^{41} - 4 q^{43} + 6 q^{47} + 2 q^{49} + 12 q^{53} + 14 q^{55}+ \cdots + 8 q^{97}+O(q^{100}) \) 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
\(5\) −3.73205 −1.66902 −0.834512 − 0.550990i \(-0.814250\pi\)
−0.834512 + 0.550990i \(0.814250\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.19615 −1.26519 −0.632594 − 0.774484i \(-0.718010\pi\)
−0.632594 + 0.774484i \(0.718010\pi\)
\(12\) 0 0
\(13\) 0.464102 0.128719 0.0643593 − 0.997927i \(-0.479500\pi\)
0.0643593 + 0.997927i \(0.479500\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.00000 −1.69775 −0.848875 − 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 0 0
\(19\) 2.73205 0.626775 0.313388 − 0.949625i \(-0.398536\pi\)
0.313388 + 0.949625i \(0.398536\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.19615 1.29199 0.645994 − 0.763343i \(-0.276443\pi\)
0.645994 + 0.763343i \(0.276443\pi\)
\(24\) 0 0
\(25\) 8.92820 1.78564
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.46410 −1.57174 −0.785872 − 0.618389i \(-0.787786\pi\)
−0.785872 + 0.618389i \(0.787786\pi\)
\(30\) 0 0
\(31\) 2.19615 0.394441 0.197220 − 0.980359i \(-0.436809\pi\)
0.197220 + 0.980359i \(0.436809\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.73205 −0.630832
\(36\) 0 0
\(37\) −6.66025 −1.09494 −0.547470 − 0.836826i \(-0.684409\pi\)
−0.547470 + 0.836826i \(0.684409\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −9.46410 −1.47804 −0.739022 − 0.673681i \(-0.764712\pi\)
−0.739022 + 0.673681i \(0.764712\pi\)
\(42\) 0 0
\(43\) −5.46410 −0.833268 −0.416634 − 0.909074i \(-0.636790\pi\)
−0.416634 + 0.909074i \(0.636790\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.26795 0.184949 0.0924747 − 0.995715i \(-0.470522\pi\)
0.0924747 + 0.995715i \(0.470522\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.53590 0.348332 0.174166 − 0.984716i \(-0.444277\pi\)
0.174166 + 0.984716i \(0.444277\pi\)
\(54\) 0 0
\(55\) 15.6603 2.11163
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.19615 −0.806670 −0.403335 − 0.915052i \(-0.632149\pi\)
−0.403335 + 0.915052i \(0.632149\pi\)
\(60\) 0 0
\(61\) −9.92820 −1.27118 −0.635588 − 0.772028i \(-0.719242\pi\)
−0.635588 + 0.772028i \(0.719242\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.73205 −0.214834
\(66\) 0 0
\(67\) 3.26795 0.399244 0.199622 − 0.979873i \(-0.436029\pi\)
0.199622 + 0.979873i \(0.436029\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.4641 1.59789 0.798947 − 0.601401i \(-0.205391\pi\)
0.798947 + 0.601401i \(0.205391\pi\)
\(72\) 0 0
\(73\) 11.7321 1.37313 0.686566 − 0.727067i \(-0.259117\pi\)
0.686566 + 0.727067i \(0.259117\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.19615 −0.478196
\(78\) 0 0
\(79\) −15.1244 −1.70162 −0.850811 − 0.525471i \(-0.823889\pi\)
−0.850811 + 0.525471i \(0.823889\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −14.5885 −1.60129 −0.800646 − 0.599138i \(-0.795510\pi\)
−0.800646 + 0.599138i \(0.795510\pi\)
\(84\) 0 0
\(85\) 26.1244 2.83358
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.92820 0.416389 0.208194 − 0.978087i \(-0.433241\pi\)
0.208194 + 0.978087i \(0.433241\pi\)
\(90\) 0 0
\(91\) 0.464102 0.0486511
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −10.1962 −1.04610
\(96\) 0 0
\(97\) −2.92820 −0.297314 −0.148657 − 0.988889i \(-0.547495\pi\)
−0.148657 + 0.988889i \(0.547495\pi\)
\(98\) 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.y.1.1 2
3.2 odd 2 9072.2.a.bp.1.2 2
4.3 odd 2 1134.2.a.m.1.1 yes 2
12.11 even 2 1134.2.a.l.1.2 ✓ 2
28.27 even 2 7938.2.a.bt.1.2 2
36.7 odd 6 1134.2.f.r.757.2 4
36.11 even 6 1134.2.f.s.757.1 4
36.23 even 6 1134.2.f.s.379.1 4
36.31 odd 6 1134.2.f.r.379.2 4
84.83 odd 2 7938.2.a.bg.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1134.2.a.l.1.2 ✓ 2 12.11 even 2
1134.2.a.m.1.1 yes 2 4.3 odd 2
1134.2.f.r.379.2 4 36.31 odd 6
1134.2.f.r.757.2 4 36.7 odd 6
1134.2.f.s.379.1 4 36.23 even 6
1134.2.f.s.757.1 4 36.11 even 6
7938.2.a.bg.1.1 2 84.83 odd 2
7938.2.a.bt.1.2 2 28.27 even 2
9072.2.a.y.1.1 2 1.1 even 1 trivial
9072.2.a.bp.1.2 2 3.2 odd 2