# Properties

 Label 450.2.e.a Level $450$ Weight $2$ Character orbit 450.e Analytic conductor $3.593$ Analytic rank $1$ 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: $$1$$ 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.

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

## 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 −1.50000 + 0.866025i −0.500000 0.866025i 0 1.73205i −2.00000 + 3.46410i 1.00000 1.50000 2.59808i 0
301.1 −0.500000 0.866025i −1.50000 0.866025i −0.500000 + 0.866025i 0 1.73205i −2.00000 3.46410i 1.00000 1.50000 + 2.59808i 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.a 2
3.b odd 2 1 1350.2.e.f 2
5.b even 2 1 450.2.e.h yes 2
5.c odd 4 2 450.2.j.d 4
9.c even 3 1 inner 450.2.e.a 2
9.c even 3 1 4050.2.a.bj 1
9.d odd 6 1 1350.2.e.f 2
9.d odd 6 1 4050.2.a.p 1
15.d odd 2 1 1350.2.e.e 2
15.e even 4 2 1350.2.j.d 4
45.h odd 6 1 1350.2.e.e 2
45.h odd 6 1 4050.2.a.t 1
45.j even 6 1 450.2.e.h yes 2
45.j even 6 1 4050.2.a.b 1
45.k odd 12 2 450.2.j.d 4
45.k odd 12 2 4050.2.c.q 2
45.l even 12 2 1350.2.j.d 4
45.l even 12 2 4050.2.c.e 2

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

 $$T_{7}^{2} + 4 T_{7} + 16$$ $$T_{11}^{2} + 3 T_{11} + 9$$ $$T_{17} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$3 + 3 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$16 + 4 T + T^{2}$$
$11$ $$9 + 3 T + T^{2}$$
$13$ $$16 + 4 T + T^{2}$$
$17$ $$( -3 + T )^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$36 - 6 T + T^{2}$$
$31$ $$64 + 8 T + T^{2}$$
$37$ $$( 8 + T )^{2}$$
$41$ $$36 - 6 T + T^{2}$$
$43$ $$1 + T + T^{2}$$
$47$ $$144 - 12 T + T^{2}$$
$53$ $$T^{2}$$
$59$ $$81 - 9 T + T^{2}$$
$61$ $$64 + 8 T + T^{2}$$
$67$ $$16 + 4 T + T^{2}$$
$71$ $$( 6 + T )^{2}$$
$73$ $$( 14 + T )^{2}$$
$79$ $$64 + 8 T + T^{2}$$
$83$ $$81 + 9 T + T^{2}$$
$89$ $$( 9 + T )^{2}$$
$97$ $$49 + 7 T + T^{2}$$