Properties

Label 336.2.s.d
Level 336
Weight 2
Character orbit 336.s
Analytic conductor 2.683
Analytic rank 0
Dimension 48
CM No

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Newspace parameters

Level: \( N \) = \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 336.s (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(2.68297350792\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48q + 6q^{6} + 48q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q + 6q^{6} + 48q^{7} - 16q^{10} + 6q^{12} - 28q^{16} - 20q^{18} + 8q^{19} - 8q^{22} - 38q^{24} - 12q^{27} - 24q^{30} + 12q^{34} - 4q^{36} + 16q^{37} - 24q^{39} + 60q^{40} + 6q^{42} - 48q^{43} + 20q^{45} + 52q^{46} + 62q^{48} + 48q^{49} + 12q^{52} + 14q^{54} - 32q^{55} - 100q^{58} - 16q^{60} + 8q^{61} - 60q^{64} - 96q^{66} - 16q^{67} - 28q^{69} - 16q^{70} - 80q^{72} - 12q^{75} + 4q^{76} - 56q^{78} + 4q^{82} + 6q^{84} - 48q^{85} + 56q^{87} + 116q^{88} + 68q^{90} - 64q^{93} + 48q^{94} + 62q^{96} + 32q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1 −1.33839 0.456860i −1.70974 + 0.277087i 1.58256 + 1.22291i −0.459458 + 0.459458i 2.41489 + 0.410265i 1.00000 −1.55938 2.35973i 2.84645 0.947494i 0.824840 0.405024i
155.2 −1.31976 0.508179i 1.45100 0.945839i 1.48351 + 1.34134i 1.69539 1.69539i −2.39561 + 0.510910i 1.00000 −1.27623 2.52413i 1.21078 2.74482i −3.09906 + 1.37594i
155.3 −1.30954 + 0.533943i 1.26449 + 1.18367i 1.42981 1.39844i 1.97913 1.97913i −2.28791 0.874897i 1.00000 −1.12571 + 2.59476i 0.197870 + 2.99347i −1.53501 + 3.64849i
155.4 −1.29139 + 0.576471i 1.71763 0.223071i 1.33536 1.48889i −2.27022 + 2.27022i −2.08953 + 1.27823i 1.00000 −0.866165 + 2.69254i 2.90048 0.766304i 1.62302 4.24045i
155.5 −1.20363 0.742471i −0.505560 1.65663i 0.897474 + 1.78733i −2.00763 + 2.00763i −0.621486 + 2.36934i 1.00000 0.246807 2.81764i −2.48882 + 1.67505i 3.90706 0.925846i
155.6 −1.15900 + 0.810376i −1.50282 0.861124i 0.686583 1.87846i −1.19856 + 1.19856i 2.43961 0.219802i 1.00000 0.726504 + 2.73353i 1.51693 + 2.58823i 0.417853 2.36042i
155.7 −0.809995 1.15927i 0.594785 + 1.62672i −0.687816 + 1.87801i 0.0563417 0.0563417i 1.40404 2.00716i 1.00000 2.73425 0.723812i −2.29246 + 1.93510i −0.110952 0.0196788i
155.8 −0.786529 + 1.17532i −1.09085 + 1.34538i −0.762743 1.84884i 1.07379 1.07379i −0.723260 2.34028i 1.00000 2.77290 + 0.557705i −0.620087 2.93522i 0.417477 + 2.10661i
155.9 −0.720078 + 1.21716i 0.399703 1.68530i −0.962977 1.75291i 2.09830 2.09830i 1.76347 + 1.70005i 1.00000 2.82699 + 0.0901270i −2.68048 1.34724i 1.04303 + 4.06491i
155.10 −0.641365 1.26042i −1.71202 0.262630i −1.17730 + 1.61678i 2.76208 2.76208i 0.767010 + 2.32631i 1.00000 2.79289 + 0.446945i 2.86205 + 0.899256i −5.25287 1.70987i
155.11 −0.277681 + 1.38668i 0.490427 + 1.66117i −1.84579 0.770111i −2.27018 + 2.27018i −2.43970 + 0.218793i 1.00000 1.58044 2.34568i −2.51896 + 1.62936i −2.51763 3.77840i
155.12 −0.103263 1.41044i −1.15087 + 1.29441i −1.97867 + 0.291291i −0.186466 + 0.186466i 1.94453 + 1.48957i 1.00000 0.615171 + 2.76072i −0.350987 2.97940i 0.282253 + 0.243743i
155.13 0.103263 + 1.41044i 1.29441 1.15087i −1.97867 + 0.291291i 0.186466 0.186466i 1.75690 + 1.70684i 1.00000 −0.615171 2.76072i 0.350987 2.97940i 0.282253 + 0.243743i
155.14 0.277681 1.38668i 1.66117 + 0.490427i −1.84579 0.770111i 2.27018 2.27018i 1.14134 2.16733i 1.00000 −1.58044 + 2.34568i 2.51896 + 1.62936i −2.51763 3.77840i
155.15 0.641365 + 1.26042i −0.262630 1.71202i −1.17730 + 1.61678i −2.76208 + 2.76208i 1.98942 1.42906i 1.00000 −2.79289 0.446945i −2.86205 + 0.899256i −5.25287 1.70987i
155.16 0.720078 1.21716i −1.68530 + 0.399703i −0.962977 1.75291i −2.09830 + 2.09830i −0.727043 + 2.33910i 1.00000 −2.82699 0.0901270i 2.68048 1.34724i 1.04303 + 4.06491i
155.17 0.786529 1.17532i 1.34538 1.09085i −0.762743 1.84884i −1.07379 + 1.07379i −0.223917 2.43923i 1.00000 −2.77290 0.557705i 0.620087 2.93522i 0.417477 + 2.10661i
155.18 0.809995 + 1.15927i 1.62672 + 0.594785i −0.687816 + 1.87801i −0.0563417 + 0.0563417i 0.628122 + 2.36759i 1.00000 −2.73425 + 0.723812i 2.29246 + 1.93510i −0.110952 0.0196788i
155.19 1.15900 0.810376i −0.861124 1.50282i 0.686583 1.87846i 1.19856 1.19856i −2.21589 1.04394i 1.00000 −0.726504 2.73353i −1.51693 + 2.58823i 0.417853 2.36042i
155.20 1.20363 + 0.742471i −1.65663 0.505560i 0.897474 + 1.78733i 2.00763 2.00763i −1.61861 1.83851i 1.00000 −0.246807 + 2.81764i 2.48882 + 1.67505i 3.90706 0.925846i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 323.24
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 \(S_{2}^{\mathrm{new}}(336, [\chi])\).