# Properties

 Label 1520.1.m.b Level $1520$ Weight $1$ Character orbit 1520.m Self dual yes Analytic conductor $0.759$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -95 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$0.758578819202$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.475.1 Artin image: $D_8$ Artin field: Galois closure of 8.2.1097440000.5

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{3} - q^{5} + q^{9} +O(q^{10})$$ $$q -\beta q^{3} - q^{5} + q^{9} -\beta q^{13} + \beta q^{15} + q^{19} + q^{25} + \beta q^{37} + 2 q^{39} - q^{45} + q^{49} + \beta q^{53} -\beta q^{57} + \beta q^{65} + \beta q^{67} -\beta q^{75} - q^{81} - q^{95} + \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{5} + 2 q^{9} + 2 q^{19} + 2 q^{25} + 4 q^{39} - 2 q^{45} + 2 q^{49} - 2 q^{81} - 2 q^{95} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1329.1
 1.41421 −1.41421
0 −1.41421 0 −1.00000 0 0 0 1.00000 0
1329.2 0 1.41421 0 −1.00000 0 0 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
95.d odd 2 1 CM by $$\Q(\sqrt{-95})$$
5.b even 2 1 inner
19.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1520.1.m.b 2
4.b odd 2 1 95.1.d.b 2
5.b even 2 1 inner 1520.1.m.b 2
12.b even 2 1 855.1.g.c 2
19.b odd 2 1 inner 1520.1.m.b 2
20.d odd 2 1 95.1.d.b 2
20.e even 4 2 475.1.c.b 2
60.h even 2 1 855.1.g.c 2
76.d even 2 1 95.1.d.b 2
76.f even 6 2 1805.1.h.b 4
76.g odd 6 2 1805.1.h.b 4
76.k even 18 6 1805.1.o.b 12
76.l odd 18 6 1805.1.o.b 12
95.d odd 2 1 CM 1520.1.m.b 2
228.b odd 2 1 855.1.g.c 2
380.d even 2 1 95.1.d.b 2
380.j odd 4 2 475.1.c.b 2
380.p odd 6 2 1805.1.h.b 4
380.s even 6 2 1805.1.h.b 4
380.ba odd 18 6 1805.1.o.b 12
380.bb even 18 6 1805.1.o.b 12
1140.p odd 2 1 855.1.g.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.1.d.b 2 4.b odd 2 1
95.1.d.b 2 20.d odd 2 1
95.1.d.b 2 76.d even 2 1
95.1.d.b 2 380.d even 2 1
475.1.c.b 2 20.e even 4 2
475.1.c.b 2 380.j odd 4 2
855.1.g.c 2 12.b even 2 1
855.1.g.c 2 60.h even 2 1
855.1.g.c 2 228.b odd 2 1
855.1.g.c 2 1140.p odd 2 1
1520.1.m.b 2 1.a even 1 1 trivial
1520.1.m.b 2 5.b even 2 1 inner
1520.1.m.b 2 19.b odd 2 1 inner
1520.1.m.b 2 95.d odd 2 1 CM
1805.1.h.b 4 76.f even 6 2
1805.1.h.b 4 76.g odd 6 2
1805.1.h.b 4 380.p odd 6 2
1805.1.h.b 4 380.s even 6 2
1805.1.o.b 12 76.k even 18 6
1805.1.o.b 12 76.l odd 18 6
1805.1.o.b 12 380.ba odd 18 6
1805.1.o.b 12 380.bb even 18 6

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1520, [\chi])$$:

 $$T_{3}^{2} - 2$$ $$T_{7}$$

## Hecke characteristic polynomials

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