Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [714,2,Mod(83,714)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(714, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([4, 4, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("714.83");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 714.w (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: | \(5.70131870432\) |
Analytic rank: | \(0\) |
Dimension: | \(80\) |
Relative dimension: | \(20\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
83.1 | 0.707107 | − | 0.707107i | −1.72967 | + | 0.0907356i | − | 1.00000i | −1.23141 | + | 0.510066i | −1.15890 | + | 1.28722i | 0.516019 | − | 2.59494i | −0.707107 | − | 0.707107i | 2.98353 | − | 0.313886i | −0.510066 | + | 1.23141i | |
83.2 | 0.707107 | − | 0.707107i | −1.72704 | + | 0.131707i | − | 1.00000i | 0.103371 | − | 0.0428177i | −1.12807 | + | 1.31433i | −0.593072 | + | 2.57842i | −0.707107 | − | 0.707107i | 2.96531 | − | 0.454927i | 0.0428177 | − | 0.103371i | |
83.3 | 0.707107 | − | 0.707107i | −1.70634 | + | 0.297343i | − | 1.00000i | 2.47278 | − | 1.02426i | −0.996310 | + | 1.41682i | 2.40380 | + | 1.10533i | −0.707107 | − | 0.707107i | 2.82317 | − | 1.01473i | 1.02426 | − | 2.47278i | |
83.4 | 0.707107 | − | 0.707107i | −1.47024 | + | 0.915636i | − | 1.00000i | −2.44731 | + | 1.01371i | −0.392165 | + | 1.68707i | −2.61269 | + | 0.416962i | −0.707107 | − | 0.707107i | 1.32322 | − | 2.69241i | −1.01371 | + | 2.44731i | |
83.5 | 0.707107 | − | 0.707107i | −1.39247 | − | 1.03006i | − | 1.00000i | 2.17160 | − | 0.899506i | −1.71299 | + | 0.256259i | 0.880980 | − | 2.49477i | −0.707107 | − | 0.707107i | 0.877938 | + | 2.86866i | 0.899506 | − | 2.17160i | |
83.6 | 0.707107 | − | 0.707107i | −1.25039 | − | 1.19855i | − | 1.00000i | 3.41749 | − | 1.41557i | −1.73166 | + | 0.0366532i | −2.56408 | − | 0.652293i | −0.707107 | − | 0.707107i | 0.126942 | + | 2.99731i | 1.41557 | − | 3.41749i | |
83.7 | 0.707107 | − | 0.707107i | −0.923536 | − | 1.46529i | − | 1.00000i | −1.11174 | + | 0.460496i | −1.68916 | − | 0.383079i | −2.37717 | + | 1.16149i | −0.707107 | − | 0.707107i | −1.29416 | + | 2.70650i | −0.460496 | + | 1.11174i | |
83.8 | 0.707107 | − | 0.707107i | −0.498655 | + | 1.65872i | − | 1.00000i | 0.0389656 | − | 0.0161401i | 0.820288 | + | 1.52549i | −2.40704 | − | 1.09824i | −0.707107 | − | 0.707107i | −2.50269 | − | 1.65425i | 0.0161401 | − | 0.0389656i | |
83.9 | 0.707107 | − | 0.707107i | −0.223046 | + | 1.71763i | − | 1.00000i | 2.82761 | − | 1.17124i | 1.05683 | + | 1.37226i | −0.153600 | + | 2.64129i | −0.707107 | − | 0.707107i | −2.90050 | − | 0.766222i | 1.17124 | − | 2.82761i | |
83.10 | 0.707107 | − | 0.707107i | −0.0756553 | + | 1.73040i | − | 1.00000i | −2.32780 | + | 0.964207i | 1.17008 | + | 1.27707i | 1.59075 | + | 2.11412i | −0.707107 | − | 0.707107i | −2.98855 | − | 0.261828i | −0.964207 | + | 2.32780i | |
83.11 | 0.707107 | − | 0.707107i | 0.0756553 | − | 1.73040i | − | 1.00000i | 2.32780 | − | 0.964207i | −1.17008 | − | 1.27707i | 2.61974 | − | 0.370078i | −0.707107 | − | 0.707107i | −2.98855 | − | 0.261828i | 0.964207 | − | 2.32780i | |
83.12 | 0.707107 | − | 0.707107i | 0.223046 | − | 1.71763i | − | 1.00000i | −2.82761 | + | 1.17124i | −1.05683 | − | 1.37226i | 1.75906 | − | 1.97628i | −0.707107 | − | 0.707107i | −2.90050 | − | 0.766222i | −1.17124 | + | 2.82761i | |
83.13 | 0.707107 | − | 0.707107i | 0.498655 | − | 1.65872i | − | 1.00000i | −0.0389656 | + | 0.0161401i | −0.820288 | − | 1.52549i | −2.47861 | − | 0.925461i | −0.707107 | − | 0.707107i | −2.50269 | − | 1.65425i | −0.0161401 | + | 0.0389656i | |
83.14 | 0.707107 | − | 0.707107i | 0.923536 | + | 1.46529i | − | 1.00000i | 1.11174 | − | 0.460496i | 1.68916 | + | 0.383079i | −0.859618 | − | 2.50221i | −0.707107 | − | 0.707107i | −1.29416 | + | 2.70650i | 0.460496 | − | 1.11174i | |
83.15 | 0.707107 | − | 0.707107i | 1.25039 | + | 1.19855i | − | 1.00000i | −3.41749 | + | 1.41557i | 1.73166 | − | 0.0366532i | −2.27432 | − | 1.35184i | −0.707107 | − | 0.707107i | 0.126942 | + | 2.99731i | −1.41557 | + | 3.41749i | |
83.16 | 0.707107 | − | 0.707107i | 1.39247 | + | 1.03006i | − | 1.00000i | −2.17160 | + | 0.899506i | 1.71299 | − | 0.256259i | −1.14112 | + | 2.38702i | −0.707107 | − | 0.707107i | 0.877938 | + | 2.86866i | −0.899506 | + | 2.17160i | |
83.17 | 0.707107 | − | 0.707107i | 1.47024 | − | 0.915636i | − | 1.00000i | 2.44731 | − | 1.01371i | 0.392165 | − | 1.68707i | −1.55261 | − | 2.14229i | −0.707107 | − | 0.707107i | 1.32322 | − | 2.69241i | 1.01371 | − | 2.44731i | |
83.18 | 0.707107 | − | 0.707107i | 1.70634 | − | 0.297343i | − | 1.00000i | −2.47278 | + | 1.02426i | 0.996310 | − | 1.41682i | 2.48133 | + | 0.918153i | −0.707107 | − | 0.707107i | 2.82317 | − | 1.01473i | −1.02426 | + | 2.47278i | |
83.19 | 0.707107 | − | 0.707107i | 1.72704 | − | 0.131707i | − | 1.00000i | −0.103371 | + | 0.0428177i | 1.12807 | − | 1.31433i | 1.40386 | − | 2.24259i | −0.707107 | − | 0.707107i | 2.96531 | − | 0.454927i | −0.0428177 | + | 0.103371i | |
83.20 | 0.707107 | − | 0.707107i | 1.72967 | − | 0.0907356i | − | 1.00000i | 1.23141 | − | 0.510066i | 1.15890 | − | 1.28722i | −1.47002 | + | 2.19978i | −0.707107 | − | 0.707107i | 2.98353 | − | 0.313886i | 0.510066 | − | 1.23141i | |
See all 80 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
51.g | odd | 8 | 1 | inner |
357.w | even | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 714.2.w.e | ✓ | 80 |
3.b | odd | 2 | 1 | 714.2.w.f | yes | 80 | |
7.b | odd | 2 | 1 | inner | 714.2.w.e | ✓ | 80 |
17.d | even | 8 | 1 | 714.2.w.f | yes | 80 | |
21.c | even | 2 | 1 | 714.2.w.f | yes | 80 | |
51.g | odd | 8 | 1 | inner | 714.2.w.e | ✓ | 80 |
119.l | odd | 8 | 1 | 714.2.w.f | yes | 80 | |
357.w | even | 8 | 1 | inner | 714.2.w.e | ✓ | 80 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
714.2.w.e | ✓ | 80 | 1.a | even | 1 | 1 | trivial |
714.2.w.e | ✓ | 80 | 7.b | odd | 2 | 1 | inner |
714.2.w.e | ✓ | 80 | 51.g | odd | 8 | 1 | inner |
714.2.w.e | ✓ | 80 | 357.w | even | 8 | 1 | inner |
714.2.w.f | yes | 80 | 3.b | odd | 2 | 1 | |
714.2.w.f | yes | 80 | 17.d | even | 8 | 1 | |
714.2.w.f | yes | 80 | 21.c | even | 2 | 1 | |
714.2.w.f | yes | 80 | 119.l | odd | 8 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(714, [\chi])\):
\( T_{5}^{80} + 16 T_{5}^{78} + 128 T_{5}^{76} - 2240 T_{5}^{74} + 60610 T_{5}^{72} + 792240 T_{5}^{70} + \cdots + 5473632256 \) |
\( T_{11}^{40} + 20 T_{11}^{38} + 168 T_{11}^{37} + 200 T_{11}^{36} + 24 T_{11}^{35} + 4532 T_{11}^{34} + \cdots + 112659579904 \) |