# Properties

 Label 1521.1.j.c Level $1521$ Weight $1$ Character orbit 1521.j Analytic conductor $0.759$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -39 Inner twists $4$

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## 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.507.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( 1 + i ) q^{2} + i q^{4} + ( -1 - i ) q^{5} +O(q^{10})$$ $$q + ( 1 + i ) q^{2} + i q^{4} + ( -1 - i ) q^{5} -2 i q^{10} + ( 1 - i ) q^{11} + q^{16} + ( 1 - i ) q^{20} + 2 q^{22} + i q^{25} + ( 1 + i ) q^{32} + ( -1 - i ) q^{41} + 2 i q^{43} + ( 1 + i ) q^{44} + ( 1 - i ) q^{47} + i q^{49} + ( -1 + i ) q^{50} -2 q^{55} + ( -1 + i ) q^{59} + i q^{64} + ( -1 - i ) q^{71} + ( -1 - i ) q^{80} -2 i q^{82} + ( 1 + i ) q^{83} + ( -2 + 2 i ) q^{86} + ( -1 + i ) q^{89} + 2 q^{94} + ( -1 + i ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{5} + O(q^{10})$$ $$2 q + 2 q^{2} - 2 q^{5} + 2 q^{11} + 2 q^{16} + 2 q^{20} + 4 q^{22} + 2 q^{32} - 2 q^{41} + 2 q^{44} + 2 q^{47} - 2 q^{50} - 4 q^{55} - 2 q^{59} - 2 q^{71} - 2 q^{80} + 2 q^{83} - 4 q^{86} - 2 q^{89} + 4 q^{94} - 2 q^{98} + O(q^{100})$$

## 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
1.00000 1.00000i 0 1.00000i −1.00000 + 1.00000i 0 0 0 0 2.00000i
775.1 1.00000 + 1.00000i 0 1.00000i −1.00000 1.00000i 0 0 0 0 2.00000i
 $$n$$: 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
39.d odd 2 1 CM by $$\Q(\sqrt{-39})$$
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.c yes 2
3.b odd 2 1 1521.1.j.a 2
13.b even 2 1 1521.1.j.a 2
13.c even 3 2 1521.1.bd.a 4
13.d odd 4 1 1521.1.j.a 2
13.d odd 4 1 inner 1521.1.j.c yes 2
13.e even 6 2 1521.1.bd.d 4
13.f odd 12 2 1521.1.bd.a 4
13.f odd 12 2 1521.1.bd.d 4
39.d odd 2 1 CM 1521.1.j.c yes 2
39.f even 4 1 1521.1.j.a 2
39.f even 4 1 inner 1521.1.j.c yes 2
39.h odd 6 2 1521.1.bd.a 4
39.i odd 6 2 1521.1.bd.d 4
39.k even 12 2 1521.1.bd.a 4
39.k even 12 2 1521.1.bd.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1521.1.j.a 2 3.b odd 2 1
1521.1.j.a 2 13.b even 2 1
1521.1.j.a 2 13.d odd 4 1
1521.1.j.a 2 39.f even 4 1
1521.1.j.c yes 2 1.a even 1 1 trivial
1521.1.j.c yes 2 13.d odd 4 1 inner
1521.1.j.c yes 2 39.d odd 2 1 CM
1521.1.j.c yes 2 39.f even 4 1 inner
1521.1.bd.a 4 13.c even 3 2
1521.1.bd.a 4 13.f odd 12 2
1521.1.bd.a 4 39.h odd 6 2
1521.1.bd.a 4 39.k even 12 2
1521.1.bd.d 4 13.e even 6 2
1521.1.bd.d 4 13.f odd 12 2
1521.1.bd.d 4 39.i odd 6 2
1521.1.bd.d 4 39.k even 12 2

## Hecke kernels

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

## Hecke characteristic polynomials

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