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Properties

Label 387.3.w.d
Level $387$
Weight $3$
Character orbit 387.w
Analytic conductor $10.545$
Analytic rank $0$
Dimension $84$
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: \(84\)
Relative dimension: \(14\) over \(\Q(\zeta_{14})\)
Twist minimal: no (minimal twist has level 129)
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.

\(\operatorname{Tr}(f)(q) = \) \( 84 q + 36 q^{4} + 56 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q + 36 q^{4} + 56 q^{8} - 8 q^{10} - 14 q^{11} + 16 q^{13} - 12 q^{14} - 4 q^{16} - 18 q^{17} + 280 q^{20} + 94 q^{23} + 118 q^{25} + 112 q^{26} + 140 q^{29} + 64 q^{31} + 280 q^{32} - 40 q^{35} + 108 q^{38} + 112 q^{40} - 36 q^{41} + 206 q^{43} + 128 q^{44} + 266 q^{46} + 82 q^{47} - 480 q^{49} + 56 q^{52} - 598 q^{53} + 630 q^{55} - 190 q^{56} + 20 q^{58} - 244 q^{59} + 336 q^{61} - 168 q^{62} - 852 q^{64} + 336 q^{65} + 594 q^{67} - 16 q^{68} - 1610 q^{70} + 546 q^{71} - 154 q^{73} + 186 q^{74} - 1442 q^{76} + 910 q^{77} + 60 q^{79} - 504 q^{82} + 714 q^{83} + 1140 q^{86} + 756 q^{88} - 532 q^{89} + 728 q^{91} - 2056 q^{92} + 364 q^{94} - 84 q^{95} + 1132 q^{97} - 1722 q^{98}+O(q^{100}) \) 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 −3.06217 − 2.44200i 0 2.52344 + 11.0559i 0.443340 − 0.920605i 0 − 9.19479i 12.4738 − 25.9021i 0 −3.60570 + 1.73641i
82.2 −2.75269 − 2.19519i 0 1.86832 + 8.18566i −1.21768 + 2.52854i 0 − 0.934517i 6.71570 − 13.9453i 0 8.90254 − 4.28724i
82.3 −2.24636 − 1.79142i 0 0.946899 + 4.14864i 3.99211 − 8.28970i 0 4.67304i 0.318301 − 0.660958i 0 −23.8180 + 11.4702i
82.4 −2.08274 − 1.66093i 0 0.689035 + 3.01886i 1.78610 − 3.70886i 0 7.96771i −1.04430 + 2.16851i 0 −9.88014 + 4.75802i
82.5 −1.22772 − 0.979073i 0 −0.341375 − 1.49566i −2.78068 + 5.77415i 0 − 8.42090i −3.77058 + 7.82968i 0 9.06721 − 4.36654i
82.6 −1.21630 − 0.969970i 0 −0.351531 − 1.54016i −1.23478 + 2.56404i 0 0.320925i −3.76633 + 7.82086i 0 3.98890 − 1.92095i
82.7 −0.404713 − 0.322748i 0 −0.830457 − 3.63847i −0.593225 + 1.23184i 0 − 0.704621i −1.73661 + 3.60610i 0 0.637660 − 0.307081i
82.8 0.266717 + 0.212700i 0 −0.864187 − 3.78625i 1.78724 − 3.71124i 0 − 9.77094i 1.16691 − 2.42311i 0 1.26607 − 0.609707i
82.9 0.991669 + 0.790830i 0 −0.532088 − 2.33123i −3.98463 + 8.27418i 0 4.90399i 3.51729 − 7.30373i 0 −10.4949 + 5.05408i
82.10 1.40655 + 1.12169i 0 −0.169876 − 0.744277i −0.938765 + 1.94937i 0 − 3.71929i 3.71822 − 7.72097i 0 −3.50701 + 1.68889i
82.11 1.59641 + 1.27309i 0 0.0376670 + 0.165030i 2.59548 − 5.38956i 0 12.1768i 3.39379 − 7.04727i 0 11.0048 − 5.29965i
82.12 2.02884 + 1.61795i 0 0.608366 + 2.66543i −0.827705 + 1.71875i 0 7.58703i 1.42545 − 2.95998i 0 −4.46013 + 2.14789i
82.13 2.66349 + 2.12407i 0 1.69246 + 7.41516i 3.92384 − 8.14795i 0 − 8.95479i −5.32991 + 11.0677i 0 27.7579 − 13.3675i
82.14 2.68212 + 2.13892i 0 1.72870 + 7.57392i −1.59374 + 3.30943i 0 − 1.84058i −5.60957 + 11.6484i 0 −11.3532 + 5.46741i
118.1 −3.06217 + 2.44200i 0 2.52344 − 11.0559i 0.443340 + 0.920605i 0 9.19479i 12.4738 + 25.9021i 0 −3.60570 − 1.73641i
118.2 −2.75269 + 2.19519i 0 1.86832 − 8.18566i −1.21768 − 2.52854i 0 0.934517i 6.71570 + 13.9453i 0 8.90254 + 4.28724i
118.3 −2.24636 + 1.79142i 0 0.946899 − 4.14864i 3.99211 + 8.28970i 0 − 4.67304i 0.318301 + 0.660958i 0 −23.8180 − 11.4702i
118.4 −2.08274 + 1.66093i 0 0.689035 − 3.01886i 1.78610 + 3.70886i 0 − 7.96771i −1.04430 − 2.16851i 0 −9.88014 − 4.75802i
118.5 −1.22772 + 0.979073i 0 −0.341375 + 1.49566i −2.78068 − 5.77415i 0 8.42090i −3.77058 − 7.82968i 0 9.06721 + 4.36654i
118.6 −1.21630 + 0.969970i 0 −0.351531 + 1.54016i −1.23478 − 2.56404i 0 − 0.320925i −3.76633 − 7.82086i 0 3.98890 + 1.92095i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.14
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.d 84
3.b odd 2 1 129.3.k.a ✓ 84
43.f odd 14 1 inner 387.3.w.d 84
129.j even 14 1 129.3.k.a ✓ 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
129.3.k.a ✓ 84 3.b odd 2 1
129.3.k.a ✓ 84 129.j even 14 1
387.3.w.d 84 1.a even 1 1 trivial
387.3.w.d 84 43.f odd 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{84} - 46 T_{2}^{82} - 56 T_{2}^{81} + 1279 T_{2}^{80} + 2296 T_{2}^{79} - 24828 T_{2}^{78} + \cdots + 21\!\cdots\!01 \) acting on \(S_{3}^{\mathrm{new}}(387, [\chi])\). Copy content Toggle raw display