Properties

Label 891.2.g.a
Level 891
Weight 2
Character orbit 891.g
Analytic conductor 7.115
Analytic rank 0
Dimension 4
CM discriminant -11
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 891.g (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.11467082010\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 2 - 2 \beta_{2} ) q^{4} + \beta_{1} q^{5} +O(q^{10})\) \( q + ( 2 - 2 \beta_{2} ) q^{4} + \beta_{1} q^{5} + \beta_{3} q^{11} -4 \beta_{2} q^{16} + 2 \beta_{3} q^{20} + \beta_{1} q^{23} + 6 \beta_{2} q^{25} + ( -5 + 5 \beta_{2} ) q^{31} -7 q^{37} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{44} + 2 \beta_{3} q^{47} + ( -7 + 7 \beta_{2} ) q^{49} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{53} + 11 q^{55} + \beta_{1} q^{59} -8 q^{64} + ( 13 - 13 \beta_{2} ) q^{67} + ( 5 \beta_{1} - 5 \beta_{3} ) q^{71} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{80} + ( 5 \beta_{1} - 5 \beta_{3} ) q^{89} + 2 \beta_{3} q^{92} -17 \beta_{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{4} + O(q^{10}) \) \( 4q + 4q^{4} - 8q^{16} + 12q^{25} - 10q^{31} - 28q^{37} - 14q^{49} + 44q^{55} - 32q^{64} + 26q^{67} - 34q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} + 10 \nu - 9 \)\()/6\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 3 \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{3} + 2 \nu^{2} + 10 \nu + 15 \)\()/6\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 5 \beta_{2}\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{3} + \beta_{1} + 4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/891\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(650\)
\(\chi(n)\) \(-1\) \(1 - \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
296.1
−1.18614 1.26217i
1.68614 + 0.396143i
−1.18614 + 1.26217i
1.68614 0.396143i
0 0 1.00000 1.73205i −2.87228 1.65831i 0 0 0 0 0
296.2 0 0 1.00000 1.73205i 2.87228 + 1.65831i 0 0 0 0 0
593.1 0 0 1.00000 + 1.73205i −2.87228 + 1.65831i 0 0 0 0 0
593.2 0 0 1.00000 + 1.73205i 2.87228 1.65831i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
33.d even 2 1 inner
99.g even 6 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 891.2.g.a 4
3.b odd 2 1 inner 891.2.g.a 4
9.c even 3 1 33.2.d.a 2
9.c even 3 1 inner 891.2.g.a 4
9.d odd 6 1 33.2.d.a 2
9.d odd 6 1 inner 891.2.g.a 4
11.b odd 2 1 CM 891.2.g.a 4
33.d even 2 1 inner 891.2.g.a 4
36.f odd 6 1 528.2.b.a 2
36.h even 6 1 528.2.b.a 2
45.h odd 6 1 825.2.f.a 2
45.j even 6 1 825.2.f.a 2
45.k odd 12 2 825.2.d.a 4
45.l even 12 2 825.2.d.a 4
72.j odd 6 1 2112.2.b.e 2
72.l even 6 1 2112.2.b.f 2
72.n even 6 1 2112.2.b.e 2
72.p odd 6 1 2112.2.b.f 2
99.g even 6 1 33.2.d.a 2
99.g even 6 1 inner 891.2.g.a 4
99.h odd 6 1 33.2.d.a 2
99.h odd 6 1 inner 891.2.g.a 4
99.m even 15 4 363.2.f.c 8
99.n odd 30 4 363.2.f.c 8
99.o odd 30 4 363.2.f.c 8
99.p even 30 4 363.2.f.c 8
396.k even 6 1 528.2.b.a 2
396.o odd 6 1 528.2.b.a 2
495.o odd 6 1 825.2.f.a 2
495.r even 6 1 825.2.f.a 2
495.bd odd 12 2 825.2.d.a 4
495.bf even 12 2 825.2.d.a 4
792.s odd 6 1 2112.2.b.f 2
792.w even 6 1 2112.2.b.e 2
792.z even 6 1 2112.2.b.f 2
792.be odd 6 1 2112.2.b.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.d.a 2 9.c even 3 1
33.2.d.a 2 9.d odd 6 1
33.2.d.a 2 99.g even 6 1
33.2.d.a 2 99.h odd 6 1
363.2.f.c 8 99.m even 15 4
363.2.f.c 8 99.n odd 30 4
363.2.f.c 8 99.o odd 30 4
363.2.f.c 8 99.p even 30 4
528.2.b.a 2 36.f odd 6 1
528.2.b.a 2 36.h even 6 1
528.2.b.a 2 396.k even 6 1
528.2.b.a 2 396.o odd 6 1
825.2.d.a 4 45.k odd 12 2
825.2.d.a 4 45.l even 12 2
825.2.d.a 4 495.bd odd 12 2
825.2.d.a 4 495.bf even 12 2
825.2.f.a 2 45.h odd 6 1
825.2.f.a 2 45.j even 6 1
825.2.f.a 2 495.o odd 6 1
825.2.f.a 2 495.r even 6 1
891.2.g.a 4 1.a even 1 1 trivial
891.2.g.a 4 3.b odd 2 1 inner
891.2.g.a 4 9.c even 3 1 inner
891.2.g.a 4 9.d odd 6 1 inner
891.2.g.a 4 11.b odd 2 1 CM
891.2.g.a 4 33.d even 2 1 inner
891.2.g.a 4 99.g even 6 1 inner
891.2.g.a 4 99.h odd 6 1 inner
2112.2.b.e 2 72.j odd 6 1
2112.2.b.e 2 72.n even 6 1
2112.2.b.e 2 792.w even 6 1
2112.2.b.e 2 792.be odd 6 1
2112.2.b.f 2 72.l even 6 1
2112.2.b.f 2 72.p odd 6 1
2112.2.b.f 2 792.s odd 6 1
2112.2.b.f 2 792.z even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(891, [\chi])\):

\( T_{2} \)
\( T_{23}^{4} - 11 T_{23}^{2} + 121 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} + 4 T^{4} )^{2} \)
$3$ \( \)
$5$ \( ( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} )( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} ) \)
$7$ \( ( 1 + 7 T^{2} + 49 T^{4} )^{2} \)
$11$ \( 1 - 11 T^{2} + 121 T^{4} \)
$13$ \( ( 1 + 13 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 17 T^{2} )^{4} \)
$19$ \( ( 1 - 19 T^{2} )^{4} \)
$23$ \( ( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4} )( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} ) \)
$29$ \( ( 1 - 29 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} )^{2} \)
$37$ \( ( 1 + 7 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 - 41 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 43 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} )( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} ) \)
$53$ \( ( 1 - 6 T + 53 T^{2} )^{2}( 1 + 6 T + 53 T^{2} )^{2} \)
$59$ \( ( 1 - 15 T + 166 T^{2} - 885 T^{3} + 3481 T^{4} )( 1 + 15 T + 166 T^{2} + 885 T^{3} + 3481 T^{4} ) \)
$61$ \( ( 1 + 61 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 3 T + 71 T^{2} )^{2}( 1 + 3 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 73 T^{2} )^{4} \)
$79$ \( ( 1 + 79 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 83 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 9 T + 89 T^{2} )^{2}( 1 + 9 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 + 17 T + 192 T^{2} + 1649 T^{3} + 9409 T^{4} )^{2} \)
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