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 $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4007,2,Mod(1,4007)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4007, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4007.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4007 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4007.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(31.9960560899\)
Analytic rank: \(1\)
Dimension: \(139\)
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.

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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:

Atkin-Lehner signs

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

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