# Properties

 Label 2151.4.a.a Level $2151$ Weight $4$ Character orbit 2151.a Self dual yes Analytic conductor $126.913$ Analytic rank $1$ Dimension $22$ 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: $$22$$ Twist minimal: no (minimal twist has level 239) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$22q + 4q^{2} + 50q^{4} + 37q^{5} - 52q^{7} + 69q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$22q + 4q^{2} + 50q^{4} + 37q^{5} - 52q^{7} + 69q^{8} - 93q^{10} + 77q^{11} - 218q^{13} + 111q^{14} - 42q^{16} + 219q^{17} - 476q^{19} + 314q^{20} - 390q^{22} + 202q^{23} - 271q^{25} + 220q^{26} - 515q^{28} + 307q^{29} - 1001q^{31} + 771q^{32} - 1297q^{34} + 430q^{35} - 922q^{37} - 49q^{38} - 1344q^{40} + 1188q^{41} - 192q^{43} + 547q^{44} - 1178q^{46} + 102q^{47} - 1952q^{49} + 471q^{50} - 1785q^{52} + 580q^{53} - 1730q^{55} + 804q^{56} - 1156q^{58} + 1528q^{59} - 1631q^{61} - 2206q^{62} + 327q^{64} - 44q^{65} - 689q^{67} - 2522q^{68} + 1175q^{70} - 341q^{71} - 2260q^{73} - 4027q^{74} - 1855q^{76} - 1578q^{77} + 396q^{79} - 6183q^{80} + 4936q^{82} - 1065q^{83} + 144q^{85} - 2915q^{86} + 1068q^{88} + 1984q^{89} - 2186q^{91} - 6720q^{92} + 174q^{94} - 2804q^{95} - 4946q^{97} - 7149q^{98} + O(q^{100})$$

## 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 −4.92623 0 16.2677 3.10245 0 −6.93638 −40.7288 0 −15.2834
1.2 −4.58869 0 13.0561 15.1251 0 −25.8403 −23.2007 0 −69.4042
1.3 −3.67099 0 5.47619 −0.939206 0 −4.16512 9.26490 0 3.44782
1.4 −3.64923 0 5.31692 17.2106 0 24.0056 9.79121 0 −62.8056
1.5 −3.32790 0 3.07490 −7.52905 0 −10.0131 16.3902 0 25.0559
1.6 −2.32829 0 −2.57908 14.1221 0 −19.8005 24.6311 0 −32.8804
1.7 −2.22859 0 −3.03338 7.56659 0 −1.62323 24.5889 0 −16.8628
1.8 −2.21535 0 −3.09221 −10.2488 0 −27.8405 24.5732 0 22.7048
1.9 −1.83829 0 −4.62067 −9.40555 0 19.7665 23.2005 0 17.2902
1.10 −0.530015 0 −7.71908 0.726105 0 27.7214 8.33135 0 −0.384847
1.11 0.594431 0 −7.64665 5.61897 0 −8.56684 −9.30086 0 3.34009
1.12 0.850702 0 −7.27631 −12.0519 0 −14.5326 −12.9956 0 −10.2525
1.13 0.976149 0 −7.04713 2.51110 0 23.2375 −14.6882 0 2.45121
1.14 1.49092 0 −5.77714 −16.6313 0 −17.5795 −20.5407 0 −24.7960
1.15 1.51884 0 −5.69312 1.30911 0 −3.41938 −20.7977 0 1.98833
1.16 2.38671 0 −2.30360 5.10344 0 12.4633 −24.5917 0 12.1804
1.17 3.24958 0 2.55975 21.6437 0 3.86879 −17.6785 0 70.3329
1.18 3.42794 0 3.75080 7.18364 0 −16.4073 −14.5660 0 24.6251
1.19 4.40109 0 11.3696 3.94300 0 −14.1222 14.8300 0 17.3535
1.20 4.43192 0 11.6419 11.7413 0 6.42321 16.1407 0 52.0364
See all 22 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.22 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.a 22
3.b odd 2 1 239.4.a.a 22

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
239.4.a.a 22 3.b odd 2 1
2151.4.a.a 22 1.a even 1 1 trivial