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Properties

Label 912.2.bb.f
Level $912$
Weight $2$
Character orbit 912.bb
Analytic conductor $7.282$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,2,Mod(31,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.bb (of order \(6\), degree \(2\), 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: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.954288.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + (\beta_{4} + \beta_{3}) q^{5} + (\beta_{5} - 2 \beta_{3} - \beta_1 - 1) q^{7} + ( - \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + (\beta_{4} + \beta_{3}) q^{5} + (\beta_{5} - 2 \beta_{3} - \beta_1 - 1) q^{7} + ( - \beta_{3} - 1) q^{9} + (\beta_{5} - \beta_1) q^{11} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{13} + (\beta_{4} + \beta_{3} - \beta_{2} + 1) q^{15} + ( - \beta_{5} + \beta_{4} + \cdots + 2 \beta_1) q^{17}+ \cdots + \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} - 2 q^{5} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{3} - 2 q^{5} - 3 q^{9} - 3 q^{13} + 2 q^{15} - 2 q^{17} + 4 q^{19} - 9 q^{21} - 12 q^{23} - 5 q^{25} - 6 q^{27} + 6 q^{29} - 2 q^{31} - 6 q^{35} - 12 q^{41} + 33 q^{43} + 4 q^{45} + 18 q^{47} - 8 q^{49} + 2 q^{51} + 36 q^{53} - 12 q^{55} - q^{57} - 2 q^{59} + 3 q^{61} - 9 q^{63} - 19 q^{67} + 16 q^{71} + 11 q^{73} - 10 q^{75} - 32 q^{77} - 9 q^{79} - 3 q^{81} - 8 q^{85} + 6 q^{89} - q^{91} - q^{93} + 26 q^{95} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 5\nu^{4} + \nu^{3} + 9\nu^{2} + 21\nu - 45 ) / 27 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 4\nu^{4} + \nu^{3} + 18\nu^{2} - 33\nu + 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{5} - 8\nu^{4} + 2\nu^{3} + 9\nu^{2} - 12\nu + 18 ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{5} - \nu^{4} + 4\nu^{3} - 3\nu^{2} + 12\nu + 27 ) / 9 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{5} - 8\beta_{3} + \beta_{2} + \beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{4} - 2\beta_{3} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{5} + 9\beta_{4} - 11\beta_{3} - 8\beta_{2} + 4\beta _1 + 16 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(1 + \beta_{3}\) \(1\) \(1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.403374 − 1.68443i
1.71903 + 0.211943i
−1.62241 + 0.606458i
0.403374 + 1.68443i
1.71903 − 0.211943i
−1.62241 − 0.606458i
0 0.500000 + 0.866025i 0 −1.66044 − 2.87597i 0 2.71781i 0 −0.500000 + 0.866025i 0
31.2 0 0.500000 + 0.866025i 0 −0.675970 − 1.17081i 0 − 1.45735i 0 −0.500000 + 0.866025i 0
31.3 0 0.500000 + 0.866025i 0 1.33641 + 2.31473i 0 3.93569i 0 −0.500000 + 0.866025i 0
559.1 0 0.500000 − 0.866025i 0 −1.66044 + 2.87597i 0 − 2.71781i 0 −0.500000 − 0.866025i 0
559.2 0 0.500000 − 0.866025i 0 −0.675970 + 1.17081i 0 1.45735i 0 −0.500000 − 0.866025i 0
559.3 0 0.500000 − 0.866025i 0 1.33641 − 2.31473i 0 − 3.93569i 0 −0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.bb.f yes 6
3.b odd 2 1 2736.2.bm.o 6
4.b odd 2 1 912.2.bb.e ✓ 6
12.b even 2 1 2736.2.bm.n 6
19.d odd 6 1 912.2.bb.e ✓ 6
57.f even 6 1 2736.2.bm.n 6
76.f even 6 1 inner 912.2.bb.f yes 6
228.n odd 6 1 2736.2.bm.o 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.bb.e ✓ 6 4.b odd 2 1
912.2.bb.e ✓ 6 19.d odd 6 1
912.2.bb.f yes 6 1.a even 1 1 trivial
912.2.bb.f yes 6 76.f even 6 1 inner
2736.2.bm.n 6 12.b even 2 1
2736.2.bm.n 6 57.f even 6 1
2736.2.bm.o 6 3.b odd 2 1
2736.2.bm.o 6 228.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{6} + 2T_{5}^{5} + 12T_{5}^{4} + 8T_{5}^{3} + 88T_{5}^{2} + 96T_{5} + 144 \) Copy content Toggle raw display
\( T_{23}^{6} + 12T_{23}^{5} + 28T_{23}^{4} - 240T_{23}^{3} - 416T_{23}^{2} + 4080T_{23} + 13872 \) Copy content Toggle raw display
\( T_{31}^{3} + T_{31}^{2} - 9T_{31} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} + 2 T^{5} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( T^{6} + 25 T^{4} + \cdots + 243 \) Copy content Toggle raw display
$11$ \( T^{6} + 16 T^{4} + \cdots + 48 \) Copy content Toggle raw display
$13$ \( T^{6} + 3 T^{5} + \cdots + 2883 \) Copy content Toggle raw display
$17$ \( T^{6} + 2 T^{5} + \cdots + 576 \) Copy content Toggle raw display
$19$ \( T^{6} - 4 T^{5} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} + 12 T^{5} + \cdots + 13872 \) Copy content Toggle raw display
$29$ \( T^{6} - 6 T^{5} + \cdots + 192 \) Copy content Toggle raw display
$31$ \( (T^{3} + T^{2} - 9 T + 3)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 217 T^{4} + \cdots + 328683 \) Copy content Toggle raw display
$41$ \( T^{6} + 12 T^{5} + \cdots + 248832 \) Copy content Toggle raw display
$43$ \( T^{6} - 33 T^{5} + \cdots + 2187 \) Copy content Toggle raw display
$47$ \( T^{6} - 18 T^{5} + \cdots + 714432 \) Copy content Toggle raw display
$53$ \( T^{6} - 36 T^{5} + \cdots + 80688 \) Copy content Toggle raw display
$59$ \( T^{6} + 2 T^{5} + \cdots + 144 \) Copy content Toggle raw display
$61$ \( T^{6} - 3 T^{5} + \cdots + 638401 \) Copy content Toggle raw display
$67$ \( T^{6} + 19 T^{5} + \cdots + 10201 \) Copy content Toggle raw display
$71$ \( T^{6} - 16 T^{5} + \cdots + 2304 \) Copy content Toggle raw display
$73$ \( T^{6} - 11 T^{5} + \cdots + 33489 \) Copy content Toggle raw display
$79$ \( T^{6} + 9 T^{5} + \cdots + 151321 \) Copy content Toggle raw display
$83$ \( T^{6} + 132 T^{4} + \cdots + 15552 \) Copy content Toggle raw display
$89$ \( T^{6} - 6 T^{5} + \cdots + 215472 \) Copy content Toggle raw display
$97$ \( T^{6} + 24 T^{5} + \cdots + 62208 \) Copy content Toggle raw display
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