# Properties

 Label 1476.1.z.a Level $1476$ Weight $1$ Character orbit 1476.z Analytic conductor $0.737$ Analytic rank $0$ Dimension $4$ Projective image $D_{10}$ 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.z (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.736619958646$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 164) Projective image: $$D_{10}$$ Projective field: Galois closure of 10.2.83809775204854016.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{10}^{3} q^{2} -\zeta_{10} q^{4} + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{5} -\zeta_{10}^{4} q^{8} +O(q^{10})$$ $$q + \zeta_{10}^{3} q^{2} -\zeta_{10} q^{4} + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{5} -\zeta_{10}^{4} q^{8} + ( \zeta_{10} - \zeta_{10}^{2} ) q^{10} + ( -\zeta_{10}^{2} + \zeta_{10}^{4} ) q^{13} + \zeta_{10}^{2} q^{16} + ( -\zeta_{10} - \zeta_{10}^{2} ) q^{17} + ( 1 + \zeta_{10}^{4} ) q^{20} + ( -\zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{25} + ( 1 - \zeta_{10}^{2} ) q^{26} - q^{32} + ( 1 - \zeta_{10}^{4} ) q^{34} + ( -1 - \zeta_{10}^{2} ) q^{37} + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{40} + \zeta_{10}^{3} q^{41} -\zeta_{10}^{4} q^{49} + ( -1 + \zeta_{10} - \zeta_{10}^{4} ) q^{50} + ( 1 + \zeta_{10}^{3} ) q^{52} + ( -\zeta_{10}^{3} - \zeta_{10}^{4} ) q^{53} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{61} -\zeta_{10}^{3} q^{64} + ( -1 + \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{65} + ( \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{68} + ( \zeta_{10} - \zeta_{10}^{4} ) q^{73} + ( 1 - \zeta_{10}^{3} ) q^{74} + ( 1 - \zeta_{10} ) q^{80} -\zeta_{10} q^{82} + ( \zeta_{10} + \zeta_{10}^{4} ) q^{85} + ( -1 + \zeta_{10}^{2} ) q^{97} + \zeta_{10}^{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} - q^{4} - 2 q^{5} + q^{8} + O(q^{10})$$ $$4 q + q^{2} - q^{4} - 2 q^{5} + q^{8} + 2 q^{10} - q^{16} + 3 q^{20} - 3 q^{25} + 5 q^{26} - 4 q^{32} + 5 q^{34} - 3 q^{37} + 2 q^{40} + q^{41} + q^{49} - 2 q^{50} + 5 q^{52} - 2 q^{61} - q^{64} - 5 q^{65} + 2 q^{73} + 3 q^{74} + 3 q^{80} - q^{82} - 5 q^{97} - q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$739$$ $$821$$ $$1441$$ $$\chi(n)$$ $$-1$$ $$1$$ $$\zeta_{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 −0.309017 + 0.951057i 0.809017 + 0.587785i −0.309017 − 0.951057i 0.809017 − 0.587785i
0.809017 0.587785i 0 0.309017 0.951057i −0.500000 + 1.53884i 0 0 −0.309017 0.951057i 0 0.500000 + 1.53884i
271.1 −0.309017 + 0.951057i 0 −0.809017 0.587785i −0.500000 0.363271i 0 0 0.809017 0.587785i 0 0.500000 0.363271i
523.1 0.809017 + 0.587785i 0 0.309017 + 0.951057i −0.500000 1.53884i 0 0 −0.309017 + 0.951057i 0 0.500000 1.53884i
1171.1 −0.309017 0.951057i 0 −0.809017 + 0.587785i −0.500000 + 0.363271i 0 0 0.809017 + 0.587785i 0 0.500000 + 0.363271i
 $$n$$: 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.f even 10 1 inner
164.l odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1476.1.z.a 4
3.b odd 2 1 164.1.l.a 4
4.b odd 2 1 CM 1476.1.z.a 4
12.b even 2 1 164.1.l.a 4
24.f even 2 1 2624.1.bx.a 4
24.h odd 2 1 2624.1.bx.a 4
41.f even 10 1 inner 1476.1.z.a 4
123.l odd 10 1 164.1.l.a 4
164.l odd 10 1 inner 1476.1.z.a 4
492.v even 10 1 164.1.l.a 4
984.bh even 10 1 2624.1.bx.a 4
984.bk odd 10 1 2624.1.bx.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.1.l.a 4 3.b odd 2 1
164.1.l.a 4 12.b even 2 1
164.1.l.a 4 123.l odd 10 1
164.1.l.a 4 492.v even 10 1
1476.1.z.a 4 1.a even 1 1 trivial
1476.1.z.a 4 4.b odd 2 1 CM
1476.1.z.a 4 41.f even 10 1 inner
1476.1.z.a 4 164.l odd 10 1 inner
2624.1.bx.a 4 24.f even 2 1
2624.1.bx.a 4 24.h odd 2 1
2624.1.bx.a 4 984.bh even 10 1
2624.1.bx.a 4 984.bk odd 10 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$1 + 3 T + 4 T^{2} + 2 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$5 + 5 T + T^{4}$$
$17$ $$5 - 5 T + T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$1 + 2 T + 4 T^{2} + 3 T^{3} + T^{4}$$
$41$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$5 + 5 T + T^{4}$$
$59$ $$T^{4}$$
$61$ $$1 + 3 T + 4 T^{2} + 2 T^{3} + T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$( -1 - T + T^{2} )^{2}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$5 + 10 T + 10 T^{2} + 5 T^{3} + T^{4}$$