Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3312,2,Mod(1,3312)] 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("3312.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3312.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: \(26.4464531494\)
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: no (minimal twist has level 1656)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.92812\) of defining polynomial
Character \(\chi\) \(=\) 3312.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.64575 q^{5} +4.81895 q^{7} -2.68303 q^{11} -3.85623 q^{13} +0.962718 q^{17} -4.21048 q^{19} +1.00000 q^{23} +8.29150 q^{25} +7.85623 q^{29} -7.85623 q^{31} +17.5687 q^{35} +11.0294 q^{37} +0.564731 q^{41} +5.57655 q^{43} -7.63790 q^{47} +16.2223 q^{49} +9.99215 q^{53} -9.78167 q^{55} +15.0732 q^{59} -0.683033 q^{61} -14.0589 q^{65} +10.7935 q^{67} +3.43527 q^{71} +0.0692033 q^{73} -12.9294 q^{77} -6.89352 q^{79} +1.31697 q^{83} +3.50983 q^{85} +18.0413 q^{89} -18.5830 q^{91} -15.3504 q^{95} -4.34640 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} - 2 q^{7} - 2 q^{11} + 4 q^{13} + 2 q^{17} - 8 q^{19} + 4 q^{23} + 12 q^{25} + 12 q^{29} - 12 q^{31} + 12 q^{35} + 14 q^{37} + 4 q^{41} - 4 q^{43} + 12 q^{47} + 28 q^{49} + 8 q^{53} - 16 q^{55}+ \cdots + 4 q^{97}+O(q^{100}) \) 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
\(7\) 4.81895 1.82139 0.910696 − 0.413076i \(-0.135546\pi\)
0.910696 + 0.413076i \(0.135546\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.68303 −0.808965 −0.404482 − 0.914546i \(-0.632548\pi\)
−0.404482 + 0.914546i \(0.632548\pi\)
\(12\) 0 0
\(13\) −3.85623 −1.06953 −0.534763 − 0.845002i \(-0.679599\pi\)
−0.534763 + 0.845002i \(0.679599\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.962718 0.233494 0.116747 − 0.993162i \(-0.462753\pi\)
0.116747 + 0.993162i \(0.462753\pi\)
\(18\) 0 0
\(19\) −4.21048 −0.965951 −0.482975 − 0.875634i \(-0.660444\pi\)
−0.482975 + 0.875634i \(0.660444\pi\)
\(20\) 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
\(29\) 7.85623 1.45887 0.729433 − 0.684052i \(-0.239784\pi\)
0.729433 + 0.684052i \(0.239784\pi\)
\(30\) 0 0
\(31\) −7.85623 −1.41102 −0.705511 − 0.708699i \(-0.749282\pi\)
−0.705511 + 0.708699i \(0.749282\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 17.5687 2.96965
\(36\) 0 0
\(37\) 11.0294 1.81323 0.906614 − 0.421961i \(-0.138658\pi\)
0.906614 + 0.421961i \(0.138658\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.564731 0.0881962 0.0440981 − 0.999027i \(-0.485959\pi\)
0.0440981 + 0.999027i \(0.485959\pi\)
\(42\) 0 0
\(43\) 5.57655 0.850416 0.425208 − 0.905096i \(-0.360201\pi\)
0.425208 + 0.905096i \(0.360201\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.63790 −1.11410 −0.557051 − 0.830478i \(-0.688067\pi\)
−0.557051 + 0.830478i \(0.688067\pi\)
\(48\) 0 0
\(49\) 16.2223 2.31747
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.99215 1.37253 0.686264 − 0.727353i \(-0.259250\pi\)
0.686264 + 0.727353i \(0.259250\pi\)
\(54\) 0 0
\(55\) −9.78167 −1.31896
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 15.0732 1.96236 0.981180 − 0.193095i \(-0.0618527\pi\)
0.981180 + 0.193095i \(0.0618527\pi\)
\(60\) 0 0
\(61\) −0.683033 −0.0874534 −0.0437267 − 0.999044i \(-0.513923\pi\)
−0.0437267 + 0.999044i \(0.513923\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −14.0589 −1.74379
\(66\) 0 0
\(67\) 10.7935 1.31863 0.659317 − 0.751865i \(-0.270845\pi\)
0.659317 + 0.751865i \(0.270845\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.43527 0.407691 0.203846 − 0.979003i \(-0.434656\pi\)
0.203846 + 0.979003i \(0.434656\pi\)
\(72\) 0 0
\(73\) 0.0692033 0.00809963 0.00404981 − 0.999992i \(-0.498711\pi\)
0.00404981 + 0.999992i \(0.498711\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −12.9294 −1.47344
\(78\) 0 0
\(79\) −6.89352 −0.775581 −0.387791 − 0.921748i \(-0.626762\pi\)
−0.387791 + 0.921748i \(0.626762\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.31697 0.144556 0.0722780 − 0.997385i \(-0.476973\pi\)
0.0722780 + 0.997385i \(0.476973\pi\)
\(84\) 0 0
\(85\) 3.50983 0.380695
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 18.0413 1.91237 0.956184 − 0.292765i \(-0.0945754\pi\)
0.956184 + 0.292765i \(0.0945754\pi\)
\(90\) 0 0
\(91\) −18.5830 −1.94803
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −15.3504 −1.57491
\(96\) 0 0
\(97\) −4.34640 −0.441310 −0.220655 − 0.975352i \(-0.570820\pi\)
−0.220655 + 0.975352i \(0.570820\pi\)
\(98\) 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 3312.2.a.bh.1.4 4
3.2 odd 2 3312.2.a.bg.1.2 4
4.3 odd 2 1656.2.a.p.1.3 yes 4
12.11 even 2 1656.2.a.o.1.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1656.2.a.o.1.1 ✓ 4 12.11 even 2
1656.2.a.p.1.3 yes 4 4.3 odd 2
3312.2.a.bg.1.2 4 3.2 odd 2
3312.2.a.bh.1.4 4 1.1 even 1 trivial