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Properties

Label 1134.2.t.f
Level $1134$
Weight $2$
Character orbit 1134.t
Analytic conductor $9.055$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1134,2,Mod(593,1134)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1134, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1134.593");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.t (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: \(9.05503558921\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 378)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_1) q^{2} + ( - \beta_{2} + 1) q^{4} - \beta_{6} q^{5} + (\beta_{4} - \beta_{2} + 1) q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_1) q^{2} + ( - \beta_{2} + 1) q^{4} - \beta_{6} q^{5} + (\beta_{4} - \beta_{2} + 1) q^{7} + \beta_{3} q^{8} + \beta_{7} q^{10} + ( - \beta_{6} + 2 \beta_{5}) q^{11} + (\beta_{7} + \beta_{2} - 2) q^{13} + ( - \beta_{6} + \beta_{3}) q^{14} - \beta_{2} q^{16} + \beta_{5} q^{17} - 2 \beta_{4} q^{19} + ( - \beta_{6} + \beta_{5}) q^{20} + (\beta_{7} - 2 \beta_{4}) q^{22} - 6 \beta_{3} q^{23} + q^{25} + ( - \beta_{6} + \beta_{5} + \cdots + \beta_1) q^{26}+ \cdots + ( - 2 \beta_{6} + 2 \beta_{5} - 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} + 4 q^{7} - 12 q^{13} - 4 q^{16} + 8 q^{25} - 4 q^{28} + 12 q^{31} - 4 q^{37} - 28 q^{43} + 24 q^{46} + 20 q^{49} + 48 q^{58} - 12 q^{61} - 8 q^{64} - 20 q^{67} + 48 q^{70} - 72 q^{73} - 20 q^{79} - 24 q^{85} + 48 q^{91} - 72 q^{94} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{7} - 2\zeta_{24}^{5} + 2\zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} - \beta_{5} + 2\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} - \beta_{6} + 2\beta_{5} + \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -2\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} - 2\beta_{6} + \beta_{5} + 2\beta_{4} ) / 6 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(1 - \beta_{2}\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
593.1
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
−0.258819 − 0.965926i
−0.866025 + 0.500000i 0 0.500000 − 0.866025i −2.44949 0 2.62132 + 0.358719i 1.00000i 0 2.12132 − 1.22474i
593.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 2.44949 0 −1.62132 − 2.09077i 1.00000i 0 −2.12132 + 1.22474i
593.3 0.866025 − 0.500000i 0 0.500000 − 0.866025i −2.44949 0 −1.62132 − 2.09077i − 1.00000i 0 −2.12132 + 1.22474i
593.4 0.866025 − 0.500000i 0 0.500000 − 0.866025i 2.44949 0 2.62132 + 0.358719i − 1.00000i 0 2.12132 − 1.22474i
1025.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −2.44949 0 2.62132 − 0.358719i − 1.00000i 0 2.12132 + 1.22474i
1025.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 2.44949 0 −1.62132 + 2.09077i − 1.00000i 0 −2.12132 − 1.22474i
1025.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i −2.44949 0 −1.62132 + 2.09077i 1.00000i 0 −2.12132 − 1.22474i
1025.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 2.44949 0 2.62132 − 0.358719i 1.00000i 0 2.12132 + 1.22474i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 593.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
63.k odd 6 1 inner
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.2.t.f 8
3.b odd 2 1 inner 1134.2.t.f 8
7.d odd 6 1 1134.2.l.e 8
9.c even 3 1 378.2.k.d ✓ 8
9.c even 3 1 1134.2.l.e 8
9.d odd 6 1 378.2.k.d ✓ 8
9.d odd 6 1 1134.2.l.e 8
21.g even 6 1 1134.2.l.e 8
63.g even 3 1 2646.2.d.d 8
63.i even 6 1 378.2.k.d ✓ 8
63.k odd 6 1 inner 1134.2.t.f 8
63.k odd 6 1 2646.2.d.d 8
63.n odd 6 1 2646.2.d.d 8
63.s even 6 1 inner 1134.2.t.f 8
63.s even 6 1 2646.2.d.d 8
63.t odd 6 1 378.2.k.d ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.k.d ✓ 8 9.c even 3 1
378.2.k.d ✓ 8 9.d odd 6 1
378.2.k.d ✓ 8 63.i even 6 1
378.2.k.d ✓ 8 63.t odd 6 1
1134.2.l.e 8 7.d odd 6 1
1134.2.l.e 8 9.c even 3 1
1134.2.l.e 8 9.d odd 6 1
1134.2.l.e 8 21.g even 6 1
1134.2.t.f 8 1.a even 1 1 trivial
1134.2.t.f 8 3.b odd 2 1 inner
1134.2.t.f 8 63.k odd 6 1 inner
1134.2.t.f 8 63.s even 6 1 inner
2646.2.d.d 8 63.g even 3 1
2646.2.d.d 8 63.k odd 6 1
2646.2.d.d 8 63.n odd 6 1
2646.2.d.d 8 63.s even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1134, [\chi])\):

\( T_{5}^{2} - 6 \) Copy content Toggle raw display
\( T_{11}^{2} + 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} - 6)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 2 T^{3} - 3 T^{2} + \cdots + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 6 T^{3} + 9 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 24 T^{2} + 576)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} - 108 T^{6} + \cdots + 104976 \) Copy content Toggle raw display
$31$ \( (T^{4} - 6 T^{3} + \cdots + 2601)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2 T^{3} + \cdots + 289)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 7 T + 49)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + 228 T^{6} + \cdots + 108243216 \) Copy content Toggle raw display
$53$ \( T^{8} - 216 T^{6} + \cdots + 1679616 \) Copy content Toggle raw display
$59$ \( (T^{4} + 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 6 T^{3} + 9 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 10 T^{3} + \cdots + 2209)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 162)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 36 T^{3} + \cdots + 7056)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 10 T^{3} + \cdots + 49)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 264 T^{6} + \cdots + 49787136 \) Copy content Toggle raw display
$89$ \( T^{8} + 324 T^{6} + \cdots + 8503056 \) Copy content Toggle raw display
$97$ \( (T^{4} + 6 T^{3} + \cdots + 441)^{2} \) Copy content Toggle raw display
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