# Properties

 Label 1575.2.a.d Level $1575$ Weight $2$ Character orbit 1575.a Self dual yes Analytic conductor $12.576$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1575.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.5764383184$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 315) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} + q^{7} + 3q^{8} + O(q^{10})$$ $$q - q^{2} - q^{4} + q^{7} + 3q^{8} - 4q^{13} - q^{14} - q^{16} - 2q^{17} + 4q^{26} - q^{28} + 8q^{29} - 4q^{31} - 5q^{32} + 2q^{34} - 8q^{37} + 4q^{41} - 8q^{43} + 12q^{47} + q^{49} + 4q^{52} - 6q^{53} + 3q^{56} - 8q^{58} + 8q^{59} + 10q^{61} + 4q^{62} + 7q^{64} - 8q^{67} + 2q^{68} - 16q^{71} - 12q^{73} + 8q^{74} - 8q^{79} - 4q^{82} - 16q^{83} + 8q^{86} - 12q^{89} - 4q^{91} - 12q^{94} - 4q^{97} - q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−1.00000 0 −1.00000 0 0 1.00000 3.00000 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.2.a.d 1
3.b odd 2 1 1575.2.a.j 1
5.b even 2 1 1575.2.a.g 1
5.c odd 4 2 315.2.d.d yes 2
15.d odd 2 1 1575.2.a.b 1
15.e even 4 2 315.2.d.b 2
20.e even 4 2 5040.2.t.r 2
35.f even 4 2 2205.2.d.c 2
60.l odd 4 2 5040.2.t.c 2
105.k odd 4 2 2205.2.d.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.d.b 2 15.e even 4 2
315.2.d.d yes 2 5.c odd 4 2
1575.2.a.b 1 15.d odd 2 1
1575.2.a.d 1 1.a even 1 1 trivial
1575.2.a.g 1 5.b even 2 1
1575.2.a.j 1 3.b odd 2 1
2205.2.d.c 2 35.f even 4 2
2205.2.d.g 2 105.k odd 4 2
5040.2.t.c 2 60.l odd 4 2
5040.2.t.r 2 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1575))$$:

 $$T_{2} + 1$$ $$T_{11}$$ $$T_{13} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-1 + T$$
$11$ $$T$$
$13$ $$4 + T$$
$17$ $$2 + T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$-8 + T$$
$31$ $$4 + T$$
$37$ $$8 + T$$
$41$ $$-4 + T$$
$43$ $$8 + T$$
$47$ $$-12 + T$$
$53$ $$6 + T$$
$59$ $$-8 + T$$
$61$ $$-10 + T$$
$67$ $$8 + T$$
$71$ $$16 + T$$
$73$ $$12 + T$$
$79$ $$8 + T$$
$83$ $$16 + T$$
$89$ $$12 + T$$
$97$ $$4 + T$$