Properties

Label 7488.2.d.m
Level $7488$
Weight $2$
Character orbit 7488.d
Analytic conductor $59.792$
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] = [7488,2,Mod(4031,7488)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7488, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("7488.4031");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7488 = 2^{6} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7488.d (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: \(59.7919810335\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.1279179096064000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 5x^{8} - 4x^{6} + 20x^{4} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 468)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{5} + \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{5} + \beta_{3} q^{7} - \beta_{2} q^{11} + q^{13} + \beta_{7} q^{17} - \beta_{10} q^{19} + (\beta_{8} + \beta_{2}) q^{23} + ( - \beta_{11} + \beta_{4} - 4) q^{25} - 3 \beta_1 q^{29} + ( - \beta_{6} + 2 \beta_{3}) q^{31} + (\beta_{9} + 2 \beta_{8}) q^{35} + (\beta_{11} + \beta_{4} - 1) q^{37} + (\beta_{5} - 2 \beta_1) q^{41} + (\beta_{10} + \beta_{6}) q^{43} + ( - 2 \beta_{8} - \beta_{2}) q^{47} + (\beta_{4} - 1) q^{49} - \beta_{7} q^{53} + (\beta_{10} + 3 \beta_{6}) q^{55} + \beta_{8} q^{59} + (\beta_{4} - 8) q^{61} + \beta_{5} q^{65} + (\beta_{10} + \beta_{6} - \beta_{3}) q^{67} + ( - \beta_{9} + \beta_{2}) q^{71} + ( - \beta_{11} + \beta_{4} - 3) q^{73} + ( - \beta_{7} - 5 \beta_1) q^{77} - 2 \beta_{10} q^{79} + ( - 2 \beta_{9} - \beta_{8} - 2 \beta_{2}) q^{83} + (\beta_{11} + \beta_{4} + 1) q^{85} + (\beta_{5} - 4 \beta_1) q^{89} + \beta_{3} q^{91} + (\beta_{9} + 4 \beta_{8} - 2 \beta_{2}) q^{95} + (\beta_{11} - \beta_{4} - 5) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{13} - 52 q^{25} - 8 q^{37} - 12 q^{49} - 96 q^{61} - 40 q^{73} + 16 q^{85} - 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 5x^{8} - 4x^{6} + 20x^{4} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{9} + 2\nu^{7} - \nu^{5} + 6\nu^{3} - 8\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{11} + 3\nu^{7} - 12\nu^{5} + 20\nu^{3} - 80\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} - 3\nu^{6} - 4\nu^{4} - 4\nu^{2} + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{10} + 5\nu^{6} - 4\nu^{4} + 4\nu^{2} ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{11} + 4\nu^{9} - 5\nu^{7} + 24\nu^{5} - 4\nu^{3} + 48\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{8} - 2\nu^{6} + 9\nu^{4} - 14\nu^{2} + 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} + \nu^{9} + 3\nu^{7} - 3\nu^{5} + 14\nu^{3} + 40\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -3\nu^{11} - 7\nu^{7} - 4\nu^{5} - 20\nu^{3} - 16\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{11} - 4\nu^{9} - 3\nu^{7} - 8\nu^{5} + 28\nu^{3} ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{10} + \nu^{8} + 3\nu^{6} + 5\nu^{4} + 22\nu^{2} + 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{10} - 4\nu^{8} + 5\nu^{6} - 8\nu^{4} + 20\nu^{2} - 24 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{8} + \beta_{7} - \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} - \beta_{6} - \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{9} + \beta_{7} + 2\beta_{5} + 2\beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{11} + \beta_{10} + 3\beta_{6} - \beta_{4} - 2\beta_{3} - 7 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{8} - 3\beta_{7} + 4\beta_{5} - \beta_{2} + 5\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -\beta_{10} + \beta_{6} + 5\beta_{4} - 8\beta_{3} + 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -5\beta_{9} + 3\beta_{7} - 2\beta_{5} - 2\beta_{2} + 19\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -9\beta_{11} + 3\beta_{10} - 7\beta_{6} + 5\beta_{4} + 2\beta_{3} - 17 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -4\beta_{9} - 5\beta_{8} + 7\beta_{7} + 4\beta_{5} + 17\beta_{2} - 33\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4\beta_{11} + 5\beta_{10} + 11\beta_{6} + 7\beta_{4} + 32\beta_{3} - 68 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 5\beta_{9} - 44\beta_{8} - 15\beta_{7} - 14\beta_{5} - 2\beta_{2} - 63\beta_1 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7488\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(5761\) \(5825\) \(6085\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4031.1
−1.31293 0.525570i
1.31293 0.525570i
0.653376 + 1.25423i
−0.653376 + 1.25423i
0.921588 1.07270i
−0.921588 1.07270i
−0.921588 + 1.07270i
0.921588 + 1.07270i
−0.653376 1.25423i
0.653376 1.25423i
1.31293 + 0.525570i
−1.31293 + 0.525570i
0 0 0 4.09430i 0 3.71352i 0 0 0
4031.2 0 0 0 4.09430i 0 3.71352i 0 0 0
4031.3 0 0 0 3.24195i 0 1.84803i 0 0 0
4031.4 0 0 0 3.24195i 0 1.84803i 0 0 0
4031.5 0 0 0 0.852353i 0 2.60664i 0 0 0
4031.6 0 0 0 0.852353i 0 2.60664i 0 0 0
4031.7 0 0 0 0.852353i 0 2.60664i 0 0 0
4031.8 0 0 0 0.852353i 0 2.60664i 0 0 0
4031.9 0 0 0 3.24195i 0 1.84803i 0 0 0
4031.10 0 0 0 3.24195i 0 1.84803i 0 0 0
4031.11 0 0 0 4.09430i 0 3.71352i 0 0 0
4031.12 0 0 0 4.09430i 0 3.71352i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4031.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7488.2.d.m 12
3.b odd 2 1 inner 7488.2.d.m 12
4.b odd 2 1 inner 7488.2.d.m 12
8.b even 2 1 468.2.c.c 12
8.d odd 2 1 468.2.c.c 12
12.b even 2 1 inner 7488.2.d.m 12
24.f even 2 1 468.2.c.c 12
24.h odd 2 1 468.2.c.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
468.2.c.c 12 8.b even 2 1
468.2.c.c 12 8.d odd 2 1
468.2.c.c 12 24.f even 2 1
468.2.c.c 12 24.h odd 2 1
7488.2.d.m 12 1.a even 1 1 trivial
7488.2.d.m 12 3.b odd 2 1 inner
7488.2.d.m 12 4.b odd 2 1 inner
7488.2.d.m 12 12.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 \(S_{2}^{\mathrm{new}}(7488, [\chi])\):

\( T_{5}^{6} + 28T_{5}^{4} + 196T_{5}^{2} + 128 \) Copy content Toggle raw display
\( T_{7}^{6} + 24T_{7}^{4} + 164T_{7}^{2} + 320 \) Copy content Toggle raw display
\( T_{11}^{6} - 38T_{11}^{4} + 156T_{11}^{2} - 40 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 28 T^{4} + 196 T^{2} + 128)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 24 T^{4} + 164 T^{2} + 320)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 38 T^{4} + 156 T^{2} - 40)^{2} \) Copy content Toggle raw display
$13$ \( (T - 1)^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} + 46 T^{4} + 444 T^{2} + 200)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 76 T^{4} + 1444 T^{2} + 80)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 88 T^{4} + 656 T^{2} - 640)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 18)^{6} \) Copy content Toggle raw display
$31$ \( (T^{6} + 124 T^{4} + 4164 T^{2} + \cdots + 23120)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 2 T^{2} - 80 T - 320)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + 52 T^{4} + 580 T^{2} + 800)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 104 T^{4} + 3344 T^{2} + \cdots + 32000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 198 T^{4} + 8156 T^{2} + \cdots - 88360)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 46 T^{4} + 444 T^{2} + 200)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 10)^{6} \) Copy content Toggle raw display
$61$ \( (T^{3} + 24 T^{2} + 164 T + 320)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 136 T^{4} + 2404 T^{2} + \cdots + 5120)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 222 T^{4} + 14636 T^{2} + \cdots - 289000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 10 T^{2} - 32 T - 256)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + 304 T^{4} + 23104 T^{2} + \cdots + 5120)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 702 T^{4} + 162476 T^{2} + \cdots - 12409960)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 124 T^{4} + 3652 T^{2} + \cdots + 8192)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 14 T^{2} - 128)^{4} \) Copy content Toggle raw display
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