# Properties

 Label 768.2.j.c Level $768$ Weight $2$ Character orbit 768.j Analytic conductor $6.133$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$6.13251087523$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{8}^{3} q^{3} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{7} - \zeta_{8}^{2} q^{9} +O(q^{10})$$ q + z^3 * q^3 + (-z^3 - z) * q^7 - z^2 * q^9 $$q + \zeta_{8}^{3} q^{3} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{7} - \zeta_{8}^{2} q^{9} + ( - \zeta_{8}^{2} + 1) q^{13} + 6 q^{17} + 6 \zeta_{8}^{3} q^{19} + (\zeta_{8}^{2} + 1) q^{21} + ( - 6 \zeta_{8}^{3} - 6 \zeta_{8}) q^{23} - 5 \zeta_{8}^{2} q^{25} + \zeta_{8} q^{27} + ( - 6 \zeta_{8}^{2} + 6) q^{29} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{31} + (5 \zeta_{8}^{2} + 5) q^{37} + (\zeta_{8}^{3} + \zeta_{8}) q^{39} + 6 \zeta_{8}^{2} q^{41} + 6 \zeta_{8} q^{43} + ( - 6 \zeta_{8}^{3} + 6 \zeta_{8}) q^{47} + 5 q^{49} + 6 \zeta_{8}^{3} q^{51} + ( - 6 \zeta_{8}^{2} - 6) q^{53} - 6 \zeta_{8}^{2} q^{57} + ( - 5 \zeta_{8}^{2} + 5) q^{61} + (\zeta_{8}^{3} - \zeta_{8}) q^{63} + 4 \zeta_{8}^{3} q^{67} + (6 \zeta_{8}^{2} + 6) q^{69} + ( - 6 \zeta_{8}^{3} - 6 \zeta_{8}) q^{71} + 5 \zeta_{8} q^{75} + (\zeta_{8}^{3} - \zeta_{8}) q^{79} - q^{81} - 12 \zeta_{8}^{3} q^{83} + (6 \zeta_{8}^{3} + 6 \zeta_{8}) q^{87} + 6 \zeta_{8}^{2} q^{89} - 2 \zeta_{8} q^{91} + (\zeta_{8}^{2} - 1) q^{93} - 12 q^{97} +O(q^{100})$$ q + z^3 * q^3 + (-z^3 - z) * q^7 - z^2 * q^9 + (-z^2 + 1) * q^13 + 6 * q^17 + 6*z^3 * q^19 + (z^2 + 1) * q^21 + (-6*z^3 - 6*z) * q^23 - 5*z^2 * q^25 + z * q^27 + (-6*z^2 + 6) * q^29 + (-z^3 + z) * q^31 + (5*z^2 + 5) * q^37 + (z^3 + z) * q^39 + 6*z^2 * q^41 + 6*z * q^43 + (-6*z^3 + 6*z) * q^47 + 5 * q^49 + 6*z^3 * q^51 + (-6*z^2 - 6) * q^53 - 6*z^2 * q^57 + (-5*z^2 + 5) * q^61 + (z^3 - z) * q^63 + 4*z^3 * q^67 + (6*z^2 + 6) * q^69 + (-6*z^3 - 6*z) * q^71 + 5*z * q^75 + (z^3 - z) * q^79 - q^81 - 12*z^3 * q^83 + (6*z^3 + 6*z) * q^87 + 6*z^2 * q^89 - 2*z * q^91 + (z^2 - 1) * q^93 - 12 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 4 q^{13} + 24 q^{17} + 4 q^{21} + 24 q^{29} + 20 q^{37} + 20 q^{49} - 24 q^{53} + 20 q^{61} + 24 q^{69} - 4 q^{81} - 4 q^{93} - 48 q^{97}+O(q^{100})$$ 4 * q + 4 * q^13 + 24 * q^17 + 4 * q^21 + 24 * q^29 + 20 * q^37 + 20 * q^49 - 24 * q^53 + 20 * q^61 + 24 * q^69 - 4 * q^81 - 4 * q^93 - 48 * q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 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
193.2 0 0.707107 0.707107i 0 0 0 1.41421i 0 1.00000i 0
577.1 0 −0.707107 0.707107i 0 0 0 1.41421i 0 1.00000i 0
577.2 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
4.b odd 2 1 inner
16.e even 4 1 inner
16.f odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.2.j.c yes 4
3.b odd 2 1 2304.2.k.c 4
4.b odd 2 1 inner 768.2.j.c yes 4
8.b even 2 1 768.2.j.b 4
8.d odd 2 1 768.2.j.b 4
12.b even 2 1 2304.2.k.c 4
16.e even 4 1 768.2.j.b 4
16.e even 4 1 inner 768.2.j.c yes 4
16.f odd 4 1 768.2.j.b 4
16.f odd 4 1 inner 768.2.j.c yes 4
24.f even 2 1 2304.2.k.b 4
24.h odd 2 1 2304.2.k.b 4
32.g even 8 1 3072.2.a.a 2
32.g even 8 1 3072.2.a.g 2
32.g even 8 2 3072.2.d.a 4
32.h odd 8 1 3072.2.a.a 2
32.h odd 8 1 3072.2.a.g 2
32.h odd 8 2 3072.2.d.a 4
48.i odd 4 1 2304.2.k.b 4
48.i odd 4 1 2304.2.k.c 4
48.k even 4 1 2304.2.k.b 4
48.k even 4 1 2304.2.k.c 4
96.o even 8 1 9216.2.a.n 2
96.o even 8 1 9216.2.a.o 2
96.p odd 8 1 9216.2.a.n 2
96.p odd 8 1 9216.2.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.2.j.b 4 8.b even 2 1
768.2.j.b 4 8.d odd 2 1
768.2.j.b 4 16.e even 4 1
768.2.j.b 4 16.f odd 4 1
768.2.j.c yes 4 1.a even 1 1 trivial
768.2.j.c yes 4 4.b odd 2 1 inner
768.2.j.c yes 4 16.e even 4 1 inner
768.2.j.c yes 4 16.f odd 4 1 inner
2304.2.k.b 4 24.f even 2 1
2304.2.k.b 4 24.h odd 2 1
2304.2.k.b 4 48.i odd 4 1
2304.2.k.b 4 48.k even 4 1
2304.2.k.c 4 3.b odd 2 1
2304.2.k.c 4 12.b even 2 1
2304.2.k.c 4 48.i odd 4 1
2304.2.k.c 4 48.k even 4 1
3072.2.a.a 2 32.g even 8 1
3072.2.a.a 2 32.h odd 8 1
3072.2.a.g 2 32.g even 8 1
3072.2.a.g 2 32.h odd 8 1
3072.2.d.a 4 32.g even 8 2
3072.2.d.a 4 32.h odd 8 2
9216.2.a.n 2 96.o even 8 1
9216.2.a.n 2 96.p odd 8 1
9216.2.a.o 2 96.o even 8 1
9216.2.a.o 2 96.p odd 8 1

## Hecke kernels

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

 $$T_{5}$$ T5 $$T_{13}^{2} - 2T_{13} + 2$$ T13^2 - 2*T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 1$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 2)^{2}$$
$11$ $$T^{4}$$
$13$ $$(T^{2} - 2 T + 2)^{2}$$
$17$ $$(T - 6)^{4}$$
$19$ $$T^{4} + 1296$$
$23$ $$(T^{2} + 72)^{2}$$
$29$ $$(T^{2} - 12 T + 72)^{2}$$
$31$ $$(T^{2} - 2)^{2}$$
$37$ $$(T^{2} - 10 T + 50)^{2}$$
$41$ $$(T^{2} + 36)^{2}$$
$43$ $$T^{4} + 1296$$
$47$ $$(T^{2} - 72)^{2}$$
$53$ $$(T^{2} + 12 T + 72)^{2}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} - 10 T + 50)^{2}$$
$67$ $$T^{4} + 256$$
$71$ $$(T^{2} + 72)^{2}$$
$73$ $$T^{4}$$
$79$ $$(T^{2} - 2)^{2}$$
$83$ $$T^{4} + 20736$$
$89$ $$(T^{2} + 36)^{2}$$
$97$ $$(T + 12)^{4}$$