Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [325,2,Mod(66,325)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(325, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("325.66");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 325 = 5^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 325.l (of order \(5\), 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: | \(2.59513806569\) |
Analytic rank: | \(0\) |
Dimension: | \(60\) |
Relative dimension: | \(15\) over \(\Q(\zeta_{5})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
66.1 | −2.11859 | − | 1.53924i | 0.771973 | + | 2.37589i | 1.50111 | + | 4.61993i | −2.18599 | − | 0.470585i | 2.02158 | − | 6.22178i | −3.55469 | 2.31251 | − | 7.11718i | −2.62186 | + | 1.90489i | 3.90686 | + | 4.36174i | ||
66.2 | −2.11550 | − | 1.53700i | −0.756399 | − | 2.32796i | 1.49493 | + | 4.60091i | 1.90238 | + | 1.17514i | −1.97790 | + | 6.08736i | 1.79715 | 2.29298 | − | 7.05707i | −2.42019 | + | 1.75837i | −2.21830 | − | 5.40995i | ||
66.3 | −1.71220 | − | 1.24398i | −0.522792 | − | 1.60899i | 0.766087 | + | 2.35777i | −1.47820 | + | 1.67777i | −1.10643 | + | 3.40525i | −2.30135 | 0.313339 | − | 0.964358i | 0.111515 | − | 0.0810203i | 4.61809 | − | 1.03382i | ||
66.4 | −1.60352 | − | 1.16503i | 0.794864 | + | 2.44634i | 0.595961 | + | 1.83418i | 0.133214 | + | 2.23210i | 1.57547 | − | 4.84880i | 4.32294 | −0.0437495 | + | 0.134647i | −2.92571 | + | 2.12566i | 2.38684 | − | 3.73441i | ||
66.5 | −1.58418 | − | 1.15097i | −0.110716 | − | 0.340750i | 0.566849 | + | 1.74458i | −1.16514 | − | 1.90852i | −0.216799 | + | 0.667240i | −0.106861 | −0.100227 | + | 0.308467i | 2.32320 | − | 1.68790i | −0.350856 | + | 4.36448i | ||
66.6 | −0.893733 | − | 0.649335i | 0.209044 | + | 0.643370i | −0.240911 | − | 0.741448i | 2.03707 | − | 0.922134i | 0.230934 | − | 0.710740i | 0.686970 | −0.948890 | + | 2.92038i | 2.05683 | − | 1.49437i | −2.41937 | − | 0.498601i | ||
66.7 | −0.430798 | − | 0.312993i | 0.962636 | + | 2.96269i | −0.530412 | − | 1.63244i | 1.23199 | + | 1.86607i | 0.512600 | − | 1.57762i | −4.48914 | −0.611542 | + | 1.88213i | −5.42381 | + | 3.94063i | 0.0533285 | − | 1.18950i | ||
66.8 | −0.111048 | − | 0.0806808i | −0.623987 | − | 1.92043i | −0.612212 | − | 1.88419i | −1.64946 | − | 1.50973i | −0.0856500 | + | 0.263604i | −1.56485 | −0.168867 | + | 0.519718i | −0.871658 | + | 0.633296i | 0.0613615 | + | 0.300732i | ||
66.9 | 0.410747 | + | 0.298425i | −0.0810801 | − | 0.249539i | −0.538378 | − | 1.65696i | −1.91546 | + | 1.15370i | 0.0411653 | − | 0.126694i | 3.54407 | 0.587124 | − | 1.80698i | 2.37136 | − | 1.72289i | −1.13106 | − | 0.0977442i | ||
66.10 | 0.443042 | + | 0.321889i | 0.814794 | + | 2.50768i | −0.525360 | − | 1.61689i | 0.585476 | − | 2.15806i | −0.446206 | + | 1.37328i | 2.12323 | 0.626157 | − | 1.92711i | −3.19751 | + | 2.32313i | 0.954045 | − | 0.767652i | ||
66.11 | 0.545716 | + | 0.396486i | −0.905162 | − | 2.78580i | −0.477429 | − | 1.46938i | 2.23054 | − | 0.157067i | 0.610570 | − | 1.87914i | 2.34324 | 0.738936 | − | 2.27421i | −4.51432 | + | 3.27985i | 1.27952 | + | 0.798666i | ||
66.12 | 1.34617 | + | 0.978052i | −0.213534 | − | 0.657191i | 0.237562 | + | 0.731141i | 1.42062 | − | 1.72680i | 0.355313 | − | 1.09354i | −2.13212 | 0.633091 | − | 1.94845i | 2.04075 | − | 1.48269i | 3.60130 | − | 0.935135i | ||
66.13 | 1.55873 | + | 1.13248i | 0.447787 | + | 1.37815i | 0.529080 | + | 1.62834i | 0.800256 | + | 2.08796i | −0.862747 | + | 2.65526i | −0.875510 | 0.171387 | − | 0.527474i | 0.728277 | − | 0.529124i | −1.11720 | + | 4.16084i | ||
66.14 | 1.82200 | + | 1.32376i | −0.674593 | − | 2.07618i | 0.949315 | + | 2.92169i | −0.435763 | + | 2.19320i | 1.51927 | − | 4.67582i | 2.98877 | −0.746085 | + | 2.29621i | −1.42841 | + | 1.03780i | −3.69723 | + | 3.41917i | ||
66.15 | 2.01610 | + | 1.46478i | 0.623234 | + | 1.91812i | 1.30104 | + | 4.00418i | −1.01154 | − | 1.99419i | −1.55312 | + | 4.78002i | 0.836184 | −1.70207 | + | 5.23842i | −0.863701 | + | 0.627516i | 0.881691 | − | 5.50217i | ||
131.1 | −0.780630 | − | 2.40253i | 0.124435 | − | 0.0904075i | −3.54474 | + | 2.57541i | −1.97204 | + | 1.05406i | −0.314345 | − | 0.228385i | 3.98052 | 4.86720 | + | 3.53623i | −0.919740 | + | 2.83067i | 4.07186 | + | 3.91506i | ||
131.2 | −0.668594 | − | 2.05772i | 0.454228 | − | 0.330016i | −2.16916 | + | 1.57599i | −0.627399 | − | 2.14625i | −0.982774 | − | 0.714027i | −1.81313 | 1.19242 | + | 0.866344i | −0.829638 | + | 2.55336i | −3.99690 | + | 2.72598i | ||
131.3 | −0.599245 | − | 1.84429i | −2.63027 | + | 1.91100i | −1.42426 | + | 1.03479i | 1.43620 | + | 1.71386i | 5.10062 | + | 3.70581i | −1.15914 | −0.375760 | − | 0.273006i | 2.33934 | − | 7.19975i | 2.30021 | − | 3.67579i | ||
131.4 | −0.344169 | − | 1.05924i | −0.769833 | + | 0.559317i | 0.614491 | − | 0.446454i | 2.16018 | − | 0.577587i | 0.857405 | + | 0.622941i | 3.14196 | −2.48648 | − | 1.80654i | −0.647243 | + | 1.99201i | −1.35527 | − | 2.08937i | ||
131.5 | −0.316435 | − | 0.973887i | 1.79577 | − | 1.30470i | 0.769710 | − | 0.559227i | 2.19613 | + | 0.420731i | −1.83888 | − | 1.33602i | −4.19384 | −2.44506 | − | 1.77644i | 0.595492 | − | 1.83274i | −0.285188 | − | 2.27192i | ||
See all 60 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
25.d | even | 5 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 325.2.l.c | ✓ | 60 |
25.d | even | 5 | 1 | inner | 325.2.l.c | ✓ | 60 |
25.d | even | 5 | 1 | 8125.2.a.j | 30 | ||
25.e | even | 10 | 1 | 8125.2.a.i | 30 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
325.2.l.c | ✓ | 60 | 1.a | even | 1 | 1 | trivial |
325.2.l.c | ✓ | 60 | 25.d | even | 5 | 1 | inner |
8125.2.a.i | 30 | 25.e | even | 10 | 1 | ||
8125.2.a.j | 30 | 25.d | even | 5 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{60} + 3 T_{2}^{59} + 24 T_{2}^{58} + 67 T_{2}^{57} + 356 T_{2}^{56} + 838 T_{2}^{55} + 3836 T_{2}^{54} + \cdots + 25 \) acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\).