# Properties

 Label 471.2.l.b Level $471$ Weight $2$ Character orbit 471.l Analytic conductor $3.761$ Analytic rank $0$ Dimension $200$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$471 = 3 \cdot 157$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 471.l (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.76095393520$$ Analytic rank: $$0$$ Dimension: $$200$$ Relative dimension: $$50$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$200 q - 6 q^{3} - 10 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$200 q - 6 q^{3} - 10 q^{9} - 36 q^{10} - 16 q^{12} - 24 q^{13} + 8 q^{15} - 208 q^{16} + 8 q^{18} - 12 q^{19} - 24 q^{22} - 20 q^{24} - 24 q^{25} + 64 q^{28} - 14 q^{30} - 48 q^{31} + 18 q^{33} + 32 q^{34} + 78 q^{36} + 44 q^{37} - 72 q^{39} + 60 q^{40} + 42 q^{42} - 24 q^{43} - 32 q^{45} - 32 q^{46} - 102 q^{48} - 54 q^{51} + 36 q^{52} + 48 q^{54} - 20 q^{55} - 6 q^{57} - 8 q^{61} - 8 q^{63} - 32 q^{66} - 16 q^{67} - 52 q^{69} + 60 q^{70} - 42 q^{72} - 64 q^{73} - 32 q^{75} + 132 q^{76} + 106 q^{78} + 40 q^{79} + 14 q^{81} + 172 q^{84} - 160 q^{85} - 88 q^{87} + 8 q^{88} + 84 q^{91} + 220 q^{93} + 128 q^{94} - 108 q^{96} - 36 q^{97} - 104 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
50.1 −1.96449 + 1.96449i 1.24725 1.20182i 5.71848i 0.309034 + 1.15333i −0.0892417 + 4.81118i 0.541767 0.541767i 7.30492 + 7.30492i 0.111255 2.99794i −2.87281 1.65862i
50.2 −1.93791 + 1.93791i 0.655068 + 1.60340i 5.51098i −1.00937 3.76703i −4.37670 1.83778i −0.357569 + 0.357569i 6.80395 + 6.80395i −2.14177 + 2.10067i 9.25622 + 5.34408i
50.3 −1.82955 + 1.82955i −1.72845 0.111624i 4.69449i 0.00122739 + 0.00458067i 3.36650 2.95806i 3.09988 3.09988i 4.92970 + 4.92970i 2.97508 + 0.385872i −0.0106261 0.00613499i
50.4 −1.73825 + 1.73825i 1.29972 + 1.14486i 4.04305i 0.867835 + 3.23881i −4.24932 + 0.269185i −2.31299 + 2.31299i 3.55133 + 3.55133i 0.378569 + 2.97602i −7.13838 4.12135i
50.5 −1.70954 + 1.70954i −1.10161 + 1.33658i 3.84504i 0.205161 + 0.765672i −0.401690 4.16818i −1.76931 + 1.76931i 3.15416 + 3.15416i −0.572902 2.94479i −1.65968 0.958215i
50.6 −1.59659 + 1.59659i −1.63950 0.558616i 3.09822i 0.704955 + 2.63093i 3.50949 1.72573i −1.39752 + 1.39752i 1.75342 + 1.75342i 2.37590 + 1.83170i −5.32605 3.07500i
50.7 −1.59610 + 1.59610i −0.929471 1.46153i 3.09507i −1.03956 3.87969i 3.81628 + 0.849227i 1.69458 1.69458i 1.74784 + 1.74784i −1.27217 + 2.71691i 7.85161 + 4.53313i
50.8 −1.52089 + 1.52089i −0.258431 1.71266i 2.62620i 0.679058 + 2.53428i 2.99781 + 2.21172i 1.37133 1.37133i 0.952382 + 0.952382i −2.86643 + 0.885212i −4.88713 2.82159i
50.9 −1.47058 + 1.47058i 1.73106 + 0.0586068i 2.32524i −0.337319 1.25889i −2.63185 + 2.45948i 0.411952 0.411952i 0.478287 + 0.478287i 2.99313 + 0.202904i 2.34736 + 1.35525i
50.10 −1.43787 + 1.43787i 1.07451 1.35846i 2.13496i −0.618816 2.30945i 0.408280 + 3.49831i −2.17897 + 2.17897i 0.194052 + 0.194052i −0.690836 2.91937i 4.21048 + 2.43092i
50.11 −1.33901 + 1.33901i −0.825865 + 1.52248i 1.58591i −0.378714 1.41338i −0.932777 3.14447i 1.10359 1.10359i −0.554467 0.554467i −1.63589 2.51473i 2.39964 + 1.38543i
50.12 −1.19074 + 1.19074i 0.685591 + 1.59059i 0.835708i 0.604819 + 2.25721i −2.71033 1.07761i 0.837234 0.837234i −1.38637 1.38637i −2.05993 + 2.18098i −3.40793 1.96757i
50.13 −1.17584 + 1.17584i −1.72956 + 0.0927798i 0.765185i −0.933382 3.48343i 1.92459 2.14278i −2.60917 + 2.60917i −1.45194 1.45194i 2.98278 0.320937i 5.19345 + 2.99844i
50.14 −1.00506 + 1.00506i 0.156162 1.72500i 0.0202720i 0.729968 + 2.72428i 1.57677 + 1.89067i −2.45817 + 2.45817i −1.98974 1.98974i −2.95123 0.538757i −3.47171 2.00439i
50.15 −0.980409 + 0.980409i 1.60565 0.649523i 0.0775977i 0.853262 + 3.18442i −0.937397 + 2.21099i 2.96274 2.96274i −2.03689 2.03689i 2.15624 2.08582i −3.95858 2.28549i
50.16 −0.850678 + 0.850678i −1.36219 1.06978i 0.552695i 0.0315701 + 0.117821i 2.06883 0.248744i 0.641782 0.641782i −2.17152 2.17152i 0.711127 + 2.91450i −0.127084 0.0733718i
50.17 −0.668208 + 0.668208i 1.73131 + 0.0507295i 1.10700i 0.363693 + 1.35732i −1.19077 + 1.12298i −2.84247 + 2.84247i −2.07612 2.07612i 2.99485 + 0.175657i −1.14999 0.663950i
50.18 −0.648600 + 0.648600i 0.734999 1.56837i 1.15864i −0.523398 1.95335i 0.540523 + 1.49396i 3.32511 3.32511i −2.04869 2.04869i −1.91955 2.30550i 1.60642 + 0.927465i
50.19 −0.640983 + 0.640983i 0.167689 + 1.72391i 1.17828i 0.280265 + 1.04596i −1.21249 0.997514i 0.268331 0.268331i −2.03722 2.03722i −2.94376 + 0.578164i −0.850090 0.490800i
50.20 −0.513152 + 0.513152i −1.60345 + 0.654944i 1.47335i −0.0432502 0.161412i 0.486727 1.15890i 2.54991 2.54991i −1.78236 1.78236i 2.14210 2.10034i 0.105023 + 0.0606349i
See next 80 embeddings (of 200 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.50 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
157.f odd 12 1 inner
471.l even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 471.2.l.b 200
3.b odd 2 1 inner 471.2.l.b 200
157.f odd 12 1 inner 471.2.l.b 200
471.l even 12 1 inner 471.2.l.b 200

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.2.l.b 200 1.a even 1 1 trivial
471.2.l.b 200 3.b odd 2 1 inner
471.2.l.b 200 157.f odd 12 1 inner
471.2.l.b 200 471.l even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$13\!\cdots\!40$$$$T_{2}^{180} +$$$$12\!\cdots\!82$$$$T_{2}^{176} +$$$$10\!\cdots\!08$$$$T_{2}^{172} +$$$$66\!\cdots\!66$$$$T_{2}^{168} +$$$$36\!\cdots\!16$$$$T_{2}^{164} +$$$$16\!\cdots\!06$$$$T_{2}^{160} +$$$$66\!\cdots\!50$$$$T_{2}^{156} +$$$$22\!\cdots\!69$$$$T_{2}^{152} +$$$$65\!\cdots\!26$$$$T_{2}^{148} +$$$$16\!\cdots\!40$$$$T_{2}^{144} +$$$$36\!\cdots\!14$$$$T_{2}^{140} +$$$$68\!\cdots\!10$$$$T_{2}^{136} +$$$$11\!\cdots\!46$$$$T_{2}^{132} +$$$$15\!\cdots\!04$$$$T_{2}^{128} +$$$$19\!\cdots\!98$$$$T_{2}^{124} +$$$$21\!\cdots\!13$$$$T_{2}^{120} +$$$$19\!\cdots\!76$$$$T_{2}^{116} +$$$$15\!\cdots\!47$$$$T_{2}^{112} +$$$$10\!\cdots\!94$$$$T_{2}^{108} +$$$$63\!\cdots\!31$$$$T_{2}^{104} +$$$$32\!\cdots\!74$$$$T_{2}^{100} +$$$$14\!\cdots\!84$$$$T_{2}^{96} +$$$$51\!\cdots\!42$$$$T_{2}^{92} +$$$$15\!\cdots\!29$$$$T_{2}^{88} +$$$$41\!\cdots\!18$$$$T_{2}^{84} +$$$$88\!\cdots\!53$$$$T_{2}^{80} +$$$$15\!\cdots\!22$$$$T_{2}^{76} +$$$$23\!\cdots\!37$$$$T_{2}^{72} +$$$$27\!\cdots\!84$$$$T_{2}^{68} +$$$$26\!\cdots\!53$$$$T_{2}^{64} +$$$$20\!\cdots\!52$$$$T_{2}^{60} +$$$$12\!\cdots\!14$$$$T_{2}^{56} +$$$$58\!\cdots\!56$$$$T_{2}^{52} +$$$$21\!\cdots\!94$$$$T_{2}^{48} +$$$$58\!\cdots\!78$$$$T_{2}^{44} +$$$$11\!\cdots\!43$$$$T_{2}^{40} +$$$$17\!\cdots\!84$$$$T_{2}^{36} +$$$$17\!\cdots\!04$$$$T_{2}^{32} +$$$$12\!\cdots\!94$$$$T_{2}^{28} +$$$$49\!\cdots\!96$$$$T_{2}^{24} +$$$$10\!\cdots\!48$$$$T_{2}^{20} +$$$$77\!\cdots\!65$$$$T_{2}^{16} +$$$$21\!\cdots\!24$$$$T_{2}^{12} +$$$$19\!\cdots\!20$$$$T_{2}^{8} +$$$$35\!\cdots\!72$$$$T_{2}^{4} +$$$$92\!\cdots\!84$$">$$T_{2}^{200} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(471, [\chi])$$.