# Properties

 Label 450.2.j.f Level $450$ Weight $2$ Character orbit 450.j Analytic conductor $3.593$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes 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 $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{24}^{2} + \zeta_{24}^{6} ) q^{2} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{3} + ( 1 - \zeta_{24}^{4} ) q^{4} + ( -\zeta_{24} - \zeta_{24}^{4} + \zeta_{24}^{7} ) q^{6} + ( \zeta_{24} + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{7} + \zeta_{24}^{6} q^{8} + ( 1 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{24}^{2} + \zeta_{24}^{6} ) q^{2} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{3} + ( 1 - \zeta_{24}^{4} ) q^{4} + ( -\zeta_{24} - \zeta_{24}^{4} + \zeta_{24}^{7} ) q^{6} + ( \zeta_{24} + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{7} + \zeta_{24}^{6} q^{8} + ( 1 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{9} + ( 2 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{11} + ( \zeta_{24}^{2} + \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{12} + ( 2 \zeta_{24} + 2 \zeta_{24}^{2} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{13} + ( -2 - 2 \zeta_{24} + \zeta_{24}^{3} + 2 \zeta_{24}^{4} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{14} -\zeta_{24}^{4} q^{16} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{17} + ( -2 \zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{18} + ( -5 + \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{19} + ( 4 + \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{21} + ( 4 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{22} + ( 2 \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{23} + ( -1 - \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{24} + ( -2 - \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{26} + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 5 \zeta_{24}^{6} ) q^{27} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{28} + ( -\zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{29} + ( -2 + 2 \zeta_{24} - \zeta_{24}^{3} + 2 \zeta_{24}^{4} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{31} + \zeta_{24}^{2} q^{32} + ( 2 \zeta_{24} - 4 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 4 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{33} + ( -2 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{34} + ( 1 + 2 \zeta_{24}^{3} - \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{36} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{37} + ( -\zeta_{24} + 5 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 5 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{38} + ( -2 \zeta_{24} + 3 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 3 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{39} + ( -9 + 9 \zeta_{24}^{4} ) q^{41} + ( -2 \zeta_{24} - 4 \zeta_{24}^{2} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 4 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{42} + ( -\zeta_{24} - 5 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 5 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{43} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{44} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{46} + ( 2 \zeta_{24} - 6 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 6 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{47} + ( -\zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{48} + ( 3 + 8 \zeta_{24} - 4 \zeta_{24}^{3} - 3 \zeta_{24}^{4} - 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{49} + ( 4 + 2 \zeta_{24} + 2 \zeta_{24}^{3} - 8 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{51} + ( \zeta_{24} + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{52} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 6 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{53} + ( \zeta_{24} - 5 \zeta_{24}^{4} - \zeta_{24}^{7} ) q^{54} + ( -\zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{56} + ( -4 \zeta_{24} + 4 \zeta_{24}^{2} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 7 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{57} + ( -2 \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{58} + ( -3 + 2 \zeta_{24} - \zeta_{24}^{3} + 3 \zeta_{24}^{4} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{59} -8 \zeta_{24}^{4} q^{61} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{62} + ( 5 \zeta_{24} + 6 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{63} - q^{64} + ( 4 - 4 \zeta_{24} + 2 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{66} + ( 6 \zeta_{24} + 7 \zeta_{24}^{2} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{67} + ( -4 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{68} + ( 2 - 2 \zeta_{24} + \zeta_{24}^{3} + 2 \zeta_{24}^{4} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{69} + ( 6 - 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{71} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{72} + \zeta_{24}^{6} q^{73} + ( 3 \zeta_{24} - 6 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 6 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{74} + ( -5 + 2 \zeta_{24} - \zeta_{24}^{3} + 5 \zeta_{24}^{4} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{76} + ( -8 \zeta_{24} - 12 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{77} + ( -3 \zeta_{24} - 4 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{78} + ( 6 \zeta_{24} - 12 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 12 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{79} + ( -7 - 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{81} -9 \zeta_{24}^{6} q^{82} + ( -\zeta_{24} + 3 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 3 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{83} + ( 4 + 3 \zeta_{24} - \zeta_{24}^{3} - 4 \zeta_{24}^{4} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{84} + ( 5 + 2 \zeta_{24} - \zeta_{24}^{3} - 5 \zeta_{24}^{4} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{86} + ( -\zeta_{24} + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{87} + ( 2 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{88} + 9 q^{89} + ( 10 + 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{91} + ( \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{92} + ( 2 \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{93} + ( 6 - 4 \zeta_{24} + 2 \zeta_{24}^{3} - 6 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{94} + ( -1 + \zeta_{24}^{3} + \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{96} + ( -4 \zeta_{24} - \zeta_{24}^{2} - 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} + \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{97} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 3 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{98} + ( 16 + 2 \zeta_{24} - 4 \zeta_{24}^{3} - 8 \zeta_{24}^{4} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{4} - 4q^{6} + 8q^{9} + O(q^{10})$$ $$8q + 4q^{4} - 4q^{6} + 8q^{9} - 8q^{14} - 4q^{16} - 40q^{19} + 32q^{21} - 8q^{24} - 16q^{26} - 8q^{31} + 4q^{36} + 16q^{39} - 36q^{41} + 12q^{49} - 20q^{54} + 8q^{56} - 12q^{59} - 32q^{61} - 8q^{64} + 48q^{66} + 24q^{69} + 48q^{71} + 16q^{74} - 20q^{76} + 8q^{79} - 56q^{81} + 16q^{84} + 20q^{86} + 72q^{89} + 80q^{91} + 24q^{94} - 4q^{96} + 96q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1 + \zeta_{24}^{2}$$ $$-1$$

## 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
 −0.965926 − 0.258819i 0.965926 + 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i
−0.866025 + 0.500000i −1.41421 + 1.00000i 0.500000 0.866025i 0 0.724745 1.57313i −0.389270 + 0.224745i 1.00000i 1.00000 2.82843i 0
49.2 −0.866025 + 0.500000i 1.41421 + 1.00000i 0.500000 0.866025i 0 −1.72474 0.158919i 3.85337 2.22474i 1.00000i 1.00000 + 2.82843i 0
49.3 0.866025 0.500000i −1.41421 1.00000i 0.500000 0.866025i 0 −1.72474 0.158919i −3.85337 + 2.22474i 1.00000i 1.00000 + 2.82843i 0
49.4 0.866025 0.500000i 1.41421 1.00000i 0.500000 0.866025i 0 0.724745 1.57313i 0.389270 0.224745i 1.00000i 1.00000 2.82843i 0
349.1 −0.866025 0.500000i −1.41421 1.00000i 0.500000 + 0.866025i 0 0.724745 + 1.57313i −0.389270 0.224745i 1.00000i 1.00000 + 2.82843i 0
349.2 −0.866025 0.500000i 1.41421 1.00000i 0.500000 + 0.866025i 0 −1.72474 + 0.158919i 3.85337 + 2.22474i 1.00000i 1.00000 2.82843i 0
349.3 0.866025 + 0.500000i −1.41421 + 1.00000i 0.500000 + 0.866025i 0 −1.72474 + 0.158919i −3.85337 2.22474i 1.00000i 1.00000 2.82843i 0
349.4 0.866025 + 0.500000i 1.41421 + 1.00000i 0.500000 + 0.866025i 0 0.724745 + 1.57313i 0.389270 + 0.224745i 1.00000i 1.00000 + 2.82843i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 349.4 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.f 8
3.b odd 2 1 1350.2.j.g 8
5.b even 2 1 inner 450.2.j.f 8
5.c odd 4 1 450.2.e.l 4
5.c odd 4 1 450.2.e.m yes 4
9.c even 3 1 inner 450.2.j.f 8
9.c even 3 1 4050.2.c.y 4
9.d odd 6 1 1350.2.j.g 8
9.d odd 6 1 4050.2.c.w 4
15.d odd 2 1 1350.2.j.g 8
15.e even 4 1 1350.2.e.k 4
15.e even 4 1 1350.2.e.n 4
45.h odd 6 1 1350.2.j.g 8
45.h odd 6 1 4050.2.c.w 4
45.j even 6 1 inner 450.2.j.f 8
45.j even 6 1 4050.2.c.y 4
45.k odd 12 1 450.2.e.l 4
45.k odd 12 1 450.2.e.m yes 4
45.k odd 12 1 4050.2.a.br 2
45.k odd 12 1 4050.2.a.bu 2
45.l even 12 1 1350.2.e.k 4
45.l even 12 1 1350.2.e.n 4
45.l even 12 1 4050.2.a.bl 2
45.l even 12 1 4050.2.a.by 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.e.l 4 5.c odd 4 1
450.2.e.l 4 45.k odd 12 1
450.2.e.m yes 4 5.c odd 4 1
450.2.e.m yes 4 45.k odd 12 1
450.2.j.f 8 1.a even 1 1 trivial
450.2.j.f 8 5.b even 2 1 inner
450.2.j.f 8 9.c even 3 1 inner
450.2.j.f 8 45.j even 6 1 inner
1350.2.e.k 4 15.e even 4 1
1350.2.e.k 4 45.l even 12 1
1350.2.e.n 4 15.e even 4 1
1350.2.e.n 4 45.l even 12 1
1350.2.j.g 8 3.b odd 2 1
1350.2.j.g 8 9.d odd 6 1
1350.2.j.g 8 15.d odd 2 1
1350.2.j.g 8 45.h odd 6 1
4050.2.a.bl 2 45.l even 12 1
4050.2.a.br 2 45.k odd 12 1
4050.2.a.bu 2 45.k odd 12 1
4050.2.a.by 2 45.l even 12 1
4050.2.c.w 4 9.d odd 6 1
4050.2.c.w 4 45.h odd 6 1
4050.2.c.y 4 9.c even 3 1
4050.2.c.y 4 45.j even 6 1

## Hecke kernels

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

 $$T_{7}^{8} - 20 T_{7}^{6} + 396 T_{7}^{4} - 80 T_{7}^{2} + 16$$ $$T_{11}^{4} + 24 T_{11}^{2} + 576$$ $$T_{19}^{2} + 10 T_{19} + 19$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ $$( 9 - 2 T^{2} + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$16 - 80 T^{2} + 396 T^{4} - 20 T^{6} + T^{8}$$
$11$ $$( 576 + 24 T^{2} + T^{4} )^{2}$$
$13$ $$16 - 80 T^{2} + 396 T^{4} - 20 T^{6} + T^{8}$$
$17$ $$( 24 + T^{2} )^{4}$$
$19$ $$( 19 + 10 T + T^{2} )^{4}$$
$23$ $$( 36 - 6 T^{2} + T^{4} )^{2}$$
$29$ $$( 36 + 6 T^{2} + T^{4} )^{2}$$
$31$ $$( 4 - 8 T + 18 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$37$ $$( 1444 + 140 T^{2} + T^{4} )^{2}$$
$41$ $$( 81 + 9 T + T^{2} )^{4}$$
$43$ $$130321 - 22382 T^{2} + 3483 T^{4} - 62 T^{6} + T^{8}$$
$47$ $$20736 - 17280 T^{2} + 14256 T^{4} - 120 T^{6} + T^{8}$$
$53$ $$( 900 + 84 T^{2} + T^{4} )^{2}$$
$59$ $$( 9 + 18 T + 33 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$61$ $$( 64 + 8 T + T^{2} )^{4}$$
$67$ $$625 - 5150 T^{2} + 42411 T^{4} - 206 T^{6} + T^{8}$$
$71$ $$( -18 - 12 T + T^{2} )^{4}$$
$73$ $$( 1 + T^{2} )^{4}$$
$79$ $$( 44944 + 848 T + 228 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$83$ $$81 - 270 T^{2} + 891 T^{4} - 30 T^{6} + T^{8}$$
$89$ $$( -9 + T )^{8}$$
$97$ $$81450625 - 1750850 T^{2} + 28611 T^{4} - 194 T^{6} + T^{8}$$