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Properties

Label 1339.2.a.g
Level $1339$
Weight $2$
Character orbit 1339.a
Self dual yes
Analytic conductor $10.692$
Analytic rank $0$
Dimension $30$
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: \(30\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 5 q^{3} + 34 q^{4} + 17 q^{5} + 3 q^{6} + 2 q^{7} + 3 q^{8} + 49 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 5 q^{3} + 34 q^{4} + 17 q^{5} + 3 q^{6} + 2 q^{7} + 3 q^{8} + 49 q^{9} + 5 q^{10} + q^{11} + 15 q^{12} - 30 q^{13} + 24 q^{14} + 6 q^{15} + 38 q^{16} + 17 q^{17} + 8 q^{18} + 9 q^{19} + 31 q^{20} + 27 q^{21} + 2 q^{22} + 14 q^{23} + 26 q^{24} + 47 q^{25} + 14 q^{27} - 6 q^{28} + 53 q^{29} + 25 q^{30} + 19 q^{31} - 4 q^{32} + q^{33} - 22 q^{34} + 9 q^{35} + 61 q^{36} + 46 q^{38} - 5 q^{39} - 35 q^{40} + 28 q^{41} - 7 q^{42} + 6 q^{43} - 12 q^{44} + 68 q^{45} + 7 q^{46} + 12 q^{47} + 13 q^{48} + 54 q^{49} - 18 q^{50} + 10 q^{51} - 34 q^{52} + 37 q^{53} + 22 q^{54} + 11 q^{55} + 67 q^{56} - 57 q^{57} - 5 q^{58} + 61 q^{59} - 102 q^{60} + 16 q^{61} - 2 q^{62} - 7 q^{63} + 29 q^{64} - 17 q^{65} - 83 q^{66} - 2 q^{67} + 57 q^{68} + 98 q^{69} - 10 q^{70} + 50 q^{71} - 8 q^{72} - 10 q^{73} - 13 q^{74} + 5 q^{75} - 19 q^{76} + 54 q^{77} - 3 q^{78} + 3 q^{79} + 76 q^{80} + 118 q^{81} + 7 q^{82} + 6 q^{83} + 44 q^{84} + 33 q^{85} - 29 q^{86} - 10 q^{87} - 13 q^{88} + 77 q^{89} + 38 q^{90} - 2 q^{91} - 3 q^{92} + 34 q^{93} - 25 q^{94} + 24 q^{95} - 28 q^{96} + 12 q^{97} - 14 q^{98} - 58 q^{99}+O(q^{100}) \) 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 −2.71030 −2.41604 5.34571 3.72865 6.54819 −2.88700 −9.06786 2.83725 −10.1057
1.2 −2.70794 2.09535 5.33295 −1.11757 −5.67408 −3.78835 −9.02545 1.39048 3.02631
1.3 −2.44933 −0.0495252 3.99920 1.48835 0.121303 2.04936 −4.89669 −2.99755 −3.64544
1.4 −2.25031 3.33769 3.06389 0.684997 −7.51084 1.16005 −2.39409 8.14018 −1.54146
1.5 −2.18260 −0.879021 2.76374 3.84044 1.91855 1.42534 −1.66693 −2.22732 −8.38214
1.6 −2.15890 −0.0701034 2.66085 −2.25465 0.151346 −1.38084 −1.42671 −2.99509 4.86756
1.7 −1.71336 −2.73021 0.935608 −2.57910 4.67784 0.316675 1.82369 4.45405 4.41893
1.8 −1.52011 −3.31256 0.310729 0.643321 5.03545 −3.98605 2.56787 7.97304 −0.977918
1.9 −1.43550 2.33343 0.0606568 2.88716 −3.34964 3.63915 2.78392 2.44491 −4.14452
1.10 −1.25596 2.77119 −0.422559 −3.18202 −3.48051 3.63505 3.04264 4.67947 3.99650
1.11 −1.13041 1.24412 −0.722174 −3.15829 −1.40637 −4.57377 3.07717 −1.45216 3.57016
1.12 −0.736639 2.43572 −1.45736 4.11528 −1.79424 −5.11924 2.54683 2.93271 −3.03147
1.13 −0.660355 0.482681 −1.56393 −1.20794 −0.318741 3.96472 2.35346 −2.76702 0.797669
1.14 −0.496151 −2.36213 −1.75383 0.249222 1.17197 2.50523 1.86247 2.57965 −0.123652
1.15 −0.319351 0.543955 −1.89802 1.90715 −0.173713 0.747001 1.24483 −2.70411 −0.609049
1.16 0.397347 −1.55160 −1.84212 0.426444 −0.616523 −5.04601 −1.52665 −0.592540 0.169446
1.17 0.430342 −0.376968 −1.81481 −3.33541 −0.162225 −1.83693 −1.64167 −2.85790 −1.43537
1.18 0.508360 −0.408426 −1.74157 4.07667 −0.207627 0.983456 −1.90207 −2.83319 2.07242
1.19 0.903670 2.25573 −1.18338 1.13079 2.03843 2.22819 −2.87673 2.08831 1.02186
1.20 0.947942 3.31104 −1.10141 4.30053 3.13867 1.61201 −2.93995 7.96298 4.07665
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.30
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.g ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1339.2.a.g ✓ 30 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1339))\):

\( T_{2}^{30} - 47 T_{2}^{28} - T_{2}^{27} + 984 T_{2}^{26} + 43 T_{2}^{25} - 12112 T_{2}^{24} + \cdots - 5184 \) Copy content Toggle raw display
\( T_{3}^{30} - 5 T_{3}^{29} - 57 T_{3}^{28} + 312 T_{3}^{27} + 1371 T_{3}^{26} - 8520 T_{3}^{25} + \cdots - 2048 \) Copy content Toggle raw display