# Properties

 Label 450.6.c.n 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 150) 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.

 $$f(q)$$ $$=$$ $$q + 4 i q^{2} -16 q^{4} + 233 i q^{7} -64 i q^{8} +O(q^{10})$$ $$q + 4 i q^{2} -16 q^{4} + 233 i q^{7} -64 i q^{8} + 498 q^{11} -809 i q^{13} -932 q^{14} + 256 q^{16} + 1002 i q^{17} + 1705 q^{19} + 1992 i q^{22} + 1554 i q^{23} + 3236 q^{26} -3728 i q^{28} + 7830 q^{29} + 977 q^{31} + 1024 i q^{32} -4008 q^{34} -4822 i q^{37} + 6820 i q^{38} + 8148 q^{41} -19469 i q^{43} -7968 q^{44} -6216 q^{46} -8418 i q^{47} -37482 q^{49} + 12944 i q^{52} + 17664 i q^{53} + 14912 q^{56} + 31320 i q^{58} + 35910 q^{59} + 3527 q^{61} + 3908 i q^{62} -4096 q^{64} + 57473 i q^{67} -16032 i q^{68} + 7548 q^{71} + 646 i q^{73} + 19288 q^{74} -27280 q^{76} + 116034 i q^{77} + 22720 q^{79} + 32592 i q^{82} + 11574 i q^{83} + 77876 q^{86} -31872 i q^{88} -78960 q^{89} + 188497 q^{91} -24864 i q^{92} + 33672 q^{94} + 54593 i q^{97} -149928 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 32q^{4} + O(q^{10})$$ $$2q - 32q^{4} + 996q^{11} - 1864q^{14} + 512q^{16} + 3410q^{19} + 6472q^{26} + 15660q^{29} + 1954q^{31} - 8016q^{34} + 16296q^{41} - 15936q^{44} - 12432q^{46} - 74964q^{49} + 29824q^{56} + 71820q^{59} + 7054q^{61} - 8192q^{64} + 15096q^{71} + 38576q^{74} - 54560q^{76} + 45440q^{79} + 155752q^{86} - 157920q^{89} + 376994q^{91} + 67344q^{94} + O(q^{100})$$

## 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 0 233.000i 64.0000i 0 0
199.2 4.00000i 0 −16.0000 0 0 233.000i 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.n 2
3.b odd 2 1 150.6.c.e 2
5.b even 2 1 inner 450.6.c.n 2
5.c odd 4 1 450.6.a.a 1
5.c odd 4 1 450.6.a.x 1
15.d odd 2 1 150.6.c.e 2
15.e even 4 1 150.6.a.c 1
15.e even 4 1 150.6.a.l yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.6.a.c 1 15.e even 4 1
150.6.a.l yes 1 15.e even 4 1
150.6.c.e 2 3.b odd 2 1
150.6.c.e 2 15.d odd 2 1
450.6.a.a 1 5.c odd 4 1
450.6.a.x 1 5.c odd 4 1
450.6.c.n 2 1.a even 1 1 trivial
450.6.c.n 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 $$S_{6}^{\mathrm{new}}(450, [\chi])$$:

 $$T_{7}^{2} + 54289$$ $$T_{11} - 498$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$54289 + T^{2}$$
$11$ $$( -498 + T )^{2}$$
$13$ $$654481 + T^{2}$$
$17$ $$1004004 + T^{2}$$
$19$ $$( -1705 + T )^{2}$$
$23$ $$2414916 + T^{2}$$
$29$ $$( -7830 + T )^{2}$$
$31$ $$( -977 + T )^{2}$$
$37$ $$23251684 + T^{2}$$
$41$ $$( -8148 + T )^{2}$$
$43$ $$379041961 + T^{2}$$
$47$ $$70862724 + T^{2}$$
$53$ $$312016896 + T^{2}$$
$59$ $$( -35910 + T )^{2}$$
$61$ $$( -3527 + T )^{2}$$
$67$ $$3303145729 + T^{2}$$
$71$ $$( -7548 + T )^{2}$$
$73$ $$417316 + T^{2}$$
$79$ $$( -22720 + T )^{2}$$
$83$ $$133957476 + T^{2}$$
$89$ $$( 78960 + T )^{2}$$
$97$ $$2980395649 + T^{2}$$