Properties

Label 1170.2.bp.a
Level $1170$
Weight $2$
Character orbit 1170.bp
Analytic conductor $9.342$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1170,2,Mod(289,1170)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1170, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1170.289");
 
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.bp (of order \(6\), degree \(2\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} + \zeta_{12}) q^{2} + ( - \zeta_{12}^{2} + 1) q^{4} + ( - \zeta_{12}^{3} - 2) q^{5} + \zeta_{12} q^{7} - \zeta_{12}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} + \zeta_{12}) q^{2} + ( - \zeta_{12}^{2} + 1) q^{4} + ( - \zeta_{12}^{3} - 2) q^{5} + \zeta_{12} q^{7} - \zeta_{12}^{3} q^{8} + (2 \zeta_{12}^{3} - \zeta_{12}^{2} - 2 \zeta_{12}) q^{10} - \zeta_{12}^{2} q^{11} + ( - 4 \zeta_{12}^{3} + \zeta_{12}) q^{13} + q^{14} - \zeta_{12}^{2} q^{16} - 4 \zeta_{12} q^{17} + (\zeta_{12}^{2} - 1) q^{19} + (2 \zeta_{12}^{2} - \zeta_{12} - 2) q^{20} - \zeta_{12} q^{22} + (4 \zeta_{12}^{3} + 3) q^{25} + ( - 4 \zeta_{12}^{2} + 1) q^{26} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{28} - 10 \zeta_{12}^{2} q^{29} - 4 q^{31} - \zeta_{12} q^{32} - 4 q^{34} + ( - \zeta_{12}^{2} - 2 \zeta_{12} + 1) q^{35} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{37} + \zeta_{12}^{3} q^{38} + (2 \zeta_{12}^{3} - 1) q^{40} - 6 \zeta_{12}^{2} q^{41} + 4 \zeta_{12} q^{43} - q^{44} - 7 \zeta_{12}^{3} q^{47} - 6 \zeta_{12}^{2} q^{49} + ( - 3 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 3 \zeta_{12}) q^{50} + ( - \zeta_{12}^{3} - 3 \zeta_{12}) q^{52} + 11 \zeta_{12}^{3} q^{53} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - \zeta_{12}) q^{55} + ( - \zeta_{12}^{2} + 1) q^{56} - 10 \zeta_{12} q^{58} + ( - 8 \zeta_{12}^{2} + 8) q^{59} + ( - 8 \zeta_{12}^{2} + 8) q^{61} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{62} - q^{64} + (8 \zeta_{12}^{3} - \zeta_{12}^{2} - 2 \zeta_{12} - 3) q^{65} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{67} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{68} + ( - \zeta_{12}^{3} - 2) q^{70} + (4 \zeta_{12}^{2} - 4) q^{71} - 8 \zeta_{12}^{3} q^{73} + (3 \zeta_{12}^{2} - 3) q^{74} + \zeta_{12}^{2} q^{76} - \zeta_{12}^{3} q^{77} - 10 q^{79} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - \zeta_{12}) q^{80} - 6 \zeta_{12} q^{82} + 18 \zeta_{12}^{3} q^{83} + (4 \zeta_{12}^{2} + 8 \zeta_{12} - 4) q^{85} + 4 q^{86} + (\zeta_{12}^{3} - \zeta_{12}) q^{88} - 3 \zeta_{12}^{2} q^{89} + ( - 3 \zeta_{12}^{2} + 4) q^{91} - 7 \zeta_{12}^{2} q^{94} + ( - 2 \zeta_{12}^{2} + \zeta_{12} + 2) q^{95} + 6 \zeta_{12} q^{97} - 6 \zeta_{12} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 8 q^{5} - 2 q^{10} - 2 q^{11} + 4 q^{14} - 2 q^{16} - 2 q^{19} - 4 q^{20} + 12 q^{25} - 4 q^{26} - 20 q^{29} - 16 q^{31} - 16 q^{34} + 2 q^{35} - 4 q^{40} - 12 q^{41} - 4 q^{44} - 12 q^{49} + 8 q^{50} + 4 q^{55} + 2 q^{56} + 16 q^{59} + 16 q^{61} - 4 q^{64} - 14 q^{65} - 8 q^{70} - 8 q^{71} - 6 q^{74} + 2 q^{76} - 40 q^{79} + 4 q^{80} - 8 q^{85} + 16 q^{86} - 6 q^{89} + 10 q^{91} - 14 q^{94} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(911\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(-1 + \zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i −2.00000 + 1.00000i 0 −0.866025 0.500000i 1.00000i 0 1.23205 1.86603i
289.2 0.866025 0.500000i 0 0.500000 0.866025i −2.00000 1.00000i 0 0.866025 + 0.500000i 1.00000i 0 −2.23205 + 0.133975i
919.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −2.00000 1.00000i 0 −0.866025 + 0.500000i 1.00000i 0 1.23205 + 1.86603i
919.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i −2.00000 + 1.00000i 0 0.866025 0.500000i 1.00000i 0 −2.23205 0.133975i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.2.bp.a 4
3.b odd 2 1 390.2.y.f 4
5.b even 2 1 inner 1170.2.bp.a 4
13.c even 3 1 inner 1170.2.bp.a 4
15.d odd 2 1 390.2.y.f 4
15.e even 4 1 1950.2.i.j 2
15.e even 4 1 1950.2.i.q 2
39.i odd 6 1 390.2.y.f 4
65.n even 6 1 inner 1170.2.bp.a 4
195.x odd 6 1 390.2.y.f 4
195.bl even 12 1 1950.2.i.j 2
195.bl even 12 1 1950.2.i.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.y.f 4 3.b odd 2 1
390.2.y.f 4 15.d odd 2 1
390.2.y.f 4 39.i odd 6 1
390.2.y.f 4 195.x odd 6 1
1170.2.bp.a 4 1.a even 1 1 trivial
1170.2.bp.a 4 5.b even 2 1 inner
1170.2.bp.a 4 13.c even 3 1 inner
1170.2.bp.a 4 65.n even 6 1 inner
1950.2.i.j 2 15.e even 4 1
1950.2.i.j 2 195.bl even 12 1
1950.2.i.q 2 15.e even 4 1
1950.2.i.q 2 195.bl even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - T_{7}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1170, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 4 T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 23T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$19$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$41$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$47$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 121)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$79$ \( (T + 10)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 324)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
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