# Properties

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

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## 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: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{2} ) q^{2} + ( -1 + \beta_{3} ) q^{3} -\beta_{2} q^{4} + ( -1 + \beta_{1} + \beta_{2} ) q^{6} + ( -2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{7} - q^{8} + ( -1 - 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{2} ) q^{2} + ( -1 + \beta_{3} ) q^{3} -\beta_{2} q^{4} + ( -1 + \beta_{1} + \beta_{2} ) q^{6} + ( -2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{7} - q^{8} + ( -1 - 2 \beta_{3} ) q^{9} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{11} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{12} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{13} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{14} + ( -1 + \beta_{2} ) q^{16} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{17} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{18} + ( 5 + 2 \beta_{1} - \beta_{3} ) q^{19} + ( 4 - 3 \beta_{1} + 2 \beta_{3} ) q^{21} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{22} + ( \beta_{1} + \beta_{3} ) q^{23} + ( 1 - \beta_{3} ) q^{24} + ( -2 + 2 \beta_{1} - \beta_{3} ) q^{26} + ( 5 + \beta_{3} ) q^{27} + ( 2 - 2 \beta_{1} + \beta_{3} ) q^{28} + ( \beta_{1} - 2 \beta_{3} ) q^{29} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{31} + \beta_{2} q^{32} + ( 4 - 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{33} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{34} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{36} + ( -4 - 6 \beta_{1} + 3 \beta_{3} ) q^{37} + ( 5 + \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{38} + ( -4 + \beta_{1} + 4 \beta_{2} - 3 \beta_{3} ) q^{39} -9 \beta_{2} q^{41} + ( 4 - \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{42} + ( -5 + \beta_{1} + 5 \beta_{2} - 2 \beta_{3} ) q^{43} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{44} + ( 2 \beta_{1} - \beta_{3} ) q^{46} + ( 6 + 2 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} ) q^{47} + ( 1 - \beta_{1} - \beta_{2} ) q^{48} + ( 4 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{49} + ( -4 - 4 \beta_{1} + 8 \beta_{2} + 2 \beta_{3} ) q^{51} + ( -2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{52} + ( -6 + 2 \beta_{1} - \beta_{3} ) q^{53} + ( 5 + \beta_{1} - 5 \beta_{2} ) q^{54} + ( 2 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{56} + ( -7 - 2 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} ) q^{57} + ( -\beta_{1} - \beta_{3} ) q^{58} + ( \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{59} + ( -8 + 8 \beta_{2} ) q^{61} + ( -2 - 2 \beta_{1} + \beta_{3} ) q^{62} + ( -2 + 3 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{63} + q^{64} + ( 8 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{66} + ( -3 \beta_{1} + 7 \beta_{2} - 3 \beta_{3} ) q^{67} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{68} + ( -4 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{69} + ( 6 + 6 \beta_{1} - 3 \beta_{3} ) q^{71} + ( 1 + 2 \beta_{3} ) q^{72} - q^{73} + ( -4 - 3 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} ) q^{74} + ( -\beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{76} + ( 4 \beta_{1} - 12 \beta_{2} + 4 \beta_{3} ) q^{77} + ( -2 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{78} + ( -2 - 6 \beta_{1} + 2 \beta_{2} + 12 \beta_{3} ) q^{79} + ( -7 + 4 \beta_{3} ) q^{81} -9 q^{82} + ( 3 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{83} + ( 2 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{84} + ( -\beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{86} + ( 2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{87} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{88} -9 q^{89} + ( 10 - 8 \beta_{1} + 4 \beta_{3} ) q^{91} + ( \beta_{1} - 2 \beta_{3} ) q^{92} + ( 4 + 3 \beta_{1} - \beta_{3} ) q^{93} + ( -2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{94} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{96} + ( 1 - 4 \beta_{1} - \beta_{2} + 8 \beta_{3} ) q^{97} + ( -3 + 8 \beta_{1} - 4 \beta_{3} ) q^{98} + ( -8 - 2 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} - 4q^{3} - 2q^{4} - 2q^{6} - 4q^{7} - 4q^{8} - 4q^{9} + O(q^{10})$$ $$4q + 2q^{2} - 4q^{3} - 2q^{4} - 2q^{6} - 4q^{7} - 4q^{8} - 4q^{9} + 2q^{12} - 4q^{13} + 4q^{14} - 2q^{16} - 2q^{18} + 20q^{19} + 16q^{21} + 4q^{24} - 8q^{26} + 20q^{27} + 8q^{28} - 4q^{31} + 2q^{32} + 24q^{33} + 2q^{36} - 16q^{37} + 10q^{38} - 8q^{39} - 18q^{41} + 8q^{42} - 10q^{43} + 12q^{47} + 2q^{48} - 6q^{49} - 4q^{52} - 24q^{53} + 10q^{54} + 4q^{56} - 20q^{57} + 6q^{59} - 16q^{61} - 8q^{62} - 20q^{63} + 4q^{64} + 24q^{66} + 14q^{67} - 12q^{69} + 24q^{71} + 4q^{72} - 4q^{73} - 8q^{74} - 10q^{76} - 24q^{77} + 8q^{78} - 4q^{79} - 28q^{81} - 36q^{82} + 6q^{83} - 8q^{84} + 10q^{86} + 12q^{87} - 36q^{89} + 40q^{91} + 16q^{93} - 12q^{94} - 2q^{96} + 2q^{97} - 12q^{98} - 48q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## 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
 −1.22474 − 0.707107i 1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i
0.500000 0.866025i −1.00000 1.41421i −0.500000 0.866025i 0 −1.72474 + 0.158919i −2.22474 + 3.85337i −1.00000 −1.00000 + 2.82843i 0
151.2 0.500000 0.866025i −1.00000 + 1.41421i −0.500000 0.866025i 0 0.724745 + 1.57313i 0.224745 0.389270i −1.00000 −1.00000 2.82843i 0
301.1 0.500000 + 0.866025i −1.00000 1.41421i −0.500000 + 0.866025i 0 0.724745 1.57313i 0.224745 + 0.389270i −1.00000 −1.00000 + 2.82843i 0
301.2 0.500000 + 0.866025i −1.00000 + 1.41421i −0.500000 + 0.866025i 0 −1.72474 0.158919i −2.22474 3.85337i −1.00000 −1.00000 2.82843i 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.m yes 4
3.b odd 2 1 1350.2.e.k 4
5.b even 2 1 450.2.e.l 4
5.c odd 4 2 450.2.j.f 8
9.c even 3 1 inner 450.2.e.m yes 4
9.c even 3 1 4050.2.a.br 2
9.d odd 6 1 1350.2.e.k 4
9.d odd 6 1 4050.2.a.by 2
15.d odd 2 1 1350.2.e.n 4
15.e even 4 2 1350.2.j.g 8
45.h odd 6 1 1350.2.e.n 4
45.h odd 6 1 4050.2.a.bl 2
45.j even 6 1 450.2.e.l 4
45.j even 6 1 4050.2.a.bu 2
45.k odd 12 2 450.2.j.f 8
45.k odd 12 2 4050.2.c.y 4
45.l even 12 2 1350.2.j.g 8
45.l even 12 2 4050.2.c.w 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.e.l 4 5.b even 2 1
450.2.e.l 4 45.j even 6 1
450.2.e.m yes 4 1.a even 1 1 trivial
450.2.e.m yes 4 9.c even 3 1 inner
450.2.j.f 8 5.c odd 4 2
450.2.j.f 8 45.k odd 12 2
1350.2.e.k 4 3.b odd 2 1
1350.2.e.k 4 9.d odd 6 1
1350.2.e.n 4 15.d odd 2 1
1350.2.e.n 4 45.h odd 6 1
1350.2.j.g 8 15.e even 4 2
1350.2.j.g 8 45.l even 12 2
4050.2.a.bl 2 45.h odd 6 1
4050.2.a.br 2 9.c even 3 1
4050.2.a.bu 2 45.j even 6 1
4050.2.a.by 2 9.d odd 6 1
4050.2.c.w 4 45.l even 12 2
4050.2.c.y 4 45.k odd 12 2

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ $$( 3 + 2 T + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$4 - 8 T + 18 T^{2} + 4 T^{3} + T^{4}$$
$11$ $$576 + 24 T^{2} + T^{4}$$
$13$ $$4 - 8 T + 18 T^{2} + 4 T^{3} + T^{4}$$
$17$ $$( -24 + T^{2} )^{2}$$
$19$ $$( 19 - 10 T + T^{2} )^{2}$$
$23$ $$36 + 6 T^{2} + T^{4}$$
$29$ $$36 + 6 T^{2} + T^{4}$$
$31$ $$4 - 8 T + 18 T^{2} + 4 T^{3} + T^{4}$$
$37$ $$( -38 + 8 T + T^{2} )^{2}$$
$41$ $$( 81 + 9 T + T^{2} )^{2}$$
$43$ $$361 + 190 T + 81 T^{2} + 10 T^{3} + T^{4}$$
$47$ $$144 - 144 T + 132 T^{2} - 12 T^{3} + T^{4}$$
$53$ $$( 30 + 12 T + T^{2} )^{2}$$
$59$ $$9 - 18 T + 33 T^{2} - 6 T^{3} + T^{4}$$
$61$ $$( 64 + 8 T + T^{2} )^{2}$$
$67$ $$25 + 70 T + 201 T^{2} - 14 T^{3} + T^{4}$$
$71$ $$( -18 - 12 T + T^{2} )^{2}$$
$73$ $$( 1 + T )^{4}$$
$79$ $$44944 - 848 T + 228 T^{2} + 4 T^{3} + T^{4}$$
$83$ $$9 - 18 T + 33 T^{2} - 6 T^{3} + T^{4}$$
$89$ $$( 9 + T )^{4}$$
$97$ $$9025 + 190 T + 99 T^{2} - 2 T^{3} + T^{4}$$
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