Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [432,2,Mod(13,432)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(432, base_ring=CyclotomicField(36))
chi = DirichletCharacter(H, H._module([0, 27, 16]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("432.13");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 432 = 2^{4} \cdot 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 432.bg (of order \(36\), degree \(12\), 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: | \(3.44953736732\) |
Analytic rank: | \(0\) |
Dimension: | \(840\) |
Relative dimension: | \(70\) over \(\Q(\zeta_{36})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{36}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
13.1 | −1.41419 | + | 0.00816009i | 0.757020 | − | 1.55786i | 1.99987 | − | 0.0230798i | 2.92668 | − | 0.256052i | −1.05786 | + | 2.20928i | 1.42579 | − | 3.91732i | −2.82800 | + | 0.0489584i | −1.85384 | − | 2.35866i | −4.13680 | + | 0.385988i |
13.2 | −1.41151 | − | 0.0874761i | 0.0727719 | + | 1.73052i | 1.98470 | + | 0.246946i | −0.723668 | + | 0.0633127i | 0.0486614 | − | 2.44901i | 0.398828 | − | 1.09577i | −2.77981 | − | 0.522179i | −2.98941 | + | 0.251867i | 1.02700 | − | 0.0260626i |
13.3 | −1.40392 | − | 0.170309i | −1.60100 | − | 0.660910i | 1.94199 | + | 0.478202i | 0.199885 | − | 0.0174877i | 2.13512 | + | 1.20053i | −0.807683 | + | 2.21909i | −2.64496 | − | 1.00210i | 2.12640 | + | 2.11623i | −0.283601 | − | 0.00949098i |
13.4 | −1.39069 | − | 0.256893i | −0.791539 | − | 1.54061i | 1.86801 | + | 0.714515i | −2.39012 | + | 0.209108i | 0.705011 | + | 2.34584i | 0.307660 | − | 0.845290i | −2.41426 | − | 1.47355i | −1.74693 | + | 2.43890i | 3.37762 | + | 0.323202i |
13.5 | −1.37047 | − | 0.349003i | 1.72507 | + | 0.155359i | 1.75639 | + | 0.956599i | 1.88606 | − | 0.165009i | −2.30994 | − | 0.814971i | −0.627115 | + | 1.72298i | −2.07323 | − | 1.92398i | 2.95173 | + | 0.536012i | −2.64238 | − | 0.432100i |
13.6 | −1.36057 | + | 0.385801i | −1.47892 | + | 0.901550i | 1.70231 | − | 1.04982i | 2.53117 | − | 0.221449i | 1.66436 | − | 1.79719i | −0.599847 | + | 1.64807i | −1.91110 | + | 2.08511i | 1.37441 | − | 2.66664i | −3.35841 | + | 1.27783i |
13.7 | −1.35642 | + | 0.400143i | 1.26410 | − | 1.18408i | 1.67977 | − | 1.08553i | −2.21064 | + | 0.193406i | −1.24086 | + | 2.11193i | −0.170100 | + | 0.467346i | −1.84412 | + | 2.14458i | 0.195921 | − | 2.99360i | 2.92117 | − | 1.14691i |
13.8 | −1.31559 | + | 0.518866i | −1.54765 | + | 0.777673i | 1.46156 | − | 1.36523i | −4.10207 | + | 0.358885i | 1.63257 | − | 1.82612i | 0.659902 | − | 1.81306i | −1.21444 | + | 2.55443i | 1.79045 | − | 2.40713i | 5.21044 | − | 2.60057i |
13.9 | −1.31389 | − | 0.523159i | 1.58196 | + | 0.705273i | 1.45261 | + | 1.37475i | −3.33491 | + | 0.291766i | −1.70955 | − | 1.75427i | 1.51056 | − | 4.15024i | −1.18936 | − | 2.56621i | 2.00518 | + | 2.23142i | 4.53434 | + | 1.36134i |
13.10 | −1.30739 | + | 0.539193i | 1.39818 | + | 1.02229i | 1.41854 | − | 1.40987i | 3.17016 | − | 0.277353i | −2.37919 | − | 0.582643i | 0.294648 | − | 0.809538i | −1.09439 | + | 2.60812i | 0.909840 | + | 2.85870i | −3.99510 | + | 2.07194i |
13.11 | −1.26954 | + | 0.623105i | −0.0426757 | − | 1.73152i | 1.22348 | − | 1.58212i | 1.79177 | − | 0.156760i | 1.13310 | + | 2.17165i | −1.76857 | + | 4.85911i | −0.567437 | + | 2.77092i | −2.99636 | + | 0.147788i | −2.17706 | + | 1.31548i |
13.12 | −1.19582 | − | 0.754986i | −0.692004 | + | 1.58781i | 0.859992 | + | 1.80566i | 2.72931 | − | 0.238784i | 2.02629 | − | 1.37628i | 0.552395 | − | 1.51769i | 0.334850 | − | 2.80854i | −2.04226 | − | 2.19754i | −3.44405 | − | 1.77505i |
13.13 | −1.19078 | − | 0.762913i | −1.38465 | + | 1.04055i | 0.835928 | + | 1.81693i | −2.45604 | + | 0.214875i | 2.44267 | − | 0.182707i | −1.20765 | + | 3.31798i | 0.390748 | − | 2.80131i | 0.834499 | − | 2.88160i | 3.08854 | + | 1.61787i |
13.14 | −1.15556 | + | 0.815280i | −1.32179 | − | 1.11931i | 0.670638 | − | 1.88421i | 0.762654 | − | 0.0667236i | 2.43997 | + | 0.215802i | 1.03298 | − | 2.83809i | 0.761196 | + | 2.72407i | 0.494277 | + | 2.95900i | −0.826895 | + | 0.698880i |
13.15 | −1.09334 | − | 0.896994i | 1.02688 | − | 1.39482i | 0.390802 | + | 1.96145i | −3.34554 | + | 0.292697i | −2.37388 | + | 0.603909i | −0.981004 | + | 2.69529i | 1.33213 | − | 2.49508i | −0.891032 | − | 2.86462i | 3.92037 | + | 2.68091i |
13.16 | −1.09259 | + | 0.897911i | 1.51775 | + | 0.834518i | 0.387511 | − | 1.96210i | −2.33873 | + | 0.204612i | −2.40761 | + | 0.451022i | −0.349949 | + | 0.961477i | 1.33840 | + | 2.49172i | 1.60716 | + | 2.53319i | 2.37155 | − | 2.32353i |
13.17 | −1.07216 | − | 0.922209i | −1.54249 | − | 0.787864i | 0.299063 | + | 1.97751i | 4.29797 | − | 0.376024i | 0.927222 | + | 2.26721i | −0.248889 | + | 0.683816i | 1.50304 | − | 2.39601i | 1.75854 | + | 2.43054i | −4.95489 | − | 3.56047i |
13.18 | −0.945342 | − | 1.05182i | 1.02940 | + | 1.39296i | −0.212657 | + | 1.98866i | 1.13926 | − | 0.0996725i | 0.492003 | − | 2.39957i | −1.44119 | + | 3.95964i | 2.29275 | − | 1.65629i | −0.880655 | + | 2.86783i | −1.18183 | − | 1.10408i |
13.19 | −0.881475 | − | 1.10589i | −0.338579 | − | 1.69864i | −0.446003 | + | 1.94964i | 0.423518 | − | 0.0370531i | −1.58006 | + | 1.87174i | 0.916962 | − | 2.51933i | 2.54923 | − | 1.22532i | −2.77073 | + | 1.15024i | −0.414298 | − | 0.435705i |
13.20 | −0.854260 | − | 1.12705i | 1.64279 | − | 0.548844i | −0.540481 | + | 1.92559i | 0.365664 | − | 0.0319915i | −2.02195 | − | 1.38265i | 0.804838 | − | 2.21127i | 2.63194 | − | 1.03580i | 2.39754 | − | 1.80328i | −0.348428 | − | 0.384793i |
See next 80 embeddings (of 840 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.e | even | 4 | 1 | inner |
27.e | even | 9 | 1 | inner |
432.bg | even | 36 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 432.2.bg.a | ✓ | 840 |
16.e | even | 4 | 1 | inner | 432.2.bg.a | ✓ | 840 |
27.e | even | 9 | 1 | inner | 432.2.bg.a | ✓ | 840 |
432.bg | even | 36 | 1 | inner | 432.2.bg.a | ✓ | 840 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
432.2.bg.a | ✓ | 840 | 1.a | even | 1 | 1 | trivial |
432.2.bg.a | ✓ | 840 | 16.e | even | 4 | 1 | inner |
432.2.bg.a | ✓ | 840 | 27.e | even | 9 | 1 | inner |
432.2.bg.a | ✓ | 840 | 432.bg | even | 36 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(432, [\chi])\).