# Properties

 Label 261.3.s.a Level $261$ Weight $3$ Character orbit 261.s Analytic conductor $7.112$ Analytic rank $0$ Dimension $48$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$261 = 3^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 261.s (of order $$28$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.11173489980$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$4$$ over $$\Q(\zeta_{28})$$ Twist minimal: no (minimal twist has level 29) Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 16q^{2} - 14q^{4} + 14q^{5} - 10q^{7} - 28q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 16q^{2} - 14q^{4} + 14q^{5} - 10q^{7} - 28q^{8} - 20q^{10} + 8q^{11} - 14q^{13} - 26q^{14} + 18q^{16} + 26q^{17} + 2q^{19} - 46q^{20} + 154q^{22} - 56q^{23} - 34q^{25} - 110q^{26} + 170q^{29} - 88q^{31} + 132q^{32} - 224q^{34} + 210q^{35} - 56q^{37} + 294q^{38} - 492q^{40} + 34q^{41} + 176q^{43} - 126q^{44} + 744q^{46} - 208q^{47} + 506q^{49} - 732q^{50} + 690q^{52} + 14q^{53} + 284q^{55} - 332q^{56} - 508q^{58} + 44q^{59} - 30q^{61} + 504q^{62} - 896q^{64} + 554q^{65} - 574q^{67} + 796q^{68} - 1066q^{70} - 224q^{71} - 22q^{73} - 820q^{74} + 514q^{76} - 436q^{77} + 564q^{79} - 1162q^{80} - 18q^{82} + 126q^{83} + 38q^{85} - 384q^{88} + 160q^{89} - 434q^{91} + 1022q^{92} - 2q^{94} + 642q^{95} + 604q^{97} + 102q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1 −2.05804 + 1.29315i 0 0.827740 1.71882i −5.87720 1.34143i 0 −9.36468 + 4.50979i −0.569384 5.05343i 0 13.8301 4.83938i
10.2 −1.44470 + 0.907767i 0 −0.472410 + 0.980970i 8.15497 + 1.86132i 0 7.53715 3.62970i −0.972146 8.62804i 0 −13.4711 + 4.71376i
10.3 1.29187 0.811733i 0 −0.725528 + 1.50658i −3.36294 0.767569i 0 −0.255710 + 0.123143i 0.968958 + 8.59974i 0 −4.96753 + 1.73821i
10.4 3.24379 2.03821i 0 4.63233 9.61914i 5.12808 + 1.17045i 0 −6.56728 + 3.16264i −2.86375 25.4165i 0 19.0200 6.65539i
19.1 −1.42815 2.27289i 0 −1.39088 + 2.88819i 6.69994 + 1.52922i 0 −4.78081 + 2.30231i −2.11889 + 0.238741i 0 −6.09279 17.4122i
19.2 −0.488049 0.776726i 0 1.37042 2.84571i 1.98497 + 0.453055i 0 9.56374 4.60566i −6.52543 + 0.735239i 0 −0.616861 1.76289i
19.3 0.448531 + 0.713833i 0 1.42716 2.96352i −4.21883 0.962920i 0 −10.1429 + 4.88457i 6.10659 0.688048i 0 −1.20491 3.44344i
19.4 1.63282 + 2.59861i 0 −2.35117 + 4.88225i 4.43970 + 1.01333i 0 10.7635 5.18344i −4.32723 + 0.487562i 0 4.61596 + 13.1917i
37.1 −1.65381 0.578694i 0 −0.727117 0.579856i 0.825315 + 1.71379i 0 −1.24782 1.56472i 4.59573 + 7.31406i 0 −0.373160 3.31188i
37.2 0.0310749 + 0.0108736i 0 −3.12648 2.49328i −3.79007 7.87017i 0 3.62612 + 4.54701i −0.140107 0.222980i 0 −0.0321993 0.285777i
37.3 2.20133 + 0.770280i 0 1.12521 + 0.897324i −2.64264 5.48750i 0 −3.22936 4.04949i −3.17747 5.05691i 0 −1.59042 14.1154i
37.4 2.32673 + 0.814157i 0 1.62348 + 1.29468i 3.83207 + 7.95737i 0 2.23777 + 2.80607i −2.52264 4.01476i 0 2.43763 + 21.6345i
55.1 −1.42815 + 2.27289i 0 −1.39088 2.88819i 6.69994 1.52922i 0 −4.78081 2.30231i −2.11889 0.238741i 0 −6.09279 + 17.4122i
55.2 −0.488049 + 0.776726i 0 1.37042 + 2.84571i 1.98497 0.453055i 0 9.56374 + 4.60566i −6.52543 0.735239i 0 −0.616861 + 1.76289i
55.3 0.448531 0.713833i 0 1.42716 + 2.96352i −4.21883 + 0.962920i 0 −10.1429 4.88457i 6.10659 + 0.688048i 0 −1.20491 + 3.44344i
55.4 1.63282 2.59861i 0 −2.35117 4.88225i 4.43970 1.01333i 0 10.7635 + 5.18344i −4.32723 0.487562i 0 4.61596 13.1917i
73.1 −2.58169 + 0.290886i 0 2.68078 0.611870i 2.49104 + 1.98654i 0 1.30161 5.70272i 3.06598 1.07283i 0 −7.00893 4.40400i
73.2 −0.415096 + 0.0467701i 0 −3.72959 + 0.851255i −0.738700 0.589093i 0 −0.577468 + 2.53005i 3.08546 1.07965i 0 0.334184 + 0.209981i
73.3 1.68783 0.190173i 0 −1.08711 + 0.248126i −0.141728 0.113024i 0 −1.55116 + 6.79606i −8.20045 + 2.86946i 0 −0.260707 0.163813i
73.4 3.56136 0.401269i 0 8.62256 1.96804i −5.24106 4.17960i 0 2.11977 9.28734i 16.3872 5.73413i 0 −20.3424 12.7820i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 253.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.f odd 28 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 261.3.s.a 48
3.b odd 2 1 29.3.f.a 48
29.f odd 28 1 inner 261.3.s.a 48
87.k even 28 1 29.3.f.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.3.f.a 48 3.b odd 2 1
29.3.f.a 48 87.k even 28 1
261.3.s.a 48 1.a even 1 1 trivial
261.3.s.a 48 29.f odd 28 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$11\!\cdots\!18$$$$T_{2}^{22} +$$$$12\!\cdots\!40$$$$T_{2}^{21} +$$$$56\!\cdots\!63$$$$T_{2}^{20} -$$$$14\!\cdots\!16$$$$T_{2}^{19} +$$$$82\!\cdots\!53$$$$T_{2}^{18} +$$$$94\!\cdots\!60$$$$T_{2}^{17} -$$$$25\!\cdots\!94$$$$T_{2}^{16} +$$$$54\!\cdots\!88$$$$T_{2}^{15} +$$$$97\!\cdots\!09$$$$T_{2}^{14} -$$$$17\!\cdots\!36$$$$T_{2}^{13} +$$$$11\!\cdots\!70$$$$T_{2}^{12} -$$$$95\!\cdots\!52$$$$T_{2}^{11} +$$$$13\!\cdots\!48$$$$T_{2}^{10} -$$$$83\!\cdots\!12$$$$T_{2}^{9} +$$$$42\!\cdots\!26$$$$T_{2}^{8} +$$$$36\!\cdots\!72$$$$T_{2}^{7} -$$$$23\!\cdots\!75$$$$T_{2}^{6} +$$$$96\!\cdots\!48$$$$T_{2}^{5} +$$$$22\!\cdots\!69$$$$T_{2}^{4} + 114085022712 T_{2}^{3} + 311190034388 T_{2}^{2} - 19931804632 T_{2} + 352951369$$">$$T_{2}^{48} - \cdots$$ acting on $$S_{3}^{\mathrm{new}}(261, [\chi])$$.