# Properties

 Label 336.4.bc.f Level $336$ Weight $4$ Character orbit 336.bc Analytic conductor $19.825$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 336.bc (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.8246417619$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 168) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q - 12q^{7} + 14q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 12q^{7} + 14q^{9} + 88q^{15} + 270q^{19} + 50q^{21} - 438q^{25} - 216q^{31} - 372q^{33} + 66q^{37} - 242q^{39} - 900q^{43} - 294q^{45} + 60q^{49} + 138q^{51} + 1384q^{57} + 108q^{61} - 1096q^{63} - 6q^{67} - 1206q^{73} + 594q^{75} + 588q^{79} - 54q^{81} - 240q^{85} + 3522q^{87} - 234q^{91} - 608q^{93} - 1988q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 0 −5.18895 0.273513i 0 2.97936 + 5.16040i 0 17.8140 5.06584i 0 26.8504 + 2.83849i 0
17.2 0 −5.12717 0.843898i 0 −8.29692 14.3707i 0 −9.36640 15.9772i 0 25.5757 + 8.65361i 0
17.3 0 −4.99832 1.42013i 0 −6.11124 10.5850i 0 −4.39164 + 17.9920i 0 22.9665 + 14.1965i 0
17.4 0 −4.88874 + 1.76075i 0 6.04693 + 10.4736i 0 −15.9632 9.39028i 0 20.7995 17.2157i 0
17.5 0 −3.72903 3.61861i 0 6.11124 + 10.5850i 0 −4.39164 + 17.9920i 0 0.811307 + 26.9878i 0
17.6 0 −3.62731 + 3.72057i 0 0.263312 + 0.456070i 0 16.6747 + 8.05944i 0 −0.685212 26.9913i 0
17.7 0 −3.29442 4.01831i 0 8.29692 + 14.3707i 0 −9.36640 15.9772i 0 −5.29359 + 26.4760i 0
17.8 0 −3.24856 + 4.05547i 0 3.20701 + 5.55470i 0 −14.8103 + 11.1200i 0 −5.89369 26.3489i 0
17.9 0 −2.83134 4.35701i 0 −2.97936 5.16040i 0 17.8140 5.06584i 0 −10.9670 + 24.6724i 0
17.10 0 −2.36622 + 4.62612i 0 −9.22876 15.9847i 0 6.70316 17.2646i 0 −15.8020 21.8928i 0
17.11 0 −0.919514 5.11415i 0 −6.04693 10.4736i 0 −15.9632 9.39028i 0 −25.3090 + 9.40505i 0
17.12 0 0.328966 + 5.18573i 0 −7.91382 13.7071i 0 −13.3812 + 12.8040i 0 −26.7836 + 3.41186i 0
17.13 0 0.726622 + 5.14510i 0 9.08545 + 15.7365i 0 18.4348 1.77707i 0 −25.9440 + 7.47709i 0
17.14 0 1.35065 + 5.01754i 0 2.40532 + 4.16613i 0 3.40309 18.2049i 0 −23.3515 + 13.5539i 0
17.15 0 1.40845 5.00163i 0 −0.263312 0.456070i 0 16.6747 + 8.05944i 0 −23.0325 14.0891i 0
17.16 0 1.88786 4.84107i 0 −3.20701 5.55470i 0 −14.8103 + 11.1200i 0 −19.8720 18.2785i 0
17.17 0 2.82323 4.36227i 0 9.22876 + 15.9847i 0 6.70316 17.2646i 0 −11.0587 24.6314i 0
17.18 0 3.06041 + 4.19928i 0 0.746155 + 1.29238i 0 −15.9029 9.49205i 0 −8.26784 + 25.7030i 0
17.19 0 4.27838 + 2.94880i 0 −5.57507 9.65631i 0 7.78587 + 16.8042i 0 9.60915 + 25.2322i 0
17.20 0 4.65546 2.30797i 0 7.91382 + 13.7071i 0 −13.3812 + 12.8040i 0 16.3465 21.4893i 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 257.24 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
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.bc.f 48
3.b odd 2 1 inner 336.4.bc.f 48
4.b odd 2 1 168.4.u.a 48
7.d odd 6 1 inner 336.4.bc.f 48
12.b even 2 1 168.4.u.a 48
21.g even 6 1 inner 336.4.bc.f 48
28.f even 6 1 168.4.u.a 48
84.j odd 6 1 168.4.u.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.u.a 48 4.b odd 2 1
168.4.u.a 48 12.b even 2 1
168.4.u.a 48 28.f even 6 1
168.4.u.a 48 84.j odd 6 1
336.4.bc.f 48 1.a even 1 1 trivial
336.4.bc.f 48 3.b odd 2 1 inner
336.4.bc.f 48 7.d odd 6 1 inner
336.4.bc.f 48 21.g even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(336, [\chi])$$:

 $$22\!\cdots\!98$$$$T_{5}^{38} +$$$$72\!\cdots\!98$$$$T_{5}^{36} +$$$$18\!\cdots\!92$$$$T_{5}^{34} +$$$$40\!\cdots\!63$$$$T_{5}^{32} +$$$$71\!\cdots\!01$$$$T_{5}^{30} +$$$$10\!\cdots\!95$$$$T_{5}^{28} +$$$$12\!\cdots\!88$$$$T_{5}^{26} +$$$$12\!\cdots\!70$$$$T_{5}^{24} +$$$$94\!\cdots\!74$$$$T_{5}^{22} +$$$$58\!\cdots\!46$$$$T_{5}^{20} +$$$$28\!\cdots\!68$$$$T_{5}^{18} +$$$$10\!\cdots\!57$$$$T_{5}^{16} +$$$$29\!\cdots\!75$$$$T_{5}^{14} +$$$$59\!\cdots\!69$$$$T_{5}^{12} +$$$$81\!\cdots\!48$$$$T_{5}^{10} +$$$$70\!\cdots\!60$$$$T_{5}^{8} +$$$$16\!\cdots\!44$$$$T_{5}^{6} +$$$$29\!\cdots\!80$$$$T_{5}^{4} +$$$$81\!\cdots\!84$$$$T_{5}^{2} +$$$$19\!\cdots\!16$$">$$T_{5}^{48} + \cdots$$ $$19\!\cdots\!07$$$$T_{13}^{18} +$$$$70\!\cdots\!32$$$$T_{13}^{16} +$$$$15\!\cdots\!68$$$$T_{13}^{14} +$$$$21\!\cdots\!68$$$$T_{13}^{12} +$$$$17\!\cdots\!56$$$$T_{13}^{10} +$$$$78\!\cdots\!56$$$$T_{13}^{8} +$$$$19\!\cdots\!04$$$$T_{13}^{6} +$$$$24\!\cdots\!92$$$$T_{13}^{4} +$$$$12\!\cdots\!60$$$$T_{13}^{2} +$$$$13\!\cdots\!64$$">$$T_{13}^{24} + \cdots$$