# Properties

 Label 38.6.e.b Level $38$ Weight $6$ Character orbit 38.e Analytic conductor $6.095$ Analytic rank $0$ Dimension $30$ 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: $$30$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$30q + 15q^{3} + 60q^{6} - 84q^{7} - 960q^{8} - 345q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q + 15q^{3} + 60q^{6} - 84q^{7} - 960q^{8} - 345q^{9} - 126q^{11} - 2610q^{13} - 864q^{14} - 5160q^{15} + 1902q^{17} + 17016q^{18} - 2400q^{19} + 3456q^{20} - 6426q^{21} + 216q^{22} - 3306q^{23} + 960q^{24} - 18060q^{25} + 456q^{26} + 13017q^{27} + 6048q^{28} + 5988q^{29} - 6840q^{31} + 51867q^{33} - 10752q^{34} + 24408q^{35} - 5520q^{36} - 8556q^{37} + 2772q^{38} + 9912q^{39} - 40527q^{41} + 27432q^{42} + 3678q^{43} + 864q^{44} - 31770q^{45} - 15744q^{46} - 133740q^{47} - 7680q^{48} - 63777q^{49} - 107580q^{50} + 45657q^{51} + 5760q^{52} - 64626q^{53} + 1476q^{54} + 142056q^{55} + 10752q^{56} + 232824q^{57} + 117264q^{58} + 169041q^{59} - 56640q^{60} + 61698q^{61} - 101616q^{62} - 188064q^{63} - 61440q^{64} - 80646q^{65} - 173772q^{66} - 72129q^{67} - 65904q^{68} - 124224q^{69} + 97632q^{70} + 417828q^{71} + 44160q^{72} + 239082q^{73} - 116112q^{74} - 174960q^{75} + 6912q^{76} + 242436q^{77} + 336624q^{78} - 145740q^{79} + 14781q^{81} - 162108q^{82} - 201630q^{83} - 54528q^{84} - 46512q^{85} - 75720q^{86} - 376512q^{87} - 8064q^{88} + 416928q^{89} + 560640q^{90} - 380316q^{91} + 269376q^{92} - 205146q^{93} + 5616q^{94} + 708168q^{95} + 858363q^{97} + 308328q^{98} - 1068531q^{99} + O(q^{100})$$

## 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 −5.14875 + 29.2000i 12.2567 10.2846i 68.5412 + 57.5129i −20.5950 116.800i 68.7063 + 119.003i −32.0000 + 55.4256i −597.784 217.576i −336.313 122.408i
5.2 −3.75877 + 1.36808i −1.36581 + 7.74591i 12.2567 10.2846i −39.1860 32.8809i −5.46326 30.9837i −9.18601 15.9106i −32.0000 + 55.4256i 170.212 + 61.9519i 192.275 + 69.9823i
5.3 −3.75877 + 1.36808i −0.196437 + 1.11405i 12.2567 10.2846i 8.74989 + 7.34203i −0.785748 4.45620i −7.74142 13.4085i −32.0000 + 55.4256i 227.143 + 82.6732i −42.9333 15.6265i
5.4 −3.75877 + 1.36808i 3.93255 22.3026i 12.2567 10.2846i 61.0402 + 51.2189i 15.7302 + 89.2104i −95.8893 166.085i −32.0000 + 55.4256i −253.595 92.3012i −299.508 109.012i
5.5 −3.75877 + 1.36808i 4.41021 25.0115i 12.2567 10.2846i −71.5678 60.0525i 17.6408 + 100.046i 91.0882 + 157.769i −32.0000 + 55.4256i −377.781 137.501i 351.164 + 127.813i
9.1 0.694593 + 3.93923i −21.8341 + 18.3210i −15.0351 + 5.47232i 33.2104 + 12.0876i −87.3364 73.2839i −42.9494 + 74.3905i −32.0000 55.4256i 98.8729 560.736i −24.5482 + 139.220i
9.2 0.694593 + 3.93923i −8.20450 + 6.88439i −15.0351 + 5.47232i −55.8231 20.3180i −32.8180 27.5376i 115.217 199.562i −32.0000 55.4256i −22.2776 + 126.342i 41.2628 234.013i
9.3 0.694593 + 3.93923i 0.893216 0.749497i −15.0351 + 5.47232i 61.2040 + 22.2764i 3.57286 + 2.99799i −47.2621 + 81.8604i −32.0000 55.4256i −41.9604 + 237.969i −45.2401 + 256.570i
9.4 0.694593 + 3.93923i 7.96877 6.68659i −15.0351 + 5.47232i −89.7666 32.6724i 31.8751 + 26.7464i −116.109 + 201.107i −32.0000 55.4256i −23.4057 + 132.740i 66.3528 376.305i
9.5 0.694593 + 3.93923i 19.8464 16.6531i −15.0351 + 5.47232i 17.3464 + 6.31358i 79.3855 + 66.6124i 61.0467 105.736i −32.0000 55.4256i 74.3569 421.699i −12.8220 + 72.7169i
17.1 0.694593 3.93923i −21.8341 18.3210i −15.0351 5.47232i 33.2104 12.0876i −87.3364 + 73.2839i −42.9494 74.3905i −32.0000 + 55.4256i 98.8729 + 560.736i −24.5482 139.220i
17.2 0.694593 3.93923i −8.20450 6.88439i −15.0351 5.47232i −55.8231 + 20.3180i −32.8180 + 27.5376i 115.217 + 199.562i −32.0000 + 55.4256i −22.2776 126.342i 41.2628 + 234.013i
17.3 0.694593 3.93923i 0.893216 + 0.749497i −15.0351 5.47232i 61.2040 22.2764i 3.57286 2.99799i −47.2621 81.8604i −32.0000 + 55.4256i −41.9604 237.969i −45.2401 256.570i
17.4 0.694593 3.93923i 7.96877 + 6.68659i −15.0351 5.47232i −89.7666 + 32.6724i 31.8751 26.7464i −116.109 201.107i −32.0000 + 55.4256i −23.4057 132.740i 66.3528 + 376.305i
17.5 0.694593 3.93923i 19.8464 + 16.6531i −15.0351 5.47232i 17.3464 6.31358i 79.3855 66.6124i 61.0467 + 105.736i −32.0000 + 55.4256i 74.3569 + 421.699i −12.8220 72.7169i
23.1 −3.75877 1.36808i −5.14875 29.2000i 12.2567 + 10.2846i 68.5412 57.5129i −20.5950 + 116.800i 68.7063 119.003i −32.0000 55.4256i −597.784 + 217.576i −336.313 + 122.408i
23.2 −3.75877 1.36808i −1.36581 7.74591i 12.2567 + 10.2846i −39.1860 + 32.8809i −5.46326 + 30.9837i −9.18601 + 15.9106i −32.0000 55.4256i 170.212 61.9519i 192.275 69.9823i
23.3 −3.75877 1.36808i −0.196437 1.11405i 12.2567 + 10.2846i 8.74989 7.34203i −0.785748 + 4.45620i −7.74142 + 13.4085i −32.0000 55.4256i 227.143 82.6732i −42.9333 + 15.6265i
23.4 −3.75877 1.36808i 3.93255 + 22.3026i 12.2567 + 10.2846i 61.0402 51.2189i 15.7302 89.2104i −95.8893 + 166.085i −32.0000 55.4256i −253.595 + 92.3012i −299.508 + 109.012i
23.5 −3.75877 1.36808i 4.41021 + 25.0115i 12.2567 + 10.2846i −71.5678 + 60.0525i 17.6408 100.046i 91.0882 157.769i −32.0000 55.4256i −377.781 + 137.501i 351.164 127.813i
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 38.6.e.b 30
19.e even 9 1 inner 38.6.e.b 30
19.e even 9 1 722.6.a.r 15
19.f odd 18 1 722.6.a.q 15

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.6.e.b 30 1.a even 1 1 trivial
38.6.e.b 30 19.e even 9 1 inner
722.6.a.q 15 19.f odd 18 1
722.6.a.r 15 19.e even 9 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$13\!\cdots\!96$$$$T_{3}^{21} -$$$$10\!\cdots\!89$$$$T_{3}^{20} +$$$$41\!\cdots\!63$$$$T_{3}^{19} +$$$$18\!\cdots\!65$$$$T_{3}^{18} -$$$$33\!\cdots\!39$$$$T_{3}^{17} +$$$$47\!\cdots\!58$$$$T_{3}^{16} -$$$$84\!\cdots\!88$$$$T_{3}^{15} -$$$$17\!\cdots\!69$$$$T_{3}^{14} +$$$$58\!\cdots\!12$$$$T_{3}^{13} +$$$$51\!\cdots\!51$$$$T_{3}^{12} +$$$$29\!\cdots\!04$$$$T_{3}^{11} -$$$$34\!\cdots\!15$$$$T_{3}^{10} +$$$$61\!\cdots\!58$$$$T_{3}^{9} +$$$$33\!\cdots\!06$$$$T_{3}^{8} +$$$$37\!\cdots\!63$$$$T_{3}^{7} +$$$$38\!\cdots\!87$$$$T_{3}^{6} +$$$$82\!\cdots\!62$$$$T_{3}^{5} +$$$$27\!\cdots\!97$$$$T_{3}^{4} -$$$$14\!\cdots\!16$$$$T_{3}^{3} +$$$$10\!\cdots\!99$$$$T_{3}^{2} -$$$$78\!\cdots\!07$$$$T_{3} +$$$$24\!\cdots\!41$$">$$T_{3}^{30} - \cdots$$ acting on $$S_{6}^{\mathrm{new}}(38, [\chi])$$.