# Properties

 Label 450.2.p.a Level $450$ Weight $2$ Character orbit 450.p 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.p (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.59326809096$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 90) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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.

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

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1 - \zeta_{24}^{4}$$ $$-\zeta_{24}^{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 0.258819 − 0.965926i −0.258819 + 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i 0.965926 − 0.258819i −0.965926 + 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i
−0.965926 0.258819i 1.33195 + 1.10721i 0.866025 + 0.500000i 0 −1.00000 1.41421i −0.283763 + 1.05902i −0.707107 0.707107i 0.548188 + 2.94949i 0
257.2 0.965926 + 0.258819i −0.599900 + 1.62484i 0.866025 + 0.500000i 0 −1.00000 + 1.41421i −1.18034 + 4.40508i 0.707107 + 0.707107i −2.28024 1.94949i 0
293.1 −0.258819 + 0.965926i −1.10721 + 1.33195i −0.866025 0.500000i 0 −1.00000 1.41421i 1.05902 + 0.283763i 0.707107 0.707107i −0.548188 2.94949i 0
293.2 0.258819 0.965926i −1.62484 0.599900i −0.866025 0.500000i 0 −1.00000 + 1.41421i 4.40508 + 1.18034i −0.707107 + 0.707107i 2.28024 + 1.94949i 0
407.1 −0.258819 0.965926i −1.10721 1.33195i −0.866025 + 0.500000i 0 −1.00000 + 1.41421i 1.05902 0.283763i 0.707107 + 0.707107i −0.548188 + 2.94949i 0
407.2 0.258819 + 0.965926i −1.62484 + 0.599900i −0.866025 + 0.500000i 0 −1.00000 1.41421i 4.40508 1.18034i −0.707107 0.707107i 2.28024 1.94949i 0
443.1 −0.965926 + 0.258819i 1.33195 1.10721i 0.866025 0.500000i 0 −1.00000 + 1.41421i −0.283763 1.05902i −0.707107 + 0.707107i 0.548188 2.94949i 0
443.2 0.965926 0.258819i −0.599900 1.62484i 0.866025 0.500000i 0 −1.00000 1.41421i −1.18034 4.40508i 0.707107 0.707107i −2.28024 + 1.94949i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 443.2 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.c odd 4 1 inner
9.d odd 6 1 inner
45.l even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.p.a 8
3.b odd 2 1 1350.2.q.g 8
5.b even 2 1 90.2.l.a 8
5.c odd 4 1 90.2.l.a 8
5.c odd 4 1 inner 450.2.p.a 8
9.c even 3 1 1350.2.q.g 8
9.d odd 6 1 inner 450.2.p.a 8
15.d odd 2 1 270.2.m.a 8
15.e even 4 1 270.2.m.a 8
15.e even 4 1 1350.2.q.g 8
20.d odd 2 1 720.2.cu.a 8
20.e even 4 1 720.2.cu.a 8
45.h odd 6 1 90.2.l.a 8
45.h odd 6 1 810.2.f.b 8
45.j even 6 1 270.2.m.a 8
45.j even 6 1 810.2.f.b 8
45.k odd 12 1 270.2.m.a 8
45.k odd 12 1 810.2.f.b 8
45.k odd 12 1 1350.2.q.g 8
45.l even 12 1 90.2.l.a 8
45.l even 12 1 inner 450.2.p.a 8
45.l even 12 1 810.2.f.b 8
180.n even 6 1 720.2.cu.a 8
180.v odd 12 1 720.2.cu.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.l.a 8 5.b even 2 1
90.2.l.a 8 5.c odd 4 1
90.2.l.a 8 45.h odd 6 1
90.2.l.a 8 45.l even 12 1
270.2.m.a 8 15.d odd 2 1
270.2.m.a 8 15.e even 4 1
270.2.m.a 8 45.j even 6 1
270.2.m.a 8 45.k odd 12 1
450.2.p.a 8 1.a even 1 1 trivial
450.2.p.a 8 5.c odd 4 1 inner
450.2.p.a 8 9.d odd 6 1 inner
450.2.p.a 8 45.l even 12 1 inner
720.2.cu.a 8 20.d odd 2 1
720.2.cu.a 8 20.e even 4 1
720.2.cu.a 8 180.n even 6 1
720.2.cu.a 8 180.v odd 12 1
810.2.f.b 8 45.h odd 6 1
810.2.f.b 8 45.j even 6 1
810.2.f.b 8 45.k odd 12 1
810.2.f.b 8 45.l even 12 1
1350.2.q.g 8 3.b odd 2 1
1350.2.q.g 8 9.c even 3 1
1350.2.q.g 8 15.e even 4 1
1350.2.q.g 8 45.k odd 12 1

## Hecke kernels

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

 $$T_{7}^{8} - \cdots$$ $$T_{11}^{4} + 12 T_{11}^{3} + 52 T_{11}^{2} + 48 T_{11} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{4} + T^{8}$$
$3$ $$81 + 108 T + 72 T^{2} + 24 T^{3} + 7 T^{4} + 8 T^{5} + 8 T^{6} + 4 T^{7} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$625 - 1000 T + 800 T^{2} - 880 T^{3} + 679 T^{4} - 176 T^{5} + 32 T^{6} - 8 T^{7} + T^{8}$$
$11$ $$( 16 + 48 T + 52 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$13$ $$20736 - 144 T^{4} + T^{8}$$
$17$ $$16 + 392 T^{4} + T^{8}$$
$19$ $$( 100 + 44 T^{2} + T^{4} )^{2}$$
$23$ $$1 - T^{4} + T^{8}$$
$29$ $$1 + 10 T^{2} + 99 T^{4} + 10 T^{6} + T^{8}$$
$31$ $$( 4 + 8 T + 18 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$37$ $$( 18 - 6 T + T^{2} )^{4}$$
$41$ $$( 841 + 174 T - 17 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$43$ $$20736 - 144 T^{4} + T^{8}$$
$47$ $$43046721 - 6561 T^{4} + T^{8}$$
$53$ $$6250000 + 8456 T^{4} + T^{8}$$
$59$ $$126247696 + 2471920 T^{2} + 37164 T^{4} + 220 T^{6} + T^{8}$$
$61$ $$( 9 - 18 T + 33 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$67$ $$390625 - 62500 T + 5000 T^{2} - 5800 T^{3} - 161 T^{4} + 232 T^{5} + 8 T^{6} + 4 T^{7} + T^{8}$$
$71$ $$( 16 + 40 T^{2} + T^{4} )^{2}$$
$73$ $$( 1600 + 320 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$79$ $$( 36 - 6 T^{2} + T^{4} )^{2}$$
$83$ $$81 - 324 T + 648 T^{2} - 864 T^{3} + 423 T^{4} + 288 T^{5} + 72 T^{6} + 12 T^{7} + T^{8}$$
$89$ $$( 361 - 70 T^{2} + T^{4} )^{2}$$
$97$ $$810000 - 324000 T + 64800 T^{2} - 47520 T^{3} + 8604 T^{4} + 1584 T^{5} + 72 T^{6} + 12 T^{7} + T^{8}$$