Properties

 Label 450.2.j.c 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.

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

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 1.73205i 0.866025 0.500000i 1.00000i −1.50000 2.59808i 0
49.2 0.866025 0.500000i 0.866025 1.50000i 0.500000 0.866025i 0 1.73205i −0.866025 + 0.500000i 1.00000i −1.50000 2.59808i 0
349.1 −0.866025 0.500000i −0.866025 1.50000i 0.500000 + 0.866025i 0 1.73205i 0.866025 + 0.500000i 1.00000i −1.50000 + 2.59808i 0
349.2 0.866025 + 0.500000i 0.866025 + 1.50000i 0.500000 + 0.866025i 0 1.73205i −0.866025 0.500000i 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.c 4
3.b odd 2 1 1350.2.j.e 4
5.b even 2 1 inner 450.2.j.c 4
5.c odd 4 1 90.2.e.a 2
5.c odd 4 1 450.2.e.e 2
9.c even 3 1 inner 450.2.j.c 4
9.c even 3 1 4050.2.c.t 2
9.d odd 6 1 1350.2.j.e 4
9.d odd 6 1 4050.2.c.a 2
15.d odd 2 1 1350.2.j.e 4
15.e even 4 1 270.2.e.b 2
15.e even 4 1 1350.2.e.b 2
20.e even 4 1 720.2.q.b 2
45.h odd 6 1 1350.2.j.e 4
45.h odd 6 1 4050.2.c.a 2
45.j even 6 1 inner 450.2.j.c 4
45.j even 6 1 4050.2.c.t 2
45.k odd 12 1 90.2.e.a 2
45.k odd 12 1 450.2.e.e 2
45.k odd 12 1 810.2.a.g 1
45.k odd 12 1 4050.2.a.n 1
45.l even 12 1 270.2.e.b 2
45.l even 12 1 810.2.a.b 1
45.l even 12 1 1350.2.e.b 2
45.l even 12 1 4050.2.a.ba 1
60.l odd 4 1 2160.2.q.b 2
180.v odd 12 1 2160.2.q.b 2
180.v odd 12 1 6480.2.a.v 1
180.x even 12 1 720.2.q.b 2
180.x even 12 1 6480.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.e.a 2 5.c odd 4 1
90.2.e.a 2 45.k odd 12 1
270.2.e.b 2 15.e even 4 1
270.2.e.b 2 45.l even 12 1
450.2.e.e 2 5.c odd 4 1
450.2.e.e 2 45.k odd 12 1
450.2.j.c 4 1.a even 1 1 trivial
450.2.j.c 4 5.b even 2 1 inner
450.2.j.c 4 9.c even 3 1 inner
450.2.j.c 4 45.j even 6 1 inner
720.2.q.b 2 20.e even 4 1
720.2.q.b 2 180.x even 12 1
810.2.a.b 1 45.l even 12 1
810.2.a.g 1 45.k odd 12 1
1350.2.e.b 2 15.e even 4 1
1350.2.e.b 2 45.l even 12 1
1350.2.j.e 4 3.b odd 2 1
1350.2.j.e 4 9.d odd 6 1
1350.2.j.e 4 15.d odd 2 1
1350.2.j.e 4 45.h odd 6 1
2160.2.q.b 2 60.l odd 4 1
2160.2.q.b 2 180.v odd 12 1
4050.2.a.n 1 45.k odd 12 1
4050.2.a.ba 1 45.l even 12 1
4050.2.c.a 2 9.d odd 6 1
4050.2.c.a 2 45.h odd 6 1
4050.2.c.t 2 9.c even 3 1
4050.2.c.t 2 45.j even 6 1
6480.2.a.g 1 180.x even 12 1
6480.2.a.v 1 180.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 $$S_{2}^{\mathrm{new}}(450, [\chi])$$:

 $$T_{7}^{4} - T_{7}^{2} + 1$$ $$T_{11}^{2} + 6 T_{11} + 36$$ $$T_{19} - 4$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$9 + 3 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$1 - T^{2} + T^{4}$$
$11$ $$( 36 + 6 T + T^{2} )^{2}$$
$13$ $$16 - 4 T^{2} + T^{4}$$
$17$ $$T^{4}$$
$19$ $$( -4 + T )^{4}$$
$23$ $$6561 - 81 T^{2} + T^{4}$$
$29$ $$( 9 - 3 T + T^{2} )^{2}$$
$31$ $$( 16 - 4 T + T^{2} )^{2}$$
$37$ $$( 64 + T^{2} )^{2}$$
$41$ $$( 9 - 3 T + T^{2} )^{2}$$
$43$ $$4096 - 64 T^{2} + T^{4}$$
$47$ $$81 - 9 T^{2} + T^{4}$$
$53$ $$( 36 + T^{2} )^{2}$$
$59$ $$( 36 - 6 T + T^{2} )^{2}$$
$61$ $$( 169 - 13 T + T^{2} )^{2}$$
$67$ $$28561 - 169 T^{2} + T^{4}$$
$71$ $$( 6 + T )^{4}$$
$73$ $$( 16 + T^{2} )^{2}$$
$79$ $$( 100 + 10 T + T^{2} )^{2}$$
$83$ $$6561 - 81 T^{2} + T^{4}$$
$89$ $$( 9 + T )^{4}$$
$97$ $$16 - 4 T^{2} + T^{4}$$