Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [144,3,Mod(5,144)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(144, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 3, 10]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("144.5");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 144 = 2^{4} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 144.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: | \(3.92371580679\) |
Analytic rank: | \(0\) |
Dimension: | \(184\) |
Relative dimension: | \(46\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −1.99959 | + | 0.0402970i | 0.538615 | + | 2.95125i | 3.99675 | − | 0.161155i | −2.08673 | − | 7.78777i | −1.19594 | − | 5.87960i | −2.60739 | + | 1.50538i | −7.98539 | + | 0.483303i | −8.41979 | + | 3.17918i | 4.48643 | + | 15.4883i |
5.2 | −1.99458 | + | 0.147176i | −2.86084 | + | 0.903108i | 3.95668 | − | 0.587107i | −0.0992523 | − | 0.370415i | 5.57325 | − | 2.22237i | 3.11010 | − | 1.79562i | −7.80549 | + | 1.75336i | 7.36879 | − | 5.16729i | 0.252482 | + | 0.724213i |
5.3 | −1.98351 | + | 0.256277i | −1.40595 | − | 2.65015i | 3.86864 | − | 1.01666i | 1.00259 | + | 3.74171i | 3.46789 | + | 4.89630i | 4.65369 | − | 2.68681i | −7.41296 | + | 3.00800i | −5.04662 | + | 7.45195i | −2.94756 | − | 7.16478i |
5.4 | −1.88582 | − | 0.666085i | 2.99994 | + | 0.0192045i | 3.11266 | + | 2.51224i | −0.345472 | − | 1.28932i | −5.64456 | − | 2.03443i | 4.11923 | − | 2.37824i | −4.19657 | − | 6.81093i | 8.99926 | + | 0.115225i | −0.207297 | + | 2.66154i |
5.5 | −1.88357 | + | 0.672432i | 2.93495 | + | 0.621365i | 3.09567 | − | 2.53315i | 1.03307 | + | 3.85549i | −5.94600 | + | 0.803167i | −11.2715 | + | 6.50760i | −4.12754 | + | 6.85298i | 8.22781 | + | 3.64735i | −4.53842 | − | 6.56740i |
5.6 | −1.84760 | − | 0.765753i | −0.120474 | + | 2.99758i | 2.82724 | + | 2.82961i | 2.37709 | + | 8.87143i | 2.51799 | − | 5.44607i | 0.152137 | − | 0.0878365i | −3.05683 | − | 7.39295i | −8.97097 | − | 0.722263i | 2.40141 | − | 18.2111i |
5.7 | −1.79350 | − | 0.885072i | 0.352522 | − | 2.97922i | 2.43329 | + | 3.17476i | 0.899691 | + | 3.35769i | −3.26907 | + | 5.03122i | −9.56176 | + | 5.52049i | −1.55423 | − | 7.84757i | −8.75146 | − | 2.10048i | 1.35820 | − | 6.81832i |
5.8 | −1.73513 | + | 0.994646i | 1.85623 | − | 2.35678i | 2.02136 | − | 3.45168i | −0.777454 | − | 2.90150i | −0.876653 | + | 5.93561i | 2.72697 | − | 1.57442i | −0.0741259 | + | 7.99966i | −2.10879 | − | 8.74946i | 4.23494 | + | 4.26119i |
5.9 | −1.61016 | − | 1.18633i | −2.90827 | + | 0.736170i | 1.18523 | + | 3.82037i | −0.295811 | − | 1.10398i | 5.55613 | + | 2.26483i | −7.30360 | + | 4.21674i | 2.62382 | − | 7.55748i | 7.91611 | − | 4.28197i | −0.833388 | + | 2.12852i |
5.10 | −1.59704 | − | 1.20393i | −1.75135 | − | 2.43573i | 1.10109 | + | 3.84547i | −2.49590 | − | 9.31484i | −0.135487 | + | 5.99847i | 7.55275 | − | 4.36058i | 2.87120 | − | 7.46701i | −2.86558 | + | 8.53161i | −7.22838 | + | 17.8811i |
5.11 | −1.58134 | + | 1.22449i | 1.74774 | + | 2.43832i | 1.00126 | − | 3.87266i | 0.973349 | + | 3.63259i | −5.74946 | − | 1.71573i | 11.7048 | − | 6.75777i | 3.15868 | + | 7.35002i | −2.89081 | + | 8.52310i | −5.98725 | − | 4.55250i |
5.12 | −1.51447 | + | 1.30628i | −2.22089 | − | 2.01684i | 0.587245 | − | 3.95666i | −1.88738 | − | 7.04379i | 5.99804 | + | 0.153334i | −8.89672 | + | 5.13652i | 4.27915 | + | 6.75935i | 0.864706 | + | 8.95836i | 12.0596 | + | 8.20216i |
5.13 | −1.26432 | + | 1.54968i | −2.98386 | − | 0.310733i | −0.803013 | − | 3.91857i | 2.23236 | + | 8.33129i | 4.25408 | − | 4.23117i | −3.69610 | + | 2.13394i | 7.08778 | + | 3.70989i | 8.80689 | + | 1.85437i | −15.7332 | − | 7.07393i |
5.14 | −1.16958 | + | 1.62237i | −1.89172 | + | 2.32839i | −1.26417 | − | 3.79498i | −1.06962 | − | 3.99189i | −1.56500 | − | 5.79230i | 2.82785 | − | 1.63266i | 7.63541 | + | 2.38759i | −1.84282 | − | 8.80931i | 7.72733 | + | 2.93351i |
5.15 | −1.09867 | − | 1.67120i | 0.894041 | + | 2.86368i | −1.58584 | + | 3.67221i | −0.835695 | − | 3.11886i | 3.80354 | − | 4.64037i | 0.108974 | − | 0.0629164i | 7.87933 | − | 1.38428i | −7.40138 | + | 5.12050i | −4.29409 | + | 4.82321i |
5.16 | −1.08605 | − | 1.67944i | 1.99667 | − | 2.23905i | −1.64101 | + | 3.64789i | 1.20857 | + | 4.51043i | −5.92880 | − | 0.921569i | 7.69955 | − | 4.44534i | 7.90861 | − | 1.20579i | −1.02665 | − | 8.94125i | 6.26242 | − | 6.92824i |
5.17 | −0.786053 | + | 1.83905i | 1.51963 | − | 2.58664i | −2.76424 | − | 2.89119i | 1.99100 | + | 7.43050i | 3.56247 | + | 4.82792i | −1.50577 | + | 0.869355i | 7.48989 | − | 2.81097i | −4.38144 | − | 7.86149i | −15.2301 | − | 2.17922i |
5.18 | −0.762335 | − | 1.84901i | −2.07303 | + | 2.16854i | −2.83769 | + | 2.81913i | −0.115759 | − | 0.432017i | 5.58999 | + | 2.17990i | 7.23679 | − | 4.17816i | 7.37588 | + | 3.09780i | −0.405106 | − | 8.99088i | −0.710557 | + | 0.543380i |
5.19 | −0.702398 | + | 1.87260i | 1.41045 | + | 2.64776i | −3.01327 | − | 2.63062i | −0.226081 | − | 0.843746i | −5.94890 | + | 0.781440i | −7.54519 | + | 4.35622i | 7.04263 | − | 3.79492i | −5.02124 | + | 7.46908i | 1.73880 | + | 0.169286i |
5.20 | −0.608739 | − | 1.90511i | 2.61769 | − | 1.46550i | −3.25887 | + | 2.31943i | −2.13498 | − | 7.96785i | −4.38543 | − | 4.09487i | −10.2471 | + | 5.91616i | 6.40256 | + | 4.79658i | 4.70460 | − | 7.67247i | −13.8800 | + | 8.91771i |
See next 80 embeddings (of 184 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.d | odd | 6 | 1 | inner |
16.e | even | 4 | 1 | inner |
144.w | odd | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 144.3.w.a | ✓ | 184 |
3.b | odd | 2 | 1 | 432.3.x.a | 184 | ||
9.c | even | 3 | 1 | 432.3.x.a | 184 | ||
9.d | odd | 6 | 1 | inner | 144.3.w.a | ✓ | 184 |
16.e | even | 4 | 1 | inner | 144.3.w.a | ✓ | 184 |
48.i | odd | 4 | 1 | 432.3.x.a | 184 | ||
144.w | odd | 12 | 1 | inner | 144.3.w.a | ✓ | 184 |
144.x | even | 12 | 1 | 432.3.x.a | 184 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
144.3.w.a | ✓ | 184 | 1.a | even | 1 | 1 | trivial |
144.3.w.a | ✓ | 184 | 9.d | odd | 6 | 1 | inner |
144.3.w.a | ✓ | 184 | 16.e | even | 4 | 1 | inner |
144.3.w.a | ✓ | 184 | 144.w | odd | 12 | 1 | inner |
432.3.x.a | 184 | 3.b | odd | 2 | 1 | ||
432.3.x.a | 184 | 9.c | even | 3 | 1 | ||
432.3.x.a | 184 | 48.i | odd | 4 | 1 | ||
432.3.x.a | 184 | 144.x | even | 12 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(144, [\chi])\).