Properties

Label 259.1.bi.a
Level $259$
Weight $1$
Character orbit 259.bi
Analytic conductor $0.129$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -7
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [259,1,Mod(34,259)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(259, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("259.34");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 259 = 7 \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 259.bi (of order \(18\), degree \(6\), 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: \(0.129257838272\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.8433463168864321.2

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{18}^{4} - \zeta_{18}^{3}) q^{2} + (\zeta_{18}^{8} - \zeta_{18}^{7} + \zeta_{18}^{6}) q^{4} + \zeta_{18}^{4} q^{7} + ( - \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18} + 1) q^{8} + \zeta_{18}^{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{18}^{4} - \zeta_{18}^{3}) q^{2} + (\zeta_{18}^{8} - \zeta_{18}^{7} + \zeta_{18}^{6}) q^{4} + \zeta_{18}^{4} q^{7} + ( - \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18} + 1) q^{8} + \zeta_{18}^{2} q^{9} + (\zeta_{18}^{2} - \zeta_{18}) q^{11} + (\zeta_{18}^{8} - \zeta_{18}^{7}) q^{14} + ( - \zeta_{18}^{7} - \zeta_{18}^{6} + \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3}) q^{16} + (\zeta_{18}^{6} - \zeta_{18}^{5}) q^{18} + (\zeta_{18}^{6} - 2 \zeta_{18}^{5} + \zeta_{18}^{4}) q^{22} + ( - \zeta_{18}^{5} - \zeta_{18}) q^{23} + \zeta_{18}^{8} q^{25} + ( - \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18}) q^{28} + ( - \zeta_{18}^{7} - \zeta_{18}^{5}) q^{29} + (\zeta_{18}^{8} - \zeta_{18}^{7} + \zeta_{18}^{6} + \zeta_{18}^{2} + \zeta_{18} - 1) q^{32} + (\zeta_{18}^{8} - \zeta_{18} + 1) q^{36} + q^{37} - q^{43} + (2 \zeta_{18}^{8} - \zeta_{18}^{7} - \zeta_{18} + 2) q^{44} + (\zeta_{18}^{8} - \zeta_{18}^{5} + \zeta_{18}^{4} + 1) q^{46} + \zeta_{18}^{8} q^{49} + ( - \zeta_{18}^{3} + \zeta_{18}^{2}) q^{50} + ( - \zeta_{18}^{7} - \zeta_{18}^{3}) q^{53} + ( - \zeta_{18}^{7} + \zeta_{18}^{6} - \zeta_{18}^{5} + \zeta_{18}^{4}) q^{56} + (\zeta_{18}^{8} + \zeta_{18}^{2} - \zeta_{18} + 1) q^{58} + \zeta_{18}^{6} q^{63} + (\zeta_{18}^{6} + \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18} + 1) q^{64} - \zeta_{18}^{4} q^{67} + \zeta_{18} q^{71} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2}) q^{72} + (\zeta_{18}^{4} - \zeta_{18}^{3}) q^{74} + (\zeta_{18}^{6} - \zeta_{18}^{5}) q^{77} + (\zeta_{18}^{6} + \zeta_{18}^{2}) q^{79} + \zeta_{18}^{4} q^{81} + ( - \zeta_{18}^{4} + \zeta_{18}^{3}) q^{86} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - \zeta_{18}) q^{88} + (\zeta_{18}^{8} - \zeta_{18}^{7} + \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} + 1) q^{92} + ( - \zeta_{18}^{3} + \zeta_{18}^{2}) q^{98} + (\zeta_{18}^{4} - \zeta_{18}^{3}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} - 3 q^{4} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} - 3 q^{4} + 3 q^{8} - 6 q^{16} - 3 q^{18} - 3 q^{22} - 3 q^{28} + 3 q^{32} + 6 q^{36} + 6 q^{37} - 6 q^{43} + 12 q^{44} + 6 q^{46} - 3 q^{50} - 3 q^{53} - 3 q^{56} + 6 q^{58} - 3 q^{63} - 3 q^{72} - 3 q^{74} - 3 q^{77} - 3 q^{79} + 3 q^{86} - 6 q^{88} + 3 q^{92} - 3 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/259\mathbb{Z}\right)^\times\).

\(n\) \(38\) \(113\)
\(\chi(n)\) \(-1\) \(-\zeta_{18}^{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1
−0.766044 + 0.642788i
−0.173648 + 0.984808i
0.939693 0.342020i
0.939693 + 0.342020i
−0.766044 0.642788i
−0.173648 0.984808i
−1.43969 1.20805i 0 0.439693 + 2.49362i 0 0 −0.939693 0.342020i 1.43969 2.49362i 0.173648 0.984808i 0
83.1 0.266044 + 1.50881i 0 −1.26604 + 0.460802i 0 0 0.766044 + 0.642788i −0.266044 0.460802i −0.939693 0.342020i 0
90.1 −0.326352 0.118782i 0 −0.673648 0.565258i 0 0 0.173648 0.984808i 0.326352 + 0.565258i 0.766044 0.642788i 0
118.1 −0.326352 + 0.118782i 0 −0.673648 + 0.565258i 0 0 0.173648 + 0.984808i 0.326352 0.565258i 0.766044 + 0.642788i 0
160.1 −1.43969 + 1.20805i 0 0.439693 2.49362i 0 0 −0.939693 + 0.342020i 1.43969 + 2.49362i 0.173648 + 0.984808i 0
181.1 0.266044 1.50881i 0 −1.26604 0.460802i 0 0 0.766044 0.642788i −0.266044 + 0.460802i −0.939693 + 0.342020i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
37.f even 9 1 inner
259.bi odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 259.1.bi.a 6
3.b odd 2 1 2331.1.hj.a 6
7.b odd 2 1 CM 259.1.bi.a 6
7.c even 3 1 1813.1.bl.a 6
7.c even 3 1 1813.1.bn.a 6
7.d odd 6 1 1813.1.bl.a 6
7.d odd 6 1 1813.1.bn.a 6
21.c even 2 1 2331.1.hj.a 6
37.f even 9 1 inner 259.1.bi.a 6
111.p odd 18 1 2331.1.hj.a 6
259.x even 9 1 1813.1.bl.a 6
259.y even 9 1 1813.1.bn.a 6
259.bh odd 18 1 1813.1.bn.a 6
259.bi odd 18 1 inner 259.1.bi.a 6
259.bj odd 18 1 1813.1.bl.a 6
777.dc even 18 1 2331.1.hj.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
259.1.bi.a 6 1.a even 1 1 trivial
259.1.bi.a 6 7.b odd 2 1 CM
259.1.bi.a 6 37.f even 9 1 inner
259.1.bi.a 6 259.bi odd 18 1 inner
1813.1.bl.a 6 7.c even 3 1
1813.1.bl.a 6 7.d odd 6 1
1813.1.bl.a 6 259.x even 9 1
1813.1.bl.a 6 259.bj odd 18 1
1813.1.bn.a 6 7.c even 3 1
1813.1.bn.a 6 7.d odd 6 1
1813.1.bn.a 6 259.y even 9 1
1813.1.bn.a 6 259.bh odd 18 1
2331.1.hj.a 6 3.b odd 2 1
2331.1.hj.a 6 21.c even 2 1
2331.1.hj.a 6 111.p odd 18 1
2331.1.hj.a 6 777.dc even 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(259, [\chi])\).

Hecke characteristic polynomials

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