Properties

Label 630.2.bt.a
Level $630$
Weight $2$
Character orbit 630.bt
Analytic conductor $5.031$
Analytic rank $0$
Dimension $192$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [630,2,Mod(317,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([2, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.317");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 630.bt (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: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(48\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 192 q + 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q + 8 q^{6} + 12 q^{15} + 96 q^{16} + 36 q^{17} + 16 q^{18} - 12 q^{21} + 36 q^{27} - 12 q^{30} - 20 q^{33} + 12 q^{41} - 4 q^{42} + 20 q^{45} - 12 q^{46} - 48 q^{50} - 24 q^{51} - 48 q^{57} + 24 q^{58} + 8 q^{60} + 12 q^{61} - 28 q^{63} + 24 q^{65} + 8 q^{72} - 88 q^{75} - 48 q^{77} - 16 q^{78} + 4 q^{81} + 40 q^{87} - 20 q^{90} - 24 q^{92} - 12 q^{93} + 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
317.1 −0.965926 0.258819i −1.73191 0.0221408i 0.866025 + 0.500000i 1.08506 + 1.95516i 1.66717 + 0.469637i −2.53684 0.751284i −0.707107 0.707107i 2.99902 + 0.0766917i −0.542058 2.16937i
317.2 −0.965926 0.258819i −1.73140 0.0476508i 0.866025 + 0.500000i −2.20409 0.376807i 1.66007 + 0.494145i −2.58006 + 0.585890i −0.707107 0.707107i 2.99546 + 0.165005i 2.03146 + 0.934428i
317.3 −0.965926 0.258819i −1.58790 + 0.691796i 0.866025 + 0.500000i 2.19402 + 0.431615i 1.71284 0.257245i 2.55054 0.703393i −0.707107 0.707107i 2.04284 2.19700i −2.00755 0.984761i
317.4 −0.965926 0.258819i −1.51438 + 0.840620i 0.866025 + 0.500000i 0.611756 2.15076i 1.68035 0.420025i −1.08314 2.41388i −0.707107 0.707107i 1.58672 2.54604i −1.14757 + 1.91914i
317.5 −0.965926 0.258819i −1.45309 0.942613i 0.866025 + 0.500000i 0.260263 2.22087i 1.15961 + 1.28658i 2.64011 0.172669i −0.707107 0.707107i 1.22296 + 2.73941i −0.826198 + 2.07783i
317.6 −0.965926 0.258819i −1.39826 1.02218i 0.866025 + 0.500000i 1.21681 + 1.87600i 1.08606 + 1.34925i 2.15901 + 1.52928i −0.707107 0.707107i 0.910289 + 2.85856i −0.689806 2.12701i
317.7 −0.965926 0.258819i −1.09766 1.33983i 0.866025 + 0.500000i −2.04805 + 0.897484i 0.713485 + 1.57827i −0.0959969 + 2.64401i −0.707107 0.707107i −0.590284 + 2.94135i 2.21055 0.336828i
317.8 −0.965926 0.258819i −1.08875 + 1.34708i 0.866025 + 0.500000i −1.18556 1.89590i 1.40030 1.01939i 0.500796 + 2.59792i −0.707107 0.707107i −0.629257 2.93326i 0.654473 + 2.13815i
317.9 −0.965926 0.258819i −0.989308 + 1.42171i 0.866025 + 0.500000i −1.72555 + 1.42213i 1.32356 1.11722i 1.90960 1.83123i −0.707107 0.707107i −1.04254 2.81303i 2.03483 0.927070i
317.10 −0.965926 0.258819i −0.624647 1.61549i 0.866025 + 0.500000i −0.455787 + 2.18912i 0.185243 + 1.72212i −0.715343 2.54721i −0.707107 0.707107i −2.21963 + 2.01823i 1.00684 1.99656i
317.11 −0.965926 0.258819i −0.196486 + 1.72087i 0.866025 + 0.500000i −0.145784 + 2.23131i 0.635185 1.61138i −2.10331 + 1.60502i −0.707107 0.707107i −2.92279 0.676254i 0.718322 2.11755i
317.12 −0.965926 0.258819i −0.186889 1.72194i 0.866025 + 0.500000i 2.17367 0.524538i −0.265150 + 1.71164i 0.387487 2.61722i −0.707107 0.707107i −2.93014 + 0.643623i −2.23537 0.0559240i
317.13 −0.965926 0.258819i 0.0615400 1.73096i 0.866025 + 0.500000i 2.23012 + 0.162985i −0.507448 + 1.65605i −1.31208 + 2.29749i −0.707107 0.707107i −2.99243 0.213046i −2.11195 0.734629i
317.14 −0.965926 0.258819i 0.241443 + 1.71514i 0.866025 + 0.500000i 2.06314 0.862238i 0.210695 1.71919i 1.66641 + 2.05501i −0.707107 0.707107i −2.88341 + 0.828216i −2.21600 + 0.298878i
317.15 −0.965926 0.258819i 0.499395 + 1.65849i 0.866025 + 0.500000i −2.06108 0.867151i −0.0531288 1.73124i −0.585433 2.58017i −0.707107 0.707107i −2.50121 + 1.65649i 1.76641 + 1.37105i
317.16 −0.965926 0.258819i 0.729061 1.57114i 0.866025 + 0.500000i −1.92875 1.13134i −1.11086 + 1.32891i 2.62221 0.352128i −0.707107 0.707107i −1.93694 2.29091i 1.57022 + 1.59198i
317.17 −0.965926 0.258819i 0.754597 1.55903i 0.866025 + 0.500000i −1.96329 + 1.07027i −1.13239 + 1.31061i −2.51410 0.824195i −0.707107 0.707107i −1.86117 2.35288i 2.17340 0.525664i
317.18 −0.965926 0.258819i 1.05933 + 1.37034i 0.866025 + 0.500000i 0.141250 + 2.23160i −0.668564 1.59782i 2.52582 + 0.787538i −0.707107 0.707107i −0.755646 + 2.90327i 0.441144 2.19212i
317.19 −0.965926 0.258819i 1.32691 + 1.11325i 0.866025 + 0.500000i −1.59332 1.56887i −0.993564 1.41874i −2.14973 + 1.54229i −0.707107 0.707107i 0.521364 + 2.95435i 1.13297 + 1.92779i
317.20 −0.965926 0.258819i 1.45299 0.942773i 0.866025 + 0.500000i 0.148657 2.23112i −1.64749 + 0.534587i −2.28350 1.33628i −0.707107 0.707107i 1.22236 2.73968i −0.721048 + 2.11662i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 317.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
63.n odd 6 1 inner
315.bx even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.bt.a 192
5.c odd 4 1 inner 630.2.bt.a 192
7.c even 3 1 630.2.cd.a yes 192
9.d odd 6 1 630.2.cd.a yes 192
35.l odd 12 1 630.2.cd.a yes 192
45.l even 12 1 630.2.cd.a yes 192
63.n odd 6 1 inner 630.2.bt.a 192
315.bx even 12 1 inner 630.2.bt.a 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.bt.a 192 1.a even 1 1 trivial
630.2.bt.a 192 5.c odd 4 1 inner
630.2.bt.a 192 63.n odd 6 1 inner
630.2.bt.a 192 315.bx even 12 1 inner
630.2.cd.a yes 192 7.c even 3 1
630.2.cd.a yes 192 9.d odd 6 1
630.2.cd.a yes 192 35.l odd 12 1
630.2.cd.a yes 192 45.l even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(630, [\chi])\).