Properties

Label 1339.2.a.e
Level $1339$
Weight $2$
Character orbit 1339.a
Self dual yes
Analytic conductor $10.692$
Analytic rank $1$
Dimension $21$
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: \(1\)
Dimension: \(21\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9} - 10 q^{10} - 5 q^{11} - 22 q^{12} - 21 q^{13} - 24 q^{14} - q^{15} - 5 q^{16} - 18 q^{17} + 10 q^{18} - 6 q^{19} - 30 q^{20} - 17 q^{21} - 8 q^{22} - 14 q^{23} - 13 q^{24} + 11 q^{25} + q^{26} - 33 q^{27} - 2 q^{28} - 36 q^{29} + 2 q^{30} + q^{31} - 9 q^{32} - 11 q^{33} - 25 q^{34} - 37 q^{35} + 3 q^{36} + 5 q^{37} - 13 q^{38} + 6 q^{39} - 2 q^{40} - 32 q^{41} + 7 q^{42} + 2 q^{43} + 4 q^{44} - 32 q^{45} - 5 q^{46} - 12 q^{47} - 18 q^{48} + 5 q^{49} - 2 q^{50} - 27 q^{51} - 15 q^{52} - 49 q^{53} - 31 q^{54} - 9 q^{55} - 53 q^{56} + 14 q^{57} - 2 q^{58} - 34 q^{59} + 25 q^{60} - 27 q^{61} - 12 q^{62} + 41 q^{63} - 70 q^{64} + 18 q^{65} - 15 q^{66} - 12 q^{67} - 20 q^{68} - 70 q^{69} + 70 q^{70} - 26 q^{71} - 14 q^{72} + 23 q^{73} + 20 q^{74} - 9 q^{75} - 36 q^{76} - 62 q^{77} + 8 q^{78} - 7 q^{79} - 41 q^{80} - 3 q^{81} - 33 q^{82} - q^{83} - 48 q^{84} + 10 q^{85} - 23 q^{86} - 7 q^{87} + 9 q^{88} - 16 q^{89} - 30 q^{90} + 2 q^{91} - 27 q^{92} + 2 q^{93} - 37 q^{94} - 35 q^{95} + 55 q^{96} - 30 q^{97} - 43 q^{98} - 42 q^{99}+O(q^{100}) \) 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 −2.50015 −2.63774 4.25075 −1.17684 6.59474 4.82219 −5.62720 3.95767 2.94227
1.2 −2.31499 1.57292 3.35919 −1.84365 −3.64130 3.43971 −3.14652 −0.525924 4.26804
1.3 −2.12469 0.347667 2.51433 −4.40399 −0.738685 −0.0796229 −1.09279 −2.87913 9.35713
1.4 −1.94515 1.90358 1.78363 1.71445 −3.70276 −2.09421 0.420881 0.623614 −3.33487
1.5 −1.91337 −1.36048 1.66097 1.75251 2.60311 −4.42851 0.648686 −1.14908 −3.35319
1.6 −1.47455 −1.76266 0.174289 0.466891 2.59913 −0.582262 2.69210 0.106981 −0.688453
1.7 −1.10076 0.263891 −0.788329 3.16171 −0.290481 1.73698 3.06928 −2.93036 −3.48028
1.8 −0.791299 0.744814 −1.37385 −0.897428 −0.589370 −1.66332 2.66972 −2.44525 0.710134
1.9 −0.784481 −1.61992 −1.38459 −4.15260 1.27079 4.61578 2.65515 −0.375870 3.25764
1.10 −0.615393 2.88796 −1.62129 −1.40822 −1.77723 −0.315755 2.22852 5.34034 0.866607
1.11 −0.0264721 −1.87769 −1.99930 2.52751 0.0497063 −1.16160 0.105870 0.525708 −0.0669085
1.12 0.106842 −3.24890 −1.98858 −2.21706 −0.347120 −0.711342 −0.426149 7.55538 −0.236876
1.13 0.267251 2.13493 −1.92858 −0.961056 0.570561 −2.53524 −1.04992 1.55791 −0.256843
1.14 0.499344 0.601343 −1.75066 −1.13095 0.300277 3.15035 −1.87287 −2.63839 −0.564734
1.15 1.33589 2.00110 −0.215389 −3.88298 2.67326 2.54624 −2.95952 1.00441 −5.18725
1.16 1.40414 1.06219 −0.0283926 0.529418 1.49147 −2.26187 −2.84815 −1.87175 0.743377
1.17 1.85697 −2.04069 1.44833 0.337566 −3.78949 −0.868388 −1.02443 1.16441 0.626849
1.18 2.08679 −1.93108 2.35469 1.87136 −4.02977 −0.548741 0.740165 0.729089 3.90513
1.19 2.21898 0.899770 2.92386 −2.41101 1.99657 −4.38690 2.05003 −2.19041 −5.34997
1.20 2.31727 −3.32603 3.36976 −2.87188 −7.70732 2.40638 3.17410 8.06248 −6.65492
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
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.e 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1339.2.a.e 21 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1339))\):

\( T_{2}^{21} + T_{2}^{20} - 28 T_{2}^{19} - 29 T_{2}^{18} + 328 T_{2}^{17} + 353 T_{2}^{16} - 2080 T_{2}^{15} - 2338 T_{2}^{14} + 7701 T_{2}^{13} + 9124 T_{2}^{12} - 16656 T_{2}^{11} - 21190 T_{2}^{10} + 19645 T_{2}^{9} + 28040 T_{2}^{8} + \cdots + 1 \) Copy content Toggle raw display
\( T_{3}^{21} + 6 T_{3}^{20} - 19 T_{3}^{19} - 163 T_{3}^{18} + 92 T_{3}^{17} + 1832 T_{3}^{16} + 376 T_{3}^{15} - 11267 T_{3}^{14} - 5517 T_{3}^{13} + 42112 T_{3}^{12} + 22875 T_{3}^{11} - 100182 T_{3}^{10} - 44449 T_{3}^{9} + \cdots - 731 \) Copy content Toggle raw display