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.
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 |
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 |