# Properties

 Label 38.6.e.a Level 38 Weight 6 Character orbit 38.e Analytic conductor 6.095 Analytic rank 0 Dimension 24 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 38.e (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.09458515289$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$4$$ over $$\Q(\zeta_{9})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 18q^{3} - 72q^{6} + 438q^{7} + 768q^{8} + 378q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 18q^{3} - 72q^{6} + 438q^{7} + 768q^{8} + 378q^{9} + 600q^{11} - 1662q^{13} - 1776q^{14} - 1206q^{15} + 4194q^{17} - 4368q^{18} + 9570q^{19} - 1344q^{20} + 2430q^{21} - 5160q^{22} - 1170q^{23} - 1152q^{24} + 5940q^{25} - 1776q^{26} - 5130q^{27} + 768q^{28} - 7554q^{29} + 294q^{31} - 45912q^{33} + 1584q^{34} + 22116q^{35} + 6048q^{36} + 28608q^{37} - 9528q^{38} - 38868q^{39} + 37278q^{41} - 9720q^{42} + 13212q^{43} + 20640q^{44} - 34344q^{45} - 29856q^{46} - 3054q^{47} - 9216q^{48} + 1422q^{49} + 16584q^{50} + 2352q^{51} - 1824q^{52} + 73806q^{53} + 136944q^{54} - 41598q^{55} + 56064q^{56} + 178248q^{57} - 52848q^{58} + 123528q^{59} + 46944q^{60} - 149172q^{61} + 58344q^{62} - 170124q^{63} - 49152q^{64} - 202674q^{65} - 84840q^{66} - 313212q^{67} + 35520q^{68} - 19938q^{69} - 88464q^{70} + 75810q^{71} + 48384q^{72} + 172872q^{73} + 4872q^{74} + 174960q^{75} + 31008q^{76} - 115056q^{77} + 101304q^{78} + 158058q^{79} - 47250q^{81} - 149112q^{82} - 148440q^{83} - 176736q^{84} - 226752q^{85} - 57528q^{86} + 182874q^{87} - 38400q^{88} - 322464q^{89} + 48816q^{90} - 27264q^{91} + 88992q^{92} + 779424q^{93} + 143712q^{94} + 634752q^{95} - 108552q^{97} - 173544q^{98} + 656436q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 3.75877 1.36808i −2.69117 + 15.2624i 12.2567 10.2846i −6.46132 5.42169i 10.7647 + 61.0494i 110.495 + 191.383i 32.0000 55.4256i 2.64806 + 0.963814i −31.7039 11.5393i
5.2 3.75877 1.36808i −0.383209 + 2.17329i 12.2567 10.2846i 53.6173 + 44.9902i 1.53284 + 8.69314i −9.54573 16.5337i 32.0000 55.4256i 223.769 + 81.4452i 263.085 + 95.7552i
5.3 3.75877 1.36808i 0.403873 2.29048i 12.2567 10.2846i −72.6237 60.9385i −1.61549 9.16190i −55.7656 96.5889i 32.0000 55.4256i 223.262 + 81.2608i −356.344 129.699i
5.4 3.75877 1.36808i 4.62861 26.2502i 12.2567 10.2846i 14.7431 + 12.3709i −18.5145 105.001i 4.52908 + 7.84460i 32.0000 55.4256i −439.302 159.893i 72.3402 + 26.3297i
9.1 −0.694593 3.93923i −22.0823 + 18.5292i −15.0351 + 5.47232i 4.74528 + 1.72714i 88.3291 + 74.1169i 104.694 181.336i 32.0000 + 55.4256i 102.098 579.027i 3.50757 19.8924i
9.2 −0.694593 3.93923i −3.81408 + 3.20039i −15.0351 + 5.47232i −0.763769 0.277989i 15.2563 + 12.8016i −52.7816 + 91.4204i 32.0000 + 55.4256i −37.8918 + 214.895i −0.564556 + 3.20175i
9.3 −0.694593 3.93923i 9.08943 7.62694i −15.0351 + 5.47232i 86.6641 + 31.5432i −36.3577 30.5078i 77.2983 133.885i 32.0000 + 55.4256i −17.7489 + 100.659i 64.0595 363.300i
9.4 −0.694593 3.93923i 15.2107 12.7633i −15.0351 + 5.47232i −77.4900 28.2040i −60.8427 51.0531i 31.0980 53.8633i 32.0000 + 55.4256i 26.2669 148.967i −57.2783 + 324.841i
17.1 −0.694593 + 3.93923i −22.0823 18.5292i −15.0351 5.47232i 4.74528 1.72714i 88.3291 74.1169i 104.694 + 181.336i 32.0000 55.4256i 102.098 + 579.027i 3.50757 + 19.8924i
17.2 −0.694593 + 3.93923i −3.81408 3.20039i −15.0351 5.47232i −0.763769 + 0.277989i 15.2563 12.8016i −52.7816 91.4204i 32.0000 55.4256i −37.8918 214.895i −0.564556 3.20175i
17.3 −0.694593 + 3.93923i 9.08943 + 7.62694i −15.0351 5.47232i 86.6641 31.5432i −36.3577 + 30.5078i 77.2983 + 133.885i 32.0000 55.4256i −17.7489 100.659i 64.0595 + 363.300i
17.4 −0.694593 + 3.93923i 15.2107 + 12.7633i −15.0351 5.47232i −77.4900 + 28.2040i −60.8427 + 51.0531i 31.0980 + 53.8633i 32.0000 55.4256i 26.2669 + 148.967i −57.2783 324.841i
23.1 3.75877 + 1.36808i −2.69117 15.2624i 12.2567 + 10.2846i −6.46132 + 5.42169i 10.7647 61.0494i 110.495 191.383i 32.0000 + 55.4256i 2.64806 0.963814i −31.7039 + 11.5393i
23.2 3.75877 + 1.36808i −0.383209 2.17329i 12.2567 + 10.2846i 53.6173 44.9902i 1.53284 8.69314i −9.54573 + 16.5337i 32.0000 + 55.4256i 223.769 81.4452i 263.085 95.7552i
23.3 3.75877 + 1.36808i 0.403873 + 2.29048i 12.2567 + 10.2846i −72.6237 + 60.9385i −1.61549 + 9.16190i −55.7656 + 96.5889i 32.0000 + 55.4256i 223.262 81.2608i −356.344 + 129.699i
23.4 3.75877 + 1.36808i 4.62861 + 26.2502i 12.2567 + 10.2846i 14.7431 12.3709i −18.5145 + 105.001i 4.52908 7.84460i 32.0000 + 55.4256i −439.302 + 159.893i 72.3402 26.3297i
25.1 −3.06418 + 2.57115i −22.4490 8.17075i 2.77837 15.7569i −4.34948 24.6671i 89.7958 32.6830i −29.2182 + 50.6074i 32.0000 + 55.4256i 251.046 + 210.652i 76.7505 + 64.4013i
25.2 −3.06418 + 2.57115i 0.577036 + 0.210024i 2.77837 15.7569i 2.13367 + 12.1006i −2.30814 + 0.840095i −76.5898 + 132.657i 32.0000 + 55.4256i −185.860 155.955i −37.6505 31.5925i
25.3 −3.06418 + 2.57115i 12.9479 + 4.71264i 2.77837 15.7569i −13.1062 74.3288i −51.7915 + 18.8506i 39.5277 68.4640i 32.0000 + 55.4256i −40.7104 34.1601i 231.270 + 194.059i
25.4 −3.06418 + 2.57115i 17.5622 + 6.39212i 2.77837 15.7569i 12.8909 + 73.1080i −70.2488 + 25.5685i 75.2582 130.351i 32.0000 + 55.4256i 81.4231 + 68.3221i −227.472 190.871i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 35.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.6.e.a 24
19.e even 9 1 inner 38.6.e.a 24
19.e even 9 1 722.6.a.o 12
19.f odd 18 1 722.6.a.p 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.6.e.a 24 1.a even 1 1 trivial
38.6.e.a 24 19.e even 9 1 inner
722.6.a.o 12 19.e even 9 1
722.6.a.p 12 19.f odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{24} - \cdots$$ acting on $$S_{6}^{\mathrm{new}}(38, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database