Properties

Label 54.4.e.b
Level $54$
Weight $4$
Character orbit 54.e
Analytic conductor $3.186$
Analytic rank $0$
Dimension $30$
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] = [54,4,Mod(7,54)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(54, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([16]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("54.7");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 54.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: \(3.18610314031\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(5\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 12 q^{5} + 30 q^{6} + 33 q^{7} + 120 q^{8} + 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 12 q^{5} + 30 q^{6} + 33 q^{7} + 120 q^{8} + 42 q^{9} - 30 q^{10} - 39 q^{11} + 24 q^{12} - 60 q^{13} - 66 q^{14} - 153 q^{15} + 102 q^{17} + 168 q^{18} - 171 q^{19} + 96 q^{20} + 78 q^{21} + 78 q^{22} + 48 q^{23} - 432 q^{25} - 468 q^{26} - 675 q^{27} + 336 q^{28} - 381 q^{29} - 486 q^{30} - 801 q^{31} + 954 q^{33} - 222 q^{34} + 624 q^{35} + 132 q^{36} - 555 q^{37} + 606 q^{38} + 363 q^{39} + 96 q^{40} + 1401 q^{41} - 420 q^{42} + 648 q^{43} + 132 q^{44} + 2583 q^{45} - 348 q^{46} + 540 q^{47} + 192 q^{48} + 15 q^{49} - 828 q^{50} - 2376 q^{51} - 240 q^{52} - 1794 q^{53} - 522 q^{54} + 3906 q^{55} - 264 q^{56} - 1656 q^{57} - 444 q^{58} - 1500 q^{59} + 1332 q^{60} - 378 q^{61} + 744 q^{62} + 840 q^{63} - 960 q^{64} + 3666 q^{65} + 1710 q^{66} + 3087 q^{67} - 24 q^{68} + 1665 q^{69} + 2118 q^{70} + 120 q^{71} + 672 q^{72} - 2604 q^{73} - 1974 q^{74} - 2643 q^{75} - 1212 q^{76} - 6504 q^{77} - 5220 q^{78} - 2625 q^{79} - 480 q^{80} - 7614 q^{81} + 3408 q^{82} - 5211 q^{83} - 1548 q^{84} - 1395 q^{85} - 1296 q^{86} + 4914 q^{87} - 912 q^{88} + 2604 q^{89} + 1512 q^{90} - 3399 q^{91} + 372 q^{92} - 222 q^{93} + 4500 q^{94} + 7545 q^{95} + 384 q^{96} + 8940 q^{97} + 4002 q^{98} + 10296 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 1.87939 0.684040i −5.00873 1.38297i 3.06418 2.57115i −2.36712 13.4246i −10.3593 + 0.827035i −10.7136 8.98975i 4.00000 6.92820i 23.1748 + 13.8539i −13.6317 23.6108i
7.2 1.87939 0.684040i −4.02802 + 3.28253i 3.06418 2.57115i 2.39299 + 13.5713i −5.32482 + 8.92448i 20.6081 + 17.2922i 4.00000 6.92820i 5.44995 26.4442i 13.7807 + 23.8688i
7.3 1.87939 0.684040i 2.51736 4.54565i 3.06418 2.57115i 0.127307 + 0.721992i 1.62168 10.2650i 1.18538 + 0.994655i 4.00000 6.92820i −14.3258 22.8860i 0.733130 + 1.26982i
7.4 1.87939 0.684040i 3.83244 + 3.50891i 3.06418 2.57115i −3.01146 17.0788i 9.60286 + 3.97304i 16.7015 + 14.0142i 4.00000 6.92820i 2.37516 + 26.8953i −17.3423 30.0378i
7.5 1.87939 0.684040i 4.39270 + 2.77565i 3.06418 2.57115i 3.05422 + 17.3214i 10.1542 + 2.21173i −19.9833 16.7680i 4.00000 6.92820i 11.5916 + 24.3851i 17.5886 + 30.4643i
13.1 −0.347296 1.96962i −4.63338 + 2.35197i −3.75877 + 1.36808i 6.77242 + 5.68273i 6.24164 + 8.30915i 30.7336 + 11.1861i 4.00000 + 6.92820i 15.9365 21.7952i 8.84076 15.3127i
13.2 −0.347296 1.96962i −3.45245 3.88337i −3.75877 + 1.36808i 0.657259 + 0.551505i −6.44973 + 8.14868i −20.5288 7.47187i 4.00000 + 6.92820i −3.16120 + 26.8143i 0.857990 1.48608i
13.3 −0.347296 1.96962i −1.19733 + 5.05632i −3.75877 + 1.36808i −9.84993 8.26508i 10.3748 + 0.602240i −26.8516 9.77320i 4.00000 + 6.92820i −24.1328 12.1082i −12.8582 + 22.2710i
13.4 −0.347296 1.96962i 3.93867 + 3.38924i −3.75877 + 1.36808i 7.73816 + 6.49309i 5.30761 8.93473i 7.54967 + 2.74785i 4.00000 + 6.92820i 4.02617 + 26.6981i 10.1015 17.4962i
13.5 −0.347296 1.96962i 4.23116 3.01618i −3.75877 + 1.36808i −14.9069 12.5084i −7.41017 7.28624i 11.7781 + 4.28689i 4.00000 + 6.92820i 8.80536 25.5238i −19.4596 + 33.7050i
25.1 −0.347296 + 1.96962i −4.63338 2.35197i −3.75877 1.36808i 6.77242 5.68273i 6.24164 8.30915i 30.7336 11.1861i 4.00000 6.92820i 15.9365 + 21.7952i 8.84076 + 15.3127i
25.2 −0.347296 + 1.96962i −3.45245 + 3.88337i −3.75877 1.36808i 0.657259 0.551505i −6.44973 8.14868i −20.5288 + 7.47187i 4.00000 6.92820i −3.16120 26.8143i 0.857990 + 1.48608i
25.3 −0.347296 + 1.96962i −1.19733 5.05632i −3.75877 1.36808i −9.84993 + 8.26508i 10.3748 0.602240i −26.8516 + 9.77320i 4.00000 6.92820i −24.1328 + 12.1082i −12.8582 22.2710i
25.4 −0.347296 + 1.96962i 3.93867 3.38924i −3.75877 1.36808i 7.73816 6.49309i 5.30761 + 8.93473i 7.54967 2.74785i 4.00000 6.92820i 4.02617 26.6981i 10.1015 + 17.4962i
25.5 −0.347296 + 1.96962i 4.23116 + 3.01618i −3.75877 1.36808i −14.9069 + 12.5084i −7.41017 + 7.28624i 11.7781 4.28689i 4.00000 6.92820i 8.80536 + 25.5238i −19.4596 33.7050i
31.1 1.87939 + 0.684040i −5.00873 + 1.38297i 3.06418 + 2.57115i −2.36712 + 13.4246i −10.3593 0.827035i −10.7136 + 8.98975i 4.00000 + 6.92820i 23.1748 13.8539i −13.6317 + 23.6108i
31.2 1.87939 + 0.684040i −4.02802 3.28253i 3.06418 + 2.57115i 2.39299 13.5713i −5.32482 8.92448i 20.6081 17.2922i 4.00000 + 6.92820i 5.44995 + 26.4442i 13.7807 23.8688i
31.3 1.87939 + 0.684040i 2.51736 + 4.54565i 3.06418 + 2.57115i 0.127307 0.721992i 1.62168 + 10.2650i 1.18538 0.994655i 4.00000 + 6.92820i −14.3258 + 22.8860i 0.733130 1.26982i
31.4 1.87939 + 0.684040i 3.83244 3.50891i 3.06418 + 2.57115i −3.01146 + 17.0788i 9.60286 3.97304i 16.7015 14.0142i 4.00000 + 6.92820i 2.37516 26.8953i −17.3423 + 30.0378i
31.5 1.87939 + 0.684040i 4.39270 2.77565i 3.06418 + 2.57115i 3.05422 17.3214i 10.1542 2.21173i −19.9833 + 16.7680i 4.00000 + 6.92820i 11.5916 24.3851i 17.5886 30.4643i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.5
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 54.4.e.b 30
3.b odd 2 1 162.4.e.b 30
27.e even 9 1 inner 54.4.e.b 30
27.e even 9 1 1458.4.a.i 15
27.f odd 18 1 162.4.e.b 30
27.f odd 18 1 1458.4.a.j 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.4.e.b 30 1.a even 1 1 trivial
54.4.e.b 30 27.e even 9 1 inner
162.4.e.b 30 3.b odd 2 1
162.4.e.b 30 27.f odd 18 1
1458.4.a.i 15 27.e even 9 1
1458.4.a.j 15 27.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{30} + 12 T_{5}^{29} + 288 T_{5}^{28} + 1077 T_{5}^{27} - 7965 T_{5}^{26} - 1743363 T_{5}^{25} + 7544988 T_{5}^{24} - 107176932 T_{5}^{23} + 5239574748 T_{5}^{22} + 56186887842 T_{5}^{21} + \cdots + 54\!\cdots\!96 \) acting on \(S_{4}^{\mathrm{new}}(54, [\chi])\). Copy content Toggle raw display