Properties

Label 1344.2.s.c
Level 1344
Weight 2
Character orbit 1344.s
Analytic conductor 10.732
Analytic rank 0
Dimension 40
CM No

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Newspace parameters

Level: \( N \) = \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1344.s (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(10.7318940317\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40q - 4q^{3} + 40q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 40q - 4q^{3} + 40q^{7} + 24q^{13} - 16q^{19} - 4q^{21} + 32q^{27} + 24q^{33} - 8q^{37} + 64q^{39} - 24q^{43} - 28q^{45} + 40q^{49} + 32q^{51} - 16q^{55} - 48q^{61} - 40q^{67} + 4q^{69} - 40q^{75} + 56q^{81} - 48q^{85} - 32q^{87} + 24q^{91} + 56q^{93} + 16q^{97} - 24q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1 0 −1.73158 0.0404130i 0 0.0573500 + 0.0573500i 0 1.00000 0 2.99673 + 0.139957i 0
239.2 0 −1.72790 0.119805i 0 0.793669 + 0.793669i 0 1.00000 0 2.97129 + 0.414021i 0
239.3 0 −1.61364 0.629429i 0 −3.08691 3.08691i 0 1.00000 0 2.20764 + 2.03134i 0
239.4 0 −1.42461 + 0.985133i 0 −0.766553 0.766553i 0 1.00000 0 1.05903 2.80686i 0
239.5 0 −1.26429 + 1.18388i 0 −2.84277 2.84277i 0 1.00000 0 0.196841 2.99354i 0
239.6 0 −1.18388 + 1.26429i 0 2.84277 + 2.84277i 0 1.00000 0 −0.196841 2.99354i 0
239.7 0 −0.985133 + 1.42461i 0 0.766553 + 0.766553i 0 1.00000 0 −1.05903 2.80686i 0
239.8 0 −0.827739 1.52146i 0 −1.23935 1.23935i 0 1.00000 0 −1.62970 + 2.51875i 0
239.9 0 −0.567369 1.63649i 0 −0.132854 0.132854i 0 1.00000 0 −2.35619 + 1.85698i 0
239.10 0 −0.408017 1.68331i 0 2.26080 + 2.26080i 0 1.00000 0 −2.66704 + 1.37363i 0
239.11 0 0.0404130 + 1.73158i 0 −0.0573500 0.0573500i 0 1.00000 0 −2.99673 + 0.139957i 0
239.12 0 0.0806676 1.73017i 0 −1.66193 1.66193i 0 1.00000 0 −2.98699 0.279137i 0
239.13 0 0.119805 + 1.72790i 0 −0.793669 0.793669i 0 1.00000 0 −2.97129 + 0.414021i 0
239.14 0 0.629429 + 1.61364i 0 3.08691 + 3.08691i 0 1.00000 0 −2.20764 + 2.03134i 0
239.15 0 0.714681 1.57773i 0 1.31983 + 1.31983i 0 1.00000 0 −1.97846 2.25515i 0
239.16 0 1.52146 + 0.827739i 0 1.23935 + 1.23935i 0 1.00000 0 1.62970 + 2.51875i 0
239.17 0 1.57773 0.714681i 0 −1.31983 1.31983i 0 1.00000 0 1.97846 2.25515i 0
239.18 0 1.63649 + 0.567369i 0 0.132854 + 0.132854i 0 1.00000 0 2.35619 + 1.85698i 0
239.19 0 1.68331 + 0.408017i 0 −2.26080 2.26080i 0 1.00000 0 2.66704 + 1.37363i 0
239.20 0 1.73017 0.0806676i 0 1.66193 + 1.66193i 0 1.00000 0 2.98699 0.279137i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 911.20
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 \(T_{5}^{40} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(1344, [\chi])\).