# Properties

 Label 450.2.f.b Level $450$ Weight $2$ Character orbit 450.f 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.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

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

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( -2 + 2 \zeta_{8}^{2} ) q^{7} + \zeta_{8}^{3} q^{8} +O(q^{10})$$ $$q + \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( -2 + 2 \zeta_{8}^{2} ) q^{7} + \zeta_{8}^{3} q^{8} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{11} + ( 3 + 3 \zeta_{8}^{2} ) q^{13} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{14} - q^{16} -4 \zeta_{8} q^{17} + 8 \zeta_{8}^{2} q^{19} + ( -2 + 2 \zeta_{8}^{2} ) q^{22} -4 \zeta_{8}^{3} q^{23} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{26} + ( -2 - 2 \zeta_{8}^{2} ) q^{28} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{29} + 4 q^{31} -\zeta_{8} q^{32} -4 \zeta_{8}^{2} q^{34} + ( 3 - 3 \zeta_{8}^{2} ) q^{37} + 8 \zeta_{8}^{3} q^{38} + ( -7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{41} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{44} + 4 q^{46} -\zeta_{8}^{2} q^{49} + ( -3 + 3 \zeta_{8}^{2} ) q^{52} + 4 \zeta_{8}^{3} q^{53} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{56} + ( -1 - \zeta_{8}^{2} ) q^{58} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{59} + 4 \zeta_{8} q^{62} -\zeta_{8}^{2} q^{64} + ( 4 - 4 \zeta_{8}^{2} ) q^{67} -4 \zeta_{8}^{3} q^{68} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{71} + ( -1 - \zeta_{8}^{2} ) q^{73} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{74} -8 q^{76} -8 \zeta_{8} q^{77} -12 \zeta_{8}^{2} q^{79} + ( 7 - 7 \zeta_{8}^{2} ) q^{82} -12 \zeta_{8}^{3} q^{83} + ( -2 - 2 \zeta_{8}^{2} ) q^{88} + ( 7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{89} -12 q^{91} + 4 \zeta_{8} q^{92} + ( 3 - 3 \zeta_{8}^{2} ) q^{97} -\zeta_{8}^{3} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{7} + O(q^{10})$$ $$4q - 8q^{7} + 12q^{13} - 4q^{16} - 8q^{22} - 8q^{28} + 16q^{31} + 12q^{37} + 16q^{46} - 12q^{52} - 4q^{58} + 16q^{67} - 4q^{73} - 32q^{76} + 28q^{82} - 8q^{88} - 48q^{91} + 12q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{8}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
−0.707107 + 0.707107i 0 1.00000i 0 0 −2.00000 2.00000i 0.707107 + 0.707107i 0 0
107.2 0.707107 0.707107i 0 1.00000i 0 0 −2.00000 2.00000i −0.707107 0.707107i 0 0
143.1 −0.707107 0.707107i 0 1.00000i 0 0 −2.00000 + 2.00000i 0.707107 0.707107i 0 0
143.2 0.707107 + 0.707107i 0 1.00000i 0 0 −2.00000 + 2.00000i −0.707107 + 0.707107i 0 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
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.f.b 4
3.b odd 2 1 inner 450.2.f.b 4
4.b odd 2 1 3600.2.w.g 4
5.b even 2 1 90.2.f.a 4
5.c odd 4 1 90.2.f.a 4
5.c odd 4 1 inner 450.2.f.b 4
12.b even 2 1 3600.2.w.g 4
15.d odd 2 1 90.2.f.a 4
15.e even 4 1 90.2.f.a 4
15.e even 4 1 inner 450.2.f.b 4
20.d odd 2 1 720.2.w.a 4
20.e even 4 1 720.2.w.a 4
20.e even 4 1 3600.2.w.g 4
40.e odd 2 1 2880.2.w.c 4
40.f even 2 1 2880.2.w.l 4
40.i odd 4 1 2880.2.w.l 4
40.k even 4 1 2880.2.w.c 4
45.h odd 6 2 810.2.m.c 8
45.j even 6 2 810.2.m.c 8
45.k odd 12 2 810.2.m.c 8
45.l even 12 2 810.2.m.c 8
60.h even 2 1 720.2.w.a 4
60.l odd 4 1 720.2.w.a 4
60.l odd 4 1 3600.2.w.g 4
120.i odd 2 1 2880.2.w.l 4
120.m even 2 1 2880.2.w.c 4
120.q odd 4 1 2880.2.w.c 4
120.w even 4 1 2880.2.w.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.f.a 4 5.b even 2 1
90.2.f.a 4 5.c odd 4 1
90.2.f.a 4 15.d odd 2 1
90.2.f.a 4 15.e even 4 1
450.2.f.b 4 1.a even 1 1 trivial
450.2.f.b 4 3.b odd 2 1 inner
450.2.f.b 4 5.c odd 4 1 inner
450.2.f.b 4 15.e even 4 1 inner
720.2.w.a 4 20.d odd 2 1
720.2.w.a 4 20.e even 4 1
720.2.w.a 4 60.h even 2 1
720.2.w.a 4 60.l odd 4 1
810.2.m.c 8 45.h odd 6 2
810.2.m.c 8 45.j even 6 2
810.2.m.c 8 45.k odd 12 2
810.2.m.c 8 45.l even 12 2
2880.2.w.c 4 40.e odd 2 1
2880.2.w.c 4 40.k even 4 1
2880.2.w.c 4 120.m even 2 1
2880.2.w.c 4 120.q odd 4 1
2880.2.w.l 4 40.f even 2 1
2880.2.w.l 4 40.i odd 4 1
2880.2.w.l 4 120.i odd 2 1
2880.2.w.l 4 120.w even 4 1
3600.2.w.g 4 4.b odd 2 1
3600.2.w.g 4 12.b even 2 1
3600.2.w.g 4 20.e even 4 1
3600.2.w.g 4 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 4 T_{7} + 8$$ acting on $$S_{2}^{\mathrm{new}}(450, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 8 + 4 T + T^{2} )^{2}$$
$11$ $$( 8 + T^{2} )^{2}$$
$13$ $$( 18 - 6 T + T^{2} )^{2}$$
$17$ $$256 + T^{4}$$
$19$ $$( 64 + T^{2} )^{2}$$
$23$ $$256 + T^{4}$$
$29$ $$( -2 + T^{2} )^{2}$$
$31$ $$( -4 + T )^{4}$$
$37$ $$( 18 - 6 T + T^{2} )^{2}$$
$41$ $$( 98 + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$256 + T^{4}$$
$59$ $$( -8 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$( 32 - 8 T + T^{2} )^{2}$$
$71$ $$( 32 + T^{2} )^{2}$$
$73$ $$( 2 + 2 T + T^{2} )^{2}$$
$79$ $$( 144 + T^{2} )^{2}$$
$83$ $$20736 + T^{4}$$
$89$ $$( -98 + T^{2} )^{2}$$
$97$ $$( 18 - 6 T + T^{2} )^{2}$$