# Properties

 Label 1476.1.l.a Level $1476$ Weight $1$ Character orbit 1476.l Analytic conductor $0.737$ Analytic rank $0$ Dimension $4$ Projective image $S_{4}$ CM/RM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.736619958646$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$S_{4}$$ Projective field: Galois closure of 4.2.7443468.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{5} +O(q^{10})$$ $$q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{5} + \zeta_{8}^{3} q^{11} + \zeta_{8} q^{17} + ( 1 - \zeta_{8}^{2} ) q^{19} + q^{25} + \zeta_{8}^{3} q^{29} - q^{31} + q^{37} + \zeta_{8}^{3} q^{41} -\zeta_{8}^{2} q^{43} -\zeta_{8} q^{47} -\zeta_{8}^{2} q^{49} + ( -1 + \zeta_{8}^{2} ) q^{55} -\zeta_{8}^{2} q^{61} + ( -1 + \zeta_{8}^{2} ) q^{67} -\zeta_{8}^{3} q^{71} -\zeta_{8}^{2} q^{73} + ( 1 + \zeta_{8}^{2} ) q^{79} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{83} + ( 1 + \zeta_{8}^{2} ) q^{85} -2 \zeta_{8}^{3} q^{95} + ( -1 + \zeta_{8}^{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + O(q^{10})$$ $$4 q + 4 q^{19} + 4 q^{25} - 4 q^{31} + 4 q^{37} - 4 q^{55} - 4 q^{67} + 4 q^{79} + 4 q^{85} - 4 q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$739$$ $$821$$ $$1441$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\zeta_{8}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
665.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
0 0 0 −1.41421 0 0 0 0 0
665.2 0 0 0 1.41421 0 0 0 0 0
1385.1 0 0 0 −1.41421 0 0 0 0 0
1385.2 0 0 0 1.41421 0 0 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 inner
41.c even 4 1 inner
123.f 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.l.a 4
3.b odd 2 1 inner 1476.1.l.a 4
41.c even 4 1 inner 1476.1.l.a 4
123.f odd 4 1 inner 1476.1.l.a 4

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1476, [\chi])$$.

## Hecke characteristic polynomials

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