# Properties

 Label 2151.4.a.h Level $2151$ Weight $4$ Character orbit 2151.a Self dual yes Analytic conductor $126.913$ Analytic rank $0$ 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: $$0$$ 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 −5.26507 0 19.7210 20.1774 0 −13.2314 −61.7120 0 −106.236
1.2 −5.24697 0 19.5307 10.0958 0 0.804440 −60.5014 0 −52.9726
1.3 −5.15423 0 18.5661 1.27174 0 −23.0732 −54.4599 0 −6.55481
1.4 −5.11618 0 18.1753 6.15957 0 24.2123 −52.0584 0 −31.5134
1.5 −5.09369 0 17.9457 −15.2658 0 −21.0984 −50.6602 0 77.7594
1.6 −4.50896 0 12.3307 −9.52144 0 −13.3640 −19.5269 0 42.9318
1.7 −4.45221 0 11.8222 12.0796 0 −15.4984 −17.0170 0 −53.7808
1.8 −4.45149 0 11.8157 14.0297 0 23.7055 −16.9858 0 −62.4530
1.9 −4.40394 0 11.3946 1.03938 0 0.641944 −14.9498 0 −4.57738
1.10 −4.10283 0 8.83322 −7.42154 0 5.67497 −3.41855 0 30.4493
1.11 −3.99709 0 7.97675 −4.04840 0 −1.10463 0.0929295 0 16.1818
1.12 −3.78499 0 6.32612 −5.39134 0 9.96233 6.33562 0 20.4061
1.13 −3.13470 0 1.82636 −4.91209 0 36.3874 19.3525 0 15.3979
1.14 −3.10672 0 1.65170 −17.8784 0 0.745718 19.7224 0 55.5430
1.15 −2.96448 0 0.788155 20.3046 0 −25.7042 21.3794 0 −60.1925
1.16 −2.96328 0 0.781040 9.17370 0 2.35613 21.3918 0 −27.1843
1.17 −2.95118 0 0.709472 −16.6350 0 −30.2397 21.5157 0 49.0928
1.18 −2.78319 0 −0.253830 19.2360 0 27.5534 22.9720 0 −53.5375
1.19 −2.12161 0 −3.49876 4.52878 0 14.7886 24.3959 0 −9.60831
1.20 −1.92673 0 −4.28770 −4.54299 0 −14.1149 23.6751 0 8.75313
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.h yes 59
3.b odd 2 1 2151.4.a.g 59

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