# Properties

 Label 990.2.n.a Level $990$ Weight $2$ Character orbit 990.n Analytic conductor $7.905$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$990 = 2 \cdot 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 990.n (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.90518980011$$ Analytic rank: $$0$$ 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 330) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} -\zeta_{10}^{3} q^{4} -\zeta_{10} q^{5} + ( -1 + \zeta_{10} - 2 \zeta_{10}^{3} ) q^{7} + \zeta_{10}^{2} q^{8} +O(q^{10})$$ $$q + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} -\zeta_{10}^{3} q^{4} -\zeta_{10} q^{5} + ( -1 + \zeta_{10} - 2 \zeta_{10}^{3} ) q^{7} + \zeta_{10}^{2} q^{8} + q^{10} + ( -2 + 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{11} + ( 1 - \zeta_{10}^{3} ) q^{13} + ( -\zeta_{10} + 3 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{14} -\zeta_{10} q^{16} + ( 4 - 4 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{17} + ( \zeta_{10} + \zeta_{10}^{3} ) q^{19} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{20} + ( 2 - 4 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{22} + ( 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{23} + \zeta_{10}^{2} q^{25} + ( -1 + \zeta_{10} + \zeta_{10}^{3} ) q^{26} + ( 1 - 3 \zeta_{10} + \zeta_{10}^{2} ) q^{28} + ( 6 - 6 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{29} + ( 6 - 8 \zeta_{10} + 8 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{31} + q^{32} + ( -4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{34} + ( -2 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{35} + ( 1 - \zeta_{10} - \zeta_{10}^{3} ) q^{37} + ( -1 - \zeta_{10}^{2} ) q^{38} -\zeta_{10}^{3} q^{40} + ( -7 \zeta_{10} + 5 \zeta_{10}^{2} - 7 \zeta_{10}^{3} ) q^{41} + ( 4 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{43} + ( 2 + \zeta_{10} + 2 \zeta_{10}^{3} ) q^{44} + ( -3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{46} + ( -\zeta_{10} + 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{47} + ( 5 - 3 \zeta_{10} + 5 \zeta_{10}^{2} ) q^{49} -\zeta_{10} q^{50} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{52} + ( 2 + 3 \zeta_{10} - 3 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{53} + ( 1 + \zeta_{10} + \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{55} + ( 2 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{56} + ( 6 \zeta_{10} + 6 \zeta_{10}^{3} ) q^{58} + ( -1 + \zeta_{10} - 13 \zeta_{10}^{3} ) q^{59} + ( 2 - 10 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{61} + ( 2 - 2 \zeta_{10} + 6 \zeta_{10}^{3} ) q^{62} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{64} + ( -1 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{65} + ( 8 + 10 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{67} + ( 4 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{68} + ( -1 + \zeta_{10} - 2 \zeta_{10}^{3} ) q^{70} + ( -6 - 4 \zeta_{10} - 6 \zeta_{10}^{2} ) q^{71} + ( -8 + 8 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{73} + ( \zeta_{10} + \zeta_{10}^{3} ) q^{74} + ( 1 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{76} + ( 5 + \zeta_{10} - 3 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{77} + ( -2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{79} + \zeta_{10}^{2} q^{80} + ( 7 - 5 \zeta_{10} + 7 \zeta_{10}^{2} ) q^{82} + ( 8 + 8 \zeta_{10}^{2} ) q^{83} + ( -4 \zeta_{10} + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{85} + ( -4 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{86} + ( -3 + 2 \zeta_{10} - 4 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{88} + ( 10 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{89} + ( -2 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{91} + ( 3 - 3 \zeta_{10} ) q^{92} + ( 1 - 2 \zeta_{10} + \zeta_{10}^{2} ) q^{94} + ( 1 - \zeta_{10} - \zeta_{10}^{3} ) q^{95} + ( -2 + 10 \zeta_{10} - 10 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{97} + ( -2 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - q^{2} - q^{4} - q^{5} - 5q^{7} - q^{8} + O(q^{10})$$ $$4q - q^{2} - q^{4} - q^{5} - 5q^{7} - q^{8} + 4q^{10} - 9q^{11} + 3q^{13} - 5q^{14} - q^{16} + 8q^{17} + 2q^{19} - q^{20} + q^{22} - 6q^{23} - q^{25} - 2q^{26} + 12q^{29} + 2q^{31} + 4q^{32} + 8q^{34} + 2q^{37} - 3q^{38} - q^{40} - 19q^{41} + 12q^{43} + 11q^{44} - 6q^{46} - 4q^{47} + 12q^{49} - q^{50} - 2q^{52} + 12q^{53} + q^{55} + 10q^{56} + 12q^{58} - 16q^{59} - 4q^{61} + 12q^{62} - q^{64} - 2q^{65} + 12q^{67} + 8q^{68} - 5q^{70} - 22q^{71} - 22q^{73} + 2q^{74} + 2q^{76} + 30q^{77} - 10q^{79} - q^{80} + 16q^{82} + 24q^{83} - 12q^{85} - 8q^{86} - 4q^{88} + 34q^{89} - 5q^{91} + 9q^{92} + q^{94} + 2q^{95} + 14q^{97} + 2q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/990\mathbb{Z}\right)^\times$$.

 $$n$$ $$397$$ $$541$$ $$551$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
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 −0.809017 0.587785i 0 0.427051 1.31433i 0.309017 + 0.951057i 0 1.00000
181.1 0.309017 + 0.951057i 0 −0.809017 + 0.587785i 0.309017 0.951057i 0 −2.92705 + 2.12663i −0.809017 0.587785i 0 1.00000
361.1 0.309017 0.951057i 0 −0.809017 0.587785i 0.309017 + 0.951057i 0 −2.92705 2.12663i −0.809017 + 0.587785i 0 1.00000
631.1 −0.809017 0.587785i 0 0.309017 + 0.951057i −0.809017 + 0.587785i 0 0.427051 + 1.31433i 0.309017 0.951057i 0 1.00000
 $$n$$: 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
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 990.2.n.a 4
3.b odd 2 1 330.2.m.d 4
11.c even 5 1 inner 990.2.n.a 4
33.f even 10 1 3630.2.a.bi 2
33.h odd 10 1 330.2.m.d 4
33.h odd 10 1 3630.2.a.bc 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.m.d 4 3.b odd 2 1
330.2.m.d 4 33.h odd 10 1
990.2.n.a 4 1.a even 1 1 trivial
990.2.n.a 4 11.c even 5 1 inner
3630.2.a.bc 2 33.h odd 10 1
3630.2.a.bi 2 33.f even 10 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(990, [\chi])$$:

 $$T_{7}^{4} + 5 T_{7}^{3} + 10 T_{7}^{2} + 25$$ $$T_{13}^{4} - 3 T_{13}^{3} + 4 T_{13}^{2} - 2 T_{13} + 1$$ $$T_{17}^{4} - 8 T_{17}^{3} + 64 T_{17}^{2} - 192 T_{17} + 256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$7$ $$25 + 10 T^{2} + 5 T^{3} + T^{4}$$
$11$ $$121 + 99 T + 41 T^{2} + 9 T^{3} + T^{4}$$
$13$ $$1 - 2 T + 4 T^{2} - 3 T^{3} + T^{4}$$
$17$ $$256 - 192 T + 64 T^{2} - 8 T^{3} + T^{4}$$
$19$ $$1 - 3 T + 4 T^{2} - 2 T^{3} + T^{4}$$
$23$ $$( -9 + 3 T + T^{2} )^{2}$$
$29$ $$1296 - 648 T + 144 T^{2} - 12 T^{3} + T^{4}$$
$31$ $$1936 - 528 T + 64 T^{2} - 2 T^{3} + T^{4}$$
$37$ $$1 - 3 T + 4 T^{2} - 2 T^{3} + T^{4}$$
$41$ $$3481 + 944 T + 186 T^{2} + 19 T^{3} + T^{4}$$
$43$ $$( 4 - 6 T + T^{2} )^{2}$$
$47$ $$1 - T + 6 T^{2} + 4 T^{3} + T^{4}$$
$53$ $$961 - 403 T + 94 T^{2} - 12 T^{3} + T^{4}$$
$59$ $$32761 + 1991 T + 186 T^{2} + 16 T^{3} + T^{4}$$
$61$ $$5776 + 1064 T + 96 T^{2} + 4 T^{3} + T^{4}$$
$67$ $$( -116 - 6 T + T^{2} )^{2}$$
$71$ $$16 - 32 T + 184 T^{2} + 22 T^{3} + T^{4}$$
$73$ $$5776 + 1368 T + 244 T^{2} + 22 T^{3} + T^{4}$$
$79$ $$400 + 200 T + 60 T^{2} + 10 T^{3} + T^{4}$$
$83$ $$4096 - 1024 T + 256 T^{2} - 24 T^{3} + T^{4}$$
$89$ $$( 61 - 17 T + T^{2} )^{2}$$
$97$ $$1936 - 1144 T + 276 T^{2} - 14 T^{3} + T^{4}$$