# Properties

 Label 1045.6.a.e Level $1045$ Weight $6$ Character orbit 1045.a Self dual yes Analytic conductor $167.601$ Analytic rank $1$ Dimension $38$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 1045.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$167.601091705$$ Analytic rank: $$1$$ Dimension: $$38$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$38 q - 24 q^{2} - 63 q^{3} + 594 q^{4} + 950 q^{5} - 67 q^{6} - 729 q^{7} - 1272 q^{8} + 3029 q^{9}+O(q^{10})$$ 38 * q - 24 * q^2 - 63 * q^3 + 594 * q^4 + 950 * q^5 - 67 * q^6 - 729 * q^7 - 1272 * q^8 + 3029 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$38 q - 24 q^{2} - 63 q^{3} + 594 q^{4} + 950 q^{5} - 67 q^{6} - 729 q^{7} - 1272 q^{8} + 3029 q^{9} - 600 q^{10} + 4598 q^{11} - 2008 q^{12} - 2663 q^{13} - 1565 q^{14} - 1575 q^{15} + 12390 q^{16} - 3311 q^{17} - 6383 q^{18} - 13718 q^{19} + 14850 q^{20} - 8179 q^{21} - 2904 q^{22} - 3412 q^{23} - 4100 q^{24} + 23750 q^{25} - 1399 q^{26} - 31596 q^{27} - 43653 q^{28} - 13633 q^{29} - 1675 q^{30} - 13789 q^{31} - 58603 q^{32} - 7623 q^{33} - 29149 q^{34} - 18225 q^{35} + 50641 q^{36} - 12103 q^{37} + 8664 q^{38} - 50960 q^{39} - 31800 q^{40} - 37885 q^{41} + 51100 q^{42} - 56119 q^{43} + 71874 q^{44} + 75725 q^{45} - 56291 q^{46} - 37532 q^{47} - 113895 q^{48} + 153501 q^{49} - 15000 q^{50} + 32882 q^{51} - 169554 q^{52} - 51511 q^{53} - 175060 q^{54} + 114950 q^{55} - 84247 q^{56} + 22743 q^{57} - 256962 q^{58} - 154267 q^{59} - 50200 q^{60} - 47165 q^{61} + 143002 q^{62} - 358780 q^{63} + 142292 q^{64} - 66575 q^{65} - 8107 q^{66} - 161712 q^{67} - 210188 q^{68} - 124602 q^{69} - 39125 q^{70} + 6118 q^{71} - 327878 q^{72} - 152182 q^{73} - 167349 q^{74} - 39375 q^{75} - 214434 q^{76} - 88209 q^{77} - 216594 q^{78} - 140433 q^{79} + 309750 q^{80} + 382874 q^{81} - 29842 q^{82} - 515287 q^{83} + 29222 q^{84} - 82775 q^{85} + 204974 q^{86} - 106764 q^{87} - 153912 q^{88} - 271610 q^{89} - 159575 q^{90} - 44332 q^{91} + 236348 q^{92} + 25202 q^{93} - 496224 q^{94} - 342950 q^{95} - 275218 q^{96} - 126390 q^{97} - 285506 q^{98} + 366509 q^{99}+O(q^{100})$$ 38 * q - 24 * q^2 - 63 * q^3 + 594 * q^4 + 950 * q^5 - 67 * q^6 - 729 * q^7 - 1272 * q^8 + 3029 * q^9 - 600 * q^10 + 4598 * q^11 - 2008 * q^12 - 2663 * q^13 - 1565 * q^14 - 1575 * q^15 + 12390 * q^16 - 3311 * q^17 - 6383 * q^18 - 13718 * q^19 + 14850 * q^20 - 8179 * q^21 - 2904 * q^22 - 3412 * q^23 - 4100 * q^24 + 23750 * q^25 - 1399 * q^26 - 31596 * q^27 - 43653 * q^28 - 13633 * q^29 - 1675 * q^30 - 13789 * q^31 - 58603 * q^32 - 7623 * q^33 - 29149 * q^34 - 18225 * q^35 + 50641 * q^36 - 12103 * q^37 + 8664 * q^38 - 50960 * q^39 - 31800 * q^40 - 37885 * q^41 + 51100 * q^42 - 56119 * q^43 + 71874 * q^44 + 75725 * q^45 - 56291 * q^46 - 37532 * q^47 - 113895 * q^48 + 153501 * q^49 - 15000 * q^50 + 32882 * q^51 - 169554 * q^52 - 51511 * q^53 - 175060 * q^54 + 114950 * q^55 - 84247 * q^56 + 22743 * q^57 - 256962 * q^58 - 154267 * q^59 - 50200 * q^60 - 47165 * q^61 + 143002 * q^62 - 358780 * q^63 + 142292 * q^64 - 66575 * q^65 - 8107 * q^66 - 161712 * q^67 - 210188 * q^68 - 124602 * q^69 - 39125 * q^70 + 6118 * q^71 - 327878 * q^72 - 152182 * q^73 - 167349 * q^74 - 39375 * q^75 - 214434 * q^76 - 88209 * q^77 - 216594 * q^78 - 140433 * q^79 + 309750 * q^80 + 382874 * q^81 - 29842 * q^82 - 515287 * q^83 + 29222 * q^84 - 82775 * q^85 + 204974 * q^86 - 106764 * q^87 - 153912 * q^88 - 271610 * q^89 - 159575 * q^90 - 44332 * q^91 + 236348 * q^92 + 25202 * q^93 - 496224 * q^94 - 342950 * q^95 - 275218 * q^96 - 126390 * q^97 - 285506 * q^98 + 366509 * q^99

## 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 −11.0901 −27.5711 90.9897 25.0000 305.766 −40.7849 −654.200 517.168 −277.252
1.2 −10.8606 27.1939 85.9518 25.0000 −295.341 −206.298 −585.946 496.509 −271.514
1.3 −10.7913 −1.89630 84.4524 25.0000 20.4636 234.566 −566.030 −239.404 −269.783
1.4 −9.91265 15.6713 66.2607 25.0000 −155.344 −122.128 −339.614 2.59022 −247.816
1.5 −9.74865 −7.53353 63.0362 25.0000 73.4417 −163.789 −302.561 −186.246 −243.716
1.6 −9.21304 −14.8344 52.8801 25.0000 136.670 144.131 −192.369 −22.9413 −230.326
1.7 −8.88488 −24.9356 46.9410 25.0000 221.550 −188.162 −132.749 378.784 −222.122
1.8 −8.25643 −6.02271 36.1686 25.0000 49.7261 24.7368 −34.4178 −206.727 −206.411
1.9 −7.95052 14.0444 31.2108 25.0000 −111.660 100.860 6.27456 −45.7550 −198.763
1.10 −7.34806 21.4317 21.9939 25.0000 −157.482 84.2288 73.5250 216.320 −183.701
1.11 −6.08152 −12.1543 4.98491 25.0000 73.9166 37.2863 164.293 −95.2732 −152.038
1.12 −4.92584 16.7465 −7.73613 25.0000 −82.4904 −110.483 195.734 37.4445 −123.146
1.13 −4.85962 −16.0436 −8.38410 25.0000 77.9658 −94.2760 196.251 14.3969 −121.490
1.14 −3.63943 11.8452 −18.7545 25.0000 −43.1098 −246.643 184.718 −102.691 −90.9858
1.15 −3.39231 −29.7744 −20.4922 25.0000 101.004 10.0128 178.070 643.513 −84.8078
1.16 −2.67362 −11.1309 −24.8518 25.0000 29.7598 158.866 152.000 −119.103 −66.8405
1.17 −2.59417 30.0518 −25.2703 25.0000 −77.9597 −144.749 148.569 660.111 −64.8544
1.18 −1.87953 7.44103 −28.4674 25.0000 −13.9857 104.152 113.650 −187.631 −46.9883
1.19 −1.77527 9.46319 −28.8484 25.0000 −16.7997 232.548 108.022 −153.448 −44.3817
1.20 −0.836459 −16.2079 −31.3003 25.0000 13.5572 −208.079 52.9482 19.6951 −20.9115
See all 38 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.38 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$11$$ $$-1$$
$$19$$ $$1$$

## 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 1045.6.a.e 38

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.6.a.e 38 1.a even 1 1 trivial