Properties

Label 4009.2.a.d
Level 4009
Weight 2
Character orbit 4009.a
Self dual Yes
Analytic conductor 32.012
Analytic rank 1
Dimension 75
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: \(1\)
Dimension: \(75\)
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.

\(\operatorname{Tr}(f)(q) = \) \(75q \) \(\mathstrut -\mathstrut 11q^{2} \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 67q^{4} \) \(\mathstrut -\mathstrut 18q^{5} \) \(\mathstrut -\mathstrut 15q^{6} \) \(\mathstrut -\mathstrut 19q^{7} \) \(\mathstrut -\mathstrut 30q^{8} \) \(\mathstrut +\mathstrut 57q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(75q \) \(\mathstrut -\mathstrut 11q^{2} \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 67q^{4} \) \(\mathstrut -\mathstrut 18q^{5} \) \(\mathstrut -\mathstrut 15q^{6} \) \(\mathstrut -\mathstrut 19q^{7} \) \(\mathstrut -\mathstrut 30q^{8} \) \(\mathstrut +\mathstrut 57q^{9} \) \(\mathstrut -\mathstrut 48q^{11} \) \(\mathstrut -\mathstrut 14q^{12} \) \(\mathstrut -\mathstrut 3q^{13} \) \(\mathstrut -\mathstrut 4q^{14} \) \(\mathstrut -\mathstrut 39q^{15} \) \(\mathstrut +\mathstrut 59q^{16} \) \(\mathstrut -\mathstrut 23q^{17} \) \(\mathstrut -\mathstrut 24q^{18} \) \(\mathstrut -\mathstrut 75q^{19} \) \(\mathstrut -\mathstrut 62q^{20} \) \(\mathstrut -\mathstrut 3q^{21} \) \(\mathstrut -\mathstrut 6q^{22} \) \(\mathstrut -\mathstrut 73q^{23} \) \(\mathstrut -\mathstrut 64q^{24} \) \(\mathstrut +\mathstrut 57q^{25} \) \(\mathstrut -\mathstrut 46q^{26} \) \(\mathstrut -\mathstrut 22q^{27} \) \(\mathstrut -\mathstrut 26q^{28} \) \(\mathstrut -\mathstrut 39q^{29} \) \(\mathstrut -\mathstrut 14q^{30} \) \(\mathstrut -\mathstrut 44q^{31} \) \(\mathstrut -\mathstrut 71q^{32} \) \(\mathstrut -\mathstrut 3q^{33} \) \(\mathstrut -\mathstrut 9q^{34} \) \(\mathstrut -\mathstrut 49q^{35} \) \(\mathstrut +\mathstrut 20q^{36} \) \(\mathstrut -\mathstrut 12q^{37} \) \(\mathstrut +\mathstrut 11q^{38} \) \(\mathstrut -\mathstrut 90q^{39} \) \(\mathstrut -\mathstrut 8q^{40} \) \(\mathstrut -\mathstrut 42q^{41} \) \(\mathstrut -\mathstrut 45q^{42} \) \(\mathstrut -\mathstrut 24q^{43} \) \(\mathstrut -\mathstrut 120q^{44} \) \(\mathstrut -\mathstrut 63q^{45} \) \(\mathstrut -\mathstrut 39q^{46} \) \(\mathstrut -\mathstrut 59q^{47} \) \(\mathstrut -\mathstrut 4q^{48} \) \(\mathstrut +\mathstrut 48q^{49} \) \(\mathstrut -\mathstrut 100q^{50} \) \(\mathstrut -\mathstrut 55q^{51} \) \(\mathstrut +\mathstrut 2q^{52} \) \(\mathstrut +\mathstrut 13q^{53} \) \(\mathstrut -\mathstrut 87q^{54} \) \(\mathstrut -\mathstrut 36q^{55} \) \(\mathstrut -\mathstrut 12q^{56} \) \(\mathstrut +\mathstrut 4q^{57} \) \(\mathstrut -\mathstrut 17q^{58} \) \(\mathstrut -\mathstrut 47q^{59} \) \(\mathstrut -\mathstrut 45q^{60} \) \(\mathstrut -\mathstrut 35q^{61} \) \(\mathstrut -\mathstrut 40q^{62} \) \(\mathstrut -\mathstrut 69q^{63} \) \(\mathstrut +\mathstrut 26q^{64} \) \(\mathstrut -\mathstrut 44q^{65} \) \(\mathstrut +\mathstrut 33q^{66} \) \(\mathstrut -\mathstrut 39q^{67} \) \(\mathstrut -\mathstrut 63q^{68} \) \(\mathstrut +\mathstrut 42q^{69} \) \(\mathstrut +\mathstrut 40q^{70} \) \(\mathstrut -\mathstrut 154q^{71} \) \(\mathstrut -\mathstrut 51q^{72} \) \(\mathstrut -\mathstrut 29q^{73} \) \(\mathstrut -\mathstrut 95q^{74} \) \(\mathstrut +\mathstrut 37q^{75} \) \(\mathstrut -\mathstrut 67q^{76} \) \(\mathstrut -\mathstrut 24q^{77} \) \(\mathstrut -\mathstrut 19q^{78} \) \(\mathstrut -\mathstrut 95q^{79} \) \(\mathstrut -\mathstrut 146q^{80} \) \(\mathstrut +\mathstrut 23q^{81} \) \(\mathstrut +\mathstrut 7q^{82} \) \(\mathstrut -\mathstrut 52q^{83} \) \(\mathstrut -\mathstrut 72q^{84} \) \(\mathstrut -\mathstrut 36q^{85} \) \(\mathstrut -\mathstrut 44q^{86} \) \(\mathstrut -\mathstrut 103q^{87} \) \(\mathstrut +\mathstrut 67q^{88} \) \(\mathstrut +\mathstrut q^{89} \) \(\mathstrut -\mathstrut 2q^{90} \) \(\mathstrut -\mathstrut 64q^{91} \) \(\mathstrut -\mathstrut 183q^{92} \) \(\mathstrut -\mathstrut 49q^{93} \) \(\mathstrut +\mathstrut 5q^{94} \) \(\mathstrut +\mathstrut 18q^{95} \) \(\mathstrut -\mathstrut 69q^{96} \) \(\mathstrut -\mathstrut 7q^{97} \) \(\mathstrut -\mathstrut 23q^{98} \) \(\mathstrut -\mathstrut 100q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.80216 −0.276714 5.85209 −4.46085 0.775398 −4.37845 −10.7942 −2.92343 12.5000
1.2 −2.73305 0.112405 5.46956 0.682101 −0.307207 −0.658651 −9.48248 −2.98737 −1.86422
1.3 −2.71128 −2.26408 5.35102 1.45359 6.13856 1.50475 −9.08554 2.12608 −3.94108
1.4 −2.62786 2.23579 4.90563 3.65029 −5.87534 −3.99765 −7.63559 1.99877 −9.59245
1.5 −2.62780 3.38775 4.90531 −4.01450 −8.90232 2.03245 −7.63457 8.47685 10.5493
1.6 −2.58872 0.285133 4.70149 −3.09212 −0.738130 5.20222 −6.99340 −2.91870 8.00463
1.7 −2.52406 −2.15147 4.37089 −1.88445 5.43045 −0.0300064 −5.98429 1.62883 4.75648
1.8 −2.42270 2.44086 3.86949 −0.894921 −5.91348 2.03761 −4.52922 2.95780 2.16813
1.9 −2.37585 1.43559 3.64464 2.52541 −3.41075 −1.13884 −3.90741 −0.939073 −5.99997
1.10 −2.33282 −0.629609 3.44205 3.06029 1.46876 3.36015 −3.36405 −2.60359 −7.13912
1.11 −2.20714 0.00654009 2.87148 −3.65103 −0.0144349 −1.05666 −1.92348 −2.99996 8.05835
1.12 −2.18144 −2.69143 2.75868 0.125286 5.87120 −3.80798 −1.65501 4.24381 −0.273304
1.13 −2.17778 −1.25453 2.74274 2.27371 2.73208 −1.62232 −1.61752 −1.42616 −4.95164
1.14 −2.06463 1.41935 2.26270 1.21401 −2.93043 2.08294 −0.542368 −0.985454 −2.50648
1.15 −1.94602 3.23698 1.78701 −0.488090 −6.29925 −2.80369 0.414483 7.47805 0.949835
1.16 −1.81819 1.88652 1.30583 −3.78374 −3.43005 2.14600 1.26214 0.558942 6.87957
1.17 −1.81267 −3.27382 1.28578 1.53613 5.93435 −3.38684 1.29465 7.71788 −2.78450
1.18 −1.72632 0.426642 0.980187 1.18315 −0.736521 −0.668631 1.76053 −2.81798 −2.04249
1.19 −1.68291 −2.73847 0.832174 −3.05253 4.60859 2.37030 1.96534 4.49922 5.13711
1.20 −1.64786 2.26384 0.715427 1.21347 −3.73048 −4.02592 2.11679 2.12497 −1.99962
See all 75 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.75
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 \(T_{2}^{75} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4009))\).