# Properties

 Label 1476.1.j.a Level $1476$ Weight $1$ Character orbit 1476.j Analytic conductor $0.737$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1476 = 2^{2} \cdot 3^{2} \cdot 41$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1476.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.736619958646$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ 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 4.2.2481156.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} - i q^{8} +O(q^{10})$$ q + z * q^2 - q^4 - z * q^8 $$q + i q^{2} - q^{4} - i q^{8} + (i + 1) q^{13} + q^{16} + (i - 1) q^{17} + q^{25} + (i - 1) q^{26} + (i + 1) q^{29} + i q^{32} + ( - i - 1) q^{34} + i q^{41} + i q^{49} + i q^{50} + ( - i - 1) q^{52} + ( - i - 1) q^{53} + (i - 1) q^{58} - q^{64} + ( - i + 1) q^{68} - i q^{73} - q^{82} + ( - i - 1) q^{89} + (i - 1) q^{97} - q^{98} +O(q^{100})$$ q + z * q^2 - q^4 - z * q^8 + (z + 1) * q^13 + q^16 + (z - 1) * q^17 + q^25 + (z - 1) * q^26 + (z + 1) * q^29 + z * q^32 + (-z - 1) * q^34 + z * q^41 + z * q^49 + z * q^50 + (-z - 1) * q^52 + (-z - 1) * q^53 + (z - 1) * q^58 - q^64 + (-z + 1) * q^68 - z * q^73 - q^82 + (-z - 1) * q^89 + (z - 1) * q^97 - q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4}+O(q^{10})$$ 2 * q - 2 * q^4 $$2 q - 2 q^{4} + 2 q^{13} + 2 q^{16} - 2 q^{17} + 2 q^{25} - 2 q^{26} + 2 q^{29} - 2 q^{34} - 2 q^{52} - 2 q^{53} - 2 q^{58} - 2 q^{64} + 2 q^{68} - 2 q^{82} - 2 q^{89} - 2 q^{97} - 2 q^{98}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^13 + 2 * q^16 - 2 * q^17 + 2 * q^25 - 2 * q^26 + 2 * q^29 - 2 * q^34 - 2 * q^52 - 2 * q^53 - 2 * q^58 - 2 * q^64 + 2 * q^68 - 2 * q^82 - 2 * q^89 - 2 * q^97 - 2 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1476\mathbb{Z}\right)^\times$$.

 $$n$$ $$739$$ $$821$$ $$1441$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
91.1
 1.00000i − 1.00000i
1.00000i 0 −1.00000 0 0 0 1.00000i 0 0
811.1 1.00000i 0 −1.00000 0 0 0 1.00000i 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
41.c even 4 1 inner
164.e odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1476.1.j.a 2
3.b odd 2 1 1476.1.j.b yes 2
4.b odd 2 1 CM 1476.1.j.a 2
12.b even 2 1 1476.1.j.b yes 2
41.c even 4 1 inner 1476.1.j.a 2
123.f odd 4 1 1476.1.j.b yes 2
164.e odd 4 1 inner 1476.1.j.a 2
492.l even 4 1 1476.1.j.b yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1476.1.j.a 2 1.a even 1 1 trivial
1476.1.j.a 2 4.b odd 2 1 CM
1476.1.j.a 2 41.c even 4 1 inner
1476.1.j.a 2 164.e odd 4 1 inner
1476.1.j.b yes 2 3.b odd 2 1
1476.1.j.b yes 2 12.b even 2 1
1476.1.j.b yes 2 123.f odd 4 1
1476.1.j.b yes 2 492.l even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{17}^{2} + 2T_{17} + 2$$ acting on $$S_{1}^{\mathrm{new}}(1476, [\chi])$$.

## Hecke characteristic polynomials

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