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Properties

Label 1339.2.a.f
Level $1339$
Weight $2$
Character orbit 1339.a
Self dual yes
Analytic conductor $10.692$
Analytic rank $0$
Dimension $28$
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] = [1339,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1339.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1339 = 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(10.6919688306\)
Analytic rank: \(0\)
Dimension: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 6 q^{2} + 5 q^{3} + 32 q^{4} + 27 q^{5} + 9 q^{6} + 6 q^{7} + 21 q^{8} + 39 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 6 q^{2} + 5 q^{3} + 32 q^{4} + 27 q^{5} + 9 q^{6} + 6 q^{7} + 21 q^{8} + 39 q^{9} + 5 q^{10} + 9 q^{11} + 15 q^{12} + 28 q^{13} + 20 q^{14} + 4 q^{15} + 32 q^{16} - 7 q^{17} + 10 q^{18} - 3 q^{19} + 53 q^{20} + 25 q^{21} - 14 q^{22} - 6 q^{23} + 10 q^{24} + 45 q^{25} + 6 q^{26} + 8 q^{27} - 12 q^{28} + 25 q^{29} - 43 q^{30} + q^{31} + 28 q^{32} + 17 q^{33} + 10 q^{34} + 11 q^{35} + 27 q^{36} + 18 q^{37} + 2 q^{38} + 5 q^{39} + 37 q^{40} + 56 q^{41} - 29 q^{42} - 30 q^{43} + 30 q^{44} + 38 q^{45} - 27 q^{46} + 40 q^{47} + 19 q^{48} + 36 q^{49} + 20 q^{50} - 18 q^{51} + 32 q^{52} + 51 q^{53} - 36 q^{54} - 5 q^{55} - 13 q^{56} + 31 q^{57} - 29 q^{58} + 29 q^{59} + 48 q^{60} + 16 q^{61} - 38 q^{62} - 23 q^{63} - 5 q^{64} + 27 q^{65} + 25 q^{66} - 2 q^{67} - 43 q^{68} + 38 q^{69} + 10 q^{70} + 34 q^{71} - 28 q^{72} + 14 q^{73} + 21 q^{74} + 19 q^{75} + 5 q^{76} + 34 q^{77} + 9 q^{78} + 3 q^{79} + 58 q^{80} + 4 q^{81} - 9 q^{82} + 38 q^{83} - 78 q^{84} - 27 q^{85} + 49 q^{86} + 8 q^{87} - 25 q^{88} + 125 q^{89} - 74 q^{90} + 6 q^{91} - 35 q^{92} + 36 q^{93} - q^{94} - 12 q^{95} + 26 q^{96} - 2 q^{97} + 18 q^{98} + 56 q^{99}+O(q^{100}) \) 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 −2.50812 −2.94446 4.29069 −0.0167965 7.38506 −1.24099 −5.74532 5.66982 0.0421278
1.2 −2.50096 2.99924 4.25481 4.07294 −7.50098 −1.09357 −5.63921 5.99543 −10.1863
1.3 −2.35134 1.07411 3.52878 3.29713 −2.52560 3.84969 −3.59467 −1.84628 −7.75266
1.4 −2.21826 −0.565867 2.92070 −1.45314 1.25524 −4.80056 −2.04235 −2.67979 3.22344
1.5 −1.86443 −1.06486 1.47608 −1.98836 1.98536 3.02740 0.976806 −1.86606 3.70715
1.6 −1.73240 1.73806 1.00120 −1.26585 −3.01101 −1.31498 1.73032 0.0208578 2.19296
1.7 −1.61074 −0.965673 0.594496 4.40810 1.55545 −3.07617 2.26391 −2.06748 −7.10033
1.8 −1.08251 −0.806331 −0.828177 1.00289 0.872859 −2.77298 3.06152 −2.34983 −1.08564
1.9 −1.04956 2.85751 −0.898418 1.97843 −2.99913 2.56542 3.04207 5.16534 −2.07648
1.10 −0.779581 −1.24905 −1.39225 1.96235 0.973735 3.90578 2.64454 −1.43988 −1.52981
1.11 −0.656140 −3.17911 −1.56948 2.97924 2.08594 −2.69809 2.34208 7.10672 −1.95480
1.12 −0.413705 3.05093 −1.82885 0.683022 −1.26219 −1.51920 1.58402 6.30816 −0.282570
1.13 −0.175714 −2.57328 −1.96912 −0.872787 0.452162 1.26960 0.697432 3.62178 0.153361
1.14 0.203882 1.83284 −1.95843 3.17177 0.373682 2.89343 −0.807052 0.359299 0.646665
1.15 0.221198 1.18074 −1.95107 −3.72276 0.261177 −1.11234 −0.873968 −1.60585 −0.823467
1.16 0.612731 −0.480212 −1.62456 −0.793154 −0.294241 3.03298 −2.22088 −2.76940 −0.485990
1.17 0.835105 −2.05058 −1.30260 3.71011 −1.71245 −2.16987 −2.75802 1.20488 3.09833
1.18 0.980046 3.26733 −1.03951 −1.81818 3.20213 3.84583 −2.97886 7.67545 −1.78190
1.19 1.56395 −2.92532 0.445953 −2.35359 −4.57507 −3.99061 −2.43046 5.55751 −3.68091
1.20 1.80754 0.896394 1.26722 3.66177 1.62027 2.54556 −1.32454 −2.19648 6.61881
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.28
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 1339.2.a.f ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1339.2.a.f ✓ 28 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1339))\):

\( T_{2}^{28} - 6 T_{2}^{27} - 26 T_{2}^{26} + 213 T_{2}^{25} + 208 T_{2}^{24} - 3290 T_{2}^{23} + \cdots - 384 \) Copy content Toggle raw display
\( T_{3}^{28} - 5 T_{3}^{27} - 49 T_{3}^{26} + 274 T_{3}^{25} + 997 T_{3}^{24} - 6568 T_{3}^{23} + \cdots + 1354752 \) Copy content Toggle raw display