Properties

Label 4004.2.m.b
Level 4004
Weight 2
Character orbit 4004.m
Analytic conductor 31.972
Analytic rank 0
Dimension 30
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: \(30\)
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.

\(\operatorname{Tr}(f)(q) = \) \(30q \) \(\mathstrut +\mathstrut 18q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(30q \) \(\mathstrut +\mathstrut 18q^{9} \) \(\mathstrut +\mathstrut 4q^{13} \) \(\mathstrut +\mathstrut 12q^{17} \) \(\mathstrut +\mathstrut 6q^{23} \) \(\mathstrut -\mathstrut 6q^{29} \) \(\mathstrut -\mathstrut 2q^{35} \) \(\mathstrut +\mathstrut 8q^{39} \) \(\mathstrut -\mathstrut 22q^{43} \) \(\mathstrut -\mathstrut 30q^{49} \) \(\mathstrut +\mathstrut 60q^{51} \) \(\mathstrut -\mathstrut 38q^{53} \) \(\mathstrut -\mathstrut 2q^{55} \) \(\mathstrut -\mathstrut 36q^{61} \) \(\mathstrut +\mathstrut 10q^{65} \) \(\mathstrut +\mathstrut 36q^{69} \) \(\mathstrut -\mathstrut 20q^{75} \) \(\mathstrut -\mathstrut 30q^{77} \) \(\mathstrut -\mathstrut 10q^{79} \) \(\mathstrut -\mathstrut 42q^{81} \) \(\mathstrut +\mathstrut 36q^{87} \) \(\mathstrut +\mathstrut 6q^{91} \) \(\mathstrut -\mathstrut 2q^{95} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.58184 0 1.48218i 0 1.00000i 0 3.66591 0
2157.2 0 −2.58184 0 1.48218i 0 1.00000i 0 3.66591 0
2157.3 0 −2.50189 0 2.36927i 0 1.00000i 0 3.25946 0
2157.4 0 −2.50189 0 2.36927i 0 1.00000i 0 3.25946 0
2157.5 0 −2.42937 0 1.20006i 0 1.00000i 0 2.90185 0
2157.6 0 −2.42937 0 1.20006i 0 1.00000i 0 2.90185 0
2157.7 0 −2.41023 0 3.42759i 0 1.00000i 0 2.80920 0
2157.8 0 −2.41023 0 3.42759i 0 1.00000i 0 2.80920 0
2157.9 0 −1.49114 0 1.37882i 0 1.00000i 0 −0.776492 0
2157.10 0 −1.49114 0 1.37882i 0 1.00000i 0 −0.776492 0
2157.11 0 −0.822838 0 0.417533i 0 1.00000i 0 −2.32294 0
2157.12 0 −0.822838 0 0.417533i 0 1.00000i 0 −2.32294 0
2157.13 0 −0.0797053 0 0.579061i 0 1.00000i 0 −2.99365 0
2157.14 0 −0.0797053 0 0.579061i 0 1.00000i 0 −2.99365 0
2157.15 0 0.0840880 0 3.49341i 0 1.00000i 0 −2.99293 0
2157.16 0 0.0840880 0 3.49341i 0 1.00000i 0 −2.99293 0
2157.17 0 0.705440 0 2.65594i 0 1.00000i 0 −2.50236 0
2157.18 0 0.705440 0 2.65594i 0 1.00000i 0 −2.50236 0
2157.19 0 0.726270 0 2.96350i 0 1.00000i 0 −2.47253 0
2157.20 0 0.726270 0 2.96350i 0 1.00000i 0 −2.47253 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2157.30
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 \(T_{3}^{15} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(4004, \chi)\).