Properties

Label 148.4.l.b
Level $148$
Weight $4$
Character orbit 148.l
Analytic conductor $8.732$
Analytic rank $0$
Dimension $216$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

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.

\(\operatorname{Tr}(f)(q) = \) \( 216 q - 6 q^{2} + 24 q^{4} + 36 q^{5} + 12 q^{6} - 976 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 216 q - 6 q^{2} + 24 q^{4} + 36 q^{5} + 12 q^{6} - 976 q^{9} + 64 q^{10} + 106 q^{12} + 128 q^{13} + 20 q^{16} + 8 q^{17} - 174 q^{18} + 406 q^{20} - 12 q^{21} - 124 q^{22} + 890 q^{24} - 1044 q^{25} + 872 q^{26} - 6 q^{28} - 680 q^{29} + 954 q^{30} + 744 q^{32} - 112 q^{33} + 250 q^{34} + 1396 q^{37} + 1120 q^{38} + 1242 q^{40} - 1428 q^{41} - 528 q^{42} - 1264 q^{44} + 192 q^{45} - 1458 q^{46} + 3816 q^{49} - 2236 q^{50} + 1448 q^{52} - 1148 q^{53} + 940 q^{54} - 46 q^{56} - 404 q^{57} - 4032 q^{58} + 4528 q^{60} + 1356 q^{61} + 1764 q^{62} - 9900 q^{65} - 5368 q^{66} - 612 q^{68} + 844 q^{69} + 926 q^{70} - 7740 q^{72} + 2740 q^{74} - 4308 q^{76} - 9840 q^{77} - 156 q^{78} + 4480 q^{80} - 4972 q^{81} + 3056 q^{82} + 7128 q^{84} + 2834 q^{86} + 3796 q^{88} - 184 q^{89} - 976 q^{90} + 10178 q^{92} + 5068 q^{93} - 524 q^{94} + 6678 q^{96} + 2296 q^{97} - 5334 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.54
Significant digits:
Format:

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 \(S_{4}^{\mathrm{new}}(148, [\chi])\). Copy content Toggle raw display