Properties

Label 5077.2.a.b
Level 5077
Weight 2
Character orbit 5077.a
Self dual Yes
Analytic conductor 40.540
Analytic rank 1
Dimension 205
CM No

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Newspace parameters

Level: \( N \) = \( 5077 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 5077.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(40.5400491062\)
Analytic rank: \(1\)
Dimension: \(205\)
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.

\(\operatorname{Tr}(f)(q) = \) \(205q \) \(\mathstrut -\mathstrut 25q^{2} \) \(\mathstrut -\mathstrut 59q^{3} \) \(\mathstrut +\mathstrut 195q^{4} \) \(\mathstrut -\mathstrut 44q^{5} \) \(\mathstrut -\mathstrut 26q^{6} \) \(\mathstrut -\mathstrut 30q^{7} \) \(\mathstrut -\mathstrut 75q^{8} \) \(\mathstrut +\mathstrut 186q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(205q \) \(\mathstrut -\mathstrut 25q^{2} \) \(\mathstrut -\mathstrut 59q^{3} \) \(\mathstrut +\mathstrut 195q^{4} \) \(\mathstrut -\mathstrut 44q^{5} \) \(\mathstrut -\mathstrut 26q^{6} \) \(\mathstrut -\mathstrut 30q^{7} \) \(\mathstrut -\mathstrut 75q^{8} \) \(\mathstrut +\mathstrut 186q^{9} \) \(\mathstrut -\mathstrut 28q^{10} \) \(\mathstrut -\mathstrut 83q^{11} \) \(\mathstrut -\mathstrut 108q^{12} \) \(\mathstrut -\mathstrut 36q^{13} \) \(\mathstrut -\mathstrut 67q^{14} \) \(\mathstrut -\mathstrut 63q^{15} \) \(\mathstrut +\mathstrut 187q^{16} \) \(\mathstrut -\mathstrut 72q^{17} \) \(\mathstrut -\mathstrut 57q^{18} \) \(\mathstrut -\mathstrut 47q^{19} \) \(\mathstrut -\mathstrut 132q^{20} \) \(\mathstrut -\mathstrut 35q^{21} \) \(\mathstrut -\mathstrut 40q^{22} \) \(\mathstrut -\mathstrut 97q^{23} \) \(\mathstrut -\mathstrut 49q^{24} \) \(\mathstrut +\mathstrut 175q^{25} \) \(\mathstrut -\mathstrut 78q^{26} \) \(\mathstrut -\mathstrut 227q^{27} \) \(\mathstrut -\mathstrut 59q^{28} \) \(\mathstrut -\mathstrut 46q^{29} \) \(\mathstrut +\mathstrut 30q^{30} \) \(\mathstrut -\mathstrut 77q^{31} \) \(\mathstrut -\mathstrut 175q^{32} \) \(\mathstrut -\mathstrut 74q^{33} \) \(\mathstrut -\mathstrut 28q^{34} \) \(\mathstrut -\mathstrut 171q^{35} \) \(\mathstrut +\mathstrut 171q^{36} \) \(\mathstrut -\mathstrut 52q^{37} \) \(\mathstrut -\mathstrut 144q^{38} \) \(\mathstrut -\mathstrut 54q^{39} \) \(\mathstrut -\mathstrut 49q^{40} \) \(\mathstrut -\mathstrut 107q^{41} \) \(\mathstrut +\mathstrut 7q^{42} \) \(\mathstrut -\mathstrut 58q^{43} \) \(\mathstrut -\mathstrut 139q^{44} \) \(\mathstrut -\mathstrut 89q^{45} \) \(\mathstrut -\mathstrut 33q^{46} \) \(\mathstrut -\mathstrut 255q^{47} \) \(\mathstrut -\mathstrut 202q^{48} \) \(\mathstrut +\mathstrut 171q^{49} \) \(\mathstrut -\mathstrut 74q^{50} \) \(\mathstrut -\mathstrut 63q^{51} \) \(\mathstrut -\mathstrut 90q^{52} \) \(\mathstrut -\mathstrut 82q^{53} \) \(\mathstrut -\mathstrut 51q^{54} \) \(\mathstrut -\mathstrut 70q^{55} \) \(\mathstrut -\mathstrut 180q^{56} \) \(\mathstrut -\mathstrut 70q^{57} \) \(\mathstrut -\mathstrut 50q^{58} \) \(\mathstrut -\mathstrut 289q^{59} \) \(\mathstrut -\mathstrut 105q^{60} \) \(\mathstrut -\mathstrut 20q^{61} \) \(\mathstrut -\mathstrut 143q^{62} \) \(\mathstrut -\mathstrut 119q^{63} \) \(\mathstrut +\mathstrut 201q^{64} \) \(\mathstrut -\mathstrut 92q^{65} \) \(\mathstrut -\mathstrut 3q^{66} \) \(\mathstrut -\mathstrut 138q^{67} \) \(\mathstrut -\mathstrut 177q^{68} \) \(\mathstrut -\mathstrut 67q^{69} \) \(\mathstrut +\mathstrut 4q^{70} \) \(\mathstrut -\mathstrut 141q^{71} \) \(\mathstrut -\mathstrut 138q^{72} \) \(\mathstrut -\mathstrut 71q^{73} \) \(\mathstrut -\mathstrut 26q^{74} \) \(\mathstrut -\mathstrut 251q^{75} \) \(\mathstrut -\mathstrut 42q^{76} \) \(\mathstrut -\mathstrut 149q^{77} \) \(\mathstrut -\mathstrut 6q^{78} \) \(\mathstrut -\mathstrut 47q^{79} \) \(\mathstrut -\mathstrut 294q^{80} \) \(\mathstrut +\mathstrut 193q^{81} \) \(\mathstrut -\mathstrut 70q^{82} \) \(\mathstrut -\mathstrut 329q^{83} \) \(\mathstrut -\mathstrut 40q^{84} \) \(\mathstrut -\mathstrut 45q^{85} \) \(\mathstrut -\mathstrut 83q^{86} \) \(\mathstrut -\mathstrut 139q^{87} \) \(\mathstrut -\mathstrut 45q^{88} \) \(\mathstrut -\mathstrut 163q^{89} \) \(\mathstrut -\mathstrut 116q^{90} \) \(\mathstrut -\mathstrut 141q^{91} \) \(\mathstrut -\mathstrut 204q^{92} \) \(\mathstrut -\mathstrut 91q^{93} \) \(\mathstrut -\mathstrut 8q^{94} \) \(\mathstrut -\mathstrut 173q^{95} \) \(\mathstrut -\mathstrut 53q^{96} \) \(\mathstrut -\mathstrut 147q^{97} \) \(\mathstrut -\mathstrut 156q^{98} \) \(\mathstrut -\mathstrut 157q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.81633 −3.09812 5.93172 −0.550838 8.72533 −3.77340 −11.0730 6.59836 1.55134
1.2 −2.81174 −2.29296 5.90586 2.02073 6.44719 2.09923 −10.9822 2.25765 −5.68176
1.3 −2.78060 0.0708990 5.73173 −3.99378 −0.197142 2.14019 −10.3764 −2.99497 11.1051
1.4 −2.77684 2.42901 5.71081 0.850994 −6.74496 0.0263654 −10.3043 2.90008 −2.36307
1.5 −2.77001 −0.453815 5.67294 −2.74837 1.25707 −3.66610 −10.1741 −2.79405 7.61300
1.6 −2.75099 0.881000 5.56794 −0.176984 −2.42362 4.01245 −9.81537 −2.22384 0.486880
1.7 −2.72893 1.44573 5.44706 1.77131 −3.94530 −0.914980 −9.40679 −0.909861 −4.83377
1.8 −2.70901 2.53540 5.33875 −3.58245 −6.86844 1.35106 −9.04473 3.42827 9.70491
1.9 −2.65566 −0.00621299 5.05254 3.61372 0.0164996 −2.37510 −8.10651 −2.99996 −9.59681
1.10 −2.63506 −2.68532 4.94352 −3.56465 7.07597 1.99537 −7.75633 4.21094 9.39304
1.11 −2.63322 −3.42055 4.93387 2.45426 9.00708 4.09681 −7.72553 8.70017 −6.46261
1.12 −2.62320 −1.94038 4.88119 −2.34759 5.09001 2.45052 −7.55796 0.765079 6.15820
1.13 −2.61107 −0.462954 4.81767 −1.81913 1.20880 2.59865 −7.35711 −2.78567 4.74986
1.14 −2.58891 3.03601 4.70247 −2.12687 −7.85998 3.03640 −6.99646 6.21738 5.50629
1.15 −2.58762 0.766219 4.69576 1.84822 −1.98268 −5.04954 −6.97561 −2.41291 −4.78248
1.16 −2.54248 2.75174 4.46418 −3.07759 −6.99622 −3.90745 −6.26513 4.57206 7.82470
1.17 −2.48215 −2.18420 4.16105 −1.67259 5.42150 2.74743 −5.36403 1.77073 4.15162
1.18 −2.47319 −1.27283 4.11668 3.46270 3.14796 5.16648 −5.23495 −1.37990 −8.56393
1.19 −2.46210 −1.32036 4.06192 0.169195 3.25085 −3.54943 −5.07665 −1.25666 −0.416575
1.20 −2.46112 −3.25745 4.05713 −3.21291 8.01698 −0.720862 −5.06284 7.61098 7.90737
See next 80 embeddings (of 205 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.205
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
\(5077\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{205} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5077))\).