Properties

Label 891.2.e.k
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: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
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.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 2) q^{2} - 2 \zeta_{6} q^{4} - \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 2) q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 2) q^{2} - 2 \zeta_{6} q^{4} - \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 2) q^{7} - 2 q^{10} + (\zeta_{6} - 1) q^{11} - 4 \zeta_{6} q^{13} - 4 \zeta_{6} q^{14} + ( - 4 \zeta_{6} + 4) q^{16} - 2 q^{17} + (2 \zeta_{6} - 2) q^{20} + 2 \zeta_{6} q^{22} + \zeta_{6} q^{23} + ( - 4 \zeta_{6} + 4) q^{25} - 8 q^{26} - 4 q^{28} - 7 \zeta_{6} q^{31} - 8 \zeta_{6} q^{32} + (4 \zeta_{6} - 4) q^{34} - 2 q^{35} + 3 q^{37} + 8 \zeta_{6} q^{41} + ( - 6 \zeta_{6} + 6) q^{43} + 2 q^{44} + 2 q^{46} + (8 \zeta_{6} - 8) q^{47} + 3 \zeta_{6} q^{49} - 8 \zeta_{6} q^{50} + (8 \zeta_{6} - 8) q^{52} - 6 q^{53} + q^{55} - 5 \zeta_{6} q^{59} + (12 \zeta_{6} - 12) q^{61} - 14 q^{62} - 8 q^{64} + (4 \zeta_{6} - 4) q^{65} + 7 \zeta_{6} q^{67} + 4 \zeta_{6} q^{68} + (4 \zeta_{6} - 4) q^{70} - 3 q^{71} + 4 q^{73} + ( - 6 \zeta_{6} + 6) q^{74} + 2 \zeta_{6} q^{77} + ( - 10 \zeta_{6} + 10) q^{79} - 4 q^{80} + 16 q^{82} + ( - 6 \zeta_{6} + 6) q^{83} + 2 \zeta_{6} q^{85} - 12 \zeta_{6} q^{86} + 15 q^{89} - 8 q^{91} + ( - 2 \zeta_{6} + 2) q^{92} + 16 \zeta_{6} q^{94} + ( - 7 \zeta_{6} + 7) q^{97} + 6 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{4} - q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{4} - q^{5} + 2 q^{7} - 4 q^{10} - q^{11} - 4 q^{13} - 4 q^{14} + 4 q^{16} - 4 q^{17} - 2 q^{20} + 2 q^{22} + q^{23} + 4 q^{25} - 16 q^{26} - 8 q^{28} - 7 q^{31} - 8 q^{32} - 4 q^{34} - 4 q^{35} + 6 q^{37} + 8 q^{41} + 6 q^{43} + 4 q^{44} + 4 q^{46} - 8 q^{47} + 3 q^{49} - 8 q^{50} - 8 q^{52} - 12 q^{53} + 2 q^{55} - 5 q^{59} - 12 q^{61} - 28 q^{62} - 16 q^{64} - 4 q^{65} + 7 q^{67} + 4 q^{68} - 4 q^{70} - 6 q^{71} + 8 q^{73} + 6 q^{74} + 2 q^{77} + 10 q^{79} - 8 q^{80} + 32 q^{82} + 6 q^{83} + 2 q^{85} - 12 q^{86} + 30 q^{89} - 16 q^{91} + 2 q^{92} + 16 q^{94} + 7 q^{97} + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
1.00000 1.73205i 0 −1.00000 1.73205i −0.500000 0.866025i 0 1.00000 1.73205i 0 0 −2.00000
595.1 1.00000 + 1.73205i 0 −1.00000 + 1.73205i −0.500000 + 0.866025i 0 1.00000 + 1.73205i 0 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.k 2
3.b odd 2 1 891.2.e.b 2
9.c even 3 1 11.2.a.a 1
9.c even 3 1 inner 891.2.e.k 2
9.d odd 6 1 99.2.a.d 1
9.d odd 6 1 891.2.e.b 2
36.f odd 6 1 176.2.a.b 1
36.h even 6 1 1584.2.a.g 1
45.h odd 6 1 2475.2.a.a 1
45.j even 6 1 275.2.a.b 1
45.k odd 12 2 275.2.b.a 2
45.l even 12 2 2475.2.c.a 2
63.g even 3 1 539.2.e.h 2
63.h even 3 1 539.2.e.h 2
63.k odd 6 1 539.2.e.g 2
63.l odd 6 1 539.2.a.a 1
63.o even 6 1 4851.2.a.t 1
63.t odd 6 1 539.2.e.g 2
72.j odd 6 1 6336.2.a.br 1
72.l even 6 1 6336.2.a.bu 1
72.n even 6 1 704.2.a.h 1
72.p odd 6 1 704.2.a.c 1
99.g even 6 1 1089.2.a.b 1
99.h odd 6 1 121.2.a.d 1
99.m even 15 4 121.2.c.e 4
99.o odd 30 4 121.2.c.a 4
117.t even 6 1 1859.2.a.b 1
144.v odd 12 2 2816.2.c.f 2
144.x even 12 2 2816.2.c.j 2
153.h even 6 1 3179.2.a.a 1
171.o odd 6 1 3971.2.a.b 1
180.p odd 6 1 4400.2.a.i 1
180.x even 12 2 4400.2.b.h 2
207.f odd 6 1 5819.2.a.a 1
252.bi even 6 1 8624.2.a.j 1
261.i even 6 1 9251.2.a.d 1
396.k even 6 1 1936.2.a.i 1
495.o odd 6 1 3025.2.a.a 1
693.bj even 6 1 5929.2.a.h 1
792.z even 6 1 7744.2.a.k 1
792.be odd 6 1 7744.2.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.2.a.a 1 9.c even 3 1
99.2.a.d 1 9.d odd 6 1
121.2.a.d 1 99.h odd 6 1
121.2.c.a 4 99.o odd 30 4
121.2.c.e 4 99.m even 15 4
176.2.a.b 1 36.f odd 6 1
275.2.a.b 1 45.j even 6 1
275.2.b.a 2 45.k odd 12 2
539.2.a.a 1 63.l odd 6 1
539.2.e.g 2 63.k odd 6 1
539.2.e.g 2 63.t odd 6 1
539.2.e.h 2 63.g even 3 1
539.2.e.h 2 63.h even 3 1
704.2.a.c 1 72.p odd 6 1
704.2.a.h 1 72.n even 6 1
891.2.e.b 2 3.b odd 2 1
891.2.e.b 2 9.d odd 6 1
891.2.e.k 2 1.a even 1 1 trivial
891.2.e.k 2 9.c even 3 1 inner
1089.2.a.b 1 99.g even 6 1
1584.2.a.g 1 36.h even 6 1
1859.2.a.b 1 117.t even 6 1
1936.2.a.i 1 396.k even 6 1
2475.2.a.a 1 45.h odd 6 1
2475.2.c.a 2 45.l even 12 2
2816.2.c.f 2 144.v odd 12 2
2816.2.c.j 2 144.x even 12 2
3025.2.a.a 1 495.o odd 6 1
3179.2.a.a 1 153.h even 6 1
3971.2.a.b 1 171.o odd 6 1
4400.2.a.i 1 180.p odd 6 1
4400.2.b.h 2 180.x even 12 2
4851.2.a.t 1 63.o even 6 1
5819.2.a.a 1 207.f odd 6 1
5929.2.a.h 1 693.bj even 6 1
6336.2.a.br 1 72.j odd 6 1
6336.2.a.bu 1 72.l even 6 1
7744.2.a.k 1 792.z even 6 1
7744.2.a.x 1 792.be odd 6 1
8624.2.a.j 1 252.bi even 6 1
9251.2.a.d 1 261.i even 6 1

Hecke kernels

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

\( T_{2}^{2} - 2T_{2} + 4 \) Copy content Toggle raw display
\( T_{5}^{2} + T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$13$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$37$ \( (T - 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$43$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$47$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$61$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$67$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$71$ \( (T + 3)^{2} \) Copy content Toggle raw display
$73$ \( (T - 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$83$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$89$ \( (T - 15)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
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