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Properties

Label 129.4.m.b
Level $129$
Weight $4$
Character orbit 129.m
Analytic conductor $7.611$
Analytic rank $0$
Dimension $144$
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] = [129,4,Mod(10,129)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(129, 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("129.10");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 129 = 3 \cdot 43 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 129.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: \(7.61124639074\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 2 q^{2} - 36 q^{3} - 98 q^{4} - 10 q^{5} - 3 q^{6} + 68 q^{7} - 28 q^{8} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 2 q^{2} - 36 q^{3} - 98 q^{4} - 10 q^{5} - 3 q^{6} + 68 q^{7} - 28 q^{8} + 108 q^{9} + 72 q^{10} - 108 q^{11} - 42 q^{12} + 194 q^{13} + 756 q^{14} + 72 q^{15} - 430 q^{16} + 114 q^{17} + 9 q^{18} + 64 q^{19} - 509 q^{20} - 138 q^{21} - 532 q^{22} - 328 q^{23} + 231 q^{24} + 1114 q^{25} + 633 q^{26} + 648 q^{27} - 1163 q^{28} + 736 q^{29} - 216 q^{30} + 2356 q^{31} + 880 q^{32} + 888 q^{33} - 856 q^{34} + 340 q^{35} - 3087 q^{36} + 636 q^{37} - 1402 q^{38} - 1188 q^{39} + 300 q^{40} - 220 q^{41} - 1260 q^{42} + 742 q^{43} + 594 q^{44} - 324 q^{45} - 1186 q^{46} + 48 q^{47} - 267 q^{48} - 5178 q^{49} - 1229 q^{50} + 684 q^{51} - 1878 q^{52} + 1820 q^{53} + 54 q^{54} - 792 q^{55} + 6087 q^{56} + 2160 q^{57} + 4329 q^{58} - 4406 q^{59} - 300 q^{60} + 3300 q^{61} + 193 q^{62} - 270 q^{63} - 10950 q^{64} - 968 q^{65} + 2436 q^{66} - 1000 q^{67} - 3453 q^{68} - 4434 q^{69} + 10563 q^{70} + 876 q^{71} + 378 q^{72} + 2804 q^{73} + 20760 q^{74} + 48 q^{75} + 833 q^{76} - 2034 q^{77} + 1026 q^{78} - 7552 q^{79} + 556 q^{80} + 972 q^{81} - 17728 q^{82} - 5762 q^{83} + 435 q^{84} + 132 q^{85} - 12355 q^{86} - 1464 q^{87} - 9684 q^{88} - 2930 q^{89} + 846 q^{90} - 4090 q^{91} - 11599 q^{92} + 2004 q^{93} + 10500 q^{94} - 1768 q^{95} + 1152 q^{96} - 1646 q^{97} - 1021 q^{98} - 2664 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −1.20383 − 5.27430i 2.19916 + 2.04052i −19.1613 + 9.22761i 7.13614 − 1.07560i 8.11491 − 14.0554i 10.7020 + 18.5364i 44.7518 + 56.1169i 0.672571 + 8.97483i −14.2637 − 36.3433i
10.2 −0.932242 − 4.08442i 2.19916 + 2.04052i −8.60564 + 4.14426i −3.54358 + 0.534109i 6.28418 − 10.8845i −5.66838 − 9.81792i 4.05276 + 5.08200i 0.672571 + 8.97483i 5.48500 + 13.9756i
10.3 −0.683746 − 2.99569i 2.19916 + 2.04052i −1.29888 + 0.625505i 5.71130 − 0.860840i 4.60909 − 7.98318i −0.737636 − 1.27762i −12.5646 − 15.7555i 0.672571 + 8.97483i −6.48389 − 16.5207i
10.4 −0.634367 − 2.77934i 2.19916 + 2.04052i −0.114570 + 0.0551742i −21.4860 + 3.23850i 4.27623 − 7.40664i 8.78686 + 15.2193i −13.9936 − 17.5474i 0.672571 + 8.97483i 22.6309 + 57.6627i
10.5 −0.210104 − 0.920524i 2.19916 + 2.04052i 6.40453 − 3.08426i 21.7982 − 3.28555i 1.41630 − 2.45310i −7.36904 − 12.7635i −8.89433 − 11.1531i 0.672571 + 8.97483i −7.60431 − 19.3755i
10.6 −0.152884 − 0.669829i 2.19916 + 2.04052i 6.78245 − 3.26626i −11.6809 + 1.76061i 1.03058 − 1.78502i −17.5164 − 30.3393i −6.65174 − 8.34101i 0.672571 + 8.97483i 2.96513 + 7.55503i
10.7 −0.0860967 − 0.377214i 2.19916 + 2.04052i 7.07287 − 3.40612i 2.96973 − 0.447614i 0.580372 − 1.00523i 15.9635 + 27.6496i −3.82369 − 4.79475i 0.672571 + 8.97483i −0.424530 − 1.08169i
10.8 0.497126 + 2.17805i 2.19916 + 2.04052i 2.71098 − 1.30554i −3.06368 + 0.461776i −3.35109 + 5.80426i 0.830249 + 1.43803i 15.3346 + 19.2289i 0.672571 + 8.97483i −2.52880 − 6.44329i
10.9 0.508911 + 2.22968i 2.19916 + 2.04052i 2.49525 − 1.20165i 9.97973 − 1.50420i −3.43054 + 5.94186i −5.75546 − 9.96874i 15.3566 + 19.2566i 0.672571 + 8.97483i 8.43269 + 21.4861i
10.10 0.819833 + 3.59192i 2.19916 + 2.04052i −5.02204 + 2.41849i −19.2100 + 2.89544i −5.52645 + 9.57208i 4.31555 + 7.47475i 5.57272 + 6.98797i 0.672571 + 8.97483i −26.1492 − 66.6271i
10.11 1.02674 + 4.49844i 2.19916 + 2.04052i −11.9741 + 5.76640i 14.3343 − 2.16054i −6.92120 + 11.9879i 10.6521 + 18.4500i −15.2192 − 19.0842i 0.672571 + 8.97483i 24.4366 + 62.2636i
10.12 1.21325 + 5.31558i 2.19916 + 2.04052i −19.5756 + 9.42713i −5.00102 + 0.753782i −8.17842 + 14.1654i −11.5209 − 19.9548i −46.6652 − 58.5163i 0.672571 + 8.97483i −10.0742 − 25.6688i
13.1 −1.20383 + 5.27430i 2.19916 − 2.04052i −19.1613 − 9.22761i 7.13614 + 1.07560i 8.11491 + 14.0554i 10.7020 − 18.5364i 44.7518 − 56.1169i 0.672571 − 8.97483i −14.2637 + 36.3433i
13.2 −0.932242 + 4.08442i 2.19916 − 2.04052i −8.60564 − 4.14426i −3.54358 − 0.534109i 6.28418 + 10.8845i −5.66838 + 9.81792i 4.05276 − 5.08200i 0.672571 − 8.97483i 5.48500 − 13.9756i
13.3 −0.683746 + 2.99569i 2.19916 − 2.04052i −1.29888 − 0.625505i 5.71130 + 0.860840i 4.60909 + 7.98318i −0.737636 + 1.27762i −12.5646 + 15.7555i 0.672571 − 8.97483i −6.48389 + 16.5207i
13.4 −0.634367 + 2.77934i 2.19916 − 2.04052i −0.114570 − 0.0551742i −21.4860 − 3.23850i 4.27623 + 7.40664i 8.78686 − 15.2193i −13.9936 + 17.5474i 0.672571 − 8.97483i 22.6309 − 57.6627i
13.5 −0.210104 + 0.920524i 2.19916 − 2.04052i 6.40453 + 3.08426i 21.7982 + 3.28555i 1.41630 + 2.45310i −7.36904 + 12.7635i −8.89433 + 11.1531i 0.672571 − 8.97483i −7.60431 + 19.3755i
13.6 −0.152884 + 0.669829i 2.19916 − 2.04052i 6.78245 + 3.26626i −11.6809 − 1.76061i 1.03058 + 1.78502i −17.5164 + 30.3393i −6.65174 + 8.34101i 0.672571 − 8.97483i 2.96513 − 7.55503i
13.7 −0.0860967 + 0.377214i 2.19916 − 2.04052i 7.07287 + 3.40612i 2.96973 + 0.447614i 0.580372 + 1.00523i 15.9635 − 27.6496i −3.82369 + 4.79475i 0.672571 − 8.97483i −0.424530 + 1.08169i
13.8 0.497126 − 2.17805i 2.19916 − 2.04052i 2.71098 + 1.30554i −3.06368 − 0.461776i −3.35109 − 5.80426i 0.830249 − 1.43803i 15.3346 − 19.2289i 0.672571 − 8.97483i −2.52880 + 6.44329i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 129.4.m.b ✓ 144
43.g even 21 1 inner 129.4.m.b ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
129.4.m.b ✓ 144 1.a even 1 1 trivial
129.4.m.b ✓ 144 43.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{144} + 2 T_{2}^{143} + 147 T_{2}^{142} + 274 T_{2}^{141} + 12316 T_{2}^{140} + \cdots + 14\!\cdots\!04 \) acting on \(S_{4}^{\mathrm{new}}(129, [\chi])\). Copy content Toggle raw display