Properties

Label 4015.2.a.g
Level 4015
Weight 2
Character orbit 4015.a
Self dual Yes
Analytic conductor 32.060
Analytic rank 1
Dimension 32
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: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \(32q \) \(\mathstrut -\mathstrut 5q^{2} \) \(\mathstrut -\mathstrut 7q^{3} \) \(\mathstrut +\mathstrut 37q^{4} \) \(\mathstrut -\mathstrut 32q^{5} \) \(\mathstrut -\mathstrut 3q^{6} \) \(\mathstrut -\mathstrut 18q^{8} \) \(\mathstrut +\mathstrut 29q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(32q \) \(\mathstrut -\mathstrut 5q^{2} \) \(\mathstrut -\mathstrut 7q^{3} \) \(\mathstrut +\mathstrut 37q^{4} \) \(\mathstrut -\mathstrut 32q^{5} \) \(\mathstrut -\mathstrut 3q^{6} \) \(\mathstrut -\mathstrut 18q^{8} \) \(\mathstrut +\mathstrut 29q^{9} \) \(\mathstrut +\mathstrut 5q^{10} \) \(\mathstrut -\mathstrut 32q^{11} \) \(\mathstrut -\mathstrut 24q^{12} \) \(\mathstrut -\mathstrut q^{13} \) \(\mathstrut -\mathstrut 5q^{14} \) \(\mathstrut +\mathstrut 7q^{15} \) \(\mathstrut +\mathstrut 47q^{16} \) \(\mathstrut -\mathstrut 30q^{17} \) \(\mathstrut -\mathstrut 11q^{18} \) \(\mathstrut +\mathstrut 16q^{19} \) \(\mathstrut -\mathstrut 37q^{20} \) \(\mathstrut +\mathstrut q^{21} \) \(\mathstrut +\mathstrut 5q^{22} \) \(\mathstrut -\mathstrut 26q^{23} \) \(\mathstrut -\mathstrut 21q^{24} \) \(\mathstrut +\mathstrut 32q^{25} \) \(\mathstrut -\mathstrut q^{26} \) \(\mathstrut -\mathstrut 31q^{27} \) \(\mathstrut -\mathstrut 24q^{28} \) \(\mathstrut -\mathstrut 10q^{29} \) \(\mathstrut +\mathstrut 3q^{30} \) \(\mathstrut -\mathstrut 2q^{31} \) \(\mathstrut -\mathstrut 31q^{32} \) \(\mathstrut +\mathstrut 7q^{33} \) \(\mathstrut -\mathstrut 14q^{34} \) \(\mathstrut +\mathstrut 38q^{36} \) \(\mathstrut -\mathstrut 28q^{37} \) \(\mathstrut -\mathstrut 63q^{38} \) \(\mathstrut -\mathstrut 2q^{39} \) \(\mathstrut +\mathstrut 18q^{40} \) \(\mathstrut -\mathstrut 62q^{41} \) \(\mathstrut -\mathstrut 9q^{42} \) \(\mathstrut +\mathstrut 8q^{43} \) \(\mathstrut -\mathstrut 37q^{44} \) \(\mathstrut -\mathstrut 29q^{45} \) \(\mathstrut +\mathstrut 19q^{46} \) \(\mathstrut -\mathstrut 21q^{47} \) \(\mathstrut -\mathstrut 79q^{48} \) \(\mathstrut +\mathstrut 34q^{49} \) \(\mathstrut -\mathstrut 5q^{50} \) \(\mathstrut +\mathstrut 17q^{51} \) \(\mathstrut +\mathstrut 15q^{52} \) \(\mathstrut -\mathstrut 32q^{53} \) \(\mathstrut +\mathstrut 5q^{54} \) \(\mathstrut +\mathstrut 32q^{55} \) \(\mathstrut -\mathstrut 52q^{56} \) \(\mathstrut -\mathstrut 57q^{57} \) \(\mathstrut +\mathstrut 4q^{58} \) \(\mathstrut -\mathstrut 37q^{59} \) \(\mathstrut +\mathstrut 24q^{60} \) \(\mathstrut +\mathstrut 15q^{61} \) \(\mathstrut -\mathstrut 22q^{62} \) \(\mathstrut +\mathstrut 5q^{63} \) \(\mathstrut +\mathstrut 70q^{64} \) \(\mathstrut +\mathstrut q^{65} \) \(\mathstrut +\mathstrut 3q^{66} \) \(\mathstrut -\mathstrut 42q^{67} \) \(\mathstrut -\mathstrut 81q^{68} \) \(\mathstrut -\mathstrut 8q^{69} \) \(\mathstrut +\mathstrut 5q^{70} \) \(\mathstrut -\mathstrut 40q^{71} \) \(\mathstrut -\mathstrut 27q^{72} \) \(\mathstrut -\mathstrut 32q^{73} \) \(\mathstrut -\mathstrut 17q^{74} \) \(\mathstrut -\mathstrut 7q^{75} \) \(\mathstrut +\mathstrut 21q^{76} \) \(\mathstrut -\mathstrut 105q^{78} \) \(\mathstrut +\mathstrut 18q^{79} \) \(\mathstrut -\mathstrut 47q^{80} \) \(\mathstrut +\mathstrut 12q^{81} \) \(\mathstrut -\mathstrut 70q^{82} \) \(\mathstrut -\mathstrut 26q^{83} \) \(\mathstrut +\mathstrut 22q^{84} \) \(\mathstrut +\mathstrut 30q^{85} \) \(\mathstrut -\mathstrut 45q^{86} \) \(\mathstrut -\mathstrut 18q^{87} \) \(\mathstrut +\mathstrut 18q^{88} \) \(\mathstrut -\mathstrut 83q^{89} \) \(\mathstrut +\mathstrut 11q^{90} \) \(\mathstrut -\mathstrut 18q^{91} \) \(\mathstrut -\mathstrut 73q^{92} \) \(\mathstrut -\mathstrut 68q^{93} \) \(\mathstrut +\mathstrut 56q^{94} \) \(\mathstrut -\mathstrut 16q^{95} \) \(\mathstrut -\mathstrut 35q^{96} \) \(\mathstrut -\mathstrut 99q^{97} \) \(\mathstrut -\mathstrut 61q^{98} \) \(\mathstrut -\mathstrut 29q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.80101 2.07520 5.84563 −1.00000 −5.81265 2.07490 −10.7716 1.30645 2.80101
1.2 −2.75666 −2.97515 5.59919 −1.00000 8.20149 1.32848 −9.92176 5.85152 2.75666
1.3 −2.62713 −1.17329 4.90182 −1.00000 3.08238 3.02645 −7.62347 −1.62339 2.62713
1.4 −2.55668 0.570205 4.53663 −1.00000 −1.45783 −4.65776 −6.48536 −2.67487 2.55668
1.5 −2.31178 −1.68436 3.34432 −1.00000 3.89386 −2.49479 −3.10778 −0.162937 2.31178
1.6 −2.06358 2.59950 2.25838 −1.00000 −5.36430 −1.01430 −0.533185 3.75743 2.06358
1.7 −2.01115 −3.15685 2.04473 −1.00000 6.34891 −3.03904 −0.0899564 6.96573 2.01115
1.8 −1.96967 2.18511 1.87958 −1.00000 −4.30394 1.92639 0.237186 1.77472 1.96967
1.9 −1.75190 −0.429918 1.06914 −1.00000 0.753172 3.79529 1.63077 −2.81517 1.75190
1.10 −1.71308 1.21665 0.934626 −1.00000 −2.08421 −4.29501 1.82507 −1.51977 1.71308
1.11 −1.50747 −0.655267 0.272451 −1.00000 0.987792 −1.30717 2.60422 −2.57063 1.50747
1.12 −0.962434 2.32544 −1.07372 −1.00000 −2.23808 2.98112 2.95825 2.40766 0.962434
1.13 −0.960547 −2.83486 −1.07735 −1.00000 2.72301 4.60653 2.95594 5.03641 0.960547
1.14 −0.763072 0.889359 −1.41772 −1.00000 −0.678645 2.40954 2.60797 −2.20904 0.763072
1.15 −0.711670 −2.33630 −1.49353 −1.00000 1.66268 −3.28922 2.48624 2.45831 0.711670
1.16 −0.193228 −1.07555 −1.96266 −1.00000 0.207825 1.51348 0.765696 −1.84320 0.193228
1.17 0.0852596 0.824276 −1.99273 −1.00000 0.0702774 0.193385 −0.340419 −2.32057 −0.0852596
1.18 0.338336 −0.770870 −1.88553 −1.00000 −0.260813 −3.60696 −1.31461 −2.40576 −0.338336
1.19 0.467946 3.03596 −1.78103 −1.00000 1.42067 −2.71430 −1.76932 6.21706 −0.467946
1.20 0.472050 −2.02631 −1.77717 −1.00000 −0.956522 −4.07098 −1.78301 1.10595 −0.472050
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.32
Significant digits:
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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 \(T_{2}^{32} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4015))\).