# Properties

 Label 675.2.a.e Level $675$ Weight $2$ Character orbit 675.a Self dual yes Analytic conductor $5.390$ Analytic rank $1$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$675 = 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 675.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.38990213644$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 27) Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{4} + q^{7}+O(q^{10})$$ q - 2 * q^4 + q^7 $$q - 2 q^{4} + q^{7} - 5 q^{13} + 4 q^{16} - 7 q^{19} - 2 q^{28} - 4 q^{31} - 11 q^{37} - 8 q^{43} - 6 q^{49} + 10 q^{52} - q^{61} - 8 q^{64} - 5 q^{67} + 7 q^{73} + 14 q^{76} + 17 q^{79} - 5 q^{91} + 19 q^{97}+O(q^{100})$$ q - 2 * q^4 + q^7 - 5 * q^13 + 4 * q^16 - 7 * q^19 - 2 * q^28 - 4 * q^31 - 11 * q^37 - 8 * q^43 - 6 * q^49 + 10 * q^52 - q^61 - 8 * q^64 - 5 * q^67 + 7 * q^73 + 14 * q^76 + 17 * q^79 - 5 * q^91 + 19 * q^97

## 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
0 0 −2.00000 0 0 1.00000 0 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
$$3$$ $$1$$
$$5$$ $$1$$

## 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 675.2.a.e 1
3.b odd 2 1 CM 675.2.a.e 1
5.b even 2 1 27.2.a.a 1
5.c odd 4 2 675.2.b.f 2
15.d odd 2 1 27.2.a.a 1
15.e even 4 2 675.2.b.f 2
20.d odd 2 1 432.2.a.e 1
35.c odd 2 1 1323.2.a.i 1
40.e odd 2 1 1728.2.a.o 1
40.f even 2 1 1728.2.a.n 1
45.h odd 6 2 81.2.c.a 2
45.j even 6 2 81.2.c.a 2
55.d odd 2 1 3267.2.a.f 1
60.h even 2 1 432.2.a.e 1
65.d even 2 1 4563.2.a.e 1
85.c even 2 1 7803.2.a.k 1
95.d odd 2 1 9747.2.a.f 1
105.g even 2 1 1323.2.a.i 1
120.i odd 2 1 1728.2.a.n 1
120.m even 2 1 1728.2.a.o 1
135.n odd 18 6 729.2.e.f 6
135.p even 18 6 729.2.e.f 6
165.d even 2 1 3267.2.a.f 1
180.n even 6 2 1296.2.i.i 2
180.p odd 6 2 1296.2.i.i 2
195.e odd 2 1 4563.2.a.e 1
255.h odd 2 1 7803.2.a.k 1
285.b even 2 1 9747.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.a.a 1 5.b even 2 1
27.2.a.a 1 15.d odd 2 1
81.2.c.a 2 45.h odd 6 2
81.2.c.a 2 45.j even 6 2
432.2.a.e 1 20.d odd 2 1
432.2.a.e 1 60.h even 2 1
675.2.a.e 1 1.a even 1 1 trivial
675.2.a.e 1 3.b odd 2 1 CM
675.2.b.f 2 5.c odd 4 2
675.2.b.f 2 15.e even 4 2
729.2.e.f 6 135.n odd 18 6
729.2.e.f 6 135.p even 18 6
1296.2.i.i 2 180.n even 6 2
1296.2.i.i 2 180.p odd 6 2
1323.2.a.i 1 35.c odd 2 1
1323.2.a.i 1 105.g even 2 1
1728.2.a.n 1 40.f even 2 1
1728.2.a.n 1 120.i odd 2 1
1728.2.a.o 1 40.e odd 2 1
1728.2.a.o 1 120.m even 2 1
3267.2.a.f 1 55.d odd 2 1
3267.2.a.f 1 165.d even 2 1
4563.2.a.e 1 65.d even 2 1
4563.2.a.e 1 195.e odd 2 1
7803.2.a.k 1 85.c even 2 1
7803.2.a.k 1 255.h odd 2 1
9747.2.a.f 1 95.d odd 2 1
9747.2.a.f 1 285.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(675))$$:

 $$T_{2}$$ T2 $$T_{7} - 1$$ T7 - 1 $$T_{11}$$ T11

## Hecke characteristic polynomials

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