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Properties

Label 684.1.q.a
Level $684$
Weight $1$
Character orbit 684.q
Analytic conductor $0.341$
Analytic rank $0$
Dimension $8$
Projective image $S_{4}$
CM/RM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [684,1,Mod(163,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.163");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 684.q (of order \(6\), degree \(2\), 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: \(0.341360468641\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
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: \(S_{4}\)
Projective field: Galois closure of 4.2.17328.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{24}^{5} q^{2} + \zeta_{24}^{10} q^{4} + (\zeta_{24}^{11} + \zeta_{24}^{5}) q^{5} - \zeta_{24}^{6} q^{7} + \zeta_{24}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{24}^{5} q^{2} + \zeta_{24}^{10} q^{4} + (\zeta_{24}^{11} + \zeta_{24}^{5}) q^{5} - \zeta_{24}^{6} q^{7} + \zeta_{24}^{3} q^{8} + ( - \zeta_{24}^{10} + \zeta_{24}^{4}) q^{10} + (\zeta_{24}^{9} + \zeta_{24}^{3}) q^{11} + \zeta_{24}^{4} q^{13} + \zeta_{24}^{11} q^{14} - \zeta_{24}^{8} q^{16} + \zeta_{24}^{6} q^{19} + ( - \zeta_{24}^{9} - \zeta_{24}^{3}) q^{20} + ( - \zeta_{24}^{8} + \zeta_{24}^{2}) q^{22} + ( - \zeta_{24}^{7} + \zeta_{24}) q^{23} - \zeta_{24}^{4} q^{25} - \zeta_{24}^{9} q^{26} + \zeta_{24}^{4} q^{28} + \zeta_{24}^{6} q^{31} - \zeta_{24} q^{32} + ( - \zeta_{24}^{11} + \zeta_{24}^{5}) q^{35} - q^{37} - \zeta_{24}^{11} q^{38} + (\zeta_{24}^{8} - \zeta_{24}^{2}) q^{40} + \zeta_{24}^{2} q^{43} + ( - \zeta_{24}^{7} - \zeta_{24}) q^{44} + ( - \zeta_{24}^{6} - 1) q^{46} + \zeta_{24}^{9} q^{50} - \zeta_{24}^{2} q^{52} + ( - \zeta_{24}^{7} - \zeta_{24}) q^{53} + ( - \zeta_{24}^{8} - 2 \zeta_{24}^{2}) q^{55} - \zeta_{24}^{9} q^{56} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{59} - \zeta_{24}^{4} q^{61} - \zeta_{24}^{11} q^{62} + \zeta_{24}^{6} q^{64} + (\zeta_{24}^{9} - \zeta_{24}^{3}) q^{65} - \zeta_{24}^{10} q^{67} + ( - \zeta_{24}^{10} - \zeta_{24}^{4}) q^{70} + \zeta_{24}^{8} q^{73} + \zeta_{24}^{5} q^{74} - \zeta_{24}^{4} q^{76} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{77} - \zeta_{24}^{2} q^{79} + (\zeta_{24}^{7} + \zeta_{24}) q^{80} - \zeta_{24}^{7} q^{86} + (\zeta_{24}^{6} - 1) q^{88} + (\zeta_{24}^{7} + \zeta_{24}) q^{89} - \zeta_{24}^{10} q^{91} + (\zeta_{24}^{11} + \zeta_{24}^{5}) q^{92} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{95} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{10} + 4 q^{13} + 4 q^{16} + 4 q^{22} - 4 q^{25} + 4 q^{28} - 8 q^{37} - 4 q^{40} - 8 q^{46} - 4 q^{61} - 4 q^{70} - 4 q^{73} - 4 q^{76} - 8 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(-\zeta_{24}^{4}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
0.258819 − 0.965926i
0.965926 + 0.258819i
−0.965926 − 0.258819i
−0.258819 + 0.965926i
0.258819 + 0.965926i
0.965926 − 0.258819i
−0.965926 + 0.258819i
−0.258819 − 0.965926i
−0.965926 + 0.258819i 0 0.866025 − 0.500000i 0.707107 − 1.22474i 0 1.00000i −0.707107 + 0.707107i 0 −0.366025 + 1.36603i
163.2 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i −0.707107 + 1.22474i 0 − 1.00000i 0.707107 + 0.707107i 0 1.36603 + 0.366025i
163.3 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0.707107 − 1.22474i 0 − 1.00000i −0.707107 − 0.707107i 0 1.36603 + 0.366025i
163.4 0.965926 − 0.258819i 0 0.866025 − 0.500000i −0.707107 + 1.22474i 0 1.00000i 0.707107 − 0.707107i 0 −0.366025 + 1.36603i
235.1 −0.965926 − 0.258819i 0 0.866025 + 0.500000i 0.707107 + 1.22474i 0 − 1.00000i −0.707107 − 0.707107i 0 −0.366025 − 1.36603i
235.2 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i −0.707107 − 1.22474i 0 1.00000i 0.707107 − 0.707107i 0 1.36603 − 0.366025i
235.3 0.258819 − 0.965926i 0 −0.866025 − 0.500000i 0.707107 + 1.22474i 0 1.00000i −0.707107 + 0.707107i 0 1.36603 − 0.366025i
235.4 0.965926 + 0.258819i 0 0.866025 + 0.500000i −0.707107 − 1.22474i 0 − 1.00000i 0.707107 + 0.707107i 0 −0.366025 − 1.36603i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
19.c even 3 1 inner
57.h odd 6 1 inner
76.g odd 6 1 inner
228.m even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.1.q.a ✓ 8
3.b odd 2 1 inner 684.1.q.a ✓ 8
4.b odd 2 1 inner 684.1.q.a ✓ 8
12.b even 2 1 inner 684.1.q.a ✓ 8
19.c even 3 1 inner 684.1.q.a ✓ 8
57.h odd 6 1 inner 684.1.q.a ✓ 8
76.g odd 6 1 inner 684.1.q.a ✓ 8
228.m even 6 1 inner 684.1.q.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.1.q.a ✓ 8 1.a even 1 1 trivial
684.1.q.a ✓ 8 3.b odd 2 1 inner
684.1.q.a ✓ 8 4.b odd 2 1 inner
684.1.q.a ✓ 8 12.b even 2 1 inner
684.1.q.a ✓ 8 19.c even 3 1 inner
684.1.q.a ✓ 8 57.h odd 6 1 inner
684.1.q.a ✓ 8 76.g odd 6 1 inner
684.1.q.a ✓ 8 228.m even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(684, [\chi])\).

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