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Properties

Label 216.4.q.a
Level $216$
Weight $4$
Character orbit 216.q
Analytic conductor $12.744$
Analytic rank $0$
Dimension $78$
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] = [216,4,Mod(25,216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(216, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 10]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("216.25");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 216.q (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: \(12.7444125612\)
Analytic rank: \(0\)
Dimension: \(78\)
Relative dimension: \(13\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 78 q + 33 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 78 q + 33 q^{7} + 6 q^{9} + 51 q^{11} - 60 q^{13} - 183 q^{15} - 102 q^{17} + 171 q^{19} + 342 q^{21} + 456 q^{23} + 480 q^{25} + 189 q^{27} - 63 q^{29} + 621 q^{31} + 732 q^{33} + 630 q^{35} + 555 q^{37} + 597 q^{39} - 987 q^{41} + 552 q^{43} - 1311 q^{45} - 216 q^{47} - 1131 q^{49} - 288 q^{51} + 954 q^{53} - 54 q^{55} + 702 q^{57} + 1752 q^{59} - 654 q^{61} - 144 q^{63} + 126 q^{65} + 2277 q^{67} - 165 q^{69} + 684 q^{73} - 1167 q^{75} + 192 q^{77} - 2745 q^{79} - 3330 q^{81} - 3567 q^{83} - 3981 q^{85} - 108 q^{87} - 1314 q^{89} + 147 q^{91} - 2406 q^{93} + 1167 q^{95} + 2046 q^{97} - 5640 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 0 −5.14635 − 0.717694i 0 7.62051 − 6.39436i 0 −26.4733 + 9.63549i 0 25.9698 + 7.38701i 0
25.2 0 −4.95983 + 1.54921i 0 −3.10944 + 2.60913i 0 15.3425 − 5.58422i 0 22.1999 − 15.3676i 0
25.3 0 −3.82100 − 3.52135i 0 −14.5761 + 12.2308i 0 5.97810 − 2.17585i 0 2.20013 + 26.9102i 0
25.4 0 −3.40900 − 3.92157i 0 10.9702 − 9.20510i 0 24.6519 − 8.97257i 0 −3.75745 + 26.7373i 0
25.5 0 −2.20849 + 4.70346i 0 −5.53324 + 4.64294i 0 −14.1673 + 5.15648i 0 −17.2451 − 20.7751i 0
25.6 0 −1.36036 + 5.01492i 0 16.4848 − 13.8323i 0 15.4759 − 5.63277i 0 −23.2988 − 13.6442i 0
25.7 0 −0.0951628 − 5.19528i 0 3.95912 − 3.32210i 0 −7.57963 + 2.75876i 0 −26.9819 + 0.988795i 0
25.8 0 2.57391 + 4.51386i 0 −4.64451 + 3.89720i 0 29.6546 − 10.7934i 0 −13.7500 + 23.2366i 0
25.9 0 2.81809 − 4.36559i 0 −9.81432 + 8.23519i 0 −8.03696 + 2.92522i 0 −11.1167 − 24.6053i 0
25.10 0 3.15436 + 4.12917i 0 5.75544 − 4.82938i 0 −13.8192 + 5.02977i 0 −7.10004 + 26.0497i 0
25.11 0 4.89500 + 1.74325i 0 −12.6811 + 10.6407i 0 −16.3558 + 5.95303i 0 20.9221 + 17.0665i 0
25.12 0 4.90684 − 1.70966i 0 −1.21843 + 1.02239i 0 30.6743 − 11.1645i 0 21.1541 − 16.7781i 0
25.13 0 4.95013 − 1.57994i 0 10.6173 − 8.90899i 0 −6.35287 + 2.31226i 0 22.0076 − 15.6418i 0
49.1 0 −5.04226 − 1.25523i 0 −0.475470 + 0.173057i 0 −3.67805 − 20.8592i 0 23.8488 + 12.6584i 0
49.2 0 −4.98381 + 1.47026i 0 −5.94819 + 2.16496i 0 5.30186 + 30.0684i 0 22.6767 − 14.6550i 0
49.3 0 −3.63180 − 3.71619i 0 17.4854 − 6.36417i 0 5.73955 + 32.5506i 0 −0.620094 + 26.9929i 0
49.4 0 −3.36992 + 3.95520i 0 14.8671 − 5.41117i 0 −1.06908 − 6.06305i 0 −4.28728 − 26.6574i 0
49.5 0 −3.06880 + 4.19315i 0 −17.7135 + 6.44718i 0 −1.46194 − 8.29106i 0 −8.16495 − 25.7358i 0
49.6 0 −2.47330 − 4.56977i 0 −10.4202 + 3.79265i 0 1.34972 + 7.65462i 0 −14.7656 + 22.6048i 0
49.7 0 0.398201 − 5.18087i 0 15.0362 − 5.47273i 0 −4.87347 − 27.6388i 0 −26.6829 − 4.12606i 0
See all 78 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.13
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 216.4.q.a ✓ 78
27.e even 9 1 inner 216.4.q.a ✓ 78
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.4.q.a ✓ 78 1.a even 1 1 trivial
216.4.q.a ✓ 78 27.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{78} - 240 T_{5}^{76} + 1403 T_{5}^{75} + 122121 T_{5}^{74} + 369327 T_{5}^{73} + \cdots + 11\!\cdots\!36 \) acting on \(S_{4}^{\mathrm{new}}(216, [\chi])\). Copy content Toggle raw display