Properties

Label 2340.1.gu.b
Level $2340$
Weight $1$
Character orbit 2340.gu
Analytic conductor $1.168$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -4
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2340,1,Mod(647,2340)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2340, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 6, 3, 10]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2340.647");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2340.gu (of order \(12\), degree \(4\), 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: \(1.16781212956\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective 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.

\(f(q)\) \(=\) \( q + \zeta_{24}^{7} q^{2} - \zeta_{24}^{2} q^{4} - \zeta_{24}^{11} q^{5} - \zeta_{24}^{9} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{24}^{7} q^{2} - \zeta_{24}^{2} q^{4} - \zeta_{24}^{11} q^{5} - \zeta_{24}^{9} q^{8} + \zeta_{24}^{6} q^{10} + \zeta_{24}^{4} q^{13} + \zeta_{24}^{4} q^{16} - \zeta_{24}^{5} q^{17} - \zeta_{24} q^{20} - \zeta_{24}^{10} q^{25} + \zeta_{24}^{11} q^{26} + ( - \zeta_{24}^{11} + \zeta_{24}^{9}) q^{29} + \zeta_{24}^{11} q^{32} + q^{34} + (\zeta_{24}^{2} + 1) q^{37} - \zeta_{24}^{8} q^{40} + ( - \zeta_{24}^{5} - \zeta_{24}^{3}) q^{41} + \zeta_{24}^{10} q^{49} + \zeta_{24}^{5} q^{50} - \zeta_{24}^{6} q^{52} + \zeta_{24}^{3} q^{53} + (\zeta_{24}^{6} - \zeta_{24}^{4}) q^{58} + ( - \zeta_{24}^{10} - \zeta_{24}^{6}) q^{61} - \zeta_{24}^{6} q^{64} + \zeta_{24}^{3} q^{65} + \zeta_{24}^{7} q^{68} + (\zeta_{24}^{4} + \zeta_{24}^{2}) q^{73} + (\zeta_{24}^{9} + \zeta_{24}^{7}) q^{74} + \zeta_{24}^{3} q^{80} + ( - \zeta_{24}^{10} + 1) q^{82} - \zeta_{24}^{4} q^{85} + ( - \zeta_{24}^{7} + \zeta_{24}) q^{89} + (\zeta_{24}^{8} + \zeta_{24}^{2}) q^{97} - \zeta_{24}^{5} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{13} + 4 q^{16} + 8 q^{34} + 8 q^{37} + 4 q^{40} - 4 q^{58} + 4 q^{73} + 8 q^{82} - 4 q^{85} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times\).

\(n\) \(937\) \(1081\) \(1171\) \(2081\)
\(\chi(n)\) \(\zeta_{24}^{6}\) \(-\zeta_{24}^{8}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
647.1
0.965926 + 0.258819i
−0.965926 0.258819i
0.258819 + 0.965926i
−0.258819 0.965926i
0.258819 0.965926i
−0.258819 + 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 + 0.965926i 0 −0.866025 0.500000i 0.965926 0.258819i 0 0 0.707107 0.707107i 0 1.00000i
647.2 0.258819 0.965926i 0 −0.866025 0.500000i −0.965926 + 0.258819i 0 0 −0.707107 + 0.707107i 0 1.00000i
1187.1 −0.965926 + 0.258819i 0 0.866025 0.500000i 0.258819 0.965926i 0 0 −0.707107 + 0.707107i 0 1.00000i
1187.2 0.965926 0.258819i 0 0.866025 0.500000i −0.258819 + 0.965926i 0 0 0.707107 0.707107i 0 1.00000i
1583.1 −0.965926 0.258819i 0 0.866025 + 0.500000i 0.258819 + 0.965926i 0 0 −0.707107 0.707107i 0 1.00000i
1583.2 0.965926 + 0.258819i 0 0.866025 + 0.500000i −0.258819 0.965926i 0 0 0.707107 + 0.707107i 0 1.00000i
2123.1 −0.258819 0.965926i 0 −0.866025 + 0.500000i 0.965926 + 0.258819i 0 0 0.707107 + 0.707107i 0 1.00000i
2123.2 0.258819 + 0.965926i 0 −0.866025 + 0.500000i −0.965926 0.258819i 0 0 −0.707107 0.707107i 0 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 647.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
3.b odd 2 1 inner
12.b even 2 1 inner
65.r odd 12 1 inner
195.bf even 12 1 inner
260.bg even 12 1 inner
780.cw odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2340.1.gu.b yes 8
3.b odd 2 1 inner 2340.1.gu.b yes 8
4.b odd 2 1 CM 2340.1.gu.b yes 8
5.c odd 4 1 2340.1.gu.a 8
12.b even 2 1 inner 2340.1.gu.b yes 8
13.e even 6 1 2340.1.gu.a 8
15.e even 4 1 2340.1.gu.a 8
20.e even 4 1 2340.1.gu.a 8
39.h odd 6 1 2340.1.gu.a 8
52.i odd 6 1 2340.1.gu.a 8
60.l odd 4 1 2340.1.gu.a 8
65.r odd 12 1 inner 2340.1.gu.b yes 8
156.r even 6 1 2340.1.gu.a 8
195.bf even 12 1 inner 2340.1.gu.b yes 8
260.bg even 12 1 inner 2340.1.gu.b yes 8
780.cw odd 12 1 inner 2340.1.gu.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2340.1.gu.a 8 5.c odd 4 1
2340.1.gu.a 8 13.e even 6 1
2340.1.gu.a 8 15.e even 4 1
2340.1.gu.a 8 20.e even 4 1
2340.1.gu.a 8 39.h odd 6 1
2340.1.gu.a 8 52.i odd 6 1
2340.1.gu.a 8 60.l odd 4 1
2340.1.gu.a 8 156.r even 6 1
2340.1.gu.b yes 8 1.a even 1 1 trivial
2340.1.gu.b yes 8 3.b odd 2 1 inner
2340.1.gu.b yes 8 4.b odd 2 1 CM
2340.1.gu.b yes 8 12.b even 2 1 inner
2340.1.gu.b yes 8 65.r odd 12 1 inner
2340.1.gu.b yes 8 195.bf even 12 1 inner
2340.1.gu.b yes 8 260.bg even 12 1 inner
2340.1.gu.b yes 8 780.cw odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{37}^{4} - 4T_{37}^{3} + 5T_{37}^{2} - 2T_{37} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2340, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + 4 T^{6} + 15 T^{4} + 4 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} - 4 T^{3} + 5 T^{2} - 2 T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 4 T^{6} + 15 T^{4} + 4 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} - 2 T^{3} + 2 T^{2} + 2 T + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4)^{2} \) Copy content Toggle raw display
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