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

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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: \(28\)
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.

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

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:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(191\) \(1\)

Hecke kernels

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