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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [522,2,Mod(17,522)] 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("522.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.16819098551\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 14x^{10} + 77x^{8} + 340x^{6} + 2772x^{4} + 18144x^{2} + 46656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} - \beta_{3} q^{4} - \beta_{4} q^{5} + ( - \beta_{2} + 1) q^{7} + \beta_{5} q^{8} - \beta_{8} q^{10} + (\beta_{8} + \beta_{7} - 2 \beta_{3}) q^{13} + (\beta_{11} - \beta_{6}) q^{14}+ \cdots + ( - \beta_{11} - 8 \beta_{6} + \cdots + 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{7} - 4 q^{10} - 12 q^{16} - 16 q^{19} + 20 q^{25} + 12 q^{37} + 4 q^{40} + 16 q^{43} - 16 q^{46} + 116 q^{49} - 32 q^{52} - 16 q^{58} + 4 q^{61} - 8 q^{70} - 52 q^{73} - 16 q^{76} - 16 q^{79}+ \cdots - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 14x^{10} + 77x^{8} + 340x^{6} + 2772x^{4} + 18144x^{2} + 46656 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -101\nu^{10} - 1162\nu^{8} - 5545\nu^{6} - 24008\nu^{4} - 297972\nu^{2} - 1354320 ) / 202176 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\nu^{10} + 118\nu^{8} + 343\nu^{6} + 2264\nu^{4} + 22140\nu^{2} + 102384 ) / 7776 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -83\nu^{11} - 802\nu^{9} - 1999\nu^{7} - 13460\nu^{5} - 134244\nu^{3} - 584496\nu ) / 606528 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13\nu^{11} + 2\nu^{9} - 223\nu^{7} + 928\nu^{5} + 4644\nu^{3} + 22032\nu ) / 93312 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 325 \nu^{11} - 882 \nu^{10} - 2210 \nu^{9} - 7812 \nu^{8} - 9113 \nu^{7} - 27738 \nu^{6} + \cdots - 7394976 ) / 2426112 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 325 \nu^{11} + 882 \nu^{10} - 2210 \nu^{9} + 7812 \nu^{8} - 9113 \nu^{7} + 27738 \nu^{6} + \cdots + 7394976 ) / 2426112 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 401 \nu^{11} + 390 \nu^{10} + 3418 \nu^{9} + 2652 \nu^{8} + 6613 \nu^{7} - 9282 \nu^{6} + \cdots + 1314144 ) / 2426112 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 401 \nu^{11} - 390 \nu^{10} + 3418 \nu^{9} - 2652 \nu^{8} + 6613 \nu^{7} + 9282 \nu^{6} + \cdots - 1314144 ) / 2426112 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 161\nu^{11} + 1894\nu^{9} + 8005\nu^{7} + 39980\nu^{5} + 350460\nu^{3} + 2606256\nu ) / 606528 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1079 \nu^{11} - 2586 \nu^{10} - 9022 \nu^{9} - 18492 \nu^{8} - 31603 \nu^{7} - 63906 \nu^{6} + \cdots - 16990560 ) / 1213056 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1079 \nu^{11} + 2586 \nu^{10} - 9022 \nu^{9} + 18492 \nu^{8} - 31603 \nu^{7} + 63906 \nu^{6} + \cdots + 16990560 ) / 1213056 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{8} + 2\beta_{7} - \beta_{6} + \beta_{5} - \beta_{2} - 3\beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{11} - 2 \beta_{10} - 3 \beta_{9} + 4 \beta_{8} + 4 \beta_{7} + 5 \beta_{6} + \cdots + 17 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{11} - 4\beta_{10} + 6\beta_{8} - 6\beta_{7} - 39\beta_{6} + 39\beta_{5} + 9\beta_{2} - \beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 18 \beta_{11} + 18 \beta_{10} + 19 \beta_{9} + 24 \beta_{8} + 24 \beta_{7} - 65 \beta_{6} + \cdots - 17 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 70\beta_{8} - 70\beta_{7} + 83\beta_{6} - 83\beta_{5} - 49\beta_{2} - 63\beta _1 - 207 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 38 \beta_{11} - 38 \beta_{10} - 51 \beta_{9} - 164 \beta_{8} - 164 \beta_{7} - 307 \beta_{6} + \cdots + 377 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 164 \beta_{11} + 164 \beta_{10} + 30 \beta_{8} - 30 \beta_{7} + 273 \beta_{6} - 273 \beta_{5} + \cdots - 905 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 258 \beta_{11} + 258 \beta_{10} - 125 \beta_{9} - 48 \beta_{8} - 48 \beta_{7} + 727 \beta_{6} + \cdots - 5825 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 936 \beta_{11} - 936 \beta_{10} + 286 \beta_{8} - 286 \beta_{7} + 4523 \beta_{6} - 4523 \beta_{5} + \cdots + 2145 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1262 \beta_{11} - 1262 \beta_{10} - 2835 \beta_{9} - 5948 \beta_{8} - 5948 \beta_{7} + \cdots + 809 \beta_{3} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/522\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(407\)
\(\chi(n)\) \(\beta_{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
2.15129 − 1.17131i
0.179336 + 2.44292i
−0.916409 − 2.27161i
0.916409 − 2.27161i
−0.179336 + 2.44292i
−2.15129 − 1.17131i
2.15129 + 1.17131i
0.179336 − 2.44292i
−0.916409 + 2.27161i
0.916409 + 2.27161i
−0.179336 − 2.44292i
−2.15129 + 1.17131i
−0.707107 − 0.707107i 0 1.00000i −2.88836 0 2.34262 0.707107 − 0.707107i 0 2.04238 + 2.04238i
17.2 −0.707107 − 0.707107i 0 1.00000i 1.05554 0 −4.88583 0.707107 − 0.707107i 0 −0.746381 − 0.746381i
17.3 −0.707107 − 0.707107i 0 1.00000i 3.24703 0 4.54321 0.707107 − 0.707107i 0 −2.29600 − 2.29600i
17.4 0.707107 + 0.707107i 0 1.00000i −3.24703 0 4.54321 −0.707107 + 0.707107i 0 −2.29600 − 2.29600i
17.5 0.707107 + 0.707107i 0 1.00000i −1.05554 0 −4.88583 −0.707107 + 0.707107i 0 −0.746381 − 0.746381i
17.6 0.707107 + 0.707107i 0 1.00000i 2.88836 0 2.34262 −0.707107 + 0.707107i 0 2.04238 + 2.04238i
215.1 −0.707107 + 0.707107i 0 − 1.00000i −2.88836 0 2.34262 0.707107 + 0.707107i 0 2.04238 − 2.04238i
215.2 −0.707107 + 0.707107i 0 − 1.00000i 1.05554 0 −4.88583 0.707107 + 0.707107i 0 −0.746381 + 0.746381i
215.3 −0.707107 + 0.707107i 0 − 1.00000i 3.24703 0 4.54321 0.707107 + 0.707107i 0 −2.29600 + 2.29600i
215.4 0.707107 − 0.707107i 0 − 1.00000i −3.24703 0 4.54321 −0.707107 − 0.707107i 0 −2.29600 + 2.29600i
215.5 0.707107 − 0.707107i 0 − 1.00000i −1.05554 0 −4.88583 −0.707107 − 0.707107i 0 −0.746381 + 0.746381i
215.6 0.707107 − 0.707107i 0 − 1.00000i 2.88836 0 2.34262 −0.707107 − 0.707107i 0 2.04238 − 2.04238i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 17.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
29.c odd 4 1 inner
87.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 522.2.g.b ✓ 12
3.b odd 2 1 inner 522.2.g.b ✓ 12
29.c odd 4 1 inner 522.2.g.b ✓ 12
87.f even 4 1 inner 522.2.g.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
522.2.g.b ✓ 12 1.a even 1 1 trivial
522.2.g.b ✓ 12 3.b odd 2 1 inner
522.2.g.b ✓ 12 29.c odd 4 1 inner
522.2.g.b ✓ 12 87.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 20T_{5}^{4} + 109T_{5}^{2} - 98 \) acting on \(S_{2}^{\mathrm{new}}(522, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} - 20 T^{4} + \cdots - 98)^{2} \) Copy content Toggle raw display
$7$ \( (T^{3} - 2 T^{2} - 23 T + 52)^{4} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( (T^{6} + 60 T^{4} + \cdots + 2304)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 2258530576 \) Copy content Toggle raw display
$19$ \( (T^{6} + 8 T^{5} + \cdots + 288)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 88 T^{4} + \cdots + 512)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 594823321 \) Copy content Toggle raw display
$31$ \( (T^{6} + 8 T^{3} + 6724 T^{2} + \cdots + 32)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 6 T^{5} + \cdots + 20402)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 12098 T^{8} + \cdots + 92236816 \) Copy content Toggle raw display
$43$ \( (T^{6} - 8 T^{5} + \cdots + 288)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 15378 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( (T^{6} + 98 T^{4} + \cdots + 6272)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 266 T^{4} + \cdots + 329672)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 2 T^{5} + \cdots + 76832)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 292 T^{4} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 188 T^{4} + \cdots - 165888)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 26 T^{5} + \cdots + 1152)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 8 T^{5} + \cdots + 23328)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 8)^{6} \) Copy content Toggle raw display
$89$ \( T^{12} + 36640 T^{8} + \cdots + 5308416 \) Copy content Toggle raw display
$97$ \( (T^{6} + 14 T^{5} + \cdots + 93312)^{2} \) Copy content Toggle raw display
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