# Properties

 Label 990.2.n.e 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} -\zeta_{10}^{3} q^{4} -\zeta_{10} q^{5} + ( -2 + 2 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{7} -\zeta_{10}^{2} q^{8} +O(q^{10})$$ $$q + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} -\zeta_{10}^{3} q^{4} -\zeta_{10} q^{5} + ( -2 + 2 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{7} -\zeta_{10}^{2} q^{8} - q^{10} + ( -1 - \zeta_{10} - \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{11} + ( -4 + 5 \zeta_{10} - 5 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{13} + ( 2 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{14} -\zeta_{10} q^{16} + ( -5 + 2 \zeta_{10} - 5 \zeta_{10}^{2} ) q^{17} + ( 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{19} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{20} + ( -2 + 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{22} + ( -5 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{23} + \zeta_{10}^{2} q^{25} + ( 1 - \zeta_{10} + 4 \zeta_{10}^{3} ) q^{26} + ( 2 + 2 \zeta_{10}^{2} ) q^{28} + ( 1 - \zeta_{10} - 2 \zeta_{10}^{3} ) q^{29} + ( 2 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{31} - q^{32} + ( -3 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{34} + ( 2 - 2 \zeta_{10}^{3} ) q^{35} + ( 1 - \zeta_{10} + 7 \zeta_{10}^{3} ) q^{37} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{38} + \zeta_{10}^{3} q^{40} + ( -2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{41} + ( 3 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{43} + ( -2 + 4 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{44} + ( -5 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{46} + ( 7 \zeta_{10} - 7 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{47} + ( -4 + 3 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{49} + \zeta_{10} q^{50} + ( -\zeta_{10} + 5 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{52} + ( 6 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{53} + ( 3 - 2 \zeta_{10} + 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{55} + ( 2 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{56} + ( -\zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{58} + ( -5 + 5 \zeta_{10} + 11 \zeta_{10}^{3} ) q^{59} + ( -1 + \zeta_{10} - 2 \zeta_{10}^{3} ) q^{62} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{64} + ( 4 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{65} + ( -2 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{67} + ( -3 - 2 \zeta_{10} + 2 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{68} + ( 2 - 2 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{70} + ( 4 + 4 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{71} + ( 12 - 12 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{73} + ( -\zeta_{10} + 8 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{74} + ( 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{76} + ( -2 \zeta_{10} - 4 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{77} + ( 1 + 4 \zeta_{10} - 4 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{79} + \zeta_{10}^{2} q^{80} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{82} + 4 \zeta_{10} q^{83} + ( 5 \zeta_{10} - 2 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{85} + ( 3 - 8 \zeta_{10} + 8 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{86} + ( 2 + \zeta_{10} + 2 \zeta_{10}^{3} ) q^{88} + ( -16 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{89} + ( -8 \zeta_{10} + 2 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{91} + ( -3 + 3 \zeta_{10} + 5 \zeta_{10}^{3} ) q^{92} + ( 7 - 7 \zeta_{10} + 7 \zeta_{10}^{2} ) q^{94} + ( 2 - 2 \zeta_{10} ) q^{95} + ( -10 - 4 \zeta_{10} + 4 \zeta_{10}^{2} + 10 \zeta_{10}^{3} ) q^{97} + ( -1 - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{2} - q^{4} - q^{5} - 4q^{7} + q^{8} + O(q^{10})$$ $$4q + q^{2} - q^{4} - q^{5} - 4q^{7} + q^{8} - 4q^{10} - q^{11} - 2q^{13} + 4q^{14} - q^{16} - 13q^{17} + 6q^{19} - q^{20} - 9q^{22} - 14q^{23} - q^{25} + 7q^{26} + 6q^{28} + q^{29} - 4q^{32} - 2q^{34} + 6q^{35} + 10q^{37} + 4q^{38} + q^{40} - 6q^{41} + 22q^{43} - q^{44} - 11q^{46} + 21q^{47} - 9q^{49} + q^{50} - 7q^{52} + 14q^{53} + 4q^{55} + 4q^{56} - q^{58} - 4q^{59} - 5q^{62} - q^{64} + 18q^{65} + 2q^{67} - 13q^{68} + 4q^{70} + 16q^{71} + 30q^{73} - 10q^{74} - 4q^{76} - 4q^{77} + 11q^{79} - q^{80} - 4q^{82} + 4q^{83} + 12q^{85} - 7q^{86} + 11q^{88} - 60q^{89} - 18q^{91} - 4q^{92} + 14q^{94} + 6q^{95} - 38q^{97} + 4q^{98} + O(q^{100})$$

## 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 −1.00000 + 3.07768i −0.309017 0.951057i 0 −1.00000
181.1 −0.309017 0.951057i 0 −0.809017 + 0.587785i 0.309017 0.951057i 0 −1.00000 + 0.726543i 0.809017 + 0.587785i 0 −1.00000
361.1 −0.309017 + 0.951057i 0 −0.809017 0.587785i 0.309017 + 0.951057i 0 −1.00000 0.726543i 0.809017 0.587785i 0 −1.00000
631.1 0.809017 + 0.587785i 0 0.309017 + 0.951057i −0.809017 + 0.587785i 0 −1.00000 3.07768i −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.e 4
3.b odd 2 1 330.2.m.b 4
11.c even 5 1 inner 990.2.n.e 4
33.f even 10 1 3630.2.a.bb 2
33.h odd 10 1 330.2.m.b 4
33.h odd 10 1 3630.2.a.bj 2

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

## Hecke kernels

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

 $$T_{7}^{4} + 4 T_{7}^{3} + 16 T_{7}^{2} + 24 T_{7} + 16$$ $$T_{13}^{4} + 2 T_{13}^{3} + 24 T_{13}^{2} + 133 T_{13} + 361$$ $$T_{17}^{4} + 13 T_{17}^{3} + 94 T_{17}^{2} + 372 T_{17} + 961$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$7$ $$16 + 24 T + 16 T^{2} + 4 T^{3} + T^{4}$$
$11$ $$121 + 11 T - 9 T^{2} + T^{3} + T^{4}$$
$13$ $$361 + 133 T + 24 T^{2} + 2 T^{3} + T^{4}$$
$17$ $$961 + 372 T + 94 T^{2} + 13 T^{3} + T^{4}$$
$19$ $$16 - 16 T + 16 T^{2} - 6 T^{3} + T^{4}$$
$23$ $$( 1 + 7 T + T^{2} )^{2}$$
$29$ $$1 + 4 T + 6 T^{2} - T^{3} + T^{4}$$
$31$ $$25 - 25 T + 10 T^{2} + T^{4}$$
$37$ $$3025 - 275 T + 60 T^{2} - 10 T^{3} + T^{4}$$
$41$ $$16 + 16 T + 16 T^{2} + 6 T^{3} + T^{4}$$
$43$ $$( -1 - 11 T + T^{2} )^{2}$$
$47$ $$2401 - 686 T + 196 T^{2} - 21 T^{3} + T^{4}$$
$53$ $$16 - 24 T + 76 T^{2} - 14 T^{3} + T^{4}$$
$59$ $$1681 - 861 T + 166 T^{2} + 4 T^{3} + T^{4}$$
$61$ $$T^{4}$$
$67$ $$( -31 - T + T^{2} )^{2}$$
$71$ $$256 + 64 T + 96 T^{2} - 16 T^{3} + T^{4}$$
$73$ $$32400 - 5400 T + 540 T^{2} - 30 T^{3} + T^{4}$$
$79$ $$841 - 406 T + 96 T^{2} - 11 T^{3} + T^{4}$$
$83$ $$256 - 64 T + 16 T^{2} - 4 T^{3} + T^{4}$$
$89$ $$( 220 + 30 T + T^{2} )^{2}$$
$97$ $$55696 + 7552 T + 744 T^{2} + 38 T^{3} + T^{4}$$