# Properties

 Label 475.2.p.a 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: yes 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{12} + \zeta_{12}^{2} ) q^{2} + ( 2 + 2 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{3} + 4 \zeta_{12}^{2} q^{6} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{12} + \zeta_{12}^{2} ) q^{2} + ( 2 + 2 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{3} + 4 \zeta_{12}^{2} q^{6} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{9} -3 q^{11} + ( 3 \zeta_{12} - 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{13} + ( -4 + 4 \zeta_{12}^{2} ) q^{16} + ( -1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{17} + ( 5 + 5 \zeta_{12}^{3} ) q^{18} + ( -5 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{19} + ( 3 - 3 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{22} + ( -6 - 3 \zeta_{12} + 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{23} + 8 \zeta_{12} q^{24} + 6 q^{26} + ( 4 - 4 \zeta_{12}^{3} ) q^{27} + ( 3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{29} + ( -5 + 10 \zeta_{12}^{2} ) q^{31} + ( -6 - 6 \zeta_{12} + 6 \zeta_{12}^{2} ) q^{33} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{34} + ( 3 - 3 \zeta_{12}^{3} ) q^{37} + ( -3 + 2 \zeta_{12} - 2 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{38} -12 \zeta_{12}^{3} q^{39} + ( 6 + 6 \zeta_{12}^{2} ) q^{41} + ( -3 + 3 \zeta_{12} - 3 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{43} + ( 6 - 12 \zeta_{12}^{2} ) q^{46} + ( -2 + \zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{47} + ( -8 \zeta_{12} + 8 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{48} -7 \zeta_{12}^{3} q^{49} + ( -8 + 4 \zeta_{12}^{2} ) q^{51} + ( -2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{53} + 8 \zeta_{12} q^{54} + ( -6 - 4 \zeta_{12} - 4 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{57} + ( -3 - 6 \zeta_{12} + 6 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{58} + ( -3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{59} + \zeta_{12}^{2} q^{61} + ( -5 - 5 \zeta_{12} - 5 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{62} + 8 \zeta_{12}^{3} q^{64} -12 \zeta_{12}^{2} q^{66} + ( 9 \zeta_{12} + 9 \zeta_{12}^{2} - 9 \zeta_{12}^{3} ) q^{67} + ( -24 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{69} + ( -9 - 9 \zeta_{12}^{2} ) q^{71} + ( 10 \zeta_{12} + 10 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{72} + ( -3 + 3 \zeta_{12} - 3 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{73} + 6 \zeta_{12} q^{74} + ( 12 + 12 \zeta_{12} - 12 \zeta_{12}^{2} ) q^{78} + ( -3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{79} + ( 1 - \zeta_{12}^{2} ) q^{81} + ( -12 + 6 \zeta_{12} + 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{82} + ( 5 + 10 \zeta_{12} - 10 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{83} + ( 12 - 6 \zeta_{12}^{2} ) q^{86} + ( -6 + 12 \zeta_{12} + 12 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{87} + ( -6 - 6 \zeta_{12}^{3} ) q^{88} + ( 3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{89} + ( 10 - 10 \zeta_{12} + 10 \zeta_{12}^{2} + 20 \zeta_{12}^{3} ) q^{93} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{94} + ( 9 - 9 \zeta_{12} - 9 \zeta_{12}^{2} ) q^{97} + ( 7 + 7 \zeta_{12} - 7 \zeta_{12}^{2} ) q^{98} + ( -15 \zeta_{12} + 15 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 4 q^{3} + 8 q^{6} + 8 q^{8} + O(q^{10})$$ $$4 q - 2 q^{2} + 4 q^{3} + 8 q^{6} + 8 q^{8} - 12 q^{11} - 6 q^{13} - 8 q^{16} - 6 q^{17} + 20 q^{18} + 6 q^{22} - 18 q^{23} + 24 q^{26} + 16 q^{27} - 12 q^{33} + 12 q^{37} - 16 q^{38} + 36 q^{41} - 18 q^{43} - 6 q^{47} + 16 q^{48} - 24 q^{51} + 4 q^{53} - 32 q^{57} + 2 q^{61} - 30 q^{62} - 24 q^{66} + 18 q^{67} - 54 q^{71} + 20 q^{72} - 18 q^{73} + 24 q^{78} + 2 q^{81} - 36 q^{82} + 36 q^{86} - 24 q^{88} + 60 q^{93} + 18 q^{97} + 14 q^{98} + O(q^{100})$$

## 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
−1.36603 0.366025i −0.732051 + 2.73205i 0 0 2.00000 3.46410i 0 2.00000 + 2.00000i −4.33013 2.50000i 0
293.1 −1.36603 + 0.366025i −0.732051 2.73205i 0 0 2.00000 + 3.46410i 0 2.00000 2.00000i −4.33013 + 2.50000i 0
407.1 0.366025 + 1.36603i 2.73205 0.732051i 0 0 2.00000 + 3.46410i 0 2.00000 + 2.00000i 4.33013 2.50000i 0
468.1 0.366025 1.36603i 2.73205 + 0.732051i 0 0 2.00000 3.46410i 0 2.00000 2.00000i 4.33013 + 2.50000i 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
95.h 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.a 4
5.b even 2 1 475.2.p.c yes 4
5.c odd 4 1 inner 475.2.p.a 4
5.c odd 4 1 475.2.p.c yes 4
19.d odd 6 1 475.2.p.c yes 4
95.h odd 6 1 inner 475.2.p.a 4
95.l even 12 1 inner 475.2.p.a 4
95.l even 12 1 475.2.p.c yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.p.a 4 1.a even 1 1 trivial
475.2.p.a 4 5.c odd 4 1 inner
475.2.p.a 4 95.h odd 6 1 inner
475.2.p.a 4 95.l even 12 1 inner
475.2.p.c yes 4 5.b even 2 1
475.2.p.c yes 4 5.c odd 4 1
475.2.p.c yes 4 19.d odd 6 1
475.2.p.c yes 4 95.l even 12 1

## Hecke kernels

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

 $$T_{2}^{4} + 2 T_{2}^{3} + 2 T_{2}^{2} + 4 T_{2} + 4$$ $$T_{3}^{4} - 4 T_{3}^{3} + 8 T_{3}^{2} - 32 T_{3} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4}$$
$3$ $$64 - 32 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 3 + T )^{4}$$
$13$ $$324 + 108 T + 18 T^{2} + 6 T^{3} + T^{4}$$
$17$ $$36 + 36 T + 18 T^{2} + 6 T^{3} + T^{4}$$
$19$ $$361 - 37 T^{2} + T^{4}$$
$23$ $$2916 + 972 T + 162 T^{2} + 18 T^{3} + T^{4}$$
$29$ $$729 + 27 T^{2} + T^{4}$$
$31$ $$( 75 + T^{2} )^{2}$$
$37$ $$( 18 - 6 T + T^{2} )^{2}$$
$41$ $$( 108 - 18 T + T^{2} )^{2}$$
$43$ $$2916 + 972 T + 162 T^{2} + 18 T^{3} + T^{4}$$
$47$ $$36 + 36 T + 18 T^{2} + 6 T^{3} + T^{4}$$
$53$ $$64 - 32 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$59$ $$729 + 27 T^{2} + T^{4}$$
$61$ $$( 1 - T + T^{2} )^{2}$$
$67$ $$26244 - 2916 T + 162 T^{2} - 18 T^{3} + T^{4}$$
$71$ $$( 243 + 27 T + T^{2} )^{2}$$
$73$ $$2916 + 972 T + 162 T^{2} + 18 T^{3} + T^{4}$$
$79$ $$729 + 27 T^{2} + T^{4}$$
$83$ $$22500 + T^{4}$$
$89$ $$729 + 27 T^{2} + T^{4}$$
$97$ $$26244 - 2916 T + 162 T^{2} - 18 T^{3} + T^{4}$$