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Properties

Label 229.10.a.a
Level $229$
Weight $10$
Character orbit 229.a
Self dual yes
Analytic conductor $117.943$
Analytic rank $1$
Dimension $83$
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] = [229,10,Mod(1,229)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(229, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("229.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 229 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 229.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: \(117.943206482\)
Analytic rank: \(1\)
Dimension: \(83\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 83 q - 113 q^{2} - 413 q^{3} + 20053 q^{4} - 3750 q^{5} - 10391 q^{6} - 16883 q^{7} - 86529 q^{8} + 458996 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 83 q - 113 q^{2} - 413 q^{3} + 20053 q^{4} - 3750 q^{5} - 10391 q^{6} - 16883 q^{7} - 86529 q^{8} + 458996 q^{9} - 17551 q^{10} - 504631 q^{11} - 172516 q^{12} - 142805 q^{13} - 645149 q^{14} - 636340 q^{15} + 4970137 q^{16} - 907429 q^{17} - 2317821 q^{18} - 2896219 q^{19} - 2630872 q^{20} - 2770420 q^{21} - 1640305 q^{22} - 4052133 q^{23} - 8158573 q^{24} + 26501155 q^{25} - 13900488 q^{26} - 10233194 q^{27} - 12284938 q^{28} - 20926861 q^{29} - 6888326 q^{30} - 13130155 q^{31} - 48378886 q^{32} - 27103662 q^{33} - 10240379 q^{34} - 81338024 q^{35} + 63677900 q^{36} - 15404546 q^{37} - 56854906 q^{38} - 74297936 q^{39} - 21224911 q^{40} - 95360448 q^{41} - 73566759 q^{42} - 116286027 q^{43} - 234936615 q^{44} - 89601488 q^{45} - 129620012 q^{46} - 104675113 q^{47} - 26953382 q^{48} + 409621362 q^{49} - 261582703 q^{50} - 375912332 q^{51} - 80618587 q^{52} - 70697066 q^{53} - 417628127 q^{54} - 17874262 q^{55} - 437878983 q^{56} - 123571322 q^{57} - 766771852 q^{58} - 1012573666 q^{59} - 1971234454 q^{60} - 647345946 q^{61} - 1194388172 q^{62} - 555585373 q^{63} + 518958419 q^{64} - 338512074 q^{65} + 675895108 q^{66} - 392635678 q^{67} - 455197554 q^{68} - 101630022 q^{69} + 1782905648 q^{70} - 1343501906 q^{71} + 989557136 q^{72} + 258965096 q^{73} + 1442922668 q^{74} + 1033741457 q^{75} + 856110679 q^{76} + 356196542 q^{77} + 2962377087 q^{78} + 13364317 q^{79} + 1517238459 q^{80} + 3496278075 q^{81} + 5656096334 q^{82} - 1934994591 q^{83} + 3595701858 q^{84} - 433920660 q^{85} - 777488307 q^{86} + 1581228292 q^{87} + 1947017746 q^{88} - 1431702022 q^{89} + 4879331944 q^{90} - 1194378542 q^{91} - 1617596606 q^{92} - 1634052086 q^{93} - 3110861041 q^{94} - 4990330756 q^{95} - 3946993916 q^{96} - 680060219 q^{97} - 3613812791 q^{98} - 9062971739 q^{99}+O(q^{100}) \) 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 −44.9385 20.4587 1507.47 593.339 −919.384 1888.67 −44734.7 −19264.4 −26663.7
1.2 −44.0474 −126.780 1428.17 1833.65 5584.34 −12625.1 −40354.9 −3609.74 −80767.3
1.3 −43.0671 246.901 1342.77 −2664.07 −10633.3 4770.85 −35779.0 41277.3 114734.
1.4 −42.1697 −47.0661 1266.28 −1838.66 1984.76 11530.1 −31807.9 −17467.8 77535.7
1.5 −40.9750 −118.418 1166.95 −1172.49 4852.18 −1951.85 −26836.4 −5660.14 48042.8
1.6 −40.5703 −223.085 1133.95 609.637 9050.64 953.313 −25232.7 30084.1 −24733.2
1.7 −40.0181 −35.4721 1089.45 1204.21 1419.53 −2650.65 −23108.3 −18424.7 −48190.0
1.8 −39.8572 67.0257 1076.60 2361.32 −2671.46 310.685 −22503.3 −15190.6 −94115.8
1.9 −39.4325 142.421 1042.92 527.108 −5616.01 6757.57 −20935.5 600.735 −20785.2
1.10 −38.8271 158.373 995.547 2413.06 −6149.18 −4498.31 −18774.8 5399.04 −93692.3
1.11 −38.0960 −41.0449 939.304 −640.824 1563.65 −11480.4 −16278.6 −17998.3 24412.8
1.12 −34.6139 229.413 686.125 720.318 −7940.88 1656.72 −6027.15 32947.3 −24933.0
1.13 −34.0602 200.862 648.099 −1479.41 −6841.39 3479.84 −4635.56 20662.4 50389.1
1.14 −33.5208 −264.603 611.643 −537.893 8869.72 −6669.34 −3340.11 50332.0 18030.6
1.15 −32.2837 −263.506 530.236 −1694.82 8506.96 11507.3 −588.710 49752.6 54714.9
1.16 −31.3487 19.3832 470.739 −2339.03 −607.636 1916.03 1293.47 −19307.3 73325.5
1.17 −31.1778 66.9513 460.057 −2375.20 −2087.40 −8362.22 1619.46 −15200.5 74053.7
1.18 −29.5210 −22.5126 359.491 1099.76 664.595 −7366.94 4502.23 −19176.2 −32466.0
1.19 −29.1594 247.924 338.272 −1458.15 −7229.32 6609.64 5065.81 41783.3 42518.9
1.20 −29.0884 −167.110 334.136 −482.894 4860.97 −1559.34 5173.78 8242.80 14046.6
See all 83 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.83
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(229\) \(1\)

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 229.10.a.a ✓ 83
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
229.10.a.a ✓ 83 1.a even 1 1 trivial