↑

Properties

Label 147.4.m.a
Level $147$
Weight $4$
Character orbit 147.m
Analytic conductor $8.673$
Analytic rank $0$
Dimension $156$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [147,4,Mod(4,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 10]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.m (of order \(21\), degree \(12\), 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: \(8.67328077084\)
Analytic rank: \(0\)
Dimension: \(156\)
Relative dimension: \(13\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 156 q - 2 q^{2} + 39 q^{3} + 44 q^{4} - 3 q^{5} + 12 q^{6} + 7 q^{7} - 24 q^{8} + 117 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 156 q - 2 q^{2} + 39 q^{3} + 44 q^{4} - 3 q^{5} + 12 q^{6} + 7 q^{7} - 24 q^{8} + 117 q^{9} + 201 q^{10} - 123 q^{11} + 132 q^{12} + 128 q^{13} + 462 q^{14} - 129 q^{15} - 212 q^{16} - 130 q^{17} + 108 q^{18} + 458 q^{19} + 5 q^{20} - 42 q^{21} - 332 q^{22} - 122 q^{23} + 351 q^{24} - 494 q^{25} - 1174 q^{26} - 702 q^{27} - 1288 q^{28} + 378 q^{29} + 435 q^{30} + 1163 q^{31} - 445 q^{32} + 219 q^{33} - 760 q^{34} - 322 q^{35} - 792 q^{36} - 1402 q^{37} - 1764 q^{38} + 522 q^{39} + 749 q^{40} - 86 q^{41} - 231 q^{42} + 348 q^{43} - 4427 q^{44} - 531 q^{45} - 2440 q^{46} - 1324 q^{47} + 4128 q^{48} - 819 q^{49} + 8130 q^{50} - 978 q^{51} - 248 q^{52} - 449 q^{53} - 54 q^{54} + 405 q^{55} - 2037 q^{56} - 270 q^{57} + 2589 q^{58} + 2013 q^{59} + 3825 q^{60} + 914 q^{61} + 1234 q^{62} - 1008 q^{63} + 6308 q^{64} + 1578 q^{65} - 825 q^{66} + 2712 q^{67} + 860 q^{68} + 732 q^{69} - 966 q^{70} - 642 q^{71} + 1431 q^{72} + 3902 q^{73} - 5592 q^{74} - 1944 q^{75} + 1327 q^{76} + 4809 q^{77} - 1272 q^{78} + 3695 q^{79} + 4520 q^{80} + 1053 q^{81} + 8965 q^{82} + 2567 q^{83} + 2352 q^{84} + 5188 q^{85} - 10888 q^{86} - 4578 q^{87} - 3870 q^{88} + 7132 q^{89} + 1674 q^{90} + 840 q^{91} - 4662 q^{92} - 1866 q^{93} - 8350 q^{94} - 14952 q^{95} + 2172 q^{96} - 2818 q^{97} - 25718 q^{98} - 2322 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −4.11685 − 2.80682i −2.19916 − 2.04052i 6.14747 + 15.6635i 11.9295 − 3.67976i 3.32622 + 14.5731i −10.7479 − 15.0825i 9.78648 − 42.8774i 0.672571 + 8.97483i −59.4402 − 18.3349i
4.2 −4.02532 − 2.74442i −2.19916 − 2.04052i 5.74864 + 14.6473i −14.3434 + 4.42434i 3.25227 + 14.2491i −10.0972 + 15.5257i 8.38544 − 36.7390i 0.672571 + 8.97483i 69.8788 + 21.5548i
4.3 −3.74438 − 2.55287i −2.19916 − 2.04052i 4.58047 + 11.6709i −2.61682 + 0.807182i 3.02528 + 13.2546i 16.8460 − 7.69486i 4.57577 − 20.0478i 0.672571 + 8.97483i 11.8590 + 3.65802i
4.4 −2.11057 − 1.43896i −2.19916 − 2.04052i −0.538832 − 1.37292i 6.11985 − 1.88772i 1.70524 + 7.47116i 3.61666 + 18.1637i −5.38566 + 23.5961i 0.672571 + 8.97483i −15.6327 − 4.82206i
4.5 −1.68383 − 1.14802i −2.19916 − 2.04052i −1.40538 − 3.58084i −13.6179 + 4.20057i 1.36046 + 5.96057i −10.0932 − 15.5283i −5.37235 + 23.5378i 0.672571 + 8.97483i 27.7527 + 8.56057i
4.6 −0.640950 − 0.436993i −2.19916 − 2.04052i −2.70287 − 6.88681i 18.6060 − 5.73920i 0.517858 + 2.26889i 12.6118 − 13.5626i −2.65803 + 11.6456i 0.672571 + 8.97483i −14.4335 − 4.45216i
4.7 −0.0432659 − 0.0294982i −2.19916 − 2.04052i −2.92173 − 7.44444i −3.49980 + 1.07954i 0.0349568 + 0.153156i 18.2783 − 2.98381i −0.186404 + 0.816691i 0.672571 + 8.97483i 0.183266 + 0.0565302i
4.8 0.758180 + 0.516918i −2.19916 − 2.04052i −2.61510 − 6.66316i −15.2135 + 4.69275i −0.612574 − 2.68386i 13.8677 + 12.2755i 3.09513 − 13.5606i 0.672571 + 8.97483i −13.9604 − 4.30620i
4.9 0.887861 + 0.605333i −2.19916 − 2.04052i −2.50086 − 6.37209i 3.25860 − 1.00515i −0.717351 − 3.14292i −18.3902 + 2.19071i 3.54976 − 15.5525i 0.672571 + 8.97483i 3.50163 + 1.08011i
4.10 2.13784 + 1.45755i −2.19916 − 2.04052i −0.476845 − 1.21498i 5.30971 − 1.63783i −1.72727 − 7.56767i −2.39700 + 18.3645i 5.35754 − 23.4729i 0.672571 + 8.97483i 13.7385 + 4.23777i
4.11 3.15659 + 2.15212i −2.19916 − 2.04052i 2.40967 + 6.13975i 12.9112 − 3.98258i −2.55038 − 11.1739i −3.21117 − 18.2397i 1.19384 − 5.23057i 0.672571 + 8.97483i 49.3263 + 15.2152i
4.12 3.71363 + 2.53191i −2.19916 − 2.04052i 4.45774 + 11.3581i −14.6509 + 4.51920i −3.00044 − 13.1458i −17.4447 + 6.21940i −4.20220 + 18.4111i 0.672571 + 8.97483i −65.8502 − 20.3121i
4.13 4.05860 + 2.76710i −2.19916 − 2.04052i 5.89261 + 15.0141i 4.01431 − 1.23825i −3.27916 − 14.3669i 15.8214 + 9.62721i −8.88551 + 38.9300i 0.672571 + 8.97483i 19.7188 + 6.08245i
16.1 −1.98724 − 5.06341i 0.224190 + 2.99161i −15.8246 + 14.6831i −3.48902 + 2.37878i 14.7022 − 7.08023i −18.4029 + 2.08126i 66.5878 + 32.0670i −8.89948 + 1.34138i 18.9783 + 12.9392i
16.2 −1.48901 − 3.79395i 0.224190 + 2.99161i −6.31245 + 5.85710i −2.37526 + 1.61943i 11.0162 − 5.30512i −1.23951 − 18.4787i 2.24435 + 1.08082i −8.89948 + 1.34138i 9.68082 + 6.60027i
16.3 −1.39677 − 3.55891i 0.224190 + 2.99161i −4.85044 + 4.50055i 3.85989 − 2.63163i 10.3337 − 4.97646i 1.48013 + 18.4610i −4.76462 − 2.29452i −8.89948 + 1.34138i −14.7571 − 10.0612i
16.4 −1.01135 − 2.57687i 0.224190 + 2.99161i 0.246973 − 0.229157i −10.9026 + 7.43329i 7.48226 − 3.60327i 18.0818 − 4.00598i −20.7930 − 10.0134i −8.89948 + 1.34138i 30.1810 + 20.5770i
16.5 −0.835252 − 2.12819i 0.224190 + 2.99161i 2.03288 − 1.88624i 16.0880 − 10.9686i 6.17945 − 2.97587i −18.5079 − 0.677446i −22.1908 − 10.6865i −8.89948 + 1.34138i −36.7808 − 25.0767i
16.6 −0.265545 − 0.676597i 0.224190 + 2.99161i 5.47715 − 5.08205i 13.3425 − 9.09677i 1.96458 − 0.946093i 17.0296 − 7.27963i −10.1318 − 4.87922i −8.89948 + 1.34138i −9.69788 − 6.61190i
16.7 −0.189972 − 0.484042i 0.224190 + 2.99161i 5.66621 − 5.25747i −4.99920 + 3.40840i 1.40548 − 0.676841i −2.11423 + 18.3992i −7.36919 − 3.54882i −8.89948 + 1.34138i 2.59952 + 1.77232i
See next 80 embeddings (of 156 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.4.m.a ✓ 156
49.g even 21 1 inner 147.4.m.a ✓ 156
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.m.a ✓ 156 1.a even 1 1 trivial
147.4.m.a ✓ 156 49.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{156} + 2 T_{2}^{155} - 72 T_{2}^{154} - 160 T_{2}^{153} + 1937 T_{2}^{152} + \cdots + 76\!\cdots\!24 \) acting on \(S_{4}^{\mathrm{new}}(147, [\chi])\). Copy content Toggle raw display