Properties

Label 7650.2.a.dj
Level $7650$
Weight $2$
Character orbit 7650.a
Self dual yes
Analytic conductor $61.086$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7650,2,Mod(1,7650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7650, 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("7650.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7650 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7650.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: \(61.0855575463\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 170)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + (\beta_{2} + \beta_1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + (\beta_{2} + \beta_1) q^{7} - q^{8} - 2 q^{11} - \beta_{2} q^{13} + ( - \beta_{2} - \beta_1) q^{14} + q^{16} - q^{17} + (\beta_{2} + 2 \beta_1) q^{19} + 2 q^{22} + ( - \beta_{2} - \beta_1 + 4) q^{23} + \beta_{2} q^{26} + (\beta_{2} + \beta_1) q^{28} + ( - 2 \beta_{2} - \beta_1 - 6) q^{29} + (2 \beta_{2} - \beta_1 + 2) q^{31} - q^{32} + q^{34} + ( - \beta_{2} - 5 \beta_1 + 4) q^{37} + ( - \beta_{2} - 2 \beta_1) q^{38} + (2 \beta_1 - 6) q^{41} + (2 \beta_{2} + 2) q^{43} - 2 q^{44} + (\beta_{2} + \beta_1 - 4) q^{46} + ( - \beta_{2} - 4 \beta_1 + 6) q^{47} + ( - 2 \beta_{2} - 1) q^{49} - \beta_{2} q^{52} + ( - 3 \beta_{2} + 2 \beta_1) q^{53} + ( - \beta_{2} - \beta_1) q^{56} + (2 \beta_{2} + \beta_1 + 6) q^{58} + (\beta_{2} - 2 \beta_1 - 2) q^{59} + ( - 3 \beta_1 - 2) q^{61} + ( - 2 \beta_{2} + \beta_1 - 2) q^{62} + q^{64} + (2 \beta_{2} - 4) q^{67} - q^{68} + (2 \beta_{2} - 3 \beta_1 + 2) q^{71} - \beta_{2} q^{73} + (\beta_{2} + 5 \beta_1 - 4) q^{74} + (\beta_{2} + 2 \beta_1) q^{76} + ( - 2 \beta_{2} - 2 \beta_1) q^{77} + ( - \beta_{2} + 3 \beta_1) q^{79} + ( - 2 \beta_1 + 6) q^{82} + (2 \beta_{2} + 4 \beta_1 + 6) q^{83} + ( - 2 \beta_{2} - 2) q^{86} + 2 q^{88} + ( - \beta_{2} + 2 \beta_1 + 2) q^{89} + (2 \beta_{2} + 2 \beta_1 - 4) q^{91} + ( - \beta_{2} - \beta_1 + 4) q^{92} + (\beta_{2} + 4 \beta_1 - 6) q^{94} + ( - \beta_{2} + 2 \beta_1) q^{97} + (2 \beta_{2} + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8} - 6 q^{11} + q^{13} + 3 q^{16} - 3 q^{17} + q^{19} + 6 q^{22} + 12 q^{23} - q^{26} - 17 q^{29} + 3 q^{31} - 3 q^{32} + 3 q^{34} + 8 q^{37} - q^{38} - 16 q^{41} + 4 q^{43} - 6 q^{44} - 12 q^{46} + 15 q^{47} - q^{49} + q^{52} + 5 q^{53} + 17 q^{58} - 9 q^{59} - 9 q^{61} - 3 q^{62} + 3 q^{64} - 14 q^{67} - 3 q^{68} + q^{71} + q^{73} - 8 q^{74} + q^{76} + 4 q^{79} + 16 q^{82} + 20 q^{83} - 4 q^{86} + 6 q^{88} + 9 q^{89} - 12 q^{91} + 12 q^{92} - 15 q^{94} + 3 q^{97} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 4 \) 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
−0.363328
−1.76156
3.12489
−1.00000 0 1.00000 0 0 −3.50466 −1.00000 0 0
1.2 −1.00000 0 1.00000 0 0 0.864641 −1.00000 0 0
1.3 −1.00000 0 1.00000 0 0 2.64002 −1.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(17\) \(1\)

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 7650.2.a.dj 3
3.b odd 2 1 850.2.a.q 3
5.b even 2 1 7650.2.a.do 3
5.c odd 4 2 1530.2.d.g 6
12.b even 2 1 6800.2.a.bk 3
15.d odd 2 1 850.2.a.p 3
15.e even 4 2 170.2.c.b 6
60.h even 2 1 6800.2.a.bp 3
60.l odd 4 2 1360.2.e.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.c.b 6 15.e even 4 2
850.2.a.p 3 15.d odd 2 1
850.2.a.q 3 3.b odd 2 1
1360.2.e.c 6 60.l odd 4 2
1530.2.d.g 6 5.c odd 4 2
6800.2.a.bk 3 12.b even 2 1
6800.2.a.bp 3 60.h even 2 1
7650.2.a.dj 3 1.a even 1 1 trivial
7650.2.a.do 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7650))\):

\( T_{7}^{3} - 10T_{7} + 8 \) Copy content Toggle raw display
\( T_{11} + 2 \) Copy content Toggle raw display
\( T_{13}^{3} - T_{13}^{2} - 8T_{13} + 4 \) Copy content Toggle raw display
\( T_{19}^{3} - T_{19}^{2} - 24T_{19} - 20 \) Copy content Toggle raw display
\( T_{23}^{3} - 12T_{23}^{2} + 38T_{23} - 32 \) Copy content Toggle raw display
\( T_{29}^{3} + 17T_{29}^{2} + 66T_{29} - 50 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 10T + 8 \) Copy content Toggle raw display
$11$ \( (T + 2)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - T^{2} - 8T + 4 \) Copy content Toggle raw display
$17$ \( (T + 1)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} - T^{2} - 24 T - 20 \) Copy content Toggle raw display
$23$ \( T^{3} - 12 T^{2} + 38 T - 32 \) Copy content Toggle raw display
$29$ \( T^{3} + 17 T^{2} + 66 T - 50 \) Copy content Toggle raw display
$31$ \( T^{3} - 3 T^{2} - 46 T - 74 \) Copy content Toggle raw display
$37$ \( T^{3} - 8 T^{2} - 122 T + 1016 \) Copy content Toggle raw display
$41$ \( T^{3} + 16 T^{2} + 60 T - 16 \) Copy content Toggle raw display
$43$ \( T^{3} - 4 T^{2} - 28 T + 32 \) Copy content Toggle raw display
$47$ \( T^{3} - 15 T^{2} - 16 T + 664 \) Copy content Toggle raw display
$53$ \( T^{3} - 5 T^{2} - 120 T + 764 \) Copy content Toggle raw display
$59$ \( T^{3} + 9 T^{2} - 16 T - 160 \) Copy content Toggle raw display
$61$ \( T^{3} + 9 T^{2} - 30 T - 34 \) Copy content Toggle raw display
$67$ \( T^{3} + 14 T^{2} + 32 T - 64 \) Copy content Toggle raw display
$71$ \( T^{3} - T^{2} - 118 T - 334 \) Copy content Toggle raw display
$73$ \( T^{3} - T^{2} - 8T + 4 \) Copy content Toggle raw display
$79$ \( T^{3} - 4 T^{2} - 74 T + 160 \) Copy content Toggle raw display
$83$ \( T^{3} - 20 T^{2} + 36 T + 128 \) Copy content Toggle raw display
$89$ \( T^{3} - 9 T^{2} - 16 T + 160 \) Copy content Toggle raw display
$97$ \( T^{3} - 3 T^{2} - 40 T + 100 \) Copy content Toggle raw display
show more
show less