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Properties

Label 7524.2.l.b
Level $7524$
Weight $2$
Character orbit 7524.l
Analytic conductor $60.079$
Analytic rank $0$
Dimension $8$
CM discriminant -627
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7524,2,Mod(2089,7524)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7524, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7524.2089");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7524 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7524.l (of order \(2\), degree \(1\), 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: \(60.0794424808\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.488455618816.6
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 14x^{5} + 105x^{4} - 238x^{3} - 426x^{2} + 548x + 3140 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{11} - \beta_1 q^{13} + \beta_{3} q^{17} - \beta_{4} q^{19} - 5 q^{25} + 7 q^{49} - \beta_{2} q^{53} + ( - 2 \beta_{5} - \beta_{2}) q^{59} + (2 \beta_{5} - \beta_{2}) q^{71} + ( - 2 \beta_{4} + 3 \beta_1) q^{79} + (2 \beta_{6} - \beta_{3}) q^{83} + ( - 4 \beta_{5} - \beta_{2}) q^{89}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 40 q^{25} + 56 q^{49}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 14x^{5} + 105x^{4} - 238x^{3} - 426x^{2} + 548x + 3140 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 107\nu^{7} + 1001\nu^{6} - 4543\nu^{5} - 13153\nu^{4} + 51884\nu^{3} + 72002\nu^{2} - 258088\nu - 983740 ) / 137550 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 39\nu^{7} - 595\nu^{6} + 1995\nu^{5} - 749\nu^{4} - 406\nu^{3} - 76426\nu^{2} + 98312\nu + 158560 ) / 45850 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 73\nu^{7} + 1120\nu^{6} - 4025\nu^{5} + 5887\nu^{4} + 4648\nu^{3} + 61978\nu^{2} - 65216\nu + 73420 ) / 68775 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 214\nu^{7} - 749\nu^{6} - 833\nu^{5} + 3955\nu^{4} + 29491\nu^{3} - 48566\nu^{2} - 285092\nu + 150790 ) / 137550 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -78\nu^{7} + 273\nu^{6} - 1239\nu^{5} + 2415\nu^{4} - 5607\nu^{3} + 6132\nu^{2} - 46236\nu + 22170 ) / 45850 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -146\nu^{7} + 511\nu^{6} - 203\nu^{5} - 770\nu^{4} - 17549\nu^{3} + 27349\nu^{2} - 18122\nu + 4465 ) / 68775 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -24\nu^{7} + 84\nu^{6} + 42\nu^{5} - 315\nu^{4} - 1302\nu^{3} + 2310\nu^{2} + 8628\nu - 4253 ) / 917 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{6} + \beta_{5} - 3\beta_{4} + 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 3\beta_{2} - \beta _1 + 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} - 36\beta_{6} + 2\beta_{5} + 12\beta_{4} - 3\beta_{3} - 9\beta_{2} - 3\beta _1 + 15 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - 10\beta_{6} + 8\beta_{4} + 2\beta_{3} - 4\beta_{2} - 8\beta _1 - 49 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 25\beta_{7} + 52\beta_{6} - 254\beta_{5} + 272\beta_{4} + 35\beta_{3} - 45\beta_{2} - 115\beta _1 - 767 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 30\beta_{7} + 248\beta_{6} - 296\beta_{5} + 304\beta_{4} + 229\beta_{3} + 132\beta_{2} - 26\beta _1 - 954 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -217\beta_{7} + 3104\beta_{6} - 2458\beta_{5} + 52\beta_{4} + 1477\beta_{3} + 1071\beta_{2} + 217\beta _1 - 3805 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7524\mathbb{Z}\right)^\times\).

\(n\) \(2377\) \(3763\) \(4105\) \(6689\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2089.1
−1.67945 − 2.15831i
2.67945 − 2.15831i
−1.67945 − 1.15831i
2.67945 − 1.15831i
−1.67945 + 2.15831i
2.67945 + 2.15831i
−1.67945 + 1.15831i
2.67945 + 1.15831i
0 0 0 0 0 0 0 0 0
2089.2 0 0 0 0 0 0 0 0 0
2089.3 0 0 0 0 0 0 0 0 0
2089.4 0 0 0 0 0 0 0 0 0
2089.5 0 0 0 0 0 0 0 0 0
2089.6 0 0 0 0 0 0 0 0 0
2089.7 0 0 0 0 0 0 0 0 0
2089.8 0 0 0 0 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2089.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
627.b odd 2 1 CM by \(\Q(\sqrt{-627}) \)
3.b odd 2 1 inner
11.b odd 2 1 inner
19.b odd 2 1 inner
33.d even 2 1 inner
57.d even 2 1 inner
209.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7524.2.l.b ✓ 8
3.b odd 2 1 inner 7524.2.l.b ✓ 8
11.b odd 2 1 inner 7524.2.l.b ✓ 8
19.b odd 2 1 inner 7524.2.l.b ✓ 8
33.d even 2 1 inner 7524.2.l.b ✓ 8
57.d even 2 1 inner 7524.2.l.b ✓ 8
209.d even 2 1 inner 7524.2.l.b ✓ 8
627.b odd 2 1 CM 7524.2.l.b ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7524.2.l.b ✓ 8 1.a even 1 1 trivial
7524.2.l.b ✓ 8 3.b odd 2 1 inner
7524.2.l.b ✓ 8 11.b odd 2 1 inner
7524.2.l.b ✓ 8 19.b odd 2 1 inner
7524.2.l.b ✓ 8 33.d even 2 1 inner
7524.2.l.b ✓ 8 57.d even 2 1 inner
7524.2.l.b ✓ 8 209.d even 2 1 inner
7524.2.l.b ✓ 8 627.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(7524, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} + 11)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 59 T^{2} + 400)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 91 T^{2} + 1600)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 19)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{4} + 109 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 145 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} + 217 T^{2} + 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} - 455 T^{2} + 47524)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 223 T^{2} + 676)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 325 T^{2} + 3364)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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