# Properties

 Label 1575.1.h.e 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.

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

## 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.e yes 2
3.b odd 2 1 inner 1575.1.h.e yes 2
5.b even 2 1 1575.1.h.d 2
5.c odd 4 2 1575.1.e.c 4
7.b odd 2 1 CM 1575.1.h.e yes 2
15.d odd 2 1 1575.1.h.d 2
15.e even 4 2 1575.1.e.c 4
21.c even 2 1 inner 1575.1.h.e yes 2
35.c odd 2 1 1575.1.h.d 2
35.f even 4 2 1575.1.e.c 4
105.g even 2 1 1575.1.h.d 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 5.b even 2 1
1575.1.h.d 2 15.d odd 2 1
1575.1.h.d 2 35.c odd 2 1
1575.1.h.d 2 105.g even 2 1
1575.1.h.e yes 2 1.a even 1 1 trivial
1575.1.h.e yes 2 3.b odd 2 1 inner
1575.1.h.e yes 2 7.b odd 2 1 CM
1575.1.h.e yes 2 21.c even 2 1 inner

## 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

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