Properties

Label 177.2.d.b
Level $177$
Weight $2$
Character orbit 177.d
Analytic conductor $1.413$
Analytic rank $0$
Dimension $6$
CM discriminant -59
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 177.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.41335211578\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.149721291.1
Defining polynomial: \(x^{6} - 7 x^{3} + 27\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -2 q^{4} + ( \beta_{2} + \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{4} ) q^{7} + \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -2 q^{4} + ( \beta_{2} + \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{4} ) q^{7} + \beta_{2} q^{9} -2 \beta_{1} q^{12} + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} ) q^{15} + 4 q^{16} + ( -1 - 2 \beta_{5} ) q^{17} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{19} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{20} + ( -2 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{21} + ( -5 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{25} + ( 4 + \beta_{5} ) q^{27} + ( 2 \beta_{1} + 2 \beta_{4} ) q^{28} + ( 3 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{29} + ( -1 - 3 \beta_{1} + 2 \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{35} -2 \beta_{2} q^{36} + ( -3 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{41} + ( -5 + \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{45} + 4 \beta_{1} q^{48} + ( 7 - 4 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{49} + ( -\beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} ) q^{51} + ( 3 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{53} + ( 7 + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{57} + ( 1 + 2 \beta_{5} ) q^{59} + ( -2 - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{60} + ( 1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{63} -8 q^{64} + ( 2 + 4 \beta_{5} ) q^{68} + ( -1 - 2 \beta_{5} ) q^{71} + ( -5 - 5 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{75} + ( -4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{76} + ( -\beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{79} + ( 4 \beta_{2} + 4 \beta_{3} ) q^{80} + ( 4 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{81} + ( 4 + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{84} + ( 10 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} ) q^{85} + ( -8 + 3 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{87} + ( 2 + 3 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} - \beta_{4} + 4 \beta_{5} ) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 12q^{4} + O(q^{10}) \) \( 6q - 12q^{4} + 3q^{15} + 24q^{16} - 15q^{21} - 30q^{25} + 21q^{27} - 33q^{45} + 42q^{49} + 39q^{57} - 6q^{60} + 12q^{63} - 48q^{64} - 24q^{75} + 30q^{84} - 51q^{87} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 7 x^{3} + 27\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} + 3 \nu^{4} - 7 \nu^{2} - 12 \nu \)\()/9\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{5} + 3 \nu^{4} + 5 \nu^{2} - 12 \nu \)\()/9\)
\(\beta_{5}\)\(=\)\( \nu^{3} - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{5} + 4\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 2 \beta_{3} + \beta_{2} + 4 \beta_{1}\)
\(\nu^{5}\)\(=\)\(-3 \beta_{4} + 3 \beta_{3} + 4 \beta_{2}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

\(n\) \(61\) \(119\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
176.1
−1.24353 1.20567i
−1.24353 + 1.20567i
−0.422380 1.67976i
−0.422380 + 1.67976i
1.66591 0.474089i
1.66591 + 0.474089i
0 −1.24353 1.20567i −2.00000 3.58579i 0 5.15952 0 0.0927106 + 2.99857i 0
176.2 0 −1.24353 + 1.20567i −2.00000 3.58579i 0 5.15952 0 0.0927106 2.99857i 0
176.3 0 −0.422380 1.67976i −2.00000 0.521533i 0 −3.59686 0 −2.64319 + 1.41899i 0
176.4 0 −0.422380 + 1.67976i −2.00000 0.521533i 0 −3.59686 0 −2.64319 1.41899i 0
176.5 0 1.66591 0.474089i −2.00000 4.10732i 0 −1.56266 0 2.55048 1.57957i 0
176.6 0 1.66591 + 0.474089i −2.00000 4.10732i 0 −1.56266 0 2.55048 + 1.57957i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 176.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.b odd 2 1 CM by \(\Q(\sqrt{-59}) \)
3.b odd 2 1 inner
177.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.2.d.b 6
3.b odd 2 1 inner 177.2.d.b 6
59.b odd 2 1 CM 177.2.d.b 6
177.d even 2 1 inner 177.2.d.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.2.d.b 6 1.a even 1 1 trivial
177.2.d.b 6 3.b odd 2 1 inner
177.2.d.b 6 59.b odd 2 1 CM
177.2.d.b 6 177.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(177, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 27 - 7 T^{3} + T^{6} \)
$5$ \( 59 + 225 T^{2} + 30 T^{4} + T^{6} \)
$7$ \( ( -29 - 21 T + T^{3} )^{2} \)
$11$ \( T^{6} \)
$13$ \( T^{6} \)
$17$ \( ( 59 + T^{2} )^{3} \)
$19$ \( ( -119 - 57 T + T^{3} )^{2} \)
$23$ \( T^{6} \)
$29$ \( 72275 + 7569 T^{2} + 174 T^{4} + T^{6} \)
$31$ \( T^{6} \)
$37$ \( T^{6} \)
$41$ \( 59 + 15129 T^{2} + 246 T^{4} + T^{6} \)
$43$ \( T^{6} \)
$47$ \( T^{6} \)
$53$ \( 488579 + 25281 T^{2} + 318 T^{4} + T^{6} \)
$59$ \( ( 59 + T^{2} )^{3} \)
$61$ \( T^{6} \)
$67$ \( T^{6} \)
$71$ \( ( 59 + T^{2} )^{3} \)
$73$ \( T^{6} \)
$79$ \( ( 835 - 237 T + T^{3} )^{2} \)
$83$ \( T^{6} \)
$89$ \( T^{6} \)
$97$ \( T^{6} \)
show more
show less