# Properties

 Label 1475.1.d.a Level $1475$ Weight $1$ Character orbit 1475.d Analytic conductor $0.736$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -59 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1475 = 5^{2} \cdot 59$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1475.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.736120893634$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 59) Projective image $$D_{3}$$ Projective field Galois closure of 3.1.59.1 Artin image $C_4\times S_3$ Artin field Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -i q^{3} - q^{4} + i q^{7} +O(q^{10})$$ $$q -i q^{3} - q^{4} + i q^{7} + i q^{12} + q^{16} -2 i q^{17} + q^{19} + q^{21} -i q^{27} -i q^{28} + q^{29} - q^{41} -i q^{48} -2 q^{51} -i q^{53} -i q^{57} - q^{59} - q^{64} + 2 i q^{68} + 2 q^{71} - q^{76} + q^{79} - q^{81} - q^{84} -i q^{87} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{16} + 2q^{19} + 2q^{21} + 2q^{29} - 2q^{41} - 4q^{51} - 2q^{59} - 2q^{64} + 4q^{71} - 2q^{76} + 2q^{79} - 2q^{81} - 2q^{84} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1475\mathbb{Z}\right)^\times$$.

 $$n$$ $$651$$ $$827$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1474.1
 1.00000i − 1.00000i
0 1.00000i −1.00000 0 0 1.00000i 0 0 0
1474.2 0 1.00000i −1.00000 0 0 1.00000i 0 0 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
59.b odd 2 1 CM by $$\Q(\sqrt{-59})$$
5.b even 2 1 inner
295.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1475.1.d.a 2
5.b even 2 1 inner 1475.1.d.a 2
5.c odd 4 1 59.1.b.a 1
5.c odd 4 1 1475.1.c.b 1
15.e even 4 1 531.1.c.a 1
20.e even 4 1 944.1.h.a 1
35.f even 4 1 2891.1.c.e 1
35.k even 12 2 2891.1.g.b 2
35.l odd 12 2 2891.1.g.d 2
40.i odd 4 1 3776.1.h.b 1
40.k even 4 1 3776.1.h.a 1
59.b odd 2 1 CM 1475.1.d.a 2
295.d odd 2 1 inner 1475.1.d.a 2
295.e even 4 1 59.1.b.a 1
295.e even 4 1 1475.1.c.b 1
295.k odd 116 28 3481.1.d.a 28
295.l even 116 28 3481.1.d.a 28
885.k odd 4 1 531.1.c.a 1
1180.l odd 4 1 944.1.h.a 1
2065.l odd 4 1 2891.1.c.e 1
2065.v even 12 2 2891.1.g.d 2
2065.x odd 12 2 2891.1.g.b 2
2360.q odd 4 1 3776.1.h.a 1
2360.w even 4 1 3776.1.h.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.1.b.a 1 5.c odd 4 1
59.1.b.a 1 295.e even 4 1
531.1.c.a 1 15.e even 4 1
531.1.c.a 1 885.k odd 4 1
944.1.h.a 1 20.e even 4 1
944.1.h.a 1 1180.l odd 4 1
1475.1.c.b 1 5.c odd 4 1
1475.1.c.b 1 295.e even 4 1
1475.1.d.a 2 1.a even 1 1 trivial
1475.1.d.a 2 5.b even 2 1 inner
1475.1.d.a 2 59.b odd 2 1 CM
1475.1.d.a 2 295.d odd 2 1 inner
2891.1.c.e 1 35.f even 4 1
2891.1.c.e 1 2065.l odd 4 1
2891.1.g.b 2 35.k even 12 2
2891.1.g.b 2 2065.x odd 12 2
2891.1.g.d 2 35.l odd 12 2
2891.1.g.d 2 2065.v even 12 2
3481.1.d.a 28 295.k odd 116 28
3481.1.d.a 28 295.l even 116 28
3776.1.h.a 1 40.k even 4 1
3776.1.h.a 1 2360.q odd 4 1
3776.1.h.b 1 40.i odd 4 1
3776.1.h.b 1 2360.w even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1475, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$( -1 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$( 1 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$1 + T^{2}$$
$59$ $$( 1 + T )^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2}$$
$71$ $$( -2 + T )^{2}$$
$73$ $$T^{2}$$
$79$ $$( -1 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$