Properties

Label 7350.2.a.q
Level $7350$
Weight $2$
Character orbit 7350.a
Self dual yes
Analytic conductor $58.690$
Analytic rank $0$
Dimension $1$
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] = [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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
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)
 
\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} + 5 q^{11} - q^{12} + q^{16} - 4 q^{17} - q^{18} - 8 q^{19} - 5 q^{22} + 4 q^{23} + q^{24} - q^{27} - 5 q^{29} - 3 q^{31} - q^{32} - 5 q^{33} + 4 q^{34} + q^{36} + 4 q^{37} + 8 q^{38} - 2 q^{43} + 5 q^{44} - 4 q^{46} - 6 q^{47} - q^{48} + 4 q^{51} + 9 q^{53} + q^{54} + 8 q^{57} + 5 q^{58} + 11 q^{59} + 6 q^{61} + 3 q^{62} + q^{64} + 5 q^{66} + 2 q^{67} - 4 q^{68} - 4 q^{69} + 2 q^{71} - q^{72} + 10 q^{73} - 4 q^{74} - 8 q^{76} + 3 q^{79} + q^{81} - 7 q^{83} + 2 q^{86} + 5 q^{87} - 5 q^{88} + 6 q^{89} + 4 q^{92} + 3 q^{93} + 6 q^{94} + q^{96} + 7 q^{97} + 5 q^{99}+O(q^{100}) \) 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
0
−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.q 1
5.b even 2 1 294.2.a.f 1
7.b odd 2 1 7350.2.a.bl 1
7.d odd 6 2 1050.2.i.l 2
15.d odd 2 1 882.2.a.d 1
20.d odd 2 1 2352.2.a.f 1
35.c odd 2 1 294.2.a.e 1
35.i odd 6 2 42.2.e.a 2
35.j even 6 2 294.2.e.b 2
35.k even 12 4 1050.2.o.a 4
40.e odd 2 1 9408.2.a.cr 1
40.f even 2 1 9408.2.a.z 1
60.h even 2 1 7056.2.a.bl 1
105.g even 2 1 882.2.a.c 1
105.o odd 6 2 882.2.g.i 2
105.p even 6 2 126.2.g.c 2
140.c even 2 1 2352.2.a.t 1
140.p odd 6 2 2352.2.q.u 2
140.s even 6 2 336.2.q.b 2
280.c odd 2 1 9408.2.a.ce 1
280.n even 2 1 9408.2.a.q 1
280.ba even 6 2 1344.2.q.s 2
280.bk odd 6 2 1344.2.q.g 2
315.q odd 6 2 1134.2.e.l 2
315.u even 6 2 1134.2.h.l 2
315.bn odd 6 2 1134.2.h.e 2
315.bq even 6 2 1134.2.e.e 2
420.o odd 2 1 7056.2.a.w 1
420.be odd 6 2 1008.2.s.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.e.a 2 35.i odd 6 2
126.2.g.c 2 105.p even 6 2
294.2.a.e 1 35.c odd 2 1
294.2.a.f 1 5.b even 2 1
294.2.e.b 2 35.j even 6 2
336.2.q.b 2 140.s even 6 2
882.2.a.c 1 105.g even 2 1
882.2.a.d 1 15.d odd 2 1
882.2.g.i 2 105.o odd 6 2
1008.2.s.k 2 420.be odd 6 2
1050.2.i.l 2 7.d odd 6 2
1050.2.o.a 4 35.k even 12 4
1134.2.e.e 2 315.bq even 6 2
1134.2.e.l 2 315.q odd 6 2
1134.2.h.e 2 315.bn odd 6 2
1134.2.h.l 2 315.u even 6 2
1344.2.q.g 2 280.bk odd 6 2
1344.2.q.s 2 280.ba even 6 2
2352.2.a.f 1 20.d odd 2 1
2352.2.a.t 1 140.c even 2 1
2352.2.q.u 2 140.p odd 6 2
7056.2.a.w 1 420.o odd 2 1
7056.2.a.bl 1 60.h even 2 1
7350.2.a.q 1 1.a even 1 1 trivial
7350.2.a.bl 1 7.b odd 2 1
9408.2.a.q 1 280.n even 2 1
9408.2.a.z 1 40.f even 2 1
9408.2.a.ce 1 280.c odd 2 1
9408.2.a.cr 1 40.e odd 2 1

Hecke kernels

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

\( T_{11} - 5 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{17} + 4 \) Copy content Toggle raw display
\( T_{19} + 8 \) Copy content Toggle raw display
\( T_{23} - 4 \) Copy content Toggle raw display
\( T_{31} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

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