# Properties

 Label 1350.2.a.x Level 1350 Weight 2 Character orbit 1350.a Self dual yes Analytic conductor 10.780 Analytic rank 0 Dimension 2 CM no Inner twists 2

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{19})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 270) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{19}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + \beta q^{7} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} + \beta q^{7} + q^{8} -\beta q^{11} + \beta q^{14} + q^{16} + 4 q^{17} + 6 q^{19} -\beta q^{22} -2 q^{23} + \beta q^{28} + 7 q^{31} + q^{32} + 4 q^{34} -2 \beta q^{37} + 6 q^{38} + 2 \beta q^{41} -2 \beta q^{43} -\beta q^{44} -2 q^{46} + 2 q^{47} + 12 q^{49} -3 q^{53} + \beta q^{56} -2 \beta q^{59} -4 q^{61} + 7 q^{62} + q^{64} + 2 \beta q^{67} + 4 q^{68} + \beta q^{73} -2 \beta q^{74} + 6 q^{76} -19 q^{77} + 2 \beta q^{82} -5 q^{83} -2 \beta q^{86} -\beta q^{88} + 2 \beta q^{89} -2 q^{92} + 2 q^{94} -\beta q^{97} + 12 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{4} + 2q^{8} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{4} + 2q^{8} + 2q^{16} + 8q^{17} + 12q^{19} - 4q^{23} + 14q^{31} + 2q^{32} + 8q^{34} + 12q^{38} - 4q^{46} + 4q^{47} + 24q^{49} - 6q^{53} - 8q^{61} + 14q^{62} + 2q^{64} + 8q^{68} + 12q^{76} - 38q^{77} - 10q^{83} - 4q^{92} + 4q^{94} + 24q^{98} + O(q^{100})$$

## 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
 −4.35890 4.35890
1.00000 0 1.00000 0 0 −4.35890 1.00000 0 0
1.2 1.00000 0 1.00000 0 0 4.35890 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.2.a.x 2
3.b odd 2 1 1350.2.a.w 2
5.b even 2 1 1350.2.a.w 2
5.c odd 4 2 270.2.c.c 4
15.d odd 2 1 inner 1350.2.a.x 2
15.e even 4 2 270.2.c.c 4
20.e even 4 2 2160.2.f.m 4
45.k odd 12 4 810.2.i.h 8
45.l even 12 4 810.2.i.h 8
60.l odd 4 2 2160.2.f.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.c.c 4 5.c odd 4 2
270.2.c.c 4 15.e even 4 2
810.2.i.h 8 45.k odd 12 4
810.2.i.h 8 45.l even 12 4
1350.2.a.w 2 3.b odd 2 1
1350.2.a.w 2 5.b even 2 1
1350.2.a.x 2 1.a even 1 1 trivial
1350.2.a.x 2 15.d odd 2 1 inner
2160.2.f.m 4 20.e even 4 2
2160.2.f.m 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1350))$$:

 $$T_{7}^{2} - 19$$ $$T_{11}^{2} - 19$$ $$T_{13}$$ $$T_{17} - 4$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{2}$$
$3$ 
$5$ 
$7$ $$1 - 5 T^{2} + 49 T^{4}$$
$11$ $$1 + 3 T^{2} + 121 T^{4}$$
$13$ $$( 1 + 13 T^{2} )^{2}$$
$17$ $$( 1 - 4 T + 17 T^{2} )^{2}$$
$19$ $$( 1 - 6 T + 19 T^{2} )^{2}$$
$23$ $$( 1 + 2 T + 23 T^{2} )^{2}$$
$29$ $$( 1 + 29 T^{2} )^{2}$$
$31$ $$( 1 - 7 T + 31 T^{2} )^{2}$$
$37$ $$1 - 2 T^{2} + 1369 T^{4}$$
$41$ $$1 + 6 T^{2} + 1681 T^{4}$$
$43$ $$1 + 10 T^{2} + 1849 T^{4}$$
$47$ $$( 1 - 2 T + 47 T^{2} )^{2}$$
$53$ $$( 1 + 3 T + 53 T^{2} )^{2}$$
$59$ $$1 + 42 T^{2} + 3481 T^{4}$$
$61$ $$( 1 + 4 T + 61 T^{2} )^{2}$$
$67$ $$1 + 58 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$1 + 127 T^{2} + 5329 T^{4}$$
$79$ $$( 1 + 79 T^{2} )^{2}$$
$83$ $$( 1 + 5 T + 83 T^{2} )^{2}$$
$89$ $$1 + 102 T^{2} + 7921 T^{4}$$
$97$ $$1 + 175 T^{2} + 9409 T^{4}$$