Properties

Label 1089.1.s.b
Level $1089$
Weight $1$
Character orbit 1089.s
Analytic conductor $0.543$
Analytic rank $0$
Dimension $16$
Projective image $S_{4}$
CM/RM no
Inner twists $16$

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Newspace parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1089.s (of order \(30\), degree \(8\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.543481798757\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{30})\)
Coefficient field: 16.0.26873856000000000000.1
Defining polynomial: \(x^{16} + 2 x^{14} - 8 x^{10} - 16 x^{8} - 32 x^{6} + 128 x^{2} + 256\)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -\beta_{7} q^{3} + \beta_{2} q^{4} + ( \beta_{3} + \beta_{15} ) q^{5} -\beta_{8} q^{6} + \beta_{6} q^{7} + ( -1 - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} + \beta_{9} - \beta_{15} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -\beta_{7} q^{3} + \beta_{2} q^{4} + ( \beta_{3} + \beta_{15} ) q^{5} -\beta_{8} q^{6} + \beta_{6} q^{7} + ( -1 - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} + \beta_{9} - \beta_{15} ) q^{9} + \beta_{13} q^{10} -\beta_{9} q^{12} + 2 \beta_{7} q^{14} + ( \beta_{2} - \beta_{11} ) q^{15} -\beta_{3} q^{16} + ( -\beta_{1} + \beta_{6} + \beta_{8} + \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{18} + ( -1 - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} + \beta_{9} - \beta_{15} ) q^{20} -\beta_{13} q^{21} -\beta_{15} q^{27} + \beta_{8} q^{28} -\beta_{6} q^{29} + \beta_{14} q^{30} + ( 1 + \beta_{2} + \beta_{3} - \beta_{7} - \beta_{9} + \beta_{15} ) q^{31} -\beta_{4} q^{32} + ( -\beta_{1} + \beta_{10} ) q^{35} + ( 1 - \beta_{5} + \beta_{11} + \beta_{15} ) q^{36} -\beta_{11} q^{37} + ( 2 + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{15} ) q^{42} - q^{45} + ( -1 + \beta_{5} + \beta_{7} - \beta_{11} - \beta_{15} ) q^{47} + \beta_{11} q^{48} + ( \beta_{3} + \beta_{15} ) q^{49} -\beta_{5} q^{53} + ( \beta_{4} - \beta_{13} ) q^{54} -2 \beta_{7} q^{58} -\beta_{2} q^{59} -\beta_{15} q^{60} + ( \beta_{1} - \beta_{4} - \beta_{6} + \beta_{12} + \beta_{13} ) q^{61} + ( \beta_{1} - \beta_{8} - \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} ) q^{62} -\beta_{14} q^{63} -\beta_{5} q^{64} + ( 1 - \beta_{9} ) q^{67} + ( -2 \beta_{2} + 2 \beta_{11} ) q^{70} -\beta_{15} q^{71} + ( -\beta_{1} + \beta_{8} + \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{73} -\beta_{12} q^{74} + ( -1 + \beta_{5} - \beta_{11} - \beta_{15} ) q^{80} -\beta_{2} q^{81} + ( \beta_{1} - \beta_{4} - \beta_{6} + \beta_{12} + \beta_{13} ) q^{83} + ( \beta_{1} - \beta_{6} - \beta_{8} - \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} ) q^{84} + \beta_{13} q^{87} -\beta_{1} q^{90} -\beta_{3} q^{93} + ( -\beta_{1} + \beta_{4} + \beta_{6} + \beta_{8} - \beta_{12} - \beta_{13} ) q^{94} + \beta_{12} q^{96} + ( -1 - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} + \beta_{9} - \beta_{15} ) q^{97} + \beta_{13} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 2q^{3} - 2q^{4} - 2q^{5} + 2q^{9} + O(q^{10}) \) \( 16q - 2q^{3} - 2q^{4} - 2q^{5} + 2q^{9} - 8q^{12} + 4q^{14} + 2q^{15} - 2q^{16} + 2q^{20} + 4q^{27} + 2q^{31} + 4q^{36} + 4q^{37} - 4q^{42} - 16q^{45} - 2q^{47} - 4q^{48} - 2q^{49} - 4q^{53} - 4q^{58} + 2q^{59} + 4q^{60} - 4q^{64} + 8q^{67} - 4q^{70} + 4q^{71} - 4q^{80} + 2q^{81} - 2q^{93} + 2q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 2 x^{14} - 8 x^{10} - 16 x^{8} - 32 x^{6} + 128 x^{2} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{4} \)\(/4\)
\(\beta_{4}\)\(=\)\( \nu^{5} \)\(/4\)
\(\beta_{5}\)\(=\)\( \nu^{6} \)\(/8\)
\(\beta_{6}\)\(=\)\( \nu^{7} \)\(/8\)
\(\beta_{7}\)\(=\)\( \nu^{8} \)\(/16\)
\(\beta_{8}\)\(=\)\( \nu^{9} \)\(/16\)
\(\beta_{9}\)\(=\)\( \nu^{10} \)\(/32\)
\(\beta_{10}\)\(=\)\( \nu^{11} \)\(/32\)
\(\beta_{11}\)\(=\)\( \nu^{12} \)\(/64\)
\(\beta_{12}\)\(=\)\( \nu^{13} \)\(/64\)
\(\beta_{13}\)\(=\)\( \nu^{15} \)\(/128\)
\(\beta_{14}\)\(=\)\((\)\( -\nu^{13} + 32 \nu^{3} \)\()/64\)
\(\beta_{15}\)\(=\)\((\)\( \nu^{14} - 32 \nu^{4} \)\()/128\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{14} + 2 \beta_{12}\)
\(\nu^{4}\)\(=\)\(4 \beta_{3}\)
\(\nu^{5}\)\(=\)\(4 \beta_{4}\)
\(\nu^{6}\)\(=\)\(8 \beta_{5}\)
\(\nu^{7}\)\(=\)\(8 \beta_{6}\)
\(\nu^{8}\)\(=\)\(16 \beta_{7}\)
\(\nu^{9}\)\(=\)\(16 \beta_{8}\)
\(\nu^{10}\)\(=\)\(32 \beta_{9}\)
\(\nu^{11}\)\(=\)\(32 \beta_{10}\)
\(\nu^{12}\)\(=\)\(64 \beta_{11}\)
\(\nu^{13}\)\(=\)\(64 \beta_{12}\)
\(\nu^{14}\)\(=\)\(128 \beta_{15} + 128 \beta_{3}\)
\(\nu^{15}\)\(=\)\(128 \beta_{13}\)

Character values

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

\(n\) \(244\) \(848\)
\(\chi(n)\) \(-\beta_{11}\) \(-1 + \beta_{9}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1
−1.40647 0.147826i
1.40647 + 0.147826i
−1.05097 0.946294i
1.05097 + 0.946294i
−0.294032 1.38331i
0.294032 + 1.38331i
−0.575212 1.29195i
0.575212 + 1.29195i
−0.294032 + 1.38331i
0.294032 1.38331i
−1.05097 + 0.946294i
1.05097 0.946294i
−0.575212 + 1.29195i
0.575212 1.29195i
−1.40647 + 0.147826i
1.40647 0.147826i
−1.40647 0.147826i −0.669131 0.743145i 0.978148 + 0.207912i 0.104528 + 0.994522i 0.831254 + 1.14412i −1.05097 0.946294i 0 −0.104528 + 0.994522i 1.41421i
40.2 1.40647 + 0.147826i −0.669131 0.743145i 0.978148 + 0.207912i 0.104528 + 0.994522i −0.831254 1.14412i 1.05097 + 0.946294i 0 −0.104528 + 0.994522i 1.41421i
94.1 −1.05097 0.946294i −0.913545 + 0.406737i 0.104528 + 0.994522i −0.669131 0.743145i 1.34500 + 0.437016i −0.575212 + 1.29195i 0 0.669131 0.743145i 1.41421i
94.2 1.05097 + 0.946294i −0.913545 + 0.406737i 0.104528 + 0.994522i −0.669131 0.743145i −1.34500 0.437016i 0.575212 1.29195i 0 0.669131 0.743145i 1.41421i
112.1 −0.294032 1.38331i 0.104528 + 0.994522i −0.913545 + 0.406737i 0.978148 + 0.207912i 1.34500 0.437016i 1.40647 + 0.147826i 0 −0.978148 + 0.207912i 1.41421i
112.2 0.294032 + 1.38331i 0.104528 + 0.994522i −0.913545 + 0.406737i 0.978148 + 0.207912i −1.34500 + 0.437016i −1.40647 0.147826i 0 −0.978148 + 0.207912i 1.41421i
403.1 −0.575212 1.29195i 0.978148 0.207912i −0.669131 + 0.743145i −0.913545 0.406737i −0.831254 1.14412i 0.294032 1.38331i 0 0.913545 0.406737i 1.41421i
403.2 0.575212 + 1.29195i 0.978148 0.207912i −0.669131 + 0.743145i −0.913545 0.406737i 0.831254 + 1.14412i −0.294032 + 1.38331i 0 0.913545 0.406737i 1.41421i
457.1 −0.294032 + 1.38331i 0.104528 0.994522i −0.913545 0.406737i 0.978148 0.207912i 1.34500 + 0.437016i 1.40647 0.147826i 0 −0.978148 0.207912i 1.41421i
457.2 0.294032 1.38331i 0.104528 0.994522i −0.913545 0.406737i 0.978148 0.207912i −1.34500 0.437016i −1.40647 + 0.147826i 0 −0.978148 0.207912i 1.41421i
475.1 −1.05097 + 0.946294i −0.913545 0.406737i 0.104528 0.994522i −0.669131 + 0.743145i 1.34500 0.437016i −0.575212 1.29195i 0 0.669131 + 0.743145i 1.41421i
475.2 1.05097 0.946294i −0.913545 0.406737i 0.104528 0.994522i −0.669131 + 0.743145i −1.34500 + 0.437016i 0.575212 + 1.29195i 0 0.669131 + 0.743145i 1.41421i
481.1 −0.575212 + 1.29195i 0.978148 + 0.207912i −0.669131 0.743145i −0.913545 + 0.406737i −0.831254 + 1.14412i 0.294032 + 1.38331i 0 0.913545 + 0.406737i 1.41421i
481.2 0.575212 1.29195i 0.978148 + 0.207912i −0.669131 0.743145i −0.913545 + 0.406737i 0.831254 1.14412i −0.294032 1.38331i 0 0.913545 + 0.406737i 1.41421i
844.1 −1.40647 + 0.147826i −0.669131 + 0.743145i 0.978148 0.207912i 0.104528 0.994522i 0.831254 1.14412i −1.05097 + 0.946294i 0 −0.104528 0.994522i 1.41421i
844.2 1.40647 0.147826i −0.669131 + 0.743145i 0.978148 0.207912i 0.104528 0.994522i −0.831254 + 1.14412i 1.05097 0.946294i 0 −0.104528 0.994522i 1.41421i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 844.2
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
11.c even 5 3 inner
11.d odd 10 3 inner
99.h odd 6 1 inner
99.m even 15 3 inner
99.o odd 30 3 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 2 T_{2}^{14} - 8 T_{2}^{10} - 16 T_{2}^{8} - 32 T_{2}^{6} + 128 T_{2}^{2} + 256 \) acting on \(S_{1}^{\mathrm{new}}(1089, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 256 + 128 T^{2} - 32 T^{6} - 16 T^{8} - 8 T^{10} + 2 T^{14} + T^{16} \)
$3$ \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
$5$ \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
$7$ \( 256 + 128 T^{2} - 32 T^{6} - 16 T^{8} - 8 T^{10} + 2 T^{14} + T^{16} \)
$11$ \( T^{16} \)
$13$ \( T^{16} \)
$17$ \( T^{16} \)
$19$ \( T^{16} \)
$23$ \( T^{16} \)
$29$ \( 256 + 128 T^{2} - 32 T^{6} - 16 T^{8} - 8 T^{10} + 2 T^{14} + T^{16} \)
$31$ \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
$37$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
$41$ \( T^{16} \)
$43$ \( T^{16} \)
$47$ \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
$53$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
$59$ \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
$61$ \( 256 + 128 T^{2} - 32 T^{6} - 16 T^{8} - 8 T^{10} + 2 T^{14} + T^{16} \)
$67$ \( ( 1 - T + T^{2} )^{8} \)
$71$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
$73$ \( ( 16 - 8 T^{2} + 4 T^{4} - 2 T^{6} + T^{8} )^{2} \)
$79$ \( T^{16} \)
$83$ \( 256 + 128 T^{2} - 32 T^{6} - 16 T^{8} - 8 T^{10} + 2 T^{14} + T^{16} \)
$89$ \( T^{16} \)
$97$ \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
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