Properties

Label 8993.2.a.a
Level $8993$
Weight $2$
Character orbit 8993.a
Self dual yes
Analytic conductor $71.809$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8993 = 17 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8993.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(71.8094665377\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{4} + 2 q^{5} - 4 q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{4} + 2 q^{5} - 4 q^{7} + 3 q^{8} - 3 q^{9} - 2 q^{10} - 2 q^{13} + 4 q^{14} - q^{16} - q^{17} + 3 q^{18} + 4 q^{19} - 2 q^{20} - q^{25} + 2 q^{26} + 4 q^{28} + 6 q^{29} + 4 q^{31} - 5 q^{32} + q^{34} - 8 q^{35} + 3 q^{36} + 2 q^{37} - 4 q^{38} + 6 q^{40} - 6 q^{41} - 4 q^{43} - 6 q^{45} + 9 q^{49} + q^{50} + 2 q^{52} - 6 q^{53} - 12 q^{56} - 6 q^{58} - 12 q^{59} + 10 q^{61} - 4 q^{62} + 12 q^{63} + 7 q^{64} - 4 q^{65} - 4 q^{67} + q^{68} + 8 q^{70} - 4 q^{71} - 9 q^{72} - 6 q^{73} - 2 q^{74} - 4 q^{76} - 12 q^{79} - 2 q^{80} + 9 q^{81} + 6 q^{82} + 4 q^{83} - 2 q^{85} + 4 q^{86} - 10 q^{89} + 6 q^{90} + 8 q^{91} + 8 q^{95} - 2 q^{97} - 9 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−1.00000 0 −1.00000 2.00000 0 −4.00000 3.00000 −3.00000 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(17\) \(1\)
\(23\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8993.2.a.a 1
23.b odd 2 1 17.2.a.a 1
69.c even 2 1 153.2.a.c 1
92.b even 2 1 272.2.a.b 1
115.c odd 2 1 425.2.a.d 1
115.e even 4 2 425.2.b.b 2
161.c even 2 1 833.2.a.a 1
161.f odd 6 2 833.2.e.b 2
161.g even 6 2 833.2.e.a 2
184.e odd 2 1 1088.2.a.i 1
184.h even 2 1 1088.2.a.h 1
253.b even 2 1 2057.2.a.e 1
276.h odd 2 1 2448.2.a.o 1
299.c odd 2 1 2873.2.a.c 1
345.h even 2 1 3825.2.a.d 1
391.c odd 2 1 289.2.a.a 1
391.f odd 4 2 289.2.b.a 2
391.h odd 8 4 289.2.c.a 4
391.k even 16 8 289.2.d.d 8
437.b even 2 1 6137.2.a.b 1
460.g even 2 1 6800.2.a.n 1
483.c odd 2 1 7497.2.a.l 1
552.b even 2 1 9792.2.a.n 1
552.h odd 2 1 9792.2.a.i 1
1173.b even 2 1 2601.2.a.g 1
1564.h even 2 1 4624.2.a.d 1
1955.d odd 2 1 7225.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.a.a 1 23.b odd 2 1
153.2.a.c 1 69.c even 2 1
272.2.a.b 1 92.b even 2 1
289.2.a.a 1 391.c odd 2 1
289.2.b.a 2 391.f odd 4 2
289.2.c.a 4 391.h odd 8 4
289.2.d.d 8 391.k even 16 8
425.2.a.d 1 115.c odd 2 1
425.2.b.b 2 115.e even 4 2
833.2.a.a 1 161.c even 2 1
833.2.e.a 2 161.g even 6 2
833.2.e.b 2 161.f odd 6 2
1088.2.a.h 1 184.h even 2 1
1088.2.a.i 1 184.e odd 2 1
2057.2.a.e 1 253.b even 2 1
2448.2.a.o 1 276.h odd 2 1
2601.2.a.g 1 1173.b even 2 1
2873.2.a.c 1 299.c odd 2 1
3825.2.a.d 1 345.h even 2 1
4624.2.a.d 1 1564.h even 2 1
6137.2.a.b 1 437.b even 2 1
6800.2.a.n 1 460.g even 2 1
7225.2.a.g 1 1955.d odd 2 1
7497.2.a.l 1 483.c odd 2 1
8993.2.a.a 1 1.a even 1 1 trivial
9792.2.a.i 1 552.h odd 2 1
9792.2.a.n 1 552.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8993))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T + 4 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T + 1 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T + 12 \) Copy content Toggle raw display
$61$ \( T - 10 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T + 4 \) Copy content Toggle raw display
$73$ \( T + 6 \) Copy content Toggle raw display
$79$ \( T + 12 \) Copy content Toggle raw display
$83$ \( T - 4 \) Copy content Toggle raw display
$89$ \( T + 10 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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