Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [671,2,Mod(21,671)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(671, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 11]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("671.21");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 671 = 11 \cdot 61 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 671.bj (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: | \(5.35796197563\) |
Analytic rank: | \(0\) |
Dimension: | \(240\) |
Relative dimension: | \(60\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
21.1 | −0.726621 | − | 2.71179i | − | 0.275133i | −5.09375 | + | 2.94088i | −2.13341 | − | 1.23173i | −0.746103 | + | 0.199918i | 0.0793007 | + | 0.295954i | 7.70593 | + | 7.70593i | 2.92430 | −1.79000 | + | 6.68036i | |||
21.2 | −0.671334 | − | 2.50545i | − | 2.02545i | −4.09455 | + | 2.36399i | 1.82518 | + | 1.05377i | −5.07467 | + | 1.35975i | 1.18834 | + | 4.43495i | 5.00344 | + | 5.00344i | −1.10246 | 1.41486 | − | 5.28033i | |||
21.3 | −0.669128 | − | 2.49722i | − | 3.34229i | −4.05632 | + | 2.34192i | 2.14227 | + | 1.23684i | −8.34643 | + | 2.23642i | −1.05755 | − | 3.94683i | 4.90629 | + | 4.90629i | −8.17091 | 1.65521 | − | 6.17733i | |||
21.4 | −0.662120 | − | 2.47107i | − | 0.560439i | −3.93572 | + | 2.27229i | 1.44440 | + | 0.833927i | −1.38488 | + | 0.371078i | −0.344015 | − | 1.28388i | 4.60300 | + | 4.60300i | 2.68591 | 1.10432 | − | 4.12138i | |||
21.5 | −0.658834 | − | 2.45880i | 2.40391i | −3.87959 | + | 2.23988i | 2.22593 | + | 1.28514i | 5.91074 | − | 1.58378i | −0.483143 | − | 1.80311i | 4.46350 | + | 4.46350i | −2.77879 | 1.69339 | − | 6.31981i | ||||
21.6 | −0.640358 | − | 2.38985i | 2.85448i | −3.56927 | + | 2.06072i | −0.196828 | − | 0.113639i | 6.82178 | − | 1.82789i | 1.10056 | + | 4.10735i | 3.71144 | + | 3.71144i | −5.14805 | −0.145539 | + | 0.543159i | ||||
21.7 | −0.592067 | − | 2.20962i | − | 1.97237i | −2.79985 | + | 1.61649i | −3.10552 | − | 1.79297i | −4.35821 | + | 1.16778i | −0.874617 | − | 3.26411i | 1.99442 | + | 1.99442i | −0.890261 | −2.12312 | + | 7.92360i | |||
21.8 | −0.578074 | − | 2.15740i | 1.37027i | −2.58816 | + | 1.49427i | 0.616533 | + | 0.355955i | 2.95622 | − | 0.792118i | −0.224432 | − | 0.837593i | 1.56124 | + | 1.56124i | 1.12236 | 0.411537 | − | 1.53588i | ||||
21.9 | −0.545851 | − | 2.03714i | − | 2.79757i | −2.11995 | + | 1.22395i | −1.59773 | − | 0.922448i | −5.69905 | + | 1.52706i | 0.747293 | + | 2.78894i | 0.667953 | + | 0.667953i | −4.82641 | −1.00704 | + | 3.75832i | |||
21.10 | −0.529457 | − | 1.97596i | − | 0.158033i | −1.89204 | + | 1.09237i | −0.468236 | − | 0.270336i | −0.312268 | + | 0.0836718i | 0.436603 | + | 1.62942i | 0.267227 | + | 0.267227i | 2.97503 | −0.286263 | + | 1.06835i | |||
21.11 | −0.521873 | − | 1.94766i | 0.552867i | −1.78897 | + | 1.03286i | −3.49214 | − | 2.01619i | 1.07680 | − | 0.288527i | 0.957365 | + | 3.57294i | 0.0937038 | + | 0.0937038i | 2.69434 | −2.10439 | + | 7.85369i | ||||
21.12 | −0.505082 | − | 1.88499i | 0.997508i | −1.56603 | + | 0.904149i | −1.66368 | − | 0.960523i | 1.88029 | − | 0.503823i | −1.28528 | − | 4.79673i | −0.264530 | − | 0.264530i | 2.00498 | −0.970286 | + | 3.62116i | ||||
21.13 | −0.485244 | − | 1.81096i | 0.156634i | −1.31205 | + | 0.757513i | 2.73556 | + | 1.57937i | 0.283658 | − | 0.0760059i | 0.685855 | + | 2.55964i | −0.642937 | − | 0.642937i | 2.97547 | 1.53277 | − | 5.72036i | ||||
21.14 | −0.462016 | − | 1.72427i | 3.01358i | −1.02758 | + | 0.593276i | −3.33285 | − | 1.92422i | 5.19622 | − | 1.39232i | −0.0492920 | − | 0.183960i | −1.02677 | − | 1.02677i | −6.08169 | −1.77804 | + | 6.63574i | ||||
21.15 | −0.450896 | − | 1.68277i | − | 1.09351i | −0.896348 | + | 0.517507i | 2.84449 | + | 1.64227i | −1.84013 | + | 0.493062i | −1.06406 | − | 3.97113i | −1.18874 | − | 1.18874i | 1.80422 | 1.48098 | − | 5.52711i | |||
21.16 | −0.390456 | − | 1.45720i | 2.15184i | −0.238929 | + | 0.137945i | 3.43790 | + | 1.98487i | 3.13566 | − | 0.840198i | 1.08358 | + | 4.04398i | −1.83919 | − | 1.83919i | −1.63041 | 1.55001 | − | 5.78472i | ||||
21.17 | −0.387783 | − | 1.44723i | 2.94302i | −0.212038 | + | 0.122420i | −0.127261 | − | 0.0734740i | 4.25922 | − | 1.14126i | −0.459320 | − | 1.71420i | −1.85949 | − | 1.85949i | −5.66139 | −0.0569840 | + | 0.212667i | ||||
21.18 | −0.375407 | − | 1.40104i | − | 2.58780i | −0.0899266 | + | 0.0519191i | −0.272209 | − | 0.157160i | −3.62561 | + | 0.971479i | 0.0181465 | + | 0.0677236i | −1.94476 | − | 1.94476i | −3.69672 | −0.117998 | + | 0.440375i | |||
21.19 | −0.358764 | − | 1.33893i | − | 1.90630i | 0.0680400 | − | 0.0392829i | 1.78909 | + | 1.03293i | −2.55239 | + | 0.683912i | −0.270943 | − | 1.01117i | −2.03733 | − | 2.03733i | −0.633976 | 0.741157 | − | 2.76603i | |||
21.20 | −0.328754 | − | 1.22693i | − | 0.178685i | 0.334784 | − | 0.193287i | −1.39664 | − | 0.806352i | −0.219233 | + | 0.0587433i | 0.860816 | + | 3.21261i | −2.14355 | − | 2.14355i | 2.96807 | −0.530182 | + | 1.97867i | |||
See next 80 embeddings (of 240 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.b | odd | 2 | 1 | inner |
61.h | odd | 12 | 1 | inner |
671.bj | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 671.2.bj.a | ✓ | 240 |
11.b | odd | 2 | 1 | inner | 671.2.bj.a | ✓ | 240 |
61.h | odd | 12 | 1 | inner | 671.2.bj.a | ✓ | 240 |
671.bj | even | 12 | 1 | inner | 671.2.bj.a | ✓ | 240 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
671.2.bj.a | ✓ | 240 | 1.a | even | 1 | 1 | trivial |
671.2.bj.a | ✓ | 240 | 11.b | odd | 2 | 1 | inner |
671.2.bj.a | ✓ | 240 | 61.h | odd | 12 | 1 | inner |
671.2.bj.a | ✓ | 240 | 671.bj | even | 12 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).