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Properties

Label 432.3.bc.a
Level $432$
Weight $3$
Character orbit 432.bc
Analytic conductor $11.771$
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] = [432,3,Mod(65,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 13]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.65");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 432.bc (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: \(11.7711474204\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(5\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 27)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 6 q^{3} - 15 q^{5} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 6 q^{3} - 15 q^{5} + 6 q^{7} + 6 q^{11} - 6 q^{13} + 9 q^{15} - 9 q^{17} + 3 q^{19} + 132 q^{21} - 120 q^{23} - 15 q^{25} + 90 q^{27} - 168 q^{29} - 39 q^{31} - 207 q^{33} + 252 q^{35} - 3 q^{37} - 15 q^{39} + 228 q^{41} + 96 q^{43} + 477 q^{45} - 399 q^{47} - 78 q^{49} - 36 q^{51} + 12 q^{55} - 192 q^{57} + 474 q^{59} + 138 q^{61} + 585 q^{63} - 411 q^{65} - 354 q^{67} + 99 q^{69} - 315 q^{71} - 66 q^{73} - 255 q^{75} + 201 q^{77} - 30 q^{79} + 36 q^{81} + 33 q^{83} - 261 q^{85} + 279 q^{87} + 72 q^{89} - 96 q^{91} + 591 q^{93} - 681 q^{95} - 582 q^{97} - 513 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1 0 −2.80515 + 1.06355i 0 −1.47839 + 4.06185i 0 −1.54363 + 8.75434i 0 6.73771 − 5.96685i 0
65.2 0 0.428198 − 2.96928i 0 2.62195 − 7.20376i 0 −0.231638 + 1.31369i 0 −8.63329 − 2.54288i 0
65.3 0 0.987122 + 2.83295i 0 0.149473 − 0.410673i 0 1.05651 − 5.99176i 0 −7.05118 + 5.59293i 0
65.4 0 2.03568 − 2.20363i 0 −2.35247 + 6.46335i 0 −1.10811 + 6.28443i 0 −0.712002 − 8.97179i 0
65.5 0 2.99958 − 0.0504108i 0 −1.19547 + 3.28452i 0 1.88718 − 10.7027i 0 8.99492 − 0.302422i 0
113.1 0 −2.80515 − 1.06355i 0 −1.47839 − 4.06185i 0 −1.54363 − 8.75434i 0 6.73771 + 5.96685i 0
113.2 0 0.428198 + 2.96928i 0 2.62195 + 7.20376i 0 −0.231638 − 1.31369i 0 −8.63329 + 2.54288i 0
113.3 0 0.987122 − 2.83295i 0 0.149473 + 0.410673i 0 1.05651 + 5.99176i 0 −7.05118 − 5.59293i 0
113.4 0 2.03568 + 2.20363i 0 −2.35247 − 6.46335i 0 −1.10811 − 6.28443i 0 −0.712002 + 8.97179i 0
113.5 0 2.99958 + 0.0504108i 0 −1.19547 − 3.28452i 0 1.88718 + 10.7027i 0 8.99492 + 0.302422i 0
209.1 0 −2.97089 + 0.416922i 0 −0.526552 + 0.0928453i 0 −2.13346 − 1.79019i 0 8.65235 − 2.47726i 0
209.2 0 −1.56284 + 2.56077i 0 −8.90104 + 1.56949i 0 1.32607 + 1.11270i 0 −4.11509 − 8.00413i 0
209.3 0 −0.859438 − 2.87426i 0 −2.41673 + 0.426135i 0 3.31426 + 2.78099i 0 −7.52273 + 4.94050i 0
209.4 0 2.15873 + 2.08324i 0 −0.0496479 + 0.00875427i 0 −7.55168 − 6.33661i 0 0.320214 + 8.99430i 0
209.5 0 2.94744 − 0.559079i 0 6.15614 − 1.08549i 0 6.21846 + 5.21791i 0 8.37486 − 3.29571i 0
257.1 0 −2.32380 − 1.89736i 0 3.98394 − 4.74788i 0 7.49258 + 2.72708i 0 1.80008 + 8.81815i 0
257.2 0 −1.47633 + 2.61160i 0 3.16671 − 3.77394i 0 3.18911 + 1.16074i 0 −4.64090 − 7.71116i 0
257.3 0 −1.10490 + 2.78912i 0 −3.46013 + 4.12362i 0 −9.89907 − 3.60297i 0 −6.55839 − 6.16340i 0
257.4 0 1.62484 − 2.52189i 0 −3.71692 + 4.42965i 0 −4.57693 − 1.66587i 0 −3.71981 − 8.19530i 0
257.5 0 2.92175 − 0.680712i 0 0.519123 − 0.618667i 0 5.56035 + 2.02380i 0 8.07326 − 3.97774i 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.3.bc.a 30
4.b odd 2 1 27.3.f.a ✓ 30
12.b even 2 1 81.3.f.a 30
27.f odd 18 1 inner 432.3.bc.a 30
36.f odd 6 1 243.3.f.c 30
36.f odd 6 1 243.3.f.d 30
36.h even 6 1 243.3.f.a 30
36.h even 6 1 243.3.f.b 30
108.j odd 18 1 81.3.f.a 30
108.j odd 18 1 243.3.f.a 30
108.j odd 18 1 243.3.f.b 30
108.j odd 18 1 729.3.b.a 30
108.l even 18 1 27.3.f.a ✓ 30
108.l even 18 1 243.3.f.c 30
108.l even 18 1 243.3.f.d 30
108.l even 18 1 729.3.b.a 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.3.f.a ✓ 30 4.b odd 2 1
27.3.f.a ✓ 30 108.l even 18 1
81.3.f.a 30 12.b even 2 1
81.3.f.a 30 108.j odd 18 1
243.3.f.a 30 36.h even 6 1
243.3.f.a 30 108.j odd 18 1
243.3.f.b 30 36.h even 6 1
243.3.f.b 30 108.j odd 18 1
243.3.f.c 30 36.f odd 6 1
243.3.f.c 30 108.l even 18 1
243.3.f.d 30 36.f odd 6 1
243.3.f.d 30 108.l even 18 1
432.3.bc.a 30 1.a even 1 1 trivial
432.3.bc.a 30 27.f odd 18 1 inner
729.3.b.a 30 108.j odd 18 1
729.3.b.a 30 108.l even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{30} + 15 T_{5}^{29} + 120 T_{5}^{28} + 975 T_{5}^{27} + 5679 T_{5}^{26} + 29592 T_{5}^{25} + \cdots + 997568747712 \) acting on \(S_{3}^{\mathrm{new}}(432, [\chi])\). Copy content Toggle raw display