Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [798,2,Mod(107,798)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(798, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 2, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("798.107");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 798.p (of order \(6\), 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: | \(6.37206208130\) |
Analytic rank: | \(0\) |
Dimension: | \(50\) |
Relative dimension: | \(25\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
107.1 | 1.00000 | −1.73110 | − | 0.0573132i | 1.00000 | 2.04210i | −1.73110 | − | 0.0573132i | 0.305109 | + | 2.62810i | 1.00000 | 2.99343 | + | 0.198430i | 2.04210i | ||||||||||
107.2 | 1.00000 | −1.60960 | − | 0.639688i | 1.00000 | 1.73455i | −1.60960 | − | 0.639688i | −0.675179 | − | 2.55815i | 1.00000 | 2.18160 | + | 2.05928i | 1.73455i | ||||||||||
107.3 | 1.00000 | −1.57026 | − | 0.730943i | 1.00000 | − | 2.50019i | −1.57026 | − | 0.730943i | 2.58513 | + | 0.563124i | 1.00000 | 1.93145 | + | 2.29554i | − | 2.50019i | ||||||||
107.4 | 1.00000 | −1.54401 | + | 0.784869i | 1.00000 | 3.23749i | −1.54401 | + | 0.784869i | 2.52933 | + | 0.776191i | 1.00000 | 1.76796 | − | 2.42370i | 3.23749i | ||||||||||
107.5 | 1.00000 | −1.53652 | + | 0.799448i | 1.00000 | − | 0.138945i | −1.53652 | + | 0.799448i | −2.56896 | − | 0.632805i | 1.00000 | 1.72177 | − | 2.45673i | − | 0.138945i | ||||||||
107.6 | 1.00000 | −1.31402 | + | 1.12843i | 1.00000 | − | 2.51691i | −1.31402 | + | 1.12843i | −0.461738 | + | 2.60515i | 1.00000 | 0.453299 | − | 2.96556i | − | 2.51691i | ||||||||
107.7 | 1.00000 | −1.27523 | + | 1.17208i | 1.00000 | − | 2.98905i | −1.27523 | + | 1.17208i | 2.60274 | − | 0.475099i | 1.00000 | 0.252435 | − | 2.98936i | − | 2.98905i | ||||||||
107.8 | 1.00000 | −1.00188 | + | 1.41288i | 1.00000 | 3.34851i | −1.00188 | + | 1.41288i | −2.58671 | − | 0.555829i | 1.00000 | −0.992484 | − | 2.83107i | 3.34851i | ||||||||||
107.9 | 1.00000 | −0.988593 | − | 1.42221i | 1.00000 | − | 0.610954i | −0.988593 | − | 1.42221i | −2.63918 | + | 0.186412i | 1.00000 | −1.04537 | + | 2.81198i | − | 0.610954i | ||||||||
107.10 | 1.00000 | −0.880492 | − | 1.49155i | 1.00000 | 3.64338i | −0.880492 | − | 1.49155i | 2.23585 | − | 1.41457i | 1.00000 | −1.44947 | + | 2.62660i | 3.64338i | ||||||||||
107.11 | 1.00000 | −0.544247 | − | 1.64432i | 1.00000 | − | 4.07740i | −0.544247 | − | 1.64432i | −1.01122 | − | 2.44488i | 1.00000 | −2.40759 | + | 1.78984i | − | 4.07740i | ||||||||
107.12 | 1.00000 | −0.423559 | − | 1.67946i | 1.00000 | 1.80606i | −0.423559 | − | 1.67946i | −2.02215 | + | 1.70614i | 1.00000 | −2.64119 | + | 1.42270i | 1.80606i | ||||||||||
107.13 | 1.00000 | 0.00876401 | + | 1.73203i | 1.00000 | − | 1.50693i | 0.00876401 | + | 1.73203i | 0.837194 | − | 2.50980i | 1.00000 | −2.99985 | + | 0.0303590i | − | 1.50693i | ||||||||
107.14 | 1.00000 | 0.309702 | − | 1.70414i | 1.00000 | − | 0.0905266i | 0.309702 | − | 1.70414i | 2.19691 | − | 1.47431i | 1.00000 | −2.80817 | − | 1.05555i | − | 0.0905266i | ||||||||
107.15 | 1.00000 | 0.584344 | − | 1.63050i | 1.00000 | − | 3.26298i | 0.584344 | − | 1.63050i | 0.0400578 | + | 2.64545i | 1.00000 | −2.31708 | − | 1.90555i | − | 3.26298i | ||||||||
107.16 | 1.00000 | 0.833647 | + | 1.51823i | 1.00000 | 0.679467i | 0.833647 | + | 1.51823i | −2.29134 | + | 1.32279i | 1.00000 | −1.61007 | + | 2.53134i | 0.679467i | ||||||||||
107.17 | 1.00000 | 0.889459 | + | 1.48622i | 1.00000 | 1.31572i | 0.889459 | + | 1.48622i | 2.50046 | + | 0.864710i | 1.00000 | −1.41773 | + | 2.64387i | 1.31572i | ||||||||||
107.18 | 1.00000 | 1.25084 | − | 1.19809i | 1.00000 | 1.95673i | 1.25084 | − | 1.19809i | 1.98940 | + | 1.74422i | 1.00000 | 0.129182 | − | 2.99722i | 1.95673i | ||||||||||
107.19 | 1.00000 | 1.26574 | + | 1.18232i | 1.00000 | 3.99497i | 1.26574 | + | 1.18232i | 1.01470 | − | 2.44344i | 1.00000 | 0.204220 | + | 2.99304i | 3.99497i | ||||||||||
107.20 | 1.00000 | 1.32297 | − | 1.11792i | 1.00000 | − | 2.52686i | 1.32297 | − | 1.11792i | −2.32425 | − | 1.26406i | 1.00000 | 0.500522 | − | 2.95795i | − | 2.52686i | ||||||||
See all 50 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
399.bm | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 798.2.p.d | yes | 50 |
3.b | odd | 2 | 1 | 798.2.p.c | ✓ | 50 | |
7.c | even | 3 | 1 | 798.2.bh.c | yes | 50 | |
19.d | odd | 6 | 1 | 798.2.bh.d | yes | 50 | |
21.h | odd | 6 | 1 | 798.2.bh.d | yes | 50 | |
57.f | even | 6 | 1 | 798.2.bh.c | yes | 50 | |
133.j | odd | 6 | 1 | 798.2.p.c | ✓ | 50 | |
399.bm | even | 6 | 1 | inner | 798.2.p.d | yes | 50 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
798.2.p.c | ✓ | 50 | 3.b | odd | 2 | 1 | |
798.2.p.c | ✓ | 50 | 133.j | odd | 6 | 1 | |
798.2.p.d | yes | 50 | 1.a | even | 1 | 1 | trivial |
798.2.p.d | yes | 50 | 399.bm | even | 6 | 1 | inner |
798.2.bh.c | yes | 50 | 7.c | even | 3 | 1 | |
798.2.bh.c | yes | 50 | 57.f | even | 6 | 1 | |
798.2.bh.d | yes | 50 | 19.d | odd | 6 | 1 | |
798.2.bh.d | yes | 50 | 21.h | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\):
\( T_{5}^{50} + 154 T_{5}^{48} + 11091 T_{5}^{46} + 496680 T_{5}^{44} + 15511693 T_{5}^{42} + 359151444 T_{5}^{40} + 6397835009 T_{5}^{38} + 89814995056 T_{5}^{36} + 1009611897357 T_{5}^{34} + \cdots + 322885837872 \) |
\( T_{11}^{50} + 3 T_{11}^{49} - 133 T_{11}^{48} - 408 T_{11}^{47} + 10271 T_{11}^{46} + 29751 T_{11}^{45} - 539483 T_{11}^{44} - 1454952 T_{11}^{43} + 21385223 T_{11}^{42} + 52190625 T_{11}^{41} + \cdots + 18\!\cdots\!03 \) |