# Properties

 Label 4002.2.a.w Level $4002$ Weight $2$ Character orbit 4002.a Self dual yes Analytic conductor $31.956$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4002 = 2 \cdot 3 \cdot 23 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4002.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$31.9561308889$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + 3 q^{5} + q^{6} + 3 q^{7} + q^{8} + q^{9} +O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} + 3 q^{5} + q^{6} + 3 q^{7} + q^{8} + q^{9} + 3 q^{10} + ( -1 - \beta ) q^{11} + q^{12} + ( 3 - \beta ) q^{13} + 3 q^{14} + 3 q^{15} + q^{16} + ( -3 - 2 \beta ) q^{17} + q^{18} + ( -1 + 2 \beta ) q^{19} + 3 q^{20} + 3 q^{21} + ( -1 - \beta ) q^{22} + q^{23} + q^{24} + 4 q^{25} + ( 3 - \beta ) q^{26} + q^{27} + 3 q^{28} - q^{29} + 3 q^{30} + ( -1 + \beta ) q^{31} + q^{32} + ( -1 - \beta ) q^{33} + ( -3 - 2 \beta ) q^{34} + 9 q^{35} + q^{36} + ( -2 - 3 \beta ) q^{37} + ( -1 + 2 \beta ) q^{38} + ( 3 - \beta ) q^{39} + 3 q^{40} + 7 \beta q^{41} + 3 q^{42} + ( -5 + 4 \beta ) q^{43} + ( -1 - \beta ) q^{44} + 3 q^{45} + q^{46} + ( 1 + 6 \beta ) q^{47} + q^{48} + 2 q^{49} + 4 q^{50} + ( -3 - 2 \beta ) q^{51} + ( 3 - \beta ) q^{52} + ( 6 + 2 \beta ) q^{53} + q^{54} + ( -3 - 3 \beta ) q^{55} + 3 q^{56} + ( -1 + 2 \beta ) q^{57} - q^{58} + ( 7 - 2 \beta ) q^{59} + 3 q^{60} + ( -4 + 6 \beta ) q^{61} + ( -1 + \beta ) q^{62} + 3 q^{63} + q^{64} + ( 9 - 3 \beta ) q^{65} + ( -1 - \beta ) q^{66} + ( -6 - 2 \beta ) q^{67} + ( -3 - 2 \beta ) q^{68} + q^{69} + 9 q^{70} + ( -3 + 5 \beta ) q^{71} + q^{72} + ( -6 - 4 \beta ) q^{73} + ( -2 - 3 \beta ) q^{74} + 4 q^{75} + ( -1 + 2 \beta ) q^{76} + ( -3 - 3 \beta ) q^{77} + ( 3 - \beta ) q^{78} + ( -5 - 7 \beta ) q^{79} + 3 q^{80} + q^{81} + 7 \beta q^{82} + ( -8 + 2 \beta ) q^{83} + 3 q^{84} + ( -9 - 6 \beta ) q^{85} + ( -5 + 4 \beta ) q^{86} - q^{87} + ( -1 - \beta ) q^{88} + ( -2 - 2 \beta ) q^{89} + 3 q^{90} + ( 9 - 3 \beta ) q^{91} + q^{92} + ( -1 + \beta ) q^{93} + ( 1 + 6 \beta ) q^{94} + ( -3 + 6 \beta ) q^{95} + q^{96} + 4 q^{97} + 2 q^{98} + ( -1 - \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{3} + 2q^{4} + 6q^{5} + 2q^{6} + 6q^{7} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{3} + 2q^{4} + 6q^{5} + 2q^{6} + 6q^{7} + 2q^{8} + 2q^{9} + 6q^{10} - 2q^{11} + 2q^{12} + 6q^{13} + 6q^{14} + 6q^{15} + 2q^{16} - 6q^{17} + 2q^{18} - 2q^{19} + 6q^{20} + 6q^{21} - 2q^{22} + 2q^{23} + 2q^{24} + 8q^{25} + 6q^{26} + 2q^{27} + 6q^{28} - 2q^{29} + 6q^{30} - 2q^{31} + 2q^{32} - 2q^{33} - 6q^{34} + 18q^{35} + 2q^{36} - 4q^{37} - 2q^{38} + 6q^{39} + 6q^{40} + 6q^{42} - 10q^{43} - 2q^{44} + 6q^{45} + 2q^{46} + 2q^{47} + 2q^{48} + 4q^{49} + 8q^{50} - 6q^{51} + 6q^{52} + 12q^{53} + 2q^{54} - 6q^{55} + 6q^{56} - 2q^{57} - 2q^{58} + 14q^{59} + 6q^{60} - 8q^{61} - 2q^{62} + 6q^{63} + 2q^{64} + 18q^{65} - 2q^{66} - 12q^{67} - 6q^{68} + 2q^{69} + 18q^{70} - 6q^{71} + 2q^{72} - 12q^{73} - 4q^{74} + 8q^{75} - 2q^{76} - 6q^{77} + 6q^{78} - 10q^{79} + 6q^{80} + 2q^{81} - 16q^{83} + 6q^{84} - 18q^{85} - 10q^{86} - 2q^{87} - 2q^{88} - 4q^{89} + 6q^{90} + 18q^{91} + 2q^{92} - 2q^{93} + 2q^{94} - 6q^{95} + 2q^{96} + 8q^{97} + 4q^{98} - 2q^{99} + O(q^{100})$$

## 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
 1.73205 −1.73205
1.00000 1.00000 1.00000 3.00000 1.00000 3.00000 1.00000 1.00000 3.00000
1.2 1.00000 1.00000 1.00000 3.00000 1.00000 3.00000 1.00000 1.00000 3.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$23$$ $$-1$$
$$29$$ $$1$$

## 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 4002.2.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.w 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4002))$$:

 $$T_{5} - 3$$ $$T_{7} - 3$$ $$T_{11}^{2} + 2 T_{11} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$( -3 + T )^{2}$$
$7$ $$( -3 + T )^{2}$$
$11$ $$-2 + 2 T + T^{2}$$
$13$ $$6 - 6 T + T^{2}$$
$17$ $$-3 + 6 T + T^{2}$$
$19$ $$-11 + 2 T + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$( 1 + T )^{2}$$
$31$ $$-2 + 2 T + T^{2}$$
$37$ $$-23 + 4 T + T^{2}$$
$41$ $$-147 + T^{2}$$
$43$ $$-23 + 10 T + T^{2}$$
$47$ $$-107 - 2 T + T^{2}$$
$53$ $$24 - 12 T + T^{2}$$
$59$ $$37 - 14 T + T^{2}$$
$61$ $$-92 + 8 T + T^{2}$$
$67$ $$24 + 12 T + T^{2}$$
$71$ $$-66 + 6 T + T^{2}$$
$73$ $$-12 + 12 T + T^{2}$$
$79$ $$-122 + 10 T + T^{2}$$
$83$ $$52 + 16 T + T^{2}$$
$89$ $$-8 + 4 T + T^{2}$$
$97$ $$( -4 + T )^{2}$$