# Properties

 Label 450.2.e.k Level $450$ Weight $2$ Character orbit 450.e Analytic conductor $3.593$ Analytic rank $0$ Dimension $4$ 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.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.59326809096$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 90) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{3} + ( -1 + \beta_{2} ) q^{4} + ( 1 + \beta_{1} - \beta_{2} ) q^{6} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{7} + q^{8} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{3} + ( -1 + \beta_{2} ) q^{4} + ( 1 + \beta_{1} - \beta_{2} ) q^{6} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{7} + q^{8} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{9} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{11} + ( -1 - \beta_{3} ) q^{12} + ( -\beta_{1} + 2 \beta_{3} ) q^{13} + ( 2 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{14} -\beta_{2} q^{16} + ( 1 - 4 \beta_{1} + 2 \beta_{3} ) q^{17} + ( -1 + 2 \beta_{3} ) q^{18} + ( -3 + 2 \beta_{1} - \beta_{3} ) q^{19} + ( -4 - 3 \beta_{1} + 2 \beta_{3} ) q^{21} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{22} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{23} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{24} + ( 2 \beta_{1} - \beta_{3} ) q^{26} + ( 5 - \beta_{3} ) q^{27} + ( -2 - 2 \beta_{1} + \beta_{3} ) q^{28} + 6 \beta_{2} q^{29} + ( -4 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{31} + ( -1 + \beta_{2} ) q^{32} + ( 1 + 3 \beta_{2} - 2 \beta_{3} ) q^{33} + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{34} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{36} -8 q^{37} + ( -\beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{38} + ( 2 - 2 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{39} + ( 1 - \beta_{2} ) q^{41} + ( 2 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{42} + ( \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{43} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{44} + ( -2 + 4 \beta_{1} - 2 \beta_{3} ) q^{46} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{47} + ( 1 + \beta_{1} - \beta_{2} ) q^{48} + ( -3 - 4 \beta_{1} + 3 \beta_{2} + 8 \beta_{3} ) q^{49} + ( 8 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{51} + ( -\beta_{1} - \beta_{3} ) q^{52} + ( -6 - 2 \beta_{1} + \beta_{3} ) q^{53} + ( -\beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{54} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{56} + ( -4 + 4 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{57} + ( 6 - 6 \beta_{2} ) q^{58} + ( -1 + 5 \beta_{1} + \beta_{2} - 10 \beta_{3} ) q^{59} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{61} + ( 4 + 2 \beta_{1} - \beta_{3} ) q^{62} + ( 6 + 2 \beta_{1} - 8 \beta_{2} - 5 \beta_{3} ) q^{63} + q^{64} + ( 3 - 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{66} + ( 7 + \beta_{1} - 7 \beta_{2} - 2 \beta_{3} ) q^{67} + ( -1 + 2 \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{68} + ( 6 - 4 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{69} + ( 2 \beta_{1} - \beta_{3} ) q^{71} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{72} + ( -5 + 8 \beta_{1} - 4 \beta_{3} ) q^{73} + 8 \beta_{2} q^{74} + ( 3 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{76} + ( 4 + \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{77} + ( -4 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{78} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{79} + ( -4 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} ) q^{81} - q^{82} -4 \beta_{2} q^{83} + ( 4 + \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{84} + ( 5 + \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{86} + ( -6 - 6 \beta_{1} + 6 \beta_{2} ) q^{87} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{88} + ( 8 + 4 \beta_{1} - 2 \beta_{3} ) q^{89} + ( -6 - 4 \beta_{1} + 2 \beta_{3} ) q^{91} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{92} + ( -2 - 2 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{93} + ( -2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{94} + ( -1 - \beta_{3} ) q^{96} + 13 \beta_{2} q^{97} + ( 3 + 8 \beta_{1} - 4 \beta_{3} ) q^{98} + ( -3 - 2 \beta_{1} + 8 \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} + 2q^{3} - 2q^{4} + 2q^{6} + 4q^{7} + 4q^{8} + 2q^{9} + O(q^{10})$$ $$4q - 2q^{2} + 2q^{3} - 2q^{4} + 2q^{6} + 4q^{7} + 4q^{8} + 2q^{9} + 2q^{11} - 4q^{12} + 4q^{14} - 2q^{16} + 4q^{17} - 4q^{18} - 12q^{19} - 16q^{21} + 2q^{22} + 4q^{23} + 2q^{24} + 20q^{27} - 8q^{28} + 12q^{29} - 8q^{31} - 2q^{32} + 10q^{33} - 2q^{34} + 2q^{36} - 32q^{37} + 6q^{38} + 2q^{41} + 8q^{42} + 10q^{43} - 4q^{44} - 8q^{46} - 4q^{47} + 2q^{48} - 6q^{49} + 26q^{51} - 24q^{53} - 10q^{54} + 4q^{56} - 18q^{57} + 12q^{58} - 2q^{59} - 4q^{61} + 16q^{62} + 8q^{63} + 4q^{64} + 4q^{66} + 14q^{67} - 2q^{68} + 8q^{69} + 2q^{72} - 20q^{73} + 16q^{74} + 6q^{76} + 8q^{77} - 12q^{78} + 14q^{81} - 4q^{82} - 8q^{83} + 8q^{84} + 10q^{86} - 12q^{87} + 2q^{88} + 32q^{89} - 24q^{91} + 4q^{92} - 16q^{93} - 4q^{94} - 4q^{96} + 26q^{97} + 12q^{98} + 4q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1 + \beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
151.1
 1.22474 − 0.707107i −1.22474 + 0.707107i 1.22474 + 0.707107i −1.22474 − 0.707107i
−0.500000 + 0.866025i −0.724745 1.57313i −0.500000 0.866025i 0 1.72474 + 0.158919i 2.22474 3.85337i 1.00000 −1.94949 + 2.28024i 0
151.2 −0.500000 + 0.866025i 1.72474 0.158919i −0.500000 0.866025i 0 −0.724745 + 1.57313i −0.224745 + 0.389270i 1.00000 2.94949 0.548188i 0
301.1 −0.500000 0.866025i −0.724745 + 1.57313i −0.500000 + 0.866025i 0 1.72474 0.158919i 2.22474 + 3.85337i 1.00000 −1.94949 2.28024i 0
301.2 −0.500000 0.866025i 1.72474 + 0.158919i −0.500000 + 0.866025i 0 −0.724745 1.57313i −0.224745 0.389270i 1.00000 2.94949 + 0.548188i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.e.k 4
3.b odd 2 1 1350.2.e.m 4
5.b even 2 1 450.2.e.n 4
5.c odd 4 2 90.2.i.b 8
9.c even 3 1 inner 450.2.e.k 4
9.c even 3 1 4050.2.a.bs 2
9.d odd 6 1 1350.2.e.m 4
9.d odd 6 1 4050.2.a.bm 2
15.d odd 2 1 1350.2.e.j 4
15.e even 4 2 270.2.i.b 8
20.e even 4 2 720.2.by.c 8
45.h odd 6 1 1350.2.e.j 4
45.h odd 6 1 4050.2.a.bz 2
45.j even 6 1 450.2.e.n 4
45.j even 6 1 4050.2.a.bq 2
45.k odd 12 2 90.2.i.b 8
45.k odd 12 2 810.2.c.f 4
45.l even 12 2 270.2.i.b 8
45.l even 12 2 810.2.c.e 4
60.l odd 4 2 2160.2.by.d 8
180.v odd 12 2 2160.2.by.d 8
180.x even 12 2 720.2.by.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.i.b 8 5.c odd 4 2
90.2.i.b 8 45.k odd 12 2
270.2.i.b 8 15.e even 4 2
270.2.i.b 8 45.l even 12 2
450.2.e.k 4 1.a even 1 1 trivial
450.2.e.k 4 9.c even 3 1 inner
450.2.e.n 4 5.b even 2 1
450.2.e.n 4 45.j even 6 1
720.2.by.c 8 20.e even 4 2
720.2.by.c 8 180.x even 12 2
810.2.c.e 4 45.l even 12 2
810.2.c.f 4 45.k odd 12 2
1350.2.e.j 4 15.d odd 2 1
1350.2.e.j 4 45.h odd 6 1
1350.2.e.m 4 3.b odd 2 1
1350.2.e.m 4 9.d odd 6 1
2160.2.by.d 8 60.l odd 4 2
2160.2.by.d 8 180.v odd 12 2
4050.2.a.bm 2 9.d odd 6 1
4050.2.a.bq 2 45.j even 6 1
4050.2.a.bs 2 9.c even 3 1
4050.2.a.bz 2 45.h odd 6 1

## Hecke kernels

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

 $$T_{7}^{4} - 4 T_{7}^{3} + 18 T_{7}^{2} + 8 T_{7} + 4$$ $$T_{11}^{4} - 2 T_{11}^{3} + 9 T_{11}^{2} + 10 T_{11} + 25$$ $$T_{17}^{2} - 2 T_{17} - 23$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$9 - 6 T + T^{2} - 2 T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$4 + 8 T + 18 T^{2} - 4 T^{3} + T^{4}$$
$11$ $$25 + 10 T + 9 T^{2} - 2 T^{3} + T^{4}$$
$13$ $$36 + 6 T^{2} + T^{4}$$
$17$ $$( -23 - 2 T + T^{2} )^{2}$$
$19$ $$( 3 + 6 T + T^{2} )^{2}$$
$23$ $$400 + 80 T + 36 T^{2} - 4 T^{3} + T^{4}$$
$29$ $$( 36 - 6 T + T^{2} )^{2}$$
$31$ $$100 + 80 T + 54 T^{2} + 8 T^{3} + T^{4}$$
$37$ $$( 8 + T )^{4}$$
$41$ $$( 1 - T + T^{2} )^{2}$$
$43$ $$361 - 190 T + 81 T^{2} - 10 T^{3} + T^{4}$$
$47$ $$4 - 8 T + 18 T^{2} + 4 T^{3} + T^{4}$$
$53$ $$( 30 + 12 T + T^{2} )^{2}$$
$59$ $$22201 - 298 T + 153 T^{2} + 2 T^{3} + T^{4}$$
$61$ $$4 - 8 T + 18 T^{2} + 4 T^{3} + T^{4}$$
$67$ $$1849 - 602 T + 153 T^{2} - 14 T^{3} + T^{4}$$
$71$ $$( -6 + T^{2} )^{2}$$
$73$ $$( -71 + 10 T + T^{2} )^{2}$$
$79$ $$2916 + 54 T^{2} + T^{4}$$
$83$ $$( 16 + 4 T + T^{2} )^{2}$$
$89$ $$( 40 - 16 T + T^{2} )^{2}$$
$97$ $$( 169 - 13 T + T^{2} )^{2}$$