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Properties

Label 7350.2.a.ds
Level $7350$
Weight $2$
Character orbit 7350.a
Self dual yes
Analytic conductor $58.690$
Analytic rank $1$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.10304.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 7x^{2} + 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1470)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. 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 \( q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} + (\beta_{2} - \beta_1) q^{11} + q^{12} + ( - 2 \beta_{2} + \beta_1) q^{13} + q^{16} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{17} - q^{18} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{19} + ( - \beta_{2} + \beta_1) q^{22} + (\beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{23} - q^{24} + (2 \beta_{2} - \beta_1) q^{26} + q^{27} + (2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{29} + ( - \beta_{2} + \beta_1 - 4) q^{31} - q^{32} + (\beta_{2} - \beta_1) q^{33} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{34} + q^{36} + (3 \beta_{2} + \beta_1 + 2) q^{37} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 3) q^{38} + ( - 2 \beta_{2} + \beta_1) q^{39} + ( - \beta_{3} - 2 \beta_{2} - 3) q^{41} + (\beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{43} + (\beta_{2} - \beta_1) q^{44} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{46} + (\beta_{3} - \beta_{2} + 2 \beta_1 + 3) q^{47} + q^{48} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{51} + ( - 2 \beta_{2} + \beta_1) q^{52} + (\beta_{3} + 2 \beta_{2} - \beta_1 - 5) q^{53} - q^{54} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{57} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 + 2) q^{58} + (\beta_{3} + 4 \beta_{2} + \beta_1 - 5) q^{59} + (3 \beta_{3} - 3) q^{61} + (\beta_{2} - \beta_1 + 4) q^{62} + q^{64} + ( - \beta_{2} + \beta_1) q^{66} + ( - \beta_{3} - \beta_{2} + 1) q^{67} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{68} + (\beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{69} + ( - 3 \beta_{3} - 2 \beta_{2} + \cdots - 5) q^{71}+ \cdots + (\beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{8} + 4 q^{9} + 4 q^{12} + 4 q^{16} + 4 q^{17} - 4 q^{18} - 12 q^{19} - 4 q^{23} - 4 q^{24} + 4 q^{27} - 8 q^{29} - 16 q^{31} - 4 q^{32} - 4 q^{34} + 4 q^{36} + 8 q^{37} + 12 q^{38} - 12 q^{41} - 4 q^{43} + 4 q^{46} + 12 q^{47} + 4 q^{48} + 4 q^{51} - 20 q^{53} - 4 q^{54} - 12 q^{57} + 8 q^{58} - 20 q^{59} - 12 q^{61} + 16 q^{62} + 4 q^{64} + 4 q^{67} + 4 q^{68} - 4 q^{69} - 20 q^{71} - 4 q^{72} - 12 q^{73} - 8 q^{74} - 12 q^{76} - 8 q^{79} + 4 q^{81} + 12 q^{82} + 8 q^{83} + 4 q^{86} - 8 q^{87} - 44 q^{89} - 4 q^{92} - 16 q^{93} - 12 q^{94} - 4 q^{96} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 7x^{2} + 8x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - \nu^{2} - 4\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - \nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 8\nu - 2 ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 4\beta_{2} + \beta _1 + 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{3} + 4\beta_{2} + 9\beta _1 + 13 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
3.16053
−0.692297
1.69230
−2.16053
−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
1.3 −1.00000 1.00000 1.00000 0 −1.00000 0 −1.00000 1.00000 0
1.4 −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.ds 4
5.b even 2 1 7350.2.a.dt 4
5.c odd 4 2 1470.2.g.j ✓ 8
7.b odd 2 1 7350.2.a.dr 4
35.c odd 2 1 7350.2.a.du 4
35.f even 4 2 1470.2.g.k yes 8
35.k even 12 4 1470.2.n.k 16
35.l odd 12 4 1470.2.n.l 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.g.j ✓ 8 5.c odd 4 2
1470.2.g.k yes 8 35.f even 4 2
1470.2.n.k 16 35.k even 12 4
1470.2.n.l 16 35.l odd 12 4
7350.2.a.dr 4 7.b odd 2 1
7350.2.a.ds 4 1.a even 1 1 trivial
7350.2.a.dt 4 5.b even 2 1
7350.2.a.du 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7350))\):

\( T_{11}^{4} - 18T_{11}^{2} - 16T_{11} + 32 \) Copy content Toggle raw display
\( T_{13}^{4} - 26T_{13}^{2} + 48T_{13} - 16 \) Copy content Toggle raw display
\( T_{17}^{4} - 4T_{17}^{3} - 32T_{17}^{2} + 8T_{17} + 124 \) Copy content Toggle raw display
\( T_{19}^{4} + 12T_{19}^{3} + 12T_{19}^{2} - 224T_{19} - 448 \) Copy content Toggle raw display
\( T_{23}^{4} + 4T_{23}^{3} - 36T_{23}^{2} + 128 \) Copy content Toggle raw display
\( T_{31}^{4} + 16T_{31}^{3} + 78T_{31}^{2} + 128T_{31} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 18 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$13$ \( T^{4} - 26 T^{2} + \cdots - 16 \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} + \cdots + 124 \) Copy content Toggle raw display
$19$ \( T^{4} + 12 T^{3} + \cdots - 448 \) Copy content Toggle raw display
$23$ \( T^{4} + 4 T^{3} + \cdots + 128 \) Copy content Toggle raw display
$29$ \( T^{4} + 8 T^{3} + \cdots + 1568 \) Copy content Toggle raw display
$31$ \( T^{4} + 16 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$37$ \( T^{4} - 8 T^{3} + \cdots + 200 \) Copy content Toggle raw display
$41$ \( T^{4} + 12 T^{3} + \cdots - 376 \) Copy content Toggle raw display
$43$ \( T^{4} + 4 T^{3} + \cdots + 736 \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} + \cdots - 1024 \) Copy content Toggle raw display
$53$ \( T^{4} + 20 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$59$ \( T^{4} + 20 T^{3} + \cdots - 3008 \) Copy content Toggle raw display
$61$ \( T^{4} + 12 T^{3} + \cdots - 648 \) Copy content Toggle raw display
$67$ \( T^{4} - 4 T^{3} + \cdots + 32 \) Copy content Toggle raw display
$71$ \( T^{4} + 20 T^{3} + \cdots + 2944 \) Copy content Toggle raw display
$73$ \( T^{4} + 12 T^{3} + \cdots - 248 \) Copy content Toggle raw display
$79$ \( T^{4} + 8 T^{3} + \cdots - 3584 \) Copy content Toggle raw display
$83$ \( T^{4} - 8 T^{3} + \cdots + 128 \) Copy content Toggle raw display
$89$ \( T^{4} + 44 T^{3} + \cdots + 5048 \) Copy content Toggle raw display
$97$ \( T^{4} - 20 T^{3} + \cdots - 10376 \) Copy content Toggle raw display
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