Properties

Label 9464.2.a.h
Level 9464
Weight 2
Character orbit 9464.a
Self dual yes
Analytic conductor 75.570
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 9464 = 2^{3} \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9464.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(75.5704204729\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{3} + 4q^{5} - q^{7} + q^{9} + O(q^{10}) \) \( q + 2q^{3} + 4q^{5} - q^{7} + q^{9} + 8q^{15} - 2q^{17} + 2q^{19} - 2q^{21} + 8q^{23} + 11q^{25} - 4q^{27} + 2q^{29} - 4q^{31} - 4q^{35} + 6q^{37} + 2q^{41} + 8q^{43} + 4q^{45} + 4q^{47} + q^{49} - 4q^{51} - 10q^{53} + 4q^{57} - 6q^{59} + 4q^{61} - q^{63} + 12q^{67} + 16q^{69} + 14q^{73} + 22q^{75} - 8q^{79} - 11q^{81} - 6q^{83} - 8q^{85} + 4q^{87} - 10q^{89} - 8q^{93} + 8q^{95} + 2q^{97} + O(q^{100}) \)

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
0 2.00000 0 4.00000 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(13\) \(1\)

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 9464.2.a.h 1
13.b even 2 1 56.2.a.b 1
39.d odd 2 1 504.2.a.h 1
52.b odd 2 1 112.2.a.a 1
65.d even 2 1 1400.2.a.a 1
65.h odd 4 2 1400.2.g.b 2
91.b odd 2 1 392.2.a.b 1
91.r even 6 2 392.2.i.a 2
91.s odd 6 2 392.2.i.e 2
104.e even 2 1 448.2.a.c 1
104.h odd 2 1 448.2.a.h 1
143.d odd 2 1 6776.2.a.h 1
156.h even 2 1 1008.2.a.m 1
208.o odd 4 2 1792.2.b.h 2
208.p even 4 2 1792.2.b.a 2
260.g odd 2 1 2800.2.a.bd 1
260.p even 4 2 2800.2.g.g 2
273.g even 2 1 3528.2.a.b 1
273.w odd 6 2 3528.2.s.a 2
273.ba even 6 2 3528.2.s.ba 2
312.b odd 2 1 4032.2.a.d 1
312.h even 2 1 4032.2.a.a 1
364.h even 2 1 784.2.a.i 1
364.x even 6 2 784.2.i.b 2
364.bl odd 6 2 784.2.i.j 2
455.h odd 2 1 9800.2.a.bj 1
728.b even 2 1 3136.2.a.c 1
728.l odd 2 1 3136.2.a.w 1
1092.d odd 2 1 7056.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.b 1 13.b even 2 1
112.2.a.a 1 52.b odd 2 1
392.2.a.b 1 91.b odd 2 1
392.2.i.a 2 91.r even 6 2
392.2.i.e 2 91.s odd 6 2
448.2.a.c 1 104.e even 2 1
448.2.a.h 1 104.h odd 2 1
504.2.a.h 1 39.d odd 2 1
784.2.a.i 1 364.h even 2 1
784.2.i.b 2 364.x even 6 2
784.2.i.j 2 364.bl odd 6 2
1008.2.a.m 1 156.h even 2 1
1400.2.a.a 1 65.d even 2 1
1400.2.g.b 2 65.h odd 4 2
1792.2.b.a 2 208.p even 4 2
1792.2.b.h 2 208.o odd 4 2
2800.2.a.bd 1 260.g odd 2 1
2800.2.g.g 2 260.p even 4 2
3136.2.a.c 1 728.b even 2 1
3136.2.a.w 1 728.l odd 2 1
3528.2.a.b 1 273.g even 2 1
3528.2.s.a 2 273.w odd 6 2
3528.2.s.ba 2 273.ba even 6 2
4032.2.a.a 1 312.h even 2 1
4032.2.a.d 1 312.b odd 2 1
6776.2.a.h 1 143.d odd 2 1
7056.2.a.c 1 1092.d odd 2 1
9464.2.a.h 1 1.a even 1 1 trivial
9800.2.a.bj 1 455.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9464))\):

\( T_{3} - 2 \)
\( T_{5} - 4 \)
\( T_{11} \)
\( T_{17} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 2 T + 3 T^{2} \)
$5$ \( 1 - 4 T + 5 T^{2} \)
$7$ \( 1 + T \)
$11$ \( 1 + 11 T^{2} \)
$13$ 1
$17$ \( 1 + 2 T + 17 T^{2} \)
$19$ \( 1 - 2 T + 19 T^{2} \)
$23$ \( 1 - 8 T + 23 T^{2} \)
$29$ \( 1 - 2 T + 29 T^{2} \)
$31$ \( 1 + 4 T + 31 T^{2} \)
$37$ \( 1 - 6 T + 37 T^{2} \)
$41$ \( 1 - 2 T + 41 T^{2} \)
$43$ \( 1 - 8 T + 43 T^{2} \)
$47$ \( 1 - 4 T + 47 T^{2} \)
$53$ \( 1 + 10 T + 53 T^{2} \)
$59$ \( 1 + 6 T + 59 T^{2} \)
$61$ \( 1 - 4 T + 61 T^{2} \)
$67$ \( 1 - 12 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 - 14 T + 73 T^{2} \)
$79$ \( 1 + 8 T + 79 T^{2} \)
$83$ \( 1 + 6 T + 83 T^{2} \)
$89$ \( 1 + 10 T + 89 T^{2} \)
$97$ \( 1 - 2 T + 97 T^{2} \)
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