Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [41,11,Mod(3,41)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(41, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([3]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("41.3");
S:= CuspForms(chi, 11);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 41 \) |
Weight: | \( k \) | \(=\) | \( 11 \) |
Character orbit: | \([\chi]\) | \(=\) | 41.e (of order \(8\), degree \(4\), 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: | \(26.0496473596\) |
Analytic rank: | \(0\) |
Dimension: | \(136\) |
Relative dimension: | \(34\) over \(\Q(\zeta_{8})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −43.4948 | + | 43.4948i | −173.386 | + | 418.590i | − | 2759.59i | 2003.37 | + | 2003.37i | −10665.1 | − | 25747.8i | 9569.84 | − | 23103.6i | 75489.1 | + | 75489.1i | −103401. | − | 103401.i | −174272. | |||
3.2 | −43.3059 | + | 43.3059i | −34.7089 | + | 83.7946i | − | 2726.80i | −932.314 | − | 932.314i | −2125.70 | − | 5131.90i | −11739.6 | + | 28341.9i | 73741.1 | + | 73741.1i | 35937.1 | + | 35937.1i | 80749.3 | |||
3.3 | −40.7233 | + | 40.7233i | 165.169 | − | 398.754i | − | 2292.78i | 3636.89 | + | 3636.89i | 9512.34 | + | 22964.8i | −288.015 | + | 695.329i | 51669.1 | + | 51669.1i | −89969.6 | − | 89969.6i | −296213. | |||
3.4 | −40.5638 | + | 40.5638i | 75.2761 | − | 181.733i | − | 2266.84i | −2614.09 | − | 2614.09i | 4318.28 | + | 10425.3i | 5340.28 | − | 12892.6i | 50414.5 | + | 50414.5i | 14393.7 | + | 14393.7i | 212075. | |||
3.5 | −35.7906 | + | 35.7906i | −5.95702 | + | 14.3815i | − | 1537.93i | 2505.68 | + | 2505.68i | −301.517 | − | 727.927i | 470.064 | − | 1134.83i | 18393.8 | + | 18393.8i | 41582.6 | + | 41582.6i | −179359. | |||
3.6 | −30.9351 | + | 30.9351i | −97.3324 | + | 234.981i | − | 889.964i | −3921.72 | − | 3921.72i | −4258.18 | − | 10280.2i | 9122.15 | − | 22022.8i | −4146.43 | − | 4146.43i | −3988.62 | − | 3988.62i | 242638. | |||
3.7 | −30.5648 | + | 30.5648i | −40.2742 | + | 97.2305i | − | 844.416i | 1121.32 | + | 1121.32i | −1740.86 | − | 4202.81i | 2970.09 | − | 7170.43i | −5488.94 | − | 5488.94i | 33922.2 | + | 33922.2i | −68546.1 | |||
3.8 | −29.6057 | + | 29.6057i | −137.161 | + | 331.137i | − | 728.991i | −1675.81 | − | 1675.81i | −5742.77 | − | 13864.3i | −7692.51 | + | 18571.4i | −8733.93 | − | 8733.93i | −49084.3 | − | 49084.3i | 99227.1 | |||
3.9 | −29.2016 | + | 29.2016i | 132.555 | − | 320.015i | − | 681.461i | −876.164 | − | 876.164i | 5474.13 | + | 13215.7i | −1512.10 | + | 3650.54i | −10002.7 | − | 10002.7i | −43084.9 | − | 43084.9i | 51170.7 | |||
3.10 | −21.1119 | + | 21.1119i | 89.4836 | − | 216.032i | 132.575i | −2323.82 | − | 2323.82i | 2671.69 | + | 6450.02i | −6803.92 | + | 16426.1i | −24417.5 | − | 24417.5i | 3091.24 | + | 3091.24i | 98120.7 | ||||
3.11 | −17.0387 | + | 17.0387i | 51.4488 | − | 124.208i | 443.364i | 3686.39 | + | 3686.39i | 1239.73 | + | 2992.98i | −11841.4 | + | 28587.6i | −25002.0 | − | 25002.0i | 28973.2 | + | 28973.2i | −125623. | ||||
3.12 | −16.4049 | + | 16.4049i | −144.811 | + | 349.605i | 485.760i | 2748.40 | + | 2748.40i | −3359.62 | − | 8110.85i | −2544.07 | + | 6141.94i | −24767.4 | − | 24767.4i | −59499.6 | − | 59499.6i | −90174.2 | ||||
3.13 | −14.7946 | + | 14.7946i | 94.8180 | − | 228.911i | 586.238i | 1671.69 | + | 1671.69i | 1983.85 | + | 4789.45i | 11564.6 | − | 27919.4i | −23822.9 | − | 23822.9i | −1655.79 | − | 1655.79i | −49463.9 | ||||
3.14 | −10.0851 | + | 10.0851i | −107.923 | + | 260.549i | 820.583i | 877.673 | + | 877.673i | −1539.24 | − | 3716.05i | 3602.01 | − | 8696.03i | −18602.7 | − | 18602.7i | −14484.3 | − | 14484.3i | −17702.8 | ||||
3.15 | −9.84548 | + | 9.84548i | −22.0412 | + | 53.2121i | 830.133i | −1874.03 | − | 1874.03i | −306.893 | − | 740.905i | 5342.50 | − | 12897.9i | −18254.8 | − | 18254.8i | 39408.2 | + | 39408.2i | 36901.4 | ||||
3.16 | −6.24980 | + | 6.24980i | −14.1891 | + | 34.2556i | 945.880i | −2714.12 | − | 2714.12i | −125.411 | − | 302.770i | −7786.00 | + | 18797.1i | −12311.4 | − | 12311.4i | 40781.8 | + | 40781.8i | 33925.5 | ||||
3.17 | −3.04539 | + | 3.04539i | 181.284 | − | 437.659i | 1005.45i | 815.394 | + | 815.394i | 780.761 | + | 1884.92i | −2428.27 | + | 5862.36i | −6180.46 | − | 6180.46i | −116928. | − | 116928.i | −4966.38 | ||||
3.18 | 4.21015 | − | 4.21015i | 35.1231 | − | 84.7947i | 988.549i | 2200.24 | + | 2200.24i | −209.125 | − | 504.871i | −2906.06 | + | 7015.84i | 8473.13 | + | 8473.13i | 35797.4 | + | 35797.4i | 18526.7 | ||||
3.19 | 6.82166 | − | 6.82166i | 129.778 | − | 313.312i | 930.930i | −4122.43 | − | 4122.43i | −1252.01 | − | 3022.61i | 4925.95 | − | 11892.3i | 13335.9 | + | 13335.9i | −39568.2 | − | 39568.2i | −56243.6 | ||||
3.20 | 7.44843 | − | 7.44843i | −179.027 | + | 432.210i | 913.042i | −2627.98 | − | 2627.98i | 1885.81 | + | 4552.76i | 6311.03 | − | 15236.2i | 14427.9 | + | 14427.9i | −113001. | − | 113001.i | −39148.7 | ||||
See next 80 embeddings (of 136 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
41.e | odd | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 41.11.e.a | ✓ | 136 |
41.e | odd | 8 | 1 | inner | 41.11.e.a | ✓ | 136 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
41.11.e.a | ✓ | 136 | 1.a | even | 1 | 1 | trivial |
41.11.e.a | ✓ | 136 | 41.e | odd | 8 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(41, [\chi])\).