# Properties

 Label 450.2.p.d Level $450$ Weight $2$ Character orbit 450.p Analytic conductor $3.593$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# 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: yes 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{24}^{7} q^{2} + ( -2 \zeta_{24} + \zeta_{24}^{5} ) q^{3} -\zeta_{24}^{2} q^{4} + ( 1 - 2 \zeta_{24}^{4} ) q^{6} + ( 2 \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{7} + ( \zeta_{24} - \zeta_{24}^{5} ) q^{8} + ( 3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{9} +O(q^{10})$$ $$q + \zeta_{24}^{7} q^{2} + ( -2 \zeta_{24} + \zeta_{24}^{5} ) q^{3} -\zeta_{24}^{2} q^{4} + ( 1 - 2 \zeta_{24}^{4} ) q^{6} + ( 2 \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{7} + ( \zeta_{24} - \zeta_{24}^{5} ) q^{8} + ( 3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{9} + ( 4 - 2 \zeta_{24}^{4} ) q^{11} + ( 2 \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{12} + ( -4 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{13} + ( -\zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{14} + \zeta_{24}^{4} q^{16} + 6 \zeta_{24}^{3} q^{17} + 3 \zeta_{24}^{5} q^{18} -2 \zeta_{24}^{6} q^{19} -3 q^{21} + ( 2 \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{22} + 3 \zeta_{24}^{5} q^{23} + ( -\zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{24} + ( 2 - 4 \zeta_{24}^{4} ) q^{26} + ( -3 \zeta_{24}^{3} + 6 \zeta_{24}^{7} ) q^{27} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{28} + ( 5 \zeta_{24}^{2} + 5 \zeta_{24}^{6} ) q^{29} + ( -10 + 10 \zeta_{24}^{4} ) q^{31} + ( -\zeta_{24}^{3} + \zeta_{24}^{7} ) q^{32} + ( -6 \zeta_{24} + 6 \zeta_{24}^{5} ) q^{33} + ( -6 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{34} -3 q^{36} + ( 2 \zeta_{24}^{3} - 4 \zeta_{24}^{7} ) q^{37} + 2 \zeta_{24} q^{38} + ( 6 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{39} + ( 5 + 5 \zeta_{24}^{4} ) q^{41} -3 \zeta_{24}^{7} q^{42} + ( 6 \zeta_{24} - 12 \zeta_{24}^{5} ) q^{43} + ( -4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{44} -3 q^{46} + 3 \zeta_{24}^{7} q^{47} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{48} -4 \zeta_{24}^{2} q^{49} + ( -6 - 6 \zeta_{24}^{4} ) q^{51} + ( 4 \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{52} + ( -3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{54} + ( 2 - \zeta_{24}^{4} ) q^{56} + ( 2 \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{57} + ( -10 \zeta_{24} + 5 \zeta_{24}^{5} ) q^{58} + ( 4 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{59} + 13 \zeta_{24}^{4} q^{61} -10 \zeta_{24}^{3} q^{62} + ( 6 \zeta_{24} - 3 \zeta_{24}^{5} ) q^{63} -\zeta_{24}^{6} q^{64} -6 \zeta_{24}^{4} q^{66} + ( -7 \zeta_{24}^{3} - 7 \zeta_{24}^{7} ) q^{67} -6 \zeta_{24}^{5} q^{68} + ( -3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{69} + ( 2 - 4 \zeta_{24}^{4} ) q^{71} -3 \zeta_{24}^{7} q^{72} + ( 8 \zeta_{24} + 8 \zeta_{24}^{5} ) q^{73} + ( 2 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{74} + ( -2 + 2 \zeta_{24}^{4} ) q^{76} + ( 6 \zeta_{24}^{3} - 6 \zeta_{24}^{7} ) q^{77} + 6 \zeta_{24}^{5} q^{78} + ( -4 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{79} + ( 9 - 9 \zeta_{24}^{4} ) q^{81} + ( -5 \zeta_{24}^{3} + 10 \zeta_{24}^{7} ) q^{82} + 3 \zeta_{24} q^{83} + 3 \zeta_{24}^{2} q^{84} + ( 6 + 6 \zeta_{24}^{4} ) q^{86} -15 \zeta_{24}^{3} q^{87} + ( 2 \zeta_{24} - 4 \zeta_{24}^{5} ) q^{88} + ( -2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{89} -6 q^{91} -3 \zeta_{24}^{7} q^{92} + ( 10 \zeta_{24} - 20 \zeta_{24}^{5} ) q^{93} -3 \zeta_{24}^{2} q^{94} + ( 2 - \zeta_{24}^{4} ) q^{96} + ( -8 \zeta_{24}^{3} + 4 \zeta_{24}^{7} ) q^{97} + ( 4 \zeta_{24} - 4 \zeta_{24}^{5} ) q^{98} + ( 6 \zeta_{24}^{2} - 12 \zeta_{24}^{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 24q^{11} + 4q^{16} - 24q^{21} - 40q^{31} - 24q^{36} + 60q^{41} - 24q^{46} - 72q^{51} + 12q^{56} + 52q^{61} - 24q^{66} - 8q^{76} + 36q^{81} + 72q^{86} - 48q^{91} + 12q^{96} + O(q^{100})$$

## 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 0.448288 + 1.67303i 0.866025 + 0.500000i 0 1.73205i −0.448288 + 1.67303i −0.707107 0.707107i −2.59808 + 1.50000i 0
257.2 0.965926 + 0.258819i −0.448288 1.67303i 0.866025 + 0.500000i 0 1.73205i 0.448288 1.67303i 0.707107 + 0.707107i −2.59808 + 1.50000i 0
293.1 −0.258819 + 0.965926i −1.67303 + 0.448288i −0.866025 0.500000i 0 1.73205i 1.67303 + 0.448288i 0.707107 0.707107i 2.59808 1.50000i 0
293.2 0.258819 0.965926i 1.67303 0.448288i −0.866025 0.500000i 0 1.73205i −1.67303 0.448288i −0.707107 + 0.707107i 2.59808 1.50000i 0
407.1 −0.258819 0.965926i −1.67303 0.448288i −0.866025 + 0.500000i 0 1.73205i 1.67303 0.448288i 0.707107 + 0.707107i 2.59808 + 1.50000i 0
407.2 0.258819 + 0.965926i 1.67303 + 0.448288i −0.866025 + 0.500000i 0 1.73205i −1.67303 + 0.448288i −0.707107 0.707107i 2.59808 + 1.50000i 0
443.1 −0.965926 + 0.258819i 0.448288 1.67303i 0.866025 0.500000i 0 1.73205i −0.448288 1.67303i −0.707107 + 0.707107i −2.59808 1.50000i 0
443.2 0.965926 0.258819i −0.448288 + 1.67303i 0.866025 0.500000i 0 1.73205i 0.448288 + 1.67303i 0.707107 0.707107i −2.59808 1.50000i 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.b even 2 1 inner
5.c odd 4 2 inner
9.d odd 6 1 inner
45.h odd 6 1 inner
45.l even 12 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.p.d 8
3.b odd 2 1 1350.2.q.b 8
5.b even 2 1 inner 450.2.p.d 8
5.c odd 4 2 inner 450.2.p.d 8
9.c even 3 1 1350.2.q.b 8
9.d odd 6 1 inner 450.2.p.d 8
15.d odd 2 1 1350.2.q.b 8
15.e even 4 2 1350.2.q.b 8
45.h odd 6 1 inner 450.2.p.d 8
45.j even 6 1 1350.2.q.b 8
45.k odd 12 2 1350.2.q.b 8
45.l even 12 2 inner 450.2.p.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.p.d 8 1.a even 1 1 trivial
450.2.p.d 8 5.b even 2 1 inner
450.2.p.d 8 5.c odd 4 2 inner
450.2.p.d 8 9.d odd 6 1 inner
450.2.p.d 8 45.h odd 6 1 inner
450.2.p.d 8 45.l even 12 2 inner
1350.2.q.b 8 3.b odd 2 1
1350.2.q.b 8 9.c even 3 1
1350.2.q.b 8 15.d odd 2 1
1350.2.q.b 8 15.e even 4 2
1350.2.q.b 8 45.j even 6 1
1350.2.q.b 8 45.k odd 12 2

## Hecke kernels

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

 $$T_{7}^{8} - 9 T_{7}^{4} + 81$$ $$T_{11}^{2} - 6 T_{11} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{4} + T^{8}$$
$3$ $$81 - 9 T^{4} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$81 - 9 T^{4} + T^{8}$$
$11$ $$( 12 - 6 T + T^{2} )^{4}$$
$13$ $$20736 - 144 T^{4} + T^{8}$$
$17$ $$( 1296 + T^{4} )^{2}$$
$19$ $$( 4 + T^{2} )^{4}$$
$23$ $$6561 - 81 T^{4} + T^{8}$$
$29$ $$( 5625 + 75 T^{2} + T^{4} )^{2}$$
$31$ $$( 100 + 10 T + T^{2} )^{4}$$
$37$ $$( 144 + T^{4} )^{2}$$
$41$ $$( 75 - 15 T + T^{2} )^{4}$$
$43$ $$136048896 - 11664 T^{4} + T^{8}$$
$47$ $$6561 - 81 T^{4} + T^{8}$$
$53$ $$T^{8}$$
$59$ $$( 2304 + 48 T^{2} + T^{4} )^{2}$$
$61$ $$( 169 - 13 T + T^{2} )^{4}$$
$67$ $$466948881 - 21609 T^{4} + T^{8}$$
$71$ $$( 12 + T^{2} )^{4}$$
$73$ $$( 36864 + T^{4} )^{2}$$
$79$ $$( 256 - 16 T^{2} + T^{4} )^{2}$$
$83$ $$6561 - 81 T^{4} + T^{8}$$
$89$ $$( -3 + T^{2} )^{4}$$
$97$ $$5308416 - 2304 T^{4} + T^{8}$$