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Properties

Label 229.10.a.b
Level $229$
Weight $10$
Character orbit 229.a
Self dual yes
Analytic conductor $117.943$
Analytic rank $0$
Dimension $88$
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: \(0\)
Dimension: \(88\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 95 q^{2} + 559 q^{3} + 23893 q^{4} + 3750 q^{5} + 7753 q^{6} + 11929 q^{7} + 73215 q^{8} + 623021 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q + 95 q^{2} + 559 q^{3} + 23893 q^{4} + 3750 q^{5} + 7753 q^{6} + 11929 q^{7} + 73215 q^{8} + 623021 q^{9} + 82449 q^{10} + 432393 q^{11} + 325148 q^{12} + 199927 q^{13} + 507331 q^{14} + 983660 q^{15} + 7263897 q^{16} + 762991 q^{17} + 1776243 q^{18} + 2837905 q^{19} + 3129128 q^{20} + 4230896 q^{21} + 1639279 q^{22} + 3783415 q^{23} + 5776019 q^{24} + 36266780 q^{25} + 5292504 q^{26} + 15275974 q^{27} + 6154742 q^{28} + 21509999 q^{29} + 15791674 q^{30} + 12728433 q^{31} + 47041530 q^{32} + 1358442 q^{33} + 21831685 q^{34} + 56719476 q^{35} + 189649100 q^{36} + 22078674 q^{37} - 2641370 q^{38} + 73762288 q^{39} + 55575089 q^{40} + 91139778 q^{41} - 11332839 q^{42} + 34141217 q^{43} + 244819673 q^{44} + 58021012 q^{45} + 76342964 q^{46} + 188105747 q^{47} + 227850586 q^{48} + 611389397 q^{49} + 307167297 q^{50} + 327668572 q^{51} + 182599589 q^{52} + 308046022 q^{53} + 58543009 q^{54} + 274945738 q^{55} + 447225657 q^{56} + 214220710 q^{57} + 503292936 q^{58} + 1033266448 q^{59} + 2676260010 q^{60} + 377337342 q^{61} + 845386648 q^{62} + 659996659 q^{63} + 2183997059 q^{64} + 995402730 q^{65} + 423749256 q^{66} + 228293056 q^{67} - 385853782 q^{68} - 149107098 q^{69} - 1561527768 q^{70} + 1152595726 q^{71} - 2904401498 q^{72} - 716108198 q^{73} - 124439538 q^{74} - 430333659 q^{75} - 1846794277 q^{76} + 106541154 q^{77} - 3350711613 q^{78} + 915220177 q^{79} + 51042047 q^{80} + 2546581456 q^{81} - 7224002386 q^{82} + 2068787245 q^{83} - 1010276164 q^{84} - 44841828 q^{85} + 489721817 q^{86} - 221392860 q^{87} - 2996687306 q^{88} + 2392100104 q^{89} - 735262610 q^{90} + 728793942 q^{91} + 481724792 q^{92} + 269944470 q^{93} - 2250585165 q^{94} + 4315932004 q^{95} + 1953595604 q^{96} + 1273586365 q^{97} + 5117800519 q^{98} + 9979387373 q^{99}+O(q^{100}) \) 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 −44.5991 245.007 1477.08 1507.62 −10927.1 8358.52 −43041.6 40345.4 −67238.5
1.2 −43.2702 −220.771 1360.31 243.744 9552.80 3655.04 −36706.7 29056.7 −10546.9
1.3 −42.7651 −201.535 1316.85 −2637.13 8618.67 −8210.05 −34419.5 20933.5 112777.
1.4 −42.0866 142.874 1259.28 −1312.97 −6013.09 −7450.82 −31450.4 730.061 55258.4
1.5 −40.7770 176.106 1150.76 53.9115 −7181.08 −6774.17 −26046.7 11330.4 −2198.35
1.6 −40.5777 65.4848 1134.55 −1459.80 −2657.22 517.417 −25261.7 −15394.7 59235.3
1.7 −38.4102 −200.556 963.345 2017.41 7703.40 7468.27 −17336.3 20539.7 −77489.3
1.8 −37.9775 −158.615 930.291 −1786.39 6023.78 −313.690 −15885.6 5475.58 67842.5
1.9 −37.9471 −62.7593 927.980 1934.43 2381.53 9326.53 −15785.2 −15744.3 −73405.8
1.10 −37.7214 271.409 910.901 628.348 −10237.9 −8655.96 −15047.1 53979.9 −23702.1
1.11 −36.2104 140.885 799.191 16.6253 −5101.50 10117.3 −10399.3 165.591 −602.009
1.12 −35.6840 −79.7952 761.348 148.705 2847.41 −5616.51 −8897.75 −13315.7 −5306.37
1.13 −35.1209 −28.5339 721.477 317.887 1002.13 2566.35 −7357.03 −18868.8 −11164.5
1.14 −33.8772 −221.098 635.667 2198.18 7490.20 −8220.07 −4189.48 29201.5 −74468.1
1.15 −33.0232 −126.346 578.531 −1208.51 4172.36 5705.40 −2197.08 −3719.58 39908.7
1.16 −31.4910 64.5690 479.685 53.2915 −2033.35 10812.9 1017.64 −15513.8 −1678.20
1.17 −31.1138 153.206 456.071 2247.63 −4766.81 −8167.19 1740.18 3788.93 −69932.3
1.18 −30.6103 131.818 424.988 −1558.99 −4034.98 789.568 2663.45 −2307.08 47721.2
1.19 −28.9772 −131.375 327.680 1686.94 3806.89 587.910 5341.08 −2423.56 −48882.8
1.20 −27.5695 121.043 248.076 635.458 −3337.08 −9297.84 7276.25 −5031.67 −17519.2
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.88
Significant digits:
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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.b ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
229.10.a.b ✓ 88 1.a even 1 1 trivial