# Properties

 Label 475.2.e.b Level $475$ Weight $2$ Character orbit 475.e Analytic conductor $3.793$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 + 2 \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{4} + 4 q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -2 + 2 \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{4} + 4 q^{7} -\zeta_{6} q^{9} + 3 q^{11} -4 q^{12} + 2 \zeta_{6} q^{13} + ( -4 + 4 \zeta_{6} ) q^{16} + ( 6 - 6 \zeta_{6} ) q^{17} + ( -2 - 3 \zeta_{6} ) q^{19} + ( -8 + 8 \zeta_{6} ) q^{21} -4 q^{27} + 8 \zeta_{6} q^{28} + 3 \zeta_{6} q^{29} -7 q^{31} + ( -6 + 6 \zeta_{6} ) q^{33} + ( 2 - 2 \zeta_{6} ) q^{36} -8 q^{37} -4 q^{39} + ( 6 - 6 \zeta_{6} ) q^{41} + ( -4 + 4 \zeta_{6} ) q^{43} + 6 \zeta_{6} q^{44} + 6 \zeta_{6} q^{47} -8 \zeta_{6} q^{48} + 9 q^{49} + 12 \zeta_{6} q^{51} + ( -4 + 4 \zeta_{6} ) q^{52} -6 \zeta_{6} q^{53} + ( 10 - 4 \zeta_{6} ) q^{57} + ( 15 - 15 \zeta_{6} ) q^{59} -5 \zeta_{6} q^{61} -4 \zeta_{6} q^{63} -8 q^{64} + 2 \zeta_{6} q^{67} + 12 q^{68} + ( 3 - 3 \zeta_{6} ) q^{71} + ( 8 - 8 \zeta_{6} ) q^{73} + ( 6 - 10 \zeta_{6} ) q^{76} + 12 q^{77} + ( -5 + 5 \zeta_{6} ) q^{79} + ( 11 - 11 \zeta_{6} ) q^{81} -12 q^{83} -16 q^{84} -6 q^{87} + 15 \zeta_{6} q^{89} + 8 \zeta_{6} q^{91} + ( 14 - 14 \zeta_{6} ) q^{93} + ( 8 - 8 \zeta_{6} ) q^{97} -3 \zeta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{4} + 8 q^{7} - q^{9} + O(q^{10})$$ $$2 q - 2 q^{3} + 2 q^{4} + 8 q^{7} - q^{9} + 6 q^{11} - 8 q^{12} + 2 q^{13} - 4 q^{16} + 6 q^{17} - 7 q^{19} - 8 q^{21} - 8 q^{27} + 8 q^{28} + 3 q^{29} - 14 q^{31} - 6 q^{33} + 2 q^{36} - 16 q^{37} - 8 q^{39} + 6 q^{41} - 4 q^{43} + 6 q^{44} + 6 q^{47} - 8 q^{48} + 18 q^{49} + 12 q^{51} - 4 q^{52} - 6 q^{53} + 16 q^{57} + 15 q^{59} - 5 q^{61} - 4 q^{63} - 16 q^{64} + 2 q^{67} + 24 q^{68} + 3 q^{71} + 8 q^{73} + 2 q^{76} + 24 q^{77} - 5 q^{79} + 11 q^{81} - 24 q^{83} - 32 q^{84} - 12 q^{87} + 15 q^{89} + 8 q^{91} + 14 q^{93} + 8 q^{97} - 3 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −1.00000 1.73205i 1.00000 1.73205i 0 0 4.00000 0 −0.500000 + 0.866025i 0
201.1 0 −1.00000 + 1.73205i 1.00000 + 1.73205i 0 0 4.00000 0 −0.500000 0.866025i 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.e.b 2
5.b even 2 1 95.2.e.a 2
5.c odd 4 2 475.2.j.a 4
15.d odd 2 1 855.2.k.b 2
19.c even 3 1 inner 475.2.e.b 2
19.c even 3 1 9025.2.a.g 1
19.d odd 6 1 9025.2.a.e 1
20.d odd 2 1 1520.2.q.c 2
95.h odd 6 1 1805.2.a.b 1
95.i even 6 1 95.2.e.a 2
95.i even 6 1 1805.2.a.a 1
95.m odd 12 2 475.2.j.a 4
285.n odd 6 1 855.2.k.b 2
380.p odd 6 1 1520.2.q.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.a 2 5.b even 2 1
95.2.e.a 2 95.i even 6 1
475.2.e.b 2 1.a even 1 1 trivial
475.2.e.b 2 19.c even 3 1 inner
475.2.j.a 4 5.c odd 4 2
475.2.j.a 4 95.m odd 12 2
855.2.k.b 2 15.d odd 2 1
855.2.k.b 2 285.n odd 6 1
1520.2.q.c 2 20.d odd 2 1
1520.2.q.c 2 380.p odd 6 1
1805.2.a.a 1 95.i even 6 1
1805.2.a.b 1 95.h odd 6 1
9025.2.a.e 1 19.d odd 6 1
9025.2.a.g 1 19.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(475, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 + 2 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -4 + T )^{2}$$
$11$ $$( -3 + T )^{2}$$
$13$ $$4 - 2 T + T^{2}$$
$17$ $$36 - 6 T + T^{2}$$
$19$ $$19 + 7 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$9 - 3 T + T^{2}$$
$31$ $$( 7 + T )^{2}$$
$37$ $$( 8 + T )^{2}$$
$41$ $$36 - 6 T + T^{2}$$
$43$ $$16 + 4 T + T^{2}$$
$47$ $$36 - 6 T + T^{2}$$
$53$ $$36 + 6 T + T^{2}$$
$59$ $$225 - 15 T + T^{2}$$
$61$ $$25 + 5 T + T^{2}$$
$67$ $$4 - 2 T + T^{2}$$
$71$ $$9 - 3 T + T^{2}$$
$73$ $$64 - 8 T + T^{2}$$
$79$ $$25 + 5 T + T^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$225 - 15 T + T^{2}$$
$97$ $$64 - 8 T + T^{2}$$