Properties

Label 4016.2.a.m
Level 4016
Weight 2
Character orbit 4016.a
Self dual Yes
Analytic conductor 32.068
Analytic rank 0
Dimension 23
CM No

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Newspace parameters

Level: \( N \) = \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4016.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0679214517\)
Analytic rank: \(0\)
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.

\(\operatorname{Tr}(f)(q) = \) \(23q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 8q^{5} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 45q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(23q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 8q^{5} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 45q^{9} \) \(\mathstrut -\mathstrut 8q^{11} \) \(\mathstrut +\mathstrut 8q^{13} \) \(\mathstrut -\mathstrut 7q^{15} \) \(\mathstrut +\mathstrut 19q^{17} \) \(\mathstrut +\mathstrut 9q^{19} \) \(\mathstrut +\mathstrut 9q^{21} \) \(\mathstrut -\mathstrut 21q^{23} \) \(\mathstrut +\mathstrut 65q^{25} \) \(\mathstrut -\mathstrut 5q^{27} \) \(\mathstrut +\mathstrut 10q^{29} \) \(\mathstrut +\mathstrut 9q^{31} \) \(\mathstrut +\mathstrut 34q^{33} \) \(\mathstrut -\mathstrut 12q^{35} \) \(\mathstrut +\mathstrut 11q^{37} \) \(\mathstrut +\mathstrut 9q^{39} \) \(\mathstrut +\mathstrut 35q^{41} \) \(\mathstrut +\mathstrut 9q^{43} \) \(\mathstrut +\mathstrut 29q^{45} \) \(\mathstrut -\mathstrut 37q^{47} \) \(\mathstrut +\mathstrut 77q^{49} \) \(\mathstrut +\mathstrut 17q^{51} \) \(\mathstrut +\mathstrut 38q^{53} \) \(\mathstrut +\mathstrut 20q^{55} \) \(\mathstrut +\mathstrut 51q^{57} \) \(\mathstrut -\mathstrut 17q^{59} \) \(\mathstrut -\mathstrut 22q^{63} \) \(\mathstrut +\mathstrut 41q^{65} \) \(\mathstrut -\mathstrut 9q^{67} \) \(\mathstrut +\mathstrut 8q^{69} \) \(\mathstrut -\mathstrut 13q^{71} \) \(\mathstrut +\mathstrut 41q^{73} \) \(\mathstrut -\mathstrut 25q^{75} \) \(\mathstrut +\mathstrut 36q^{77} \) \(\mathstrut +\mathstrut 36q^{79} \) \(\mathstrut +\mathstrut 127q^{81} \) \(\mathstrut -\mathstrut 29q^{83} \) \(\mathstrut +\mathstrut 34q^{85} \) \(\mathstrut -\mathstrut 10q^{87} \) \(\mathstrut +\mathstrut 36q^{89} \) \(\mathstrut +\mathstrut 6q^{91} \) \(\mathstrut +\mathstrut 36q^{93} \) \(\mathstrut -\mathstrut 25q^{95} \) \(\mathstrut +\mathstrut 40q^{97} \) \(\mathstrut -\mathstrut 19q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 0 −3.32587 0 −0.316897 0 −0.268063 0 8.06143 0
1.2 0 −3.26165 0 −4.10224 0 1.61221 0 7.63839 0
1.3 0 −3.15136 0 4.04418 0 −3.52087 0 6.93109 0
1.4 0 −2.67736 0 4.29688 0 4.01755 0 4.16827 0
1.5 0 −2.53334 0 1.50187 0 −0.478852 0 3.41783 0
1.6 0 −2.51727 0 −0.472191 0 −4.16870 0 3.33662 0
1.7 0 −1.96174 0 0.807414 0 −3.34299 0 0.848432 0
1.8 0 −1.63778 0 2.87336 0 2.53418 0 −0.317671 0
1.9 0 −1.39383 0 −3.54571 0 3.34862 0 −1.05724 0
1.10 0 −0.452441 0 1.91039 0 −1.96124 0 −2.79530 0
1.11 0 −0.259421 0 −1.96371 0 4.76268 0 −2.93270 0
1.12 0 0.0812979 0 4.08000 0 −4.55548 0 −2.99339 0
1.13 0 0.201914 0 −1.79819 0 −3.34181 0 −2.95923 0
1.14 0 0.347951 0 −1.66668 0 −3.92094 0 −2.87893 0
1.15 0 1.15579 0 −4.09592 0 0.621313 0 −1.66414 0
1.16 0 1.19399 0 −1.76720 0 4.20476 0 −1.57440 0
1.17 0 1.28390 0 3.72143 0 0.978625 0 −1.35160 0
1.18 0 2.13165 0 0.257423 0 −0.578263 0 1.54395 0
1.19 0 2.21536 0 3.09704 0 4.67984 0 1.90784 0
1.20 0 2.74485 0 2.14962 0 3.54966 0 4.53421 0
See all 23 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.23
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
\(2\) \(1\)
\(251\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{23} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4016))\).