# Properties

 Label 450.2.j.e 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 18) 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.

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

## 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 1.50000 + 0.866025i −1.73205 + 1.00000i 1.00000i −1.50000 + 2.59808i 0
49.2 0.866025 0.500000i 0.866025 + 1.50000i 0.500000 0.866025i 0 1.50000 + 0.866025i 1.73205 1.00000i 1.00000i −1.50000 + 2.59808i 0
349.1 −0.866025 0.500000i −0.866025 + 1.50000i 0.500000 + 0.866025i 0 1.50000 0.866025i −1.73205 1.00000i 1.00000i −1.50000 2.59808i 0
349.2 0.866025 + 0.500000i 0.866025 1.50000i 0.500000 + 0.866025i 0 1.50000 0.866025i 1.73205 + 1.00000i 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.e 4
3.b odd 2 1 1350.2.j.a 4
5.b even 2 1 inner 450.2.j.e 4
5.c odd 4 1 18.2.c.a 2
5.c odd 4 1 450.2.e.i 2
9.c even 3 1 inner 450.2.j.e 4
9.c even 3 1 4050.2.c.c 2
9.d odd 6 1 1350.2.j.a 4
9.d odd 6 1 4050.2.c.r 2
15.d odd 2 1 1350.2.j.a 4
15.e even 4 1 54.2.c.a 2
15.e even 4 1 1350.2.e.c 2
20.e even 4 1 144.2.i.c 2
35.f even 4 1 882.2.f.d 2
35.k even 12 1 882.2.e.g 2
35.k even 12 1 882.2.h.b 2
35.l odd 12 1 882.2.e.i 2
35.l odd 12 1 882.2.h.c 2
40.i odd 4 1 576.2.i.g 2
40.k even 4 1 576.2.i.a 2
45.h odd 6 1 1350.2.j.a 4
45.h odd 6 1 4050.2.c.r 2
45.j even 6 1 inner 450.2.j.e 4
45.j even 6 1 4050.2.c.c 2
45.k odd 12 1 18.2.c.a 2
45.k odd 12 1 162.2.a.c 1
45.k odd 12 1 450.2.e.i 2
45.k odd 12 1 4050.2.a.c 1
45.l even 12 1 54.2.c.a 2
45.l even 12 1 162.2.a.b 1
45.l even 12 1 1350.2.e.c 2
45.l even 12 1 4050.2.a.v 1
60.l odd 4 1 432.2.i.b 2
105.k odd 4 1 2646.2.f.g 2
105.w odd 12 1 2646.2.e.c 2
105.w odd 12 1 2646.2.h.i 2
105.x even 12 1 2646.2.e.b 2
105.x even 12 1 2646.2.h.h 2
120.q odd 4 1 1728.2.i.f 2
120.w even 4 1 1728.2.i.e 2
180.v odd 12 1 432.2.i.b 2
180.v odd 12 1 1296.2.a.f 1
180.x even 12 1 144.2.i.c 2
180.x even 12 1 1296.2.a.g 1
315.bs even 12 1 882.2.h.b 2
315.bt odd 12 1 882.2.h.c 2
315.bu odd 12 1 2646.2.h.i 2
315.bv even 12 1 2646.2.h.h 2
315.bw odd 12 1 2646.2.e.c 2
315.bx even 12 1 2646.2.e.b 2
315.cb even 12 1 882.2.f.d 2
315.cb even 12 1 7938.2.a.x 1
315.cf odd 12 1 2646.2.f.g 2
315.cf odd 12 1 7938.2.a.i 1
315.cg even 12 1 882.2.e.g 2
315.ch odd 12 1 882.2.e.i 2
360.bo even 12 1 576.2.i.a 2
360.bo even 12 1 5184.2.a.o 1
360.br even 12 1 1728.2.i.e 2
360.br even 12 1 5184.2.a.q 1
360.bt odd 12 1 1728.2.i.f 2
360.bt odd 12 1 5184.2.a.p 1
360.bu odd 12 1 576.2.i.g 2
360.bu odd 12 1 5184.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.2.c.a 2 5.c odd 4 1
18.2.c.a 2 45.k odd 12 1
54.2.c.a 2 15.e even 4 1
54.2.c.a 2 45.l even 12 1
144.2.i.c 2 20.e even 4 1
144.2.i.c 2 180.x even 12 1
162.2.a.b 1 45.l even 12 1
162.2.a.c 1 45.k odd 12 1
432.2.i.b 2 60.l odd 4 1
432.2.i.b 2 180.v odd 12 1
450.2.e.i 2 5.c odd 4 1
450.2.e.i 2 45.k odd 12 1
450.2.j.e 4 1.a even 1 1 trivial
450.2.j.e 4 5.b even 2 1 inner
450.2.j.e 4 9.c even 3 1 inner
450.2.j.e 4 45.j even 6 1 inner
576.2.i.a 2 40.k even 4 1
576.2.i.a 2 360.bo even 12 1
576.2.i.g 2 40.i odd 4 1
576.2.i.g 2 360.bu odd 12 1
882.2.e.g 2 35.k even 12 1
882.2.e.g 2 315.cg even 12 1
882.2.e.i 2 35.l odd 12 1
882.2.e.i 2 315.ch odd 12 1
882.2.f.d 2 35.f even 4 1
882.2.f.d 2 315.cb even 12 1
882.2.h.b 2 35.k even 12 1
882.2.h.b 2 315.bs even 12 1
882.2.h.c 2 35.l odd 12 1
882.2.h.c 2 315.bt odd 12 1
1296.2.a.f 1 180.v odd 12 1
1296.2.a.g 1 180.x even 12 1
1350.2.e.c 2 15.e even 4 1
1350.2.e.c 2 45.l even 12 1
1350.2.j.a 4 3.b odd 2 1
1350.2.j.a 4 9.d odd 6 1
1350.2.j.a 4 15.d odd 2 1
1350.2.j.a 4 45.h odd 6 1
1728.2.i.e 2 120.w even 4 1
1728.2.i.e 2 360.br even 12 1
1728.2.i.f 2 120.q odd 4 1
1728.2.i.f 2 360.bt odd 12 1
2646.2.e.b 2 105.x even 12 1
2646.2.e.b 2 315.bx even 12 1
2646.2.e.c 2 105.w odd 12 1
2646.2.e.c 2 315.bw odd 12 1
2646.2.f.g 2 105.k odd 4 1
2646.2.f.g 2 315.cf odd 12 1
2646.2.h.h 2 105.x even 12 1
2646.2.h.h 2 315.bv even 12 1
2646.2.h.i 2 105.w odd 12 1
2646.2.h.i 2 315.bu odd 12 1
4050.2.a.c 1 45.k odd 12 1
4050.2.a.v 1 45.l even 12 1
4050.2.c.c 2 9.c even 3 1
4050.2.c.c 2 45.j even 6 1
4050.2.c.r 2 9.d odd 6 1
4050.2.c.r 2 45.h odd 6 1
5184.2.a.o 1 360.bo even 12 1
5184.2.a.p 1 360.bt odd 12 1
5184.2.a.q 1 360.br even 12 1
5184.2.a.r 1 360.bu odd 12 1
7938.2.a.i 1 315.cf odd 12 1
7938.2.a.x 1 315.cb even 12 1

## Hecke kernels

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

 $$T_{7}^{4} - 4 T_{7}^{2} + 16$$ $$T_{11}^{2} - 3 T_{11} + 9$$ $$T_{19} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$9 + 3 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$16 - 4 T^{2} + T^{4}$$
$11$ $$( 9 - 3 T + T^{2} )^{2}$$
$13$ $$16 - 4 T^{2} + T^{4}$$
$17$ $$( 9 + T^{2} )^{2}$$
$19$ $$( -1 + T )^{4}$$
$23$ $$1296 - 36 T^{2} + T^{4}$$
$29$ $$( 36 - 6 T + T^{2} )^{2}$$
$31$ $$( 16 - 4 T + T^{2} )^{2}$$
$37$ $$( 16 + T^{2} )^{2}$$
$41$ $$( 81 + 9 T + T^{2} )^{2}$$
$43$ $$1 - T^{2} + T^{4}$$
$47$ $$1296 - 36 T^{2} + T^{4}$$
$53$ $$( 144 + T^{2} )^{2}$$
$59$ $$( 9 - 3 T + T^{2} )^{2}$$
$61$ $$( 64 + 8 T + T^{2} )^{2}$$
$67$ $$625 - 25 T^{2} + T^{4}$$
$71$ $$( 12 + T )^{4}$$
$73$ $$( 121 + T^{2} )^{2}$$
$79$ $$( 16 + 4 T + T^{2} )^{2}$$
$83$ $$20736 - 144 T^{2} + T^{4}$$
$89$ $$( 6 + T )^{4}$$
$97$ $$625 - 25 T^{2} + T^{4}$$