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Properties

Label 38.9.d.a
Level $38$
Weight $9$
Character orbit 38.d
Analytic conductor $15.480$
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] = [38,9,Mod(27,38)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(38, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("38.27");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 38.d (of order \(6\), degree \(2\), 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: \(15.4803871823\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 168 q^{3} + 1536 q^{4} - 558 q^{5} + 896 q^{6} + 11992 q^{7} + 18448 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 168 q^{3} + 1536 q^{4} - 558 q^{5} + 896 q^{6} + 11992 q^{7} + 18448 q^{9} + 33516 q^{11} + 60870 q^{13} - 71424 q^{14} - 351930 q^{15} - 196608 q^{16} + 3546 q^{17} - 727738 q^{19} - 142848 q^{20} - 674568 q^{21} - 681600 q^{22} + 726 q^{23} - 114688 q^{24} - 277874 q^{25} - 960000 q^{26} + 767488 q^{28} - 254286 q^{29} - 71168 q^{30} - 2796150 q^{33} - 2740224 q^{34} - 690672 q^{35} - 2361344 q^{36} - 6829056 q^{38} - 1599924 q^{39} - 5153976 q^{41} + 4153600 q^{42} + 5692338 q^{43} + 2145024 q^{44} + 42653840 q^{45} + 11113938 q^{47} - 2752512 q^{48} + 17359944 q^{49} + 62169786 q^{51} + 7791360 q^{52} + 10399050 q^{53} - 24215680 q^{54} + 40181572 q^{55} - 27798558 q^{57} - 10126848 q^{58} - 58809492 q^{59} - 45047040 q^{60} - 45012614 q^{61} + 5892096 q^{62} + 62488724 q^{63} - 50331648 q^{64} + 45107968 q^{66} - 93677268 q^{67} + 907776 q^{68} - 291840 q^{70} + 5993046 q^{71} + 11698176 q^{72} + 23359860 q^{73} - 31474944 q^{74} - 1303552 q^{76} + 35611680 q^{77} + 94099200 q^{78} - 93221166 q^{79} - 9142272 q^{80} - 115510580 q^{81} + 6074880 q^{82} - 194283156 q^{83} - 15300674 q^{85} + 58611456 q^{86} + 89779380 q^{87} - 49833126 q^{89} - 63377664 q^{90} - 101275308 q^{91} - 92928 q^{92} - 157549992 q^{93} - 87570126 q^{95} - 29360128 q^{96} - 178814556 q^{97} - 88409088 q^{98} - 54444376 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1 −9.79796 − 5.65685i −127.651 − 73.6994i 64.0000 + 110.851i 381.154 − 660.178i 833.814 + 1444.21i 2763.12 − 1448.15i 7582.71 + 13133.6i −7469.06 + 4312.26i
27.2 −9.79796 − 5.65685i −57.0846 − 32.9578i 64.0000 + 110.851i −412.457 + 714.397i 372.875 + 645.838i 2843.86 − 1448.15i −1108.07 − 1919.23i 8082.48 − 4666.42i
27.3 −9.79796 − 5.65685i −31.2989 − 18.0704i 64.0000 + 110.851i 117.295 − 203.161i 204.444 + 354.107i −867.592 − 1448.15i −2627.42 − 4550.82i −2298.50 + 1327.04i
27.4 −9.79796 − 5.65685i 58.9458 + 34.0324i 64.0000 + 110.851i −530.398 + 918.676i −385.033 − 666.896i −2153.04 − 1448.15i −964.092 − 1669.86i 10393.6 − 6000.77i
27.5 −9.79796 − 5.65685i 65.0070 + 37.5318i 64.0000 + 110.851i 393.618 − 681.767i −424.624 − 735.470i −1847.62 − 1448.15i −463.230 − 802.337i −7713.31 + 4453.28i
27.6 −9.79796 − 5.65685i 99.7890 + 57.6132i 64.0000 + 110.851i −88.7120 + 153.654i −651.819 − 1128.98i 4081.69 − 1448.15i 3358.06 + 5816.33i 1738.39 − 1003.66i
27.7 9.79796 + 5.65685i −107.051 − 61.8057i 64.0000 + 110.851i 333.136 − 577.008i −699.252 − 1211.14i −377.866 1448.15i 4359.40 + 7550.70i 6528.10 − 3769.00i
27.8 9.79796 + 5.65685i −36.3650 − 20.9954i 64.0000 + 110.851i 106.719 − 184.842i −237.535 − 411.423i 680.248 1448.15i −2398.89 − 4155.00i 2091.25 − 1207.38i
27.9 9.79796 + 5.65685i −3.23455 − 1.86747i 64.0000 + 110.851i −499.152 + 864.556i −21.1280 − 36.5948i 3106.83 1448.15i −3273.53 − 5669.91i −9781.33 + 5647.26i
27.10 9.79796 + 5.65685i 30.3133 + 17.5014i 64.0000 + 110.851i −189.461 + 328.157i 198.006 + 342.956i −4482.18 1448.15i −2667.90 − 4620.94i −3712.67 + 2143.51i
27.11 9.79796 + 5.65685i 61.4665 + 35.4877i 64.0000 + 110.851i 219.968 − 380.996i 401.497 + 695.414i 2502.16 1448.15i −761.746 − 1319.38i 4310.48 − 2488.66i
27.12 9.79796 + 5.65685i 131.163 + 75.7272i 64.0000 + 110.851i −110.709 + 191.754i 856.755 + 1483.94i −253.604 1448.15i 8188.71 + 14183.3i −2169.45 + 1252.53i
31.1 −9.79796 + 5.65685i −127.651 + 73.6994i 64.0000 − 110.851i 381.154 + 660.178i 833.814 − 1444.21i 2763.12 1448.15i 7582.71 − 13133.6i −7469.06 − 4312.26i
31.2 −9.79796 + 5.65685i −57.0846 + 32.9578i 64.0000 − 110.851i −412.457 − 714.397i 372.875 − 645.838i 2843.86 1448.15i −1108.07 + 1919.23i 8082.48 + 4666.42i
31.3 −9.79796 + 5.65685i −31.2989 + 18.0704i 64.0000 − 110.851i 117.295 + 203.161i 204.444 − 354.107i −867.592 1448.15i −2627.42 + 4550.82i −2298.50 − 1327.04i
31.4 −9.79796 + 5.65685i 58.9458 − 34.0324i 64.0000 − 110.851i −530.398 − 918.676i −385.033 + 666.896i −2153.04 1448.15i −964.092 + 1669.86i 10393.6 + 6000.77i
31.5 −9.79796 + 5.65685i 65.0070 − 37.5318i 64.0000 − 110.851i 393.618 + 681.767i −424.624 + 735.470i −1847.62 1448.15i −463.230 + 802.337i −7713.31 − 4453.28i
31.6 −9.79796 + 5.65685i 99.7890 − 57.6132i 64.0000 − 110.851i −88.7120 − 153.654i −651.819 + 1128.98i 4081.69 1448.15i 3358.06 − 5816.33i 1738.39 + 1003.66i
31.7 9.79796 − 5.65685i −107.051 + 61.8057i 64.0000 − 110.851i 333.136 + 577.008i −699.252 + 1211.14i −377.866 − 1448.15i 4359.40 − 7550.70i 6528.10 + 3769.00i
31.8 9.79796 − 5.65685i −36.3650 + 20.9954i 64.0000 − 110.851i 106.719 + 184.842i −237.535 + 411.423i 680.248 − 1448.15i −2398.89 + 4155.00i 2091.25 + 1207.38i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.9.d.a ✓ 24
19.d odd 6 1 inner 38.9.d.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.9.d.a ✓ 24 1.a even 1 1 trivial
38.9.d.a ✓ 24 19.d odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(38, [\chi])\).