Properties

Label 1339.4.a.d
Level $1339$
Weight $4$
Character orbit 1339.a
Self dual yes
Analytic conductor $79.004$
Analytic rank $0$
Dimension $81$
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] = [1339,4,Mod(1,1339)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1339, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1339.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1339 = 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1339.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: \(79.0035574977\)
Analytic rank: \(0\)
Dimension: \(81\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 81 q + 25 q^{2} + 17 q^{3} + 353 q^{4} + 141 q^{5} + 56 q^{6} + 63 q^{7} + 264 q^{8} + 890 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 81 q + 25 q^{2} + 17 q^{3} + 353 q^{4} + 141 q^{5} + 56 q^{6} + 63 q^{7} + 264 q^{8} + 890 q^{9} + 40 q^{10} + 160 q^{11} + 204 q^{12} + 1053 q^{13} + 287 q^{14} + 261 q^{15} + 1649 q^{16} + 179 q^{17} + 702 q^{18} + 524 q^{19} + 1156 q^{20} + 609 q^{21} - 66 q^{22} + 536 q^{23} + 769 q^{24} + 2460 q^{25} + 325 q^{26} + 179 q^{27} + 755 q^{28} + 1300 q^{29} + 1178 q^{30} + 28 q^{31} + 2637 q^{32} + 1556 q^{33} - 349 q^{34} + 9 q^{35} + 3657 q^{36} + 759 q^{37} + 1454 q^{38} + 221 q^{39} - 1620 q^{40} + 3166 q^{41} + 670 q^{42} + 105 q^{43} + 714 q^{44} + 3816 q^{45} + 1257 q^{46} + 2059 q^{47} + 1248 q^{48} + 4700 q^{49} + 1620 q^{50} + 745 q^{51} + 4589 q^{52} + 2534 q^{53} + 4031 q^{54} + 1356 q^{55} + 3334 q^{56} + 1370 q^{57} + 1409 q^{58} + 4790 q^{59} + 227 q^{60} + 1786 q^{61} + 1728 q^{62} + 3060 q^{63} + 8484 q^{64} + 1833 q^{65} - 19 q^{66} + 1420 q^{67} + 1024 q^{68} + 4858 q^{69} - 553 q^{70} + 2907 q^{71} + 5420 q^{72} + 3628 q^{73} - 412 q^{74} + 720 q^{75} + 4527 q^{76} + 3626 q^{77} + 728 q^{78} + 1370 q^{79} + 8161 q^{80} + 11993 q^{81} + 791 q^{82} + 5738 q^{83} + 6171 q^{84} + 3899 q^{85} + 2724 q^{86} + 1982 q^{87} + 295 q^{88} + 10940 q^{89} + 3532 q^{90} + 819 q^{91} + 9521 q^{92} + 7934 q^{93} + 4544 q^{94} + 2422 q^{95} + 4799 q^{96} + 4400 q^{97} + 7112 q^{98} + 4376 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −5.49150 2.90028 22.1566 5.78569 −15.9269 −13.7516 −77.7408 −18.5884 −31.7721
1.2 −5.48493 5.86488 22.0845 15.9853 −32.1685 10.3210 −77.2526 7.39676 −87.6785
1.3 −5.16274 −8.76065 18.6539 10.7511 45.2289 −33.6084 −55.0031 49.7489 −55.5053
1.4 −5.11659 −4.25777 18.1795 −12.5358 21.7853 −16.5271 −52.0845 −8.87137 64.1405
1.5 −5.01493 −4.02441 17.1495 5.04403 20.1821 31.0445 −45.8841 −10.8041 −25.2955
1.6 −4.98675 3.24508 16.8677 1.77219 −16.1824 −7.78574 −44.2211 −16.4694 −8.83745
1.7 −4.81639 −10.0473 15.1976 17.0888 48.3918 23.6945 −34.6664 73.9488 −82.3063
1.8 −4.56725 2.74545 12.8598 −6.65464 −12.5392 −13.0337 −22.1958 −19.4625 30.3934
1.9 −4.38698 6.27446 11.2456 −5.24908 −27.5259 27.3853 −14.2382 12.3689 23.0276
1.10 −4.25477 −2.33039 10.1031 −10.1776 9.91529 6.77364 −8.94819 −21.5693 43.3035
1.11 −4.23979 10.0611 9.97578 0.270687 −42.6568 4.36470 −8.37689 74.2252 −1.14765
1.12 −4.22150 8.06806 9.82110 −19.4222 −34.0594 7.23234 −7.68778 38.0936 81.9908
1.13 −4.14109 2.03460 9.14866 21.3331 −8.42547 −34.9102 −4.75673 −22.8604 −88.3426
1.14 −3.96488 −5.53216 7.72029 11.3747 21.9343 20.9003 1.10900 3.60475 −45.0995
1.15 −3.86408 −8.10434 6.93114 −6.45482 31.3159 13.3220 4.13016 38.6804 24.9420
1.16 −3.54273 3.14722 4.55090 19.9871 −11.1497 31.0178 12.2192 −17.0950 −70.8088
1.17 −3.45822 9.42402 3.95926 17.1535 −32.5903 −5.97598 13.9738 61.8121 −59.3204
1.18 −3.45037 −1.33681 3.90505 9.92993 4.61250 2.13656 14.1291 −25.2129 −34.2619
1.19 −3.27427 −0.503529 2.72086 6.03692 1.64869 −19.6821 17.2853 −26.7465 −19.7665
1.20 −3.18679 −6.03709 2.15566 −21.7124 19.2390 −5.02844 18.6247 9.44643 69.1929
See all 81 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.81
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \(-1\)
\(103\) \(-1\)

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 1339.4.a.d 81
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1339.4.a.d 81 1.a even 1 1 trivial