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Properties

Label 8007.2.a.i
Level $8007$
Weight $2$
Character orbit 8007.a
Self dual yes
Analytic conductor $63.936$
Analytic rank $0$
Dimension $63$
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: \(0\)
Dimension: \(63\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 63 q + 10 q^{2} + 63 q^{3} + 70 q^{4} + 19 q^{5} + 10 q^{6} + 11 q^{7} + 27 q^{8} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 63 q + 10 q^{2} + 63 q^{3} + 70 q^{4} + 19 q^{5} + 10 q^{6} + 11 q^{7} + 27 q^{8} + 63 q^{9} + 4 q^{10} + 23 q^{11} + 70 q^{12} + 10 q^{13} + 18 q^{14} + 19 q^{15} + 72 q^{16} + 63 q^{17} + 10 q^{18} + 6 q^{19} + 48 q^{20} + 11 q^{21} + 21 q^{22} + 44 q^{23} + 27 q^{24} + 110 q^{25} + 41 q^{26} + 63 q^{27} + 26 q^{28} + 35 q^{29} + 4 q^{30} + q^{31} + 54 q^{32} + 23 q^{33} + 10 q^{34} + 47 q^{35} + 70 q^{36} + 40 q^{37} + 38 q^{38} + 10 q^{39} - 10 q^{40} + 35 q^{41} + 18 q^{42} + 27 q^{43} + 46 q^{44} + 19 q^{45} + 8 q^{46} + 29 q^{47} + 72 q^{48} + 114 q^{49} + 27 q^{50} + 63 q^{51} - q^{52} + 75 q^{53} + 10 q^{54} + 5 q^{55} + 24 q^{56} + 6 q^{57} + 41 q^{58} + 105 q^{59} + 48 q^{60} + 5 q^{61} + 22 q^{62} + 11 q^{63} + 61 q^{64} + 49 q^{65} + 21 q^{66} + 4 q^{67} + 70 q^{68} + 44 q^{69} - 16 q^{70} + 16 q^{71} + 27 q^{72} + 39 q^{73} + 54 q^{74} + 110 q^{75} + 6 q^{76} + 88 q^{77} + 41 q^{78} + 16 q^{79} + 102 q^{80} + 63 q^{81} - 29 q^{82} + 73 q^{83} + 26 q^{84} + 19 q^{85} + 46 q^{86} + 35 q^{87} + 18 q^{88} + 88 q^{89} + 4 q^{90} - 15 q^{91} + 110 q^{92} + q^{93} - 8 q^{94} + 28 q^{95} + 54 q^{96} + 70 q^{97} + 33 q^{98} + 23 q^{99}+O(q^{100}) \) 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 −2.76200 1.00000 5.62867 −0.159585 −2.76200 −3.17690 −10.0224 1.00000 0.440774
1.2 −2.70479 1.00000 5.31589 4.10743 −2.70479 3.26601 −8.96878 1.00000 −11.1097
1.3 −2.58914 1.00000 4.70363 −1.26310 −2.58914 −1.23370 −7.00007 1.00000 3.27035
1.4 −2.40365 1.00000 3.77751 −0.410000 −2.40365 −0.100135 −4.27252 1.00000 0.985496
1.5 −2.34343 1.00000 3.49167 2.63222 −2.34343 1.69426 −3.49562 1.00000 −6.16842
1.6 −2.27277 1.00000 3.16549 3.41153 −2.27277 4.87508 −2.64888 1.00000 −7.75362
1.7 −2.21840 1.00000 2.92128 −1.40773 −2.21840 0.0138217 −2.04376 1.00000 3.12290
1.8 −2.18826 1.00000 2.78847 −0.00677076 −2.18826 −4.11828 −1.72538 1.00000 0.0148162
1.9 −2.12676 1.00000 2.52310 −0.0337885 −2.12676 4.05166 −1.11251 1.00000 0.0718600
1.10 −2.11232 1.00000 2.46191 −3.85779 −2.11232 0.0837724 −0.975695 1.00000 8.14890
1.11 −2.02903 1.00000 2.11694 4.33739 −2.02903 −3.04221 −0.237280 1.00000 −8.80067
1.12 −1.99602 1.00000 1.98408 2.66347 −1.99602 −1.66715 0.0317759 1.00000 −5.31632
1.13 −1.80525 1.00000 1.25895 −4.24051 −1.80525 −2.73260 1.33779 1.00000 7.65520
1.14 −1.76021 1.00000 1.09834 1.37447 −1.76021 3.49395 1.58711 1.00000 −2.41936
1.15 −1.74167 1.00000 1.03340 −2.69066 −1.74167 0.494465 1.68349 1.00000 4.68623
1.16 −1.25126 1.00000 −0.434343 3.57483 −1.25126 −4.07525 3.04600 1.00000 −4.47305
1.17 −1.23503 1.00000 −0.474692 0.598945 −1.23503 −4.57610 3.05633 1.00000 −0.739717
1.18 −1.22360 1.00000 −0.502806 1.43997 −1.22360 −1.77841 3.06243 1.00000 −1.76195
1.19 −1.05868 1.00000 −0.879191 −2.81588 −1.05868 4.49493 3.04815 1.00000 2.98112
1.20 −1.04403 1.00000 −0.910007 3.07941 −1.04403 2.83741 3.03813 1.00000 −3.21499
See all 63 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.63
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.i ✓ 63
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8007.2.a.i ✓ 63 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{63} - 10 T_{2}^{62} - 48 T_{2}^{61} + 791 T_{2}^{60} + 293 T_{2}^{59} - 29052 T_{2}^{58} + \cdots + 59195 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8007))\). Copy content Toggle raw display