Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [756,2,Mod(103,756)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(756, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([9, 14, 15]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("756.103");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 756.bt (of order \(18\), degree \(6\), 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: | \(6.03669039281\) |
Analytic rank: | \(0\) |
Dimension: | \(840\) |
Relative dimension: | \(140\) over \(\Q(\zeta_{18})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
103.1 | −1.41414 | + | 0.0147396i | −0.326297 | + | 1.70104i | 1.99957 | − | 0.0416875i | −1.34679 | − | 0.237475i | 0.436356 | − | 2.41031i | 1.86287 | + | 1.87875i | −2.82704 | + | 0.0884245i | −2.78706 | − | 1.11009i | 1.90804 | + | 0.315971i |
103.2 | −1.41407 | + | 0.0203430i | 1.63923 | − | 0.559403i | 1.99917 | − | 0.0575327i | −1.49077 | − | 0.262863i | −2.30660 | + | 0.824380i | 1.66611 | − | 2.05525i | −2.82579 | + | 0.122024i | 2.37414 | − | 1.83398i | 2.11340 | + | 0.341379i |
103.3 | −1.41387 | + | 0.0311086i | 0.755283 | − | 1.55870i | 1.99806 | − | 0.0879671i | 3.35278 | + | 0.591186i | −1.01938 | + | 2.22730i | 2.13149 | + | 1.56741i | −2.82227 | + | 0.186531i | −1.85910 | − | 2.35452i | −4.75879 | − | 0.731561i |
103.4 | −1.41356 | + | 0.0430901i | −1.54969 | + | 0.773606i | 1.99629 | − | 0.121821i | −0.175857 | − | 0.0310083i | 2.15724 | − | 1.16031i | −2.42990 | + | 1.04671i | −2.81662 | + | 0.258221i | 1.80307 | − | 2.39770i | 0.249920 | + | 0.0362544i |
103.5 | −1.41121 | + | 0.0920983i | −0.696355 | − | 1.58590i | 1.98304 | − | 0.259940i | 0.824097 | + | 0.145310i | 1.12876 | + | 2.17391i | −2.46179 | + | 0.969313i | −2.77454 | + | 0.549465i | −2.03018 | + | 2.20870i | −1.17636 | − | 0.129166i |
103.6 | −1.40532 | + | 0.158328i | 1.69106 | − | 0.374584i | 1.94986 | − | 0.445005i | 3.97187 | + | 0.700348i | −2.31718 | + | 0.794154i | −2.15958 | − | 1.52847i | −2.66973 | + | 0.934094i | 2.71937 | − | 1.26689i | −5.69265 | − | 0.355355i |
103.7 | −1.39958 | + | 0.202941i | −1.70851 | − | 0.284574i | 1.91763 | − | 0.568062i | 2.54587 | + | 0.448906i | 2.44895 | + | 0.0515557i | 2.36953 | − | 1.17699i | −2.56859 | + | 1.18421i | 2.83804 | + | 0.972395i | −3.65425 | − | 0.111617i |
103.8 | −1.39646 | + | 0.223410i | 0.897457 | + | 1.48141i | 1.90018 | − | 0.623965i | −1.37715 | − | 0.242829i | −1.58422 | − | 1.86822i | −1.27459 | + | 2.31850i | −2.51411 | + | 1.29586i | −1.38914 | + | 2.65900i | 1.97738 | + | 0.0314303i |
103.9 | −1.39221 | + | 0.248524i | −1.49487 | − | 0.874854i | 1.87647 | − | 0.691994i | −2.90630 | − | 0.512459i | 2.29859 | + | 0.846466i | −1.25872 | − | 2.32715i | −2.44046 | + | 1.42975i | 1.46926 | + | 2.61558i | 4.17352 | − | 0.00883825i |
103.10 | −1.36742 | − | 0.360769i | −1.03034 | + | 1.39226i | 1.73969 | + | 0.986648i | 1.98405 | + | 0.349842i | 1.91120 | − | 1.53209i | 1.38264 | − | 2.25573i | −2.02294 | − | 1.97679i | −0.876779 | − | 2.86902i | −2.58683 | − | 1.19417i |
103.11 | −1.36642 | − | 0.364532i | −0.0919920 | − | 1.72961i | 1.73423 | + | 0.996212i | −2.28806 | − | 0.403446i | −0.504797 | + | 2.39691i | 2.63964 | − | 0.179668i | −2.00655 | − | 1.99343i | −2.98307 | + | 0.318220i | 2.97939 | + | 1.38535i |
103.12 | −1.35942 | − | 0.389832i | 1.54409 | + | 0.784711i | 1.69606 | + | 1.05989i | 0.874454 | + | 0.154190i | −1.79317 | − | 1.66869i | 2.54554 | + | 0.721260i | −1.89249 | − | 2.10202i | 1.76846 | + | 2.42334i | −1.12865 | − | 0.550499i |
103.13 | −1.34560 | − | 0.435164i | −1.53529 | − | 0.801804i | 1.62126 | + | 1.17111i | −2.59686 | − | 0.457897i | 1.71696 | + | 1.74701i | 0.683168 | + | 2.55603i | −1.67194 | − | 2.28136i | 1.71422 | + | 2.46200i | 3.29507 | + | 1.74621i |
103.14 | −1.34481 | − | 0.437584i | 1.72493 | − | 0.156906i | 1.61704 | + | 1.17694i | −4.40150 | − | 0.776103i | −2.38837 | − | 0.543792i | −2.15811 | + | 1.53054i | −1.65961 | − | 2.29035i | 2.95076 | − | 0.541305i | 5.57958 | + | 2.96974i |
103.15 | −1.33467 | + | 0.467620i | −0.284726 | + | 1.70849i | 1.56266 | − | 1.24823i | 1.73945 | + | 0.306711i | −0.418909 | − | 2.41340i | −0.667920 | − | 2.56006i | −1.50193 | + | 2.39671i | −2.83786 | − | 0.972903i | −2.46500 | + | 0.404043i |
103.16 | −1.31489 | + | 0.520649i | 1.19189 | + | 1.25674i | 1.45785 | − | 1.36919i | −3.13702 | − | 0.553141i | −2.22152 | − | 1.03191i | 0.170788 | − | 2.64023i | −1.20404 | + | 2.55935i | −0.158782 | + | 2.99580i | 4.41281 | − | 0.905969i |
103.17 | −1.30594 | − | 0.542696i | 0.375808 | + | 1.69079i | 1.41096 | + | 1.41746i | 3.92095 | + | 0.691370i | 0.426801 | − | 2.41202i | −1.73044 | + | 2.00139i | −1.07338 | − | 2.61684i | −2.71754 | + | 1.27083i | −4.74533 | − | 3.03077i |
103.18 | −1.28497 | − | 0.590645i | −0.974544 | − | 1.43187i | 1.30228 | + | 1.51792i | 2.54084 | + | 0.448019i | 0.406527 | + | 2.41552i | −0.471692 | − | 2.60336i | −0.776834 | − | 2.71966i | −1.10053 | + | 2.79085i | −3.00028 | − | 2.07643i |
103.19 | −1.27998 | − | 0.601365i | −1.63693 | + | 0.566086i | 1.27672 | + | 1.53948i | −1.53946 | − | 0.271447i | 2.43567 | + | 0.259812i | −1.93149 | − | 1.80814i | −0.708397 | − | 2.73828i | 2.35909 | − | 1.85329i | 1.80724 | + | 1.27322i |
103.20 | −1.27697 | + | 0.607749i | 1.12163 | − | 1.31983i | 1.26128 | − | 1.55215i | −2.50783 | − | 0.442198i | −0.630164 | + | 2.36704i | 1.61158 | + | 2.09829i | −0.667298 | + | 2.74858i | −0.483880 | − | 2.96072i | 3.47116 | − | 0.959460i |
See next 80 embeddings (of 840 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
189.x | odd | 18 | 1 | inner |
756.bt | even | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 756.2.bt.a | ✓ | 840 |
4.b | odd | 2 | 1 | inner | 756.2.bt.a | ✓ | 840 |
7.d | odd | 6 | 1 | 756.2.cd.a | yes | 840 | |
27.e | even | 9 | 1 | 756.2.cd.a | yes | 840 | |
28.f | even | 6 | 1 | 756.2.cd.a | yes | 840 | |
108.j | odd | 18 | 1 | 756.2.cd.a | yes | 840 | |
189.x | odd | 18 | 1 | inner | 756.2.bt.a | ✓ | 840 |
756.bt | even | 18 | 1 | inner | 756.2.bt.a | ✓ | 840 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
756.2.bt.a | ✓ | 840 | 1.a | even | 1 | 1 | trivial |
756.2.bt.a | ✓ | 840 | 4.b | odd | 2 | 1 | inner |
756.2.bt.a | ✓ | 840 | 189.x | odd | 18 | 1 | inner |
756.2.bt.a | ✓ | 840 | 756.bt | even | 18 | 1 | inner |
756.2.cd.a | yes | 840 | 7.d | odd | 6 | 1 | |
756.2.cd.a | yes | 840 | 27.e | even | 9 | 1 | |
756.2.cd.a | yes | 840 | 28.f | even | 6 | 1 | |
756.2.cd.a | yes | 840 | 108.j | odd | 18 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(756, [\chi])\).