# Properties

 Label 4029.2.a.l Level 4029 Weight 2 Character orbit 4029.a Self dual yes Analytic conductor 32.172 Analytic rank 0 Dimension 32 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4029 = 3 \cdot 17 \cdot 79$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4029.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.1717269744$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q - q^{2} - 32q^{3} + 41q^{4} - q^{5} + q^{6} + 4q^{7} - 3q^{8} + 32q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - q^{2} - 32q^{3} + 41q^{4} - q^{5} + q^{6} + 4q^{7} - 3q^{8} + 32q^{9} + 17q^{10} + 8q^{11} - 41q^{12} + 17q^{13} + q^{14} + q^{15} + 55q^{16} - 32q^{17} - q^{18} + 48q^{19} - 7q^{20} - 4q^{21} - 4q^{22} - 19q^{23} + 3q^{24} + 63q^{25} + 27q^{26} - 32q^{27} + 17q^{28} - 15q^{29} - 17q^{30} + 20q^{31} + 13q^{32} - 8q^{33} + q^{34} + 22q^{35} + 41q^{36} + 6q^{37} + 11q^{38} - 17q^{39} + 47q^{40} + q^{41} - q^{42} + 40q^{43} + 22q^{44} - q^{45} + 5q^{46} - 5q^{47} - 55q^{48} + 88q^{49} + 17q^{50} + 32q^{51} + 23q^{52} - 34q^{53} + q^{54} + 48q^{55} - 48q^{57} - 9q^{58} + 41q^{59} + 7q^{60} + 20q^{61} + 15q^{62} + 4q^{63} + 93q^{64} - 58q^{65} + 4q^{66} + 52q^{67} - 41q^{68} + 19q^{69} + 25q^{70} + q^{71} - 3q^{72} + 19q^{73} + 12q^{74} - 63q^{75} + 128q^{76} - 20q^{77} - 27q^{78} + 32q^{79} - 16q^{80} + 32q^{81} - 5q^{82} + 31q^{83} - 17q^{84} + q^{85} - 62q^{86} + 15q^{87} + 35q^{88} + 18q^{89} + 17q^{90} + 48q^{91} - 75q^{92} - 20q^{93} + 29q^{94} + 5q^{95} - 13q^{96} + 17q^{97} + 30q^{98} + 8q^{99} + O(q^{100})$$

## 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 −2.77670 −1.00000 5.71004 −0.322981 2.77670 −2.70395 −10.3017 1.00000 0.896820
1.2 −2.65510 −1.00000 5.04957 −3.36488 2.65510 2.92058 −8.09693 1.00000 8.93412
1.3 −2.57711 −1.00000 4.64151 2.80201 2.57711 −2.75409 −6.80748 1.00000 −7.22109
1.4 −2.50856 −1.00000 4.29286 −2.69434 2.50856 3.74400 −5.75176 1.00000 6.75889
1.5 −2.27866 −1.00000 3.19228 −3.13196 2.27866 −0.0802156 −2.71679 1.00000 7.13667
1.6 −2.16757 −1.00000 2.69838 2.47527 2.16757 5.12027 −1.51379 1.00000 −5.36532
1.7 −2.03180 −1.00000 2.12821 1.66964 2.03180 −1.52463 −0.260503 1.00000 −3.39237
1.8 −1.67650 −1.00000 0.810659 −3.83953 1.67650 −4.72769 1.99393 1.00000 6.43697
1.9 −1.61915 −1.00000 0.621643 1.62143 1.61915 3.15810 2.23177 1.00000 −2.62534
1.10 −1.60231 −1.00000 0.567397 −0.489899 1.60231 −0.574101 2.29547 1.00000 0.784971
1.11 −1.34067 −1.00000 −0.202611 1.16053 1.34067 −4.05917 2.95297 1.00000 −1.55588
1.12 −1.17408 −1.00000 −0.621542 3.86856 1.17408 3.06255 3.07789 1.00000 −4.54199
1.13 −0.857766 −1.00000 −1.26424 −4.41756 0.857766 0.00991750 2.79995 1.00000 3.78923
1.14 −0.510062 −1.00000 −1.73984 −1.60726 0.510062 −0.772958 1.90755 1.00000 0.819801
1.15 −0.249798 −1.00000 −1.93760 0.601573 0.249798 −2.69175 0.983604 1.00000 −0.150272
1.16 −0.0772262 −1.00000 −1.99404 3.97384 0.0772262 0.969209 0.308444 1.00000 −0.306884
1.17 −0.0717307 −1.00000 −1.99485 −1.92367 0.0717307 3.89464 0.286554 1.00000 0.137987
1.18 0.403182 −1.00000 −1.83744 2.23661 −0.403182 −0.778796 −1.54719 1.00000 0.901761
1.19 0.437930 −1.00000 −1.80822 −1.50297 −0.437930 −4.33772 −1.66773 1.00000 −0.658196
1.20 0.937640 −1.00000 −1.12083 0.347083 −0.937640 4.09662 −2.92622 1.00000 0.325439
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 4029.2.a.l 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4029.2.a.l 32 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$17$$ $$1$$
$$79$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4029))$$:

 $$T_{2}^{32} + \cdots$$ $$T_{5}^{32} + \cdots$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database