# Properties

 Label 1520.2.d.b Level $1520$ Weight $2$ Character orbit 1520.d Analytic conductor $12.137$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 95) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - 2 i ) q^{5} -2 i q^{7} + 3 q^{9} +O(q^{10})$$ $$q + ( -1 - 2 i ) q^{5} -2 i q^{7} + 3 q^{9} + 4 q^{11} + 2 i q^{13} + 4 i q^{17} + q^{19} -6 i q^{23} + ( -3 + 4 i ) q^{25} + 6 q^{29} + 4 q^{31} + ( -4 + 2 i ) q^{35} -10 i q^{37} -10 q^{41} + 2 i q^{43} + ( -3 - 6 i ) q^{45} + 6 i q^{47} + 3 q^{49} -10 i q^{53} + ( -4 - 8 i ) q^{55} + 2 q^{61} -6 i q^{63} + ( 4 - 2 i ) q^{65} -8 i q^{67} -4 q^{71} -4 i q^{73} -8 i q^{77} + 4 q^{79} + 9 q^{81} -18 i q^{83} + ( 8 - 4 i ) q^{85} + 2 q^{89} + 4 q^{91} + ( -1 - 2 i ) q^{95} + 6 i q^{97} + 12 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{5} + 6q^{9} + O(q^{10})$$ $$2q - 2q^{5} + 6q^{9} + 8q^{11} + 2q^{19} - 6q^{25} + 12q^{29} + 8q^{31} - 8q^{35} - 20q^{41} - 6q^{45} + 6q^{49} - 8q^{55} + 4q^{61} + 8q^{65} - 8q^{71} + 8q^{79} + 18q^{81} + 16q^{85} + 4q^{89} + 8q^{91} - 2q^{95} + 24q^{99} + O(q^{100})$$

## 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
 1.00000i − 1.00000i
0 0 0 −1.00000 2.00000i 0 2.00000i 0 3.00000 0
609.2 0 0 0 −1.00000 + 2.00000i 0 2.00000i 0 3.00000 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 1520.2.d.b 2
4.b odd 2 1 95.2.b.a 2
5.b even 2 1 inner 1520.2.d.b 2
5.c odd 4 1 7600.2.a.i 1
5.c odd 4 1 7600.2.a.l 1
12.b even 2 1 855.2.c.b 2
20.d odd 2 1 95.2.b.a 2
20.e even 4 1 475.2.a.a 1
20.e even 4 1 475.2.a.c 1
60.h even 2 1 855.2.c.b 2
60.l odd 4 1 4275.2.a.e 1
60.l odd 4 1 4275.2.a.p 1
76.d even 2 1 1805.2.b.c 2
380.d even 2 1 1805.2.b.c 2
380.j odd 4 1 9025.2.a.c 1
380.j odd 4 1 9025.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.b.a 2 4.b odd 2 1
95.2.b.a 2 20.d odd 2 1
475.2.a.a 1 20.e even 4 1
475.2.a.c 1 20.e even 4 1
855.2.c.b 2 12.b even 2 1
855.2.c.b 2 60.h even 2 1
1520.2.d.b 2 1.a even 1 1 trivial
1520.2.d.b 2 5.b even 2 1 inner
1805.2.b.c 2 76.d even 2 1
1805.2.b.c 2 380.d even 2 1
4275.2.a.e 1 60.l odd 4 1
4275.2.a.p 1 60.l odd 4 1
7600.2.a.i 1 5.c odd 4 1
7600.2.a.l 1 5.c odd 4 1
9025.2.a.c 1 380.j odd 4 1
9025.2.a.h 1 380.j odd 4 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$5 + 2 T + T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$( -4 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$16 + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$36 + T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$( -4 + T )^{2}$$
$37$ $$100 + T^{2}$$
$41$ $$( 10 + T )^{2}$$
$43$ $$4 + T^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$100 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$64 + T^{2}$$
$71$ $$( 4 + T )^{2}$$
$73$ $$16 + T^{2}$$
$79$ $$( -4 + T )^{2}$$
$83$ $$324 + T^{2}$$
$89$ $$( -2 + T )^{2}$$
$97$ $$36 + T^{2}$$