# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 450.j (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.59326809096$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 90) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-\zeta_{12}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.866025 + 0.500000i 0.866025 + 1.50000i 0.500000 0.866025i 0 −1.50000 0.866025i −3.46410 + 2.00000i 1.00000i −1.50000 + 2.59808i 0
49.2 0.866025 0.500000i −0.866025 1.50000i 0.500000 0.866025i 0 −1.50000 0.866025i 3.46410 2.00000i 1.00000i −1.50000 + 2.59808i 0
349.1 −0.866025 0.500000i 0.866025 1.50000i 0.500000 + 0.866025i 0 −1.50000 + 0.866025i −3.46410 2.00000i 1.00000i −1.50000 2.59808i 0
349.2 0.866025 + 0.500000i −0.866025 + 1.50000i 0.500000 + 0.866025i 0 −1.50000 + 0.866025i 3.46410 + 2.00000i 1.00000i −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
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.j.a 4
3.b odd 2 1 1350.2.j.c 4
5.b even 2 1 inner 450.2.j.a 4
5.c odd 4 1 90.2.e.b 2
5.c odd 4 1 450.2.e.d 2
9.c even 3 1 inner 450.2.j.a 4
9.c even 3 1 4050.2.c.p 2
9.d odd 6 1 1350.2.j.c 4
9.d odd 6 1 4050.2.c.d 2
15.d odd 2 1 1350.2.j.c 4
15.e even 4 1 270.2.e.a 2
15.e even 4 1 1350.2.e.g 2
20.e even 4 1 720.2.q.c 2
45.h odd 6 1 1350.2.j.c 4
45.h odd 6 1 4050.2.c.d 2
45.j even 6 1 inner 450.2.j.a 4
45.j even 6 1 4050.2.c.p 2
45.k odd 12 1 90.2.e.b 2
45.k odd 12 1 450.2.e.d 2
45.k odd 12 1 810.2.a.a 1
45.k odd 12 1 4050.2.a.bi 1
45.l even 12 1 270.2.e.a 2
45.l even 12 1 810.2.a.e 1
45.l even 12 1 1350.2.e.g 2
45.l even 12 1 4050.2.a.q 1
60.l odd 4 1 2160.2.q.d 2
180.v odd 12 1 2160.2.q.d 2
180.v odd 12 1 6480.2.a.l 1
180.x even 12 1 720.2.q.c 2
180.x even 12 1 6480.2.a.z 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.e.b 2 5.c odd 4 1
90.2.e.b 2 45.k odd 12 1
270.2.e.a 2 15.e even 4 1
270.2.e.a 2 45.l even 12 1
450.2.e.d 2 5.c odd 4 1
450.2.e.d 2 45.k odd 12 1
450.2.j.a 4 1.a even 1 1 trivial
450.2.j.a 4 5.b even 2 1 inner
450.2.j.a 4 9.c even 3 1 inner
450.2.j.a 4 45.j even 6 1 inner
720.2.q.c 2 20.e even 4 1
720.2.q.c 2 180.x even 12 1
810.2.a.a 1 45.k odd 12 1
810.2.a.e 1 45.l even 12 1
1350.2.e.g 2 15.e even 4 1
1350.2.e.g 2 45.l even 12 1
1350.2.j.c 4 3.b odd 2 1
1350.2.j.c 4 9.d odd 6 1
1350.2.j.c 4 15.d odd 2 1
1350.2.j.c 4 45.h odd 6 1
2160.2.q.d 2 60.l odd 4 1
2160.2.q.d 2 180.v odd 12 1
4050.2.a.q 1 45.l even 12 1
4050.2.a.bi 1 45.k odd 12 1
4050.2.c.d 2 9.d odd 6 1
4050.2.c.d 2 45.h odd 6 1
4050.2.c.p 2 9.c even 3 1
4050.2.c.p 2 45.j even 6 1
6480.2.a.l 1 180.v odd 12 1
6480.2.a.z 1 180.x even 12 1

## Hecke kernels

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

 $$T_{7}^{4} - 16 T_{7}^{2} + 256$$ $$T_{11}^{2} + 3 T_{11} + 9$$ $$T_{19} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$9 + 3 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$256 - 16 T^{2} + T^{4}$$
$11$ $$( 9 + 3 T + T^{2} )^{2}$$
$13$ $$256 - 16 T^{2} + T^{4}$$
$17$ $$( 9 + T^{2} )^{2}$$
$19$ $$( 5 + T )^{4}$$
$23$ $$1296 - 36 T^{2} + T^{4}$$
$29$ $$( 36 - 6 T + T^{2} )^{2}$$
$31$ $$( 4 + 2 T + T^{2} )^{2}$$
$37$ $$( 16 + T^{2} )^{2}$$
$41$ $$( 9 - 3 T + T^{2} )^{2}$$
$43$ $$14641 - 121 T^{2} + T^{4}$$
$47$ $$T^{4}$$
$53$ $$( 36 + T^{2} )^{2}$$
$59$ $$( 9 + 3 T + T^{2} )^{2}$$
$61$ $$( 100 - 10 T + T^{2} )^{2}$$
$67$ $$625 - 25 T^{2} + T^{4}$$
$71$ $$( -6 + T )^{4}$$
$73$ $$( 49 + T^{2} )^{2}$$
$79$ $$( 196 - 14 T + T^{2} )^{2}$$
$83$ $$20736 - 144 T^{2} + T^{4}$$
$89$ $$( 6 + T )^{4}$$
$97$ $$14641 - 121 T^{2} + T^{4}$$