# Properties

 Label 475.2.e.a Level $475$ Weight $2$ Character orbit 475.e Analytic conductor $3.793$ Analytic rank $1$ 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: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} -2 \zeta_{6} q^{4} -4 q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} -2 \zeta_{6} q^{4} -4 q^{7} + 3 \zeta_{6} q^{9} - q^{11} + 2 \zeta_{6} q^{13} + ( 8 - 8 \zeta_{6} ) q^{14} + ( 4 - 4 \zeta_{6} ) q^{16} + ( 2 - 2 \zeta_{6} ) q^{17} -6 q^{18} + ( -2 - 3 \zeta_{6} ) q^{19} + ( 2 - 2 \zeta_{6} ) q^{22} -6 \zeta_{6} q^{23} -4 q^{26} + 8 \zeta_{6} q^{28} -9 \zeta_{6} q^{29} -7 q^{31} + 8 \zeta_{6} q^{32} + 4 \zeta_{6} q^{34} + ( 6 - 6 \zeta_{6} ) q^{36} + 2 q^{37} + ( 10 - 4 \zeta_{6} ) q^{38} + ( -2 + 2 \zeta_{6} ) q^{41} + ( -2 + 2 \zeta_{6} ) q^{43} + 2 \zeta_{6} q^{44} + 12 q^{46} -6 \zeta_{6} q^{47} + 9 q^{49} + ( 4 - 4 \zeta_{6} ) q^{52} -4 \zeta_{6} q^{53} + 18 q^{58} + ( -9 + 9 \zeta_{6} ) q^{59} + 7 \zeta_{6} q^{61} + ( 14 - 14 \zeta_{6} ) q^{62} -12 \zeta_{6} q^{63} -8 q^{64} + 10 \zeta_{6} q^{67} -4 q^{68} + ( -1 + \zeta_{6} ) q^{71} + ( -10 + 10 \zeta_{6} ) q^{73} + ( -4 + 4 \zeta_{6} ) q^{74} + ( -6 + 10 \zeta_{6} ) q^{76} + 4 q^{77} + ( -1 + \zeta_{6} ) q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} -4 \zeta_{6} q^{82} -6 q^{83} -4 \zeta_{6} q^{86} + 11 \zeta_{6} q^{89} -8 \zeta_{6} q^{91} + ( -12 + 12 \zeta_{6} ) q^{92} + 12 q^{94} + ( 6 - 6 \zeta_{6} ) q^{97} + ( -18 + 18 \zeta_{6} ) q^{98} -3 \zeta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{4} - 8 q^{7} + 3 q^{9} + O(q^{10})$$ $$2 q - 2 q^{2} - 2 q^{4} - 8 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} + 8 q^{14} + 4 q^{16} + 2 q^{17} - 12 q^{18} - 7 q^{19} + 2 q^{22} - 6 q^{23} - 8 q^{26} + 8 q^{28} - 9 q^{29} - 14 q^{31} + 8 q^{32} + 4 q^{34} + 6 q^{36} + 4 q^{37} + 16 q^{38} - 2 q^{41} - 2 q^{43} + 2 q^{44} + 24 q^{46} - 6 q^{47} + 18 q^{49} + 4 q^{52} - 4 q^{53} + 36 q^{58} - 9 q^{59} + 7 q^{61} + 14 q^{62} - 12 q^{63} - 16 q^{64} + 10 q^{67} - 8 q^{68} - q^{71} - 10 q^{73} - 4 q^{74} - 2 q^{76} + 8 q^{77} - q^{79} - 9 q^{81} - 4 q^{82} - 12 q^{83} - 4 q^{86} + 11 q^{89} - 8 q^{91} - 12 q^{92} + 24 q^{94} + 6 q^{97} - 18 q^{98} - 3 q^{99} + O(q^{100})$$

## 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
−1.00000 1.73205i 0 −1.00000 + 1.73205i 0 0 −4.00000 0 1.50000 2.59808i 0
201.1 −1.00000 + 1.73205i 0 −1.00000 1.73205i 0 0 −4.00000 0 1.50000 + 2.59808i 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.a 2
5.b even 2 1 475.2.e.c 2
5.c odd 4 2 95.2.i.a 4
15.e even 4 2 855.2.be.a 4
19.c even 3 1 inner 475.2.e.a 2
19.c even 3 1 9025.2.a.i 1
19.d odd 6 1 9025.2.a.a 1
95.h odd 6 1 9025.2.a.j 1
95.i even 6 1 475.2.e.c 2
95.i even 6 1 9025.2.a.b 1
95.l even 12 2 1805.2.b.a 2
95.m odd 12 2 95.2.i.a 4
95.m odd 12 2 1805.2.b.b 2
285.v even 12 2 855.2.be.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.i.a 4 5.c odd 4 2
95.2.i.a 4 95.m odd 12 2
475.2.e.a 2 1.a even 1 1 trivial
475.2.e.a 2 19.c even 3 1 inner
475.2.e.c 2 5.b even 2 1
475.2.e.c 2 95.i even 6 1
855.2.be.a 4 15.e even 4 2
855.2.be.a 4 285.v even 12 2
1805.2.b.a 2 95.l even 12 2
1805.2.b.b 2 95.m odd 12 2
9025.2.a.a 1 19.d odd 6 1
9025.2.a.b 1 95.i even 6 1
9025.2.a.i 1 19.c even 3 1
9025.2.a.j 1 95.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 2 T_{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(475, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 4 + T )^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$4 - 2 T + T^{2}$$
$17$ $$4 - 2 T + T^{2}$$
$19$ $$19 + 7 T + T^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$81 + 9 T + T^{2}$$
$31$ $$( 7 + T )^{2}$$
$37$ $$( -2 + T )^{2}$$
$41$ $$4 + 2 T + T^{2}$$
$43$ $$4 + 2 T + T^{2}$$
$47$ $$36 + 6 T + T^{2}$$
$53$ $$16 + 4 T + T^{2}$$
$59$ $$81 + 9 T + T^{2}$$
$61$ $$49 - 7 T + T^{2}$$
$67$ $$100 - 10 T + T^{2}$$
$71$ $$1 + T + T^{2}$$
$73$ $$100 + 10 T + T^{2}$$
$79$ $$1 + T + T^{2}$$
$83$ $$( 6 + T )^{2}$$
$89$ $$121 - 11 T + T^{2}$$
$97$ $$36 - 6 T + T^{2}$$