# Properties

 Label 450.2.c.c 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 50) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} + 2 i q^{7} -i q^{8} +O(q^{10})$$ $$q + i q^{2} - q^{4} + 2 i q^{7} -i q^{8} + 3 q^{11} + 4 i q^{13} -2 q^{14} + q^{16} + 3 i q^{17} -5 q^{19} + 3 i q^{22} + 6 i q^{23} -4 q^{26} -2 i q^{28} + 2 q^{31} + i q^{32} -3 q^{34} + 2 i q^{37} -5 i q^{38} + 3 q^{41} + 4 i q^{43} -3 q^{44} -6 q^{46} -12 i q^{47} + 3 q^{49} -4 i q^{52} + 6 i q^{53} + 2 q^{56} + 2 q^{61} + 2 i q^{62} - q^{64} -13 i q^{67} -3 i q^{68} -12 q^{71} -11 i q^{73} -2 q^{74} + 5 q^{76} + 6 i q^{77} + 10 q^{79} + 3 i q^{82} -9 i q^{83} -4 q^{86} -3 i q^{88} + 15 q^{89} -8 q^{91} -6 i q^{92} + 12 q^{94} + 2 i q^{97} + 3 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} + 6q^{11} - 4q^{14} + 2q^{16} - 10q^{19} - 8q^{26} + 4q^{31} - 6q^{34} + 6q^{41} - 6q^{44} - 12q^{46} + 6q^{49} + 4q^{56} + 4q^{61} - 2q^{64} - 24q^{71} - 4q^{74} + 10q^{76} + 20q^{79} - 8q^{86} + 30q^{89} - 16q^{91} + 24q^{94} + O(q^{100})$$

## 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.c 2
3.b odd 2 1 50.2.b.a 2
4.b odd 2 1 3600.2.f.f 2
5.b even 2 1 inner 450.2.c.c 2
5.c odd 4 1 450.2.a.c 1
5.c odd 4 1 450.2.a.g 1
12.b even 2 1 400.2.c.c 2
15.d odd 2 1 50.2.b.a 2
15.e even 4 1 50.2.a.a 1
15.e even 4 1 50.2.a.b yes 1
20.d odd 2 1 3600.2.f.f 2
20.e even 4 1 3600.2.a.l 1
20.e even 4 1 3600.2.a.bc 1
21.c even 2 1 2450.2.c.m 2
24.f even 2 1 1600.2.c.h 2
24.h odd 2 1 1600.2.c.i 2
60.h even 2 1 400.2.c.c 2
60.l odd 4 1 400.2.a.d 1
60.l odd 4 1 400.2.a.f 1
105.g even 2 1 2450.2.c.m 2
105.k odd 4 1 2450.2.a.g 1
105.k odd 4 1 2450.2.a.bd 1
120.i odd 2 1 1600.2.c.i 2
120.m even 2 1 1600.2.c.h 2
120.q odd 4 1 1600.2.a.i 1
120.q odd 4 1 1600.2.a.p 1
120.w even 4 1 1600.2.a.j 1
120.w even 4 1 1600.2.a.q 1
165.l odd 4 1 6050.2.a.h 1
165.l odd 4 1 6050.2.a.bi 1
195.s even 4 1 8450.2.a.d 1
195.s even 4 1 8450.2.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.a.a 1 15.e even 4 1
50.2.a.b yes 1 15.e even 4 1
50.2.b.a 2 3.b odd 2 1
50.2.b.a 2 15.d odd 2 1
400.2.a.d 1 60.l odd 4 1
400.2.a.f 1 60.l odd 4 1
400.2.c.c 2 12.b even 2 1
400.2.c.c 2 60.h even 2 1
450.2.a.c 1 5.c odd 4 1
450.2.a.g 1 5.c odd 4 1
450.2.c.c 2 1.a even 1 1 trivial
450.2.c.c 2 5.b even 2 1 inner
1600.2.a.i 1 120.q odd 4 1
1600.2.a.j 1 120.w even 4 1
1600.2.a.p 1 120.q odd 4 1
1600.2.a.q 1 120.w even 4 1
1600.2.c.h 2 24.f even 2 1
1600.2.c.h 2 120.m even 2 1
1600.2.c.i 2 24.h odd 2 1
1600.2.c.i 2 120.i odd 2 1
2450.2.a.g 1 105.k odd 4 1
2450.2.a.bd 1 105.k odd 4 1
2450.2.c.m 2 21.c even 2 1
2450.2.c.m 2 105.g even 2 1
3600.2.a.l 1 20.e even 4 1
3600.2.a.bc 1 20.e even 4 1
3600.2.f.f 2 4.b odd 2 1
3600.2.f.f 2 20.d odd 2 1
6050.2.a.h 1 165.l odd 4 1
6050.2.a.bi 1 165.l odd 4 1
8450.2.a.d 1 195.s even 4 1
8450.2.a.v 1 195.s even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 4$$ $$T_{11} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$( -3 + T )^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$9 + T^{2}$$
$19$ $$( 5 + T )^{2}$$
$23$ $$36 + T^{2}$$
$29$ $$T^{2}$$
$31$ $$( -2 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( -3 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$144 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$169 + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$121 + T^{2}$$
$79$ $$( -10 + T )^{2}$$
$83$ $$81 + T^{2}$$
$89$ $$( -15 + T )^{2}$$
$97$ $$4 + T^{2}$$