# Properties

 Label 990.2.n.i Level $990$ Weight $2$ Character orbit 990.n Analytic conductor $7.905$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.2769390625.1 Defining polynomial: $$x^{8} - x^{7} - 2 x^{6} + 5 x^{5} + 21 x^{4} + 75 x^{3} - 198 x^{2} - 87 x + 841$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( -1 - \beta_{2} + \beta_{3} - \beta_{5} ) q^{4} + \beta_{5} q^{5} + ( -\beta_{1} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{7} -\beta_{3} q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( -1 - \beta_{2} + \beta_{3} - \beta_{5} ) q^{4} + \beta_{5} q^{5} + ( -\beta_{1} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{7} -\beta_{3} q^{8} + q^{10} + ( -2 + \beta_{1} + 2 \beta_{3} - \beta_{7} ) q^{11} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{13} + ( 1 - \beta_{1} - \beta_{6} ) q^{14} + \beta_{5} q^{16} + ( 1 + \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{17} + ( -3 + \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{19} + \beta_{2} q^{20} + ( -2 \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{22} + ( 2 \beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{23} -\beta_{3} q^{25} + ( -2 \beta_{1} - \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{26} + ( \beta_{1} + \beta_{2} + \beta_{7} ) q^{28} + ( -5 - \beta_{1} + \beta_{4} - 4 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{29} + ( \beta_{1} + 4 \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{31} + q^{32} + ( 3 + 2 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{34} + ( \beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{35} + ( -2 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{37} + ( 2 - 3 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{38} + ( -1 - \beta_{2} + \beta_{3} - \beta_{5} ) q^{40} + ( 3 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 4 \beta_{7} ) q^{41} + ( 1 + 2 \beta_{1} + 3 \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{43} + ( -\beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{44} + ( -1 - \beta_{1} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{46} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{47} + ( -8 + 7 \beta_{3} - \beta_{4} + 4 \beta_{5} + \beta_{6} ) q^{49} + \beta_{5} q^{50} + ( 1 - 2 \beta_{1} + \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{52} + ( -\beta_{1} - 4 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{53} + ( 2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{6} ) q^{55} + ( -1 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{56} + ( -4 - \beta_{1} - 5 \beta_{2} - \beta_{6} ) q^{58} + ( 6 - \beta_{2} + \beta_{3} + 6 \beta_{5} ) q^{59} + ( -4 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{7} ) q^{61} + ( -5 - \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + \beta_{4} - 4 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{62} + \beta_{2} q^{64} + ( -1 + 2 \beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{65} + ( 4 - 4 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 4 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{67} + ( -1 + 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{68} + ( -\beta_{1} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{70} + ( -2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{71} + ( 2 - 6 \beta_{2} + 6 \beta_{3} + 2 \beta_{5} ) q^{73} + ( 1 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{74} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{76} + ( -8 + 3 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{77} + ( 3 + 4 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{79} -\beta_{3} q^{80} + ( -4 + \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{82} + 8 \beta_{5} q^{83} + ( -\beta_{1} - 3 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{85} + ( -1 - 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{86} + ( -\beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{7} ) q^{88} + ( 5 - 4 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{89} + ( -4 + 2 \beta_{1} - 3 \beta_{2} - 10 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{91} + ( 1 - \beta_{2} - 2 \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{92} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{94} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{95} + ( -2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{97} + ( 4 - 2 \beta_{1} - 8 \beta_{2} + \beta_{4} - 7 \beta_{5} + \beta_{6} + \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{2} - 2q^{4} - 2q^{5} + 3q^{7} - 2q^{8} + O(q^{10})$$ $$8q - 2q^{2} - 2q^{4} - 2q^{5} + 3q^{7} - 2q^{8} + 8q^{10} - 10q^{11} + q^{13} + 3q^{14} - 2q^{16} - 3q^{17} - 12q^{19} - 2q^{20} + 10q^{22} - 2q^{25} + q^{26} - 2q^{28} - 27q^{29} - 6q^{31} + 8q^{32} + 22q^{34} - 2q^{35} - 4q^{37} + 13q^{38} - 2q^{40} + 23q^{41} - 2q^{43} - 10q^{44} - 5q^{46} - 5q^{47} - 53q^{49} - 2q^{50} + q^{52} + 18q^{53} + 5q^{55} - 2q^{56} - 27q^{58} + 40q^{59} - 20q^{61} - 11q^{62} - 2q^{64} - 4q^{65} + 18q^{67} - 3q^{68} + 3q^{70} + 10q^{71} + 36q^{73} - 4q^{74} - 2q^{76} - 50q^{77} - 11q^{79} - 2q^{80} - 12q^{82} - 16q^{83} - 8q^{85} + 13q^{86} + 5q^{88} + 46q^{89} - 39q^{91} + 5q^{92} + 15q^{94} - 12q^{95} + 16q^{97} + 62q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 2 x^{6} + 5 x^{5} + 21 x^{4} + 75 x^{3} - 198 x^{2} - 87 x + 841$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$3622 \nu^{7} - 168690 \nu^{6} + 210024 \nu^{5} - 119321 \nu^{4} + 621001 \nu^{3} - 2170527 \nu^{2} - 10517880 \nu + 20083080$$$$)/11583209$$ $$\beta_{3}$$ $$=$$ $$($$$$4915 \nu^{7} - 90523 \nu^{6} + 209410 \nu^{5} + 136080 \nu^{4} + 66153 \nu^{3} - 2218842 \nu^{2} - 4202349 \nu + 23823587$$$$)/11583209$$ $$\beta_{4}$$ $$=$$ $$($$$$908 \nu^{7} - 3211 \nu^{6} + 11167 \nu^{5} + 3050 \nu^{4} + 96671 \nu^{3} - 39405 \nu^{2} - 93476 \nu + 949895$$$$)/399421$$ $$\beta_{5}$$ $$=$$ $$($$$$-31462 \nu^{7} + 137254 \nu^{6} - 28223 \nu^{5} + 415469 \nu^{4} - 1154253 \nu^{3} + 298519 \nu^{2} + 11658276 \nu - 7703821$$$$)/11583209$$ $$\beta_{6}$$ $$=$$ $$($$$$-3648 \nu^{7} + 3143 \nu^{6} - 19751 \nu^{5} + 17019 \nu^{4} - 91661 \nu^{3} - 187200 \nu^{2} + 360035 \nu - 912398$$$$)/399421$$ $$\beta_{7}$$ $$=$$ $$($$$$5692 \nu^{7} - 7492 \nu^{6} + 4739 \nu^{5} - 18791 \nu^{4} + 84213 \nu^{3} + 337956 \nu^{2} - 703386 \nu + 105038$$$$)/399421$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} + 5 \beta_{5} - 6 \beta_{3} + 6 \beta_{2} + 5$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + 5 \beta_{4} - 6 \beta_{3} + \beta_{2} - \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$9 \beta_{7} + 11 \beta_{6} + 21 \beta_{5} + 11 \beta_{4} + 10 \beta_{3} - 9 \beta_{1} - 10$$ $$\nu^{5}$$ $$=$$ $$-8 \beta_{7} - 20 \beta_{6} + 21 \beta_{5} - 8 \beta_{4} + 21 \beta_{2} - 47$$ $$\nu^{6}$$ $$=$$ $$-29 \beta_{7} - 29 \beta_{6} - 60 \beta_{5} + 60 \beta_{3} - 128 \beta_{2} - 59 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$9 \beta_{7} - 324 \beta_{5} - 31 \beta_{4} + 557 \beta_{3} - 557 \beta_{2} + 31 \beta_{1} - 324$$

## Character values

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

 $$n$$ $$397$$ $$541$$ $$551$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{2} + \beta_{3} - \beta_{5}$$ $$1$$

## 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
 1.88778 + 1.74766i −1.07877 − 2.33544i 1.77873 − 1.10367i −2.08774 + 0.152618i 1.77873 + 1.10367i −2.08774 − 0.152618i 1.88778 − 1.74766i −1.07877 + 2.33544i
−0.809017 + 0.587785i 0 0.309017 0.951057i −0.809017 0.587785i 0 −1.38778 + 4.27116i 0.309017 + 0.951057i 0 1.00000
91.2 −0.809017 + 0.587785i 0 0.309017 0.951057i −0.809017 0.587785i 0 1.57877 4.85894i 0.309017 + 0.951057i 0 1.00000
181.1 0.309017 + 0.951057i 0 −0.809017 + 0.587785i 0.309017 0.951057i 0 −1.27873 + 0.929049i −0.809017 0.587785i 0 1.00000
181.2 0.309017 + 0.951057i 0 −0.809017 + 0.587785i 0.309017 0.951057i 0 2.58774 1.88011i −0.809017 0.587785i 0 1.00000
361.1 0.309017 0.951057i 0 −0.809017 0.587785i 0.309017 + 0.951057i 0 −1.27873 0.929049i −0.809017 + 0.587785i 0 1.00000
361.2 0.309017 0.951057i 0 −0.809017 0.587785i 0.309017 + 0.951057i 0 2.58774 + 1.88011i −0.809017 + 0.587785i 0 1.00000
631.1 −0.809017 0.587785i 0 0.309017 + 0.951057i −0.809017 + 0.587785i 0 −1.38778 4.27116i 0.309017 0.951057i 0 1.00000
631.2 −0.809017 0.587785i 0 0.309017 + 0.951057i −0.809017 + 0.587785i 0 1.57877 + 4.85894i 0.309017 0.951057i 0 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 631.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.i 8
3.b odd 2 1 330.2.m.f 8
11.c even 5 1 inner 990.2.n.i 8
33.f even 10 1 3630.2.a.bs 4
33.h odd 10 1 330.2.m.f 8
33.h odd 10 1 3630.2.a.bq 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.m.f 8 3.b odd 2 1
330.2.m.f 8 33.h odd 10 1
990.2.n.i 8 1.a even 1 1 trivial
990.2.n.i 8 11.c even 5 1 inner
3630.2.a.bq 4 33.h odd 10 1
3630.2.a.bs 4 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 $$S_{2}^{\mathrm{new}}(990, [\chi])$$:

 $$T_{7}^{8} - \cdots$$ $$T_{13}^{8} - \cdots$$ $$T_{17}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$7$ $$13456 + 7192 T + 808 T^{2} - 896 T^{3} + 505 T^{4} - 76 T^{5} + 38 T^{6} - 3 T^{7} + T^{8}$$
$11$ $$14641 + 13310 T + 3509 T^{2} - 330 T^{3} - 349 T^{4} - 30 T^{5} + 29 T^{6} + 10 T^{7} + T^{8}$$
$13$ $$14641 + 7381 T - 898 T^{2} - 2097 T^{3} + 1505 T^{4} - 123 T^{5} + 62 T^{6} - T^{7} + T^{8}$$
$17$ $$256 - 192 T + 608 T^{2} - 1104 T^{3} + 1105 T^{4} + 276 T^{5} + 38 T^{6} + 3 T^{7} + T^{8}$$
$19$ $$16 + 112 T + 2488 T^{2} - 2226 T^{3} + 575 T^{4} + 309 T^{5} + 88 T^{6} + 12 T^{7} + T^{8}$$
$23$ $$( 55 - 130 T - 65 T^{2} + T^{4} )^{2}$$
$29$ $$524176 + 143352 T + 122528 T^{2} + 62604 T^{3} + 18505 T^{4} + 3204 T^{5} + 378 T^{6} + 27 T^{7} + T^{8}$$
$31$ $$26896 - 43296 T + 32112 T^{2} - 9178 T^{3} + 2955 T^{4} + 263 T^{5} + 42 T^{6} + 6 T^{7} + T^{8}$$
$37$ $$3031081 - 111424 T + 161147 T^{2} - 26832 T^{3} + 4250 T^{4} + 192 T^{5} + 32 T^{6} + 4 T^{7} + T^{8}$$
$41$ $$3254416 - 4387328 T + 2420328 T^{2} - 313946 T^{3} + 45705 T^{4} - 4536 T^{5} + 408 T^{6} - 23 T^{7} + T^{8}$$
$43$ $$( -4 - 54 T - 39 T^{2} + T^{3} + T^{4} )^{2}$$
$47$ $$555025 + 182525 T + 84900 T^{2} + 7925 T^{3} + 585 T^{4} - 45 T^{5} + 30 T^{6} + 5 T^{7} + T^{8}$$
$53$ $$1936 - 4488 T + 4668 T^{2} - 1826 T^{3} + 1275 T^{4} - 311 T^{5} + 138 T^{6} - 18 T^{7} + T^{8}$$
$59$ $$( 3025 - 825 T + 190 T^{2} - 20 T^{3} + T^{4} )^{2}$$
$61$ $$1638400 - 614400 T + 204800 T^{2} - 32000 T^{3} + 6160 T^{4} - 280 T^{5} + 160 T^{6} + 20 T^{7} + T^{8}$$
$67$ $$( 76 - 74 T - 89 T^{2} - 9 T^{3} + T^{4} )^{2}$$
$71$ $$1638400 + 819200 T + 153600 T^{2} - 16000 T^{3} + 9360 T^{4} - 960 T^{5} + 120 T^{6} - 10 T^{7} + T^{8}$$
$73$ $$( 5776 - 1672 T + 244 T^{2} - 18 T^{3} + T^{4} )^{2}$$
$79$ $$82373776 - 13486936 T + 1077372 T^{2} + 37522 T^{3} + 24005 T^{4} + 1678 T^{5} + 222 T^{6} + 11 T^{7} + T^{8}$$
$83$ $$( 4096 + 512 T + 64 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$89$ $$( -6724 + 1178 T + 79 T^{2} - 23 T^{3} + T^{4} )^{2}$$
$97$ $$1290496 + 518016 T + 389312 T^{2} + 31168 T^{3} + 2800 T^{4} + 272 T^{5} + 112 T^{6} - 16 T^{7} + T^{8}$$