# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 450.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

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

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} + 2 i q^{7} -i q^{8} +O(q^{10})$$ $$q + i q^{2} - q^{4} + 2 i q^{7} -i q^{8} -6 q^{11} + 4 i q^{13} -2 q^{14} + q^{16} + 6 i q^{17} + 4 q^{19} -6 i q^{22} -4 q^{26} -2 i q^{28} -6 q^{29} -4 q^{31} + i q^{32} -6 q^{34} + 8 i q^{37} + 4 i q^{38} -8 i q^{43} + 6 q^{44} + 3 q^{49} -4 i q^{52} -6 i q^{53} + 2 q^{56} -6 i q^{58} + 6 q^{59} + 2 q^{61} -4 i q^{62} - q^{64} -4 i q^{67} -6 i q^{68} + 12 q^{71} + 10 i q^{73} -8 q^{74} -4 q^{76} -12 i q^{77} + 4 q^{79} + 12 i q^{83} + 8 q^{86} + 6 i q^{88} + 12 q^{89} -8 q^{91} + 2 i q^{97} + 3 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} - 12q^{11} - 4q^{14} + 2q^{16} + 8q^{19} - 8q^{26} - 12q^{29} - 8q^{31} - 12q^{34} + 12q^{44} + 6q^{49} + 4q^{56} + 12q^{59} + 4q^{61} - 2q^{64} + 24q^{71} - 16q^{74} - 8q^{76} + 8q^{79} + 16q^{86} + 24q^{89} - 16q^{91} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 − 1.00000i 1.00000i
1.00000i 0 −1.00000 0 0 2.00000i 1.00000i 0 0
199.2 1.00000i 0 −1.00000 0 0 2.00000i 1.00000i 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.c.a 2
3.b odd 2 1 450.2.c.d 2
4.b odd 2 1 3600.2.f.u 2
5.b even 2 1 inner 450.2.c.a 2
5.c odd 4 1 90.2.a.b yes 1
5.c odd 4 1 450.2.a.a 1
12.b even 2 1 3600.2.f.a 2
15.d odd 2 1 450.2.c.d 2
15.e even 4 1 90.2.a.a 1
15.e even 4 1 450.2.a.e 1
20.d odd 2 1 3600.2.f.u 2
20.e even 4 1 720.2.a.b 1
20.e even 4 1 3600.2.a.bj 1
35.f even 4 1 4410.2.a.bf 1
40.i odd 4 1 2880.2.a.bf 1
40.k even 4 1 2880.2.a.u 1
45.k odd 12 2 810.2.e.e 2
45.l even 12 2 810.2.e.h 2
60.h even 2 1 3600.2.f.a 2
60.l odd 4 1 720.2.a.g 1
60.l odd 4 1 3600.2.a.ba 1
105.k odd 4 1 4410.2.a.k 1
120.q odd 4 1 2880.2.a.h 1
120.w even 4 1 2880.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.a.a 1 15.e even 4 1
90.2.a.b yes 1 5.c odd 4 1
450.2.a.a 1 5.c odd 4 1
450.2.a.e 1 15.e even 4 1
450.2.c.a 2 1.a even 1 1 trivial
450.2.c.a 2 5.b even 2 1 inner
450.2.c.d 2 3.b odd 2 1
450.2.c.d 2 15.d odd 2 1
720.2.a.b 1 20.e even 4 1
720.2.a.g 1 60.l odd 4 1
810.2.e.e 2 45.k odd 12 2
810.2.e.h 2 45.l even 12 2
2880.2.a.h 1 120.q odd 4 1
2880.2.a.k 1 120.w even 4 1
2880.2.a.u 1 40.k even 4 1
2880.2.a.bf 1 40.i odd 4 1
3600.2.a.ba 1 60.l odd 4 1
3600.2.a.bj 1 20.e even 4 1
3600.2.f.a 2 12.b even 2 1
3600.2.f.a 2 60.h even 2 1
3600.2.f.u 2 4.b odd 2 1
3600.2.f.u 2 20.d odd 2 1
4410.2.a.k 1 105.k odd 4 1
4410.2.a.bf 1 35.f even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 4$$ $$T_{11} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$( 6 + T )^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$36 + T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$64 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( -6 + T )^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$( -4 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( -12 + T )^{2}$$
$97$ $$4 + T^{2}$$