Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2664,2,Mod(1,2664)] 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("2664.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2664 = 2^{3} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2664.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,-2,0,-5,0,0,0,3] 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: \(21.2721470985\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 888)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.81361\) of defining polynomial
Character \(\chi\) \(=\) 2664.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.52444 q^{5} +2.91638 q^{7} +1.28917 q^{11} +3.28917 q^{13} -1.23527 q^{17} +1.28917 q^{19} +4.01916 q^{23} +1.37279 q^{25} -1.47556 q^{29} -2.57834 q^{31} -7.36222 q^{35} -1.00000 q^{37} -10.4111 q^{41} +4.00000 q^{43} +5.83276 q^{47} +1.50528 q^{49} +2.54359 q^{53} -3.25443 q^{55} +2.05390 q^{59} +14.4111 q^{61} -8.30330 q^{65} +13.2544 q^{67} -3.25443 q^{71} +3.86751 q^{73} +3.75971 q^{77} -4.78389 q^{79} +4.54359 q^{83} +3.11836 q^{85} +6.76473 q^{89} +9.59247 q^{91} -3.25443 q^{95} +11.0489 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{5} - 5 q^{7} + 3 q^{11} + 9 q^{13} + q^{17} + 3 q^{19} - 9 q^{23} + 17 q^{25} - 10 q^{29} - 6 q^{31} - 4 q^{35} - 3 q^{37} - 2 q^{41} + 12 q^{43} - 10 q^{47} + 20 q^{49} - 19 q^{53} + 16 q^{55}+ \cdots + 22 q^{97}+O(q^{100}) \) 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
\(5\) −2.52444 −1.12896 −0.564481 − 0.825446i \(-0.690924\pi\)
−0.564481 + 0.825446i \(0.690924\pi\)
\(6\) 0 0
\(7\) 2.91638 1.10229 0.551144 − 0.834410i \(-0.314191\pi\)
0.551144 + 0.834410i \(0.314191\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.28917 0.388699 0.194349 − 0.980932i \(-0.437740\pi\)
0.194349 + 0.980932i \(0.437740\pi\)
\(12\) 0 0
\(13\) 3.28917 0.912251 0.456126 − 0.889915i \(-0.349237\pi\)
0.456126 + 0.889915i \(0.349237\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.23527 −0.299597 −0.149798 − 0.988717i \(-0.547862\pi\)
−0.149798 + 0.988717i \(0.547862\pi\)
\(18\) 0 0
\(19\) 1.28917 0.295756 0.147878 − 0.989006i \(-0.452756\pi\)
0.147878 + 0.989006i \(0.452756\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.01916 0.838052 0.419026 − 0.907974i \(-0.362372\pi\)
0.419026 + 0.907974i \(0.362372\pi\)
\(24\) 0 0
\(25\) 1.37279 0.274557
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.47556 −0.274005 −0.137002 − 0.990571i \(-0.543747\pi\)
−0.137002 + 0.990571i \(0.543747\pi\)
\(30\) 0 0
\(31\) −2.57834 −0.463083 −0.231542 − 0.972825i \(-0.574377\pi\)
−0.231542 + 0.972825i \(0.574377\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −7.36222 −1.24444
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.4111 −1.62594 −0.812970 − 0.582305i \(-0.802151\pi\)
−0.812970 + 0.582305i \(0.802151\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.83276 0.850796 0.425398 − 0.905006i \(-0.360134\pi\)
0.425398 + 0.905006i \(0.360134\pi\)
\(48\) 0 0
\(49\) 1.50528 0.215040
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.54359 0.349390 0.174695 − 0.984623i \(-0.444106\pi\)
0.174695 + 0.984623i \(0.444106\pi\)
\(54\) 0 0
\(55\) −3.25443 −0.438827
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.05390 0.267395 0.133697 − 0.991022i \(-0.457315\pi\)
0.133697 + 0.991022i \(0.457315\pi\)
\(60\) 0 0
\(61\) 14.4111 1.84515 0.922576 − 0.385815i \(-0.126080\pi\)
0.922576 + 0.385815i \(0.126080\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −8.30330 −1.02990
\(66\) 0 0
\(67\) 13.2544 1.61929 0.809643 − 0.586923i \(-0.199661\pi\)
0.809643 + 0.586923i \(0.199661\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.25443 −0.386229 −0.193115 − 0.981176i \(-0.561859\pi\)
−0.193115 + 0.981176i \(0.561859\pi\)
\(72\) 0 0
\(73\) 3.86751 0.452657 0.226329 − 0.974051i \(-0.427328\pi\)
0.226329 + 0.974051i \(0.427328\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.75971 0.428458
\(78\) 0 0
\(79\) −4.78389 −0.538229 −0.269115 − 0.963108i \(-0.586731\pi\)
−0.269115 + 0.963108i \(0.586731\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.54359 0.498724 0.249362 − 0.968410i \(-0.419779\pi\)
0.249362 + 0.968410i \(0.419779\pi\)
\(84\) 0 0
\(85\) 3.11836 0.338234
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.76473 0.717060 0.358530 − 0.933518i \(-0.383278\pi\)
0.358530 + 0.933518i \(0.383278\pi\)
\(90\) 0 0
\(91\) 9.59247 1.00556
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.25443 −0.333897
\(96\) 0 0
\(97\) 11.0489 1.12184 0.560922 − 0.827869i \(-0.310447\pi\)
0.560922 + 0.827869i \(0.310447\pi\)
\(98\) 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 2664.2.a.m.1.2 3
3.2 odd 2 888.2.a.i.1.2 ✓ 3
4.3 odd 2 5328.2.a.bl.1.2 3
12.11 even 2 1776.2.a.s.1.2 3
24.5 odd 2 7104.2.a.bw.1.2 3
24.11 even 2 7104.2.a.bq.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.a.i.1.2 ✓ 3 3.2 odd 2
1776.2.a.s.1.2 3 12.11 even 2
2664.2.a.m.1.2 3 1.1 even 1 trivial
5328.2.a.bl.1.2 3 4.3 odd 2
7104.2.a.bq.1.2 3 24.11 even 2
7104.2.a.bw.1.2 3 24.5 odd 2