# Properties

 Label 450.6.c.e 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.

 $$f(q)$$ $$=$$ $$q -4 i q^{2} -16 q^{4} + 47 i q^{7} + 64 i q^{8} +O(q^{10})$$ $$q -4 i q^{2} -16 q^{4} + 47 i q^{7} + 64 i q^{8} -222 q^{11} -101 i q^{13} + 188 q^{14} + 256 q^{16} -162 i q^{17} -1685 q^{19} + 888 i q^{22} + 306 i q^{23} -404 q^{26} -752 i q^{28} + 7890 q^{29} -8593 q^{31} -1024 i q^{32} -648 q^{34} + 8642 i q^{37} + 6740 i q^{38} + 18168 q^{41} -14351 i q^{43} + 3552 q^{44} + 1224 q^{46} + 1098 i q^{47} + 14598 q^{49} + 1616 i q^{52} + 17916 i q^{53} -3008 q^{56} -31560 i q^{58} + 17610 q^{59} -21853 q^{61} + 34372 i q^{62} -4096 q^{64} + 107 i q^{67} + 2592 i q^{68} + 40728 q^{71} -34706 i q^{73} + 34568 q^{74} + 26960 q^{76} -10434 i q^{77} + 69160 q^{79} -72672 i q^{82} -108534 i q^{83} -57404 q^{86} -14208 i q^{88} + 35040 q^{89} + 4747 q^{91} -4896 i q^{92} + 4392 q^{94} -823 i q^{97} -58392 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 32q^{4} + O(q^{10})$$ $$2q - 32q^{4} - 444q^{11} + 376q^{14} + 512q^{16} - 3370q^{19} - 808q^{26} + 15780q^{29} - 17186q^{31} - 1296q^{34} + 36336q^{41} + 7104q^{44} + 2448q^{46} + 29196q^{49} - 6016q^{56} + 35220q^{59} - 43706q^{61} - 8192q^{64} + 81456q^{71} + 69136q^{74} + 53920q^{76} + 138320q^{79} - 114808q^{86} + 70080q^{89} + 9494q^{91} + 8784q^{94} + O(q^{100})$$

## 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 47.0000i 64.0000i 0 0
199.2 4.00000i 0 −16.0000 0 0 47.0000i 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.e 2
3.b odd 2 1 150.6.c.g 2
5.b even 2 1 inner 450.6.c.e 2
5.c odd 4 1 450.6.a.i 1
5.c odd 4 1 450.6.a.p 1
15.d odd 2 1 150.6.c.g 2
15.e even 4 1 150.6.a.a 1
15.e even 4 1 150.6.a.m yes 1

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

 $$T_{7}^{2} + 2209$$ $$T_{11} + 222$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$2209 + T^{2}$$
$11$ $$( 222 + T )^{2}$$
$13$ $$10201 + T^{2}$$
$17$ $$26244 + T^{2}$$
$19$ $$( 1685 + T )^{2}$$
$23$ $$93636 + T^{2}$$
$29$ $$( -7890 + T )^{2}$$
$31$ $$( 8593 + T )^{2}$$
$37$ $$74684164 + T^{2}$$
$41$ $$( -18168 + T )^{2}$$
$43$ $$205951201 + T^{2}$$
$47$ $$1205604 + T^{2}$$
$53$ $$320983056 + T^{2}$$
$59$ $$( -17610 + T )^{2}$$
$61$ $$( 21853 + T )^{2}$$
$67$ $$11449 + T^{2}$$
$71$ $$( -40728 + T )^{2}$$
$73$ $$1204506436 + T^{2}$$
$79$ $$( -69160 + T )^{2}$$
$83$ $$11779629156 + T^{2}$$
$89$ $$( -35040 + T )^{2}$$
$97$ $$677329 + T^{2}$$