Properties

Label 1344.2.s.d
Level 1344
Weight 2
Character orbit 1344.s
Analytic conductor 10.732
Analytic rank 0
Dimension 48
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: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48q - 48q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q - 48q^{7} - 8q^{19} + 12q^{27} + 16q^{37} + 24q^{39} + 48q^{43} + 20q^{45} + 48q^{49} + 32q^{55} + 8q^{61} + 16q^{67} - 28q^{69} + 12q^{75} - 48q^{85} - 56q^{87} - 64q^{93} - 32q^{99} + O(q^{100}) \)

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 −1.71763 0.223071i 0 −2.27022 2.27022i 0 −1.00000 0 2.90048 + 0.766304i 0
239.2 0 −1.66117 + 0.490427i 0 2.27018 + 2.27018i 0 −1.00000 0 2.51896 1.62936i 0
239.3 0 −1.62672 + 0.594785i 0 −0.0563417 0.0563417i 0 −1.00000 0 2.29246 1.93510i 0
239.4 0 −1.45100 0.945839i 0 1.69539 + 1.69539i 0 −1.00000 0 1.21078 + 2.74482i 0
239.5 0 −1.34538 1.09085i 0 −1.07379 1.07379i 0 −1.00000 0 0.620087 + 2.93522i 0
239.6 0 −1.29441 1.15087i 0 0.186466 + 0.186466i 0 −1.00000 0 0.350987 + 2.97940i 0
239.7 0 −1.26449 + 1.18367i 0 1.97913 + 1.97913i 0 −1.00000 0 0.197870 2.99347i 0
239.8 0 −1.18367 + 1.26449i 0 −1.97913 1.97913i 0 −1.00000 0 −0.197870 2.99347i 0
239.9 0 −0.594785 + 1.62672i 0 0.0563417 + 0.0563417i 0 −1.00000 0 −2.29246 1.93510i 0
239.10 0 −0.490427 + 1.66117i 0 −2.27018 2.27018i 0 −1.00000 0 −2.51896 1.62936i 0
239.11 0 −0.399703 1.68530i 0 2.09830 + 2.09830i 0 −1.00000 0 −2.68048 + 1.34724i 0
239.12 0 −0.277087 1.70974i 0 0.459458 + 0.459458i 0 −1.00000 0 −2.84645 + 0.947494i 0
239.13 0 0.223071 + 1.71763i 0 2.27022 + 2.27022i 0 −1.00000 0 −2.90048 + 0.766304i 0
239.14 0 0.262630 1.71202i 0 −2.76208 2.76208i 0 −1.00000 0 −2.86205 0.899256i 0
239.15 0 0.505560 1.65663i 0 −2.00763 2.00763i 0 −1.00000 0 −2.48882 1.67505i 0
239.16 0 0.861124 1.50282i 0 1.19856 + 1.19856i 0 −1.00000 0 −1.51693 2.58823i 0
239.17 0 0.945839 + 1.45100i 0 −1.69539 1.69539i 0 −1.00000 0 −1.21078 + 2.74482i 0
239.18 0 1.09085 + 1.34538i 0 1.07379 + 1.07379i 0 −1.00000 0 −0.620087 + 2.93522i 0
239.19 0 1.15087 + 1.29441i 0 −0.186466 0.186466i 0 −1.00000 0 −0.350987 + 2.97940i 0
239.20 0 1.50282 0.861124i 0 −1.19856 1.19856i 0 −1.00000 0 1.51693 2.58823i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 911.24
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 \(T_{5}^{48} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(1344, [\chi])\).