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Properties

Label 1014.2.e.j
Level $1014$
Weight $2$
Character orbit 1014.e
Analytic conductor $8.097$
Analytic rank $0$
Dimension $4$
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] = [1014,2,Mod(529,1014)]
 
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, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1014.529");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.e (of order \(3\), 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 \(\Q(\zeta_{3})\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_1 q^{3} + (\beta_1 - 1) q^{4} - \beta_{3} q^{5} + (\beta_1 - 1) q^{6} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{7} - q^{8} + (\beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_1 q^{3} + (\beta_1 - 1) q^{4} - \beta_{3} q^{5} + (\beta_1 - 1) q^{6} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{7} - q^{8} + (\beta_1 - 1) q^{9} - \beta_{2} q^{10} + ( - \beta_{2} - 3 \beta_1) q^{11} - q^{12} + ( - \beta_{3} - 3) q^{14} - \beta_{2} q^{15} - \beta_1 q^{16} + (3 \beta_{3} - 3 \beta_{2}) q^{17} - q^{18} + (\beta_{3} - \beta_{2} + 3 \beta_1 - 3) q^{19} + (\beta_{3} - \beta_{2}) q^{20} + ( - \beta_{3} - 3) q^{21} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{22} + ( - 3 \beta_{2} + 3 \beta_1) q^{23} - \beta_1 q^{24} - 2 q^{25} - q^{27} + ( - \beta_{2} - 3 \beta_1) q^{28} + 3 \beta_1 q^{29} + (\beta_{3} - \beta_{2}) q^{30} + ( - 2 \beta_{3} + 6) q^{31} + ( - \beta_1 + 1) q^{32} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{33} + 3 \beta_{3} q^{34} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 3) q^{35} - \beta_1 q^{36} - 3 \beta_1 q^{37} + (\beta_{3} - 3) q^{38} + \beta_{3} q^{40} + (2 \beta_{2} - 3 \beta_1) q^{41} + ( - \beta_{2} - 3 \beta_1) q^{42} + ( - 3 \beta_{3} + 3 \beta_{2} + \cdots - 1) q^{43}+ \cdots + (\beta_{3} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{6} - 6 q^{7} - 4 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{6} - 6 q^{7} - 4 q^{8} - 2 q^{9} - 6 q^{11} - 4 q^{12} - 12 q^{14} - 2 q^{16} - 4 q^{18} - 6 q^{19} - 12 q^{21} + 6 q^{22} + 6 q^{23} - 2 q^{24} - 8 q^{25} - 4 q^{27} - 6 q^{28} + 6 q^{29} + 24 q^{31} + 2 q^{32} + 6 q^{33} + 6 q^{35} - 2 q^{36} - 6 q^{37} - 12 q^{38} - 6 q^{41} - 6 q^{42} - 2 q^{43} + 12 q^{44} - 6 q^{46} - 12 q^{47} + 2 q^{48} - 10 q^{49} - 4 q^{50} + 12 q^{53} - 2 q^{54} + 6 q^{55} + 6 q^{56} - 12 q^{57} - 6 q^{58} - 20 q^{61} + 12 q^{62} - 6 q^{63} + 4 q^{64} + 12 q^{66} - 18 q^{67} - 6 q^{69} + 12 q^{70} - 6 q^{71} + 2 q^{72} + 6 q^{74} - 4 q^{75} - 6 q^{76} + 48 q^{77} - 8 q^{79} - 2 q^{81} + 6 q^{82} - 12 q^{83} + 6 q^{84} - 18 q^{85} - 4 q^{86} - 6 q^{87} + 6 q^{88} - 12 q^{89} - 12 q^{92} + 12 q^{93} - 6 q^{94} - 6 q^{95} + 4 q^{96} - 12 q^{97} + 10 q^{98} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

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

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1
0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i −1.73205 −0.500000 − 0.866025i −2.36603 − 4.09808i −1.00000 −0.500000 − 0.866025i −0.866025 + 1.50000i
529.2 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i 1.73205 −0.500000 − 0.866025i −0.633975 − 1.09808i −1.00000 −0.500000 − 0.866025i 0.866025 − 1.50000i
991.1 0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i −1.73205 −0.500000 + 0.866025i −2.36603 + 4.09808i −1.00000 −0.500000 + 0.866025i −0.866025 − 1.50000i
991.2 0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.73205 −0.500000 + 0.866025i −0.633975 + 1.09808i −1.00000 −0.500000 + 0.866025i 0.866025 + 1.50000i
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1014.2.e.j 4
13.b even 2 1 1014.2.e.h 4
13.c even 3 1 1014.2.a.h 2
13.c even 3 1 inner 1014.2.e.j 4
13.d odd 4 1 78.2.i.b ✓ 4
13.d odd 4 1 1014.2.i.f 4
13.e even 6 1 1014.2.a.j 2
13.e even 6 1 1014.2.e.h 4
13.f odd 12 1 78.2.i.b ✓ 4
13.f odd 12 2 1014.2.b.d 4
13.f odd 12 1 1014.2.i.f 4
39.f even 4 1 234.2.l.a 4
39.h odd 6 1 3042.2.a.s 2
39.i odd 6 1 3042.2.a.v 2
39.k even 12 1 234.2.l.a 4
39.k even 12 2 3042.2.b.l 4
52.f even 4 1 624.2.bv.d 4
52.i odd 6 1 8112.2.a.bx 2
52.j odd 6 1 8112.2.a.bq 2
52.l even 12 1 624.2.bv.d 4
65.f even 4 1 1950.2.y.a 4
65.g odd 4 1 1950.2.bc.c 4
65.k even 4 1 1950.2.y.h 4
65.o even 12 1 1950.2.y.h 4
65.s odd 12 1 1950.2.bc.c 4
65.t even 12 1 1950.2.y.a 4
156.l odd 4 1 1872.2.by.k 4
156.v odd 12 1 1872.2.by.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.i.b ✓ 4 13.d odd 4 1
78.2.i.b ✓ 4 13.f odd 12 1
234.2.l.a 4 39.f even 4 1
234.2.l.a 4 39.k even 12 1
624.2.bv.d 4 52.f even 4 1
624.2.bv.d 4 52.l even 12 1
1014.2.a.h 2 13.c even 3 1
1014.2.a.j 2 13.e even 6 1
1014.2.b.d 4 13.f odd 12 2
1014.2.e.h 4 13.b even 2 1
1014.2.e.h 4 13.e even 6 1
1014.2.e.j 4 1.a even 1 1 trivial
1014.2.e.j 4 13.c even 3 1 inner
1014.2.i.f 4 13.d odd 4 1
1014.2.i.f 4 13.f odd 12 1
1872.2.by.k 4 156.l odd 4 1
1872.2.by.k 4 156.v odd 12 1
1950.2.y.a 4 65.f even 4 1
1950.2.y.a 4 65.t even 12 1
1950.2.y.h 4 65.k even 4 1
1950.2.y.h 4 65.o even 12 1
1950.2.bc.c 4 65.g odd 4 1
1950.2.bc.c 4 65.s odd 12 1
3042.2.a.s 2 39.h odd 6 1
3042.2.a.v 2 39.i odd 6 1
3042.2.b.l 4 39.k even 12 2
8112.2.a.bq 2 52.j odd 6 1
8112.2.a.bx 2 52.i odd 6 1

Hecke kernels

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

\( T_{5}^{2} - 3 \) Copy content Toggle raw display
\( T_{7}^{4} + 6T_{7}^{3} + 30T_{7}^{2} + 36T_{7} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 6 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$23$ \( T^{4} - 6 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$29$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 12 T + 24)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$43$ \( T^{4} + 2 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$47$ \( (T^{2} + 6 T + 6)^{2} \) Copy content Toggle raw display
$53$ \( (T - 3)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 192 T^{2} + 36864 \) Copy content Toggle raw display
$61$ \( T^{4} + 20 T^{3} + \cdots + 5329 \) Copy content Toggle raw display
$67$ \( T^{4} + 18 T^{3} + \cdots + 6084 \) Copy content Toggle raw display
$71$ \( T^{4} + 6 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$73$ \( (T^{2} - 147)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T - 104)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 6 T - 66)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 12 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$97$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
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