Properties

Label 1520.2.bq.h
Level $1520$
Weight $2$
Character orbit 1520.bq
Analytic conductor $12.137$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1520,2,Mod(31,1520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1520, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1520.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.bq (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: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 2) q^{3} + (\zeta_{6} - 1) q^{5} + ( - 2 \zeta_{6} + 1) q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 2) q^{3} + (\zeta_{6} - 1) q^{5} + ( - 2 \zeta_{6} + 1) q^{7} - \zeta_{6} q^{9} + (6 \zeta_{6} - 3) q^{11} + 2 \zeta_{6} q^{15} + ( - 3 \zeta_{6} + 5) q^{19} + ( - 2 \zeta_{6} - 2) q^{21} + ( - 5 \zeta_{6} + 10) q^{23} - \zeta_{6} q^{25} + 4 q^{27} + ( - 2 \zeta_{6} + 4) q^{29} + 4 q^{31} + (6 \zeta_{6} + 6) q^{33} + (\zeta_{6} + 1) q^{35} + ( - 2 \zeta_{6} + 1) q^{37} + ( - 5 \zeta_{6} - 5) q^{41} + q^{45} + ( - 2 \zeta_{6} + 4) q^{47} + 4 q^{49} + ( - 3 \zeta_{6} + 6) q^{53} + ( - 3 \zeta_{6} - 3) q^{55} + ( - 10 \zeta_{6} + 4) q^{57} + (12 \zeta_{6} - 12) q^{59} - 10 \zeta_{6} q^{61} + (\zeta_{6} - 2) q^{63} - 4 \zeta_{6} q^{67} + ( - 20 \zeta_{6} + 10) q^{69} + (6 \zeta_{6} - 6) q^{71} + ( - 2 \zeta_{6} + 2) q^{73} - 2 q^{75} + 9 q^{77} + ( - 4 \zeta_{6} + 4) q^{79} + ( - 11 \zeta_{6} + 11) q^{81} + (12 \zeta_{6} - 6) q^{83} + ( - 8 \zeta_{6} + 4) q^{87} + (9 \zeta_{6} - 18) q^{89} + ( - 8 \zeta_{6} + 8) q^{93} + (5 \zeta_{6} - 2) q^{95} + ( - 3 \zeta_{6} + 6) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - q^{5} - q^{9} + 2 q^{15} + 7 q^{19} - 6 q^{21} + 15 q^{23} - q^{25} + 8 q^{27} + 6 q^{29} + 8 q^{31} + 18 q^{33} + 3 q^{35} - 15 q^{41} + 2 q^{45} + 6 q^{47} + 8 q^{49} + 9 q^{53} - 9 q^{55} - 2 q^{57} - 12 q^{59} - 10 q^{61} - 3 q^{63} - 4 q^{67} - 6 q^{71} + 2 q^{73} - 4 q^{75} + 18 q^{77} + 4 q^{79} + 11 q^{81} - 27 q^{89} + 8 q^{93} + q^{95} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(-1\) \(\zeta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.500000 0.866025i
0.500000 + 0.866025i
0 1.00000 + 1.73205i 0 −0.500000 0.866025i 0 1.73205i 0 −0.500000 + 0.866025i 0
1471.1 0 1.00000 1.73205i 0 −0.500000 + 0.866025i 0 1.73205i 0 −0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1520.2.bq.h yes 2
4.b odd 2 1 1520.2.bq.b 2
19.d odd 6 1 1520.2.bq.b 2
76.f even 6 1 inner 1520.2.bq.h yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1520.2.bq.b 2 4.b odd 2 1
1520.2.bq.b 2 19.d odd 6 1
1520.2.bq.h yes 2 1.a even 1 1 trivial
1520.2.bq.h yes 2 76.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1520, [\chi])\):

\( T_{3}^{2} - 2T_{3} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 3 \) Copy content Toggle raw display
\( T_{11}^{2} + 27 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 3 \) Copy content Toggle raw display
$11$ \( T^{2} + 27 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 7T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} - 15T + 75 \) Copy content Toggle raw display
$29$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 3 \) Copy content Toggle raw display
$41$ \( T^{2} + 15T + 75 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$53$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$73$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$83$ \( T^{2} + 108 \) Copy content Toggle raw display
$89$ \( T^{2} + 27T + 243 \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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