Properties

Label 7500.2.a.f
Level $7500$
Weight $2$
Character orbit 7500.a
Self dual yes
Analytic conductor $59.888$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7500,2,Mod(1,7500)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7500.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7500.a (trivial)

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: yes
Analytic conductor: \(59.8878015160\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.5125.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 6x^{2} + 7x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 300)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( - \beta_{2} - \beta_1 - 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + ( - \beta_{2} - \beta_1 - 1) q^{7} + q^{9} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{11} + ( - \beta_{3} + \beta_1 - 2) q^{13} + (\beta_{2} + \beta_1 - 1) q^{17} + ( - \beta_{3} - \beta_1 - 1) q^{19} + ( - \beta_{2} - \beta_1 - 1) q^{21} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{23} + q^{27} + ( - \beta_1 + 2) q^{29} + (3 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{31} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{33} + (3 \beta_1 - 2) q^{37} + ( - \beta_{3} + \beta_1 - 2) q^{39} + (2 \beta_{2} + 4 \beta_1 - 1) q^{41} + (\beta_{2} + 2 \beta_1 + 1) q^{43} + (2 \beta_{3} - 6 \beta_{2} - \beta_1 - 6) q^{47} + (2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 1) q^{49} + (\beta_{2} + \beta_1 - 1) q^{51} + ( - 2 \beta_{3} - 3 \beta_{2} - \beta_1 - 1) q^{53} + ( - \beta_{3} - \beta_1 - 1) q^{57} + ( - \beta_{3} - 7 \beta_{2} + \beta_1 - 4) q^{59} + ( - 2 \beta_{3} + 8 \beta_{2} - 2 \beta_1 - 1) q^{61} + ( - \beta_{2} - \beta_1 - 1) q^{63} + (2 \beta_{3} + \beta_{2} + 2 \beta_1 - 9) q^{67} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{69} + (3 \beta_{2} + \beta_1 + 6) q^{71} + (2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{73} + ( - \beta_{3} - 3 \beta_{2} + 1) q^{77} + ( - \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 3) q^{79} + q^{81} + ( - 2 \beta_{3} + 4 \beta_{2} - 6 \beta_1 + 1) q^{83} + ( - \beta_1 + 2) q^{87} + ( - 3 \beta_{3} - 6 \beta_{2} + \beta_1 - 8) q^{89} + (4 \beta_{2} + \beta_1 - 1) q^{91} + (3 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{93} + ( - 4 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 3) q^{97} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{7} + 4 q^{9} - q^{11} - 5 q^{13} - 4 q^{17} - 5 q^{19} - 4 q^{21} - 9 q^{23} + 4 q^{27} + 6 q^{29} + 11 q^{31} - q^{33} - 2 q^{37} - 5 q^{39} + 6 q^{43} - 16 q^{47} - 4 q^{49} - 4 q^{51} + 2 q^{53} - 5 q^{57} + q^{59} - 22 q^{61} - 4 q^{63} - 36 q^{67} - 9 q^{69} + 20 q^{71} - 12 q^{73} + 11 q^{77} - 3 q^{79} + 4 q^{81} - 14 q^{83} + 6 q^{87} - 15 q^{89} - 10 q^{91} + 11 q^{93} + 12 q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 6x^{2} + 7x + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta _1 + 4 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.70636
2.12233
−1.70636
−1.12233
0 1.00000 0 0 0 −4.32440 0 1.00000 0
1.2 0 1.00000 0 0 0 −1.50430 0 1.00000 0
1.3 0 1.00000 0 0 0 0.0883282 0 1.00000 0
1.4 0 1.00000 0 0 0 1.74037 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7500.2.a.f 4
5.b even 2 1 7500.2.a.e 4
5.c odd 4 2 7500.2.d.c 8
25.d even 5 2 1500.2.m.a 8
25.e even 10 2 300.2.m.b 8
25.f odd 20 4 1500.2.o.b 16
75.h odd 10 2 900.2.n.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.m.b 8 25.e even 10 2
900.2.n.b 8 75.h odd 10 2
1500.2.m.a 8 25.d even 5 2
1500.2.o.b 16 25.f odd 20 4
7500.2.a.e 4 5.b even 2 1
7500.2.a.f 4 1.a even 1 1 trivial
7500.2.d.c 8 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 4T_{7}^{3} - 4T_{7}^{2} - 11T_{7} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7500))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 4 T^{3} - 4 T^{2} - 11 T + 1 \) Copy content Toggle raw display
$11$ \( T^{4} + T^{3} - 24 T^{2} + 46 T - 19 \) Copy content Toggle raw display
$13$ \( T^{4} + 5 T^{3} - 10 T^{2} - 60 T - 45 \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} - 4 T^{2} - 21 T - 9 \) Copy content Toggle raw display
$19$ \( T^{4} + 5 T^{3} - 5 T^{2} - 15 T + 5 \) Copy content Toggle raw display
$23$ \( T^{4} + 9 T^{3} - 14 T^{2} - 126 T + 171 \) Copy content Toggle raw display
$29$ \( T^{4} - 6 T^{3} + 6 T^{2} + 9 T + 1 \) Copy content Toggle raw display
$31$ \( T^{4} - 11 T^{3} - 34 T^{2} + \cdots + 981 \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} - 66 T^{2} + \cdots + 1021 \) Copy content Toggle raw display
$41$ \( T^{4} - 130 T^{2} - 160 T + 2705 \) Copy content Toggle raw display
$43$ \( T^{4} - 6 T^{3} - 19 T^{2} + 64 T + 131 \) Copy content Toggle raw display
$47$ \( T^{4} + 16 T^{3} - 14 T^{2} + \cdots - 4099 \) Copy content Toggle raw display
$53$ \( T^{4} - 2 T^{3} - 76 T^{2} + 177 T - 99 \) Copy content Toggle raw display
$59$ \( T^{4} - T^{3} - 159 T^{2} - 211 T + 3701 \) Copy content Toggle raw display
$61$ \( T^{4} + 22 T^{3} + 4 T^{2} + \cdots - 7909 \) Copy content Toggle raw display
$67$ \( T^{4} + 36 T^{3} + 421 T^{2} + \cdots - 639 \) Copy content Toggle raw display
$71$ \( T^{4} - 20 T^{3} + 120 T^{2} - 215 T + 5 \) Copy content Toggle raw display
$73$ \( T^{4} + 12 T^{3} - 61 T^{2} + 38 T + 11 \) Copy content Toggle raw display
$79$ \( T^{4} + 3 T^{3} - 146 T^{2} + \cdots - 639 \) Copy content Toggle raw display
$83$ \( T^{4} + 14 T^{3} - 224 T^{2} + \cdots - 6849 \) Copy content Toggle raw display
$89$ \( T^{4} + 15 T^{3} - 150 T^{2} + \cdots - 9875 \) Copy content Toggle raw display
$97$ \( T^{4} - 12 T^{3} - 206 T^{2} + \cdots - 8019 \) Copy content Toggle raw display
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