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Properties

Label 229.6.a.a
Level $229$
Weight $6$
Character orbit 229.a
Self dual yes
Analytic conductor $36.728$
Analytic rank $1$
Dimension $45$
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] = [229,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("229.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 229 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(36.7278947372\)
Analytic rank: \(1\)
Dimension: \(45\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 45 q - 25 q^{2} - 65 q^{3} + 629 q^{4} - 150 q^{5} - 155 q^{6} - 283 q^{7} - 1185 q^{8} + 2912 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 45 q - 25 q^{2} - 65 q^{3} + 629 q^{4} - 150 q^{5} - 155 q^{6} - 283 q^{7} - 1185 q^{8} + 2912 q^{9} - 631 q^{10} - 4125 q^{11} - 2500 q^{12} - 845 q^{13} - 3945 q^{14} - 3805 q^{15} + 7097 q^{16} - 4213 q^{17} - 7041 q^{18} - 8356 q^{19} - 5912 q^{20} - 9073 q^{21} - 3421 q^{22} - 5434 q^{23} - 11053 q^{24} + 22645 q^{25} - 15756 q^{26} - 19976 q^{27} - 19194 q^{28} - 20317 q^{29} - 2846 q^{30} - 4863 q^{31} - 40310 q^{32} - 9816 q^{33} - 23319 q^{34} - 46449 q^{35} + 27020 q^{36} - 26840 q^{37} - 12822 q^{38} - 53438 q^{39} - 31471 q^{40} - 53694 q^{41} - 31887 q^{42} - 44854 q^{43} - 124087 q^{44} - 53993 q^{45} - 52944 q^{46} - 95394 q^{47} - 58646 q^{48} + 55734 q^{49} - 115863 q^{50} - 134465 q^{51} - 11627 q^{52} - 78819 q^{53} - 77063 q^{54} - 49492 q^{55} - 220055 q^{56} - 76598 q^{57} - 52352 q^{58} - 285908 q^{59} - 624214 q^{60} - 188207 q^{61} - 326768 q^{62} - 254407 q^{63} - 290253 q^{64} - 270229 q^{65} - 588908 q^{66} - 263700 q^{67} - 427586 q^{68} - 275856 q^{69} - 559512 q^{70} - 366149 q^{71} - 883144 q^{72} - 116944 q^{73} - 407948 q^{74} - 659308 q^{75} - 448329 q^{76} - 331802 q^{77} - 456729 q^{78} - 329159 q^{79} - 397221 q^{80} - 74199 q^{81} - 294314 q^{82} - 515383 q^{83} - 328710 q^{84} - 125925 q^{85} - 485479 q^{86} - 9125 q^{87} - 34206 q^{88} - 224478 q^{89} - 417056 q^{90} - 194812 q^{91} - 240390 q^{92} + 697 q^{93} - 48733 q^{94} - 661931 q^{95} - 117884 q^{96} - 33963 q^{97} - 39011 q^{98} - 867317 q^{99}+O(q^{100}) \) 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 −10.9730 26.5629 88.4070 −45.6966 −291.475 −95.3748 −618.955 462.589 501.429
1.2 −10.5028 −29.5272 78.3095 106.162 310.120 60.6214 −486.381 628.857 −1115.01
1.3 −10.3467 12.7944 75.0547 −46.2522 −132.380 218.712 −445.475 −79.3029 478.559
1.4 −10.3375 −16.4773 74.8648 −40.8596 170.335 126.070 −443.116 28.5027 422.388
1.5 −9.18259 2.34835 52.3200 −100.748 −21.5640 −90.0030 −186.590 −237.485 925.125
1.6 −9.16632 −16.2032 52.0213 31.1898 148.524 −67.8755 −183.522 19.5434 −285.895
1.7 −9.06805 −11.2861 50.2296 50.2845 102.343 199.797 −165.307 −115.625 −455.982
1.8 −8.92261 20.2481 47.6129 47.0371 −180.666 −53.4987 −139.308 166.986 −419.693
1.9 −8.34819 −22.5690 37.6922 −88.5938 188.410 −230.734 −47.5198 266.358 739.598
1.10 −7.55832 21.6350 25.1281 −11.7797 −163.524 −223.648 51.9397 225.073 89.0345
1.11 −7.29150 −2.83795 21.1660 96.3180 20.6929 45.7191 78.9960 −234.946 −702.303
1.12 −6.84886 16.9363 14.9068 5.21653 −115.994 71.5857 117.069 43.8381 −35.7272
1.13 −6.60759 −18.9698 11.6602 65.2964 125.345 −105.857 134.397 116.854 −431.452
1.14 −5.91798 −1.30065 3.02254 −14.2490 7.69723 −97.8587 171.488 −241.308 84.3254
1.15 −5.46526 10.2987 −2.13092 54.3160 −56.2853 187.553 186.534 −136.936 −296.851
1.16 −4.77103 −7.78442 −9.23730 −90.7999 37.1397 183.360 196.744 −182.403 433.209
1.17 −3.15986 −9.93205 −22.0153 −41.8173 31.3839 −26.3827 170.681 −144.354 132.137
1.18 −2.63432 18.5494 −25.0604 11.1239 −48.8651 32.5937 150.315 101.081 −29.3038
1.19 −2.34991 −19.7544 −26.4779 −69.8239 46.4212 −165.203 137.418 147.238 164.080
1.20 −1.96186 14.1596 −28.1511 −74.7459 −27.7792 108.020 118.008 −42.5049 146.641
See all 45 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.45
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 229.6.a.a ✓ 45
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
229.6.a.a ✓ 45 1.a even 1 1 trivial