# Properties

 Label 1350.2.e.l 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: $$\Q(\sqrt{-3}, \sqrt{-11})$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{1} ) q^{4} + \beta_{3} q^{7} - q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{1} ) q^{4} + \beta_{3} q^{7} - q^{8} + ( -\beta_{1} - \beta_{3} ) q^{11} + ( -2 - 2 \beta_{2} + 2 \beta_{3} ) q^{13} + ( -1 - \beta_{2} + \beta_{3} ) q^{14} -\beta_{1} q^{16} + ( -4 + \beta_{2} ) q^{17} -\beta_{2} q^{19} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{22} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{23} + ( -2 - 2 \beta_{2} ) q^{26} + ( -1 - \beta_{2} ) q^{28} + ( -2 \beta_{1} + \beta_{3} ) q^{29} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{31} + ( 1 - \beta_{1} ) q^{32} + ( -5 \beta_{1} + \beta_{3} ) q^{34} + 4 q^{37} + ( \beta_{1} - \beta_{3} ) q^{38} + ( -3 + 3 \beta_{1} ) q^{41} + ( -9 \beta_{1} + \beta_{3} ) q^{43} + ( 2 + \beta_{2} ) q^{44} + ( 1 - \beta_{2} ) q^{46} + ( 4 \beta_{1} + \beta_{3} ) q^{47} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{49} -2 \beta_{3} q^{52} + ( -2 - 4 \beta_{2} ) q^{53} -\beta_{3} q^{56} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{58} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{59} + ( -2 \beta_{1} + 3 \beta_{3} ) q^{61} -2 \beta_{2} q^{62} + q^{64} + ( -7 + 7 \beta_{1} ) q^{67} + ( 4 - 5 \beta_{1} - \beta_{2} + \beta_{3} ) q^{68} + 6 q^{71} + ( 4 - 3 \beta_{2} ) q^{73} + 4 \beta_{1} q^{74} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{76} + ( 10 - 8 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{77} -2 \beta_{1} q^{79} -3 q^{82} + ( -4 \beta_{1} - \beta_{3} ) q^{83} + ( 8 - 9 \beta_{1} - \beta_{2} + \beta_{3} ) q^{86} + ( \beta_{1} + \beta_{3} ) q^{88} + ( -9 - 3 \beta_{2} ) q^{89} + ( -18 - 2 \beta_{2} ) q^{91} + ( 2 \beta_{1} - \beta_{3} ) q^{92} + ( -5 + 4 \beta_{1} - \beta_{2} + \beta_{3} ) q^{94} + ( -5 \beta_{1} - \beta_{3} ) q^{97} + ( -2 - \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} - 2q^{4} + q^{7} - 4q^{8} + O(q^{10})$$ $$4q + 2q^{2} - 2q^{4} + q^{7} - 4q^{8} - 3q^{11} - 2q^{13} - q^{14} - 2q^{16} - 18q^{17} + 2q^{19} + 3q^{22} + 3q^{23} - 4q^{26} - 2q^{28} - 3q^{29} + 2q^{31} + 2q^{32} - 9q^{34} + 16q^{37} + q^{38} - 6q^{41} - 17q^{43} + 6q^{44} + 6q^{46} + 9q^{47} - 3q^{49} - 2q^{52} - q^{56} + 3q^{58} + 3q^{59} - q^{61} + 4q^{62} + 4q^{64} - 14q^{67} + 9q^{68} + 24q^{71} + 22q^{73} + 8q^{74} - q^{76} + 18q^{77} - 4q^{79} - 12q^{82} - 9q^{83} + 17q^{86} + 3q^{88} - 30q^{89} - 68q^{91} + 3q^{92} - 9q^{94} - 11q^{97} - 6q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 3$$$$)/6$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 5 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{3} + \nu^{2} + 2 \nu - 9$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 2 \beta_{1} + 2$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 1$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 11$$$$)/3$$

## Character values

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

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

## 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
 −1.18614 + 1.26217i 1.68614 − 0.396143i −1.18614 − 1.26217i 1.68614 + 0.396143i
0.500000 0.866025i 0 −0.500000 0.866025i 0 0 −1.18614 + 2.05446i −1.00000 0 0
451.2 0.500000 0.866025i 0 −0.500000 0.866025i 0 0 1.68614 2.92048i −1.00000 0 0
901.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 −1.18614 2.05446i −1.00000 0 0
901.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 1.68614 + 2.92048i −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.l 4
3.b odd 2 1 450.2.e.j 4
5.b even 2 1 270.2.e.c 4
5.c odd 4 2 1350.2.j.f 8
9.c even 3 1 inner 1350.2.e.l 4
9.c even 3 1 4050.2.a.bo 2
9.d odd 6 1 450.2.e.j 4
9.d odd 6 1 4050.2.a.bw 2
15.d odd 2 1 90.2.e.c 4
15.e even 4 2 450.2.j.g 8
20.d odd 2 1 2160.2.q.f 4
45.h odd 6 1 90.2.e.c 4
45.h odd 6 1 810.2.a.i 2
45.j even 6 1 270.2.e.c 4
45.j even 6 1 810.2.a.k 2
45.k odd 12 2 1350.2.j.f 8
45.k odd 12 2 4050.2.c.ba 4
45.l even 12 2 450.2.j.g 8
45.l even 12 2 4050.2.c.v 4
60.h even 2 1 720.2.q.f 4
180.n even 6 1 720.2.q.f 4
180.n even 6 1 6480.2.a.be 2
180.p odd 6 1 2160.2.q.f 4
180.p odd 6 1 6480.2.a.bn 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.e.c 4 15.d odd 2 1
90.2.e.c 4 45.h odd 6 1
270.2.e.c 4 5.b even 2 1
270.2.e.c 4 45.j even 6 1
450.2.e.j 4 3.b odd 2 1
450.2.e.j 4 9.d odd 6 1
450.2.j.g 8 15.e even 4 2
450.2.j.g 8 45.l even 12 2
720.2.q.f 4 60.h even 2 1
720.2.q.f 4 180.n even 6 1
810.2.a.i 2 45.h odd 6 1
810.2.a.k 2 45.j even 6 1
1350.2.e.l 4 1.a even 1 1 trivial
1350.2.e.l 4 9.c even 3 1 inner
1350.2.j.f 8 5.c odd 4 2
1350.2.j.f 8 45.k odd 12 2
2160.2.q.f 4 20.d odd 2 1
2160.2.q.f 4 180.p odd 6 1
4050.2.a.bo 2 9.c even 3 1
4050.2.a.bw 2 9.d odd 6 1
4050.2.c.v 4 45.l even 12 2
4050.2.c.ba 4 45.k odd 12 2
6480.2.a.be 2 180.n even 6 1
6480.2.a.bn 2 180.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 $$S_{2}^{\mathrm{new}}(1350, [\chi])$$:

 $$T_{7}^{4} - T_{7}^{3} + 9 T_{7}^{2} + 8 T_{7} + 64$$ $$T_{11}^{4} + 3 T_{11}^{3} + 15 T_{11}^{2} - 18 T_{11} + 36$$ $$T_{17}^{2} + 9 T_{17} + 12$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ 
$5$ 
$7$ $$1 - T - 5 T^{2} + 8 T^{3} - 20 T^{4} + 56 T^{5} - 245 T^{6} - 343 T^{7} + 2401 T^{8}$$
$11$ $$1 + 3 T - 7 T^{2} - 18 T^{3} + 36 T^{4} - 198 T^{5} - 847 T^{6} + 3993 T^{7} + 14641 T^{8}$$
$13$ $$1 + 2 T + 10 T^{2} - 64 T^{3} - 185 T^{4} - 832 T^{5} + 1690 T^{6} + 4394 T^{7} + 28561 T^{8}$$
$17$ $$( 1 + 9 T + 46 T^{2} + 153 T^{3} + 289 T^{4} )^{2}$$
$19$ $$( 1 - T + 30 T^{2} - 19 T^{3} + 361 T^{4} )^{2}$$
$23$ $$1 - 3 T - 31 T^{2} + 18 T^{3} + 864 T^{4} + 414 T^{5} - 16399 T^{6} - 36501 T^{7} + 279841 T^{8}$$
$29$ $$1 + 3 T - 43 T^{2} - 18 T^{3} + 1602 T^{4} - 522 T^{5} - 36163 T^{6} + 73167 T^{7} + 707281 T^{8}$$
$31$ $$1 - 2 T - 26 T^{2} + 64 T^{3} - 185 T^{4} + 1984 T^{5} - 24986 T^{6} - 59582 T^{7} + 923521 T^{8}$$
$37$ $$( 1 - 4 T + 37 T^{2} )^{4}$$
$41$ $$( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 + 17 T + 139 T^{2} + 1088 T^{3} + 8224 T^{4} + 46784 T^{5} + 257011 T^{6} + 1351619 T^{7} + 3418801 T^{8}$$
$47$ $$1 - 9 T - 25 T^{2} - 108 T^{3} + 5220 T^{4} - 5076 T^{5} - 55225 T^{6} - 934407 T^{7} + 4879681 T^{8}$$
$53$ $$( 1 - 26 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$1 - 3 T - 103 T^{2} + 18 T^{3} + 8532 T^{4} + 1062 T^{5} - 358543 T^{6} - 616137 T^{7} + 12117361 T^{8}$$
$61$ $$1 + T - 47 T^{2} - 74 T^{3} - 1478 T^{4} - 4514 T^{5} - 174887 T^{6} + 226981 T^{7} + 13845841 T^{8}$$
$67$ $$( 1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 6 T + 71 T^{2} )^{4}$$
$73$ $$( 1 - 11 T + 102 T^{2} - 803 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$( 1 + 2 T - 75 T^{2} + 158 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$1 + 9 T - 97 T^{2} + 108 T^{3} + 18072 T^{4} + 8964 T^{5} - 668233 T^{6} + 5146083 T^{7} + 47458321 T^{8}$$
$89$ $$( 1 + 15 T + 160 T^{2} + 1335 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 + 11 T - 95 T^{2} + 242 T^{3} + 25510 T^{4} + 23474 T^{5} - 893855 T^{6} + 10039403 T^{7} + 88529281 T^{8}$$