Properties

Label 7500.2.d.d
Level $7500$
Weight $2$
Character orbit 7500.d
Analytic conductor $59.888$
Analytic rank $0$
Dimension $8$
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] = [7500,2,Mod(1249,7500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7500, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7500.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7500.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: \(59.8878015160\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.324000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 300)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + (\beta_{7} + 2 \beta_{5} + 2 \beta_{3} + \beta_1) q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} + (\beta_{7} + 2 \beta_{5} + 2 \beta_{3} + \beta_1) q^{7} - q^{9} + \beta_{2} q^{11} + ( - \beta_{7} + 2 \beta_{5} + 2 \beta_1) q^{13} + (3 \beta_{7} - \beta_1) q^{17} + (\beta_{6} - 3 \beta_{2}) q^{19} + ( - \beta_{4} - \beta_{2} - 1) q^{21} + ( - 4 \beta_{5} - 2 \beta_{3} - 3 \beta_1) q^{23} - \beta_{5} q^{27} + ( - 3 \beta_{6} - \beta_{4} - 3 \beta_{2} + 2) q^{29} + ( - \beta_{6} - 2 \beta_{4} + \beta_{2} - 1) q^{31} + (\beta_{7} + \beta_{3}) q^{33} + ( - 3 \beta_{7} - 3 \beta_{3} + \beta_1) q^{37} + (\beta_{6} - 2 \beta_{4} + \beta_{2} - 1) q^{39} + ( - 6 \beta_{6} + 2 \beta_{4} + 2 \beta_{2} + 3) q^{41} + ( - 2 \beta_{7} - 8 \beta_{5} - \beta_{3} + 2 \beta_1) q^{43} + ( - \beta_{7} - \beta_{3} - 5 \beta_1) q^{47} + ( - \beta_{6} - 3 \beta_{4} - 5 \beta_{2} + 1) q^{49} + (2 \beta_{6} + \beta_{4} - 3 \beta_{2} - 3) q^{51} + (3 \beta_{7} + \beta_1) q^{53} + ( - 3 \beta_{7} + \beta_{5} - 2 \beta_{3}) q^{57} + ( - 2 \beta_{6} + 6 \beta_{4} - \beta_{2} - 1) q^{59} + (2 \beta_{6} + 4 \beta_{4} - 2 \beta_{2} - 3) q^{61} + ( - \beta_{7} - 2 \beta_{5} - 2 \beta_{3} - \beta_1) q^{63} + ( - 2 \beta_{7} - 2 \beta_{5} - 5 \beta_{3} + 4 \beta_1) q^{67} + ( - \beta_{6} + 3 \beta_{4} + 2) q^{69} + ( - 4 \beta_{6} - 3 \beta_{4} + \beta_{2} - 2) q^{71} + ( - 6 \beta_{7} - 5 \beta_{5} - 5 \beta_{3} - 4 \beta_1) q^{73} + (2 \beta_{7} + 4 \beta_{5} + 4 \beta_{3} + \beta_1) q^{77} + ( - 8 \beta_{6} + \beta_{4} + 6 \beta_{2} + 6) q^{79} + q^{81} + ( - 3 \beta_{5} + 2 \beta_1) q^{83} + ( - 3 \beta_{7} - 2 \beta_{5} - 7 \beta_{3} - \beta_1) q^{87} + (\beta_{6} - 2 \beta_{4} + 3 \beta_{2} - 3) q^{89} + (\beta_{6} - \beta_{4} - \beta_{2} - 3) q^{91} + (\beta_{7} - 4 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{93} + (4 \beta_{7} + 5 \beta_{5} - 2 \beta_{3}) q^{97} - \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{9} - 2 q^{11} + 10 q^{19} - 8 q^{21} + 8 q^{29} - 18 q^{31} - 10 q^{39} + 8 q^{49} - 8 q^{51} - 2 q^{59} - 4 q^{61} + 18 q^{69} - 40 q^{71} + 6 q^{79} + 8 q^{81} - 30 q^{89} - 20 q^{91} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 4\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 5\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} + 6\nu^{4} + 9\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} + 7\nu^{5} + 14\nu^{3} + 7\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 4\beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 5\beta_{3} + 10\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} - 6\beta_{4} + 15\beta_{2} - 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{7} - 7\beta_{5} + 21\beta_{3} - 35\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7500\mathbb{Z}\right)^\times\).

\(n\) \(2501\) \(3751\) \(6877\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
0.209057i
1.95630i
1.82709i
1.33826i
1.33826i
1.82709i
1.95630i
0.209057i
0 1.00000i 0 0 0 4.78339i 0 −1.00000 0
1249.2 0 1.00000i 0 0 0 0.511170i 0 −1.00000 0
1249.3 0 1.00000i 0 0 0 0.547318i 0 −1.00000 0
1249.4 0 1.00000i 0 0 0 0.747238i 0 −1.00000 0
1249.5 0 1.00000i 0 0 0 0.747238i 0 −1.00000 0
1249.6 0 1.00000i 0 0 0 0.547318i 0 −1.00000 0
1249.7 0 1.00000i 0 0 0 0.511170i 0 −1.00000 0
1249.8 0 1.00000i 0 0 0 4.78339i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1249.8
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 7500.2.d.d 8
5.b even 2 1 inner 7500.2.d.d 8
5.c odd 4 1 7500.2.a.d 4
5.c odd 4 1 7500.2.a.g 4
25.d even 5 2 1500.2.o.a 16
25.e even 10 2 1500.2.o.a 16
25.f odd 20 2 300.2.m.a 8
25.f odd 20 2 1500.2.m.b 8
75.l even 20 2 900.2.n.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.m.a 8 25.f odd 20 2
900.2.n.a 8 75.l even 20 2
1500.2.m.b 8 25.f odd 20 2
1500.2.o.a 16 25.d even 5 2
1500.2.o.a 16 25.e even 10 2
7500.2.a.d 4 5.c odd 4 1
7500.2.a.g 4 5.c odd 4 1
7500.2.d.d 8 1.a even 1 1 trivial
7500.2.d.d 8 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 24T_{7}^{6} + 26T_{7}^{4} + 9T_{7}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(7500, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 24 T^{6} + 26 T^{4} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{4} + T^{3} - 4 T^{2} - 4 T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 45 T^{6} + 290 T^{4} + \cdots + 25 \) Copy content Toggle raw display
$17$ \( T^{8} + 84 T^{6} + 2186 T^{4} + \cdots + 73441 \) Copy content Toggle raw display
$19$ \( (T^{4} - 5 T^{3} - 25 T^{2} + 95 T + 145)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 89 T^{6} + 2346 T^{4} + \cdots + 961 \) Copy content Toggle raw display
$29$ \( (T^{4} - 4 T^{3} - 94 T^{2} + 571 T - 599)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 9 T^{3} + 6 T^{2} - 126 T - 279)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 96 T^{6} + 2546 T^{4} + \cdots + 32761 \) Copy content Toggle raw display
$41$ \( (T^{4} - 70 T^{2} + 240 T - 155)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 354 T^{6} + 40151 T^{4} + \cdots + 5621641 \) Copy content Toggle raw display
$47$ \( T^{8} + 224 T^{6} + 16146 T^{4} + \cdots + 2627641 \) Copy content Toggle raw display
$53$ \( T^{8} + 96 T^{6} + 2546 T^{4} + \cdots + 32041 \) Copy content Toggle raw display
$59$ \( (T^{4} + T^{3} - 139 T^{2} + 41 T + 2341)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 2 T^{3} - 96 T^{2} + 398 T - 449)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 334 T^{6} + 30891 T^{4} + \cdots + 6046681 \) Copy content Toggle raw display
$71$ \( (T^{4} + 20 T^{3} + 50 T^{2} - 875 T - 3875)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 426 T^{6} + 42111 T^{4} + \cdots + 2653641 \) Copy content Toggle raw display
$79$ \( (T^{4} - 3 T^{3} - 186 T^{2} + 1278 T - 1359)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 84 T^{6} + 2006 T^{4} + \cdots + 7921 \) Copy content Toggle raw display
$89$ \( (T^{4} + 15 T^{3} + 30 T^{2} - 90 T + 45)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 396 T^{6} + \cdots + 18139081 \) Copy content Toggle raw display
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