# Properties

 Label 450.6.c.a Level $450$ Weight $6$ Character orbit 450.c Analytic conductor $72.173$ 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: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 450.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$72.1727189158$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ 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.

 $$f(q)$$ $$=$$ $$q -4 i q^{2} -16 q^{4} + 142 i q^{7} + 64 i q^{8} +O(q^{10})$$ $$q -4 i q^{2} -16 q^{4} + 142 i q^{7} + 64 i q^{8} -777 q^{11} + 884 i q^{13} + 568 q^{14} + 256 q^{16} -27 i q^{17} -1145 q^{19} + 3108 i q^{22} -1854 i q^{23} + 3536 q^{26} -2272 i q^{28} -4920 q^{29} + 1802 q^{31} -1024 i q^{32} -108 q^{34} -13178 i q^{37} + 4580 i q^{38} + 15123 q^{41} + 7844 i q^{43} + 12432 q^{44} -7416 q^{46} -6732 i q^{47} -3357 q^{49} -14144 i q^{52} -3414 i q^{53} -9088 q^{56} + 19680 i q^{58} + 33960 q^{59} + 47402 q^{61} -7208 i q^{62} -4096 q^{64} + 13177 i q^{67} + 432 i q^{68} + 7548 q^{71} -59821 i q^{73} -52712 q^{74} + 18320 q^{76} -110334 i q^{77} -75830 q^{79} -60492 i q^{82} -46299 i q^{83} + 31376 q^{86} -49728 i q^{88} -30585 q^{89} -125528 q^{91} + 29664 i q^{92} -26928 q^{94} -104018 i q^{97} + 13428 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 32q^{4} + O(q^{10})$$ $$2q - 32q^{4} - 1554q^{11} + 1136q^{14} + 512q^{16} - 2290q^{19} + 7072q^{26} - 9840q^{29} + 3604q^{31} - 216q^{34} + 30246q^{41} + 24864q^{44} - 14832q^{46} - 6714q^{49} - 18176q^{56} + 67920q^{59} + 94804q^{61} - 8192q^{64} + 15096q^{71} - 105424q^{74} + 36640q^{76} - 151660q^{79} + 62752q^{86} - 61170q^{89} - 251056q^{91} - 53856q^{94} + O(q^{100})$$

## 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
4.00000i 0 −16.0000 0 0 142.000i 64.0000i 0 0
199.2 4.00000i 0 −16.0000 0 0 142.000i 64.0000i 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 450.6.c.a 2
3.b odd 2 1 50.6.b.c 2
5.b even 2 1 inner 450.6.c.a 2
5.c odd 4 1 450.6.a.j 1
5.c odd 4 1 450.6.a.n 1
12.b even 2 1 400.6.c.g 2
15.d odd 2 1 50.6.b.c 2
15.e even 4 1 50.6.a.a 1
15.e even 4 1 50.6.a.f yes 1
60.h even 2 1 400.6.c.g 2
60.l odd 4 1 400.6.a.e 1
60.l odd 4 1 400.6.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.6.a.a 1 15.e even 4 1
50.6.a.f yes 1 15.e even 4 1
50.6.b.c 2 3.b odd 2 1
50.6.b.c 2 15.d odd 2 1
400.6.a.e 1 60.l odd 4 1
400.6.a.j 1 60.l odd 4 1
400.6.c.g 2 12.b even 2 1
400.6.c.g 2 60.h even 2 1
450.6.a.j 1 5.c odd 4 1
450.6.a.n 1 5.c odd 4 1
450.6.c.a 2 1.a even 1 1 trivial
450.6.c.a 2 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(450, [\chi])$$:

 $$T_{7}^{2} + 20164$$ $$T_{11} + 777$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$20164 + T^{2}$$
$11$ $$( 777 + T )^{2}$$
$13$ $$781456 + T^{2}$$
$17$ $$729 + T^{2}$$
$19$ $$( 1145 + T )^{2}$$
$23$ $$3437316 + T^{2}$$
$29$ $$( 4920 + T )^{2}$$
$31$ $$( -1802 + T )^{2}$$
$37$ $$173659684 + T^{2}$$
$41$ $$( -15123 + T )^{2}$$
$43$ $$61528336 + T^{2}$$
$47$ $$45319824 + T^{2}$$
$53$ $$11655396 + T^{2}$$
$59$ $$( -33960 + T )^{2}$$
$61$ $$( -47402 + T )^{2}$$
$67$ $$173633329 + T^{2}$$
$71$ $$( -7548 + T )^{2}$$
$73$ $$3578552041 + T^{2}$$
$79$ $$( 75830 + T )^{2}$$
$83$ $$2143597401 + T^{2}$$
$89$ $$( 30585 + T )^{2}$$
$97$ $$10819744324 + T^{2}$$