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Properties

Label 3103.2.a.d
Level $3103$
Weight $2$
Character orbit 3103.a
Self dual yes
Analytic conductor $24.778$
Analytic rank $1$
Dimension $54$
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: \(54\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 54 q - 9 q^{2} - 15 q^{3} + 55 q^{4} - 9 q^{5} - q^{6} - 10 q^{7} - 36 q^{8} + 47 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 54 q - 9 q^{2} - 15 q^{3} + 55 q^{4} - 9 q^{5} - q^{6} - 10 q^{7} - 36 q^{8} + 47 q^{9} - 24 q^{10} - 20 q^{11} - 57 q^{12} - 23 q^{13} - 21 q^{14} - 6 q^{15} + 49 q^{16} - 80 q^{17} - 19 q^{18} - 22 q^{19} - 14 q^{20} + 13 q^{22} - 32 q^{23} + 25 q^{24} + 45 q^{25} - 30 q^{26} - 66 q^{27} + 3 q^{28} + 54 q^{29} + 21 q^{30} - 10 q^{31} - 86 q^{32} - 41 q^{33} + 12 q^{34} - 27 q^{35} + 22 q^{36} - 52 q^{37} - 46 q^{38} - 14 q^{39} - 28 q^{40} - 115 q^{41} - 40 q^{42} - 38 q^{43} - 40 q^{44} - 60 q^{45} - 5 q^{46} - 52 q^{47} - 138 q^{48} + 36 q^{49} - 44 q^{50} + q^{51} - 63 q^{52} - 69 q^{53} + 49 q^{54} - 46 q^{56} - 49 q^{57} - 9 q^{58} - 13 q^{59} - 25 q^{60} - 66 q^{61} - 25 q^{62} - 15 q^{63} + 54 q^{64} - 103 q^{65} - 71 q^{66} + 7 q^{67} - 176 q^{68} - 67 q^{69} - 7 q^{70} - 12 q^{71} + 15 q^{72} - 107 q^{73} - 25 q^{74} - 49 q^{75} - 28 q^{76} - 118 q^{77} - 36 q^{78} + 8 q^{79} - 27 q^{80} - 10 q^{81} + 11 q^{82} - 33 q^{83} + 19 q^{84} - 59 q^{85} - 8 q^{86} - 15 q^{87} + 72 q^{88} - 80 q^{89} - 100 q^{90} + 34 q^{91} - 9 q^{92} - 35 q^{93} - 8 q^{94} - 36 q^{95} + 38 q^{96} - 66 q^{97} - 41 q^{98} + 53 q^{99}+O(q^{100}) \) 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 −2.77795 −2.53780 5.71700 −1.19218 7.04987 −3.72262 −10.3256 3.44040 3.31181
1.2 −2.73196 0.166451 5.46363 3.41362 −0.454739 2.82852 −9.46250 −2.97229 −9.32590
1.3 −2.71865 1.63206 5.39107 −1.71107 −4.43701 0.358613 −9.21914 −0.336372 4.65180
1.4 −2.71362 −2.01478 5.36375 1.41156 5.46736 4.90734 −9.12794 1.05935 −3.83043
1.5 −2.60662 −2.75630 4.79447 4.28301 7.18464 −2.19699 −7.28413 4.59720 −11.1642
1.6 −2.49954 1.39517 4.24768 −4.20587 −3.48728 3.72857 −5.61815 −1.05349 10.5127
1.7 −2.42081 −1.25448 3.86034 −0.0173400 3.03687 −4.45901 −4.50353 −1.42627 0.0419768
1.8 −2.33049 −0.803706 3.43118 −4.10721 1.87303 0.0986952 −3.33534 −2.35406 9.57181
1.9 −2.30282 −3.01656 3.30299 −2.77869 6.94660 2.41795 −3.00057 6.09963 6.39884
1.10 −2.06553 −1.32653 2.26639 1.58398 2.73997 1.37849 −0.550243 −1.24033 −3.27176
1.11 −2.04255 1.87748 2.17200 3.39533 −3.83483 1.56571 −0.351308 0.524921 −6.93512
1.12 −1.97254 1.35748 1.89091 1.41519 −2.67768 −3.29823 0.215193 −1.15724 −2.79151
1.13 −1.89868 2.07906 1.60498 −0.797222 −3.94747 −0.232704 0.750018 1.32250 1.51367
1.14 −1.86968 0.757129 1.49569 1.93465 −1.41559 0.215874 0.942898 −2.42676 −3.61718
1.15 −1.63356 −3.28172 0.668510 0.596830 5.36089 −0.769667 2.17507 7.76972 −0.974956
1.16 −1.53126 1.53374 0.344769 1.97910 −2.34856 −2.83106 2.53459 −0.647638 −3.03053
1.17 −1.45139 −1.98452 0.106538 −1.22401 2.88031 −2.86732 2.74815 0.938305 1.77652
1.18 −1.41586 3.24196 0.00465871 −0.988899 −4.59016 −0.0432056 2.82512 7.51029 1.40014
1.19 −1.41417 −0.136549 −0.000109363 0 −1.69174 0.193105 3.71338 2.82850 −2.98135 2.39242
1.20 −1.16306 −2.65439 −0.647283 1.90198 3.08723 −1.55674 3.07896 4.04581 −2.21212
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.54
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.d ✓ 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3103.2.a.d ✓ 54 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3103))\):

\( T_{2}^{54} + 9 T_{2}^{53} - 41 T_{2}^{52} - 588 T_{2}^{51} + 307 T_{2}^{50} + 17782 T_{2}^{49} + \cdots + 229 \) Copy content Toggle raw display
\( T_{5}^{54} + 9 T_{5}^{53} - 117 T_{5}^{52} - 1255 T_{5}^{51} + 5796 T_{5}^{50} + 80714 T_{5}^{49} + \cdots - 36291106400 \) Copy content Toggle raw display