Properties

Label 176.2.a.a
Level $176$
Weight $2$
Character orbit 176.a
Self dual yes
Analytic conductor $1.405$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 176.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.40536707557\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 44)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - 3q^{5} - 2q^{7} - 2q^{9} + O(q^{10}) \) \( q - q^{3} - 3q^{5} - 2q^{7} - 2q^{9} + q^{11} - 4q^{13} + 3q^{15} + 6q^{17} - 8q^{19} + 2q^{21} + 3q^{23} + 4q^{25} + 5q^{27} - 5q^{31} - q^{33} + 6q^{35} - q^{37} + 4q^{39} + 10q^{43} + 6q^{45} - 3q^{49} - 6q^{51} - 6q^{53} - 3q^{55} + 8q^{57} - 3q^{59} - 4q^{61} + 4q^{63} + 12q^{65} + q^{67} - 3q^{69} - 15q^{71} - 4q^{73} - 4q^{75} - 2q^{77} - 2q^{79} + q^{81} - 6q^{83} - 18q^{85} - 9q^{89} + 8q^{91} + 5q^{93} + 24q^{95} - 7q^{97} - 2q^{99} + O(q^{100}) \)

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 −1.00000 0 −3.00000 0 −2.00000 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(-1\)

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 176.2.a.a 1
3.b odd 2 1 1584.2.a.p 1
4.b odd 2 1 44.2.a.a 1
5.b even 2 1 4400.2.a.v 1
5.c odd 4 2 4400.2.b.k 2
7.b odd 2 1 8624.2.a.w 1
8.b even 2 1 704.2.a.i 1
8.d odd 2 1 704.2.a.f 1
11.b odd 2 1 1936.2.a.c 1
12.b even 2 1 396.2.a.c 1
16.e even 4 2 2816.2.c.k 2
16.f odd 4 2 2816.2.c.e 2
20.d odd 2 1 1100.2.a.b 1
20.e even 4 2 1100.2.b.c 2
24.f even 2 1 6336.2.a.j 1
24.h odd 2 1 6336.2.a.i 1
28.d even 2 1 2156.2.a.a 1
28.f even 6 2 2156.2.i.c 2
28.g odd 6 2 2156.2.i.b 2
36.f odd 6 2 3564.2.i.j 2
36.h even 6 2 3564.2.i.a 2
44.c even 2 1 484.2.a.a 1
44.g even 10 4 484.2.e.b 4
44.h odd 10 4 484.2.e.a 4
52.b odd 2 1 7436.2.a.d 1
60.h even 2 1 9900.2.a.h 1
60.l odd 4 2 9900.2.c.g 2
88.b odd 2 1 7744.2.a.bc 1
88.g even 2 1 7744.2.a.m 1
132.d odd 2 1 4356.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.a.a 1 4.b odd 2 1
176.2.a.a 1 1.a even 1 1 trivial
396.2.a.c 1 12.b even 2 1
484.2.a.a 1 44.c even 2 1
484.2.e.a 4 44.h odd 10 4
484.2.e.b 4 44.g even 10 4
704.2.a.f 1 8.d odd 2 1
704.2.a.i 1 8.b even 2 1
1100.2.a.b 1 20.d odd 2 1
1100.2.b.c 2 20.e even 4 2
1584.2.a.p 1 3.b odd 2 1
1936.2.a.c 1 11.b odd 2 1
2156.2.a.a 1 28.d even 2 1
2156.2.i.b 2 28.g odd 6 2
2156.2.i.c 2 28.f even 6 2
2816.2.c.e 2 16.f odd 4 2
2816.2.c.k 2 16.e even 4 2
3564.2.i.a 2 36.h even 6 2
3564.2.i.j 2 36.f odd 6 2
4356.2.a.j 1 132.d odd 2 1
4400.2.a.v 1 5.b even 2 1
4400.2.b.k 2 5.c odd 4 2
6336.2.a.i 1 24.h odd 2 1
6336.2.a.j 1 24.f even 2 1
7436.2.a.d 1 52.b odd 2 1
7744.2.a.m 1 88.g even 2 1
7744.2.a.bc 1 88.b odd 2 1
8624.2.a.w 1 7.b odd 2 1
9900.2.a.h 1 60.h even 2 1
9900.2.c.g 2 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(176))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 1 + T \)
$5$ \( 3 + T \)
$7$ \( 2 + T \)
$11$ \( -1 + T \)
$13$ \( 4 + T \)
$17$ \( -6 + T \)
$19$ \( 8 + T \)
$23$ \( -3 + T \)
$29$ \( T \)
$31$ \( 5 + T \)
$37$ \( 1 + T \)
$41$ \( T \)
$43$ \( -10 + T \)
$47$ \( T \)
$53$ \( 6 + T \)
$59$ \( 3 + T \)
$61$ \( 4 + T \)
$67$ \( -1 + T \)
$71$ \( 15 + T \)
$73$ \( 4 + T \)
$79$ \( 2 + T \)
$83$ \( 6 + T \)
$89$ \( 9 + T \)
$97$ \( 7 + T \)
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