# Properties

 Label 2151.4.a.b Level $2151$ Weight $4$ Character orbit 2151.a Self dual yes Analytic conductor $126.913$ Analytic rank $1$ Dimension $28$ 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: $$28$$ Twist minimal: no (minimal twist has level 717) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28q + 5q^{2} + 103q^{4} - 6q^{5} - 68q^{7} + 39q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q + 5q^{2} + 103q^{4} - 6q^{5} - 68q^{7} + 39q^{8} - 88q^{10} + 110q^{11} - 82q^{13} - 126q^{14} + 271q^{16} - 100q^{17} - 292q^{19} + 52q^{20} - 351q^{22} + 276q^{23} + 386q^{25} - 84q^{26} - 1010q^{28} + 38q^{29} - 432q^{31} + 452q^{32} - 524q^{34} + 166q^{35} - 936q^{37} + 41q^{38} - 1183q^{40} - 1054q^{41} - 1804q^{43} + 341q^{44} - 888q^{46} + 560q^{47} + 1074q^{49} + 1054q^{50} - 632q^{52} + 160q^{53} - 842q^{55} - 509q^{56} - 1266q^{58} - 846q^{59} - 2220q^{61} - 82q^{62} - 1565q^{64} - 296q^{65} - 4752q^{67} + 1719q^{68} - 5601q^{70} + 802q^{71} - 2732q^{73} + 4581q^{74} - 5614q^{76} + 1008q^{77} - 3172q^{79} + 732q^{80} - 9709q^{82} + 4780q^{83} - 4624q^{85} + 2009q^{86} - 9331q^{88} - 4372q^{89} - 7398q^{91} + 6138q^{92} - 7068q^{94} + 3160q^{95} - 4846q^{97} + 3772q^{98} + O(q^{100})$$

## 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 −5.30430 0 20.1356 1.74933 0 −31.0971 −64.3712 0 −9.27899
1.2 −4.86739 0 15.6914 12.8877 0 25.7021 −37.4372 0 −62.7292
1.3 −4.68120 0 13.9136 9.12054 0 −4.35896 −27.6829 0 −42.6951
1.4 −4.21516 0 9.76757 −14.1283 0 −29.9315 −7.45059 0 59.5530
1.5 −3.99686 0 7.97490 10.1568 0 5.65927 0.100320 0 −40.5954
1.6 −3.64270 0 5.26927 −14.5917 0 6.09065 9.94722 0 53.1530
1.7 −3.55218 0 4.61801 −1.20455 0 9.27993 12.0135 0 4.27878
1.8 −3.15491 0 1.95344 −4.03944 0 6.03808 19.0763 0 12.7440
1.9 −1.61692 0 −5.38557 −9.90974 0 −20.2386 21.6434 0 16.0233
1.10 −1.48544 0 −5.79347 10.1836 0 8.86157 20.4894 0 −15.1271
1.11 −1.36717 0 −6.13085 15.7404 0 15.1668 19.3193 0 −21.5198
1.12 −0.915825 0 −7.16127 −7.54581 0 24.7091 13.8851 0 6.91064
1.13 −0.889333 0 −7.20909 −6.34700 0 −11.3318 13.5259 0 5.64460
1.14 0.410040 0 −7.83187 19.1512 0 −11.0347 −6.49170 0 7.85278
1.15 0.609904 0 −7.62802 11.6127 0 −26.3682 −9.53159 0 7.08263
1.16 0.709237 0 −7.49698 −22.0711 0 −0.837973 −10.9910 0 −15.6536
1.17 1.15681 0 −6.66179 10.4070 0 11.9554 −16.9609 0 12.0389
1.18 1.62942 0 −5.34499 −14.4675 0 −35.3781 −21.7446 0 −23.5737
1.19 2.22027 0 −3.07038 −14.7142 0 26.0557 −24.5793 0 −32.6696
1.20 3.21293 0 2.32293 16.5200 0 −15.7443 −18.2400 0 53.0777
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.28 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.b 28
3.b odd 2 1 717.4.a.b 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
717.4.a.b 28 3.b odd 2 1
2151.4.a.b 28 1.a even 1 1 trivial