Properties

Label 8009.2.a.a
Level 8009
Weight 2
Character orbit 8009.a
Self dual Yes
Analytic conductor 63.952
Analytic rank 1
Dimension 306
CM No

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 8009 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8009.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.9521869788\)
Analytic rank: \(1\)
Dimension: \(306\)
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.

\(\operatorname{Tr}(f)(q) = \) \(306q \) \(\mathstrut -\mathstrut 13q^{2} \) \(\mathstrut -\mathstrut 25q^{3} \) \(\mathstrut +\mathstrut 253q^{4} \) \(\mathstrut -\mathstrut 25q^{5} \) \(\mathstrut -\mathstrut 49q^{6} \) \(\mathstrut -\mathstrut 102q^{7} \) \(\mathstrut -\mathstrut 33q^{8} \) \(\mathstrut +\mathstrut 251q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(306q \) \(\mathstrut -\mathstrut 13q^{2} \) \(\mathstrut -\mathstrut 25q^{3} \) \(\mathstrut +\mathstrut 253q^{4} \) \(\mathstrut -\mathstrut 25q^{5} \) \(\mathstrut -\mathstrut 49q^{6} \) \(\mathstrut -\mathstrut 102q^{7} \) \(\mathstrut -\mathstrut 33q^{8} \) \(\mathstrut +\mathstrut 251q^{9} \) \(\mathstrut -\mathstrut 61q^{10} \) \(\mathstrut -\mathstrut 43q^{11} \) \(\mathstrut -\mathstrut 50q^{12} \) \(\mathstrut -\mathstrut 89q^{13} \) \(\mathstrut -\mathstrut 40q^{14} \) \(\mathstrut -\mathstrut 61q^{15} \) \(\mathstrut +\mathstrut 151q^{16} \) \(\mathstrut -\mathstrut 52q^{17} \) \(\mathstrut -\mathstrut 57q^{18} \) \(\mathstrut -\mathstrut 185q^{19} \) \(\mathstrut -\mathstrut 66q^{20} \) \(\mathstrut -\mathstrut 63q^{21} \) \(\mathstrut -\mathstrut 55q^{22} \) \(\mathstrut -\mathstrut 62q^{23} \) \(\mathstrut -\mathstrut 131q^{24} \) \(\mathstrut +\mathstrut 209q^{25} \) \(\mathstrut -\mathstrut 57q^{26} \) \(\mathstrut -\mathstrut 88q^{27} \) \(\mathstrut -\mathstrut 182q^{28} \) \(\mathstrut -\mathstrut 67q^{29} \) \(\mathstrut -\mathstrut 68q^{30} \) \(\mathstrut -\mathstrut 240q^{31} \) \(\mathstrut -\mathstrut 64q^{32} \) \(\mathstrut -\mathstrut 52q^{33} \) \(\mathstrut -\mathstrut 128q^{34} \) \(\mathstrut -\mathstrut 99q^{35} \) \(\mathstrut +\mathstrut 106q^{36} \) \(\mathstrut -\mathstrut 49q^{37} \) \(\mathstrut -\mathstrut 45q^{38} \) \(\mathstrut -\mathstrut 190q^{39} \) \(\mathstrut -\mathstrut 158q^{40} \) \(\mathstrut -\mathstrut 72q^{41} \) \(\mathstrut -\mathstrut 36q^{42} \) \(\mathstrut -\mathstrut 141q^{43} \) \(\mathstrut -\mathstrut 80q^{44} \) \(\mathstrut -\mathstrut 100q^{45} \) \(\mathstrut -\mathstrut 91q^{46} \) \(\mathstrut -\mathstrut 105q^{47} \) \(\mathstrut -\mathstrut 85q^{48} \) \(\mathstrut +\mathstrut 116q^{49} \) \(\mathstrut -\mathstrut 51q^{50} \) \(\mathstrut -\mathstrut 145q^{51} \) \(\mathstrut -\mathstrut 237q^{52} \) \(\mathstrut -\mathstrut 48q^{53} \) \(\mathstrut -\mathstrut 156q^{54} \) \(\mathstrut -\mathstrut 420q^{55} \) \(\mathstrut -\mathstrut 116q^{56} \) \(\mathstrut -\mathstrut 35q^{57} \) \(\mathstrut -\mathstrut 43q^{58} \) \(\mathstrut -\mathstrut 139q^{59} \) \(\mathstrut -\mathstrut 73q^{60} \) \(\mathstrut -\mathstrut 233q^{61} \) \(\mathstrut -\mathstrut 58q^{62} \) \(\mathstrut -\mathstrut 252q^{63} \) \(\mathstrut -\mathstrut 3q^{64} \) \(\mathstrut -\mathstrut 45q^{65} \) \(\mathstrut -\mathstrut 127q^{66} \) \(\mathstrut -\mathstrut 108q^{67} \) \(\mathstrut -\mathstrut 85q^{68} \) \(\mathstrut -\mathstrut 164q^{69} \) \(\mathstrut -\mathstrut 56q^{70} \) \(\mathstrut -\mathstrut 131q^{71} \) \(\mathstrut -\mathstrut 117q^{72} \) \(\mathstrut -\mathstrut 118q^{73} \) \(\mathstrut -\mathstrut 47q^{74} \) \(\mathstrut -\mathstrut 112q^{75} \) \(\mathstrut -\mathstrut 389q^{76} \) \(\mathstrut -\mathstrut 36q^{77} \) \(\mathstrut +\mathstrut 9q^{78} \) \(\mathstrut -\mathstrut 382q^{79} \) \(\mathstrut -\mathstrut 119q^{80} \) \(\mathstrut +\mathstrut 102q^{81} \) \(\mathstrut -\mathstrut 131q^{82} \) \(\mathstrut -\mathstrut 59q^{83} \) \(\mathstrut -\mathstrut 144q^{84} \) \(\mathstrut -\mathstrut 140q^{85} \) \(\mathstrut -\mathstrut 38q^{86} \) \(\mathstrut -\mathstrut 301q^{87} \) \(\mathstrut -\mathstrut 131q^{88} \) \(\mathstrut -\mathstrut 98q^{89} \) \(\mathstrut -\mathstrut 138q^{90} \) \(\mathstrut -\mathstrut 176q^{91} \) \(\mathstrut -\mathstrut 97q^{92} \) \(\mathstrut -\mathstrut 60q^{93} \) \(\mathstrut -\mathstrut 342q^{94} \) \(\mathstrut -\mathstrut 154q^{95} \) \(\mathstrut -\mathstrut 243q^{96} \) \(\mathstrut -\mathstrut 109q^{97} \) \(\mathstrut -\mathstrut 21q^{98} \) \(\mathstrut -\mathstrut 173q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.77719 −1.72357 5.71279 −0.359137 4.78667 −0.363859 −10.3111 −0.0293152 0.997391
1.2 −2.76073 2.52985 5.62166 0.0311099 −6.98425 0.584590 −9.99844 3.40015 −0.0858861
1.3 −2.72646 1.03232 5.43357 3.59624 −2.81456 2.40174 −9.36147 −1.93433 −9.80500
1.4 −2.72316 −0.885304 5.41558 2.13071 2.41082 −0.787494 −9.30115 −2.21624 −5.80224
1.5 −2.70598 −0.529374 5.32233 3.62605 1.43248 −4.04119 −8.99016 −2.71976 −9.81202
1.6 −2.66884 3.40767 5.12268 0.00216831 −9.09450 −2.11405 −8.33393 8.61218 −0.00578687
1.7 −2.66627 −0.991276 5.10897 −2.13375 2.64301 1.36601 −8.28935 −2.01737 5.68914
1.8 −2.64630 2.23042 5.00289 −0.684253 −5.90235 0.773793 −7.94655 1.97477 1.81074
1.9 −2.63809 −1.56073 4.95951 2.27175 4.11735 −0.895063 −7.80745 −0.564115 −5.99307
1.10 −2.63380 −2.43471 4.93693 −0.366932 6.41255 4.18826 −7.73529 2.92781 0.966426
1.11 −2.60787 1.96510 4.80101 3.32724 −5.12474 0.944759 −7.30469 0.861625 −8.67702
1.12 −2.59381 0.956385 4.72785 −4.27828 −2.48068 1.62544 −7.07552 −2.08533 11.0970
1.13 −2.58891 −2.43363 4.70248 −1.53347 6.30046 1.06858 −6.99648 2.92255 3.97002
1.14 −2.58035 2.52546 4.65823 0.386293 −6.51658 −3.83630 −6.85917 3.37795 −0.996774
1.15 −2.56583 −0.0859559 4.58350 0.644754 0.220548 −0.863168 −6.62882 −2.99261 −1.65433
1.16 −2.56252 1.99928 4.56650 −3.64858 −5.12319 −3.78692 −6.57671 0.997123 9.34955
1.17 −2.55906 0.342269 4.54881 −2.51856 −0.875888 −3.84856 −6.52258 −2.88285 6.44516
1.18 −2.52963 0.577486 4.39902 −1.57582 −1.46082 −2.02955 −6.06864 −2.66651 3.98624
1.19 −2.52232 −2.77220 4.36210 3.44597 6.99239 −1.87628 −5.95798 4.68511 −8.69185
1.20 −2.50471 3.07842 4.27357 4.01883 −7.71055 −2.99037 −5.69464 6.47666 −10.0660
See next 80 embeddings (of 306 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.306
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(8009\) \(1\)