Properties

Label 891.2.e.g
Level 891
Weight 2
Character orbit 891.e
Analytic conductor 7.115
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(7.11467082010\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{2} + \zeta_{6} q^{4} -2 \zeta_{6} q^{5} + ( -4 + 4 \zeta_{6} ) q^{7} + 3 q^{8} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{2} + \zeta_{6} q^{4} -2 \zeta_{6} q^{5} + ( -4 + 4 \zeta_{6} ) q^{7} + 3 q^{8} -2 q^{10} + ( 1 - \zeta_{6} ) q^{11} + 2 \zeta_{6} q^{13} + 4 \zeta_{6} q^{14} + ( 1 - \zeta_{6} ) q^{16} + 2 q^{17} + ( 2 - 2 \zeta_{6} ) q^{20} -\zeta_{6} q^{22} + 8 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} + 2 q^{26} -4 q^{28} + ( -6 + 6 \zeta_{6} ) q^{29} + 8 \zeta_{6} q^{31} + 5 \zeta_{6} q^{32} + ( 2 - 2 \zeta_{6} ) q^{34} + 8 q^{35} + 6 q^{37} -6 \zeta_{6} q^{40} -2 \zeta_{6} q^{41} + q^{44} + 8 q^{46} + ( 8 - 8 \zeta_{6} ) q^{47} -9 \zeta_{6} q^{49} -\zeta_{6} q^{50} + ( -2 + 2 \zeta_{6} ) q^{52} -6 q^{53} -2 q^{55} + ( -12 + 12 \zeta_{6} ) q^{56} + 6 \zeta_{6} q^{58} -4 \zeta_{6} q^{59} + ( -6 + 6 \zeta_{6} ) q^{61} + 8 q^{62} + 7 q^{64} + ( 4 - 4 \zeta_{6} ) q^{65} + 4 \zeta_{6} q^{67} + 2 \zeta_{6} q^{68} + ( 8 - 8 \zeta_{6} ) q^{70} -14 q^{73} + ( 6 - 6 \zeta_{6} ) q^{74} + 4 \zeta_{6} q^{77} + ( 4 - 4 \zeta_{6} ) q^{79} -2 q^{80} -2 q^{82} + ( 12 - 12 \zeta_{6} ) q^{83} -4 \zeta_{6} q^{85} + ( 3 - 3 \zeta_{6} ) q^{88} + 6 q^{89} -8 q^{91} + ( -8 + 8 \zeta_{6} ) q^{92} -8 \zeta_{6} q^{94} + ( -2 + 2 \zeta_{6} ) q^{97} -9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + q^{4} - 2q^{5} - 4q^{7} + 6q^{8} + O(q^{10}) \) \( 2q + q^{2} + q^{4} - 2q^{5} - 4q^{7} + 6q^{8} - 4q^{10} + q^{11} + 2q^{13} + 4q^{14} + q^{16} + 4q^{17} + 2q^{20} - q^{22} + 8q^{23} + q^{25} + 4q^{26} - 8q^{28} - 6q^{29} + 8q^{31} + 5q^{32} + 2q^{34} + 16q^{35} + 12q^{37} - 6q^{40} - 2q^{41} + 2q^{44} + 16q^{46} + 8q^{47} - 9q^{49} - q^{50} - 2q^{52} - 12q^{53} - 4q^{55} - 12q^{56} + 6q^{58} - 4q^{59} - 6q^{61} + 16q^{62} + 14q^{64} + 4q^{65} + 4q^{67} + 2q^{68} + 8q^{70} - 28q^{73} + 6q^{74} + 4q^{77} + 4q^{79} - 4q^{80} - 4q^{82} + 12q^{83} - 4q^{85} + 3q^{88} + 12q^{89} - 16q^{91} - 8q^{92} - 8q^{94} - 2q^{97} - 18q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(244\) \(650\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
298.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i 0 0.500000 + 0.866025i −1.00000 1.73205i 0 −2.00000 + 3.46410i 3.00000 0 −2.00000
595.1 0.500000 + 0.866025i 0 0.500000 0.866025i −1.00000 + 1.73205i 0 −2.00000 3.46410i 3.00000 0 −2.00000
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 891.2.e.g 2
3.b odd 2 1 891.2.e.e 2
9.c even 3 1 99.2.a.b 1
9.c even 3 1 inner 891.2.e.g 2
9.d odd 6 1 33.2.a.a 1
9.d odd 6 1 891.2.e.e 2
36.f odd 6 1 1584.2.a.o 1
36.h even 6 1 528.2.a.g 1
45.h odd 6 1 825.2.a.a 1
45.j even 6 1 2475.2.a.g 1
45.k odd 12 2 2475.2.c.d 2
45.l even 12 2 825.2.c.a 2
63.l odd 6 1 4851.2.a.b 1
63.o even 6 1 1617.2.a.j 1
72.j odd 6 1 2112.2.a.bb 1
72.l even 6 1 2112.2.a.j 1
72.n even 6 1 6336.2.a.x 1
72.p odd 6 1 6336.2.a.n 1
99.g even 6 1 363.2.a.b 1
99.h odd 6 1 1089.2.a.j 1
99.n odd 30 4 363.2.e.e 4
99.p even 30 4 363.2.e.g 4
117.n odd 6 1 5577.2.a.a 1
153.i odd 6 1 9537.2.a.m 1
396.o odd 6 1 5808.2.a.t 1
495.r even 6 1 9075.2.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 9.d odd 6 1
99.2.a.b 1 9.c even 3 1
363.2.a.b 1 99.g even 6 1
363.2.e.e 4 99.n odd 30 4
363.2.e.g 4 99.p even 30 4
528.2.a.g 1 36.h even 6 1
825.2.a.a 1 45.h odd 6 1
825.2.c.a 2 45.l even 12 2
891.2.e.e 2 3.b odd 2 1
891.2.e.e 2 9.d odd 6 1
891.2.e.g 2 1.a even 1 1 trivial
891.2.e.g 2 9.c even 3 1 inner
1089.2.a.j 1 99.h odd 6 1
1584.2.a.o 1 36.f odd 6 1
1617.2.a.j 1 63.o even 6 1
2112.2.a.j 1 72.l even 6 1
2112.2.a.bb 1 72.j odd 6 1
2475.2.a.g 1 45.j even 6 1
2475.2.c.d 2 45.k odd 12 2
4851.2.a.b 1 63.l odd 6 1
5577.2.a.a 1 117.n odd 6 1
5808.2.a.t 1 396.o odd 6 1
6336.2.a.n 1 72.p odd 6 1
6336.2.a.x 1 72.n even 6 1
9075.2.a.q 1 495.r even 6 1
9537.2.a.m 1 153.i odd 6 1

Hecke kernels

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

\( T_{2}^{2} - T_{2} + 1 \)
\( T_{5}^{2} + 2 T_{5} + 4 \)
\( T_{7}^{2} + 4 T_{7} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T - T^{2} - 2 T^{3} + 4 T^{4} \)
$3$ 1
$5$ \( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \)
$7$ \( ( 1 - T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \)
$11$ \( 1 - T + T^{2} \)
$13$ \( ( 1 - 7 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \)
$17$ \( ( 1 - 2 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + 19 T^{2} )^{2} \)
$23$ \( 1 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4} \)
$29$ \( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} \)
$31$ \( 1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - 6 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 2 T - 37 T^{2} + 82 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 43 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 8 T + 17 T^{2} - 376 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 + 6 T + 53 T^{2} )^{2} \)
$59$ \( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 6 T - 25 T^{2} + 366 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 14 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \)
$83$ \( 1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 - 6 T + 89 T^{2} )^{2} \)
$97$ \( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \)
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