Properties

Label 270.4.a.d
Level $270$
Weight $4$
Character orbit 270.a
Self dual yes
Analytic conductor $15.931$
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] = [270,4,Mod(1,270)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(270, 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("270.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 270.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: \(15.9305157015\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} + 5 q^{5} - 13 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 4 q^{4} + 5 q^{5} - 13 q^{7} - 8 q^{8} - 10 q^{10} + 30 q^{11} - 61 q^{13} + 26 q^{14} + 16 q^{16} - 12 q^{17} - 49 q^{19} + 20 q^{20} - 60 q^{22} - 18 q^{23} + 25 q^{25} + 122 q^{26} - 52 q^{28} + 186 q^{29} - 160 q^{31} - 32 q^{32} + 24 q^{34} - 65 q^{35} - 91 q^{37} + 98 q^{38} - 40 q^{40} - 378 q^{41} - 268 q^{43} + 120 q^{44} + 36 q^{46} - 144 q^{47} - 174 q^{49} - 50 q^{50} - 244 q^{52} - 570 q^{53} + 150 q^{55} + 104 q^{56} - 372 q^{58} - 204 q^{59} - 877 q^{61} + 320 q^{62} + 64 q^{64} - 305 q^{65} - 187 q^{67} - 48 q^{68} + 130 q^{70} + 606 q^{71} + 431 q^{73} + 182 q^{74} - 196 q^{76} - 390 q^{77} + 1151 q^{79} + 80 q^{80} + 756 q^{82} - 102 q^{83} - 60 q^{85} + 536 q^{86} - 240 q^{88} - 984 q^{89} + 793 q^{91} - 72 q^{92} + 288 q^{94} - 245 q^{95} - 265 q^{97} + 348 q^{98}+O(q^{100}) \) 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 5.00000 0 −13.0000 −8.00000 0 −10.0000
\(n\): 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 270.4.a.d 1
3.b odd 2 1 270.4.a.h yes 1
4.b odd 2 1 2160.4.a.q 1
5.b even 2 1 1350.4.a.w 1
5.c odd 4 2 1350.4.c.o 2
9.c even 3 2 810.4.e.r 2
9.d odd 6 2 810.4.e.j 2
12.b even 2 1 2160.4.a.g 1
15.d odd 2 1 1350.4.a.i 1
15.e even 4 2 1350.4.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.d 1 1.a even 1 1 trivial
270.4.a.h yes 1 3.b odd 2 1
810.4.e.j 2 9.d odd 6 2
810.4.e.r 2 9.c even 3 2
1350.4.a.i 1 15.d odd 2 1
1350.4.a.w 1 5.b even 2 1
1350.4.c.f 2 15.e even 4 2
1350.4.c.o 2 5.c odd 4 2
2160.4.a.g 1 12.b even 2 1
2160.4.a.q 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(270))\):

\( T_{7} + 13 \) Copy content Toggle raw display
\( T_{11} - 30 \) 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
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T + 13 \) Copy content Toggle raw display
$11$ \( T - 30 \) Copy content Toggle raw display
$13$ \( T + 61 \) Copy content Toggle raw display
$17$ \( T + 12 \) Copy content Toggle raw display
$19$ \( T + 49 \) Copy content Toggle raw display
$23$ \( T + 18 \) Copy content Toggle raw display
$29$ \( T - 186 \) Copy content Toggle raw display
$31$ \( T + 160 \) Copy content Toggle raw display
$37$ \( T + 91 \) Copy content Toggle raw display
$41$ \( T + 378 \) Copy content Toggle raw display
$43$ \( T + 268 \) Copy content Toggle raw display
$47$ \( T + 144 \) Copy content Toggle raw display
$53$ \( T + 570 \) Copy content Toggle raw display
$59$ \( T + 204 \) Copy content Toggle raw display
$61$ \( T + 877 \) Copy content Toggle raw display
$67$ \( T + 187 \) Copy content Toggle raw display
$71$ \( T - 606 \) Copy content Toggle raw display
$73$ \( T - 431 \) Copy content Toggle raw display
$79$ \( T - 1151 \) Copy content Toggle raw display
$83$ \( T + 102 \) Copy content Toggle raw display
$89$ \( T + 984 \) Copy content Toggle raw display
$97$ \( T + 265 \) Copy content Toggle raw display
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