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Properties

Label 1170.2.b.d
Level $1170$
Weight $2$
Character orbit 1170.b
Analytic conductor $9.342$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1170,2,Mod(181,1170)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1170.181");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(9.34249703649\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{4} - \beta_1 q^{5} + (\beta_{2} - \beta_1) q^{7} + \beta_1 q^{8} - q^{10} - \beta_{3} q^{13} + (\beta_{3} - 1) q^{14} + q^{16} + (\beta_{3} - 1) q^{17} + (\beta_{2} - \beta_1) q^{19}+ \cdots + ( - 2 \beta_{2} + 7 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{10} - 4 q^{14} + 4 q^{16} - 4 q^{17} + 20 q^{23} - 4 q^{25} - 4 q^{29} - 4 q^{35} - 4 q^{38} + 4 q^{40} + 32 q^{43} - 28 q^{49} - 24 q^{53} + 4 q^{56} - 16 q^{61} + 24 q^{62} - 4 q^{64}+ \cdots - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 10\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 7 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{2} + 5\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times\).

\(n\) \(911\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
− 2.30278i
1.30278i
− 1.30278i
2.30278i
− 1.00000i 0 −1.00000 − 1.00000i 0 − 4.60555i 1.00000i 0 −1.00000
181.2 − 1.00000i 0 −1.00000 − 1.00000i 0 2.60555i 1.00000i 0 −1.00000
181.3 1.00000i 0 −1.00000 1.00000i 0 − 2.60555i − 1.00000i 0 −1.00000
181.4 1.00000i 0 −1.00000 1.00000i 0 4.60555i − 1.00000i 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.2.b.d 4
3.b odd 2 1 390.2.b.c ✓ 4
12.b even 2 1 3120.2.g.q 4
13.b even 2 1 inner 1170.2.b.d 4
15.d odd 2 1 1950.2.b.k 4
15.e even 4 1 1950.2.f.m 4
15.e even 4 1 1950.2.f.n 4
39.d odd 2 1 390.2.b.c ✓ 4
39.f even 4 1 5070.2.a.z 2
39.f even 4 1 5070.2.a.bf 2
156.h even 2 1 3120.2.g.q 4
195.e odd 2 1 1950.2.b.k 4
195.s even 4 1 1950.2.f.m 4
195.s even 4 1 1950.2.f.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.b.c ✓ 4 3.b odd 2 1
390.2.b.c ✓ 4 39.d odd 2 1
1170.2.b.d 4 1.a even 1 1 trivial
1170.2.b.d 4 13.b even 2 1 inner
1950.2.b.k 4 15.d odd 2 1
1950.2.b.k 4 195.e odd 2 1
1950.2.f.m 4 15.e even 4 1
1950.2.f.m 4 195.s even 4 1
1950.2.f.n 4 15.e even 4 1
1950.2.f.n 4 195.s even 4 1
3120.2.g.q 4 12.b even 2 1
3120.2.g.q 4 156.h even 2 1
5070.2.a.z 2 39.f even 4 1
5070.2.a.bf 2 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1170, [\chi])\):

\( T_{7}^{4} + 28T_{7}^{2} + 144 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{17}^{2} + 2T_{17} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 28T^{2} + 144 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 13)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 2 T - 12)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 28T^{2} + 144 \) Copy content Toggle raw display
$23$ \( (T^{2} - 10 T + 12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2 T - 12)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 112T^{2} + 2304 \) Copy content Toggle raw display
$41$ \( T^{4} + 136T^{2} + 1296 \) Copy content Toggle raw display
$43$ \( (T - 8)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 112T^{2} + 2304 \) Copy content Toggle raw display
$53$ \( (T + 6)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 112T^{2} + 2304 \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T - 36)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 136T^{2} + 1296 \) Copy content Toggle raw display
$71$ \( T^{4} + 112T^{2} + 2304 \) Copy content Toggle raw display
$73$ \( T^{4} + 76T^{2} + 144 \) Copy content Toggle raw display
$79$ \( (T^{2} - 208)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 304T^{2} + 2304 \) Copy content Toggle raw display
$89$ \( T^{4} + 232T^{2} + 144 \) Copy content Toggle raw display
$97$ \( T^{4} + 76T^{2} + 144 \) Copy content Toggle raw display
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