# Properties

 Label 450.2.h.a Level $450$ Weight $2$ Character orbit 450.h Analytic conductor $3.593$ Analytic rank $1$ 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.h (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.59326809096$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 50) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} -\zeta_{10}^{3} q^{4} + ( -2 + 2 \zeta_{10} + \zeta_{10}^{3} ) q^{5} -3 q^{7} + \zeta_{10}^{2} q^{8} +O(q^{10})$$ $$q + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} -\zeta_{10}^{3} q^{4} + ( -2 + 2 \zeta_{10} + \zeta_{10}^{3} ) q^{5} -3 q^{7} + \zeta_{10}^{2} q^{8} + ( -2 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{10} + ( 1 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{11} -\zeta_{10} q^{13} + ( 3 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{14} -\zeta_{10} q^{16} + ( -3 \zeta_{10} - 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{17} + ( -3 \zeta_{10} + 4 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{19} + ( 2 - \zeta_{10} + 2 \zeta_{10}^{2} ) q^{20} + ( 2 - 2 \zeta_{10} + \zeta_{10}^{3} ) q^{22} + ( -5 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{23} -5 \zeta_{10} q^{25} + q^{26} + 3 \zeta_{10}^{3} q^{28} + ( -4 + 4 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{29} + ( 2 \zeta_{10} + \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{31} + q^{32} + ( 3 + 3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{34} + ( 6 - 6 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{35} + ( -7 + 4 \zeta_{10} - 7 \zeta_{10}^{2} ) q^{37} + ( 3 - 4 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{38} + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{40} + ( -4 - \zeta_{10} - 4 \zeta_{10}^{2} ) q^{41} + ( -3 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{43} + ( 2 \zeta_{10} - 3 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{44} + ( 2 - 2 \zeta_{10} - 5 \zeta_{10}^{3} ) q^{46} + ( -8 + 8 \zeta_{10} + \zeta_{10}^{3} ) q^{47} + 2 q^{49} + 5 q^{50} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{52} + ( 4 - 4 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{53} + ( 4 \zeta_{10} - 7 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{55} -3 \zeta_{10}^{2} q^{56} + ( -4 \zeta_{10} + 7 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{58} + ( 4 - 2 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{59} + ( 4 - 7 \zeta_{10} + 7 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{61} + ( -2 - \zeta_{10} - 2 \zeta_{10}^{2} ) q^{62} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{64} + ( 1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{65} + ( 2 \zeta_{10} - 9 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{67} + ( -6 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{68} + ( 6 \zeta_{10} - 3 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{70} -3 \zeta_{10}^{3} q^{71} + ( 2 + 4 \zeta_{10} - 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{73} + ( 3 + 7 \zeta_{10}^{2} - 7 \zeta_{10}^{3} ) q^{74} + ( 1 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{76} + ( -3 + 9 \zeta_{10} - 9 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{77} + ( 2 - 2 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{79} + ( 1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{80} + ( 5 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{82} + ( 4 \zeta_{10} + 7 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{83} + ( 12 + 9 \zeta_{10}^{2} - 9 \zeta_{10}^{3} ) q^{85} + ( 3 - 5 \zeta_{10} + 5 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{86} + ( -2 + 3 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{88} + ( -2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{89} + 3 \zeta_{10} q^{91} + ( 2 \zeta_{10} + 3 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{92} + ( -8 \zeta_{10} + 7 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{94} + ( 5 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{95} + ( -9 + 9 \zeta_{10} + 5 \zeta_{10}^{3} ) q^{97} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - q^{2} - q^{4} - 5q^{5} - 12q^{7} - q^{8} + O(q^{10})$$ $$4q - q^{2} - q^{4} - 5q^{5} - 12q^{7} - q^{8} - 5q^{10} - 3q^{11} - q^{13} + 3q^{14} - q^{16} - 3q^{17} - 10q^{19} + 5q^{20} + 7q^{22} - 9q^{23} - 5q^{25} + 4q^{26} + 3q^{28} - 15q^{29} + 3q^{31} + 4q^{32} + 12q^{34} + 15q^{35} - 17q^{37} + 5q^{38} - 13q^{41} - 16q^{43} + 7q^{44} + q^{46} - 23q^{47} + 8q^{49} + 20q^{50} - q^{52} + 16q^{53} + 15q^{55} + 3q^{56} - 15q^{58} + 10q^{59} - 2q^{61} - 7q^{62} - q^{64} + 5q^{65} + 13q^{67} - 18q^{68} + 15q^{70} - 3q^{71} + 14q^{73} - 2q^{74} + 10q^{76} + 9q^{77} + 10q^{79} + 5q^{80} + 12q^{82} + q^{83} + 30q^{85} - q^{86} - 3q^{88} - 10q^{89} + 3q^{91} + q^{92} - 23q^{94} + 30q^{95} - 22q^{97} - 2q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
91.1
 0.809017 − 0.587785i −0.309017 + 0.951057i −0.309017 − 0.951057i 0.809017 + 0.587785i
−0.809017 0.587785i 0 0.309017 + 0.951057i −0.690983 2.12663i 0 −3.00000 0.309017 0.951057i 0 −0.690983 + 2.12663i
181.1 0.309017 + 0.951057i 0 −0.809017 + 0.587785i −1.80902 + 1.31433i 0 −3.00000 −0.809017 0.587785i 0 −1.80902 1.31433i
271.1 0.309017 0.951057i 0 −0.809017 0.587785i −1.80902 1.31433i 0 −3.00000 −0.809017 + 0.587785i 0 −1.80902 + 1.31433i
361.1 −0.809017 + 0.587785i 0 0.309017 0.951057i −0.690983 + 2.12663i 0 −3.00000 0.309017 + 0.951057i 0 −0.690983 2.12663i
 $$n$$: 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
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.h.a 4
3.b odd 2 1 50.2.d.a 4
12.b even 2 1 400.2.u.c 4
15.d odd 2 1 250.2.d.a 4
15.e even 4 2 250.2.e.b 8
25.d even 5 1 inner 450.2.h.a 4
75.h odd 10 1 250.2.d.a 4
75.h odd 10 1 1250.2.a.d 2
75.j odd 10 1 50.2.d.a 4
75.j odd 10 1 1250.2.a.a 2
75.l even 20 2 250.2.e.b 8
75.l even 20 2 1250.2.b.b 4
300.n even 10 1 400.2.u.c 4
300.n even 10 1 10000.2.a.n 2
300.r even 10 1 10000.2.a.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.d.a 4 3.b odd 2 1
50.2.d.a 4 75.j odd 10 1
250.2.d.a 4 15.d odd 2 1
250.2.d.a 4 75.h odd 10 1
250.2.e.b 8 15.e even 4 2
250.2.e.b 8 75.l even 20 2
400.2.u.c 4 12.b even 2 1
400.2.u.c 4 300.n even 10 1
450.2.h.a 4 1.a even 1 1 trivial
450.2.h.a 4 25.d even 5 1 inner
1250.2.a.a 2 75.j odd 10 1
1250.2.a.d 2 75.h odd 10 1
1250.2.b.b 4 75.l even 20 2
10000.2.a.a 2 300.r even 10 1
10000.2.a.n 2 300.n even 10 1

## Hecke kernels

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

 $$T_{7} + 3$$ $$T_{11}^{4} + 3 T_{11}^{3} + 19 T_{11}^{2} + 7 T_{11} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$25 + 25 T + 15 T^{2} + 5 T^{3} + T^{4}$$
$7$ $$( 3 + T )^{4}$$
$11$ $$1 + 7 T + 19 T^{2} + 3 T^{3} + T^{4}$$
$13$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$17$ $$81 - 108 T + 54 T^{2} + 3 T^{3} + T^{4}$$
$19$ $$25 + 25 T + 40 T^{2} + 10 T^{3} + T^{4}$$
$23$ $$121 - 11 T + 31 T^{2} + 9 T^{3} + T^{4}$$
$29$ $$25 - 25 T + 85 T^{2} + 15 T^{3} + T^{4}$$
$31$ $$1 - 7 T + 19 T^{2} - 3 T^{3} + T^{4}$$
$37$ $$3721 + 1098 T + 184 T^{2} + 17 T^{3} + T^{4}$$
$41$ $$121 + 77 T + 69 T^{2} + 13 T^{3} + T^{4}$$
$43$ $$( 11 + 8 T + T^{2} )^{2}$$
$47$ $$5041 + 1207 T + 249 T^{2} + 23 T^{3} + T^{4}$$
$53$ $$256 + 64 T + 96 T^{2} - 16 T^{3} + T^{4}$$
$59$ $$400 - 200 T + 60 T^{2} - 10 T^{3} + T^{4}$$
$61$ $$361 - 247 T + 64 T^{2} + 2 T^{3} + T^{4}$$
$67$ $$3481 - 177 T + 79 T^{2} - 13 T^{3} + T^{4}$$
$71$ $$81 + 27 T + 9 T^{2} + 3 T^{3} + T^{4}$$
$73$ $$1936 - 704 T + 136 T^{2} - 14 T^{3} + T^{4}$$
$79$ $$400 + 40 T^{2} - 10 T^{3} + T^{4}$$
$83$ $$3721 + 1159 T + 141 T^{2} - T^{3} + T^{4}$$
$89$ $$400 + 200 T + 60 T^{2} + 10 T^{3} + T^{4}$$
$97$ $$10201 + 2323 T + 304 T^{2} + 22 T^{3} + T^{4}$$