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Properties

Label 129.2.n.b
Level $129$
Weight $2$
Character orbit 129.n
Analytic conductor $1.030$
Analytic rank $0$
Dimension $144$
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] = [129,2,Mod(5,129)]
 
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([21, 25]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("129.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 129 = 3 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 129.n (of order \(42\), 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: \(1.03007018607\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(12\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 14 q^{3} - 48 q^{4} - 4 q^{6} - 42 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 14 q^{3} - 48 q^{4} - 4 q^{6} - 42 q^{7} - 26 q^{10} - 4 q^{12} - 48 q^{13} - 19 q^{15} - 8 q^{16} + 16 q^{18} - 10 q^{19} - 15 q^{21} + 14 q^{22} - 11 q^{24} - 16 q^{25} - 14 q^{27} - 42 q^{28} - 6 q^{30} - 36 q^{31} + 7 q^{33} + 80 q^{34} + 48 q^{36} - 6 q^{37} + 7 q^{39} + 118 q^{40} + 46 q^{43} + 78 q^{46} - 41 q^{48} - 54 q^{49} + 7 q^{51} - 4 q^{52} - 52 q^{54} - 22 q^{55} - 110 q^{57} + 18 q^{58} - 9 q^{60} - 58 q^{61} - 82 q^{63} + 72 q^{64} - 127 q^{66} + 10 q^{67} - 15 q^{69} - 196 q^{70} + 179 q^{72} + 42 q^{73} + 119 q^{75} + 4 q^{76} + 62 q^{78} - 30 q^{79} + 200 q^{81} - 28 q^{82} + 111 q^{84} - 18 q^{87} - 14 q^{88} + 131 q^{90} - 38 q^{91} + 156 q^{93} + 28 q^{94} + 304 q^{96} + 72 q^{97} + 121 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −0.608139 + 2.66443i −0.739274 − 1.56636i −4.92742 − 2.37292i −0.999992 − 2.54794i 4.62303 − 1.01718i −0.457900 + 0.264368i 5.91110 − 7.41228i −1.90695 + 2.31593i 7.39694 − 1.11491i
5.2 −0.560235 + 2.45455i −0.790747 + 1.54101i −3.90902 − 1.88248i 1.36058 + 3.46671i −3.33949 − 2.80426i 2.34383 − 1.35321i 3.67113 − 4.60345i −1.74944 − 2.43710i −9.27146 + 1.39745i
5.3 −0.437293 + 1.91590i 1.65136 − 0.522508i −1.67753 − 0.807855i 0.258500 + 0.658648i 0.278949 + 3.39233i −0.427704 + 0.246935i −0.169191 + 0.212159i 2.45397 − 1.72570i −1.37495 + 0.207240i
5.4 −0.306235 + 1.34170i −1.69962 − 0.333615i 0.0955536 + 0.0460162i 0.503180 + 1.28208i 0.968094 − 2.17822i −3.19967 + 1.84733i −1.80710 + 2.26604i 2.77740 + 1.13404i −1.87426 + 0.282499i
5.5 −0.234198 + 1.02609i 0.968186 + 1.43618i 0.803929 + 0.387152i −0.485964 − 1.23822i −1.70040 + 0.657093i 2.66652 − 1.53951i −1.89795 + 2.37995i −1.12523 + 2.78098i 1.38433 − 0.208654i
5.6 −0.0804380 + 0.352422i 0.736839 − 1.56750i 1.68421 + 0.811071i −1.30036 − 3.31326i 0.493153 + 0.385765i −0.812758 + 0.469246i −0.872078 + 1.09355i −1.91414 − 2.31000i 1.27226 − 0.191763i
5.7 0.0804380 − 0.352422i 0.274201 − 1.71021i 1.68421 + 0.811071i 1.30036 + 3.31326i −0.580659 − 0.234200i −0.812758 + 0.469246i 0.872078 − 1.09355i −2.84963 − 0.937883i 1.27226 − 0.191763i
5.8 0.234198 − 1.02609i −1.60898 + 0.641230i 0.803929 + 0.387152i 0.485964 + 1.23822i 0.281139 + 1.80113i 2.66652 − 1.53951i 1.89795 − 2.37995i 2.17765 − 2.06346i 1.38433 − 0.208654i
5.9 0.306235 − 1.34170i 1.59222 + 0.681783i 0.0955536 + 0.0460162i −0.503180 − 1.28208i 1.40234 − 1.92750i −3.19967 + 1.84733i 1.80710 − 2.26604i 2.07034 + 2.17110i −1.87426 + 0.282499i
5.10 0.437293 − 1.91590i −1.07008 − 1.36196i −1.67753 − 0.807855i −0.258500 − 0.658648i −3.07732 + 1.45459i −0.427704 + 0.246935i 0.169191 − 0.212159i −0.709870 + 2.91480i −1.37495 + 0.207240i
5.11 0.560235 − 2.45455i −0.214738 + 1.71869i −3.90902 − 1.88248i −1.36058 − 3.46671i 4.09830 + 1.48995i 2.34383 − 1.35321i −3.67113 + 4.60345i −2.90778 − 0.738134i −9.27146 + 1.39745i
5.12 0.608139 − 2.66443i 1.49318 − 0.877737i −4.92742 − 2.37292i 0.999992 + 2.54794i −1.43061 − 4.51225i −0.457900 + 0.264368i −5.91110 + 7.41228i 1.45915 − 2.62123i 7.39694 − 1.11491i
20.1 −1.60638 − 2.01433i 1.34415 + 1.09236i −1.03205 + 4.52170i −0.0116054 + 0.154863i 0.0411734 − 4.46231i 2.00737 − 1.15896i 6.12349 − 2.94892i 0.613480 + 2.93660i 0.330588 − 0.225391i
20.2 −1.50765 − 1.89054i −1.59375 − 0.678193i −0.856069 + 3.75068i −0.198318 + 2.64637i 1.12068 + 4.03553i 0.0463085 − 0.0267363i 4.02422 − 1.93796i 2.08011 + 2.16175i 5.30205 − 3.61487i
20.3 −1.22242 − 1.53287i −1.02339 + 1.39738i −0.410329 + 1.79777i 0.256779 − 3.42648i 3.39302 − 0.139460i −1.51706 + 0.875874i −0.275559 + 0.132702i −0.905335 − 2.86013i −5.56624 + 3.79500i
20.4 −0.732899 − 0.919027i −0.297099 + 1.70638i 0.137573 − 0.602748i −0.292695 + 3.90574i 1.78595 − 0.977562i −2.04471 + 1.18051i −2.77291 + 1.33536i −2.82346 − 1.01393i 3.80400 − 2.59352i
20.5 −0.606050 − 0.759962i −0.431951 − 1.67732i 0.234796 − 1.02871i 0.0149759 − 0.199839i −1.01292 + 1.34481i 0.0942007 − 0.0543868i −2.67561 + 1.28851i −2.62684 + 1.44905i −0.160946 + 0.109731i
20.6 −0.553934 − 0.694612i 1.71160 + 0.265385i 0.269400 − 1.18032i 0.147069 − 1.96250i −0.763774 − 1.33590i −1.81332 + 1.04692i −2.57001 + 1.23765i 2.85914 + 0.908467i −1.44464 + 0.984941i
20.7 0.553934 + 0.694612i 1.07418 − 1.35872i 0.269400 − 1.18032i −0.147069 + 1.96250i 1.53881 − 0.00650406i −1.81332 + 1.04692i 2.57001 − 1.23765i −0.692263 − 2.91904i −1.44464 + 0.984941i
20.8 0.606050 + 0.759962i 0.824228 + 1.52337i 0.234796 − 1.02871i −0.0149759 + 0.199839i −0.658179 + 1.54962i 0.0942007 − 0.0543868i 2.67561 − 1.28851i −1.64130 + 2.51120i −0.160946 + 0.109731i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
43.h odd 42 1 inner
129.n even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 129.2.n.b ✓ 144
3.b odd 2 1 inner 129.2.n.b ✓ 144
43.h odd 42 1 inner 129.2.n.b ✓ 144
129.n even 42 1 inner 129.2.n.b ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
129.2.n.b ✓ 144 1.a even 1 1 trivial
129.2.n.b ✓ 144 3.b odd 2 1 inner
129.2.n.b ✓ 144 43.h odd 42 1 inner
129.2.n.b ✓ 144 129.n even 42 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{144} + 48 T_{2}^{142} + 1232 T_{2}^{140} + 22460 T_{2}^{138} + 326216 T_{2}^{136} + \cdots + 31\!\cdots\!21 \) acting on \(S_{2}^{\mathrm{new}}(129, [\chi])\). Copy content Toggle raw display