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Properties

Label 429.2.bg.b
Level $429$
Weight $2$
Character orbit 429.bg
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(16,429)]
 
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, 12, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("429.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.bg (of order \(15\), 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 \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 14 q^{3} + 8 q^{4} - 6 q^{5} - 24 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q + 14 q^{3} + 8 q^{4} - 6 q^{5} - 24 q^{8} + 14 q^{9} - 24 q^{10} + q^{11} + 104 q^{12} - 15 q^{13} - 30 q^{14} - 2 q^{15} + 24 q^{16} + 4 q^{17} - 28 q^{20} + 7 q^{22} + 8 q^{23} - 18 q^{24} - 42 q^{25} + 7 q^{26} - 28 q^{27} - 6 q^{28} - 2 q^{29} + 6 q^{30} + 12 q^{31} - 6 q^{32} + 6 q^{33} + 64 q^{34} - 8 q^{35} + 8 q^{36} + 4 q^{37} - 6 q^{38} + 12 q^{39} - 36 q^{40} - 26 q^{41} - 20 q^{42} - 76 q^{43} + 12 q^{44} - 2 q^{45} + 18 q^{46} + 50 q^{47} - 6 q^{48} + 32 q^{49} + 17 q^{50} - 18 q^{51} - 39 q^{52} - 54 q^{53} - 36 q^{55} + 12 q^{56} + 48 q^{58} + 20 q^{59} + 56 q^{60} + 34 q^{61} + 33 q^{62} - 68 q^{64} - 44 q^{65} - 4 q^{66} - 36 q^{67} + 4 q^{68} + 3 q^{69} - 92 q^{70} + 22 q^{71} - 18 q^{72} - 34 q^{73} - 4 q^{74} - 4 q^{75} + 4 q^{76} + 32 q^{77} - 18 q^{78} - 20 q^{79} + 30 q^{80} + 14 q^{81} - 41 q^{82} + 56 q^{83} + 9 q^{84} + 6 q^{85} - 86 q^{86} - 52 q^{87} - 70 q^{88} - 96 q^{89} - 12 q^{90} - 7 q^{91} - 126 q^{92} - 6 q^{93} - 10 q^{94} + 52 q^{95} - 88 q^{96} - 5 q^{97} + 104 q^{98} - 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −1.60050 + 1.77754i −0.104528 + 0.994522i −0.388975 − 3.70085i −1.06190 + 3.26819i −1.60050 − 1.77754i 0.204434 + 1.94506i 3.33076 + 2.41994i −0.978148 − 0.207912i −4.10976 − 7.11831i
16.2 −1.53199 + 1.70145i −0.104528 + 0.994522i −0.338874 − 3.22417i 0.485300 − 1.49360i −1.53199 − 1.70145i 0.0971269 + 0.924101i 2.30039 + 1.67133i −0.978148 − 0.207912i 1.79781 + 3.11390i
16.3 −1.22841 + 1.36429i −0.104528 + 0.994522i −0.143231 − 1.36275i −0.972545 + 2.99319i −1.22841 − 1.36429i −0.369323 − 3.51387i −0.935298 − 0.679534i −0.978148 − 0.207912i −2.88888 − 5.00368i
16.4 −1.12855 + 1.25338i −0.104528 + 0.994522i −0.0882831 − 0.839957i 0.636771 − 1.95978i −1.12855 − 1.25338i −0.222872 − 2.12049i −1.57654 − 1.14542i −0.978148 − 0.207912i 1.73772 + 3.00982i
16.5 −0.802761 + 0.891557i −0.104528 + 0.994522i 0.0586092 + 0.557629i 0.833954 − 2.56665i −0.802761 − 0.891557i 0.259398 + 2.46800i −2.48538 − 1.80573i −0.978148 − 0.207912i 1.61885 + 2.80392i
16.6 −0.471127 + 0.523240i −0.104528 + 0.994522i 0.157238 + 1.49602i −0.263224 + 0.810122i −0.471127 − 0.523240i 0.496767 + 4.72642i −1.99610 − 1.45025i −0.978148 − 0.207912i −0.299876 − 0.519400i
16.7 0.0112957 − 0.0125452i −0.104528 + 0.994522i 0.209027 + 1.98876i 1.04169 − 3.20599i 0.0112957 + 0.0125452i −0.462856 − 4.40378i 0.0546248 + 0.0396872i −0.978148 − 0.207912i −0.0284531 − 0.0492821i
16.8 0.0581605 − 0.0645938i −0.104528 + 0.994522i 0.208267 + 1.98153i −0.913448 + 2.81130i 0.0581605 + 0.0645938i −0.0279312 − 0.265748i 0.280746 + 0.203974i −0.978148 − 0.207912i 0.128466 + 0.222510i
16.9 0.433437 − 0.481381i −0.104528 + 0.994522i 0.165197 + 1.57175i −0.543246 + 1.67194i 0.433437 + 0.481381i 0.0903479 + 0.859603i 1.87631 + 1.36322i −0.978148 − 0.207912i 0.569377 + 0.986190i
16.10 0.682919 − 0.758458i −0.104528 + 0.994522i 0.100176 + 0.953113i 0.0526100 − 0.161917i 0.682919 + 0.758458i −0.171511 − 1.63181i 2.44268 + 1.77471i −0.978148 − 0.207912i −0.0868789 − 0.150479i
16.11 0.932419 − 1.03556i −0.104528 + 0.994522i 0.00608545 + 0.0578992i 0.554150 − 1.70550i 0.932419 + 1.03556i 0.357902 + 3.40521i 2.32033 + 1.68582i −0.978148 − 0.207912i −1.24944 − 2.16409i
16.12 1.38006 − 1.53272i −0.104528 + 0.994522i −0.235585 − 2.24144i −1.05420 + 3.24451i 1.38006 + 1.53272i 0.100634 + 0.957465i −0.423465 − 0.307666i −0.978148 − 0.207912i 3.51804 + 6.09342i
16.13 1.49760 − 1.66326i −0.104528 + 0.994522i −0.314550 − 2.99275i 0.954688 − 2.93823i 1.49760 + 1.66326i −0.278334 − 2.64817i −1.82740 − 1.32769i −0.978148 − 0.207912i −3.45728 − 5.98819i
16.14 1.76744 − 1.96294i −0.104528 + 0.994522i −0.520238 − 4.94974i 0.0584231 − 0.179808i 1.76744 + 1.96294i −0.0737832 − 0.702000i −6.36167 − 4.62202i −0.978148 − 0.207912i −0.249693 − 0.432481i
256.1 −0.265449 − 2.52558i −0.978148 − 0.207912i −4.35177 + 0.924997i −0.205708 − 0.149455i −0.265449 + 2.52558i −0.893413 + 0.189901i 1.92183 + 5.91479i 0.913545 + 0.406737i −0.322856 + 0.559203i
256.2 −0.245821 − 2.33883i −0.978148 − 0.207912i −3.45339 + 0.734041i 1.76043 + 1.27903i −0.245821 + 2.33883i 2.44940 − 0.520637i 1.11227 + 3.42322i 0.913545 + 0.406737i 2.55867 − 4.43175i
256.3 −0.190759 − 1.81495i −0.978148 − 0.207912i −1.30136 + 0.276613i −3.21129 − 2.33314i −0.190759 + 1.81495i 3.99809 − 0.849821i −0.377595 − 1.16212i 0.913545 + 0.406737i −3.62195 + 6.27341i
256.4 −0.176325 − 1.67762i −0.978148 − 0.207912i −0.827032 + 0.175791i 0.993113 + 0.721539i −0.176325 + 1.67762i −1.11199 + 0.236360i −0.601801 − 1.85215i 0.913545 + 0.406737i 1.03536 − 1.79330i
256.5 −0.121483 − 1.15584i −0.978148 − 0.207912i 0.635096 − 0.134994i −2.64351 − 1.92062i −0.121483 + 1.15584i −2.84394 + 0.604498i −0.951465 − 2.92831i 0.913545 + 0.406737i −1.89878 + 3.28879i
256.6 −0.0891258 − 0.847975i −0.978148 − 0.207912i 1.24518 − 0.264670i 1.84578 + 1.34104i −0.0891258 + 0.847975i −2.53940 + 0.539767i −0.862376 − 2.65412i 0.913545 + 0.406737i 0.972662 − 1.68470i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
13.c even 3 1 inner
143.q even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.bg.b ✓ 112
11.c even 5 1 inner 429.2.bg.b ✓ 112
13.c even 3 1 inner 429.2.bg.b ✓ 112
143.q even 15 1 inner 429.2.bg.b ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bg.b ✓ 112 1.a even 1 1 trivial
429.2.bg.b ✓ 112 11.c even 5 1 inner
429.2.bg.b ✓ 112 13.c even 3 1 inner
429.2.bg.b ✓ 112 143.q even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{112} - 18 T_{2}^{110} + 8 T_{2}^{109} + 101 T_{2}^{108} - 162 T_{2}^{107} + 652 T_{2}^{106} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\). Copy content Toggle raw display