Properties

Label 891.2.e.j
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: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 99)
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{2} + \zeta_{6} q^{4} + 4 \zeta_{6} q^{5} + ( 2 - 2 \zeta_{6} ) q^{7} + 3 q^{8} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{2} + \zeta_{6} q^{4} + 4 \zeta_{6} q^{5} + ( 2 - 2 \zeta_{6} ) q^{7} + 3 q^{8} + 4 q^{10} + ( 1 - \zeta_{6} ) q^{11} + 2 \zeta_{6} q^{13} -2 \zeta_{6} q^{14} + ( 1 - \zeta_{6} ) q^{16} + 2 q^{17} -6 q^{19} + ( -4 + 4 \zeta_{6} ) q^{20} -\zeta_{6} q^{22} -4 \zeta_{6} q^{23} + ( -11 + 11 \zeta_{6} ) q^{25} + 2 q^{26} + 2 q^{28} + ( 6 - 6 \zeta_{6} ) q^{29} -4 \zeta_{6} q^{31} + 5 \zeta_{6} q^{32} + ( 2 - 2 \zeta_{6} ) q^{34} + 8 q^{35} -6 q^{37} + ( -6 + 6 \zeta_{6} ) q^{38} + 12 \zeta_{6} q^{40} + 10 \zeta_{6} q^{41} + ( -6 + 6 \zeta_{6} ) q^{43} + q^{44} -4 q^{46} + ( 8 - 8 \zeta_{6} ) q^{47} + 3 \zeta_{6} q^{49} + 11 \zeta_{6} q^{50} + ( -2 + 2 \zeta_{6} ) q^{52} + 4 q^{55} + ( 6 - 6 \zeta_{6} ) q^{56} -6 \zeta_{6} q^{58} -4 \zeta_{6} q^{59} + ( 6 - 6 \zeta_{6} ) q^{61} -4 q^{62} + 7 q^{64} + ( -8 + 8 \zeta_{6} ) q^{65} -8 \zeta_{6} q^{67} + 2 \zeta_{6} q^{68} + ( 8 - 8 \zeta_{6} ) q^{70} -2 q^{73} + ( -6 + 6 \zeta_{6} ) q^{74} -6 \zeta_{6} q^{76} -2 \zeta_{6} q^{77} + ( 10 - 10 \zeta_{6} ) q^{79} + 4 q^{80} + 10 q^{82} + ( -12 + 12 \zeta_{6} ) q^{83} + 8 \zeta_{6} q^{85} + 6 \zeta_{6} q^{86} + ( 3 - 3 \zeta_{6} ) q^{88} + 4 q^{91} + ( 4 - 4 \zeta_{6} ) q^{92} -8 \zeta_{6} q^{94} -24 \zeta_{6} q^{95} + ( -2 + 2 \zeta_{6} ) q^{97} + 3 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{4} + 4 q^{5} + 2 q^{7} + 6 q^{8} + O(q^{10}) \) \( 2 q + q^{2} + q^{4} + 4 q^{5} + 2 q^{7} + 6 q^{8} + 8 q^{10} + q^{11} + 2 q^{13} - 2 q^{14} + q^{16} + 4 q^{17} - 12 q^{19} - 4 q^{20} - q^{22} - 4 q^{23} - 11 q^{25} + 4 q^{26} + 4 q^{28} + 6 q^{29} - 4 q^{31} + 5 q^{32} + 2 q^{34} + 16 q^{35} - 12 q^{37} - 6 q^{38} + 12 q^{40} + 10 q^{41} - 6 q^{43} + 2 q^{44} - 8 q^{46} + 8 q^{47} + 3 q^{49} + 11 q^{50} - 2 q^{52} + 8 q^{55} + 6 q^{56} - 6 q^{58} - 4 q^{59} + 6 q^{61} - 8 q^{62} + 14 q^{64} - 8 q^{65} - 8 q^{67} + 2 q^{68} + 8 q^{70} - 4 q^{73} - 6 q^{74} - 6 q^{76} - 2 q^{77} + 10 q^{79} + 8 q^{80} + 20 q^{82} - 12 q^{83} + 8 q^{85} + 6 q^{86} + 3 q^{88} + 8 q^{91} + 4 q^{92} - 8 q^{94} - 24 q^{95} - 2 q^{97} + 6 q^{98} + O(q^{100}) \)

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 2.00000 + 3.46410i 0 1.00000 1.73205i 3.00000 0 4.00000
595.1 0.500000 + 0.866025i 0 0.500000 0.866025i 2.00000 3.46410i 0 1.00000 + 1.73205i 3.00000 0 4.00000
\(n\): 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.j 2
3.b odd 2 1 891.2.e.c 2
9.c even 3 1 99.2.a.a 1
9.c even 3 1 inner 891.2.e.j 2
9.d odd 6 1 99.2.a.c yes 1
9.d odd 6 1 891.2.e.c 2
36.f odd 6 1 1584.2.a.b 1
36.h even 6 1 1584.2.a.r 1
45.h odd 6 1 2475.2.a.c 1
45.j even 6 1 2475.2.a.j 1
45.k odd 12 2 2475.2.c.b 2
45.l even 12 2 2475.2.c.g 2
63.l odd 6 1 4851.2.a.g 1
63.o even 6 1 4851.2.a.o 1
72.j odd 6 1 6336.2.a.b 1
72.l even 6 1 6336.2.a.f 1
72.n even 6 1 6336.2.a.cl 1
72.p odd 6 1 6336.2.a.cm 1
99.g even 6 1 1089.2.a.d 1
99.h odd 6 1 1089.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.a.a 1 9.c even 3 1
99.2.a.c yes 1 9.d odd 6 1
891.2.e.c 2 3.b odd 2 1
891.2.e.c 2 9.d odd 6 1
891.2.e.j 2 1.a even 1 1 trivial
891.2.e.j 2 9.c even 3 1 inner
1089.2.a.d 1 99.g even 6 1
1089.2.a.h 1 99.h odd 6 1
1584.2.a.b 1 36.f odd 6 1
1584.2.a.r 1 36.h even 6 1
2475.2.a.c 1 45.h odd 6 1
2475.2.a.j 1 45.j even 6 1
2475.2.c.b 2 45.k odd 12 2
2475.2.c.g 2 45.l even 12 2
4851.2.a.g 1 63.l odd 6 1
4851.2.a.o 1 63.o even 6 1
6336.2.a.b 1 72.j odd 6 1
6336.2.a.f 1 72.l even 6 1
6336.2.a.cl 1 72.n even 6 1
6336.2.a.cm 1 72.p odd 6 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 16 - 4 T + T^{2} \)
$7$ \( 4 - 2 T + T^{2} \)
$11$ \( 1 - T + T^{2} \)
$13$ \( 4 - 2 T + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( ( 6 + T )^{2} \)
$23$ \( 16 + 4 T + T^{2} \)
$29$ \( 36 - 6 T + T^{2} \)
$31$ \( 16 + 4 T + T^{2} \)
$37$ \( ( 6 + T )^{2} \)
$41$ \( 100 - 10 T + T^{2} \)
$43$ \( 36 + 6 T + T^{2} \)
$47$ \( 64 - 8 T + T^{2} \)
$53$ \( T^{2} \)
$59$ \( 16 + 4 T + T^{2} \)
$61$ \( 36 - 6 T + T^{2} \)
$67$ \( 64 + 8 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 2 + T )^{2} \)
$79$ \( 100 - 10 T + T^{2} \)
$83$ \( 144 + 12 T + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 4 + 2 T + T^{2} \)
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