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Properties

Label 324.3.j.a
Level $324$
Weight $3$
Character orbit 324.j
Analytic conductor $8.828$
Analytic rank $0$
Dimension $204$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [324,3,Mod(19,324)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(324, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 16]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 324.j (of order \(18\), degree \(6\), not 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: \(8.82836056527\)
Analytic rank: \(0\)
Dimension: \(204\)
Relative dimension: \(34\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 204 q + 6 q^{2} - 6 q^{4} + 12 q^{5} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 204 q + 6 q^{2} - 6 q^{4} + 12 q^{5} + 3 q^{8} - 3 q^{10} - 12 q^{13} - 39 q^{14} - 6 q^{16} + 6 q^{17} + 69 q^{20} - 6 q^{22} - 12 q^{25} + 174 q^{26} - 12 q^{28} - 60 q^{29} + 96 q^{32} + 6 q^{34} - 6 q^{37} - 72 q^{38} + 69 q^{40} + 192 q^{41} + 219 q^{44} - 3 q^{46} - 12 q^{49} + 165 q^{50} + 21 q^{52} + 24 q^{53} - 99 q^{56} - 141 q^{58} - 12 q^{61} - 294 q^{62} - 3 q^{64} + 156 q^{65} - 375 q^{68} - 165 q^{70} - 6 q^{73} - 447 q^{74} - 54 q^{76} - 132 q^{77} - 798 q^{80} - 12 q^{82} + 138 q^{85} - 606 q^{86} - 198 q^{88} + 114 q^{89} - 723 q^{92} - 357 q^{94} + 168 q^{97} - 510 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.99738 − 0.102316i 0 3.97906 + 0.408727i −1.35104 − 7.66211i 0 −0.787463 + 0.938462i −7.90589 − 1.22351i 0 1.91458 + 15.4424i
19.2 −1.99036 − 0.196104i 0 3.92309 + 0.780636i 0.479555 + 2.71969i 0 −6.88504 + 8.20527i −7.65528 − 2.32308i 0 −0.421146 − 5.50722i
19.3 −1.97794 + 0.296237i 0 3.82449 − 1.17188i 0.00321173 + 0.0182146i 0 0.446686 − 0.532339i −7.21745 + 3.45086i 0 −0.0117484 − 0.0350759i
19.4 −1.91833 − 0.565700i 0 3.35997 + 2.17040i 1.66005 + 9.41462i 0 4.14643 − 4.94152i −5.21773 − 6.06427i 0 2.14133 − 18.9994i
19.5 −1.70561 + 1.04446i 0 1.81819 − 3.56289i 0.00321173 + 0.0182146i 0 −0.446686 + 0.532339i 0.620191 + 7.97592i 0 −0.0245024 − 0.0277124i
19.6 −1.63904 − 1.14610i 0 1.37291 + 3.75701i −1.09499 − 6.21002i 0 5.65814 − 6.74311i 2.05564 − 7.73139i 0 −5.32256 + 11.4335i
19.7 −1.48574 − 1.33887i 0 0.414837 + 3.97843i 0.542224 + 3.07510i 0 2.98812 − 3.56110i 4.71027 − 6.46632i 0 3.31157 − 5.29477i
19.8 −1.46432 + 1.36227i 0 0.288439 − 3.98959i −1.35104 − 7.66211i 0 0.787463 − 0.938462i 5.01253 + 6.23495i 0 12.4162 + 9.37927i
19.9 −1.43960 − 1.38836i 0 0.144916 + 3.99737i 0.133306 + 0.756014i 0 −3.08121 + 3.67204i 5.34117 − 5.95583i 0 0.857712 − 1.27344i
19.10 −1.39865 + 1.42960i 0 −0.0875399 − 3.99904i 0.479555 + 2.71969i 0 6.88504 − 8.20527i 5.83949 + 5.46812i 0 −4.55882 − 3.11833i
19.11 −1.10590 + 1.66643i 0 −1.55397 − 3.68581i 1.66005 + 9.41462i 0 −4.14643 + 4.94152i 7.86067 + 1.48655i 0 −17.5246 − 7.64527i
19.12 −0.747520 − 1.85505i 0 −2.88243 + 2.77337i −0.463207 − 2.62698i 0 −4.34885 + 5.18275i 7.29942 + 3.27390i 0 −4.52692 + 2.82299i
19.13 −0.518880 + 1.93152i 0 −3.46153 − 2.00445i −1.09499 − 6.21002i 0 −5.65814 + 6.74311i 5.66776 − 5.64593i 0 12.5629 + 1.10726i
19.14 −0.376263 − 1.96429i 0 −3.71685 + 1.47818i 1.26933 + 7.19876i 0 2.67622 − 3.18940i 4.30208 + 6.74478i 0 13.6628 − 5.20197i
19.15 −0.343802 − 1.97023i 0 −3.76360 + 1.35474i −0.608002 − 3.44815i 0 −5.71739 + 6.81372i 3.96308 + 6.94939i 0 −6.58462 + 2.38339i
19.16 −0.277531 + 1.98065i 0 −3.84595 − 1.09938i 0.542224 + 3.07510i 0 −2.98812 + 3.56110i 3.24486 − 7.31238i 0 −6.24119 + 0.220520i
19.17 −0.210380 + 1.98890i 0 −3.91148 − 0.836851i 0.133306 + 0.756014i 0 3.08121 − 3.67204i 2.48731 − 7.60350i 0 −1.53168 + 0.106082i
19.18 −0.103865 − 1.99730i 0 −3.97842 + 0.414901i −1.50670 − 8.54492i 0 7.53806 − 8.98351i 1.24190 + 7.90302i 0 −16.9103 + 3.89685i
19.19 0.443343 − 1.95024i 0 −3.60689 − 1.72925i 0.890579 + 5.05073i 0 −0.0265129 + 0.0315969i −4.97155 + 6.26767i 0 10.2450 + 0.502357i
19.20 0.619770 + 1.90155i 0 −3.23177 + 2.35705i −0.463207 − 2.62698i 0 4.34885 − 5.18275i −6.48499 − 4.68454i 0 4.70825 − 2.50894i
See next 80 embeddings (of 204 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
27.e even 9 1 inner
108.j odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.3.j.a 204
3.b odd 2 1 108.3.j.a ✓ 204
4.b odd 2 1 inner 324.3.j.a 204
12.b even 2 1 108.3.j.a ✓ 204
27.e even 9 1 inner 324.3.j.a 204
27.f odd 18 1 108.3.j.a ✓ 204
108.j odd 18 1 inner 324.3.j.a 204
108.l even 18 1 108.3.j.a ✓ 204
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.j.a ✓ 204 3.b odd 2 1
108.3.j.a ✓ 204 12.b even 2 1
108.3.j.a ✓ 204 27.f odd 18 1
108.3.j.a ✓ 204 108.l even 18 1
324.3.j.a 204 1.a even 1 1 trivial
324.3.j.a 204 4.b odd 2 1 inner
324.3.j.a 204 27.e even 9 1 inner
324.3.j.a 204 108.j odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(324, [\chi])\).