Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,19,Mod(2,9)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 19, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.2");
S:= CuspForms(chi, 19);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 19 \) |
Character orbit: | \([\chi]\) | \(=\) | 9.d (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: | \(18.4847523939\) |
Analytic rank: | \(0\) |
Dimension: | \(34\) |
Relative dimension: | \(17\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −826.948 | − | 477.438i | −4479.77 | − | 19166.4i | 324823. | + | 562610.i | 2.90361e6 | − | 1.67640e6i | −5.44626e6 | + | 1.79885e7i | −4.86570e6 | + | 8.42763e6i | − | 3.70017e8i | −3.47284e8 | + | 1.71722e8i | −3.20151e9 | |||
2.2 | −778.562 | − | 449.503i | 19565.6 | + | 2146.79i | 273034. | + | 472909.i | −2.42540e6 | + | 1.40030e6i | −1.42680e7 | − | 1.04662e7i | −3.53698e7 | + | 6.12622e7i | − | 2.55250e8i | 3.78203e8 | + | 8.40066e7i | 2.51777e9 | |||
2.3 | −645.469 | − | 372.662i | −1755.59 | + | 19604.6i | 146682. | + | 254060.i | 690555. | − | 398692.i | 8.43904e6 | − | 1.19999e7i | 1.15930e7 | − | 2.00797e7i | − | 2.32684e7i | −3.81256e8 | − | 6.88350e7i | −5.94309e8 | |||
2.4 | −633.468 | − | 365.733i | −19383.5 | − | 3420.35i | 136449. | + | 236336.i | −2.15900e6 | + | 1.24650e6i | 1.10279e7 | + | 9.25588e6i | 1.48825e7 | − | 2.57772e7i | − | 7.86595e6i | 3.64023e8 | + | 1.32597e8i | 1.82355e9 | |||
2.5 | −462.218 | − | 266.862i | 18990.6 | − | 5174.86i | 11358.3 | + | 19673.1i | 1.29042e6 | − | 745022.i | −1.01587e7 | − | 2.67594e6i | 2.69368e7 | − | 4.66559e7i | 1.27788e8i | 3.33862e8 | − | 1.96547e8i | −7.95271e8 | ||||
2.6 | −307.242 | − | 177.386i | 3077.23 | − | 19441.0i | −68140.4 | − | 118023.i | −996976. | + | 575604.i | −4.39401e6 | + | 5.42722e6i | −8.05977e6 | + | 1.39599e7i | 1.41350e8i | −3.68482e8 | − | 1.19649e8i | 4.08417e8 | ||||
2.7 | −301.302 | − | 173.957i | −18971.4 | + | 5244.55i | −70549.9 | − | 122196.i | 2.15861e6 | − | 1.24627e6i | 6.62846e6 | + | 1.72002e6i | −3.75558e7 | + | 6.50485e7i | 1.40294e8i | 3.32410e8 | − | 1.98993e8i | −8.67191e8 | ||||
2.8 | −77.3719 | − | 44.6707i | 14059.8 | + | 13774.7i | −127081. | − | 220111.i | 10314.3 | − | 5954.97i | −472508. | − | 1.69384e6i | −5.95096e6 | + | 1.03074e7i | 4.61275e7i | 7.93549e6 | + | 3.87339e8i | −1.06405e6 | ||||
2.9 | 56.6635 | + | 32.7147i | −11098.7 | + | 16255.4i | −128931. | − | 223316.i | −2.62400e6 | + | 1.51497e6i | −1.16068e6 | + | 557998.i | −1.47703e6 | + | 2.55830e6i | − | 3.40238e7i | −1.41057e8 | − | 3.60829e8i | −1.98247e8 | |||
2.10 | 82.9990 | + | 47.9195i | −16416.3 | − | 10859.3i | −126479. | − | 219069.i | 1.05847e6 | − | 611105.i | −842168. | − | 1.68797e6i | 3.06063e7 | − | 5.30117e7i | − | 4.93669e7i | 1.51572e8 | + | 3.56540e8i | 1.17135e8 | |||
2.11 | 336.967 | + | 194.548i | 16767.8 | − | 10308.3i | −55374.0 | − | 95910.6i | 2.58095e6 | − | 1.49011e6i | 7.65567e6 | − | 211404.i | −1.88244e7 | + | 3.26048e7i | − | 1.45091e8i | 1.74899e8 | − | 3.45695e8i | 1.15959e9 | |||
2.12 | 406.376 | + | 234.621i | 18030.6 | − | 7894.22i | −20977.9 | − | 36334.9i | −2.94106e6 | + | 1.69802e6i | 9.17934e6 | + | 1.02233e6i | 2.46134e7 | − | 4.26317e7i | − | 1.42696e8i | 2.62783e8 | − | 2.84675e8i | −1.59357e9 | |||
2.13 | 437.639 | + | 252.671i | −7960.89 | − | 18001.2i | −3386.81 | − | 5866.13i | −693411. | + | 400341.i | 1.06440e6 | − | 9.88953e6i | −2.35552e7 | + | 4.07987e7i | − | 1.35895e8i | −2.60669e8 | + | 2.86612e8i | −4.04618e8 | |||
2.14 | 451.478 | + | 260.661i | −6615.47 | + | 18538.0i | 4816.35 | + | 8342.15i | 2.21110e6 | − | 1.27658e6i | −7.81886e6 | + | 6.64509e6i | 2.07433e7 | − | 3.59285e7i | − | 1.31640e8i | −2.99892e8 | − | 2.45275e8i | 1.33102e9 | |||
2.15 | 689.327 | + | 397.983i | −19476.2 | + | 2845.44i | 185709. | + | 321657.i | −768821. | + | 443879.i | −1.45579e7 | − | 5.78978e6i | −1.10203e7 | + | 1.90877e7i | 8.69784e7i | 3.71227e8 | − | 1.10837e8i | −7.06626e8 | ||||
2.16 | 709.707 | + | 409.750i | 13445.3 | + | 14375.1i | 204718. | + | 354581.i | −432139. | + | 249496.i | 3.65207e6 | + | 1.57113e7i | −1.14976e7 | + | 1.99144e7i | 1.20705e8i | −2.58659e7 | + | 3.86556e8i | −4.08923e8 | ||||
2.17 | 859.924 | + | 496.477i | 4123.08 | − | 19246.3i | 361908. | + | 626842.i | 1.23289e6 | − | 711809.i | 1.31009e7 | − | 1.45034e7i | 3.09522e7 | − | 5.36108e7i | 4.58419e8i | −3.53421e8 | − | 1.58708e8i | 1.41359e9 | ||||
5.1 | −826.948 | + | 477.438i | −4479.77 | + | 19166.4i | 324823. | − | 562610.i | 2.90361e6 | + | 1.67640e6i | −5.44626e6 | − | 1.79885e7i | −4.86570e6 | − | 8.42763e6i | 3.70017e8i | −3.47284e8 | − | 1.71722e8i | −3.20151e9 | ||||
5.2 | −778.562 | + | 449.503i | 19565.6 | − | 2146.79i | 273034. | − | 472909.i | −2.42540e6 | − | 1.40030e6i | −1.42680e7 | + | 1.04662e7i | −3.53698e7 | − | 6.12622e7i | 2.55250e8i | 3.78203e8 | − | 8.40066e7i | 2.51777e9 | ||||
5.3 | −645.469 | + | 372.662i | −1755.59 | − | 19604.6i | 146682. | − | 254060.i | 690555. | + | 398692.i | 8.43904e6 | + | 1.19999e7i | 1.15930e7 | + | 2.00797e7i | 2.32684e7i | −3.81256e8 | + | 6.88350e7i | −5.94309e8 | ||||
See all 34 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 9.19.d.a | ✓ | 34 |
3.b | odd | 2 | 1 | 27.19.d.a | 34 | ||
9.c | even | 3 | 1 | 27.19.d.a | 34 | ||
9.c | even | 3 | 1 | 81.19.b.a | 34 | ||
9.d | odd | 6 | 1 | inner | 9.19.d.a | ✓ | 34 |
9.d | odd | 6 | 1 | 81.19.b.a | 34 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
9.19.d.a | ✓ | 34 | 1.a | even | 1 | 1 | trivial |
9.19.d.a | ✓ | 34 | 9.d | odd | 6 | 1 | inner |
27.19.d.a | 34 | 3.b | odd | 2 | 1 | ||
27.19.d.a | 34 | 9.c | even | 3 | 1 | ||
81.19.b.a | 34 | 9.c | even | 3 | 1 | ||
81.19.b.a | 34 | 9.d | odd | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{19}^{\mathrm{new}}(9, [\chi])\).