# Properties

 Label 1176.3.d.a Level $1176$ Weight $3$ Character orbit 1176.d Analytic conductor $32.044$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1176 = 2^{3} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1176.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.0436790888$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta ) q^{3} + 2 \beta q^{5} + ( -7 + 2 \beta ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta ) q^{3} + 2 \beta q^{5} + ( -7 + 2 \beta ) q^{9} -2 \beta q^{11} -10 q^{13} + ( 16 - 2 \beta ) q^{15} -8 \beta q^{17} -2 q^{19} + 4 \beta q^{23} -7 q^{25} + ( 23 + 5 \beta ) q^{27} -6 \beta q^{29} + 22 q^{31} + ( -16 + 2 \beta ) q^{33} -6 q^{37} + ( 10 + 10 \beta ) q^{39} + 12 \beta q^{41} + 82 q^{43} + ( -32 - 14 \beta ) q^{45} + 24 \beta q^{47} + ( -64 + 8 \beta ) q^{51} + 22 \beta q^{53} + 32 q^{55} + ( 2 + 2 \beta ) q^{57} + 26 \beta q^{59} + 86 q^{61} -20 \beta q^{65} + 2 q^{67} + ( 32 - 4 \beta ) q^{69} + 44 \beta q^{71} -82 q^{73} + ( 7 + 7 \beta ) q^{75} + 10 q^{79} + ( 17 - 28 \beta ) q^{81} -26 \beta q^{83} + 128 q^{85} + ( -48 + 6 \beta ) q^{87} -12 \beta q^{89} + ( -22 - 22 \beta ) q^{93} -4 \beta q^{95} + 94 q^{97} + ( 32 + 14 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 14q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 14q^{9} - 20q^{13} + 32q^{15} - 4q^{19} - 14q^{25} + 46q^{27} + 44q^{31} - 32q^{33} - 12q^{37} + 20q^{39} + 164q^{43} - 64q^{45} - 128q^{51} + 64q^{55} + 4q^{57} + 172q^{61} + 4q^{67} + 64q^{69} - 164q^{73} + 14q^{75} + 20q^{79} + 34q^{81} + 256q^{85} - 96q^{87} - 44q^{93} + 188q^{97} + 64q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times$$.

 $$n$$ $$295$$ $$589$$ $$785$$ $$1081$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
785.1
 1.41421i − 1.41421i
0 −1.00000 2.82843i 0 5.65685i 0 0 0 −7.00000 + 5.65685i 0
785.2 0 −1.00000 + 2.82843i 0 5.65685i 0 0 0 −7.00000 5.65685i 0
 $$n$$: e.g. 2-40 or 990-1000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1176.3.d.a 2
3.b odd 2 1 inner 1176.3.d.a 2
7.b odd 2 1 24.3.e.a 2
21.c even 2 1 24.3.e.a 2
28.d even 2 1 48.3.e.b 2
35.c odd 2 1 600.3.l.b 2
35.f even 4 2 600.3.c.a 4
56.e even 2 1 192.3.e.d 2
56.h odd 2 1 192.3.e.c 2
63.l odd 6 2 648.3.m.d 4
63.o even 6 2 648.3.m.d 4
84.h odd 2 1 48.3.e.b 2
105.g even 2 1 600.3.l.b 2
105.k odd 4 2 600.3.c.a 4
112.j even 4 2 768.3.h.c 4
112.l odd 4 2 768.3.h.d 4
140.c even 2 1 1200.3.l.n 2
140.j odd 4 2 1200.3.c.i 4
168.e odd 2 1 192.3.e.d 2
168.i even 2 1 192.3.e.c 2
252.s odd 6 2 1296.3.q.e 4
252.bi even 6 2 1296.3.q.e 4
336.v odd 4 2 768.3.h.c 4
336.y even 4 2 768.3.h.d 4
420.o odd 2 1 1200.3.l.n 2
420.w even 4 2 1200.3.c.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.e.a 2 7.b odd 2 1
24.3.e.a 2 21.c even 2 1
48.3.e.b 2 28.d even 2 1
48.3.e.b 2 84.h odd 2 1
192.3.e.c 2 56.h odd 2 1
192.3.e.c 2 168.i even 2 1
192.3.e.d 2 56.e even 2 1
192.3.e.d 2 168.e odd 2 1
600.3.c.a 4 35.f even 4 2
600.3.c.a 4 105.k odd 4 2
600.3.l.b 2 35.c odd 2 1
600.3.l.b 2 105.g even 2 1
648.3.m.d 4 63.l odd 6 2
648.3.m.d 4 63.o even 6 2
768.3.h.c 4 112.j even 4 2
768.3.h.c 4 336.v odd 4 2
768.3.h.d 4 112.l odd 4 2
768.3.h.d 4 336.y even 4 2
1176.3.d.a 2 1.a even 1 1 trivial
1176.3.d.a 2 3.b odd 2 1 inner
1200.3.c.i 4 140.j odd 4 2
1200.3.c.i 4 420.w even 4 2
1200.3.l.n 2 140.c even 2 1
1200.3.l.n 2 420.o odd 2 1
1296.3.q.e 4 252.s odd 6 2
1296.3.q.e 4 252.bi even 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1176, [\chi])$$:

 $$T_{5}^{2} + 32$$ $$T_{13} + 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + 2 T + T^{2}$$
$5$ $$32 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$32 + T^{2}$$
$13$ $$( 10 + T )^{2}$$
$17$ $$512 + T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$128 + T^{2}$$
$29$ $$288 + T^{2}$$
$31$ $$( -22 + T )^{2}$$
$37$ $$( 6 + T )^{2}$$
$41$ $$1152 + T^{2}$$
$43$ $$( -82 + T )^{2}$$
$47$ $$4608 + T^{2}$$
$53$ $$3872 + T^{2}$$
$59$ $$5408 + T^{2}$$
$61$ $$( -86 + T )^{2}$$
$67$ $$( -2 + T )^{2}$$
$71$ $$15488 + T^{2}$$
$73$ $$( 82 + T )^{2}$$
$79$ $$( -10 + T )^{2}$$
$83$ $$5408 + T^{2}$$
$89$ $$1152 + T^{2}$$
$97$ $$( -94 + T )^{2}$$