# Properties

 Label 475.3.c.b Level $475$ Weight $3$ Character orbit 475.c Analytic conductor $12.943$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 475.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.9428125571$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-13})$$ Defining polynomial: $$x^{2} + 13$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-13}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} -\beta q^{3} -9 q^{4} + 13 q^{6} + 5 q^{7} -5 \beta q^{8} -4 q^{9} +O(q^{10})$$ $$q + \beta q^{2} -\beta q^{3} -9 q^{4} + 13 q^{6} + 5 q^{7} -5 \beta q^{8} -4 q^{9} -10 q^{11} + 9 \beta q^{12} + \beta q^{13} + 5 \beta q^{14} + 29 q^{16} -15 q^{17} -4 \beta q^{18} + ( -6 - 5 \beta ) q^{19} -5 \beta q^{21} -10 \beta q^{22} -35 q^{23} -65 q^{24} -13 q^{26} -5 \beta q^{27} -45 q^{28} -5 \beta q^{29} + 10 \beta q^{31} + 9 \beta q^{32} + 10 \beta q^{33} -15 \beta q^{34} + 36 q^{36} -6 \beta q^{37} + ( 65 - 6 \beta ) q^{38} + 13 q^{39} -10 \beta q^{41} + 65 q^{42} + 20 q^{43} + 90 q^{44} -35 \beta q^{46} -10 q^{47} -29 \beta q^{48} -24 q^{49} + 15 \beta q^{51} -9 \beta q^{52} -21 \beta q^{53} + 65 q^{54} -25 \beta q^{56} + ( -65 + 6 \beta ) q^{57} + 65 q^{58} -5 \beta q^{59} -40 q^{61} -130 q^{62} -20 q^{63} - q^{64} -130 q^{66} + 11 \beta q^{67} + 135 q^{68} + 35 \beta q^{69} -30 \beta q^{71} + 20 \beta q^{72} -105 q^{73} + 78 q^{74} + ( 54 + 45 \beta ) q^{76} -50 q^{77} + 13 \beta q^{78} + 10 \beta q^{79} -101 q^{81} + 130 q^{82} + 40 q^{83} + 45 \beta q^{84} + 20 \beta q^{86} -65 q^{87} + 50 \beta q^{88} + 5 \beta q^{91} + 315 q^{92} + 130 q^{93} -10 \beta q^{94} + 117 q^{96} + 34 \beta q^{97} -24 \beta q^{98} + 40 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 18q^{4} + 26q^{6} + 10q^{7} - 8q^{9} + O(q^{10})$$ $$2q - 18q^{4} + 26q^{6} + 10q^{7} - 8q^{9} - 20q^{11} + 58q^{16} - 30q^{17} - 12q^{19} - 70q^{23} - 130q^{24} - 26q^{26} - 90q^{28} + 72q^{36} + 130q^{38} + 26q^{39} + 130q^{42} + 40q^{43} + 180q^{44} - 20q^{47} - 48q^{49} + 130q^{54} - 130q^{57} + 130q^{58} - 80q^{61} - 260q^{62} - 40q^{63} - 2q^{64} - 260q^{66} + 270q^{68} - 210q^{73} + 156q^{74} + 108q^{76} - 100q^{77} - 202q^{81} + 260q^{82} + 80q^{83} - 130q^{87} + 630q^{92} + 260q^{93} + 234q^{96} + 80q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
151.1
 − 3.60555i 3.60555i
3.60555i 3.60555i −9.00000 0 13.0000 5.00000 18.0278i −4.00000 0
151.2 3.60555i 3.60555i −9.00000 0 13.0000 5.00000 18.0278i −4.00000 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 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.3.c.b 2
5.b even 2 1 19.3.b.b 2
5.c odd 4 2 475.3.d.b 4
15.d odd 2 1 171.3.c.b 2
19.b odd 2 1 inner 475.3.c.b 2
20.d odd 2 1 304.3.e.d 2
40.e odd 2 1 1216.3.e.h 2
40.f even 2 1 1216.3.e.g 2
60.h even 2 1 2736.3.o.d 2
95.d odd 2 1 19.3.b.b 2
95.g even 4 2 475.3.d.b 4
95.h odd 6 2 361.3.d.b 4
95.i even 6 2 361.3.d.b 4
95.o odd 18 6 361.3.f.d 12
95.p even 18 6 361.3.f.d 12
285.b even 2 1 171.3.c.b 2
380.d even 2 1 304.3.e.d 2
760.b odd 2 1 1216.3.e.g 2
760.p even 2 1 1216.3.e.h 2
1140.p odd 2 1 2736.3.o.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.b 2 5.b even 2 1
19.3.b.b 2 95.d odd 2 1
171.3.c.b 2 15.d odd 2 1
171.3.c.b 2 285.b even 2 1
304.3.e.d 2 20.d odd 2 1
304.3.e.d 2 380.d even 2 1
361.3.d.b 4 95.h odd 6 2
361.3.d.b 4 95.i even 6 2
361.3.f.d 12 95.o odd 18 6
361.3.f.d 12 95.p even 18 6
475.3.c.b 2 1.a even 1 1 trivial
475.3.c.b 2 19.b odd 2 1 inner
475.3.d.b 4 5.c odd 4 2
475.3.d.b 4 95.g even 4 2
1216.3.e.g 2 40.f even 2 1
1216.3.e.g 2 760.b odd 2 1
1216.3.e.h 2 40.e odd 2 1
1216.3.e.h 2 760.p even 2 1
2736.3.o.d 2 60.h even 2 1
2736.3.o.d 2 1140.p odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(475, [\chi])$$:

 $$T_{2}^{2} + 13$$ $$T_{7} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$13 + T^{2}$$
$3$ $$13 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -5 + T )^{2}$$
$11$ $$( 10 + T )^{2}$$
$13$ $$13 + T^{2}$$
$17$ $$( 15 + T )^{2}$$
$19$ $$361 + 12 T + T^{2}$$
$23$ $$( 35 + T )^{2}$$
$29$ $$325 + T^{2}$$
$31$ $$1300 + T^{2}$$
$37$ $$468 + T^{2}$$
$41$ $$1300 + T^{2}$$
$43$ $$( -20 + T )^{2}$$
$47$ $$( 10 + T )^{2}$$
$53$ $$5733 + T^{2}$$
$59$ $$325 + T^{2}$$
$61$ $$( 40 + T )^{2}$$
$67$ $$1573 + T^{2}$$
$71$ $$11700 + T^{2}$$
$73$ $$( 105 + T )^{2}$$
$79$ $$1300 + T^{2}$$
$83$ $$( -40 + T )^{2}$$
$89$ $$T^{2}$$
$97$ $$15028 + T^{2}$$