Properties

Label 1350.4.a.x
Level $1350$
Weight $4$
Character orbit 1350.a
Self dual yes
Analytic conductor $79.653$
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] = [1350,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1350.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(79.6525785077\)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + 14 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} + 14 q^{7} + 8 q^{8} - 22 q^{11} - 30 q^{13} + 28 q^{14} + 16 q^{16} - 7 q^{17} - 81 q^{19} - 44 q^{22} - 151 q^{23} - 60 q^{26} + 56 q^{28} + 270 q^{29} - 113 q^{31} + 32 q^{32} - 14 q^{34} - 88 q^{37} - 162 q^{38} - 406 q^{41} - 442 q^{43} - 88 q^{44} - 302 q^{46} - 56 q^{47} - 147 q^{49} - 120 q^{52} + 141 q^{53} + 112 q^{56} + 540 q^{58} + 274 q^{59} + 41 q^{61} - 226 q^{62} + 64 q^{64} + 328 q^{67} - 28 q^{68} + 390 q^{71} + 626 q^{73} - 176 q^{74} - 324 q^{76} - 308 q^{77} - 1215 q^{79} - 812 q^{82} - 505 q^{83} - 884 q^{86} - 176 q^{88} - 514 q^{89} - 420 q^{91} - 604 q^{92} - 112 q^{94} + 1816 q^{97} - 294 q^{98}+O(q^{100}) \) 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
2.00000 0 4.00000 0 0 14.0000 8.00000 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 1350.4.a.x 1
3.b odd 2 1 1350.4.a.j 1
5.b even 2 1 1350.4.a.d 1
5.c odd 4 2 270.4.c.b yes 2
15.d odd 2 1 1350.4.a.s 1
15.e even 4 2 270.4.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.c.a 2 15.e even 4 2
270.4.c.b yes 2 5.c odd 4 2
1350.4.a.d 1 5.b even 2 1
1350.4.a.j 1 3.b odd 2 1
1350.4.a.s 1 15.d odd 2 1
1350.4.a.x 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1350))\):

\( T_{7} - 14 \) Copy content Toggle raw display
\( T_{11} + 22 \) Copy content Toggle raw display
\( T_{17} + 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 14 \) Copy content Toggle raw display
$11$ \( T + 22 \) Copy content Toggle raw display
$13$ \( T + 30 \) Copy content Toggle raw display
$17$ \( T + 7 \) Copy content Toggle raw display
$19$ \( T + 81 \) Copy content Toggle raw display
$23$ \( T + 151 \) Copy content Toggle raw display
$29$ \( T - 270 \) Copy content Toggle raw display
$31$ \( T + 113 \) Copy content Toggle raw display
$37$ \( T + 88 \) Copy content Toggle raw display
$41$ \( T + 406 \) Copy content Toggle raw display
$43$ \( T + 442 \) Copy content Toggle raw display
$47$ \( T + 56 \) Copy content Toggle raw display
$53$ \( T - 141 \) Copy content Toggle raw display
$59$ \( T - 274 \) Copy content Toggle raw display
$61$ \( T - 41 \) Copy content Toggle raw display
$67$ \( T - 328 \) Copy content Toggle raw display
$71$ \( T - 390 \) Copy content Toggle raw display
$73$ \( T - 626 \) Copy content Toggle raw display
$79$ \( T + 1215 \) Copy content Toggle raw display
$83$ \( T + 505 \) Copy content Toggle raw display
$89$ \( T + 514 \) Copy content Toggle raw display
$97$ \( T - 1816 \) Copy content Toggle raw display
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