# Properties

 Label 450.2.l.b Level $450$ Weight $2$ Character orbit 450.l Analytic conductor $3.593$ Analytic rank $0$ Dimension $8$ 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.l (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.59326809096$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{20})$$ Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 50) Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{20}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{20} q^{2} + \zeta_{20}^{2} q^{4} + ( 1 - \zeta_{20} - \zeta_{20}^{3} - \zeta_{20}^{4} ) q^{5} + ( 1 - 2 \zeta_{20}^{2} + 2 \zeta_{20}^{5} - 2 \zeta_{20}^{6} ) q^{7} + \zeta_{20}^{3} q^{8} +O(q^{10})$$ $$q + \zeta_{20} q^{2} + \zeta_{20}^{2} q^{4} + ( 1 - \zeta_{20} - \zeta_{20}^{3} - \zeta_{20}^{4} ) q^{5} + ( 1 - 2 \zeta_{20}^{2} + 2 \zeta_{20}^{5} - 2 \zeta_{20}^{6} ) q^{7} + \zeta_{20}^{3} q^{8} + ( \zeta_{20} - \zeta_{20}^{2} - \zeta_{20}^{4} - \zeta_{20}^{5} ) q^{10} + ( 1 + 2 \zeta_{20} - \zeta_{20}^{2} - 4 \zeta_{20}^{3} + 3 \zeta_{20}^{5} - \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{11} + ( 4 \zeta_{20} + \zeta_{20}^{2} - 3 \zeta_{20}^{3} + 3 \zeta_{20}^{5} - \zeta_{20}^{6} - 4 \zeta_{20}^{7} ) q^{13} + ( \zeta_{20} - 2 \zeta_{20}^{3} + 2 \zeta_{20}^{6} - 2 \zeta_{20}^{7} ) q^{14} + \zeta_{20}^{4} q^{16} + ( -1 + 2 \zeta_{20}^{3} - \zeta_{20}^{6} ) q^{17} + ( 1 + \zeta_{20} + \zeta_{20}^{2} - \zeta_{20}^{4} - \zeta_{20}^{5} - \zeta_{20}^{6} ) q^{19} + ( \zeta_{20}^{2} - \zeta_{20}^{3} - \zeta_{20}^{5} - \zeta_{20}^{6} ) q^{20} + ( 1 + \zeta_{20} + \zeta_{20}^{2} - \zeta_{20}^{3} - 3 \zeta_{20}^{4} + 2 \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{22} + ( -2 \zeta_{20}^{2} + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{5} - \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{23} + ( -2 \zeta_{20} + 2 \zeta_{20}^{2} - 2 \zeta_{20}^{3} - \zeta_{20}^{4} + 2 \zeta_{20}^{5} + 2 \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{25} + ( 4 + \zeta_{20}^{3} + \zeta_{20}^{4} - \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{26} + ( 2 - \zeta_{20}^{2} - 2 \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{28} + ( -3 + \zeta_{20} + 4 \zeta_{20}^{2} + \zeta_{20}^{3} - 3 \zeta_{20}^{4} ) q^{29} + ( -1 - 2 \zeta_{20} + 3 \zeta_{20}^{2} - 3 \zeta_{20}^{4} + 2 \zeta_{20}^{5} + \zeta_{20}^{6} ) q^{31} + \zeta_{20}^{5} q^{32} + ( -\zeta_{20} + 2 \zeta_{20}^{4} - \zeta_{20}^{7} ) q^{34} + ( 1 - \zeta_{20} - 4 \zeta_{20}^{2} + \zeta_{20}^{3} + \zeta_{20}^{4} + 4 \zeta_{20}^{5} - 4 \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{35} + ( -2 - 5 \zeta_{20} + 4 \zeta_{20}^{3} - \zeta_{20}^{4} - 4 \zeta_{20}^{5} + 2 \zeta_{20}^{6} + 5 \zeta_{20}^{7} ) q^{37} + ( \zeta_{20} + \zeta_{20}^{2} + \zeta_{20}^{3} - \zeta_{20}^{5} - \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{38} + ( \zeta_{20}^{3} - \zeta_{20}^{4} - \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{40} + ( -4 \zeta_{20} + 4 \zeta_{20}^{2} + 2 \zeta_{20}^{3} + \zeta_{20}^{4} + 2 \zeta_{20}^{5} + 4 \zeta_{20}^{6} - 4 \zeta_{20}^{7} ) q^{41} + ( 3 - 6 \zeta_{20}^{2} + 3 \zeta_{20}^{3} + \zeta_{20}^{4} - 5 \zeta_{20}^{5} - 5 \zeta_{20}^{6} + 3 \zeta_{20}^{7} ) q^{43} + ( 1 + \zeta_{20} + \zeta_{20}^{3} - 3 \zeta_{20}^{5} - \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{44} + ( -2 + 2 \zeta_{20}^{2} - 2 \zeta_{20}^{3} - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{46} + ( -4 \zeta_{20} + 3 \zeta_{20}^{2} + 4 \zeta_{20}^{3} - 6 \zeta_{20}^{4} + 6 \zeta_{20}^{6} + 4 \zeta_{20}^{7} ) q^{47} + ( -4 + 8 \zeta_{20} - 4 \zeta_{20}^{4} + 4 \zeta_{20}^{5} + 4 \zeta_{20}^{6} - 8 \zeta_{20}^{7} ) q^{49} + ( -2 + 2 \zeta_{20}^{3} - 4 \zeta_{20}^{4} - \zeta_{20}^{5} + 4 \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{50} + ( 1 + 4 \zeta_{20} - \zeta_{20}^{2} + 2 \zeta_{20}^{4} + \zeta_{20}^{5} - \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{52} + ( 2 - 2 \zeta_{20} - 2 \zeta_{20}^{2} + 2 \zeta_{20}^{3} + 2 \zeta_{20}^{4} - 4 \zeta_{20}^{6} ) q^{53} + ( 1 + 2 \zeta_{20} - 5 \zeta_{20}^{2} - 6 \zeta_{20}^{3} + 3 \zeta_{20}^{4} + 4 \zeta_{20}^{5} - \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{55} + ( -2 + 2 \zeta_{20} + 2 \zeta_{20}^{2} - \zeta_{20}^{3} - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{6} - 2 \zeta_{20}^{7} ) q^{56} + ( -3 \zeta_{20} + \zeta_{20}^{2} + 4 \zeta_{20}^{3} + \zeta_{20}^{4} - 3 \zeta_{20}^{5} ) q^{58} + ( 4 \zeta_{20} + 4 \zeta_{20}^{7} ) q^{59} + ( -3 - 3 \zeta_{20} + 3 \zeta_{20}^{2} + 6 \zeta_{20}^{3} - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{6} + 4 \zeta_{20}^{7} ) q^{61} + ( -\zeta_{20} - 2 \zeta_{20}^{2} + 3 \zeta_{20}^{3} - 3 \zeta_{20}^{5} + 2 \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{62} + \zeta_{20}^{6} q^{64} + ( -6 + 2 \zeta_{20} - 2 \zeta_{20}^{2} - 6 \zeta_{20}^{3} - 2 \zeta_{20}^{4} - \zeta_{20}^{6} - 2 \zeta_{20}^{7} ) q^{65} + ( 1 - \zeta_{20} + 2 \zeta_{20}^{2} + \zeta_{20}^{3} + 2 \zeta_{20}^{4} - \zeta_{20}^{5} + \zeta_{20}^{6} ) q^{67} + ( 1 - 2 \zeta_{20}^{2} + \zeta_{20}^{4} + 2 \zeta_{20}^{5} - \zeta_{20}^{6} ) q^{68} + ( -2 + \zeta_{20} + \zeta_{20}^{2} - 4 \zeta_{20}^{3} - \zeta_{20}^{4} + \zeta_{20}^{5} + 6 \zeta_{20}^{6} - 4 \zeta_{20}^{7} ) q^{70} + ( 6 - 2 \zeta_{20} - \zeta_{20}^{2} - 2 \zeta_{20}^{3} + 6 \zeta_{20}^{4} ) q^{71} + ( 4 - 6 \zeta_{20} - 4 \zeta_{20}^{2} + 8 \zeta_{20}^{4} - 4 \zeta_{20}^{6} ) q^{73} + ( -5 - 2 \zeta_{20} - \zeta_{20}^{4} - \zeta_{20}^{5} + \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{74} + ( 1 + \zeta_{20}^{3} + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{76} + ( -1 - 7 \zeta_{20}^{2} + 6 \zeta_{20}^{4} + 3 \zeta_{20}^{5} - 3 \zeta_{20}^{6} - 3 \zeta_{20}^{7} ) q^{77} + ( -2 - 6 \zeta_{20} + 6 \zeta_{20}^{2} - 2 \zeta_{20}^{3} - 2 \zeta_{20}^{4} - 4 \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{79} + ( 1 - \zeta_{20}^{2} + 2 \zeta_{20}^{4} - \zeta_{20}^{5} - \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{80} + ( 4 - 8 \zeta_{20}^{2} + 4 \zeta_{20}^{3} + 6 \zeta_{20}^{4} + \zeta_{20}^{5} - 2 \zeta_{20}^{6} + 4 \zeta_{20}^{7} ) q^{82} + ( -5 - 5 \zeta_{20} - \zeta_{20}^{3} - 5 \zeta_{20}^{5} - 5 \zeta_{20}^{6} ) q^{83} + ( -2 + 4 \zeta_{20}^{3} - \zeta_{20}^{4} - \zeta_{20}^{5} - 3 \zeta_{20}^{6} ) q^{85} + ( -3 + 3 \zeta_{20} + 3 \zeta_{20}^{2} - 6 \zeta_{20}^{3} + \zeta_{20}^{5} - 2 \zeta_{20}^{6} - 5 \zeta_{20}^{7} ) q^{86} + ( -2 + \zeta_{20} + 3 \zeta_{20}^{2} - \zeta_{20}^{4} - \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{88} + ( -6 + 6 \zeta_{20}^{2} + 8 \zeta_{20}^{6} ) q^{89} + ( -2 \zeta_{20} + \zeta_{20}^{2} - 5 \zeta_{20}^{3} + 4 \zeta_{20}^{4} - 5 \zeta_{20}^{5} + \zeta_{20}^{6} - 2 \zeta_{20}^{7} ) q^{91} + ( 1 - 2 \zeta_{20} - \zeta_{20}^{2} + 2 \zeta_{20}^{3} - \zeta_{20}^{4} - 2 \zeta_{20}^{5} + \zeta_{20}^{6} ) q^{92} + ( -4 + 3 \zeta_{20}^{3} - 6 \zeta_{20}^{5} + 4 \zeta_{20}^{6} + 6 \zeta_{20}^{7} ) q^{94} + ( -2 - 2 \zeta_{20} + 2 \zeta_{20}^{2} - 5 \zeta_{20}^{4} - 4 \zeta_{20}^{5} + \zeta_{20}^{6} + 4 \zeta_{20}^{7} ) q^{95} + ( -3 - 2 \zeta_{20} + \zeta_{20}^{2} + 2 \zeta_{20}^{3} + \zeta_{20}^{4} + 2 \zeta_{20}^{6} - 6 \zeta_{20}^{7} ) q^{97} + ( 8 - 4 \zeta_{20} + 8 \zeta_{20}^{4} - 4 \zeta_{20}^{5} - 4 \zeta_{20}^{6} + 4 \zeta_{20}^{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{4} + 10q^{5} + O(q^{10})$$ $$8q + 2q^{4} + 10q^{5} + 4q^{11} + 4q^{14} - 2q^{16} - 10q^{17} + 10q^{19} + 20q^{22} - 10q^{23} + 10q^{25} + 28q^{26} + 10q^{28} - 10q^{29} + 6q^{31} - 4q^{34} - 10q^{35} - 10q^{37} + 14q^{41} + 6q^{44} - 8q^{46} + 30q^{47} - 16q^{49} - 10q^{55} - 4q^{56} - 14q^{61} + 2q^{64} - 50q^{65} + 10q^{67} + 34q^{71} - 36q^{74} - 40q^{77} - 50q^{83} - 20q^{85} - 22q^{86} - 10q^{88} - 20q^{89} - 4q^{91} + 10q^{92} - 24q^{94} - 20q^{97} + 40q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$\zeta_{20}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −0.951057 + 0.309017i 0.951057 − 0.309017i −0.587785 + 0.809017i 0.587785 − 0.809017i −0.587785 − 0.809017i 0.587785 + 0.809017i −0.951057 − 0.309017i 0.951057 + 0.309017i
−0.951057 + 0.309017i 0 0.809017 0.587785i 2.22982 0.166977i 0 5.07768i −0.587785 + 0.809017i 0 −2.06909 + 0.847859i
19.2 0.951057 0.309017i 0 0.809017 0.587785i −0.847859 + 2.06909i 0 1.07768i 0.587785 0.809017i 0 −0.166977 + 2.22982i
109.1 −0.587785 + 0.809017i 0 −0.309017 0.951057i 1.44575 1.70582i 0 1.27346i 0.951057 + 0.309017i 0 0.530249 + 2.17229i
109.2 0.587785 0.809017i 0 −0.309017 0.951057i 2.17229 + 0.530249i 0 2.72654i −0.951057 0.309017i 0 1.70582 1.44575i
289.1 −0.587785 0.809017i 0 −0.309017 + 0.951057i 1.44575 + 1.70582i 0 1.27346i 0.951057 0.309017i 0 0.530249 2.17229i
289.2 0.587785 + 0.809017i 0 −0.309017 + 0.951057i 2.17229 0.530249i 0 2.72654i −0.951057 + 0.309017i 0 1.70582 + 1.44575i
379.1 −0.951057 0.309017i 0 0.809017 + 0.587785i 2.22982 + 0.166977i 0 5.07768i −0.587785 0.809017i 0 −2.06909 0.847859i
379.2 0.951057 + 0.309017i 0 0.809017 + 0.587785i −0.847859 2.06909i 0 1.07768i 0.587785 + 0.809017i 0 −0.166977 2.22982i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 379.2 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.e even 10 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} + 36 T_{7}^{6} + 286 T_{7}^{4} + 596 T_{7}^{2} + 361$$ acting on $$S_{2}^{\mathrm{new}}(450, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$625 - 1250 T + 1125 T^{2} - 650 T^{3} + 305 T^{4} - 130 T^{5} + 45 T^{6} - 10 T^{7} + T^{8}$$
$7$ $$361 + 596 T^{2} + 286 T^{4} + 36 T^{6} + T^{8}$$
$11$ $$3481 - 2596 T + 3597 T^{2} - 818 T^{3} + 425 T^{4} + 58 T^{5} - 3 T^{6} - 4 T^{7} + T^{8}$$
$13$ $$1681 - 3280 T + 5329 T^{2} + 3410 T^{3} + 421 T^{4} - 110 T^{5} - 31 T^{6} + T^{8}$$
$17$ $$1 - 20 T + 116 T^{2} - 70 T^{3} + 46 T^{4} + 80 T^{5} + 41 T^{6} + 10 T^{7} + T^{8}$$
$19$ $$25 - 100 T + 375 T^{2} - 600 T^{3} + 510 T^{4} - 220 T^{5} + 60 T^{6} - 10 T^{7} + T^{8}$$
$23$ $$1 - 10 T + 59 T^{2} - 100 T^{3} - 19 T^{4} + 100 T^{5} + 59 T^{6} + 10 T^{7} + T^{8}$$
$29$ $$9025 - 6650 T + 25 T^{2} + 2650 T^{3} + 1785 T^{4} + 470 T^{5} + 105 T^{6} + 10 T^{7} + T^{8}$$
$31$ $$361 - 494 T + 1867 T^{2} + 168 T^{3} - 175 T^{4} - 48 T^{5} + 67 T^{6} - 6 T^{7} + T^{8}$$
$37$ $$58081 - 21690 T - 244 T^{2} - 1690 T^{3} + 1286 T^{4} - 140 T^{5} - 9 T^{6} + 10 T^{7} + T^{8}$$
$41$ $$28504921 - 3384926 T + 874047 T^{2} - 13228 T^{3} + 3705 T^{4} - 212 T^{5} + 127 T^{6} - 14 T^{7} + T^{8}$$
$43$ $$3682561 + 473344 T^{2} + 16686 T^{4} + 224 T^{6} + T^{8}$$
$47$ $$8637721 - 5025690 T + 1438691 T^{2} - 222240 T^{3} + 19761 T^{4} - 2040 T^{5} + 331 T^{6} - 30 T^{7} + T^{8}$$
$53$ $$4096 - 10240 T + 8704 T^{2} - 2560 T^{3} + 256 T^{4} + 160 T^{5} - 16 T^{6} + T^{8}$$
$59$ $$1638400 + 102400 T^{2} + 2560 T^{4} + T^{8}$$
$61$ $$32761 - 35114 T + 26027 T^{2} - 8532 T^{3} + 1250 T^{4} + 702 T^{5} + 152 T^{6} + 14 T^{7} + T^{8}$$
$67$ $$3481 - 4130 T - 179 T^{2} + 1490 T^{3} - 99 T^{4} - 110 T^{5} + 61 T^{6} - 10 T^{7} + T^{8}$$
$71$ $$1042441 - 843346 T + 494407 T^{2} - 169528 T^{3} + 37905 T^{4} - 5552 T^{5} + 567 T^{6} - 34 T^{7} + T^{8}$$
$73$ $$92416 + 194560 T + 170944 T^{2} + 74880 T^{3} + 12496 T^{4} - 2080 T^{5} - 36 T^{6} + T^{8}$$
$79$ $$22278400 + 6041600 T + 1364800 T^{2} + 142400 T^{3} + 10960 T^{4} - 80 T^{5} + 20 T^{6} + T^{8}$$
$83$ $$17131321 + 2276450 T + 333209 T^{2} + 376050 T^{3} + 112861 T^{4} + 15450 T^{5} + 1169 T^{6} + 50 T^{7} + T^{8}$$
$89$ $$( 400 + 400 T + 160 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$97$ $$130321 - 299630 T + 496201 T^{2} + 7280 T^{3} - 9594 T^{4} - 1070 T^{5} + 76 T^{6} + 20 T^{7} + T^{8}$$