Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [760,2,Mod(267,760)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(760, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 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("760.267");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 760 = 2^{3} \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 760.w (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: | \(6.06863055362\) |
Analytic rank: | \(0\) |
Dimension: | \(104\) |
Relative dimension: | \(52\) 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 | −1.40565 | − | 0.155404i | 2.13203 | − | 2.13203i | 1.95170 | + | 0.436886i | −1.55346 | + | 1.60834i | −3.32821 | + | 2.66556i | 0.758554 | − | 0.758554i | −2.67551 | − | 0.917409i | − | 6.09111i | 2.43357 | − | 2.01934i | |
267.2 | −1.38362 | − | 0.292576i | 0.685526 | − | 0.685526i | 1.82880 | + | 0.809627i | 2.17301 | + | 0.527280i | −1.14907 | + | 0.747938i | 1.03632 | − | 1.03632i | −2.29348 | − | 1.65528i | 2.06011i | −2.85235 | − | 1.36532i | ||
267.3 | −1.37178 | + | 0.343844i | 0.929890 | − | 0.929890i | 1.76354 | − | 0.943354i | −1.47116 | − | 1.68395i | −0.955864 | + | 1.59534i | 3.32714 | − | 3.32714i | −2.09482 | + | 1.90045i | 1.27061i | 2.59712 | + | 1.80415i | ||
267.4 | −1.34678 | + | 0.431504i | −2.17446 | + | 2.17446i | 1.62761 | − | 1.16228i | 1.96091 | − | 1.07464i | 1.99022 | − | 3.86680i | 0.483709 | − | 0.483709i | −1.69050 | + | 2.26765i | − | 6.45655i | −2.17719 | + | 2.29344i | |
267.5 | −1.34262 | − | 0.444279i | −1.69762 | + | 1.69762i | 1.60523 | + | 1.19299i | 0.143240 | + | 2.23148i | 3.03347 | − | 1.52503i | 3.45743 | − | 3.45743i | −1.62519 | − | 2.31490i | − | 2.76384i | 0.799082 | − | 3.05965i | |
267.6 | −1.30931 | + | 0.534525i | 1.64962 | − | 1.64962i | 1.42856 | − | 1.39972i | 1.08792 | + | 1.95357i | −1.27809 | + | 3.04162i | −1.60504 | + | 1.60504i | −1.12225 | + | 2.59626i | − | 2.44250i | −2.46865 | − | 1.97630i | |
267.7 | −1.29709 | + | 0.563528i | −1.05307 | + | 1.05307i | 1.36487 | − | 1.46189i | −1.60536 | − | 1.55654i | 0.772487 | − | 1.95935i | −1.06078 | + | 1.06078i | −0.946544 | + | 2.66534i | 0.782105i | 2.95945 | + | 1.11430i | ||
267.8 | −1.29605 | − | 0.565915i | −0.899454 | + | 0.899454i | 1.35948 | + | 1.46691i | −0.710484 | − | 2.12019i | 1.67475 | − | 0.656720i | 0.152896 | − | 0.152896i | −0.931806 | − | 2.67053i | 1.38197i | −0.279027 | + | 3.14994i | ||
267.9 | −1.24129 | − | 0.677637i | 0.0295384 | − | 0.0295384i | 1.08162 | + | 1.68229i | −0.945240 | + | 2.02646i | −0.0566821 | + | 0.0166495i | −3.50400 | + | 3.50400i | −0.202618 | − | 2.82116i | 2.99825i | 2.54652 | − | 1.87490i | ||
267.10 | −1.18702 | + | 0.768758i | −0.696739 | + | 0.696739i | 0.818024 | − | 1.82506i | 2.11169 | − | 0.735369i | 0.291418 | − | 1.36267i | −3.48083 | + | 3.48083i | 0.432019 | + | 2.79524i | 2.02911i | −1.94129 | + | 2.49627i | ||
267.11 | −1.14719 | + | 0.827016i | 0.0947206 | − | 0.0947206i | 0.632090 | − | 1.89749i | 0.772974 | + | 2.09822i | −0.0303271 | + | 0.186998i | 2.71510 | − | 2.71510i | 0.844126 | + | 2.69953i | 2.98206i | −2.62201 | − | 1.76779i | ||
267.12 | −1.08661 | − | 0.905136i | −1.87646 | + | 1.87646i | 0.361456 | + | 1.96707i | 2.04208 | − | 0.911004i | 3.73743 | − | 0.340533i | −0.531458 | + | 0.531458i | 1.38770 | − | 2.46461i | − | 4.04218i | −3.04353 | − | 0.858448i | |
267.13 | −1.03366 | − | 0.965164i | 1.76063 | − | 1.76063i | 0.136915 | + | 1.99531i | −2.15452 | − | 0.598366i | −3.51920 | + | 0.120599i | −2.14797 | + | 2.14797i | 1.78428 | − | 2.19462i | − | 3.19965i | 1.64952 | + | 2.69798i | |
267.14 | −0.965164 | − | 1.03366i | 1.76063 | − | 1.76063i | −0.136915 | + | 1.99531i | 2.15452 | + | 0.598366i | −3.51920 | − | 0.120599i | 2.14797 | − | 2.14797i | 2.19462 | − | 1.78428i | − | 3.19965i | −1.46096 | − | 2.80457i | |
267.15 | −0.905136 | − | 1.08661i | −1.87646 | + | 1.87646i | −0.361456 | + | 1.96707i | −2.04208 | + | 0.911004i | 3.73743 | + | 0.340533i | 0.531458 | − | 0.531458i | 2.46461 | − | 1.38770i | − | 4.04218i | 2.83827 | + | 1.39436i | |
267.16 | −0.897815 | + | 1.09267i | 1.75587 | − | 1.75587i | −0.387856 | − | 1.96203i | −0.899297 | − | 2.04726i | 0.342141 | + | 3.49504i | 1.30028 | − | 1.30028i | 2.49208 | + | 1.33774i | − | 3.16619i | 3.04438 | + | 0.855424i | |
267.17 | −0.881214 | + | 1.10610i | −0.667159 | + | 0.667159i | −0.446923 | − | 1.94943i | −2.22363 | − | 0.235482i | −0.150036 | − | 1.32586i | −0.840214 | + | 0.840214i | 2.55010 | + | 1.22352i | 2.10980i | 2.21997 | − | 2.25206i | ||
267.18 | −0.677637 | − | 1.24129i | 0.0295384 | − | 0.0295384i | −1.08162 | + | 1.68229i | 0.945240 | − | 2.02646i | −0.0566821 | − | 0.0166495i | 3.50400 | − | 3.50400i | 2.82116 | + | 0.202618i | 2.99825i | −3.15595 | + | 0.199882i | ||
267.19 | −0.598568 | + | 1.28129i | −1.10517 | + | 1.10517i | −1.28343 | − | 1.53388i | 1.15323 | − | 1.91574i | −0.754530 | − | 2.07757i | 1.04961 | − | 1.04961i | 2.73358 | − | 0.726323i | 0.557193i | 1.76434 | + | 2.62433i | ||
267.20 | −0.565915 | − | 1.29605i | −0.899454 | + | 0.899454i | −1.35948 | + | 1.46691i | 0.710484 | + | 2.12019i | 1.67475 | + | 0.656720i | −0.152896 | + | 0.152896i | 2.67053 | + | 0.931806i | 1.38197i | 2.34580 | − | 2.12067i | ||
See next 80 embeddings (of 104 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
8.d | odd | 2 | 1 | inner |
40.k | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 760.2.w.c | ✓ | 104 |
5.c | odd | 4 | 1 | inner | 760.2.w.c | ✓ | 104 |
8.d | odd | 2 | 1 | inner | 760.2.w.c | ✓ | 104 |
40.k | even | 4 | 1 | inner | 760.2.w.c | ✓ | 104 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
760.2.w.c | ✓ | 104 | 1.a | even | 1 | 1 | trivial |
760.2.w.c | ✓ | 104 | 5.c | odd | 4 | 1 | inner |
760.2.w.c | ✓ | 104 | 8.d | odd | 2 | 1 | inner |
760.2.w.c | ✓ | 104 | 40.k | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\):
\( T_{3}^{52} + 4 T_{3}^{51} + 8 T_{3}^{50} + 320 T_{3}^{48} + 1276 T_{3}^{47} + 2544 T_{3}^{46} + \cdots + 102400 \) |
\( T_{7}^{104} + 3304 T_{7}^{100} + 4690210 T_{7}^{96} + 3747630204 T_{7}^{92} + 1859385007855 T_{7}^{88} + \cdots + 46\!\cdots\!56 \) |