# Properties

 Label 6001.2.a.a Level $6001$ Weight $2$ Character orbit 6001.a Self dual yes Analytic conductor $47.918$ Analytic rank $1$ Dimension $113$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6001 = 17 \cdot 353$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6001.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$47.9182262530$$ Analytic rank: $$1$$ Dimension: $$113$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$113q - 11q^{2} - 11q^{3} + 103q^{4} - 19q^{5} - 13q^{6} + 11q^{7} - 36q^{8} + 94q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$113q - 11q^{2} - 11q^{3} + 103q^{4} - 19q^{5} - 13q^{6} + 11q^{7} - 36q^{8} + 94q^{9} - 5q^{10} - 40q^{11} - 19q^{12} - 18q^{13} - 48q^{14} - 63q^{15} + 79q^{16} - 113q^{17} - 32q^{18} - 46q^{19} - 56q^{20} - 46q^{21} + 14q^{22} - 35q^{23} - 42q^{24} + 88q^{25} - 89q^{26} - 41q^{27} + 20q^{28} - 51q^{29} - 18q^{30} - 57q^{31} - 93q^{32} - 40q^{33} + 11q^{34} - 69q^{35} + 18q^{36} + 16q^{37} - 74q^{38} - 51q^{39} + 2q^{40} - 87q^{41} - 23q^{42} - 32q^{43} - 110q^{44} - 17q^{45} - 17q^{46} - 161q^{47} - 36q^{48} + 56q^{49} - 69q^{50} + 11q^{51} - 49q^{52} - 48q^{53} - 38q^{54} - 79q^{55} - 171q^{56} + 20q^{57} + 13q^{58} - 174q^{59} - 146q^{60} - 34q^{61} - 34q^{62} - 14q^{63} + 62q^{64} - 22q^{65} - 60q^{66} - 50q^{67} - 103q^{68} - 59q^{69} - 58q^{70} - 189q^{71} - 123q^{72} - 4q^{73} - 24q^{74} - 106q^{75} - 92q^{76} - 78q^{77} - 42q^{78} + 8q^{79} - 150q^{80} + 13q^{81} + 6q^{82} - 109q^{83} - 114q^{84} + 19q^{85} - 116q^{86} - 106q^{87} + 54q^{88} - 170q^{89} - q^{90} - 43q^{91} - 94q^{92} - 69q^{93} - 35q^{94} - 78q^{95} - 44q^{96} - 3q^{97} - 68q^{98} - 119q^{99} + O(q^{100})$$

## 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 −2.78095 −2.69588 5.73371 2.30800 7.49712 4.50681 −10.3833 4.26777 −6.41845
1.2 −2.76448 −0.301476 5.64235 −1.88471 0.833425 2.57260 −10.0692 −2.90911 5.21024
1.3 −2.72078 −0.920111 5.40267 −2.98260 2.50342 −1.43517 −9.25792 −2.15340 8.11501
1.4 −2.69808 2.45523 5.27961 −1.37053 −6.62441 5.08280 −8.84865 3.02818 3.69780
1.5 −2.64384 −0.876989 4.98991 2.40505 2.31862 1.36446 −7.90487 −2.23089 −6.35858
1.6 −2.63611 2.68277 4.94905 −3.99469 −7.07208 −3.60242 −7.77401 4.19728 10.5304
1.7 −2.63080 1.97535 4.92113 2.13953 −5.19675 1.65541 −7.68492 0.902000 −5.62867
1.8 −2.58651 −0.844986 4.69006 −0.771482 2.18557 −2.98561 −6.95787 −2.28600 1.99545
1.9 −2.50694 0.525806 4.28474 −1.87738 −1.31816 4.48031 −5.72770 −2.72353 4.70647
1.10 −2.49624 1.48745 4.23122 −3.58200 −3.71302 1.79069 −5.56966 −0.787505 8.94153
1.11 −2.49146 −2.38691 4.20738 −2.02282 5.94689 1.32918 −5.49960 2.69734 5.03978
1.12 −2.49070 2.79631 4.20359 2.44678 −6.96477 −1.91297 −5.48850 4.81934 −6.09421
1.13 −2.30912 −2.56184 3.33205 −3.64906 5.91560 −1.49745 −3.07586 3.56301 8.42614
1.14 −2.26295 −1.98147 3.12095 3.45141 4.48397 −1.45595 −2.53665 0.926221 −7.81036
1.15 −2.22225 −0.120604 2.93840 1.74504 0.268012 −3.61581 −2.08537 −2.98545 −3.87792
1.16 −2.20707 2.35834 2.87116 2.79748 −5.20501 −1.23289 −1.92271 2.56174 −6.17423
1.17 −2.18121 0.0139647 2.75766 −0.236603 −0.0304599 2.78463 −1.65261 −2.99980 0.516079
1.18 −2.15299 2.37258 2.63537 −0.231510 −5.10813 3.32403 −1.36794 2.62912 0.498439
1.19 −2.14256 −2.11238 2.59055 1.25903 4.52589 1.44603 −1.26528 1.46215 −2.69753
1.20 −2.09226 2.92082 2.37755 −1.12068 −6.11112 −1.50686 −0.789928 5.53120 2.34475
See next 80 embeddings (of 113 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.113 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$17$$ $$1$$
$$353$$ $$1$$

## 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 6001.2.a.a 113

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6001.2.a.a 113 1.a even 1 1 trivial

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database