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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [522,2,Mod(175,522)] 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("522.175"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-6,2,-6,-2] 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: \(4.16819098551\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
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: \( x^{12} - 4 x^{11} + 6 x^{10} - 10 x^{9} + 22 x^{8} - 18 x^{7} - 3 x^{6} - 54 x^{5} + 198 x^{4} + \cdots + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + (\beta_{9} + \beta_1) q^{3} + (\beta_{6} - 1) q^{4} + ( - \beta_{11} + \beta_1) q^{5} - \beta_{9} q^{6} + ( - \beta_{10} - \beta_{7} + \beta_{6} + \cdots - 1) q^{7} + q^{8} + (\beta_{9} - \beta_{7} - \beta_{6} + \cdots - 1) q^{9}+ \cdots + (\beta_{11} + 3 \beta_{10} + 7 \beta_{9} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{2} + 2 q^{3} - 6 q^{4} - 2 q^{5} + 2 q^{6} + 8 q^{7} + 12 q^{8} - 8 q^{9} + 4 q^{10} - 4 q^{12} - 6 q^{13} + 8 q^{14} + 14 q^{15} - 6 q^{16} + 32 q^{17} + 4 q^{18} + 4 q^{19} - 2 q^{20} - 10 q^{21}+ \cdots - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 6 x^{10} - 10 x^{9} + 22 x^{8} - 18 x^{7} - 3 x^{6} - 54 x^{5} + 198 x^{4} + \cdots + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{11} + 4 \nu^{10} - 6 \nu^{9} + 10 \nu^{8} - 22 \nu^{7} + 18 \nu^{6} + 3 \nu^{5} + 54 \nu^{4} + \cdots + 972 ) / 243 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5 \nu^{11} - 11 \nu^{10} - 3 \nu^{9} - 8 \nu^{8} + 56 \nu^{7} + 6 \nu^{6} - 57 \nu^{5} - 315 \nu^{4} + \cdots - 1701 ) / 243 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4 \nu^{11} + 13 \nu^{10} - 18 \nu^{9} + 55 \nu^{8} - 85 \nu^{7} - 6 \nu^{6} + 6 \nu^{5} + \cdots + 3888 ) / 243 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 8 \nu^{11} + 14 \nu^{10} - 15 \nu^{9} + 74 \nu^{8} - 113 \nu^{7} - 33 \nu^{6} - 24 \nu^{5} + \cdots + 4617 ) / 243 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13 \nu^{11} - 31 \nu^{10} + 18 \nu^{9} - 73 \nu^{8} + 148 \nu^{7} + 6 \nu^{6} - 51 \nu^{5} + \cdots - 5589 ) / 243 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 16 \nu^{11} + 34 \nu^{10} - 30 \nu^{9} + 106 \nu^{8} - 178 \nu^{7} - 21 \nu^{6} + 30 \nu^{5} + \cdots + 7047 ) / 243 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6 \nu^{11} - 14 \nu^{10} + 13 \nu^{9} - 32 \nu^{8} + 56 \nu^{7} + 2 \nu^{6} - 13 \nu^{5} - 303 \nu^{4} + \cdots - 2106 ) / 81 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 22 \nu^{11} - 46 \nu^{10} + 33 \nu^{9} - 109 \nu^{8} + 217 \nu^{7} + 18 \nu^{6} - 60 \nu^{5} + \cdots - 8019 ) / 243 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 28 \nu^{11} + 64 \nu^{10} - 66 \nu^{9} + 190 \nu^{8} - 298 \nu^{7} - 30 \nu^{6} + 21 \nu^{5} + \cdots + 11664 ) / 243 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11 \nu^{11} - 25 \nu^{10} + 22 \nu^{9} - 73 \nu^{8} + 124 \nu^{7} + 14 \nu^{6} - 19 \nu^{5} + \cdots - 4860 ) / 81 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 40 \nu^{11} + 91 \nu^{10} - 75 \nu^{9} + 232 \nu^{8} - 433 \nu^{7} - 21 \nu^{6} + 90 \nu^{5} + \cdots + 16524 ) / 243 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + 2\beta_{8} + \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{11} - 3\beta_{10} - \beta_{8} - 2\beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} + 3\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 5 \beta_{11} + 3 \beta_{9} - \beta_{8} - 3 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} + \cdots + 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4 \beta_{11} + 6 \beta_{10} + 9 \beta_{9} + 2 \beta_{8} + 9 \beta_{7} + \beta_{6} + \beta_{5} - 3 \beta_{4} + \cdots + 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 8 \beta_{11} + 3 \beta_{9} - 4 \beta_{8} + 6 \beta_{7} + 28 \beta_{6} + 10 \beta_{5} - 18 \beta_{4} + \cdots - 14 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2 \beta_{11} + 12 \beta_{10} + 6 \beta_{9} - 4 \beta_{8} + 12 \beta_{7} + 19 \beta_{6} - 20 \beta_{5} + \cdots - 11 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13 \beta_{11} + 18 \beta_{10} + 12 \beta_{9} + 11 \beta_{8} + 24 \beta_{7} + 37 \beta_{6} - 20 \beta_{5} + \cdots + 106 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 43 \beta_{11} - 6 \beta_{10} - 21 \beta_{9} + 98 \beta_{8} + 21 \beta_{7} + 115 \beta_{6} + 4 \beta_{5} + \cdots - 53 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 80 \beta_{11} - 147 \beta_{10} - 72 \beta_{9} - 43 \beta_{8} - 77 \beta_{6} - 92 \beta_{5} + 3 \beta_{4} + \cdots - 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 140 \beta_{11} - 24 \beta_{10} + 69 \beta_{9} + 56 \beta_{8} - 177 \beta_{7} - 224 \beta_{6} + \cdots + 187 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 184 \beta_{11} + 60 \beta_{10} + 243 \beta_{9} + 200 \beta_{8} + 288 \beta_{7} - 248 \beta_{6} + \cdots - 20 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(379\) \(407\)
\(\chi(n)\) \(1\) \(-1 + \beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
175.1
−0.538988 − 1.64605i
1.16678 − 1.28009i
−1.55661 + 0.759577i
1.73173 − 0.0331860i
−0.452567 + 1.67188i
1.64965 + 0.527869i
−0.538988 + 1.64605i
1.16678 + 1.28009i
−1.55661 − 0.759577i
1.73173 + 0.0331860i
−0.452567 − 1.67188i
1.64965 − 0.527869i
−0.500000 − 0.866025i −1.69502 + 0.356250i −0.500000 + 0.866025i −2.14966 + 3.72332i 1.15603 + 1.28980i 1.95690 + 3.38944i 1.00000 2.74617 − 1.20770i 4.29932
175.2 −0.500000 − 0.866025i −0.525196 + 1.65051i −0.500000 + 0.866025i 0.353519 − 0.612313i 1.69198 − 0.370419i −0.134930 − 0.233706i 1.00000 −2.44834 − 1.73368i −0.707038
175.3 −0.500000 − 0.866025i −0.120494 − 1.72785i −0.500000 + 0.866025i −1.19682 + 2.07295i −1.43612 + 0.968278i 0.279091 + 0.483399i 1.00000 −2.97096 + 0.416391i 2.39363
175.4 −0.500000 − 0.866025i 0.837126 + 1.51632i −0.500000 + 0.866025i 1.26263 − 2.18694i 0.894606 − 1.48313i 2.51863 + 4.36239i 1.00000 −1.59844 + 2.53870i −2.52526
175.5 −0.500000 − 0.866025i 1.22161 − 1.22787i −0.500000 + 0.866025i 1.58186 − 2.73987i −1.67417 − 0.444006i 0.567358 + 0.982693i 1.00000 −0.0153503 − 2.99996i −3.16372
175.6 −0.500000 − 0.866025i 1.28197 + 1.16471i −0.500000 + 0.866025i −0.851534 + 1.47490i 0.367679 − 1.69258i −1.18704 − 2.05602i 1.00000 0.286916 + 2.98625i 1.70307
349.1 −0.500000 + 0.866025i −1.69502 − 0.356250i −0.500000 − 0.866025i −2.14966 − 3.72332i 1.15603 − 1.28980i 1.95690 − 3.38944i 1.00000 2.74617 + 1.20770i 4.29932
349.2 −0.500000 + 0.866025i −0.525196 − 1.65051i −0.500000 − 0.866025i 0.353519 + 0.612313i 1.69198 + 0.370419i −0.134930 + 0.233706i 1.00000 −2.44834 + 1.73368i −0.707038
349.3 −0.500000 + 0.866025i −0.120494 + 1.72785i −0.500000 − 0.866025i −1.19682 − 2.07295i −1.43612 − 0.968278i 0.279091 − 0.483399i 1.00000 −2.97096 − 0.416391i 2.39363
349.4 −0.500000 + 0.866025i 0.837126 − 1.51632i −0.500000 − 0.866025i 1.26263 + 2.18694i 0.894606 + 1.48313i 2.51863 − 4.36239i 1.00000 −1.59844 − 2.53870i −2.52526
349.5 −0.500000 + 0.866025i 1.22161 + 1.22787i −0.500000 − 0.866025i 1.58186 + 2.73987i −1.67417 + 0.444006i 0.567358 − 0.982693i 1.00000 −0.0153503 + 2.99996i −3.16372
349.6 −0.500000 + 0.866025i 1.28197 − 1.16471i −0.500000 − 0.866025i −0.851534 − 1.47490i 0.367679 + 1.69258i −1.18704 + 2.05602i 1.00000 0.286916 − 2.98625i 1.70307
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 175.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 522.2.e.h ✓ 12
3.b odd 2 1 1566.2.e.h 12
9.c even 3 1 inner 522.2.e.h ✓ 12
9.c even 3 1 4698.2.a.bh 6
9.d odd 6 1 1566.2.e.h 12
9.d odd 6 1 4698.2.a.be 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
522.2.e.h ✓ 12 1.a even 1 1 trivial
522.2.e.h ✓ 12 9.c even 3 1 inner
1566.2.e.h 12 3.b odd 2 1
1566.2.e.h 12 9.d odd 6 1
4698.2.a.be 6 9.d odd 6 1
4698.2.a.bh 6 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(522, [\chi])\):

\( T_{5}^{12} + 2 T_{5}^{11} + 24 T_{5}^{10} + 12 T_{5}^{9} + 351 T_{5}^{8} + 204 T_{5}^{7} + 2322 T_{5}^{6} + \cdots + 9801 \) Copy content Toggle raw display
\( T_{7}^{12} - 8 T_{7}^{11} + 56 T_{7}^{10} - 160 T_{7}^{9} + 516 T_{7}^{8} - 696 T_{7}^{7} + 2928 T_{7}^{6} + \cdots + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} - 2 T^{11} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} + 2 T^{11} + \cdots + 9801 \) Copy content Toggle raw display
$7$ \( T^{12} - 8 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{12} + 48 T^{10} + \cdots + 6561 \) Copy content Toggle raw display
$13$ \( T^{12} + 6 T^{11} + \cdots + 121 \) Copy content Toggle raw display
$17$ \( (T^{6} - 16 T^{5} + \cdots - 24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 2 T^{5} + \cdots - 1233)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 2 T^{11} + \cdots + 788544 \) Copy content Toggle raw display
$29$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$31$ \( T^{12} + 8 T^{11} + \cdots + 531441 \) Copy content Toggle raw display
$37$ \( (T^{6} - 90 T^{4} + \cdots - 344)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 6410244096 \) Copy content Toggle raw display
$43$ \( T^{12} - 2 T^{11} + \cdots + 63001 \) Copy content Toggle raw display
$47$ \( T^{12} + 14 T^{11} + \cdots + 463761 \) Copy content Toggle raw display
$53$ \( (T^{6} - 20 T^{5} + \cdots + 66189)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 96 T^{10} + \cdots + 5645376 \) Copy content Toggle raw display
$61$ \( T^{12} + 22 T^{11} + \cdots + 7744 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 158703437376 \) Copy content Toggle raw display
$71$ \( (T^{6} - 4 T^{5} + \cdots - 8232)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 8 T^{5} + \cdots + 14696)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} - 4 T^{11} + \cdots + 81 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 153958464 \) Copy content Toggle raw display
$89$ \( (T^{6} - 10 T^{5} + \cdots + 227064)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 309750128704 \) Copy content Toggle raw display
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