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Properties

Label 648.2.v.b
Level $648$
Weight $2$
Character orbit 648.v
Analytic conductor $5.174$
Analytic rank $0$
Dimension $192$
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] = [648,2,Mod(35,648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(648, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 9, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("648.35");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 648.v (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: \(5.17430605098\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(32\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 216)
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.

\(\operatorname{Tr}(f)(q) = \) \( 192 q + 6 q^{2} - 6 q^{4} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q + 6 q^{2} - 6 q^{4} + 9 q^{8} - 3 q^{10} + 30 q^{11} - 9 q^{14} - 6 q^{16} + 18 q^{17} - 6 q^{19} + 27 q^{20} - 18 q^{22} - 12 q^{25} - 12 q^{28} + 36 q^{32} + 12 q^{34} + 18 q^{35} + 102 q^{38} + 9 q^{40} - 18 q^{41} - 42 q^{43} + 81 q^{44} - 3 q^{46} - 12 q^{49} - 57 q^{50} + 21 q^{52} + 69 q^{56} - 33 q^{58} + 84 q^{59} - 90 q^{62} - 51 q^{64} + 12 q^{65} + 30 q^{67} - 63 q^{68} - 33 q^{70} - 6 q^{73} - 51 q^{74} + 30 q^{76} - 12 q^{82} + 72 q^{83} - 42 q^{86} - 78 q^{88} - 144 q^{89} - 6 q^{91} + 3 q^{92} - 33 q^{94} - 42 q^{97} - 162 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1 −1.41380 − 0.0343398i 0 1.99764 + 0.0970990i 2.25586 + 0.821067i 0 −1.95474 − 0.344674i −2.82092 − 0.205877i 0 −3.16114 − 1.23829i
35.2 −1.41205 − 0.0781964i 0 1.98777 + 0.220834i 0.479395 + 0.174486i 0 −1.63176 − 0.287723i −2.78956 − 0.467266i 0 −0.663286 − 0.283869i
35.3 −1.39125 − 0.253795i 0 1.87118 + 0.706186i 0.715441 + 0.260399i 0 3.32699 + 0.586638i −2.42406 − 1.45738i 0 −0.929272 − 0.543857i
35.4 −1.35595 + 0.401734i 0 1.67722 − 1.08946i −1.93709 − 0.705042i 0 −0.744451 − 0.131267i −1.83656 + 2.15106i 0 2.90984 + 0.177811i
35.5 −1.19259 − 0.760091i 0 0.844523 + 1.81295i −2.35728 − 0.857979i 0 −2.09025 − 0.368567i 0.370840 − 2.80401i 0 2.15912 + 2.81496i
35.6 −1.16920 + 0.795599i 0 0.734044 − 1.86042i 3.21529 + 1.17027i 0 −2.85909 − 0.504135i 0.621910 + 2.75921i 0 −4.69037 + 1.18981i
35.7 −1.12537 − 0.856465i 0 0.532936 + 1.92769i 0.588722 + 0.214277i 0 4.57427 + 0.806567i 1.05124 − 2.62581i 0 −0.479012 − 0.745362i
35.8 −0.955487 + 1.04261i 0 −0.174090 − 1.99241i 3.00936 + 1.09532i 0 4.58614 + 0.808660i 2.24366 + 1.72221i 0 −4.01740 + 2.09104i
35.9 −0.860856 + 1.12202i 0 −0.517853 − 1.93179i −3.00936 − 1.09532i 0 −4.58614 − 0.808660i 2.61331 + 1.08196i 0 3.81960 − 2.43365i
35.10 −0.844953 − 1.13404i 0 −0.572109 + 1.91643i 4.00434 + 1.45746i 0 1.46723 + 0.258712i 2.65672 − 0.970494i 0 −1.73065 − 5.77258i
35.11 −0.821807 − 1.15093i 0 −0.649266 + 1.89168i −0.426624 − 0.155279i 0 −1.28150 − 0.225962i 2.71076 − 0.807339i 0 0.171889 + 0.618623i
35.12 −0.580483 + 1.28959i 0 −1.32608 − 1.49717i −3.21529 − 1.17027i 0 2.85909 + 0.504135i 2.70050 − 0.841014i 0 3.37559 − 3.46708i
35.13 −0.505867 − 1.32064i 0 −1.48820 + 1.33614i −3.21396 − 1.16979i 0 −0.199218 − 0.0351275i 2.51739 + 1.28947i 0 0.0809654 + 4.83625i
35.14 −0.160171 + 1.40511i 0 −1.94869 − 0.450118i 1.93709 + 0.705042i 0 0.744451 + 0.131267i 0.944592 − 2.66604i 0 −1.30093 + 2.60890i
35.15 −0.143175 − 1.40695i 0 −1.95900 + 0.402879i −2.25103 − 0.819307i 0 4.05746 + 0.715440i 0.847309 + 2.69853i 0 −0.830431 + 3.28438i
35.16 −0.133842 − 1.40787i 0 −1.96417 + 0.376862i 2.47648 + 0.901367i 0 −2.55949 − 0.451307i 0.793459 + 2.71485i 0 0.937547 − 3.60720i
35.17 0.279321 + 1.38635i 0 −1.84396 + 0.774477i −2.25586 − 0.821067i 0 1.95474 + 0.344674i −1.58876 − 2.34005i 0 0.508180 − 3.35677i
35.18 0.315008 − 1.37868i 0 −1.80154 − 0.868594i 0.437371 + 0.159190i 0 −3.46177 − 0.610403i −1.76502 + 2.21014i 0 0.357248 − 0.552850i
35.19 0.322208 + 1.37702i 0 −1.79236 + 0.887374i −0.479395 − 0.174486i 0 1.63176 + 0.287723i −1.79945 − 2.18220i 0 0.0858049 − 0.716357i
35.20 0.491528 + 1.32605i 0 −1.51680 + 1.30358i −0.715441 − 0.260399i 0 −3.32699 − 0.586638i −2.47416 − 1.37060i 0 −0.00635759 − 1.07670i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
27.f odd 18 1 inner
216.v even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.2.v.b 192
3.b odd 2 1 216.2.v.b ✓ 192
8.d odd 2 1 inner 648.2.v.b 192
12.b even 2 1 864.2.bh.b 192
24.f even 2 1 216.2.v.b ✓ 192
24.h odd 2 1 864.2.bh.b 192
27.e even 9 1 216.2.v.b ✓ 192
27.f odd 18 1 inner 648.2.v.b 192
108.j odd 18 1 864.2.bh.b 192
216.r odd 18 1 216.2.v.b ✓ 192
216.t even 18 1 864.2.bh.b 192
216.v even 18 1 inner 648.2.v.b 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.v.b ✓ 192 3.b odd 2 1
216.2.v.b ✓ 192 24.f even 2 1
216.2.v.b ✓ 192 27.e even 9 1
216.2.v.b ✓ 192 216.r odd 18 1
648.2.v.b 192 1.a even 1 1 trivial
648.2.v.b 192 8.d odd 2 1 inner
648.2.v.b 192 27.f odd 18 1 inner
648.2.v.b 192 216.v even 18 1 inner
864.2.bh.b 192 12.b even 2 1
864.2.bh.b 192 24.h odd 2 1
864.2.bh.b 192 108.j odd 18 1
864.2.bh.b 192 216.t even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{192} + 6 T_{5}^{190} - 9 T_{5}^{188} + 13251 T_{5}^{186} + 71217 T_{5}^{184} + \cdots + 23\!\cdots\!36 \) acting on \(S_{2}^{\mathrm{new}}(648, [\chi])\). Copy content Toggle raw display