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Properties

Label 1339.4.a.c
Level $1339$
Weight $4$
Character orbit 1339.a
Self dual yes
Analytic conductor $79.004$
Analytic rank $0$
Dimension $80$
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: \(80\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - q^{2} + 17 q^{3} + 341 q^{4} + 97 q^{5} + 20 q^{6} + 3 q^{7} - 18 q^{8} + 877 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - q^{2} + 17 q^{3} + 341 q^{4} + 97 q^{5} + 20 q^{6} + 3 q^{7} - 18 q^{8} + 877 q^{9} + 40 q^{10} + 6 q^{11} + 204 q^{12} - 1040 q^{13} + 279 q^{14} + 183 q^{15} + 1537 q^{16} + 237 q^{17} - 110 q^{18} + 198 q^{19} + 812 q^{20} + 831 q^{21} + 234 q^{22} + 128 q^{23} - 37 q^{24} + 2285 q^{25} + 13 q^{26} + 935 q^{27} + 187 q^{28} + 1622 q^{29} + 542 q^{30} + 762 q^{31} - 35 q^{32} + 68 q^{33} + 2501 q^{34} + 1345 q^{35} + 5933 q^{36} + 823 q^{37} + 818 q^{38} - 221 q^{39} + 2916 q^{40} + 1336 q^{41} + 2128 q^{42} + 305 q^{43} + 2368 q^{44} + 2762 q^{45} - 557 q^{46} + 1375 q^{47} + 2538 q^{48} + 6447 q^{49} + 2344 q^{50} + 773 q^{51} - 4433 q^{52} + 3544 q^{53} + 1011 q^{54} + 1116 q^{55} + 5922 q^{56} - 1126 q^{57} - 2003 q^{58} + 3448 q^{59} + 6067 q^{60} + 3636 q^{61} + 3332 q^{62} + 514 q^{63} + 8160 q^{64} - 1261 q^{65} + 5893 q^{66} + 1770 q^{67} + 3644 q^{68} + 6854 q^{69} + 2817 q^{70} + 2827 q^{71} - 2534 q^{72} + 538 q^{73} + 4202 q^{74} + 5312 q^{75} + 2127 q^{76} + 4710 q^{77} - 260 q^{78} + 18 q^{79} + 7809 q^{80} + 11384 q^{81} + 1987 q^{82} - 520 q^{83} + 2827 q^{84} + 813 q^{85} + 2508 q^{86} + 3510 q^{87} + 3079 q^{88} + 5094 q^{89} + 368 q^{90} - 39 q^{91} + 3017 q^{92} - 66 q^{93} + 3692 q^{94} + 1498 q^{95} + 5207 q^{96} + 590 q^{97} - 926 q^{98} + 434 q^{99}+O(q^{100}) \) 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 −5.54707 −0.724941 22.7700 2.60725 4.02130 −30.7700 −81.9301 −26.4745 −14.4626
1.2 −5.52599 −7.31433 22.5365 6.90913 40.4189 13.4995 −80.3288 26.4994 −38.1798
1.3 −5.37149 1.52913 20.8529 −9.10023 −8.21371 11.7265 −69.0393 −24.6618 48.8818
1.4 −5.33758 9.27174 20.4898 18.1454 −49.4887 −31.5400 −66.6652 58.9652 −96.8526
1.5 −5.15017 10.0039 18.5243 7.61853 −51.5216 34.0526 −54.2018 73.0772 −39.2367
1.6 −5.11843 −9.63131 18.1984 −15.8140 49.2972 −25.7294 −52.1996 65.7622 80.9428
1.7 −5.01485 3.39088 17.1487 −3.06313 −17.0047 −6.61777 −45.8792 −15.5019 15.3611
1.8 −4.97277 7.61404 16.7284 −15.7706 −37.8628 −8.98791 −43.4043 30.9736 78.4235
1.9 −4.80239 −7.39982 15.0630 −0.257358 35.5368 16.9071 −33.9192 27.7573 1.23593
1.10 −4.63168 3.67474 13.4525 16.0736 −17.0202 −25.3863 −25.2542 −13.4963 −74.4477
1.11 −4.41141 2.22391 11.4605 8.20971 −9.81057 18.2484 −15.2657 −22.0542 −36.2164
1.12 −4.35315 −2.75296 10.9499 6.37869 11.9841 −7.95377 −12.8413 −19.4212 −27.7674
1.13 −4.33733 −4.13638 10.8125 −15.6640 17.9409 21.0386 −12.1986 −9.89033 67.9400
1.14 −3.88047 −6.12906 7.05808 18.4109 23.7836 26.6816 3.65508 10.5653 −71.4432
1.15 −3.87080 9.69631 6.98306 −16.2200 −37.5325 −11.9118 3.93636 67.0185 62.7844
1.16 −3.85094 2.58446 6.82977 −15.4988 −9.95263 −12.4893 4.50648 −20.3205 59.6851
1.17 −3.80412 −6.86813 6.47134 9.65199 26.1272 −7.35821 5.81519 20.1712 −36.7173
1.18 −3.62765 −9.18874 5.15984 19.3624 33.3335 −17.8721 10.3031 57.4330 −70.2402
1.19 −3.56881 3.46452 4.73638 −4.09682 −12.3642 26.5786 11.6472 −14.9971 14.6208
1.20 −3.23777 −1.43901 2.48313 −5.39302 4.65917 0.914381 17.8623 −24.9293 17.4613
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.80
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.c ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1339.4.a.c ✓ 80 1.a even 1 1 trivial