Properties

Label 8047.2.a.b
Level 8047
Weight 2
Character orbit 8047.a
Self dual Yes
Analytic conductor 64.256
Analytic rank 1
Dimension 142
CM No

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Newspace parameters

Level: \( N \) = \( 8047 = 13 \cdot 619 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8047.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.2556185065\)
Analytic rank: \(1\)
Dimension: \(142\)
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.

\(\operatorname{Tr}(f)(q) = \) \(142q \) \(\mathstrut -\mathstrut 13q^{2} \) \(\mathstrut -\mathstrut 26q^{3} \) \(\mathstrut +\mathstrut 129q^{4} \) \(\mathstrut -\mathstrut 37q^{5} \) \(\mathstrut -\mathstrut 15q^{6} \) \(\mathstrut -\mathstrut 14q^{7} \) \(\mathstrut -\mathstrut 39q^{8} \) \(\mathstrut +\mathstrut 98q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(142q \) \(\mathstrut -\mathstrut 13q^{2} \) \(\mathstrut -\mathstrut 26q^{3} \) \(\mathstrut +\mathstrut 129q^{4} \) \(\mathstrut -\mathstrut 37q^{5} \) \(\mathstrut -\mathstrut 15q^{6} \) \(\mathstrut -\mathstrut 14q^{7} \) \(\mathstrut -\mathstrut 39q^{8} \) \(\mathstrut +\mathstrut 98q^{9} \) \(\mathstrut -\mathstrut 25q^{10} \) \(\mathstrut -\mathstrut 25q^{11} \) \(\mathstrut -\mathstrut 62q^{12} \) \(\mathstrut +\mathstrut 142q^{13} \) \(\mathstrut -\mathstrut 57q^{14} \) \(\mathstrut -\mathstrut 14q^{15} \) \(\mathstrut +\mathstrut 111q^{16} \) \(\mathstrut -\mathstrut 141q^{17} \) \(\mathstrut -\mathstrut 29q^{18} \) \(\mathstrut -\mathstrut 3q^{19} \) \(\mathstrut -\mathstrut 87q^{20} \) \(\mathstrut -\mathstrut 19q^{21} \) \(\mathstrut -\mathstrut 24q^{22} \) \(\mathstrut -\mathstrut 69q^{23} \) \(\mathstrut -\mathstrut 40q^{24} \) \(\mathstrut +\mathstrut 87q^{25} \) \(\mathstrut -\mathstrut 13q^{26} \) \(\mathstrut -\mathstrut 95q^{27} \) \(\mathstrut -\mathstrut 34q^{28} \) \(\mathstrut -\mathstrut 147q^{29} \) \(\mathstrut -\mathstrut 2q^{30} \) \(\mathstrut -\mathstrut 21q^{31} \) \(\mathstrut -\mathstrut 66q^{32} \) \(\mathstrut -\mathstrut 62q^{33} \) \(\mathstrut -\mathstrut 6q^{34} \) \(\mathstrut -\mathstrut 59q^{35} \) \(\mathstrut +\mathstrut 74q^{36} \) \(\mathstrut -\mathstrut 37q^{37} \) \(\mathstrut -\mathstrut 76q^{38} \) \(\mathstrut -\mathstrut 26q^{39} \) \(\mathstrut -\mathstrut 61q^{40} \) \(\mathstrut -\mathstrut 97q^{41} \) \(\mathstrut -\mathstrut 29q^{42} \) \(\mathstrut -\mathstrut 33q^{43} \) \(\mathstrut -\mathstrut 57q^{44} \) \(\mathstrut -\mathstrut 86q^{45} \) \(\mathstrut -\mathstrut q^{46} \) \(\mathstrut -\mathstrut 102q^{47} \) \(\mathstrut -\mathstrut 141q^{48} \) \(\mathstrut +\mathstrut 70q^{49} \) \(\mathstrut -\mathstrut 28q^{50} \) \(\mathstrut -\mathstrut 13q^{51} \) \(\mathstrut +\mathstrut 129q^{52} \) \(\mathstrut -\mathstrut 137q^{53} \) \(\mathstrut -\mathstrut 29q^{54} \) \(\mathstrut -\mathstrut 24q^{55} \) \(\mathstrut -\mathstrut 130q^{56} \) \(\mathstrut -\mathstrut 65q^{57} \) \(\mathstrut -\mathstrut 15q^{58} \) \(\mathstrut -\mathstrut 56q^{59} \) \(\mathstrut +\mathstrut 11q^{60} \) \(\mathstrut -\mathstrut 77q^{61} \) \(\mathstrut -\mathstrut 150q^{62} \) \(\mathstrut -\mathstrut 32q^{63} \) \(\mathstrut +\mathstrut 73q^{64} \) \(\mathstrut -\mathstrut 37q^{65} \) \(\mathstrut -\mathstrut 32q^{66} \) \(\mathstrut -\mathstrut 9q^{67} \) \(\mathstrut -\mathstrut 226q^{68} \) \(\mathstrut -\mathstrut 113q^{69} \) \(\mathstrut +\mathstrut 6q^{70} \) \(\mathstrut -\mathstrut 18q^{71} \) \(\mathstrut -\mathstrut 82q^{72} \) \(\mathstrut -\mathstrut 117q^{73} \) \(\mathstrut -\mathstrut 70q^{74} \) \(\mathstrut -\mathstrut 83q^{75} \) \(\mathstrut +\mathstrut 40q^{76} \) \(\mathstrut -\mathstrut 214q^{77} \) \(\mathstrut -\mathstrut 15q^{78} \) \(\mathstrut -\mathstrut 52q^{79} \) \(\mathstrut -\mathstrut 161q^{80} \) \(\mathstrut -\mathstrut 10q^{81} \) \(\mathstrut -\mathstrut 36q^{82} \) \(\mathstrut -\mathstrut 74q^{83} \) \(\mathstrut +\mathstrut 53q^{84} \) \(\mathstrut +\mathstrut 2q^{85} \) \(\mathstrut +\mathstrut 17q^{86} \) \(\mathstrut -\mathstrut 49q^{87} \) \(\mathstrut -\mathstrut 29q^{88} \) \(\mathstrut -\mathstrut 171q^{89} \) \(\mathstrut -\mathstrut 57q^{90} \) \(\mathstrut -\mathstrut 14q^{91} \) \(\mathstrut -\mathstrut 187q^{92} \) \(\mathstrut -\mathstrut 39q^{93} \) \(\mathstrut +\mathstrut 13q^{94} \) \(\mathstrut -\mathstrut 150q^{95} \) \(\mathstrut -\mathstrut 47q^{96} \) \(\mathstrut -\mathstrut 126q^{97} \) \(\mathstrut -\mathstrut 85q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.78532 −0.218444 5.75799 3.30077 0.608437 −0.196953 −10.4672 −2.95228 −9.19368
1.2 −2.78511 −1.73890 5.75683 −2.18885 4.84303 2.36535 −10.4632 0.0237834 6.09618
1.3 −2.74745 −2.94528 5.54847 1.95253 8.09201 3.66595 −9.74923 5.67469 −5.36448
1.4 −2.73051 1.50429 5.45570 −3.60561 −4.10747 2.19221 −9.43585 −0.737125 9.84515
1.5 −2.66381 1.33027 5.09589 −0.440221 −3.54360 2.49453 −8.24687 −1.23037 1.17267
1.6 −2.65986 −2.58294 5.07483 0.599360 6.87024 −3.90786 −8.17861 3.67156 −1.59421
1.7 −2.65232 2.65488 5.03481 −0.675897 −7.04159 −2.02772 −8.04928 4.04839 1.79269
1.8 −2.56842 0.368799 4.59680 −0.479572 −0.947232 4.01594 −6.66967 −2.86399 1.23174
1.9 −2.55403 1.62979 4.52309 1.02400 −4.16254 −2.90866 −6.44404 −0.343777 −2.61533
1.10 −2.52587 −3.25300 4.38004 −2.46689 8.21667 −3.38874 −6.01169 7.58202 6.23106
1.11 −2.52091 −0.134089 4.35498 2.66858 0.338025 −2.31114 −5.93668 −2.98202 −6.72724
1.12 −2.51095 −2.93259 4.30485 −3.06715 7.36357 3.82085 −5.78735 5.60007 7.70146
1.13 −2.50524 −0.199696 4.27622 −1.81098 0.500286 −1.67033 −5.70247 −2.96012 4.53694
1.14 −2.45771 2.98686 4.04034 −1.52748 −7.34083 5.18217 −5.01456 5.92132 3.75410
1.15 −2.43267 −0.839960 3.91786 −3.64522 2.04334 −2.93716 −4.66552 −2.29447 8.86761
1.16 −2.41564 2.60884 3.83533 1.90553 −6.30203 −0.0638029 −4.43349 3.80607 −4.60307
1.17 −2.27884 −1.20174 3.19313 −1.58295 2.73859 1.25281 −2.71895 −1.55581 3.60730
1.18 −2.23848 1.32587 3.01078 3.90699 −2.96792 −1.03210 −2.26261 −1.24208 −8.74571
1.19 −2.23735 −0.938044 3.00574 3.28954 2.09873 1.35040 −2.25019 −2.12007 −7.35985
1.20 −2.15511 1.98414 2.64449 −3.20612 −4.27605 1.44494 −1.38895 0.936828 6.90953
See next 80 embeddings (of 142 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.142
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(13\) \(-1\)
\(619\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{142} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8047))\).