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Properties

Label 931.2.i.f
Level $931$
Weight $2$
Character orbit 931.i
Analytic conductor $7.434$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [931,2,Mod(411,931)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(931, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.411");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.43407242818\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 9x^{10} + 57x^{8} - 178x^{6} + 405x^{4} - 456x^{2} + 361 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 133)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + (\beta_{9} - \beta_{6} + \beta_{2}) q^{2} + (\beta_{8} + \beta_{7} - \beta_{3}) q^{3} + ( - \beta_{11} - \beta_{9} + \beta_{2}) q^{4} + ( - \beta_{5} - \beta_1) q^{5} + ( - \beta_{8} + \beta_{7} + \cdots + \beta_1) q^{6}+ \cdots + (3 \beta_{11} + \beta_{9} - 3 \beta_{6} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} - \beta_{6} + \beta_{2}) q^{2} + (\beta_{8} + \beta_{7} - \beta_{3}) q^{3} + ( - \beta_{11} - \beta_{9} + \beta_{2}) q^{4} + ( - \beta_{5} - \beta_1) q^{5} + ( - \beta_{8} + \beta_{7} + \cdots + \beta_1) q^{6}+ \cdots + (5 \beta_{11} - 5 \beta_{9} + \cdots + 4 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{9} + 6 q^{11} - 24 q^{16} - 18 q^{18} - 18 q^{22} + 24 q^{25} + 6 q^{30} - 24 q^{36} + 54 q^{37} - 6 q^{39} + 36 q^{43} - 18 q^{46} - 18 q^{51} - 48 q^{57} + 12 q^{58} + 36 q^{60} - 12 q^{64} + 72 q^{65} + 36 q^{71} - 54 q^{72} + 18 q^{78} - 18 q^{81} - 12 q^{85} + 54 q^{86} + 6 q^{92} - 120 q^{93} - 42 q^{95} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 9x^{10} + 57x^{8} - 178x^{6} + 405x^{4} - 456x^{2} + 361 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\nu^{10} - 230\nu^{8} + 1691\nu^{6} - 7525\nu^{4} + 15271\nu^{2} - 15637 ) / 6327 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\nu^{11} - 230\nu^{9} + 1691\nu^{7} - 7525\nu^{5} + 15271\nu^{3} - 15637\nu ) / 6327 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -8\nu^{10} + 63\nu^{8} - 399\nu^{6} + 1063\nu^{4} - 2835\nu^{2} + 3192 ) / 2109 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8\nu^{11} - 63\nu^{9} + 399\nu^{7} - 1063\nu^{5} + 2835\nu^{3} - 3192\nu ) / 2109 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -40\nu^{10} + 463\nu^{8} - 2698\nu^{6} + 9533\nu^{4} - 15914\nu^{2} + 19475 ) / 6327 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -40\nu^{11} + 463\nu^{9} - 2698\nu^{7} + 9533\nu^{5} - 15914\nu^{3} + 19475\nu ) / 6327 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -56\nu^{11} + 404\nu^{9} - 2090\nu^{7} + 3223\nu^{5} - 2011\nu^{3} - 6479\nu ) / 6327 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 61\nu^{10} - 337\nu^{8} + 1900\nu^{6} - 2042\nu^{4} + 3917\nu^{2} + 5890 ) / 6327 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -61\nu^{11} + 337\nu^{9} - 1900\nu^{7} + 2042\nu^{5} - 3917\nu^{3} - 5890\nu ) / 6327 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 67\nu^{10} - 634\nu^{8} + 3781\nu^{6} - 10748\nu^{4} + 17282\nu^{2} - 9158 ) / 6327 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} - 3\beta_{4} - \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + 3\beta_{5} - \beta_{3} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{11} - 7\beta_{9} + 6\beta_{6} - 11\beta_{4} - 6\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{10} - 7\beta_{8} + 6\beta_{7} + 11\beta_{5} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 30\beta_{11} - 9\beta_{9} + 39\beta_{6} + 9\beta_{2} - 46 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 9\beta_{10} - 30\beta_{8} + 39\beta_{7} + 39\beta_{3} - 46\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -57\beta_{11} + 145\beta_{9} + 57\beta_{6} + 207\beta_{4} + 202\beta_{2} - 207 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -145\beta_{10} + 57\beta_{8} + 57\beta_{7} - 207\beta_{5} + 145\beta_{3} - 207\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -1015\beta_{11} + 1015\beta_{9} - 699\beta_{6} + 968\beta_{4} + 699\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -1015\beta_{10} + 1015\beta_{8} - 699\beta_{7} - 968\beta_{5} - 316\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/931\mathbb{Z}\right)^\times\).

\(n\) \(248\) \(344\)
\(\chi(n)\) \(\beta_{4}\) \(\beta_{4}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
411.1
−1.91299 − 1.10447i
1.91299 + 1.10447i
1.41051 + 0.814356i
−1.41051 − 0.814356i
−1.04925 − 0.605787i
1.04925 + 0.605787i
−1.04925 + 0.605787i
1.04925 − 0.605787i
1.41051 − 0.814356i
−1.41051 + 0.814356i
−1.91299 + 1.10447i
1.91299 − 1.10447i
− 1.28558i −0.755500 + 1.30856i 0.347296 2.20893i 1.68226 + 0.971252i 0 − 3.01763i 0.358441 + 0.620838i 2.83975
411.2 − 1.28558i 0.755500 − 1.30856i 0.347296 − 2.20893i −1.68226 − 0.971252i 0 − 3.01763i 0.358441 + 0.620838i −2.83975
411.3 − 0.684040i −1.60397 + 2.77815i 1.53209 − 1.62871i 1.90037 + 1.09718i 0 − 2.41609i −3.64543 − 6.31407i −1.11410
411.4 − 0.684040i 1.60397 − 2.77815i 1.53209 1.62871i −1.90037 − 1.09718i 0 − 2.41609i −3.64543 − 6.31407i 1.11410
411.5 1.96962i −0.778785 + 1.34889i −1.87939 1.21157i −2.65680 − 1.53391i 0 0.237565i 0.286989 + 0.497079i −2.38633
411.6 1.96962i 0.778785 − 1.34889i −1.87939 − 1.21157i 2.65680 + 1.53391i 0 0.237565i 0.286989 + 0.497079i 2.38633
521.1 − 1.96962i −0.778785 − 1.34889i −1.87939 − 1.21157i −2.65680 + 1.53391i 0 − 0.237565i 0.286989 − 0.497079i −2.38633
521.2 − 1.96962i 0.778785 + 1.34889i −1.87939 1.21157i 2.65680 − 1.53391i 0 − 0.237565i 0.286989 − 0.497079i 2.38633
521.3 0.684040i −1.60397 − 2.77815i 1.53209 1.62871i 1.90037 − 1.09718i 0 2.41609i −3.64543 + 6.31407i −1.11410
521.4 0.684040i 1.60397 + 2.77815i 1.53209 − 1.62871i −1.90037 + 1.09718i 0 2.41609i −3.64543 + 6.31407i 1.11410
521.5 1.28558i −0.755500 − 1.30856i 0.347296 − 2.20893i 1.68226 − 0.971252i 0 3.01763i 0.358441 − 0.620838i 2.83975
521.6 1.28558i 0.755500 + 1.30856i 0.347296 2.20893i −1.68226 + 0.971252i 0 3.01763i 0.358441 − 0.620838i −2.83975
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 411.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
133.i even 6 1 inner
133.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.i.f 12
7.b odd 2 1 inner 931.2.i.f 12
7.c even 3 1 133.2.p.c ✓ 12
7.c even 3 1 931.2.s.f 12
7.d odd 6 1 133.2.p.c ✓ 12
7.d odd 6 1 931.2.s.f 12
19.d odd 6 1 931.2.s.f 12
133.i even 6 1 inner 931.2.i.f 12
133.j odd 6 1 inner 931.2.i.f 12
133.n odd 6 1 133.2.p.c ✓ 12
133.p even 6 1 931.2.s.f 12
133.s even 6 1 133.2.p.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.p.c ✓ 12 7.c even 3 1
133.2.p.c ✓ 12 7.d odd 6 1
133.2.p.c ✓ 12 133.n odd 6 1
133.2.p.c ✓ 12 133.s even 6 1
931.2.i.f 12 1.a even 1 1 trivial
931.2.i.f 12 7.b odd 2 1 inner
931.2.i.f 12 133.i even 6 1 inner
931.2.i.f 12 133.j odd 6 1 inner
931.2.s.f 12 7.c even 3 1
931.2.s.f 12 7.d odd 6 1
931.2.s.f 12 19.d odd 6 1
931.2.s.f 12 133.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(931, [\chi])\):

\( T_{2}^{6} + 6T_{2}^{4} + 9T_{2}^{2} + 3 \) Copy content Toggle raw display
\( T_{3}^{12} + 15T_{3}^{10} + 171T_{3}^{8} + 696T_{3}^{6} + 2061T_{3}^{4} + 3078T_{3}^{2} + 3249 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 6 T^{4} + 9 T^{2} + 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + 15 T^{10} + \cdots + 3249 \) Copy content Toggle raw display
$5$ \( (T^{6} + 9 T^{4} + 24 T^{2} + 19)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} - 3 T^{5} + \cdots + 289)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 60 T^{10} + \cdots + 3249 \) Copy content Toggle raw display
$17$ \( T^{12} - 21 T^{10} + \cdots + 361 \) Copy content Toggle raw display
$19$ \( T^{12} - 45 T^{10} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( (T^{6} + 21 T^{4} + \cdots + 289)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 39 T^{4} + \cdots + 867)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 271359729 \) Copy content Toggle raw display
$37$ \( (T^{6} - 27 T^{5} + \cdots + 7803)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 6089149089 \) Copy content Toggle raw display
$43$ \( (T^{6} - 18 T^{5} + 255 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 84 T^{10} + \cdots + 1478656 \) Copy content Toggle raw display
$53$ \( (T^{6} + 123 T^{4} + \cdots + 15123)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 108 T^{10} + \cdots + 263169 \) Copy content Toggle raw display
$61$ \( T^{12} - 45 T^{10} + \cdots + 361 \) Copy content Toggle raw display
$67$ \( (T^{6} + 267 T^{4} + \cdots + 116427)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 18 T^{5} + \cdots + 189003)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 2442237561 \) Copy content Toggle raw display
$79$ \( (T^{6} + 33 T^{4} + 27 T^{2} + 3)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 72 T^{4} + \cdots + 6859)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 1111489449984 \) Copy content Toggle raw display
$97$ \( T^{12} + 72 T^{10} + \cdots + 263169 \) Copy content Toggle raw display
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