# Properties

 Label 450.6.c.m 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 18) 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.

 $$f(q)$$ $$=$$ $$q + 4 i q^{2} -16 q^{4} + 148 i q^{7} -64 i q^{8} +O(q^{10})$$ $$q + 4 i q^{2} -16 q^{4} + 148 i q^{7} -64 i q^{8} + 384 q^{11} -334 i q^{13} -592 q^{14} + 256 q^{16} -576 i q^{17} + 664 q^{19} + 1536 i q^{22} -3840 i q^{23} + 1336 q^{26} -2368 i q^{28} -96 q^{29} -4564 q^{31} + 1024 i q^{32} + 2304 q^{34} -5798 i q^{37} + 2656 i q^{38} -6720 q^{41} -14872 i q^{43} -6144 q^{44} + 15360 q^{46} + 19200 i q^{47} -5097 q^{49} + 5344 i q^{52} + 7776 i q^{53} + 9472 q^{56} -384 i q^{58} + 13056 q^{59} + 42782 q^{61} -18256 i q^{62} -4096 q^{64} -36656 i q^{67} + 9216 i q^{68} + 64512 q^{71} -16810 i q^{73} + 23192 q^{74} -10624 q^{76} + 56832 i q^{77} -28076 q^{79} -26880 i q^{82} -66432 i q^{83} + 59488 q^{86} -24576 i q^{88} + 81792 q^{89} + 49432 q^{91} + 61440 i q^{92} -76800 q^{94} + 29938 i q^{97} -20388 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 32q^{4} + O(q^{10})$$ $$2q - 32q^{4} + 768q^{11} - 1184q^{14} + 512q^{16} + 1328q^{19} + 2672q^{26} - 192q^{29} - 9128q^{31} + 4608q^{34} - 13440q^{41} - 12288q^{44} + 30720q^{46} - 10194q^{49} + 18944q^{56} + 26112q^{59} + 85564q^{61} - 8192q^{64} + 129024q^{71} + 46384q^{74} - 21248q^{76} - 56152q^{79} + 118976q^{86} + 163584q^{89} + 98864q^{91} - 153600q^{94} + O(q^{100})$$

## 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 0 148.000i 64.0000i 0 0
199.2 4.00000i 0 −16.0000 0 0 148.000i 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.m 2
3.b odd 2 1 450.6.c.c 2
5.b even 2 1 inner 450.6.c.m 2
5.c odd 4 1 18.6.a.a 1
5.c odd 4 1 450.6.a.v 1
15.d odd 2 1 450.6.c.c 2
15.e even 4 1 18.6.a.c yes 1
15.e even 4 1 450.6.a.k 1
20.e even 4 1 144.6.a.a 1
35.f even 4 1 882.6.a.k 1
40.i odd 4 1 576.6.a.bh 1
40.k even 4 1 576.6.a.bi 1
45.k odd 12 2 162.6.c.l 2
45.l even 12 2 162.6.c.a 2
60.l odd 4 1 144.6.a.l 1
105.k odd 4 1 882.6.a.l 1
120.q odd 4 1 576.6.a.b 1
120.w even 4 1 576.6.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.6.a.a 1 5.c odd 4 1
18.6.a.c yes 1 15.e even 4 1
144.6.a.a 1 20.e even 4 1
144.6.a.l 1 60.l odd 4 1
162.6.c.a 2 45.l even 12 2
162.6.c.l 2 45.k odd 12 2
450.6.a.k 1 15.e even 4 1
450.6.a.v 1 5.c odd 4 1
450.6.c.c 2 3.b odd 2 1
450.6.c.c 2 15.d odd 2 1
450.6.c.m 2 1.a even 1 1 trivial
450.6.c.m 2 5.b even 2 1 inner
576.6.a.a 1 120.w even 4 1
576.6.a.b 1 120.q odd 4 1
576.6.a.bh 1 40.i odd 4 1
576.6.a.bi 1 40.k even 4 1
882.6.a.k 1 35.f even 4 1
882.6.a.l 1 105.k odd 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 21904$$ $$T_{11} - 384$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$21904 + T^{2}$$
$11$ $$( -384 + T )^{2}$$
$13$ $$111556 + T^{2}$$
$17$ $$331776 + T^{2}$$
$19$ $$( -664 + T )^{2}$$
$23$ $$14745600 + T^{2}$$
$29$ $$( 96 + T )^{2}$$
$31$ $$( 4564 + T )^{2}$$
$37$ $$33616804 + T^{2}$$
$41$ $$( 6720 + T )^{2}$$
$43$ $$221176384 + T^{2}$$
$47$ $$368640000 + T^{2}$$
$53$ $$60466176 + T^{2}$$
$59$ $$( -13056 + T )^{2}$$
$61$ $$( -42782 + T )^{2}$$
$67$ $$1343662336 + T^{2}$$
$71$ $$( -64512 + T )^{2}$$
$73$ $$282576100 + T^{2}$$
$79$ $$( 28076 + T )^{2}$$
$83$ $$4413210624 + T^{2}$$
$89$ $$( -81792 + T )^{2}$$
$97$ $$896283844 + T^{2}$$