Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [308,2,Mod(219,308)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(308, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 2, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("308.219");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 308 = 2^{2} \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 308.n (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: | \(2.45939238226\) |
Analytic rank: | \(0\) |
Dimension: | \(80\) |
Relative dimension: | \(40\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
219.1 | −1.41411 | − | 0.0168770i | 2.56217 | − | 1.47927i | 1.99943 | + | 0.0477320i | −1.86389 | + | 3.22836i | −3.64817 | + | 2.04861i | 2.46730 | − | 0.955225i | −2.82661 | − | 0.101243i | 2.87648 | − | 4.98222i | 2.69024 | − | 4.53381i |
219.2 | −1.40099 | + | 0.192974i | 0.581000 | − | 0.335441i | 1.92552 | − | 0.540707i | −0.251848 | + | 0.436213i | −0.749242 | + | 0.582066i | −1.57544 | + | 2.12556i | −2.59329 | + | 1.12910i | −1.27496 | + | 2.20829i | 0.268657 | − | 0.659728i |
219.3 | −1.35934 | + | 0.390127i | 0.349115 | − | 0.201562i | 1.69560 | − | 1.06063i | 1.01720 | − | 1.76184i | −0.395931 | + | 0.410190i | 2.11437 | + | 1.59042i | −1.89112 | + | 2.10326i | −1.41875 | + | 2.45734i | −0.695377 | + | 2.79178i |
219.4 | −1.31677 | − | 0.515860i | −0.751169 | + | 0.433688i | 1.46778 | + | 1.35854i | −0.818239 | + | 1.41723i | 1.21284 | − | 0.183570i | 2.64271 | + | 0.126788i | −1.23191 | − | 2.54605i | −1.12383 | + | 1.94653i | 1.80853 | − | 1.44407i |
219.5 | −1.27563 | + | 0.610541i | −0.910192 | + | 0.525500i | 1.25448 | − | 1.55765i | −1.88689 | + | 3.26820i | 0.840232 | − | 1.22605i | −1.60297 | − | 2.10487i | −0.649247 | + | 2.75290i | −0.947700 | + | 1.64147i | 0.411618 | − | 5.32105i |
219.6 | −1.27184 | − | 0.618406i | −1.51941 | + | 0.877234i | 1.23515 | + | 1.57303i | 1.94446 | − | 3.36791i | 2.47494 | − | 0.176086i | −1.65729 | + | 2.06238i | −0.598141 | − | 2.76446i | 0.0390796 | − | 0.0676878i | −4.55578 | + | 3.08097i |
219.7 | −1.20943 | + | 0.732996i | −2.75123 | + | 1.58842i | 0.925433 | − | 1.77301i | 0.319329 | − | 0.553095i | 2.16311 | − | 3.93773i | 2.28868 | + | 1.32739i | 0.180368 | + | 2.82267i | 3.54618 | − | 6.14217i | 0.0192105 | + | 0.902995i |
219.8 | −1.17147 | − | 0.792242i | 1.51941 | − | 0.877234i | 0.744706 | + | 1.85618i | 1.94446 | − | 3.36791i | −2.47494 | − | 0.176086i | 1.65729 | − | 2.06238i | 0.598141 | − | 2.76446i | 0.0390796 | − | 0.0676878i | −4.94609 | + | 2.40493i |
219.9 | −1.10513 | − | 0.882428i | 0.751169 | − | 0.433688i | 0.442641 | + | 1.95040i | −0.818239 | + | 1.41723i | −1.21284 | − | 0.183570i | −2.64271 | − | 0.126788i | 1.23191 | − | 2.54605i | −1.12383 | + | 1.94653i | 2.15487 | − | 0.844193i |
219.10 | −1.09680 | + | 0.892770i | 1.70506 | − | 0.984418i | 0.405924 | − | 1.95837i | 0.622247 | − | 1.07776i | −0.991247 | + | 2.60193i | 1.08267 | − | 2.41409i | 1.30316 | + | 2.51033i | 0.438159 | − | 0.758913i | 0.279716 | + | 1.73761i |
219.11 | −1.06558 | + | 0.929809i | −1.31926 | + | 0.761674i | 0.270911 | − | 1.98157i | 1.45192 | − | 2.51479i | 0.697560 | − | 2.03828i | −2.53420 | − | 0.760154i | 1.55380 | + | 2.36341i | −0.339706 | + | 0.588387i | 0.791147 | + | 4.02971i |
219.12 | −0.775952 | + | 1.18233i | 1.59924 | − | 0.923319i | −0.795796 | − | 1.83486i | −1.03428 | + | 1.79143i | −0.149265 | + | 2.60727i | −0.278241 | + | 2.63108i | 2.78690 | + | 0.482873i | 0.205037 | − | 0.355134i | −1.31550 | − | 2.61292i |
219.13 | −0.721672 | − | 1.21622i | −2.56217 | + | 1.47927i | −0.958378 | + | 1.75542i | −1.86389 | + | 3.22836i | 3.64817 | + | 2.04861i | −2.46730 | + | 0.955225i | 2.82661 | − | 0.101243i | 2.87648 | − | 4.98222i | 5.27151 | − | 0.0629139i |
219.14 | −0.635949 | + | 1.26316i | −1.59924 | + | 0.923319i | −1.19114 | − | 1.60661i | −1.03428 | + | 1.79143i | −0.149265 | − | 2.60727i | −0.278241 | + | 2.63108i | 2.78690 | − | 0.482873i | 0.205037 | − | 0.355134i | −1.60511 | − | 2.44572i |
219.15 | −0.533373 | − | 1.30978i | −0.581000 | + | 0.335441i | −1.43103 | + | 1.39720i | −0.251848 | + | 0.436213i | 0.749242 | + | 0.582066i | 1.57544 | − | 2.12556i | 2.59329 | + | 1.12910i | −1.27496 | + | 2.20829i | 0.705670 | + | 0.0972001i |
219.16 | −0.341809 | − | 1.37229i | −0.349115 | + | 0.201562i | −1.76633 | + | 0.938119i | 1.01720 | − | 1.76184i | 0.395931 | + | 0.410190i | −2.11437 | − | 1.59042i | 1.89112 | + | 2.10326i | −1.41875 | + | 2.45734i | −2.76544 | − | 0.793675i |
219.17 | −0.272449 | + | 1.38772i | 1.31926 | − | 0.761674i | −1.85154 | − | 0.756168i | 1.45192 | − | 2.51479i | 0.697560 | + | 2.03828i | −2.53420 | − | 0.760154i | 1.55380 | − | 2.36341i | −0.339706 | + | 0.588387i | 3.09426 | + | 2.70001i |
219.18 | −0.224763 | + | 1.39624i | −1.70506 | + | 0.984418i | −1.89896 | − | 0.627646i | 0.622247 | − | 1.07776i | −0.991247 | − | 2.60193i | 1.08267 | − | 2.41409i | 1.30316 | − | 2.51033i | 0.438159 | − | 0.758913i | 1.36496 | + | 1.11105i |
219.19 | −0.109073 | − | 1.41000i | 0.910192 | − | 0.525500i | −1.97621 | + | 0.307586i | −1.88689 | + | 3.26820i | −0.840232 | − | 1.22605i | 1.60297 | + | 2.10487i | 0.649247 | + | 2.75290i | −0.947700 | + | 1.64147i | 4.81397 | + | 2.30405i |
219.20 | −0.0300795 | + | 1.41389i | 2.75123 | − | 1.58842i | −1.99819 | − | 0.0850584i | 0.319329 | − | 0.553095i | 2.16311 | + | 3.93773i | 2.28868 | + | 1.32739i | 0.180368 | − | 2.82267i | 3.54618 | − | 6.14217i | 0.772412 | + | 0.468134i |
See all 80 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
11.b | odd | 2 | 1 | inner |
28.g | odd | 6 | 1 | inner |
44.c | even | 2 | 1 | inner |
77.h | odd | 6 | 1 | inner |
308.n | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 308.2.n.c | ✓ | 80 |
4.b | odd | 2 | 1 | inner | 308.2.n.c | ✓ | 80 |
7.c | even | 3 | 1 | inner | 308.2.n.c | ✓ | 80 |
11.b | odd | 2 | 1 | inner | 308.2.n.c | ✓ | 80 |
28.g | odd | 6 | 1 | inner | 308.2.n.c | ✓ | 80 |
44.c | even | 2 | 1 | inner | 308.2.n.c | ✓ | 80 |
77.h | odd | 6 | 1 | inner | 308.2.n.c | ✓ | 80 |
308.n | even | 6 | 1 | inner | 308.2.n.c | ✓ | 80 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
308.2.n.c | ✓ | 80 | 1.a | even | 1 | 1 | trivial |
308.2.n.c | ✓ | 80 | 4.b | odd | 2 | 1 | inner |
308.2.n.c | ✓ | 80 | 7.c | even | 3 | 1 | inner |
308.2.n.c | ✓ | 80 | 11.b | odd | 2 | 1 | inner |
308.2.n.c | ✓ | 80 | 28.g | odd | 6 | 1 | inner |
308.2.n.c | ✓ | 80 | 44.c | even | 2 | 1 | inner |
308.2.n.c | ✓ | 80 | 77.h | odd | 6 | 1 | inner |
308.2.n.c | ✓ | 80 | 308.n | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(308, [\chi])\):
\( T_{3}^{40} - 34 T_{3}^{38} + 689 T_{3}^{36} - 9098 T_{3}^{34} + 88474 T_{3}^{32} - 643760 T_{3}^{30} + 3651971 T_{3}^{28} - 16307078 T_{3}^{26} + 58317871 T_{3}^{24} - 166885326 T_{3}^{22} + 383836347 T_{3}^{20} + \cdots + 257049 \) |
\( T_{43}^{20} - 404 T_{43}^{18} + 69400 T_{43}^{16} - 6601616 T_{43}^{14} + 379126059 T_{43}^{12} - 13424006278 T_{43}^{10} + 286439448279 T_{43}^{8} - 3429281218448 T_{43}^{6} + \cdots + 1770219606016 \) |