Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [528,2,Mod(133,528)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(528, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("528.133");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 528 = 2^{4} \cdot 3 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 528.t (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: | \(4.21610122672\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(20\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
133.1 | −1.40318 | + | 0.176320i | −0.707107 | + | 0.707107i | 1.93782 | − | 0.494818i | −0.177505 | − | 0.177505i | 0.867520 | − | 1.11687i | − | 0.297421i | −2.63186 | + | 1.03600i | − | 1.00000i | 0.280370 | + | 0.217774i | ||
133.2 | −1.39548 | − | 0.229436i | 0.707107 | − | 0.707107i | 1.89472 | + | 0.640345i | −1.44191 | − | 1.44191i | −1.14899 | + | 0.824517i | 3.86773i | −2.49712 | − | 1.32830i | − | 1.00000i | 1.68133 | + | 2.34298i | |||
133.3 | −1.27899 | − | 0.603477i | 0.707107 | − | 0.707107i | 1.27163 | + | 1.54368i | 0.360029 | + | 0.360029i | −1.33111 | + | 0.477660i | − | 3.61419i | −0.694825 | − | 2.74175i | − | 1.00000i | −0.243204 | − | 0.677743i | ||
133.4 | −1.09834 | − | 0.890868i | −0.707107 | + | 0.707107i | 0.412708 | + | 1.95695i | −1.78499 | − | 1.78499i | 1.40658 | − | 0.146706i | 4.29781i | 1.29009 | − | 2.51707i | − | 1.00000i | 0.370337 | + | 3.55072i | |||
133.5 | −1.07628 | + | 0.917404i | −0.707107 | + | 0.707107i | 0.316740 | − | 1.97476i | 2.94133 | + | 2.94133i | 0.112340 | − | 1.40974i | 4.25298i | 1.47075 | + | 2.41597i | − | 1.00000i | −5.86407 | − | 0.467295i | |||
133.6 | −0.899005 | + | 1.09169i | −0.707107 | + | 0.707107i | −0.383579 | − | 1.96287i | −0.245391 | − | 0.245391i | −0.136250 | − | 1.40763i | − | 2.09093i | 2.48769 | + | 1.34588i | − | 1.00000i | 0.488500 | − | 0.0472835i | ||
133.7 | −0.611904 | − | 1.27498i | −0.707107 | + | 0.707107i | −1.25115 | + | 1.56033i | 2.47469 | + | 2.47469i | 1.33423 | + | 0.468865i | 0.523941i | 2.75497 | + | 0.640412i | − | 1.00000i | 1.64090 | − | 4.66945i | |||
133.8 | −0.610414 | + | 1.27569i | 0.707107 | − | 0.707107i | −1.25479 | − | 1.55740i | −1.71726 | − | 1.71726i | 0.470424 | + | 1.33368i | 0.688369i | 2.75271 | − | 0.650069i | − | 1.00000i | 3.23893 | − | 1.14246i | |||
133.9 | −0.532522 | − | 1.31012i | 0.707107 | − | 0.707107i | −1.43284 | + | 1.39534i | −1.78100 | − | 1.78100i | −1.30295 | − | 0.549847i | 3.86812i | 2.59108 | + | 1.13415i | − | 1.00000i | −1.38491 | + | 3.28176i | |||
133.10 | 0.112225 | + | 1.40975i | 0.707107 | − | 0.707107i | −1.97481 | + | 0.316420i | 0.198134 | + | 0.198134i | 1.07620 | + | 0.917491i | − | 2.99121i | −0.667697 | − | 2.74849i | − | 1.00000i | −0.257084 | + | 0.301556i | ||
133.11 | 0.116618 | − | 1.40940i | 0.707107 | − | 0.707107i | −1.97280 | − | 0.328724i | 2.29623 | + | 2.29623i | −0.914133 | − | 1.07906i | 2.56341i | −0.693367 | + | 2.74212i | − | 1.00000i | 3.50408 | − | 2.96851i | |||
133.12 | 0.138138 | + | 1.40745i | −0.707107 | + | 0.707107i | −1.96184 | + | 0.388846i | −1.48034 | − | 1.48034i | −1.09290 | − | 0.897539i | − | 0.896581i | −0.818286 | − | 2.70747i | − | 1.00000i | 1.87902 | − | 2.28800i | ||
133.13 | 0.392209 | − | 1.35874i | −0.707107 | + | 0.707107i | −1.69234 | − | 1.06582i | 1.57974 | + | 1.57974i | 0.683440 | + | 1.23811i | − | 2.84773i | −2.11192 | + | 1.88143i | − | 1.00000i | 2.76604 | − | 1.52686i | ||
133.14 | 0.747359 | + | 1.20061i | 0.707107 | − | 0.707107i | −0.882909 | + | 1.79457i | 2.98823 | + | 2.98823i | 1.37742 | + | 0.320494i | 1.35380i | −2.81442 | + | 0.281161i | − | 1.00000i | −1.35440 | + | 5.82096i | |||
133.15 | 0.757184 | − | 1.19443i | 0.707107 | − | 0.707107i | −0.853344 | − | 1.80881i | −0.248435 | − | 0.248435i | −0.309182 | − | 1.38000i | − | 1.46803i | −2.80665 | − | 0.350341i | − | 1.00000i | −0.484850 | + | 0.108628i | ||
133.16 | 1.06297 | + | 0.932783i | −0.707107 | + | 0.707107i | 0.259832 | + | 1.98305i | −1.50956 | − | 1.50956i | −1.41121 | + | 0.0920596i | 2.25600i | −1.57356 | + | 2.35030i | − | 1.00000i | −0.196532 | − | 3.01271i | |||
133.17 | 1.37459 | − | 0.332432i | −0.707107 | + | 0.707107i | 1.77898 | − | 0.913914i | 1.37452 | + | 1.37452i | −0.736915 | + | 1.20704i | 2.32592i | 2.14154 | − | 1.84764i | − | 1.00000i | 2.34633 | + | 1.43246i | |||
133.18 | 1.38370 | + | 0.292167i | 0.707107 | − | 0.707107i | 1.82928 | + | 0.808547i | −2.65097 | − | 2.65097i | 1.18502 | − | 0.771833i | − | 3.16552i | 2.29495 | + | 1.65325i | − | 1.00000i | −2.89363 | − | 4.44268i | ||
133.19 | 1.40742 | − | 0.138462i | 0.707107 | − | 0.707107i | 1.96166 | − | 0.389747i | 0.582740 | + | 0.582740i | 0.897288 | − | 1.09310i | 4.89753i | 2.70691 | − | 0.820152i | − | 1.00000i | 0.900847 | + | 0.739473i | |||
133.20 | 1.41369 | + | 0.0384766i | −0.707107 | + | 0.707107i | 1.99704 | + | 0.108788i | −1.75827 | − | 1.75827i | −1.02684 | + | 0.972423i | − | 1.52399i | 2.81901 | + | 0.230632i | − | 1.00000i | −2.41800 | − | 2.55331i | ||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 528.2.t.b | ✓ | 40 |
4.b | odd | 2 | 1 | 2112.2.t.b | 40 | ||
16.e | even | 4 | 1 | inner | 528.2.t.b | ✓ | 40 |
16.f | odd | 4 | 1 | 2112.2.t.b | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
528.2.t.b | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
528.2.t.b | ✓ | 40 | 16.e | even | 4 | 1 | inner |
2112.2.t.b | 40 | 4.b | odd | 2 | 1 | ||
2112.2.t.b | 40 | 16.f | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{40} + 48 T_{5}^{37} + 664 T_{5}^{36} + 656 T_{5}^{35} + 1152 T_{5}^{34} + 23200 T_{5}^{33} + 183656 T_{5}^{32} + 334592 T_{5}^{31} + 563840 T_{5}^{30} + 4020032 T_{5}^{29} + 25164320 T_{5}^{28} + \cdots + 16777216 \)
acting on \(S_{2}^{\mathrm{new}}(528, [\chi])\).