Properties

Label 7350.2.a.de
Level $7350$
Weight $2$
Character orbit 7350.a
Self dual yes
Analytic conductor $58.690$
Analytic rank $1$
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] = [7350,2,Mod(1,7350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7350, 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("7350.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7350.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: \(58.6900454856\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} + (3 \beta - 2) q^{11} + q^{12} + ( - 2 \beta - 3) q^{13} + q^{16} + ( - 4 \beta - 1) q^{17} - q^{18} + (\beta + 4) q^{19} + ( - 3 \beta + 2) q^{22} + ( - \beta - 3) q^{23} - q^{24} + (2 \beta + 3) q^{26} + q^{27} + (4 \beta - 5) q^{29} + ( - 3 \beta + 3) q^{31} - q^{32} + (3 \beta - 2) q^{33} + (4 \beta + 1) q^{34} + q^{36} + (3 \beta + 4) q^{37} + ( - \beta - 4) q^{38} + ( - 2 \beta - 3) q^{39} + 7 q^{41} + (\beta + 5) q^{43} + (3 \beta - 2) q^{44} + (\beta + 3) q^{46} - 8 q^{47} + q^{48} + ( - 4 \beta - 1) q^{51} + ( - 2 \beta - 3) q^{52} + (4 \beta + 3) q^{53} - q^{54} + (\beta + 4) q^{57} + ( - 4 \beta + 5) q^{58} + ( - \beta - 3) q^{59} + ( - 4 \beta - 3) q^{61} + (3 \beta - 3) q^{62} + q^{64} + ( - 3 \beta + 2) q^{66} + (4 \beta - 4) q^{67} + ( - 4 \beta - 1) q^{68} + ( - \beta - 3) q^{69} - 2 q^{71} - q^{72} + (3 \beta - 4) q^{73} + ( - 3 \beta - 4) q^{74} + (\beta + 4) q^{76} + (2 \beta + 3) q^{78} + ( - 3 \beta + 2) q^{79} + q^{81} - 7 q^{82} + (\beta + 7) q^{83} + ( - \beta - 5) q^{86} + (4 \beta - 5) q^{87} + ( - 3 \beta + 2) q^{88} + ( - 8 \beta + 4) q^{89} + ( - \beta - 3) q^{92} + ( - 3 \beta + 3) q^{93} + 8 q^{94} - q^{96} - 14 q^{97} + (3 \beta - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9} - 4 q^{11} + 2 q^{12} - 6 q^{13} + 2 q^{16} - 2 q^{17} - 2 q^{18} + 8 q^{19} + 4 q^{22} - 6 q^{23} - 2 q^{24} + 6 q^{26} + 2 q^{27} - 10 q^{29} + 6 q^{31} - 2 q^{32} - 4 q^{33} + 2 q^{34} + 2 q^{36} + 8 q^{37} - 8 q^{38} - 6 q^{39} + 14 q^{41} + 10 q^{43} - 4 q^{44} + 6 q^{46} - 16 q^{47} + 2 q^{48} - 2 q^{51} - 6 q^{52} + 6 q^{53} - 2 q^{54} + 8 q^{57} + 10 q^{58} - 6 q^{59} - 6 q^{61} - 6 q^{62} + 2 q^{64} + 4 q^{66} - 8 q^{67} - 2 q^{68} - 6 q^{69} - 4 q^{71} - 2 q^{72} - 8 q^{73} - 8 q^{74} + 8 q^{76} + 6 q^{78} + 4 q^{79} + 2 q^{81} - 14 q^{82} + 14 q^{83} - 10 q^{86} - 10 q^{87} + 4 q^{88} + 8 q^{89} - 6 q^{92} + 6 q^{93} + 16 q^{94} - 2 q^{96} - 28 q^{97} - 4 q^{99}+O(q^{100}) \) 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
−1.00000 1.00000 1.00000 0 −1.00000 0 −1.00000 1.00000 0
1.2 −1.00000 1.00000 1.00000 0 −1.00000 0 −1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)

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 7350.2.a.de yes 2
5.b even 2 1 7350.2.a.di yes 2
7.b odd 2 1 7350.2.a.db 2
35.c odd 2 1 7350.2.a.dk yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7350.2.a.db 2 7.b odd 2 1
7350.2.a.de yes 2 1.a even 1 1 trivial
7350.2.a.di yes 2 5.b even 2 1
7350.2.a.dk yes 2 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7350))\):

\( T_{11}^{2} + 4T_{11} - 14 \) Copy content Toggle raw display
\( T_{13}^{2} + 6T_{13} + 1 \) Copy content Toggle raw display
\( T_{17}^{2} + 2T_{17} - 31 \) Copy content Toggle raw display
\( T_{19}^{2} - 8T_{19} + 14 \) Copy content Toggle raw display
\( T_{23}^{2} + 6T_{23} + 7 \) Copy content Toggle raw display
\( T_{31}^{2} - 6T_{31} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4T - 14 \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T - 31 \) Copy content Toggle raw display
$19$ \( T^{2} - 8T + 14 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 7 \) Copy content Toggle raw display
$29$ \( T^{2} + 10T - 7 \) Copy content Toggle raw display
$31$ \( T^{2} - 6T - 9 \) Copy content Toggle raw display
$37$ \( T^{2} - 8T - 2 \) Copy content Toggle raw display
$41$ \( (T - 7)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 23 \) Copy content Toggle raw display
$47$ \( (T + 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 6T - 23 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T + 7 \) Copy content Toggle raw display
$61$ \( T^{2} + 6T - 23 \) Copy content Toggle raw display
$67$ \( T^{2} + 8T - 16 \) Copy content Toggle raw display
$71$ \( (T + 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 8T - 2 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T - 14 \) Copy content Toggle raw display
$83$ \( T^{2} - 14T + 47 \) Copy content Toggle raw display
$89$ \( T^{2} - 8T - 112 \) Copy content Toggle raw display
$97$ \( (T + 14)^{2} \) Copy content Toggle raw display
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