Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9680,2,Mod(1,9680)] 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("9680.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-2,0,-6,0,6,0,10,0,0,0,6,0,2,0,-11] 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: \(77.2951891566\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.45753625.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 13x^{4} + 11x^{3} + 41x^{2} - 30x - 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.05921\) of defining polynomial
Character \(\chi\) \(=\) 9680.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.89070 q^{3} -1.00000 q^{5} -2.74075 q^{7} +0.574737 q^{9} -0.196821 q^{13} +1.89070 q^{15} +1.46809 q^{17} -2.68154 q^{19} +5.18193 q^{21} -1.81736 q^{23} +1.00000 q^{25} +4.58544 q^{27} -1.37946 q^{29} -5.90411 q^{31} +2.74075 q^{35} +4.57623 q^{37} +0.372129 q^{39} -1.23428 q^{41} +2.96514 q^{43} -0.574737 q^{45} -9.30107 q^{47} +0.511708 q^{49} -2.77571 q^{51} +2.69388 q^{53} +5.06997 q^{57} -14.2092 q^{59} -7.54027 q^{61} -1.57521 q^{63} +0.196821 q^{65} -10.6500 q^{67} +3.43607 q^{69} +6.87672 q^{71} -10.4314 q^{73} -1.89070 q^{75} +1.87328 q^{79} -10.3939 q^{81} -5.79397 q^{83} -1.46809 q^{85} +2.60815 q^{87} +17.3060 q^{89} +0.539438 q^{91} +11.1629 q^{93} +2.68154 q^{95} -18.3188 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9} + 6 q^{13} + 2 q^{15} - 11 q^{17} - 11 q^{19} - 2 q^{21} - 18 q^{23} + 6 q^{25} + q^{27} + 6 q^{29} - q^{31} - 6 q^{35} + 4 q^{37} + 27 q^{39} + 4 q^{41} + 3 q^{43}+ \cdots + 20 q^{97}+O(q^{100}) \) 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
\(3\) −1.89070 −1.09159 −0.545797 − 0.837917i \(-0.683773\pi\)
−0.545797 + 0.837917i \(0.683773\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.74075 −1.03591 −0.517953 − 0.855409i \(-0.673306\pi\)
−0.517953 + 0.855409i \(0.673306\pi\)
\(8\) 0 0
\(9\) 0.574737 0.191579
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −0.196821 −0.0545884 −0.0272942 − 0.999627i \(-0.508689\pi\)
−0.0272942 + 0.999627i \(0.508689\pi\)
\(14\) 0 0
\(15\) 1.89070 0.488176
\(16\) 0 0
\(17\) 1.46809 0.356063 0.178032 − 0.984025i \(-0.443027\pi\)
0.178032 + 0.984025i \(0.443027\pi\)
\(18\) 0 0
\(19\) −2.68154 −0.615187 −0.307593 − 0.951518i \(-0.599524\pi\)
−0.307593 + 0.951518i \(0.599524\pi\)
\(20\) 0 0
\(21\) 5.18193 1.13079
\(22\) 0 0
\(23\) −1.81736 −0.378945 −0.189473 − 0.981886i \(-0.560678\pi\)
−0.189473 + 0.981886i \(0.560678\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.58544 0.882468
\(28\) 0 0
\(29\) −1.37946 −0.256160 −0.128080 − 0.991764i \(-0.540881\pi\)
−0.128080 + 0.991764i \(0.540881\pi\)
\(30\) 0 0
\(31\) −5.90411 −1.06041 −0.530205 − 0.847870i \(-0.677885\pi\)
−0.530205 + 0.847870i \(0.677885\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.74075 0.463271
\(36\) 0 0
\(37\) 4.57623 0.752328 0.376164 − 0.926553i \(-0.377243\pi\)
0.376164 + 0.926553i \(0.377243\pi\)
\(38\) 0 0
\(39\) 0.372129 0.0595884
\(40\) 0 0
\(41\) −1.23428 −0.192762 −0.0963812 − 0.995344i \(-0.530727\pi\)
−0.0963812 + 0.995344i \(0.530727\pi\)
\(42\) 0 0
\(43\) 2.96514 0.452180 0.226090 − 0.974106i \(-0.427406\pi\)
0.226090 + 0.974106i \(0.427406\pi\)
\(44\) 0 0
\(45\) −0.574737 −0.0856768
\(46\) 0 0
\(47\) −9.30107 −1.35670 −0.678350 − 0.734739i \(-0.737305\pi\)
−0.678350 + 0.734739i \(0.737305\pi\)
\(48\) 0 0
\(49\) 0.511708 0.0731012
\(50\) 0 0
\(51\) −2.77571 −0.388677
\(52\) 0 0
\(53\) 2.69388 0.370032 0.185016 − 0.982735i \(-0.440766\pi\)
0.185016 + 0.982735i \(0.440766\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.06997 0.671535
\(58\) 0 0
\(59\) −14.2092 −1.84988 −0.924942 − 0.380108i \(-0.875887\pi\)
−0.924942 + 0.380108i \(0.875887\pi\)
\(60\) 0 0
\(61\) −7.54027 −0.965433 −0.482716 − 0.875777i \(-0.660350\pi\)
−0.482716 + 0.875777i \(0.660350\pi\)
\(62\) 0 0
\(63\) −1.57521 −0.198458
\(64\) 0 0
\(65\) 0.196821 0.0244127
\(66\) 0 0
\(67\) −10.6500 −1.30111 −0.650553 − 0.759461i \(-0.725463\pi\)
−0.650553 + 0.759461i \(0.725463\pi\)
\(68\) 0 0
\(69\) 3.43607 0.413655
\(70\) 0 0
\(71\) 6.87672 0.816116 0.408058 − 0.912956i \(-0.366206\pi\)
0.408058 + 0.912956i \(0.366206\pi\)
\(72\) 0 0
\(73\) −10.4314 −1.22090 −0.610452 − 0.792053i \(-0.709012\pi\)
−0.610452 + 0.792053i \(0.709012\pi\)
\(74\) 0 0
\(75\) −1.89070 −0.218319
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.87328 0.210761 0.105380 − 0.994432i \(-0.466394\pi\)
0.105380 + 0.994432i \(0.466394\pi\)
\(80\) 0 0
\(81\) −10.3939 −1.15488
\(82\) 0 0
\(83\) −5.79397 −0.635971 −0.317986 − 0.948096i \(-0.603006\pi\)
−0.317986 + 0.948096i \(0.603006\pi\)
\(84\) 0 0
\(85\) −1.46809 −0.159236
\(86\) 0 0
\(87\) 2.60815 0.279623
\(88\) 0 0
\(89\) 17.3060 1.83443 0.917214 − 0.398396i \(-0.130433\pi\)
0.917214 + 0.398396i \(0.130433\pi\)
\(90\) 0 0
\(91\) 0.539438 0.0565484
\(92\) 0 0
\(93\) 11.1629 1.15754
\(94\) 0 0
\(95\) 2.68154 0.275120
\(96\) 0 0
\(97\) −18.3188 −1.86000 −0.929998 − 0.367564i \(-0.880192\pi\)
−0.929998 + 0.367564i \(0.880192\pi\)
\(98\) 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 9680.2.a.cy.1.2 6
4.3 odd 2 4840.2.a.be.1.5 6
11.2 odd 10 880.2.bo.j.81.1 12
11.6 odd 10 880.2.bo.j.641.1 12
11.10 odd 2 9680.2.a.cx.1.2 6
44.35 even 10 440.2.y.b.81.3 ✓ 12
44.39 even 10 440.2.y.b.201.3 yes 12
44.43 even 2 4840.2.a.bf.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.b.81.3 ✓ 12 44.35 even 10
440.2.y.b.201.3 yes 12 44.39 even 10
880.2.bo.j.81.1 12 11.2 odd 10
880.2.bo.j.641.1 12 11.6 odd 10
4840.2.a.be.1.5 6 4.3 odd 2
4840.2.a.bf.1.5 6 44.43 even 2
9680.2.a.cx.1.2 6 11.10 odd 2
9680.2.a.cy.1.2 6 1.1 even 1 trivial