# Properties

 Label 4019.2.a.a Level 4019 Weight 2 Character orbit 4019.a Self dual yes Analytic conductor 32.092 Analytic rank 1 Dimension 149 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4019$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4019.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.0918765724$$ Analytic rank: $$1$$ Dimension: $$149$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$149q - 8q^{2} - 12q^{3} + 124q^{4} - 36q^{5} - 45q^{6} - 32q^{7} - 21q^{8} + 115q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$149q - 8q^{2} - 12q^{3} + 124q^{4} - 36q^{5} - 45q^{6} - 32q^{7} - 21q^{8} + 115q^{9} - 58q^{10} - 33q^{11} - 33q^{12} - 107q^{13} - 28q^{14} - 24q^{15} + 74q^{16} - 39q^{17} - 33q^{18} - 93q^{19} - 63q^{20} - 113q^{21} - 38q^{22} - 11q^{23} - 130q^{24} + 85q^{25} - 33q^{26} - 30q^{27} - 94q^{28} - 85q^{29} - 16q^{30} - 129q^{31} - 35q^{32} - 64q^{33} - 78q^{34} - 27q^{35} + 79q^{36} - 135q^{37} - 11q^{38} - 73q^{39} - 146q^{40} - 101q^{41} + 4q^{42} - 55q^{43} - 82q^{44} - 168q^{45} - 113q^{46} - 40q^{47} - 65q^{48} + 27q^{49} - 5q^{50} - 49q^{51} - 177q^{52} - 32q^{53} - 155q^{54} - 128q^{55} - 44q^{56} - 47q^{57} - 46q^{58} - 53q^{59} - 11q^{60} - 347q^{61} - 11q^{62} - 73q^{63} + q^{64} - 31q^{65} - 37q^{66} - 40q^{67} - 80q^{68} - 175q^{69} - 61q^{70} - 31q^{71} - 68q^{72} - 193q^{73} - 33q^{74} - 56q^{75} - 248q^{76} - 84q^{77} + 40q^{78} - 111q^{79} - 54q^{80} + 49q^{81} - 74q^{82} - 24q^{83} - 159q^{84} - 258q^{85} - q^{86} - 66q^{87} - 97q^{88} - 76q^{89} - 75q^{90} - 134q^{91} + 31q^{92} - 97q^{93} - 111q^{94} - 14q^{95} - 216q^{96} - 140q^{97} - 13q^{98} - 116q^{99} + O(q^{100})$$

## 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 −2.78495 3.42897 5.75594 1.05066 −9.54951 −0.902468 −10.4601 8.75783 −2.92604
1.2 −2.73191 −1.70544 5.46336 2.32218 4.65913 1.19146 −9.46160 −0.0914604 −6.34400
1.3 −2.70774 1.77949 5.33184 1.68400 −4.81839 −4.00894 −9.02175 0.166582 −4.55984
1.4 −2.68954 1.48905 5.23361 2.84645 −4.00487 −1.97182 −8.69693 −0.782716 −7.65564
1.5 −2.62051 −0.192294 4.86706 −1.97989 0.503908 −4.52578 −7.51315 −2.96302 5.18832
1.6 −2.59547 1.08147 4.73647 0.749609 −2.80691 2.15869 −7.10242 −1.83043 −1.94559
1.7 −2.58406 1.99817 4.67739 −2.81422 −5.16341 −2.01468 −6.91855 0.992695 7.27214
1.8 −2.57089 0.190612 4.60945 −1.40475 −0.490042 1.52623 −6.70860 −2.96367 3.61144
1.9 −2.51991 −1.75801 4.34994 3.61499 4.43004 3.76242 −5.92164 0.0906134 −9.10944
1.10 −2.46575 −2.43164 4.07992 2.59428 5.99581 −0.656341 −5.12858 2.91285 −6.39684
1.11 −2.45850 −1.74268 4.04423 −0.525405 4.28438 −0.912805 −5.02575 0.0369224 1.29171
1.12 −2.45807 −2.34751 4.04212 −0.597755 5.77034 −0.214441 −5.01969 2.51079 1.46933
1.13 −2.42490 −1.02548 3.88013 −4.00476 2.48668 −1.16451 −4.55913 −1.94839 9.71114
1.14 −2.40917 −3.09144 3.80411 −1.16894 7.44782 2.97947 −4.34641 6.55703 2.81617
1.15 −2.36640 0.537231 3.59984 −0.969490 −1.27130 3.20883 −3.78587 −2.71138 2.29420
1.16 −2.35135 2.82086 3.52886 0.184417 −6.63284 2.16509 −3.59489 4.95726 −0.433630
1.17 −2.30650 −1.73394 3.31993 −0.392279 3.99934 −3.42422 −3.04443 0.00656301 0.904791
1.18 −2.27033 2.56855 3.15439 −3.84625 −5.83145 2.23399 −2.62085 3.59745 8.73225
1.19 −2.23088 1.97807 2.97683 1.53013 −4.41285 −1.35303 −2.17920 0.912770 −3.41354
1.20 −2.21437 0.176199 2.90346 3.35713 −0.390170 0.00990688 −2.00059 −2.96895 −7.43394
See next 80 embeddings (of 149 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.149 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 4019.2.a.a 149

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4019.2.a.a 149 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$4019$$ $$1$$