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 1.4
Root \(-0.444728\) 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)
 
\(f(q)\) \(=\) \(q-0.719585 q^{3} -1.00000 q^{5} +4.18418 q^{7} -2.48220 q^{9} +6.05056 q^{13} +0.719585 q^{15} -6.52180 q^{17} +0.739455 q^{19} -3.01088 q^{21} +5.13093 q^{23} +1.00000 q^{25} +3.94491 q^{27} -2.08036 q^{29} +1.12698 q^{31} -4.18418 q^{35} +7.11676 q^{37} -4.35390 q^{39} +4.02170 q^{41} +6.29891 q^{43} +2.48220 q^{45} -9.24147 q^{47} +10.5074 q^{49} +4.69299 q^{51} +7.77015 q^{53} -0.532101 q^{57} -6.16416 q^{59} -14.6869 q^{61} -10.3860 q^{63} -6.05056 q^{65} +5.53268 q^{67} -3.69214 q^{69} -8.87252 q^{71} +8.09351 q^{73} -0.719585 q^{75} -0.570482 q^{79} +4.60789 q^{81} -15.2224 q^{83} +6.52180 q^{85} +1.49700 q^{87} -1.33867 q^{89} +25.3167 q^{91} -0.810957 q^{93} -0.739455 q^{95} +9.96643 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9} + 6 q^{13} + 2 q^{15} - 11 q^{17} - 11 q^{19} - 2 q^{21} - 18 q^{23} + 6 q^{25} + q^{27} + 6 q^{29} - q^{31} - 6 q^{35} + 4 q^{37} + 27 q^{39} + 4 q^{41} + 3 q^{43}+ \cdots + 20 q^{97}+O(q^{100}) \) 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
\(3\) −0.719585 −0.415453 −0.207726 − 0.978187i \(-0.566606\pi\)
−0.207726 + 0.978187i \(0.566606\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.18418 1.58147 0.790736 − 0.612157i \(-0.209698\pi\)
0.790736 + 0.612157i \(0.209698\pi\)
\(8\) 0 0
\(9\) −2.48220 −0.827399
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 6.05056 1.67812 0.839062 − 0.544035i \(-0.183104\pi\)
0.839062 + 0.544035i \(0.183104\pi\)
\(14\) 0 0
\(15\) 0.719585 0.185796
\(16\) 0 0
\(17\) −6.52180 −1.58177 −0.790885 − 0.611965i \(-0.790379\pi\)
−0.790885 + 0.611965i \(0.790379\pi\)
\(18\) 0 0
\(19\) 0.739455 0.169642 0.0848212 − 0.996396i \(-0.472968\pi\)
0.0848212 + 0.996396i \(0.472968\pi\)
\(20\) 0 0
\(21\) −3.01088 −0.657027
\(22\) 0 0
\(23\) 5.13093 1.06987 0.534936 − 0.844893i \(-0.320336\pi\)
0.534936 + 0.844893i \(0.320336\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 3.94491 0.759198
\(28\) 0 0
\(29\) −2.08036 −0.386313 −0.193157 − 0.981168i \(-0.561873\pi\)
−0.193157 + 0.981168i \(0.561873\pi\)
\(30\) 0 0
\(31\) 1.12698 0.202411 0.101206 − 0.994866i \(-0.467730\pi\)
0.101206 + 0.994866i \(0.467730\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.18418 −0.707256
\(36\) 0 0
\(37\) 7.11676 1.16999 0.584994 − 0.811037i \(-0.301097\pi\)
0.584994 + 0.811037i \(0.301097\pi\)
\(38\) 0 0
\(39\) −4.35390 −0.697182
\(40\) 0 0
\(41\) 4.02170 0.628085 0.314042 − 0.949409i \(-0.398317\pi\)
0.314042 + 0.949409i \(0.398317\pi\)
\(42\) 0 0
\(43\) 6.29891 0.960575 0.480288 − 0.877111i \(-0.340532\pi\)
0.480288 + 0.877111i \(0.340532\pi\)
\(44\) 0 0
\(45\) 2.48220 0.370024
\(46\) 0 0
\(47\) −9.24147 −1.34801 −0.674003 − 0.738728i \(-0.735427\pi\)
−0.674003 + 0.738728i \(0.735427\pi\)
\(48\) 0 0
\(49\) 10.5074 1.50106
\(50\) 0 0
\(51\) 4.69299 0.657151
\(52\) 0 0
\(53\) 7.77015 1.06731 0.533656 − 0.845702i \(-0.320818\pi\)
0.533656 + 0.845702i \(0.320818\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.532101 −0.0704785
\(58\) 0 0
\(59\) −6.16416 −0.802505 −0.401253 − 0.915967i \(-0.631425\pi\)
−0.401253 + 0.915967i \(0.631425\pi\)
\(60\) 0 0
\(61\) −14.6869 −1.88046 −0.940230 − 0.340541i \(-0.889390\pi\)
−0.940230 + 0.340541i \(0.889390\pi\)
\(62\) 0 0
\(63\) −10.3860 −1.30851
\(64\) 0 0
\(65\) −6.05056 −0.750480
\(66\) 0 0
\(67\) 5.53268 0.675924 0.337962 − 0.941160i \(-0.390262\pi\)
0.337962 + 0.941160i \(0.390262\pi\)
\(68\) 0 0
\(69\) −3.69214 −0.444481
\(70\) 0 0
\(71\) −8.87252 −1.05297 −0.526487 − 0.850183i \(-0.676491\pi\)
−0.526487 + 0.850183i \(0.676491\pi\)
\(72\) 0 0
\(73\) 8.09351 0.947274 0.473637 − 0.880720i \(-0.342941\pi\)
0.473637 + 0.880720i \(0.342941\pi\)
\(74\) 0 0
\(75\) −0.719585 −0.0830906
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −0.570482 −0.0641842 −0.0320921 − 0.999485i \(-0.510217\pi\)
−0.0320921 + 0.999485i \(0.510217\pi\)
\(80\) 0 0
\(81\) 4.60789 0.511988
\(82\) 0 0
\(83\) −15.2224 −1.67088 −0.835439 − 0.549584i \(-0.814786\pi\)
−0.835439 + 0.549584i \(0.814786\pi\)
\(84\) 0 0
\(85\) 6.52180 0.707389
\(86\) 0 0
\(87\) 1.49700 0.160495
\(88\) 0 0
\(89\) −1.33867 −0.141899 −0.0709495 − 0.997480i \(-0.522603\pi\)
−0.0709495 + 0.997480i \(0.522603\pi\)
\(90\) 0 0
\(91\) 25.3167 2.65391
\(92\) 0 0
\(93\) −0.810957 −0.0840923
\(94\) 0 0
\(95\) −0.739455 −0.0758664
\(96\) 0 0
\(97\) 9.96643 1.01194 0.505969 − 0.862552i \(-0.331135\pi\)
0.505969 + 0.862552i \(0.331135\pi\)
\(98\) 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.1.4 6
4.3 odd 2 4840.2.a.be.1.3 6
11.7 odd 10 880.2.bo.j.401.2 12
11.8 odd 10 880.2.bo.j.801.2 12
11.10 odd 2 9680.2.a.cx.1.4 6
44.7 even 10 440.2.y.b.401.2 yes 12
44.19 even 10 440.2.y.b.361.2 ✓ 12
44.43 even 2 4840.2.a.bf.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.b.361.2 ✓ 12 44.19 even 10
440.2.y.b.401.2 yes 12 44.7 even 10
880.2.bo.j.401.2 12 11.7 odd 10
880.2.bo.j.801.2 12 11.8 odd 10
4840.2.a.be.1.3 6 4.3 odd 2
4840.2.a.bf.1.3 6 44.43 even 2
9680.2.a.cx.1.4 6 11.10 odd 2
9680.2.a.cy.1.4 6 1.1 even 1 trivial