Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [17,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("17.2");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 17 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
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: | \(2.72652493682\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(7\) 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 | −6.32994 | − | 6.32994i | 26.8751 | − | 11.1320i | 48.1363i | −24.9886 | − | 60.3278i | −240.583 | − | 99.6526i | −43.4369 | + | 104.866i | 102.142 | − | 102.142i | 426.521 | − | 426.521i | −223.695 | + | 540.048i | ||
2.2 | −5.57898 | − | 5.57898i | −9.69582 | + | 4.01614i | 30.2500i | 21.4113 | + | 51.6915i | 76.4987 | + | 31.6868i | −20.6531 | + | 49.8610i | −9.76339 | + | 9.76339i | −93.9474 | + | 93.9474i | 168.932 | − | 407.839i | ||
2.3 | −1.63520 | − | 1.63520i | 3.76372 | − | 1.55898i | − | 26.6522i | −7.23148 | − | 17.4583i | −8.70370 | − | 3.60519i | 93.1665 | − | 224.924i | −95.9083 | + | 95.9083i | −160.092 | + | 160.092i | −16.7230 | + | 40.3729i | |
2.4 | 0.226068 | + | 0.226068i | −20.1254 | + | 8.33623i | − | 31.8978i | −31.6611 | − | 76.4366i | −6.43429 | − | 2.66517i | −77.8766 | + | 188.011i | 14.4453 | − | 14.4453i | 163.714 | − | 163.714i | 10.1223 | − | 24.4375i | |
2.5 | 2.65830 | + | 2.65830i | 15.7332 | − | 6.51691i | − | 17.8669i | 13.8556 | + | 33.4503i | 59.1474 | + | 24.4997i | −40.7626 | + | 98.4095i | 132.561 | − | 132.561i | 33.2366 | − | 33.2366i | −52.0887 | + | 125.753i | |
2.6 | 5.25239 | + | 5.25239i | −18.5982 | + | 7.70364i | 23.1753i | 32.1442 | + | 77.6030i | −138.148 | − | 57.2227i | 39.9387 | − | 96.4206i | 46.3508 | − | 46.3508i | 114.721 | − | 114.721i | −238.767 | + | 576.436i | ||
2.7 | 7.23579 | + | 7.23579i | 8.11856 | − | 3.36282i | 72.7132i | −35.2492 | − | 85.0992i | 83.0768 | + | 34.4115i | 7.61170 | − | 18.3763i | −294.592 | + | 294.592i | −117.224 | + | 117.224i | 360.704 | − | 870.815i | ||
8.1 | −7.43061 | + | 7.43061i | −9.60499 | + | 23.1885i | − | 78.4279i | 51.4729 | + | 21.3208i | −100.934 | − | 243.676i | −84.7940 | + | 35.1228i | 344.988 | + | 344.988i | −273.623 | − | 273.623i | −540.902 | + | 224.049i | |
8.2 | −5.77048 | + | 5.77048i | 8.99350 | − | 21.7122i | − | 34.5969i | 33.8199 | + | 14.0087i | 73.3931 | + | 177.187i | 191.069 | − | 79.1434i | 14.9851 | + | 14.9851i | −218.711 | − | 218.711i | −275.994 | + | 114.320i | |
8.3 | −3.68811 | + | 3.68811i | −0.537605 | + | 1.29789i | 4.79575i | −80.0401 | − | 33.1537i | −2.80402 | − | 6.76951i | −92.4561 | + | 38.2966i | −135.707 | − | 135.707i | 170.431 | + | 170.431i | 417.471 | − | 172.922i | ||
8.4 | −0.441843 | + | 0.441843i | −0.853199 | + | 2.05980i | 31.6095i | 70.2501 | + | 29.0985i | −0.533130 | − | 1.28709i | −64.4124 | + | 26.6805i | −28.1054 | − | 28.1054i | 168.312 | + | 168.312i | −43.8965 | + | 18.1825i | ||
8.5 | 2.46953 | − | 2.46953i | −10.5837 | + | 25.5512i | 19.8029i | −30.5219 | − | 12.6426i | 36.9628 | + | 89.2361i | 151.735 | − | 62.8506i | 127.929 | + | 127.929i | −369.025 | − | 369.025i | −106.596 | + | 44.1535i | ||
8.6 | 3.81385 | − | 3.81385i | 6.88457 | − | 16.6208i | 2.90907i | −27.9569 | − | 11.5801i | −37.1326 | − | 89.6461i | 62.5721 | − | 25.9182i | 133.138 | + | 133.138i | −57.0277 | − | 57.0277i | −150.788 | + | 62.4585i | ||
8.7 | 7.21923 | − | 7.21923i | −2.36967 | + | 5.72090i | − | 72.2346i | 34.6952 | + | 14.3712i | 24.1933 | + | 58.4077i | −123.701 | + | 51.2388i | −290.463 | − | 290.463i | 144.714 | + | 144.714i | 354.222 | − | 146.724i | |
9.1 | −6.32994 | + | 6.32994i | 26.8751 | + | 11.1320i | − | 48.1363i | −24.9886 | + | 60.3278i | −240.583 | + | 99.6526i | −43.4369 | − | 104.866i | 102.142 | + | 102.142i | 426.521 | + | 426.521i | −223.695 | − | 540.048i | |
9.2 | −5.57898 | + | 5.57898i | −9.69582 | − | 4.01614i | − | 30.2500i | 21.4113 | − | 51.6915i | 76.4987 | − | 31.6868i | −20.6531 | − | 49.8610i | −9.76339 | − | 9.76339i | −93.9474 | − | 93.9474i | 168.932 | + | 407.839i | |
9.3 | −1.63520 | + | 1.63520i | 3.76372 | + | 1.55898i | 26.6522i | −7.23148 | + | 17.4583i | −8.70370 | + | 3.60519i | 93.1665 | + | 224.924i | −95.9083 | − | 95.9083i | −160.092 | − | 160.092i | −16.7230 | − | 40.3729i | ||
9.4 | 0.226068 | − | 0.226068i | −20.1254 | − | 8.33623i | 31.8978i | −31.6611 | + | 76.4366i | −6.43429 | + | 2.66517i | −77.8766 | − | 188.011i | 14.4453 | + | 14.4453i | 163.714 | + | 163.714i | 10.1223 | + | 24.4375i | ||
9.5 | 2.65830 | − | 2.65830i | 15.7332 | + | 6.51691i | 17.8669i | 13.8556 | − | 33.4503i | 59.1474 | − | 24.4997i | −40.7626 | − | 98.4095i | 132.561 | + | 132.561i | 33.2366 | + | 33.2366i | −52.0887 | − | 125.753i | ||
9.6 | 5.25239 | − | 5.25239i | −18.5982 | − | 7.70364i | − | 23.1753i | 32.1442 | − | 77.6030i | −138.148 | + | 57.2227i | 39.9387 | + | 96.4206i | 46.3508 | + | 46.3508i | 114.721 | + | 114.721i | −238.767 | − | 576.436i | |
See all 28 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 | 17.6.d.a | ✓ | 28 |
17.d | even | 8 | 1 | inner | 17.6.d.a | ✓ | 28 |
17.e | odd | 16 | 2 | 289.6.a.j | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
17.6.d.a | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
17.6.d.a | ✓ | 28 | 17.d | even | 8 | 1 | inner |
289.6.a.j | 28 | 17.e | odd | 16 | 2 |
Hecke kernels
This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(17, [\chi])\).