# Properties

 Label 450.2.e.g 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: 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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 −0.500000 + 0.866025i −1.00000 1.50000 2.59808i 0
301.1 0.500000 + 0.866025i 1.50000 + 0.866025i −0.500000 + 0.866025i 0 1.73205i −0.500000 0.866025i −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.g 2
3.b odd 2 1 1350.2.e.a 2
5.b even 2 1 450.2.e.b 2
5.c odd 4 2 90.2.i.a 4
9.c even 3 1 inner 450.2.e.g 2
9.c even 3 1 4050.2.a.j 1
9.d odd 6 1 1350.2.e.a 2
9.d odd 6 1 4050.2.a.be 1
15.d odd 2 1 1350.2.e.i 2
15.e even 4 2 270.2.i.a 4
20.e even 4 2 720.2.by.a 4
45.h odd 6 1 1350.2.e.i 2
45.h odd 6 1 4050.2.a.g 1
45.j even 6 1 450.2.e.b 2
45.j even 6 1 4050.2.a.x 1
45.k odd 12 2 90.2.i.a 4
45.k odd 12 2 810.2.c.b 2
45.l even 12 2 270.2.i.a 4
45.l even 12 2 810.2.c.c 2
60.l odd 4 2 2160.2.by.b 4
180.v odd 12 2 2160.2.by.b 4
180.x even 12 2 720.2.by.a 4

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

## Hecke kernels

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

 $$T_{7}^{2} + T_{7} + 1$$ $$T_{11}^{2} - 2 T_{11} + 4$$ $$T_{17} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$3 - 3 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T + T^{2}$$
$11$ $$4 - 2 T + T^{2}$$
$13$ $$36 + 6 T + T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$( -6 + T )^{2}$$
$23$ $$1 - T + T^{2}$$
$29$ $$81 + 9 T + T^{2}$$
$31$ $$4 - 2 T + T^{2}$$
$37$ $$( 2 + T )^{2}$$
$41$ $$121 - 11 T + T^{2}$$
$43$ $$16 + 4 T + T^{2}$$
$47$ $$49 + 7 T + T^{2}$$
$53$ $$T^{2}$$
$59$ $$16 - 4 T + T^{2}$$
$61$ $$49 - 7 T + T^{2}$$
$67$ $$121 + 11 T + T^{2}$$
$71$ $$( 6 + T )^{2}$$
$73$ $$( -4 + T )^{2}$$
$79$ $$144 - 12 T + T^{2}$$
$83$ $$121 + 11 T + T^{2}$$
$89$ $$( -1 + T )^{2}$$
$97$ $$64 + 8 T + T^{2}$$