# Properties

 Label 1521.1.j.b Level $1521$ Weight $1$ Character orbit 1521.j Analytic conductor $0.759$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1521.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.759077884215$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 117) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.6591.1 Artin image: $C_4\wr C_2$ Artin field: Galois closure of 8.0.1601613.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -i q^{4} + ( 1 - i ) q^{7} +O(q^{10})$$ $$q -i q^{4} + ( 1 - i ) q^{7} - q^{16} + ( 1 + i ) q^{19} -i q^{25} + ( -1 - i ) q^{28} + ( -1 - i ) q^{31} + ( -1 + i ) q^{37} -i q^{49} + i q^{64} + ( 1 + i ) q^{67} + ( 1 - i ) q^{73} + ( 1 - i ) q^{76} + ( -1 - i ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{7} + O(q^{10})$$ $$2 q + 2 q^{7} - 2 q^{16} + 2 q^{19} - 2 q^{28} - 2 q^{31} - 2 q^{37} + 2 q^{67} + 2 q^{73} + 2 q^{76} - 2 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 − 1.00000i 1.00000i
0 0 1.00000i 0 0 1.00000 + 1.00000i 0 0 0
775.1 0 0 1.00000i 0 0 1.00000 1.00000i 0 0 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
13.d odd 4 1 inner
39.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.1.j.b 2
3.b odd 2 1 CM 1521.1.j.b 2
13.b even 2 1 117.1.j.a 2
13.c even 3 2 1521.1.bd.b 4
13.d odd 4 1 117.1.j.a 2
13.d odd 4 1 inner 1521.1.j.b 2
13.e even 6 2 1521.1.bd.c 4
13.f odd 12 2 1521.1.bd.b 4
13.f odd 12 2 1521.1.bd.c 4
39.d odd 2 1 117.1.j.a 2
39.f even 4 1 117.1.j.a 2
39.f even 4 1 inner 1521.1.j.b 2
39.h odd 6 2 1521.1.bd.c 4
39.i odd 6 2 1521.1.bd.b 4
39.k even 12 2 1521.1.bd.b 4
39.k even 12 2 1521.1.bd.c 4
52.b odd 2 1 1872.1.bd.a 2
52.f even 4 1 1872.1.bd.a 2
65.d even 2 1 2925.1.s.a 2
65.f even 4 1 2925.1.t.b 2
65.g odd 4 1 2925.1.s.a 2
65.h odd 4 1 2925.1.t.a 2
65.h odd 4 1 2925.1.t.b 2
65.k even 4 1 2925.1.t.a 2
117.n odd 6 2 1053.1.bb.a 4
117.t even 6 2 1053.1.bb.a 4
117.y odd 12 2 1053.1.bb.a 4
117.z even 12 2 1053.1.bb.a 4
156.h even 2 1 1872.1.bd.a 2
156.l odd 4 1 1872.1.bd.a 2
195.e odd 2 1 2925.1.s.a 2
195.j odd 4 1 2925.1.t.a 2
195.n even 4 1 2925.1.s.a 2
195.s even 4 1 2925.1.t.a 2
195.s even 4 1 2925.1.t.b 2
195.u odd 4 1 2925.1.t.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.1.j.a 2 13.b even 2 1
117.1.j.a 2 13.d odd 4 1
117.1.j.a 2 39.d odd 2 1
117.1.j.a 2 39.f even 4 1
1053.1.bb.a 4 117.n odd 6 2
1053.1.bb.a 4 117.t even 6 2
1053.1.bb.a 4 117.y odd 12 2
1053.1.bb.a 4 117.z even 12 2
1521.1.j.b 2 1.a even 1 1 trivial
1521.1.j.b 2 3.b odd 2 1 CM
1521.1.j.b 2 13.d odd 4 1 inner
1521.1.j.b 2 39.f even 4 1 inner
1521.1.bd.b 4 13.c even 3 2
1521.1.bd.b 4 13.f odd 12 2
1521.1.bd.b 4 39.i odd 6 2
1521.1.bd.b 4 39.k even 12 2
1521.1.bd.c 4 13.e even 6 2
1521.1.bd.c 4 13.f odd 12 2
1521.1.bd.c 4 39.h odd 6 2
1521.1.bd.c 4 39.k even 12 2
1872.1.bd.a 2 52.b odd 2 1
1872.1.bd.a 2 52.f even 4 1
1872.1.bd.a 2 156.h even 2 1
1872.1.bd.a 2 156.l odd 4 1
2925.1.s.a 2 65.d even 2 1
2925.1.s.a 2 65.g odd 4 1
2925.1.s.a 2 195.e odd 2 1
2925.1.s.a 2 195.n even 4 1
2925.1.t.a 2 65.h odd 4 1
2925.1.t.a 2 65.k even 4 1
2925.1.t.a 2 195.j odd 4 1
2925.1.t.a 2 195.s even 4 1
2925.1.t.b 2 65.f even 4 1
2925.1.t.b 2 65.h odd 4 1
2925.1.t.b 2 195.s even 4 1
2925.1.t.b 2 195.u odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{1}^{\mathrm{new}}(1521, [\chi])$$.

## Hecke characteristic polynomials

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