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 = [2,0,0,0,2,0,-2,0,0,0,-4,0,-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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) 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)
 
\(f(q)\) \(=\) \(q+2.73205 q^{5} -2.73205 q^{7} -5.46410 q^{11} +1.46410 q^{13} -2.73205 q^{17} -1.26795 q^{19} -1.00000 q^{23} +2.46410 q^{25} -1.46410 q^{29} -9.46410 q^{31} -7.46410 q^{35} -7.46410 q^{37} +6.92820 q^{41} -2.73205 q^{43} -0.535898 q^{47} +0.464102 q^{49} +3.80385 q^{53} -14.9282 q^{55} +7.46410 q^{59} -14.3923 q^{61} +4.00000 q^{65} +4.19615 q^{67} +2.92820 q^{71} -6.00000 q^{73} +14.9282 q^{77} +15.1244 q^{79} +4.39230 q^{83} -7.46410 q^{85} +4.19615 q^{89} -4.00000 q^{91} -3.46410 q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{7} - 4 q^{11} - 4 q^{13} - 2 q^{17} - 6 q^{19} - 2 q^{23} - 2 q^{25} + 4 q^{29} - 12 q^{31} - 8 q^{35} - 8 q^{37} - 2 q^{43} - 8 q^{47} - 6 q^{49} + 18 q^{53} - 16 q^{55} + 8 q^{59}+ \cdots - 4 q^{97}+O(q^{100}) \) 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
\(5\) 2.73205 1.22181 0.610905 − 0.791704i \(-0.290806\pi\)
0.610905 + 0.791704i \(0.290806\pi\)
\(6\) 0 0
\(7\) −2.73205 −1.03262 −0.516309 − 0.856402i \(-0.672694\pi\)
−0.516309 + 0.856402i \(0.672694\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.46410 −1.64749 −0.823744 − 0.566961i \(-0.808119\pi\)
−0.823744 + 0.566961i \(0.808119\pi\)
\(12\) 0 0
\(13\) 1.46410 0.406069 0.203034 − 0.979172i \(-0.434920\pi\)
0.203034 + 0.979172i \(0.434920\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.73205 −0.662620 −0.331310 − 0.943522i \(-0.607491\pi\)
−0.331310 + 0.943522i \(0.607491\pi\)
\(18\) 0 0
\(19\) −1.26795 −0.290887 −0.145444 − 0.989367i \(-0.546461\pi\)
−0.145444 + 0.989367i \(0.546461\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 2.46410 0.492820
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.46410 −0.271877 −0.135938 − 0.990717i \(-0.543405\pi\)
−0.135938 + 0.990717i \(0.543405\pi\)
\(30\) 0 0
\(31\) −9.46410 −1.69980 −0.849901 − 0.526942i \(-0.823339\pi\)
−0.849901 + 0.526942i \(0.823339\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −7.46410 −1.26166
\(36\) 0 0
\(37\) −7.46410 −1.22709 −0.613545 − 0.789659i \(-0.710257\pi\)
−0.613545 + 0.789659i \(0.710257\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.92820 1.08200 0.541002 − 0.841021i \(-0.318045\pi\)
0.541002 + 0.841021i \(0.318045\pi\)
\(42\) 0 0
\(43\) −2.73205 −0.416634 −0.208317 − 0.978061i \(-0.566799\pi\)
−0.208317 + 0.978061i \(0.566799\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.535898 −0.0781688 −0.0390844 − 0.999236i \(-0.512444\pi\)
−0.0390844 + 0.999236i \(0.512444\pi\)
\(48\) 0 0
\(49\) 0.464102 0.0663002
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.80385 0.522499 0.261249 − 0.965271i \(-0.415866\pi\)
0.261249 + 0.965271i \(0.415866\pi\)
\(54\) 0 0
\(55\) −14.9282 −2.01292
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.46410 0.971743 0.485872 − 0.874030i \(-0.338502\pi\)
0.485872 + 0.874030i \(0.338502\pi\)
\(60\) 0 0
\(61\) −14.3923 −1.84275 −0.921373 − 0.388680i \(-0.872931\pi\)
−0.921373 + 0.388680i \(0.872931\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 4.19615 0.512642 0.256321 − 0.966592i \(-0.417490\pi\)
0.256321 + 0.966592i \(0.417490\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.92820 0.347514 0.173757 − 0.984789i \(-0.444409\pi\)
0.173757 + 0.984789i \(0.444409\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 14.9282 1.70123
\(78\) 0 0
\(79\) 15.1244 1.70162 0.850811 − 0.525471i \(-0.176111\pi\)
0.850811 + 0.525471i \(0.176111\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.39230 0.482118 0.241059 − 0.970510i \(-0.422505\pi\)
0.241059 + 0.970510i \(0.422505\pi\)
\(84\) 0 0
\(85\) −7.46410 −0.809595
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.19615 0.444791 0.222396 − 0.974957i \(-0.428612\pi\)
0.222396 + 0.974957i \(0.428612\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.46410 −0.355409
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 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.m.1.2 yes 2
3.2 odd 2 1656.2.a.k.1.1 ✓ 2
4.3 odd 2 3312.2.a.bd.1.2 2
12.11 even 2 3312.2.a.x.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1656.2.a.k.1.1 ✓ 2 3.2 odd 2
1656.2.a.m.1.2 yes 2 1.1 even 1 trivial
3312.2.a.x.1.1 2 12.11 even 2
3312.2.a.bd.1.2 2 4.3 odd 2