Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [17,16,Mod(2,17)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(17, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([7]))
N = Newforms(chi, 16, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("17.2");
S:= CuspForms(chi, 16);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 17 \) |
Weight: | \( k \) | \(=\) | \( 16 \) |
Character orbit: | \([\chi]\) | \(=\) | 17.d (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: | \(24.2578958670\) |
Analytic rank: | \(0\) |
Dimension: | \(84\) |
Relative dimension: | \(21\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −245.820 | − | 245.820i | −1672.50 | + | 692.772i | 88087.0i | −25565.3 | − | 61720.1i | 581431. | + | 240837.i | 1.16718e6 | − | 2.81783e6i | 1.35985e7 | − | 1.35985e7i | −7.82889e6 | + | 7.82889e6i | −8.88757e6 | + | 2.14565e7i | ||
2.2 | −212.195 | − | 212.195i | 5316.82 | − | 2202.30i | 57285.2i | 29926.6 | + | 72249.3i | −1.59552e6 | − | 660884.i | −50910.9 | + | 122910.i | 5.20242e6 | − | 5.20242e6i | 1.32722e7 | − | 1.32722e7i | 8.98064e6 | − | 2.16812e7i | ||
2.3 | −200.374 | − | 200.374i | 476.536 | − | 197.388i | 47531.5i | 16556.1 | + | 39969.9i | −135037. | − | 55934.1i | −1.20371e6 | + | 2.90602e6i | 2.95821e6 | − | 2.95821e6i | −9.95808e6 | + | 9.95808e6i | 4.69153e6 | − | 1.13263e7i | ||
2.4 | −182.282 | − | 182.282i | −5695.73 | + | 2359.25i | 33685.3i | 87740.6 | + | 211825.i | 1.46828e6 | + | 608180.i | −90193.3 | + | 217746.i | 167216. | − | 167216.i | 1.67290e7 | − | 1.67290e7i | 2.26183e7 | − | 5.46053e7i | ||
2.5 | −170.250 | − | 170.250i | −4130.31 | + | 1710.83i | 25201.8i | −124080. | − | 299556.i | 994451. | + | 411915.i | −905698. | + | 2.18655e6i | −1.28814e6 | + | 1.28814e6i | 3.98630e6 | − | 3.98630e6i | −2.98746e7 | + | 7.21238e7i | ||
2.6 | −129.673 | − | 129.673i | 4061.21 | − | 1682.21i | 862.087i | −105067. | − | 253655.i | −744765. | − | 308492.i | 456097. | − | 1.10112e6i | −4.13733e6 | + | 4.13733e6i | 3.51739e6 | − | 3.51739e6i | −1.92678e7 | + | 4.65166e7i | ||
2.7 | −123.094 | − | 123.094i | −906.241 | + | 375.377i | − | 2463.77i | 17515.8 | + | 42286.8i | 157759. | + | 65346.1i | 987054. | − | 2.38296e6i | −4.33682e6 | + | 4.33682e6i | −9.46584e6 | + | 9.46584e6i | 3.04917e6 | − | 7.36134e6i | |
2.8 | −89.2325 | − | 89.2325i | 2770.30 | − | 1147.50i | − | 16843.1i | 120460. | + | 290817.i | −349595. | − | 144807.i | 148211. | − | 357812.i | −4.42692e6 | + | 4.42692e6i | −3.78838e6 | + | 3.78838e6i | 1.52013e7 | − | 3.66993e7i | |
2.9 | −40.7523 | − | 40.7523i | −3075.62 | + | 1273.97i | − | 29446.5i | 8583.59 | + | 20722.6i | 177256. | + | 73421.8i | −428729. | + | 1.03504e6i | −2.53539e6 | + | 2.53539e6i | −2.30974e6 | + | 2.30974e6i | 494694. | − | 1.19430e6i | |
2.10 | −27.3745 | − | 27.3745i | −6006.82 | + | 2488.11i | − | 31269.3i | −53706.7 | − | 129659.i | 232545. | + | 96323.1i | 789571. | − | 1.90619e6i | −1.75299e6 | + | 1.75299e6i | 1.97450e7 | − | 1.97450e7i | −2.07917e6 | + | 5.01956e6i | |
2.11 | −22.5360 | − | 22.5360i | 2542.58 | − | 1053.17i | − | 31752.3i | −22347.0 | − | 53950.4i | −81033.7 | − | 33565.3i | −1.46128e6 | + | 3.52785e6i | −1.45403e6 | + | 1.45403e6i | −4.79067e6 | + | 4.79067e6i | −712213. | + | 1.71944e6i | |
2.12 | 15.5628 | + | 15.5628i | 6353.60 | − | 2631.75i | − | 32283.6i | 16618.5 | + | 40120.5i | 139837. | + | 57922.5i | 389638. | − | 940668.i | 1.01239e6 | − | 1.01239e6i | 2.32959e7 | − | 2.32959e7i | −365758. | + | 883018.i | |
2.13 | 50.7927 | + | 50.7927i | −431.133 | + | 178.581i | − | 27608.2i | −62169.6 | − | 150091.i | −30969.0 | − | 12827.8i | 775764. | − | 1.87286e6i | 3.06667e6 | − | 3.06667e6i | −9.99222e6 | + | 9.99222e6i | 4.46575e6 | − | 1.07813e7i | |
2.14 | 74.8074 | + | 74.8074i | −4152.39 | + | 1719.98i | − | 21575.7i | 90601.2 | + | 218731.i | −439296. | − | 181962.i | −964925. | + | 2.32953e6i | 4.06531e6 | − | 4.06531e6i | 4.13780e6 | − | 4.13780e6i | −9.58503e6 | + | 2.31403e7i | |
2.15 | 115.076 | + | 115.076i | 833.789 | − | 345.367i | − | 6282.82i | 89161.7 | + | 215255.i | 135693. | + | 56205.9i | 1.12431e6 | − | 2.71432e6i | 4.49383e6 | − | 4.49383e6i | −9.57028e6 | + | 9.57028e6i | −1.45104e7 | + | 3.50312e7i | |
2.16 | 122.476 | + | 122.476i | 2916.40 | − | 1208.01i | − | 2767.50i | −62547.2 | − | 151002.i | 505140. | + | 209236.i | −281393. | + | 679342.i | 4.35223e6 | − | 4.35223e6i | −3.10010e6 | + | 3.10010e6i | 1.08336e7 | − | 2.61546e7i | |
2.17 | 160.402 | + | 160.402i | −3356.58 | + | 1390.34i | 18689.4i | −110139. | − | 265899.i | −761415. | − | 315388.i | −944161. | + | 2.27941e6i | 2.25823e6 | − | 2.25823e6i | −812613. | + | 812613.i | 2.49841e7 | − | 6.03171e7i | ||
2.18 | 175.124 | + | 175.124i | −5939.27 | + | 2460.13i | 28568.5i | 6790.58 | + | 16393.9i | −1.47093e6 | − | 609280.i | 856416. | − | 2.06757e6i | 735433. | − | 735433.i | 1.90765e7 | − | 1.90765e7i | −1.68177e6 | + | 4.06015e6i | ||
2.19 | 183.168 | + | 183.168i | 4576.47 | − | 1895.63i | 34333.1i | 63067.3 | + | 152258.i | 1.18548e6 | + | 491043.i | −951603. | + | 2.29737e6i | −286685. | + | 286685.i | 7.20441e6 | − | 7.20441e6i | −1.63369e7 | + | 3.94407e7i | ||
2.20 | 225.107 | + | 225.107i | 5049.07 | − | 2091.39i | 68578.6i | −91778.2 | − | 221572.i | 1.60737e6 | + | 665794.i | 1.36178e6 | − | 3.28763e6i | −8.06122e6 | + | 8.06122e6i | 1.09729e7 | − | 1.09729e7i | 2.92176e7 | − | 7.05375e7i | ||
See all 84 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
17.d | even | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 17.16.d.a | ✓ | 84 |
17.d | even | 8 | 1 | inner | 17.16.d.a | ✓ | 84 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
17.16.d.a | ✓ | 84 | 1.a | even | 1 | 1 | trivial |
17.16.d.a | ✓ | 84 | 17.d | even | 8 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{16}^{\mathrm{new}}(17, [\chi])\).