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Properties

Label 3103.2.a.f
Level $3103$
Weight $2$
Character orbit 3103.a
Self dual yes
Analytic conductor $24.778$
Analytic rank $0$
Dimension $65$
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: \(0\)
Dimension: \(65\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 65 q + 14 q^{2} + 11 q^{3} + 66 q^{4} + 14 q^{5} + 11 q^{6} + 6 q^{7} + 39 q^{8} + 80 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 65 q + 14 q^{2} + 11 q^{3} + 66 q^{4} + 14 q^{5} + 11 q^{6} + 6 q^{7} + 39 q^{8} + 80 q^{9} + 9 q^{10} + 8 q^{11} + 13 q^{12} + 21 q^{13} - q^{14} + 9 q^{15} + 56 q^{16} + 92 q^{17} + 36 q^{18} - 3 q^{19} + 27 q^{20} + 24 q^{21} + 3 q^{22} + 19 q^{23} + 25 q^{24} + 83 q^{25} + 10 q^{26} + 32 q^{27} + 3 q^{28} + 65 q^{29} - 2 q^{30} + 25 q^{31} + 57 q^{32} + 59 q^{33} + 31 q^{35} + 103 q^{36} + 39 q^{37} + 33 q^{38} - 8 q^{39} + 17 q^{40} + 68 q^{41} + 12 q^{42} + 9 q^{43} + 36 q^{44} + 6 q^{45} + 28 q^{46} + 47 q^{47} + 40 q^{48} + 75 q^{49} + 4 q^{50} + 39 q^{51} + 53 q^{52} + 69 q^{53} + 7 q^{54} + 47 q^{55} - 10 q^{56} + 12 q^{57} + 14 q^{58} + 29 q^{59} + 76 q^{60} + 49 q^{61} + 52 q^{62} - 11 q^{63} + 21 q^{64} + 56 q^{65} + 83 q^{66} + 11 q^{67} + 100 q^{68} + 10 q^{69} - 41 q^{70} + 26 q^{71} + 78 q^{72} + 73 q^{73} + 10 q^{74} + 24 q^{75} - 31 q^{76} + 102 q^{77} - 22 q^{78} + 2 q^{79} + 118 q^{80} + 89 q^{81} + 6 q^{82} + 21 q^{83} + 85 q^{84} + 5 q^{85} - 65 q^{86} + 11 q^{87} - 6 q^{88} + 103 q^{89} + 16 q^{90} - 42 q^{91} + 54 q^{92} - 38 q^{93} + 61 q^{94} + 56 q^{95} - 50 q^{96} + 110 q^{97} + 46 q^{98} + 9 q^{99}+O(q^{100}) \) 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 −2.61887 2.25311 4.85849 0.327233 −5.90060 4.33404 −7.48603 2.07650 −0.856980
1.2 −2.60242 −0.301309 4.77260 0.799813 0.784134 0.292668 −7.21548 −2.90921 −2.08145
1.3 −2.57396 0.317662 4.62526 3.26543 −0.817649 −3.89083 −6.75730 −2.89909 −8.40509
1.4 −2.40676 −2.57943 3.79249 −3.99348 6.20808 −3.01307 −4.31408 3.65348 9.61134
1.5 −2.38893 −2.34532 3.70698 −1.07385 5.60281 2.00254 −4.07785 2.50053 2.56534
1.6 −2.38836 2.92830 3.70426 3.25866 −6.99382 2.11466 −4.07038 5.57492 −7.78286
1.7 −2.08330 −2.21360 2.34012 3.69451 4.61159 3.02114 −0.708579 1.90004 −7.69676
1.8 −2.03623 2.62703 2.14625 −3.11074 −5.34925 −4.80826 −0.297791 3.90130 6.33419
1.9 −2.02468 0.143584 2.09931 −2.13661 −0.290710 −1.55739 −0.201079 −2.97938 4.32595
1.10 −1.93050 −2.98635 1.72683 1.56589 5.76515 −1.86293 0.527361 5.91830 −3.02295
1.11 −1.90183 1.89453 1.61697 −0.995421 −3.60309 2.27479 0.728453 0.589257 1.89313
1.12 −1.78107 −0.0679805 1.17222 0.0711609 0.121078 −0.823844 1.47433 −2.99538 −0.126743
1.13 −1.75818 −1.43861 1.09121 2.62122 2.52934 3.58222 1.59782 −0.930411 −4.60859
1.14 −1.66260 −1.56052 0.764247 −3.39501 2.59452 −1.32945 2.05457 −0.564791 5.64456
1.15 −1.45771 2.79082 0.124921 0.505787 −4.06821 −3.30541 2.73332 4.78866 −0.737291
1.16 −1.32841 0.840443 −0.235338 2.59462 −1.11645 2.48860 2.96944 −2.29365 −3.44671
1.17 −1.29282 −2.93537 −0.328629 −2.31771 3.79489 0.518460 3.01049 5.61637 2.99637
1.18 −1.27775 2.32197 −0.367361 −3.47453 −2.96689 4.32636 3.02489 2.39153 4.43957
1.19 −1.27730 2.26963 −0.368511 4.39820 −2.89899 −0.304005 3.02529 2.15122 −5.61781
1.20 −1.21017 0.877246 −0.535498 −2.69888 −1.06161 0.923882 3.06837 −2.23044 3.26609
See all 65 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.65
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.f ✓ 65
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3103.2.a.f ✓ 65 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3103))\):

\( T_{2}^{65} - 14 T_{2}^{64} + 883 T_{2}^{62} - 3001 T_{2}^{61} - 23619 T_{2}^{60} + 139751 T_{2}^{59} + \cdots + 471 \) Copy content Toggle raw display
\( T_{5}^{65} - 14 T_{5}^{64} - 106 T_{5}^{63} + 2340 T_{5}^{62} + 2141 T_{5}^{61} - 181444 T_{5}^{60} + \cdots + 106018502369280 \) Copy content Toggle raw display