# Properties

 Label 9900.2.a.bd Level 9900 Weight 2 Character orbit 9900.a Self dual yes Analytic conductor 79.052 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9900.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$79.0518980011$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 220) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 4q^{7} + O(q^{10})$$ $$q + 4q^{7} + q^{11} + 4q^{13} - 4q^{19} - 6q^{23} + 6q^{29} + 8q^{31} - 2q^{37} - 6q^{41} - 8q^{43} + 6q^{47} + 9q^{49} - 6q^{53} + 12q^{59} + 2q^{61} + 10q^{67} + 12q^{71} + 16q^{73} + 4q^{77} + 8q^{79} - 6q^{89} + 16q^{91} - 14q^{97} + O(q^{100})$$

## 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
0 0 0 0 0 4.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 9900.2.a.bd 1
3.b odd 2 1 1100.2.a.e 1
5.b even 2 1 1980.2.a.a 1
5.c odd 4 2 9900.2.c.m 2
12.b even 2 1 4400.2.a.e 1
15.d odd 2 1 220.2.a.a 1
15.e even 4 2 1100.2.b.a 2
20.d odd 2 1 7920.2.a.o 1
60.h even 2 1 880.2.a.j 1
60.l odd 4 2 4400.2.b.f 2
120.i odd 2 1 3520.2.a.bd 1
120.m even 2 1 3520.2.a.d 1
165.d even 2 1 2420.2.a.b 1
660.g odd 2 1 9680.2.a.bb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.2.a.a 1 15.d odd 2 1
880.2.a.j 1 60.h even 2 1
1100.2.a.e 1 3.b odd 2 1
1100.2.b.a 2 15.e even 4 2
1980.2.a.a 1 5.b even 2 1
2420.2.a.b 1 165.d even 2 1
3520.2.a.d 1 120.m even 2 1
3520.2.a.bd 1 120.i odd 2 1
4400.2.a.e 1 12.b even 2 1
4400.2.b.f 2 60.l odd 4 2
7920.2.a.o 1 20.d odd 2 1
9680.2.a.bb 1 660.g odd 2 1
9900.2.a.bd 1 1.a even 1 1 trivial
9900.2.c.m 2 5.c odd 4 2

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9900))$$:

 $$T_{7} - 4$$ $$T_{13} - 4$$ $$T_{17}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ 1
$7$ $$1 - 4 T + 7 T^{2}$$
$11$ $$1 - T$$
$13$ $$1 - 4 T + 13 T^{2}$$
$17$ $$1 + 17 T^{2}$$
$19$ $$1 + 4 T + 19 T^{2}$$
$23$ $$1 + 6 T + 23 T^{2}$$
$29$ $$1 - 6 T + 29 T^{2}$$
$31$ $$1 - 8 T + 31 T^{2}$$
$37$ $$1 + 2 T + 37 T^{2}$$
$41$ $$1 + 6 T + 41 T^{2}$$
$43$ $$1 + 8 T + 43 T^{2}$$
$47$ $$1 - 6 T + 47 T^{2}$$
$53$ $$1 + 6 T + 53 T^{2}$$
$59$ $$1 - 12 T + 59 T^{2}$$
$61$ $$1 - 2 T + 61 T^{2}$$
$67$ $$1 - 10 T + 67 T^{2}$$
$71$ $$1 - 12 T + 71 T^{2}$$
$73$ $$1 - 16 T + 73 T^{2}$$
$79$ $$1 - 8 T + 79 T^{2}$$
$83$ $$1 + 83 T^{2}$$
$89$ $$1 + 6 T + 89 T^{2}$$
$97$ $$1 + 14 T + 97 T^{2}$$