Properties

Label 176.2.a.d
Level $176$
Weight $2$
Character orbit 176.a
Self dual yes
Analytic conductor $1.405$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 88)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{3} + ( 2 - \beta ) q^{5} + 2 \beta q^{7} + ( 1 + \beta ) q^{9} +O(q^{10})\) \( q -\beta q^{3} + ( 2 - \beta ) q^{5} + 2 \beta q^{7} + ( 1 + \beta ) q^{9} + q^{11} + ( -2 + 2 \beta ) q^{13} + ( 4 - \beta ) q^{15} + 2 q^{17} + 4 q^{19} + ( -8 - 2 \beta ) q^{21} + ( -4 - \beta ) q^{23} + ( 3 - 3 \beta ) q^{25} + ( -4 + \beta ) q^{27} + ( -2 + 2 \beta ) q^{29} + ( 4 - \beta ) q^{31} -\beta q^{33} + ( -8 + 2 \beta ) q^{35} + ( -6 + \beta ) q^{37} -8 q^{39} + ( 2 + 2 \beta ) q^{41} + ( 4 - 2 \beta ) q^{43} -2 q^{45} -8 q^{47} + ( 9 + 4 \beta ) q^{49} -2 \beta q^{51} + ( 6 - 4 \beta ) q^{53} + ( 2 - \beta ) q^{55} -4 \beta q^{57} + 5 \beta q^{59} + ( -2 - 2 \beta ) q^{61} + ( 8 + 4 \beta ) q^{63} + ( -12 + 4 \beta ) q^{65} + ( -8 + \beta ) q^{67} + ( 4 + 5 \beta ) q^{69} + ( 4 - 3 \beta ) q^{71} + ( 2 - 2 \beta ) q^{73} + 12 q^{75} + 2 \beta q^{77} + ( 8 - 2 \beta ) q^{79} -7 q^{81} + ( -4 - 2 \beta ) q^{83} + ( 4 - 2 \beta ) q^{85} -8 q^{87} + ( -2 - 3 \beta ) q^{89} + 16 q^{91} + ( 4 - 3 \beta ) q^{93} + ( 8 - 4 \beta ) q^{95} + ( 14 - \beta ) q^{97} + ( 1 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + 3q^{5} + 2q^{7} + 3q^{9} + O(q^{10}) \) \( 2q - q^{3} + 3q^{5} + 2q^{7} + 3q^{9} + 2q^{11} - 2q^{13} + 7q^{15} + 4q^{17} + 8q^{19} - 18q^{21} - 9q^{23} + 3q^{25} - 7q^{27} - 2q^{29} + 7q^{31} - q^{33} - 14q^{35} - 11q^{37} - 16q^{39} + 6q^{41} + 6q^{43} - 4q^{45} - 16q^{47} + 22q^{49} - 2q^{51} + 8q^{53} + 3q^{55} - 4q^{57} + 5q^{59} - 6q^{61} + 20q^{63} - 20q^{65} - 15q^{67} + 13q^{69} + 5q^{71} + 2q^{73} + 24q^{75} + 2q^{77} + 14q^{79} - 14q^{81} - 10q^{83} + 6q^{85} - 16q^{87} - 7q^{89} + 32q^{91} + 5q^{93} + 12q^{95} + 27q^{97} + 3q^{99} + O(q^{100}) \)

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
2.56155
−1.56155
0 −2.56155 0 −0.561553 0 5.12311 0 3.56155 0
1.2 0 1.56155 0 3.56155 0 −3.12311 0 −0.561553 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.d 2
3.b odd 2 1 1584.2.a.t 2
4.b odd 2 1 88.2.a.b 2
5.b even 2 1 4400.2.a.bp 2
5.c odd 4 2 4400.2.b.v 4
7.b odd 2 1 8624.2.a.cb 2
8.b even 2 1 704.2.a.p 2
8.d odd 2 1 704.2.a.m 2
11.b odd 2 1 1936.2.a.r 2
12.b even 2 1 792.2.a.h 2
16.e even 4 2 2816.2.c.p 4
16.f odd 4 2 2816.2.c.w 4
20.d odd 2 1 2200.2.a.o 2
20.e even 4 2 2200.2.b.g 4
24.f even 2 1 6336.2.a.cu 2
24.h odd 2 1 6336.2.a.cx 2
28.d even 2 1 4312.2.a.n 2
44.c even 2 1 968.2.a.j 2
44.g even 10 4 968.2.i.q 8
44.h odd 10 4 968.2.i.r 8
88.b odd 2 1 7744.2.a.cl 2
88.g even 2 1 7744.2.a.by 2
132.d odd 2 1 8712.2.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.a.b 2 4.b odd 2 1
176.2.a.d 2 1.a even 1 1 trivial
704.2.a.m 2 8.d odd 2 1
704.2.a.p 2 8.b even 2 1
792.2.a.h 2 12.b even 2 1
968.2.a.j 2 44.c even 2 1
968.2.i.q 8 44.g even 10 4
968.2.i.r 8 44.h odd 10 4
1584.2.a.t 2 3.b odd 2 1
1936.2.a.r 2 11.b odd 2 1
2200.2.a.o 2 20.d odd 2 1
2200.2.b.g 4 20.e even 4 2
2816.2.c.p 4 16.e even 4 2
2816.2.c.w 4 16.f odd 4 2
4312.2.a.n 2 28.d even 2 1
4400.2.a.bp 2 5.b even 2 1
4400.2.b.v 4 5.c odd 4 2
6336.2.a.cu 2 24.f even 2 1
6336.2.a.cx 2 24.h odd 2 1
7744.2.a.by 2 88.g even 2 1
7744.2.a.cl 2 88.b odd 2 1
8624.2.a.cb 2 7.b odd 2 1
8712.2.a.bb 2 132.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} - 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(176))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -4 + T + T^{2} \)
$5$ \( -2 - 3 T + T^{2} \)
$7$ \( -16 - 2 T + T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( -16 + 2 T + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( 16 + 9 T + T^{2} \)
$29$ \( -16 + 2 T + T^{2} \)
$31$ \( 8 - 7 T + T^{2} \)
$37$ \( 26 + 11 T + T^{2} \)
$41$ \( -8 - 6 T + T^{2} \)
$43$ \( -8 - 6 T + T^{2} \)
$47$ \( ( 8 + T )^{2} \)
$53$ \( -52 - 8 T + T^{2} \)
$59$ \( -100 - 5 T + T^{2} \)
$61$ \( -8 + 6 T + T^{2} \)
$67$ \( 52 + 15 T + T^{2} \)
$71$ \( -32 - 5 T + T^{2} \)
$73$ \( -16 - 2 T + T^{2} \)
$79$ \( 32 - 14 T + T^{2} \)
$83$ \( 8 + 10 T + T^{2} \)
$89$ \( -26 + 7 T + T^{2} \)
$97$ \( 178 - 27 T + T^{2} \)
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