↑

Properties

Label 47.20.a.a
Level $47$
Weight $20$
Character orbit 47.a
Self dual yes
Analytic conductor $107.544$
Analytic rank $1$
Dimension $34$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [47,20,Mod(1,47)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(47, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 20, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("47.1");
 
S:= CuspForms(chi, 20);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 47 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 47.a (trivial)

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: yes
Analytic conductor: \(107.543847381\)
Analytic rank: \(1\)
Dimension: \(34\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 34 q - 1481 q^{2} - 74552 q^{3} + 8752837 q^{4} + 28914 q^{5} - 43599872 q^{6} - 203565056 q^{7} - 994215087 q^{8} + 10020983718 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 34 q - 1481 q^{2} - 74552 q^{3} + 8752837 q^{4} + 28914 q^{5} - 43599872 q^{6} - 203565056 q^{7} - 994215087 q^{8} + 10020983718 q^{9} - 10197084160 q^{10} - 7963915630 q^{11} - 12629269764 q^{12} - 159160177690 q^{13} + 404118350082 q^{14} - 59651276056 q^{15} + 1400499411089 q^{16} - 2004886737784 q^{17} - 4449273908039 q^{18} - 1058821844658 q^{19} + 5114247081432 q^{20} + 2403861756792 q^{21} - 3900401557590 q^{22} - 17333732320340 q^{23} + 32877217250016 q^{24} + 85478486158774 q^{25} - 52056718761868 q^{26} - 137248515015920 q^{27} - 361374372214712 q^{28} - 66840103484258 q^{29} - 884984566401484 q^{30} - 481560705870844 q^{31} - 19\!\cdots\!67 q^{32}+ \cdots + 10\!\cdots\!22 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −1422.63 37168.2 1.49960e6 8.12681e6 −5.28767e7 −1.25537e8 −1.38751e9 2.19212e8 −1.15615e10
1.2 −1327.75 14414.1 1.23862e6 −3.85808e6 −1.91383e7 −1.85248e8 −9.48452e8 −9.54495e8 5.12255e9
1.3 −1322.20 −58648.5 1.22393e6 −2.69135e6 7.75453e7 −1.70403e8 −9.25073e8 2.27739e9 3.55852e9
1.4 −1264.48 −5659.55 1.07462e6 330758. 7.15639e6 4.35067e6 −6.95884e8 −1.13023e9 −4.18237e8
1.5 −1238.82 34394.5 1.01039e6 846752. −4.26087e7 8.83998e7 −6.02196e8 2.07219e7 −1.04898e9
1.6 −1022.36 −52765.0 520940. −4.07696e6 5.39451e7 1.19233e8 3.42261e6 1.62189e9 4.16814e9
1.7 −891.031 −62968.5 269647. 4.92517e6 5.61069e7 6.71748e7 2.26892e8 2.80278e9 −4.38848e9
1.8 −849.515 −2993.86 197388. 3.98265e6 2.54333e6 −1.37997e8 2.77706e8 −1.15330e9 −3.38332e9
1.9 −814.888 29976.0 139755. −5.47265e6 −2.44271e7 1.27912e8 3.13352e8 −2.63703e8 4.45960e9
1.10 −785.206 57709.1 92260.1 5.56461e6 −4.53135e7 −7.74136e7 3.39231e8 2.16808e9 −4.36937e9
1.11 −776.505 −19140.9 78672.3 1.24083e6 1.48630e7 1.82231e8 3.46023e8 −7.95886e8 −9.63512e8
1.12 −646.326 −20007.8 −106551. −5.04611e6 1.29315e7 −1.24817e8 4.07727e8 −7.61951e8 3.26143e9
1.13 −480.216 −36068.5 −293681. 5.02224e6 1.73207e7 6.92578e7 3.92802e8 1.38678e8 −2.41176e9
1.14 −410.297 54747.7 −355944. 1.29343e6 −2.24628e7 1.30811e8 3.61157e8 1.83505e9 −5.30689e8
1.15 −368.710 20882.6 −388341. 2.99874e6 −7.69963e6 −1.36719e8 3.36495e8 −7.26177e8 −1.10566e9
1.16 −354.041 60953.1 −398943. −6.70395e6 −2.15799e7 −9.58551e6 3.26862e8 2.55302e9 2.37347e9
1.17 12.9624 −54491.1 −524120. 3.94872e6 −706337. −4.65728e7 −1.35899e7 1.80702e9 5.11850e7
1.18 43.4102 −24477.6 −522404. −8.30265e6 −1.06258e6 −4.43921e7 −4.54371e7 −5.63108e8 −3.60420e8
1.19 56.7339 27252.4 −521069. 6.54449e6 1.54614e6 5.21870e7 −5.93072e7 −4.19567e8 3.71295e8
1.20 69.9663 19831.2 −519393. −5.62385e6 1.38752e6 −4.99804e7 −7.30225e7 −7.68983e8 −3.93480e8
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.34
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(47\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 47.20.a.a ✓ 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
47.20.a.a ✓ 34 1.a even 1 1 trivial