# Properties

 Label 475.1.m.a Level $475$ Weight $1$ Character orbit 475.m Analytic conductor $0.237$ Analytic rank $0$ Dimension $4$ Projective image $D_{10}$ CM discriminant -19 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 475.m (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.237055881001$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{10}$$ Projective field: Galois closure of 10.0.1889113616943359375.4

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{10}^{3} q^{4} -\zeta_{10} q^{5} + ( \zeta_{10} + \zeta_{10}^{4} ) q^{7} + \zeta_{10} q^{9} +O(q^{10})$$ $$q + \zeta_{10}^{3} q^{4} -\zeta_{10} q^{5} + ( \zeta_{10} + \zeta_{10}^{4} ) q^{7} + \zeta_{10} q^{9} + ( -1 + \zeta_{10}^{3} ) q^{11} -\zeta_{10} q^{16} + ( 1 - \zeta_{10}^{4} ) q^{17} + \zeta_{10}^{2} q^{19} -\zeta_{10}^{4} q^{20} + ( -1 - \zeta_{10}^{3} ) q^{23} + \zeta_{10}^{2} q^{25} + ( -\zeta_{10}^{2} + \zeta_{10}^{4} ) q^{28} + ( 1 - \zeta_{10}^{2} ) q^{35} + \zeta_{10}^{4} q^{36} + ( -\zeta_{10}^{2} - \zeta_{10}^{3} ) q^{43} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{44} -\zeta_{10}^{2} q^{45} + ( -1 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{49} + ( \zeta_{10} - \zeta_{10}^{4} ) q^{55} + ( \zeta_{10} - \zeta_{10}^{2} ) q^{61} + ( -1 + \zeta_{10}^{2} ) q^{63} -\zeta_{10}^{4} q^{64} + ( \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{68} - q^{76} + ( -\zeta_{10} - \zeta_{10}^{2} ) q^{77} + \zeta_{10}^{2} q^{80} + \zeta_{10}^{2} q^{81} + ( 1 - \zeta_{10}^{4} ) q^{83} + ( -1 - \zeta_{10} ) q^{85} + ( \zeta_{10} - \zeta_{10}^{3} ) q^{92} -\zeta_{10}^{3} q^{95} + ( -\zeta_{10} + \zeta_{10}^{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{4} - q^{5} + q^{9} + O(q^{10})$$ $$4 q + q^{4} - q^{5} + q^{9} - 3 q^{11} - q^{16} + 5 q^{17} - q^{19} + q^{20} - 5 q^{23} - q^{25} + 5 q^{35} - q^{36} - 2 q^{44} + q^{45} - 6 q^{49} + 2 q^{55} + 2 q^{61} - 5 q^{63} + q^{64} - 4 q^{76} - q^{80} - q^{81} + 5 q^{83} - 5 q^{85} - q^{95} - 2 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$\zeta_{10}^{3}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
94.1
 −0.309017 + 0.951057i 0.809017 + 0.587785i 0.809017 − 0.587785i −0.309017 − 0.951057i
0 0 0.809017 0.587785i 0.309017 0.951057i 0 1.90211i 0 −0.309017 + 0.951057i 0
189.1 0 0 −0.309017 + 0.951057i −0.809017 0.587785i 0 1.17557i 0 0.809017 + 0.587785i 0
284.1 0 0 −0.309017 0.951057i −0.809017 + 0.587785i 0 1.17557i 0 0.809017 0.587785i 0
379.1 0 0 0.809017 + 0.587785i 0.309017 + 0.951057i 0 1.90211i 0 −0.309017 0.951057i 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
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
25.e even 10 1 inner
475.m odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.1.m.a 4
5.b even 2 1 2375.1.m.a 4
5.c odd 4 2 2375.1.o.b 8
19.b odd 2 1 CM 475.1.m.a 4
25.d even 5 1 2375.1.m.a 4
25.e even 10 1 inner 475.1.m.a 4
25.f odd 20 2 2375.1.o.b 8
95.d odd 2 1 2375.1.m.a 4
95.g even 4 2 2375.1.o.b 8
475.m odd 10 1 inner 475.1.m.a 4
475.o odd 10 1 2375.1.m.a 4
475.v even 20 2 2375.1.o.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.1.m.a 4 1.a even 1 1 trivial
475.1.m.a 4 19.b odd 2 1 CM
475.1.m.a 4 25.e even 10 1 inner
475.1.m.a 4 475.m odd 10 1 inner
2375.1.m.a 4 5.b even 2 1
2375.1.m.a 4 25.d even 5 1
2375.1.m.a 4 95.d odd 2 1
2375.1.m.a 4 475.o odd 10 1
2375.1.o.b 8 5.c odd 4 2
2375.1.o.b 8 25.f odd 20 2
2375.1.o.b 8 95.g even 4 2
2375.1.o.b 8 475.v even 20 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(475, [\chi])$$.

## Hecke characteristic polynomials

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