# Properties

 Label 2151.4.a.g Level $2151$ Weight $4$ Character orbit 2151.a Self dual yes Analytic conductor $126.913$ Analytic rank $1$ Dimension $59$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2151 = 3^{2} \cdot 239$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2151.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$126.913108422$$ Analytic rank: $$1$$ Dimension: $$59$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$59q - 8q^{2} + 238q^{4} - 80q^{5} - 10q^{7} - 96q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$59q - 8q^{2} + 238q^{4} - 80q^{5} - 10q^{7} - 96q^{8} - 36q^{10} - 132q^{11} + 104q^{13} - 280q^{14} + 822q^{16} - 408q^{17} + 20q^{19} - 800q^{20} - 2q^{22} - 276q^{23} + 1477q^{25} - 780q^{26} + 224q^{28} - 696q^{29} - 380q^{31} - 896q^{32} - 72q^{34} - 700q^{35} + 224q^{37} - 988q^{38} - 258q^{40} - 2706q^{41} - 156q^{43} - 1584q^{44} + 428q^{46} - 1316q^{47} + 2135q^{49} - 1400q^{50} + 1092q^{52} - 1484q^{53} - 992q^{55} - 3360q^{56} - 120q^{58} - 3186q^{59} - 254q^{61} - 1240q^{62} + 3054q^{64} - 5120q^{65} + 288q^{67} - 9420q^{68} + 1108q^{70} - 4468q^{71} - 1770q^{73} - 6214q^{74} + 720q^{76} - 6352q^{77} - 746q^{79} - 7040q^{80} + 276q^{82} - 5484q^{83} + 588q^{85} - 10152q^{86} + 1186q^{88} - 11570q^{89} + 1768q^{91} - 15366q^{92} - 2142q^{94} - 5736q^{95} + 2390q^{97} - 6912q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 −5.54707 0 22.7699 −11.8059 0 −16.2858 −81.9299 0 65.4880
1.2 −5.50845 0 22.3430 10.5610 0 7.83920 −79.0077 0 −58.1747
1.3 −5.43867 0 21.5792 −5.07597 0 21.4842 −73.8527 0 27.6066
1.4 −5.06856 0 17.6903 14.8628 0 −19.8824 −49.1160 0 −75.3330
1.5 −4.88066 0 15.8208 −22.1582 0 19.1765 −38.1708 0 108.146
1.6 −4.81368 0 15.1715 −10.3877 0 20.9834 −34.5214 0 50.0029
1.7 −4.79889 0 15.0293 −17.2694 0 7.94711 −33.7329 0 82.8740
1.8 −4.59357 0 13.1009 7.15514 0 33.8683 −23.4314 0 −32.8677
1.9 −4.52883 0 12.5103 −9.53238 0 −26.9020 −20.4262 0 43.1705
1.10 −4.39292 0 11.2977 17.5164 0 15.6908 −14.4866 0 −76.9482
1.11 −4.26856 0 10.2206 4.08830 0 −28.0668 −9.47876 0 −17.4512
1.12 −3.66557 0 5.43642 19.0359 0 −9.80549 9.39697 0 −69.7775
1.13 −3.55261 0 4.62104 −5.70782 0 19.6278 12.0041 0 20.2777
1.14 −3.36448 0 3.31971 −9.29992 0 3.75868 15.7467 0 31.2894
1.15 −3.23820 0 2.48591 −18.9858 0 −17.9377 17.8557 0 61.4797
1.16 −3.14305 0 1.87878 8.40099 0 6.53557 19.2393 0 −26.4048
1.17 −3.12258 0 1.75053 10.0516 0 −27.9204 19.5145 0 −31.3869
1.18 −2.94849 0 0.693582 3.01150 0 −30.9852 21.5429 0 −8.87936
1.19 −2.83641 0 0.0452483 6.73903 0 0.810661 22.5630 0 −19.1147
1.20 −2.81951 0 −0.0503577 −16.9496 0 31.0725 22.6981 0 47.7895
See all 59 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.59 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$239$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2151.4.a.g 59
3.b odd 2 1 2151.4.a.h yes 59

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2151.4.a.g 59 1.a even 1 1 trivial
2151.4.a.h yes 59 3.b odd 2 1