# Properties

 Label 1350.2.c.a Level 1350 Weight 2 Character orbit 1350.c Analytic conductor 10.780 Analytic rank 1 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.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.7798042729$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 270) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} + 2 i q^{7} -i q^{8} +O(q^{10})$$ $$q + i q^{2} - q^{4} + 2 i q^{7} -i q^{8} -3 q^{11} + i q^{13} -2 q^{14} + q^{16} -3 i q^{17} -8 q^{19} -3 i q^{22} -3 i q^{23} - q^{26} -2 i q^{28} + 9 q^{29} -7 q^{31} + i q^{32} + 3 q^{34} + 2 i q^{37} -8 i q^{38} -12 q^{41} + 7 i q^{43} + 3 q^{44} + 3 q^{46} -3 i q^{47} + 3 q^{49} -i q^{52} -12 i q^{53} + 2 q^{56} + 9 i q^{58} -12 q^{59} -10 q^{61} -7 i q^{62} - q^{64} -4 i q^{67} + 3 i q^{68} -2 i q^{73} -2 q^{74} + 8 q^{76} -6 i q^{77} + q^{79} -12 i q^{82} -18 i q^{83} -7 q^{86} + 3 i q^{88} -2 q^{91} + 3 i q^{92} + 3 q^{94} + 14 i q^{97} + 3 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} - 6q^{11} - 4q^{14} + 2q^{16} - 16q^{19} - 2q^{26} + 18q^{29} - 14q^{31} + 6q^{34} - 24q^{41} + 6q^{44} + 6q^{46} + 6q^{49} + 4q^{56} - 24q^{59} - 20q^{61} - 2q^{64} - 4q^{74} + 16q^{76} + 2q^{79} - 14q^{86} - 4q^{91} + 6q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1027$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 1.00000i 1.00000i
1.00000i 0 −1.00000 0 0 2.00000i 1.00000i 0 0
649.2 1.00000i 0 −1.00000 0 0 2.00000i 1.00000i 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.2.c.a 2
3.b odd 2 1 1350.2.c.l 2
5.b even 2 1 inner 1350.2.c.a 2
5.c odd 4 1 270.2.a.d yes 1
5.c odd 4 1 1350.2.a.c 1
15.d odd 2 1 1350.2.c.l 2
15.e even 4 1 270.2.a.a 1
15.e even 4 1 1350.2.a.p 1
20.e even 4 1 2160.2.a.p 1
40.i odd 4 1 8640.2.a.z 1
40.k even 4 1 8640.2.a.f 1
45.k odd 12 2 810.2.e.a 2
45.l even 12 2 810.2.e.k 2
60.l odd 4 1 2160.2.a.a 1
120.q odd 4 1 8640.2.a.bo 1
120.w even 4 1 8640.2.a.by 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.a.a 1 15.e even 4 1
270.2.a.d yes 1 5.c odd 4 1
810.2.e.a 2 45.k odd 12 2
810.2.e.k 2 45.l even 12 2
1350.2.a.c 1 5.c odd 4 1
1350.2.a.p 1 15.e even 4 1
1350.2.c.a 2 1.a even 1 1 trivial
1350.2.c.a 2 5.b even 2 1 inner
1350.2.c.l 2 3.b odd 2 1
1350.2.c.l 2 15.d odd 2 1
2160.2.a.a 1 60.l odd 4 1
2160.2.a.p 1 20.e even 4 1
8640.2.a.f 1 40.k even 4 1
8640.2.a.z 1 40.i odd 4 1
8640.2.a.bo 1 120.q odd 4 1
8640.2.a.by 1 120.w even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 4$$ $$T_{11} + 3$$ $$T_{13}^{2} + 1$$ $$T_{17}^{2} + 9$$ $$T_{29} - 9$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ 
$5$ 
$7$ $$1 - 10 T^{2} + 49 T^{4}$$
$11$ $$( 1 + 3 T + 11 T^{2} )^{2}$$
$13$ $$1 - 25 T^{2} + 169 T^{4}$$
$17$ $$1 - 25 T^{2} + 289 T^{4}$$
$19$ $$( 1 + 8 T + 19 T^{2} )^{2}$$
$23$ $$1 - 37 T^{2} + 529 T^{4}$$
$29$ $$( 1 - 9 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 7 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$
$41$ $$( 1 + 12 T + 41 T^{2} )^{2}$$
$43$ $$1 - 37 T^{2} + 1849 T^{4}$$
$47$ $$1 - 85 T^{2} + 2209 T^{4}$$
$53$ $$1 + 38 T^{2} + 2809 T^{4}$$
$59$ $$( 1 + 12 T + 59 T^{2} )^{2}$$
$61$ $$( 1 + 10 T + 61 T^{2} )^{2}$$
$67$ $$1 - 118 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$1 - 142 T^{2} + 5329 T^{4}$$
$79$ $$( 1 - T + 79 T^{2} )^{2}$$
$83$ $$1 + 158 T^{2} + 6889 T^{4}$$
$89$ $$( 1 + 89 T^{2} )^{2}$$
$97$ $$1 + 2 T^{2} + 9409 T^{4}$$