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Properties

Label 387.3.bn.b
Level $387$
Weight $3$
Character orbit 387.bn
Analytic conductor $10.545$
Analytic rank $0$
Dimension $72$
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] = [387,3,Mod(19,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 19]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 387.bn (of order \(42\), degree \(12\), 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: \(10.5449862307\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(6\) over \(\Q(\zeta_{42})\)
Twist minimal: no (minimal twist has level 43)
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 14 q^{2} + 12 q^{4} + 11 q^{5} - 30 q^{7} + 42 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 14 q^{2} + 12 q^{4} + 11 q^{5} - 30 q^{7} + 42 q^{8} - 13 q^{10} + 42 q^{11} - 24 q^{13} + 108 q^{14} - 40 q^{16} + 7 q^{17} - 38 q^{19} + 55 q^{20} - 98 q^{22} - 30 q^{23} + 49 q^{25} + 79 q^{26} + 66 q^{28} - 27 q^{29} + 330 q^{31} - 56 q^{32} + 109 q^{34} + 31 q^{35} + 69 q^{37} - 262 q^{38} + 239 q^{40} + 94 q^{41} - 19 q^{43} + 64 q^{44} - 9 q^{46} + 66 q^{47} - 6 q^{49} + 495 q^{50} - 452 q^{52} - 16 q^{53} + 328 q^{55} + 1015 q^{56} - 420 q^{58} + 245 q^{59} - 50 q^{61} + 191 q^{62} - 306 q^{64} + 182 q^{65} + 599 q^{67} - 757 q^{68} - 287 q^{70} - 367 q^{71} + 486 q^{73} - 1656 q^{74} + 746 q^{76} - 79 q^{77} + 261 q^{79} - 138 q^{80} + 364 q^{82} + 220 q^{83} + 284 q^{86} - 490 q^{88} + 564 q^{89} - 145 q^{91} + 406 q^{92} - 1666 q^{94} + 353 q^{95} - 99 q^{97} + 500 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −2.58929 − 2.06489i 0 1.55056 + 6.79347i 5.29180 + 7.76164i 0 −3.46414 + 2.00002i 4.26510 − 8.85658i 0 2.32493 − 31.0241i
19.2 −1.91832 − 1.52981i 0 0.449547 + 1.96959i −4.13909 − 6.07092i 0 −8.82356 + 5.09429i −2.10762 + 4.37651i 0 −1.34726 + 17.9780i
19.3 −1.00539 − 0.801775i 0 −0.522109 − 2.28751i −0.680335 − 0.997867i 0 −2.49921 + 1.44292i −3.54095 + 7.35286i 0 −0.116061 + 1.54873i
19.4 0.420451 + 0.335299i 0 −0.825730 − 3.61776i 0.334173 + 0.490142i 0 3.51870 − 2.03152i 1.79918 − 3.73604i 0 −0.0238404 + 0.318129i
19.5 2.24704 + 1.79196i 0 0.948008 + 4.15349i −3.03177 − 4.44679i 0 5.12110 − 2.95667i −0.324610 + 0.674059i 0 1.15594 − 15.4249i
19.6 2.50874 + 2.00065i 0 1.40107 + 6.13849i 1.97904 + 2.90271i 0 −9.27426 + 5.35450i −3.19708 + 6.63880i 0 −0.842433 + 11.2415i
28.1 −2.24052 − 1.78675i 0 0.937348 + 4.10679i 0.381243 − 0.0285702i 0 −0.491781 − 0.283930i 0.264104 − 0.548417i 0 −0.905230 − 0.617175i
28.2 −0.776193 − 0.618993i 0 −0.670761 − 2.93880i 8.46795 − 0.634585i 0 −4.17794 − 2.41213i −3.02147 + 6.27415i 0 −6.96556 − 4.74904i
28.3 −0.191549 − 0.152755i 0 −0.876727 − 3.84119i −3.10855 + 0.232953i 0 −6.38587 − 3.68688i −0.844033 + 1.75265i 0 0.631024 + 0.430225i
28.4 0.507473 + 0.404696i 0 −0.796334 − 3.48897i −4.78317 + 0.358449i 0 4.33114 + 2.50058i 2.13436 − 4.43204i 0 −2.57239 − 1.75383i
28.5 1.90617 + 1.52012i 0 0.432633 + 1.89549i 6.44074 − 0.482667i 0 −0.913021 − 0.527133i 2.17468 − 4.51576i 0 13.0108 + 8.87064i
28.6 2.89798 + 2.31106i 0 2.16720 + 9.49513i −3.85009 + 0.288525i 0 −4.47290 − 2.58243i −9.23030 + 19.1669i 0 −11.8243 − 8.06167i
46.1 −1.31976 + 2.74051i 0 −3.27466 − 4.10629i 1.51671 − 4.91705i 0 −4.74658 + 2.74044i 3.71320 − 0.847514i 0 11.4735 + 10.6459i
46.2 −1.25989 + 2.61618i 0 −2.76312 − 3.46484i −1.10235 + 3.57375i 0 6.79133 − 3.92097i 1.22212 − 0.278940i 0 −7.96072 − 7.38647i
46.3 −0.304460 + 0.632217i 0 2.18696 + 2.74236i −1.20897 + 3.91938i 0 −0.834967 + 0.482068i −5.13606 + 1.17227i 0 −2.10982 − 1.95763i
46.4 0.402412 − 0.835617i 0 1.95764 + 2.45480i 2.67009 − 8.65623i 0 8.78777 − 5.07362i 6.45589 − 1.47352i 0 −6.15881 − 5.71454i
46.5 1.03664 − 2.15260i 0 −1.06511 − 1.33560i −0.257183 + 0.833768i 0 −1.23199 + 0.711287i 5.33806 − 1.21838i 0 1.52816 + 1.41793i
46.6 1.23133 − 2.55689i 0 −2.52755 − 3.16944i −1.18875 + 3.85384i 0 −8.83061 + 5.09836i −0.149040 + 0.0340174i 0 8.39010 + 7.78488i
55.1 −1.47143 − 3.05545i 0 −4.67672 + 5.86443i 0.262700 − 0.283123i 0 4.87516 − 2.81467i 11.5749 + 2.64189i 0 −1.25161 − 0.386071i
55.2 −0.704959 − 1.46386i 0 0.848033 − 1.06340i −0.650793 + 0.701388i 0 −11.1429 + 6.43335i −8.49061 − 1.93793i 0 1.48552 + 0.458221i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.h odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.3.bn.b 72
3.b odd 2 1 43.3.h.a ✓ 72
43.h odd 42 1 inner 387.3.bn.b 72
129.n even 42 1 43.3.h.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.3.h.a ✓ 72 3.b odd 2 1
43.3.h.a ✓ 72 129.n even 42 1
387.3.bn.b 72 1.a even 1 1 trivial
387.3.bn.b 72 43.h odd 42 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} - 14 T_{2}^{71} + 68 T_{2}^{70} - 70 T_{2}^{69} - 248 T_{2}^{68} - 1624 T_{2}^{67} + \cdots + 3385979050609 \) acting on \(S_{3}^{\mathrm{new}}(387, [\chi])\). Copy content Toggle raw display