# Properties

 Label 450.2.e.j Level $450$ Weight $2$ Character orbit 450.e Analytic conductor $3.593$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 450.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.59326809096$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{2} ) q^{2} + ( -\beta_{1} + \beta_{3} ) q^{3} -\beta_{2} q^{4} + \beta_{1} q^{6} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{7} + q^{8} + ( -3 + \beta_{1} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{2} ) q^{2} + ( -\beta_{1} + \beta_{3} ) q^{3} -\beta_{2} q^{4} + \beta_{1} q^{6} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{7} + q^{8} + ( -3 + \beta_{1} - \beta_{3} ) q^{9} + ( 1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{11} -\beta_{3} q^{12} + ( -2 + 4 \beta_{1} - 2 \beta_{3} ) q^{13} + ( 1 - 2 \beta_{1} + \beta_{3} ) q^{14} + ( -1 + \beta_{2} ) q^{16} + ( 4 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{17} + ( 3 - \beta_{1} - 3 \beta_{2} ) q^{18} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{19} + ( 3 - \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{21} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{22} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{23} + ( -\beta_{1} + \beta_{3} ) q^{24} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{26} + ( 3 + 2 \beta_{1} - 2 \beta_{3} ) q^{27} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{28} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{29} + ( -2 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{31} -\beta_{2} q^{32} + ( 3 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{33} + ( -5 + \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{34} + ( 3 \beta_{2} + \beta_{3} ) q^{36} + 4 q^{37} + ( -1 + \beta_{1} - 2 \beta_{3} ) q^{38} + ( 12 - 2 \beta_{1} - 6 \beta_{2} ) q^{39} + 3 \beta_{2} q^{41} + ( -6 + \beta_{1} + 3 \beta_{2} ) q^{42} + ( -9 + \beta_{1} + 8 \beta_{2} - 2 \beta_{3} ) q^{43} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{44} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{46} + ( -4 - \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{47} + \beta_{1} q^{48} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{49} + ( 3 - 5 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{51} + ( -2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{52} + ( 2 - 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{53} + ( -3 - 2 \beta_{1} + 3 \beta_{2} ) q^{54} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{56} + ( 3 - \beta_{1} - 6 \beta_{2} ) q^{57} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{58} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{59} + ( -2 + 3 \beta_{1} - \beta_{2} - 6 \beta_{3} ) q^{61} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{62} + ( -3 - 2 \beta_{1} + 5 \beta_{3} ) q^{63} + q^{64} + ( -6 + \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{66} -7 \beta_{2} q^{67} + ( 1 - 2 \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{68} + ( -6 + \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{69} -6 q^{71} + ( -3 + \beta_{1} - \beta_{3} ) q^{72} + ( 4 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{73} + ( -4 + 4 \beta_{2} ) q^{74} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{76} + ( -2 + 4 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{77} + ( -6 + 12 \beta_{2} + 2 \beta_{3} ) q^{78} + ( -2 + 2 \beta_{2} ) q^{79} + ( 6 - 5 \beta_{1} + 5 \beta_{3} ) q^{81} -3 q^{82} + ( 4 + \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{83} + ( 3 - 6 \beta_{2} - \beta_{3} ) q^{84} + ( 1 - 2 \beta_{1} - 9 \beta_{2} + \beta_{3} ) q^{86} + ( -3 - \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{87} + ( 1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{88} + ( 9 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{89} + ( -18 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{91} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{92} + ( 12 - 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{93} + ( -1 + 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{94} -\beta_{3} q^{96} + ( -5 - \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{97} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{98} + ( -6 - \beta_{1} + 3 \beta_{2} + 5 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} - 2q^{3} - 2q^{4} + q^{6} + q^{7} + 4q^{8} - 10q^{9} + O(q^{10})$$ $$4q - 2q^{2} - 2q^{3} - 2q^{4} + q^{6} + q^{7} + 4q^{8} - 10q^{9} + 3q^{11} + q^{12} - 2q^{13} + q^{14} - 2q^{16} + 18q^{17} + 5q^{18} + 2q^{19} + 16q^{21} + 3q^{22} - 3q^{23} - 2q^{24} + 4q^{26} + 16q^{27} - 2q^{28} + 3q^{29} + 2q^{31} - 2q^{32} + 15q^{33} - 9q^{34} + 5q^{36} + 16q^{37} - q^{38} + 34q^{39} + 6q^{41} - 17q^{42} - 17q^{43} - 6q^{44} + 6q^{46} - 9q^{47} + q^{48} - 3q^{49} - 9q^{51} - 2q^{52} - 8q^{54} + q^{56} - q^{57} + 3q^{58} - 3q^{59} - q^{61} - 4q^{62} - 19q^{63} + 4q^{64} - 18q^{66} - 14q^{67} - 9q^{68} - 15q^{69} - 24q^{71} - 10q^{72} + 22q^{73} - 8q^{74} - q^{76} - 18q^{77} - 2q^{78} - 4q^{79} + 14q^{81} - 12q^{82} + 9q^{83} + q^{84} - 17q^{86} - 18q^{87} + 3q^{88} + 30q^{89} - 68q^{91} - 3q^{92} + 32q^{93} - 9q^{94} + q^{96} - 11q^{97} + 6q^{98} - 24q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 3$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu + 3$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{3} + 2 \beta_{1} + 3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
151.1
 1.68614 + 0.396143i −1.18614 − 1.26217i −1.18614 + 1.26217i 1.68614 − 0.396143i
−0.500000 + 0.866025i −0.500000 1.65831i −0.500000 0.866025i 0 1.68614 + 0.396143i −1.18614 + 2.05446i 1.00000 −2.50000 + 1.65831i 0
151.2 −0.500000 + 0.866025i −0.500000 + 1.65831i −0.500000 0.866025i 0 −1.18614 1.26217i 1.68614 2.92048i 1.00000 −2.50000 1.65831i 0
301.1 −0.500000 0.866025i −0.500000 1.65831i −0.500000 + 0.866025i 0 −1.18614 + 1.26217i 1.68614 + 2.92048i 1.00000 −2.50000 + 1.65831i 0
301.2 −0.500000 0.866025i −0.500000 + 1.65831i −0.500000 + 0.866025i 0 1.68614 0.396143i −1.18614 2.05446i 1.00000 −2.50000 1.65831i 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 450.2.e.j 4
3.b odd 2 1 1350.2.e.l 4
5.b even 2 1 90.2.e.c 4
5.c odd 4 2 450.2.j.g 8
9.c even 3 1 inner 450.2.e.j 4
9.c even 3 1 4050.2.a.bw 2
9.d odd 6 1 1350.2.e.l 4
9.d odd 6 1 4050.2.a.bo 2
15.d odd 2 1 270.2.e.c 4
15.e even 4 2 1350.2.j.f 8
20.d odd 2 1 720.2.q.f 4
45.h odd 6 1 270.2.e.c 4
45.h odd 6 1 810.2.a.k 2
45.j even 6 1 90.2.e.c 4
45.j even 6 1 810.2.a.i 2
45.k odd 12 2 450.2.j.g 8
45.k odd 12 2 4050.2.c.v 4
45.l even 12 2 1350.2.j.f 8
45.l even 12 2 4050.2.c.ba 4
60.h even 2 1 2160.2.q.f 4
180.n even 6 1 2160.2.q.f 4
180.n even 6 1 6480.2.a.bn 2
180.p odd 6 1 720.2.q.f 4
180.p odd 6 1 6480.2.a.be 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.e.c 4 5.b even 2 1
90.2.e.c 4 45.j even 6 1
270.2.e.c 4 15.d odd 2 1
270.2.e.c 4 45.h odd 6 1
450.2.e.j 4 1.a even 1 1 trivial
450.2.e.j 4 9.c even 3 1 inner
450.2.j.g 8 5.c odd 4 2
450.2.j.g 8 45.k odd 12 2
720.2.q.f 4 20.d odd 2 1
720.2.q.f 4 180.p odd 6 1
810.2.a.i 2 45.j even 6 1
810.2.a.k 2 45.h odd 6 1
1350.2.e.l 4 3.b odd 2 1
1350.2.e.l 4 9.d odd 6 1
1350.2.j.f 8 15.e even 4 2
1350.2.j.f 8 45.l even 12 2
2160.2.q.f 4 60.h even 2 1
2160.2.q.f 4 180.n even 6 1
4050.2.a.bo 2 9.d odd 6 1
4050.2.a.bw 2 9.c even 3 1
4050.2.c.v 4 45.k odd 12 2
4050.2.c.ba 4 45.l even 12 2
6480.2.a.be 2 180.p odd 6 1
6480.2.a.bn 2 180.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(450, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$( 3 + T + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$64 + 8 T + 9 T^{2} - T^{3} + T^{4}$$
$11$ $$36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4}$$
$13$ $$1024 - 64 T + 36 T^{2} + 2 T^{3} + T^{4}$$
$17$ $$( 12 - 9 T + T^{2} )^{2}$$
$19$ $$( -8 - T + T^{2} )^{2}$$
$23$ $$36 - 18 T + 15 T^{2} + 3 T^{3} + T^{4}$$
$29$ $$36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4}$$
$31$ $$1024 + 64 T + 36 T^{2} - 2 T^{3} + T^{4}$$
$37$ $$( -4 + T )^{4}$$
$41$ $$( 9 - 3 T + T^{2} )^{2}$$
$43$ $$4096 + 1088 T + 225 T^{2} + 17 T^{3} + T^{4}$$
$47$ $$144 + 108 T + 69 T^{2} + 9 T^{3} + T^{4}$$
$53$ $$( -132 + T^{2} )^{2}$$
$59$ $$36 - 18 T + 15 T^{2} + 3 T^{3} + T^{4}$$
$61$ $$5476 - 74 T + 75 T^{2} + T^{3} + T^{4}$$
$67$ $$( 49 + 7 T + T^{2} )^{2}$$
$71$ $$( 6 + T )^{4}$$
$73$ $$( -44 - 11 T + T^{2} )^{2}$$
$79$ $$( 4 + 2 T + T^{2} )^{2}$$
$83$ $$144 - 108 T + 69 T^{2} - 9 T^{3} + T^{4}$$
$89$ $$( -18 - 15 T + T^{2} )^{2}$$
$97$ $$484 + 242 T + 99 T^{2} + 11 T^{3} + T^{4}$$