Properties

Label 4004.2.m.c
Level 4004
Weight 2
Character orbit 4004.m
Analytic conductor 31.972
Analytic rank 0
Dimension 36
CM No

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Newspace parameters

Level: \( N \) = \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4004.m (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(31.9721009693\)
Analytic rank: \(0\)
Dimension: \(36\)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut -\mathstrut 80q^{25} \) \(\mathstrut +\mathstrut 8q^{27} \) \(\mathstrut +\mathstrut 8q^{29} \) \(\mathstrut -\mathstrut 24q^{39} \) \(\mathstrut +\mathstrut 32q^{43} \) \(\mathstrut -\mathstrut 36q^{49} \) \(\mathstrut -\mathstrut 20q^{51} \) \(\mathstrut +\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 32q^{61} \) \(\mathstrut -\mathstrut 24q^{65} \) \(\mathstrut +\mathstrut 80q^{69} \) \(\mathstrut -\mathstrut 36q^{75} \) \(\mathstrut +\mathstrut 36q^{77} \) \(\mathstrut +\mathstrut 16q^{79} \) \(\mathstrut +\mathstrut 132q^{81} \) \(\mathstrut +\mathstrut 8q^{91} \) \(\mathstrut +\mathstrut 56q^{95} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2157.1 0 −3.27485 0 2.17871i 0 1.00000i 0 7.72466 0
2157.2 0 −3.27485 0 2.17871i 0 1.00000i 0 7.72466 0
2157.3 0 −3.10315 0 1.71558i 0 1.00000i 0 6.62954 0
2157.4 0 −3.10315 0 1.71558i 0 1.00000i 0 6.62954 0
2157.5 0 −2.75761 0 1.98236i 0 1.00000i 0 4.60441 0
2157.6 0 −2.75761 0 1.98236i 0 1.00000i 0 4.60441 0
2157.7 0 −2.10949 0 3.39046i 0 1.00000i 0 1.44995 0
2157.8 0 −2.10949 0 3.39046i 0 1.00000i 0 1.44995 0
2157.9 0 −1.64159 0 4.02832i 0 1.00000i 0 −0.305195 0
2157.10 0 −1.64159 0 4.02832i 0 1.00000i 0 −0.305195 0
2157.11 0 −1.08723 0 3.34996i 0 1.00000i 0 −1.81792 0
2157.12 0 −1.08723 0 3.34996i 0 1.00000i 0 −1.81792 0
2157.13 0 −1.07570 0 3.26058i 0 1.00000i 0 −1.84286 0
2157.14 0 −1.07570 0 3.26058i 0 1.00000i 0 −1.84286 0
2157.15 0 −0.819032 0 1.69723i 0 1.00000i 0 −2.32919 0
2157.16 0 −0.819032 0 1.69723i 0 1.00000i 0 −2.32919 0
2157.17 0 −0.722868 0 0.383791i 0 1.00000i 0 −2.47746 0
2157.18 0 −0.722868 0 0.383791i 0 1.00000i 0 −2.47746 0
2157.19 0 −0.189991 0 1.01340i 0 1.00000i 0 −2.96390 0
2157.20 0 −0.189991 0 1.01340i 0 1.00000i 0 −2.96390 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2157.36
Significant digits:
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Inner twists

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{18} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(4004, \chi)\).