Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [52,5,Mod(23,52)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(52, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 5]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("52.23");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 52 = 2^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 52.i (of order \(6\), degree \(2\), minimal) |
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: | no |
Analytic conductor: | \(5.37523808036\) |
Analytic rank: | \(0\) |
Dimension: | \(52\) |
Relative dimension: | \(26\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
23.1 | −3.99628 | − | 0.172396i | −12.9573 | + | 7.48092i | 15.9406 | + | 1.37789i | 28.0008i | 53.0709 | − | 27.6621i | −33.5372 | + | 58.0882i | −63.4654 | − | 8.25451i | 71.4283 | − | 123.717i | 4.82722 | − | 111.899i | ||
23.2 | −3.93752 | + | 0.704243i | 9.57461 | − | 5.52790i | 15.0081 | − | 5.54594i | − | 20.6402i | −33.8072 | + | 28.5091i | 0.499659 | − | 0.865435i | −55.1889 | + | 32.4066i | 20.6154 | − | 35.7070i | 14.5357 | + | 81.2710i | |
23.3 | −3.89412 | + | 0.914238i | −2.36372 | + | 1.36470i | 14.3283 | − | 7.12031i | 7.63478i | 7.95696 | − | 7.47529i | 28.3503 | − | 49.1041i | −49.2866 | + | 40.8268i | −36.7752 | + | 63.6965i | −6.98000 | − | 29.7307i | ||
23.4 | −3.65004 | − | 1.63623i | 10.1376 | − | 5.85294i | 10.6455 | + | 11.9446i | 40.5931i | −46.5793 | + | 4.77603i | −26.4101 | + | 45.7437i | −19.3124 | − | 61.0166i | 28.0138 | − | 48.5213i | 66.4196 | − | 148.166i | ||
23.5 | −3.50457 | − | 1.92822i | −4.16271 | + | 2.40334i | 8.56396 | + | 13.5151i | − | 43.7800i | 19.2226 | − | 0.396065i | −27.5011 | + | 47.6332i | −3.95290 | − | 63.8778i | −28.9479 | + | 50.1392i | −84.4173 | + | 153.430i | |
23.6 | −3.23747 | − | 2.34921i | −3.63166 | + | 2.09674i | 4.96240 | + | 15.2110i | 7.96541i | 16.6831 | + | 1.74341i | 35.6880 | − | 61.8134i | 19.6682 | − | 60.9029i | −31.7074 | + | 54.9188i | 18.7124 | − | 25.7878i | ||
23.7 | −2.73881 | + | 2.91529i | 2.36372 | − | 1.36470i | −0.997804 | − | 15.9689i | 7.63478i | −2.49531 | + | 10.6286i | −28.3503 | + | 49.1041i | 49.2866 | + | 40.8268i | −36.7752 | + | 63.6965i | −22.2576 | − | 20.9102i | ||
23.8 | −2.57865 | + | 3.05787i | −9.57461 | + | 5.52790i | −2.70112 | − | 15.7704i | − | 20.6402i | 7.78597 | − | 43.5324i | −0.499659 | + | 0.865435i | 55.1889 | + | 32.4066i | 20.6154 | − | 35.7070i | 63.1149 | + | 53.2238i | |
23.9 | −1.84884 | + | 3.54708i | 12.9573 | − | 7.48092i | −9.16356 | − | 13.1160i | 28.0008i | 2.57936 | + | 59.7917i | 33.5372 | − | 58.0882i | 63.4654 | − | 8.25451i | 71.4283 | − | 123.717i | −99.3211 | − | 51.7691i | ||
23.10 | −1.80419 | − | 3.57000i | 11.6983 | − | 6.75402i | −9.48982 | + | 12.8819i | − | 26.8049i | −45.2178 | − | 29.5775i | 5.90736 | − | 10.2318i | 63.1098 | + | 10.6373i | 50.7336 | − | 87.8731i | −95.6934 | + | 48.3610i | |
23.11 | −1.10537 | − | 3.84424i | −15.4197 | + | 8.90255i | −13.5563 | + | 8.49861i | − | 7.00386i | 51.2680 | + | 49.4363i | 18.9216 | − | 32.7731i | 47.6554 | + | 42.7195i | 118.011 | − | 204.401i | −26.9245 | + | 7.74186i | |
23.12 | −1.02671 | − | 3.86599i | 0.384527 | − | 0.222007i | −13.8917 | + | 7.93852i | 20.9340i | −1.25307 | − | 1.25864i | −11.6885 | + | 20.2451i | 44.9530 | + | 45.5546i | −40.4014 | + | 69.9773i | 80.9306 | − | 21.4932i | ||
23.13 | −0.408003 | + | 3.97914i | −10.1376 | + | 5.85294i | −15.6671 | − | 3.24700i | 40.5931i | −19.1535 | − | 42.7269i | 26.4101 | − | 45.7437i | 19.3124 | − | 61.0166i | 28.0138 | − | 48.5213i | −161.526 | − | 16.5621i | ||
23.14 | −0.0823987 | + | 3.99915i | 4.16271 | − | 2.40334i | −15.9864 | − | 0.659050i | − | 43.7800i | 9.26832 | + | 16.8453i | 27.5011 | − | 47.6332i | 3.95290 | − | 63.8778i | −28.9479 | + | 50.1392i | 175.083 | + | 3.60742i | |
23.15 | 0.415743 | + | 3.97834i | 3.63166 | − | 2.09674i | −15.6543 | + | 3.30793i | 7.96541i | 9.85137 | + | 13.5763i | −35.6880 | + | 61.8134i | −19.6682 | − | 60.9029i | −31.7074 | + | 54.9188i | −31.6891 | + | 3.31156i | ||
23.16 | 1.20110 | − | 3.81541i | −4.08378 | + | 2.35777i | −13.1147 | − | 9.16535i | − | 26.8316i | 4.09085 | + | 18.4132i | −20.7689 | + | 35.9729i | −50.7217 | + | 39.0296i | −29.3818 | + | 50.8908i | −102.374 | − | 32.2273i | |
23.17 | 1.65700 | − | 3.64065i | 9.17215 | − | 5.29554i | −10.5087 | − | 12.0651i | 8.58415i | −4.08095 | − | 42.1673i | 32.7370 | − | 56.7022i | −61.3379 | + | 18.2665i | 15.5856 | − | 26.9950i | 31.2519 | + | 14.2240i | ||
23.18 | 2.18962 | + | 3.34747i | −11.6983 | + | 6.75402i | −6.41114 | + | 14.6594i | − | 26.8049i | −48.2237 | − | 24.3710i | −5.90736 | + | 10.2318i | −63.1098 | + | 10.6373i | 50.7336 | − | 87.8731i | 89.7286 | − | 58.6925i | |
23.19 | 2.47042 | − | 3.14596i | −7.98357 | + | 4.60932i | −3.79408 | − | 15.5436i | 41.7731i | −5.22203 | + | 36.5029i | −5.71487 | + | 9.89844i | −58.2726 | − | 26.4633i | 1.99158 | − | 3.44951i | 131.416 | + | 103.197i | ||
23.20 | 2.77652 | + | 2.87940i | 15.4197 | − | 8.90255i | −0.581861 | + | 15.9894i | − | 7.00386i | 68.4471 | + | 19.6813i | −18.9216 | + | 32.7731i | −47.6554 | + | 42.7195i | 118.011 | − | 204.401i | 20.1669 | − | 19.4464i | |
See all 52 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
13.e | even | 6 | 1 | inner |
52.i | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 52.5.i.a | ✓ | 52 |
4.b | odd | 2 | 1 | inner | 52.5.i.a | ✓ | 52 |
13.e | even | 6 | 1 | inner | 52.5.i.a | ✓ | 52 |
52.i | odd | 6 | 1 | inner | 52.5.i.a | ✓ | 52 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
52.5.i.a | ✓ | 52 | 1.a | even | 1 | 1 | trivial |
52.5.i.a | ✓ | 52 | 4.b | odd | 2 | 1 | inner |
52.5.i.a | ✓ | 52 | 13.e | even | 6 | 1 | inner |
52.5.i.a | ✓ | 52 | 52.i | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(52, [\chi])\).