# Properties

 Label 9747.2.a.f Level $9747$ Weight $2$ Character orbit 9747.a Self dual yes Analytic conductor $77.830$ Analytic rank $0$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9747 = 3^{3} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9747.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$77.8301868501$$ Analytic rank: $$0$$ 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 $$q - 2 q^{4} - q^{7} - 5 q^{13} + 4 q^{16} - 5 q^{25} + 2 q^{28} + 4 q^{31} - 11 q^{37} + 8 q^{43} - 6 q^{49} + 10 q^{52} - q^{61} - 8 q^{64} - 5 q^{67} - 7 q^{73} - 17 q^{79} + 5 q^{91} + 19 q^{97}+O(q^{100})$$ q - 2 * q^4 - q^7 - 5 * q^13 + 4 * q^16 - 5 * q^25 + 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 - 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
$$3$$ $$1$$
$$19$$ $$-1$$

## 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 9747.2.a.f 1
3.b odd 2 1 CM 9747.2.a.f 1
19.b odd 2 1 27.2.a.a 1
57.d even 2 1 27.2.a.a 1
76.d even 2 1 432.2.a.e 1
95.d odd 2 1 675.2.a.e 1
95.g even 4 2 675.2.b.f 2
133.c even 2 1 1323.2.a.i 1
152.b even 2 1 1728.2.a.o 1
152.g odd 2 1 1728.2.a.n 1
171.l even 6 2 81.2.c.a 2
171.o odd 6 2 81.2.c.a 2
209.d even 2 1 3267.2.a.f 1
228.b odd 2 1 432.2.a.e 1
247.d odd 2 1 4563.2.a.e 1
285.b even 2 1 675.2.a.e 1
285.j odd 4 2 675.2.b.f 2
323.c odd 2 1 7803.2.a.k 1
399.h odd 2 1 1323.2.a.i 1
456.l odd 2 1 1728.2.a.o 1
456.p even 2 1 1728.2.a.n 1
513.bw even 18 6 729.2.e.f 6
513.ca odd 18 6 729.2.e.f 6
627.b odd 2 1 3267.2.a.f 1
684.w even 6 2 1296.2.i.i 2
684.bh odd 6 2 1296.2.i.i 2
741.d even 2 1 4563.2.a.e 1
969.h even 2 1 7803.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.a.a 1 19.b odd 2 1
27.2.a.a 1 57.d even 2 1
81.2.c.a 2 171.l even 6 2
81.2.c.a 2 171.o odd 6 2
432.2.a.e 1 76.d even 2 1
432.2.a.e 1 228.b odd 2 1
675.2.a.e 1 95.d odd 2 1
675.2.a.e 1 285.b even 2 1
675.2.b.f 2 95.g even 4 2
675.2.b.f 2 285.j odd 4 2
729.2.e.f 6 513.bw even 18 6
729.2.e.f 6 513.ca odd 18 6
1296.2.i.i 2 684.w even 6 2
1296.2.i.i 2 684.bh odd 6 2
1323.2.a.i 1 133.c even 2 1
1323.2.a.i 1 399.h odd 2 1
1728.2.a.n 1 152.g odd 2 1
1728.2.a.n 1 456.p even 2 1
1728.2.a.o 1 152.b even 2 1
1728.2.a.o 1 456.l odd 2 1
3267.2.a.f 1 209.d even 2 1
3267.2.a.f 1 627.b odd 2 1
4563.2.a.e 1 247.d odd 2 1
4563.2.a.e 1 741.d even 2 1
7803.2.a.k 1 323.c odd 2 1
7803.2.a.k 1 969.h even 2 1
9747.2.a.f 1 1.a even 1 1 trivial
9747.2.a.f 1 3.b odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9747))$$:

 $$T_{2}$$ T2 $$T_{5}$$ T5 $$T_{7} + 1$$ T7 + 1 $$T_{13} + 5$$ T13 + 5

## Hecke characteristic polynomials

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