# Properties

 Label 75.3.c.a Level $75$ Weight $3$ Character orbit 75.c Self dual yes Analytic conductor $2.044$ Analytic rank $0$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 75.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.04360198270$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3q^{3} + 4q^{4} + 11q^{7} + 9q^{9} + O(q^{10})$$ $$q - 3q^{3} + 4q^{4} + 11q^{7} + 9q^{9} - 12q^{12} - q^{13} + 16q^{16} - 37q^{19} - 33q^{21} - 27q^{27} + 44q^{28} - 13q^{31} + 36q^{36} + 26q^{37} + 3q^{39} - 61q^{43} - 48q^{48} + 72q^{49} - 4q^{52} + 111q^{57} + 47q^{61} + 99q^{63} + 64q^{64} - 109q^{67} - 46q^{73} - 148q^{76} - 142q^{79} + 81q^{81} - 132q^{84} - 11q^{91} + 39q^{93} - 169q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$$.

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 0
0 −3.00000 4.00000 0 0 11.0000 0 9.00000 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.3.c.a 1
3.b odd 2 1 CM 75.3.c.a 1
4.b odd 2 1 1200.3.l.d 1
5.b even 2 1 75.3.c.b yes 1
5.c odd 4 2 75.3.d.a 2
12.b even 2 1 1200.3.l.d 1
15.d odd 2 1 75.3.c.b yes 1
15.e even 4 2 75.3.d.a 2
20.d odd 2 1 1200.3.l.c 1
20.e even 4 2 1200.3.c.a 2
60.h even 2 1 1200.3.l.c 1
60.l odd 4 2 1200.3.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.c.a 1 1.a even 1 1 trivial
75.3.c.a 1 3.b odd 2 1 CM
75.3.c.b yes 1 5.b even 2 1
75.3.c.b yes 1 15.d odd 2 1
75.3.d.a 2 5.c odd 4 2
75.3.d.a 2 15.e even 4 2
1200.3.c.a 2 20.e even 4 2
1200.3.c.a 2 60.l odd 4 2
1200.3.l.c 1 20.d odd 2 1
1200.3.l.c 1 60.h even 2 1
1200.3.l.d 1 4.b odd 2 1
1200.3.l.d 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(75, [\chi])$$:

 $$T_{2}$$ $$T_{7} - 11$$ $$T_{13} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$3 + T$$
$5$ $$T$$
$7$ $$-11 + T$$
$11$ $$T$$
$13$ $$1 + T$$
$17$ $$T$$
$19$ $$37 + T$$
$23$ $$T$$
$29$ $$T$$
$31$ $$13 + T$$
$37$ $$-26 + T$$
$41$ $$T$$
$43$ $$61 + T$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$-47 + T$$
$67$ $$109 + T$$
$71$ $$T$$
$73$ $$46 + T$$
$79$ $$142 + T$$
$83$ $$T$$
$89$ $$T$$
$97$ $$169 + T$$