# Properties

 Label 4011.2.a.l Level 4011 Weight 2 Character orbit 4011.a Self dual yes Analytic conductor 32.028 Analytic rank 0 Dimension 28 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4011 = 3 \cdot 7 \cdot 191$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4011.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.0279962507$$ Analytic rank: $$0$$ Dimension: $$28$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28q - 6q^{2} - 28q^{3} + 34q^{4} + 8q^{5} + 6q^{6} + 28q^{7} - 15q^{8} + 28q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q - 6q^{2} - 28q^{3} + 34q^{4} + 8q^{5} + 6q^{6} + 28q^{7} - 15q^{8} + 28q^{9} + 4q^{10} - 8q^{11} - 34q^{12} + 26q^{13} - 6q^{14} - 8q^{15} + 62q^{16} + 9q^{17} - 6q^{18} + 25q^{19} + 20q^{20} - 28q^{21} + 3q^{22} - 30q^{23} + 15q^{24} + 42q^{25} + 25q^{26} - 28q^{27} + 34q^{28} - 5q^{29} - 4q^{30} + 18q^{31} - 26q^{32} + 8q^{33} + 30q^{34} + 8q^{35} + 34q^{36} + 36q^{37} - 2q^{38} - 26q^{39} + 28q^{40} + 21q^{41} + 6q^{42} + 8q^{43} - 20q^{44} + 8q^{45} + 24q^{46} + 6q^{47} - 62q^{48} + 28q^{49} - 48q^{50} - 9q^{51} + 54q^{52} - 12q^{53} + 6q^{54} + 15q^{55} - 15q^{56} - 25q^{57} + 19q^{58} + 33q^{59} - 20q^{60} + 48q^{61} + 28q^{63} + 75q^{64} + 21q^{65} - 3q^{66} + 27q^{67} + 19q^{68} + 30q^{69} + 4q^{70} - 45q^{71} - 15q^{72} + 61q^{73} - 31q^{74} - 42q^{75} + 63q^{76} - 8q^{77} - 25q^{78} + 35q^{79} + 84q^{80} + 28q^{81} + 11q^{82} + 43q^{83} - 34q^{84} + 43q^{85} - q^{86} + 5q^{87} - 27q^{88} + 25q^{89} + 4q^{90} + 26q^{91} - 102q^{92} - 18q^{93} + 55q^{94} - 43q^{95} + 26q^{96} + 40q^{97} - 6q^{98} - 8q^{99} + O(q^{100})$$

## 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 −2.77095 −1.00000 5.67816 3.12132 2.77095 1.00000 −10.1920 1.00000 −8.64902
1.2 −2.75611 −1.00000 5.59616 −2.11611 2.75611 1.00000 −9.91144 1.00000 5.83224
1.3 −2.60224 −1.00000 4.77168 −2.58506 2.60224 1.00000 −7.21258 1.00000 6.72695
1.4 −2.55579 −1.00000 4.53209 4.15915 2.55579 1.00000 −6.47150 1.00000 −10.6299
1.5 −2.32219 −1.00000 3.39258 −2.33514 2.32219 1.00000 −3.23383 1.00000 5.42264
1.6 −2.23874 −1.00000 3.01194 1.19635 2.23874 1.00000 −2.26547 1.00000 −2.67832
1.7 −1.81456 −1.00000 1.29264 0.616523 1.81456 1.00000 1.28355 1.00000 −1.11872
1.8 −1.80294 −1.00000 1.25060 −3.82317 1.80294 1.00000 1.35113 1.00000 6.89294
1.9 −1.70611 −1.00000 0.910828 3.20896 1.70611 1.00000 1.85825 1.00000 −5.47486
1.10 −1.53559 −1.00000 0.358037 −0.986091 1.53559 1.00000 2.52138 1.00000 1.51423
1.11 −0.815618 −1.00000 −1.33477 3.08846 0.815618 1.00000 2.71990 1.00000 −2.51900
1.12 −0.786373 −1.00000 −1.38162 2.33863 0.786373 1.00000 2.65921 1.00000 −1.83903
1.13 −0.482188 −1.00000 −1.76749 −4.13435 0.482188 1.00000 1.81664 1.00000 1.99354
1.14 −0.386952 −1.00000 −1.85027 3.15570 0.386952 1.00000 1.48987 1.00000 −1.22110
1.15 −0.268740 −1.00000 −1.92778 −1.86903 0.268740 1.00000 1.05555 1.00000 0.502284
1.16 −0.0484532 −1.00000 −1.99765 1.78925 0.0484532 1.00000 0.193699 1.00000 −0.0866947
1.17 0.343679 −1.00000 −1.88188 −1.00728 −0.343679 1.00000 −1.33412 1.00000 −0.346179
1.18 0.544491 −1.00000 −1.70353 2.42913 −0.544491 1.00000 −2.01654 1.00000 1.32264
1.19 0.602953 −1.00000 −1.63645 −1.53490 −0.602953 1.00000 −2.19261 1.00000 −0.925472
1.20 1.03548 −1.00000 −0.927780 −1.23903 −1.03548 1.00000 −3.03166 1.00000 −1.28299
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

## 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 4011.2.a.l 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4011.2.a.l 28 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$7$$ $$-1$$
$$191$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{28} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4011))$$.