Properties

Label 1089.1.h.a
Level $1089$
Weight $1$
Character orbit 1089.h
Analytic conductor $0.543$
Analytic rank $0$
Dimension $4$
Projective image $S_{4}$
CM/RM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1089.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.543481798757\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.107811.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{3} ) q^{2} + ( 1 - \beta_{2} ) q^{3} + ( 1 - \beta_{2} ) q^{4} + ( 1 - \beta_{2} ) q^{5} -\beta_{3} q^{6} + ( -\beta_{1} + \beta_{3} ) q^{7} -\beta_{2} q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{3} ) q^{2} + ( 1 - \beta_{2} ) q^{3} + ( 1 - \beta_{2} ) q^{4} + ( 1 - \beta_{2} ) q^{5} -\beta_{3} q^{6} + ( -\beta_{1} + \beta_{3} ) q^{7} -\beta_{2} q^{9} -\beta_{3} q^{10} -\beta_{2} q^{12} + ( -2 + 2 \beta_{2} ) q^{14} -\beta_{2} q^{15} + \beta_{2} q^{16} -\beta_{1} q^{18} -\beta_{2} q^{20} + \beta_{3} q^{21} - q^{27} + \beta_{3} q^{28} + ( \beta_{1} - \beta_{3} ) q^{29} -\beta_{1} q^{30} + ( -1 + \beta_{2} ) q^{31} + \beta_{1} q^{32} + \beta_{3} q^{35} - q^{36} - q^{37} + 2 \beta_{2} q^{42} - q^{45} + \beta_{2} q^{47} + q^{48} + ( 1 - \beta_{2} ) q^{49} + q^{53} + ( -\beta_{1} + \beta_{3} ) q^{54} + ( 2 - 2 \beta_{2} ) q^{58} + ( -1 + \beta_{2} ) q^{59} - q^{60} + ( -\beta_{1} + \beta_{3} ) q^{61} + \beta_{3} q^{62} + \beta_{1} q^{63} + q^{64} + ( 1 - \beta_{2} ) q^{67} + 2 \beta_{2} q^{70} - q^{71} -\beta_{3} q^{73} + ( -\beta_{1} + \beta_{3} ) q^{74} + q^{80} + ( -1 + \beta_{2} ) q^{81} + ( -\beta_{1} + \beta_{3} ) q^{83} + \beta_{1} q^{84} -\beta_{3} q^{87} + ( -\beta_{1} + \beta_{3} ) q^{90} + \beta_{2} q^{93} + \beta_{1} q^{94} + ( \beta_{1} - \beta_{3} ) q^{96} -\beta_{2} q^{97} -\beta_{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{9} + O(q^{10}) \) \( 4q + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{9} - 2q^{12} - 4q^{14} - 2q^{15} + 2q^{16} - 2q^{20} - 4q^{27} - 2q^{31} - 4q^{36} - 4q^{37} + 4q^{42} - 4q^{45} + 2q^{47} + 4q^{48} + 2q^{49} + 4q^{53} + 4q^{58} - 2q^{59} - 4q^{60} + 4q^{64} + 2q^{67} + 4q^{70} - 4q^{71} + 4q^{80} - 2q^{81} + 2q^{93} - 2q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(-1\) \(-1 + \beta_{2}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
241.1
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 0.707107i 0.500000 + 0.866025i 0.500000 + 0.866025i 0.500000 + 0.866025i 1.41421i 1.22474 + 0.707107i 0 −0.500000 + 0.866025i 1.41421i
241.2 1.22474 + 0.707107i 0.500000 + 0.866025i 0.500000 + 0.866025i 0.500000 + 0.866025i 1.41421i −1.22474 0.707107i 0 −0.500000 + 0.866025i 1.41421i
967.1 −1.22474 + 0.707107i 0.500000 0.866025i 0.500000 0.866025i 0.500000 0.866025i 1.41421i 1.22474 0.707107i 0 −0.500000 0.866025i 1.41421i
967.2 1.22474 0.707107i 0.500000 0.866025i 0.500000 0.866025i 0.500000 0.866025i 1.41421i −1.22474 + 0.707107i 0 −0.500000 0.866025i 1.41421i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
11.b odd 2 1 inner
99.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.1.h.a 4
3.b odd 2 1 3267.1.h.a 4
9.c even 3 1 inner 1089.1.h.a 4
9.d odd 6 1 3267.1.h.a 4
11.b odd 2 1 inner 1089.1.h.a 4
11.c even 5 4 1089.1.s.b 16
11.d odd 10 4 1089.1.s.b 16
33.d even 2 1 3267.1.h.a 4
33.f even 10 4 3267.1.w.b 16
33.h odd 10 4 3267.1.w.b 16
99.g even 6 1 3267.1.h.a 4
99.h odd 6 1 inner 1089.1.h.a 4
99.m even 15 4 1089.1.s.b 16
99.n odd 30 4 3267.1.w.b 16
99.o odd 30 4 1089.1.s.b 16
99.p even 30 4 3267.1.w.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.1.h.a 4 1.a even 1 1 trivial
1089.1.h.a 4 9.c even 3 1 inner
1089.1.h.a 4 11.b odd 2 1 inner
1089.1.h.a 4 99.h odd 6 1 inner
1089.1.s.b 16 11.c even 5 4
1089.1.s.b 16 11.d odd 10 4
1089.1.s.b 16 99.m even 15 4
1089.1.s.b 16 99.o odd 30 4
3267.1.h.a 4 3.b odd 2 1
3267.1.h.a 4 9.d odd 6 1
3267.1.h.a 4 33.d even 2 1
3267.1.h.a 4 99.g even 6 1
3267.1.w.b 16 33.f even 10 4
3267.1.w.b 16 33.h odd 10 4
3267.1.w.b 16 99.n odd 30 4
3267.1.w.b 16 99.p even 30 4

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1089, [\chi])\).

Hecke characteristic polynomials

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