# Properties

 Label 1350.2.e.a Level 1350 Weight 2 Character orbit 1350.e 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.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.7798042729$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 90) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -1 + \zeta_{6} ) q^{7} + q^{8} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -1 + \zeta_{6} ) q^{7} + q^{8} + ( -2 + 2 \zeta_{6} ) q^{11} -6 \zeta_{6} q^{13} -\zeta_{6} q^{14} + ( -1 + \zeta_{6} ) q^{16} -2 q^{17} + 6 q^{19} -2 \zeta_{6} q^{22} -\zeta_{6} q^{23} + 6 q^{26} + q^{28} + ( 9 - 9 \zeta_{6} ) q^{29} + 2 \zeta_{6} q^{31} -\zeta_{6} q^{32} + ( 2 - 2 \zeta_{6} ) q^{34} -2 q^{37} + ( -6 + 6 \zeta_{6} ) q^{38} -11 \zeta_{6} q^{41} + ( -4 + 4 \zeta_{6} ) q^{43} + 2 q^{44} + q^{46} + ( 7 - 7 \zeta_{6} ) q^{47} + 6 \zeta_{6} q^{49} + ( -6 + 6 \zeta_{6} ) q^{52} + ( -1 + \zeta_{6} ) q^{56} + 9 \zeta_{6} q^{58} -4 \zeta_{6} q^{59} + ( 7 - 7 \zeta_{6} ) q^{61} -2 q^{62} + q^{64} -11 \zeta_{6} q^{67} + 2 \zeta_{6} q^{68} + 6 q^{71} + 4 q^{73} + ( 2 - 2 \zeta_{6} ) q^{74} -6 \zeta_{6} q^{76} -2 \zeta_{6} q^{77} + ( 12 - 12 \zeta_{6} ) q^{79} + 11 q^{82} + ( 11 - 11 \zeta_{6} ) q^{83} -4 \zeta_{6} q^{86} + ( -2 + 2 \zeta_{6} ) q^{88} - q^{89} + 6 q^{91} + ( -1 + \zeta_{6} ) q^{92} + 7 \zeta_{6} q^{94} + ( -8 + 8 \zeta_{6} ) q^{97} -6 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} - q^{7} + 2q^{8} + O(q^{10})$$ $$2q - q^{2} - q^{4} - q^{7} + 2q^{8} - 2q^{11} - 6q^{13} - q^{14} - q^{16} - 4q^{17} + 12q^{19} - 2q^{22} - q^{23} + 12q^{26} + 2q^{28} + 9q^{29} + 2q^{31} - q^{32} + 2q^{34} - 4q^{37} - 6q^{38} - 11q^{41} - 4q^{43} + 4q^{44} + 2q^{46} + 7q^{47} + 6q^{49} - 6q^{52} - q^{56} + 9q^{58} - 4q^{59} + 7q^{61} - 4q^{62} + 2q^{64} - 11q^{67} + 2q^{68} + 12q^{71} + 8q^{73} + 2q^{74} - 6q^{76} - 2q^{77} + 12q^{79} + 22q^{82} + 11q^{83} - 4q^{86} - 2q^{88} - 2q^{89} + 12q^{91} - q^{92} + 7q^{94} - 8q^{97} - 12q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$1001$$ $$1027$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 −0.500000 + 0.866025i 1.00000 0 0
901.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −0.500000 0.866025i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.2.e.a 2
3.b odd 2 1 450.2.e.g 2
5.b even 2 1 1350.2.e.i 2
5.c odd 4 2 270.2.i.a 4
9.c even 3 1 inner 1350.2.e.a 2
9.c even 3 1 4050.2.a.be 1
9.d odd 6 1 450.2.e.g 2
9.d odd 6 1 4050.2.a.j 1
15.d odd 2 1 450.2.e.b 2
15.e even 4 2 90.2.i.a 4
20.e even 4 2 2160.2.by.b 4
45.h odd 6 1 450.2.e.b 2
45.h odd 6 1 4050.2.a.x 1
45.j even 6 1 1350.2.e.i 2
45.j even 6 1 4050.2.a.g 1
45.k odd 12 2 270.2.i.a 4
45.k odd 12 2 810.2.c.c 2
45.l even 12 2 90.2.i.a 4
45.l even 12 2 810.2.c.b 2
60.l odd 4 2 720.2.by.a 4
180.v odd 12 2 720.2.by.a 4
180.x even 12 2 2160.2.by.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.i.a 4 15.e even 4 2
90.2.i.a 4 45.l even 12 2
270.2.i.a 4 5.c odd 4 2
270.2.i.a 4 45.k odd 12 2
450.2.e.b 2 15.d odd 2 1
450.2.e.b 2 45.h odd 6 1
450.2.e.g 2 3.b odd 2 1
450.2.e.g 2 9.d odd 6 1
720.2.by.a 4 60.l odd 4 2
720.2.by.a 4 180.v odd 12 2
810.2.c.b 2 45.l even 12 2
810.2.c.c 2 45.k odd 12 2
1350.2.e.a 2 1.a even 1 1 trivial
1350.2.e.a 2 9.c even 3 1 inner
1350.2.e.i 2 5.b even 2 1
1350.2.e.i 2 45.j even 6 1
2160.2.by.b 4 20.e even 4 2
2160.2.by.b 4 180.x even 12 2
4050.2.a.g 1 45.j even 6 1
4050.2.a.j 1 9.d odd 6 1
4050.2.a.x 1 45.h odd 6 1
4050.2.a.be 1 9.c even 3 1

## Hecke kernels

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

 $$T_{7}^{2} + T_{7} + 1$$ $$T_{11}^{2} + 2 T_{11} + 4$$ $$T_{17} + 2$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ 
$5$ 
$7$ $$( 1 - 4 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )$$
$11$ $$1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4}$$
$13$ $$1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4}$$
$17$ $$( 1 + 2 T + 17 T^{2} )^{2}$$
$19$ $$( 1 - 6 T + 19 T^{2} )^{2}$$
$23$ $$1 + T - 22 T^{2} + 23 T^{3} + 529 T^{4}$$
$29$ $$1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4}$$
$31$ $$1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4}$$
$37$ $$( 1 + 2 T + 37 T^{2} )^{2}$$
$41$ $$1 + 11 T + 80 T^{2} + 451 T^{3} + 1681 T^{4}$$
$43$ $$1 + 4 T - 27 T^{2} + 172 T^{3} + 1849 T^{4}$$
$47$ $$1 - 7 T + 2 T^{2} - 329 T^{3} + 2209 T^{4}$$
$53$ $$( 1 + 53 T^{2} )^{2}$$
$59$ $$1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4}$$
$61$ $$1 - 7 T - 12 T^{2} - 427 T^{3} + 3721 T^{4}$$
$67$ $$( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$
$71$ $$( 1 - 6 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 4 T + 73 T^{2} )^{2}$$
$79$ $$1 - 12 T + 65 T^{2} - 948 T^{3} + 6241 T^{4}$$
$83$ $$1 - 11 T + 38 T^{2} - 913 T^{3} + 6889 T^{4}$$
$89$ $$( 1 + T + 89 T^{2} )^{2}$$
$97$ $$1 + 8 T - 33 T^{2} + 776 T^{3} + 9409 T^{4}$$