↑

Properties

Label 47.16.a.a
Level $47$
Weight $16$
Character orbit 47.a
Self dual yes
Analytic conductor $67.066$
Analytic rank $1$
Dimension $26$
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] = [47,16,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, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("47.1");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 47 \)
Weight: \( k \) \(=\) \( 16 \)
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: \(67.0659473970\)
Analytic rank: \(1\)
Dimension: \(26\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 26 q - 473 q^{2} - 1745 q^{3} + 352357 q^{4} - 204476 q^{5} - 747008 q^{6} - 8691077 q^{7} - 17976447 q^{8} + 89941305 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q - 473 q^{2} - 1745 q^{3} + 352357 q^{4} - 204476 q^{5} - 747008 q^{6} - 8691077 q^{7} - 17976447 q^{8} + 89941305 q^{9} - 51585280 q^{10} + 3001662 q^{11} + 512888508 q^{12} - 508793550 q^{13} - 942073166 q^{14} + 434322614 q^{15} + 7184267409 q^{16} - 4325904793 q^{17} - 9157469975 q^{18} - 4303587332 q^{19} - 32919037912 q^{20} - 45132645408 q^{21} - 50162299142 q^{22} - 13266095576 q^{23} - 110793074064 q^{24} + 65244286550 q^{25} + 10349295428 q^{26} - 77660429552 q^{27} - 239645313896 q^{28} - 66416799998 q^{29} + 507306134084 q^{30} + 103672859796 q^{31} + 368590250641 q^{32} + 570965925392 q^{33} - 100184286950 q^{34} + 581363635284 q^{35} + 2931853687289 q^{36} - 560172981803 q^{37} + 2624915279530 q^{38} - 1035242881236 q^{39} + 241463579696 q^{40} - 228932453550 q^{41} + 2858157268962 q^{42} - 2794438909830 q^{43} + 1064714773938 q^{44} - 3913841787756 q^{45} - 9400875205784 q^{46} + 13172201132038 q^{47} + 15759485913912 q^{48} + 1120374064245 q^{49} - 7441418685467 q^{50} - 42740308387651 q^{51} - 44596665749220 q^{52} - 28744292825331 q^{53} - 95850311037158 q^{54} - 39514788196568 q^{55} - 49201538022872 q^{56} - 40736796192480 q^{57} - 63400902535764 q^{58} - 10165773170481 q^{59} - 167491035794256 q^{60} - 46855749858739 q^{61} + 31884013995940 q^{62} - 37101898831500 q^{63} + 112051931566497 q^{64} + 52029013079708 q^{65} - 354971424918056 q^{66} + 13467823312288 q^{67} - 437586159999366 q^{68} + 59943985913034 q^{69} - 441355039314724 q^{70} - 307074659343603 q^{71} - 11\!\cdots\!83 q^{72}+ \cdots - 34\!\cdots\!42 q^{99}+O(q^{100}) \) 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 −355.528 6788.70 93632.3 −76456.2 −2.41357e6 −1.76519e6 −2.16390e7 3.17376e7 2.71823e7
1.2 −324.510 −4906.76 72538.5 −103828. 1.59229e6 1.65037e6 −1.29059e7 9.72741e6 3.36933e7
1.3 −303.654 655.104 59437.9 −317945. −198925. −1.82342e6 −8.09843e6 −1.39197e7 9.65454e7
1.4 −283.487 4555.66 47596.8 116516. −1.29147e6 1.90427e6 −4.20378e6 6.40514e6 −3.30309e7
1.5 −266.117 −5452.20 38050.5 251447. 1.45093e6 3.15065e6 −1.40577e6 1.53776e7 −6.69144e7
1.6 −249.350 4602.78 29407.6 −40629.9 −1.14771e6 −2.65716e6 837924. 6.83671e6 1.01311e7
1.7 −222.041 −4587.97 16534.4 253588. 1.01872e6 −2.59846e6 3.60454e6 6.70057e6 −5.63071e7
1.8 −216.679 −512.428 14181.6 36328.6 111032. 1.12252e6 4.02727e6 −1.40863e7 −7.87164e6
1.9 −149.354 −3524.31 −10461.4 −183953. 526370. −798325. 6.45648e6 −1.92812e6 2.74741e7
1.10 −115.171 1674.59 −19503.5 168673. −192865. 3.55424e6 6.02019e6 −1.15446e7 −1.94263e7
1.11 −93.1031 3829.62 −24099.8 95229.3 −356549. −3.54992e6 5.29457e6 317065. −8.86614e6
1.12 −47.1111 −3464.85 −30548.5 −202066. 163233. −3.99751e6 2.98291e6 −2.34374e6 9.51955e6
1.13 −27.7131 2918.51 −32000.0 −196167. −80881.1 1.73557e6 1.79492e6 −5.83119e6 5.43642e6
1.14 16.9368 −4901.78 −32481.1 172362. −83020.6 −664194. −1.10511e6 9.67859e6 2.91925e6
1.15 16.9433 6363.08 −32480.9 89644.0 107812. −695454. −1.10553e6 2.61399e7 1.51887e6
1.16 23.5685 −2899.22 −32212.5 −252228. −68330.1 1.71804e6 −1.53149e6 −5.94345e6 −5.94462e6
1.17 78.0158 −1203.14 −26681.5 332823. −93864.3 81775.3 −4.63801e6 −1.29014e7 2.59654e7
1.18 117.665 −4829.16 −18922.8 24578.1 −568225. −1.04772e6 −6.08223e6 8.97185e6 2.89199e6
1.19 145.987 −7375.81 −11455.7 −254205. −1.07678e6 815164. −6.45610e6 4.00537e7 −3.71107e7
1.20 180.170 4814.12 −306.823 234624. 867360. −3.90024e6 −5.95909e6 8.82688e6 4.22722e7
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.26
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 47.16.a.a ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
47.16.a.a ✓ 26 1.a even 1 1 trivial