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Properties

Label 229.4.a.b
Level $229$
Weight $4$
Character orbit 229.a
Self dual yes
Analytic conductor $13.511$
Analytic rank $0$
Dimension $31$
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] = [229,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("229.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 229 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(13.5114373913\)
Analytic rank: \(0\)
Dimension: \(31\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 31 q + 11 q^{2} + 19 q^{3} + 149 q^{4} + 30 q^{5} + 31 q^{6} + 29 q^{7} + 135 q^{8} + 338 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 31 q + 11 q^{2} + 19 q^{3} + 149 q^{4} + 30 q^{5} + 31 q^{6} + 29 q^{7} + 135 q^{8} + 338 q^{9} + 13 q^{10} + 369 q^{11} + 128 q^{12} + 65 q^{13} + 237 q^{14} + 260 q^{15} + 737 q^{16} + 185 q^{17} + 453 q^{18} + 397 q^{19} + 352 q^{20} + 404 q^{21} + 145 q^{22} + 299 q^{23} + 467 q^{24} + 863 q^{25} + 762 q^{26} + 670 q^{27} + 538 q^{28} + 761 q^{29} + 142 q^{30} + 277 q^{31} + 1306 q^{32} + 78 q^{33} + 323 q^{34} + 1248 q^{35} + 2000 q^{36} + 410 q^{37} + 456 q^{38} + 808 q^{39} + 841 q^{40} + 732 q^{41} + 693 q^{42} + 1053 q^{43} + 2813 q^{44} + 640 q^{45} + 1658 q^{46} + 1287 q^{47} + 802 q^{48} + 2544 q^{49} + 1917 q^{50} + 2212 q^{51} + 961 q^{52} + 1146 q^{53} + 1429 q^{54} + 610 q^{55} + 2209 q^{56} + 2050 q^{57} - 1154 q^{58} + 3730 q^{59} - 3886 q^{60} - 918 q^{61} - 2198 q^{62} - 469 q^{63} + 571 q^{64} + 1538 q^{65} - 6452 q^{66} - 1250 q^{67} - 1838 q^{68} - 1554 q^{69} - 7892 q^{70} + 4534 q^{71} - 5500 q^{72} - 1796 q^{73} - 1952 q^{74} - 1663 q^{75} - 4645 q^{76} + 706 q^{77} - 5637 q^{78} + 605 q^{79} - 1989 q^{80} + 1095 q^{81} - 14774 q^{82} + 3665 q^{83} - 6834 q^{84} - 1404 q^{85} - 1381 q^{86} - 1316 q^{87} - 8614 q^{88} + 2682 q^{89} - 12908 q^{90} - 2518 q^{91} + 42 q^{92} - 3602 q^{93} - 5015 q^{94} + 2452 q^{95} - 9236 q^{96} - 3529 q^{97} - 2969 q^{98} + 7669 q^{99}+O(q^{100}) \) 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 −5.34099 7.13544 20.5262 −2.38275 −38.1103 −12.7399 −66.9023 23.9145 12.7262
1.2 −4.85023 0.217457 15.5248 −8.77475 −1.05472 −31.7994 −36.4968 −26.9527 42.5596
1.3 −4.82118 −9.21332 15.2438 1.11404 44.4191 −6.48518 −34.9235 57.8853 −5.37097
1.4 −4.62232 −5.88106 13.3658 12.6265 27.1841 28.3559 −24.8024 7.58681 −58.3639
1.5 −4.47237 7.90230 12.0021 9.54784 −35.3420 6.59527 −17.8987 35.4463 −42.7014
1.6 −3.75338 −2.66690 6.08788 −10.0822 10.0099 −8.48462 7.17691 −19.8877 37.8424
1.7 −3.65463 0.921327 5.35630 −4.60383 −3.36711 28.9944 9.66173 −26.1512 16.8253
1.8 −2.88450 −5.88205 0.320348 15.3082 16.9668 12.1154 22.1520 7.59846 −44.1565
1.9 −2.37384 4.18191 −2.36489 12.7133 −9.92718 −15.4888 24.6046 −9.51161 −30.1794
1.10 −2.34939 7.83071 −2.48035 18.1134 −18.3974 33.7446 24.6225 34.3200 −42.5554
1.11 −2.22528 3.69690 −3.04814 −16.9655 −8.22662 −30.5225 24.5852 −13.3330 37.7529
1.12 −0.773463 8.51802 −7.40175 −6.68052 −6.58838 17.6163 11.9127 45.5567 5.16714
1.13 −0.574614 −4.96890 −7.66982 6.51738 2.85519 −32.8464 9.00409 −2.31008 −3.74498
1.14 −0.538616 −3.93427 −7.70989 −13.3983 2.11906 −9.67038 8.46161 −11.5216 7.21652
1.15 0.326266 0.491876 −7.89355 −21.3626 0.160483 16.2135 −5.18553 −26.7581 −6.96991
1.16 0.382988 −0.630231 −7.85332 18.9013 −0.241371 16.7361 −6.07163 −26.6028 7.23895
1.17 0.773939 9.25080 −7.40102 15.1765 7.15956 −25.3663 −11.9195 58.5772 11.7457
1.18 0.924652 −2.55490 −7.14502 −6.54440 −2.36239 −15.1979 −14.0039 −20.4725 −6.05129
1.19 1.96874 −6.79733 −4.12405 −4.85319 −13.3822 7.61828 −23.8691 19.2036 −9.55469
1.20 2.11062 −9.41426 −3.54527 −18.7710 −19.8700 −27.5847 −24.3677 61.6283 −39.6185
See all 31 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.31
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 229.4.a.b ✓ 31
3.b odd 2 1 2061.4.a.e 31
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
229.4.a.b ✓ 31 1.a even 1 1 trivial
2061.4.a.e 31 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{31} - 11 T_{2}^{30} - 138 T_{2}^{29} + 1858 T_{2}^{28} + 7372 T_{2}^{27} - 138771 T_{2}^{26} + \cdots - 852531609600 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(229))\). Copy content Toggle raw display