Properties

Label 276.1.h.d
Level $276$
Weight $1$
Character orbit 276.h
Analytic conductor $0.138$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -23
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [276,1,Mod(275,276)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(276, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("276.275");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 276 = 2^{2} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 276.h (of order \(2\), degree \(1\), 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: \(0.137741943487\)
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: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.914112.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{6}^{2} q^{2} - \zeta_{6} q^{3} - \zeta_{6} q^{4} - q^{6} - q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6}^{2} q^{2} - \zeta_{6} q^{3} - \zeta_{6} q^{4} - q^{6} - q^{8} + \zeta_{6}^{2} q^{9} + \zeta_{6}^{2} q^{12} + q^{13} + \zeta_{6}^{2} q^{16} + \zeta_{6} q^{18} + q^{23} + \zeta_{6} q^{24} - q^{25} - \zeta_{6}^{2} q^{26} + q^{27} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{29} + (\zeta_{6}^{2} + \zeta_{6}) q^{31} + \zeta_{6} q^{32} + q^{36} - \zeta_{6} q^{39} + (\zeta_{6}^{2} + \zeta_{6}) q^{41} - \zeta_{6}^{2} q^{46} + q^{47} + q^{48} - q^{49} + \zeta_{6}^{2} q^{50} - \zeta_{6} q^{52} - \zeta_{6}^{2} q^{54} + ( - \zeta_{6} - 1) q^{58} - q^{59} + (\zeta_{6} + 1) q^{62} + q^{64} - \zeta_{6} q^{69} - q^{71} - \zeta_{6}^{2} q^{72} - q^{73} + \zeta_{6} q^{75} - q^{78} - \zeta_{6} q^{81} + (\zeta_{6} + 1) q^{82} + (\zeta_{6}^{2} - 1) q^{87} - \zeta_{6} q^{92} + ( - \zeta_{6}^{2} + 1) q^{93} - \zeta_{6}^{2} q^{94} - \zeta_{6}^{2} q^{96} + \zeta_{6}^{2} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} - q^{4} - 2 q^{6} - 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{3} - q^{4} - 2 q^{6} - 2 q^{8} - q^{9} - q^{12} + 2 q^{13} - q^{16} + q^{18} + 2 q^{23} + q^{24} - 2 q^{25} + q^{26} + 2 q^{27} + q^{32} + 2 q^{36} - q^{39} + q^{46} + 2 q^{47} + 2 q^{48} - 2 q^{49} - q^{50} - q^{52} + q^{54} - 3 q^{58} - 4 q^{59} + 3 q^{62} + 2 q^{64} - q^{69} - 2 q^{71} + q^{72} - 2 q^{73} + q^{75} - 2 q^{78} - q^{81} + 3 q^{82} - 3 q^{87} - q^{92} + 3 q^{93} + q^{94} + q^{96} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/276\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(139\) \(185\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
275.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0 −1.00000 0 −1.00000 −0.500000 + 0.866025i 0
275.2 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0 −1.00000 0 −1.00000 −0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 CM by \(\Q(\sqrt{-23}) \)
12.b even 2 1 inner
276.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 276.1.h.d yes 2
3.b odd 2 1 276.1.h.c 2
4.b odd 2 1 276.1.h.c 2
12.b even 2 1 inner 276.1.h.d yes 2
23.b odd 2 1 CM 276.1.h.d yes 2
69.c even 2 1 276.1.h.c 2
92.b even 2 1 276.1.h.c 2
276.h odd 2 1 inner 276.1.h.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.1.h.c 2 3.b odd 2 1
276.1.h.c 2 4.b odd 2 1
276.1.h.c 2 69.c even 2 1
276.1.h.c 2 92.b even 2 1
276.1.h.d yes 2 1.a even 1 1 trivial
276.1.h.d yes 2 12.b even 2 1 inner
276.1.h.d yes 2 23.b odd 2 1 CM
276.1.h.d yes 2 276.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(276, [\chi])\):

\( T_{13} - 1 \) Copy content Toggle raw display
\( T_{47} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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