Properties

Label 7600.2.a.bf
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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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: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) 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 380)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} + 2 q^{7} + (2 \beta + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{3} + 2 q^{7} + (2 \beta + 1) q^{9} + 2 \beta q^{11} + (\beta + 1) q^{13} + 2 \beta q^{17} - q^{19} + (2 \beta + 2) q^{21} - 2 \beta q^{23} + 4 q^{27} + 2 \beta q^{29} + (2 \beta - 2) q^{31} + (2 \beta + 6) q^{33} + ( - \beta - 5) q^{37} + (2 \beta + 4) q^{39} - 6 q^{41} + ( - 4 \beta + 2) q^{43} + (4 \beta + 6) q^{47} - 3 q^{49} + (2 \beta + 6) q^{51} + (\beta + 9) q^{53} + ( - \beta - 1) q^{57} - 4 \beta q^{59} + (6 \beta + 2) q^{61} + (4 \beta + 2) q^{63} + (\beta + 5) q^{67} + ( - 2 \beta - 6) q^{69} + ( - 2 \beta + 6) q^{71} + ( - 2 \beta + 4) q^{73} + 4 \beta q^{77} + ( - 4 \beta + 4) q^{79} + ( - 2 \beta + 1) q^{81} + 2 \beta q^{83} + (2 \beta + 6) q^{87} + ( - 2 \beta - 12) q^{89} + (2 \beta + 2) q^{91} + 4 q^{93} + (9 \beta + 1) q^{97} + (2 \beta + 12) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 4 q^{7} + 2 q^{9} + 2 q^{13} - 2 q^{19} + 4 q^{21} + 8 q^{27} - 4 q^{31} + 12 q^{33} - 10 q^{37} + 8 q^{39} - 12 q^{41} + 4 q^{43} + 12 q^{47} - 6 q^{49} + 12 q^{51} + 18 q^{53} - 2 q^{57} + 4 q^{61} + 4 q^{63} + 10 q^{67} - 12 q^{69} + 12 q^{71} + 8 q^{73} + 8 q^{79} + 2 q^{81} + 12 q^{87} - 24 q^{89} + 4 q^{91} + 8 q^{93} + 2 q^{97} + 24 q^{99}+O(q^{100}) \) 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
−1.73205
1.73205
0 −0.732051 0 0 0 2.00000 0 −2.46410 0
1.2 0 2.73205 0 0 0 2.00000 0 4.46410 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.bf 2
4.b odd 2 1 1900.2.a.d 2
5.b even 2 1 1520.2.a.l 2
20.d odd 2 1 380.2.a.d 2
20.e even 4 2 1900.2.c.e 4
40.e odd 2 1 6080.2.a.z 2
40.f even 2 1 6080.2.a.bj 2
60.h even 2 1 3420.2.a.h 2
380.d even 2 1 7220.2.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.a.d 2 20.d odd 2 1
1520.2.a.l 2 5.b even 2 1
1900.2.a.d 2 4.b odd 2 1
1900.2.c.e 4 20.e even 4 2
3420.2.a.h 2 60.h even 2 1
6080.2.a.z 2 40.e odd 2 1
6080.2.a.bj 2 40.f even 2 1
7220.2.a.h 2 380.d even 2 1
7600.2.a.bf 2 1.a even 1 1 trivial

Hecke kernels

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

\( T_{3}^{2} - 2T_{3} - 2 \) Copy content Toggle raw display
\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 12 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 12 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$17$ \( T^{2} - 12 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 12 \) Copy content Toggle raw display
$29$ \( T^{2} - 12 \) Copy content Toggle raw display
$31$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
$37$ \( T^{2} + 10T + 22 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 44 \) Copy content Toggle raw display
$47$ \( T^{2} - 12T - 12 \) Copy content Toggle raw display
$53$ \( T^{2} - 18T + 78 \) Copy content Toggle raw display
$59$ \( T^{2} - 48 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T - 104 \) Copy content Toggle raw display
$67$ \( T^{2} - 10T + 22 \) Copy content Toggle raw display
$71$ \( T^{2} - 12T + 24 \) Copy content Toggle raw display
$73$ \( T^{2} - 8T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T - 32 \) Copy content Toggle raw display
$83$ \( T^{2} - 12 \) Copy content Toggle raw display
$89$ \( T^{2} + 24T + 132 \) Copy content Toggle raw display
$97$ \( T^{2} - 2T - 242 \) Copy content Toggle raw display
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