# Properties

 Label 450.2.j.g 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: 8.0.303595776.1 Defining polynomial: $$x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + ( \beta_{1} - \beta_{3} - \beta_{5} - \beta_{7} ) q^{3} -\beta_{4} q^{4} + ( \beta_{2} - \beta_{6} ) q^{6} + ( 2 \beta_{1} - \beta_{3} - \beta_{5} - \beta_{7} ) q^{7} + ( \beta_{3} + \beta_{5} ) q^{8} + ( 3 + \beta_{6} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{2} + ( \beta_{1} - \beta_{3} - \beta_{5} - \beta_{7} ) q^{3} -\beta_{4} q^{4} + ( \beta_{2} - \beta_{6} ) q^{6} + ( 2 \beta_{1} - \beta_{3} - \beta_{5} - \beta_{7} ) q^{7} + ( \beta_{3} + \beta_{5} ) q^{8} + ( 3 + \beta_{6} ) q^{9} + ( 1 - \beta_{2} + 2 \beta_{4} - \beta_{6} ) q^{11} + ( \beta_{1} - \beta_{5} ) q^{12} + ( -2 \beta_{1} + 2 \beta_{3} + 4 \beta_{7} ) q^{13} + ( -1 + \beta_{2} - 2 \beta_{6} ) q^{14} + ( -1 - \beta_{4} ) q^{16} + ( -\beta_{1} - 5 \beta_{3} - 4 \beta_{5} - \beta_{7} ) q^{17} + ( 2 \beta_{3} - \beta_{7} ) q^{18} + ( -2 \beta_{2} + \beta_{4} + \beta_{6} ) q^{19} + ( 3 - 3 \beta_{4} + \beta_{6} ) q^{21} + ( -\beta_{1} + \beta_{3} - \beta_{5} + 2 \beta_{7} ) q^{22} + ( \beta_{1} - \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{23} -\beta_{6} q^{24} + ( 2 - 4 \beta_{2} + 2 \beta_{4} + 2 \beta_{6} ) q^{26} + ( 2 \beta_{1} - 5 \beta_{3} - 5 \beta_{5} - 2 \beta_{7} ) q^{27} + ( \beta_{1} - \beta_{5} + \beta_{7} ) q^{28} + ( -2 - \beta_{2} - \beta_{4} - \beta_{6} ) q^{29} + ( -2 + 2 \beta_{2} - 2 \beta_{4} - 4 \beta_{6} ) q^{31} + \beta_{5} q^{32} + ( \beta_{1} + \beta_{3} + 5 \beta_{5} - 2 \beta_{7} ) q^{33} + ( 5 + \beta_{2} + 4 \beta_{4} + \beta_{6} ) q^{34} + ( \beta_{2} - 3 \beta_{4} ) q^{36} + ( -4 \beta_{3} - 4 \beta_{5} ) q^{37} + ( -2 \beta_{1} + \beta_{5} + \beta_{7} ) q^{38} + ( -12 + 2 \beta_{2} - 6 \beta_{4} - 2 \beta_{6} ) q^{39} -3 \beta_{4} q^{41} + ( 5 \beta_{3} + 3 \beta_{5} - \beta_{7} ) q^{42} + ( -2 \beta_{1} - 8 \beta_{3} + \beta_{5} + \beta_{7} ) q^{43} + ( 2 - 2 \beta_{2} + \beta_{4} + \beta_{6} ) q^{44} + ( 1 + 2 \beta_{2} - \beta_{4} - \beta_{6} ) q^{46} + ( -2 \beta_{1} + 5 \beta_{3} + \beta_{5} + \beta_{7} ) q^{47} + ( \beta_{3} + \beta_{7} ) q^{48} + ( 1 - \beta_{2} - \beta_{4} + 2 \beta_{6} ) q^{49} + ( 3 - \beta_{2} + 6 \beta_{4} + 5 \beta_{6} ) q^{51} + ( -4 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{7} ) q^{52} + ( -4 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} - 4 \beta_{7} ) q^{53} + ( 3 + 2 \beta_{2} + 3 \beta_{4} - 2 \beta_{6} ) q^{54} + ( -\beta_{2} + \beta_{4} - \beta_{6} ) q^{56} + ( -2 \beta_{3} + 3 \beta_{5} + \beta_{7} ) q^{57} + ( -\beta_{1} + \beta_{3} + 2 \beta_{5} + 2 \beta_{7} ) q^{58} + ( 1 - \beta_{2} - \beta_{4} + 2 \beta_{6} ) q^{59} + ( -2 - 3 \beta_{2} + \beta_{4} - 3 \beta_{6} ) q^{61} + ( 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{7} ) q^{62} + ( 5 \beta_{1} - 5 \beta_{3} - 8 \beta_{5} - 2 \beta_{7} ) q^{63} - q^{64} + ( -6 + 2 \beta_{2} - 3 \beta_{4} - \beta_{6} ) q^{66} + 7 \beta_{5} q^{67} + ( \beta_{1} - \beta_{3} - 5 \beta_{5} - 2 \beta_{7} ) q^{68} + ( 6 + \beta_{2} + 3 \beta_{4} + \beta_{6} ) q^{69} -6 q^{71} + ( \beta_{1} + 2 \beta_{3} + 2 \beta_{5} - \beta_{7} ) q^{72} + ( 3 \beta_{1} + 7 \beta_{3} + 4 \beta_{5} + 3 \beta_{7} ) q^{73} + ( 4 + 4 \beta_{4} ) q^{74} + ( 1 - \beta_{2} + \beta_{4} + 2 \beta_{6} ) q^{76} + ( 2 \beta_{1} - 2 \beta_{3} + 8 \beta_{5} - 4 \beta_{7} ) q^{77} + ( 2 \beta_{1} - 6 \beta_{3} + 4 \beta_{5} ) q^{78} + ( 2 + 2 \beta_{4} ) q^{79} + ( 6 + 5 \beta_{6} ) q^{81} + ( 3 \beta_{3} + 3 \beta_{5} ) q^{82} + ( -2 \beta_{1} + 5 \beta_{3} + \beta_{5} + \beta_{7} ) q^{83} + ( -3 + \beta_{2} - 6 \beta_{4} ) q^{84} + ( 1 - \beta_{2} + 9 \beta_{4} + 2 \beta_{6} ) q^{86} + ( \beta_{1} + 4 \beta_{3} + 5 \beta_{5} + \beta_{7} ) q^{87} + ( -2 \beta_{1} + 2 \beta_{3} + \beta_{5} + \beta_{7} ) q^{88} + ( -9 + 6 \beta_{2} - 3 \beta_{4} - 3 \beta_{6} ) q^{89} + ( -18 + 4 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} ) q^{91} + ( 2 \beta_{1} + \beta_{3} - \beta_{5} - \beta_{7} ) q^{92} + ( 2 \beta_{1} + 10 \beta_{3} + 4 \beta_{5} - 2 \beta_{7} ) q^{93} + ( 1 - \beta_{2} - 4 \beta_{4} + 2 \beta_{6} ) q^{94} -\beta_{2} q^{96} + ( -2 \beta_{1} + 6 \beta_{3} + \beta_{5} + \beta_{7} ) q^{97} + ( -\beta_{1} + \beta_{3} + 2 \beta_{5} - \beta_{7} ) q^{98} + ( 6 - 4 \beta_{2} + 3 \beta_{4} - \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{4} + 2q^{6} + 20q^{9} + O(q^{10})$$ $$8q + 4q^{4} + 2q^{6} + 20q^{9} + 6q^{11} - 2q^{14} - 4q^{16} - 4q^{19} + 32q^{21} + 4q^{24} + 8q^{26} - 6q^{29} + 4q^{31} + 18q^{34} + 10q^{36} - 68q^{39} + 12q^{41} + 12q^{44} + 12q^{46} + 6q^{49} - 18q^{51} + 16q^{54} + 2q^{56} + 6q^{59} - 2q^{61} - 8q^{64} - 36q^{66} + 30q^{69} - 48q^{71} + 16q^{74} - 2q^{76} + 8q^{79} + 28q^{81} - 2q^{84} - 34q^{86} - 60q^{89} - 136q^{91} + 18q^{94} + 2q^{96} + 48q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 32 \nu^{4} + 16 \nu^{2} + 45$$$$)/144$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 32 \nu^{5} + 16 \nu^{3} + 45 \nu$$$$)/432$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{6} - 16 \nu^{4} - 80 \nu^{2} - 225$$$$)/144$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 13 \nu$$$$)/48$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} - 13$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{7} + 16 \nu^{5} + 80 \nu^{3} + 225 \nu$$$$)/144$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - 2 \beta_{4} - \beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{7} - 3 \beta_{5} - 3 \beta_{3} - 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + 5 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$-\beta_{7} + 15 \beta_{3}$$ $$\nu^{6}$$ $$=$$ $$-16 \beta_{6} - 13$$ $$\nu^{7}$$ $$=$$ $$48 \beta_{5} - 13 \beta_{1}$$

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$\beta_{4}$$ $$-1$$

## 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
 −0.396143 + 1.68614i 1.26217 − 1.18614i −1.26217 + 1.18614i 0.396143 − 1.68614i −0.396143 − 1.68614i 1.26217 + 1.18614i −1.26217 − 1.18614i 0.396143 + 1.68614i
−0.866025 + 0.500000i −1.65831 0.500000i 0.500000 0.866025i 0 1.68614 0.396143i −2.05446 + 1.18614i 1.00000i 2.50000 + 1.65831i 0
49.2 −0.866025 + 0.500000i 1.65831 0.500000i 0.500000 0.866025i 0 −1.18614 + 1.26217i 2.92048 1.68614i 1.00000i 2.50000 1.65831i 0
49.3 0.866025 0.500000i −1.65831 + 0.500000i 0.500000 0.866025i 0 −1.18614 + 1.26217i −2.92048 + 1.68614i 1.00000i 2.50000 1.65831i 0
49.4 0.866025 0.500000i 1.65831 + 0.500000i 0.500000 0.866025i 0 1.68614 0.396143i 2.05446 1.18614i 1.00000i 2.50000 + 1.65831i 0
349.1 −0.866025 0.500000i −1.65831 + 0.500000i 0.500000 + 0.866025i 0 1.68614 + 0.396143i −2.05446 1.18614i 1.00000i 2.50000 1.65831i 0
349.2 −0.866025 0.500000i 1.65831 + 0.500000i 0.500000 + 0.866025i 0 −1.18614 1.26217i 2.92048 + 1.68614i 1.00000i 2.50000 + 1.65831i 0
349.3 0.866025 + 0.500000i −1.65831 0.500000i 0.500000 + 0.866025i 0 −1.18614 1.26217i −2.92048 1.68614i 1.00000i 2.50000 + 1.65831i 0
349.4 0.866025 + 0.500000i 1.65831 0.500000i 0.500000 + 0.866025i 0 1.68614 + 0.396143i 2.05446 + 1.18614i 1.00000i 2.50000 1.65831i 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.g 8
3.b odd 2 1 1350.2.j.f 8
5.b even 2 1 inner 450.2.j.g 8
5.c odd 4 1 90.2.e.c 4
5.c odd 4 1 450.2.e.j 4
9.c even 3 1 inner 450.2.j.g 8
9.c even 3 1 4050.2.c.v 4
9.d odd 6 1 1350.2.j.f 8
9.d odd 6 1 4050.2.c.ba 4
15.d odd 2 1 1350.2.j.f 8
15.e even 4 1 270.2.e.c 4
15.e even 4 1 1350.2.e.l 4
20.e even 4 1 720.2.q.f 4
45.h odd 6 1 1350.2.j.f 8
45.h odd 6 1 4050.2.c.ba 4
45.j even 6 1 inner 450.2.j.g 8
45.j even 6 1 4050.2.c.v 4
45.k odd 12 1 90.2.e.c 4
45.k odd 12 1 450.2.e.j 4
45.k odd 12 1 810.2.a.i 2
45.k odd 12 1 4050.2.a.bw 2
45.l even 12 1 270.2.e.c 4
45.l even 12 1 810.2.a.k 2
45.l even 12 1 1350.2.e.l 4
45.l even 12 1 4050.2.a.bo 2
60.l odd 4 1 2160.2.q.f 4
180.v odd 12 1 2160.2.q.f 4
180.v odd 12 1 6480.2.a.bn 2
180.x even 12 1 720.2.q.f 4
180.x even 12 1 6480.2.a.be 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.e.c 4 5.c odd 4 1
90.2.e.c 4 45.k odd 12 1
270.2.e.c 4 15.e even 4 1
270.2.e.c 4 45.l even 12 1
450.2.e.j 4 5.c odd 4 1
450.2.e.j 4 45.k odd 12 1
450.2.j.g 8 1.a even 1 1 trivial
450.2.j.g 8 5.b even 2 1 inner
450.2.j.g 8 9.c even 3 1 inner
450.2.j.g 8 45.j even 6 1 inner
720.2.q.f 4 20.e even 4 1
720.2.q.f 4 180.x even 12 1
810.2.a.i 2 45.k odd 12 1
810.2.a.k 2 45.l even 12 1
1350.2.e.l 4 15.e even 4 1
1350.2.e.l 4 45.l even 12 1
1350.2.j.f 8 3.b odd 2 1
1350.2.j.f 8 9.d odd 6 1
1350.2.j.f 8 15.d odd 2 1
1350.2.j.f 8 45.h odd 6 1
2160.2.q.f 4 60.l odd 4 1
2160.2.q.f 4 180.v odd 12 1
4050.2.a.bo 2 45.l even 12 1
4050.2.a.bw 2 45.k odd 12 1
4050.2.c.v 4 9.c even 3 1
4050.2.c.v 4 45.j even 6 1
4050.2.c.ba 4 9.d odd 6 1
4050.2.c.ba 4 45.h odd 6 1
6480.2.a.be 2 180.x even 12 1
6480.2.a.bn 2 180.v odd 12 1

## Hecke kernels

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

 $$T_{7}^{8} - 17 T_{7}^{6} + 225 T_{7}^{4} - 1088 T_{7}^{2} + 4096$$ $$T_{11}^{4} - 3 T_{11}^{3} + 15 T_{11}^{2} + 18 T_{11} + 36$$ $$T_{19}^{2} + T_{19} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ $$( 9 - 5 T^{2} + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$4096 - 1088 T^{2} + 225 T^{4} - 17 T^{6} + T^{8}$$
$11$ $$( 36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4} )^{2}$$
$13$ $$1048576 - 69632 T^{2} + 3600 T^{4} - 68 T^{6} + T^{8}$$
$17$ $$( 144 + 57 T^{2} + T^{4} )^{2}$$
$19$ $$( -8 + T + T^{2} )^{4}$$
$23$ $$1296 - 756 T^{2} + 405 T^{4} - 21 T^{6} + T^{8}$$
$29$ $$( 36 - 18 T + 15 T^{2} + 3 T^{3} + T^{4} )^{2}$$
$31$ $$( 1024 + 64 T + 36 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$37$ $$( 16 + T^{2} )^{4}$$
$41$ $$( 9 - 3 T + T^{2} )^{4}$$
$43$ $$16777216 - 659456 T^{2} + 21825 T^{4} - 161 T^{6} + T^{8}$$
$47$ $$20736 - 8208 T^{2} + 3105 T^{4} - 57 T^{6} + T^{8}$$
$53$ $$( 132 + T^{2} )^{4}$$
$59$ $$( 36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4} )^{2}$$
$61$ $$( 5476 - 74 T + 75 T^{2} + T^{3} + T^{4} )^{2}$$
$67$ $$( 2401 - 49 T^{2} + T^{4} )^{2}$$
$71$ $$( 6 + T )^{8}$$
$73$ $$( 1936 + 209 T^{2} + T^{4} )^{2}$$
$79$ $$( 4 - 2 T + T^{2} )^{4}$$
$83$ $$20736 - 8208 T^{2} + 3105 T^{4} - 57 T^{6} + T^{8}$$
$89$ $$( -18 + 15 T + T^{2} )^{4}$$
$97$ $$234256 - 37268 T^{2} + 5445 T^{4} - 77 T^{6} + T^{8}$$