# Properties

 Label 4017.2.a.d Level 4017 Weight 2 Character orbit 4017.a Self dual yes Analytic conductor 32.076 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4017 = 3 \cdot 13 \cdot 103$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4017.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.0759064919$$ Analytic rank: $$1$$ 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.

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

## 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
−2.41421 −1.00000 3.82843 −3.41421 2.41421 2.82843 −4.41421 1.00000 8.24264
1.2 0.414214 −1.00000 −1.82843 −0.585786 −0.414214 −2.82843 −1.58579 1.00000 −0.242641
 $$n$$: 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 4017.2.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4017.2.a.d 2 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$13$$ $$-1$$
$$103$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4017))$$:

 $$T_{2}^{2} + 2 T_{2} - 1$$ $$T_{23}^{2} - 4 T_{23} - 28$$