# Properties

 Label 304.5.r.d Level $304$ Weight $5$ Character orbit 304.r Analytic conductor $31.424$ Analytic rank $0$ Dimension $40$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 304.r (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$31.4244687775$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$20$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 152) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40q - 12q^{3} + 32q^{7} + 624q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q - 12q^{3} + 32q^{7} + 624q^{9} - 24q^{11} + 264q^{13} - 624q^{15} + 216q^{17} + 652q^{19} - 216q^{21} + 1296q^{23} - 3044q^{25} + 288q^{29} - 6660q^{33} - 360q^{35} - 3184q^{39} + 1260q^{41} - 632q^{43} + 256q^{45} - 1248q^{47} + 16696q^{49} + 8064q^{51} - 3672q^{53} + 3408q^{55} - 4552q^{57} - 12492q^{59} + 2720q^{61} - 12472q^{63} - 16260q^{67} + 504q^{71} + 9220q^{73} - 14688q^{77} + 28944q^{79} - 1660q^{81} + 39192q^{83} - 18632q^{85} + 34400q^{87} + 3456q^{89} - 54432q^{91} - 17208q^{93} - 44520q^{95} - 30540q^{97} + 10096q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1 0 −14.4186 8.32460i 0 10.1950 17.6583i 0 −80.1603 0 98.0979 + 169.911i 0
65.2 0 −13.5253 7.80881i 0 14.4872 25.0925i 0 26.1842 0 81.4552 + 141.085i 0
65.3 0 −12.4606 7.19410i 0 −20.6420 + 35.7530i 0 −1.47481 0 63.0102 + 109.137i 0
65.4 0 −10.6119 6.12677i 0 3.27153 5.66646i 0 87.4966 0 34.5747 + 59.8851i 0
65.5 0 −9.48048 5.47356i 0 −15.3284 + 26.5496i 0 −33.8970 0 19.4197 + 33.6359i 0
65.6 0 −7.37289 4.25674i 0 7.14367 12.3732i 0 35.7108 0 −4.26032 7.37909i 0
65.7 0 −5.77348 3.33332i 0 −1.87999 + 3.25623i 0 40.7434 0 −18.2780 31.6584i 0
65.8 0 −5.09915 2.94399i 0 0.570982 0.988970i 0 −80.7369 0 −23.1658 40.1243i 0
65.9 0 −2.52583 1.45829i 0 22.1213 38.3151i 0 11.1904 0 −36.2468 62.7813i 0
65.10 0 −1.93870 1.11931i 0 −15.6195 + 27.0538i 0 6.74912 0 −37.9943 65.8080i 0
65.11 0 −0.620551 0.358275i 0 16.9339 29.3305i 0 −38.8265 0 −40.2433 69.7034i 0
65.12 0 0.962404 + 0.555644i 0 −13.4131 + 23.2322i 0 62.8585 0 −39.8825 69.0785i 0
65.13 0 5.28375 + 3.05057i 0 1.28449 2.22480i 0 6.16432 0 −21.8880 37.9111i 0
65.14 0 5.59346 + 3.22938i 0 −4.68830 + 8.12037i 0 −33.5286 0 −19.6422 34.0212i 0
65.15 0 7.81020 + 4.50922i 0 15.3964 26.6673i 0 70.8202 0 0.166103 + 0.287698i 0
65.16 0 9.62676 + 5.55801i 0 −23.6855 + 41.0244i 0 −82.2157 0 21.2830 + 36.8632i 0
65.17 0 10.7629 + 6.21398i 0 20.5828 35.6505i 0 −44.5677 0 36.7272 + 63.6134i 0
65.18 0 11.3222 + 6.53690i 0 −19.8361 + 34.3572i 0 76.8642 0 44.9621 + 77.8766i 0
65.19 0 12.1979 + 7.04247i 0 0.573898 0.994020i 0 40.4604 0 58.6927 + 101.659i 0
65.20 0 14.2678 + 8.23749i 0 2.53183 4.38526i 0 −53.8347 0 95.2125 + 164.913i 0
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 145.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.5.r.d 40
4.b odd 2 1 152.5.n.a 40
19.d odd 6 1 inner 304.5.r.d 40
76.f even 6 1 152.5.n.a 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.5.n.a 40 4.b odd 2 1
152.5.n.a 40 76.f even 6 1
304.5.r.d 40 1.a even 1 1 trivial
304.5.r.d 40 19.d odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$10\!\cdots\!92$$$$T_{3}^{31} -$$$$20\!\cdots\!28$$$$T_{3}^{30} -$$$$26\!\cdots\!80$$$$T_{3}^{29} +$$$$39\!\cdots\!58$$$$T_{3}^{28} +$$$$50\!\cdots\!56$$$$T_{3}^{27} -$$$$60\!\cdots\!28$$$$T_{3}^{26} -$$$$77\!\cdots\!64$$$$T_{3}^{25} +$$$$74\!\cdots\!21$$$$T_{3}^{24} +$$$$95\!\cdots\!44$$$$T_{3}^{23} -$$$$71\!\cdots\!70$$$$T_{3}^{22} -$$$$95\!\cdots\!00$$$$T_{3}^{21} +$$$$54\!\cdots\!45$$$$T_{3}^{20} +$$$$76\!\cdots\!28$$$$T_{3}^{19} -$$$$30\!\cdots\!34$$$$T_{3}^{18} -$$$$47\!\cdots\!64$$$$T_{3}^{17} +$$$$12\!\cdots\!81$$$$T_{3}^{16} +$$$$23\!\cdots\!96$$$$T_{3}^{15} -$$$$25\!\cdots\!76$$$$T_{3}^{14} -$$$$84\!\cdots\!52$$$$T_{3}^{13} -$$$$62\!\cdots\!34$$$$T_{3}^{12} +$$$$22\!\cdots\!48$$$$T_{3}^{11} +$$$$30\!\cdots\!20$$$$T_{3}^{10} -$$$$36\!\cdots\!28$$$$T_{3}^{9} -$$$$96\!\cdots\!81$$$$T_{3}^{8} +$$$$31\!\cdots\!24$$$$T_{3}^{7} +$$$$17\!\cdots\!38$$$$T_{3}^{6} +$$$$21\!\cdots\!88$$$$T_{3}^{5} -$$$$92\!\cdots\!17$$$$T_{3}^{4} -$$$$30\!\cdots\!28$$$$T_{3}^{3} +$$$$14\!\cdots\!94$$$$T_{3}^{2} +$$$$37\!\cdots\!16$$$$T_{3} +$$$$16\!\cdots\!21$$">$$T_{3}^{40} + \cdots$$ acting on $$S_{5}^{\mathrm{new}}(304, [\chi])$$.