Properties

Label 1350.2.a.x
Level $1350$
Weight $2$
Character orbit 1350.a
Self dual yes
Analytic conductor $10.780$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + \beta q^{7} + q^{8} - \beta q^{11} + \beta q^{14} + q^{16} + 4 q^{17} + 6 q^{19} - \beta q^{22} - 2 q^{23} + \beta q^{28} + 7 q^{31} + q^{32} + 4 q^{34} - 2 \beta q^{37} + 6 q^{38} + 2 \beta q^{41} - 2 \beta q^{43} - \beta q^{44} - 2 q^{46} + 2 q^{47} + 12 q^{49} - 3 q^{53} + \beta q^{56} - 2 \beta q^{59} - 4 q^{61} + 7 q^{62} + q^{64} + 2 \beta q^{67} + 4 q^{68} + \beta q^{73} - 2 \beta q^{74} + 6 q^{76} - 19 q^{77} + 2 \beta q^{82} - 5 q^{83} - 2 \beta q^{86} - \beta q^{88} + 2 \beta q^{89} - 2 q^{92} + 2 q^{94} - \beta q^{97} + 12 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 2 q^{16} + 8 q^{17} + 12 q^{19} - 4 q^{23} + 14 q^{31} + 2 q^{32} + 8 q^{34} + 12 q^{38} - 4 q^{46} + 4 q^{47} + 24 q^{49} - 6 q^{53} - 8 q^{61} + 14 q^{62} + 2 q^{64} + 8 q^{68} + 12 q^{76} - 38 q^{77} - 10 q^{83} - 4 q^{92} + 4 q^{94} + 24 q^{98}+O(q^{100}) \) 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
−4.35890
4.35890
1.00000 0 1.00000 0 0 −4.35890 1.00000 0 0
1.2 1.00000 0 1.00000 0 0 4.35890 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.2.a.x 2
3.b odd 2 1 1350.2.a.w 2
5.b even 2 1 1350.2.a.w 2
5.c odd 4 2 270.2.c.c 4
15.d odd 2 1 inner 1350.2.a.x 2
15.e even 4 2 270.2.c.c 4
20.e even 4 2 2160.2.f.m 4
45.k odd 12 4 810.2.i.h 8
45.l even 12 4 810.2.i.h 8
60.l odd 4 2 2160.2.f.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.c.c 4 5.c odd 4 2
270.2.c.c 4 15.e even 4 2
810.2.i.h 8 45.k odd 12 4
810.2.i.h 8 45.l even 12 4
1350.2.a.w 2 3.b odd 2 1
1350.2.a.w 2 5.b even 2 1
1350.2.a.x 2 1.a even 1 1 trivial
1350.2.a.x 2 15.d odd 2 1 inner
2160.2.f.m 4 20.e even 4 2
2160.2.f.m 4 60.l odd 4 2

Hecke kernels

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

\( T_{7}^{2} - 19 \) Copy content Toggle raw display
\( T_{11}^{2} - 19 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{17} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 19 \) Copy content Toggle raw display
$11$ \( T^{2} - 19 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 4)^{2} \) Copy content Toggle raw display
$19$ \( (T - 6)^{2} \) Copy content Toggle raw display
$23$ \( (T + 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 7)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 76 \) Copy content Toggle raw display
$41$ \( T^{2} - 76 \) Copy content Toggle raw display
$43$ \( T^{2} - 76 \) Copy content Toggle raw display
$47$ \( (T - 2)^{2} \) Copy content Toggle raw display
$53$ \( (T + 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 76 \) Copy content Toggle raw display
$61$ \( (T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 76 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 19 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( (T + 5)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 76 \) Copy content Toggle raw display
$97$ \( T^{2} - 19 \) Copy content Toggle raw display
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