Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [770,2,Mod(139,770)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(770, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 5, 7]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("770.139");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 770.w (of order \(10\), 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: | \(6.14848095564\) |
Analytic rank: | \(0\) |
Dimension: | \(96\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
139.1 | −0.809017 | − | 0.587785i | −0.968385 | + | 2.98038i | 0.309017 | + | 0.951057i | −0.264979 | + | 2.22031i | 2.53526 | − | 1.84198i | −2.11931 | + | 1.58384i | 0.309017 | − | 0.951057i | −5.51786 | − | 4.00896i | 1.51944 | − | 1.64052i |
139.2 | −0.809017 | − | 0.587785i | −0.934488 | + | 2.87606i | 0.309017 | + | 0.951057i | −0.226592 | − | 2.22456i | 2.44652 | − | 1.77750i | 2.25369 | − | 1.38595i | 0.309017 | − | 0.951057i | −4.97140 | − | 3.61193i | −1.12425 | + | 1.93289i |
139.3 | −0.809017 | − | 0.587785i | −0.822398 | + | 2.53108i | 0.309017 | + | 0.951057i | −0.114957 | + | 2.23311i | 2.15307 | − | 1.56429i | 0.953236 | − | 2.46806i | 0.309017 | − | 0.951057i | −3.30298 | − | 2.39976i | 1.40559 | − | 1.73905i |
139.4 | −0.809017 | − | 0.587785i | −0.732958 | + | 2.25581i | 0.309017 | + | 0.951057i | −1.93690 | − | 1.11733i | 1.91891 | − | 1.39417i | −1.96220 | + | 1.77476i | 0.309017 | − | 0.951057i | −2.12441 | − | 1.54347i | 0.910238 | + | 2.04242i |
139.5 | −0.809017 | − | 0.587785i | −0.704096 | + | 2.16698i | 0.309017 | + | 0.951057i | 2.13221 | − | 0.673563i | 1.84335 | − | 1.33927i | 1.07060 | + | 2.41946i | 0.309017 | − | 0.951057i | −1.77302 | − | 1.28817i | −2.12090 | − | 0.708357i |
139.6 | −0.809017 | − | 0.587785i | −0.497687 | + | 1.53172i | 0.309017 | + | 0.951057i | 0.911178 | − | 2.04200i | 1.30296 | − | 0.946658i | −2.43957 | − | 1.02395i | 0.309017 | − | 0.951057i | 0.328565 | + | 0.238717i | −1.93741 | + | 1.11643i |
139.7 | −0.809017 | − | 0.587785i | −0.463552 | + | 1.42667i | 0.309017 | + | 0.951057i | −2.23591 | − | 0.0266895i | 1.21359 | − | 0.881728i | −1.05163 | − | 2.42777i | 0.309017 | − | 0.951057i | 0.606556 | + | 0.440689i | 1.79320 | + | 1.33583i |
139.8 | −0.809017 | − | 0.587785i | −0.407445 | + | 1.25399i | 0.309017 | + | 0.951057i | 0.820886 | + | 2.07994i | 1.06670 | − | 0.775006i | 2.54546 | − | 0.721566i | 0.309017 | − | 0.951057i | 1.02058 | + | 0.741496i | 0.558447 | − | 2.16521i |
139.9 | −0.809017 | − | 0.587785i | −0.403178 | + | 1.24085i | 0.309017 | + | 0.951057i | 1.44012 | − | 1.71057i | 1.05553 | − | 0.766890i | 2.09063 | − | 1.62150i | 0.309017 | − | 0.951057i | 1.04989 | + | 0.762787i | −2.17053 | + | 0.537397i |
139.10 | −0.809017 | − | 0.587785i | −0.266549 | + | 0.820354i | 0.309017 | + | 0.951057i | 1.93580 | + | 1.11923i | 0.697835 | − | 0.507006i | −2.50422 | − | 0.853746i | 0.309017 | − | 0.951057i | 1.82512 | + | 1.32603i | −0.908230 | − | 2.04331i |
139.11 | −0.809017 | − | 0.587785i | −0.208297 | + | 0.641072i | 0.309017 | + | 0.951057i | −1.84225 | + | 1.26732i | 0.545328 | − | 0.396204i | 0.482283 | + | 2.60142i | 0.309017 | − | 0.951057i | 2.05947 | + | 1.49629i | 2.23533 | + | 0.0575660i |
139.12 | −0.809017 | − | 0.587785i | −0.0561872 | + | 0.172927i | 0.309017 | + | 0.951057i | 1.35138 | + | 1.78151i | 0.147100 | − | 0.106874i | −1.13715 | + | 2.38891i | 0.309017 | − | 0.951057i | 2.40030 | + | 1.74392i | −0.0461423 | − | 2.23559i |
139.13 | −0.809017 | − | 0.587785i | 0.0561872 | − | 0.172927i | 0.309017 | + | 0.951057i | −1.35138 | − | 1.78151i | −0.147100 | + | 0.106874i | 2.32414 | + | 1.26426i | 0.309017 | − | 0.951057i | 2.40030 | + | 1.74392i | 0.0461423 | + | 2.23559i |
139.14 | −0.809017 | − | 0.587785i | 0.208297 | − | 0.641072i | 0.309017 | + | 0.951057i | 1.84225 | − | 1.26732i | −0.545328 | + | 0.396204i | 1.13890 | + | 2.38807i | 0.309017 | − | 0.951057i | 2.05947 | + | 1.49629i | −2.23533 | − | 0.0575660i |
139.15 | −0.809017 | − | 0.587785i | 0.266549 | − | 0.820354i | 0.309017 | + | 0.951057i | −1.93580 | − | 1.11923i | −0.697835 | + | 0.507006i | 1.52414 | − | 2.16264i | 0.309017 | − | 0.951057i | 1.82512 | + | 1.32603i | 0.908230 | + | 2.04331i |
139.16 | −0.809017 | − | 0.587785i | 0.403178 | − | 1.24085i | 0.309017 | + | 0.951057i | −1.44012 | + | 1.71057i | −1.05553 | + | 0.766890i | −2.64445 | − | 0.0829819i | 0.309017 | − | 0.951057i | 1.04989 | + | 0.762787i | 2.17053 | − | 0.537397i |
139.17 | −0.809017 | − | 0.587785i | 0.407445 | − | 1.25399i | 0.309017 | + | 0.951057i | −0.820886 | − | 2.07994i | −1.06670 | + | 0.775006i | −2.48344 | + | 0.912422i | 0.309017 | − | 0.951057i | 1.02058 | + | 0.741496i | −0.558447 | + | 2.16521i |
139.18 | −0.809017 | − | 0.587785i | 0.463552 | − | 1.42667i | 0.309017 | + | 0.951057i | 2.23591 | + | 0.0266895i | −1.21359 | + | 0.881728i | −0.576225 | − | 2.58224i | 0.309017 | − | 0.951057i | 0.606556 | + | 0.440689i | −1.79320 | − | 1.33583i |
139.19 | −0.809017 | − | 0.587785i | 0.497687 | − | 1.53172i | 0.309017 | + | 0.951057i | −0.911178 | + | 2.04200i | −1.30296 | + | 0.946658i | 1.37179 | − | 2.26234i | 0.309017 | − | 0.951057i | 0.328565 | + | 0.238717i | 1.93741 | − | 1.11643i |
139.20 | −0.809017 | − | 0.587785i | 0.704096 | − | 2.16698i | 0.309017 | + | 0.951057i | −2.13221 | + | 0.673563i | −1.84335 | + | 1.33927i | 0.555990 | + | 2.58667i | 0.309017 | − | 0.951057i | −1.77302 | − | 1.28817i | 2.12090 | + | 0.708357i |
See all 96 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
55.h | odd | 10 | 1 | inner |
385.v | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 770.2.w.a | ✓ | 96 |
5.b | even | 2 | 1 | 770.2.w.b | yes | 96 | |
7.b | odd | 2 | 1 | inner | 770.2.w.a | ✓ | 96 |
11.d | odd | 10 | 1 | 770.2.w.b | yes | 96 | |
35.c | odd | 2 | 1 | 770.2.w.b | yes | 96 | |
55.h | odd | 10 | 1 | inner | 770.2.w.a | ✓ | 96 |
77.l | even | 10 | 1 | 770.2.w.b | yes | 96 | |
385.v | even | 10 | 1 | inner | 770.2.w.a | ✓ | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
770.2.w.a | ✓ | 96 | 1.a | even | 1 | 1 | trivial |
770.2.w.a | ✓ | 96 | 7.b | odd | 2 | 1 | inner |
770.2.w.a | ✓ | 96 | 55.h | odd | 10 | 1 | inner |
770.2.w.a | ✓ | 96 | 385.v | even | 10 | 1 | inner |
770.2.w.b | yes | 96 | 5.b | even | 2 | 1 | |
770.2.w.b | yes | 96 | 11.d | odd | 10 | 1 | |
770.2.w.b | yes | 96 | 35.c | odd | 2 | 1 | |
770.2.w.b | yes | 96 | 77.l | even | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{37}^{48} - 86 T_{37}^{46} + 605 T_{37}^{45} + 23337 T_{37}^{44} - 52030 T_{37}^{43} + \cdots + 88\!\cdots\!56 \) acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\).