# Properties

 Label 1520.2.d.d Level $1520$ Weight $2$ Character orbit 1520.d Analytic conductor $12.137$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1520 = 2^{4} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1520.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.1372611072$$ 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: $$1$$ Twist minimal: no (minimal twist has level 760) 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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( 2 \zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{7} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( 2 \zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{7} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{11} + ( -2 \zeta_{8} + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{13} + ( -1 + \zeta_{8} + 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{15} + \zeta_{8}^{2} q^{17} - q^{19} + ( -3 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{21} + ( 3 \zeta_{8} + \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{23} + ( 4 + 3 \zeta_{8}^{2} ) q^{25} + ( \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{27} + ( -1 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{29} + ( -2 - \zeta_{8} + \zeta_{8}^{3} ) q^{31} + ( -\zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{33} + ( -1 + \zeta_{8} + 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{35} + ( -2 \zeta_{8} + 8 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{37} + ( 3 + \zeta_{8} - \zeta_{8}^{3} ) q^{39} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{41} + ( 3 \zeta_{8} - 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{43} + ( -6 - 2 \zeta_{8}^{2} ) q^{45} -8 \zeta_{8}^{2} q^{47} + ( 4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{49} + ( -1 - \zeta_{8} + \zeta_{8}^{3} ) q^{51} + ( 2 \zeta_{8} - \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{53} + ( -3 - \zeta_{8}^{2} ) q^{55} + ( -\zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{57} + ( 13 + \zeta_{8} - \zeta_{8}^{3} ) q^{59} + ( 2 + \zeta_{8} - \zeta_{8}^{3} ) q^{61} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{63} + ( 2 + \zeta_{8} - 6 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{65} + ( 5 \zeta_{8} - \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{67} + ( -7 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{69} + ( 4 - 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{71} + ( -2 \zeta_{8} - 11 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{73} + ( -3 + \zeta_{8} + 4 \zeta_{8}^{2} + 7 \zeta_{8}^{3} ) q^{75} + ( -\zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{77} + ( 2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{79} + ( -1 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{81} + ( -6 \zeta_{8} + 6 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{83} + ( \zeta_{8} + 2 \zeta_{8}^{3} ) q^{85} + ( -3 \zeta_{8} - 5 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{87} + ( 8 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{89} + ( 3 + \zeta_{8} - \zeta_{8}^{3} ) q^{91} + ( -3 \zeta_{8} - 4 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{93} + ( -2 \zeta_{8} + \zeta_{8}^{3} ) q^{95} + ( 4 \zeta_{8} + 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{97} + 4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 4q^{15} - 4q^{19} - 12q^{21} + 16q^{25} - 4q^{29} - 8q^{31} - 4q^{35} + 12q^{39} - 24q^{45} + 16q^{49} - 4q^{51} - 12q^{55} + 52q^{59} + 8q^{61} + 8q^{65} - 28q^{69} + 16q^{71} - 12q^{75} + 8q^{79} - 4q^{81} + 32q^{89} + 12q^{91} + 16q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$401$$ $$1141$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 2.41421i 0 2.12132 0.707107i 0 2.41421i 0 −2.82843 0
609.2 0 0.414214i 0 −2.12132 0.707107i 0 0.414214i 0 2.82843 0
609.3 0 0.414214i 0 −2.12132 + 0.707107i 0 0.414214i 0 2.82843 0
609.4 0 2.41421i 0 2.12132 + 0.707107i 0 2.41421i 0 −2.82843 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 1520.2.d.d 4
4.b odd 2 1 760.2.d.c 4
5.b even 2 1 inner 1520.2.d.d 4
5.c odd 4 1 7600.2.a.x 2
5.c odd 4 1 7600.2.a.bc 2
20.d odd 2 1 760.2.d.c 4
20.e even 4 1 3800.2.a.l 2
20.e even 4 1 3800.2.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.d.c 4 4.b odd 2 1
760.2.d.c 4 20.d odd 2 1
1520.2.d.d 4 1.a even 1 1 trivial
1520.2.d.d 4 5.b even 2 1 inner
3800.2.a.l 2 20.e even 4 1
3800.2.a.p 2 20.e even 4 1
7600.2.a.x 2 5.c odd 4 1
7600.2.a.bc 2 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1520, [\chi])$$:

 $$T_{3}^{4} + 6 T_{3}^{2} + 1$$ $$T_{7}^{4} + 6 T_{7}^{2} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 + 6 T^{2} + T^{4}$$
$5$ $$25 - 8 T^{2} + T^{4}$$
$7$ $$1 + 6 T^{2} + T^{4}$$
$11$ $$( -2 + T^{2} )^{2}$$
$13$ $$49 + 18 T^{2} + T^{4}$$
$17$ $$( 1 + T^{2} )^{2}$$
$19$ $$( 1 + T )^{4}$$
$23$ $$289 + 38 T^{2} + T^{4}$$
$29$ $$( -7 + 2 T + T^{2} )^{2}$$
$31$ $$( 2 + 4 T + T^{2} )^{2}$$
$37$ $$3136 + 144 T^{2} + T^{4}$$
$41$ $$( -50 + T^{2} )^{2}$$
$43$ $$196 + 44 T^{2} + T^{4}$$
$47$ $$( 64 + T^{2} )^{2}$$
$53$ $$49 + 18 T^{2} + T^{4}$$
$59$ $$( 167 - 26 T + T^{2} )^{2}$$
$61$ $$( 2 - 4 T + T^{2} )^{2}$$
$67$ $$2401 + 102 T^{2} + T^{4}$$
$71$ $$( -34 - 8 T + T^{2} )^{2}$$
$73$ $$12769 + 258 T^{2} + T^{4}$$
$79$ $$( -4 - 4 T + T^{2} )^{2}$$
$83$ $$1296 + 216 T^{2} + T^{4}$$
$89$ $$( 46 - 16 T + T^{2} )^{2}$$
$97$ $$784 + 72 T^{2} + T^{4}$$