Properties

Label 20.8.e.b
Level 20
Weight 8
Character orbit 20.e
Analytic conductor 6.248
Analytic rank 0
Dimension 36
CM No

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Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 20.e (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(6.24770050968\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \(36q \) \(\mathstrut +\mathstrut 14q^{2} \) \(\mathstrut -\mathstrut 560q^{5} \) \(\mathstrut +\mathstrut 104q^{6} \) \(\mathstrut -\mathstrut 3052q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(36q \) \(\mathstrut +\mathstrut 14q^{2} \) \(\mathstrut -\mathstrut 560q^{5} \) \(\mathstrut +\mathstrut 104q^{6} \) \(\mathstrut -\mathstrut 3052q^{8} \) \(\mathstrut +\mathstrut 390q^{10} \) \(\mathstrut +\mathstrut 10960q^{12} \) \(\mathstrut -\mathstrut 28564q^{13} \) \(\mathstrut -\mathstrut 12104q^{16} \) \(\mathstrut -\mathstrut 27012q^{17} \) \(\mathstrut +\mathstrut 95298q^{18} \) \(\mathstrut +\mathstrut 97180q^{20} \) \(\mathstrut -\mathstrut 2352q^{21} \) \(\mathstrut -\mathstrut 191320q^{22} \) \(\mathstrut -\mathstrut 112580q^{25} \) \(\mathstrut +\mathstrut 604628q^{26} \) \(\mathstrut -\mathstrut 142800q^{28} \) \(\mathstrut -\mathstrut 252760q^{30} \) \(\mathstrut -\mathstrut 73976q^{32} \) \(\mathstrut -\mathstrut 248080q^{33} \) \(\mathstrut -\mathstrut 328580q^{36} \) \(\mathstrut -\mathstrut 108732q^{37} \) \(\mathstrut -\mathstrut 406560q^{38} \) \(\mathstrut -\mathstrut 680660q^{40} \) \(\mathstrut +\mathstrut 2019472q^{41} \) \(\mathstrut +\mathstrut 1616840q^{42} \) \(\mathstrut -\mathstrut 827660q^{45} \) \(\mathstrut -\mathstrut 2690616q^{46} \) \(\mathstrut +\mathstrut 549280q^{48} \) \(\mathstrut +\mathstrut 1343690q^{50} \) \(\mathstrut -\mathstrut 6025332q^{52} \) \(\mathstrut +\mathstrut 5672196q^{53} \) \(\mathstrut +\mathstrut 4064064q^{56} \) \(\mathstrut -\mathstrut 1796160q^{57} \) \(\mathstrut +\mathstrut 3463736q^{58} \) \(\mathstrut +\mathstrut 3280240q^{60} \) \(\mathstrut -\mathstrut 5624928q^{61} \) \(\mathstrut +\mathstrut 2750760q^{62} \) \(\mathstrut +\mathstrut 75940q^{65} \) \(\mathstrut -\mathstrut 2175120q^{66} \) \(\mathstrut -\mathstrut 5673084q^{68} \) \(\mathstrut -\mathstrut 11417280q^{70} \) \(\mathstrut +\mathstrut 4328964q^{72} \) \(\mathstrut -\mathstrut 22930604q^{73} \) \(\mathstrut -\mathstrut 4045600q^{76} \) \(\mathstrut +\mathstrut 10982160q^{77} \) \(\mathstrut +\mathstrut 27127080q^{78} \) \(\mathstrut +\mathstrut 34488560q^{80} \) \(\mathstrut +\mathstrut 21593004q^{81} \) \(\mathstrut -\mathstrut 13289192q^{82} \) \(\mathstrut -\mathstrut 28299060q^{85} \) \(\mathstrut -\mathstrut 2245656q^{86} \) \(\mathstrut -\mathstrut 38612320q^{88} \) \(\mathstrut -\mathstrut 38642970q^{90} \) \(\mathstrut -\mathstrut 22809360q^{92} \) \(\mathstrut +\mathstrut 31343120q^{93} \) \(\mathstrut +\mathstrut 49033664q^{96} \) \(\mathstrut -\mathstrut 11169292q^{97} \) \(\mathstrut +\mathstrut 45095638q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −11.3062 + 0.411421i −2.83352 + 2.83352i 127.661 9.30323i −150.277 235.673i 30.8707 33.2022i 411.746 + 411.746i −1439.54 + 157.707i 2170.94i 1796.03 + 2602.74i
3.2 −11.1445 1.94961i −31.8653 + 31.8653i 120.398 + 43.4547i 92.5871 + 263.728i 417.246 292.996i −789.211 789.211i −1257.05 719.009i 156.211i −517.666 3119.62i
3.3 −9.99887 5.29363i 51.5668 51.5668i 71.9550 + 105.861i 278.010 28.9057i −788.585 + 242.634i 645.384 + 645.384i −159.082 1439.39i 3131.26i −2932.80 1182.66i
3.4 −9.35451 + 6.36342i 56.5129 56.5129i 47.0138 119.053i −240.823 + 141.877i −169.035 + 888.266i −764.897 764.897i 317.795 + 1412.85i 4200.41i 1349.96 2859.65i
3.5 −6.47794 9.27558i −53.6136 + 53.6136i −44.0727 + 120.173i 17.4122 278.966i 844.603 + 149.992i 308.666 + 308.666i 1400.18 369.675i 3561.84i −2700.36 + 1645.61i
3.6 −6.36342 + 9.35451i −56.5129 + 56.5129i −47.0138 119.053i −240.823 + 141.877i −169.035 888.266i 764.897 + 764.897i 1412.85 + 317.795i 4200.41i 205.270 3155.61i
3.7 −5.78656 9.72192i 13.4355 13.4355i −61.0315 + 112.513i −222.545 + 169.112i −208.364 52.8736i −9.12405 9.12405i 1447.00 57.7198i 1825.97i 2931.86 + 1184.99i
3.8 −0.411421 + 11.3062i 2.83352 2.83352i −127.661 9.30323i −150.277 235.673i 30.8707 + 33.2022i −411.746 411.746i 157.707 1439.54i 2170.94i 2726.40 1602.11i
3.9 0.138397 11.3129i 19.3526 19.3526i −127.962 3.13132i 191.974 203.153i −216.255 221.612i −939.327 939.327i −53.1337 + 1447.18i 1437.95i −2271.67 2199.89i
3.10 1.94961 + 11.1445i 31.8653 31.8653i −120.398 + 43.4547i 92.5871 + 263.728i 417.246 + 292.996i 789.211 + 789.211i −719.009 1257.05i 156.211i −2758.60 + 1546.00i
3.11 3.68683 10.6961i −32.6647 + 32.6647i −100.815 78.8696i 145.214 + 238.826i 228.957 + 469.815i 669.751 + 669.751i −1215.29 + 787.548i 53.0351i 3089.89 672.721i
3.12 5.29363 + 9.99887i −51.5668 + 51.5668i −71.9550 + 105.861i 278.010 28.9057i −788.585 242.634i −645.384 645.384i −1439.39 159.082i 3131.26i 1760.71 + 2626.77i
3.13 6.52813 9.24032i 41.8246 41.8246i −42.7671 120.644i −251.552 121.846i −113.436 659.509i 1013.70 + 1013.70i −1393.98 392.398i 1311.59i −2768.06 + 1528.99i
3.14 9.24032 6.52813i −41.8246 + 41.8246i 42.7671 120.644i −251.552 121.846i −113.436 + 659.509i −1013.70 1013.70i −392.398 1393.98i 1311.59i −3119.85 + 516.264i
3.15 9.27558 + 6.47794i 53.6136 53.6136i 44.0727 + 120.173i 17.4122 278.966i 844.603 149.992i −308.666 308.666i −369.675 + 1400.18i 3561.84i 1968.63 2474.77i
3.16 9.72192 + 5.78656i −13.4355 + 13.4355i 61.0315 + 112.513i −222.545 + 169.112i −208.364 + 52.8736i 9.12405 + 9.12405i −57.7198 + 1447.00i 1825.97i −3142.14 + 356.325i
3.17 10.6961 3.68683i 32.6647 32.6647i 100.815 78.8696i 145.214 + 238.826i 228.957 469.815i −669.751 669.751i 787.548 1215.29i 53.0351i 2433.74 + 2019.13i
3.18 11.3129 0.138397i −19.3526 + 19.3526i 127.962 3.13132i 191.974 203.153i −216.255 + 221.612i 939.327 + 939.327i 1447.18 53.1337i 1437.95i 2143.66 2324.81i
7.1 −11.3062 0.411421i −2.83352 2.83352i 127.661 + 9.30323i −150.277 + 235.673i 30.8707 + 33.2022i 411.746 411.746i −1439.54 157.707i 2170.94i 1796.03 2602.74i
7.2 −11.1445 + 1.94961i −31.8653 31.8653i 120.398 43.4547i 92.5871 263.728i 417.246 + 292.996i −789.211 + 789.211i −1257.05 + 719.009i 156.211i −517.666 + 3119.62i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.18
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 \(T_{3}^{36} + \cdots\) acting on \(S_{8}^{\mathrm{new}}(20, [\chi])\).