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Properties

Label 229.6.a.b
Level $229$
Weight $6$
Character orbit 229.a
Self dual yes
Analytic conductor $36.728$
Analytic rank $0$
Dimension $50$
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: \(0\)
Dimension: \(50\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 50 q + 27 q^{2} + 43 q^{3} + 869 q^{4} + 150 q^{5} + 349 q^{6} + 305 q^{7} + 1311 q^{8} + 4937 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 50 q + 27 q^{2} + 43 q^{3} + 869 q^{4} + 150 q^{5} + 349 q^{6} + 305 q^{7} + 1311 q^{8} + 4937 q^{9} + 369 q^{10} + 3619 q^{11} + 956 q^{12} + 1183 q^{13} + 1935 q^{14} + 3395 q^{15} + 16057 q^{16} + 1567 q^{17} + 5595 q^{18} + 7528 q^{19} + 8488 q^{20} + 6803 q^{21} + 3355 q^{22} + 9378 q^{23} + 13139 q^{24} + 38270 q^{25} + 12636 q^{26} + 15016 q^{27} + 4326 q^{28} + 30143 q^{29} + 22354 q^{30} + 22045 q^{31} + 52874 q^{32} + 16320 q^{33} + 4425 q^{34} + 66251 q^{35} + 124220 q^{36} + 540 q^{37} + 24722 q^{38} + 43906 q^{39} + 16529 q^{40} + 57252 q^{41} + 3393 q^{42} + 36502 q^{43} + 123721 q^{44} + 18907 q^{45} + 44392 q^{46} + 37146 q^{47} + 51946 q^{48} + 139769 q^{49} + 111637 q^{50} + 136039 q^{51} + 85717 q^{52} + 56013 q^{53} + 86233 q^{54} + 47308 q^{55} + 62185 q^{56} + 27370 q^{57} + 213164 q^{58} + 272880 q^{59} + 612330 q^{60} + 129177 q^{61} + 452460 q^{62} + 167539 q^{63} + 620259 q^{64} + 219025 q^{65} + 639192 q^{66} + 363450 q^{67} + 305210 q^{68} + 227598 q^{69} + 598592 q^{70} + 418379 q^{71} + 816958 q^{72} + 250180 q^{73} + 436174 q^{74} + 231816 q^{75} + 564379 q^{76} + 140636 q^{77} + 295707 q^{78} + 133481 q^{79} + 404287 q^{80} + 698206 q^{81} + 370270 q^{82} + 265573 q^{83} + 694532 q^{84} + 261547 q^{85} + 320253 q^{86} + 310023 q^{87} + 255270 q^{88} + 433298 q^{89} - 251810 q^{90} + 253416 q^{91} + 172768 q^{92} + 258009 q^{93} + 1215 q^{94} + 265329 q^{95} - 108076 q^{96} + 293 q^{97} + 317415 q^{98} + 714383 q^{99}+O(q^{100}) \) 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 −11.1242 −14.8778 91.7475 −20.7747 165.503 −136.720 −664.643 −21.6515 231.102
1.2 −10.7223 4.77221 82.9674 63.9565 −51.1690 76.9174 −546.486 −220.226 −685.760
1.3 −10.0048 5.43439 68.0953 60.1348 −54.3698 −183.776 −361.125 −213.467 −601.634
1.4 −9.66764 27.4764 61.4633 79.2121 −265.632 120.444 −284.841 511.951 −765.794
1.5 −9.65010 10.3911 61.1245 −55.8371 −100.275 −15.6586 −281.055 −135.026 538.834
1.6 −9.28372 −29.1782 54.1875 −60.1853 270.882 87.9472 −205.983 608.367 558.744
1.7 −8.24398 −5.56632 35.9632 −16.9412 45.8886 38.0591 −32.6724 −212.016 139.663
1.8 −8.00946 −23.1812 32.1515 53.8830 185.669 −224.220 −1.21360 294.369 −431.574
1.9 −7.06159 18.3415 17.8660 −20.3727 −129.520 206.558 99.8085 93.4107 143.864
1.10 −6.97849 29.3627 16.6993 −75.4864 −204.907 46.7598 106.776 619.168 526.781
1.11 −6.66207 −12.0225 12.3832 −93.3701 80.0949 174.426 130.688 −98.4591 622.038
1.12 −6.38437 −22.2444 8.76014 −14.9030 142.016 61.2210 148.372 251.814 95.1463
1.13 −5.85856 2.00049 2.32275 −31.4703 −11.7200 −130.452 173.866 −238.998 184.371
1.14 −5.23758 4.81132 −4.56771 75.9820 −25.1997 69.9851 191.526 −219.851 −397.962
1.15 −4.82304 21.1358 −8.73829 95.9806 −101.939 −125.505 196.482 203.722 −462.918
1.16 −4.68724 −29.1341 −10.0298 1.03505 136.558 10.4986 197.004 605.794 −4.85151
1.17 −4.45881 16.6305 −12.1190 −92.1989 −74.1522 −89.0949 196.718 33.5735 411.098
1.18 −3.98085 −22.5006 −16.1528 52.5931 89.5716 231.901 191.689 263.277 −209.366
1.19 −3.25457 −2.53713 −21.4078 36.4212 8.25727 −224.178 173.819 −236.563 −118.535
1.20 −2.54892 29.9772 −25.5030 42.4176 −76.4095 109.918 146.571 655.632 −108.119
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.50
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 229.6.a.b ✓ 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
229.6.a.b ✓ 50 1.a even 1 1 trivial