# 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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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