Properties

Label 4011.2.a.k
Level 4011
Weight 2
Character orbit 4011.a
Self dual Yes
Analytic conductor 32.028
Analytic rank 0
Dimension 27
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: \(27\)
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.

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

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 −2.65342 1.00000 5.04066 4.09935 −2.65342 1.00000 −8.06816 1.00000 −10.8773
1.2 −2.52420 1.00000 4.37157 −0.338133 −2.52420 1.00000 −5.98631 1.00000 0.853514
1.3 −2.42203 1.00000 3.86625 1.50568 −2.42203 1.00000 −4.52011 1.00000 −3.64681
1.4 −1.70789 1.00000 0.916895 2.36693 −1.70789 1.00000 1.84983 1.00000 −4.04246
1.5 −1.69492 1.00000 0.872752 −3.19478 −1.69492 1.00000 1.91059 1.00000 5.41490
1.6 −1.64011 1.00000 0.689959 −1.40502 −1.64011 1.00000 2.14861 1.00000 2.30438
1.7 −1.28891 1.00000 −0.338712 0.716613 −1.28891 1.00000 3.01439 1.00000 −0.923650
1.8 −1.05296 1.00000 −0.891266 4.03535 −1.05296 1.00000 3.04440 1.00000 −4.24908
1.9 −0.768548 1.00000 −1.40933 3.25090 −0.768548 1.00000 2.62024 1.00000 −2.49847
1.10 −0.667457 1.00000 −1.55450 −3.08787 −0.667457 1.00000 2.37248 1.00000 2.06102
1.11 −0.105370 1.00000 −1.98890 −1.28706 −0.105370 1.00000 0.420311 1.00000 0.135618
1.12 −0.0599798 1.00000 −1.99640 3.32796 −0.0599798 1.00000 0.239704 1.00000 −0.199610
1.13 0.267433 1.00000 −1.92848 0.0366447 0.267433 1.00000 −1.05060 1.00000 0.00980000
1.14 0.374739 1.00000 −1.85957 2.21215 0.374739 1.00000 −1.44633 1.00000 0.828982
1.15 0.640778 1.00000 −1.58940 −2.69509 0.640778 1.00000 −2.30001 1.00000 −1.72696
1.16 1.25182 1.00000 −0.432954 2.68317 1.25182 1.00000 −3.04561 1.00000 3.35884
1.17 1.25557 1.00000 −0.423542 3.43059 1.25557 1.00000 −3.04293 1.00000 4.30735
1.18 1.40876 1.00000 −0.0154013 −2.50452 1.40876 1.00000 −2.83921 1.00000 −3.52826
1.19 1.44666 1.00000 0.0928390 2.29705 1.44666 1.00000 −2.75902 1.00000 3.32306
1.20 1.74433 1.00000 1.04268 −1.87140 1.74433 1.00000 −1.66989 1.00000 −3.26434
See all 27 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.27
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 \(T_{2}^{27} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4011))\).