Properties

Label 38.7.f.a
Level $38$
Weight $7$
Character orbit 38.f
Analytic conductor $8.742$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 38.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.74205517755\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(10\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60q + 30q^{3} - 240q^{6} + 432q^{7} + 690q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 60q + 30q^{3} - 240q^{6} + 432q^{7} + 690q^{9} - 3360q^{11} + 9660q^{13} + 2496q^{14} - 37464q^{15} + 7704q^{17} + 46704q^{19} + 21504q^{20} + 78732q^{21} - 20304q^{22} - 54120q^{23} + 7680q^{24} - 41328q^{25} - 25536q^{26} - 124830q^{27} + 65280q^{28} - 21840q^{29} + 30780q^{31} - 54582q^{33} - 70560q^{34} - 56796q^{35} - 22080q^{36} + 52080q^{38} + 496344q^{39} + 83394q^{41} + 38880q^{42} - 294384q^{43} - 8448q^{44} - 391824q^{45} + 738720q^{46} + 639264q^{47} + 61440q^{48} - 85530q^{49} - 689472q^{50} - 215478q^{51} - 456960q^{52} - 1189428q^{53} - 907200q^{54} - 316512q^{55} + 517776q^{57} + 440640q^{58} + 957390q^{59} + 586752q^{60} + 1390404q^{61} + 978384q^{62} + 2002152q^{63} + 983040q^{64} - 878940q^{65} - 976992q^{66} - 2920854q^{67} - 78144q^{68} - 2667600q^{69} - 643392q^{70} + 1585980q^{71} + 706560q^{72} + 1734648q^{73} - 1167552q^{74} - 411264q^{76} - 1481400q^{77} - 2170608q^{78} + 4991952q^{79} + 40002q^{81} + 2299392q^{82} - 30672q^{83} + 1313280q^{84} + 815976q^{85} - 1424976q^{86} - 125892q^{87} - 6804636q^{89} - 2283456q^{90} - 3838704q^{91} - 244992q^{92} - 3271128q^{93} - 4631640q^{95} + 2112390q^{97} + 7177728q^{98} + 21467982q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −3.63616 + 4.33340i −14.5675 40.0239i −5.55674 31.5138i 14.4780 82.1091i 226.409 + 82.4062i −61.1665 105.943i 156.767 + 90.5097i −831.251 + 697.502i 303.167 + 361.301i
3.2 −3.63616 + 4.33340i −7.01359 19.2697i −5.55674 31.5138i −32.9273 + 186.740i 109.006 + 39.6749i −111.397 192.946i 156.767 + 90.5097i 236.316 198.293i −689.491 821.703i
3.3 −3.63616 + 4.33340i 0.501194 + 1.37702i −5.55674 31.5138i 6.94044 39.3612i −7.78959 2.83518i 159.899 + 276.953i 156.767 + 90.5097i 556.801 467.212i 145.331 + 173.199i
3.4 −3.63616 + 4.33340i 10.0117 + 27.5069i −5.55674 31.5138i 26.4694 150.115i −155.603 56.6347i −232.836 403.285i 156.767 + 90.5097i −97.9501 + 82.1899i 554.263 + 660.545i
3.5 −3.63616 + 4.33340i 12.5750 + 34.5495i −5.55674 31.5138i −24.6848 + 139.995i −195.442 71.1350i 136.699 + 236.770i 156.767 + 90.5097i −477.093 + 400.329i −516.895 616.012i
3.6 3.63616 4.33340i −14.7348 40.4836i −5.55674 31.5138i 40.1148 227.502i −229.010 83.3528i 276.187 + 478.370i −156.767 90.5097i −863.360 + 724.445i −839.995 1001.07i
3.7 3.63616 4.33340i −10.8314 29.7590i −5.55674 31.5138i −11.6174 + 65.8855i −168.342 61.2716i −205.116 355.271i −156.767 90.5097i −209.833 + 176.071i 243.266 + 289.913i
3.8 3.63616 4.33340i −1.10346 3.03174i −5.55674 31.5138i −30.9885 + 175.744i −17.1501 6.24213i 294.726 + 510.481i −156.767 90.5097i 550.473 461.901i 648.892 + 773.320i
3.9 3.63616 4.33340i 4.25575 + 11.6926i −5.55674 31.5138i 13.0241 73.8634i 66.1433 + 24.0742i −97.4265 168.748i −156.767 90.5097i 439.841 369.071i −272.722 325.017i
3.10 3.63616 4.33340i 16.5102 + 45.3614i −5.55674 31.5138i −20.2573 + 114.885i 256.603 + 93.3957i −72.0498 124.794i −156.767 90.5097i −1226.62 + 1029.26i 424.183 + 505.522i
13.1 −3.63616 4.33340i −14.5675 + 40.0239i −5.55674 + 31.5138i 14.4780 + 82.1091i 226.409 82.4062i −61.1665 + 105.943i 156.767 90.5097i −831.251 697.502i 303.167 361.301i
13.2 −3.63616 4.33340i −7.01359 + 19.2697i −5.55674 + 31.5138i −32.9273 186.740i 109.006 39.6749i −111.397 + 192.946i 156.767 90.5097i 236.316 + 198.293i −689.491 + 821.703i
13.3 −3.63616 4.33340i 0.501194 1.37702i −5.55674 + 31.5138i 6.94044 + 39.3612i −7.78959 + 2.83518i 159.899 276.953i 156.767 90.5097i 556.801 + 467.212i 145.331 173.199i
13.4 −3.63616 4.33340i 10.0117 27.5069i −5.55674 + 31.5138i 26.4694 + 150.115i −155.603 + 56.6347i −232.836 + 403.285i 156.767 90.5097i −97.9501 82.1899i 554.263 660.545i
13.5 −3.63616 4.33340i 12.5750 34.5495i −5.55674 + 31.5138i −24.6848 139.995i −195.442 + 71.1350i 136.699 236.770i 156.767 90.5097i −477.093 400.329i −516.895 + 616.012i
13.6 3.63616 + 4.33340i −14.7348 + 40.4836i −5.55674 + 31.5138i 40.1148 + 227.502i −229.010 + 83.3528i 276.187 478.370i −156.767 + 90.5097i −863.360 724.445i −839.995 + 1001.07i
13.7 3.63616 + 4.33340i −10.8314 + 29.7590i −5.55674 + 31.5138i −11.6174 65.8855i −168.342 + 61.2716i −205.116 + 355.271i −156.767 + 90.5097i −209.833 176.071i 243.266 289.913i
13.8 3.63616 + 4.33340i −1.10346 + 3.03174i −5.55674 + 31.5138i −30.9885 175.744i −17.1501 + 6.24213i 294.726 510.481i −156.767 + 90.5097i 550.473 + 461.901i 648.892 773.320i
13.9 3.63616 + 4.33340i 4.25575 11.6926i −5.55674 + 31.5138i 13.0241 + 73.8634i 66.1433 24.0742i −97.4265 + 168.748i −156.767 + 90.5097i 439.841 + 369.071i −272.722 + 325.017i
13.10 3.63616 + 4.33340i 16.5102 45.3614i −5.55674 + 31.5138i −20.2573 114.885i 256.603 93.3957i −72.0498 + 124.794i −156.767 + 90.5097i −1226.62 1029.26i 424.183 505.522i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.7.f.a 60
19.f odd 18 1 inner 38.7.f.a 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.7.f.a 60 1.a even 1 1 trivial
38.7.f.a 60 19.f odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(38, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database