Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [210,3,Mod(17,210)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(210, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 3, 2]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("210.17");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 210.w (of order \(12\), 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: | \(5.72208555157\) |
Analytic rank: | \(0\) |
Dimension: | \(64\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | 1.36603 | + | 0.366025i | −2.99378 | − | 0.193092i | 1.73205 | + | 1.00000i | 4.84166 | + | 1.24831i | −4.01890 | − | 1.35957i | −0.400482 | − | 6.98853i | 2.00000 | + | 2.00000i | 8.92543 | + | 1.15615i | 6.15692 | + | 3.47740i |
17.2 | 1.36603 | + | 0.366025i | −2.98701 | − | 0.278846i | 1.73205 | + | 1.00000i | −4.38913 | − | 2.39490i | −3.97827 | − | 1.47423i | 5.69479 | + | 4.07055i | 2.00000 | + | 2.00000i | 8.84449 | + | 1.66583i | −5.11906 | − | 4.87803i |
17.3 | 1.36603 | + | 0.366025i | −2.51561 | + | 1.63453i | 1.73205 | + | 1.00000i | −2.05899 | − | 4.55638i | −4.03467 | + | 1.31204i | −6.80611 | − | 1.63612i | 2.00000 | + | 2.00000i | 3.65660 | − | 8.22370i | −1.14488 | − | 6.97777i |
17.4 | 1.36603 | + | 0.366025i | −2.20828 | − | 2.03064i | 1.73205 | + | 1.00000i | −4.28576 | + | 2.57532i | −2.27330 | − | 3.58219i | −1.16075 | − | 6.90309i | 2.00000 | + | 2.00000i | 0.752986 | + | 8.96845i | −6.79709 | + | 1.94926i |
17.5 | 1.36603 | + | 0.366025i | −1.72513 | + | 2.45437i | 1.73205 | + | 1.00000i | 2.34013 | + | 4.41857i | −3.25493 | + | 2.72129i | −6.07917 | + | 3.47039i | 2.00000 | + | 2.00000i | −3.04785 | − | 8.46821i | 1.57936 | + | 6.89243i |
17.6 | 1.36603 | + | 0.366025i | −1.66769 | − | 2.49375i | 1.73205 | + | 1.00000i | 1.09236 | + | 4.87922i | −1.36533 | − | 4.01695i | −1.72310 | + | 6.78461i | 2.00000 | + | 2.00000i | −3.43763 | + | 8.31761i | −0.293729 | + | 7.06496i |
17.7 | 1.36603 | + | 0.366025i | −0.962278 | + | 2.84148i | 1.73205 | + | 1.00000i | 3.83488 | − | 3.20838i | −2.35455 | + | 3.52932i | 6.92770 | − | 1.00349i | 2.00000 | + | 2.00000i | −7.14804 | − | 5.46859i | 6.41289 | − | 2.97907i |
17.8 | 1.36603 | + | 0.366025i | −0.489601 | − | 2.95978i | 1.73205 | + | 1.00000i | 1.47500 | − | 4.77749i | 0.414547 | − | 4.22234i | 6.99874 | + | 0.132847i | 2.00000 | + | 2.00000i | −8.52058 | + | 2.89822i | 3.76357 | − | 5.98628i |
17.9 | 1.36603 | + | 0.366025i | 0.145428 | + | 2.99647i | 1.73205 | + | 1.00000i | −4.92543 | + | 0.860317i | −0.898127 | + | 4.14649i | 1.68496 | + | 6.79418i | 2.00000 | + | 2.00000i | −8.95770 | + | 0.871542i | −7.04316 | − | 0.627617i |
17.10 | 1.36603 | + | 0.366025i | 1.22328 | − | 2.73927i | 1.73205 | + | 1.00000i | 4.98956 | − | 0.322873i | 2.67368 | − | 3.29415i | −4.85698 | − | 5.04081i | 2.00000 | + | 2.00000i | −6.00715 | − | 6.70180i | 6.93405 | + | 1.38525i |
17.11 | 1.36603 | + | 0.366025i | 1.97985 | + | 2.25393i | 1.73205 | + | 1.00000i | 3.75994 | + | 3.29588i | 1.87953 | + | 3.80360i | 5.65896 | − | 4.12022i | 2.00000 | + | 2.00000i | −1.16038 | + | 8.92488i | 3.92980 | + | 5.87849i |
17.12 | 1.36603 | + | 0.366025i | 2.23964 | + | 1.99599i | 1.73205 | + | 1.00000i | −2.72423 | + | 4.19268i | 2.32883 | + | 3.54635i | −5.35730 | − | 4.50548i | 2.00000 | + | 2.00000i | 1.03201 | + | 8.94064i | −5.25600 | + | 4.73017i |
17.13 | 1.36603 | + | 0.366025i | 2.32144 | + | 1.90024i | 1.73205 | + | 1.00000i | 0.695196 | − | 4.95143i | 2.47561 | + | 3.44548i | −2.48720 | + | 6.54323i | 2.00000 | + | 2.00000i | 1.77817 | + | 8.82259i | 2.76201 | − | 6.50933i |
17.14 | 1.36603 | + | 0.366025i | 2.39979 | − | 1.80027i | 1.73205 | + | 1.00000i | −2.39011 | + | 4.39174i | 3.93713 | − | 1.58084i | 6.65191 | + | 2.17994i | 2.00000 | + | 2.00000i | 2.51802 | − | 8.64058i | −4.87243 | + | 5.12439i |
17.15 | 1.36603 | + | 0.366025i | 2.92260 | − | 0.677069i | 1.73205 | + | 1.00000i | 4.99957 | − | 0.0656422i | 4.24017 | + | 0.144852i | −2.70652 | + | 6.45560i | 2.00000 | + | 2.00000i | 8.08316 | − | 3.95760i | 6.85357 | + | 1.74030i |
17.16 | 1.36603 | + | 0.366025i | 2.95132 | − | 0.538273i | 1.73205 | + | 1.00000i | −2.52260 | − | 4.31700i | 4.22859 | + | 0.344962i | 0.692615 | − | 6.96565i | 2.00000 | + | 2.00000i | 8.42052 | − | 3.17723i | −1.86581 | − | 6.82047i |
47.1 | −0.366025 | − | 1.36603i | −2.99350 | + | 0.197383i | −1.73205 | + | 1.00000i | −3.67935 | + | 3.38562i | 1.36533 | + | 4.01695i | 6.78461 | − | 1.72310i | 2.00000 | + | 2.00000i | 8.92208 | − | 1.18173i | 5.97157 | + | 3.78686i |
47.2 | −0.366025 | − | 1.36603i | −2.86273 | + | 0.897104i | −1.73205 | + | 1.00000i | −4.37317 | − | 2.42391i | 2.27330 | + | 3.58219i | −6.90309 | − | 1.16075i | 2.00000 | + | 2.00000i | 7.39041 | − | 5.13633i | −1.71043 | + | 6.86108i |
47.3 | −0.366025 | − | 1.36603i | −2.80804 | − | 1.05588i | −1.73205 | + | 1.00000i | 4.87492 | − | 1.11135i | −0.414547 | + | 4.22234i | 0.132847 | + | 6.99874i | 2.00000 | + | 2.00000i | 6.77022 | + | 5.92993i | −3.30248 | − | 6.25249i |
47.4 | −0.366025 | − | 1.36603i | −1.76063 | − | 2.42903i | −1.73205 | + | 1.00000i | 2.77440 | + | 4.15965i | −2.67368 | + | 3.29415i | −5.04081 | − | 4.85698i | 2.00000 | + | 2.00000i | −2.80035 | + | 8.55325i | 4.66669 | − | 5.31244i |
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
15.e | even | 4 | 1 | inner |
105.w | odd | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 210.3.w.b | yes | 64 |
3.b | odd | 2 | 1 | 210.3.w.a | ✓ | 64 | |
5.c | odd | 4 | 1 | 210.3.w.a | ✓ | 64 | |
7.d | odd | 6 | 1 | inner | 210.3.w.b | yes | 64 |
15.e | even | 4 | 1 | inner | 210.3.w.b | yes | 64 |
21.g | even | 6 | 1 | 210.3.w.a | ✓ | 64 | |
35.k | even | 12 | 1 | 210.3.w.a | ✓ | 64 | |
105.w | odd | 12 | 1 | inner | 210.3.w.b | yes | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
210.3.w.a | ✓ | 64 | 3.b | odd | 2 | 1 | |
210.3.w.a | ✓ | 64 | 5.c | odd | 4 | 1 | |
210.3.w.a | ✓ | 64 | 21.g | even | 6 | 1 | |
210.3.w.a | ✓ | 64 | 35.k | even | 12 | 1 | |
210.3.w.b | yes | 64 | 1.a | even | 1 | 1 | trivial |
210.3.w.b | yes | 64 | 7.d | odd | 6 | 1 | inner |
210.3.w.b | yes | 64 | 15.e | even | 4 | 1 | inner |
210.3.w.b | yes | 64 | 105.w | odd | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{17}^{64} - 1232998 T_{17}^{60} + 1668936 T_{17}^{59} - 749871912 T_{17}^{57} + 986760210559 T_{17}^{56} - 2057794750128 T_{17}^{55} + 1392673686048 T_{17}^{54} + \cdots + 47\!\cdots\!00 \)
acting on \(S_{3}^{\mathrm{new}}(210, [\chi])\).