# Properties

 Label 1764.3.d.g Level $1764$ Weight $3$ Character orbit 1764.d 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.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$48.0655186332$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.3288334336.1 Defining polynomial: $$x^{8} + 12 x^{6} + 30 x^{4} + 12 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{3}\cdot 7^{6}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{5} +O(q^{10})$$ $$q -\beta_{1} q^{5} -\beta_{3} q^{11} + \beta_{6} q^{13} + 3 \beta_{1} q^{17} + ( \beta_{5} - 2 \beta_{6} ) q^{19} + ( -\beta_{3} - \beta_{4} ) q^{23} + ( -3 + \beta_{2} ) q^{25} + ( \beta_{3} - \beta_{4} ) q^{29} + ( 5 \beta_{5} + 4 \beta_{6} ) q^{31} + ( 16 + 3 \beta_{2} ) q^{37} + ( \beta_{1} + \beta_{7} ) q^{41} + ( 22 - 2 \beta_{2} ) q^{43} + ( -2 \beta_{1} + \beta_{7} ) q^{47} + 2 \beta_{3} q^{53} + ( -\beta_{5} + 18 \beta_{6} ) q^{55} + ( -2 \beta_{1} - \beta_{7} ) q^{59} + ( 12 \beta_{5} - 13 \beta_{6} ) q^{61} + \beta_{3} q^{65} + ( -32 + 6 \beta_{2} ) q^{67} + ( \beta_{3} + 2 \beta_{4} ) q^{71} + ( 22 \beta_{5} + 21 \beta_{6} ) q^{73} + ( 54 + 8 \beta_{2} ) q^{79} + ( 2 \beta_{1} - 2 \beta_{7} ) q^{83} + ( 84 - 3 \beta_{2} ) q^{85} + ( 9 \beta_{1} - 2 \beta_{7} ) q^{89} + ( -2 \beta_{3} + \beta_{4} ) q^{95} + ( 2 \beta_{5} + 27 \beta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 24q^{25} + 128q^{37} + 176q^{43} - 256q^{67} + 432q^{79} + 672q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 12 x^{6} + 30 x^{4} + 12 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 13 \nu^{5} + 43 \nu^{3} + 55 \nu$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$7 \nu^{6} + 84 \nu^{4} + 203 \nu^{2} + 42$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-6 \nu^{6} - 71 \nu^{4} - 174 \nu^{2} - 57$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$-17 \nu^{6} - 193 \nu^{4} - 395 \nu^{2} - 39$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$-2 \nu^{7} - 23 \nu^{5} - 49 \nu^{3} - 4 \nu$$ $$\beta_{6}$$ $$=$$ $$($$$$-9 \nu^{7} - 107 \nu^{5} - 259 \nu^{3} - 81 \nu$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$43 \nu^{7} + 510 \nu^{5} + 1212 \nu^{3} + 307 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + 28 \beta_{6} - 7 \beta_{5} + 24 \beta_{1}$$$$)/98$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} - 11 \beta_{3} - 7 \beta_{2} - 147$$$$)/49$$ $$\nu^{3}$$ $$=$$ $$($$$$-13 \beta_{7} - 280 \beta_{6} + 21 \beta_{5} - 116 \beta_{1}$$$$)/98$$ $$\nu^{4}$$ $$=$$ $$($$$$14 \beta_{3} + 12 \beta_{2} + 147$$$$)/7$$ $$\nu^{5}$$ $$=$$ $$($$$$125 \beta_{7} + 2464 \beta_{6} + 21 \beta_{5} + 844 \beta_{1}$$$$)/98$$ $$\nu^{6}$$ $$=$$ $$($$$$-29 \beta_{4} - 857 \beta_{3} - 791 \beta_{2} - 8379$$$$)/49$$ $$\nu^{7}$$ $$=$$ $$($$$$-1121 \beta_{7} - 21532 \beta_{6} - 791 \beta_{5} - 6912 \beta_{1}$$$$)/98$$

## Character values

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

 $$n$$ $$785$$ $$883$$ $$1081$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
685.1
 1.67825i 0.595859i 2.95189i 0.338766i − 2.95189i − 0.338766i − 1.67825i − 0.595859i
0 0 0 6.15626i 0 0 0 0 0
685.2 0 0 0 6.15626i 0 0 0 0 0
685.3 0 0 0 4.25447i 0 0 0 0 0
685.4 0 0 0 4.25447i 0 0 0 0 0
685.5 0 0 0 4.25447i 0 0 0 0 0
685.6 0 0 0 4.25447i 0 0 0 0 0
685.7 0 0 0 6.15626i 0 0 0 0 0
685.8 0 0 0 6.15626i 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 685.8 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.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.3.d.g 8
3.b odd 2 1 inner 1764.3.d.g 8
7.b odd 2 1 inner 1764.3.d.g 8
7.c even 3 2 1764.3.z.n 16
7.d odd 6 2 1764.3.z.n 16
21.c even 2 1 inner 1764.3.d.g 8
21.g even 6 2 1764.3.z.n 16
21.h odd 6 2 1764.3.z.n 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.3.d.g 8 1.a even 1 1 trivial
1764.3.d.g 8 3.b odd 2 1 inner
1764.3.d.g 8 7.b odd 2 1 inner
1764.3.d.g 8 21.c even 2 1 inner
1764.3.z.n 16 7.c even 3 2
1764.3.z.n 16 7.d odd 6 2
1764.3.z.n 16 21.g even 6 2
1764.3.z.n 16 21.h odd 6 2

## Hecke kernels

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

 $$T_{5}^{4} + 56 T_{5}^{2} + 686$$ $$T_{11}^{4} - 364 T_{11}^{2} + 1372$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 686 + 56 T^{2} + T^{4} )^{2}$$
$7$ $$T^{8}$$
$11$ $$( 1372 - 364 T^{2} + T^{4} )^{2}$$
$13$ $$( 2 + 20 T^{2} + T^{4} )^{2}$$
$17$ $$( 55566 + 504 T^{2} + T^{4} )^{2}$$
$19$ $$( 4232 + 136 T^{2} + T^{4} )^{2}$$
$23$ $$( 396508 - 1652 T^{2} + T^{4} )^{2}$$
$29$ $$( 725788 - 2100 T^{2} + T^{4} )^{2}$$
$31$ $$( 223112 + 1160 T^{2} + T^{4} )^{2}$$
$37$ $$( -626 - 32 T + T^{2} )^{4}$$
$41$ $$( 3655694 + 4928 T^{2} + T^{4} )^{2}$$
$43$ $$( 92 - 44 T + T^{2} )^{4}$$
$47$ $$( 6061496 + 4928 T^{2} + T^{4} )^{2}$$
$53$ $$( 21952 - 1456 T^{2} + T^{4} )^{2}$$
$59$ $$( 2636984 + 5152 T^{2} + T^{4} )^{2}$$
$61$ $$( 25589858 + 10388 T^{2} + T^{4} )^{2}$$
$67$ $$( -2504 + 64 T + T^{2} )^{4}$$
$71$ $$( 1318492 - 5964 T^{2} + T^{4} )^{2}$$
$73$ $$( 122649122 + 24484 T^{2} + T^{4} )^{2}$$
$79$ $$( -3356 - 108 T + T^{2} )^{4}$$
$83$ $$( 86940896 + 19264 T^{2} + T^{4} )^{2}$$
$89$ $$( 104876366 + 22792 T^{2} + T^{4} )^{2}$$
$97$ $$( 235298 + 14308 T^{2} + T^{4} )^{2}$$