Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [930,2,Mod(377,930)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(930, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 3, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("930.377");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 930.bf (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: | \(7.42608738798\) |
Analytic rank: | \(0\) |
Dimension: | \(256\) |
Relative dimension: | \(64\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
377.1 | −0.707107 | + | 0.707107i | −1.72168 | − | 0.189281i | − | 1.00000i | 0.213393 | + | 2.22586i | 1.35125 | − | 1.08357i | 3.30534 | − | 0.885662i | 0.707107 | + | 0.707107i | 2.92835 | + | 0.651761i | −1.72481 | − | 1.42303i | |
377.2 | −0.707107 | + | 0.707107i | −1.70669 | + | 0.295334i | − | 1.00000i | 2.22324 | + | 0.239148i | 0.997976 | − | 1.41564i | −3.38718 | + | 0.907593i | 0.707107 | + | 0.707107i | 2.82556 | − | 1.00809i | −1.74117 | + | 1.40297i | |
377.3 | −0.707107 | + | 0.707107i | −1.65429 | − | 0.513158i | − | 1.00000i | 1.83992 | − | 1.27071i | 1.53262 | − | 0.806900i | 3.00096 | − | 0.804106i | 0.707107 | + | 0.707107i | 2.47334 | + | 1.69782i | −0.402493 | + | 2.19955i | |
377.4 | −0.707107 | + | 0.707107i | −1.64704 | − | 0.535974i | − | 1.00000i | −0.601583 | + | 2.15362i | 1.54362 | − | 0.785641i | −4.77476 | + | 1.27939i | 0.707107 | + | 0.707107i | 2.42546 | + | 1.76554i | −1.09746 | − | 1.94823i | |
377.5 | −0.707107 | + | 0.707107i | −1.64517 | + | 0.541693i | − | 1.00000i | −2.17658 | − | 0.512363i | 0.780273 | − | 1.54634i | 0.122813 | − | 0.0329078i | 0.707107 | + | 0.707107i | 2.41314 | − | 1.78235i | 1.90137 | − | 1.17678i | |
377.6 | −0.707107 | + | 0.707107i | −1.45410 | − | 0.941054i | − | 1.00000i | −1.41418 | − | 1.73208i | 1.69363 | − | 0.362781i | −1.47385 | + | 0.394917i | 0.707107 | + | 0.707107i | 1.22883 | + | 2.73678i | 2.22474 | + | 0.224784i | |
377.7 | −0.707107 | + | 0.707107i | −1.38753 | + | 1.03670i | − | 1.00000i | −0.924875 | + | 2.03583i | 0.248075 | − | 1.71419i | −0.101233 | + | 0.0271252i | 0.707107 | + | 0.707107i | 0.850495 | − | 2.87692i | −0.785564 | − | 2.09354i | |
377.8 | −0.707107 | + | 0.707107i | −1.34147 | + | 1.09565i | − | 1.00000i | 2.02568 | + | 0.946905i | 0.173818 | − | 1.72331i | 2.29933 | − | 0.616104i | 0.707107 | + | 0.707107i | 0.599084 | − | 2.93957i | −2.10193 | + | 0.762808i | |
377.9 | −0.707107 | + | 0.707107i | −1.27826 | − | 1.16878i | − | 1.00000i | −2.23600 | − | 0.0178176i | 1.73032 | − | 0.0774131i | 4.17555 | − | 1.11884i | 0.707107 | + | 0.707107i | 0.267899 | + | 2.98801i | 1.59369 | − | 1.56849i | |
377.10 | −0.707107 | + | 0.707107i | −1.00727 | + | 1.40904i | − | 1.00000i | −0.926393 | − | 2.03514i | −0.284092 | − | 1.70859i | 3.45336 | − | 0.925324i | 0.707107 | + | 0.707107i | −0.970795 | − | 2.83858i | 2.09412 | + | 0.784002i | |
377.11 | −0.707107 | + | 0.707107i | −1.00673 | − | 1.40943i | − | 1.00000i | 0.813777 | − | 2.08273i | 1.70848 | + | 0.284755i | −1.29509 | + | 0.347018i | 0.707107 | + | 0.707107i | −0.972997 | + | 2.83783i | 0.897286 | + | 2.04814i | |
377.12 | −0.707107 | + | 0.707107i | −0.792487 | − | 1.54012i | − | 1.00000i | −0.947623 | + | 2.02534i | 1.64940 | + | 0.528655i | −1.77478 | + | 0.475552i | 0.707107 | + | 0.707107i | −1.74393 | + | 2.44105i | −0.762062 | − | 2.10220i | |
377.13 | −0.707107 | + | 0.707107i | −0.537394 | − | 1.64657i | − | 1.00000i | 0.858744 | + | 2.06460i | 1.54430 | + | 0.784309i | 2.17271 | − | 0.582177i | 0.707107 | + | 0.707107i | −2.42242 | + | 1.76972i | −2.06711 | − | 0.852667i | |
377.14 | −0.707107 | + | 0.707107i | −0.390383 | + | 1.68748i | − | 1.00000i | −1.57881 | − | 1.58347i | −0.917189 | − | 1.46927i | −4.83994 | + | 1.29686i | 0.707107 | + | 0.707107i | −2.69520 | − | 1.31753i | 2.23607 | + | 0.00329236i | |
377.15 | −0.707107 | + | 0.707107i | −0.320052 | + | 1.70222i | − | 1.00000i | 1.97048 | − | 1.05698i | −0.977343 | − | 1.42997i | 1.65949 | − | 0.444658i | 0.707107 | + | 0.707107i | −2.79513 | − | 1.08960i | −0.645943 | + | 2.14074i | |
377.16 | −0.707107 | + | 0.707107i | −0.136833 | + | 1.72664i | − | 1.00000i | −2.04548 | + | 0.903327i | −1.12416 | − | 1.31767i | −0.382918 | + | 0.102603i | 0.707107 | + | 0.707107i | −2.96255 | − | 0.472523i | 0.807626 | − | 2.08512i | |
377.17 | −0.707107 | + | 0.707107i | −0.134174 | − | 1.72685i | − | 1.00000i | 2.18104 | + | 0.493020i | 1.31594 | + | 1.12619i | −3.79793 | + | 1.01765i | 0.707107 | + | 0.707107i | −2.96399 | + | 0.463396i | −1.89085 | + | 1.19361i | |
377.18 | −0.707107 | + | 0.707107i | 0.321845 | − | 1.70189i | − | 1.00000i | −0.575509 | − | 2.16074i | 0.975836 | + | 1.43099i | 2.59908 | − | 0.696422i | 0.707107 | + | 0.707107i | −2.79283 | − | 1.09549i | 1.93482 | + | 1.12093i | |
377.19 | −0.707107 | + | 0.707107i | 0.609255 | + | 1.62136i | − | 1.00000i | 1.73683 | − | 1.40835i | −1.57728 | − | 0.715666i | −3.28936 | + | 0.881380i | 0.707107 | + | 0.707107i | −2.25762 | + | 1.97564i | −0.232271 | + | 2.22397i | |
377.20 | −0.707107 | + | 0.707107i | 0.661860 | − | 1.60061i | − | 1.00000i | −2.12466 | + | 0.697001i | 0.663795 | + | 1.59981i | 1.92566 | − | 0.515980i | 0.707107 | + | 0.707107i | −2.12388 | − | 2.11875i | 1.00951 | − | 1.99522i | |
See next 80 embeddings (of 256 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
15.e | even | 4 | 1 | inner |
31.c | even | 3 | 1 | inner |
93.h | odd | 6 | 1 | inner |
155.o | odd | 12 | 1 | inner |
465.be | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 930.2.bf.a | ✓ | 256 |
3.b | odd | 2 | 1 | inner | 930.2.bf.a | ✓ | 256 |
5.c | odd | 4 | 1 | inner | 930.2.bf.a | ✓ | 256 |
15.e | even | 4 | 1 | inner | 930.2.bf.a | ✓ | 256 |
31.c | even | 3 | 1 | inner | 930.2.bf.a | ✓ | 256 |
93.h | odd | 6 | 1 | inner | 930.2.bf.a | ✓ | 256 |
155.o | odd | 12 | 1 | inner | 930.2.bf.a | ✓ | 256 |
465.be | even | 12 | 1 | inner | 930.2.bf.a | ✓ | 256 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
930.2.bf.a | ✓ | 256 | 1.a | even | 1 | 1 | trivial |
930.2.bf.a | ✓ | 256 | 3.b | odd | 2 | 1 | inner |
930.2.bf.a | ✓ | 256 | 5.c | odd | 4 | 1 | inner |
930.2.bf.a | ✓ | 256 | 15.e | even | 4 | 1 | inner |
930.2.bf.a | ✓ | 256 | 31.c | even | 3 | 1 | inner |
930.2.bf.a | ✓ | 256 | 93.h | odd | 6 | 1 | inner |
930.2.bf.a | ✓ | 256 | 155.o | odd | 12 | 1 | inner |
930.2.bf.a | ✓ | 256 | 465.be | even | 12 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(930, [\chi])\).