# Properties

 Label 475.3.d.a Level $475$ Weight $3$ Character orbit 475.d Analytic conductor $12.943$ Analytic rank $0$ Dimension $2$ CM discriminant -19 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 475.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.9428125571$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$5$$ Twist minimal: no (minimal twist has level 19) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -4 q^{4} -5 i q^{7} -9 q^{9} +O(q^{10})$$ $$q -4 q^{4} -5 i q^{7} -9 q^{9} + 3 q^{11} + 16 q^{16} + 15 i q^{17} + 19 q^{19} + 30 i q^{23} + 20 i q^{28} + 36 q^{36} + 85 i q^{43} -12 q^{44} + 75 i q^{47} + 24 q^{49} + 103 q^{61} + 45 i q^{63} -64 q^{64} -60 i q^{68} + 25 i q^{73} -76 q^{76} -15 i q^{77} + 81 q^{81} -90 i q^{83} -120 i q^{92} -27 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4} - 18 q^{9} + O(q^{10})$$ $$2 q - 8 q^{4} - 18 q^{9} + 6 q^{11} + 32 q^{16} + 38 q^{19} + 72 q^{36} - 24 q^{44} + 48 q^{49} + 206 q^{61} - 128 q^{64} - 152 q^{76} + 162 q^{81} - 54 q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
474.1
 1.00000i − 1.00000i
0 0 −4.00000 0 0 5.00000i 0 −9.00000 0
474.2 0 0 −4.00000 0 0 5.00000i 0 −9.00000 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 CM by $$\Q(\sqrt{-19})$$
5.b even 2 1 inner
95.d 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.d.a 2
5.b even 2 1 inner 475.3.d.a 2
5.c odd 4 1 19.3.b.a 1
5.c odd 4 1 475.3.c.a 1
15.e even 4 1 171.3.c.a 1
19.b odd 2 1 CM 475.3.d.a 2
20.e even 4 1 304.3.e.a 1
40.i odd 4 1 1216.3.e.a 1
40.k even 4 1 1216.3.e.b 1
60.l odd 4 1 2736.3.o.a 1
95.d odd 2 1 inner 475.3.d.a 2
95.g even 4 1 19.3.b.a 1
95.g even 4 1 475.3.c.a 1
95.l even 12 2 361.3.d.a 2
95.m odd 12 2 361.3.d.a 2
95.q odd 36 6 361.3.f.a 6
95.r even 36 6 361.3.f.a 6
285.j odd 4 1 171.3.c.a 1
380.j odd 4 1 304.3.e.a 1
760.t even 4 1 1216.3.e.a 1
760.y odd 4 1 1216.3.e.b 1
1140.w even 4 1 2736.3.o.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.a 1 5.c odd 4 1
19.3.b.a 1 95.g even 4 1
171.3.c.a 1 15.e even 4 1
171.3.c.a 1 285.j odd 4 1
304.3.e.a 1 20.e even 4 1
304.3.e.a 1 380.j odd 4 1
361.3.d.a 2 95.l even 12 2
361.3.d.a 2 95.m odd 12 2
361.3.f.a 6 95.q odd 36 6
361.3.f.a 6 95.r even 36 6
475.3.c.a 1 5.c odd 4 1
475.3.c.a 1 95.g even 4 1
475.3.d.a 2 1.a even 1 1 trivial
475.3.d.a 2 5.b even 2 1 inner
475.3.d.a 2 19.b odd 2 1 CM
475.3.d.a 2 95.d odd 2 1 inner
1216.3.e.a 1 40.i odd 4 1
1216.3.e.a 1 760.t even 4 1
1216.3.e.b 1 40.k even 4 1
1216.3.e.b 1 760.y odd 4 1
2736.3.o.a 1 60.l odd 4 1
2736.3.o.a 1 1140.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{3}^{\mathrm{new}}(475, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$25 + T^{2}$$
$11$ $$( -3 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$225 + T^{2}$$
$19$ $$( -19 + T )^{2}$$
$23$ $$900 + T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$7225 + T^{2}$$
$47$ $$5625 + T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$( -103 + T )^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$625 + T^{2}$$
$79$ $$T^{2}$$
$83$ $$8100 + T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$