Properties

Label 2511.1.m.a
Level $2511$
Weight $1$
Character orbit 2511.m
Analytic conductor $1.253$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -31
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2511,1,Mod(433,2511)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2511, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2511.433");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2511 = 3^{4} \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2511.m (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: \(1.25315224672\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 31)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.31.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{2} - \zeta_{6} q^{5} - \zeta_{6}^{2} q^{7} - q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6}^{2} q^{2} - \zeta_{6} q^{5} - \zeta_{6}^{2} q^{7} - q^{8} + q^{10} + \zeta_{6} q^{14} - \zeta_{6}^{2} q^{16} - q^{19} - \zeta_{6} q^{31} - q^{35} - \zeta_{6}^{2} q^{38} + \zeta_{6} q^{40} - \zeta_{6} q^{41} - 2 \zeta_{6}^{2} q^{47} + \zeta_{6}^{2} q^{56} - \zeta_{6} q^{59} + q^{62} + q^{64} - 2 \zeta_{6} q^{67} - \zeta_{6}^{2} q^{70} + q^{71} - q^{80} + q^{82} + 2 \zeta_{6} q^{94} + \zeta_{6} q^{95} - \zeta_{6}^{2} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{5} + q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{5} + q^{7} - 2 q^{8} + 2 q^{10} + q^{14} + q^{16} - 2 q^{19} - q^{31} - 2 q^{35} + q^{38} + q^{40} - q^{41} + 2 q^{47} - q^{56} - q^{59} + 2 q^{62} + 2 q^{64} - 2 q^{67} + q^{70} + 2 q^{71} - 2 q^{80} + 2 q^{82} + 2 q^{94} + q^{95} + q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2511\mathbb{Z}\right)^\times\).

\(n\) \(406\) \(1055\)
\(\chi(n)\) \(-1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 0.866025i 0 0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i −1.00000 0 1.00000
1270.1 −0.500000 + 0.866025i 0 0 −0.500000 0.866025i 0 0.500000 0.866025i −1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 CM by \(\Q(\sqrt{-31}) \)
9.c even 3 1 inner
279.m odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2511.1.m.a 2
3.b odd 2 1 2511.1.m.e 2
9.c even 3 1 279.1.d.b 1
9.c even 3 1 inner 2511.1.m.a 2
9.d odd 6 1 31.1.b.a 1
9.d odd 6 1 2511.1.m.e 2
31.b odd 2 1 CM 2511.1.m.a 2
36.h even 6 1 496.1.e.a 1
45.h odd 6 1 775.1.d.b 1
45.l even 12 2 775.1.c.a 2
63.i even 6 1 1519.1.n.a 2
63.j odd 6 1 1519.1.n.b 2
63.n odd 6 1 1519.1.n.b 2
63.o even 6 1 1519.1.c.a 1
63.s even 6 1 1519.1.n.a 2
72.j odd 6 1 1984.1.e.a 1
72.l even 6 1 1984.1.e.b 1
93.c even 2 1 2511.1.m.e 2
99.g even 6 1 3751.1.d.b 1
99.n odd 30 4 3751.1.t.c 4
99.p even 30 4 3751.1.t.a 4
279.m odd 6 1 279.1.d.b 1
279.m odd 6 1 inner 2511.1.m.a 2
279.o even 6 1 961.1.e.a 2
279.p odd 6 1 961.1.e.a 2
279.q odd 6 1 961.1.e.a 2
279.r even 6 1 961.1.e.a 2
279.s even 6 1 31.1.b.a 1
279.s even 6 1 2511.1.m.e 2
279.bd odd 30 4 961.1.h.a 8
279.be even 30 4 961.1.h.a 8
279.bf odd 30 4 961.1.f.a 4
279.bg even 30 4 961.1.f.a 4
279.bh even 30 4 961.1.h.a 8
279.bi odd 30 4 961.1.h.a 8
1116.bk odd 6 1 496.1.e.a 1
1395.ba even 6 1 775.1.d.b 1
1395.cg odd 12 2 775.1.c.a 2
1953.z odd 6 1 1519.1.n.a 2
1953.bf even 6 1 1519.1.n.b 2
1953.cs even 6 1 1519.1.n.b 2
1953.cu odd 6 1 1519.1.c.a 1
1953.dw odd 6 1 1519.1.n.a 2
2232.bi even 6 1 1984.1.e.a 1
2232.bp odd 6 1 1984.1.e.b 1
3069.ba odd 6 1 3751.1.d.b 1
3069.gm even 30 4 3751.1.t.c 4
3069.ib odd 30 4 3751.1.t.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.1.b.a 1 9.d odd 6 1
31.1.b.a 1 279.s even 6 1
279.1.d.b 1 9.c even 3 1
279.1.d.b 1 279.m odd 6 1
496.1.e.a 1 36.h even 6 1
496.1.e.a 1 1116.bk odd 6 1
775.1.c.a 2 45.l even 12 2
775.1.c.a 2 1395.cg odd 12 2
775.1.d.b 1 45.h odd 6 1
775.1.d.b 1 1395.ba even 6 1
961.1.e.a 2 279.o even 6 1
961.1.e.a 2 279.p odd 6 1
961.1.e.a 2 279.q odd 6 1
961.1.e.a 2 279.r even 6 1
961.1.f.a 4 279.bf odd 30 4
961.1.f.a 4 279.bg even 30 4
961.1.h.a 8 279.bd odd 30 4
961.1.h.a 8 279.be even 30 4
961.1.h.a 8 279.bh even 30 4
961.1.h.a 8 279.bi odd 30 4
1519.1.c.a 1 63.o even 6 1
1519.1.c.a 1 1953.cu odd 6 1
1519.1.n.a 2 63.i even 6 1
1519.1.n.a 2 63.s even 6 1
1519.1.n.a 2 1953.z odd 6 1
1519.1.n.a 2 1953.dw odd 6 1
1519.1.n.b 2 63.j odd 6 1
1519.1.n.b 2 63.n odd 6 1
1519.1.n.b 2 1953.bf even 6 1
1519.1.n.b 2 1953.cs even 6 1
1984.1.e.a 1 72.j odd 6 1
1984.1.e.a 1 2232.bi even 6 1
1984.1.e.b 1 72.l even 6 1
1984.1.e.b 1 2232.bp odd 6 1
2511.1.m.a 2 1.a even 1 1 trivial
2511.1.m.a 2 9.c even 3 1 inner
2511.1.m.a 2 31.b odd 2 1 CM
2511.1.m.a 2 279.m odd 6 1 inner
2511.1.m.e 2 3.b odd 2 1
2511.1.m.e 2 9.d odd 6 1
2511.1.m.e 2 93.c even 2 1
2511.1.m.e 2 279.s even 6 1
3751.1.d.b 1 99.g even 6 1
3751.1.d.b 1 3069.ba odd 6 1
3751.1.t.a 4 99.p even 30 4
3751.1.t.a 4 3069.ib odd 30 4
3751.1.t.c 4 99.n odd 30 4
3751.1.t.c 4 3069.gm even 30 4

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2511, [\chi])\):

\( T_{2}^{2} + T_{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} - T_{7} + 1 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

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