Properties

Label 229.8.a.a
Level $229$
Weight $8$
Character orbit 229.a
Self dual yes
Analytic conductor $71.536$
Analytic rank $1$
Dimension $64$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(1\)
Dimension: \(64\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 49 q^{2} - 149 q^{3} + 3685 q^{4} - 750 q^{5} - 2159 q^{6} - 1727 q^{7} - 9345 q^{8} + 40469 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 49 q^{2} - 149 q^{3} + 3685 q^{4} - 750 q^{5} - 2159 q^{6} - 1727 q^{7} - 9345 q^{8} + 40469 q^{9} - 4767 q^{10} - 37341 q^{11} - 21196 q^{12} - 15379 q^{13} - 20309 q^{14} - 72685 q^{15} + 175577 q^{16} - 46427 q^{17} - 101205 q^{18} - 154876 q^{19} - 211048 q^{20} - 192679 q^{21} - 74665 q^{22} - 234254 q^{23} - 433981 q^{24} + 894268 q^{25} - 330072 q^{26} - 339080 q^{27} - 234954 q^{28} - 984835 q^{29} - 786518 q^{30} - 530083 q^{31} - 1575054 q^{32} - 247728 q^{33} - 399363 q^{34} - 2276697 q^{35} + 1334180 q^{36} - 312252 q^{37} - 20258 q^{38} - 1635830 q^{39} - 1579439 q^{40} - 1936248 q^{41} - 117591 q^{42} - 1155214 q^{43} - 4021111 q^{44} - 1148135 q^{45} - 3505156 q^{46} - 2845966 q^{47} - 2245502 q^{48} + 5040601 q^{49} - 5973423 q^{50} - 4382309 q^{51} - 5480107 q^{52} - 4733921 q^{53} - 4293431 q^{54} - 2142700 q^{55} - 2940639 q^{56} - 4098914 q^{57} + 3846688 q^{58} - 12762764 q^{59} + 16907130 q^{60} + 7053051 q^{61} + 12246424 q^{62} - 741485 q^{63} + 22841571 q^{64} - 9468081 q^{65} + 10841688 q^{66} - 9032298 q^{67} + 3205610 q^{68} - 4508814 q^{69} - 2348456 q^{70} - 26446589 q^{71} - 25225238 q^{72} - 15073288 q^{73} - 26555526 q^{74} - 23440332 q^{75} - 39229357 q^{76} - 21863532 q^{77} - 37567917 q^{78} - 19440071 q^{79} - 44462737 q^{80} - 10066988 q^{81} - 66111898 q^{82} - 32430347 q^{83} - 113541160 q^{84} - 34041391 q^{85} - 75796135 q^{86} - 66802881 q^{87} - 92879346 q^{88} - 56039558 q^{89} - 126309134 q^{90} - 60602568 q^{91} - 86381764 q^{92} - 53315277 q^{93} - 79371909 q^{94} - 96246623 q^{95} - 166200220 q^{96} - 36487673 q^{97} - 112358237 q^{98} - 137902009 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −22.3614 70.2006 372.031 371.547 −1569.78 −1164.59 −5456.86 2741.13 −8308.30
1.2 −22.1804 2.55311 363.971 −408.637 −56.6291 −773.516 −5233.93 −2180.48 9063.73
1.3 −21.5465 −65.8686 336.254 −56.5868 1419.24 1001.03 −4487.15 2151.67 1219.25
1.4 −21.1177 63.4421 317.957 −22.1263 −1339.75 1017.87 −4011.45 1837.90 467.255
1.5 −20.5994 26.3400 296.334 204.460 −542.588 −191.516 −3467.57 −1493.20 −4211.75
1.6 −19.6676 −71.7639 258.814 −449.104 1411.42 849.145 −2572.80 2963.06 8832.80
1.7 −19.2888 −25.9609 244.057 −96.0755 500.754 −1294.56 −2238.60 −1513.03 1853.18
1.8 −18.4838 32.8648 213.649 176.472 −607.465 −1472.53 −1583.12 −1106.91 −3261.86
1.9 −16.7965 18.8316 154.123 −522.855 −316.305 1527.53 −438.780 −1832.37 8782.15
1.10 −16.3019 −72.4049 137.754 245.722 1180.34 −980.480 −159.002 3055.47 −4005.75
1.11 −15.7547 −80.4131 120.212 281.633 1266.89 1042.03 122.701 4279.26 −4437.05
1.12 −15.5398 −75.2696 113.485 202.467 1169.67 874.841 225.566 3478.51 −3146.30
1.13 −15.4355 26.8764 110.255 194.102 −414.851 501.614 273.900 −1464.66 −2996.06
1.14 −15.3514 52.4289 107.666 −189.180 −804.858 −466.491 312.152 561.790 2904.19
1.15 −14.6596 −9.45906 86.9026 11.4195 138.666 1413.89 602.470 −2097.53 −167.404
1.16 −13.7968 −43.6368 62.3515 −245.563 602.047 −846.187 905.739 −282.833 3387.98
1.17 −13.0108 68.8598 41.2821 477.746 −895.924 −931.981 1128.27 2554.67 −6215.88
1.18 −12.9010 82.3706 38.4367 −80.9826 −1062.67 1479.79 1155.46 4597.91 1044.76
1.19 −10.9834 −52.3468 −7.36490 −125.995 574.946 737.928 1486.77 553.190 1383.86
1.20 −10.8696 −21.3041 −9.85160 −144.259 231.567 −1584.63 1498.39 −1733.14 1568.04
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.64
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.a 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
229.8.a.a 64 1.a even 1 1 trivial