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 1.2
Root \(1.27733\) 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)
 
\(f(q)\) \(=\) \(q-1.64575 q^{5} +4.65684 q^{7} +3.56576 q^{11} -0.554669 q^{13} -5.21151 q^{17} +6.20042 q^{19} -1.00000 q^{23} -2.29150 q^{25} +4.55467 q^{29} +4.55467 q^{31} -7.66401 q^{35} +3.54358 q^{37} +7.84617 q^{41} -9.33194 q^{43} -11.3137 q^{47} +14.6862 q^{49} -3.66794 q^{53} -5.86836 q^{55} +11.1599 q^{59} -1.56576 q^{61} +0.912847 q^{65} +8.38259 q^{67} +3.84617 q^{71} -8.97769 q^{73} +16.6052 q^{77} +9.76618 q^{79} -0.434239 q^{83} +8.57685 q^{85} +7.02935 q^{89} -2.58301 q^{91} -10.2043 q^{95} +4.02218 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} + 2 q^{7} + 2 q^{11} + 4 q^{13} + 2 q^{17} + 8 q^{19} - 4 q^{23} + 12 q^{25} + 12 q^{29} + 12 q^{31} - 12 q^{35} + 14 q^{37} + 4 q^{41} + 4 q^{43} - 12 q^{47} + 28 q^{49} + 8 q^{53} + 16 q^{55}+ \cdots + 4 q^{97}+O(q^{100}) \) 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
\(5\) −1.64575 −0.736002 −0.368001 − 0.929825i \(-0.619958\pi\)
−0.368001 + 0.929825i \(0.619958\pi\)
\(6\) 0 0
\(7\) 4.65684 1.76012 0.880061 − 0.474861i \(-0.157502\pi\)
0.880061 + 0.474861i \(0.157502\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.56576 1.07512 0.537559 − 0.843226i \(-0.319347\pi\)
0.537559 + 0.843226i \(0.319347\pi\)
\(12\) 0 0
\(13\) −0.554669 −0.153837 −0.0769187 − 0.997037i \(-0.524508\pi\)
−0.0769187 + 0.997037i \(0.524508\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.21151 −1.26398 −0.631989 − 0.774978i \(-0.717761\pi\)
−0.631989 + 0.774978i \(0.717761\pi\)
\(18\) 0 0
\(19\) 6.20042 1.42247 0.711237 − 0.702952i \(-0.248135\pi\)
0.711237 + 0.702952i \(0.248135\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −2.29150 −0.458301
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.55467 0.845781 0.422890 − 0.906181i \(-0.361016\pi\)
0.422890 + 0.906181i \(0.361016\pi\)
\(30\) 0 0
\(31\) 4.55467 0.818043 0.409021 − 0.912525i \(-0.365870\pi\)
0.409021 + 0.912525i \(0.365870\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −7.66401 −1.29545
\(36\) 0 0
\(37\) 3.54358 0.582560 0.291280 − 0.956638i \(-0.405919\pi\)
0.291280 + 0.956638i \(0.405919\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.84617 1.22537 0.612683 − 0.790329i \(-0.290090\pi\)
0.612683 + 0.790329i \(0.290090\pi\)
\(42\) 0 0
\(43\) −9.33194 −1.42311 −0.711554 − 0.702632i \(-0.752008\pi\)
−0.711554 + 0.702632i \(0.752008\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.3137 −1.65027 −0.825135 − 0.564935i \(-0.808901\pi\)
−0.825135 + 0.564935i \(0.808901\pi\)
\(48\) 0 0
\(49\) 14.6862 2.09803
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.66794 −0.503830 −0.251915 − 0.967749i \(-0.581060\pi\)
−0.251915 + 0.967749i \(0.581060\pi\)
\(54\) 0 0
\(55\) −5.86836 −0.791289
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.1599 1.45289 0.726445 − 0.687225i \(-0.241171\pi\)
0.726445 + 0.687225i \(0.241171\pi\)
\(60\) 0 0
\(61\) −1.56576 −0.200475 −0.100238 − 0.994964i \(-0.531960\pi\)
−0.100238 + 0.994964i \(0.531960\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.912847 0.113225
\(66\) 0 0
\(67\) 8.38259 1.02410 0.512048 − 0.858957i \(-0.328887\pi\)
0.512048 + 0.858957i \(0.328887\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.84617 0.456457 0.228228 − 0.973608i \(-0.426707\pi\)
0.228228 + 0.973608i \(0.426707\pi\)
\(72\) 0 0
\(73\) −8.97769 −1.05076 −0.525380 − 0.850868i \(-0.676077\pi\)
−0.525380 + 0.850868i \(0.676077\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 16.6052 1.89234
\(78\) 0 0
\(79\) 9.76618 1.09878 0.549391 − 0.835566i \(-0.314860\pi\)
0.549391 + 0.835566i \(0.314860\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −0.434239 −0.0476639 −0.0238320 − 0.999716i \(-0.507587\pi\)
−0.0238320 + 0.999716i \(0.507587\pi\)
\(84\) 0 0
\(85\) 8.57685 0.930290
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.02935 0.745109 0.372555 − 0.928010i \(-0.378482\pi\)
0.372555 + 0.928010i \(0.378482\pi\)
\(90\) 0 0
\(91\) −2.58301 −0.270773
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −10.2043 −1.04694
\(96\) 0 0
\(97\) 4.02218 0.408391 0.204195 − 0.978930i \(-0.434542\pi\)
0.204195 + 0.978930i \(0.434542\pi\)
\(98\) 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.1.2 yes 4
3.2 odd 2 1656.2.a.o.1.4 ✓ 4
4.3 odd 2 3312.2.a.bh.1.1 4
12.11 even 2 3312.2.a.bg.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1656.2.a.o.1.4 ✓ 4 3.2 odd 2
1656.2.a.p.1.2 yes 4 1.1 even 1 trivial
3312.2.a.bg.1.3 4 12.11 even 2
3312.2.a.bh.1.1 4 4.3 odd 2