Properties

Label 8045.2.a.c
Level 8045
Weight 2
Character orbit 8045.a
Self dual Yes
Analytic conductor 64.240
Analytic rank 1
Dimension 127
CM No

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Newspace parameters

Level: \( N \) = \( 8045 = 5 \cdot 1609 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8045.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.2396484261\)
Analytic rank: \(1\)
Dimension: \(127\)
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.

\(\operatorname{Tr}(f)(q) = \) \(127q \) \(\mathstrut -\mathstrut 20q^{2} \) \(\mathstrut -\mathstrut 31q^{3} \) \(\mathstrut +\mathstrut 116q^{4} \) \(\mathstrut +\mathstrut 127q^{5} \) \(\mathstrut -\mathstrut 17q^{6} \) \(\mathstrut -\mathstrut 63q^{7} \) \(\mathstrut -\mathstrut 57q^{8} \) \(\mathstrut +\mathstrut 122q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(127q \) \(\mathstrut -\mathstrut 20q^{2} \) \(\mathstrut -\mathstrut 31q^{3} \) \(\mathstrut +\mathstrut 116q^{4} \) \(\mathstrut +\mathstrut 127q^{5} \) \(\mathstrut -\mathstrut 17q^{6} \) \(\mathstrut -\mathstrut 63q^{7} \) \(\mathstrut -\mathstrut 57q^{8} \) \(\mathstrut +\mathstrut 122q^{9} \) \(\mathstrut -\mathstrut 20q^{10} \) \(\mathstrut -\mathstrut 32q^{11} \) \(\mathstrut -\mathstrut 65q^{12} \) \(\mathstrut -\mathstrut 49q^{13} \) \(\mathstrut -\mathstrut 4q^{14} \) \(\mathstrut -\mathstrut 31q^{15} \) \(\mathstrut +\mathstrut 98q^{16} \) \(\mathstrut -\mathstrut 53q^{17} \) \(\mathstrut -\mathstrut 60q^{18} \) \(\mathstrut -\mathstrut 126q^{19} \) \(\mathstrut +\mathstrut 116q^{20} \) \(\mathstrut -\mathstrut 24q^{21} \) \(\mathstrut -\mathstrut 46q^{22} \) \(\mathstrut -\mathstrut 121q^{23} \) \(\mathstrut -\mathstrut 51q^{24} \) \(\mathstrut +\mathstrut 127q^{25} \) \(\mathstrut -\mathstrut 9q^{26} \) \(\mathstrut -\mathstrut 112q^{27} \) \(\mathstrut -\mathstrut 123q^{28} \) \(\mathstrut -\mathstrut 6q^{29} \) \(\mathstrut -\mathstrut 17q^{30} \) \(\mathstrut -\mathstrut 54q^{31} \) \(\mathstrut -\mathstrut 119q^{32} \) \(\mathstrut -\mathstrut 57q^{33} \) \(\mathstrut -\mathstrut 53q^{34} \) \(\mathstrut -\mathstrut 63q^{35} \) \(\mathstrut +\mathstrut 133q^{36} \) \(\mathstrut -\mathstrut 61q^{37} \) \(\mathstrut -\mathstrut 27q^{38} \) \(\mathstrut -\mathstrut 33q^{39} \) \(\mathstrut -\mathstrut 57q^{40} \) \(\mathstrut -\mathstrut 8q^{41} \) \(\mathstrut -\mathstrut 46q^{42} \) \(\mathstrut -\mathstrut 208q^{43} \) \(\mathstrut -\mathstrut 54q^{44} \) \(\mathstrut +\mathstrut 122q^{45} \) \(\mathstrut -\mathstrut 34q^{46} \) \(\mathstrut -\mathstrut 116q^{47} \) \(\mathstrut -\mathstrut 106q^{48} \) \(\mathstrut +\mathstrut 110q^{49} \) \(\mathstrut -\mathstrut 20q^{50} \) \(\mathstrut -\mathstrut 54q^{51} \) \(\mathstrut -\mathstrut 142q^{52} \) \(\mathstrut -\mathstrut 60q^{53} \) \(\mathstrut -\mathstrut 62q^{54} \) \(\mathstrut -\mathstrut 32q^{55} \) \(\mathstrut +\mathstrut 33q^{56} \) \(\mathstrut -\mathstrut 56q^{57} \) \(\mathstrut -\mathstrut 87q^{58} \) \(\mathstrut -\mathstrut 53q^{59} \) \(\mathstrut -\mathstrut 65q^{60} \) \(\mathstrut -\mathstrut 76q^{61} \) \(\mathstrut -\mathstrut 84q^{62} \) \(\mathstrut -\mathstrut 215q^{63} \) \(\mathstrut +\mathstrut 67q^{64} \) \(\mathstrut -\mathstrut 49q^{65} \) \(\mathstrut +\mathstrut 9q^{66} \) \(\mathstrut -\mathstrut 145q^{67} \) \(\mathstrut -\mathstrut 133q^{68} \) \(\mathstrut +\mathstrut q^{69} \) \(\mathstrut -\mathstrut 4q^{70} \) \(\mathstrut -\mathstrut 4q^{71} \) \(\mathstrut -\mathstrut 167q^{72} \) \(\mathstrut -\mathstrut 155q^{73} \) \(\mathstrut -\mathstrut 14q^{74} \) \(\mathstrut -\mathstrut 31q^{75} \) \(\mathstrut -\mathstrut 199q^{76} \) \(\mathstrut -\mathstrut 97q^{77} \) \(\mathstrut -\mathstrut 24q^{78} \) \(\mathstrut -\mathstrut 73q^{79} \) \(\mathstrut +\mathstrut 98q^{80} \) \(\mathstrut +\mathstrut 127q^{81} \) \(\mathstrut -\mathstrut 69q^{82} \) \(\mathstrut -\mathstrut 225q^{83} \) \(\mathstrut -\mathstrut 59q^{84} \) \(\mathstrut -\mathstrut 53q^{85} \) \(\mathstrut +\mathstrut 30q^{86} \) \(\mathstrut -\mathstrut 179q^{87} \) \(\mathstrut -\mathstrut 119q^{88} \) \(\mathstrut -\mathstrut 25q^{89} \) \(\mathstrut -\mathstrut 60q^{90} \) \(\mathstrut -\mathstrut 160q^{91} \) \(\mathstrut -\mathstrut 188q^{92} \) \(\mathstrut -\mathstrut 44q^{93} \) \(\mathstrut -\mathstrut 32q^{94} \) \(\mathstrut -\mathstrut 126q^{95} \) \(\mathstrut -\mathstrut 43q^{96} \) \(\mathstrut -\mathstrut 72q^{97} \) \(\mathstrut -\mathstrut 111q^{98} \) \(\mathstrut -\mathstrut 141q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.82452 −1.42855 5.97791 1.00000 4.03498 −1.55315 −11.2357 −0.959233 −2.82452
1.2 −2.74468 −3.15686 5.53326 1.00000 8.66457 −4.01431 −9.69765 6.96577 −2.74468
1.3 −2.72254 3.22119 5.41223 1.00000 −8.76982 −3.61577 −9.28994 7.37607 −2.72254
1.4 −2.71791 −0.717872 5.38703 1.00000 1.95111 −2.87590 −9.20565 −2.48466 −2.71791
1.5 −2.69637 1.21455 5.27040 1.00000 −3.27487 2.45597 −8.81819 −1.52487 −2.69637
1.6 −2.69512 −2.02504 5.26369 1.00000 5.45774 1.98108 −8.79604 1.10080 −2.69512
1.7 −2.68498 2.75441 5.20913 1.00000 −7.39554 −2.94460 −8.61647 4.58677 −2.68498
1.8 −2.61813 0.911081 4.85463 1.00000 −2.38533 −2.60374 −7.47379 −2.16993 −2.61813
1.9 −2.53777 0.243430 4.44030 1.00000 −0.617771 4.53173 −6.19292 −2.94074 −2.53777
1.10 −2.52291 −2.56612 4.36506 1.00000 6.47407 −1.96305 −5.96681 3.58496 −2.52291
1.11 −2.51055 −0.644369 4.30288 1.00000 1.61772 −0.493808 −5.78150 −2.58479 −2.51055
1.12 −2.49863 2.41476 4.24316 1.00000 −6.03361 1.44699 −5.60483 2.83108 −2.49863
1.13 −2.47054 −3.37029 4.10358 1.00000 8.32645 4.71800 −5.19700 8.35885 −2.47054
1.14 −2.44032 0.296251 3.95518 1.00000 −0.722948 −5.11077 −4.77127 −2.91224 −2.44032
1.15 −2.41609 −2.71250 3.83748 1.00000 6.55363 −4.44734 −4.43951 4.35764 −2.41609
1.16 −2.35235 −3.18470 3.53353 1.00000 7.49152 −1.35488 −3.60740 7.14233 −2.35235
1.17 −2.31881 0.00298266 3.37687 1.00000 −0.00691620 −0.956165 −3.19269 −2.99999 −2.31881
1.18 −2.26786 −1.17957 3.14318 1.00000 2.67510 2.15242 −2.59257 −1.60862 −2.26786
1.19 −2.26133 1.69211 3.11360 1.00000 −3.82641 −1.09235 −2.51822 −0.136771 −2.26133
1.20 −2.24197 1.72389 3.02644 1.00000 −3.86491 −1.20930 −2.30126 −0.0282099 −2.24197
See next 80 embeddings (of 127 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.127
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(1609\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{127} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8045))\).