# Properties

 Label 450.2.e.f Level $450$ Weight $2$ Character orbit 450.e Analytic conductor $3.593$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{2} + ( 1 - 2 \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + ( -1 - \zeta_{6} ) q^{6} + ( 2 - 2 \zeta_{6} ) q^{7} - q^{8} -3 q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{2} + ( 1 - 2 \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + ( -1 - \zeta_{6} ) q^{6} + ( 2 - 2 \zeta_{6} ) q^{7} - q^{8} -3 q^{9} + ( -2 + \zeta_{6} ) q^{12} -4 \zeta_{6} q^{13} -2 \zeta_{6} q^{14} + ( -1 + \zeta_{6} ) q^{16} + 6 q^{17} + ( -3 + 3 \zeta_{6} ) q^{18} -7 q^{19} + ( -2 - 2 \zeta_{6} ) q^{21} + ( -1 + 2 \zeta_{6} ) q^{24} -4 q^{26} + ( -3 + 6 \zeta_{6} ) q^{27} -2 q^{28} + ( 6 - 6 \zeta_{6} ) q^{29} + 10 \zeta_{6} q^{31} + \zeta_{6} q^{32} + ( 6 - 6 \zeta_{6} ) q^{34} + 3 \zeta_{6} q^{36} -2 q^{37} + ( -7 + 7 \zeta_{6} ) q^{38} + ( -8 + 4 \zeta_{6} ) q^{39} -9 \zeta_{6} q^{41} + ( -4 + 2 \zeta_{6} ) q^{42} + ( -1 + \zeta_{6} ) q^{43} + ( 6 - 6 \zeta_{6} ) q^{47} + ( 1 + \zeta_{6} ) q^{48} + 3 \zeta_{6} q^{49} + ( 6 - 12 \zeta_{6} ) q^{51} + ( -4 + 4 \zeta_{6} ) q^{52} + 12 q^{53} + ( 3 + 3 \zeta_{6} ) q^{54} + ( -2 + 2 \zeta_{6} ) q^{56} + ( -7 + 14 \zeta_{6} ) q^{57} -6 \zeta_{6} q^{58} + 9 \zeta_{6} q^{59} + ( 4 - 4 \zeta_{6} ) q^{61} + 10 q^{62} + ( -6 + 6 \zeta_{6} ) q^{63} + q^{64} -13 \zeta_{6} q^{67} -6 \zeta_{6} q^{68} + 6 q^{71} + 3 q^{72} + q^{73} + ( -2 + 2 \zeta_{6} ) q^{74} + 7 \zeta_{6} q^{76} + ( -4 + 8 \zeta_{6} ) q^{78} + ( -2 + 2 \zeta_{6} ) q^{79} + 9 q^{81} -9 q^{82} + ( 9 - 9 \zeta_{6} ) q^{83} + ( -2 + 4 \zeta_{6} ) q^{84} + \zeta_{6} q^{86} + ( -6 - 6 \zeta_{6} ) q^{87} + 15 q^{89} -8 q^{91} + ( 20 - 10 \zeta_{6} ) q^{93} -6 \zeta_{6} q^{94} + ( 2 - \zeta_{6} ) q^{96} + ( 17 - 17 \zeta_{6} ) q^{97} + 3 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{4} - 3q^{6} + 2q^{7} - 2q^{8} - 6q^{9} + O(q^{10})$$ $$2q + q^{2} - q^{4} - 3q^{6} + 2q^{7} - 2q^{8} - 6q^{9} - 3q^{12} - 4q^{13} - 2q^{14} - q^{16} + 12q^{17} - 3q^{18} - 14q^{19} - 6q^{21} - 8q^{26} - 4q^{28} + 6q^{29} + 10q^{31} + q^{32} + 6q^{34} + 3q^{36} - 4q^{37} - 7q^{38} - 12q^{39} - 9q^{41} - 6q^{42} - q^{43} + 6q^{47} + 3q^{48} + 3q^{49} - 4q^{52} + 24q^{53} + 9q^{54} - 2q^{56} - 6q^{58} + 9q^{59} + 4q^{61} + 20q^{62} - 6q^{63} + 2q^{64} - 13q^{67} - 6q^{68} + 12q^{71} + 6q^{72} + 2q^{73} - 2q^{74} + 7q^{76} - 2q^{79} + 18q^{81} - 18q^{82} + 9q^{83} + q^{86} - 18q^{87} + 30q^{89} - 16q^{91} + 30q^{93} - 6q^{94} + 3q^{96} + 17q^{97} + 6q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 1.73205i −0.500000 0.866025i 0 −1.50000 0.866025i 1.00000 1.73205i −1.00000 −3.00000 0
301.1 0.500000 + 0.866025i 1.73205i −0.500000 + 0.866025i 0 −1.50000 + 0.866025i 1.00000 + 1.73205i −1.00000 −3.00000 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.f yes 2
3.b odd 2 1 1350.2.e.d 2
5.b even 2 1 450.2.e.c 2
5.c odd 4 2 450.2.j.b 4
9.c even 3 1 inner 450.2.e.f yes 2
9.c even 3 1 4050.2.a.d 1
9.d odd 6 1 1350.2.e.d 2
9.d odd 6 1 4050.2.a.u 1
15.d odd 2 1 1350.2.e.h 2
15.e even 4 2 1350.2.j.b 4
45.h odd 6 1 1350.2.e.h 2
45.h odd 6 1 4050.2.a.o 1
45.j even 6 1 450.2.e.c 2
45.j even 6 1 4050.2.a.bg 1
45.k odd 12 2 450.2.j.b 4
45.k odd 12 2 4050.2.c.h 2
45.l even 12 2 1350.2.j.b 4
45.l even 12 2 4050.2.c.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.e.c 2 5.b even 2 1
450.2.e.c 2 45.j even 6 1
450.2.e.f yes 2 1.a even 1 1 trivial
450.2.e.f yes 2 9.c even 3 1 inner
450.2.j.b 4 5.c odd 4 2
450.2.j.b 4 45.k odd 12 2
1350.2.e.d 2 3.b odd 2 1
1350.2.e.d 2 9.d odd 6 1
1350.2.e.h 2 15.d odd 2 1
1350.2.e.h 2 45.h odd 6 1
1350.2.j.b 4 15.e even 4 2
1350.2.j.b 4 45.l even 12 2
4050.2.a.d 1 9.c even 3 1
4050.2.a.o 1 45.h odd 6 1
4050.2.a.u 1 9.d odd 6 1
4050.2.a.bg 1 45.j even 6 1
4050.2.c.h 2 45.k odd 12 2
4050.2.c.m 2 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 $$S_{2}^{\mathrm{new}}(450, [\chi])$$:

 $$T_{7}^{2} - 2 T_{7} + 4$$ $$T_{11}$$ $$T_{17} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$3 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 - 2 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$16 + 4 T + T^{2}$$
$17$ $$( -6 + T )^{2}$$
$19$ $$( 7 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$36 - 6 T + T^{2}$$
$31$ $$100 - 10 T + T^{2}$$
$37$ $$( 2 + T )^{2}$$
$41$ $$81 + 9 T + T^{2}$$
$43$ $$1 + T + T^{2}$$
$47$ $$36 - 6 T + T^{2}$$
$53$ $$( -12 + T )^{2}$$
$59$ $$81 - 9 T + T^{2}$$
$61$ $$16 - 4 T + T^{2}$$
$67$ $$169 + 13 T + T^{2}$$
$71$ $$( -6 + T )^{2}$$
$73$ $$( -1 + T )^{2}$$
$79$ $$4 + 2 T + T^{2}$$
$83$ $$81 - 9 T + T^{2}$$
$89$ $$( -15 + T )^{2}$$
$97$ $$289 - 17 T + T^{2}$$