Properties

Label 6047.2.a.a
Level 6047
Weight 2
Character orbit 6047.a
Self dual Yes
Analytic conductor 48.286
Analytic rank 1
Dimension 217
CM No

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Newspace parameters

Level: \( N \) = \( 6047 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6047.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2855381023\)
Analytic rank: \(1\)
Dimension: \(217\)
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.

\(\operatorname{Tr}(f)(q) = \) \(217q \) \(\mathstrut -\mathstrut 20q^{2} \) \(\mathstrut -\mathstrut 27q^{3} \) \(\mathstrut +\mathstrut 184q^{4} \) \(\mathstrut -\mathstrut 19q^{5} \) \(\mathstrut -\mathstrut 17q^{6} \) \(\mathstrut -\mathstrut 48q^{7} \) \(\mathstrut -\mathstrut 57q^{8} \) \(\mathstrut +\mathstrut 152q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(217q \) \(\mathstrut -\mathstrut 20q^{2} \) \(\mathstrut -\mathstrut 27q^{3} \) \(\mathstrut +\mathstrut 184q^{4} \) \(\mathstrut -\mathstrut 19q^{5} \) \(\mathstrut -\mathstrut 17q^{6} \) \(\mathstrut -\mathstrut 48q^{7} \) \(\mathstrut -\mathstrut 57q^{8} \) \(\mathstrut +\mathstrut 152q^{9} \) \(\mathstrut -\mathstrut 46q^{10} \) \(\mathstrut -\mathstrut 32q^{11} \) \(\mathstrut -\mathstrut 72q^{12} \) \(\mathstrut -\mathstrut 80q^{13} \) \(\mathstrut -\mathstrut 22q^{14} \) \(\mathstrut -\mathstrut 43q^{15} \) \(\mathstrut +\mathstrut 122q^{16} \) \(\mathstrut -\mathstrut 61q^{17} \) \(\mathstrut -\mathstrut 88q^{18} \) \(\mathstrut -\mathstrut 43q^{19} \) \(\mathstrut -\mathstrut 41q^{20} \) \(\mathstrut -\mathstrut 61q^{21} \) \(\mathstrut -\mathstrut 93q^{22} \) \(\mathstrut -\mathstrut 60q^{23} \) \(\mathstrut -\mathstrut 41q^{24} \) \(\mathstrut +\mathstrut 26q^{25} \) \(\mathstrut -\mathstrut 9q^{26} \) \(\mathstrut -\mathstrut 93q^{27} \) \(\mathstrut -\mathstrut 126q^{28} \) \(\mathstrut -\mathstrut 47q^{29} \) \(\mathstrut -\mathstrut 36q^{30} \) \(\mathstrut -\mathstrut 100q^{31} \) \(\mathstrut -\mathstrut 114q^{32} \) \(\mathstrut -\mathstrut 133q^{33} \) \(\mathstrut -\mathstrut 75q^{34} \) \(\mathstrut -\mathstrut 37q^{35} \) \(\mathstrut +\mathstrut 75q^{36} \) \(\mathstrut -\mathstrut 264q^{37} \) \(\mathstrut -\mathstrut 35q^{38} \) \(\mathstrut -\mathstrut 47q^{39} \) \(\mathstrut -\mathstrut 118q^{40} \) \(\mathstrut -\mathstrut 72q^{41} \) \(\mathstrut -\mathstrut 64q^{42} \) \(\mathstrut -\mathstrut 107q^{43} \) \(\mathstrut -\mathstrut 59q^{44} \) \(\mathstrut -\mathstrut 69q^{45} \) \(\mathstrut -\mathstrut 111q^{46} \) \(\mathstrut -\mathstrut 54q^{47} \) \(\mathstrut -\mathstrut 135q^{48} \) \(\mathstrut +\mathstrut 33q^{49} \) \(\mathstrut -\mathstrut 42q^{50} \) \(\mathstrut -\mathstrut 26q^{51} \) \(\mathstrut -\mathstrut 173q^{52} \) \(\mathstrut -\mathstrut 103q^{53} \) \(\mathstrut -\mathstrut 28q^{54} \) \(\mathstrut -\mathstrut 78q^{55} \) \(\mathstrut -\mathstrut 44q^{56} \) \(\mathstrut -\mathstrut 205q^{57} \) \(\mathstrut -\mathstrut 189q^{58} \) \(\mathstrut -\mathstrut 38q^{59} \) \(\mathstrut -\mathstrut 105q^{60} \) \(\mathstrut -\mathstrut 108q^{61} \) \(\mathstrut -\mathstrut 14q^{62} \) \(\mathstrut -\mathstrut 116q^{63} \) \(\mathstrut +\mathstrut 39q^{64} \) \(\mathstrut -\mathstrut 146q^{65} \) \(\mathstrut +\mathstrut 5q^{66} \) \(\mathstrut -\mathstrut 206q^{67} \) \(\mathstrut -\mathstrut 62q^{68} \) \(\mathstrut -\mathstrut 55q^{69} \) \(\mathstrut -\mathstrut 125q^{70} \) \(\mathstrut -\mathstrut 78q^{71} \) \(\mathstrut -\mathstrut 225q^{72} \) \(\mathstrut -\mathstrut 326q^{73} \) \(\mathstrut +\mathstrut 3q^{74} \) \(\mathstrut -\mathstrut 95q^{75} \) \(\mathstrut -\mathstrut 84q^{76} \) \(\mathstrut -\mathstrut 79q^{77} \) \(\mathstrut -\mathstrut 86q^{78} \) \(\mathstrut -\mathstrut 117q^{79} \) \(\mathstrut -\mathstrut 39q^{80} \) \(\mathstrut +\mathstrut q^{81} \) \(\mathstrut -\mathstrut 96q^{82} \) \(\mathstrut -\mathstrut 23q^{83} \) \(\mathstrut -\mathstrut 57q^{84} \) \(\mathstrut -\mathstrut 224q^{85} \) \(\mathstrut -\mathstrut 7q^{86} \) \(\mathstrut -\mathstrut 45q^{87} \) \(\mathstrut -\mathstrut 250q^{88} \) \(\mathstrut -\mathstrut 104q^{89} \) \(\mathstrut -\mathstrut 36q^{90} \) \(\mathstrut -\mathstrut 96q^{91} \) \(\mathstrut -\mathstrut 137q^{92} \) \(\mathstrut -\mathstrut 155q^{93} \) \(\mathstrut -\mathstrut 48q^{94} \) \(\mathstrut -\mathstrut 38q^{95} \) \(\mathstrut -\mathstrut 33q^{96} \) \(\mathstrut -\mathstrut 447q^{97} \) \(\mathstrut -\mathstrut 46q^{98} \) \(\mathstrut -\mathstrut 94q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.79459 −1.21209 5.80974 −0.914621 3.38731 −4.48023 −10.6467 −1.53083 2.55599
1.2 −2.78935 0.603668 5.78045 3.11422 −1.68384 0.146622 −10.5450 −2.63558 −8.68663
1.3 −2.77438 −3.26218 5.69718 2.18536 9.05052 2.86186 −10.2574 7.64181 −6.06301
1.4 −2.74777 2.83106 5.55022 −0.268060 −7.77911 0.663291 −9.75518 5.01493 0.736566
1.5 −2.70650 0.217632 5.32513 0.669687 −0.589021 −0.581450 −8.99946 −2.95264 −1.81251
1.6 −2.67338 −1.23035 5.14695 0.526879 3.28919 −0.672464 −8.41297 −1.48624 −1.40855
1.7 −2.65722 0.109936 5.06084 0.753739 −0.292126 3.98656 −8.13334 −2.98791 −2.00285
1.8 −2.65451 −2.99510 5.04641 −2.08040 7.95052 −1.11556 −8.08670 5.97065 5.52243
1.9 −2.63907 −2.61338 4.96471 2.16033 6.89691 −4.93155 −7.82409 3.82978 −5.70128
1.10 −2.62503 1.59617 4.89080 2.83370 −4.19001 −4.08027 −7.58846 −0.452230 −7.43856
1.11 −2.59272 3.24215 4.72220 −1.12445 −8.40598 −0.119273 −7.05789 7.51153 2.91540
1.12 −2.57504 1.64822 4.63081 −3.44910 −4.24422 −0.596125 −6.77442 −0.283374 8.88154
1.13 −2.54610 −2.31345 4.48264 −3.02578 5.89027 −2.15804 −6.32106 2.35204 7.70396
1.14 −2.50842 0.797647 4.29216 −3.82675 −2.00083 −2.73995 −5.74970 −2.36376 9.59909
1.15 −2.50438 2.01082 4.27191 3.19046 −5.03585 0.790850 −5.68973 1.04339 −7.99013
1.16 −2.47362 −1.15677 4.11880 −1.98795 2.86140 3.96077 −5.24112 −1.66189 4.91743
1.17 −2.46440 −0.383920 4.07326 2.75346 0.946133 −1.51513 −5.10935 −2.85261 −6.78563
1.18 −2.46358 2.70933 4.06925 0.207418 −6.67465 −2.27877 −5.09776 4.34044 −0.510992
1.19 −2.45739 0.0299595 4.03876 −2.30057 −0.0736221 −3.67314 −5.01004 −2.99910 5.65341
1.20 −2.44753 −2.68992 3.99041 −1.41647 6.58366 −0.890122 −4.87158 4.23566 3.46686
See next 80 embeddings (of 217 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.217
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(6047\) \(1\)