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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 $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [38,6,Mod(5,38)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(38, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([16]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("38.5");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 38.e (of order \(9\), degree \(6\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 5.5
Significant digits:
Format:

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 \( T_{3}^{30} - 15 T_{3}^{29} + 285 T_{3}^{28} - 7489 T_{3}^{27} - 27969 T_{3}^{26} + 35547 T_{3}^{25} + \cdots + 24\!\cdots\!41 \) acting on \(S_{6}^{\mathrm{new}}(38, [\chi])\). Copy content Toggle raw display