Properties

Label 1014.2.i.d
Level $1014$
Weight $2$
Character orbit 1014.i
Analytic conductor $8.097$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1014,2,Mod(361,1014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1014, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("1014.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.i (of order \(6\), degree \(2\), not 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: \(8.09683076496\)
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 78)
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.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - \zeta_{12}) q^{2} + \zeta_{12}^{2} q^{3} + ( - \zeta_{12}^{2} + 1) q^{4} + 2 \zeta_{12}^{3} q^{5} - \zeta_{12} q^{6} + 4 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + (\zeta_{12}^{2} - 1) q^{9} + \cdots + 4 \zeta_{12}^{3} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 2 q^{4} - 2 q^{9} - 4 q^{10} + 4 q^{12} - 16 q^{14} - 2 q^{16} + 4 q^{17} - 8 q^{22} + 4 q^{25} - 4 q^{27} - 12 q^{29} + 4 q^{30} - 16 q^{35} + 2 q^{36} - 32 q^{38} - 8 q^{40} - 8 q^{42}+ \cdots - 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(1 - \zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 2.00000i −0.866025 0.500000i 3.46410 + 2.00000i 1.00000i −0.500000 + 0.866025i −1.00000 1.73205i
361.2 0.866025 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 2.00000i 0.866025 + 0.500000i −3.46410 2.00000i 1.00000i −0.500000 + 0.866025i −1.00000 1.73205i
823.1 −0.866025 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 2.00000i −0.866025 + 0.500000i 3.46410 2.00000i 1.00000i −0.500000 0.866025i −1.00000 + 1.73205i
823.2 0.866025 + 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 2.00000i 0.866025 0.500000i −3.46410 + 2.00000i 1.00000i −0.500000 0.866025i −1.00000 + 1.73205i
\(n\): 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
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1014.2.i.d 4
13.b even 2 1 inner 1014.2.i.d 4
13.c even 3 1 1014.2.b.b 2
13.c even 3 1 inner 1014.2.i.d 4
13.d odd 4 1 1014.2.e.c 2
13.d odd 4 1 1014.2.e.f 2
13.e even 6 1 1014.2.b.b 2
13.e even 6 1 inner 1014.2.i.d 4
13.f odd 12 1 78.2.a.a 1
13.f odd 12 1 1014.2.a.d 1
13.f odd 12 1 1014.2.e.c 2
13.f odd 12 1 1014.2.e.f 2
39.h odd 6 1 3042.2.b.g 2
39.i odd 6 1 3042.2.b.g 2
39.k even 12 1 234.2.a.c 1
39.k even 12 1 3042.2.a.f 1
52.l even 12 1 624.2.a.h 1
52.l even 12 1 8112.2.a.v 1
65.o even 12 1 1950.2.e.i 2
65.s odd 12 1 1950.2.a.w 1
65.t even 12 1 1950.2.e.i 2
91.bc even 12 1 3822.2.a.j 1
104.u even 12 1 2496.2.a.b 1
104.x odd 12 1 2496.2.a.t 1
117.w odd 12 1 2106.2.e.q 2
117.x even 12 1 2106.2.e.j 2
117.bb odd 12 1 2106.2.e.q 2
117.bc even 12 1 2106.2.e.j 2
143.o even 12 1 9438.2.a.t 1
156.v odd 12 1 1872.2.a.c 1
195.bc odd 12 1 5850.2.e.bb 2
195.bh even 12 1 5850.2.a.d 1
195.bn odd 12 1 5850.2.e.bb 2
312.bo even 12 1 7488.2.a.bz 1
312.bq odd 12 1 7488.2.a.bk 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.a.a 1 13.f odd 12 1
234.2.a.c 1 39.k even 12 1
624.2.a.h 1 52.l even 12 1
1014.2.a.d 1 13.f odd 12 1
1014.2.b.b 2 13.c even 3 1
1014.2.b.b 2 13.e even 6 1
1014.2.e.c 2 13.d odd 4 1
1014.2.e.c 2 13.f odd 12 1
1014.2.e.f 2 13.d odd 4 1
1014.2.e.f 2 13.f odd 12 1
1014.2.i.d 4 1.a even 1 1 trivial
1014.2.i.d 4 13.b even 2 1 inner
1014.2.i.d 4 13.c even 3 1 inner
1014.2.i.d 4 13.e even 6 1 inner
1872.2.a.c 1 156.v odd 12 1
1950.2.a.w 1 65.s odd 12 1
1950.2.e.i 2 65.o even 12 1
1950.2.e.i 2 65.t even 12 1
2106.2.e.j 2 117.x even 12 1
2106.2.e.j 2 117.bc even 12 1
2106.2.e.q 2 117.w odd 12 1
2106.2.e.q 2 117.bb odd 12 1
2496.2.a.b 1 104.u even 12 1
2496.2.a.t 1 104.x odd 12 1
3042.2.a.f 1 39.k even 12 1
3042.2.b.g 2 39.h odd 6 1
3042.2.b.g 2 39.i odd 6 1
3822.2.a.j 1 91.bc even 12 1
5850.2.a.d 1 195.bh even 12 1
5850.2.e.bb 2 195.bc odd 12 1
5850.2.e.bb 2 195.bn odd 12 1
7488.2.a.bk 1 312.bq odd 12 1
7488.2.a.bz 1 312.bo even 12 1
8112.2.a.v 1 52.l even 12 1
9438.2.a.t 1 143.o even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1014, [\chi])\):

\( T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{4} - 16T_{7}^{2} + 256 \) 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
$5$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$11$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$41$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
$43$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$53$ \( (T + 10)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$61$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 256 T^{2} + 65536 \) Copy content Toggle raw display
$71$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$73$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$79$ \( (T - 8)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 196 T^{2} + 38416 \) Copy content Toggle raw display
$97$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
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