Properties

Label 6005.2.a.e
Level 6005
Weight 2
Character orbit 6005.a
Self dual Yes
Analytic conductor 47.950
Analytic rank 1
Dimension 88
CM No

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Newspace parameters

Level: \( N \) = \( 6005 = 5 \cdot 1201 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6005.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(47.9501664138\)
Analytic rank: \(1\)
Dimension: \(88\)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \(88q \) \(\mathstrut -\mathstrut 14q^{2} \) \(\mathstrut -\mathstrut 34q^{3} \) \(\mathstrut +\mathstrut 66q^{4} \) \(\mathstrut +\mathstrut 88q^{5} \) \(\mathstrut -\mathstrut q^{6} \) \(\mathstrut -\mathstrut 35q^{7} \) \(\mathstrut -\mathstrut 39q^{8} \) \(\mathstrut +\mathstrut 72q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(88q \) \(\mathstrut -\mathstrut 14q^{2} \) \(\mathstrut -\mathstrut 34q^{3} \) \(\mathstrut +\mathstrut 66q^{4} \) \(\mathstrut +\mathstrut 88q^{5} \) \(\mathstrut -\mathstrut q^{6} \) \(\mathstrut -\mathstrut 35q^{7} \) \(\mathstrut -\mathstrut 39q^{8} \) \(\mathstrut +\mathstrut 72q^{9} \) \(\mathstrut -\mathstrut 14q^{10} \) \(\mathstrut -\mathstrut 26q^{11} \) \(\mathstrut -\mathstrut 64q^{12} \) \(\mathstrut -\mathstrut 31q^{13} \) \(\mathstrut -\mathstrut 17q^{14} \) \(\mathstrut -\mathstrut 34q^{15} \) \(\mathstrut +\mathstrut 34q^{16} \) \(\mathstrut -\mathstrut 31q^{17} \) \(\mathstrut -\mathstrut 42q^{18} \) \(\mathstrut -\mathstrut 56q^{19} \) \(\mathstrut +\mathstrut 66q^{20} \) \(\mathstrut -\mathstrut q^{21} \) \(\mathstrut -\mathstrut 49q^{22} \) \(\mathstrut -\mathstrut 74q^{23} \) \(\mathstrut -\mathstrut 3q^{24} \) \(\mathstrut +\mathstrut 88q^{25} \) \(\mathstrut -\mathstrut q^{26} \) \(\mathstrut -\mathstrut 130q^{27} \) \(\mathstrut -\mathstrut 57q^{28} \) \(\mathstrut -\mathstrut 6q^{29} \) \(\mathstrut -\mathstrut q^{30} \) \(\mathstrut -\mathstrut 37q^{31} \) \(\mathstrut -\mathstrut 87q^{32} \) \(\mathstrut -\mathstrut 43q^{33} \) \(\mathstrut -\mathstrut 35q^{34} \) \(\mathstrut -\mathstrut 35q^{35} \) \(\mathstrut +\mathstrut 53q^{36} \) \(\mathstrut -\mathstrut 67q^{37} \) \(\mathstrut -\mathstrut 40q^{38} \) \(\mathstrut -\mathstrut 21q^{39} \) \(\mathstrut -\mathstrut 39q^{40} \) \(\mathstrut +\mathstrut 2q^{41} \) \(\mathstrut -\mathstrut 15q^{42} \) \(\mathstrut -\mathstrut 136q^{43} \) \(\mathstrut -\mathstrut 15q^{44} \) \(\mathstrut +\mathstrut 72q^{45} \) \(\mathstrut -\mathstrut 16q^{46} \) \(\mathstrut -\mathstrut 139q^{47} \) \(\mathstrut -\mathstrut 71q^{48} \) \(\mathstrut +\mathstrut 41q^{49} \) \(\mathstrut -\mathstrut 14q^{50} \) \(\mathstrut -\mathstrut 71q^{51} \) \(\mathstrut -\mathstrut 71q^{52} \) \(\mathstrut -\mathstrut 75q^{53} \) \(\mathstrut +\mathstrut 26q^{54} \) \(\mathstrut -\mathstrut 26q^{55} \) \(\mathstrut -\mathstrut 22q^{56} \) \(\mathstrut -\mathstrut 34q^{57} \) \(\mathstrut -\mathstrut 65q^{58} \) \(\mathstrut -\mathstrut 41q^{59} \) \(\mathstrut -\mathstrut 64q^{60} \) \(\mathstrut -\mathstrut 11q^{61} \) \(\mathstrut -\mathstrut 30q^{62} \) \(\mathstrut -\mathstrut 114q^{63} \) \(\mathstrut -\mathstrut 33q^{64} \) \(\mathstrut -\mathstrut 31q^{65} \) \(\mathstrut +\mathstrut 24q^{66} \) \(\mathstrut -\mathstrut 209q^{67} \) \(\mathstrut -\mathstrut 42q^{68} \) \(\mathstrut -\mathstrut 22q^{69} \) \(\mathstrut -\mathstrut 17q^{70} \) \(\mathstrut -\mathstrut 43q^{71} \) \(\mathstrut -\mathstrut 80q^{72} \) \(\mathstrut -\mathstrut 50q^{73} \) \(\mathstrut +\mathstrut 9q^{74} \) \(\mathstrut -\mathstrut 34q^{75} \) \(\mathstrut -\mathstrut 62q^{76} \) \(\mathstrut -\mathstrut 49q^{77} \) \(\mathstrut -\mathstrut 19q^{78} \) \(\mathstrut -\mathstrut 77q^{79} \) \(\mathstrut +\mathstrut 34q^{80} \) \(\mathstrut +\mathstrut 72q^{81} \) \(\mathstrut -\mathstrut 107q^{82} \) \(\mathstrut -\mathstrut 113q^{83} \) \(\mathstrut +\mathstrut 19q^{84} \) \(\mathstrut -\mathstrut 31q^{85} \) \(\mathstrut +\mathstrut 14q^{86} \) \(\mathstrut -\mathstrut 87q^{87} \) \(\mathstrut -\mathstrut 107q^{88} \) \(\mathstrut -\mathstrut 5q^{89} \) \(\mathstrut -\mathstrut 42q^{90} \) \(\mathstrut -\mathstrut 159q^{91} \) \(\mathstrut -\mathstrut 100q^{92} \) \(\mathstrut -\mathstrut 82q^{93} \) \(\mathstrut -\mathstrut 31q^{94} \) \(\mathstrut -\mathstrut 56q^{95} \) \(\mathstrut +\mathstrut 58q^{96} \) \(\mathstrut -\mathstrut 105q^{97} \) \(\mathstrut -\mathstrut 29q^{98} \) \(\mathstrut -\mathstrut 68q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.76833 −0.546396 5.66368 1.00000 1.51261 −1.48260 −10.1423 −2.70145 −2.76833
1.2 −2.70073 −2.65354 5.29393 1.00000 7.16648 −4.13839 −8.89601 4.04126 −2.70073
1.3 −2.69844 0.716329 5.28159 1.00000 −1.93297 1.67359 −8.85517 −2.48687 −2.69844
1.4 −2.69778 −3.22929 5.27799 1.00000 8.71190 2.27707 −8.84328 7.42831 −2.69778
1.5 −2.63715 2.33888 4.95457 1.00000 −6.16798 −0.829927 −7.79166 2.47036 −2.63715
1.6 −2.56191 2.09483 4.56337 1.00000 −5.36676 −0.643691 −6.56712 1.38832 −2.56191
1.7 −2.49547 −2.51949 4.22735 1.00000 6.28731 2.97292 −5.55828 3.34784 −2.49547
1.8 −2.45453 −0.943055 4.02473 1.00000 2.31476 −0.315218 −4.96977 −2.11065 −2.45453
1.9 −2.32346 1.00159 3.39848 1.00000 −2.32717 3.48702 −3.24933 −1.99681 −2.32346
1.10 −2.28766 −2.82727 3.23337 1.00000 6.46782 −3.61501 −2.82152 4.99347 −2.28766
1.11 −2.26517 2.21267 3.13101 1.00000 −5.01209 −3.04532 −2.56193 1.89592 −2.26517
1.12 −2.25928 −0.374349 3.10434 1.00000 0.845758 −3.16273 −2.49501 −2.85986 −2.25928
1.13 −2.21323 −3.30657 2.89841 1.00000 7.31821 −1.75850 −1.98838 7.93340 −2.21323
1.14 −2.21193 0.112500 2.89262 1.00000 −0.248842 −5.20479 −1.97441 −2.98734 −2.21193
1.15 −2.17289 −1.65848 2.72145 1.00000 3.60369 5.14399 −1.56762 −0.249442 −2.17289
1.16 −2.07691 −0.413168 2.31355 1.00000 0.858111 2.69878 −0.651210 −2.82929 −2.07691
1.17 −1.97194 2.67165 1.88854 1.00000 −5.26832 −0.758910 0.219801 4.13769 −1.97194
1.18 −1.97101 −1.60435 1.88489 1.00000 3.16220 2.33314 0.226875 −0.426059 −1.97101
1.19 −1.93565 −2.84784 1.74674 1.00000 5.51242 −0.215804 0.490231 5.11019 −1.93565
1.20 −1.89204 0.756696 1.57982 1.00000 −1.43170 1.91014 0.794996 −2.42741 −1.89204
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.88
Significant digits:
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Inner twists

This newform does not have CM; other inner twists have not been computed.

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(1201\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{88} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6005))\).