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Properties

Label 3103.2.a.e
Level $3103$
Weight $2$
Character orbit 3103.a
Self dual yes
Analytic conductor $24.778$
Analytic rank $1$
Dimension $58$
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] = [3103,2,Mod(1,3103)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3103, 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("3103.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3103 = 29 \cdot 107 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3103.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: \(24.7775797472\)
Analytic rank: \(1\)
Dimension: \(58\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 58 q - 11 q^{2} - 5 q^{3} + 55 q^{4} - 18 q^{5} - 11 q^{6} - 14 q^{7} - 30 q^{8} + 39 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 58 q - 11 q^{2} - 5 q^{3} + 55 q^{4} - 18 q^{5} - 11 q^{6} - 14 q^{7} - 30 q^{8} + 39 q^{9} - q^{10} - 12 q^{11} - 15 q^{12} - 31 q^{13} - 9 q^{14} - 19 q^{15} + 53 q^{16} - 74 q^{17} - 39 q^{18} - q^{19} - 53 q^{20} - 16 q^{21} - 27 q^{22} - 37 q^{23} - 41 q^{24} + 28 q^{25} - 10 q^{26} - 14 q^{27} - 37 q^{28} - 58 q^{29} - 40 q^{30} - 9 q^{31} - 56 q^{32} - 67 q^{33} + 4 q^{34} - 45 q^{35} + 30 q^{36} + 15 q^{37} - 63 q^{38} - 12 q^{39} - 23 q^{40} - 86 q^{41} - 24 q^{42} - 19 q^{43} + 8 q^{44} - 32 q^{45} + 30 q^{46} - 43 q^{47} + 16 q^{48} + 12 q^{49} - 43 q^{50} - 21 q^{51} - 67 q^{52} - 71 q^{53} - 51 q^{54} - 17 q^{55} - 58 q^{56} - 72 q^{57} + 11 q^{58} - 43 q^{59} - 8 q^{60} + 21 q^{61} - 90 q^{62} - 47 q^{63} + 54 q^{64} - 74 q^{65} + 33 q^{66} - 33 q^{67} - 128 q^{68} - 14 q^{69} + 11 q^{70} + 2 q^{71} - 123 q^{72} - 15 q^{73} + 4 q^{74} - 2 q^{75} + 5 q^{76} - 52 q^{77} + 44 q^{78} - 12 q^{79} - 58 q^{80} + 26 q^{81} - 8 q^{82} - 91 q^{83} + 27 q^{84} + 15 q^{85} - 15 q^{86} + 5 q^{87} - 96 q^{88} - 103 q^{89} + 108 q^{90} - 46 q^{91} - 108 q^{92} - 48 q^{93} - 37 q^{94} - 36 q^{95} - 56 q^{96} - 64 q^{97} - 65 q^{98} - 31 q^{99}+O(q^{100}) \) 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 −2.81809 2.12477 5.94163 1.56497 −5.98778 −3.58626 −11.1079 1.51463 −4.41024
1.2 −2.79912 −1.61763 5.83507 −3.36896 4.52794 1.95903 −10.7348 −0.383273 9.43012
1.3 −2.63203 3.22905 4.92761 −1.93995 −8.49898 3.49826 −7.70556 7.42679 5.10603
1.4 −2.58085 −3.33848 4.66076 −0.721857 8.61611 −0.370947 −6.86701 8.14547 1.86300
1.5 −2.53930 0.329413 4.44803 −3.05544 −0.836478 −4.89229 −6.21627 −2.89149 7.75868
1.6 −2.49986 0.177442 4.24928 −0.635609 −0.443579 2.29228 −5.62287 −2.96851 1.58893
1.7 −2.45519 −1.14579 4.02797 2.83189 2.81314 0.0713324 −4.97905 −1.68716 −6.95284
1.8 −2.36100 −1.93025 3.57432 2.28055 4.55733 −1.37832 −3.71697 0.725882 −5.38439
1.9 −2.35832 1.71183 3.56169 1.13735 −4.03706 1.94951 −3.68297 −0.0696211 −2.68223
1.10 −2.25258 2.51872 3.07411 2.90766 −5.67361 −3.67083 −2.41952 3.34393 −6.54974
1.11 −2.08215 0.847255 2.33534 −0.867713 −1.76411 4.31212 −0.698220 −2.28216 1.80671
1.12 −1.87709 2.46200 1.52346 −3.78098 −4.62138 2.05719 0.894505 3.06142 7.09724
1.13 −1.87280 0.300360 1.50739 −3.24095 −0.562516 −1.03098 0.922557 −2.90978 6.06967
1.14 −1.86858 −1.13083 1.49158 2.70485 2.11305 −3.34236 0.950018 −1.72121 −5.05423
1.15 −1.71703 −3.11142 0.948176 −3.45177 5.34238 −3.98130 1.80601 6.68091 5.92677
1.16 −1.68963 −1.43837 0.854860 −2.83523 2.43032 −1.59045 1.93487 −0.931092 4.79049
1.17 −1.67795 −1.98426 0.815505 1.87267 3.32948 1.10408 1.98752 0.937281 −3.14224
1.18 −1.56935 −2.53697 0.462866 −2.31017 3.98140 3.87462 2.41230 3.43623 3.62548
1.19 −1.31801 1.57054 −0.262841 1.13372 −2.06999 1.70188 2.98245 −0.533418 −1.49426
1.20 −1.25139 2.66301 −0.434020 −0.253228 −3.33247 −3.35041 3.04591 4.09163 0.316888
See all 58 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.58
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(29\) \( +1 \)
\(107\) \( +1 \)

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 3103.2.a.e ✓ 58
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3103.2.a.e ✓ 58 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3103))\):

\( T_{2}^{58} + 11 T_{2}^{57} - 25 T_{2}^{56} - 694 T_{2}^{55} - 848 T_{2}^{54} + 19739 T_{2}^{53} + \cdots - 137 \) Copy content Toggle raw display
\( T_{5}^{58} + 18 T_{5}^{57} + 3 T_{5}^{56} - 1788 T_{5}^{55} - 7737 T_{5}^{54} + 74964 T_{5}^{53} + \cdots - 14266693616 \) Copy content Toggle raw display