Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3240,2,Mod(1,3240)] 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("3240.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3240.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,3,0,-5,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(25.8715302549\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 360)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.08613\) of defining polynomial
Character \(\chi\) \(=\) 3240.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} +1.43807 q^{7} -1.35194 q^{11} -5.52420 q^{13} +4.82032 q^{17} -0.648061 q^{19} -8.90645 q^{23} +1.00000 q^{25} -7.17226 q^{29} -4.64806 q^{31} +1.43807 q^{35} +1.35194 q^{37} +0.351939 q^{41} +4.82032 q^{43} -9.49389 q^{47} -4.93196 q^{49} -8.17226 q^{53} -1.35194 q^{55} +1.46838 q^{59} +6.69646 q^{61} -5.52420 q^{65} +12.4307 q^{67} +2.22808 q^{71} -4.34452 q^{73} -1.94418 q^{77} -13.0484 q^{79} -5.26581 q^{83} +4.82032 q^{85} -11.0000 q^{89} -7.94418 q^{91} -0.648061 q^{95} +17.5800 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} - 5 q^{7} - 2 q^{11} + 2 q^{17} - 4 q^{19} - 7 q^{23} + 3 q^{25} - 7 q^{29} - 16 q^{31} - 5 q^{35} + 2 q^{37} - q^{41} + 2 q^{43} - 13 q^{47} + 10 q^{49} - 10 q^{53} - 2 q^{55} - 6 q^{59}+ \cdots + 30 q^{97}+O(q^{100}) \) 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
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.43807 0.543539 0.271770 − 0.962362i \(-0.412391\pi\)
0.271770 + 0.962362i \(0.412391\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.35194 −0.407625 −0.203813 − 0.979010i \(-0.565333\pi\)
−0.203813 + 0.979010i \(0.565333\pi\)
\(12\) 0 0
\(13\) −5.52420 −1.53214 −0.766069 − 0.642759i \(-0.777790\pi\)
−0.766069 + 0.642759i \(0.777790\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.82032 1.16910 0.584550 − 0.811358i \(-0.301271\pi\)
0.584550 + 0.811358i \(0.301271\pi\)
\(18\) 0 0
\(19\) −0.648061 −0.148675 −0.0743377 − 0.997233i \(-0.523684\pi\)
−0.0743377 + 0.997233i \(0.523684\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −8.90645 −1.85712 −0.928562 − 0.371178i \(-0.878954\pi\)
−0.928562 + 0.371178i \(0.878954\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.17226 −1.33186 −0.665928 − 0.746016i \(-0.731964\pi\)
−0.665928 + 0.746016i \(0.731964\pi\)
\(30\) 0 0
\(31\) −4.64806 −0.834816 −0.417408 − 0.908719i \(-0.637061\pi\)
−0.417408 + 0.908719i \(0.637061\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.43807 0.243078
\(36\) 0 0
\(37\) 1.35194 0.222257 0.111129 − 0.993806i \(-0.464553\pi\)
0.111129 + 0.993806i \(0.464553\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.351939 0.0549637 0.0274818 − 0.999622i \(-0.491251\pi\)
0.0274818 + 0.999622i \(0.491251\pi\)
\(42\) 0 0
\(43\) 4.82032 0.735092 0.367546 − 0.930005i \(-0.380198\pi\)
0.367546 + 0.930005i \(0.380198\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.49389 −1.38483 −0.692413 − 0.721501i \(-0.743452\pi\)
−0.692413 + 0.721501i \(0.743452\pi\)
\(48\) 0 0
\(49\) −4.93196 −0.704565
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.17226 −1.12255 −0.561273 − 0.827631i \(-0.689688\pi\)
−0.561273 + 0.827631i \(0.689688\pi\)
\(54\) 0 0
\(55\) −1.35194 −0.182295
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.46838 0.191167 0.0955835 − 0.995421i \(-0.469528\pi\)
0.0955835 + 0.995421i \(0.469528\pi\)
\(60\) 0 0
\(61\) 6.69646 0.857394 0.428697 − 0.903448i \(-0.358973\pi\)
0.428697 + 0.903448i \(0.358973\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.52420 −0.685193
\(66\) 0 0
\(67\) 12.4307 1.51865 0.759323 − 0.650714i \(-0.225530\pi\)
0.759323 + 0.650714i \(0.225530\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.22808 0.264424 0.132212 − 0.991221i \(-0.457792\pi\)
0.132212 + 0.991221i \(0.457792\pi\)
\(72\) 0 0
\(73\) −4.34452 −0.508488 −0.254244 − 0.967140i \(-0.581827\pi\)
−0.254244 + 0.967140i \(0.581827\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.94418 −0.221560
\(78\) 0 0
\(79\) −13.0484 −1.46806 −0.734030 − 0.679117i \(-0.762363\pi\)
−0.734030 + 0.679117i \(0.762363\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −5.26581 −0.577998 −0.288999 − 0.957329i \(-0.593322\pi\)
−0.288999 + 0.957329i \(0.593322\pi\)
\(84\) 0 0
\(85\) 4.82032 0.522837
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −11.0000 −1.16600 −0.582999 − 0.812473i \(-0.698121\pi\)
−0.582999 + 0.812473i \(0.698121\pi\)
\(90\) 0 0
\(91\) −7.94418 −0.832777
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.648061 −0.0664896
\(96\) 0 0
\(97\) 17.5800 1.78498 0.892490 − 0.451067i \(-0.148956\pi\)
0.892490 + 0.451067i \(0.148956\pi\)
\(98\) 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 3240.2.a.r.1.3 3
3.2 odd 2 3240.2.a.q.1.3 3
4.3 odd 2 6480.2.a.bx.1.1 3
9.2 odd 6 1080.2.q.d.361.1 6
9.4 even 3 360.2.q.d.241.1 yes 6
9.5 odd 6 1080.2.q.d.721.1 6
9.7 even 3 360.2.q.d.121.1 ✓ 6
12.11 even 2 6480.2.a.bu.1.1 3
36.7 odd 6 720.2.q.j.481.3 6
36.11 even 6 2160.2.q.j.1441.3 6
36.23 even 6 2160.2.q.j.721.3 6
36.31 odd 6 720.2.q.j.241.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.q.d.121.1 ✓ 6 9.7 even 3
360.2.q.d.241.1 yes 6 9.4 even 3
720.2.q.j.241.3 6 36.31 odd 6
720.2.q.j.481.3 6 36.7 odd 6
1080.2.q.d.361.1 6 9.2 odd 6
1080.2.q.d.721.1 6 9.5 odd 6
2160.2.q.j.721.3 6 36.23 even 6
2160.2.q.j.1441.3 6 36.11 even 6
3240.2.a.q.1.3 3 3.2 odd 2
3240.2.a.r.1.3 3 1.1 even 1 trivial
6480.2.a.bu.1.1 3 12.11 even 2
6480.2.a.bx.1.1 3 4.3 odd 2