Properties

Label 177.4.d.a
Level $177$
Weight $4$
Character orbit 177.d
Analytic conductor $10.443$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-59}) \)
Defining polynomial: \(x^{2} - x + 15\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-59})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 4 - \beta ) q^{3} -8 q^{4} + ( -1 + 2 \beta ) q^{5} + 29 q^{7} + ( 1 - 7 \beta ) q^{9} +O(q^{10})\) \( q + ( 4 - \beta ) q^{3} -8 q^{4} + ( -1 + 2 \beta ) q^{5} + 29 q^{7} + ( 1 - 7 \beta ) q^{9} + ( -32 + 8 \beta ) q^{12} + ( 26 + 7 \beta ) q^{15} + 64 q^{16} + ( 8 - 16 \beta ) q^{17} + 119 q^{19} + ( 8 - 16 \beta ) q^{20} + ( 116 - 29 \beta ) q^{21} + 66 q^{25} + ( -101 - 22 \beta ) q^{27} -232 q^{28} + ( 35 - 70 \beta ) q^{29} + ( -29 + 58 \beta ) q^{35} + ( -8 + 56 \beta ) q^{36} + ( -1 + 2 \beta ) q^{41} + ( 209 - 5 \beta ) q^{45} + ( 256 - 64 \beta ) q^{48} + 498 q^{49} + ( -208 - 56 \beta ) q^{51} + ( -91 + 182 \beta ) q^{53} + ( 476 - 119 \beta ) q^{57} + ( -59 + 118 \beta ) q^{59} + ( -208 - 56 \beta ) q^{60} + ( 29 - 203 \beta ) q^{63} -512 q^{64} + ( -64 + 128 \beta ) q^{68} + ( -154 + 308 \beta ) q^{71} + ( 264 - 66 \beta ) q^{75} -952 q^{76} -835 q^{79} + ( -64 + 128 \beta ) q^{80} + ( -734 + 35 \beta ) q^{81} + ( -928 + 232 \beta ) q^{84} + 472 q^{85} + ( -910 - 245 \beta ) q^{87} + ( -119 + 238 \beta ) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 7q^{3} - 16q^{4} + 58q^{7} - 5q^{9} + O(q^{10}) \) \( 2q + 7q^{3} - 16q^{4} + 58q^{7} - 5q^{9} - 56q^{12} + 59q^{15} + 128q^{16} + 238q^{19} + 203q^{21} + 132q^{25} - 224q^{27} - 464q^{28} + 40q^{36} + 413q^{45} + 448q^{48} + 996q^{49} - 472q^{51} + 833q^{57} - 472q^{60} - 145q^{63} - 1024q^{64} + 462q^{75} - 1904q^{76} - 1670q^{79} - 1433q^{81} - 1624q^{84} + 944q^{85} - 2065q^{87} + O(q^{100}) \)

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
0.500000 + 3.84057i
0.500000 3.84057i
0 3.50000 3.84057i −8.00000 7.68115i 0 29.0000 0 −2.50000 26.8840i 0
176.2 0 3.50000 + 3.84057i −8.00000 7.68115i 0 29.0000 0 −2.50000 + 26.8840i 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.a 2
3.b odd 2 1 inner 177.4.d.a 2
59.b odd 2 1 CM 177.4.d.a 2
177.d even 2 1 inner 177.4.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.4.d.a 2 1.a even 1 1 trivial
177.4.d.a 2 3.b odd 2 1 inner
177.4.d.a 2 59.b odd 2 1 CM
177.4.d.a 2 177.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 \(S_{4}^{\mathrm{new}}(177, [\chi])\):

\( T_{2} \)
\( T_{5}^{2} + 59 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 27 - 7 T + T^{2} \)
$5$ \( 59 + T^{2} \)
$7$ \( ( -29 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( 3776 + T^{2} \)
$19$ \( ( -119 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( 72275 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( 59 + T^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 488579 + T^{2} \)
$59$ \( 205379 + T^{2} \)
$61$ \( T^{2} \)
$67$ \( T^{2} \)
$71$ \( 1399244 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( ( 835 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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