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Properties

Label 8007.2.a.d
Level $8007$
Weight $2$
Character orbit 8007.a
Self dual yes
Analytic conductor $63.936$
Analytic rank $1$
Dimension $40$
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] = [8007,2,Mod(1,8007)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8007, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8007.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8007 = 3 \cdot 17 \cdot 157 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8007.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: \(63.9362168984\)
Analytic rank: \(1\)
Dimension: \(40\)
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.

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

Display 100 coefficients 10 coefficients

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 −2.69841 1.00000 5.28144 2.43860 −2.69841 −1.23437 −8.85467 1.00000 −6.58035
1.2 −2.61866 1.00000 4.85739 0.391872 −2.61866 2.70632 −7.48254 1.00000 −1.02618
1.3 −2.59865 1.00000 4.75296 −3.62998 −2.59865 4.35344 −7.15398 1.00000 9.43304
1.4 −2.49435 1.00000 4.22179 −2.73091 −2.49435 0.916542 −5.54193 1.00000 6.81185
1.5 −2.23744 1.00000 3.00614 1.45299 −2.23744 −4.68307 −2.25118 1.00000 −3.25099
1.6 −2.17139 1.00000 2.71494 −0.590074 −2.17139 −3.96431 −1.55242 1.00000 1.28128
1.7 −2.12487 1.00000 2.51508 2.60620 −2.12487 −0.251909 −1.09447 1.00000 −5.53783
1.8 −1.86400 1.00000 1.47451 −2.73802 −1.86400 2.48904 0.979516 1.00000 5.10367
1.9 −1.85562 1.00000 1.44333 −0.427574 −1.85562 2.91879 1.03298 1.00000 0.793415
1.10 −1.83549 1.00000 1.36903 −0.437870 −1.83549 1.24463 1.15814 1.00000 0.803707
1.11 −1.72962 1.00000 0.991576 0.136397 −1.72962 −2.36016 1.74419 1.00000 −0.235915
1.12 −1.55034 1.00000 0.403543 −3.10689 −1.55034 −2.24451 2.47505 1.00000 4.81672
1.13 −1.32783 1.00000 −0.236868 2.41386 −1.32783 0.553664 2.97018 1.00000 −3.20519
1.14 −1.21037 1.00000 −0.535004 3.43189 −1.21037 −1.42621 3.06829 1.00000 −4.15386
1.15 −1.15648 1.00000 −0.662543 −2.13083 −1.15648 1.76837 3.07919 1.00000 2.46428
1.16 −0.802287 1.00000 −1.35634 0.612480 −0.802287 −2.13461 2.69274 1.00000 −0.491385
1.17 −0.656002 1.00000 −1.56966 0.401662 −0.656002 −2.40151 2.34170 1.00000 −0.263491
1.18 −0.524942 1.00000 −1.72444 −0.0309405 −0.524942 4.53114 1.95511 1.00000 0.0162420
1.19 −0.481635 1.00000 −1.76803 −2.37410 −0.481635 −1.68399 1.81481 1.00000 1.14345
1.20 −0.422786 1.00000 −1.82125 −3.08259 −0.422786 −4.30268 1.61557 1.00000 1.30328
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.40
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(17\) \(-1\)
\(157\) \(1\)

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 8007.2.a.d ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8007.2.a.d ✓ 40 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + 7 T_{2}^{39} - 30 T_{2}^{38} - 309 T_{2}^{37} + 232 T_{2}^{36} + 6145 T_{2}^{35} + \cdots - 521 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8007))\). Copy content Toggle raw display