Properties

Label 4009.2.a.f
Level 4009
Weight 2
Character orbit 4009.a
Self dual Yes
Analytic conductor 32.012
Analytic rank 0
Dimension 83
CM No

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Newspace parameters

Level: \( N \) = \( 4009 = 19 \cdot 211 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4009.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0120261703\)
Analytic rank: \(0\)
Dimension: \(83\)
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.

\(\operatorname{Tr}(f)(q) = \) \(83q \) \(\mathstrut +\mathstrut 11q^{2} \) \(\mathstrut +\mathstrut 95q^{4} \) \(\mathstrut +\mathstrut 15q^{5} \) \(\mathstrut +\mathstrut 23q^{6} \) \(\mathstrut +\mathstrut 19q^{7} \) \(\mathstrut +\mathstrut 30q^{8} \) \(\mathstrut +\mathstrut 101q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(83q \) \(\mathstrut +\mathstrut 11q^{2} \) \(\mathstrut +\mathstrut 95q^{4} \) \(\mathstrut +\mathstrut 15q^{5} \) \(\mathstrut +\mathstrut 23q^{6} \) \(\mathstrut +\mathstrut 19q^{7} \) \(\mathstrut +\mathstrut 30q^{8} \) \(\mathstrut +\mathstrut 101q^{9} \) \(\mathstrut +\mathstrut 9q^{10} \) \(\mathstrut +\mathstrut 56q^{11} \) \(\mathstrut -\mathstrut 2q^{12} \) \(\mathstrut -\mathstrut 5q^{13} \) \(\mathstrut +\mathstrut 6q^{14} \) \(\mathstrut +\mathstrut 19q^{15} \) \(\mathstrut +\mathstrut 123q^{16} \) \(\mathstrut +\mathstrut 19q^{17} \) \(\mathstrut +\mathstrut 40q^{18} \) \(\mathstrut -\mathstrut 83q^{19} \) \(\mathstrut +\mathstrut 49q^{20} \) \(\mathstrut +\mathstrut 9q^{21} \) \(\mathstrut +\mathstrut 18q^{22} \) \(\mathstrut +\mathstrut 74q^{23} \) \(\mathstrut +\mathstrut 38q^{24} \) \(\mathstrut +\mathstrut 98q^{25} \) \(\mathstrut +\mathstrut 28q^{26} \) \(\mathstrut +\mathstrut 6q^{27} \) \(\mathstrut +\mathstrut 50q^{28} \) \(\mathstrut +\mathstrut 16q^{29} \) \(\mathstrut +\mathstrut 56q^{30} \) \(\mathstrut +\mathstrut 24q^{31} \) \(\mathstrut +\mathstrut 81q^{32} \) \(\mathstrut +\mathstrut 13q^{33} \) \(\mathstrut +\mathstrut 9q^{34} \) \(\mathstrut +\mathstrut 71q^{35} \) \(\mathstrut +\mathstrut 156q^{36} \) \(\mathstrut -\mathstrut 6q^{37} \) \(\mathstrut -\mathstrut 11q^{38} \) \(\mathstrut +\mathstrut 126q^{39} \) \(\mathstrut +\mathstrut q^{40} \) \(\mathstrut -\mathstrut q^{42} \) \(\mathstrut +\mathstrut 34q^{43} \) \(\mathstrut +\mathstrut 140q^{44} \) \(\mathstrut +\mathstrut 42q^{45} \) \(\mathstrut +\mathstrut 34q^{46} \) \(\mathstrut +\mathstrut 53q^{47} \) \(\mathstrut +\mathstrut 16q^{48} \) \(\mathstrut +\mathstrut 118q^{49} \) \(\mathstrut +\mathstrut 51q^{50} \) \(\mathstrut +\mathstrut 57q^{51} \) \(\mathstrut +\mathstrut 32q^{52} \) \(\mathstrut +\mathstrut q^{53} \) \(\mathstrut +\mathstrut 53q^{54} \) \(\mathstrut +\mathstrut 60q^{55} \) \(\mathstrut -\mathstrut 2q^{56} \) \(\mathstrut -\mathstrut 2q^{58} \) \(\mathstrut +\mathstrut 44q^{59} \) \(\mathstrut -\mathstrut 9q^{60} \) \(\mathstrut +\mathstrut 21q^{61} \) \(\mathstrut +\mathstrut 28q^{62} \) \(\mathstrut +\mathstrut 83q^{63} \) \(\mathstrut +\mathstrut 154q^{64} \) \(\mathstrut +\mathstrut 44q^{65} \) \(\mathstrut +\mathstrut 17q^{66} \) \(\mathstrut +\mathstrut 5q^{67} \) \(\mathstrut +\mathstrut 63q^{68} \) \(\mathstrut -\mathstrut 36q^{69} \) \(\mathstrut -\mathstrut 48q^{70} \) \(\mathstrut +\mathstrut 193q^{71} \) \(\mathstrut +\mathstrut 135q^{72} \) \(\mathstrut +\mathstrut 54q^{73} \) \(\mathstrut +\mathstrut 127q^{74} \) \(\mathstrut +\mathstrut 5q^{75} \) \(\mathstrut -\mathstrut 95q^{76} \) \(\mathstrut +\mathstrut 54q^{77} \) \(\mathstrut +\mathstrut 45q^{78} \) \(\mathstrut +\mathstrut 54q^{79} \) \(\mathstrut +\mathstrut 45q^{80} \) \(\mathstrut +\mathstrut 147q^{81} \) \(\mathstrut -\mathstrut 35q^{82} \) \(\mathstrut +\mathstrut 84q^{83} \) \(\mathstrut +\mathstrut 12q^{84} \) \(\mathstrut +\mathstrut 28q^{85} \) \(\mathstrut +\mathstrut 60q^{86} \) \(\mathstrut +\mathstrut 51q^{87} \) \(\mathstrut +\mathstrut 23q^{88} \) \(\mathstrut -\mathstrut 24q^{89} \) \(\mathstrut +\mathstrut 31q^{90} \) \(\mathstrut +\mathstrut 28q^{91} \) \(\mathstrut +\mathstrut 108q^{92} \) \(\mathstrut +\mathstrut 39q^{93} \) \(\mathstrut -\mathstrut 49q^{94} \) \(\mathstrut -\mathstrut 15q^{95} \) \(\mathstrut +\mathstrut 25q^{96} \) \(\mathstrut -\mathstrut 22q^{97} \) \(\mathstrut -\mathstrut 67q^{98} \) \(\mathstrut +\mathstrut 132q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.76212 −1.26441 5.62933 −0.901770 3.49245 2.01058 −10.0247 −1.40127 2.49080
1.2 −2.70959 1.36959 5.34187 3.33123 −3.71102 3.43608 −9.05510 −1.12423 −9.02626
1.3 −2.67158 −3.12490 5.13731 1.64338 8.34840 −0.166545 −8.38157 6.76498 −4.39041
1.4 −2.57059 1.65554 4.60795 −0.615415 −4.25572 −3.21164 −6.70397 −0.259188 1.58198
1.5 −2.54651 2.15718 4.48471 0.595796 −5.49327 3.08751 −6.32733 1.65341 −1.51720
1.6 −2.52441 2.53119 4.37265 −2.23047 −6.38976 −2.32994 −5.98955 3.40692 5.63062
1.7 −2.39927 −1.08640 3.75651 4.25844 2.60656 3.20578 −4.21434 −1.81974 −10.2171
1.8 −2.36270 −3.38908 3.58234 0.417258 8.00738 4.46721 −3.73860 8.48588 −0.985855
1.9 −2.35956 −1.58207 3.56753 −1.28760 3.73299 −1.93582 −3.69869 −0.497055 3.03817
1.10 −2.30870 −1.28180 3.33008 1.74419 2.95929 −4.60684 −3.07074 −1.35699 −4.02679
1.11 −2.26307 −2.97799 3.12147 −3.73061 6.73939 −2.90027 −2.53796 5.86843 8.44263
1.12 −2.15681 −0.442776 2.65181 −2.84036 0.954981 0.461477 −1.40583 −2.80395 6.12610
1.13 −2.15273 1.11488 2.63426 −2.91632 −2.40003 1.19746 −1.36540 −1.75705 6.27807
1.14 −1.98014 0.389885 1.92094 2.34447 −0.772025 −3.40620 0.156544 −2.84799 −4.64237
1.15 −1.90192 0.698740 1.61729 0.858137 −1.32894 1.83968 0.727887 −2.51176 −1.63210
1.16 −1.77808 −1.50294 1.16155 1.25808 2.67234 4.62687 1.49083 −0.741168 −2.23696
1.17 −1.73978 2.70968 1.02684 3.17028 −4.71425 1.26105 1.69309 4.34237 −5.51560
1.18 −1.71565 −0.772476 0.943456 −1.16427 1.32530 −0.902518 1.81266 −2.40328 1.99748
1.19 −1.59861 3.13645 0.555538 2.23749 −5.01395 4.84396 2.30912 6.83733 −3.57686
1.20 −1.56314 0.463979 0.443399 3.91464 −0.725264 −3.83453 2.43318 −2.78472 −6.11913
See all 83 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.83
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
\(19\) \(1\)
\(211\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{83} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4009))\).