# Properties

 Label 1350.2.j.d 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 450) 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + 4 \zeta_{12} q^{7} -\zeta_{12}^{3} q^{8} +O(q^{10})$$ $$q -\zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + 4 \zeta_{12} q^{7} -\zeta_{12}^{3} q^{8} + ( 3 - 3 \zeta_{12}^{2} ) q^{11} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{13} -4 \zeta_{12}^{2} q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + 3 \zeta_{12}^{3} q^{17} + 4 q^{19} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{22} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{23} -4 q^{26} + 4 \zeta_{12}^{3} q^{28} + ( 6 - 6 \zeta_{12}^{2} ) q^{29} -8 \zeta_{12}^{2} q^{31} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{32} + ( 3 - 3 \zeta_{12}^{2} ) q^{34} + 8 \zeta_{12}^{3} q^{37} -4 \zeta_{12} q^{38} -6 \zeta_{12}^{2} q^{41} -\zeta_{12} q^{43} + 3 q^{44} + 6 q^{46} + 12 \zeta_{12} q^{47} + 9 \zeta_{12}^{2} q^{49} + 4 \zeta_{12} q^{52} + ( 4 - 4 \zeta_{12}^{2} ) q^{56} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{58} + 9 \zeta_{12}^{2} q^{59} + ( -8 + 8 \zeta_{12}^{2} ) q^{61} + 8 \zeta_{12}^{3} q^{62} - q^{64} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{67} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{68} + 6 q^{71} -14 \zeta_{12}^{3} q^{73} + ( 8 - 8 \zeta_{12}^{2} ) q^{74} + 4 \zeta_{12}^{2} q^{76} + ( 12 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{77} + ( 8 - 8 \zeta_{12}^{2} ) q^{79} + 6 \zeta_{12}^{3} q^{82} + 9 \zeta_{12} q^{83} + \zeta_{12}^{2} q^{86} -3 \zeta_{12} q^{88} -9 q^{89} + 16 q^{91} -6 \zeta_{12} q^{92} -12 \zeta_{12}^{2} q^{94} + 7 \zeta_{12} q^{97} -9 \zeta_{12}^{3} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} + O(q^{10})$$ $$4q + 2q^{4} + 6q^{11} - 8q^{14} - 2q^{16} + 16q^{19} - 16q^{26} + 12q^{29} - 16q^{31} + 6q^{34} - 12q^{41} + 12q^{44} + 24q^{46} + 18q^{49} + 8q^{56} + 18q^{59} - 16q^{61} - 4q^{64} + 24q^{71} + 16q^{74} + 8q^{76} + 16q^{79} + 2q^{86} - 36q^{89} + 64q^{91} - 24q^{94} + O(q^{100})$$

## 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 3.46410 2.00000i 1.00000i 0 0
199.2 0.866025 0.500000i 0 0.500000 0.866025i 0 0 −3.46410 + 2.00000i 1.00000i 0 0
1099.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 3.46410 + 2.00000i 1.00000i 0 0
1099.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 −3.46410 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
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.d 4
3.b odd 2 1 450.2.j.d 4
5.b even 2 1 inner 1350.2.j.d 4
5.c odd 4 1 1350.2.e.e 2
5.c odd 4 1 1350.2.e.f 2
9.c even 3 1 inner 1350.2.j.d 4
9.c even 3 1 4050.2.c.e 2
9.d odd 6 1 450.2.j.d 4
9.d odd 6 1 4050.2.c.q 2
15.d odd 2 1 450.2.j.d 4
15.e even 4 1 450.2.e.a 2
15.e even 4 1 450.2.e.h yes 2
45.h odd 6 1 450.2.j.d 4
45.h odd 6 1 4050.2.c.q 2
45.j even 6 1 inner 1350.2.j.d 4
45.j even 6 1 4050.2.c.e 2
45.k odd 12 1 1350.2.e.e 2
45.k odd 12 1 1350.2.e.f 2
45.k odd 12 1 4050.2.a.p 1
45.k odd 12 1 4050.2.a.t 1
45.l even 12 1 450.2.e.a 2
45.l even 12 1 450.2.e.h yes 2
45.l even 12 1 4050.2.a.b 1
45.l even 12 1 4050.2.a.bj 1

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

## Hecke kernels

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

 $$T_{7}^{4} - 16 T_{7}^{2} + 256$$ $$T_{11}^{2} - 3 T_{11} + 9$$ $$T_{19} - 4$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ 
$5$ 
$7$ $$( 1 - 13 T^{2} + 49 T^{4} )( 1 + 11 T^{2} + 49 T^{4} )$$
$11$ $$( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4} )( 1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4} )$$
$17$ $$( 1 - 25 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 4 T + 19 T^{2} )^{4}$$
$23$ $$1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8}$$
$29$ $$( 1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4} )^{2}$$
$31$ $$( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 10 T^{2} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 + 85 T^{2} + 5376 T^{4} + 157165 T^{6} + 3418801 T^{8}$$
$47$ $$1 - 50 T^{2} + 291 T^{4} - 110450 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 - 53 T^{2} )^{4}$$
$59$ $$( 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 + 118 T^{2} + 9435 T^{4} + 529702 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 - 6 T + 71 T^{2} )^{4}$$
$73$ $$( 1 + 50 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$1 + 85 T^{2} + 336 T^{4} + 585565 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 + 9 T + 89 T^{2} )^{4}$$
$97$ $$1 + 145 T^{2} + 11616 T^{4} + 1364305 T^{6} + 88529281 T^{8}$$