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Properties

Label 47.13.b.b
Level $47$
Weight $13$
Character orbit 47.b
Analytic conductor $42.958$
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] = [47,13,Mod(46,47)]
 
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([1]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("47.46");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 47 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 47.b (of order \(2\), degree \(1\), 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: \(42.9577094120\)
Analytic rank: \(0\)
Dimension: \(42\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 42 q - 2 q^{2} - 322 q^{3} + 72802 q^{4} + 47356 q^{6} + 87838 q^{7} - 756634 q^{8} + 5810696 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q - 2 q^{2} - 322 q^{3} + 72802 q^{4} + 47356 q^{6} + 87838 q^{7} - 756634 q^{8} + 5810696 q^{9} - 2309916 q^{12} - 26624624 q^{14} + 47673026 q^{16} + 40537438 q^{17} - 214555406 q^{18} + 182105932 q^{21} + 2879827412 q^{24} - 3665531190 q^{25} + 1891781516 q^{27} + 1514668224 q^{28} - 6782468506 q^{32} - 2115270736 q^{34} + 3621955566 q^{36} + 1975024798 q^{37} - 63157096648 q^{42} - 30178286646 q^{47} - 79407419300 q^{48} + 36430571512 q^{49} + 94169808682 q^{50} + 154429710850 q^{51} + 15908736958 q^{53} - 174789209048 q^{54} + 60301133664 q^{55} - 47652279048 q^{56} + 62315133694 q^{59} - 206698011410 q^{61} - 476430210176 q^{63} + 9104082738 q^{64} - 326012422320 q^{65} - 204453583576 q^{68} + 368664572590 q^{71} - 1493491380382 q^{72} + 479677458164 q^{74} + 837826741646 q^{75} + 567058700542 q^{79} - 2115275352586 q^{81} + 2157705851748 q^{83} + 462297170448 q^{84} - 1440684666002 q^{89} + 113761335034 q^{94} - 1693968725808 q^{95} + 14192043918948 q^{96} + 4512642522478 q^{97} + 5512304692434 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1 −124.969 −964.605 11521.1 30622.2i 120545. 63150.7 −927907. 399022. − 3.82681e6i
46.2 −124.969 −964.605 11521.1 − 30622.2i 120545. 63150.7 −927907. 399022. 3.82681e6i
46.3 −103.524 1234.81 6621.17 − 23335.9i −127833. 129287. −261415. 993323. 2.41582e6i
46.4 −103.524 1234.81 6621.17 23335.9i −127833. 129287. −261415. 993323. − 2.41582e6i
46.5 −101.661 −56.6243 6239.03 7709.74i 5756.50 98185.1 −217863. −528235. − 783783.i
46.6 −101.661 −56.6243 6239.03 − 7709.74i 5756.50 98185.1 −217863. −528235. 783783.i
46.7 −96.3839 −902.973 5193.86 − 4600.30i 87032.0 −172950. −105816. 283919. 443395.i
46.8 −96.3839 −902.973 5193.86 4600.30i 87032.0 −172950. −105816. 283919. − 443395.i
46.9 −78.7334 405.115 2102.95 27289.6i −31896.1 −125273. 156919. −367323. − 2.14861e6i
46.10 −78.7334 405.115 2102.95 − 27289.6i −31896.1 −125273. 156919. −367323. 2.14861e6i
46.11 −64.2325 1060.18 29.8138 − 4882.53i −68097.7 −71795.8 261181. 592532. 313617.i
46.12 −64.2325 1060.18 29.8138 4882.53i −68097.7 −71795.8 261181. 592532. − 313617.i
46.13 −54.0429 −779.782 −1175.37 22984.3i 42141.6 34625.3 284880. 76618.2 − 1.24214e6i
46.14 −54.0429 −779.782 −1175.37 − 22984.3i 42141.6 34625.3 284880. 76618.2 1.24214e6i
46.15 −40.5298 563.625 −2453.34 12333.1i −22843.6 205614. 265443. −213767. − 499858.i
46.16 −40.5298 563.625 −2453.34 − 12333.1i −22843.6 205614. 265443. −213767. 499858.i
46.17 −35.6690 −240.545 −2823.72 13164.9i 8579.99 −30291.7 246820. −473579. − 469578.i
46.18 −35.6690 −240.545 −2823.72 − 13164.9i 8579.99 −30291.7 246820. −473579. 469578.i
46.19 −3.46553 −1245.02 −4083.99 − 16111.9i 4314.66 −113146. 28348.0 1.01864e6 55836.3i
46.20 −3.46553 −1245.02 −4083.99 16111.9i 4314.66 −113146. 28348.0 1.01864e6 − 55836.3i
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.42
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
47.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 47.13.b.b ✓ 42
47.b odd 2 1 inner 47.13.b.b ✓ 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
47.13.b.b ✓ 42 1.a even 1 1 trivial
47.13.b.b ✓ 42 47.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{21} + T_{2}^{20} - 61208 T_{2}^{19} + 67628 T_{2}^{18} + 1579394700 T_{2}^{17} + \cdots + 15\!\cdots\!00 \) acting on \(S_{13}^{\mathrm{new}}(47, [\chi])\). Copy content Toggle raw display