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Properties

Label 38.6.e.a
Level $38$
Weight $6$
Character orbit 38.e
Analytic conductor $6.095$
Analytic rank $0$
Dimension $24$
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: \(24\)
Relative dimension: \(4\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 18 q^{3} - 72 q^{6} + 438 q^{7} + 768 q^{8} + 378 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 18 q^{3} - 72 q^{6} + 438 q^{7} + 768 q^{8} + 378 q^{9} + 600 q^{11} - 1662 q^{13} - 1776 q^{14} - 1206 q^{15} + 4194 q^{17} - 4368 q^{18} + 9570 q^{19} - 1344 q^{20} + 2430 q^{21} - 5160 q^{22} - 1170 q^{23} - 1152 q^{24} + 5940 q^{25} - 1776 q^{26} - 5130 q^{27} + 768 q^{28} - 7554 q^{29} + 294 q^{31} - 45912 q^{33} + 1584 q^{34} + 22116 q^{35} + 6048 q^{36} + 28608 q^{37} - 9528 q^{38} - 38868 q^{39} + 37278 q^{41} - 9720 q^{42} + 13212 q^{43} + 20640 q^{44} - 34344 q^{45} - 29856 q^{46} - 3054 q^{47} - 9216 q^{48} + 1422 q^{49} + 16584 q^{50} + 2352 q^{51} - 1824 q^{52} + 73806 q^{53} + 136944 q^{54} - 41598 q^{55} + 56064 q^{56} + 178248 q^{57} - 52848 q^{58} + 123528 q^{59} + 46944 q^{60} - 149172 q^{61} + 58344 q^{62} - 170124 q^{63} - 49152 q^{64} - 202674 q^{65} - 84840 q^{66} - 313212 q^{67} + 35520 q^{68} - 19938 q^{69} - 88464 q^{70} + 75810 q^{71} + 48384 q^{72} + 172872 q^{73} + 4872 q^{74} + 174960 q^{75} + 31008 q^{76} - 115056 q^{77} + 101304 q^{78} + 158058 q^{79} - 47250 q^{81} - 149112 q^{82} - 148440 q^{83} - 176736 q^{84} - 226752 q^{85} - 57528 q^{86} + 182874 q^{87} - 38400 q^{88} - 322464 q^{89} + 48816 q^{90} - 27264 q^{91} + 88992 q^{92} + 779424 q^{93} + 143712 q^{94} + 634752 q^{95} - 108552 q^{97} - 173544 q^{98} + 656436 q^{99}+O(q^{100}) \) 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 −2.69117 + 15.2624i 12.2567 − 10.2846i −6.46132 − 5.42169i 10.7647 + 61.0494i 110.495 + 191.383i 32.0000 − 55.4256i 2.64806 + 0.963814i −31.7039 − 11.5393i
5.2 3.75877 − 1.36808i −0.383209 + 2.17329i 12.2567 − 10.2846i 53.6173 + 44.9902i 1.53284 + 8.69314i −9.54573 − 16.5337i 32.0000 − 55.4256i 223.769 + 81.4452i 263.085 + 95.7552i
5.3 3.75877 − 1.36808i 0.403873 − 2.29048i 12.2567 − 10.2846i −72.6237 − 60.9385i −1.61549 − 9.16190i −55.7656 − 96.5889i 32.0000 − 55.4256i 223.262 + 81.2608i −356.344 − 129.699i
5.4 3.75877 − 1.36808i 4.62861 − 26.2502i 12.2567 − 10.2846i 14.7431 + 12.3709i −18.5145 − 105.001i 4.52908 + 7.84460i 32.0000 − 55.4256i −439.302 − 159.893i 72.3402 + 26.3297i
9.1 −0.694593 − 3.93923i −22.0823 + 18.5292i −15.0351 + 5.47232i 4.74528 + 1.72714i 88.3291 + 74.1169i 104.694 − 181.336i 32.0000 + 55.4256i 102.098 − 579.027i 3.50757 − 19.8924i
9.2 −0.694593 − 3.93923i −3.81408 + 3.20039i −15.0351 + 5.47232i −0.763769 − 0.277989i 15.2563 + 12.8016i −52.7816 + 91.4204i 32.0000 + 55.4256i −37.8918 + 214.895i −0.564556 + 3.20175i
9.3 −0.694593 − 3.93923i 9.08943 − 7.62694i −15.0351 + 5.47232i 86.6641 + 31.5432i −36.3577 − 30.5078i 77.2983 − 133.885i 32.0000 + 55.4256i −17.7489 + 100.659i 64.0595 − 363.300i
9.4 −0.694593 − 3.93923i 15.2107 − 12.7633i −15.0351 + 5.47232i −77.4900 − 28.2040i −60.8427 − 51.0531i 31.0980 − 53.8633i 32.0000 + 55.4256i 26.2669 − 148.967i −57.2783 + 324.841i
17.1 −0.694593 + 3.93923i −22.0823 − 18.5292i −15.0351 − 5.47232i 4.74528 − 1.72714i 88.3291 − 74.1169i 104.694 + 181.336i 32.0000 − 55.4256i 102.098 + 579.027i 3.50757 + 19.8924i
17.2 −0.694593 + 3.93923i −3.81408 − 3.20039i −15.0351 − 5.47232i −0.763769 + 0.277989i 15.2563 − 12.8016i −52.7816 − 91.4204i 32.0000 − 55.4256i −37.8918 − 214.895i −0.564556 − 3.20175i
17.3 −0.694593 + 3.93923i 9.08943 + 7.62694i −15.0351 − 5.47232i 86.6641 − 31.5432i −36.3577 + 30.5078i 77.2983 + 133.885i 32.0000 − 55.4256i −17.7489 − 100.659i 64.0595 + 363.300i
17.4 −0.694593 + 3.93923i 15.2107 + 12.7633i −15.0351 − 5.47232i −77.4900 + 28.2040i −60.8427 + 51.0531i 31.0980 + 53.8633i 32.0000 − 55.4256i 26.2669 + 148.967i −57.2783 − 324.841i
23.1 3.75877 + 1.36808i −2.69117 − 15.2624i 12.2567 + 10.2846i −6.46132 + 5.42169i 10.7647 − 61.0494i 110.495 − 191.383i 32.0000 + 55.4256i 2.64806 − 0.963814i −31.7039 + 11.5393i
23.2 3.75877 + 1.36808i −0.383209 − 2.17329i 12.2567 + 10.2846i 53.6173 − 44.9902i 1.53284 − 8.69314i −9.54573 + 16.5337i 32.0000 + 55.4256i 223.769 − 81.4452i 263.085 − 95.7552i
23.3 3.75877 + 1.36808i 0.403873 + 2.29048i 12.2567 + 10.2846i −72.6237 + 60.9385i −1.61549 + 9.16190i −55.7656 + 96.5889i 32.0000 + 55.4256i 223.262 − 81.2608i −356.344 + 129.699i
23.4 3.75877 + 1.36808i 4.62861 + 26.2502i 12.2567 + 10.2846i 14.7431 − 12.3709i −18.5145 + 105.001i 4.52908 − 7.84460i 32.0000 + 55.4256i −439.302 + 159.893i 72.3402 − 26.3297i
25.1 −3.06418 + 2.57115i −22.4490 − 8.17075i 2.77837 − 15.7569i −4.34948 − 24.6671i 89.7958 − 32.6830i −29.2182 + 50.6074i 32.0000 + 55.4256i 251.046 + 210.652i 76.7505 + 64.4013i
25.2 −3.06418 + 2.57115i 0.577036 + 0.210024i 2.77837 − 15.7569i 2.13367 + 12.1006i −2.30814 + 0.840095i −76.5898 + 132.657i 32.0000 + 55.4256i −185.860 − 155.955i −37.6505 − 31.5925i
25.3 −3.06418 + 2.57115i 12.9479 + 4.71264i 2.77837 − 15.7569i −13.1062 − 74.3288i −51.7915 + 18.8506i 39.5277 − 68.4640i 32.0000 + 55.4256i −40.7104 − 34.1601i 231.270 + 194.059i
25.4 −3.06418 + 2.57115i 17.5622 + 6.39212i 2.77837 − 15.7569i 12.8909 + 73.1080i −70.2488 + 25.5685i 75.2582 − 130.351i 32.0000 + 55.4256i 81.4231 + 68.3221i −227.472 − 190.871i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.4
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.a ✓ 24
19.e even 9 1 inner 38.6.e.a ✓ 24
19.e even 9 1 722.6.a.o 12
19.f odd 18 1 722.6.a.p 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.6.e.a ✓ 24 1.a even 1 1 trivial
38.6.e.a ✓ 24 19.e even 9 1 inner
722.6.a.o 12 19.e even 9 1
722.6.a.p 12 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 18 T_{3}^{23} - 27 T_{3}^{22} + 4140 T_{3}^{21} + 110430 T_{3}^{20} - 13323294 T_{3}^{19} + \cdots + 73\!\cdots\!21 \) acting on \(S_{6}^{\mathrm{new}}(38, [\chi])\). Copy content Toggle raw display