# Properties

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

## Newform invariants

 Self dual: no Analytic conductor: $$10.7798042729$$ Analytic rank: $$1$$ 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 450) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -4 + 4 \zeta_{6} ) q^{7} - q^{8} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -4 + 4 \zeta_{6} ) q^{7} - q^{8} + ( 3 - 3 \zeta_{6} ) q^{11} -4 \zeta_{6} q^{13} + 4 \zeta_{6} q^{14} + ( -1 + \zeta_{6} ) q^{16} -3 q^{17} -4 q^{19} -3 \zeta_{6} q^{22} + 6 \zeta_{6} q^{23} -4 q^{26} + 4 q^{28} + ( -6 + 6 \zeta_{6} ) q^{29} -8 \zeta_{6} q^{31} + \zeta_{6} q^{32} + ( -3 + 3 \zeta_{6} ) q^{34} -8 q^{37} + ( -4 + 4 \zeta_{6} ) q^{38} -6 \zeta_{6} q^{41} + ( -1 + \zeta_{6} ) q^{43} -3 q^{44} + 6 q^{46} + ( -12 + 12 \zeta_{6} ) q^{47} -9 \zeta_{6} q^{49} + ( -4 + 4 \zeta_{6} ) q^{52} + ( 4 - 4 \zeta_{6} ) q^{56} + 6 \zeta_{6} q^{58} -9 \zeta_{6} q^{59} + ( -8 + 8 \zeta_{6} ) q^{61} -8 q^{62} + q^{64} -4 \zeta_{6} q^{67} + 3 \zeta_{6} q^{68} + 6 q^{71} -14 q^{73} + ( -8 + 8 \zeta_{6} ) q^{74} + 4 \zeta_{6} q^{76} + 12 \zeta_{6} q^{77} + ( -8 + 8 \zeta_{6} ) q^{79} -6 q^{82} + ( 9 - 9 \zeta_{6} ) q^{83} + \zeta_{6} q^{86} + ( -3 + 3 \zeta_{6} ) q^{88} + 9 q^{89} + 16 q^{91} + ( 6 - 6 \zeta_{6} ) q^{92} + 12 \zeta_{6} q^{94} + ( -7 + 7 \zeta_{6} ) q^{97} -9 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{4} - 4q^{7} - 2q^{8} + O(q^{10})$$ $$2q + q^{2} - q^{4} - 4q^{7} - 2q^{8} + 3q^{11} - 4q^{13} + 4q^{14} - q^{16} - 6q^{17} - 8q^{19} - 3q^{22} + 6q^{23} - 8q^{26} + 8q^{28} - 6q^{29} - 8q^{31} + q^{32} - 3q^{34} - 16q^{37} - 4q^{38} - 6q^{41} - q^{43} - 6q^{44} + 12q^{46} - 12q^{47} - 9q^{49} - 4q^{52} + 4q^{56} + 6q^{58} - 9q^{59} - 8q^{61} - 16q^{62} + 2q^{64} - 4q^{67} + 3q^{68} + 12q^{71} - 28q^{73} - 8q^{74} + 4q^{76} + 12q^{77} - 8q^{79} - 12q^{82} + 9q^{83} + q^{86} - 3q^{88} + 18q^{89} + 32q^{91} + 6q^{92} + 12q^{94} - 7q^{97} - 18q^{98} + O(q^{100})$$

## 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 −2.00000 + 3.46410i −1.00000 0 0
901.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 −2.00000 3.46410i −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.f 2
3.b odd 2 1 450.2.e.a 2
5.b even 2 1 1350.2.e.e 2
5.c odd 4 2 1350.2.j.d 4
9.c even 3 1 inner 1350.2.e.f 2
9.c even 3 1 4050.2.a.p 1
9.d odd 6 1 450.2.e.a 2
9.d odd 6 1 4050.2.a.bj 1
15.d odd 2 1 450.2.e.h yes 2
15.e even 4 2 450.2.j.d 4
45.h odd 6 1 450.2.e.h yes 2
45.h odd 6 1 4050.2.a.b 1
45.j even 6 1 1350.2.e.e 2
45.j even 6 1 4050.2.a.t 1
45.k odd 12 2 1350.2.j.d 4
45.k odd 12 2 4050.2.c.e 2
45.l even 12 2 450.2.j.d 4
45.l even 12 2 4050.2.c.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.e.a 2 3.b odd 2 1
450.2.e.a 2 9.d odd 6 1
450.2.e.h yes 2 15.d odd 2 1
450.2.e.h yes 2 45.h odd 6 1
450.2.j.d 4 15.e even 4 2
450.2.j.d 4 45.l even 12 2
1350.2.e.e 2 5.b even 2 1
1350.2.e.e 2 45.j even 6 1
1350.2.e.f 2 1.a even 1 1 trivial
1350.2.e.f 2 9.c even 3 1 inner
1350.2.j.d 4 5.c odd 4 2
1350.2.j.d 4 45.k odd 12 2
4050.2.a.b 1 45.h odd 6 1
4050.2.a.p 1 9.c even 3 1
4050.2.a.t 1 45.j even 6 1
4050.2.a.bj 1 9.d odd 6 1
4050.2.c.e 2 45.k odd 12 2
4050.2.c.q 2 45.l even 12 2

## Hecke kernels

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

 $$T_{7}^{2} + 4 T_{7} + 16$$ $$T_{11}^{2} - 3 T_{11} + 9$$ $$T_{17} + 3$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ 
$5$ 
$7$ $$( 1 - T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )$$
$11$ $$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$
$13$ $$1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4}$$
$17$ $$( 1 + 3 T + 17 T^{2} )^{2}$$
$19$ $$( 1 + 4 T + 19 T^{2} )^{2}$$
$23$ $$1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4}$$
$29$ $$1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4}$$
$31$ $$1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4}$$
$37$ $$( 1 + 8 T + 37 T^{2} )^{2}$$
$41$ $$1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4}$$
$43$ $$1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4}$$
$47$ $$1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4}$$
$53$ $$( 1 + 53 T^{2} )^{2}$$
$59$ $$1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4}$$
$61$ $$1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4}$$
$67$ $$1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 6 T + 71 T^{2} )^{2}$$
$73$ $$( 1 + 14 T + 73 T^{2} )^{2}$$
$79$ $$1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4}$$
$83$ $$1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4}$$
$89$ $$( 1 - 9 T + 89 T^{2} )^{2}$$
$97$ $$1 + 7 T - 48 T^{2} + 679 T^{3} + 9409 T^{4}$$