Properties

Label 1350.2.a.c
Level $1350$
Weight $2$
Character orbit 1350.a
Self dual yes
Analytic conductor $10.780$
Analytic rank $1$
Dimension $1$
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] = [1350,2,Mod(1,1350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1350, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1350.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1350.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: \(10.7798042729\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 270)
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)
 
\(f(q)\) \(=\) \( q - q^{2} + q^{4} - 2 q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - 2 q^{7} - q^{8} - 3 q^{11} + q^{13} + 2 q^{14} + q^{16} + 3 q^{17} + 8 q^{19} + 3 q^{22} - 3 q^{23} - q^{26} - 2 q^{28} - 9 q^{29} - 7 q^{31} - q^{32} - 3 q^{34} - 2 q^{37} - 8 q^{38} - 12 q^{41} + 7 q^{43} - 3 q^{44} + 3 q^{46} + 3 q^{47} - 3 q^{49} + q^{52} - 12 q^{53} + 2 q^{56} + 9 q^{58} + 12 q^{59} - 10 q^{61} + 7 q^{62} + q^{64} + 4 q^{67} + 3 q^{68} - 2 q^{73} + 2 q^{74} + 8 q^{76} + 6 q^{77} - q^{79} + 12 q^{82} - 18 q^{83} - 7 q^{86} + 3 q^{88} - 2 q^{91} - 3 q^{92} - 3 q^{94} - 14 q^{97} + 3 q^{98}+O(q^{100}) \) 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 0 1.00000 0 0 −2.00000 −1.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( +1 \)
\(5\) \( +1 \)

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 1350.2.a.c 1
3.b odd 2 1 1350.2.a.p 1
5.b even 2 1 270.2.a.d yes 1
5.c odd 4 2 1350.2.c.a 2
15.d odd 2 1 270.2.a.a 1
15.e even 4 2 1350.2.c.l 2
20.d odd 2 1 2160.2.a.p 1
40.e odd 2 1 8640.2.a.f 1
40.f even 2 1 8640.2.a.z 1
45.h odd 6 2 810.2.e.k 2
45.j even 6 2 810.2.e.a 2
60.h even 2 1 2160.2.a.a 1
120.i odd 2 1 8640.2.a.by 1
120.m even 2 1 8640.2.a.bo 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.a.a 1 15.d odd 2 1
270.2.a.d yes 1 5.b even 2 1
810.2.e.a 2 45.j even 6 2
810.2.e.k 2 45.h odd 6 2
1350.2.a.c 1 1.a even 1 1 trivial
1350.2.a.p 1 3.b odd 2 1
1350.2.c.a 2 5.c odd 4 2
1350.2.c.l 2 15.e even 4 2
2160.2.a.a 1 60.h even 2 1
2160.2.a.p 1 20.d odd 2 1
8640.2.a.f 1 40.e odd 2 1
8640.2.a.z 1 40.f even 2 1
8640.2.a.bo 1 120.m even 2 1
8640.2.a.by 1 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1350))\):

\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{11} + 3 \) Copy content Toggle raw display
\( T_{13} - 1 \) Copy content Toggle raw display
\( T_{17} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T + 3 \) Copy content Toggle raw display
$13$ \( T - 1 \) Copy content Toggle raw display
$17$ \( T - 3 \) Copy content Toggle raw display
$19$ \( T - 8 \) Copy content Toggle raw display
$23$ \( T + 3 \) Copy content Toggle raw display
$29$ \( T + 9 \) Copy content Toggle raw display
$31$ \( T + 7 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T + 12 \) Copy content Toggle raw display
$43$ \( T - 7 \) Copy content Toggle raw display
$47$ \( T - 3 \) Copy content Toggle raw display
$53$ \( T + 12 \) Copy content Toggle raw display
$59$ \( T - 12 \) Copy content Toggle raw display
$61$ \( T + 10 \) Copy content Toggle raw display
$67$ \( T - 4 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 2 \) Copy content Toggle raw display
$79$ \( T + 1 \) Copy content Toggle raw display
$83$ \( T + 18 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T + 14 \) Copy content Toggle raw display
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