# Properties

 Label 1764.3.z.k Level $1764$ Weight $3$ Character orbit 1764.z 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.z (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: 8.0.339738624.1 Defining polynomial: $$x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$7^{4}$$ Twist minimal: no (minimal twist has level 196) 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{2} - 2 \beta_{6} ) q^{5} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{2} - 2 \beta_{6} ) q^{5} + ( -6 \beta_{4} + \beta_{7} ) q^{11} + ( -3 \beta_{1} + 4 \beta_{3} + 4 \beta_{6} ) q^{13} -3 \beta_{3} q^{17} + ( 5 \beta_{1} - 5 \beta_{2} - \beta_{6} ) q^{19} + ( 26 - 26 \beta_{4} ) q^{23} + ( 27 \beta_{4} + 2 \beta_{7} ) q^{25} + ( -8 - 2 \beta_{5} ) q^{29} + ( -2 \beta_{2} + 8 \beta_{3} ) q^{31} + ( 32 - 32 \beta_{4} ) q^{37} + ( 3 \beta_{1} + 5 \beta_{3} + 5 \beta_{6} ) q^{41} + ( 10 - 7 \beta_{5} ) q^{43} + ( 12 \beta_{1} - 12 \beta_{2} - 2 \beta_{6} ) q^{47} + ( 6 \beta_{4} - 2 \beta_{7} ) q^{53} + ( 14 \beta_{1} - 6 \beta_{3} - 6 \beta_{6} ) q^{55} + ( 11 \beta_{2} + 3 \beta_{3} ) q^{59} + ( -9 \beta_{1} + 9 \beta_{2} - 8 \beta_{6} ) q^{61} + ( 24 - 24 \beta_{4} - 14 \beta_{5} - 14 \beta_{7} ) q^{65} + ( -28 \beta_{4} + 6 \beta_{7} ) q^{67} + ( -56 + 4 \beta_{5} ) q^{71} + ( 11 \beta_{2} - 3 \beta_{3} ) q^{73} + ( -60 + 60 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{79} + ( -13 \beta_{1} - 33 \beta_{3} - 33 \beta_{6} ) q^{83} + ( -54 + 6 \beta_{5} ) q^{85} + ( 11 \beta_{1} - 11 \beta_{2} - 25 \beta_{6} ) q^{89} + ( -62 \beta_{4} + 12 \beta_{7} ) q^{95} + ( -22 \beta_{1} - 33 \beta_{3} - 33 \beta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 24q^{11} + 104q^{23} + 108q^{25} - 64q^{29} + 128q^{37} + 80q^{43} + 24q^{53} + 96q^{65} - 112q^{67} - 448q^{71} - 240q^{79} - 432q^{85} - 248q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + 8 \nu$$$$)/14$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} - 41 \nu$$$$)/7$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{6} + 7 \nu^{4} - 28 \nu^{2} + 16$$$$)/14$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{6} - 20$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$-12 \nu^{7} + 49 \nu^{5} - 168 \nu^{3} + 96 \nu$$$$)/14$$ $$\beta_{7}$$ $$=$$ $$($$$$-3 \nu^{6} + 14 \nu^{4} - 42 \nu^{2} + 24$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{3} + 2 \beta_{2}$$$$)/7$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + \beta_{5} - 14 \beta_{4} + 14$$$$)/7$$ $$\nu^{3}$$ $$=$$ $$\beta_{1}$$ $$\nu^{4}$$ $$=$$ $$($$$$4 \beta_{7} - 42 \beta_{4}$$$$)/7$$ $$\nu^{5}$$ $$=$$ $$($$$$2 \beta_{6} - 24 \beta_{2} + 24 \beta_{1}$$$$)/7$$ $$\nu^{6}$$ $$=$$ $$-2 \beta_{5} - 20$$ $$\nu^{7}$$ $$=$$ $$($$$$-8 \beta_{3} - 82 \beta_{2}$$$$)/7$$

## Character values

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

 $$n$$ $$785$$ $$883$$ $$1081$$ $$\chi(n)$$ $$1$$ $$1$$ $$1 - \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
325.1
 0.662827 + 0.382683i −1.60021 − 0.923880i 1.60021 + 0.923880i −0.662827 − 0.382683i 0.662827 − 0.382683i −1.60021 + 0.923880i 1.60021 − 0.923880i −0.662827 + 0.382683i
0 0 0 −7.33820 + 4.23671i 0 0 0 0 0
325.2 0 0 0 −4.91434 + 2.83730i 0 0 0 0 0
325.3 0 0 0 4.91434 2.83730i 0 0 0 0 0
325.4 0 0 0 7.33820 4.23671i 0 0 0 0 0
901.1 0 0 0 −7.33820 4.23671i 0 0 0 0 0
901.2 0 0 0 −4.91434 2.83730i 0 0 0 0 0
901.3 0 0 0 4.91434 + 2.83730i 0 0 0 0 0
901.4 0 0 0 7.33820 + 4.23671i 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 901.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d 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.z.k 8
3.b odd 2 1 196.3.h.c 8
7.b odd 2 1 inner 1764.3.z.k 8
7.c even 3 1 1764.3.d.e 4
7.c even 3 1 inner 1764.3.z.k 8
7.d odd 6 1 1764.3.d.e 4
7.d odd 6 1 inner 1764.3.z.k 8
12.b even 2 1 784.3.s.g 8
21.c even 2 1 196.3.h.c 8
21.g even 6 1 196.3.b.b 4
21.g even 6 1 196.3.h.c 8
21.h odd 6 1 196.3.b.b 4
21.h odd 6 1 196.3.h.c 8
84.h odd 2 1 784.3.s.g 8
84.j odd 6 1 784.3.c.d 4
84.j odd 6 1 784.3.s.g 8
84.n even 6 1 784.3.c.d 4
84.n even 6 1 784.3.s.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.3.b.b 4 21.g even 6 1
196.3.b.b 4 21.h odd 6 1
196.3.h.c 8 3.b odd 2 1
196.3.h.c 8 21.c even 2 1
196.3.h.c 8 21.g even 6 1
196.3.h.c 8 21.h odd 6 1
784.3.c.d 4 84.j odd 6 1
784.3.c.d 4 84.n even 6 1
784.3.s.g 8 12.b even 2 1
784.3.s.g 8 84.h odd 2 1
784.3.s.g 8 84.j odd 6 1
784.3.s.g 8 84.n even 6 1
1764.3.d.e 4 7.c even 3 1
1764.3.d.e 4 7.d odd 6 1
1764.3.z.k 8 1.a even 1 1 trivial
1764.3.z.k 8 7.b odd 2 1 inner
1764.3.z.k 8 7.c even 3 1 inner
1764.3.z.k 8 7.d odd 6 1 inner

## Hecke kernels

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

 $$T_{5}^{8} - 104 T_{5}^{6} + 8504 T_{5}^{4} - 240448 T_{5}^{2} + 5345344$$ $$T_{11}^{4} + 12 T_{11}^{3} + 206 T_{11}^{2} - 744 T_{11} + 3844$$ $$T_{13}^{4} + 776 T_{13}^{2} + 150152$$ $$T_{19}^{8} - 1060 T_{19}^{6} + 1077998 T_{19}^{4} - 48338120 T_{19}^{2} + 2079542404$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$5345344 - 240448 T^{2} + 8504 T^{4} - 104 T^{6} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$( 3844 - 744 T + 206 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$13$ $$( 150152 + 776 T^{2} + T^{4} )^{2}$$
$17$ $$26244 - 29160 T^{2} + 32238 T^{4} - 180 T^{6} + T^{8}$$
$19$ $$2079542404 - 48338120 T^{2} + 1077998 T^{4} - 1060 T^{6} + T^{8}$$
$23$ $$( 676 - 26 T + T^{2} )^{4}$$
$29$ $$( -328 + 16 T + T^{2} )^{4}$$
$31$ $$94450499584 - 481890304 T^{2} + 2151296 T^{4} - 1568 T^{6} + T^{8}$$
$37$ $$( 1024 - 32 T + T^{2} )^{4}$$
$41$ $$( 132098 + 740 T^{2} + T^{4} )^{2}$$
$43$ $$( -4702 - 20 T + T^{2} )^{4}$$
$47$ $$1420787713024 - 7189950976 T^{2} + 35193056 T^{4} - 6032 T^{6} + T^{8}$$
$53$ $$( 126736 + 4272 T + 500 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$59$ $$111931731844 - 1591176872 T^{2} + 22284974 T^{4} - 4756 T^{6} + T^{8}$$
$61$ $$8688298598464 - 11625302848 T^{2} + 12607544 T^{4} - 3944 T^{6} + T^{8}$$
$67$ $$( 7529536 - 153664 T + 5880 T^{2} + 56 T^{3} + T^{4} )^{2}$$
$71$ $$( 1568 + 112 T + T^{2} )^{4}$$
$73$ $$2755075464964 - 8770605128 T^{2} + 26260814 T^{4} - 5284 T^{6} + T^{8}$$
$79$ $$( 10291264 + 384960 T + 11192 T^{2} + 120 T^{3} + T^{4} )^{2}$$
$83$ $$( 102330818 + 25108 T^{2} + T^{4} )^{2}$$
$89$ $$6573541223354884 - 1584251966120 T^{2} + 300734222 T^{4} - 19540 T^{6} + T^{8}$$
$97$ $$( 310652738 + 35332 T^{2} + T^{4} )^{2}$$