# 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 $4$

# Related objects

## 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: 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 177.6.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

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