# Properties

 Label 1350.2.j.e Level 1350 Weight 2 Character orbit 1350.j Analytic conductor 10.780 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$1350 = 2 \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1350.j (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.7798042729$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 90) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10})$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + ( 6 - 6 \zeta_{12}^{2} ) q^{11} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{13} + \zeta_{12}^{2} q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + 4 q^{19} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{22} + ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{23} -2 q^{26} + \zeta_{12}^{3} q^{28} + ( -3 + 3 \zeta_{12}^{2} ) q^{29} + 4 \zeta_{12}^{2} q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} + 8 \zeta_{12}^{3} q^{37} + 4 \zeta_{12} q^{38} -3 \zeta_{12}^{2} q^{41} + 8 \zeta_{12} q^{43} + 6 q^{44} + 9 q^{46} -3 \zeta_{12} q^{47} -6 \zeta_{12}^{2} q^{49} -2 \zeta_{12} q^{52} + 6 \zeta_{12}^{3} q^{53} + ( -1 + \zeta_{12}^{2} ) q^{56} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{58} -6 \zeta_{12}^{2} q^{59} + ( 13 - 13 \zeta_{12}^{2} ) q^{61} + 4 \zeta_{12}^{3} q^{62} - q^{64} + ( -13 \zeta_{12} + 13 \zeta_{12}^{3} ) q^{67} + 6 q^{71} + 4 \zeta_{12}^{3} q^{73} + ( -8 + 8 \zeta_{12}^{2} ) q^{74} + 4 \zeta_{12}^{2} q^{76} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{77} + ( -10 + 10 \zeta_{12}^{2} ) q^{79} -3 \zeta_{12}^{3} q^{82} + 9 \zeta_{12} q^{83} + 8 \zeta_{12}^{2} q^{86} + 6 \zeta_{12} q^{88} + 9 q^{89} -2 q^{91} + 9 \zeta_{12} q^{92} -3 \zeta_{12}^{2} q^{94} -2 \zeta_{12} q^{97} -6 \zeta_{12}^{3} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} + O(q^{10})$$ $$4q + 2q^{4} + 12q^{11} + 2q^{14} - 2q^{16} + 16q^{19} - 8q^{26} - 6q^{29} + 8q^{31} - 6q^{41} + 24q^{44} + 36q^{46} - 12q^{49} - 2q^{56} - 12q^{59} + 26q^{61} - 4q^{64} + 24q^{71} - 16q^{74} + 8q^{76} - 20q^{79} + 16q^{86} + 36q^{89} - 8q^{91} - 6q^{94} + O(q^{100})$$

## Character values

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

 $$n$$ $$1001$$ $$1027$$ $$\chi(n)$$ $$-\zeta_{12}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 −0.866025 + 0.500000i 1.00000i 0 0
199.2 0.866025 0.500000i 0 0.500000 0.866025i 0 0 0.866025 0.500000i 1.00000i 0 0
1099.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 −0.866025 0.500000i 1.00000i 0 0
1099.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 0.866025 + 0.500000i 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
9.c even 3 1 inner
45.j even 6 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{7}^{4} - T_{7}^{2} + 1$$ $$T_{11}^{2} - 6 T_{11} + 36$$ $$T_{19} - 4$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ 
$5$ 
$7$ $$( 1 + 2 T^{2} + 49 T^{4} )( 1 + 11 T^{2} + 49 T^{4} )$$
$11$ $$( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 - T^{2} + 169 T^{4} )( 1 + 23 T^{2} + 169 T^{4} )$$
$17$ $$( 1 - 17 T^{2} )^{4}$$
$19$ $$( 1 - 4 T + 19 T^{2} )^{4}$$
$23$ $$1 - 35 T^{2} + 696 T^{4} - 18515 T^{6} + 279841 T^{8}$$
$29$ $$( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 10 T^{2} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 61 T^{2} + 1849 T^{4} )( 1 + 83 T^{2} + 1849 T^{4} )$$
$47$ $$1 + 85 T^{2} + 5016 T^{4} + 187765 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 - 70 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 14 T + 61 T^{2} )^{2}( 1 + T + 61 T^{2} )^{2}$$
$67$ $$1 - 35 T^{2} - 3264 T^{4} - 157115 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 - 6 T + 71 T^{2} )^{4}$$
$73$ $$( 1 - 130 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$1 + 85 T^{2} + 336 T^{4} + 585565 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 - 9 T + 89 T^{2} )^{4}$$
$97$ $$1 + 190 T^{2} + 26691 T^{4} + 1787710 T^{6} + 88529281 T^{8}$$