# Properties

 Label 2166.4.a.m Level $2166$ Weight $4$ Character orbit 2166.a Self dual yes Analytic conductor $127.798$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{313})$$ Defining polynomial: $$x^{2} - x - 78$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{313})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{2} + 3 q^{3} + 4 q^{4} + ( -7 - \beta ) q^{5} -6 q^{6} + ( 3 + 3 \beta ) q^{7} -8 q^{8} + 9 q^{9} +O(q^{10})$$ $$q -2 q^{2} + 3 q^{3} + 4 q^{4} + ( -7 - \beta ) q^{5} -6 q^{6} + ( 3 + 3 \beta ) q^{7} -8 q^{8} + 9 q^{9} + ( 14 + 2 \beta ) q^{10} + ( -5 + \beta ) q^{11} + 12 q^{12} + 24 q^{13} + ( -6 - 6 \beta ) q^{14} + ( -21 - 3 \beta ) q^{15} + 16 q^{16} + ( 59 - 7 \beta ) q^{17} -18 q^{18} + ( -28 - 4 \beta ) q^{20} + ( 9 + 9 \beta ) q^{21} + ( 10 - 2 \beta ) q^{22} + ( -64 - 10 \beta ) q^{23} -24 q^{24} + ( 2 + 15 \beta ) q^{25} -48 q^{26} + 27 q^{27} + ( 12 + 12 \beta ) q^{28} + ( -192 - 6 \beta ) q^{29} + ( 42 + 6 \beta ) q^{30} + ( 84 + 24 \beta ) q^{31} -32 q^{32} + ( -15 + 3 \beta ) q^{33} + ( -118 + 14 \beta ) q^{34} + ( -255 - 27 \beta ) q^{35} + 36 q^{36} -84 q^{37} + 72 q^{39} + ( 56 + 8 \beta ) q^{40} + ( -96 - 6 \beta ) q^{41} + ( -18 - 18 \beta ) q^{42} + ( -61 - 45 \beta ) q^{43} + ( -20 + 4 \beta ) q^{44} + ( -63 - 9 \beta ) q^{45} + ( 128 + 20 \beta ) q^{46} + ( 25 + 43 \beta ) q^{47} + 48 q^{48} + ( 368 + 27 \beta ) q^{49} + ( -4 - 30 \beta ) q^{50} + ( 177 - 21 \beta ) q^{51} + 96 q^{52} + ( 96 - 30 \beta ) q^{53} -54 q^{54} + ( -43 - 3 \beta ) q^{55} + ( -24 - 24 \beta ) q^{56} + ( 384 + 12 \beta ) q^{58} + ( -24 - 60 \beta ) q^{59} + ( -84 - 12 \beta ) q^{60} + ( 129 - 45 \beta ) q^{61} + ( -168 - 48 \beta ) q^{62} + ( 27 + 27 \beta ) q^{63} + 64 q^{64} + ( -168 - 24 \beta ) q^{65} + ( 30 - 6 \beta ) q^{66} + ( -84 - 36 \beta ) q^{67} + ( 236 - 28 \beta ) q^{68} + ( -192 - 30 \beta ) q^{69} + ( 510 + 54 \beta ) q^{70} + ( 864 + 24 \beta ) q^{71} -72 q^{72} + ( -713 + 9 \beta ) q^{73} + 168 q^{74} + ( 6 + 45 \beta ) q^{75} + ( 219 - 9 \beta ) q^{77} -144 q^{78} + ( -588 + 84 \beta ) q^{79} + ( -112 - 16 \beta ) q^{80} + 81 q^{81} + ( 192 + 12 \beta ) q^{82} + ( -172 + 74 \beta ) q^{83} + ( 36 + 36 \beta ) q^{84} + ( 133 - 3 \beta ) q^{85} + ( 122 + 90 \beta ) q^{86} + ( -576 - 18 \beta ) q^{87} + ( 40 - 8 \beta ) q^{88} + ( -264 - 78 \beta ) q^{89} + ( 126 + 18 \beta ) q^{90} + ( 72 + 72 \beta ) q^{91} + ( -256 - 40 \beta ) q^{92} + ( 252 + 72 \beta ) q^{93} + ( -50 - 86 \beta ) q^{94} -96 q^{96} + ( 264 + 84 \beta ) q^{97} + ( -736 - 54 \beta ) q^{98} + ( -45 + 9 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} + 6q^{3} + 8q^{4} - 15q^{5} - 12q^{6} + 9q^{7} - 16q^{8} + 18q^{9} + O(q^{10})$$ $$2q - 4q^{2} + 6q^{3} + 8q^{4} - 15q^{5} - 12q^{6} + 9q^{7} - 16q^{8} + 18q^{9} + 30q^{10} - 9q^{11} + 24q^{12} + 48q^{13} - 18q^{14} - 45q^{15} + 32q^{16} + 111q^{17} - 36q^{18} - 60q^{20} + 27q^{21} + 18q^{22} - 138q^{23} - 48q^{24} + 19q^{25} - 96q^{26} + 54q^{27} + 36q^{28} - 390q^{29} + 90q^{30} + 192q^{31} - 64q^{32} - 27q^{33} - 222q^{34} - 537q^{35} + 72q^{36} - 168q^{37} + 144q^{39} + 120q^{40} - 198q^{41} - 54q^{42} - 167q^{43} - 36q^{44} - 135q^{45} + 276q^{46} + 93q^{47} + 96q^{48} + 763q^{49} - 38q^{50} + 333q^{51} + 192q^{52} + 162q^{53} - 108q^{54} - 89q^{55} - 72q^{56} + 780q^{58} - 108q^{59} - 180q^{60} + 213q^{61} - 384q^{62} + 81q^{63} + 128q^{64} - 360q^{65} + 54q^{66} - 204q^{67} + 444q^{68} - 414q^{69} + 1074q^{70} + 1752q^{71} - 144q^{72} - 1417q^{73} + 336q^{74} + 57q^{75} + 429q^{77} - 288q^{78} - 1092q^{79} - 240q^{80} + 162q^{81} + 396q^{82} - 270q^{83} + 108q^{84} + 263q^{85} + 334q^{86} - 1170q^{87} + 72q^{88} - 606q^{89} + 270q^{90} + 216q^{91} - 552q^{92} + 576q^{93} - 186q^{94} - 192q^{96} + 612q^{97} - 1526q^{98} - 81q^{99} + O(q^{100})$$

## 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
 9.34590 −8.34590
−2.00000 3.00000 4.00000 −16.3459 −6.00000 31.0377 −8.00000 9.00000 32.6918
1.2 −2.00000 3.00000 4.00000 1.34590 −6.00000 −22.0377 −8.00000 9.00000 −2.69181
 $$n$$: 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.m 2
19.b odd 2 1 2166.4.a.o yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2166.4.a.m 2 1.a even 1 1 trivial
2166.4.a.o yes 2 19.b odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 15 T_{5} - 22$$ $$T_{13} - 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T )^{2}$$
$3$ $$( -3 + T )^{2}$$
$5$ $$-22 + 15 T + T^{2}$$
$7$ $$-684 - 9 T + T^{2}$$
$11$ $$-58 + 9 T + T^{2}$$
$13$ $$( -24 + T )^{2}$$
$17$ $$-754 - 111 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$-3064 + 138 T + T^{2}$$
$29$ $$35208 + 390 T + T^{2}$$
$31$ $$-35856 - 192 T + T^{2}$$
$37$ $$( 84 + T )^{2}$$
$41$ $$6984 + 198 T + T^{2}$$
$43$ $$-151484 + 167 T + T^{2}$$
$47$ $$-142522 - 93 T + T^{2}$$
$53$ $$-63864 - 162 T + T^{2}$$
$59$ $$-278784 + 108 T + T^{2}$$
$61$ $$-147114 - 213 T + T^{2}$$
$67$ $$-91008 + 204 T + T^{2}$$
$71$ $$722304 - 1752 T + T^{2}$$
$73$ $$495634 + 1417 T + T^{2}$$
$79$ $$-254016 + 1092 T + T^{2}$$
$83$ $$-410272 + 270 T + T^{2}$$
$89$ $$-384264 + 606 T + T^{2}$$
$97$ $$-458496 - 612 T + T^{2}$$