Properties

 Label 1475.1.c.b Level $1475$ Weight $1$ Character orbit 1475.c Self dual yes Analytic conductor $0.736$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -59 Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1475 = 5^{2} \cdot 59$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1475.c (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: yes Analytic conductor: $$0.736120893634$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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 $D_6$ Artin field Galois closure of 6.2.435125.1

$q$-expansion

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
176.1
 0
0 1.00000 1.00000 0 0 1.00000 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 $$\Q(\sqrt{-59})$$

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1475.1.c.b 1
5.b even 2 1 59.1.b.a 1
5.c odd 4 2 1475.1.d.a 2
15.d odd 2 1 531.1.c.a 1
20.d odd 2 1 944.1.h.a 1
35.c odd 2 1 2891.1.c.e 1
35.i odd 6 2 2891.1.g.b 2
35.j even 6 2 2891.1.g.d 2
40.e odd 2 1 3776.1.h.a 1
40.f even 2 1 3776.1.h.b 1
59.b odd 2 1 CM 1475.1.c.b 1
295.d odd 2 1 59.1.b.a 1
295.e even 4 2 1475.1.d.a 2
295.h odd 58 28 3481.1.d.a 28
295.j even 58 28 3481.1.d.a 28
885.c even 2 1 531.1.c.a 1
1180.h even 2 1 944.1.h.a 1
2065.h even 2 1 2891.1.c.e 1
2065.n even 6 2 2891.1.g.b 2
2065.p odd 6 2 2891.1.g.d 2
2360.f even 2 1 3776.1.h.a 1
2360.l odd 2 1 3776.1.h.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.1.b.a 1 5.b even 2 1
59.1.b.a 1 295.d odd 2 1
531.1.c.a 1 15.d odd 2 1
531.1.c.a 1 885.c even 2 1
944.1.h.a 1 20.d odd 2 1
944.1.h.a 1 1180.h even 2 1
1475.1.c.b 1 1.a even 1 1 trivial
1475.1.c.b 1 59.b odd 2 1 CM
1475.1.d.a 2 5.c odd 4 2
1475.1.d.a 2 295.e even 4 2
2891.1.c.e 1 35.c odd 2 1
2891.1.c.e 1 2065.h even 2 1
2891.1.g.b 2 35.i odd 6 2
2891.1.g.b 2 2065.n even 6 2
2891.1.g.d 2 35.j even 6 2
2891.1.g.d 2 2065.p odd 6 2
3481.1.d.a 28 295.h odd 58 28
3481.1.d.a 28 295.j even 58 28
3776.1.h.a 1 40.e odd 2 1
3776.1.h.a 1 2360.f even 2 1
3776.1.h.b 1 40.f even 2 1
3776.1.h.b 1 2360.l odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1475, [\chi])$$:

 $$T_{2}$$ $$T_{3} - 1$$

Hecke characteristic polynomials

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