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Properties

Label 429.2.bn.a
Level $429$
Weight $2$
Character orbit 429.bn
Analytic conductor $3.426$
Analytic rank $0$
Dimension $112$
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] = [429,2,Mod(4,429)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(429, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 6, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("429.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.bn (of order \(30\), degree \(8\), 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: \(3.42558224671\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(14\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q - 14 q^{3} - 20 q^{4} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q - 14 q^{3} - 20 q^{4} + 14 q^{9} + 21 q^{11} + 120 q^{12} - 15 q^{13} + 30 q^{14} - 6 q^{15} - 8 q^{16} - 12 q^{19} - 66 q^{20} - 17 q^{22} + 12 q^{23} + 14 q^{25} + 9 q^{26} + 28 q^{27} + 18 q^{28} - 2 q^{29} + 10 q^{30} - 30 q^{32} - 24 q^{33} + 16 q^{35} - 20 q^{36} + 38 q^{38} - 12 q^{39} - 60 q^{40} - 36 q^{41} + 12 q^{43} - 6 q^{45} + 54 q^{46} - 42 q^{48} - 40 q^{49} - 51 q^{50} - 30 q^{51} - 15 q^{52} - 22 q^{53} + 22 q^{55} - 76 q^{56} + 132 q^{58} + 72 q^{59} + 34 q^{61} + 17 q^{62} - 84 q^{64} + 28 q^{65} - 24 q^{66} - 48 q^{67} - 12 q^{68} - 7 q^{69} + 30 q^{71} + 20 q^{74} + 32 q^{75} - 48 q^{76} - 136 q^{77} - 14 q^{78} - 36 q^{79} + 18 q^{80} + 14 q^{81} - 5 q^{82} - 27 q^{84} - 66 q^{85} + 52 q^{87} + 52 q^{88} - 96 q^{89} - 20 q^{90} - 125 q^{91} - 10 q^{92} - 18 q^{93} + 22 q^{94} + 72 q^{95} - 93 q^{97} + 24 q^{98}+O(q^{100}) \) 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 −1.05844 − 2.37729i −0.669131 − 0.743145i −3.19297 + 3.54616i 0.696965 + 0.959290i −1.05844 + 2.37729i 1.37323 + 1.23646i 6.86002 + 2.22895i −0.104528 + 0.994522i 1.54282 − 2.67224i
4.2 −0.866606 − 1.94643i −0.669131 − 0.743145i −1.69932 + 1.88729i 1.80442 + 2.48357i −0.866606 + 1.94643i −3.61054 − 3.25094i 1.09341 + 0.355271i −0.104528 + 0.994522i 3.27037 − 5.66445i
4.3 −0.763963 − 1.71589i −0.669131 − 0.743145i −1.02237 + 1.13546i −1.07133 − 1.47456i −0.763963 + 1.71589i 1.14179 + 1.02807i −0.843308 − 0.274007i −0.104528 + 0.994522i −1.71172 + 2.96479i
4.4 −0.688193 − 1.54571i −0.669131 − 0.743145i −0.577339 + 0.641200i 1.57124 + 2.16263i −0.688193 + 1.54571i 1.25124 + 1.12663i −1.82992 − 0.594578i −0.104528 + 0.994522i 2.26148 − 3.91699i
4.5 −0.672174 − 1.50973i −0.669131 − 0.743145i −0.489197 + 0.543308i −2.52085 − 3.46966i −0.672174 + 1.50973i −2.16497 − 1.94935i −1.99436 − 0.648008i −0.104528 + 0.994522i −3.54378 + 6.13802i
4.6 −0.177986 − 0.399764i −0.669131 − 0.743145i 1.21013 − 1.34398i 0.0311778 + 0.0429126i −0.177986 + 0.399764i −0.860670 − 0.774951i −1.58502 − 0.515004i −0.104528 + 0.994522i 0.0116057 − 0.0201016i
4.7 −0.0803440 − 0.180456i −0.669131 − 0.743145i 1.31215 − 1.45729i 0.296705 + 0.408379i −0.0803440 + 0.180456i 3.31863 + 2.98811i −0.744131 − 0.241783i −0.104528 + 0.994522i 0.0498559 − 0.0863529i
4.8 0.250968 + 0.563684i −0.669131 − 0.743145i 1.08351 − 1.20336i −1.79971 − 2.47709i 0.250968 − 0.563684i −2.23596 − 2.01327i 2.12390 + 0.690096i −0.104528 + 0.994522i 0.944626 − 1.63614i
4.9 0.275867 + 0.619607i −0.669131 − 0.743145i 1.03045 − 1.14443i −1.77061 − 2.43703i 0.275867 − 0.619607i 3.22654 + 2.90519i 2.28346 + 0.741942i −0.104528 + 0.994522i 1.02155 − 1.76938i
4.10 0.386229 + 0.867484i −0.669131 − 0.743145i 0.734905 − 0.816195i 2.09746 + 2.88691i 0.386229 − 0.867484i 0.681566 + 0.613685i 2.79809 + 0.909153i −0.104528 + 0.994522i −1.69425 + 2.93453i
4.11 0.534181 + 1.19979i −0.669131 − 0.743145i 0.184114 − 0.204479i 1.16055 + 1.59736i 0.534181 − 1.19979i −2.62438 − 2.36301i 2.84179 + 0.923354i −0.104528 + 0.994522i −1.29655 + 2.24569i
4.12 0.883617 + 1.98464i −0.669131 − 0.743145i −1.81974 + 2.02103i −1.20025 − 1.65200i 0.883617 − 1.98464i −2.60068 − 2.34166i −1.48670 − 0.483058i −0.104528 + 0.994522i 2.21806 − 3.84180i
4.13 0.896442 + 2.01344i −0.669131 − 0.743145i −1.91208 + 2.12358i 0.00844073 + 0.0116177i 0.896442 − 2.01344i 1.50438 + 1.35455i −1.79754 − 0.584056i −0.104528 + 0.994522i −0.0158249 + 0.0274095i
4.14 1.08040 + 2.42662i −0.669131 − 0.743145i −3.38297 + 3.75717i 2.34307 + 3.22496i 1.08040 − 2.42662i 1.59982 + 1.44048i −7.71968 − 2.50828i −0.104528 + 0.994522i −5.29430 + 9.17000i
49.1 −1.97287 − 1.77638i 0.104528 − 0.994522i 0.527630 + 5.02007i 1.64057 + 0.533054i −1.97287 + 1.77638i −2.55155 + 0.268179i 4.75574 − 6.54571i −0.978148 − 0.207912i −2.28973 − 3.96592i
49.2 −1.80408 − 1.62440i 0.104528 − 0.994522i 0.406968 + 3.87204i −2.99760 − 0.973979i −1.80408 + 1.62440i 1.24847 − 0.131220i 2.70169 − 3.71856i −0.978148 − 0.207912i 3.82577 + 6.62643i
49.3 −1.54435 − 1.39054i 0.104528 − 0.994522i 0.242362 + 2.30592i 1.99851 + 0.649356i −1.54435 + 1.39054i 2.22364 − 0.233714i 0.389193 − 0.535678i −0.978148 − 0.207912i −2.18345 − 3.78184i
49.4 −1.38170 − 1.24409i 0.104528 − 0.994522i 0.152285 + 1.44889i −1.73448 − 0.563568i −1.38170 + 1.24409i −4.66758 + 0.490582i −0.593557 + 0.816960i −0.978148 − 0.207912i 1.69541 + 2.93654i
49.5 −1.04336 − 0.939450i 0.104528 − 0.994522i −0.00301284 − 0.0286652i 1.68655 + 0.547992i −1.04336 + 0.939450i −0.0876448 + 0.00921184i −1.67427 + 2.30444i −0.978148 − 0.207912i −1.24487 − 2.15618i
49.6 −0.280336 − 0.252415i 0.104528 − 0.994522i −0.194182 − 1.84752i −2.49942 − 0.812112i −0.280336 + 0.252415i −1.73582 + 0.182442i −0.855366 + 1.17731i −0.978148 − 0.207912i 0.495688 + 0.858557i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
13.e even 6 1 inner
143.u even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.bn.a ✓ 112
11.c even 5 1 inner 429.2.bn.a ✓ 112
13.e even 6 1 inner 429.2.bn.a ✓ 112
143.u even 30 1 inner 429.2.bn.a ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bn.a ✓ 112 1.a even 1 1 trivial
429.2.bn.a ✓ 112 11.c even 5 1 inner
429.2.bn.a ✓ 112 13.e even 6 1 inner
429.2.bn.a ✓ 112 143.u even 30 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{112} + 24 T_{2}^{110} + 247 T_{2}^{108} + 6 T_{2}^{107} + 1054 T_{2}^{106} - 96 T_{2}^{105} + \cdots + 9150625 \) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\). Copy content Toggle raw display