# Properties

 Label 475.2.g.a Level $475$ Weight $2$ Character orbit 475.g Analytic conductor $3.793$ Analytic rank $0$ Dimension $4$ CM discriminant -19 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{19})$$ Defining polynomial: $$x^{4} - 9 x^{2} + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 95) Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 \beta_{1} q^{4} + ( -2 + 2 \beta_{1} + \beta_{3} ) q^{7} + 3 \beta_{1} q^{9} +O(q^{10})$$ $$q -2 \beta_{1} q^{4} + ( -2 + 2 \beta_{1} + \beta_{3} ) q^{7} + 3 \beta_{1} q^{9} + ( \beta_{2} + \beta_{3} ) q^{11} -4 q^{16} + ( 3 - 3 \beta_{1} + \beta_{3} ) q^{17} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{19} + ( -3 - \beta_{1} - 2 \beta_{2} ) q^{23} + ( 2 + 4 \beta_{1} - 2 \beta_{2} ) q^{28} + 6 q^{36} + ( 2 - \beta_{1} + 3 \beta_{2} ) q^{43} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{44} + ( -7 + 7 \beta_{1} + \beta_{3} ) q^{47} + ( 3 - 10 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{49} + ( \beta_{2} + \beta_{3} ) q^{61} + ( -3 - 6 \beta_{1} + 3 \beta_{2} ) q^{63} + 8 \beta_{1} q^{64} + ( -8 - 6 \beta_{1} - 2 \beta_{2} ) q^{68} + ( 7 + 4 \beta_{1} + 3 \beta_{2} ) q^{73} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{76} + ( 11 - 11 \beta_{1} - 3 \beta_{3} ) q^{77} -9 q^{81} + ( 7 + 9 \beta_{1} - 2 \beta_{2} ) q^{83} + ( -2 + 2 \beta_{1} - 4 \beta_{3} ) q^{92} + ( 3 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{7} + O(q^{10})$$ $$4 q - 6 q^{7} - 16 q^{16} + 14 q^{17} - 8 q^{23} + 12 q^{28} + 24 q^{36} + 2 q^{43} - 26 q^{47} - 18 q^{63} - 28 q^{68} + 22 q^{73} + 38 q^{77} - 36 q^{81} + 32 q^{83} - 16 q^{92} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 9 x^{2} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - 4 \nu$$$$)/5$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 5$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} - 5 \nu^{2} + 9 \nu + 25$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} - \beta_{1} + 10$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3} + 2 \beta_{2} + 7 \beta_{1}$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
18.1
 −2.17945 + 0.500000i 2.17945 + 0.500000i −2.17945 − 0.500000i 2.17945 − 0.500000i
0 0 2.00000i 0 0 −3.67945 + 3.67945i 0 3.00000i 0
18.2 0 0 2.00000i 0 0 0.679449 0.679449i 0 3.00000i 0
132.1 0 0 2.00000i 0 0 −3.67945 3.67945i 0 3.00000i 0
132.2 0 0 2.00000i 0 0 0.679449 + 0.679449i 0 3.00000i 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 $$\Q(\sqrt{-19})$$
5.c odd 4 1 inner
95.g even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.g.a 4
5.b even 2 1 95.2.g.a 4
5.c odd 4 1 95.2.g.a 4
5.c odd 4 1 inner 475.2.g.a 4
15.d odd 2 1 855.2.p.a 4
15.e even 4 1 855.2.p.a 4
19.b odd 2 1 CM 475.2.g.a 4
95.d odd 2 1 95.2.g.a 4
95.g even 4 1 95.2.g.a 4
95.g even 4 1 inner 475.2.g.a 4
285.b even 2 1 855.2.p.a 4
285.j odd 4 1 855.2.p.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.g.a 4 5.b even 2 1
95.2.g.a 4 5.c odd 4 1
95.2.g.a 4 95.d odd 2 1
95.2.g.a 4 95.g even 4 1
475.2.g.a 4 1.a even 1 1 trivial
475.2.g.a 4 5.c odd 4 1 inner
475.2.g.a 4 19.b odd 2 1 CM
475.2.g.a 4 95.g even 4 1 inner
855.2.p.a 4 15.d odd 2 1
855.2.p.a 4 15.e even 4 1
855.2.p.a 4 285.b even 2 1
855.2.p.a 4 285.j odd 4 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

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$25 - 30 T + 18 T^{2} + 6 T^{3} + T^{4}$$
$11$ $$( -19 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$225 - 210 T + 98 T^{2} - 14 T^{3} + T^{4}$$
$19$ $$( 19 + T^{2} )^{2}$$
$23$ $$900 - 240 T + 32 T^{2} + 8 T^{3} + T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$7225 + 170 T + 2 T^{2} - 2 T^{3} + T^{4}$$
$47$ $$5625 + 1950 T + 338 T^{2} + 26 T^{3} + T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( -19 + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$625 + 550 T + 242 T^{2} - 22 T^{3} + T^{4}$$
$79$ $$T^{4}$$
$83$ $$8100 - 2880 T + 512 T^{2} - 32 T^{3} + T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$