# Properties

 Label 2166.4.a.q Level $2166$ Weight $4$ Character orbit 2166.a Self dual yes Analytic conductor $127.798$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2166 = 2 \cdot 3 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2166.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$127.798137072$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{30})$$ Defining polynomial: $$x^{2} - 30$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 114) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{30}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 3 q^{3} + 4 q^{4} + ( 4 + \beta ) q^{5} + 6 q^{6} + ( 9 - \beta ) q^{7} + 8 q^{8} + 9 q^{9} +O(q^{10})$$ $$q + 2 q^{2} + 3 q^{3} + 4 q^{4} + ( 4 + \beta ) q^{5} + 6 q^{6} + ( 9 - \beta ) q^{7} + 8 q^{8} + 9 q^{9} + ( 8 + 2 \beta ) q^{10} + ( 10 - \beta ) q^{11} + 12 q^{12} + ( 5 + 4 \beta ) q^{13} + ( 18 - 2 \beta ) q^{14} + ( 12 + 3 \beta ) q^{15} + 16 q^{16} + 12 q^{17} + 18 q^{18} + ( 16 + 4 \beta ) q^{20} + ( 27 - 3 \beta ) q^{21} + ( 20 - 2 \beta ) q^{22} + ( 40 + \beta ) q^{23} + 24 q^{24} + ( 11 + 8 \beta ) q^{25} + ( 10 + 8 \beta ) q^{26} + 27 q^{27} + ( 36 - 4 \beta ) q^{28} + ( 32 + 2 \beta ) q^{29} + ( 24 + 6 \beta ) q^{30} + ( 109 - 11 \beta ) q^{31} + 32 q^{32} + ( 30 - 3 \beta ) q^{33} + 24 q^{34} + ( -84 + 5 \beta ) q^{35} + 36 q^{36} + ( 43 - 20 \beta ) q^{37} + ( 15 + 12 \beta ) q^{39} + ( 32 + 8 \beta ) q^{40} + ( 188 + 6 \beta ) q^{41} + ( 54 - 6 \beta ) q^{42} + ( 127 + 33 \beta ) q^{43} + ( 40 - 4 \beta ) q^{44} + ( 36 + 9 \beta ) q^{45} + ( 80 + 2 \beta ) q^{46} + ( 130 - 44 \beta ) q^{47} + 48 q^{48} + ( -142 - 18 \beta ) q^{49} + ( 22 + 16 \beta ) q^{50} + 36 q^{51} + ( 20 + 16 \beta ) q^{52} + ( -152 - 3 \beta ) q^{53} + 54 q^{54} + ( -80 + 6 \beta ) q^{55} + ( 72 - 8 \beta ) q^{56} + ( 64 + 4 \beta ) q^{58} + ( 270 + \beta ) q^{59} + ( 48 + 12 \beta ) q^{60} + ( 323 + 22 \beta ) q^{61} + ( 218 - 22 \beta ) q^{62} + ( 81 - 9 \beta ) q^{63} + 64 q^{64} + ( 500 + 21 \beta ) q^{65} + ( 60 - 6 \beta ) q^{66} + ( 195 - 35 \beta ) q^{67} + 48 q^{68} + ( 120 + 3 \beta ) q^{69} + ( -168 + 10 \beta ) q^{70} + ( -266 + 34 \beta ) q^{71} + 72 q^{72} + ( -435 - 50 \beta ) q^{73} + ( 86 - 40 \beta ) q^{74} + ( 33 + 24 \beta ) q^{75} + ( 210 - 19 \beta ) q^{77} + ( 30 + 24 \beta ) q^{78} + ( 881 - 7 \beta ) q^{79} + ( 64 + 16 \beta ) q^{80} + 81 q^{81} + ( 376 + 12 \beta ) q^{82} + ( -912 - 12 \beta ) q^{83} + ( 108 - 12 \beta ) q^{84} + ( 48 + 12 \beta ) q^{85} + ( 254 + 66 \beta ) q^{86} + ( 96 + 6 \beta ) q^{87} + ( 80 - 8 \beta ) q^{88} + ( 30 + 77 \beta ) q^{89} + ( 72 + 18 \beta ) q^{90} + ( -435 + 31 \beta ) q^{91} + ( 160 + 4 \beta ) q^{92} + ( 327 - 33 \beta ) q^{93} + ( 260 - 88 \beta ) q^{94} + 96 q^{96} + ( 802 - 62 \beta ) q^{97} + ( -284 - 36 \beta ) q^{98} + ( 90 - 9 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{2} + 6q^{3} + 8q^{4} + 8q^{5} + 12q^{6} + 18q^{7} + 16q^{8} + 18q^{9} + O(q^{10})$$ $$2q + 4q^{2} + 6q^{3} + 8q^{4} + 8q^{5} + 12q^{6} + 18q^{7} + 16q^{8} + 18q^{9} + 16q^{10} + 20q^{11} + 24q^{12} + 10q^{13} + 36q^{14} + 24q^{15} + 32q^{16} + 24q^{17} + 36q^{18} + 32q^{20} + 54q^{21} + 40q^{22} + 80q^{23} + 48q^{24} + 22q^{25} + 20q^{26} + 54q^{27} + 72q^{28} + 64q^{29} + 48q^{30} + 218q^{31} + 64q^{32} + 60q^{33} + 48q^{34} - 168q^{35} + 72q^{36} + 86q^{37} + 30q^{39} + 64q^{40} + 376q^{41} + 108q^{42} + 254q^{43} + 80q^{44} + 72q^{45} + 160q^{46} + 260q^{47} + 96q^{48} - 284q^{49} + 44q^{50} + 72q^{51} + 40q^{52} - 304q^{53} + 108q^{54} - 160q^{55} + 144q^{56} + 128q^{58} + 540q^{59} + 96q^{60} + 646q^{61} + 436q^{62} + 162q^{63} + 128q^{64} + 1000q^{65} + 120q^{66} + 390q^{67} + 96q^{68} + 240q^{69} - 336q^{70} - 532q^{71} + 144q^{72} - 870q^{73} + 172q^{74} + 66q^{75} + 420q^{77} + 60q^{78} + 1762q^{79} + 128q^{80} + 162q^{81} + 752q^{82} - 1824q^{83} + 216q^{84} + 96q^{85} + 508q^{86} + 192q^{87} + 160q^{88} + 60q^{89} + 144q^{90} - 870q^{91} + 320q^{92} + 654q^{93} + 520q^{94} + 192q^{96} + 1604q^{97} - 568q^{98} + 180q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −5.47723 5.47723
2.00000 3.00000 4.00000 −6.95445 6.00000 19.9545 8.00000 9.00000 −13.9089
1.2 2.00000 3.00000 4.00000 14.9545 6.00000 −1.95445 8.00000 9.00000 29.9089
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$19$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2166.4.a.q 2
19.b odd 2 1 2166.4.a.k 2
19.c even 3 2 114.4.e.c 4
57.h odd 6 2 342.4.g.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.e.c 4 19.c even 3 2
342.4.g.e 4 57.h odd 6 2
2166.4.a.k 2 19.b odd 2 1
2166.4.a.q 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2166))$$:

 $$T_{5}^{2} - 8 T_{5} - 104$$ $$T_{13}^{2} - 10 T_{13} - 1895$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T )^{2}$$
$3$ $$( -3 + T )^{2}$$
$5$ $$-104 - 8 T + T^{2}$$
$7$ $$-39 - 18 T + T^{2}$$
$11$ $$-20 - 20 T + T^{2}$$
$13$ $$-1895 - 10 T + T^{2}$$
$17$ $$( -12 + T )^{2}$$
$19$ $$T^{2}$$
$23$ $$1480 - 80 T + T^{2}$$
$29$ $$544 - 64 T + T^{2}$$
$31$ $$-2639 - 218 T + T^{2}$$
$37$ $$-46151 - 86 T + T^{2}$$
$41$ $$31024 - 376 T + T^{2}$$
$43$ $$-114551 - 254 T + T^{2}$$
$47$ $$-215420 - 260 T + T^{2}$$
$53$ $$22024 + 304 T + T^{2}$$
$59$ $$72780 - 540 T + T^{2}$$
$61$ $$46249 - 646 T + T^{2}$$
$67$ $$-108975 - 390 T + T^{2}$$
$71$ $$-67964 + 532 T + T^{2}$$
$73$ $$-110775 + 870 T + T^{2}$$
$79$ $$770281 - 1762 T + T^{2}$$
$83$ $$814464 + 1824 T + T^{2}$$
$89$ $$-710580 - 60 T + T^{2}$$
$97$ $$181924 - 1604 T + T^{2}$$