Properties

Label 7600.2.a.bg
Level $7600$
Weight $2$
Character orbit 7600.a
Self dual yes
Analytic conductor $60.686$
Analytic rank $0$
Dimension $2$
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] = [7600,2,Mod(1,7600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("7600.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7600.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: \(60.6863055362\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} + ( - \beta + 3) q^{7} + 2 \beta q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{3} + ( - \beta + 3) q^{7} + 2 \beta q^{9} - \beta q^{11} + (2 \beta - 3) q^{13} - q^{17} + q^{19} + (2 \beta + 1) q^{21} + (3 \beta + 5) q^{23} + ( - \beta + 1) q^{27} + ( - 2 \beta - 3) q^{29} + (3 \beta - 2) q^{31} + ( - \beta - 2) q^{33} - 6 \beta q^{37} + ( - \beta + 1) q^{39} + 3 \beta q^{41} + (3 \beta + 6) q^{43} + ( - 6 \beta + 4) q^{49} + ( - \beta - 1) q^{51} + (6 \beta + 3) q^{53} + (\beta + 1) q^{57} + (7 \beta + 3) q^{59} + ( - 3 \beta + 10) q^{61} + (6 \beta - 4) q^{63} + (3 \beta + 9) q^{67} + (8 \beta + 11) q^{69} + ( - \beta + 12) q^{71} + (6 \beta - 3) q^{73} + ( - 3 \beta + 2) q^{77} + ( - 6 \beta - 2) q^{79} + ( - 6 \beta - 1) q^{81} + ( - 6 \beta + 6) q^{83} + ( - 5 \beta - 7) q^{87} - 5 \beta q^{89} + (9 \beta - 13) q^{91} + (\beta + 4) q^{93} + (4 \beta + 6) q^{97} - 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 6 q^{7} - 6 q^{13} - 2 q^{17} + 2 q^{19} + 2 q^{21} + 10 q^{23} + 2 q^{27} - 6 q^{29} - 4 q^{31} - 4 q^{33} + 2 q^{39} + 12 q^{43} + 8 q^{49} - 2 q^{51} + 6 q^{53} + 2 q^{57} + 6 q^{59} + 20 q^{61} - 8 q^{63} + 18 q^{67} + 22 q^{69} + 24 q^{71} - 6 q^{73} + 4 q^{77} - 4 q^{79} - 2 q^{81} + 12 q^{83} - 14 q^{87} - 26 q^{91} + 8 q^{93} + 12 q^{97} - 8 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
0 −0.414214 0 0 0 4.41421 0 −2.82843 0
1.2 0 2.41421 0 0 0 1.58579 0 2.82843 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( -1 \)
\(19\) \( -1 \)

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 7600.2.a.bg 2
4.b odd 2 1 950.2.a.g 2
5.b even 2 1 7600.2.a.v 2
5.c odd 4 2 1520.2.d.e 4
12.b even 2 1 8550.2.a.bn 2
20.d odd 2 1 950.2.a.f 2
20.e even 4 2 190.2.b.a 4
60.h even 2 1 8550.2.a.cb 2
60.l odd 4 2 1710.2.d.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.b.a 4 20.e even 4 2
950.2.a.f 2 20.d odd 2 1
950.2.a.g 2 4.b odd 2 1
1520.2.d.e 4 5.c odd 4 2
1710.2.d.c 4 60.l odd 4 2
7600.2.a.v 2 5.b even 2 1
7600.2.a.bg 2 1.a even 1 1 trivial
8550.2.a.bn 2 12.b even 2 1
8550.2.a.cb 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7600))\):

\( T_{3}^{2} - 2T_{3} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 6T_{7} + 7 \) Copy content Toggle raw display
\( T_{11}^{2} - 2 \) Copy content Toggle raw display
\( T_{13}^{2} + 6T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 6T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - 2 \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 1 \) Copy content Toggle raw display
$17$ \( (T + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 10T + 7 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 1 \) Copy content Toggle raw display
$31$ \( T^{2} + 4T - 14 \) Copy content Toggle raw display
$37$ \( T^{2} - 72 \) Copy content Toggle raw display
$41$ \( T^{2} - 18 \) Copy content Toggle raw display
$43$ \( T^{2} - 12T + 18 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 6T - 63 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T - 89 \) Copy content Toggle raw display
$61$ \( T^{2} - 20T + 82 \) Copy content Toggle raw display
$67$ \( T^{2} - 18T + 63 \) Copy content Toggle raw display
$71$ \( T^{2} - 24T + 142 \) Copy content Toggle raw display
$73$ \( T^{2} + 6T - 63 \) Copy content Toggle raw display
$79$ \( T^{2} + 4T - 68 \) Copy content Toggle raw display
$83$ \( T^{2} - 12T - 36 \) Copy content Toggle raw display
$89$ \( T^{2} - 50 \) Copy content Toggle raw display
$97$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
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