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Properties

Label 387.3.w.b
Level $387$
Weight $3$
Character orbit 387.w
Analytic conductor $10.545$
Analytic rank $0$
Dimension $42$
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(82,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 11]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.82");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 387.w (of order \(14\), 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: \(10.5449862307\)
Analytic rank: \(0\)
Dimension: \(42\)
Relative dimension: \(7\) over \(\Q(\zeta_{14})\)
Twist minimal: no (minimal twist has level 43)
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 42 q + 7 q^{2} + 5 q^{4} + 7 q^{5} - 21 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q + 7 q^{2} + 5 q^{4} + 7 q^{5} - 21 q^{8} - 5 q^{10} + 24 q^{11} - 34 q^{13} - 69 q^{14} - 39 q^{16} - 22 q^{17} - 49 q^{19} - 133 q^{20} + 77 q^{22} - 42 q^{23} + 10 q^{25} - 49 q^{26} + 105 q^{28} - 63 q^{29} - 152 q^{31} - 343 q^{32} + 161 q^{34} - 58 q^{35} + 289 q^{38} - 101 q^{40} - 133 q^{41} - 79 q^{43} - 148 q^{44} - 504 q^{46} - 6 q^{47} - 302 q^{49} - 267 q^{52} + 394 q^{53} - 637 q^{55} - 355 q^{56} + 165 q^{58} + 46 q^{59} - 175 q^{61} + 91 q^{62} + 725 q^{64} - 161 q^{65} - 756 q^{67} + 586 q^{68} + 1526 q^{70} - 266 q^{71} - 252 q^{73} - 204 q^{74} + 994 q^{76} - 791 q^{77} - 178 q^{79} + 245 q^{82} - 238 q^{83} - 365 q^{86} - 119 q^{88} - 252 q^{89} - 224 q^{91} + 764 q^{92} + 133 q^{94} - 11 q^{95} - 491 q^{97} + 553 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1 −2.31044 − 1.84251i 0 1.05318 + 4.61430i −3.37568 + 7.00967i 0 7.33610i 0.939799 − 1.95151i 0 20.7147 − 9.97566i
82.2 −1.96310 − 1.56552i 0 0.512821 + 2.24681i 2.07465 − 4.30805i 0 − 12.3653i −1.84704 + 3.83541i 0 −10.8171 + 5.20922i
82.3 −0.405381 − 0.323281i 0 −0.830260 − 3.63761i 0.149470 − 0.310378i 0 9.37466i −1.73928 + 3.61164i 0 −0.160932 + 0.0775006i
82.4 0.290487 + 0.231656i 0 −0.859365 − 3.76513i 3.26645 − 6.78286i 0 1.66643i 1.26741 − 2.63181i 0 2.52015 − 1.21364i
82.5 1.12245 + 0.895127i 0 −0.431434 − 1.89023i −2.85407 + 5.92653i 0 − 7.32927i 3.69939 − 7.68187i 0 −8.50856 + 4.09751i
82.6 2.20426 + 1.75784i 0 0.878675 + 3.84973i 1.34756 − 2.79824i 0 − 7.15415i 0.0627165 − 0.130232i 0 7.88921 − 3.79924i
82.7 2.96269 + 2.36266i 0 2.30525 + 10.0999i −0.910321 + 1.89030i 0 8.82371i −10.4564 + 21.7129i 0 −7.16314 + 3.44959i
118.1 −2.31044 + 1.84251i 0 1.05318 − 4.61430i −3.37568 − 7.00967i 0 − 7.33610i 0.939799 + 1.95151i 0 20.7147 + 9.97566i
118.2 −1.96310 + 1.56552i 0 0.512821 − 2.24681i 2.07465 + 4.30805i 0 12.3653i −1.84704 − 3.83541i 0 −10.8171 − 5.20922i
118.3 −0.405381 + 0.323281i 0 −0.830260 + 3.63761i 0.149470 + 0.310378i 0 − 9.37466i −1.73928 − 3.61164i 0 −0.160932 − 0.0775006i
118.4 0.290487 − 0.231656i 0 −0.859365 + 3.76513i 3.26645 + 6.78286i 0 − 1.66643i 1.26741 + 2.63181i 0 2.52015 + 1.21364i
118.5 1.12245 − 0.895127i 0 −0.431434 + 1.89023i −2.85407 − 5.92653i 0 7.32927i 3.69939 + 7.68187i 0 −8.50856 − 4.09751i
118.6 2.20426 − 1.75784i 0 0.878675 − 3.84973i 1.34756 + 2.79824i 0 7.15415i 0.0627165 + 0.130232i 0 7.88921 + 3.79924i
118.7 2.96269 − 2.36266i 0 2.30525 − 10.0999i −0.910321 − 1.89030i 0 − 8.82371i −10.4564 − 21.7129i 0 −7.16314 − 3.44959i
199.1 −3.62198 + 0.826694i 0 8.83146 − 4.25301i 2.82431 − 2.25231i 0 9.33217i −16.8530 + 13.4399i 0 −8.36764 + 10.4927i
199.2 −2.11328 + 0.482341i 0 0.629407 − 0.303106i 3.78878 − 3.02145i 0 − 10.9446i 5.59495 − 4.46183i 0 −6.54936 + 8.21264i
199.3 −1.71853 + 0.392244i 0 −0.804371 + 0.387365i −5.40310 + 4.30882i 0 6.65286i 6.74303 − 5.37738i 0 7.59529 − 9.52420i
199.4 0.811544 − 0.185230i 0 −2.97958 + 1.43489i 3.71827 − 2.96522i 0 0.589128i −4.75551 + 3.79239i 0 2.46829 − 3.09514i
199.5 0.844666 − 0.192789i 0 −2.92758 + 1.40985i −0.831553 + 0.663142i 0 0.302383i −4.91050 + 3.91600i 0 −0.574538 + 0.720448i
199.6 2.63550 − 0.601536i 0 2.98015 − 1.43516i −7.00305 + 5.58475i 0 − 3.35098i −1.46315 + 1.16682i 0 −15.0971 + 18.9312i
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.7
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.f odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.3.w.b 42
3.b odd 2 1 43.3.f.a ✓ 42
43.f odd 14 1 inner 387.3.w.b 42
129.j even 14 1 43.3.f.a ✓ 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.3.f.a ✓ 42 3.b odd 2 1
43.3.f.a ✓ 42 129.j even 14 1
387.3.w.b 42 1.a even 1 1 trivial
387.3.w.b 42 43.f odd 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{42} - 7 T_{2}^{41} + 8 T_{2}^{40} + 84 T_{2}^{39} - 246 T_{2}^{38} - 476 T_{2}^{37} + \cdots + 103657489047 \) acting on \(S_{3}^{\mathrm{new}}(387, [\chi])\). Copy content Toggle raw display