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Properties

Label 18.13.d.a
Level $18$
Weight $13$
Character orbit 18.d
Analytic conductor $16.452$
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] = [18,13,Mod(5,18)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(18, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("18.5");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 18.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: \(16.4518887110\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 780 q^{3} + 24576 q^{4} + 31968 q^{5} - 108288 q^{6} - 68640 q^{7} - 420900 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 780 q^{3} + 24576 q^{4} + 31968 q^{5} - 108288 q^{6} - 68640 q^{7} - 420900 q^{9} + 7514316 q^{11} - 1228800 q^{12} + 850080 q^{13} + 19920384 q^{14} - 16182576 q^{15} - 50331648 q^{16} + 15513600 q^{18} - 47432760 q^{19} + 65470464 q^{20} + 266986872 q^{21} - 67334400 q^{22} - 825178320 q^{23} - 20447232 q^{24} + 860585244 q^{25} - 1215311040 q^{27} - 281149440 q^{28} + 1080101736 q^{29} + 1936226304 q^{30} + 1011666000 q^{31} - 243568260 q^{33} - 377786880 q^{34} + 1343594496 q^{36} - 8737330560 q^{37} + 431965440 q^{38} - 10480976088 q^{39} + 18115081596 q^{41} - 1234106880 q^{42} - 1282315020 q^{43} + 20685971592 q^{45} + 16593325056 q^{46} - 50877684360 q^{47} + 754974720 q^{48} - 38339469252 q^{49} + 30185561088 q^{50} + 61461493884 q^{51} - 1740963840 q^{52} - 31969117440 q^{54} - 82896170256 q^{55} + 40796946432 q^{56} - 28717611060 q^{57} + 1465228800 q^{58} + 320300511084 q^{59} + 73244344320 q^{60} - 72089045952 q^{61} - 446050377840 q^{63} - 206158430208 q^{64} + 19622324736 q^{65} - 63727308288 q^{66} - 182374358460 q^{67} + 206677647360 q^{68} + 679646167056 q^{69} + 75234037248 q^{70} + 1777336320 q^{72} - 404658904200 q^{73} - 520170287616 q^{74} - 174862718820 q^{75} - 48571146240 q^{76} + 1796923904400 q^{77} + 627848701440 q^{78} + 127716024792 q^{79} - 701940271284 q^{81} - 966809364480 q^{82} - 2614888438920 q^{83} - 380438740992 q^{84} - 170761904352 q^{85} + 1801163858688 q^{86} + 4059123231240 q^{87} + 137900851200 q^{88} - 1768593996288 q^{90} - 563012378112 q^{91} - 1689965199360 q^{92} - 3669147604320 q^{93} + 155881668096 q^{94} + 9139492713240 q^{95} + 412316860416 q^{96} - 995790575220 q^{97} - 7132624985448 q^{99}+O(q^{100}) \) 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 −39.1918 + 22.6274i −673.314 − 279.444i 1024.00 − 1773.62i 3545.56 + 2047.03i 32711.5 − 4283.41i 39462.2 + 68350.6i 92681.9i 375263. + 376308.i −185276.
5.2 −39.1918 + 22.6274i −392.782 + 614.136i 1024.00 − 1773.62i 23697.2 + 13681.6i 1497.56 − 32956.8i −88333.4 − 152998.i 92681.9i −222885. − 482444.i −1.23831e6
5.3 −39.1918 + 22.6274i −50.6040 − 727.242i 1024.00 − 1773.62i −24528.3 − 14161.4i 18438.9 + 27356.9i −63059.5 − 109222.i 92681.9i −526319. + 73602.7i 1.28175e6
5.4 −39.1918 + 22.6274i 365.111 + 630.980i 1024.00 − 1773.62i −4386.86 − 2532.75i −28586.8 − 16467.7i 80633.1 + 139661.i 92681.9i −264829. + 460755.i 229239.
5.5 −39.1918 + 22.6274i 532.269 − 498.127i 1024.00 − 1773.62i 9029.09 + 5212.95i −9589.27 + 31566.4i 51557.7 + 89300.6i 92681.9i 35179.7 − 530275.i −471822.
5.6 −39.1918 + 22.6274i 683.233 + 254.230i 1024.00 − 1773.62i 635.371 + 366.832i −32529.7 + 5496.06i −100955. − 174859.i 92681.9i 402175. + 347397.i −33201.8
5.7 39.1918 − 22.6274i −703.796 + 190.032i 1024.00 − 1773.62i 12341.5 + 7125.34i −23283.1 + 23372.8i −53184.3 − 92117.9i − 92681.9i 459217. − 267487.i 644912.
5.8 39.1918 − 22.6274i −696.965 − 213.731i 1024.00 − 1773.62i −26548.6 − 15327.8i −32151.5 + 7393.98i 50051.9 + 86692.4i − 92681.9i 440079. + 297927.i −1.38732e6
5.9 39.1918 − 22.6274i −302.392 − 663.325i 1024.00 − 1773.62i 19890.6 + 11483.8i −26860.6 − 19154.6i 107133. + 185559.i − 92681.9i −348560. + 401168.i 1.03940e6
5.10 39.1918 − 22.6274i −137.198 + 715.973i 1024.00 − 1773.62i −8377.54 − 4836.78i 10823.6 + 31164.7i −16171.4 − 28009.7i − 92681.9i −493794. − 196460.i −437775.
5.11 39.1918 − 22.6274i 416.005 − 598.649i 1024.00 − 1773.62i −4293.14 − 2478.64i 2758.12 − 32875.3i −36354.5 − 62967.8i − 92681.9i −185321. − 498082.i −224341.
5.12 39.1918 − 22.6274i 570.433 + 453.925i 1024.00 − 1773.62i 14979.2 + 8648.26i 32627.4 + 4882.73i −5099.55 − 8832.67i − 92681.9i 119346. + 517867.i 782751.
11.1 −39.1918 − 22.6274i −673.314 + 279.444i 1024.00 + 1773.62i 3545.56 − 2047.03i 32711.5 + 4283.41i 39462.2 − 68350.6i − 92681.9i 375263. − 376308.i −185276.
11.2 −39.1918 − 22.6274i −392.782 − 614.136i 1024.00 + 1773.62i 23697.2 − 13681.6i 1497.56 + 32956.8i −88333.4 + 152998.i − 92681.9i −222885. + 482444.i −1.23831e6
11.3 −39.1918 − 22.6274i −50.6040 + 727.242i 1024.00 + 1773.62i −24528.3 + 14161.4i 18438.9 − 27356.9i −63059.5 + 109222.i − 92681.9i −526319. − 73602.7i 1.28175e6
11.4 −39.1918 − 22.6274i 365.111 − 630.980i 1024.00 + 1773.62i −4386.86 + 2532.75i −28586.8 + 16467.7i 80633.1 − 139661.i − 92681.9i −264829. − 460755.i 229239.
11.5 −39.1918 − 22.6274i 532.269 + 498.127i 1024.00 + 1773.62i 9029.09 − 5212.95i −9589.27 − 31566.4i 51557.7 − 89300.6i − 92681.9i 35179.7 + 530275.i −471822.
11.6 −39.1918 − 22.6274i 683.233 − 254.230i 1024.00 + 1773.62i 635.371 − 366.832i −32529.7 − 5496.06i −100955. + 174859.i − 92681.9i 402175. − 347397.i −33201.8
11.7 39.1918 + 22.6274i −703.796 − 190.032i 1024.00 + 1773.62i 12341.5 − 7125.34i −23283.1 − 23372.8i −53184.3 + 92117.9i 92681.9i 459217. + 267487.i 644912.
11.8 39.1918 + 22.6274i −696.965 + 213.731i 1024.00 + 1773.62i −26548.6 + 15327.8i −32151.5 − 7393.98i 50051.9 − 86692.4i 92681.9i 440079. − 297927.i −1.38732e6
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 18.13.d.a ✓ 24
3.b odd 2 1 54.13.d.a 24
9.c even 3 1 54.13.d.a 24
9.c even 3 1 162.13.b.c 24
9.d odd 6 1 inner 18.13.d.a ✓ 24
9.d odd 6 1 162.13.b.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.13.d.a ✓ 24 1.a even 1 1 trivial
18.13.d.a ✓ 24 9.d odd 6 1 inner
54.13.d.a 24 3.b odd 2 1
54.13.d.a 24 9.c even 3 1
162.13.b.c 24 9.c even 3 1
162.13.b.c 24 9.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{13}^{\mathrm{new}}(18, [\chi])\).