# Properties

 Label 1350.2.e.k 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 450) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{2} -\beta_{1} q^{4} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + q^{8} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{2} -\beta_{1} q^{4} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + q^{8} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{11} + ( -2 \beta_{1} + \beta_{2} ) q^{13} + ( -2 \beta_{1} + \beta_{2} ) q^{14} + ( -1 + \beta_{1} ) q^{16} -2 \beta_{3} q^{17} + ( 5 + \beta_{3} ) q^{19} -2 \beta_{2} q^{22} -\beta_{2} q^{23} + ( 2 - \beta_{3} ) q^{26} + ( 2 - \beta_{3} ) q^{28} + ( \beta_{2} - \beta_{3} ) q^{29} + ( -2 \beta_{1} - \beta_{2} ) q^{31} -\beta_{1} q^{32} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{34} + ( -4 - 3 \beta_{3} ) q^{37} + ( -5 + 5 \beta_{1} + \beta_{2} - \beta_{3} ) q^{38} + 9 \beta_{1} q^{41} + ( -5 + 5 \beta_{1} - \beta_{2} + \beta_{3} ) q^{43} + 2 \beta_{3} q^{44} + \beta_{3} q^{46} + ( -6 + 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{47} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{49} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{52} + ( 6 - \beta_{3} ) q^{53} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{56} -\beta_{2} q^{58} + ( -3 \beta_{1} - \beta_{2} ) q^{59} + ( -8 + 8 \beta_{1} ) q^{61} + ( 2 + \beta_{3} ) q^{62} + q^{64} + ( 7 \beta_{1} - 3 \beta_{2} ) q^{67} + 2 \beta_{2} q^{68} + ( -6 - 3 \beta_{3} ) q^{71} - q^{73} + ( 4 - 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{74} + ( -5 \beta_{1} - \beta_{2} ) q^{76} + ( 12 \beta_{1} - 4 \beta_{2} ) q^{77} + ( -2 + 2 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{79} -9 q^{82} + ( -3 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{83} + ( -5 \beta_{1} + \beta_{2} ) q^{86} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{88} + 9 q^{89} + ( 10 - 4 \beta_{3} ) q^{91} + ( \beta_{2} - \beta_{3} ) q^{92} + ( -6 \beta_{1} - 2 \beta_{2} ) q^{94} + ( 1 - \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{97} + ( 3 - 4 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} - 2q^{4} - 4q^{7} + 4q^{8} + O(q^{10})$$ $$4q - 2q^{2} - 2q^{4} - 4q^{7} + 4q^{8} - 4q^{13} - 4q^{14} - 2q^{16} + 20q^{19} + 8q^{26} + 8q^{28} - 4q^{31} - 2q^{32} - 16q^{37} - 10q^{38} + 18q^{41} - 10q^{43} - 12q^{47} - 6q^{49} - 4q^{52} + 24q^{53} - 4q^{56} - 6q^{59} - 16q^{61} + 8q^{62} + 4q^{64} + 14q^{67} - 24q^{71} - 4q^{73} + 8q^{74} - 10q^{76} + 24q^{77} - 4q^{79} - 36q^{82} - 6q^{83} - 10q^{86} + 36q^{89} + 40q^{91} - 12q^{94} + 2q^{97} + 12q^{98} + O(q^{100})$$

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 −2.22474 + 3.85337i 1.00000 0 0
451.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 0.224745 0.389270i 1.00000 0 0
901.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −2.22474 3.85337i 1.00000 0 0
901.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 0.224745 + 0.389270i 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.k 4
3.b odd 2 1 450.2.e.m yes 4
5.b even 2 1 1350.2.e.n 4
5.c odd 4 2 1350.2.j.g 8
9.c even 3 1 inner 1350.2.e.k 4
9.c even 3 1 4050.2.a.by 2
9.d odd 6 1 450.2.e.m yes 4
9.d odd 6 1 4050.2.a.br 2
15.d odd 2 1 450.2.e.l 4
15.e even 4 2 450.2.j.f 8
45.h odd 6 1 450.2.e.l 4
45.h odd 6 1 4050.2.a.bu 2
45.j even 6 1 1350.2.e.n 4
45.j even 6 1 4050.2.a.bl 2
45.k odd 12 2 1350.2.j.g 8
45.k odd 12 2 4050.2.c.w 4
45.l even 12 2 450.2.j.f 8
45.l even 12 2 4050.2.c.y 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.e.l 4 15.d odd 2 1
450.2.e.l 4 45.h odd 6 1
450.2.e.m yes 4 3.b odd 2 1
450.2.e.m yes 4 9.d odd 6 1
450.2.j.f 8 15.e even 4 2
450.2.j.f 8 45.l even 12 2
1350.2.e.k 4 1.a even 1 1 trivial
1350.2.e.k 4 9.c even 3 1 inner
1350.2.e.n 4 5.b even 2 1
1350.2.e.n 4 45.j even 6 1
1350.2.j.g 8 5.c odd 4 2
1350.2.j.g 8 45.k odd 12 2
4050.2.a.bl 2 45.j even 6 1
4050.2.a.br 2 9.d odd 6 1
4050.2.a.bu 2 45.h odd 6 1
4050.2.a.by 2 9.c even 3 1
4050.2.c.w 4 45.k odd 12 2
4050.2.c.y 4 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 $$S_{2}^{\mathrm{new}}(1350, [\chi])$$:

 $$T_{7}^{4} + 4 T_{7}^{3} + 18 T_{7}^{2} - 8 T_{7} + 4$$ $$T_{11}^{4} + 24 T_{11}^{2} + 576$$ $$T_{17}^{2} - 24$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ 
$5$ 
$7$ $$1 + 4 T + 4 T^{2} - 8 T^{3} - 17 T^{4} - 56 T^{5} + 196 T^{6} + 1372 T^{7} + 2401 T^{8}$$
$11$ $$1 + 2 T^{2} - 117 T^{4} + 242 T^{6} + 14641 T^{8}$$
$13$ $$1 + 4 T - 8 T^{2} - 8 T^{3} + 199 T^{4} - 104 T^{5} - 1352 T^{6} + 8788 T^{7} + 28561 T^{8}$$
$17$ $$( 1 + 10 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 10 T + 57 T^{2} - 190 T^{3} + 361 T^{4} )^{2}$$
$23$ $$1 - 40 T^{2} + 1071 T^{4} - 21160 T^{6} + 279841 T^{8}$$
$29$ $$1 - 52 T^{2} + 1863 T^{4} - 43732 T^{6} + 707281 T^{8}$$
$31$ $$1 + 4 T - 44 T^{2} - 8 T^{3} + 2143 T^{4} - 248 T^{5} - 42284 T^{6} + 119164 T^{7} + 923521 T^{8}$$
$37$ $$( 1 + 8 T + 36 T^{2} + 296 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 + 10 T - 5 T^{2} + 190 T^{3} + 4876 T^{4} + 8170 T^{5} - 9245 T^{6} + 795070 T^{7} + 3418801 T^{8}$$
$47$ $$1 + 12 T + 38 T^{2} + 144 T^{3} + 2259 T^{4} + 6768 T^{5} + 83942 T^{6} + 1245876 T^{7} + 4879681 T^{8}$$
$53$ $$( 1 - 12 T + 136 T^{2} - 636 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$1 + 6 T - 85 T^{2} + 18 T^{3} + 9036 T^{4} + 1062 T^{5} - 295885 T^{6} + 1232274 T^{7} + 12117361 T^{8}$$
$61$ $$( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 14 T + 67 T^{2} )^{2}( 1 + 14 T + 129 T^{2} + 938 T^{3} + 4489 T^{4} )$$
$71$ $$( 1 + 12 T + 124 T^{2} + 852 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + T + 73 T^{2} )^{4}$$
$79$ $$1 + 4 T + 70 T^{2} - 848 T^{3} - 4589 T^{4} - 66992 T^{5} + 436870 T^{6} + 1972156 T^{7} + 38950081 T^{8}$$
$83$ $$1 + 6 T - 133 T^{2} + 18 T^{3} + 18684 T^{4} + 1494 T^{5} - 916237 T^{6} + 3430722 T^{7} + 47458321 T^{8}$$
$89$ $$( 1 - 9 T + 89 T^{2} )^{4}$$
$97$ $$1 - 2 T - 95 T^{2} + 190 T^{3} + 4 T^{4} + 18430 T^{5} - 893855 T^{6} - 1825346 T^{7} + 88529281 T^{8}$$