Properties

Label 1170.4.a.b
Level $1170$
Weight $4$
Character orbit 1170.a
Self dual yes
Analytic conductor $69.032$
Analytic rank $0$
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] = [1170,4,Mod(1,1170)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1170, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1170.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1170.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: \(69.0322347067\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} - 5 q^{5} - 12 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 4 q^{4} - 5 q^{5} - 12 q^{7} - 8 q^{8} + 10 q^{10} + 48 q^{11} + 13 q^{13} + 24 q^{14} + 16 q^{16} + 62 q^{17} - 32 q^{19} - 20 q^{20} - 96 q^{22} + 8 q^{23} + 25 q^{25} - 26 q^{26} - 48 q^{28} + 58 q^{29} - 124 q^{31} - 32 q^{32} - 124 q^{34} + 60 q^{35} - 162 q^{37} + 64 q^{38} + 40 q^{40} - 74 q^{41} - 396 q^{43} + 192 q^{44} - 16 q^{46} + 164 q^{47} - 199 q^{49} - 50 q^{50} + 52 q^{52} - 270 q^{53} - 240 q^{55} + 96 q^{56} - 116 q^{58} + 416 q^{59} + 70 q^{61} + 248 q^{62} + 64 q^{64} - 65 q^{65} + 448 q^{67} + 248 q^{68} - 120 q^{70} + 1092 q^{71} + 10 q^{73} + 324 q^{74} - 128 q^{76} - 576 q^{77} + 328 q^{79} - 80 q^{80} + 148 q^{82} + 144 q^{83} - 310 q^{85} + 792 q^{86} - 384 q^{88} + 502 q^{89} - 156 q^{91} + 32 q^{92} - 328 q^{94} + 160 q^{95} + 1042 q^{97} + 398 q^{98}+O(q^{100}) \) 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 −12.0000 −8.00000 0 10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)
\(13\) \(-1\)

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 1170.4.a.b 1
3.b odd 2 1 390.4.a.j 1
15.d odd 2 1 1950.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.4.a.j 1 3.b odd 2 1
1170.4.a.b 1 1.a even 1 1 trivial
1950.4.a.f 1 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1170))\):

\( T_{7} + 12 \) Copy content Toggle raw display
\( T_{11} - 48 \) 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
$7$ \( T + 12 \) Copy content Toggle raw display
$11$ \( T - 48 \) Copy content Toggle raw display
$13$ \( T - 13 \) Copy content Toggle raw display
$17$ \( T - 62 \) Copy content Toggle raw display
$19$ \( T + 32 \) Copy content Toggle raw display
$23$ \( T - 8 \) Copy content Toggle raw display
$29$ \( T - 58 \) Copy content Toggle raw display
$31$ \( T + 124 \) Copy content Toggle raw display
$37$ \( T + 162 \) Copy content Toggle raw display
$41$ \( T + 74 \) Copy content Toggle raw display
$43$ \( T + 396 \) Copy content Toggle raw display
$47$ \( T - 164 \) Copy content Toggle raw display
$53$ \( T + 270 \) Copy content Toggle raw display
$59$ \( T - 416 \) Copy content Toggle raw display
$61$ \( T - 70 \) Copy content Toggle raw display
$67$ \( T - 448 \) Copy content Toggle raw display
$71$ \( T - 1092 \) Copy content Toggle raw display
$73$ \( T - 10 \) Copy content Toggle raw display
$79$ \( T - 328 \) Copy content Toggle raw display
$83$ \( T - 144 \) Copy content Toggle raw display
$89$ \( T - 502 \) Copy content Toggle raw display
$97$ \( T - 1042 \) Copy content Toggle raw display
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