Properties

Label 1002.2.a.h
Level $1002$
Weight $2$
Character orbit 1002.a
Self dual yes
Analytic conductor $8.001$
Analytic rank $1$
Dimension $3$
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] = [1002,2,Mod(1,1002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1002, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1002 = 2 \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1002.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: \(8.00101028253\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + (\beta_{2} - 1) q^{5} - q^{6} + ( - \beta_{2} + \beta_1 - 2) q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} + (\beta_{2} - 1) q^{5} - q^{6} + ( - \beta_{2} + \beta_1 - 2) q^{7} + q^{8} + q^{9} + (\beta_{2} - 1) q^{10} + ( - \beta_{2} - 3 \beta_1 + 1) q^{11} - q^{12} + ( - \beta_{2} + \beta_1 - 3) q^{13} + ( - \beta_{2} + \beta_1 - 2) q^{14} + ( - \beta_{2} + 1) q^{15} + q^{16} + ( - \beta_{2} - \beta_1 - 5) q^{17} + q^{18} + (3 \beta_{2} + \beta_1 - 1) q^{19} + (\beta_{2} - 1) q^{20} + (\beta_{2} - \beta_1 + 2) q^{21} + ( - \beta_{2} - 3 \beta_1 + 1) q^{22} + (\beta_{2} - \beta_1 - 1) q^{23} - q^{24} + ( - 3 \beta_{2} - \beta_1 - 1) q^{25} + ( - \beta_{2} + \beta_1 - 3) q^{26} - q^{27} + ( - \beta_{2} + \beta_1 - 2) q^{28} + (\beta_{2} + 3 \beta_1 - 1) q^{29} + ( - \beta_{2} + 1) q^{30} + (2 \beta_{2} + 6 \beta_1 - 5) q^{31} + q^{32} + (\beta_{2} + 3 \beta_1 - 1) q^{33} + ( - \beta_{2} - \beta_1 - 5) q^{34} + (\beta_1 - 2) q^{35} + q^{36} + ( - 3 \beta_{2} - 2 \beta_1 - 1) q^{37} + (3 \beta_{2} + \beta_1 - 1) q^{38} + (\beta_{2} - \beta_1 + 3) q^{39} + (\beta_{2} - 1) q^{40} + (2 \beta_{2} - 2 \beta_1 - 4) q^{41} + (\beta_{2} - \beta_1 + 2) q^{42} + ( - 3 \beta_{2} + \beta_1 - 1) q^{43} + ( - \beta_{2} - 3 \beta_1 + 1) q^{44} + (\beta_{2} - 1) q^{45} + (\beta_{2} - \beta_1 - 1) q^{46} + (3 \beta_{2} - 3 \beta_1) q^{47} - q^{48} + (4 \beta_{2} - 6 \beta_1 + 4) q^{49} + ( - 3 \beta_{2} - \beta_1 - 1) q^{50} + (\beta_{2} + \beta_1 + 5) q^{51} + ( - \beta_{2} + \beta_1 - 3) q^{52} + (4 \beta_{2} + 3 \beta_1 - 6) q^{53} - q^{54} + (3 \beta_{2} + \beta_1 - 1) q^{55} + ( - \beta_{2} + \beta_1 - 2) q^{56} + ( - 3 \beta_{2} - \beta_1 + 1) q^{57} + (\beta_{2} + 3 \beta_1 - 1) q^{58} + ( - \beta_{2} + 4 \beta_1 - 3) q^{59} + ( - \beta_{2} + 1) q^{60} + ( - 4 \beta_{2} - 4 \beta_1) q^{61} + (2 \beta_{2} + 6 \beta_1 - 5) q^{62} + ( - \beta_{2} + \beta_1 - 2) q^{63} + q^{64} + ( - \beta_{2} + \beta_1 - 1) q^{65} + (\beta_{2} + 3 \beta_1 - 1) q^{66} + ( - 3 \beta_{2} - 6 \beta_1 + 3) q^{67} + ( - \beta_{2} - \beta_1 - 5) q^{68} + ( - \beta_{2} + \beta_1 + 1) q^{69} + (\beta_1 - 2) q^{70} + (3 \beta_{2} - \beta_1 - 1) q^{71} + q^{72} + ( - 5 \beta_{2} + 3 \beta_1 + 1) q^{73} + ( - 3 \beta_{2} - 2 \beta_1 - 1) q^{74} + (3 \beta_{2} + \beta_1 + 1) q^{75} + (3 \beta_{2} + \beta_1 - 1) q^{76} + ( - 3 \beta_{2} + 5 \beta_1 - 7) q^{77} + (\beta_{2} - \beta_1 + 3) q^{78} + (2 \beta_{2} + 2 \beta_1 + 10) q^{79} + (\beta_{2} - 1) q^{80} + q^{81} + (2 \beta_{2} - 2 \beta_1 - 4) q^{82} + ( - 3 \beta_1 + 2) q^{83} + (\beta_{2} - \beta_1 + 2) q^{84} + ( - 3 \beta_{2} + \beta_1 + 3) q^{85} + ( - 3 \beta_{2} + \beta_1 - 1) q^{86} + ( - \beta_{2} - 3 \beta_1 + 1) q^{87} + ( - \beta_{2} - 3 \beta_1 + 1) q^{88} + ( - 5 \beta_{2} - 3 \beta_1 - 8) q^{89} + (\beta_{2} - 1) q^{90} + (5 \beta_{2} - 7 \beta_1 + 13) q^{91} + (\beta_{2} - \beta_1 - 1) q^{92} + ( - 2 \beta_{2} - 6 \beta_1 + 5) q^{93} + (3 \beta_{2} - 3 \beta_1) q^{94} + ( - 7 \beta_{2} - 3 \beta_1 + 9) q^{95} - q^{96} + (3 \beta_{2} - \beta_1 + 2) q^{97} + (4 \beta_{2} - 6 \beta_1 + 4) q^{98} + ( - \beta_{2} - 3 \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 5 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 5 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} - 3 q^{12} - 8 q^{13} - 5 q^{14} + 3 q^{15} + 3 q^{16} - 16 q^{17} + 3 q^{18} - 2 q^{19} - 3 q^{20} + 5 q^{21} - 4 q^{23} - 3 q^{24} - 4 q^{25} - 8 q^{26} - 3 q^{27} - 5 q^{28} + 3 q^{30} - 9 q^{31} + 3 q^{32} - 16 q^{34} - 5 q^{35} + 3 q^{36} - 5 q^{37} - 2 q^{38} + 8 q^{39} - 3 q^{40} - 14 q^{41} + 5 q^{42} - 2 q^{43} - 3 q^{45} - 4 q^{46} - 3 q^{47} - 3 q^{48} + 6 q^{49} - 4 q^{50} + 16 q^{51} - 8 q^{52} - 15 q^{53} - 3 q^{54} - 2 q^{55} - 5 q^{56} + 2 q^{57} - 5 q^{59} + 3 q^{60} - 4 q^{61} - 9 q^{62} - 5 q^{63} + 3 q^{64} - 2 q^{65} + 3 q^{67} - 16 q^{68} + 4 q^{69} - 5 q^{70} - 4 q^{71} + 3 q^{72} + 6 q^{73} - 5 q^{74} + 4 q^{75} - 2 q^{76} - 16 q^{77} + 8 q^{78} + 32 q^{79} - 3 q^{80} + 3 q^{81} - 14 q^{82} + 3 q^{83} + 5 q^{84} + 10 q^{85} - 2 q^{86} - 27 q^{89} - 3 q^{90} + 32 q^{91} - 4 q^{92} + 9 q^{93} - 3 q^{94} + 24 q^{95} - 3 q^{96} + 5 q^{97} + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0.311108
2.17009
−1.48119
1.00000 −1.00000 1.00000 −3.21432 −1.00000 0.525428 1.00000 1.00000 −3.21432
1.2 1.00000 −1.00000 1.00000 −0.460811 −1.00000 −0.369102 1.00000 1.00000 −0.460811
1.3 1.00000 −1.00000 1.00000 0.675131 −1.00000 −5.15633 1.00000 1.00000 0.675131
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(167\) \(-1\)

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 1002.2.a.h 3
3.b odd 2 1 3006.2.a.o 3
4.b odd 2 1 8016.2.a.l 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1002.2.a.h 3 1.a even 1 1 trivial
3006.2.a.o 3 3.b odd 2 1
8016.2.a.l 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} + 3T_{5}^{2} - T_{5} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1002))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 3T^{2} - T - 1 \) Copy content Toggle raw display
$7$ \( T^{3} + 5T^{2} - T - 1 \) Copy content Toggle raw display
$11$ \( T^{3} - 28T + 52 \) Copy content Toggle raw display
$13$ \( T^{3} + 8 T^{2} + 12 T + 4 \) Copy content Toggle raw display
$17$ \( T^{3} + 16 T^{2} + 80 T + 124 \) Copy content Toggle raw display
$19$ \( T^{3} + 2 T^{2} - 32 T + 52 \) Copy content Toggle raw display
$23$ \( T^{3} + 4 T^{2} - 4 T - 20 \) Copy content Toggle raw display
$29$ \( T^{3} - 28T - 52 \) Copy content Toggle raw display
$31$ \( T^{3} + 9 T^{2} - 85 T - 725 \) Copy content Toggle raw display
$37$ \( T^{3} + 5 T^{2} - 29 T - 107 \) Copy content Toggle raw display
$41$ \( T^{3} + 14 T^{2} + 28 T - 152 \) Copy content Toggle raw display
$43$ \( T^{3} + 2 T^{2} - 44 T - 20 \) Copy content Toggle raw display
$47$ \( T^{3} + 3 T^{2} - 81 T - 351 \) Copy content Toggle raw display
$53$ \( T^{3} + 15 T^{2} + 5 T - 139 \) Copy content Toggle raw display
$59$ \( T^{3} + 5 T^{2} - 57 T + 25 \) Copy content Toggle raw display
$61$ \( T^{3} + 4 T^{2} - 80 T - 64 \) Copy content Toggle raw display
$67$ \( T^{3} - 3 T^{2} - 117 T + 621 \) Copy content Toggle raw display
$71$ \( T^{3} + 4 T^{2} - 40 T - 68 \) Copy content Toggle raw display
$73$ \( T^{3} - 6 T^{2} - 148 T + 740 \) Copy content Toggle raw display
$79$ \( T^{3} - 32 T^{2} + 320 T - 992 \) Copy content Toggle raw display
$83$ \( T^{3} - 3 T^{2} - 27 T + 31 \) Copy content Toggle raw display
$89$ \( T^{3} + 27 T^{2} + 143 T - 439 \) Copy content Toggle raw display
$97$ \( T^{3} - 5 T^{2} - 37 T + 61 \) Copy content Toggle raw display
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