# Properties

 Label 4001.2.a.b Level 4001 Weight 2 Character orbit 4001.a Self dual yes Analytic conductor 31.948 Analytic rank 0 Dimension 184 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4001$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4001.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$31.9481458487$$ Analytic rank: $$0$$ Dimension: $$184$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$184q + 3q^{2} + 28q^{3} + 217q^{4} + 15q^{5} + 31q^{6} + 49q^{7} + 6q^{8} + 210q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$184q + 3q^{2} + 28q^{3} + 217q^{4} + 15q^{5} + 31q^{6} + 49q^{7} + 6q^{8} + 210q^{9} + 46q^{10} + 25q^{11} + 61q^{12} + 52q^{13} + 28q^{14} + 59q^{15} + 279q^{16} + 16q^{17} - 2q^{18} + 86q^{19} + 26q^{20} + 22q^{21} + 54q^{22} + 55q^{23} + 72q^{24} + 241q^{25} + 32q^{26} + 97q^{27} + 75q^{28} + 27q^{29} - 10q^{30} + 276q^{31} + 20q^{33} + 122q^{34} + 30q^{35} + 278q^{36} + 42q^{37} + 14q^{38} + 113q^{39} + 115q^{40} + 39q^{41} + 15q^{42} + 65q^{43} + 32q^{44} + 54q^{45} + 65q^{46} + 82q^{47} + 117q^{48} + 297q^{49} + 4q^{50} + 45q^{51} + 136q^{52} + 21q^{53} + 93q^{54} + 252q^{55} + 74q^{56} + 14q^{57} + 54q^{58} + 95q^{59} + 58q^{60} + 131q^{61} + 14q^{62} + 88q^{63} + 368q^{64} - 9q^{65} + 52q^{66} + 90q^{67} + 27q^{68} + 101q^{69} + 18q^{70} + 117q^{71} - 15q^{72} + 72q^{73} + 7q^{74} + 150q^{75} + 148q^{76} + 7q^{77} + 22q^{78} + 287q^{79} + 43q^{80} + 244q^{81} + 86q^{82} + 25q^{83} + 14q^{84} + 41q^{85} + 25q^{86} + 82q^{87} + 115q^{88} + 48q^{89} + 78q^{90} + 272q^{91} + 69q^{92} + 44q^{93} + 161q^{94} + 37q^{95} + 129q^{96} + 106q^{97} - 46q^{98} + 53q^{99} + O(q^{100})$$

## 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 −2.80114 −3.24231 5.84639 −0.362239 9.08218 0.683027 −10.7743 7.51260 1.01468
1.2 −2.79357 0.695080 5.80404 −1.43429 −1.94176 3.16464 −10.6269 −2.51686 4.00680
1.3 −2.77961 −0.970175 5.72623 0.973772 2.69671 0.392591 −10.3575 −2.05876 −2.70671
1.4 −2.76351 −0.782711 5.63697 4.06144 2.16303 1.66325 −10.0508 −2.38736 −11.2238
1.5 −2.73134 2.60660 5.46019 −2.78013 −7.11951 −4.77010 −9.45095 3.79438 7.59346
1.6 −2.71583 2.95270 5.37572 2.81232 −8.01903 −1.71576 −9.16786 5.71846 −7.63778
1.7 −2.69288 −1.22667 5.25159 2.75099 3.30328 −4.02180 −8.75615 −1.49527 −7.40807
1.8 −2.68105 −2.59933 5.18802 −3.48579 6.96892 −5.16960 −8.54725 3.75650 9.34557
1.9 −2.66682 0.260886 5.11190 −1.12952 −0.695735 −1.67133 −8.29887 −2.93194 3.01221
1.10 −2.65266 3.05290 5.03661 −1.94746 −8.09830 2.48037 −8.05510 6.32018 5.16596
1.11 −2.64594 −1.99415 5.00101 −3.20075 5.27640 3.40947 −7.94051 0.976617 8.46900
1.12 −2.63285 1.55445 4.93191 −4.32053 −4.09264 −0.850828 −7.71930 −0.583679 11.3753
1.13 −2.61467 2.84813 4.83651 0.542328 −7.44692 4.00529 −7.41654 5.11183 −1.41801
1.14 −2.61059 −0.259768 4.81518 −4.09316 0.678147 2.08780 −7.34928 −2.93252 10.6856
1.15 −2.55462 2.40673 4.52611 3.51168 −6.14829 3.52515 −6.45325 2.79234 −8.97103
1.16 −2.43211 0.645473 3.91518 −1.15456 −1.56986 1.52754 −4.65795 −2.58336 2.80801
1.17 −2.42514 −1.95794 3.88128 1.35530 4.74827 4.91337 −4.56237 0.833523 −3.28679
1.18 −2.40164 −1.91938 3.76787 −2.41584 4.60966 −1.95333 −4.24578 0.684021 5.80197
1.19 −2.38954 1.02374 3.70989 −0.691971 −2.44626 1.59189 −4.08584 −1.95196 1.65349
1.20 −2.38517 0.919938 3.68905 3.44498 −2.19421 −3.87561 −4.02868 −2.15371 −8.21687
See next 80 embeddings (of 184 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.184 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 4001.2.a.b 184

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4001.2.a.b 184 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$4001$$ $$-1$$