Properties

Label 4007.2.a.a
Level 4007
Weight 2
Character orbit 4007.a
Self dual Yes
Analytic conductor 31.996
Analytic rank 1
Dimension 139
CM No

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Newspace parameters

Level: \( N \) = \( 4007 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4007.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.9960560899\)
Analytic rank: \(1\)
Dimension: \(139\)
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.

\(\operatorname{Tr}(f)(q) = \) \(139q \) \(\mathstrut -\mathstrut 13q^{2} \) \(\mathstrut -\mathstrut 22q^{3} \) \(\mathstrut +\mathstrut 113q^{4} \) \(\mathstrut -\mathstrut 16q^{5} \) \(\mathstrut -\mathstrut 15q^{6} \) \(\mathstrut -\mathstrut 44q^{7} \) \(\mathstrut -\mathstrut 36q^{8} \) \(\mathstrut +\mathstrut 87q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(139q \) \(\mathstrut -\mathstrut 13q^{2} \) \(\mathstrut -\mathstrut 22q^{3} \) \(\mathstrut +\mathstrut 113q^{4} \) \(\mathstrut -\mathstrut 16q^{5} \) \(\mathstrut -\mathstrut 15q^{6} \) \(\mathstrut -\mathstrut 44q^{7} \) \(\mathstrut -\mathstrut 36q^{8} \) \(\mathstrut +\mathstrut 87q^{9} \) \(\mathstrut -\mathstrut 40q^{10} \) \(\mathstrut -\mathstrut 17q^{11} \) \(\mathstrut -\mathstrut 59q^{12} \) \(\mathstrut -\mathstrut 89q^{13} \) \(\mathstrut -\mathstrut 15q^{14} \) \(\mathstrut -\mathstrut 29q^{15} \) \(\mathstrut +\mathstrut 73q^{16} \) \(\mathstrut -\mathstrut 58q^{17} \) \(\mathstrut -\mathstrut 51q^{18} \) \(\mathstrut -\mathstrut 37q^{19} \) \(\mathstrut -\mathstrut 24q^{20} \) \(\mathstrut -\mathstrut 37q^{21} \) \(\mathstrut -\mathstrut 99q^{22} \) \(\mathstrut -\mathstrut 42q^{23} \) \(\mathstrut -\mathstrut 27q^{24} \) \(\mathstrut -\mathstrut 11q^{25} \) \(\mathstrut +\mathstrut 2q^{26} \) \(\mathstrut -\mathstrut 73q^{27} \) \(\mathstrut -\mathstrut 113q^{28} \) \(\mathstrut -\mathstrut 57q^{29} \) \(\mathstrut -\mathstrut 29q^{30} \) \(\mathstrut -\mathstrut 51q^{31} \) \(\mathstrut -\mathstrut 80q^{32} \) \(\mathstrut -\mathstrut 78q^{33} \) \(\mathstrut -\mathstrut 28q^{34} \) \(\mathstrut -\mathstrut 34q^{35} \) \(\mathstrut +\mathstrut 28q^{36} \) \(\mathstrut -\mathstrut 117q^{37} \) \(\mathstrut -\mathstrut 31q^{38} \) \(\mathstrut -\mathstrut 36q^{39} \) \(\mathstrut -\mathstrut 107q^{40} \) \(\mathstrut -\mathstrut 60q^{41} \) \(\mathstrut -\mathstrut 41q^{42} \) \(\mathstrut -\mathstrut 109q^{43} \) \(\mathstrut -\mathstrut 21q^{44} \) \(\mathstrut -\mathstrut 62q^{45} \) \(\mathstrut -\mathstrut 92q^{46} \) \(\mathstrut -\mathstrut 26q^{47} \) \(\mathstrut -\mathstrut 90q^{48} \) \(\mathstrut -\mathstrut 7q^{49} \) \(\mathstrut -\mathstrut 22q^{50} \) \(\mathstrut -\mathstrut 47q^{51} \) \(\mathstrut -\mathstrut 182q^{52} \) \(\mathstrut -\mathstrut 83q^{53} \) \(\mathstrut -\mathstrut 19q^{54} \) \(\mathstrut -\mathstrut 53q^{55} \) \(\mathstrut -\mathstrut 23q^{56} \) \(\mathstrut -\mathstrut 201q^{57} \) \(\mathstrut -\mathstrut 112q^{58} \) \(\mathstrut +\mathstrut 14q^{59} \) \(\mathstrut -\mathstrut 64q^{60} \) \(\mathstrut -\mathstrut 73q^{61} \) \(\mathstrut -\mathstrut 21q^{62} \) \(\mathstrut -\mathstrut 94q^{63} \) \(\mathstrut +\mathstrut 14q^{64} \) \(\mathstrut -\mathstrut 123q^{65} \) \(\mathstrut -\mathstrut 10q^{66} \) \(\mathstrut -\mathstrut 135q^{67} \) \(\mathstrut -\mathstrut 84q^{68} \) \(\mathstrut -\mathstrut 50q^{69} \) \(\mathstrut -\mathstrut 35q^{70} \) \(\mathstrut -\mathstrut 29q^{71} \) \(\mathstrut -\mathstrut 143q^{72} \) \(\mathstrut -\mathstrut 266q^{73} \) \(\mathstrut -\mathstrut 53q^{74} \) \(\mathstrut -\mathstrut 32q^{75} \) \(\mathstrut -\mathstrut 66q^{76} \) \(\mathstrut -\mathstrut 69q^{77} \) \(\mathstrut -\mathstrut 59q^{78} \) \(\mathstrut -\mathstrut 124q^{79} \) \(\mathstrut -\mathstrut 20q^{80} \) \(\mathstrut -\mathstrut 33q^{81} \) \(\mathstrut -\mathstrut 93q^{82} \) \(\mathstrut -\mathstrut 28q^{83} \) \(\mathstrut -\mathstrut 4q^{84} \) \(\mathstrut -\mathstrut 179q^{85} \) \(\mathstrut +\mathstrut 6q^{86} \) \(\mathstrut -\mathstrut 40q^{87} \) \(\mathstrut -\mathstrut 259q^{88} \) \(\mathstrut -\mathstrut 41q^{89} \) \(\mathstrut +\mathstrut 2q^{90} \) \(\mathstrut -\mathstrut 50q^{91} \) \(\mathstrut -\mathstrut 77q^{92} \) \(\mathstrut -\mathstrut 60q^{93} \) \(\mathstrut -\mathstrut 48q^{94} \) \(\mathstrut -\mathstrut 37q^{95} \) \(\mathstrut +\mathstrut 3q^{96} \) \(\mathstrut -\mathstrut 220q^{97} \) \(\mathstrut -\mathstrut 9q^{98} \) \(\mathstrut -\mathstrut 35q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.81570 −0.263068 5.92814 2.38290 0.740719 −2.49985 −11.0604 −2.93080 −6.70951
1.2 −2.77109 1.88840 5.67894 −0.563279 −5.23292 0.553316 −10.1947 0.566050 1.56090
1.3 −2.72706 −2.54460 5.43687 −1.96557 6.93928 −3.80223 −9.37255 3.47499 5.36024
1.4 −2.70090 −3.24744 5.29488 −0.141100 8.77102 1.56207 −8.89915 7.54587 0.381096
1.5 −2.64078 1.79793 4.97369 −0.236310 −4.74793 −1.11879 −7.85286 0.232555 0.624041
1.6 −2.62057 −0.101978 4.86739 0.441388 0.267240 0.132029 −7.51419 −2.98960 −1.15669
1.7 −2.60765 2.41443 4.79982 2.62983 −6.29599 −4.53793 −7.30095 2.82948 −6.85767
1.8 −2.53714 −2.64168 4.43706 4.10471 6.70230 −0.283767 −6.18314 3.97846 −10.4142
1.9 −2.52380 −1.27675 4.36956 2.82369 3.22226 3.70991 −5.98028 −1.36991 −7.12642
1.10 −2.51027 −1.47560 4.30145 −0.0269498 3.70415 2.66833 −5.77725 −0.822611 0.0676513
1.11 −2.46958 0.828101 4.09882 −3.19925 −2.04506 −2.79800 −5.18321 −2.31425 7.90080
1.12 −2.42814 −1.84708 3.89588 −2.88367 4.48497 −1.14516 −4.60348 0.411691 7.00198
1.13 −2.41992 2.90998 3.85601 −2.04833 −7.04192 0.763772 −4.49140 5.46800 4.95680
1.14 −2.39846 −2.50987 3.75261 0.204658 6.01982 −4.29057 −4.20358 3.29943 −0.490864
1.15 −2.38393 −0.401819 3.68311 −0.901205 0.957907 −4.24761 −4.01240 −2.83854 2.14841
1.16 −2.37198 1.06086 3.62628 2.75167 −2.51634 1.78055 −3.85749 −1.87457 −6.52689
1.17 −2.24444 −1.81943 3.03751 1.76577 4.08359 0.689140 −2.32862 0.310309 −3.96317
1.18 −2.23150 1.82610 2.97957 2.21560 −4.07492 3.73360 −2.18591 0.334626 −4.94411
1.19 −2.19482 2.00637 2.81722 −1.97221 −4.40361 2.24566 −1.79364 1.02552 4.32863
1.20 −2.15727 1.62818 2.65379 2.01382 −3.51242 −1.64574 −1.41041 −0.349027 −4.34434
See next 80 embeddings (of 139 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.139
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(4007\) \(1\)