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Properties

Label 243.4.e.d
Level $243$
Weight $4$
Character orbit 243.e
Analytic conductor $14.337$
Analytic rank $0$
Dimension $48$
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] = [243,4,Mod(28,243)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(243, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("243.28");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 243 = 3^{5} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 243.e (of order \(9\), 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: \(14.3374641314\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{9})\)
Twist minimal: no (minimal twist has level 27)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 3 q^{2} + 3 q^{4} + 15 q^{5} + 3 q^{7} - 75 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 3 q^{2} + 3 q^{4} + 15 q^{5} + 3 q^{7} - 75 q^{8} - 3 q^{10} + 102 q^{11} + 3 q^{13} + 285 q^{14} + 27 q^{16} - 207 q^{17} - 3 q^{19} - 84 q^{20} - 51 q^{22} - 435 q^{23} - 213 q^{25} + 1914 q^{26} - 12 q^{28} - 429 q^{29} - 51 q^{31} - 999 q^{32} + 225 q^{34} - 1257 q^{35} - 3 q^{37} + 2049 q^{38} - 555 q^{40} + 1284 q^{41} - 1023 q^{43} - 2211 q^{44} - 3 q^{46} - 3093 q^{47} + 1191 q^{49} + 579 q^{50} + 2781 q^{52} + 2736 q^{53} - 12 q^{55} + 1866 q^{56} + 1785 q^{58} - 5286 q^{59} - 105 q^{61} - 2118 q^{62} - 195 q^{64} + 3693 q^{65} - 1401 q^{67} + 7596 q^{68} - 1491 q^{70} - 3105 q^{71} - 219 q^{73} - 12549 q^{74} - 2001 q^{76} + 4263 q^{77} - 1401 q^{79} + 9870 q^{80} - 12 q^{82} + 5043 q^{83} - 702 q^{85} - 16131 q^{86} + 843 q^{88} - 5202 q^{89} + 267 q^{91} + 13335 q^{92} + 3864 q^{94} + 4935 q^{95} + 1812 q^{97} - 4392 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1 −0.812863 − 4.60998i 0 −13.0736 + 4.75840i −2.40097 − 2.01465i 0 28.6635 + 10.4327i 13.8388 + 23.9695i 0 −7.33583 + 12.7060i
28.2 −0.723057 − 4.10066i 0 −8.77506 + 3.19386i 11.4150 + 9.57830i 0 −4.10372 − 1.49363i 2.78612 + 4.82571i 0 31.0237 − 53.7346i
28.3 −0.316340 − 1.79405i 0 4.39899 − 1.60110i −10.3630 − 8.69559i 0 −21.4798 − 7.81801i −11.5509 − 20.0068i 0 −12.3221 + 21.3425i
28.4 −0.200319 − 1.13606i 0 6.26703 − 2.28101i 5.88313 + 4.93653i 0 −12.1897 − 4.43667i −8.46114 − 14.6551i 0 4.42972 − 7.67249i
28.5 0.146052 + 0.828300i 0 6.85279 − 2.49421i −8.28596 − 6.95275i 0 25.0820 + 9.12911i 6.43113 + 11.1390i 0 4.54878 − 7.87873i
28.6 0.512603 + 2.90711i 0 −0.671007 + 0.244226i 13.0560 + 10.9553i 0 12.0565 + 4.38819i 10.7539 + 18.6263i 0 −25.1558 + 43.5711i
28.7 0.553542 + 3.13930i 0 −2.03123 + 0.739306i −5.49264 − 4.60887i 0 −4.56436 − 1.66129i 9.30563 + 16.1178i 0 11.4282 − 19.7942i
28.8 0.921634 + 5.22685i 0 −18.9530 + 6.89831i 4.22440 + 3.54469i 0 −23.1381 − 8.42157i −32.2942 − 55.9352i 0 −14.6342 + 25.3472i
55.1 −4.89591 + 1.78197i 0 14.6662 − 12.3064i 2.43912 + 13.8329i 0 −7.89031 − 6.62076i −29.0343 + 50.2889i 0 −36.5916 − 63.3784i
55.2 −3.46620 + 1.26159i 0 4.29454 − 3.60355i −0.280549 − 1.59107i 0 26.0189 + 21.8324i 4.41508 − 7.64714i 0 2.97972 + 5.16102i
55.3 −2.99674 + 1.09072i 0 1.66242 − 1.39493i −2.48939 − 14.1180i 0 −24.6679 − 20.6988i 9.29592 − 16.1010i 0 22.8590 + 39.5929i
55.4 −0.605602 + 0.220421i 0 −5.81019 + 4.87533i −0.0893021 − 0.506458i 0 −7.35639 − 6.17274i 5.02191 − 8.69820i 0 0.165716 + 0.287028i
55.5 0.280832 − 0.102214i 0 −6.05994 + 5.08489i 3.48193 + 19.7470i 0 16.4627 + 13.8138i −2.37749 + 4.11794i 0 2.99626 + 5.18968i
55.6 2.19021 − 0.797172i 0 −1.96681 + 1.65035i −1.08379 − 6.14650i 0 −2.07054 − 1.73739i −12.3152 + 21.3306i 0 −7.27356 − 12.5982i
55.7 3.20688 − 1.16721i 0 2.79337 − 2.34391i −1.90706 − 10.8155i 0 −3.37516 − 2.83209i −7.42861 + 12.8667i 0 −18.7397 − 32.4581i
55.8 4.73349 − 1.72285i 0 13.3094 − 11.1679i 2.70490 + 15.3402i 0 4.31842 + 3.62358i 23.6100 − 40.8938i 0 39.2325 + 67.9527i
109.1 −3.31592 + 2.78239i 0 1.86447 − 10.5739i 4.03535 − 1.46875i 0 −3.54847 − 20.1244i 5.92381 + 10.2603i 0 −9.29428 + 16.0982i
109.2 −2.87878 + 2.41559i 0 1.06315 − 6.02945i −5.73755 + 2.08830i 0 2.99660 + 16.9946i −3.52788 − 6.11047i 0 11.4727 − 19.8713i
109.3 −1.48645 + 1.24728i 0 −0.735357 + 4.17042i 2.54549 − 0.926483i 0 −2.06947 − 11.7366i −11.8703 − 20.5600i 0 −2.62817 + 4.55212i
109.4 0.759876 − 0.637611i 0 −1.21832 + 6.90945i −0.135765 + 0.0494143i 0 4.40626 + 24.9892i 7.44756 + 12.8996i 0 −0.0716572 + 0.124114i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 243.4.e.d 48
3.b odd 2 1 243.4.e.a 48
9.c even 3 1 27.4.e.a ✓ 48
9.c even 3 1 243.4.e.c 48
9.d odd 6 1 81.4.e.a 48
9.d odd 6 1 243.4.e.b 48
27.e even 9 1 27.4.e.a ✓ 48
27.e even 9 1 243.4.e.c 48
27.e even 9 1 inner 243.4.e.d 48
27.e even 9 1 729.4.a.d 24
27.f odd 18 1 81.4.e.a 48
27.f odd 18 1 243.4.e.a 48
27.f odd 18 1 243.4.e.b 48
27.f odd 18 1 729.4.a.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.4.e.a ✓ 48 9.c even 3 1
27.4.e.a ✓ 48 27.e even 9 1
81.4.e.a 48 9.d odd 6 1
81.4.e.a 48 27.f odd 18 1
243.4.e.a 48 3.b odd 2 1
243.4.e.a 48 27.f odd 18 1
243.4.e.b 48 9.d odd 6 1
243.4.e.b 48 27.f odd 18 1
243.4.e.c 48 9.c even 3 1
243.4.e.c 48 27.e even 9 1
243.4.e.d 48 1.a even 1 1 trivial
243.4.e.d 48 27.e even 9 1 inner
729.4.a.c 24 27.f odd 18 1
729.4.a.d 24 27.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} - 3 T_{2}^{47} + 3 T_{2}^{46} + 57 T_{2}^{45} - 198 T_{2}^{44} + 1080 T_{2}^{43} + \cdots + 18\!\cdots\!16 \) acting on \(S_{4}^{\mathrm{new}}(243, [\chi])\). Copy content Toggle raw display