Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [148,4,Mod(23,148)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(148, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 5]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("148.23");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 148 = 2^{2} \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 148.l (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: | \(8.73228268085\) |
Analytic rank: | \(0\) |
Dimension: | \(216\) |
Relative dimension: | \(54\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
23.1 | −2.82270 | + | 0.179922i | −1.45851 | − | 2.52622i | 7.93526 | − | 1.01573i | 4.41612 | − | 16.4812i | 4.57147 | + | 6.86834i | 9.31107 | − | 5.37575i | −22.2161 | + | 4.29483i | 9.24548 | − | 16.0136i | −9.50006 | + | 47.3160i |
23.2 | −2.81607 | − | 0.264069i | 0.742909 | + | 1.28676i | 7.86054 | + | 1.48727i | −5.53085 | + | 20.6414i | −1.75229 | − | 3.81978i | −27.5623 | + | 15.9131i | −21.7431 | − | 6.26400i | 12.3962 | − | 21.4708i | 21.0260 | − | 56.6672i |
23.3 | −2.81163 | + | 0.307777i | 4.11399 | + | 7.12564i | 7.81055 | − | 1.73071i | 1.69680 | − | 6.33255i | −13.7601 | − | 18.7685i | −15.1822 | + | 8.76547i | −21.4277 | + | 7.27003i | −20.3498 | + | 35.2469i | −2.82177 | + | 18.3270i |
23.4 | −2.79751 | + | 0.417061i | −3.28290 | − | 5.68614i | 7.65212 | − | 2.33346i | −2.65116 | + | 9.89426i | 11.5554 | + | 14.5379i | 13.5888 | − | 7.84550i | −20.4337 | + | 9.71929i | −8.05481 | + | 13.9513i | 3.29013 | − | 28.7850i |
23.5 | −2.76966 | − | 0.573572i | 2.40316 | + | 4.16239i | 7.34203 | + | 3.17720i | −1.66205 | + | 6.20287i | −4.26850 | − | 12.9068i | 24.4859 | − | 14.1370i | −18.5126 | − | 13.0109i | 1.94968 | − | 3.37694i | 8.16112 | − | 16.2265i |
23.6 | −2.74699 | + | 0.673811i | −4.72921 | − | 8.19123i | 7.09196 | − | 3.70191i | 0.497508 | − | 1.85673i | 18.5104 | + | 19.3147i | −27.8597 | + | 16.0848i | −16.9872 | + | 14.9478i | −31.2308 | + | 54.0934i | −0.115571 | + | 5.43565i |
23.7 | −2.61514 | − | 1.07753i | −2.17326 | − | 3.76419i | 5.67787 | + | 5.63576i | 1.21252 | − | 4.52519i | 1.62734 | + | 12.1856i | −5.63584 | + | 3.25385i | −8.77570 | − | 20.8563i | 4.05391 | − | 7.02158i | −8.04692 | + | 10.5274i |
23.8 | −2.58771 | + | 1.14182i | 1.44716 | + | 2.50655i | 5.39251 | − | 5.90939i | 0.472983 | − | 1.76520i | −6.60686 | − | 4.83385i | 7.55483 | − | 4.36178i | −7.20682 | + | 21.4490i | 9.31146 | − | 16.1279i | 0.791587 | + | 5.10788i |
23.9 | −2.32311 | + | 1.61343i | 0.944053 | + | 1.63515i | 2.79369 | − | 7.49635i | 0.490534 | − | 1.83070i | −4.83134 | − | 2.27547i | −11.6435 | + | 6.72237i | 5.60480 | + | 21.9223i | 11.7175 | − | 20.2954i | 1.81414 | + | 5.04436i |
23.10 | −2.30739 | − | 1.63584i | 3.41198 | + | 5.90972i | 2.64809 | + | 7.54902i | 5.00825 | − | 18.6911i | 1.79457 | − | 19.2175i | 10.3132 | − | 5.95432i | 6.23878 | − | 21.7503i | −9.78319 | + | 16.9450i | −42.1315 | + | 34.9349i |
23.11 | −2.30604 | − | 1.63774i | −3.67401 | − | 6.36358i | 2.63562 | + | 7.55338i | −3.52095 | + | 13.1404i | −1.94947 | + | 20.6917i | 12.1059 | − | 6.98933i | 6.29264 | − | 21.7348i | −13.4968 | + | 23.3771i | 29.6399 | − | 24.5358i |
23.12 | −2.25719 | − | 1.70443i | 0.931174 | + | 1.61284i | 2.18981 | + | 7.69446i | 1.49542 | − | 5.58098i | 0.647145 | − | 5.22761i | −19.9021 | + | 11.4905i | 8.17190 | − | 21.1002i | 11.7658 | − | 20.3790i | −12.8879 | + | 10.0485i |
23.13 | −2.14735 | + | 1.84089i | −1.51011 | − | 2.61559i | 1.22222 | − | 7.90608i | −3.80362 | + | 14.1953i | 8.05777 | + | 2.83663i | 1.72485 | − | 0.995841i | 11.9297 | + | 19.2271i | 8.93912 | − | 15.4830i | −17.9643 | − | 37.4843i |
23.14 | −2.13815 | + | 1.85157i | 4.77709 | + | 8.27416i | 1.14338 | − | 7.91787i | −5.11711 | + | 19.0973i | −25.5343 | − | 8.84629i | 14.8825 | − | 8.59244i | 12.2158 | + | 19.0467i | −32.1411 | + | 55.6701i | −24.4188 | − | 50.3076i |
23.15 | −1.94845 | + | 2.05026i | −4.30868 | − | 7.46285i | −0.407103 | − | 7.98963i | 1.88050 | − | 7.01810i | 23.6960 | + | 5.70708i | 30.0880 | − | 17.3713i | 17.1740 | + | 14.7327i | −23.6294 | + | 40.9274i | 10.7249 | + | 17.5299i |
23.16 | −1.75806 | − | 2.21567i | 1.53192 | + | 2.65336i | −1.81843 | + | 7.79059i | −3.15538 | + | 11.7760i | 3.18577 | − | 8.05900i | 11.0228 | − | 6.36402i | 20.4583 | − | 9.66732i | 8.80646 | − | 15.2532i | 31.6392 | − | 13.7117i |
23.17 | −1.73244 | − | 2.23577i | 4.77385 | + | 8.26855i | −1.99733 | + | 7.74665i | −2.46153 | + | 9.18654i | 10.2162 | − | 24.9980i | −12.1954 | + | 7.04103i | 20.7800 | − | 8.95500i | −32.0793 | + | 55.5630i | 24.8034 | − | 10.4117i |
23.18 | −1.61715 | + | 2.32052i | −1.88621 | − | 3.26700i | −2.76964 | − | 7.50527i | 4.55685 | − | 17.0064i | 10.6314 | + | 0.906257i | −11.4344 | + | 6.60166i | 21.8951 | + | 5.71013i | 6.38446 | − | 11.0582i | 32.0946 | + | 38.0762i |
23.19 | −1.46909 | + | 2.41697i | 4.05055 | + | 7.01576i | −3.68353 | − | 7.10152i | 4.80220 | − | 17.9221i | −22.9076 | − | 0.516735i | 7.83551 | − | 4.52383i | 22.5756 | + | 1.52982i | −19.3139 | + | 33.4527i | 36.2623 | + | 37.9360i |
23.20 | −1.25502 | − | 2.53474i | −3.84046 | − | 6.65188i | −4.84984 | + | 6.36231i | 1.72327 | − | 6.43134i | −12.0409 | + | 18.0828i | −15.7965 | + | 9.12014i | 22.2135 | + | 4.30827i | −15.9983 | + | 27.7099i | −18.4645 | + | 3.70341i |
See next 80 embeddings (of 216 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
37.g | odd | 12 | 1 | inner |
148.l | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 148.4.l.b | ✓ | 216 |
4.b | odd | 2 | 1 | inner | 148.4.l.b | ✓ | 216 |
37.g | odd | 12 | 1 | inner | 148.4.l.b | ✓ | 216 |
148.l | even | 12 | 1 | inner | 148.4.l.b | ✓ | 216 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
148.4.l.b | ✓ | 216 | 1.a | even | 1 | 1 | trivial |
148.4.l.b | ✓ | 216 | 4.b | odd | 2 | 1 | inner |
148.4.l.b | ✓ | 216 | 37.g | odd | 12 | 1 | inner |
148.4.l.b | ✓ | 216 | 148.l | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{216} + 1946 T_{3}^{214} + 1953831 T_{3}^{212} + 1339848450 T_{3}^{210} + 702434331807 T_{3}^{208} + 299152712121836 T_{3}^{206} + \cdots + 12\!\cdots\!56 \)
acting on \(S_{4}^{\mathrm{new}}(148, [\chi])\).