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
Inner twists 1

Related objects

Downloads

Learn more about

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: \(139\)
Coefficient ring index: multiple of None
Twist minimal: yes
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.

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

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 admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4007.2.a.a 139
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4007.2.a.a 139 1.a even 1 1 trivial

Atkin-Lehner signs

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