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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [672,2,Mod(257,672)] 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("672.257"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.bc (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{14} - 3x^{12} + 14x^{10} - 74x^{8} + 126x^{6} - 243x^{4} - 729x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{10} q^{3} + \beta_{8} q^{5} + \beta_{9} q^{7} + (\beta_{11} + \beta_{6} - \beta_{4} + \cdots + 1) q^{9} + ( - 2 \beta_{14} - 2 \beta_{12} + \cdots - \beta_1) q^{11} + ( - \beta_{11} - \beta_{6} + 4 \beta_{2} + 2) q^{13}+ \cdots + (4 \beta_{14} - 3 \beta_{13} + \cdots - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{9} + 26 q^{21} - 12 q^{25} - 66 q^{33} + 16 q^{37} - 18 q^{45} - 12 q^{49} + 60 q^{57} - 84 q^{61} - 48 q^{73} + 14 q^{81} + 104 q^{85} - 22 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{14} - 3x^{12} + 14x^{10} - 74x^{8} + 126x^{6} - 243x^{4} - 729x^{2} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13\nu^{14} - 58\nu^{12} + 87\nu^{10} - 493\nu^{8} - 377\nu^{6} + 3915\nu^{4} - 14094\nu^{2} + 21141 ) / 34992 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{14} - 7\nu^{12} + 15\nu^{10} - \nu^{8} + 22\nu^{6} + 1143\nu^{4} + 81\nu^{2} + 2187 ) / 4374 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{14} + 13\nu^{12} + 3\nu^{10} - 83\nu^{8} - 307\nu^{6} - 441\nu^{4} + 7209\nu^{2} ) / 8748 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{14} + 4\nu^{12} + 93\nu^{10} - 137\nu^{8} + 53\nu^{6} - 99\nu^{4} - 1377\nu^{2} + 12393 ) / 8748 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4\nu^{14} - 13\nu^{12} - 3\nu^{10} + 83\nu^{8} + 307\nu^{6} + 441\nu^{4} + 1539\nu^{2} ) / 8748 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{15} + 4\nu^{13} + 15\nu^{11} - 17\nu^{9} - 145\nu^{7} + 915\nu^{5} - 999\nu^{3} + 1215\nu ) / 8748 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -7\nu^{14} + 46\nu^{12} - 45\nu^{10} + 55\nu^{8} + 659\nu^{6} + 471\nu^{4} + 6426\nu^{2} - 18711 ) / 11664 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -2\nu^{15} + 14\nu^{13} - 33\nu^{11} - 37\nu^{9} + 397\nu^{7} - 789\nu^{5} + 4725\nu^{3} + 243\nu ) / 8748 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 29 \nu^{15} + 146 \nu^{13} - 435 \nu^{11} + 377 \nu^{9} - 2291 \nu^{7} - 7047 \nu^{5} + \cdots - 105705 \nu ) / 104976 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 8\nu^{14} - 8\nu^{12} + 57\nu^{10} + 31\nu^{8} - 835\nu^{6} + 1413\nu^{4} - 4293\nu^{2} + 3645 ) / 8748 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13\nu^{15} - 58\nu^{13} + 87\nu^{11} - 493\nu^{9} - 377\nu^{7} + 3915\nu^{5} - 14094\nu^{3} + 21141\nu ) / 34992 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 4\nu^{15} + 2\nu^{13} + 9\nu^{11} + 11\nu^{9} - 293\nu^{7} - 291\nu^{5} + 1431\nu^{3} + 2673\nu ) / 8748 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( \nu^{15} - \nu^{13} - 3\nu^{11} + 14\nu^{9} - 74\nu^{7} + 126\nu^{5} - 243\nu^{3} - 729\nu ) / 2187 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -29\nu^{15} + 128\nu^{13} - 255\nu^{11} + 269\nu^{9} + 1345\nu^{7} - 12195\nu^{5} + 19278\nu^{3} - 54675\nu ) / 52488 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{10} + \beta_{9} - \beta_{7} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + 2\beta_{8} - \beta_{5} + 2\beta_{3} + 2\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{15} + \beta_{14} - 4\beta_{13} - 2\beta_{12} + 2\beta_{10} + 2\beta_{9} + 4\beta_{7} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{11} + 4\beta_{8} + 8\beta_{6} + 4\beta_{5} - 4\beta_{4} - 2\beta_{3} + 4\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 6\beta_{14} - 6\beta_{13} - 6\beta_{12} - 18\beta_{10} + 12\beta_{9} - 11\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 12\beta_{11} - 6\beta_{8} + 7\beta_{6} - 17\beta_{4} + 12\beta_{3} - 54\beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -14\beta_{15} + 29\beta_{14} - 11\beta_{13} - 67\beta_{12} - 29\beta_{10} - 11\beta_{9} - 7\beta_{7} + 11\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 31\beta_{11} - 4\beta_{8} + 60\beta_{6} + 89\beta_{5} - 4\beta_{3} - 52\beta_{2} - 112 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 52 \beta_{15} - 203 \beta_{14} + 128 \beta_{13} - 104 \beta_{12} - 64 \beta_{10} - 64 \beta_{9} + \cdots - 168 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 116\beta_{11} + 232\beta_{8} - 76\beta_{6} + 40\beta_{5} - 40\beta_{4} - 116\beta_{3} + 40\beta_{2} + 301 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 348\beta_{15} - 120\beta_{10} + 696\beta_{7} + 145\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 348\beta_{11} + 348\beta_{8} + 841\beta_{6} + 145\beta_{4} - 696\beta_{3} + 684\beta_{2} + 348 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 406 \beta_{15} + 1247 \beta_{14} + 841 \beta_{13} + 191 \beta_{12} - 1247 \beta_{10} + 841 \beta_{9} + \cdots - 841 \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-\beta_{2}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
−1.05965 + 1.37009i
0.0465589 + 1.73142i
−1.71636 + 0.232633i
1.47618 + 0.906034i
−1.47618 − 0.906034i
1.71636 − 0.232633i
−0.0465589 − 1.73142i
1.05965 − 1.37009i
−1.05965 − 1.37009i
0.0465589 − 1.73142i
−1.71636 − 0.232633i
1.47618 − 0.906034i
−1.47618 + 0.906034i
1.71636 + 0.232633i
−0.0465589 + 1.73142i
1.05965 + 1.37009i
0 −1.71636 + 0.232633i 0 −1.60273 + 2.77600i 0 −2.44053 + 1.02167i 0 2.89176 − 0.798564i 0
257.2 0 −1.47618 − 0.906034i 0 0.825391 − 1.42962i 0 0.542044 − 2.58963i 0 1.35821 + 2.67493i 0
257.3 0 −1.05965 + 1.37009i 0 1.60273 − 2.77600i 0 −2.44053 + 1.02167i 0 −0.754305 − 2.90362i 0
257.4 0 −0.0465589 − 1.73142i 0 −0.825391 + 1.42962i 0 −0.542044 + 2.58963i 0 −2.99566 + 0.161227i 0
257.5 0 0.0465589 + 1.73142i 0 −0.825391 + 1.42962i 0 0.542044 − 2.58963i 0 −2.99566 + 0.161227i 0
257.6 0 1.05965 − 1.37009i 0 1.60273 − 2.77600i 0 2.44053 − 1.02167i 0 −0.754305 − 2.90362i 0
257.7 0 1.47618 + 0.906034i 0 0.825391 − 1.42962i 0 −0.542044 + 2.58963i 0 1.35821 + 2.67493i 0
257.8 0 1.71636 − 0.232633i 0 −1.60273 + 2.77600i 0 2.44053 − 1.02167i 0 2.89176 − 0.798564i 0
353.1 0 −1.71636 − 0.232633i 0 −1.60273 − 2.77600i 0 −2.44053 − 1.02167i 0 2.89176 + 0.798564i 0
353.2 0 −1.47618 + 0.906034i 0 0.825391 + 1.42962i 0 0.542044 + 2.58963i 0 1.35821 − 2.67493i 0
353.3 0 −1.05965 − 1.37009i 0 1.60273 + 2.77600i 0 −2.44053 − 1.02167i 0 −0.754305 + 2.90362i 0
353.4 0 −0.0465589 + 1.73142i 0 −0.825391 − 1.42962i 0 −0.542044 − 2.58963i 0 −2.99566 − 0.161227i 0
353.5 0 0.0465589 − 1.73142i 0 −0.825391 − 1.42962i 0 0.542044 + 2.58963i 0 −2.99566 − 0.161227i 0
353.6 0 1.05965 + 1.37009i 0 1.60273 + 2.77600i 0 2.44053 + 1.02167i 0 −0.754305 + 2.90362i 0
353.7 0 1.47618 − 0.906034i 0 0.825391 + 1.42962i 0 −0.542044 − 2.58963i 0 1.35821 − 2.67493i 0
353.8 0 1.71636 + 0.232633i 0 −1.60273 − 2.77600i 0 2.44053 + 1.02167i 0 2.89176 + 0.798564i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 257.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
7.d odd 6 1 inner
12.b even 2 1 inner
21.g even 6 1 inner
28.f even 6 1 inner
84.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.bc.d ✓ 16
3.b odd 2 1 inner 672.2.bc.d ✓ 16
4.b odd 2 1 inner 672.2.bc.d ✓ 16
7.d odd 6 1 inner 672.2.bc.d ✓ 16
12.b even 2 1 inner 672.2.bc.d ✓ 16
21.g even 6 1 inner 672.2.bc.d ✓ 16
28.f even 6 1 inner 672.2.bc.d ✓ 16
84.j odd 6 1 inner 672.2.bc.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.bc.d ✓ 16 1.a even 1 1 trivial
672.2.bc.d ✓ 16 3.b odd 2 1 inner
672.2.bc.d ✓ 16 4.b odd 2 1 inner
672.2.bc.d ✓ 16 7.d odd 6 1 inner
672.2.bc.d ✓ 16 12.b even 2 1 inner
672.2.bc.d ✓ 16 21.g even 6 1 inner
672.2.bc.d ✓ 16 28.f even 6 1 inner
672.2.bc.d ✓ 16 84.j odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 13T_{5}^{6} + 141T_{5}^{4} + 364T_{5}^{2} + 784 \) acting on \(S_{2}^{\mathrm{new}}(672, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( (T^{8} + 13 T^{6} + \cdots + 784)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 3 T^{6} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 33 T^{6} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 47 T^{2} + 196)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 13 T^{6} + \cdots + 784)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 48 T^{6} + \cdots + 3969)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 69 T^{6} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 172 T^{2} + 7168)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} - 40 T^{6} + \cdots + 117649)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 4 T^{3} + \cdots + 2809)^{4} \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( (T^{4} - 85 T^{2} + 1792)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + 23 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 39 T^{6} + \cdots + 63504)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 47 T^{6} + \cdots + 38416)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 21 T^{3} + \cdots + 1024)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 144 T^{6} + \cdots + 321489)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 100)^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} + 12 T^{3} + \cdots + 49)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + 120 T^{6} + \cdots + 9529569)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 44 T^{2} + 256)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} + 265 T^{6} + \cdots + 51380224)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 348 T^{2} + 28224)^{4} \) Copy content Toggle raw display
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