# Properties

 Label 450.2.e.i 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 18) 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.

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

## 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 1.50000 0.866025i 1.00000 1.73205i −1.00000 1.50000 + 2.59808i 0
301.1 0.500000 + 0.866025i 1.50000 0.866025i −0.500000 + 0.866025i 0 1.50000 + 0.866025i 1.00000 + 1.73205i −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.i 2
3.b odd 2 1 1350.2.e.c 2
5.b even 2 1 18.2.c.a 2
5.c odd 4 2 450.2.j.e 4
9.c even 3 1 inner 450.2.e.i 2
9.c even 3 1 4050.2.a.c 1
9.d odd 6 1 1350.2.e.c 2
9.d odd 6 1 4050.2.a.v 1
15.d odd 2 1 54.2.c.a 2
15.e even 4 2 1350.2.j.a 4
20.d odd 2 1 144.2.i.c 2
35.c odd 2 1 882.2.f.d 2
35.i odd 6 1 882.2.e.g 2
35.i odd 6 1 882.2.h.b 2
35.j even 6 1 882.2.e.i 2
35.j even 6 1 882.2.h.c 2
40.e odd 2 1 576.2.i.a 2
40.f even 2 1 576.2.i.g 2
45.h odd 6 1 54.2.c.a 2
45.h odd 6 1 162.2.a.b 1
45.j even 6 1 18.2.c.a 2
45.j even 6 1 162.2.a.c 1
45.k odd 12 2 450.2.j.e 4
45.k odd 12 2 4050.2.c.c 2
45.l even 12 2 1350.2.j.a 4
45.l even 12 2 4050.2.c.r 2
60.h even 2 1 432.2.i.b 2
105.g even 2 1 2646.2.f.g 2
105.o odd 6 1 2646.2.e.b 2
105.o odd 6 1 2646.2.h.h 2
105.p even 6 1 2646.2.e.c 2
105.p even 6 1 2646.2.h.i 2
120.i odd 2 1 1728.2.i.e 2
120.m even 2 1 1728.2.i.f 2
180.n even 6 1 432.2.i.b 2
180.n even 6 1 1296.2.a.f 1
180.p odd 6 1 144.2.i.c 2
180.p odd 6 1 1296.2.a.g 1
315.q odd 6 1 882.2.h.b 2
315.r even 6 1 882.2.h.c 2
315.u even 6 1 2646.2.e.c 2
315.v odd 6 1 2646.2.e.b 2
315.z even 6 1 2646.2.f.g 2
315.z even 6 1 7938.2.a.i 1
315.bg odd 6 1 882.2.f.d 2
315.bg odd 6 1 7938.2.a.x 1
315.bn odd 6 1 882.2.e.g 2
315.bo even 6 1 882.2.e.i 2
315.bq even 6 1 2646.2.h.i 2
315.br odd 6 1 2646.2.h.h 2
360.z odd 6 1 576.2.i.a 2
360.z odd 6 1 5184.2.a.o 1
360.bd even 6 1 1728.2.i.f 2
360.bd even 6 1 5184.2.a.p 1
360.bh odd 6 1 1728.2.i.e 2
360.bh odd 6 1 5184.2.a.q 1
360.bk even 6 1 576.2.i.g 2
360.bk even 6 1 5184.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.2.c.a 2 5.b even 2 1
18.2.c.a 2 45.j even 6 1
54.2.c.a 2 15.d odd 2 1
54.2.c.a 2 45.h odd 6 1
144.2.i.c 2 20.d odd 2 1
144.2.i.c 2 180.p odd 6 1
162.2.a.b 1 45.h odd 6 1
162.2.a.c 1 45.j even 6 1
432.2.i.b 2 60.h even 2 1
432.2.i.b 2 180.n even 6 1
450.2.e.i 2 1.a even 1 1 trivial
450.2.e.i 2 9.c even 3 1 inner
450.2.j.e 4 5.c odd 4 2
450.2.j.e 4 45.k odd 12 2
576.2.i.a 2 40.e odd 2 1
576.2.i.a 2 360.z odd 6 1
576.2.i.g 2 40.f even 2 1
576.2.i.g 2 360.bk even 6 1
882.2.e.g 2 35.i odd 6 1
882.2.e.g 2 315.bn odd 6 1
882.2.e.i 2 35.j even 6 1
882.2.e.i 2 315.bo even 6 1
882.2.f.d 2 35.c odd 2 1
882.2.f.d 2 315.bg odd 6 1
882.2.h.b 2 35.i odd 6 1
882.2.h.b 2 315.q odd 6 1
882.2.h.c 2 35.j even 6 1
882.2.h.c 2 315.r even 6 1
1296.2.a.f 1 180.n even 6 1
1296.2.a.g 1 180.p odd 6 1
1350.2.e.c 2 3.b odd 2 1
1350.2.e.c 2 9.d odd 6 1
1350.2.j.a 4 15.e even 4 2
1350.2.j.a 4 45.l even 12 2
1728.2.i.e 2 120.i odd 2 1
1728.2.i.e 2 360.bh odd 6 1
1728.2.i.f 2 120.m even 2 1
1728.2.i.f 2 360.bd even 6 1
2646.2.e.b 2 105.o odd 6 1
2646.2.e.b 2 315.v odd 6 1
2646.2.e.c 2 105.p even 6 1
2646.2.e.c 2 315.u even 6 1
2646.2.f.g 2 105.g even 2 1
2646.2.f.g 2 315.z even 6 1
2646.2.h.h 2 105.o odd 6 1
2646.2.h.h 2 315.br odd 6 1
2646.2.h.i 2 105.p even 6 1
2646.2.h.i 2 315.bq even 6 1
4050.2.a.c 1 9.c even 3 1
4050.2.a.v 1 9.d odd 6 1
4050.2.c.c 2 45.k odd 12 2
4050.2.c.r 2 45.l even 12 2
5184.2.a.o 1 360.z odd 6 1
5184.2.a.p 1 360.bd even 6 1
5184.2.a.q 1 360.bh odd 6 1
5184.2.a.r 1 360.bk even 6 1
7938.2.a.i 1 315.z even 6 1
7938.2.a.x 1 315.bg odd 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$3 - 3 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 - 2 T + T^{2}$$
$11$ $$9 - 3 T + T^{2}$$
$13$ $$4 - 2 T + T^{2}$$
$17$ $$( -3 + T )^{2}$$
$19$ $$( 1 + T )^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$36 + 6 T + T^{2}$$
$31$ $$16 - 4 T + T^{2}$$
$37$ $$( -4 + T )^{2}$$
$41$ $$81 + 9 T + T^{2}$$
$43$ $$1 + T + T^{2}$$
$47$ $$36 + 6 T + T^{2}$$
$53$ $$( 12 + T )^{2}$$
$59$ $$9 + 3 T + T^{2}$$
$61$ $$64 + 8 T + T^{2}$$
$67$ $$25 - 5 T + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$( 11 + T )^{2}$$
$79$ $$16 - 4 T + T^{2}$$
$83$ $$144 - 12 T + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$25 - 5 T + T^{2}$$