Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [16,14,Mod(5,16)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(16, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("16.5");
S:= CuspForms(chi, 14);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 16 = 2^{4} \) |
Weight: | \( k \) | \(=\) | \( 14 \) |
Character orbit: | \([\chi]\) | \(=\) | 16.e (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: | \(17.1569486323\) |
Analytic rank: | \(0\) |
Dimension: | \(50\) |
Relative dimension: | \(25\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −90.1629 | + | 7.91484i | 1174.50 | − | 1174.50i | 8066.71 | − | 1427.25i | −33114.8 | − | 33114.8i | −96600.1 | + | 115192.i | − | 548741.i | −716022. | + | 192532.i | − | 1.16456e6i | 3.24783e6 | + | 2.72363e6i | ||
5.2 | −86.5802 | − | 26.3795i | 913.885 | − | 913.885i | 6800.25 | + | 4567.88i | 15715.7 | + | 15715.7i | −103232. | + | 55016.5i | 356100.i | −468268. | − | 574875.i | − | 76047.5i | −946097. | − | 1.77524e6i | |||
5.3 | −85.7364 | − | 29.0046i | −924.404 | + | 924.404i | 6509.46 | + | 4973.51i | 42023.8 | + | 42023.8i | 106067. | − | 52443.1i | − | 291398.i | −413843. | − | 615215.i | − | 114722.i | −2.38408e6 | − | 4.82186e6i | ||
5.4 | −85.4847 | + | 29.7382i | −660.650 | + | 660.650i | 6423.28 | − | 5084.32i | −6982.48 | − | 6982.48i | 36829.0 | − | 76122.0i | 294350.i | −397894. | + | 625648.i | 721406.i | 804541. | + | 389249.i | ||||
5.5 | −81.6959 | − | 38.9586i | −1316.54 | + | 1316.54i | 5156.45 | + | 6365.52i | −43811.8 | − | 43811.8i | 158847. | − | 56265.6i | − | 156812.i | −173269. | − | 720925.i | − | 1.87225e6i | 1.87240e6 | + | 5.28609e6i | ||
5.6 | −71.4691 | + | 55.5353i | 293.446 | − | 293.446i | 2023.65 | − | 7938.12i | 22716.6 | + | 22716.6i | −4675.68 | + | 37268.9i | − | 343566.i | 296217. | + | 679714.i | 1.42210e6i | −2.88511e6 | − | 361960.i | |||
5.7 | −56.2421 | + | 70.9142i | 1621.89 | − | 1621.89i | −1865.65 | − | 7976.73i | 8037.57 | + | 8037.57i | 23796.5 | + | 206234.i | 516249.i | 670591. | + | 316327.i | − | 3.66675e6i | −1.02203e6 | + | 117928.i | |||
5.8 | −55.8189 | − | 71.2478i | 375.766 | − | 375.766i | −1960.51 | + | 7953.95i | −11477.0 | − | 11477.0i | −47747.4 | − | 5797.69i | 9991.15i | 676135. | − | 304298.i | 1.31192e6i | −177079. | + | 1.45835e6i | ||||
5.9 | −40.5224 | + | 80.9316i | −1639.97 | + | 1639.97i | −4907.86 | − | 6559.10i | 9558.16 | + | 9558.16i | −66269.9 | − | 199181.i | − | 133901.i | 729717. | − | 131411.i | − | 3.78468e6i | −1.16088e6 | + | 386238.i | ||
5.10 | −30.3106 | + | 85.2834i | −26.4760 | + | 26.4760i | −6354.53 | − | 5169.99i | −38066.2 | − | 38066.2i | −1455.46 | − | 3060.47i | 23704.0i | 633524. | − | 385231.i | 1.59292e6i | 4.40022e6 | − | 2.09260e6i | ||||
5.11 | −21.6798 | − | 87.8748i | −1056.56 | + | 1056.56i | −7251.97 | + | 3810.23i | 11923.3 | + | 11923.3i | 115752. | + | 69939.2i | 170872.i | 492044. | + | 554660.i | − | 638332.i | 789265. | − | 1.30626e6i | |||
5.12 | −18.7534 | − | 88.5455i | 1610.25 | − | 1610.25i | −7488.62 | + | 3321.06i | 39780.2 | + | 39780.2i | −172778. | − | 112383.i | − | 439933.i | 434502. | + | 600803.i | − | 3.59146e6i | 2.77634e6 | − | 4.26837e6i | ||
5.13 | −0.844839 | + | 90.5057i | 381.223 | − | 381.223i | −8190.57 | − | 152.926i | 29472.6 | + | 29472.6i | 34180.8 | + | 34824.9i | − | 167372.i | 20760.3 | − | 741165.i | 1.30366e6i | −2.69234e6 | + | 2.64254e6i | |||
5.14 | 20.0123 | − | 88.2695i | 983.557 | − | 983.557i | −7391.01 | − | 3532.96i | −35371.2 | − | 35371.2i | −67134.8 | − | 106501.i | 334186.i | −459764. | + | 581698.i | − | 340445.i | −3.83006e6 | + | 2.41434e6i | |||
5.15 | 28.6247 | − | 85.8640i | −336.873 | + | 336.873i | −6553.25 | − | 4915.66i | −6822.00 | − | 6822.00i | 19282.4 | + | 38568.2i | − | 509047.i | −609663. | + | 421979.i | 1.36736e6i | −781042. | + | 390487.i | |||
5.16 | 34.4647 | + | 83.6910i | −787.355 | + | 787.355i | −5816.37 | + | 5768.77i | 13376.5 | + | 13376.5i | −93030.5 | − | 38758.6i | 547213.i | −683254. | − | 287958.i | 354466.i | −658474. | + | 1.58051e6i | ||||
5.17 | 37.4414 | + | 82.4023i | 1317.13 | − | 1317.13i | −5388.28 | + | 6170.52i | −24836.8 | − | 24836.8i | 157850. | + | 59219.5i | − | 73212.9i | −710210. | − | 212974.i | − | 1.87536e6i | 1.11668e6 | − | 2.97653e6i | ||
5.18 | 54.8815 | − | 71.9724i | 242.259 | − | 242.259i | −2168.05 | − | 7899.90i | 42421.3 | + | 42421.3i | −4140.43 | − | 30731.5i | 421955.i | −687560. | − | 277518.i | 1.47694e6i | 5.38130e6 | − | 725019.i | ||||
5.19 | 58.4516 | + | 69.1043i | −1009.54 | + | 1009.54i | −1358.81 | + | 8078.52i | −19923.8 | − | 19923.8i | −128773. | − | 10754.3i | − | 484985.i | −637685. | + | 378303.i | − | 444038.i | 212241. | − | 2.54140e6i | ||
5.20 | 64.2863 | − | 63.7124i | −1618.47 | + | 1618.47i | 73.4667 | − | 8191.67i | −23191.8 | − | 23191.8i | −928.945 | + | 207162.i | 280152.i | −517188. | − | 531293.i | − | 3.64459e6i | −2.96853e6 | − | 13311.3i | |||
See all 50 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 16.14.e.a | ✓ | 50 |
4.b | odd | 2 | 1 | 64.14.e.a | 50 | ||
8.b | even | 2 | 1 | 128.14.e.b | 50 | ||
8.d | odd | 2 | 1 | 128.14.e.a | 50 | ||
16.e | even | 4 | 1 | inner | 16.14.e.a | ✓ | 50 |
16.e | even | 4 | 1 | 128.14.e.b | 50 | ||
16.f | odd | 4 | 1 | 64.14.e.a | 50 | ||
16.f | odd | 4 | 1 | 128.14.e.a | 50 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
16.14.e.a | ✓ | 50 | 1.a | even | 1 | 1 | trivial |
16.14.e.a | ✓ | 50 | 16.e | even | 4 | 1 | inner |
64.14.e.a | 50 | 4.b | odd | 2 | 1 | ||
64.14.e.a | 50 | 16.f | odd | 4 | 1 | ||
128.14.e.a | 50 | 8.d | odd | 2 | 1 | ||
128.14.e.a | 50 | 16.f | odd | 4 | 1 | ||
128.14.e.b | 50 | 8.b | even | 2 | 1 | ||
128.14.e.b | 50 | 16.e | even | 4 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{14}^{\mathrm{new}}(16, [\chi])\).