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Properties

Label 912.3.be.f
Level $912$
Weight $3$
Character orbit 912.be
Analytic conductor $24.850$
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] = [912,3,Mod(145,912)]
 
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([0, 0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.145");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 912.be (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: \(24.8502001097\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.92607408.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 20x^{4} - 35x^{3} + 94x^{2} - 77x + 43 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 57)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{3} + (\beta_{5} - 2 \beta_{3} - \beta_{2} + \cdots + 2) q^{5}+ \cdots + 3 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{3} + (\beta_{5} - 2 \beta_{3} - \beta_{2} + \cdots + 2) q^{5}+ \cdots + (3 \beta_{5} + 6 \beta_{4} + \cdots - 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 9 q^{3} + 4 q^{5} + 22 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 9 q^{3} + 4 q^{5} + 22 q^{7} + 9 q^{9} + 36 q^{11} - 3 q^{13} + 12 q^{15} + 38 q^{17} + 10 q^{19} + 33 q^{21} - 54 q^{23} - 21 q^{25} - 102 q^{29} + 54 q^{33} + 24 q^{35} - 6 q^{39} + 96 q^{41} - 107 q^{43} + 24 q^{45} + 50 q^{47} - 48 q^{49} + 114 q^{51} - 90 q^{53} - 148 q^{55} - 3 q^{57} + 27 q^{61} + 33 q^{63} + 39 q^{67} - 84 q^{71} - 77 q^{73} + 260 q^{77} - 9 q^{79} - 27 q^{81} + 348 q^{83} + 68 q^{85} - 204 q^{87} - 72 q^{89} + 393 q^{91} + 129 q^{93} - 104 q^{95} - 228 q^{97} + 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 20x^{4} - 35x^{3} + 94x^{2} - 77x + 43 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 2\nu - 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 26\nu^{3} + 34\nu^{2} - 56\nu + 34 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 14\nu^{4} - 13\nu^{3} + 109\nu^{2} + 41\nu + 63 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} - 9\nu^{4} + 33\nu^{3} - 121\nu^{2} + 271\nu - 98 ) / 23 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{3} - 8\beta_{2} - 6\beta _1 - 15 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{4} - \beta_{3} - 3\beta_{2} - 11\beta _1 + 33 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -13\beta_{5} - 3\beta_{4} - 15\beta_{3} + 61\beta_{2} + 29\beta _1 + 196 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
0.500000 + 2.93068i
0.500000 − 2.69511i
0.500000 + 0.630453i
0.500000 − 2.93068i
0.500000 + 2.69511i
0.500000 − 0.630453i
0 1.50000 − 0.866025i 0 −3.20750 − 5.55555i 0 2.26281 0 1.50000 − 2.59808i 0
145.2 0 1.50000 − 0.866025i 0 2.32722 + 4.03087i 0 10.6817 0 1.50000 − 2.59808i 0
145.3 0 1.50000 − 0.866025i 0 2.88028 + 4.98878i 0 −1.94451 0 1.50000 − 2.59808i 0
673.1 0 1.50000 + 0.866025i 0 −3.20750 + 5.55555i 0 2.26281 0 1.50000 + 2.59808i 0
673.2 0 1.50000 + 0.866025i 0 2.32722 − 4.03087i 0 10.6817 0 1.50000 + 2.59808i 0
673.3 0 1.50000 + 0.866025i 0 2.88028 − 4.98878i 0 −1.94451 0 1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.3.be.f 6
4.b odd 2 1 57.3.g.b ✓ 6
12.b even 2 1 171.3.p.c 6
19.d odd 6 1 inner 912.3.be.f 6
76.f even 6 1 57.3.g.b ✓ 6
228.n odd 6 1 171.3.p.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.3.g.b ✓ 6 4.b odd 2 1
57.3.g.b ✓ 6 76.f even 6 1
171.3.p.c 6 12.b even 2 1
171.3.p.c 6 228.n odd 6 1
912.3.be.f 6 1.a even 1 1 trivial
912.3.be.f 6 19.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{6} - 4T_{5}^{5} + 56T_{5}^{4} - 184T_{5}^{3} + 2288T_{5}^{2} - 6880T_{5} + 29584 \) Copy content Toggle raw display
\( T_{7}^{3} - 11T_{7}^{2} - T_{7} + 47 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 3)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} - 4 T^{5} + \cdots + 29584 \) Copy content Toggle raw display
$7$ \( (T^{3} - 11 T^{2} + \cdots + 47)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} - 18 T^{2} + \cdots + 1084)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 3 T^{5} + \cdots + 2883 \) Copy content Toggle raw display
$17$ \( T^{6} - 38 T^{5} + \cdots + 52186176 \) Copy content Toggle raw display
$19$ \( T^{6} - 10 T^{5} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{6} + 54 T^{5} + \cdots + 21049744 \) Copy content Toggle raw display
$29$ \( T^{6} + 102 T^{5} + \cdots + 487228608 \) Copy content Toggle raw display
$31$ \( T^{6} + 2345 T^{4} + \cdots + 112326483 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 3652564347 \) Copy content Toggle raw display
$41$ \( T^{6} - 96 T^{5} + \cdots + 1354752 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 1477402969 \) Copy content Toggle raw display
$47$ \( T^{6} - 50 T^{5} + \cdots + 5798464 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 6327041328 \) Copy content Toggle raw display
$59$ \( T^{6} - 1048 T^{4} + \cdots + 25509168 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 1215986641 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 11787475467 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 149905029888 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 434693631969 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 32898206883 \) Copy content Toggle raw display
$83$ \( (T^{3} - 174 T^{2} + \cdots + 1174072)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 591631797168 \) Copy content Toggle raw display
$97$ \( T^{6} + 228 T^{5} + \cdots + 68659968 \) Copy content Toggle raw display
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