# Properties

 Label 1575.1.h.a Level $1575$ Weight $1$ Character orbit 1575.h Self dual yes Analytic conductor $0.786$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -7 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1575.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.786027394897$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 175) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.175.1 Artin image: $D_6$ Artin field: Galois closure of 6.0.4134375.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{7} + q^{8} + O(q^{10})$$ $$q - q^{2} - q^{7} + q^{8} + q^{11} + q^{14} - q^{16} - q^{22} - q^{23} + q^{29} + q^{37} + q^{43} + q^{46} + q^{49} + 2 q^{53} - q^{56} - q^{58} + q^{64} + q^{67} + q^{71} - q^{74} - q^{77} - q^{79} - q^{86} + q^{88} - q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$1226$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1126.1
 0
−1.00000 0 0 0 0 −1.00000 1.00000 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.1.h.a 1
3.b odd 2 1 175.1.d.b yes 1
5.b even 2 1 1575.1.h.c 1
5.c odd 4 2 1575.1.e.a 2
7.b odd 2 1 CM 1575.1.h.a 1
12.b even 2 1 2800.1.f.b 1
15.d odd 2 1 175.1.d.a 1
15.e even 4 2 175.1.c.a 2
21.c even 2 1 175.1.d.b yes 1
21.g even 6 2 1225.1.i.a 2
21.h odd 6 2 1225.1.i.a 2
35.c odd 2 1 1575.1.h.c 1
35.f even 4 2 1575.1.e.a 2
60.h even 2 1 2800.1.f.a 1
60.l odd 4 2 2800.1.p.a 2
84.h odd 2 1 2800.1.f.b 1
105.g even 2 1 175.1.d.a 1
105.k odd 4 2 175.1.c.a 2
105.o odd 6 2 1225.1.i.b 2
105.p even 6 2 1225.1.i.b 2
105.w odd 12 4 1225.1.j.a 4
105.x even 12 4 1225.1.j.a 4
420.o odd 2 1 2800.1.f.a 1
420.w even 4 2 2800.1.p.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.1.c.a 2 15.e even 4 2
175.1.c.a 2 105.k odd 4 2
175.1.d.a 1 15.d odd 2 1
175.1.d.a 1 105.g even 2 1
175.1.d.b yes 1 3.b odd 2 1
175.1.d.b yes 1 21.c even 2 1
1225.1.i.a 2 21.g even 6 2
1225.1.i.a 2 21.h odd 6 2
1225.1.i.b 2 105.o odd 6 2
1225.1.i.b 2 105.p even 6 2
1225.1.j.a 4 105.w odd 12 4
1225.1.j.a 4 105.x even 12 4
1575.1.e.a 2 5.c odd 4 2
1575.1.e.a 2 35.f even 4 2
1575.1.h.a 1 1.a even 1 1 trivial
1575.1.h.a 1 7.b odd 2 1 CM
1575.1.h.c 1 5.b even 2 1
1575.1.h.c 1 35.c odd 2 1
2800.1.f.a 1 60.h even 2 1
2800.1.f.a 1 420.o odd 2 1
2800.1.f.b 1 12.b even 2 1
2800.1.f.b 1 84.h odd 2 1
2800.1.p.a 2 60.l odd 4 2
2800.1.p.a 2 420.w even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1575, [\chi])$$:

 $$T_{2} + 1$$ $$T_{37} - 1$$

## Hecke characteristic polynomials

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