# Properties

 Label 760.2.d.d Level $760$ Weight $2$ Character orbit 760.d Analytic conductor $6.069$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$760 = 2^{3} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 760.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.06863055362$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{3} + ( 1 + 2 \zeta_{8}^{2} ) q^{5} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{7} + q^{9} +O(q^{10})$$ $$q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{3} + ( 1 + 2 \zeta_{8}^{2} ) q^{5} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{7} + q^{9} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{11} + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{13} + ( -\zeta_{8} + 3 \zeta_{8}^{3} ) q^{15} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{17} + q^{19} + ( 4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{21} + ( -4 \zeta_{8} - 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{23} + ( -3 + 4 \zeta_{8}^{2} ) q^{25} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{27} + ( 2 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{29} + ( 4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{31} + ( 2 \zeta_{8} + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{33} + ( 4 + 2 \zeta_{8} - 2 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{35} + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{37} + ( 2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{39} + ( -2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{41} + ( 2 \zeta_{8} + 6 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{43} + ( 1 + 2 \zeta_{8}^{2} ) q^{45} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{47} + ( -5 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{49} + 4 q^{51} + ( 7 \zeta_{8} + 2 \zeta_{8}^{2} + 7 \zeta_{8}^{3} ) q^{53} + ( 2 + 6 \zeta_{8} + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{55} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{57} + ( 4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{59} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{61} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{63} + ( -4 + \zeta_{8} + 2 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{65} + ( 7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{67} + ( 8 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{69} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{71} + ( -2 \zeta_{8} + 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{73} + ( -7 \zeta_{8} + \zeta_{8}^{3} ) q^{75} + ( -8 \zeta_{8} - 12 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{77} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{79} -5 q^{81} + ( -4 \zeta_{8} - 10 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{83} + ( 2 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{85} + ( 2 \zeta_{8} - 8 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{87} + ( 2 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{89} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{91} + ( 4 \zeta_{8} + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{93} + ( 1 + 2 \zeta_{8}^{2} ) q^{95} + ( 5 \zeta_{8} - 6 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{97} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{5} + 4 q^{9} + O(q^{10})$$ $$4 q + 4 q^{5} + 4 q^{9} + 8 q^{11} + 4 q^{19} + 16 q^{21} - 12 q^{25} + 8 q^{29} + 16 q^{31} + 16 q^{35} + 8 q^{39} - 8 q^{41} + 4 q^{45} - 20 q^{49} + 16 q^{51} + 8 q^{55} + 16 q^{59} - 16 q^{65} + 32 q^{69} - 20 q^{81} + 8 q^{89} + 4 q^{95} + 8 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/760\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$381$$ $$401$$ $$457$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
609.1
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
0 1.41421i 0 1.00000 2.00000i 0 4.82843i 0 1.00000 0
609.2 0 1.41421i 0 1.00000 + 2.00000i 0 0.828427i 0 1.00000 0
609.3 0 1.41421i 0 1.00000 2.00000i 0 0.828427i 0 1.00000 0
609.4 0 1.41421i 0 1.00000 + 2.00000i 0 4.82843i 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 760.2.d.d 4
4.b odd 2 1 1520.2.d.g 4
5.b even 2 1 inner 760.2.d.d 4
5.c odd 4 1 3800.2.a.m 2
5.c odd 4 1 3800.2.a.o 2
20.d odd 2 1 1520.2.d.g 4
20.e even 4 1 7600.2.a.z 2
20.e even 4 1 7600.2.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.d.d 4 1.a even 1 1 trivial
760.2.d.d 4 5.b even 2 1 inner
1520.2.d.g 4 4.b odd 2 1
1520.2.d.g 4 20.d odd 2 1
3800.2.a.m 2 5.c odd 4 1
3800.2.a.o 2 5.c odd 4 1
7600.2.a.z 2 20.e even 4 1
7600.2.a.bb 2 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 2$$ acting on $$S_{2}^{\mathrm{new}}(760, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 2 + T^{2} )^{2}$$
$5$ $$( 5 - 2 T + T^{2} )^{2}$$
$7$ $$16 + 24 T^{2} + T^{4}$$
$11$ $$( -4 - 4 T + T^{2} )^{2}$$
$13$ $$4 + 12 T^{2} + T^{4}$$
$17$ $$( 8 + T^{2} )^{2}$$
$19$ $$( -1 + T )^{4}$$
$23$ $$784 + 72 T^{2} + T^{4}$$
$29$ $$( -28 - 4 T + T^{2} )^{2}$$
$31$ $$( 8 - 8 T + T^{2} )^{2}$$
$37$ $$4 + 12 T^{2} + T^{4}$$
$41$ $$( -4 + 4 T + T^{2} )^{2}$$
$43$ $$784 + 88 T^{2} + T^{4}$$
$47$ $$16 + 24 T^{2} + T^{4}$$
$53$ $$8836 + 204 T^{2} + T^{4}$$
$59$ $$( 8 - 8 T + T^{2} )^{2}$$
$61$ $$( -32 + T^{2} )^{2}$$
$67$ $$( 98 + T^{2} )^{2}$$
$71$ $$( -72 + T^{2} )^{2}$$
$73$ $$64 + 48 T^{2} + T^{4}$$
$79$ $$( -72 + T^{2} )^{2}$$
$83$ $$4624 + 264 T^{2} + T^{4}$$
$89$ $$( -124 - 4 T + T^{2} )^{2}$$
$97$ $$196 + 172 T^{2} + T^{4}$$