# Properties

 Label 475.2.p.b Level $475$ Weight $2$ Character orbit 475.p Analytic conductor $3.793$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.p (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{3} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{4} + ( -2 - 2 \zeta_{12}^{3} ) q^{7} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{3} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{4} + ( -2 - 2 \zeta_{12}^{3} ) q^{7} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{9} - q^{11} + ( 2 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{12} + ( -2 + \zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{13} + ( 4 - 4 \zeta_{12}^{2} ) q^{16} + ( 3 + 3 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{17} + ( -5 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{19} + ( -4 - 4 \zeta_{12}^{2} ) q^{21} + ( 4 \zeta_{12} + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{23} + ( -4 \zeta_{12} - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{28} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{29} + ( -5 + 10 \zeta_{12}^{2} ) q^{31} + ( -1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{33} + ( 6 - 6 \zeta_{12}^{2} ) q^{36} + ( -2 - 4 \zeta_{12} + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{37} + 6 \zeta_{12}^{3} q^{39} + ( -6 - 6 \zeta_{12}^{2} ) q^{41} + ( -8 + 8 \zeta_{12} + 8 \zeta_{12}^{2} ) q^{43} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{44} + ( 5 \zeta_{12} - 5 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{47} + ( 8 - 4 \zeta_{12} - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{48} + \zeta_{12}^{3} q^{49} + ( 12 - 6 \zeta_{12}^{2} ) q^{51} + ( 2 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{52} + ( 2 - \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{53} + ( -7 - 8 \zeta_{12} + 8 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{57} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{59} + 7 \zeta_{12}^{2} q^{61} + ( -6 \zeta_{12} - 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{63} -8 \zeta_{12}^{3} q^{64} + ( 2 + \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{67} + ( 6 - 6 \zeta_{12}^{3} ) q^{68} + ( 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{69} + ( 5 + 5 \zeta_{12}^{2} ) q^{71} + ( 2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{73} + ( -10 + 6 \zeta_{12}^{2} ) q^{76} + ( 2 + 2 \zeta_{12}^{3} ) q^{77} + ( 3 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{79} + ( -9 + 9 \zeta_{12}^{2} ) q^{81} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{84} + ( 3 + 3 \zeta_{12}^{3} ) q^{87} + ( -7 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{89} + ( 8 - 4 \zeta_{12}^{2} ) q^{91} + ( 8 + 8 \zeta_{12} - 8 \zeta_{12}^{2} ) q^{92} + ( -15 + 15 \zeta_{12} + 15 \zeta_{12}^{2} ) q^{93} + ( -4 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{97} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{3} - 8 q^{7} + O(q^{10})$$ $$4 q + 6 q^{3} - 8 q^{7} - 4 q^{11} - 6 q^{13} + 8 q^{16} + 6 q^{17} - 24 q^{21} + 8 q^{23} - 8 q^{28} - 6 q^{33} + 12 q^{36} - 36 q^{41} - 16 q^{43} - 10 q^{47} + 24 q^{48} + 36 q^{51} + 12 q^{52} + 6 q^{53} - 12 q^{57} + 14 q^{61} - 12 q^{63} + 6 q^{67} + 24 q^{68} + 30 q^{71} + 4 q^{73} - 28 q^{76} + 8 q^{77} - 18 q^{81} + 12 q^{87} + 24 q^{91} + 16 q^{92} - 30 q^{93} - 24 q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$\zeta_{12}^{3}$$ $$\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i
0 0.633975 2.36603i −1.73205 1.00000i 0 0 −2.00000 2.00000i 0 −2.59808 1.50000i 0
293.1 0 0.633975 + 2.36603i −1.73205 + 1.00000i 0 0 −2.00000 + 2.00000i 0 −2.59808 + 1.50000i 0
407.1 0 2.36603 0.633975i 1.73205 1.00000i 0 0 −2.00000 2.00000i 0 2.59808 1.50000i 0
468.1 0 2.36603 + 0.633975i 1.73205 + 1.00000i 0 0 −2.00000 + 2.00000i 0 2.59808 + 1.50000i 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
5.c odd 4 1 inner
19.d odd 6 1 inner
95.l even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.p.b 4
5.b even 2 1 95.2.l.b 4
5.c odd 4 1 95.2.l.b 4
5.c odd 4 1 inner 475.2.p.b 4
15.d odd 2 1 855.2.cj.b 4
15.e even 4 1 855.2.cj.b 4
19.d odd 6 1 inner 475.2.p.b 4
95.h odd 6 1 95.2.l.b 4
95.l even 12 1 95.2.l.b 4
95.l even 12 1 inner 475.2.p.b 4
285.q even 6 1 855.2.cj.b 4
285.w odd 12 1 855.2.cj.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.l.b 4 5.b even 2 1
95.2.l.b 4 5.c odd 4 1
95.2.l.b 4 95.h odd 6 1
95.2.l.b 4 95.l even 12 1
475.2.p.b 4 1.a even 1 1 trivial
475.2.p.b 4 5.c odd 4 1 inner
475.2.p.b 4 19.d odd 6 1 inner
475.2.p.b 4 95.l even 12 1 inner
855.2.cj.b 4 15.d odd 2 1
855.2.cj.b 4 15.e even 4 1
855.2.cj.b 4 285.q even 6 1
855.2.cj.b 4 285.w odd 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(475, [\chi])$$:

 $$T_{2}$$ $$T_{3}^{4} - 6 T_{3}^{3} + 18 T_{3}^{2} - 36 T_{3} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$36 - 36 T + 18 T^{2} - 6 T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 8 + 4 T + T^{2} )^{2}$$
$11$ $$( 1 + T )^{4}$$
$13$ $$36 + 36 T + 18 T^{2} + 6 T^{3} + T^{4}$$
$17$ $$324 - 108 T + 18 T^{2} - 6 T^{3} + T^{4}$$
$19$ $$361 - 37 T^{2} + T^{4}$$
$23$ $$1024 - 256 T + 32 T^{2} - 8 T^{3} + T^{4}$$
$29$ $$9 + 3 T^{2} + T^{4}$$
$31$ $$( 75 + T^{2} )^{2}$$
$37$ $$576 + T^{4}$$
$41$ $$( 108 + 18 T + T^{2} )^{2}$$
$43$ $$16384 + 2048 T + 128 T^{2} + 16 T^{3} + T^{4}$$
$47$ $$2500 + 500 T + 50 T^{2} + 10 T^{3} + T^{4}$$
$53$ $$36 - 36 T + 18 T^{2} - 6 T^{3} + T^{4}$$
$59$ $$9 + 3 T^{2} + T^{4}$$
$61$ $$( 49 - 7 T + T^{2} )^{2}$$
$67$ $$36 - 36 T + 18 T^{2} - 6 T^{3} + T^{4}$$
$71$ $$( 75 - 15 T + T^{2} )^{2}$$
$73$ $$64 - 32 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$79$ $$729 + 27 T^{2} + T^{4}$$
$83$ $$T^{4}$$
$89$ $$21609 + 147 T^{2} + T^{4}$$
$97$ $$9216 + 2304 T + 288 T^{2} + 24 T^{3} + T^{4}$$