# Properties

 Label 1350.2.e.b Level 1350 Weight 2 Character orbit 1350.e Analytic conductor 10.780 Analytic rank 0 Dimension 2 CM no Inner twists 2

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## 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -1 + \zeta_{6} ) q^{7} + q^{8} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -1 + \zeta_{6} ) q^{7} + q^{8} + ( 6 - 6 \zeta_{6} ) q^{11} + 2 \zeta_{6} q^{13} -\zeta_{6} q^{14} + ( -1 + \zeta_{6} ) q^{16} -4 q^{19} + 6 \zeta_{6} q^{22} -9 \zeta_{6} q^{23} -2 q^{26} + q^{28} + ( 3 - 3 \zeta_{6} ) q^{29} + 4 \zeta_{6} q^{31} -\zeta_{6} q^{32} -8 q^{37} + ( 4 - 4 \zeta_{6} ) q^{38} -3 \zeta_{6} q^{41} + ( 8 - 8 \zeta_{6} ) q^{43} -6 q^{44} + 9 q^{46} + ( 3 - 3 \zeta_{6} ) q^{47} + 6 \zeta_{6} q^{49} + ( 2 - 2 \zeta_{6} ) q^{52} + 6 q^{53} + ( -1 + \zeta_{6} ) q^{56} + 3 \zeta_{6} q^{58} + 6 \zeta_{6} q^{59} + ( 13 - 13 \zeta_{6} ) q^{61} -4 q^{62} + q^{64} -13 \zeta_{6} q^{67} + 6 q^{71} + 4 q^{73} + ( 8 - 8 \zeta_{6} ) q^{74} + 4 \zeta_{6} q^{76} + 6 \zeta_{6} q^{77} + ( 10 - 10 \zeta_{6} ) q^{79} + 3 q^{82} + ( 9 - 9 \zeta_{6} ) q^{83} + 8 \zeta_{6} q^{86} + ( 6 - 6 \zeta_{6} ) q^{88} -9 q^{89} -2 q^{91} + ( -9 + 9 \zeta_{6} ) q^{92} + 3 \zeta_{6} q^{94} + ( 2 - 2 \zeta_{6} ) q^{97} -6 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} - q^{7} + 2q^{8} + O(q^{10})$$ $$2q - q^{2} - q^{4} - q^{7} + 2q^{8} + 6q^{11} + 2q^{13} - q^{14} - q^{16} - 8q^{19} + 6q^{22} - 9q^{23} - 4q^{26} + 2q^{28} + 3q^{29} + 4q^{31} - q^{32} - 16q^{37} + 4q^{38} - 3q^{41} + 8q^{43} - 12q^{44} + 18q^{46} + 3q^{47} + 6q^{49} + 2q^{52} + 12q^{53} - q^{56} + 3q^{58} + 6q^{59} + 13q^{61} - 8q^{62} + 2q^{64} - 13q^{67} + 12q^{71} + 8q^{73} + 8q^{74} + 4q^{76} + 6q^{77} + 10q^{79} + 6q^{82} + 9q^{83} + 8q^{86} + 6q^{88} - 18q^{89} - 4q^{91} - 9q^{92} + 3q^{94} + 2q^{97} - 12q^{98} + O(q^{100})$$

## 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.b 2
3.b odd 2 1 450.2.e.e 2
5.b even 2 1 270.2.e.b 2
5.c odd 4 2 1350.2.j.e 4
9.c even 3 1 inner 1350.2.e.b 2
9.c even 3 1 4050.2.a.ba 1
9.d odd 6 1 450.2.e.e 2
9.d odd 6 1 4050.2.a.n 1
15.d odd 2 1 90.2.e.a 2
15.e even 4 2 450.2.j.c 4
20.d odd 2 1 2160.2.q.b 2
45.h odd 6 1 90.2.e.a 2
45.h odd 6 1 810.2.a.g 1
45.j even 6 1 270.2.e.b 2
45.j even 6 1 810.2.a.b 1
45.k odd 12 2 1350.2.j.e 4
45.k odd 12 2 4050.2.c.a 2
45.l even 12 2 450.2.j.c 4
45.l even 12 2 4050.2.c.t 2
60.h even 2 1 720.2.q.b 2
180.n even 6 1 720.2.q.b 2
180.n even 6 1 6480.2.a.g 1
180.p odd 6 1 2160.2.q.b 2
180.p odd 6 1 6480.2.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.e.a 2 15.d odd 2 1
90.2.e.a 2 45.h odd 6 1
270.2.e.b 2 5.b even 2 1
270.2.e.b 2 45.j even 6 1
450.2.e.e 2 3.b odd 2 1
450.2.e.e 2 9.d odd 6 1
450.2.j.c 4 15.e even 4 2
450.2.j.c 4 45.l even 12 2
720.2.q.b 2 60.h even 2 1
720.2.q.b 2 180.n even 6 1
810.2.a.b 1 45.j even 6 1
810.2.a.g 1 45.h odd 6 1
1350.2.e.b 2 1.a even 1 1 trivial
1350.2.e.b 2 9.c even 3 1 inner
1350.2.j.e 4 5.c odd 4 2
1350.2.j.e 4 45.k odd 12 2
2160.2.q.b 2 20.d odd 2 1
2160.2.q.b 2 180.p odd 6 1
4050.2.a.n 1 9.d odd 6 1
4050.2.a.ba 1 9.c even 3 1
4050.2.c.a 2 45.k odd 12 2
4050.2.c.t 2 45.l even 12 2
6480.2.a.g 1 180.n even 6 1
6480.2.a.v 1 180.p odd 6 1

## Hecke kernels

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

 $$T_{7}^{2} + T_{7} + 1$$ $$T_{11}^{2} - 6 T_{11} + 36$$ $$T_{17}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ 
$5$ 
$7$ $$( 1 - 4 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )$$
$11$ $$1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 7 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )$$
$17$ $$( 1 + 17 T^{2} )^{2}$$
$19$ $$( 1 + 4 T + 19 T^{2} )^{2}$$
$23$ $$1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4}$$
$29$ $$1 - 3 T - 20 T^{2} - 87 T^{3} + 841 T^{4}$$
$31$ $$( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$
$37$ $$( 1 + 8 T + 37 T^{2} )^{2}$$
$41$ $$1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4}$$
$43$ $$( 1 - 13 T + 43 T^{2} )( 1 + 5 T + 43 T^{2} )$$
$47$ $$1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4}$$
$53$ $$( 1 - 6 T + 53 T^{2} )^{2}$$
$59$ $$1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4}$$
$61$ $$( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} )$$
$67$ $$1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 6 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 4 T + 73 T^{2} )^{2}$$
$79$ $$1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4}$$
$83$ $$1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4}$$
$89$ $$( 1 + 9 T + 89 T^{2} )^{2}$$
$97$ $$1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4}$$
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