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Properties

Label 8007.2.a.g
Level $8007$
Weight $2$
Character orbit 8007.a
Self dual yes
Analytic conductor $63.936$
Analytic rank $0$
Dimension $56$
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: \(56\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + q^{2} - 56 q^{3} + 61 q^{4} + q^{5} - q^{6} + 19 q^{7} + 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + q^{2} - 56 q^{3} + 61 q^{4} + q^{5} - q^{6} + 19 q^{7} + 56 q^{9} + 8 q^{10} - 7 q^{11} - 61 q^{12} + 8 q^{13} - 8 q^{14} - q^{15} + 71 q^{16} - 56 q^{17} + q^{18} - 2 q^{19} - 4 q^{20} - 19 q^{21} + 47 q^{22} + 16 q^{23} + 85 q^{25} - 11 q^{26} - 56 q^{27} + 52 q^{28} + 17 q^{29} - 8 q^{30} + 23 q^{31} + 11 q^{32} + 7 q^{33} - q^{34} - 41 q^{35} + 61 q^{36} + 58 q^{37} - 22 q^{38} - 8 q^{39} + 38 q^{40} - q^{41} + 8 q^{42} + 27 q^{43} + 2 q^{44} + q^{45} + 46 q^{46} + 5 q^{47} - 71 q^{48} + 59 q^{49} - 4 q^{50} + 56 q^{51} + 25 q^{52} + 15 q^{53} - q^{54} + 9 q^{55} - 36 q^{56} + 2 q^{57} + 89 q^{58} - 61 q^{59} + 4 q^{60} + 47 q^{61} + 8 q^{62} + 19 q^{63} + 88 q^{64} + 39 q^{65} - 47 q^{66} + 20 q^{67} - 61 q^{68} - 16 q^{69} + 36 q^{70} - 2 q^{71} + 93 q^{73} + 48 q^{74} - 85 q^{75} + 38 q^{76} + 26 q^{77} + 11 q^{78} + 72 q^{79} + 42 q^{80} + 56 q^{81} + 33 q^{82} - 11 q^{83} - 52 q^{84} - q^{85} - 4 q^{86} - 17 q^{87} + 130 q^{88} - 6 q^{89} + 8 q^{90} + 37 q^{91} + 132 q^{92} - 23 q^{93} - 32 q^{94} + 12 q^{95} - 11 q^{96} + 100 q^{97} + 42 q^{98} - 7 q^{99}+O(q^{100}) \) 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 −2.76474 −1.00000 5.64378 −1.85589 2.76474 1.33970 −10.0741 1.00000 5.13105
1.2 −2.71794 −1.00000 5.38720 2.54834 2.71794 4.22064 −9.20622 1.00000 −6.92625
1.3 −2.65517 −1.00000 5.04993 −2.53504 2.65517 2.76662 −8.09809 1.00000 6.73097
1.4 −2.51234 −1.00000 4.31187 −0.437106 2.51234 0.108032 −5.80821 1.00000 1.09816
1.5 −2.49549 −1.00000 4.22749 2.28895 2.49549 −3.97589 −5.55870 1.00000 −5.71206
1.6 −2.45801 −1.00000 4.04182 −3.57546 2.45801 3.37294 −5.01881 1.00000 8.78852
1.7 −2.36108 −1.00000 3.57471 4.34691 2.36108 −0.239132 −3.71802 1.00000 −10.2634
1.8 −2.27923 −1.00000 3.19488 1.61810 2.27923 2.49401 −2.72340 1.00000 −3.68801
1.9 −2.11176 −1.00000 2.45954 −3.52613 2.11176 −3.47105 −0.970439 1.00000 7.44635
1.10 −1.95968 −1.00000 1.84033 −3.13690 1.95968 −0.0823024 0.312893 1.00000 6.14731
1.11 −1.93055 −1.00000 1.72702 −2.51216 1.93055 −1.67367 0.526999 1.00000 4.84984
1.12 −1.84970 −1.00000 1.42137 0.505373 1.84970 −1.72444 1.07028 1.00000 −0.934786
1.13 −1.84393 −1.00000 1.40007 2.87244 1.84393 −1.74244 1.10623 1.00000 −5.29656
1.14 −1.75940 −1.00000 1.09547 −0.617389 1.75940 3.20947 1.59142 1.00000 1.08623
1.15 −1.59327 −1.00000 0.538495 2.65587 1.59327 −0.355214 2.32857 1.00000 −4.23151
1.16 −1.50676 −1.00000 0.270331 −1.85890 1.50676 4.43102 2.60620 1.00000 2.80092
1.17 −1.26654 −1.00000 −0.395877 2.14407 1.26654 1.86831 3.03447 1.00000 −2.71555
1.18 −1.23565 −1.00000 −0.473178 −0.721310 1.23565 −3.93183 3.05597 1.00000 0.891284
1.19 −1.15978 −1.00000 −0.654916 −0.00396327 1.15978 3.25506 3.07911 1.00000 0.00459651
1.20 −0.882919 −1.00000 −1.22045 −0.454468 0.882919 1.65699 2.84340 1.00000 0.401259
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.56
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.g ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8007.2.a.g ✓ 56 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} - T_{2}^{55} - 86 T_{2}^{54} + 85 T_{2}^{53} + 3478 T_{2}^{52} - 3397 T_{2}^{51} + \cdots + 110519 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8007))\). Copy content Toggle raw display