# Properties

 Label 4017.2.a.k Level 4017 Weight 2 Character orbit 4017.a Self dual yes Analytic conductor 32.076 Analytic rank 0 Dimension 32 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: $$0$$ Dimension: $$32$$ Coefficient ring index: multiple of None Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 5q^{2} + 32q^{3} + 41q^{4} + 7q^{5} + 5q^{6} + 25q^{7} + 12q^{8} + 32q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 5q^{2} + 32q^{3} + 41q^{4} + 7q^{5} + 5q^{6} + 25q^{7} + 12q^{8} + 32q^{9} + 2q^{10} + 17q^{11} + 41q^{12} + 32q^{13} - 10q^{14} + 7q^{15} + 51q^{16} + 2q^{17} + 5q^{18} + 36q^{19} - 2q^{20} + 25q^{21} - 3q^{22} + 37q^{23} + 12q^{24} + 43q^{25} + 5q^{26} + 32q^{27} + 54q^{28} + 2q^{29} + 2q^{30} + 44q^{31} + 19q^{32} + 17q^{33} + 27q^{34} - 10q^{35} + 41q^{36} + 46q^{37} - 6q^{38} + 32q^{39} - 6q^{40} + 5q^{41} - 10q^{42} + 19q^{43} + 37q^{44} + 7q^{45} + 23q^{46} + 50q^{47} + 51q^{48} + 67q^{49} - 4q^{50} + 2q^{51} + 41q^{52} + 5q^{54} + 18q^{55} - 54q^{56} + 36q^{57} + 27q^{58} + 26q^{59} - 2q^{60} + 23q^{61} + 27q^{62} + 25q^{63} + 70q^{64} + 7q^{65} - 3q^{66} + 30q^{67} - 22q^{68} + 37q^{69} + 59q^{70} + 34q^{71} + 12q^{72} + 54q^{73} + 18q^{74} + 43q^{75} + 40q^{76} - 5q^{77} + 5q^{78} + 35q^{79} - 46q^{80} + 32q^{81} + 23q^{83} + 54q^{84} + 59q^{85} - 5q^{86} + 2q^{87} - 13q^{88} + 16q^{89} + 2q^{90} + 25q^{91} + 101q^{92} + 44q^{93} - 16q^{94} - q^{95} + 19q^{96} + 44q^{97} - 44q^{98} + 17q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 −2.81093 1.00000 5.90132 −1.40450 −2.81093 4.16883 −10.9663 1.00000 3.94794
1.2 −2.59984 1.00000 4.75917 1.47347 −2.59984 1.66117 −7.17341 1.00000 −3.83079
1.3 −2.39948 1.00000 3.75753 −1.39162 −2.39948 −0.807842 −4.21716 1.00000 3.33917
1.4 −2.23055 1.00000 2.97534 2.00994 −2.23055 −4.12307 −2.17553 1.00000 −4.48326
1.5 −2.21491 1.00000 2.90581 3.81514 −2.21491 4.67656 −2.00629 1.00000 −8.45019
1.6 −2.13224 1.00000 2.54643 −4.09989 −2.13224 4.42610 −1.16513 1.00000 8.74193
1.7 −1.74675 1.00000 1.05112 2.78436 −1.74675 −1.73589 1.65745 1.00000 −4.86356
1.8 −1.60531 1.00000 0.577033 −3.03050 −1.60531 3.88777 2.28431 1.00000 4.86490
1.9 −1.46454 1.00000 0.144881 1.31201 −1.46454 2.70261 2.71690 1.00000 −1.92149
1.10 −1.16220 1.00000 −0.649300 0.840727 −1.16220 −4.11692 3.07901 1.00000 −0.977089
1.11 −1.05981 1.00000 −0.876804 −3.52707 −1.05981 −1.13469 3.04886 1.00000 3.73802
1.12 −0.869920 1.00000 −1.24324 3.55798 −0.869920 3.07477 2.82136 1.00000 −3.09516
1.13 −0.628086 1.00000 −1.60551 −0.0316739 −0.628086 −0.526505 2.26457 1.00000 0.0198939
1.14 −0.166477 1.00000 −1.97229 3.74884 −0.166477 −0.361248 0.661293 1.00000 −0.624094
1.15 −0.0300119 1.00000 −1.99910 −0.348908 −0.0300119 −0.368688 0.120021 1.00000 0.0104714
1.16 0.135928 1.00000 −1.98152 −2.49648 0.135928 4.30504 −0.541201 1.00000 −0.339342
1.17 0.262075 1.00000 −1.93132 0.855445 0.262075 3.34709 −1.03030 1.00000 0.224191
1.18 0.397144 1.00000 −1.84228 −2.35712 0.397144 −3.48052 −1.52594 1.00000 −0.936118
1.19 0.921096 1.00000 −1.15158 −1.44579 0.921096 −0.134596 −2.90291 1.00000 −1.33171
1.20 1.15421 1.00000 −0.667790 1.41297 1.15421 4.47597 −3.07920 1.00000 1.63087
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.32 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.k 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4017.2.a.k 32 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4017))$$:

 $$T_{2}^{32} - \cdots$$ $$T_{23}^{32} - \cdots$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database