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Properties

Label 54.6.e.b
Level $54$
Weight $6$
Character orbit 54.e
Analytic conductor $8.661$
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] = [54,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("54.7");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(8.66072626990\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 87 q^{5} + 84 q^{6} + 15 q^{7} - 1536 q^{8} + 96 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 87 q^{5} + 84 q^{6} + 15 q^{7} - 1536 q^{8} + 96 q^{9} - 300 q^{10} + 582 q^{11} + 240 q^{12} - 2508 q^{13} + 60 q^{14} - 2880 q^{15} - 1734 q^{17} + 2760 q^{18} - 3249 q^{19} - 2784 q^{20} - 15960 q^{21} + 2328 q^{22} + 10866 q^{23} + 4761 q^{25} + 12168 q^{26} + 108 q^{27} + 9408 q^{28} + 5487 q^{29} - 15444 q^{30} + 234 q^{31} - 22536 q^{33} + 10452 q^{34} - 13821 q^{35} + 4560 q^{36} - 20535 q^{37} + 34932 q^{38} + 43104 q^{39} + 5568 q^{40} + 12102 q^{41} - 16836 q^{42} + 24831 q^{43} - 5808 q^{44} - 19800 q^{45} + 18012 q^{46} - 33471 q^{47} - 8448 q^{48} + 31821 q^{49} - 10404 q^{50} + 76599 q^{51} - 40128 q^{52} + 155832 q^{53} + 74808 q^{54} + 63198 q^{55} + 960 q^{56} - 104418 q^{57} - 28884 q^{58} - 33144 q^{59} - 30528 q^{60} + 90108 q^{61} - 46128 q^{62} - 375135 q^{63} - 98304 q^{64} - 18141 q^{65} + 66240 q^{66} + 75474 q^{67} - 45168 q^{68} - 253620 q^{69} - 21156 q^{70} - 67812 q^{71} - 12288 q^{72} - 147747 q^{73} - 97644 q^{74} + 109974 q^{75} + 139728 q^{76} + 134733 q^{77} - 22392 q^{78} - 126843 q^{79} + 38400 q^{80} + 324972 q^{81} + 285288 q^{82} + 205290 q^{83} + 43056 q^{84} - 418419 q^{85} + 99324 q^{86} + 442935 q^{87} + 34944 q^{88} - 204756 q^{89} - 429948 q^{90} - 261780 q^{91} - 181824 q^{92} - 1815 q^{93} + 3024 q^{94} - 269043 q^{95} - 30720 q^{96} + 404913 q^{97} - 229800 q^{98} + 850995 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −3.75877 + 1.36808i −14.9994 − 4.24460i 12.2567 − 10.2846i 14.7184 + 83.4722i 62.1864 − 4.56599i −25.4020 − 21.3148i −32.0000 + 55.4256i 206.967 + 127.333i −169.520 − 293.617i
7.2 −3.75877 + 1.36808i −13.2051 + 8.28410i 12.2567 − 10.2846i −4.66574 − 26.4607i 38.3015 − 49.2036i −72.3766 − 60.7312i −32.0000 + 55.4256i 105.747 − 218.784i 53.7379 + 93.0767i
7.3 −3.75877 + 1.36808i −10.4341 − 11.5814i 12.2567 − 10.2846i −18.4394 − 104.575i 55.0637 + 29.2573i 56.2488 + 47.1984i −32.0000 + 55.4256i −25.2594 + 241.684i 212.377 + 367.847i
7.4 −3.75877 + 1.36808i −3.88245 + 15.0972i 12.2567 − 10.2846i 3.35425 + 19.0229i −6.06098 − 62.0586i 196.530 + 164.909i −32.0000 + 55.4256i −212.853 − 117.229i −38.6327 − 66.9138i
7.5 −3.75877 + 1.36808i 5.13118 − 14.7197i 12.2567 − 10.2846i 5.93697 + 33.6702i 0.850882 + 62.3480i 116.375 + 97.6502i −32.0000 + 55.4256i −190.342 − 151.059i −68.3792 − 118.436i
7.6 −3.75877 + 1.36808i 6.96111 + 13.9479i 12.2567 − 10.2846i −7.89130 − 44.7538i −45.2470 − 42.9035i −107.569 − 90.2612i −32.0000 + 55.4256i −146.086 + 194.185i 90.8884 + 157.423i
7.7 −3.75877 + 1.36808i 12.3483 − 9.51423i 12.2567 − 10.2846i −8.71968 − 49.4518i −33.3980 + 52.6552i −67.1772 − 56.3683i −32.0000 + 55.4256i 61.9587 − 234.968i 100.429 + 173.949i
7.8 −3.75877 + 1.36808i 15.4043 + 2.38879i 12.2567 − 10.2846i 12.3324 + 69.9408i −61.1694 + 12.0955i −22.8871 − 19.2046i −32.0000 + 55.4256i 231.587 + 73.5954i −142.039 − 246.020i
13.1 0.694593 + 3.93923i −15.5777 − 0.577959i −15.0351 + 5.47232i 34.6328 + 29.0604i −8.54347 − 61.7658i −181.817 − 66.1758i −32.0000 − 55.4256i 242.332 + 18.0066i −90.4198 + 156.612i
13.2 0.694593 + 3.93923i −15.0584 + 4.03035i −15.0351 + 5.47232i −58.2647 − 48.8899i −26.3359 − 56.5192i 157.702 + 57.3990i −32.0000 − 55.4256i 210.513 − 121.381i 152.118 − 263.477i
13.3 0.694593 + 3.93923i −8.06745 − 13.3385i −15.0351 + 5.47232i 19.6712 + 16.5061i 46.9399 − 41.0444i 48.6364 + 17.7022i −32.0000 − 55.4256i −112.832 + 215.216i −51.3579 + 88.9546i
13.4 0.694593 + 3.93923i −4.93330 + 14.7872i −15.0351 + 5.47232i 71.5535 + 60.0405i −61.6770 − 9.16232i 184.493 + 67.1499i −32.0000 − 55.4256i −194.325 − 145.900i −186.813 + 323.570i
13.5 0.694593 + 3.93923i 2.43088 + 15.3978i −15.0351 + 5.47232i −35.9157 − 30.1369i −58.9668 + 20.2710i −88.1841 − 32.0964i −32.0000 − 55.4256i −231.182 + 74.8603i 93.7693 − 162.413i
13.6 0.694593 + 3.93923i 5.89450 − 14.4310i −15.0351 + 5.47232i −4.16094 − 3.49144i 60.9415 + 13.1961i 89.1101 + 32.4334i −32.0000 − 55.4256i −173.510 − 170.127i 10.8634 − 18.8160i
13.7 0.694593 + 3.93923i 12.4440 − 9.38869i −15.0351 + 5.47232i −42.2567 − 35.4576i 45.6277 + 42.4984i −182.493 − 66.4220i −32.0000 − 55.4256i 66.7049 − 233.665i 110.325 − 191.088i
13.8 0.694593 + 3.93923i 15.3808 + 2.53607i −15.0351 + 5.47232i 75.6428 + 63.4718i 0.693219 + 62.3500i −112.339 − 40.8882i −32.0000 − 55.4256i 230.137 + 78.0134i −197.489 + 342.061i
25.1 0.694593 − 3.93923i −15.5777 + 0.577959i −15.0351 − 5.47232i 34.6328 − 29.0604i −8.54347 + 61.7658i −181.817 + 66.1758i −32.0000 + 55.4256i 242.332 − 18.0066i −90.4198 − 156.612i
25.2 0.694593 − 3.93923i −15.0584 − 4.03035i −15.0351 − 5.47232i −58.2647 + 48.8899i −26.3359 + 56.5192i 157.702 − 57.3990i −32.0000 + 55.4256i 210.513 + 121.381i 152.118 + 263.477i
25.3 0.694593 − 3.93923i −8.06745 + 13.3385i −15.0351 − 5.47232i 19.6712 − 16.5061i 46.9399 + 41.0444i 48.6364 − 17.7022i −32.0000 + 55.4256i −112.832 − 215.216i −51.3579 − 88.9546i
25.4 0.694593 − 3.93923i −4.93330 − 14.7872i −15.0351 − 5.47232i 71.5535 − 60.0405i −61.6770 + 9.16232i 184.493 − 67.1499i −32.0000 + 55.4256i −194.325 + 145.900i −186.813 − 323.570i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.8
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 54.6.e.b ✓ 48
3.b odd 2 1 162.6.e.b 48
27.e even 9 1 inner 54.6.e.b ✓ 48
27.f odd 18 1 162.6.e.b 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.6.e.b ✓ 48 1.a even 1 1 trivial
54.6.e.b ✓ 48 27.e even 9 1 inner
162.6.e.b 48 3.b odd 2 1
162.6.e.b 48 27.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{48} - 87 T_{5}^{47} + 1404 T_{5}^{46} + 421953 T_{5}^{45} - 31639707 T_{5}^{44} + \cdots + 31\!\cdots\!44 \) acting on \(S_{6}^{\mathrm{new}}(54, [\chi])\). Copy content Toggle raw display