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Properties

Label 2925.2.c.x
Level $2925$
Weight $2$
Character orbit 2925.c
Analytic conductor $23.356$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2925,2,Mod(2224,2925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2925, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2925.2224");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2925.c (of order \(2\), degree \(1\), 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: \(23.3562425912\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5089536.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 975)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + (\beta_{3} - 2) q^{4} + ( - \beta_{5} + 2 \beta_{4}) q^{7} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{8} + ( - \beta_{3} + \beta_1 + 1) q^{11} - \beta_{4} q^{13} + (\beta_{3} - 2 \beta_1 - 4) q^{14}+ \cdots + (2 \beta_{5} - 15 \beta_{4} - 3 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4} + 2 q^{11} - 18 q^{14} + 10 q^{16} - 12 q^{19} - 2 q^{26} - 26 q^{29} + 14 q^{31} + 38 q^{34} - 34 q^{44} + 52 q^{46} + 4 q^{49} + 18 q^{56} + 6 q^{59} - 10 q^{61} + 30 q^{64} + 36 q^{71}+ \cdots + 10 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 24\nu^{4} - 6\nu^{3} - \nu^{2} + 6\nu + 285 ) / 131 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} - 11\nu^{4} + 101\nu^{3} - 136\nu^{2} + 292\nu - 147 ) / 393 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{5} - 13\nu^{4} + 36\nu^{3} + 6\nu^{2} - 36\nu - 7 ) / 131 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 23\nu^{5} - 28\nu^{4} + 7\nu^{3} + 154\nu^{2} + 386\nu - 267 ) / 393 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -23\nu^{5} + 28\nu^{4} - 7\nu^{3} - 23\nu^{2} - 386\nu + 267 ) / 131 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 3\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 2\beta_{4} + 2\beta_{3} + 2\beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 6\beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{5} - 12\beta_{4} + 9\beta_{3} - 9\beta_{2} + 7\beta _1 - 12 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(2251\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2224.1
1.66044 + 1.66044i
0.675970 − 0.675970i
−1.33641 − 1.33641i
−1.33641 + 1.33641i
0.675970 + 0.675970i
1.66044 − 1.66044i
− 2.51414i 0 −4.32088 0 0 − 0.514137i 5.83502i 0 0
2224.2 − 2.08613i 0 −2.35194 0 0 − 4.08613i 0.734191i 0 0
2224.3 − 0.571993i 0 1.67282 0 0 1.42801i − 2.10083i 0 0
2224.4 0.571993i 0 1.67282 0 0 − 1.42801i 2.10083i 0 0
2224.5 2.08613i 0 −2.35194 0 0 4.08613i − 0.734191i 0 0
2224.6 2.51414i 0 −4.32088 0 0 0.514137i − 5.83502i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2224.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2925.2.c.x 6
3.b odd 2 1 975.2.c.j 6
5.b even 2 1 inner 2925.2.c.x 6
5.c odd 4 1 2925.2.a.bg 3
5.c odd 4 1 2925.2.a.bi 3
15.d odd 2 1 975.2.c.j 6
15.e even 4 1 975.2.a.n ✓ 3
15.e even 4 1 975.2.a.p yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.a.n ✓ 3 15.e even 4 1
975.2.a.p yes 3 15.e even 4 1
975.2.c.j 6 3.b odd 2 1
975.2.c.j 6 15.d odd 2 1
2925.2.a.bg 3 5.c odd 4 1
2925.2.a.bi 3 5.c odd 4 1
2925.2.c.x 6 1.a even 1 1 trivial
2925.2.c.x 6 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2925, [\chi])\):

\( T_{2}^{6} + 11T_{2}^{4} + 31T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{6} + 19T_{7}^{4} + 39T_{7}^{2} + 9 \) Copy content Toggle raw display
\( T_{11}^{3} - T_{11}^{2} - 11T_{11} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 11 T^{4} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 19 T^{4} + \cdots + 9 \) Copy content Toggle raw display
$11$ \( (T^{3} - T^{2} - 11 T + 9)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} + 59 T^{4} + \cdots + 729 \) Copy content Toggle raw display
$19$ \( (T^{3} + 6 T^{2} - 12 T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 80 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$29$ \( (T^{3} + 13 T^{2} + \cdots - 129)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 7 T^{2} + \cdots + 223)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 76 T^{4} + \cdots + 11664 \) Copy content Toggle raw display
$41$ \( (T^{3} - 24 T + 36)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 168 T^{4} + \cdots + 26896 \) Copy content Toggle raw display
$47$ \( T^{6} + 299 T^{4} + \cdots + 826281 \) Copy content Toggle raw display
$53$ \( T^{6} + 75 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$59$ \( (T^{3} - 3 T^{2} + \cdots + 117)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 5 T^{2} + \cdots - 113)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 327 T^{4} + \cdots + 744769 \) Copy content Toggle raw display
$71$ \( (T^{3} - 18 T^{2} + \cdots - 36)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 244 T^{4} + \cdots + 318096 \) Copy content Toggle raw display
$79$ \( (T^{3} + 4 T^{2} + \cdots + 164)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 323 T^{4} + \cdots + 114921 \) Copy content Toggle raw display
$89$ \( (T^{3} + 16 T^{2} + \cdots + 48)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 316 T^{4} + \cdots + 46656 \) Copy content Toggle raw display
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