Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [380,2,Mod(267,380)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(380, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("380.267");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 380 = 2^{2} \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 380.k (of order \(4\), degree \(2\), 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: | \(3.03431527681\) |
Analytic rank: | \(0\) |
Dimension: | \(52\) |
Relative dimension: | \(26\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
267.1 | −1.41374 | + | 0.0365245i | −0.389264 | − | 0.389264i | 1.99733 | − | 0.103273i | −1.49421 | + | 1.66354i | 0.564536 | + | 0.536101i | −2.85353 | + | 2.85353i | −2.81994 | + | 0.218952i | − | 2.69695i | 2.05166 | − | 2.40638i | |
267.2 | −1.38561 | − | 0.282971i | 0.321015 | + | 0.321015i | 1.83985 | + | 0.784178i | 1.77712 | + | 1.35715i | −0.353965 | − | 0.535642i | 1.91243 | − | 1.91243i | −2.32743 | − | 1.60719i | − | 2.79390i | −2.07836 | − | 2.38336i | |
267.3 | −1.38463 | − | 0.287770i | 1.15541 | + | 1.15541i | 1.83438 | + | 0.796907i | −0.283261 | − | 2.21805i | −1.26732 | − | 1.93230i | −3.26325 | + | 3.26325i | −2.31060 | − | 1.63130i | − | 0.330062i | −0.246078 | + | 3.15269i | |
267.4 | −1.34620 | + | 0.433304i | −0.497882 | − | 0.497882i | 1.62450 | − | 1.16662i | −2.01079 | − | 0.978129i | 0.885982 | + | 0.454514i | 2.91849 | − | 2.91849i | −1.68139 | + | 2.27441i | − | 2.50423i | 3.13074 | + | 0.445474i | |
267.5 | −1.34536 | + | 0.435901i | −1.82161 | − | 1.82161i | 1.61998 | − | 1.17289i | 2.11542 | + | 0.724569i | 3.24476 | + | 1.65668i | −1.36053 | + | 1.36053i | −1.66819 | + | 2.28411i | 3.63653i | −3.16184 | − | 0.0526904i | ||
267.6 | −1.01972 | + | 0.979881i | 0.582309 | + | 0.582309i | 0.0796647 | − | 1.99841i | 1.46731 | + | 1.68731i | −1.16439 | − | 0.0231993i | 0.972933 | − | 0.972933i | 1.87697 | + | 2.11589i | − | 2.32183i | −3.14961 | − | 0.282794i | |
267.7 | −0.923868 | − | 1.07073i | 1.14885 | + | 1.14885i | −0.292934 | + | 1.97843i | 1.95382 | − | 1.08746i | 0.168724 | − | 2.29149i | 2.70883 | − | 2.70883i | 2.38900 | − | 1.51416i | − | 0.360306i | −2.96946 | − | 1.08735i | |
267.8 | −0.830627 | − | 1.14458i | −1.85040 | − | 1.85040i | −0.620117 | + | 1.90143i | 1.13686 | + | 1.92550i | −0.580936 | + | 3.65493i | 1.96775 | − | 1.96775i | 2.69143 | − | 0.869611i | 3.84798i | 1.25958 | − | 2.90060i | ||
267.9 | −0.780925 | + | 1.17905i | −2.27237 | − | 2.27237i | −0.780312 | − | 1.84150i | 0.440177 | − | 2.19231i | 4.45378 | − | 0.904682i | 2.19634 | − | 2.19634i | 2.78058 | + | 0.518047i | 7.32729i | 2.24110 | + | 2.23102i | ||
267.10 | −0.600761 | + | 1.28027i | 1.93295 | + | 1.93295i | −1.27817 | − | 1.53827i | 2.18132 | + | 0.491792i | −3.63593 | + | 1.31345i | −3.06024 | + | 3.06024i | 2.73727 | − | 0.712271i | 4.47257i | −1.94007 | + | 2.49722i | ||
267.11 | −0.424847 | − | 1.34889i | 2.38733 | + | 2.38733i | −1.63901 | + | 1.14614i | −2.23504 | − | 0.0679064i | 2.20600 | − | 4.23450i | 0.160827 | − | 0.160827i | 2.24235 | + | 1.72391i | 8.39872i | 0.857951 | + | 3.04367i | ||
267.12 | −0.378581 | − | 1.36260i | −1.58108 | − | 1.58108i | −1.71335 | + | 1.03171i | −1.74843 | − | 1.39392i | −1.55581 | + | 2.75294i | −1.94368 | + | 1.94368i | 2.05445 | + | 1.94403i | 1.99961i | −1.23743 | + | 2.91012i | ||
267.13 | −0.291109 | − | 1.38393i | 1.18694 | + | 1.18694i | −1.83051 | + | 0.805748i | 1.27222 | − | 1.83887i | 1.29711 | − | 1.98816i | −1.53689 | + | 1.53689i | 1.64798 | + | 2.29873i | − | 0.182360i | −2.91522 | − | 1.22536i | |
267.14 | 0.0237221 | + | 1.41401i | −1.56852 | − | 1.56852i | −1.99887 | + | 0.0670868i | −1.80219 | + | 1.32368i | 2.18070 | − | 2.25511i | 2.36081 | − | 2.36081i | −0.142279 | − | 2.82485i | 1.92048i | −1.91445 | − | 2.51692i | ||
267.15 | 0.0413255 | + | 1.41361i | 0.135093 | + | 0.135093i | −1.99658 | + | 0.116836i | −0.869041 | − | 2.06028i | −0.185386 | + | 0.196551i | −0.485902 | + | 0.485902i | −0.247670 | − | 2.81756i | − | 2.96350i | 2.87652 | − | 1.31363i | |
267.16 | 0.413585 | + | 1.35239i | 2.24210 | + | 2.24210i | −1.65789 | + | 1.11865i | −0.349685 | + | 2.20856i | −2.10488 | + | 3.95948i | 2.40464 | − | 2.40464i | −2.19853 | − | 1.77945i | 7.05400i | −3.13144 | + | 0.440517i | ||
267.17 | 0.436150 | − | 1.34528i | −1.71881 | − | 1.71881i | −1.61955 | − | 1.17349i | −0.923046 | + | 2.03666i | −3.06193 | + | 1.56262i | −0.323011 | + | 0.323011i | −2.28503 | + | 1.66692i | 2.90860i | 2.33729 | + | 2.13004i | ||
267.18 | 0.655429 | + | 1.25316i | −0.124098 | − | 0.124098i | −1.14082 | + | 1.64272i | −0.0211435 | + | 2.23597i | 0.0741774 | − | 0.236853i | −2.60801 | + | 2.60801i | −2.80632 | − | 0.352951i | − | 2.96920i | −2.81589 | + | 1.43902i | |
267.19 | 0.683080 | − | 1.23831i | −0.494605 | − | 0.494605i | −1.06680 | − | 1.69172i | 1.31079 | − | 1.81158i | −0.950327 | + | 0.274617i | 0.327577 | − | 0.327577i | −2.82358 | + | 0.165443i | − | 2.51073i | −1.34791 | − | 2.86062i | |
267.20 | 1.08061 | + | 0.912296i | 1.34016 | + | 1.34016i | 0.335432 | + | 1.97167i | −2.21942 | − | 0.272360i | 0.225567 | + | 2.67081i | −0.697742 | + | 0.697742i | −1.43628 | + | 2.43662i | 0.592060i | −2.14985 | − | 2.31908i | ||
See all 52 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
20.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 380.2.k.c | ✓ | 52 |
4.b | odd | 2 | 1 | 380.2.k.d | yes | 52 | |
5.c | odd | 4 | 1 | 380.2.k.d | yes | 52 | |
20.e | even | 4 | 1 | inner | 380.2.k.c | ✓ | 52 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
380.2.k.c | ✓ | 52 | 1.a | even | 1 | 1 | trivial |
380.2.k.c | ✓ | 52 | 20.e | even | 4 | 1 | inner |
380.2.k.d | yes | 52 | 4.b | odd | 2 | 1 | |
380.2.k.d | yes | 52 | 5.c | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{52} + 2 T_{3}^{51} + 2 T_{3}^{50} + 364 T_{3}^{48} + 768 T_{3}^{47} + 808 T_{3}^{46} - 496 T_{3}^{45} + 51686 T_{3}^{44} + 112644 T_{3}^{43} + 122716 T_{3}^{42} - 124792 T_{3}^{41} + 3705020 T_{3}^{40} + \cdots + 6553600 \)
acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\).