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Properties

Label 47.16.a.b
Level $47$
Weight $16$
Character orbit 47.a
Self dual yes
Analytic conductor $67.066$
Analytic rank $0$
Dimension $31$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(31\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 31 q + 39 q^{2} + 7003 q^{3} + 516197 q^{4} + 108024 q^{5} + 372736 q^{6} + 1191439 q^{7} + 7189377 q^{8} + 161685840 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 31 q + 39 q^{2} + 7003 q^{3} + 516197 q^{4} + 108024 q^{5} + 372736 q^{6} + 1191439 q^{7} + 7189377 q^{8} + 161685840 q^{9} + 48414720 q^{10} + 119924688 q^{11} - 80275171 q^{12} + 1150834930 q^{13} - 143011845 q^{14} - 562554966 q^{15} + 4516839585 q^{16} - 1029128825 q^{17} + 4078844490 q^{18} + 12843388206 q^{19} + 34780170952 q^{20} + 37634724968 q^{21} + 68044259144 q^{22} + 53528673200 q^{23} + 76462934219 q^{24} + 311955580969 q^{25} + 94130523296 q^{26} + 58232474896 q^{27} - 16924799444 q^{28} + 32912207086 q^{29} - 536284860814 q^{30} + 70967194440 q^{31} - 501878202930 q^{32} + 468149509416 q^{33} - 1077770989170 q^{34} - 463865299076 q^{35} - 1277881289119 q^{36} + 2407288859917 q^{37} - 1757757677190 q^{38} - 2627724826576 q^{39} - 877235935122 q^{40} - 740376983712 q^{41} - 2011675111195 q^{42} + 4968238096048 q^{43} + 2946565547142 q^{44} + 1578264779592 q^{45} - 130880145254 q^{46} - 15705316734353 q^{47} + 4289668910868 q^{48} + 24644259314864 q^{49} + 40505354735957 q^{50} + 38865190339193 q^{51} + 54455288088120 q^{52} + 34996285019277 q^{53} + 48662735340607 q^{54} + 41868084951228 q^{55} + 86001900741880 q^{56} + 70322590265756 q^{57} + 50738698254424 q^{58} - 8517371539409 q^{59} + 90802085710930 q^{60} + 51785900408893 q^{61} + 57225684015256 q^{62} - 31380583680408 q^{63} + 134021017860153 q^{64} + 18396321926936 q^{65} + 275094420223536 q^{66} + 220735097857354 q^{67} + 331084597877383 q^{68} + 184205274850634 q^{69} + 356334746644826 q^{70} - 3190759645815 q^{71} + 508218376107897 q^{72} + 495087868356732 q^{73} + 10\!\cdots\!06 q^{74}+ \cdots + 60\!\cdots\!88 q^{99}+O(q^{100}) \) 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 −347.076 −1874.82 87693.7 90434.8 650704. −2.21582e6 −1.90634e7 −1.08340e7 −3.13877e7
1.2 −332.319 65.1353 77668.2 233646. −21645.7 1.29938e6 −1.49212e7 −1.43447e7 −7.76451e7
1.3 −315.859 2794.74 66998.8 −176988. −882742. 3.54010e6 −1.08121e7 −6.53836e6 5.59031e7
1.4 −289.463 −6747.13 51020.9 −74362.3 1.95305e6 −2.87314e6 −5.28354e6 3.11749e7 2.15251e7
1.5 −282.454 4071.68 47012.5 278899. −1.15006e6 −1.61170e6 −4.02343e6 2.22967e6 −7.87763e7
1.6 −247.636 −825.877 28555.6 −47441.4 204517. −1.98716e6 1.04315e6 −1.36668e7 1.17482e7
1.7 −204.619 −5125.24 9100.99 −295777. 1.04872e6 2.17347e6 4.84272e6 1.19192e7 6.05217e7
1.8 −196.484 7384.46 5838.09 −80125.8 −1.45093e6 1.79954e6 5.29131e6 4.01813e7 1.57435e7
1.9 −170.815 −1577.45 −3590.07 −60656.0 269454. 2.05410e6 6.21052e6 −1.18605e7 1.03610e7
1.10 −166.769 4467.25 −4956.24 −275415. −744996. 1.29132e6 6.29122e6 5.60738e6 4.59305e7
1.11 −134.892 5744.45 −14572.1 305143. −774882. 708924. 6.38581e6 1.86498e7 −4.11614e7
1.12 −123.497 4333.29 −17516.4 −212106. −535150. −2.34291e6 6.20999e6 4.42852e6 2.61946e7
1.13 −111.846 −454.935 −20258.4 232320. 50882.9 −1.90772e6 5.93081e6 −1.41419e7 −2.59841e7
1.14 −102.011 −7055.84 −22361.7 −24594.6 719773. −479464. 5.62384e6 3.54359e7 2.50892e6
1.15 −60.9091 −5454.73 −29058.1 155531. 332242. 2.12237e6 3.76577e6 1.54051e7 −9.47324e6
1.16 5.75205 478.006 −32734.9 46524.6 2749.51 −836503. −376776. −1.41204e7 267612.
1.17 63.6476 −3563.02 −28717.0 −66305.0 −226777. 4.10132e6 −3.91337e6 −1.65382e6 −4.22015e6
1.18 64.0654 5551.79 −28663.6 47228.5 355678. 2.41499e6 −3.93564e6 1.64735e7 3.02571e6
1.19 120.653 273.905 −18210.9 −255795. 33047.3 −1.04487e6 −6.15074e6 −1.42739e7 −3.08624e7
1.20 125.645 7161.26 −16981.2 −317241. 899779. −3.11557e6 −6.25076e6 3.69347e7 −3.98599e7
See all 31 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.31
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.b ✓ 31
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
47.16.a.b ✓ 31 1.a even 1 1 trivial