Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6664,2,Mod(1,6664)] 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("6664.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6664 = 2^{3} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6664.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,0,3,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(53.2123079070\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 8x^{5} + 21x^{4} + 25x^{3} - 41x^{2} - 28x + 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 952)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.474430\) of defining polynomial
Character \(\chi\) \(=\) 6664.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.474430 q^{3} -1.63521 q^{5} -2.77492 q^{9} -0.241103 q^{11} -1.90325 q^{13} -0.775795 q^{15} -1.00000 q^{17} +5.11412 q^{19} +3.66413 q^{23} -2.32607 q^{25} -2.73980 q^{27} +2.50385 q^{29} +2.41583 q^{31} -0.114387 q^{33} +3.35512 q^{37} -0.902958 q^{39} +9.52792 q^{41} +9.54468 q^{43} +4.53758 q^{45} +8.63945 q^{47} -0.474430 q^{51} -11.6158 q^{53} +0.394255 q^{55} +2.42629 q^{57} +2.07093 q^{59} -10.6760 q^{61} +3.11222 q^{65} -12.5662 q^{67} +1.73837 q^{69} +0.576299 q^{71} -15.5443 q^{73} -1.10356 q^{75} -8.46606 q^{79} +7.02491 q^{81} -1.02923 q^{83} +1.63521 q^{85} +1.18790 q^{87} -11.4269 q^{89} +1.14614 q^{93} -8.36268 q^{95} +3.69602 q^{97} +0.669041 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 3 q^{3} - 2 q^{5} + 4 q^{9} - 12 q^{11} - 2 q^{13} - 6 q^{15} - 7 q^{17} + 3 q^{19} - 18 q^{23} + 15 q^{25} + 18 q^{27} + 5 q^{29} + 10 q^{31} - 21 q^{33} + 11 q^{37} - 12 q^{39} - 15 q^{43} - 13 q^{45}+ \cdots - 20 q^{99}+O(q^{100}) \) 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
\(3\) 0.474430 0.273913 0.136956 − 0.990577i \(-0.456268\pi\)
0.136956 + 0.990577i \(0.456268\pi\)
\(4\) 0 0
\(5\) −1.63521 −0.731290 −0.365645 − 0.930754i \(-0.619152\pi\)
−0.365645 + 0.930754i \(0.619152\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −2.77492 −0.924972
\(10\) 0 0
\(11\) −0.241103 −0.0726953 −0.0363477 − 0.999339i \(-0.511572\pi\)
−0.0363477 + 0.999339i \(0.511572\pi\)
\(12\) 0 0
\(13\) −1.90325 −0.527866 −0.263933 − 0.964541i \(-0.585020\pi\)
−0.263933 + 0.964541i \(0.585020\pi\)
\(14\) 0 0
\(15\) −0.775795 −0.200310
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) 5.11412 1.17326 0.586630 − 0.809855i \(-0.300454\pi\)
0.586630 + 0.809855i \(0.300454\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.66413 0.764023 0.382012 − 0.924158i \(-0.375231\pi\)
0.382012 + 0.924158i \(0.375231\pi\)
\(24\) 0 0
\(25\) −2.32607 −0.465215
\(26\) 0 0
\(27\) −2.73980 −0.527274
\(28\) 0 0
\(29\) 2.50385 0.464953 0.232477 − 0.972602i \(-0.425317\pi\)
0.232477 + 0.972602i \(0.425317\pi\)
\(30\) 0 0
\(31\) 2.41583 0.433896 0.216948 − 0.976183i \(-0.430390\pi\)
0.216948 + 0.976183i \(0.430390\pi\)
\(32\) 0 0
\(33\) −0.114387 −0.0199122
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.35512 0.551579 0.275789 − 0.961218i \(-0.411061\pi\)
0.275789 + 0.961218i \(0.411061\pi\)
\(38\) 0 0
\(39\) −0.902958 −0.144589
\(40\) 0 0
\(41\) 9.52792 1.48801 0.744006 − 0.668173i \(-0.232924\pi\)
0.744006 + 0.668173i \(0.232924\pi\)
\(42\) 0 0
\(43\) 9.54468 1.45555 0.727775 − 0.685816i \(-0.240555\pi\)
0.727775 + 0.685816i \(0.240555\pi\)
\(44\) 0 0
\(45\) 4.53758 0.676423
\(46\) 0 0
\(47\) 8.63945 1.26019 0.630097 − 0.776517i \(-0.283015\pi\)
0.630097 + 0.776517i \(0.283015\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −0.474430 −0.0664335
\(52\) 0 0
\(53\) −11.6158 −1.59556 −0.797779 − 0.602949i \(-0.793992\pi\)
−0.797779 + 0.602949i \(0.793992\pi\)
\(54\) 0 0
\(55\) 0.394255 0.0531614
\(56\) 0 0
\(57\) 2.42629 0.321370
\(58\) 0 0
\(59\) 2.07093 0.269612 0.134806 − 0.990872i \(-0.456959\pi\)
0.134806 + 0.990872i \(0.456959\pi\)
\(60\) 0 0
\(61\) −10.6760 −1.36693 −0.683463 − 0.729985i \(-0.739527\pi\)
−0.683463 + 0.729985i \(0.739527\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.11222 0.386023
\(66\) 0 0
\(67\) −12.5662 −1.53521 −0.767604 − 0.640924i \(-0.778551\pi\)
−0.767604 + 0.640924i \(0.778551\pi\)
\(68\) 0 0
\(69\) 1.73837 0.209275
\(70\) 0 0
\(71\) 0.576299 0.0683942 0.0341971 − 0.999415i \(-0.489113\pi\)
0.0341971 + 0.999415i \(0.489113\pi\)
\(72\) 0 0
\(73\) −15.5443 −1.81932 −0.909659 − 0.415355i \(-0.863657\pi\)
−0.909659 + 0.415355i \(0.863657\pi\)
\(74\) 0 0
\(75\) −1.10356 −0.127428
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.46606 −0.952507 −0.476253 − 0.879308i \(-0.658005\pi\)
−0.476253 + 0.879308i \(0.658005\pi\)
\(80\) 0 0
\(81\) 7.02491 0.780545
\(82\) 0 0
\(83\) −1.02923 −0.112972 −0.0564862 − 0.998403i \(-0.517990\pi\)
−0.0564862 + 0.998403i \(0.517990\pi\)
\(84\) 0 0
\(85\) 1.63521 0.177364
\(86\) 0 0
\(87\) 1.18790 0.127356
\(88\) 0 0
\(89\) −11.4269 −1.21125 −0.605624 − 0.795751i \(-0.707076\pi\)
−0.605624 + 0.795751i \(0.707076\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.14614 0.118850
\(94\) 0 0
\(95\) −8.36268 −0.857993
\(96\) 0 0
\(97\) 3.69602 0.375274 0.187637 − 0.982238i \(-0.439917\pi\)
0.187637 + 0.982238i \(0.439917\pi\)
\(98\) 0 0
\(99\) 0.669041 0.0672411
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 6664.2.a.y.1.4 7
7.3 odd 6 952.2.q.e.681.4 yes 14
7.5 odd 6 952.2.q.e.137.4 ✓ 14
7.6 odd 2 6664.2.a.v.1.4 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.q.e.137.4 ✓ 14 7.5 odd 6
952.2.q.e.681.4 yes 14 7.3 odd 6
6664.2.a.v.1.4 7 7.6 odd 2
6664.2.a.y.1.4 7 1.1 even 1 trivial