Properties

Label 6005.2.a.g
Level 6005
Weight 2
Character orbit 6005.a
Self dual Yes
Analytic conductor 47.950
Analytic rank 0
Dimension 113
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: \(0\)
Dimension: \(113\)
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.

\(\operatorname{Tr}(f)(q) = \) \(113q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 6q^{3} \) \(\mathstrut +\mathstrut 141q^{4} \) \(\mathstrut -\mathstrut 113q^{5} \) \(\mathstrut +\mathstrut 7q^{6} \) \(\mathstrut +\mathstrut 7q^{7} \) \(\mathstrut -\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut 141q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(113q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 6q^{3} \) \(\mathstrut +\mathstrut 141q^{4} \) \(\mathstrut -\mathstrut 113q^{5} \) \(\mathstrut +\mathstrut 7q^{6} \) \(\mathstrut +\mathstrut 7q^{7} \) \(\mathstrut -\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut 141q^{9} \) \(\mathstrut +\mathstrut 3q^{10} \) \(\mathstrut +\mathstrut 38q^{11} \) \(\mathstrut -\mathstrut 4q^{12} \) \(\mathstrut +\mathstrut 17q^{13} \) \(\mathstrut +\mathstrut 23q^{14} \) \(\mathstrut -\mathstrut 6q^{15} \) \(\mathstrut +\mathstrut 193q^{16} \) \(\mathstrut -\mathstrut 11q^{17} \) \(\mathstrut -\mathstrut 3q^{18} \) \(\mathstrut +\mathstrut 76q^{19} \) \(\mathstrut -\mathstrut 141q^{20} \) \(\mathstrut +\mathstrut 19q^{21} \) \(\mathstrut +\mathstrut 41q^{22} \) \(\mathstrut -\mathstrut 28q^{23} \) \(\mathstrut +\mathstrut 29q^{24} \) \(\mathstrut +\mathstrut 113q^{25} \) \(\mathstrut +\mathstrut 21q^{26} \) \(\mathstrut +\mathstrut 18q^{27} \) \(\mathstrut +\mathstrut 29q^{28} \) \(\mathstrut +\mathstrut 24q^{29} \) \(\mathstrut -\mathstrut 7q^{30} \) \(\mathstrut +\mathstrut 59q^{31} \) \(\mathstrut -\mathstrut 22q^{32} \) \(\mathstrut +\mathstrut 3q^{33} \) \(\mathstrut +\mathstrut 55q^{34} \) \(\mathstrut -\mathstrut 7q^{35} \) \(\mathstrut +\mathstrut 232q^{36} \) \(\mathstrut +\mathstrut 41q^{37} \) \(\mathstrut -\mathstrut 6q^{38} \) \(\mathstrut +\mathstrut 55q^{39} \) \(\mathstrut +\mathstrut 12q^{40} \) \(\mathstrut +\mathstrut 24q^{41} \) \(\mathstrut +\mathstrut 17q^{42} \) \(\mathstrut +\mathstrut 136q^{43} \) \(\mathstrut +\mathstrut 85q^{44} \) \(\mathstrut -\mathstrut 141q^{45} \) \(\mathstrut +\mathstrut 84q^{46} \) \(\mathstrut -\mathstrut 91q^{47} \) \(\mathstrut -\mathstrut 19q^{48} \) \(\mathstrut +\mathstrut 198q^{49} \) \(\mathstrut -\mathstrut 3q^{50} \) \(\mathstrut +\mathstrut 97q^{51} \) \(\mathstrut +\mathstrut 45q^{52} \) \(\mathstrut +\mathstrut 9q^{53} \) \(\mathstrut +\mathstrut 54q^{54} \) \(\mathstrut -\mathstrut 38q^{55} \) \(\mathstrut +\mathstrut 98q^{56} \) \(\mathstrut +\mathstrut 22q^{57} \) \(\mathstrut +\mathstrut 69q^{58} \) \(\mathstrut +\mathstrut 59q^{59} \) \(\mathstrut +\mathstrut 4q^{60} \) \(\mathstrut +\mathstrut 51q^{61} \) \(\mathstrut -\mathstrut 30q^{62} \) \(\mathstrut -\mathstrut 22q^{63} \) \(\mathstrut +\mathstrut 298q^{64} \) \(\mathstrut -\mathstrut 17q^{65} \) \(\mathstrut +\mathstrut 76q^{66} \) \(\mathstrut +\mathstrut 201q^{67} \) \(\mathstrut -\mathstrut 34q^{68} \) \(\mathstrut +\mathstrut 42q^{69} \) \(\mathstrut -\mathstrut 23q^{70} \) \(\mathstrut +\mathstrut 69q^{71} \) \(\mathstrut -\mathstrut 7q^{72} \) \(\mathstrut +\mathstrut 30q^{73} \) \(\mathstrut +\mathstrut 35q^{74} \) \(\mathstrut +\mathstrut 6q^{75} \) \(\mathstrut +\mathstrut 170q^{76} \) \(\mathstrut -\mathstrut 37q^{77} \) \(\mathstrut -\mathstrut 11q^{78} \) \(\mathstrut +\mathstrut 143q^{79} \) \(\mathstrut -\mathstrut 193q^{80} \) \(\mathstrut +\mathstrut 197q^{81} \) \(\mathstrut +\mathstrut 55q^{82} \) \(\mathstrut -\mathstrut 15q^{83} \) \(\mathstrut +\mathstrut 83q^{84} \) \(\mathstrut +\mathstrut 11q^{85} \) \(\mathstrut +\mathstrut 78q^{86} \) \(\mathstrut -\mathstrut 51q^{87} \) \(\mathstrut +\mathstrut 113q^{88} \) \(\mathstrut +\mathstrut 53q^{89} \) \(\mathstrut +\mathstrut 3q^{90} \) \(\mathstrut +\mathstrut 217q^{91} \) \(\mathstrut -\mathstrut 40q^{92} \) \(\mathstrut +\mathstrut 36q^{93} \) \(\mathstrut +\mathstrut 81q^{94} \) \(\mathstrut -\mathstrut 76q^{95} \) \(\mathstrut +\mathstrut 66q^{96} \) \(\mathstrut +\mathstrut 63q^{97} \) \(\mathstrut -\mathstrut 62q^{98} \) \(\mathstrut +\mathstrut 184q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.81643 −3.03025 5.93229 −1.00000 8.53449 1.00907 −11.0750 6.18240 2.81643
1.2 −2.80393 2.29330 5.86204 −1.00000 −6.43026 2.82627 −10.8289 2.25923 2.80393
1.3 −2.78685 −1.57697 5.76652 −1.00000 4.39478 −0.289151 −10.4967 −0.513163 2.78685
1.4 −2.77794 1.80495 5.71693 −1.00000 −5.01404 −4.50639 −10.3254 0.257853 2.77794
1.5 −2.68650 2.64233 5.21726 −1.00000 −7.09862 −4.41903 −8.64317 3.98193 2.68650
1.6 −2.67625 1.85244 5.16230 −1.00000 −4.95758 4.71146 −8.46311 0.431523 2.67625
1.7 −2.66122 −2.12417 5.08210 −1.00000 5.65288 −3.49276 −8.20215 1.51209 2.66122
1.8 −2.62680 −1.30643 4.90009 −1.00000 3.43174 −4.40647 −7.61796 −1.29324 2.62680
1.9 −2.61532 0.118458 4.83989 −1.00000 −0.309805 −0.279288 −7.42721 −2.98597 2.61532
1.10 −2.57539 −2.35580 4.63261 −1.00000 6.06709 3.76299 −6.78000 2.54978 2.57539
1.11 −2.56745 −0.556289 4.59182 −1.00000 1.42824 3.15883 −6.65437 −2.69054 2.56745
1.12 −2.49553 −3.37079 4.22769 −1.00000 8.41191 −2.54255 −5.55927 8.36220 2.49553
1.13 −2.48400 0.156242 4.17024 −1.00000 −0.388105 1.49577 −5.39087 −2.97559 2.48400
1.14 −2.47572 3.00665 4.12920 −1.00000 −7.44362 1.05496 −5.27130 6.03992 2.47572
1.15 −2.24487 −2.47232 3.03945 −1.00000 5.55004 −1.67086 −2.33343 3.11237 2.24487
1.16 −2.21162 0.635696 2.89127 −1.00000 −1.40592 −2.43179 −1.97116 −2.59589 2.21162
1.17 −2.18171 2.09909 2.75987 −1.00000 −4.57961 −2.68153 −1.65781 1.40617 2.18171
1.18 −2.16728 2.60896 2.69711 −1.00000 −5.65435 4.04826 −1.51083 3.80667 2.16728
1.19 −2.09533 −3.35381 2.39043 −1.00000 7.02736 3.72693 −0.818074 8.24805 2.09533
1.20 −2.08489 −0.988264 2.34676 −1.00000 2.06042 4.76657 −0.722948 −2.02333 2.08489
See next 80 embeddings (of 113 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.113
Significant digits:
Format:

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 \(T_{2}^{113} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6005))\).