# Properties

 Label 1350.2.a.h Level $1350$ Weight $2$ Character orbit 1350.a Self dual yes Analytic conductor $10.780$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 54) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + q^{7} - q^{8} + O(q^{10})$$ $$q - q^{2} + q^{4} + q^{7} - q^{8} + 3 q^{11} + 4 q^{13} - q^{14} + q^{16} + 2 q^{19} - 3 q^{22} - 6 q^{23} - 4 q^{26} + q^{28} - 6 q^{29} + 5 q^{31} - q^{32} - 2 q^{37} - 2 q^{38} + 6 q^{41} + 10 q^{43} + 3 q^{44} + 6 q^{46} + 6 q^{47} - 6 q^{49} + 4 q^{52} + 9 q^{53} - q^{56} + 6 q^{58} - 12 q^{59} + 8 q^{61} - 5 q^{62} + q^{64} - 14 q^{67} + 7 q^{73} + 2 q^{74} + 2 q^{76} + 3 q^{77} + 8 q^{79} - 6 q^{82} - 3 q^{83} - 10 q^{86} - 3 q^{88} + 18 q^{89} + 4 q^{91} - 6 q^{92} - 6 q^{94} + q^{97} + 6 q^{98} + O(q^{100})$$

## 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 1.00000 −1.00000 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$

## 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.h 1
3.b odd 2 1 1350.2.a.r 1
5.b even 2 1 54.2.a.b yes 1
5.c odd 4 2 1350.2.c.k 2
15.d odd 2 1 54.2.a.a 1
15.e even 4 2 1350.2.c.b 2
20.d odd 2 1 432.2.a.b 1
35.c odd 2 1 2646.2.a.bd 1
40.e odd 2 1 1728.2.a.z 1
40.f even 2 1 1728.2.a.y 1
45.h odd 6 2 162.2.c.c 2
45.j even 6 2 162.2.c.b 2
55.d odd 2 1 6534.2.a.b 1
60.h even 2 1 432.2.a.g 1
65.d even 2 1 9126.2.a.r 1
105.g even 2 1 2646.2.a.a 1
120.i odd 2 1 1728.2.a.c 1
120.m even 2 1 1728.2.a.d 1
165.d even 2 1 6534.2.a.bc 1
180.n even 6 2 1296.2.i.c 2
180.p odd 6 2 1296.2.i.o 2
195.e odd 2 1 9126.2.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.a.a 1 15.d odd 2 1
54.2.a.b yes 1 5.b even 2 1
162.2.c.b 2 45.j even 6 2
162.2.c.c 2 45.h odd 6 2
432.2.a.b 1 20.d odd 2 1
432.2.a.g 1 60.h even 2 1
1296.2.i.c 2 180.n even 6 2
1296.2.i.o 2 180.p odd 6 2
1350.2.a.h 1 1.a even 1 1 trivial
1350.2.a.r 1 3.b odd 2 1
1350.2.c.b 2 15.e even 4 2
1350.2.c.k 2 5.c odd 4 2
1728.2.a.c 1 120.i odd 2 1
1728.2.a.d 1 120.m even 2 1
1728.2.a.y 1 40.f even 2 1
1728.2.a.z 1 40.e odd 2 1
2646.2.a.a 1 105.g even 2 1
2646.2.a.bd 1 35.c odd 2 1
6534.2.a.b 1 55.d odd 2 1
6534.2.a.bc 1 165.d even 2 1
9126.2.a.r 1 65.d even 2 1
9126.2.a.u 1 195.e odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-1 + T$$
$11$ $$-3 + T$$
$13$ $$-4 + T$$
$17$ $$T$$
$19$ $$-2 + T$$
$23$ $$6 + T$$
$29$ $$6 + T$$
$31$ $$-5 + T$$
$37$ $$2 + T$$
$41$ $$-6 + T$$
$43$ $$-10 + T$$
$47$ $$-6 + T$$
$53$ $$-9 + T$$
$59$ $$12 + T$$
$61$ $$-8 + T$$
$67$ $$14 + T$$
$71$ $$T$$
$73$ $$-7 + T$$
$79$ $$-8 + T$$
$83$ $$3 + T$$
$89$ $$-18 + T$$
$97$ $$-1 + T$$