# Properties

 Label 1575.1.h.d Level $1575$ Weight $1$ Character orbit 1575.h Self dual yes Analytic conductor $0.786$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -7 Inner twists $4$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.5788125.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + 2 q^{4} - q^{7} -\beta q^{8} +O(q^{10})$$ $$q -\beta q^{2} + 2 q^{4} - q^{7} -\beta q^{8} -\beta q^{11} + \beta q^{14} + q^{16} + 3 q^{22} + \beta q^{23} -2 q^{28} + \beta q^{29} + q^{37} + q^{43} -2 \beta q^{44} -3 q^{46} + q^{49} + \beta q^{56} -3 q^{58} - q^{64} - q^{67} + \beta q^{71} -\beta q^{74} + \beta q^{77} - q^{79} -\beta q^{86} + 3 q^{88} + 2 \beta q^{92} -\beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4} - 2 q^{7} + O(q^{10})$$ $$2 q + 4 q^{4} - 2 q^{7} + 2 q^{16} + 6 q^{22} - 4 q^{28} + 2 q^{37} + 2 q^{43} - 6 q^{46} + 2 q^{49} - 6 q^{58} - 2 q^{64} - 2 q^{67} - 2 q^{79} + 6 q^{88} + O(q^{100})$$

## 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
 1.73205 −1.73205
−1.73205 0 2.00000 0 0 −1.00000 −1.73205 0 0
1126.2 1.73205 0 2.00000 0 0 −1.00000 1.73205 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 $$\Q(\sqrt{-7})$$
3.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.1.h.d 2
3.b odd 2 1 inner 1575.1.h.d 2
5.b even 2 1 1575.1.h.e yes 2
5.c odd 4 2 1575.1.e.c 4
7.b odd 2 1 CM 1575.1.h.d 2
15.d odd 2 1 1575.1.h.e yes 2
15.e even 4 2 1575.1.e.c 4
21.c even 2 1 inner 1575.1.h.d 2
35.c odd 2 1 1575.1.h.e yes 2
35.f even 4 2 1575.1.e.c 4
105.g even 2 1 1575.1.h.e yes 2
105.k odd 4 2 1575.1.e.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.1.e.c 4 5.c odd 4 2
1575.1.e.c 4 15.e even 4 2
1575.1.e.c 4 35.f even 4 2
1575.1.e.c 4 105.k odd 4 2
1575.1.h.d 2 1.a even 1 1 trivial
1575.1.h.d 2 3.b odd 2 1 inner
1575.1.h.d 2 7.b odd 2 1 CM
1575.1.h.d 2 21.c even 2 1 inner
1575.1.h.e yes 2 5.b even 2 1
1575.1.h.e yes 2 15.d odd 2 1
1575.1.h.e yes 2 35.c odd 2 1
1575.1.h.e yes 2 105.g even 2 1

## Hecke kernels

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

 $$T_{2}^{2} - 3$$ $$T_{37} - 1$$

## Hecke characteristic polynomials

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