Properties

Label 2015.1.bq.e
Level $2015$
Weight $1$
Character orbit 2015.bq
Analytic conductor $1.006$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2015,1,Mod(464,2015)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2015, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2015.464");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2015 = 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2015.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: \(1.00561600046\)
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: yes
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.4060225.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{12} q^{2} - \zeta_{12}^{4} q^{3} - \zeta_{12}^{3} q^{5} + \zeta_{12}^{5} q^{6} - \zeta_{12}^{5} q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12} q^{2} - \zeta_{12}^{4} q^{3} - \zeta_{12}^{3} q^{5} + \zeta_{12}^{5} q^{6} - \zeta_{12}^{5} q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{4} q^{10} - \zeta_{12} q^{11} + q^{13} - q^{14} - \zeta_{12} q^{15} - \zeta_{12}^{4} q^{16} - \zeta_{12}^{2} q^{17} + \zeta_{12}^{2} q^{19} - \zeta_{12}^{3} q^{21} + \zeta_{12}^{2} q^{22} - \zeta_{12}^{4} q^{23} + \zeta_{12} q^{24} - q^{25} - \zeta_{12} q^{26} + q^{27} - \zeta_{12} q^{29} + \zeta_{12}^{2} q^{30} - q^{31} + \zeta_{12}^{5} q^{33} + \zeta_{12}^{3} q^{34} - \zeta_{12}^{2} q^{35} + \zeta_{12}^{4} q^{37} - \zeta_{12}^{3} q^{38} - \zeta_{12}^{4} q^{39} + q^{40} - \zeta_{12}^{4} q^{41} + \zeta_{12}^{4} q^{42} - \zeta_{12}^{2} q^{43} + \zeta_{12}^{5} q^{46} - \zeta_{12}^{2} q^{48} + \zeta_{12} q^{50} - q^{51} - \zeta_{12} q^{54} + \zeta_{12}^{4} q^{55} + \zeta_{12}^{2} q^{56} + q^{57} + \zeta_{12}^{2} q^{58} + \zeta_{12}^{2} q^{59} - \zeta_{12}^{5} q^{61} + \zeta_{12} q^{62} - q^{64} - \zeta_{12}^{3} q^{65} + q^{66} + \zeta_{12} q^{67} - \zeta_{12}^{2} q^{69} + \zeta_{12}^{3} q^{70} + \zeta_{12}^{2} q^{71} - \zeta_{12}^{5} q^{74} + \zeta_{12}^{4} q^{75} - q^{77} + \zeta_{12}^{5} q^{78} + \zeta_{12}^{3} q^{79} - \zeta_{12} q^{80} - \zeta_{12}^{4} q^{81} + \zeta_{12}^{5} q^{82} - q^{83} + \zeta_{12}^{5} q^{85} + \zeta_{12}^{3} q^{86} + \zeta_{12}^{5} q^{87} - \zeta_{12}^{4} q^{88} + \zeta_{12} q^{89} - \zeta_{12}^{5} q^{91} + \zeta_{12}^{4} q^{93} - \zeta_{12}^{5} q^{95} + \zeta_{12}^{5} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 2 q^{10} + 4 q^{13} - 4 q^{14} + 2 q^{16} - 2 q^{17} + 2 q^{19} + 2 q^{22} + 2 q^{23} - 4 q^{25} + 4 q^{27} + 2 q^{30} - 4 q^{31} - 2 q^{35} - 2 q^{37} + 2 q^{39} + 4 q^{40} + 2 q^{41} - 2 q^{42} - 2 q^{43} - 2 q^{48} - 4 q^{51} - 2 q^{55} + 2 q^{56} + 4 q^{57} + 2 q^{58} + 2 q^{59} - 4 q^{64} + 4 q^{66} - 2 q^{69} + 2 q^{71} - 2 q^{75} - 4 q^{77} + 2 q^{81} - 8 q^{83} + 2 q^{88} - 2 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2015\mathbb{Z}\right)^\times\).

\(n\) \(716\) \(807\) \(1861\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
464.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i 0.500000 0.866025i 0 1.00000i −0.866025 + 0.500000i 0.866025 0.500000i 1.00000i 0 −0.500000 + 0.866025i
464.2 0.866025 + 0.500000i 0.500000 0.866025i 0 1.00000i 0.866025 0.500000i −0.866025 + 0.500000i 1.00000i 0 −0.500000 + 0.866025i
1394.1 −0.866025 + 0.500000i 0.500000 + 0.866025i 0 1.00000i −0.866025 0.500000i 0.866025 + 0.500000i 1.00000i 0 −0.500000 0.866025i
1394.2 0.866025 0.500000i 0.500000 + 0.866025i 0 1.00000i 0.866025 + 0.500000i −0.866025 0.500000i 1.00000i 0 −0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
155.c odd 2 1 inner
2015.bq odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2015.1.bq.e yes 4
5.b even 2 1 2015.1.bq.c 4
13.c even 3 1 inner 2015.1.bq.e yes 4
31.b odd 2 1 2015.1.bq.c 4
65.n even 6 1 2015.1.bq.c 4
155.c odd 2 1 inner 2015.1.bq.e yes 4
403.p odd 6 1 2015.1.bq.c 4
2015.bq odd 6 1 inner 2015.1.bq.e yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2015.1.bq.c 4 5.b even 2 1
2015.1.bq.c 4 31.b odd 2 1
2015.1.bq.c 4 65.n even 6 1
2015.1.bq.c 4 403.p odd 6 1
2015.1.bq.e yes 4 1.a even 1 1 trivial
2015.1.bq.e yes 4 13.c even 3 1 inner
2015.1.bq.e yes 4 155.c odd 2 1 inner
2015.1.bq.e yes 4 2015.bq odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2015, [\chi])\):

\( T_{2}^{4} - T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{2} - T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$13$ \( (T - 1)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$31$ \( (T + 1)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$67$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$71$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$83$ \( (T + 2)^{4} \) Copy content Toggle raw display
$89$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$97$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
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