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Properties

Label 387.2.y.d
Level $387$
Weight $2$
Character orbit 387.y
Analytic conductor $3.090$
Analytic rank $0$
Dimension $48$
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] = [387,2,Mod(10,387)]
 
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, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.10");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 387.y (of order \(21\), 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: \(3.09021055822\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(4\) over \(\Q(\zeta_{21})\)
Twist minimal: no (minimal twist has level 129)
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2 q^{2} - 2 q^{4} - q^{5} - 16 q^{7} + 14 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2 q^{2} - 2 q^{4} - q^{5} - 16 q^{7} + 14 q^{8} - 2 q^{10} + 15 q^{11} - 3 q^{13} - 60 q^{14} - 38 q^{16} - 3 q^{17} + 9 q^{19} + 61 q^{20} + 16 q^{22} + 2 q^{23} - 25 q^{25} + 15 q^{26} - 39 q^{28} + 52 q^{29} + 6 q^{31} - 20 q^{32} - 62 q^{34} - 50 q^{35} + 5 q^{37} - 52 q^{38} + 154 q^{40} + 23 q^{41} - 31 q^{43} + 30 q^{44} + 18 q^{46} + 6 q^{47} - 48 q^{49} - 29 q^{50} + 8 q^{52} - 61 q^{53} - 41 q^{55} + 21 q^{56} - 57 q^{58} + 28 q^{59} + 71 q^{61} + 61 q^{62} - 90 q^{64} + 37 q^{65} - 48 q^{67} + 57 q^{68} + 117 q^{70} + 6 q^{71} - 7 q^{73} - 102 q^{74} + 39 q^{76} - 33 q^{77} + 30 q^{79} - 80 q^{80} - 38 q^{82} - 71 q^{83} + 44 q^{85} - 55 q^{86} + 112 q^{88} - 32 q^{89} + 117 q^{91} + 11 q^{92} - 2 q^{94} - 25 q^{95} - 17 q^{97} + 23 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −0.317286 − 1.39012i 0 −0.0298257 + 0.0143633i 0.0923750 − 0.0139233i 0 2.21829 + 3.84220i −1.74860 − 2.19268i 0 −0.0486643 − 0.123995i
10.2 −0.154092 − 0.675123i 0 1.36989 − 0.659705i −1.24535 + 0.187706i 0 −2.01310 − 3.48679i −1.51999 − 1.90600i 0 0.318623 + 0.811839i
10.3 0.303422 + 1.32938i 0 0.126757 − 0.0610427i −2.76477 + 0.416723i 0 0.605872 + 1.04940i 1.81995 + 2.28214i 0 −1.39287 − 3.54899i
10.4 0.330548 + 1.44823i 0 −0.186161 + 0.0896504i 3.62547 − 0.546451i 0 −1.15432 − 1.99934i 1.66098 + 2.08281i 0 1.98978 + 5.06987i
100.1 −2.46573 + 1.18743i 0 3.42286 − 4.29212i 3.58166 − 1.10480i 0 −1.04805 + 1.81527i −2.12526 + 9.31136i 0 −7.51954 + 6.97711i
100.2 −1.30495 + 0.628433i 0 0.0609987 − 0.0764900i −1.14041 + 0.351770i 0 0.969971 − 1.68004i 0.613063 − 2.68600i 0 1.26712 − 1.17572i
100.3 0.180484 − 0.0869164i 0 −1.22196 + 1.53229i −0.694074 + 0.214093i 0 −1.32610 + 2.29688i −0.176514 + 0.773360i 0 −0.106661 + 0.0989667i
100.4 2.26929 − 1.09283i 0 2.70841 − 3.39624i 1.86451 − 0.575125i 0 −1.60016 + 2.77155i 1.31371 − 5.75575i 0 3.60259 − 3.34272i
109.1 −1.54856 − 1.94183i 0 −0.927628 + 4.06420i −0.248151 + 3.31135i 0 −0.737686 − 1.27771i 4.85301 − 2.33709i 0 6.81435 − 4.64595i
109.2 −0.953738 − 1.19595i 0 −0.0756378 + 0.331391i 0.271564 − 3.62377i 0 −2.48045 − 4.29627i −2.28792 + 1.10180i 0 −4.59285 + 3.13135i
109.3 0.155436 + 0.194911i 0 0.431212 − 1.88926i −0.192366 + 2.56695i 0 0.511003 + 0.885083i 0.884487 − 0.425947i 0 −0.530227 + 0.361502i
109.4 1.31656 + 1.65091i 0 −0.547138 + 2.39717i 0.185329 − 2.47304i 0 1.72027 + 2.97960i −0.872888 + 0.420361i 0 4.32676 − 2.94994i
154.1 −1.46561 − 0.705802i 0 0.402884 + 0.505201i 0.729821 − 0.677175i 0 −0.750503 + 1.29991i 0.490052 + 2.14706i 0 −1.54759 + 0.477367i
154.2 −0.311018 − 0.149778i 0 −1.17268 − 1.47050i −2.25911 + 2.09615i 0 −0.178704 + 0.309525i 0.298107 + 1.30609i 0 1.01658 − 0.313574i
154.3 1.33842 + 0.644548i 0 0.128942 + 0.161688i 0.424426 − 0.393809i 0 2.53449 − 4.38987i −0.592762 − 2.59706i 0 0.821888 − 0.253519i
154.4 2.16009 + 1.04025i 0 2.33692 + 2.93040i 0.319581 − 0.296528i 0 −2.16948 + 3.75766i 0.932622 + 4.08608i 0 0.998787 − 0.308085i
181.1 −1.57513 − 1.97516i 0 −0.975151 + 4.27241i −0.458266 − 0.312440i 0 0.543321 − 0.941060i 5.42241 − 2.61129i 0 0.104712 + 1.39728i
181.2 −0.393914 − 0.493952i 0 0.356221 − 1.56071i −1.89349 − 1.29096i 0 0.505220 − 0.875066i −2.04968 + 0.987073i 0 0.108199 + 1.44382i
181.3 0.311351 + 0.390422i 0 0.389552 − 1.70674i 2.85876 + 1.94907i 0 −1.89820 + 3.28778i 1.68747 − 0.812641i 0 0.129119 + 1.72297i
181.4 1.56451 + 1.96183i 0 −0.956057 + 4.18876i −3.11638 − 2.12471i 0 −2.24289 + 3.88480i −5.19184 + 2.50026i 0 −0.707276 − 9.43794i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.2.y.d 48
3.b odd 2 1 129.2.m.b ✓ 48
43.g even 21 1 inner 387.2.y.d 48
129.n even 42 1 5547.2.a.ba 24
129.o odd 42 1 129.2.m.b ✓ 48
129.o odd 42 1 5547.2.a.bb 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
129.2.m.b ✓ 48 3.b odd 2 1
129.2.m.b ✓ 48 129.o odd 42 1
387.2.y.d 48 1.a even 1 1 trivial
387.2.y.d 48 43.g even 21 1 inner
5547.2.a.ba 24 129.n even 42 1
5547.2.a.bb 24 129.o odd 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} + 2 T_{2}^{47} + 11 T_{2}^{46} + 8 T_{2}^{45} + 86 T_{2}^{44} + 102 T_{2}^{43} + 806 T_{2}^{42} + \cdots + 4096 \) acting on \(S_{2}^{\mathrm{new}}(387, [\chi])\). Copy content Toggle raw display