Properties

Label 4015.2.a.b
Level 4015
Weight 2
Character orbit 4015.a
Self dual Yes
Analytic conductor 32.060
Analytic rank 1
Dimension 23
CM No

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Newspace parameters

Level: \( N \) = \( 4015 = 5 \cdot 11 \cdot 73 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4015.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0599364115\)
Analytic rank: \(1\)
Dimension: \(23\)
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.

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

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 −2.71506 1.57103 5.37156 1.00000 −4.26545 1.27925 −9.15401 −0.531858 −2.71506
1.2 −2.35657 2.73189 3.55343 1.00000 −6.43790 −2.31150 −3.66076 4.46323 −2.35657
1.3 −2.32698 −0.285099 3.41485 1.00000 0.663421 −4.31195 −3.29232 −2.91872 −2.32698
1.4 −2.28085 −0.867872 3.20226 1.00000 1.97948 0.729352 −2.74217 −2.24680 −2.28085
1.5 −2.06292 0.357225 2.25565 1.00000 −0.736927 1.84250 −0.527390 −2.87239 −2.06292
1.6 −1.83447 −2.15866 1.36527 1.00000 3.95999 0.444043 1.16439 1.65981 −1.83447
1.7 −1.19799 1.69638 −0.564821 1.00000 −2.03224 −2.11615 3.07263 −0.122309 −1.19799
1.8 −1.16229 1.36223 −0.649085 1.00000 −1.58330 3.38995 3.07900 −1.14434 −1.16229
1.9 −1.09501 −2.94583 −0.800943 1.00000 3.22573 2.47152 3.06707 5.67794 −1.09501
1.10 −1.08357 −1.29689 −0.825886 1.00000 1.40527 −5.06508 3.06203 −1.31806 −1.08357
1.11 −0.654679 2.96699 −1.57140 1.00000 −1.94243 −1.18438 2.33812 5.80304 −0.654679
1.12 −0.182665 1.60195 −1.96663 1.00000 −0.292620 −3.46851 0.724566 −0.433762 −0.182665
1.13 0.152616 −0.242697 −1.97671 1.00000 −0.0370393 2.57515 −0.606909 −2.94110 0.152616
1.14 0.152825 −1.31847 −1.97664 1.00000 −0.201495 −0.959999 −0.607731 −1.26164 0.152825
1.15 0.430868 −1.25921 −1.81435 1.00000 −0.542554 4.34543 −1.64348 −1.41438 0.430868
1.16 0.772478 −3.03398 −1.40328 1.00000 −2.34368 1.44909 −2.62896 6.20504 0.772478
1.17 1.18208 −1.23608 −0.602686 1.00000 −1.46115 −1.06876 −3.07658 −1.47210 1.18208
1.18 1.32991 2.72450 −0.231346 1.00000 3.62333 −4.53918 −2.96748 4.42288 1.32991
1.19 1.55677 0.976240 0.423526 1.00000 1.51978 −0.0535957 −2.45420 −2.04696 1.55677
1.20 1.79465 0.618778 1.22075 1.00000 1.11049 1.73339 −1.39847 −2.61711 1.79465
See all 23 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.23
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(11\) \(-1\)
\(73\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{23} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4015))\).