Properties

Label 8035.2.a.d
Level 8035
Weight 2
Character orbit 8035.a
Self dual Yes
Analytic conductor 64.160
Analytic rank 1
Dimension 140
CM No

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Newspace parameters

Level: \( N \) = \( 8035 = 5 \cdot 1607 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8035.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1597980241\)
Analytic rank: \(1\)
Dimension: \(140\)
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.

\(\operatorname{Tr}(f)(q) = \) \(140q \) \(\mathstrut -\mathstrut 20q^{2} \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut +\mathstrut 144q^{4} \) \(\mathstrut -\mathstrut 140q^{5} \) \(\mathstrut -\mathstrut 15q^{7} \) \(\mathstrut -\mathstrut 63q^{8} \) \(\mathstrut +\mathstrut 134q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(140q \) \(\mathstrut -\mathstrut 20q^{2} \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut +\mathstrut 144q^{4} \) \(\mathstrut -\mathstrut 140q^{5} \) \(\mathstrut -\mathstrut 15q^{7} \) \(\mathstrut -\mathstrut 63q^{8} \) \(\mathstrut +\mathstrut 134q^{9} \) \(\mathstrut +\mathstrut 20q^{10} \) \(\mathstrut -\mathstrut 26q^{11} \) \(\mathstrut -\mathstrut 31q^{12} \) \(\mathstrut -\mathstrut 32q^{13} \) \(\mathstrut -\mathstrut 37q^{14} \) \(\mathstrut +\mathstrut 12q^{15} \) \(\mathstrut +\mathstrut 152q^{16} \) \(\mathstrut -\mathstrut 69q^{17} \) \(\mathstrut -\mathstrut 64q^{18} \) \(\mathstrut +\mathstrut 37q^{19} \) \(\mathstrut -\mathstrut 144q^{20} \) \(\mathstrut -\mathstrut 43q^{21} \) \(\mathstrut -\mathstrut 25q^{22} \) \(\mathstrut -\mathstrut 63q^{23} \) \(\mathstrut -\mathstrut 5q^{24} \) \(\mathstrut +\mathstrut 140q^{25} \) \(\mathstrut -\mathstrut 16q^{26} \) \(\mathstrut -\mathstrut 48q^{27} \) \(\mathstrut -\mathstrut 52q^{28} \) \(\mathstrut -\mathstrut 136q^{29} \) \(\mathstrut +\mathstrut 25q^{31} \) \(\mathstrut -\mathstrut 151q^{32} \) \(\mathstrut -\mathstrut 48q^{33} \) \(\mathstrut +\mathstrut 29q^{34} \) \(\mathstrut +\mathstrut 15q^{35} \) \(\mathstrut +\mathstrut 120q^{36} \) \(\mathstrut -\mathstrut 82q^{37} \) \(\mathstrut -\mathstrut 69q^{38} \) \(\mathstrut -\mathstrut 26q^{39} \) \(\mathstrut +\mathstrut 63q^{40} \) \(\mathstrut -\mathstrut 11q^{41} \) \(\mathstrut -\mathstrut 35q^{42} \) \(\mathstrut -\mathstrut 54q^{43} \) \(\mathstrut -\mathstrut 83q^{44} \) \(\mathstrut -\mathstrut 134q^{45} \) \(\mathstrut +\mathstrut 25q^{46} \) \(\mathstrut -\mathstrut 39q^{47} \) \(\mathstrut -\mathstrut 83q^{48} \) \(\mathstrut +\mathstrut 215q^{49} \) \(\mathstrut -\mathstrut 20q^{50} \) \(\mathstrut -\mathstrut 75q^{51} \) \(\mathstrut -\mathstrut 56q^{52} \) \(\mathstrut -\mathstrut 196q^{53} \) \(\mathstrut -\mathstrut 29q^{54} \) \(\mathstrut +\mathstrut 26q^{55} \) \(\mathstrut -\mathstrut 132q^{56} \) \(\mathstrut -\mathstrut 110q^{57} \) \(\mathstrut -\mathstrut 29q^{58} \) \(\mathstrut -\mathstrut 31q^{59} \) \(\mathstrut +\mathstrut 31q^{60} \) \(\mathstrut -\mathstrut 18q^{61} \) \(\mathstrut -\mathstrut 107q^{62} \) \(\mathstrut -\mathstrut 67q^{63} \) \(\mathstrut +\mathstrut 165q^{64} \) \(\mathstrut +\mathstrut 32q^{65} \) \(\mathstrut -\mathstrut 16q^{66} \) \(\mathstrut -\mathstrut 50q^{67} \) \(\mathstrut -\mathstrut 201q^{68} \) \(\mathstrut -\mathstrut 46q^{69} \) \(\mathstrut +\mathstrut 37q^{70} \) \(\mathstrut -\mathstrut 84q^{71} \) \(\mathstrut -\mathstrut 200q^{72} \) \(\mathstrut -\mathstrut 70q^{73} \) \(\mathstrut -\mathstrut 101q^{74} \) \(\mathstrut -\mathstrut 12q^{75} \) \(\mathstrut +\mathstrut 118q^{76} \) \(\mathstrut -\mathstrut 166q^{77} \) \(\mathstrut -\mathstrut 106q^{78} \) \(\mathstrut -\mathstrut 35q^{79} \) \(\mathstrut -\mathstrut 152q^{80} \) \(\mathstrut +\mathstrut 116q^{81} \) \(\mathstrut -\mathstrut 72q^{82} \) \(\mathstrut -\mathstrut 66q^{83} \) \(\mathstrut -\mathstrut 60q^{84} \) \(\mathstrut +\mathstrut 69q^{85} \) \(\mathstrut -\mathstrut 66q^{86} \) \(\mathstrut -\mathstrut 75q^{87} \) \(\mathstrut -\mathstrut 101q^{88} \) \(\mathstrut +\mathstrut 8q^{89} \) \(\mathstrut +\mathstrut 64q^{90} \) \(\mathstrut +\mathstrut 2q^{91} \) \(\mathstrut -\mathstrut 197q^{92} \) \(\mathstrut -\mathstrut 134q^{93} \) \(\mathstrut +\mathstrut 65q^{94} \) \(\mathstrut -\mathstrut 37q^{95} \) \(\mathstrut +\mathstrut 6q^{96} \) \(\mathstrut -\mathstrut 73q^{97} \) \(\mathstrut -\mathstrut 151q^{98} \) \(\mathstrut -\mathstrut 51q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.82578 1.42747 5.98503 −1.00000 −4.03370 3.78558 −11.2608 −0.962340 2.82578
1.2 −2.79540 −1.02268 5.81427 −1.00000 2.85879 3.00451 −10.6624 −1.95413 2.79540
1.3 −2.76648 3.20166 5.65342 −1.00000 −8.85734 −4.04586 −10.1071 7.25064 2.76648
1.4 −2.72603 −2.40516 5.43123 −1.00000 6.55652 −0.808497 −9.35364 2.78477 2.72603
1.5 −2.70698 2.06859 5.32774 −1.00000 −5.59964 3.95188 −9.00814 1.27907 2.70698
1.6 −2.70543 0.0377636 5.31934 −1.00000 −0.102167 −3.03479 −8.98024 −2.99857 2.70543
1.7 −2.70407 −1.94382 5.31198 −1.00000 5.25623 −4.62711 −8.95580 0.778451 2.70407
1.8 −2.69188 −1.94222 5.24620 −1.00000 5.22822 1.57451 −8.73838 0.772219 2.69188
1.9 −2.67655 −2.96805 5.16395 −1.00000 7.94414 4.47636 −8.46847 5.80930 2.67655
1.10 −2.66914 −3.32233 5.12432 −1.00000 8.86776 −1.95943 −8.33924 8.03786 2.66914
1.11 −2.56068 −0.755855 4.55711 −1.00000 1.93551 −1.57635 −6.54794 −2.42868 2.56068
1.12 −2.54398 1.20144 4.47181 −1.00000 −3.05645 −0.416723 −6.28824 −1.55653 2.54398
1.13 −2.54024 0.460298 4.45282 −1.00000 −1.16927 −2.08486 −6.23076 −2.78813 2.54024
1.14 −2.53622 2.11151 4.43241 −1.00000 −5.35524 2.12387 −6.16912 1.45846 2.53622
1.15 −2.44538 −0.0835630 3.97986 −1.00000 0.204343 5.18435 −4.84151 −2.99302 2.44538
1.16 −2.42652 0.577409 3.88800 −1.00000 −1.40110 3.20436 −4.58127 −2.66660 2.42652
1.17 −2.41030 −2.44694 3.80955 −1.00000 5.89785 −4.60731 −4.36157 2.98749 2.41030
1.18 −2.40990 2.64302 3.80764 −1.00000 −6.36944 −3.14385 −4.35625 3.98558 2.40990
1.19 −2.36612 −3.07539 3.59854 −1.00000 7.27675 3.28740 −3.78235 6.45802 2.36612
1.20 −2.31791 3.17051 3.37269 −1.00000 −7.34893 3.65453 −3.18176 7.05211 2.31791
See next 80 embeddings (of 140 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.140
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(1607\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{140} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8035))\).