# Properties

 Label 4031.2.a.a Level 4031 Weight 2 Character orbit 4031.a Self dual yes Analytic conductor 32.188 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4031 = 29 \cdot 139$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4031.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.1876970548$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.41421 1.41421
−0.414214 −0.414214 −1.82843 2.00000 0.171573 1.82843 1.58579 −2.82843 −0.828427
1.2 2.41421 2.41421 3.82843 2.00000 5.82843 −3.82843 4.41421 2.82843 4.82843
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 4031.2.a.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4031.2.a.a 2 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$29$$ $$-1$$
$$139$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2 T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4031))$$.