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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 $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [38,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("38.5");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(11.8706309684\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 45 q^{3} + 360 q^{6} + 1806 q^{7} - 7680 q^{8} - 3105 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 45 q^{3} + 360 q^{6} + 1806 q^{7} - 7680 q^{8} - 3105 q^{9} + 9786 q^{11} - 11130 q^{13} + 6144 q^{14} - 1818 q^{15} - 65250 q^{17} + 29040 q^{18} + 105906 q^{19} - 47616 q^{20} + 172854 q^{21} + 102336 q^{22} + 101004 q^{23} + 23040 q^{24} - 132756 q^{25} - 92112 q^{26} + 43659 q^{27} + 96384 q^{28} - 273912 q^{29} + 58710 q^{31} - 406965 q^{33} + 84960 q^{34} + 356100 q^{35} - 198720 q^{36} - 2598852 q^{37} - 219336 q^{38} + 1769844 q^{39} + 43377 q^{41} - 79056 q^{42} + 1126698 q^{43} + 818688 q^{44} - 912546 q^{45} - 530880 q^{46} + 588306 q^{47} - 368640 q^{48} - 2778285 q^{49} + 1665288 q^{50} + 3972045 q^{51} - 430080 q^{52} + 6520866 q^{53} - 486648 q^{54} - 1666818 q^{55} - 1849344 q^{56} - 6947844 q^{57} - 4573152 q^{58} - 970467 q^{59} + 4634496 q^{60} - 4037670 q^{61} + 755952 q^{62} + 7946472 q^{63} - 3932160 q^{64} + 1497690 q^{65} + 6499272 q^{66} - 4817025 q^{67} + 1472064 q^{68} + 10299876 q^{69} + 2848800 q^{70} - 25171290 q^{71} + 3179520 q^{72} + 1347750 q^{73} - 3123936 q^{74} + 8817984 q^{75} + 1583232 q^{76} - 11476764 q^{77} - 930960 q^{78} + 10777986 q^{79} - 5365035 q^{81} + 347016 q^{82} + 9745314 q^{83} + 877056 q^{84} + 29258556 q^{85} - 9338208 q^{86} + 25598802 q^{87} + 5010432 q^{88} - 20353740 q^{89} - 30112128 q^{90} + 19748724 q^{91} - 14097792 q^{92} - 18738174 q^{93} + 12376608 q^{94} - 20427204 q^{95} - 23005029 q^{97} + 14869008 q^{98} - 45819771 q^{99}+O(q^{100}) \) 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 −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 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 38.8.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 \( T_{3}^{30} - 45 T_{3}^{29} + 2565 T_{3}^{28} - 99603 T_{3}^{27} - 5241915 T_{3}^{26} + \cdots + 10\!\cdots\!61 \) acting on \(S_{8}^{\mathrm{new}}(38, [\chi])\). Copy content Toggle raw display