# Properties

 Label 429.2.be.a Level $429$ Weight $2$ Character orbit 429.be Analytic conductor $3.426$ Analytic rank $0$ Dimension $184$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.be (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$184$$ Relative dimension: $$46$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$184q - 12q^{6} + 12q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$184q - 12q^{6} + 12q^{7} - 48q^{10} - 16q^{13} + 16q^{15} + 80q^{16} - 8q^{18} - 4q^{19} - 24q^{24} - 48q^{27} - 40q^{28} + 48q^{30} - 20q^{31} + 16q^{34} - 36q^{36} + 44q^{37} + 48q^{39} - 80q^{40} + 20q^{42} - 84q^{43} - 4q^{45} - 64q^{46} + 44q^{48} - 60q^{49} - 200q^{52} - 4q^{54} - 64q^{57} - 48q^{58} - 148q^{60} - 48q^{61} + 40q^{66} + 48q^{67} - 12q^{69} + 24q^{70} - 128q^{72} + 108q^{73} - 60q^{75} - 24q^{76} + 148q^{78} + 32q^{79} + 16q^{81} - 48q^{82} + 116q^{84} + 104q^{85} - 24q^{87} + 72q^{88} + 60q^{91} + 36q^{93} + 16q^{94} - 72q^{96} + 4q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
89.1 −2.56597 0.687549i 0.227829 1.71700i 4.37942 + 2.52846i 2.70172 2.70172i −1.76513 + 4.24913i 0.243586 + 0.909075i −5.74218 5.74218i −2.89619 0.782365i −8.79010 + 5.07497i
89.2 −2.48352 0.665458i −1.51130 0.846155i 3.99300 + 2.30536i −0.0572884 + 0.0572884i 3.19026 + 3.10715i −0.793036 2.95965i −4.74647 4.74647i 1.56804 + 2.55758i 0.180400 0.104154i
89.3 −2.44763 0.655840i 1.31899 1.12261i 3.82870 + 2.21050i −1.62249 + 1.62249i −3.96466 + 1.88269i 0.335823 + 1.25331i −4.33792 4.33792i 0.479486 2.96143i 5.03534 2.90715i
89.4 −2.43755 0.653139i 0.235409 + 1.71598i 3.78299 + 2.18411i 1.30401 1.30401i 0.546952 4.33653i −1.18861 4.43596i −4.22588 4.22588i −2.88917 + 0.807913i −4.03028 + 2.32688i
89.5 −2.39352 0.641342i 1.64395 + 0.545381i 3.58556 + 2.07013i −0.0323011 + 0.0323011i −3.58504 2.35971i 0.0457503 + 0.170743i −3.75010 3.75010i 2.40512 + 1.79315i 0.0980293 0.0565972i
89.6 −2.25191 0.603398i −1.65469 + 0.511873i 2.97497 + 1.71760i 2.28764 2.28764i 4.03507 0.154258i 0.850397 + 3.17372i −2.36594 2.36594i 2.47597 1.69398i −6.53192 + 3.77121i
89.7 −2.20878 0.591842i −1.22709 1.22240i 2.79640 + 1.61450i −2.05423 + 2.05423i 1.98690 + 3.42626i 0.902198 + 3.36705i −1.98722 1.98722i 0.0114791 + 2.99998i 5.75313 3.32157i
89.8 −1.99225 0.533822i −1.33180 + 1.10738i 1.95204 + 1.12701i −0.191278 + 0.191278i 3.24443 1.49524i −0.125579 0.468666i −0.370477 0.370477i 0.547407 2.94963i 0.483182 0.278965i
89.9 −1.76077 0.471796i 1.54331 + 0.786256i 1.14566 + 0.661445i 1.75250 1.75250i −2.34645 2.11254i 0.531771 + 1.98460i 0.872774 + 0.872774i 1.76360 + 2.42687i −3.91258 + 2.25893i
89.10 −1.72773 0.462944i 0.702909 + 1.58301i 1.03868 + 0.599682i −2.41452 + 2.41452i −0.481591 3.06042i −0.301061 1.12357i 1.01263 + 1.01263i −2.01184 + 2.22542i 5.28942 3.05385i
89.11 −1.52845 0.409546i −0.446060 1.67363i 0.436367 + 0.251936i −0.731227 + 0.731227i −0.00364908 + 2.74073i −0.466780 1.74204i 1.67402 + 1.67402i −2.60206 + 1.49308i 1.41711 0.818170i
89.12 −1.52005 0.407295i 1.56613 0.739746i 0.412602 + 0.238216i 1.90285 1.90285i −2.68189 + 0.486569i −1.32203 4.93388i 1.69535 + 1.69535i 1.90555 2.31708i −3.66744 + 2.11740i
89.13 −1.50551 0.403401i 0.00525696 + 1.73204i 0.371782 + 0.214648i −0.722785 + 0.722785i 0.690793 2.60973i 1.16889 + 4.36236i 1.73109 + 1.73109i −2.99994 + 0.0182105i 1.37973 0.796589i
89.14 −1.32901 0.356107i 1.71883 0.213571i −0.0925961 0.0534604i −2.48553 + 2.48553i −2.36040 0.328250i −0.825366 3.08031i 2.04983 + 2.04983i 2.90877 0.734187i 4.18841 2.41818i
89.15 −1.22945 0.329429i −1.72326 0.174256i −0.329037 0.189970i 0.992009 0.992009i 2.06125 + 0.781932i −0.778711 2.90619i 2.14199 + 2.14199i 2.93927 + 0.600579i −1.54642 + 0.892825i
89.16 −0.954637 0.255794i 1.35903 1.07380i −0.886149 0.511619i −1.59904 + 1.59904i −1.57205 + 0.677454i 0.743538 + 2.77492i 2.11277 + 2.11277i 0.693924 2.91864i 1.93553 1.11748i
89.17 −0.810471 0.217165i −1.43084 0.976061i −1.12235 0.647988i 1.96327 1.96327i 0.947689 + 1.10180i 1.13192 + 4.22438i 1.95552 + 1.95552i 1.09461 + 2.79318i −2.01753 + 1.16482i
89.18 −0.781729 0.209464i 0.746974 + 1.56270i −1.16483 0.672513i 2.62100 2.62100i −0.256602 1.37807i 0.0507104 + 0.189254i 1.91424 + 1.91424i −1.88406 + 2.33459i −2.59791 + 1.49990i
89.19 −0.745893 0.199861i −0.815569 + 1.52802i −1.21564 0.701849i −0.458199 + 0.458199i 0.913720 0.976739i −0.672164 2.50855i 1.85853 + 1.85853i −1.66969 2.49241i 0.433344 0.250191i
89.20 −0.283022 0.0758356i −1.01211 1.40557i −1.65770 0.957074i −2.69708 + 2.69708i 0.179857 + 0.474562i −0.867145 3.23623i 0.810959 + 0.810959i −0.951270 + 2.84519i 0.967866 0.558798i
See next 80 embeddings (of 184 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 353.46 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.f odd 12 1 inner
39.k even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.be.a 184
3.b odd 2 1 inner 429.2.be.a 184
13.f odd 12 1 inner 429.2.be.a 184
39.k even 12 1 inner 429.2.be.a 184

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.be.a 184 1.a even 1 1 trivial
429.2.be.a 184 3.b odd 2 1 inner
429.2.be.a 184 13.f odd 12 1 inner
429.2.be.a 184 39.k even 12 1 inner

## Hecke kernels

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