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Properties

Label 2513.2.a.e
Level $2513$
Weight $2$
Character orbit 2513.a
Self dual yes
Analytic conductor $20.066$
Analytic rank $0$
Dimension $57$
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] = [2513,2,Mod(1,2513)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2513, 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("2513.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2513 = 7 \cdot 359 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2513.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: \(20.0664060279\)
Analytic rank: \(0\)
Dimension: \(57\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 57 q + 8 q^{2} + 12 q^{3} + 74 q^{4} + 3 q^{5} + 3 q^{6} + 57 q^{7} + 15 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 57 q + 8 q^{2} + 12 q^{3} + 74 q^{4} + 3 q^{5} + 3 q^{6} + 57 q^{7} + 15 q^{8} + 81 q^{9} + 20 q^{10} + 13 q^{11} + 25 q^{12} + 26 q^{13} + 8 q^{14} + 29 q^{15} + 96 q^{16} + 18 q^{17} + 19 q^{18} + 13 q^{19} - 5 q^{20} + 12 q^{21} + 51 q^{22} + 33 q^{23} + q^{24} + 104 q^{25} - 23 q^{26} + 36 q^{27} + 74 q^{28} + 6 q^{29} + 10 q^{30} + 35 q^{31} + 28 q^{32} + 2 q^{33} - 11 q^{34} + 3 q^{35} + 83 q^{36} + 62 q^{37} - 21 q^{38} + 45 q^{39} + 45 q^{40} + 3 q^{42} + 79 q^{43} + 22 q^{44} - 20 q^{45} + 40 q^{46} + 31 q^{47} + 30 q^{48} + 57 q^{49} - 13 q^{50} + 26 q^{51} + 20 q^{52} + 19 q^{53} - 26 q^{54} + 66 q^{55} + 15 q^{56} + 23 q^{57} + 60 q^{58} - 12 q^{59} + 13 q^{60} + 20 q^{61} - 7 q^{62} + 81 q^{63} + 107 q^{64} + 11 q^{65} - 52 q^{66} + 79 q^{67} - 31 q^{68} - 29 q^{69} + 20 q^{70} + 37 q^{71} + 63 q^{72} + 56 q^{73} + 23 q^{74} + 30 q^{75} + 9 q^{76} + 13 q^{77} - 50 q^{78} + 156 q^{79} - 63 q^{80} + 105 q^{81} - 11 q^{82} - 19 q^{83} + 25 q^{84} + 29 q^{85} + 12 q^{86} + 46 q^{87} + 103 q^{88} - 43 q^{89} - 12 q^{90} + 26 q^{91} + 50 q^{92} - 19 q^{93} - 33 q^{94} + 64 q^{95} + 7 q^{96} + 26 q^{97} + 8 q^{98} + 69 q^{99}+O(q^{100}) \) 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 −2.80756 −0.475860 5.88242 −3.13135 1.33601 1.00000 −10.9001 −2.77356 8.79147
1.2 −2.75461 1.67917 5.58790 2.60979 −4.62546 1.00000 −9.88327 −0.180395 −7.18897
1.3 −2.59980 −2.06408 4.75896 1.06026 5.36620 1.00000 −7.17274 1.26044 −2.75647
1.4 −2.55345 1.30586 4.52008 −4.17694 −3.33443 1.00000 −6.43490 −1.29474 10.6656
1.5 −2.51710 −2.29739 4.33581 1.64870 5.78276 1.00000 −5.87947 2.27798 −4.14995
1.6 −2.38209 3.01758 3.67436 −3.89438 −7.18814 1.00000 −3.98847 6.10578 9.27676
1.7 −2.28263 1.19388 3.21040 −1.15375 −2.72519 1.00000 −2.76288 −1.57465 2.63358
1.8 −2.24538 3.34048 3.04172 3.03019 −7.50063 1.00000 −2.33905 8.15878 −6.80393
1.9 −2.17223 1.23930 2.71856 −1.26633 −2.69204 1.00000 −1.56088 −1.46414 2.75076
1.10 −2.16634 −2.74427 2.69302 −3.55550 5.94502 1.00000 −1.50133 4.53101 7.70243
1.11 −1.95347 2.63013 1.81605 2.26074 −5.13788 1.00000 0.359348 3.91758 −4.41628
1.12 −1.81748 −2.37118 1.30322 2.24528 4.30956 1.00000 1.26637 2.62247 −4.08075
1.13 −1.75276 1.71013 1.07217 3.48782 −2.99745 1.00000 1.62626 −0.0754564 −6.11332
1.14 −1.62043 −1.30479 0.625788 −0.299885 2.11432 1.00000 2.22681 −1.29752 0.485943
1.15 −1.60270 −1.83109 0.568657 0.218884 2.93469 1.00000 2.29402 0.352879 −0.350806
1.16 −1.57244 0.0463509 0.472577 2.06889 −0.0728841 1.00000 2.40179 −2.99785 −3.25321
1.17 −1.25396 0.346378 −0.427590 2.28277 −0.434344 1.00000 3.04410 −2.88002 −2.86250
1.18 −1.20186 −3.00113 −0.555523 −3.64715 3.60695 1.00000 3.07139 6.00680 4.38338
1.19 −0.905859 2.73099 −1.17942 −2.39444 −2.47390 1.00000 2.88011 4.45833 2.16903
1.20 −0.861680 0.208491 −1.25751 −2.60928 −0.179652 1.00000 2.80693 −2.95653 2.24837
See all 57 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.57
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \( -1 \)
\(359\) \( +1 \)

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 2513.2.a.e ✓ 57
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2513.2.a.e ✓ 57 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{57} - 8 T_{2}^{56} - 62 T_{2}^{55} + 651 T_{2}^{54} + 1457 T_{2}^{53} - 24694 T_{2}^{52} + \cdots + 22353 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2513))\). Copy content Toggle raw display