# Properties

 Label 1350.2.a.j Level 1350 Weight 2 Character orbit 1350.a Self dual yes Analytic conductor 10.780 Analytic rank 1 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$1350 = 2 \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1350.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$10.7798042729$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 270) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + 4q^{7} - q^{8} + O(q^{10})$$ $$q - q^{2} + q^{4} + 4q^{7} - q^{8} - 5q^{11} - 3q^{13} - 4q^{14} + q^{16} - q^{17} - 6q^{19} + 5q^{22} - q^{23} + 3q^{26} + 4q^{28} - 9q^{29} - 5q^{31} - q^{32} + q^{34} - 2q^{37} + 6q^{38} - 2q^{41} + q^{43} - 5q^{44} + q^{46} + 13q^{47} + 9q^{49} - 3q^{52} - 4q^{56} + 9q^{58} - 4q^{59} + 8q^{61} + 5q^{62} + q^{64} - 4q^{67} - q^{68} - 6q^{71} - 2q^{73} + 2q^{74} - 6q^{76} - 20q^{77} + 9q^{79} + 2q^{82} - 4q^{83} - q^{86} + 5q^{88} - 14q^{89} - 12q^{91} - q^{92} - 13q^{94} - 10q^{97} - 9q^{98} + O(q^{100})$$

## 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 4.00000 −1.00000 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 1350.2.a.j 1
3.b odd 2 1 1350.2.a.v 1
5.b even 2 1 1350.2.a.l 1
5.c odd 4 2 270.2.c.a 2
15.d odd 2 1 1350.2.a.b 1
15.e even 4 2 270.2.c.b yes 2
20.e even 4 2 2160.2.f.d 2
45.k odd 12 4 810.2.i.d 4
45.l even 12 4 810.2.i.c 4
60.l odd 4 2 2160.2.f.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.c.a 2 5.c odd 4 2
270.2.c.b yes 2 15.e even 4 2
810.2.i.c 4 45.l even 12 4
810.2.i.d 4 45.k odd 12 4
1350.2.a.b 1 15.d odd 2 1
1350.2.a.j 1 1.a even 1 1 trivial
1350.2.a.l 1 5.b even 2 1
1350.2.a.v 1 3.b odd 2 1
2160.2.f.d 2 20.e even 4 2
2160.2.f.e 2 60.l odd 4 2

## Atkin-Lehner signs

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1350))$$:

 $$T_{7} - 4$$ $$T_{11} + 5$$ $$T_{13} + 3$$ $$T_{17} + 1$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ 
$5$ 
$7$ $$1 - 4 T + 7 T^{2}$$
$11$ $$1 + 5 T + 11 T^{2}$$
$13$ $$1 + 3 T + 13 T^{2}$$
$17$ $$1 + T + 17 T^{2}$$
$19$ $$1 + 6 T + 19 T^{2}$$
$23$ $$1 + T + 23 T^{2}$$
$29$ $$1 + 9 T + 29 T^{2}$$
$31$ $$1 + 5 T + 31 T^{2}$$
$37$ $$1 + 2 T + 37 T^{2}$$
$41$ $$1 + 2 T + 41 T^{2}$$
$43$ $$1 - T + 43 T^{2}$$
$47$ $$1 - 13 T + 47 T^{2}$$
$53$ $$1 + 53 T^{2}$$
$59$ $$1 + 4 T + 59 T^{2}$$
$61$ $$1 - 8 T + 61 T^{2}$$
$67$ $$1 + 4 T + 67 T^{2}$$
$71$ $$1 + 6 T + 71 T^{2}$$
$73$ $$1 + 2 T + 73 T^{2}$$
$79$ $$1 - 9 T + 79 T^{2}$$
$83$ $$1 + 4 T + 83 T^{2}$$
$89$ $$1 + 14 T + 89 T^{2}$$
$97$ $$1 + 10 T + 97 T^{2}$$