Properties

Label 177.6.d.a
Level $177$
Weight $6$
Character orbit 177.d
Analytic conductor $28.388$
Analytic rank $0$
Dimension $6$
CM discriminant -59
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 177.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.3879361069\)
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.

\(f(q)\) \(=\) \( q + ( 4 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} ) q^{3} -32 q^{4} + ( 21 \beta_{1} - 24 \beta_{2} - 10 \beta_{3} - 7 \beta_{4} ) q^{5} + ( -67 \beta_{1} - 29 \beta_{2} + 29 \beta_{3} - 9 \beta_{4} ) q^{7} + ( -101 \beta_{1} + 22 \beta_{2} + 44 \beta_{3} + 22 \beta_{4} ) q^{9} +O(q^{10})\) \( q + ( 4 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} ) q^{3} -32 q^{4} + ( 21 \beta_{1} - 24 \beta_{2} - 10 \beta_{3} - 7 \beta_{4} ) q^{5} + ( -67 \beta_{1} - 29 \beta_{2} + 29 \beta_{3} - 9 \beta_{4} ) q^{7} + ( -101 \beta_{1} + 22 \beta_{2} + 44 \beta_{3} + 22 \beta_{4} ) q^{9} + ( -128 \beta_{2} - 96 \beta_{3} + 96 \beta_{4} ) q^{12} + ( 616 - 65 \beta_{1} - 131 \beta_{2} - 262 \beta_{3} - 131 \beta_{4} + 181 \beta_{5} ) q^{15} + 1024 q^{16} + ( 89 + 178 \beta_{5} ) q^{17} + ( 587 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + 595 \beta_{4} ) q^{19} + ( -672 \beta_{1} + 768 \beta_{2} + 320 \beta_{3} + 224 \beta_{4} ) q^{20} + ( -767 - 584 \beta_{1} - 281 \beta_{2} - 562 \beta_{3} - 281 \beta_{4} - 509 \beta_{5} ) q^{21} + ( -3125 + 1580 \beta_{1} + 243 \beta_{2} - 243 \beta_{3} + 1094 \beta_{4} ) q^{25} + ( -2411 - 839 \beta_{5} ) q^{27} + ( 2144 \beta_{1} + 928 \beta_{2} - 928 \beta_{3} + 288 \beta_{4} ) q^{28} + ( 1887 \beta_{1} - 1855 \beta_{2} - 597 \beta_{3} - 629 \beta_{4} ) q^{29} + ( 719 + 4185 \beta_{1} - 563 \beta_{2} + 2227 \beta_{3} - 1395 \beta_{4} + 1438 \beta_{5} ) q^{35} + ( 3232 \beta_{1} - 704 \beta_{2} - 1408 \beta_{3} - 704 \beta_{4} ) q^{36} + ( 4932 \beta_{1} + 1819 \beta_{2} + 5107 \beta_{3} - 1644 \beta_{4} ) q^{41} + ( 13690 - 251 \beta_{2} + 3477 \beta_{3} - 3477 \beta_{4} + 331 \beta_{5} ) q^{45} + ( 4096 \beta_{2} + 3072 \beta_{3} - 3072 \beta_{4} ) q^{48} + ( 16807 + 9449 \beta_{1} + 3486 \beta_{2} - 3486 \beta_{3} + 2477 \beta_{4} ) q^{49} + ( -2314 \beta_{2} + 1869 \beta_{3} - 1869 \beta_{4} ) q^{51} + ( -1056 \beta_{1} - 8863 \beta_{2} - 9567 \beta_{3} + 352 \beta_{4} ) q^{53} + ( -20378 + 13633 \beta_{1} - 635 \beta_{2} - 1270 \beta_{3} - 635 \beta_{4} + 2911 \beta_{5} ) q^{57} + ( 3481 + 6962 \beta_{5} ) q^{59} + ( -19712 + 2080 \beta_{1} + 4192 \beta_{2} + 8384 \beta_{3} + 4192 \beta_{4} - 5792 \beta_{5} ) q^{60} + ( 28921 + 4567 \beta_{2} - 6882 \beta_{3} + 6882 \beta_{4} - 2402 \beta_{5} ) q^{63} -32768 q^{64} + ( -2848 - 5696 \beta_{5} ) q^{68} + ( -7741 - 15482 \beta_{5} ) q^{71} + ( -28205 + 28321 \beta_{1} - 11164 \beta_{2} - 6703 \beta_{3} + 10711 \beta_{4} + 9358 \beta_{5} ) q^{75} + ( -18784 \beta_{1} + 128 \beta_{2} - 128 \beta_{3} - 19040 \beta_{4} ) q^{76} + ( 32126 \beta_{1} + 8789 \beta_{2} - 8789 \beta_{3} + 14548 \beta_{4} ) q^{79} + ( 21504 \beta_{1} - 24576 \beta_{2} - 10240 \beta_{3} - 7168 \beta_{4} ) q^{80} + ( 2941 \beta_{2} - 14784 \beta_{3} + 14784 \beta_{4} ) q^{81} + ( 24544 + 18688 \beta_{1} + 8992 \beta_{2} + 17984 \beta_{3} + 8992 \beta_{4} + 16288 \beta_{5} ) q^{84} + ( 16999 \beta_{1} + 1424 \beta_{2} - 1424 \beta_{3} + 14151 \beta_{4} ) q^{85} + ( 44797 - 9158 \beta_{1} - 10565 \beta_{2} - 21130 \beta_{3} - 10565 \beta_{4} + 15661 \beta_{5} ) q^{87} + ( -22558 - 17310 \beta_{1} + 41063 \beta_{2} + 29523 \beta_{3} + 5770 \beta_{4} - 45116 \beta_{5} ) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 192q^{4} + O(q^{10}) \) \( 6q - 192q^{4} + 3153q^{15} + 6144q^{16} - 3075q^{21} - 18750q^{25} - 11949q^{27} + 81147q^{45} + 100842q^{49} - 131001q^{57} - 100896q^{60} + 180732q^{63} - 196608q^{64} - 197304q^{75} + 98400q^{84} + 221799q^{87} + O(q^{100}) \)

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
−0.422380 1.67976i
−0.422380 + 1.67976i
1.66591 0.474089i
1.66591 + 0.474089i
−1.24353 + 1.20567i
−1.24353 1.20567i
0 −14.7009 5.18489i −32.0000 61.6839i 0 145.431 0 189.234 + 152.445i 0
176.2 0 −14.7009 + 5.18489i −32.0000 61.6839i 0 145.431 0 189.234 152.445i 0
176.3 0 2.86021 15.3238i −32.0000 49.9128i 0 −258.614 0 −226.638 87.6587i 0
176.4 0 2.86021 + 15.3238i −32.0000 49.9128i 0 −258.614 0 −226.638 + 87.6587i 0
176.5 0 11.8407 10.1389i −32.0000 111.597i 0 113.183 0 37.4045 240.104i 0
176.6 0 11.8407 + 10.1389i −32.0000 111.597i 0 113.183 0 37.4045 + 240.104i 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 177.6.d.a 6
3.b odd 2 1 inner 177.6.d.a 6
59.b odd 2 1 CM 177.6.d.a 6
177.d even 2 1 inner 177.6.d.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.6.d.a 6 1.a even 1 1 trivial
177.6.d.a 6 3.b odd 2 1 inner
177.6.d.a 6 59.b odd 2 1 CM
177.6.d.a 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 \(S_{6}^{\mathrm{new}}(177, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 14348907 + 3983 T^{3} + T^{6} \)
$5$ \( 118050879299 + 87890625 T^{2} + 18750 T^{4} + T^{6} \)
$7$ \( ( 4256881 - 50421 T + T^{3} )^{2} \)
$11$ \( T^{6} \)
$13$ \( T^{6} \)
$17$ \( ( 467339 + T^{2} )^{3} \)
$19$ \( ( 5936692111 - 7428297 T + T^{3} )^{2} \)
$23$ \( T^{6} \)
$29$ \( \)\(27\!\cdots\!75\)\( + 3786365099701809 T^{2} + 123066894 T^{4} + T^{6} \)
$31$ \( T^{6} \)
$37$ \( T^{6} \)
$41$ \( \)\(33\!\cdots\!79\)\( + 120803933791371609 T^{2} + 695137206 T^{4} + T^{6} \)
$43$ \( T^{6} \)
$47$ \( T^{6} \)
$53$ \( \)\(97\!\cdots\!39\)\( + 1573987233289617441 T^{2} + 2509172958 T^{4} + T^{6} \)
$59$ \( ( 714924299 + T^{2} )^{3} \)
$61$ \( T^{6} \)
$67$ \( T^{6} \)
$71$ \( ( 3535461779 + T^{2} )^{3} \)
$73$ \( T^{6} \)
$79$ \( ( -14391730713575 - 9231169197 T + T^{3} )^{2} \)
$83$ \( T^{6} \)
$89$ \( T^{6} \)
$97$ \( T^{6} \)
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