# Properties

 Label 768.2.d.h Level $768$ Weight $2$ Character orbit 768.d Analytic conductor $6.133$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 768.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.13251087523$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 96) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -i q^{3} + 2 i q^{5} + 4 q^{7} - q^{9} +O(q^{10})$$ $$q -i q^{3} + 2 i q^{5} + 4 q^{7} - q^{9} + 4 i q^{11} + 2 i q^{13} + 2 q^{15} -6 q^{17} + 4 i q^{19} -4 i q^{21} + q^{25} + i q^{27} -2 i q^{29} + 4 q^{31} + 4 q^{33} + 8 i q^{35} -2 i q^{37} + 2 q^{39} -2 q^{41} + 4 i q^{43} -2 i q^{45} + 8 q^{47} + 9 q^{49} + 6 i q^{51} + 10 i q^{53} -8 q^{55} + 4 q^{57} -4 i q^{59} -6 i q^{61} -4 q^{63} -4 q^{65} -4 i q^{67} + 16 q^{71} + 6 q^{73} -i q^{75} + 16 i q^{77} + 4 q^{79} + q^{81} -12 i q^{83} -12 i q^{85} -2 q^{87} -10 q^{89} + 8 i q^{91} -4 i q^{93} -8 q^{95} -14 q^{97} -4 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 8q^{7} - 2q^{9} + O(q^{10})$$ $$2q + 8q^{7} - 2q^{9} + 4q^{15} - 12q^{17} + 2q^{25} + 8q^{31} + 8q^{33} + 4q^{39} - 4q^{41} + 16q^{47} + 18q^{49} - 16q^{55} + 8q^{57} - 8q^{63} - 8q^{65} + 32q^{71} + 12q^{73} + 8q^{79} + 2q^{81} - 4q^{87} - 20q^{89} - 16q^{95} - 28q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$511$$ $$517$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
385.1
 1.00000i − 1.00000i
0 1.00000i 0 2.00000i 0 4.00000 0 −1.00000 0
385.2 0 1.00000i 0 2.00000i 0 4.00000 0 −1.00000 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
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.2.d.h 2
3.b odd 2 1 2304.2.d.s 2
4.b odd 2 1 768.2.d.a 2
8.b even 2 1 inner 768.2.d.h 2
8.d odd 2 1 768.2.d.a 2
12.b even 2 1 2304.2.d.c 2
16.e even 4 1 96.2.a.b yes 1
16.e even 4 1 192.2.a.a 1
16.f odd 4 1 96.2.a.a 1
16.f odd 4 1 192.2.a.c 1
24.f even 2 1 2304.2.d.c 2
24.h odd 2 1 2304.2.d.s 2
48.i odd 4 1 288.2.a.b 1
48.i odd 4 1 576.2.a.g 1
48.k even 4 1 288.2.a.c 1
48.k even 4 1 576.2.a.h 1
80.i odd 4 1 2400.2.f.r 2
80.i odd 4 1 4800.2.f.e 2
80.j even 4 1 2400.2.f.a 2
80.j even 4 1 4800.2.f.bh 2
80.k odd 4 1 2400.2.a.r 1
80.k odd 4 1 4800.2.a.f 1
80.q even 4 1 2400.2.a.q 1
80.q even 4 1 4800.2.a.co 1
80.s even 4 1 2400.2.f.a 2
80.s even 4 1 4800.2.f.bh 2
80.t odd 4 1 2400.2.f.r 2
80.t odd 4 1 4800.2.f.e 2
112.j even 4 1 4704.2.a.t 1
112.j even 4 1 9408.2.a.bj 1
112.l odd 4 1 4704.2.a.e 1
112.l odd 4 1 9408.2.a.ct 1
144.u even 12 2 2592.2.i.q 2
144.v odd 12 2 2592.2.i.b 2
144.w odd 12 2 2592.2.i.w 2
144.x even 12 2 2592.2.i.h 2
240.t even 4 1 7200.2.a.e 1
240.z odd 4 1 7200.2.f.x 2
240.bb even 4 1 7200.2.f.f 2
240.bd odd 4 1 7200.2.f.x 2
240.bf even 4 1 7200.2.f.f 2
240.bm odd 4 1 7200.2.a.bx 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.a.a 1 16.f odd 4 1
96.2.a.b yes 1 16.e even 4 1
192.2.a.a 1 16.e even 4 1
192.2.a.c 1 16.f odd 4 1
288.2.a.b 1 48.i odd 4 1
288.2.a.c 1 48.k even 4 1
576.2.a.g 1 48.i odd 4 1
576.2.a.h 1 48.k even 4 1
768.2.d.a 2 4.b odd 2 1
768.2.d.a 2 8.d odd 2 1
768.2.d.h 2 1.a even 1 1 trivial
768.2.d.h 2 8.b even 2 1 inner
2304.2.d.c 2 12.b even 2 1
2304.2.d.c 2 24.f even 2 1
2304.2.d.s 2 3.b odd 2 1
2304.2.d.s 2 24.h odd 2 1
2400.2.a.q 1 80.q even 4 1
2400.2.a.r 1 80.k odd 4 1
2400.2.f.a 2 80.j even 4 1
2400.2.f.a 2 80.s even 4 1
2400.2.f.r 2 80.i odd 4 1
2400.2.f.r 2 80.t odd 4 1
2592.2.i.b 2 144.v odd 12 2
2592.2.i.h 2 144.x even 12 2
2592.2.i.q 2 144.u even 12 2
2592.2.i.w 2 144.w odd 12 2
4704.2.a.e 1 112.l odd 4 1
4704.2.a.t 1 112.j even 4 1
4800.2.a.f 1 80.k odd 4 1
4800.2.a.co 1 80.q even 4 1
4800.2.f.e 2 80.i odd 4 1
4800.2.f.e 2 80.t odd 4 1
4800.2.f.bh 2 80.j even 4 1
4800.2.f.bh 2 80.s even 4 1
7200.2.a.e 1 240.t even 4 1
7200.2.a.bx 1 240.bm odd 4 1
7200.2.f.f 2 240.bb even 4 1
7200.2.f.f 2 240.bf even 4 1
7200.2.f.x 2 240.z odd 4 1
7200.2.f.x 2 240.bd odd 4 1
9408.2.a.bj 1 112.j even 4 1
9408.2.a.ct 1 112.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(768, [\chi])$$:

 $$T_{5}^{2} + 4$$ $$T_{7} - 4$$ $$T_{23}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + T^{2}$$
$5$ $$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$
$7$ $$( 1 - 4 T + 7 T^{2} )^{2}$$
$11$ $$1 - 6 T^{2} + 121 T^{4}$$
$13$ $$1 - 22 T^{2} + 169 T^{4}$$
$17$ $$( 1 + 6 T + 17 T^{2} )^{2}$$
$19$ $$1 - 22 T^{2} + 361 T^{4}$$
$23$ $$( 1 + 23 T^{2} )^{2}$$
$29$ $$1 - 54 T^{2} + 841 T^{4}$$
$31$ $$( 1 - 4 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$
$41$ $$( 1 + 2 T + 41 T^{2} )^{2}$$
$43$ $$1 - 70 T^{2} + 1849 T^{4}$$
$47$ $$( 1 - 8 T + 47 T^{2} )^{2}$$
$53$ $$1 - 6 T^{2} + 2809 T^{4}$$
$59$ $$1 - 102 T^{2} + 3481 T^{4}$$
$61$ $$1 - 86 T^{2} + 3721 T^{4}$$
$67$ $$1 - 118 T^{2} + 4489 T^{4}$$
$71$ $$( 1 - 16 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 6 T + 73 T^{2} )^{2}$$
$79$ $$( 1 - 4 T + 79 T^{2} )^{2}$$
$83$ $$1 - 22 T^{2} + 6889 T^{4}$$
$89$ $$( 1 + 10 T + 89 T^{2} )^{2}$$
$97$ $$( 1 + 14 T + 97 T^{2} )^{2}$$