# Properties

 Label 1350.2.e.m Level 1350 Weight 2 Character orbit 1350.e Analytic conductor 10.780 Analytic rank 0 Dimension 4 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{2} -\beta_{1} q^{4} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} - q^{8} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{2} -\beta_{1} q^{4} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} - q^{8} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{11} + \beta_{2} q^{13} + ( -2 \beta_{1} + \beta_{2} ) q^{14} + ( -1 + \beta_{1} ) q^{16} + ( -1 - 2 \beta_{3} ) q^{17} + ( -3 - \beta_{3} ) q^{19} + ( \beta_{1} + \beta_{2} ) q^{22} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{23} + \beta_{3} q^{26} + ( -2 + \beta_{3} ) q^{28} + ( -6 + 6 \beta_{1} ) q^{29} + ( -4 \beta_{1} + \beta_{2} ) q^{31} + \beta_{1} q^{32} + ( -1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{34} -8 q^{37} + ( -3 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{38} -\beta_{1} q^{41} + ( 5 - 5 \beta_{1} + \beta_{2} - \beta_{3} ) q^{43} + ( 1 + \beta_{3} ) q^{44} + ( -2 - 2 \beta_{3} ) q^{46} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{47} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{49} + ( -\beta_{2} + \beta_{3} ) q^{52} + ( 6 - \beta_{3} ) q^{53} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{56} + 6 \beta_{1} q^{58} + ( \beta_{1} + 5 \beta_{2} ) q^{59} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{61} + ( -4 + \beta_{3} ) q^{62} + q^{64} + ( 7 \beta_{1} - \beta_{2} ) q^{67} + ( \beta_{1} + 2 \beta_{2} ) q^{68} + \beta_{3} q^{71} + ( -5 - 4 \beta_{3} ) q^{73} + ( -8 + 8 \beta_{1} ) q^{74} + ( 3 \beta_{1} + \beta_{2} ) q^{76} + ( -4 \beta_{1} + \beta_{2} ) q^{77} + ( 3 \beta_{2} - 3 \beta_{3} ) q^{79} - q^{82} + ( 4 - 4 \beta_{1} ) q^{83} + ( -5 \beta_{1} + \beta_{2} ) q^{86} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{88} + ( -8 + 2 \beta_{3} ) q^{89} + ( -6 + 2 \beta_{3} ) q^{91} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{92} + ( -2 \beta_{1} - \beta_{2} ) q^{94} + ( 13 - 13 \beta_{1} ) q^{97} + ( -3 + 4 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} - 2q^{4} + 4q^{7} - 4q^{8} + O(q^{10})$$ $$4q + 2q^{2} - 2q^{4} + 4q^{7} - 4q^{8} - 2q^{11} - 4q^{14} - 2q^{16} - 4q^{17} - 12q^{19} + 2q^{22} - 4q^{23} - 8q^{28} - 12q^{29} - 8q^{31} + 2q^{32} - 2q^{34} - 32q^{37} - 6q^{38} - 2q^{41} + 10q^{43} + 4q^{44} - 8q^{46} + 4q^{47} - 6q^{49} + 24q^{53} - 4q^{56} + 12q^{58} + 2q^{59} - 4q^{61} - 16q^{62} + 4q^{64} + 14q^{67} + 2q^{68} - 20q^{73} - 16q^{74} + 6q^{76} - 8q^{77} - 4q^{82} + 8q^{83} - 10q^{86} + 2q^{88} - 32q^{89} - 24q^{91} - 4q^{92} - 4q^{94} + 26q^{97} - 12q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 4 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{3} + 4 \beta_{2}$$$$)/3$$

## Character values

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

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

## 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
 1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i
0.500000 0.866025i 0 −0.500000 0.866025i 0 0 −0.224745 + 0.389270i −1.00000 0 0
451.2 0.500000 0.866025i 0 −0.500000 0.866025i 0 0 2.22474 3.85337i −1.00000 0 0
901.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 −0.224745 0.389270i −1.00000 0 0
901.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 2.22474 + 3.85337i −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.m 4
3.b odd 2 1 450.2.e.k 4
5.b even 2 1 1350.2.e.j 4
5.c odd 4 2 270.2.i.b 8
9.c even 3 1 inner 1350.2.e.m 4
9.c even 3 1 4050.2.a.bm 2
9.d odd 6 1 450.2.e.k 4
9.d odd 6 1 4050.2.a.bs 2
15.d odd 2 1 450.2.e.n 4
15.e even 4 2 90.2.i.b 8
20.e even 4 2 2160.2.by.d 8
45.h odd 6 1 450.2.e.n 4
45.h odd 6 1 4050.2.a.bq 2
45.j even 6 1 1350.2.e.j 4
45.j even 6 1 4050.2.a.bz 2
45.k odd 12 2 270.2.i.b 8
45.k odd 12 2 810.2.c.e 4
45.l even 12 2 90.2.i.b 8
45.l even 12 2 810.2.c.f 4
60.l odd 4 2 720.2.by.c 8
180.v odd 12 2 720.2.by.c 8
180.x even 12 2 2160.2.by.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.i.b 8 15.e even 4 2
90.2.i.b 8 45.l even 12 2
270.2.i.b 8 5.c odd 4 2
270.2.i.b 8 45.k odd 12 2
450.2.e.k 4 3.b odd 2 1
450.2.e.k 4 9.d odd 6 1
450.2.e.n 4 15.d odd 2 1
450.2.e.n 4 45.h odd 6 1
720.2.by.c 8 60.l odd 4 2
720.2.by.c 8 180.v odd 12 2
810.2.c.e 4 45.k odd 12 2
810.2.c.f 4 45.l even 12 2
1350.2.e.j 4 5.b even 2 1
1350.2.e.j 4 45.j even 6 1
1350.2.e.m 4 1.a even 1 1 trivial
1350.2.e.m 4 9.c even 3 1 inner
2160.2.by.d 8 20.e even 4 2
2160.2.by.d 8 180.x even 12 2
4050.2.a.bm 2 9.c even 3 1
4050.2.a.bq 2 45.h odd 6 1
4050.2.a.bs 2 9.d odd 6 1
4050.2.a.bz 2 45.j even 6 1

## Hecke kernels

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

 $$T_{7}^{4} - 4 T_{7}^{3} + 18 T_{7}^{2} + 8 T_{7} + 4$$ $$T_{11}^{4} + 2 T_{11}^{3} + 9 T_{11}^{2} - 10 T_{11} + 25$$ $$T_{17}^{2} + 2 T_{17} - 23$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ 
$5$ 
$7$ $$1 - 4 T + 4 T^{2} + 8 T^{3} - 17 T^{4} + 56 T^{5} + 196 T^{6} - 1372 T^{7} + 2401 T^{8}$$
$11$ $$1 + 2 T - 13 T^{2} - 10 T^{3} + 124 T^{4} - 110 T^{5} - 1573 T^{6} + 2662 T^{7} + 14641 T^{8}$$
$13$ $$1 - 20 T^{2} + 231 T^{4} - 3380 T^{6} + 28561 T^{8}$$
$17$ $$( 1 + 2 T + 11 T^{2} + 34 T^{3} + 289 T^{4} )^{2}$$
$19$ $$( 1 + 6 T + 41 T^{2} + 114 T^{3} + 361 T^{4} )^{2}$$
$23$ $$1 + 4 T - 10 T^{2} - 80 T^{3} - 221 T^{4} - 1840 T^{5} - 5290 T^{6} + 48668 T^{7} + 279841 T^{8}$$
$29$ $$( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} )^{2}$$
$31$ $$1 + 8 T - 8 T^{2} + 80 T^{3} + 2239 T^{4} + 2480 T^{5} - 7688 T^{6} + 238328 T^{7} + 923521 T^{8}$$
$37$ $$( 1 + 8 T + 37 T^{2} )^{4}$$
$41$ $$( 1 + T - 40 T^{2} + 41 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 - 10 T - 5 T^{2} - 190 T^{3} + 4876 T^{4} - 8170 T^{5} - 9245 T^{6} - 795070 T^{7} + 3418801 T^{8}$$
$47$ $$1 - 4 T - 76 T^{2} + 8 T^{3} + 5503 T^{4} + 376 T^{5} - 167884 T^{6} - 415292 T^{7} + 4879681 T^{8}$$
$53$ $$( 1 - 12 T + 136 T^{2} - 636 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$1 - 2 T + 35 T^{2} + 298 T^{3} - 2756 T^{4} + 17582 T^{5} + 121835 T^{6} - 410758 T^{7} + 12117361 T^{8}$$
$61$ $$1 + 4 T - 104 T^{2} - 8 T^{3} + 9703 T^{4} - 488 T^{5} - 386984 T^{6} + 907924 T^{7} + 13845841 T^{8}$$
$67$ $$1 - 14 T + 19 T^{2} - 602 T^{3} + 13708 T^{4} - 40334 T^{5} + 85291 T^{6} - 4210682 T^{7} + 20151121 T^{8}$$
$71$ $$( 1 + 136 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + 10 T + 75 T^{2} + 730 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$1 - 104 T^{2} + 4575 T^{4} - 649064 T^{6} + 38950081 T^{8}$$
$83$ $$( 1 - 4 T - 67 T^{2} - 332 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 16 T + 218 T^{2} + 1424 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 13 T + 72 T^{2} - 1261 T^{3} + 9409 T^{4} )^{2}$$