Properties

Label 1014.2.a.a
Level $1014$
Weight $2$
Character orbit 1014.a
Self dual yes
Analytic conductor $8.097$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1014,2,Mod(1,1014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1014, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1014.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.a (trivial)

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: yes
Analytic conductor: \(8.09683076496\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - 2 q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - 2 q^{7} - q^{8} + q^{9} + q^{10} + 2 q^{11} - q^{12} + 2 q^{14} + q^{15} + q^{16} + 5 q^{17} - q^{18} - 2 q^{19} - q^{20} + 2 q^{21} - 2 q^{22} + 6 q^{23} + q^{24} - 4 q^{25} - q^{27} - 2 q^{28} - 9 q^{29} - q^{30} - 4 q^{31} - q^{32} - 2 q^{33} - 5 q^{34} + 2 q^{35} + q^{36} - 11 q^{37} + 2 q^{38} + q^{40} + 5 q^{41} - 2 q^{42} + 10 q^{43} + 2 q^{44} - q^{45} - 6 q^{46} + 2 q^{47} - q^{48} - 3 q^{49} + 4 q^{50} - 5 q^{51} - q^{53} + q^{54} - 2 q^{55} + 2 q^{56} + 2 q^{57} + 9 q^{58} - 8 q^{59} + q^{60} - 11 q^{61} + 4 q^{62} - 2 q^{63} + q^{64} + 2 q^{66} + 2 q^{67} + 5 q^{68} - 6 q^{69} - 2 q^{70} - 14 q^{71} - q^{72} - 13 q^{73} + 11 q^{74} + 4 q^{75} - 2 q^{76} - 4 q^{77} - 4 q^{79} - q^{80} + q^{81} - 5 q^{82} + 6 q^{83} + 2 q^{84} - 5 q^{85} - 10 q^{86} + 9 q^{87} - 2 q^{88} + 2 q^{89} + q^{90} + 6 q^{92} + 4 q^{93} - 2 q^{94} + 2 q^{95} + q^{96} - 2 q^{97} + 3 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−1.00000 −1.00000 1.00000 −1.00000 1.00000 −2.00000 −1.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( +1 \)
\(13\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1014.2.a.a 1
3.b odd 2 1 3042.2.a.m 1
4.b odd 2 1 8112.2.a.x 1
13.b even 2 1 1014.2.a.e 1
13.c even 3 2 78.2.e.b 2
13.d odd 4 2 1014.2.b.a 2
13.e even 6 2 1014.2.e.d 2
13.f odd 12 4 1014.2.i.e 4
39.d odd 2 1 3042.2.a.d 1
39.f even 4 2 3042.2.b.d 2
39.i odd 6 2 234.2.h.b 2
52.b odd 2 1 8112.2.a.bb 1
52.j odd 6 2 624.2.q.b 2
65.n even 6 2 1950.2.i.b 2
65.q odd 12 4 1950.2.z.b 4
156.p even 6 2 1872.2.t.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.b 2 13.c even 3 2
234.2.h.b 2 39.i odd 6 2
624.2.q.b 2 52.j odd 6 2
1014.2.a.a 1 1.a even 1 1 trivial
1014.2.a.e 1 13.b even 2 1
1014.2.b.a 2 13.d odd 4 2
1014.2.e.d 2 13.e even 6 2
1014.2.i.e 4 13.f odd 12 4
1872.2.t.i 2 156.p even 6 2
1950.2.i.b 2 65.n even 6 2
1950.2.z.b 4 65.q odd 12 4
3042.2.a.d 1 39.d odd 2 1
3042.2.a.m 1 3.b odd 2 1
3042.2.b.d 2 39.f even 4 2
8112.2.a.x 1 4.b odd 2 1
8112.2.a.bb 1 52.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1014))\):

\( T_{5} + 1 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T - 2 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 5 \) Copy content Toggle raw display
$19$ \( T + 2 \) Copy content Toggle raw display
$23$ \( T - 6 \) Copy content Toggle raw display
$29$ \( T + 9 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T + 11 \) Copy content Toggle raw display
$41$ \( T - 5 \) Copy content Toggle raw display
$43$ \( T - 10 \) Copy content Toggle raw display
$47$ \( T - 2 \) Copy content Toggle raw display
$53$ \( T + 1 \) Copy content Toggle raw display
$59$ \( T + 8 \) Copy content Toggle raw display
$61$ \( T + 11 \) Copy content Toggle raw display
$67$ \( T - 2 \) Copy content Toggle raw display
$71$ \( T + 14 \) Copy content Toggle raw display
$73$ \( T + 13 \) Copy content Toggle raw display
$79$ \( T + 4 \) Copy content Toggle raw display
$83$ \( T - 6 \) Copy content Toggle raw display
$89$ \( T - 2 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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