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Properties

Label 126.4.e.b
Level $126$
Weight $4$
Character orbit 126.e
Analytic conductor $7.434$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [126,4,Mod(25,126)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(126, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.25");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 126.e (of order \(3\), 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: \(7.43424066072\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{2} + 2 q^{3} + 96 q^{4} + 10 q^{5} + 4 q^{6} + 17 q^{7} + 192 q^{8} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{2} + 2 q^{3} + 96 q^{4} + 10 q^{5} + 4 q^{6} + 17 q^{7} + 192 q^{8} + 32 q^{9} + 20 q^{10} - 4 q^{11} + 8 q^{12} + 80 q^{13} + 34 q^{14} - 126 q^{15} + 384 q^{16} + 92 q^{17} + 64 q^{18} + 54 q^{19} + 40 q^{20} + 230 q^{21} - 8 q^{22} + 131 q^{23} + 16 q^{24} - 178 q^{25} + 160 q^{26} + 92 q^{27} + 68 q^{28} - 278 q^{29} - 252 q^{30} - 220 q^{31} + 768 q^{32} - 396 q^{33} + 184 q^{34} - 493 q^{35} + 128 q^{36} - 21 q^{37} + 108 q^{38} - 17 q^{39} + 80 q^{40} + 465 q^{41} + 460 q^{42} + 159 q^{43} - 16 q^{44} - 870 q^{45} + 262 q^{46} - 678 q^{47} + 32 q^{48} - 207 q^{49} - 356 q^{50} - 444 q^{51} + 320 q^{52} - 78 q^{53} + 184 q^{54} - 1532 q^{55} + 136 q^{56} - 1970 q^{57} - 556 q^{58} - 1622 q^{59} - 504 q^{60} - 1978 q^{61} - 440 q^{62} - 3784 q^{63} + 1536 q^{64} - 624 q^{65} - 792 q^{66} - 80 q^{67} + 368 q^{68} - 2049 q^{69} - 986 q^{70} + 980 q^{71} + 256 q^{72} + 1510 q^{73} - 42 q^{74} - 43 q^{75} + 216 q^{76} - 350 q^{77} - 34 q^{78} + 812 q^{79} + 160 q^{80} + 1292 q^{81} + 930 q^{82} - 7 q^{83} + 920 q^{84} - 581 q^{85} + 318 q^{86} + 3336 q^{87} - 32 q^{88} + 675 q^{89} - 1740 q^{90} - 232 q^{91} + 524 q^{92} - 443 q^{93} - 1356 q^{94} + 2438 q^{95} + 64 q^{96} + 2836 q^{97} - 414 q^{98} + 6429 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 2.00000 −5.17368 − 0.482721i 4.00000 −1.28937 + 2.23325i −10.3474 − 0.965443i −3.69263 − 18.1484i 8.00000 26.5340 + 4.99489i −2.57874 + 4.46651i
25.2 2.00000 −5.16888 + 0.531643i 4.00000 1.73163 − 2.99928i −10.3378 + 1.06329i −7.22460 + 17.0530i 8.00000 26.4347 − 5.49600i 3.46326 − 5.99855i
25.3 2.00000 −3.80574 − 3.53785i 4.00000 −7.73394 + 13.3956i −7.61147 − 7.07570i 16.9604 + 7.43937i 8.00000 1.96725 + 26.9282i −15.4679 + 26.7912i
25.4 2.00000 −2.56421 + 4.51938i 4.00000 8.32069 − 14.4119i −5.12842 + 9.03877i 9.51058 − 15.8918i 8.00000 −13.8497 − 23.1773i 16.6414 − 28.8237i
25.5 2.00000 −2.42222 − 4.59705i 4.00000 9.32266 − 16.1473i −4.84443 − 9.19410i −17.5785 + 5.83062i 8.00000 −15.2657 + 22.2701i 18.6453 − 32.2946i
25.6 2.00000 −1.15587 + 5.06596i 4.00000 −5.67646 + 9.83192i −2.31175 + 10.1319i −18.1638 + 3.61619i 8.00000 −24.3279 − 11.7112i −11.3529 + 19.6638i
25.7 2.00000 1.14289 − 5.06891i 4.00000 4.24036 − 7.34452i 2.28578 − 10.1378i 16.0321 + 9.27204i 8.00000 −24.3876 − 11.5864i 8.48072 − 14.6890i
25.8 2.00000 2.30858 + 4.65516i 4.00000 −5.06718 + 8.77661i 4.61715 + 9.31031i 14.4394 − 11.5975i 8.00000 −16.3409 + 21.4936i −10.1344 + 17.5532i
25.9 2.00000 3.59981 − 3.74718i 4.00000 −1.86554 + 3.23121i 7.19962 − 7.49436i 11.0357 − 14.8733i 8.00000 −1.08273 − 26.9783i −3.73108 + 6.46242i
25.10 2.00000 3.90115 + 3.43235i 4.00000 5.69975 − 9.87226i 7.80230 + 6.86470i 8.79532 + 16.2985i 8.00000 3.43795 + 26.7802i 11.3995 − 19.7445i
25.11 2.00000 5.14207 − 0.747742i 4.00000 5.11758 − 8.86391i 10.2841 − 1.49548i −14.3519 − 11.7057i 8.00000 25.8818 − 7.68988i 10.2352 − 17.7278i
25.12 2.00000 5.19610 − 0.0230458i 4.00000 −7.80019 + 13.5103i 10.3922 − 0.0460917i −7.26216 + 17.0370i 8.00000 26.9989 − 0.239497i −15.6004 + 27.0206i
121.1 2.00000 −5.17368 + 0.482721i 4.00000 −1.28937 − 2.23325i −10.3474 + 0.965443i −3.69263 + 18.1484i 8.00000 26.5340 − 4.99489i −2.57874 − 4.46651i
121.2 2.00000 −5.16888 − 0.531643i 4.00000 1.73163 + 2.99928i −10.3378 − 1.06329i −7.22460 − 17.0530i 8.00000 26.4347 + 5.49600i 3.46326 + 5.99855i
121.3 2.00000 −3.80574 + 3.53785i 4.00000 −7.73394 − 13.3956i −7.61147 + 7.07570i 16.9604 − 7.43937i 8.00000 1.96725 − 26.9282i −15.4679 − 26.7912i
121.4 2.00000 −2.56421 − 4.51938i 4.00000 8.32069 + 14.4119i −5.12842 − 9.03877i 9.51058 + 15.8918i 8.00000 −13.8497 + 23.1773i 16.6414 + 28.8237i
121.5 2.00000 −2.42222 + 4.59705i 4.00000 9.32266 + 16.1473i −4.84443 + 9.19410i −17.5785 − 5.83062i 8.00000 −15.2657 − 22.2701i 18.6453 + 32.2946i
121.6 2.00000 −1.15587 − 5.06596i 4.00000 −5.67646 − 9.83192i −2.31175 − 10.1319i −18.1638 − 3.61619i 8.00000 −24.3279 + 11.7112i −11.3529 − 19.6638i
121.7 2.00000 1.14289 + 5.06891i 4.00000 4.24036 + 7.34452i 2.28578 + 10.1378i 16.0321 − 9.27204i 8.00000 −24.3876 + 11.5864i 8.48072 + 14.6890i
121.8 2.00000 2.30858 − 4.65516i 4.00000 −5.06718 − 8.77661i 4.61715 − 9.31031i 14.4394 + 11.5975i 8.00000 −16.3409 − 21.4936i −10.1344 − 17.5532i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.4.e.b ✓ 24
3.b odd 2 1 378.4.e.a 24
7.c even 3 1 126.4.h.a yes 24
9.c even 3 1 126.4.h.a yes 24
9.d odd 6 1 378.4.h.b 24
21.h odd 6 1 378.4.h.b 24
63.h even 3 1 inner 126.4.e.b ✓ 24
63.j odd 6 1 378.4.e.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.e.b ✓ 24 1.a even 1 1 trivial
126.4.e.b ✓ 24 63.h even 3 1 inner
126.4.h.a yes 24 7.c even 3 1
126.4.h.a yes 24 9.c even 3 1
378.4.e.a 24 3.b odd 2 1
378.4.e.a 24 63.j odd 6 1
378.4.h.b 24 9.d odd 6 1
378.4.h.b 24 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 10 T_{5}^{23} + 889 T_{5}^{22} - 5740 T_{5}^{21} + 459811 T_{5}^{20} + \cdots + 80\!\cdots\!04 \) acting on \(S_{4}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display