Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1656,2,Mod(1,1656)] 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("1656.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1656, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1656 = 2^{3} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1656.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,4,0,2,0,0,0,2,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(13.2232265747\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{13 +4 \sqrt{7}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 5x^{2} + 6x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.92812\) of defining polynomial
Character \(\chi\) \(=\) 1656.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.64575 q^{5} +3.17320 q^{7} +0.962718 q^{11} +5.85623 q^{13} +2.68303 q^{17} -5.50198 q^{19} -1.00000 q^{23} +8.29150 q^{25} -1.85623 q^{29} -1.85623 q^{31} +11.5687 q^{35} -6.67519 q^{37} -9.14774 q^{41} +7.57655 q^{43} -8.34640 q^{47} +3.06920 q^{49} -5.99215 q^{53} +3.50983 q^{55} -8.80133 q^{59} +1.03728 q^{61} +21.3504 q^{65} -1.08102 q^{67} -13.1477 q^{71} +13.2223 q^{73} +3.05490 q^{77} -4.53927 q^{79} -3.03728 q^{83} +9.78167 q^{85} -3.10400 q^{89} +18.5830 q^{91} -20.0589 q^{95} +11.6379 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} + 2 q^{7} + 2 q^{11} + 4 q^{13} + 2 q^{17} + 8 q^{19} - 4 q^{23} + 12 q^{25} + 12 q^{29} + 12 q^{31} - 12 q^{35} + 14 q^{37} + 4 q^{41} + 4 q^{43} - 12 q^{47} + 28 q^{49} + 8 q^{53} + 16 q^{55}+ \cdots + 4 q^{97}+O(q^{100}) \) 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
\(5\) 3.64575 1.63043 0.815215 − 0.579159i \(-0.196619\pi\)
0.815215 + 0.579159i \(0.196619\pi\)
\(6\) 0 0
\(7\) 3.17320 1.19936 0.599679 − 0.800241i \(-0.295295\pi\)
0.599679 + 0.800241i \(0.295295\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.962718 0.290271 0.145135 − 0.989412i \(-0.453638\pi\)
0.145135 + 0.989412i \(0.453638\pi\)
\(12\) 0 0
\(13\) 5.85623 1.62423 0.812113 − 0.583500i \(-0.198317\pi\)
0.812113 + 0.583500i \(0.198317\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.68303 0.650731 0.325366 − 0.945588i \(-0.394513\pi\)
0.325366 + 0.945588i \(0.394513\pi\)
\(18\) 0 0
\(19\) −5.50198 −1.26224 −0.631121 − 0.775684i \(-0.717405\pi\)
−0.631121 + 0.775684i \(0.717405\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 8.29150 1.65830
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.85623 −0.344694 −0.172347 − 0.985036i \(-0.555135\pi\)
−0.172347 + 0.985036i \(0.555135\pi\)
\(30\) 0 0
\(31\) −1.85623 −0.333389 −0.166695 − 0.986009i \(-0.553309\pi\)
−0.166695 + 0.986009i \(0.553309\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 11.5687 1.95547
\(36\) 0 0
\(37\) −6.67519 −1.09739 −0.548697 − 0.836021i \(-0.684876\pi\)
−0.548697 + 0.836021i \(0.684876\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −9.14774 −1.42864 −0.714318 − 0.699821i \(-0.753263\pi\)
−0.714318 + 0.699821i \(0.753263\pi\)
\(42\) 0 0
\(43\) 7.57655 1.15541 0.577706 − 0.816245i \(-0.303948\pi\)
0.577706 + 0.816245i \(0.303948\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.34640 −1.21745 −0.608724 − 0.793382i \(-0.708318\pi\)
−0.608724 + 0.793382i \(0.708318\pi\)
\(48\) 0 0
\(49\) 3.06920 0.438458
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.99215 −0.823086 −0.411543 − 0.911390i \(-0.635010\pi\)
−0.411543 + 0.911390i \(0.635010\pi\)
\(54\) 0 0
\(55\) 3.50983 0.473266
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.80133 −1.14584 −0.572918 − 0.819613i \(-0.694189\pi\)
−0.572918 + 0.819613i \(0.694189\pi\)
\(60\) 0 0
\(61\) 1.03728 0.132810 0.0664051 − 0.997793i \(-0.478847\pi\)
0.0664051 + 0.997793i \(0.478847\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 21.3504 2.64819
\(66\) 0 0
\(67\) −1.08102 −0.132068 −0.0660338 − 0.997817i \(-0.521035\pi\)
−0.0660338 + 0.997817i \(0.521035\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −13.1477 −1.56035 −0.780175 − 0.625562i \(-0.784870\pi\)
−0.780175 + 0.625562i \(0.784870\pi\)
\(72\) 0 0
\(73\) 13.2223 1.54755 0.773777 − 0.633459i \(-0.218365\pi\)
0.773777 + 0.633459i \(0.218365\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.05490 0.348138
\(78\) 0 0
\(79\) −4.53927 −0.510707 −0.255354 − 0.966848i \(-0.582192\pi\)
−0.255354 + 0.966848i \(0.582192\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.03728 −0.333385 −0.166692 − 0.986009i \(-0.553309\pi\)
−0.166692 + 0.986009i \(0.553309\pi\)
\(84\) 0 0
\(85\) 9.78167 1.06097
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.10400 −0.329023 −0.164512 − 0.986375i \(-0.552605\pi\)
−0.164512 + 0.986375i \(0.552605\pi\)
\(90\) 0 0
\(91\) 18.5830 1.94803
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −20.0589 −2.05800
\(96\) 0 0
\(97\) 11.6379 1.18165 0.590825 − 0.806800i \(-0.298802\pi\)
0.590825 + 0.806800i \(0.298802\pi\)
\(98\) 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 1656.2.a.p.1.4 yes 4
3.2 odd 2 1656.2.a.o.1.2 ✓ 4
4.3 odd 2 3312.2.a.bh.1.3 4
12.11 even 2 3312.2.a.bg.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1656.2.a.o.1.2 ✓ 4 3.2 odd 2
1656.2.a.p.1.4 yes 4 1.1 even 1 trivial
3312.2.a.bg.1.1 4 12.11 even 2
3312.2.a.bh.1.3 4 4.3 odd 2