Properties

Label 177.4.d.b
Level $177$
Weight $4$
Character orbit 177.d
Analytic conductor $10.443$
Analytic rank $0$
Dimension $4$
CM discriminant -59
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 177.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.4433380710\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-59})\)
Defining polynomial: \(x^{4} - x^{3} - 14 x^{2} - 15 x + 225\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 - \beta_{2} - \beta_{3} ) q^{3} -8 q^{4} + ( 4 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{5} + ( -16 - 2 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{7} + ( 3 + \beta_{1} - \beta_{2} + 6 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -2 - \beta_{2} - \beta_{3} ) q^{3} -8 q^{4} + ( 4 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{5} + ( -16 - 2 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{7} + ( 3 + \beta_{1} - \beta_{2} + 6 \beta_{3} ) q^{9} + ( 16 + 8 \beta_{2} + 8 \beta_{3} ) q^{12} + ( -46 - 15 \beta_{1} + \beta_{2} + 22 \beta_{3} ) q^{15} + 64 q^{16} + ( 8 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} ) q^{17} + ( -52 + 10 \beta_{1} + 5 \beta_{2} + 20 \beta_{3} ) q^{19} + ( -32 \beta_{1} + 24 \beta_{2} + 32 \beta_{3} ) q^{20} + ( 98 + 3 \beta_{1} + \beta_{2} + 22 \beta_{3} ) q^{21} + ( -231 - 14 \beta_{1} - 7 \beta_{2} - 28 \beta_{3} ) q^{25} + ( -112 + 11 \beta_{1} + 11 \beta_{2} - 11 \beta_{3} ) q^{27} + ( 128 + 16 \beta_{1} + 8 \beta_{2} + 32 \beta_{3} ) q^{28} + ( -44 \beta_{1} + 9 \beta_{2} + 44 \beta_{3} ) q^{29} + ( -120 \beta_{1} + 11 \beta_{2} + 120 \beta_{3} ) q^{35} + ( -24 - 8 \beta_{1} + 8 \beta_{2} - 48 \beta_{3} ) q^{36} + ( 88 \beta_{1} - 87 \beta_{2} - 88 \beta_{3} ) q^{41} + ( 30 + 101 \beta_{1} + 46 \beta_{2} - 30 \beta_{3} ) q^{45} + ( -128 - 64 \beta_{2} - 64 \beta_{3} ) q^{48} + ( 309 + 58 \beta_{1} + 29 \beta_{2} + 116 \beta_{3} ) q^{49} + ( 104 - 72 \beta_{1} + 16 \beta_{2} + 16 \beta_{3} ) q^{51} + ( 100 \beta_{1} - 9 \beta_{2} - 100 \beta_{3} ) q^{53} + ( -226 - 15 \beta_{1} + 127 \beta_{2} + 22 \beta_{3} ) q^{57} + ( -59 \beta_{1} - 59 \beta_{2} + 59 \beta_{3} ) q^{59} + ( 368 + 120 \beta_{1} - 8 \beta_{2} - 176 \beta_{3} ) q^{60} + ( -510 + 29 \beta_{1} - 44 \beta_{2} - 138 \beta_{3} ) q^{63} -512 q^{64} + ( -64 \beta_{1} - 64 \beta_{2} + 64 \beta_{3} ) q^{68} + ( -154 \beta_{1} - 154 \beta_{2} + 154 \beta_{3} ) q^{71} + ( 924 + 21 \beta_{1} + 126 \beta_{2} + 273 \beta_{3} ) q^{75} + ( 416 - 80 \beta_{1} - 40 \beta_{2} - 160 \beta_{3} ) q^{76} + ( 344 - 98 \beta_{1} - 49 \beta_{2} - 196 \beta_{3} ) q^{79} + ( 256 \beta_{1} - 192 \beta_{2} - 256 \beta_{3} ) q^{80} + ( 367 - 99 \beta_{1} + 134 \beta_{2} + 134 \beta_{3} ) q^{81} + ( -784 - 24 \beta_{1} - 8 \beta_{2} - 176 \beta_{3} ) q^{84} + ( -320 - 112 \beta_{1} - 56 \beta_{2} - 224 \beta_{3} ) q^{85} + ( 170 + 237 \beta_{1} - 35 \beta_{2} - 194 \beta_{3} ) q^{87} + ( 72 \beta_{1} + 341 \beta_{2} - 72 \beta_{3} ) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 7q^{3} - 32q^{4} - 58q^{7} + 5q^{9} + O(q^{10}) \) \( 4q - 7q^{3} - 32q^{4} - 58q^{7} + 5q^{9} + 56q^{12} - 191q^{15} + 256q^{16} - 238q^{19} + 367q^{21} - 882q^{25} - 448q^{27} + 464q^{28} - 40q^{36} + 49q^{45} - 448q^{48} + 1062q^{49} + 472q^{51} - 911q^{57} + 1528q^{60} - 1931q^{63} - 2048q^{64} + 3402q^{75} + 1904q^{76} + 1670q^{79} + 1433q^{81} - 2936q^{84} - 944q^{85} + 637q^{87} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 14 x^{2} - 15 x + 225\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 14 \nu^{2} + 56 \nu - 155 \)\()/70\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{3} - 7 \nu^{2} + 7 \nu + 5 \)\()/35\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 14 \nu + 15 \)\()/14\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-3 \beta_{3} - 5 \beta_{2} + 5 \beta_{1} + 15\)\()/2\)
\(\nu^{3}\)\(=\)\(-7 \beta_{3} + 7 \beta_{2} + 7 \beta_{1} + 22\)

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
3.57603 1.48727i
3.57603 + 1.48727i
−3.07603 + 2.35330i
−3.07603 2.35330i
0 −5.07603 1.11080i −8.00000 22.0271i 0 −34.4562 0 24.5322 + 11.2769i 0
176.2 0 −5.07603 + 1.11080i −8.00000 22.0271i 0 −34.4562 0 24.5322 11.2769i 0
176.3 0 1.57603 4.95138i −8.00000 14.3460i 0 5.45620 0 −22.0322 15.6071i 0
176.4 0 1.57603 + 4.95138i −8.00000 14.3460i 0 5.45620 0 −22.0322 + 15.6071i 0
\(n\): e.g. 2-40 or 990-1000
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 177.4.d.b 4
3.b odd 2 1 inner 177.4.d.b 4
59.b odd 2 1 CM 177.4.d.b 4
177.d even 2 1 inner 177.4.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.4.d.b 4 1.a even 1 1 trivial
177.4.d.b 4 3.b odd 2 1 inner
177.4.d.b 4 59.b odd 2 1 CM
177.4.d.b 4 177.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(177, [\chi])\):

\( T_{2} \)
\( T_{5}^{4} + 691 T_{5}^{2} + 99856 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 729 + 189 T + 22 T^{2} + 7 T^{3} + T^{4} \)
$5$ \( 99856 + 691 T^{2} + T^{4} \)
$7$ \( ( -188 + 29 T + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( 3776 + T^{2} )^{2} \)
$19$ \( ( -6416 + 119 T + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( 795664 + 74059 T^{2} + T^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( 42726543616 + 413467 T^{2} + T^{4} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( 1759634704 + 404683 T^{2} + T^{4} \)
$59$ \( ( 205379 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( ( 1399244 + T^{2} )^{2} \)
$73$ \( T^{4} \)
$79$ \( ( -781892 - 835 T + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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