Properties

Label 429.2.bv.a
Level $429$
Weight $2$
Character orbit 429.bv
Analytic conductor $3.426$
Analytic rank $0$
Dimension $832$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.bv (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.42558224671\)
Analytic rank: \(0\)
Dimension: \(832\)
Relative dimension: \(52\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 832q - 6q^{3} - 36q^{4} - 4q^{6} - 24q^{7} - 6q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 832q - 6q^{3} - 36q^{4} - 4q^{6} - 24q^{7} - 6q^{9} - 96q^{10} - 40q^{13} + 12q^{15} - 68q^{16} - 20q^{18} - 32q^{19} - 84q^{21} - 24q^{22} - 72q^{24} - 114q^{30} - 8q^{31} + 14q^{33} - 24q^{34} + 78q^{36} - 24q^{37} - 72q^{39} - 112q^{40} + 2q^{42} - 96q^{43} + 4q^{45} - 16q^{46} + 14q^{48} - 36q^{49} + 8q^{52} + 32q^{54} - 24q^{55} - 12q^{57} + 112q^{58} - 120q^{60} + 12q^{61} + 28q^{63} - 300q^{66} - 32q^{67} - 18q^{69} - 140q^{70} + 204q^{72} - 24q^{73} + 66q^{75} + 176q^{76} - 28q^{78} - 224q^{79} - 58q^{81} - 264q^{82} + 20q^{84} - 168q^{85} + 72q^{87} - 120q^{88} - 146q^{93} - 4q^{94} - 10q^{96} - 88q^{97} + 40q^{99} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1 −2.76683 0.145003i 1.66622 + 0.472982i 5.64525 + 0.593340i −0.523203 0.266585i −4.54156 1.55027i 0.0260847 + 0.0322120i −10.0604 1.59341i 2.55258 + 1.57618i 1.40895 + 0.813460i
20.2 −2.62774 0.137714i −0.224523 + 1.71744i 4.89702 + 0.514697i 1.89968 + 0.967935i 0.826504 4.48206i −0.815502 1.00706i −7.59931 1.20361i −2.89918 0.771208i −4.85857 2.80510i
20.3 −2.50925 0.131504i 0.691538 1.58801i 4.29002 + 0.450899i −2.60114 1.32535i −1.94407 + 3.89378i −2.59163 3.20040i −5.74193 0.909433i −2.04355 2.19634i 6.35263 + 3.66769i
20.4 −2.37518 0.124478i −1.72714 0.130360i 3.63696 + 0.382260i 1.22228 + 0.622785i 4.08604 + 0.524621i −1.12395 1.38796i −3.89254 0.616518i 2.96601 + 0.450301i −2.82563 1.63138i
20.5 −2.28988 0.120007i 1.09330 1.34339i 3.24009 + 0.340548i 3.86458 + 1.96910i −2.66474 + 2.94499i −1.55450 1.91965i −2.84897 0.451233i −0.609383 2.93746i −8.61310 4.97278i
20.6 −2.28609 0.119809i −0.750909 1.56081i 3.22281 + 0.338731i 0.125105 + 0.0637444i 1.52965 + 3.65812i 0.00715252 + 0.00883262i −2.80496 0.444262i −1.87227 + 2.34406i −0.278365 0.160714i
20.7 −2.16363 0.113391i −1.53088 + 0.810191i 2.67939 + 0.281615i 2.44832 + 1.24748i 3.40412 1.57936i 2.26128 + 2.79244i −1.48541 0.235267i 1.68718 2.48061i −5.15579 2.97670i
20.8 −2.15849 0.113122i 1.68832 0.386733i 2.65724 + 0.279288i 0.408417 + 0.208099i −3.68798 + 0.643774i 2.50194 + 3.08964i −1.43436 0.227180i 2.70088 1.30586i −0.858025 0.495381i
20.9 −2.11349 0.110763i −0.0744693 + 1.73045i 2.46552 + 0.259137i −1.61438 0.822569i 0.349060 3.64904i 0.621753 + 0.767801i −1.00148 0.158619i −2.98891 0.257731i 3.32087 + 1.91730i
20.10 −1.99540 0.104575i 1.06644 + 1.36481i 1.98165 + 0.208280i −2.39161 1.21859i −1.98525 2.83487i −0.280750 0.346697i 0.0146712 + 0.00232369i −0.725421 + 2.91097i 4.64480 + 2.68168i
20.11 −1.79980 0.0943236i −1.44075 0.961374i 1.24134 + 0.130471i −2.65618 1.35339i 2.50238 + 1.86618i 1.02884 + 1.27051i 1.33830 + 0.211966i 1.15152 + 2.77020i 4.65295 + 2.68638i
20.12 −1.53416 0.0804022i 1.22300 + 1.22648i 0.358152 + 0.0376433i 3.02311 + 1.54035i −1.77768 1.97996i 2.44154 + 3.01505i 2.48827 + 0.394104i −0.00852784 + 2.99999i −4.51410 2.60622i
20.13 −1.49304 0.0782468i −1.51095 + 0.846781i 0.233997 + 0.0245940i −2.86498 1.45978i 2.32216 1.14605i −1.76998 2.18574i 2.60592 + 0.412737i 1.56592 2.55888i 4.16330 + 2.40368i
20.14 −1.40019 0.0733811i −0.460922 1.66960i −0.0338826 0.00356121i 2.36606 + 1.20557i 0.522864 + 2.37158i 0.593855 + 0.733350i 2.81689 + 0.446151i −2.57510 + 1.53911i −3.22448 1.86166i
20.15 −1.36547 0.0715611i 0.862643 + 1.50195i −0.129662 0.0136280i 1.20967 + 0.616356i −1.07043 2.11259i −2.59780 3.20801i 2.87709 + 0.455686i −1.51169 + 2.59129i −1.60765 0.928179i
20.16 −1.33300 0.0698593i 1.66414 0.480245i −0.217048 0.0228127i −0.706650 0.360056i −2.25184 + 0.523909i −1.22329 1.51064i 2.92451 + 0.463197i 2.53873 1.59839i 0.916807 + 0.529319i
20.17 −1.20280 0.0630360i −1.28110 + 1.16567i −0.546295 0.0574179i −0.505038 0.257330i 1.61438 1.32131i 1.25810 + 1.55363i 3.03270 + 0.480333i 0.282417 2.98668i 0.591237 + 0.341351i
20.18 −1.03397 0.0541882i 1.05441 1.37413i −0.922880 0.0969986i 0.459701 + 0.234229i −1.16469 + 1.36367i −0.356420 0.440142i 2.99427 + 0.474245i −0.776456 2.89778i −0.462625 0.267097i
20.19 −1.03370 0.0541738i −1.71154 0.265772i −0.923448 0.0970583i 2.13376 + 1.08720i 1.75482 + 0.367448i −2.13395 2.63521i 2.99405 + 0.474211i 2.85873 + 0.909757i −2.14676 1.23943i
20.20 −0.719787 0.0377225i −0.704391 1.58235i −1.47237 0.154753i −2.50517 1.27645i 0.447321 + 1.16553i −3.13449 3.87078i 2.47776 + 0.392439i −2.00767 + 2.22919i 1.75504 + 1.01327i
See next 80 embeddings (of 832 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 422.52
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.c even 5 1 inner
13.f odd 12 1 inner
33.h odd 10 1 inner
39.k even 12 1 inner
143.x odd 60 1 inner
429.bv even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.bv.a 832
3.b odd 2 1 inner 429.2.bv.a 832
11.c even 5 1 inner 429.2.bv.a 832
13.f odd 12 1 inner 429.2.bv.a 832
33.h odd 10 1 inner 429.2.bv.a 832
39.k even 12 1 inner 429.2.bv.a 832
143.x odd 60 1 inner 429.2.bv.a 832
429.bv even 60 1 inner 429.2.bv.a 832
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bv.a 832 1.a even 1 1 trivial
429.2.bv.a 832 3.b odd 2 1 inner
429.2.bv.a 832 11.c even 5 1 inner
429.2.bv.a 832 13.f odd 12 1 inner
429.2.bv.a 832 33.h odd 10 1 inner
429.2.bv.a 832 39.k even 12 1 inner
429.2.bv.a 832 143.x odd 60 1 inner
429.2.bv.a 832 429.bv even 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(429, [\chi])\).