Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [41,10,Mod(9,41)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(41, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("41.9");
S:= CuspForms(chi, 10);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 41 \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 41.c (of order \(4\), 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: | \(21.1164692827\) |
Analytic rank: | \(0\) |
Dimension: | \(62\) |
Relative dimension: | \(31\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
9.1 | − | 44.0476i | −107.424 | − | 107.424i | −1428.19 | 1964.96i | −4731.77 | + | 4731.77i | −6990.74 | − | 6990.74i | 40355.9i | 3396.85i | 86551.8 | |||||||||||
9.2 | − | 43.5939i | 116.221 | + | 116.221i | −1388.43 | − | 347.886i | 5066.51 | − | 5066.51i | 4266.58 | + | 4266.58i | 38206.9i | 7331.44i | −15165.7 | ||||||||||
9.3 | − | 40.6130i | −147.297 | − | 147.297i | −1137.42 | − | 1955.60i | −5982.18 | + | 5982.18i | 6867.80 | + | 6867.80i | 25400.1i | 23709.9i | −79422.8 | ||||||||||
9.4 | − | 35.0737i | 75.3048 | + | 75.3048i | −718.166 | 217.529i | 2641.22 | − | 2641.22i | −4539.45 | − | 4539.45i | 7231.03i | − | 8341.37i | 7629.54 | ||||||||||
9.5 | − | 34.0948i | −25.2088 | − | 25.2088i | −650.456 | − | 1459.50i | −859.489 | + | 859.489i | −3092.91 | − | 3092.91i | 4720.64i | − | 18412.0i | −49761.4 | |||||||||
9.6 | − | 32.5005i | 7.26533 | + | 7.26533i | −544.284 | 1433.15i | 236.127 | − | 236.127i | 5548.45 | + | 5548.45i | 1049.26i | − | 19577.4i | 46578.0 | ||||||||||
9.7 | − | 28.6799i | 172.163 | + | 172.163i | −310.538 | 2765.16i | 4937.61 | − | 4937.61i | −1759.10 | − | 1759.10i | − | 5777.92i | 39597.0i | 79304.4 | ||||||||||
9.8 | − | 28.6591i | −133.710 | − | 133.710i | −309.346 | 1241.77i | −3832.03 | + | 3832.03i | 1725.41 | + | 1725.41i | − | 5807.88i | 16074.0i | 35588.0 | ||||||||||
9.9 | − | 25.7433i | 159.796 | + | 159.796i | −150.720 | − | 1915.98i | 4113.68 | − | 4113.68i | −929.116 | − | 929.116i | − | 9300.56i | 31386.3i | −49323.6 | |||||||||
9.10 | − | 18.8163i | −187.426 | − | 187.426i | 157.945 | − | 836.977i | −3526.68 | + | 3526.68i | −7855.38 | − | 7855.38i | − | 12605.9i | 50574.3i | −15748.8 | |||||||||
9.11 | − | 18.0564i | −83.0950 | − | 83.0950i | 185.966 | − | 415.379i | −1500.40 | + | 1500.40i | 2030.90 | + | 2030.90i | − | 12602.8i | − | 5873.43i | −7500.27 | ||||||||
9.12 | − | 12.4598i | 32.3176 | + | 32.3176i | 356.754 | 1203.69i | 402.670 | − | 402.670i | −5019.01 | − | 5019.01i | − | 10824.5i | − | 17594.1i | 14997.8 | |||||||||
9.13 | − | 12.1812i | 95.7860 | + | 95.7860i | 363.618 | − | 100.810i | 1166.79 | − | 1166.79i | 7261.36 | + | 7261.36i | − | 10666.1i | − | 1333.07i | −1227.99 | ||||||||
9.14 | − | 7.65294i | −33.9566 | − | 33.9566i | 453.432 | − | 2680.21i | −259.868 | + | 259.868i | 1109.50 | + | 1109.50i | − | 7388.40i | − | 17376.9i | −20511.5 | ||||||||
9.15 | − | 3.11710i | −103.776 | − | 103.776i | 502.284 | 2428.58i | −323.482 | + | 323.482i | −43.4878 | − | 43.4878i | − | 3161.63i | 1856.11i | 7570.15 | ||||||||||
9.16 | − | 2.91537i | 83.6684 | + | 83.6684i | 503.501 | − | 244.198i | 243.925 | − | 243.925i | −6830.06 | − | 6830.06i | − | 2960.56i | − | 5682.18i | −711.928 | ||||||||
9.17 | 5.80705i | −164.374 | − | 164.374i | 478.278 | − | 651.280i | 954.527 | − | 954.527i | 5590.28 | + | 5590.28i | 5750.59i | 34354.6i | 3782.01 | |||||||||||
9.18 | 6.02559i | 182.688 | + | 182.688i | 475.692 | 583.218i | −1100.80 | + | 1100.80i | −581.237 | − | 581.237i | 5951.43i | 47066.7i | −3514.23 | ||||||||||||
9.19 | 11.0747i | −54.4217 | − | 54.4217i | 389.352 | 201.486i | 602.701 | − | 602.701i | 523.559 | + | 523.559i | 9982.16i | − | 13759.6i | −2231.39 | |||||||||||
9.20 | 13.9877i | 64.9677 | + | 64.9677i | 316.344 | 2181.64i | −908.750 | + | 908.750i | 4287.58 | + | 4287.58i | 11586.6i | − | 11241.4i | −30516.2 | |||||||||||
See all 62 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
41.c | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 41.10.c.a | ✓ | 62 |
41.c | even | 4 | 1 | inner | 41.10.c.a | ✓ | 62 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
41.10.c.a | ✓ | 62 | 1.a | even | 1 | 1 | trivial |
41.10.c.a | ✓ | 62 | 41.c | even | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(41, [\chi])\).