# Properties

 Label 1764.3.bk.d Level $1764$ Weight $3$ Character orbit 1764.bk Analytic conductor $48.066$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1764.bk (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$48.0655186332$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.12745506816.5 Defining polynomial: $$x^{8} - 8 x^{6} + 55 x^{4} - 72 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{4}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 252) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} +O(q^{10})$$ $$q + \beta_{1} q^{5} + ( 2 \beta_{5} - \beta_{7} ) q^{11} + ( -2 - 2 \beta_{3} ) q^{13} + ( 5 \beta_{5} - 2 \beta_{7} ) q^{17} + ( 10 \beta_{2} - 3 \beta_{6} ) q^{19} + ( 2 \beta_{1} - 3 \beta_{4} ) q^{23} + ( -9 + 9 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} ) q^{25} + ( -10 \beta_{1} - \beta_{4} - 10 \beta_{5} - \beta_{7} ) q^{29} + ( -10 + 10 \beta_{2} - \beta_{3} - \beta_{6} ) q^{31} + ( -8 \beta_{2} - 6 \beta_{6} ) q^{37} + ( -3 \beta_{1} - 3 \beta_{5} ) q^{41} + ( -4 + 6 \beta_{3} ) q^{43} + ( -6 \beta_{1} + 14 \beta_{4} ) q^{47} -11 \beta_{7} q^{53} + ( -26 - \beta_{3} ) q^{55} + ( -14 \beta_{5} - 2 \beta_{7} ) q^{59} + ( 68 \beta_{2} + \beta_{6} ) q^{61} + ( -6 \beta_{1} - 8 \beta_{4} ) q^{65} + ( -42 + 42 \beta_{2} - 14 \beta_{3} - 14 \beta_{6} ) q^{67} + ( 22 \beta_{1} - 15 \beta_{4} + 22 \beta_{5} - 15 \beta_{7} ) q^{71} + ( -36 + 36 \beta_{2} + \beta_{3} + \beta_{6} ) q^{73} + ( -58 \beta_{2} + 16 \beta_{6} ) q^{79} + ( 22 \beta_{1} - 20 \beta_{4} + 22 \beta_{5} - 20 \beta_{7} ) q^{83} + ( -68 - 4 \beta_{3} ) q^{85} + ( 3 \beta_{1} + 36 \beta_{4} ) q^{89} + ( -16 \beta_{5} - 12 \beta_{7} ) q^{95} + ( 24 + 17 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 16q^{13} + 40q^{19} - 36q^{25} - 40q^{31} - 32q^{37} - 32q^{43} - 208q^{55} + 272q^{61} - 168q^{67} - 144q^{73} - 232q^{79} - 544q^{85} + 192q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 8 x^{6} + 55 x^{4} - 72 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-8 \nu^{6} + 55 \nu^{4} - 440 \nu^{2} + 576$$$$)/495$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{6} - 296$$$$)/55$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - 203 \nu$$$$)/55$$ $$\beta_{5}$$ $$=$$ $$($$$$16 \nu^{7} - 110 \nu^{5} + 880 \nu^{3} - 1152 \nu$$$$)/495$$ $$\beta_{6}$$ $$=$$ $$($$$$-46 \nu^{6} + 440 \nu^{4} - 2530 \nu^{2} + 3312$$$$)/495$$ $$\beta_{7}$$ $$=$$ $$($$$$-31 \nu^{7} + 275 \nu^{5} - 1705 \nu^{3} + 2232 \nu$$$$)/495$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + \beta_{3} - 8 \beta_{2} + 8$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{7} + 5 \beta_{5} + 2 \beta_{4} + 5 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$4 \beta_{6} - 23 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$($$$$16 \beta_{7} + 31 \beta_{5}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-55 \beta_{3} - 296$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-110 \beta_{4} - 203 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$883$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
557.1
 −2.23256 + 1.28897i −1.00781 + 0.581861i 1.00781 − 0.581861i 2.23256 − 1.28897i −2.23256 − 1.28897i −1.00781 − 0.581861i 1.00781 + 0.581861i 2.23256 + 1.28897i
0 0 0 −4.46512 + 2.57794i 0 0 0 0 0
557.2 0 0 0 −2.01563 + 1.16372i 0 0 0 0 0
557.3 0 0 0 2.01563 1.16372i 0 0 0 0 0
557.4 0 0 0 4.46512 2.57794i 0 0 0 0 0
1745.1 0 0 0 −4.46512 2.57794i 0 0 0 0 0
1745.2 0 0 0 −2.01563 1.16372i 0 0 0 0 0
1745.3 0 0 0 2.01563 + 1.16372i 0 0 0 0 0
1745.4 0 0 0 4.46512 + 2.57794i 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1745.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.3.bk.d 8
3.b odd 2 1 inner 1764.3.bk.d 8
7.b odd 2 1 1764.3.bk.e 8
7.c even 3 1 252.3.c.a 4
7.c even 3 1 inner 1764.3.bk.d 8
7.d odd 6 1 1764.3.c.f 4
7.d odd 6 1 1764.3.bk.e 8
21.c even 2 1 1764.3.bk.e 8
21.g even 6 1 1764.3.c.f 4
21.g even 6 1 1764.3.bk.e 8
21.h odd 6 1 252.3.c.a 4
21.h odd 6 1 inner 1764.3.bk.d 8
28.g odd 6 1 1008.3.d.c 4
56.k odd 6 1 4032.3.d.e 4
56.p even 6 1 4032.3.d.h 4
63.g even 3 1 2268.3.bg.c 8
63.h even 3 1 2268.3.bg.c 8
63.j odd 6 1 2268.3.bg.c 8
63.n odd 6 1 2268.3.bg.c 8
84.n even 6 1 1008.3.d.c 4
168.s odd 6 1 4032.3.d.h 4
168.v even 6 1 4032.3.d.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.3.c.a 4 7.c even 3 1
252.3.c.a 4 21.h odd 6 1
1008.3.d.c 4 28.g odd 6 1
1008.3.d.c 4 84.n even 6 1
1764.3.c.f 4 7.d odd 6 1
1764.3.c.f 4 21.g even 6 1
1764.3.bk.d 8 1.a even 1 1 trivial
1764.3.bk.d 8 3.b odd 2 1 inner
1764.3.bk.d 8 7.c even 3 1 inner
1764.3.bk.d 8 21.h odd 6 1 inner
1764.3.bk.e 8 7.b odd 2 1
1764.3.bk.e 8 7.d odd 6 1
1764.3.bk.e 8 21.c even 2 1
1764.3.bk.e 8 21.g even 6 1
2268.3.bg.c 8 63.g even 3 1
2268.3.bg.c 8 63.h even 3 1
2268.3.bg.c 8 63.j odd 6 1
2268.3.bg.c 8 63.n odd 6 1
4032.3.d.e 4 56.k odd 6 1
4032.3.d.e 4 168.v even 6 1
4032.3.d.h 4 56.p even 6 1
4032.3.d.h 4 168.s odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1764, [\chi])$$:

 $$T_{5}^{8} - 32 T_{5}^{6} + 880 T_{5}^{4} - 4608 T_{5}^{2} + 20736$$ $$T_{13}^{2} + 4 T_{13} - 108$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$20736 - 4608 T^{2} + 880 T^{4} - 32 T^{6} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$8503056 - 338256 T^{2} + 10540 T^{4} - 116 T^{6} + T^{8}$$
$13$ $$( -108 + 4 T + T^{2} )^{4}$$
$17$ $$14666178816 - 85257216 T^{2} + 374512 T^{4} - 704 T^{6} + T^{8}$$
$19$ $$( 23104 + 3040 T + 552 T^{2} - 20 T^{3} + T^{4} )^{2}$$
$23$ $$3111696 - 543312 T^{2} + 93100 T^{4} - 308 T^{6} + T^{8}$$
$29$ $$( 1127844 + 3476 T^{2} + T^{4} )^{2}$$
$31$ $$( 5184 + 1440 T + 328 T^{2} + 20 T^{3} + T^{4} )^{2}$$
$37$ $$( 891136 - 15104 T + 1200 T^{2} + 16 T^{3} + T^{4} )^{2}$$
$41$ $$( 11664 + 288 T^{2} + T^{4} )^{2}$$
$43$ $$( -992 + 8 T + T^{2} )^{4}$$
$47$ $$19007367745536 - 26995534848 T^{2} + 33981120 T^{4} - 6192 T^{6} + T^{8}$$
$53$ $$( 4743684 - 2178 T^{2} + T^{4} )^{2}$$
$59$ $$14281868906496 - 26786515968 T^{2} + 46460608 T^{4} - 7088 T^{6} + T^{8}$$
$61$ $$( 21123216 - 625056 T + 13900 T^{2} - 136 T^{3} + T^{4} )^{2}$$
$67$ $$( 13868176 - 312816 T + 10780 T^{2} + 84 T^{3} + T^{4} )^{2}$$
$71$ $$( 32695524 + 15668 T^{2} + T^{4} )^{2}$$
$73$ $$( 1607824 + 91296 T + 3916 T^{2} + 72 T^{3} + T^{4} )^{2}$$
$79$ $$( 14470416 - 441264 T + 17260 T^{2} + 116 T^{3} + T^{4} )^{2}$$
$83$ $$( 15116544 + 19328 T^{2} + T^{4} )^{2}$$
$89$ $$361242176885473536 - 29772832905216 T^{2} + 1852781040 T^{4} - 49536 T^{6} + T^{8}$$
$97$ $$( -7516 - 48 T + T^{2} )^{4}$$