# Properties

 Label 177.2.d.a Level $177$ Weight $2$ Character orbit 177.d Analytic conductor $1.413$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# 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: 6.0.19298288.1 Defining polynomial: $$x^{6} - x^{5} + 3 x^{4} - 2 x^{3} + 9 x^{2} - 9 x + 27$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} + \beta_{2} + \beta_{5} ) q^{2} -\beta_{1} q^{3} + ( 2 - \beta_{2} ) q^{4} -\beta_{4} q^{5} + ( -1 - \beta_{2} - \beta_{4} - \beta_{5} ) q^{6} + \beta_{2} q^{7} + ( -1 + 2 \beta_{2} ) q^{8} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} + \beta_{2} + \beta_{5} ) q^{2} -\beta_{1} q^{3} + ( 2 - \beta_{2} ) q^{4} -\beta_{4} q^{5} + ( -1 - \beta_{2} - \beta_{4} - \beta_{5} ) q^{6} + \beta_{2} q^{7} + ( -1 + 2 \beta_{2} ) q^{8} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{9} + ( -2 \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{10} + ( 3 + \beta_{1} + \beta_{5} ) q^{11} + ( -1 - 2 \beta_{1} - \beta_{3} + \beta_{5} ) q^{12} + ( \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{13} + ( 1 - 2 \beta_{2} ) q^{14} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{15} + ( -1 - \beta_{1} - 3 \beta_{2} - \beta_{5} ) q^{16} + ( \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{17} + ( 3 - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{18} + ( 1 - \beta_{1} - 3 \beta_{2} - \beta_{5} ) q^{19} + ( \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{20} + ( 1 + \beta_{3} - \beta_{5} ) q^{21} + ( -1 + 4 \beta_{1} + 5 \beta_{2} + 4 \beta_{5} ) q^{22} + ( 3 + \beta_{2} ) q^{23} + ( 2 + \beta_{1} + 2 \beta_{3} - 2 \beta_{5} ) q^{24} + ( -2 \beta_{1} - 2 \beta_{5} ) q^{25} + ( -\beta_{1} - 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{26} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{27} + ( -3 + \beta_{1} + 3 \beta_{2} + \beta_{5} ) q^{28} + ( \beta_{1} - \beta_{3} - \beta_{5} ) q^{29} + ( -6 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{30} + ( \beta_{3} - 2 \beta_{4} ) q^{31} + ( -2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{5} ) q^{32} + ( -2 - 4 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{33} + ( -\beta_{3} + 2 \beta_{4} ) q^{34} + ( -\beta_{1} - \beta_{3} + \beta_{5} ) q^{35} + ( -4 + 3 \beta_{1} + 4 \beta_{2} + \beta_{3} + \beta_{4} ) q^{36} + ( -\beta_{1} - 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{37} + ( -6 + 5 \beta_{2} ) q^{38} + ( 1 - 3 \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{39} + ( -2 \beta_{1} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{40} + ( -\beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{41} + ( -2 - \beta_{1} - 2 \beta_{3} + 2 \beta_{5} ) q^{42} + ( 3 \beta_{1} + \beta_{4} - 3 \beta_{5} ) q^{43} + ( 8 + \beta_{1} - 3 \beta_{2} + \beta_{5} ) q^{44} + ( 2 + \beta_{1} + 3 \beta_{2} - \beta_{3} + 4 \beta_{5} ) q^{45} + ( -2 + 3 \beta_{1} + \beta_{2} + 3 \beta_{5} ) q^{46} + ( -8 + \beta_{1} + \beta_{2} + \beta_{5} ) q^{47} + ( -1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{48} + ( -4 - \beta_{1} - \beta_{2} - \beta_{5} ) q^{49} + ( -4 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{5} ) q^{50} + ( 2 - \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{51} + ( 4 \beta_{1} + 3 \beta_{3} - 4 \beta_{5} ) q^{52} + ( 2 \beta_{1} - \beta_{4} - 2 \beta_{5} ) q^{53} + ( 1 - 2 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{54} + ( -\beta_{1} - 5 \beta_{4} + \beta_{5} ) q^{55} + ( 6 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{5} ) q^{56} + ( -1 + \beta_{2} - 2 \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{57} + ( 2 \beta_{3} + 3 \beta_{4} ) q^{58} + ( 1 + \beta_{1} + 3 \beta_{2} + \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{59} + ( 5 - 3 \beta_{1} - 7 \beta_{2} + 2 \beta_{4} - 4 \beta_{5} ) q^{60} + ( 2 \beta_{1} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{61} + ( -5 \beta_{1} - 4 \beta_{3} - 3 \beta_{4} + 5 \beta_{5} ) q^{62} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{63} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{5} ) q^{64} + ( -1 + 3 \beta_{1} + 6 \beta_{2} + 3 \beta_{5} ) q^{65} + ( -3 - 3 \beta_{1} - 4 \beta_{2} + \beta_{3} - 4 \beta_{4} - 5 \beta_{5} ) q^{66} + ( -\beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{67} + ( 3 \beta_{1} + 2 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{68} + ( 1 - 3 \beta_{1} + \beta_{3} - \beta_{5} ) q^{69} + ( 2 \beta_{1} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{70} + ( -3 \beta_{1} - 3 \beta_{4} + 3 \beta_{5} ) q^{71} + ( 5 - 3 \beta_{1} - 5 \beta_{2} + \beta_{3} + \beta_{4} ) q^{72} + ( 2 \beta_{1} + 3 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{73} + ( 6 \beta_{1} + 7 \beta_{3} + 2 \beta_{4} - 6 \beta_{5} ) q^{74} + ( 4 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{75} + ( 9 - 4 \beta_{1} - 10 \beta_{2} - 4 \beta_{5} ) q^{76} + ( -2 + \beta_{1} + 3 \beta_{2} + \beta_{5} ) q^{77} + ( -2 + 2 \beta_{1} + 9 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{78} + ( -1 - 4 \beta_{1} - 4 \beta_{5} ) q^{79} + ( 4 \beta_{1} + 3 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} ) q^{80} + ( -3 - 2 \beta_{1} - 5 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{81} + ( -2 \beta_{1} - 3 \beta_{4} + 2 \beta_{5} ) q^{82} + ( 8 + \beta_{1} - 3 \beta_{2} + \beta_{5} ) q^{83} + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{84} + ( 1 + 4 \beta_{1} + 5 \beta_{2} + 4 \beta_{5} ) q^{85} + ( -\beta_{1} + \beta_{3} + 7 \beta_{4} + \beta_{5} ) q^{86} + ( 5 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{87} + ( -7 + \beta_{1} + 6 \beta_{2} + \beta_{5} ) q^{88} + ( -4 - \beta_{1} - 5 \beta_{2} - \beta_{5} ) q^{89} + ( 6 + 7 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{90} + ( -2 \beta_{1} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{91} + ( 3 + \beta_{1} + \beta_{5} ) q^{92} + ( 1 - 2 \beta_{1} - 6 \beta_{2} + \beta_{3} + 3 \beta_{4} - 4 \beta_{5} ) q^{93} + ( 11 - 7 \beta_{1} - 8 \beta_{2} - 7 \beta_{5} ) q^{94} + ( 4 \beta_{1} + 3 \beta_{3} + \beta_{4} - 4 \beta_{5} ) q^{95} + ( 3 + 4 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{96} + ( \beta_{1} - 4 \beta_{3} - \beta_{4} - \beta_{5} ) q^{97} + ( 1 - 5 \beta_{1} - 4 \beta_{2} - 5 \beta_{5} ) q^{98} + ( -3 + 5 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 4q^{2} - q^{3} + 12q^{4} - 7q^{6} - 6q^{8} - 5q^{9} + O(q^{10})$$ $$6q - 4q^{2} - q^{3} + 12q^{4} - 7q^{6} - 6q^{8} - 5q^{9} + 20q^{11} - 7q^{12} + 6q^{14} + 3q^{15} - 8q^{16} + 17q^{18} + 4q^{19} + 5q^{21} + 2q^{22} + 18q^{23} + 11q^{24} - 4q^{25} + 2q^{27} - 16q^{28} - 37q^{30} - 16q^{32} - 16q^{33} - 21q^{36} - 36q^{38} + 8q^{39} - 11q^{42} + 50q^{44} + 17q^{45} - 6q^{46} - 46q^{47} - q^{48} - 26q^{49} - 28q^{50} + 14q^{51} + 8q^{54} + 32q^{56} - 3q^{57} + 10q^{59} + 23q^{60} + 11q^{63} - 10q^{64} - 26q^{66} + 2q^{69} + 27q^{72} + 26q^{75} + 46q^{76} - 10q^{77} - 8q^{78} - 14q^{79} - 21q^{81} + 50q^{83} + 5q^{84} + 14q^{85} + 29q^{87} - 40q^{88} - 26q^{89} + 45q^{90} + 20q^{92} + 52q^{94} + 23q^{96} - 4q^{98} - 14q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 3 x^{4} - 2 x^{3} + 9 x^{2} - 9 x + 27$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{4} + \nu^{3} - \nu - 3$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} - \nu^{4} - 6 \nu^{3} + 7 \nu^{2} - 9 \nu$$$$)/18$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{5} + 4 \nu^{4} + 3 \nu^{3} + 11 \nu^{2} - 6 \nu + 27$$$$)/18$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 3 \nu^{3} + 2 \nu^{2} - 9 \nu + 9$$$$)/9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} - 1$$ $$\nu^{3}$$ $$=$$ $$-\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{5} + \beta_{4} - \beta_{3} - 5 \beta_{2} - 2 \beta_{1} - 3$$ $$\nu^{5}$$ $$=$$ $$-7 \beta_{5} + 4 \beta_{3} - 6 \beta_{2} - 6 \beta_{1} + 4$$

## 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
 0.321037 + 1.70204i 0.321037 − 1.70204i −1.16170 + 1.28470i −1.16170 − 1.28470i 1.34067 + 1.09664i 1.34067 − 1.09664i
−2.47283 −0.321037 1.70204i 4.11491 2.50682i 0.793871 + 4.20886i −2.11491 −5.22982 −2.79387 + 1.09283i 6.19895i
176.2 −2.47283 −0.321037 + 1.70204i 4.11491 2.50682i 0.793871 4.20886i −2.11491 −5.22982 −2.79387 1.09283i 6.19895i
176.3 −1.46260 1.16170 1.28470i 0.139194 0.594299i −1.69910 + 1.87900i 1.86081 2.72161 −0.300896 2.98487i 0.869221i
176.4 −1.46260 1.16170 + 1.28470i 0.139194 0.594299i −1.69910 1.87900i 1.86081 2.72161 −0.300896 + 2.98487i 0.869221i
176.5 1.93543 −1.34067 1.09664i 1.74590 3.21911i −2.59477 2.12247i 0.254102 −0.491797 0.594767 + 2.94045i 6.23037i
176.6 1.93543 −1.34067 + 1.09664i 1.74590 3.21911i −2.59477 + 2.12247i 0.254102 −0.491797 0.594767 2.94045i 6.23037i
 $$n$$: 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
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.a 6
3.b odd 2 1 177.2.d.c yes 6
59.b odd 2 1 177.2.d.c yes 6
177.d even 2 1 inner 177.2.d.a 6

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + 2 T_{2}^{2} - 4 T_{2} - 7$$ acting on $$S_{2}^{\mathrm{new}}(177, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -7 - 4 T + 2 T^{2} + T^{3} )^{2}$$
$3$ $$27 + 9 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{5} + T^{6}$$
$5$ $$23 + 71 T^{2} + 17 T^{4} + T^{6}$$
$7$ $$( 1 - 4 T + T^{3} )^{2}$$
$11$ $$( -14 + 27 T - 10 T^{2} + T^{3} )^{2}$$
$13$ $$4508 + 997 T^{2} + 58 T^{4} + T^{6}$$
$17$ $$368 + 197 T^{2} + 27 T^{4} + T^{6}$$
$19$ $$( -37 - 26 T - 2 T^{2} + T^{3} )^{2}$$
$23$ $$( -14 + 23 T - 9 T^{2} + T^{3} )^{2}$$
$29$ $$1127 + 614 T^{2} + 54 T^{4} + T^{6}$$
$31$ $$18032 + 3589 T^{2} + 115 T^{4} + T^{6}$$
$37$ $$220892 + 13537 T^{2} + 223 T^{4} + T^{6}$$
$41$ $$23 + 282 T^{2} + 58 T^{4} + T^{6}$$
$43$ $$288512 + 13565 T^{2} + 206 T^{4} + T^{6}$$
$47$ $$( 406 + 171 T + 23 T^{2} + T^{3} )^{2}$$
$53$ $$3887 + 3015 T^{2} + 121 T^{4} + T^{6}$$
$59$ $$205379 - 34810 T + 5015 T^{2} - 732 T^{3} + 85 T^{4} - 10 T^{5} + T^{6}$$
$61$ $$4508 + 9945 T^{2} + 199 T^{4} + T^{6}$$
$67$ $$4508 + 997 T^{2} + 58 T^{4} + T^{6}$$
$71$ $$67068 + 11421 T^{2} + 306 T^{4} + T^{6}$$
$73$ $$72128 + 13261 T^{2} + 247 T^{4} + T^{6}$$
$79$ $$( -347 - 85 T + 7 T^{2} + T^{3} )^{2}$$
$83$ $$( -14 + 151 T - 25 T^{2} + T^{3} )^{2}$$
$89$ $$( -518 - 25 T + 13 T^{2} + T^{3} )^{2}$$
$97$ $$18032 + 7349 T^{2} + 342 T^{4} + T^{6}$$