Properties

 Label 768.1.i.b Level $768$ Weight $1$ Character orbit 768.i Analytic conductor $0.383$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -3 Inner twists $8$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 768.i (of order $$4$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.383281929702$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.18432.2

$q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{8} q^{3} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{7} + \zeta_{8}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{8} q^{3} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{7} + \zeta_{8}^{2} q^{9} + ( 1 + \zeta_{8}^{2} ) q^{13} + ( 1 - \zeta_{8}^{2} ) q^{21} -\zeta_{8}^{2} q^{25} + \zeta_{8}^{3} q^{27} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{31} + ( -1 + \zeta_{8}^{2} ) q^{37} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{39} - q^{49} + ( -1 - \zeta_{8}^{2} ) q^{61} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{63} -2 \zeta_{8} q^{67} -\zeta_{8}^{3} q^{75} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{79} - q^{81} -2 \zeta_{8}^{3} q^{91} + ( -1 - \zeta_{8}^{2} ) q^{93} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 4q^{13} + 4q^{21} - 4q^{37} - 4q^{49} - 4q^{61} - 4q^{81} - 4q^{93} + O(q^{100})$$

Character values

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

 $$n$$ $$257$$ $$511$$ $$517$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\zeta_{8}^{2}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
0 −0.707107 0.707107i 0 0 0 1.41421i 0 1.00000i 0
65.2 0 0.707107 + 0.707107i 0 0 0 1.41421i 0 1.00000i 0
449.1 0 −0.707107 + 0.707107i 0 0 0 1.41421i 0 1.00000i 0
449.2 0 0.707107 0.707107i 0 0 0 1.41421i 0 1.00000i 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 CM by $$\Q(\sqrt{-3})$$
4.b odd 2 1 inner
12.b even 2 1 inner
16.e even 4 1 inner
16.f odd 4 1 inner
48.i odd 4 1 inner
48.k even 4 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.1.i.b yes 4
3.b odd 2 1 CM 768.1.i.b yes 4
4.b odd 2 1 inner 768.1.i.b yes 4
8.b even 2 1 768.1.i.a 4
8.d odd 2 1 768.1.i.a 4
12.b even 2 1 inner 768.1.i.b yes 4
16.e even 4 1 768.1.i.a 4
16.e even 4 1 inner 768.1.i.b yes 4
16.f odd 4 1 768.1.i.a 4
16.f odd 4 1 inner 768.1.i.b yes 4
24.f even 2 1 768.1.i.a 4
24.h odd 2 1 768.1.i.a 4
32.g even 8 1 3072.1.e.a 2
32.g even 8 1 3072.1.e.b 2
32.g even 8 2 3072.1.h.a 4
32.h odd 8 1 3072.1.e.a 2
32.h odd 8 1 3072.1.e.b 2
32.h odd 8 2 3072.1.h.a 4
48.i odd 4 1 768.1.i.a 4
48.i odd 4 1 inner 768.1.i.b yes 4
48.k even 4 1 768.1.i.a 4
48.k even 4 1 inner 768.1.i.b yes 4
96.o even 8 1 3072.1.e.a 2
96.o even 8 1 3072.1.e.b 2
96.o even 8 2 3072.1.h.a 4
96.p odd 8 1 3072.1.e.a 2
96.p odd 8 1 3072.1.e.b 2
96.p odd 8 2 3072.1.h.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.1.i.a 4 8.b even 2 1
768.1.i.a 4 8.d odd 2 1
768.1.i.a 4 16.e even 4 1
768.1.i.a 4 16.f odd 4 1
768.1.i.a 4 24.f even 2 1
768.1.i.a 4 24.h odd 2 1
768.1.i.a 4 48.i odd 4 1
768.1.i.a 4 48.k even 4 1
768.1.i.b yes 4 1.a even 1 1 trivial
768.1.i.b yes 4 3.b odd 2 1 CM
768.1.i.b yes 4 4.b odd 2 1 inner
768.1.i.b yes 4 12.b even 2 1 inner
768.1.i.b yes 4 16.e even 4 1 inner
768.1.i.b yes 4 16.f odd 4 1 inner
768.1.i.b yes 4 48.i odd 4 1 inner
768.1.i.b yes 4 48.k even 4 1 inner
3072.1.e.a 2 32.g even 8 1
3072.1.e.a 2 32.h odd 8 1
3072.1.e.a 2 96.o even 8 1
3072.1.e.a 2 96.p odd 8 1
3072.1.e.b 2 32.g even 8 1
3072.1.e.b 2 32.h odd 8 1
3072.1.e.b 2 96.o even 8 1
3072.1.e.b 2 96.p odd 8 1
3072.1.h.a 4 32.g even 8 2
3072.1.h.a 4 32.h odd 8 2
3072.1.h.a 4 96.o even 8 2
3072.1.h.a 4 96.p odd 8 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{2} - 2 T_{13} + 2$$ acting on $$S_{1}^{\mathrm{new}}(768, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 2 + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$( 2 - 2 T + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$( -2 + T^{2} )^{2}$$
$37$ $$( 2 + 2 T + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( 2 + 2 T + T^{2} )^{2}$$
$67$ $$16 + T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$( -2 + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$