# Properties

 Label 38.8.e.a Level 38 Weight 8 Character orbit 38.e Analytic conductor 11.871 Analytic rank 0 Dimension 30 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 38.e (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.8706309684$$ Analytic rank: $$0$$ Dimension: $$30$$ Relative dimension: $$5$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$30q + 45q^{3} + 360q^{6} + 1806q^{7} - 7680q^{8} - 3105q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q + 45q^{3} + 360q^{6} + 1806q^{7} - 7680q^{8} - 3105q^{9} + 9786q^{11} - 11130q^{13} + 6144q^{14} - 1818q^{15} - 65250q^{17} + 29040q^{18} + 105906q^{19} - 47616q^{20} + 172854q^{21} + 102336q^{22} + 101004q^{23} + 23040q^{24} - 132756q^{25} - 92112q^{26} + 43659q^{27} + 96384q^{28} - 273912q^{29} + 58710q^{31} - 406965q^{33} + 84960q^{34} + 356100q^{35} - 198720q^{36} - 2598852q^{37} - 219336q^{38} + 1769844q^{39} + 43377q^{41} - 79056q^{42} + 1126698q^{43} + 818688q^{44} - 912546q^{45} - 530880q^{46} + 588306q^{47} - 368640q^{48} - 2778285q^{49} + 1665288q^{50} + 3972045q^{51} - 430080q^{52} + 6520866q^{53} - 486648q^{54} - 1666818q^{55} - 1849344q^{56} - 6947844q^{57} - 4573152q^{58} - 970467q^{59} + 4634496q^{60} - 4037670q^{61} + 755952q^{62} + 7946472q^{63} - 3932160q^{64} + 1497690q^{65} + 6499272q^{66} - 4817025q^{67} + 1472064q^{68} + 10299876q^{69} + 2848800q^{70} - 25171290q^{71} + 3179520q^{72} + 1347750q^{73} - 3123936q^{74} + 8817984q^{75} + 1583232q^{76} - 11476764q^{77} - 930960q^{78} + 10777986q^{79} - 5365035q^{81} + 347016q^{82} + 9745314q^{83} + 877056q^{84} + 29258556q^{85} - 9338208q^{86} + 25598802q^{87} + 5010432q^{88} - 20353740q^{89} - 30112128q^{90} + 19748724q^{91} - 14097792q^{92} - 18738174q^{93} + 12376608q^{94} - 20427204q^{95} - 23005029q^{97} + 14869008q^{98} - 45819771q^{99} + O(q^{100})$$

## 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 −7.51754 + 2.73616i −11.6345 + 65.9825i 49.0268 41.1384i −116.114 97.4309i −93.0759 527.860i −468.313 811.141i −256.000 + 443.405i −2163.22 787.347i 1139.48 + 414.735i
5.2 −7.51754 + 2.73616i −7.18109 + 40.7260i 49.0268 41.1384i −24.6197 20.6584i −57.4487 325.808i 764.198 + 1323.63i −256.000 + 443.405i 448.070 + 163.084i 241.604 + 87.9368i
5.3 −7.51754 + 2.73616i 5.10191 28.9344i 49.0268 41.1384i 128.722 + 108.010i 40.8153 + 231.475i −158.544 274.607i −256.000 + 443.405i 1243.94 + 452.756i −1263.21 459.769i
5.4 −7.51754 + 2.73616i 7.10410 40.2894i 49.0268 41.1384i −241.265 202.445i 56.8328 + 322.315i −369.505 640.002i −256.000 + 443.405i 482.343 + 175.558i 2367.64 + 861.750i
5.5 −7.51754 + 2.73616i 11.5048 65.2472i 49.0268 41.1384i 158.287 + 132.818i 92.0387 + 521.978i 668.912 + 1158.59i −256.000 + 443.405i −2069.73 753.319i −1553.34 565.369i
9.1 1.38919 + 7.87846i −46.9776 + 39.4189i −60.1403 + 21.8893i −248.833 90.5677i −375.821 315.351i 81.2847 140.789i −256.000 443.405i 273.278 1549.84i 367.859 2086.23i
9.2 1.38919 + 7.87846i −21.2048 + 17.7929i −60.1403 + 21.8893i 194.773 + 70.8916i −169.638 142.344i −485.627 + 841.131i −256.000 443.405i −246.714 + 1399.18i −287.941 + 1632.99i
9.3 1.38919 + 7.87846i −9.71662 + 8.15321i −60.1403 + 21.8893i 424.315 + 154.438i −77.7330 65.2257i 847.859 1468.54i −256.000 443.405i −351.831 + 1995.33i −627.282 + 3557.49i
9.4 1.38919 + 7.87846i 28.3430 23.7826i −60.1403 + 21.8893i −54.5194 19.8434i 226.744 + 190.261i −356.201 + 616.958i −256.000 443.405i −142.054 + 805.630i 80.5982 457.095i
9.5 1.38919 + 7.87846i 45.5653 38.2339i −60.1403 + 21.8893i −199.214 72.5079i 364.523 + 305.871i 458.368 793.917i −256.000 443.405i 234.604 1330.50i 294.506 1670.23i
17.1 1.38919 7.87846i −46.9776 39.4189i −60.1403 21.8893i −248.833 + 90.5677i −375.821 + 315.351i 81.2847 + 140.789i −256.000 + 443.405i 273.278 + 1549.84i 367.859 + 2086.23i
17.2 1.38919 7.87846i −21.2048 17.7929i −60.1403 21.8893i 194.773 70.8916i −169.638 + 142.344i −485.627 841.131i −256.000 + 443.405i −246.714 1399.18i −287.941 1632.99i
17.3 1.38919 7.87846i −9.71662 8.15321i −60.1403 21.8893i 424.315 154.438i −77.7330 + 65.2257i 847.859 + 1468.54i −256.000 + 443.405i −351.831 1995.33i −627.282 3557.49i
17.4 1.38919 7.87846i 28.3430 + 23.7826i −60.1403 21.8893i −54.5194 + 19.8434i 226.744 190.261i −356.201 616.958i −256.000 + 443.405i −142.054 805.630i 80.5982 + 457.095i
17.5 1.38919 7.87846i 45.5653 + 38.2339i −60.1403 21.8893i −199.214 + 72.5079i 364.523 305.871i 458.368 + 793.917i −256.000 + 443.405i 234.604 + 1330.50i 294.506 + 1670.23i
23.1 −7.51754 2.73616i −11.6345 65.9825i 49.0268 + 41.1384i −116.114 + 97.4309i −93.0759 + 527.860i −468.313 + 811.141i −256.000 443.405i −2163.22 + 787.347i 1139.48 414.735i
23.2 −7.51754 2.73616i −7.18109 40.7260i 49.0268 + 41.1384i −24.6197 + 20.6584i −57.4487 + 325.808i 764.198 1323.63i −256.000 443.405i 448.070 163.084i 241.604 87.9368i
23.3 −7.51754 2.73616i 5.10191 + 28.9344i 49.0268 + 41.1384i 128.722 108.010i 40.8153 231.475i −158.544 + 274.607i −256.000 443.405i 1243.94 452.756i −1263.21 + 459.769i
23.4 −7.51754 2.73616i 7.10410 + 40.2894i 49.0268 + 41.1384i −241.265 + 202.445i 56.8328 322.315i −369.505 + 640.002i −256.000 443.405i 482.343 175.558i 2367.64 861.750i
23.5 −7.51754 2.73616i 11.5048 + 65.2472i 49.0268 + 41.1384i 158.287 132.818i 92.0387 521.978i 668.912 1158.59i −256.000 443.405i −2069.73 + 753.319i −1553.34 + 565.369i
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 35.5 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 38.8.e.a 30
19.e even 9 1 inner 38.8.e.a 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.8.e.a 30 1.a even 1 1 trivial
38.8.e.a 30 19.e even 9 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{30} - \cdots$$ acting on $$S_{8}^{\mathrm{new}}(38, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database