Properties

Label 7245.2.a.ba
Level $7245$
Weight $2$
Character orbit 7245.a
Self dual yes
Analytic conductor $57.852$
Analytic rank $0$
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] = [7245,2,Mod(1,7245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7245, 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("7245.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7245 = 3^{2} \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7245.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: \(57.8516162644\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 805)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - 2 \beta_1 + 1) q^{4} + q^{5} - q^{7} + (2 \beta_{2} - \beta_1 + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - 2 \beta_1 + 1) q^{4} + q^{5} - q^{7} + (2 \beta_{2} - \beta_1 + 2) q^{8} + ( - \beta_1 + 1) q^{10} + (\beta_1 + 2) q^{11} + ( - 3 \beta_{2} - \beta_1 + 1) q^{13} + (\beta_1 - 1) q^{14} + ( - \beta_{2} + \beta_1) q^{16} + (2 \beta_{2} - 3 \beta_1) q^{17} + (\beta_{2} - 2 \beta_1 - 1) q^{19} + (\beta_{2} - 2 \beta_1 + 1) q^{20} + ( - \beta_{2} - \beta_1) q^{22} + q^{23} + q^{25} + (\beta_{2} - 2 \beta_1 + 6) q^{26} + ( - \beta_{2} + 2 \beta_1 - 1) q^{28} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{29} + (\beta_{2} - 1) q^{31} + ( - 5 \beta_{2} + 3 \beta_1 - 5) q^{32} + (3 \beta_{2} - 3 \beta_1 + 4) q^{34} - q^{35} + 3 \beta_1 q^{37} + (2 \beta_{2} - \beta_1 + 2) q^{38} + (2 \beta_{2} - \beta_1 + 2) q^{40} + (4 \beta_{2} - 6 \beta_1 + 3) q^{41} + ( - \beta_1 + 1) q^{43} + (\beta_{2} - 3 \beta_1 - 1) q^{44} + ( - \beta_1 + 1) q^{46} + (5 \beta_{2} - 4 \beta_1 - 1) q^{47} + q^{49} + ( - \beta_1 + 1) q^{50} + (8 \beta_{2} - 6 \beta_1 + 7) q^{52} + ( - 5 \beta_{2} + 10 \beta_1 - 2) q^{53} + (\beta_1 + 2) q^{55} + ( - 2 \beta_{2} + \beta_1 - 2) q^{56} + (2 \beta_{2} - 6 \beta_1 + 10) q^{58} + ( - 3 \beta_{2} + 3 \beta_1 + 5) q^{59} + ( - 5 \beta_{2} + 4 \beta_1 - 6) q^{61} + (\beta_1 - 2) q^{62} + ( - \beta_{2} + 6 \beta_1 - 6) q^{64} + ( - 3 \beta_{2} - \beta_1 + 1) q^{65} + ( - 5 \beta_{2} - \beta_1 + 3) q^{67} + ( - \beta_{2} - \beta_1 + 7) q^{68} + (\beta_1 - 1) q^{70} + ( - \beta_{2} - 6 \beta_1 + 7) q^{71} + (5 \beta_{2} - 4 \beta_1) q^{73} + ( - 3 \beta_{2} + 3 \beta_1 - 6) q^{74} + ( - \beta_{2} + \beta_1 + 4) q^{76} + ( - \beta_1 - 2) q^{77} + (3 \beta_{2} + 3 \beta_1 - 6) q^{79} + ( - \beta_{2} + \beta_1) q^{80} + (6 \beta_{2} - 9 \beta_1 + 11) q^{82} + (7 \beta_{2} - 3 \beta_1 + 13) q^{83} + (2 \beta_{2} - 3 \beta_1) q^{85} + (\beta_{2} - 2 \beta_1 + 3) q^{86} + (5 \beta_{2} + 4) q^{88} + ( - 4 \beta_{2} + \beta_1 - 4) q^{89} + (3 \beta_{2} + \beta_1 - 1) q^{91} + (\beta_{2} - 2 \beta_1 + 1) q^{92} + (4 \beta_{2} - 3 \beta_1 + 2) q^{94} + (\beta_{2} - 2 \beta_1 - 1) q^{95} + (8 \beta_{2} + 3) q^{97} + ( - \beta_1 + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 3 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} + 3 q^{5} - 3 q^{7} + 3 q^{8} + 2 q^{10} + 7 q^{11} + 5 q^{13} - 2 q^{14} + 2 q^{16} - 5 q^{17} - 6 q^{19} + 3 q^{23} + 3 q^{25} + 15 q^{26} + 12 q^{29} - 4 q^{31} - 7 q^{32} + 6 q^{34} - 3 q^{35} + 3 q^{37} + 3 q^{38} + 3 q^{40} - q^{41} + 2 q^{43} - 7 q^{44} + 2 q^{46} - 12 q^{47} + 3 q^{49} + 2 q^{50} + 7 q^{52} + 9 q^{53} + 7 q^{55} - 3 q^{56} + 22 q^{58} + 21 q^{59} - 9 q^{61} - 5 q^{62} - 11 q^{64} + 5 q^{65} + 13 q^{67} + 21 q^{68} - 2 q^{70} + 16 q^{71} - 9 q^{73} - 12 q^{74} + 14 q^{76} - 7 q^{77} - 18 q^{79} + 2 q^{80} + 18 q^{82} + 29 q^{83} - 5 q^{85} + 6 q^{86} + 7 q^{88} - 7 q^{89} - 5 q^{91} - q^{94} - 6 q^{95} + q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) 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
1.80194
0.445042
−1.24698
−0.801938 0 −1.35690 1.00000 0 −1.00000 2.69202 0 −0.801938
1.2 0.554958 0 −1.69202 1.00000 0 −1.00000 −2.04892 0 0.554958
1.3 2.24698 0 3.04892 1.00000 0 −1.00000 2.35690 0 2.24698
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)
\(23\) \(-1\)

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 7245.2.a.ba 3
3.b odd 2 1 805.2.a.f 3
15.d odd 2 1 4025.2.a.k 3
21.c even 2 1 5635.2.a.r 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.a.f 3 3.b odd 2 1
4025.2.a.k 3 15.d odd 2 1
5635.2.a.r 3 21.c even 2 1
7245.2.a.ba 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7245))\):

\( T_{2}^{3} - 2T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{3} - 7T_{11}^{2} + 14T_{11} - 7 \) Copy content Toggle raw display
\( T_{13}^{3} - 5T_{13}^{2} - 22T_{13} + 97 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 2T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( (T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 7 T^{2} + 14 T - 7 \) Copy content Toggle raw display
$13$ \( T^{3} - 5 T^{2} - 22 T + 97 \) Copy content Toggle raw display
$17$ \( T^{3} + 5 T^{2} - 8 T - 41 \) Copy content Toggle raw display
$19$ \( T^{3} + 6 T^{2} + 5 T - 13 \) Copy content Toggle raw display
$23$ \( (T - 1)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 12 T^{2} + 20 T + 104 \) Copy content Toggle raw display
$31$ \( T^{3} + 4 T^{2} + 3 T - 1 \) Copy content Toggle raw display
$37$ \( T^{3} - 3 T^{2} - 18 T + 27 \) Copy content Toggle raw display
$41$ \( T^{3} + T^{2} - 65 T - 169 \) Copy content Toggle raw display
$43$ \( T^{3} - 2T^{2} - T + 1 \) Copy content Toggle raw display
$47$ \( T^{3} + 12 T^{2} - T - 41 \) Copy content Toggle raw display
$53$ \( T^{3} - 9 T^{2} - 148 T + 1373 \) Copy content Toggle raw display
$59$ \( T^{3} - 21 T^{2} + 126 T - 203 \) Copy content Toggle raw display
$61$ \( T^{3} + 9 T^{2} - 22 T - 211 \) Copy content Toggle raw display
$67$ \( T^{3} - 13 T^{2} - 16 T + 377 \) Copy content Toggle raw display
$71$ \( T^{3} - 16 T^{2} - 15 T + 463 \) Copy content Toggle raw display
$73$ \( T^{3} + 9 T^{2} - 22 T - 29 \) Copy content Toggle raw display
$79$ \( T^{3} + 18 T^{2} + 45 T - 351 \) Copy content Toggle raw display
$83$ \( T^{3} - 29 T^{2} + 194 T + 211 \) Copy content Toggle raw display
$89$ \( T^{3} + 7 T^{2} - 14 T - 91 \) Copy content Toggle raw display
$97$ \( T^{3} - T^{2} - 149 T - 83 \) Copy content Toggle raw display
show more
show less