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Properties

Label 529.8.a.k
Level $529$
Weight $8$
Character orbit 529.a
Self dual yes
Analytic conductor $165.252$
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] = [529,8,Mod(1,529)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(529, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("529.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 529 = 23^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 529.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: \(165.251678481\)
Analytic rank: \(0\)
Dimension: \(65\)
Twist minimal: no (minimal twist has level 23)
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.

\(\operatorname{Tr}(f)(q) = \) \( 65 q + 8 q^{2} + 14 q^{3} + 3776 q^{4} + 1181 q^{5} + 1082 q^{6} + 3628 q^{7} - 5757 q^{8} + 38263 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 65 q + 8 q^{2} + 14 q^{3} + 3776 q^{4} + 1181 q^{5} + 1082 q^{6} + 3628 q^{7} - 5757 q^{8} + 38263 q^{9} + 6879 q^{10} + 26147 q^{11} + 35689 q^{12} - 100 q^{13} + 55020 q^{14} + 92935 q^{15} + 176888 q^{16} + 89552 q^{17} + 190670 q^{18} + 169530 q^{19} + 210098 q^{20} + 190951 q^{21} + 218663 q^{22} + 190835 q^{24} + 721210 q^{25} + 186894 q^{26} - 670321 q^{27} + 366019 q^{28} + 340963 q^{29} + 671873 q^{30} - 175163 q^{31} - 406068 q^{32} + 726837 q^{33} + 1110895 q^{34} - 596883 q^{35} + 361164 q^{36} + 790283 q^{37} + 1963838 q^{38} - 21993 q^{39} + 1600645 q^{40} - 623859 q^{41} + 2696872 q^{42} + 1189188 q^{43} + 3885217 q^{44} + 2158369 q^{45} + 1365507 q^{47} + 1218748 q^{48} + 5854235 q^{49} - 2728396 q^{50} + 4554663 q^{51} + 7887791 q^{52} + 4362562 q^{53} + 5072916 q^{54} - 3989544 q^{55} + 12949007 q^{56} + 551326 q^{57} - 7443768 q^{58} + 13201460 q^{59} + 19296285 q^{60} + 8838644 q^{61} - 14615728 q^{62} + 12479203 q^{63} - 9711751 q^{64} + 13544545 q^{65} + 4907285 q^{66} + 5760553 q^{67} + 17044359 q^{68} + 6122411 q^{70} + 4531745 q^{71} + 10759455 q^{72} - 2535486 q^{73} + 21404597 q^{74} + 22917428 q^{75} + 21958812 q^{76} + 11733492 q^{77} - 35437508 q^{78} + 27234683 q^{79} + 28932289 q^{80} + 3995985 q^{81} - 11682773 q^{82} + 39797339 q^{83} + 33971997 q^{84} - 19766357 q^{85} + 43836707 q^{86} - 13972389 q^{87} + 33099818 q^{88} + 41235321 q^{89} + 17210275 q^{90} + 30292506 q^{91} + 47965734 q^{93} - 9981701 q^{94} - 5998448 q^{95} - 48738628 q^{96} + 13984256 q^{97} + 56728191 q^{98} + 81303822 q^{99}+O(q^{100}) \) 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 −22.2450 −21.1615 366.842 −276.800 470.738 −915.622 −5313.05 −1739.19 6157.42
1.2 −21.4744 55.6054 333.151 467.411 −1194.09 775.571 −4405.49 904.960 −10037.4
1.3 −21.1812 31.2436 320.644 258.424 −661.778 −1468.28 −4080.44 −1210.84 −5473.73
1.4 −19.7339 −56.3928 261.425 −195.031 1112.85 1005.58 −2632.99 993.149 3848.70
1.5 −19.5827 47.5792 255.484 129.745 −931.731 −65.2068 −2496.48 76.7800 −2540.76
1.6 −19.2999 −10.3617 244.486 402.837 199.979 −1437.92 −2248.17 −2079.64 −7774.72
1.7 −18.7709 −24.2008 224.348 −516.812 454.271 750.443 −1808.53 −1601.32 9701.04
1.8 −17.9923 −66.1588 195.721 36.4705 1190.35 −328.468 −1218.46 2189.99 −656.186
1.9 −17.6822 69.7192 184.662 −167.250 −1232.79 75.3096 −1001.91 2673.76 2957.35
1.10 −17.1110 −49.0517 164.787 −235.358 839.324 760.433 −629.455 219.067 4027.21
1.11 −17.0166 30.8411 161.564 −525.842 −524.809 −254.675 −571.147 −1235.83 8948.03
1.12 −16.5789 64.1589 146.860 −77.5813 −1063.68 1527.82 −312.680 1929.36 1286.21
1.13 −16.1206 −90.2943 131.875 346.475 1455.60 204.913 −62.4603 5966.06 −5585.39
1.14 −14.5181 −62.5740 82.7745 242.540 908.454 649.054 656.588 1728.51 −3521.21
1.15 −13.6685 32.5199 58.8282 283.224 −444.499 −1401.09 945.476 −1129.46 −3871.25
1.16 −13.3625 23.0798 50.5569 241.413 −308.404 1442.10 1034.83 −1654.32 −3225.88
1.17 −12.5291 −36.6297 28.9784 291.166 458.937 −950.506 1240.65 −845.264 −3648.05
1.18 −11.4685 −17.7230 3.52642 116.293 203.257 705.153 1427.52 −1872.89 −1333.71
1.19 −11.2545 50.2769 −1.33538 −420.176 −565.843 −429.788 1455.61 340.767 4728.88
1.20 −9.94357 72.7469 −29.1255 −130.748 −723.363 −514.778 1562.39 3105.11 1300.10
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
\(23\) \( +1 \)

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 529.8.a.k 65
23.b odd 2 1 529.8.a.j 65
23.c even 11 2 23.8.c.a ✓ 130
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.8.c.a ✓ 130 23.c even 11 2
529.8.a.j 65 23.b odd 2 1
529.8.a.k 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 \(S_{8}^{\mathrm{new}}(\Gamma_0(529))\):

\( T_{2}^{65} - 8 T_{2}^{64} - 6016 T_{2}^{63} + 49535 T_{2}^{62} + 17146698 T_{2}^{61} + \cdots - 13\!\cdots\!36 \) Copy content Toggle raw display
\( T_{5}^{65} - 1181 T_{5}^{64} - 2202287 T_{5}^{63} + 3045718614 T_{5}^{62} + 2137830747407 T_{5}^{61} + \cdots - 90\!\cdots\!75 \) Copy content Toggle raw display