# Properties

 Label 475.2.p.d 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{2} + ( 1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{3} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{4} + 6 \zeta_{12}^{2} q^{6} + ( 2 + 2 \zeta_{12}^{3} ) q^{7} + ( 2 - 4 \zeta_{12} - 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{8} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{2} + ( 1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{3} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{4} + 6 \zeta_{12}^{2} q^{6} + ( 2 + 2 \zeta_{12}^{3} ) q^{7} + ( 2 - 4 \zeta_{12} - 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{8} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{9} -5 q^{11} + ( -4 - 8 \zeta_{12} + 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{12} + ( -4 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{14} + ( 4 - 4 \zeta_{12}^{2} ) q^{16} + ( 2 + 2 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{17} + ( -3 + 6 \zeta_{12} + 6 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{18} + ( 5 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{19} + ( 4 + 4 \zeta_{12}^{2} ) q^{21} + ( -5 + 5 \zeta_{12} - 5 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{22} + ( 3 \zeta_{12} + 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{23} -12 \zeta_{12} q^{24} + ( -8 \zeta_{12} - 8 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{28} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{29} + ( 3 - 6 \zeta_{12}^{2} ) q^{31} + ( -5 - 5 \zeta_{12} - 5 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{33} + ( 4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{34} + ( -12 + 12 \zeta_{12}^{2} ) q^{36} + ( -1 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{37} + ( -7 + 8 \zeta_{12} + 8 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{38} + ( -2 - 2 \zeta_{12}^{2} ) q^{41} + ( -12 \zeta_{12} + 12 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{42} + ( 7 - 7 \zeta_{12} - 7 \zeta_{12}^{2} ) q^{43} + ( 20 \zeta_{12} - 20 \zeta_{12}^{3} ) q^{44} + ( -6 + 12 \zeta_{12}^{2} ) q^{46} + ( 6 \zeta_{12} - 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{47} + ( 8 - 4 \zeta_{12} - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{48} + \zeta_{12}^{3} q^{49} + ( 8 - 4 \zeta_{12}^{2} ) q^{51} + ( 6 - 3 \zeta_{12} - 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{53} + ( 8 - 16 \zeta_{12}^{2} ) q^{56} + ( 7 + 8 \zeta_{12} - 8 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{57} + ( 3 - 3 \zeta_{12}^{3} ) q^{58} + ( -3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{59} + 5 \zeta_{12}^{2} q^{61} + ( 9 + 9 \zeta_{12} - 9 \zeta_{12}^{2} ) q^{62} + ( 6 \zeta_{12} + 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{63} -8 \zeta_{12}^{3} q^{64} -30 \zeta_{12}^{2} q^{66} + ( 8 + 4 \zeta_{12} - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{67} + ( -8 + 8 \zeta_{12}^{3} ) q^{68} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{69} + ( -3 - 3 \zeta_{12}^{2} ) q^{71} + ( -12 - 6 \zeta_{12} + 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{72} + ( -1 + \zeta_{12} + \zeta_{12}^{2} ) q^{73} -6 \zeta_{12} q^{74} + ( -20 + 12 \zeta_{12}^{2} ) q^{76} + ( -10 - 10 \zeta_{12}^{3} ) q^{77} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{79} + ( -9 + 9 \zeta_{12}^{2} ) q^{81} + ( 6 \zeta_{12} - 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{82} + ( -5 + 5 \zeta_{12}^{3} ) q^{83} + ( -32 \zeta_{12} + 16 \zeta_{12}^{3} ) q^{84} + ( 28 - 14 \zeta_{12}^{2} ) q^{86} + ( -3 - 3 \zeta_{12}^{3} ) q^{87} + ( -10 + 20 \zeta_{12} + 20 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{88} + ( -3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{89} + ( -12 - 12 \zeta_{12} + 12 \zeta_{12}^{2} ) q^{92} + ( 9 - 9 \zeta_{12} - 9 \zeta_{12}^{2} ) q^{93} + ( 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{94} + ( 3 - 3 \zeta_{12} + 3 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{97} + ( -1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{98} + ( -15 \zeta_{12} + 15 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{2} + 6 q^{3} + 12 q^{6} + 8 q^{7} + O(q^{10})$$ $$4 q + 6 q^{2} + 6 q^{3} + 12 q^{6} + 8 q^{7} - 20 q^{11} + 8 q^{16} + 4 q^{17} + 24 q^{21} - 30 q^{22} + 6 q^{23} - 16 q^{28} - 30 q^{33} - 24 q^{36} - 12 q^{38} - 12 q^{41} + 24 q^{42} + 14 q^{43} - 12 q^{47} + 24 q^{48} + 24 q^{51} + 18 q^{53} + 12 q^{57} + 12 q^{58} + 10 q^{61} + 18 q^{62} + 12 q^{63} - 60 q^{66} + 24 q^{67} - 32 q^{68} - 18 q^{71} - 36 q^{72} - 2 q^{73} - 56 q^{76} - 40 q^{77} - 18 q^{81} - 12 q^{82} - 20 q^{83} + 84 q^{86} - 12 q^{87} - 24 q^{92} + 18 q^{93} + 18 q^{97} - 6 q^{98} + O(q^{100})$$

## 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
2.36603 + 0.633975i 0.633975 2.36603i 3.46410 + 2.00000i 0 3.00000 5.19615i 2.00000 + 2.00000i 3.46410 + 3.46410i −2.59808 1.50000i 0
293.1 2.36603 0.633975i 0.633975 + 2.36603i 3.46410 2.00000i 0 3.00000 + 5.19615i 2.00000 2.00000i 3.46410 3.46410i −2.59808 + 1.50000i 0
407.1 0.633975 + 2.36603i 2.36603 0.633975i −3.46410 + 2.00000i 0 3.00000 + 5.19615i 2.00000 + 2.00000i −3.46410 3.46410i 2.59808 1.50000i 0
468.1 0.633975 2.36603i 2.36603 + 0.633975i −3.46410 2.00000i 0 3.00000 5.19615i 2.00000 2.00000i −3.46410 + 3.46410i 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.d 4
5.b even 2 1 95.2.l.a 4
5.c odd 4 1 95.2.l.a 4
5.c odd 4 1 inner 475.2.p.d 4
15.d odd 2 1 855.2.cj.d 4
15.e even 4 1 855.2.cj.d 4
19.d odd 6 1 inner 475.2.p.d 4
95.h odd 6 1 95.2.l.a 4
95.l even 12 1 95.2.l.a 4
95.l even 12 1 inner 475.2.p.d 4
285.q even 6 1 855.2.cj.d 4
285.w odd 12 1 855.2.cj.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.l.a 4 5.b even 2 1
95.2.l.a 4 5.c odd 4 1
95.2.l.a 4 95.h odd 6 1
95.2.l.a 4 95.l even 12 1
475.2.p.d 4 1.a even 1 1 trivial
475.2.p.d 4 5.c odd 4 1 inner
475.2.p.d 4 19.d odd 6 1 inner
475.2.p.d 4 95.l even 12 1 inner
855.2.cj.d 4 15.d odd 2 1
855.2.cj.d 4 15.e even 4 1
855.2.cj.d 4 285.q even 6 1
855.2.cj.d 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 $$S_{2}^{\mathrm{new}}(475, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$36 - 36 T + 18 T^{2} - 6 T^{3} + T^{4}$$
$3$ $$36 - 36 T + 18 T^{2} - 6 T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 8 - 4 T + T^{2} )^{2}$$
$11$ $$( 5 + T )^{4}$$
$13$ $$T^{4}$$
$17$ $$64 - 32 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$19$ $$361 - 37 T^{2} + T^{4}$$
$23$ $$324 - 108 T + 18 T^{2} - 6 T^{3} + T^{4}$$
$29$ $$9 + 3 T^{2} + T^{4}$$
$31$ $$( 27 + T^{2} )^{2}$$
$37$ $$36 + T^{4}$$
$41$ $$( 12 + 6 T + T^{2} )^{2}$$
$43$ $$9604 - 1372 T + 98 T^{2} - 14 T^{3} + T^{4}$$
$47$ $$5184 + 864 T + 72 T^{2} + 12 T^{3} + T^{4}$$
$53$ $$2916 - 972 T + 162 T^{2} - 18 T^{3} + T^{4}$$
$59$ $$729 + 27 T^{2} + T^{4}$$
$61$ $$( 25 - 5 T + T^{2} )^{2}$$
$67$ $$9216 - 2304 T + 288 T^{2} - 24 T^{3} + T^{4}$$
$71$ $$( 27 + 9 T + T^{2} )^{2}$$
$73$ $$4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4}$$
$79$ $$9 + 3 T^{2} + T^{4}$$
$83$ $$( 50 + 10 T + T^{2} )^{2}$$
$89$ $$729 + 27 T^{2} + T^{4}$$
$97$ $$2916 - 972 T + 162 T^{2} - 18 T^{3} + T^{4}$$