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Properties

Label 229.8.a.b
Level $229$
Weight $8$
Character orbit 229.a
Self dual yes
Analytic conductor $71.536$
Analytic rank $0$
Dimension $69$
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] = [229,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("229.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 229 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(71.5361708359\)
Analytic rank: \(0\)
Dimension: \(69\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 69 q + 55 q^{2} + 175 q^{3} + 4645 q^{4} + 750 q^{5} + 865 q^{6} + 2389 q^{7} + 10623 q^{8} + 58694 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 69 q + 55 q^{2} + 175 q^{3} + 4645 q^{4} + 750 q^{5} + 865 q^{6} + 2389 q^{7} + 10623 q^{8} + 58694 q^{9} + 5233 q^{10} + 47843 q^{11} + 20276 q^{12} + 10985 q^{13} + 62011 q^{14} + 35315 q^{15} + 318937 q^{16} + 51833 q^{17} + 126243 q^{18} + 146920 q^{19} + 76952 q^{20} + 140717 q^{21} + 74407 q^{22} + 106422 q^{23} + 146627 q^{24} + 1284893 q^{25} + 408120 q^{26} + 605704 q^{27} + 423606 q^{28} + 478505 q^{29} - 30518 q^{30} + 304065 q^{31} + 1406834 q^{32} + 614760 q^{33} + 543933 q^{34} + 1667803 q^{35} + 4833380 q^{36} + 700808 q^{37} + 1406414 q^{38} + 2160586 q^{39} + 340561 q^{40} + 2612538 q^{41} + 1364169 q^{42} + 2343094 q^{43} + 6882441 q^{44} + 2132365 q^{45} + 972300 q^{46} + 3383414 q^{47} + 3062914 q^{48} + 9158316 q^{49} + 5401577 q^{50} + 9413395 q^{51} - 418219 q^{52} + 2412175 q^{53} + 4524553 q^{54} + 3181300 q^{55} + 12864801 q^{56} + 1827262 q^{57} - 6598324 q^{58} + 8059568 q^{59} - 30266566 q^{60} - 1901917 q^{61} - 11797196 q^{62} + 4553273 q^{63} + 14008179 q^{64} + 4456773 q^{65} - 14560316 q^{66} - 253056 q^{67} + 12479838 q^{68} + 6047112 q^{69} + 6770688 q^{70} + 19992307 q^{71} + 34551980 q^{72} + 1009564 q^{73} + 20504672 q^{74} + 34365872 q^{75} + 34793343 q^{76} + 16602674 q^{77} + 39794751 q^{78} + 22604489 q^{79} + 45180411 q^{80} + 82865049 q^{81} + 98260022 q^{82} + 43087881 q^{83} + 83913558 q^{84} + 26329761 q^{85} + 58073901 q^{86} + 31722163 q^{87} + 81769722 q^{88} + 39050906 q^{89} + 131306212 q^{90} + 69177412 q^{91} + 66642190 q^{92} + 39131539 q^{93} + 102280663 q^{94} + 94948037 q^{95} + 208509652 q^{96} + 41800599 q^{97} + 113665141 q^{98} + 123953011 q^{99}+O(q^{100}) \) 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 −21.4309 −43.8110 331.283 529.950 938.907 −199.365 −4356.52 −267.601 −11357.3
1.2 −20.9421 −81.8766 310.573 −5.17768 1714.67 −1085.11 −3823.46 4516.77 108.432
1.3 −20.9223 17.4869 309.741 −214.529 −365.865 599.249 −3802.43 −1881.21 4488.44
1.4 −20.8591 −26.5232 307.103 164.221 553.252 1265.94 −3735.94 −1483.52 −3425.51
1.5 −19.8334 73.0675 265.365 −428.835 −1449.18 612.539 −2724.42 3151.86 8505.28
1.6 −19.0544 −23.6396 235.069 236.074 450.437 −515.433 −2040.13 −1628.17 −4498.24
1.7 −18.6856 77.5987 221.152 −75.9997 −1449.98 −94.3328 −1740.61 3834.56 1420.10
1.8 −18.2671 47.8546 205.685 485.082 −874.163 1363.77 −1419.09 103.062 −8861.02
1.9 −18.0777 69.8211 198.804 −452.808 −1262.21 −1793.29 −1279.97 2687.99 8185.74
1.10 −17.3801 −56.8158 174.067 −427.560 987.464 −547.526 −800.656 1041.04 7431.03
1.11 −17.0514 −29.8380 162.749 −248.997 508.779 494.263 −592.515 −1296.69 4245.74
1.12 −16.3696 18.6424 139.965 −253.148 −305.169 −163.599 −195.857 −1839.46 4143.94
1.13 −15.9500 92.0929 126.402 292.964 −1468.88 −240.070 25.4940 6294.09 −4672.76
1.14 −15.1138 −32.2684 100.428 491.760 487.699 −386.506 416.712 −1145.75 −7432.38
1.15 −14.1616 28.2785 72.5504 353.095 −400.469 −182.046 785.254 −1387.33 −5000.38
1.16 −12.5924 38.1653 30.5688 −134.659 −480.593 −738.569 1226.89 −730.413 1695.68
1.17 −10.7045 2.75857 −13.4136 327.438 −29.5292 −1703.21 1513.76 −2179.39 −3505.06
1.18 −10.6565 39.1013 −14.4384 −304.532 −416.684 −256.518 1517.90 −658.092 3245.25
1.19 −10.4514 −80.5560 −18.7686 −310.429 841.921 414.009 1533.93 4302.26 3244.41
1.20 −9.92344 −90.8222 −29.5253 −302.267 901.269 −1001.78 1563.19 6061.67 2999.53
See all 69 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.69
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 229.8.a.b ✓ 69
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
229.8.a.b ✓ 69 1.a even 1 1 trivial