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Properties

Label 1339.4.a.a
Level $1339$
Weight $4$
Character orbit 1339.a
Self dual yes
Analytic conductor $79.004$
Analytic rank $1$
Dimension $72$
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: \(1\)
Dimension: \(72\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - q^{2} - 19 q^{3} + 261 q^{4} - 73 q^{5} - 52 q^{6} + 3 q^{7} - 18 q^{8} + 517 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - q^{2} - 19 q^{3} + 261 q^{4} - 73 q^{5} - 52 q^{6} + 3 q^{7} - 18 q^{8} + 517 q^{9} - 100 q^{10} - 104 q^{11} - 228 q^{12} - 936 q^{13} - 337 q^{14} + 63 q^{15} + 833 q^{16} - 307 q^{17} - 110 q^{18} - 182 q^{19} - 428 q^{20} - 177 q^{21} - 74 q^{22} - 516 q^{23} - 901 q^{24} + 1285 q^{25} + 13 q^{26} - 817 q^{27} + 187 q^{28} - 1458 q^{29} - 178 q^{30} - 620 q^{31} - 35 q^{32} - 196 q^{33} - 1619 q^{34} - 1155 q^{35} - 615 q^{36} - 1031 q^{37} - 1614 q^{38} + 247 q^{39} - 3300 q^{40} - 1124 q^{41} - 2190 q^{42} - 383 q^{43} - 1902 q^{44} - 1828 q^{45} - 1109 q^{46} - 1531 q^{47} - 4464 q^{48} + 1191 q^{49} - 352 q^{50} - 1267 q^{51} - 3393 q^{52} - 3092 q^{53} - 1581 q^{54} - 2264 q^{55} - 1470 q^{56} - 502 q^{57} - 2003 q^{58} - 2452 q^{59} - 1909 q^{60} - 1764 q^{61} - 2232 q^{62} - 1086 q^{63} + 2272 q^{64} + 949 q^{65} - 2539 q^{66} + 832 q^{67} - 1796 q^{68} - 4738 q^{69} - 3251 q^{70} - 2995 q^{71} - 2534 q^{72} + 16 q^{73} - 3408 q^{74} - 3060 q^{75} - 3729 q^{76} - 5454 q^{77} + 676 q^{78} - 3594 q^{79} - 1631 q^{80} + 16 q^{81} - 1293 q^{82} - 2832 q^{83} - 4565 q^{84} - 887 q^{85} - 3982 q^{86} - 1530 q^{87} - 1757 q^{88} - 8104 q^{89} - 5932 q^{90} - 39 q^{91} - 5371 q^{92} - 3042 q^{93} - 3464 q^{94} - 4582 q^{95} - 3991 q^{96} - 2514 q^{97} - 2886 q^{98} - 2752 q^{99}+O(q^{100}) \) 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 −5.34804 −8.53831 20.6015 9.32891 45.6632 −1.96973 −67.3935 45.9027 −49.8914
1.2 −5.34054 6.25250 20.5214 −8.11558 −33.3918 4.20933 −66.8711 12.0938 43.3416
1.3 −5.30480 −0.703776 20.1409 20.1510 3.73339 13.6578 −64.4051 −26.5047 −106.897
1.4 −5.22780 −4.00253 19.3299 −17.6609 20.9244 9.66433 −59.2302 −10.9798 92.3278
1.5 −5.11041 0.0856874 18.1162 6.41522 −0.437897 31.2200 −51.6981 −26.9927 −32.7844
1.6 −5.01150 7.02874 17.1151 10.8758 −35.2245 −2.45123 −45.6803 22.4032 −54.5042
1.7 −4.92395 −2.93214 16.2453 −11.5849 14.4377 −19.5705 −40.5992 −18.4026 57.0434
1.8 −4.58197 7.92871 12.9944 2.27386 −36.3291 −10.1480 −22.8843 35.8644 −10.4187
1.9 −4.46382 −4.96293 11.9257 10.6442 22.1536 −24.9925 −17.5236 −2.36933 −47.5139
1.10 −4.36957 −0.400103 11.0932 −5.49935 1.74828 −22.6968 −13.5159 −26.8399 24.0298
1.11 −4.19802 −5.00905 9.62337 0.437026 21.0281 20.8068 −6.81493 −1.90941 −1.83464
1.12 −4.15650 −8.86915 9.27649 −4.80618 36.8646 −10.5877 −5.30572 51.6619 19.9769
1.13 −4.00571 7.68042 8.04570 −16.7343 −30.7655 34.7970 −0.183055 31.9889 67.0328
1.14 −3.67846 4.19977 5.53108 −12.6221 −15.4487 −32.5017 9.08183 −9.36197 46.4300
1.15 −3.59728 2.67027 4.94044 4.14791 −9.60572 19.2068 11.0061 −19.8697 −14.9212
1.16 −3.57610 2.78833 4.78852 18.8510 −9.97135 −3.89968 11.4846 −19.2252 −67.4133
1.17 −3.43190 5.27441 3.77794 −2.98040 −18.1013 9.93284 14.4897 0.819407 10.2284
1.18 −2.96121 −9.16520 0.768757 −15.5507 27.1401 19.3083 21.4132 57.0009 46.0489
1.19 −2.80828 8.88240 −0.113587 13.3487 −24.9442 −20.7109 22.7852 51.8971 −37.4868
1.20 −2.79494 −2.53814 −0.188321 −13.1782 7.09393 33.1779 22.8859 −20.5579 36.8323
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.72
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.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1339.4.a.a ✓ 72 1.a even 1 1 trivial