Properties

Label 1002.2.a.j
Level $1002$
Weight $2$
Character orbit 1002.a
Self dual yes
Analytic conductor $8.001$
Analytic rank $0$
Dimension $5$
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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.11256624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 16x^{3} + 20x^{2} + 31x - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 16x^{3} + 20x^{2} + 31x - 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + \nu^{2} - 11\nu - 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - 14\nu^{2} + 8\nu + 17 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} - 2\beta_{2} + 11\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} + 28\beta_{2} - 8\beta _1 + 81 \) 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
0.308735
−1.14410
2.50259
3.20970
−3.87693
−1.00000 1.00000 1.00000 −3.45234 −1.00000 −2.53613 −1.00000 1.00000 3.45234
1.2 −1.00000 1.00000 1.00000 −2.84552 −1.00000 4.19124 −1.00000 1.00000 2.84552
1.3 −1.00000 1.00000 1.00000 −0.368511 −1.00000 4.85901 −1.00000 1.00000 0.368511
1.4 −1.00000 1.00000 1.00000 1.65109 −1.00000 0.854515 −1.00000 1.00000 −1.65109
1.5 −1.00000 1.00000 1.00000 4.01528 −1.00000 1.63136 −1.00000 1.00000 −4.01528
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.j 5
3.b odd 2 1 3006.2.a.t 5
4.b odd 2 1 8016.2.a.r 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1002.2.a.j 5 1.a even 1 1 trivial
3006.2.a.t 5 3.b odd 2 1
8016.2.a.r 5 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} + T_{5}^{4} - 19T_{5}^{3} - 21T_{5}^{2} + 60T_{5} + 24 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1002))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{5} \) Copy content Toggle raw display
$3$ \( (T - 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} + T^{4} - 19 T^{3} - 21 T^{2} + \cdots + 24 \) Copy content Toggle raw display
$7$ \( T^{5} - 9 T^{4} + 15 T^{3} + 49 T^{2} + \cdots + 72 \) Copy content Toggle raw display
$11$ \( T^{5} + 2 T^{4} - 28 T^{3} - 60 T^{2} + \cdots + 288 \) Copy content Toggle raw display
$13$ \( T^{5} - 6 T^{4} - 2 T^{3} + 52 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$17$ \( T^{5} - 4 T^{4} - 10 T^{3} + 24 T^{2} + \cdots - 24 \) Copy content Toggle raw display
$19$ \( T^{5} - 12 T^{4} + 28 T^{3} + \cdots - 176 \) Copy content Toggle raw display
$23$ \( T^{5} - 2 T^{4} - 28 T^{3} + 60 T^{2} + \cdots - 288 \) Copy content Toggle raw display
$29$ \( T^{5} + 6 T^{4} - 84 T^{3} + \cdots - 1728 \) Copy content Toggle raw display
$31$ \( T^{5} - 9 T^{4} - 5 T^{3} + 229 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$37$ \( T^{5} - 5 T^{4} - 67 T^{3} + 415 T^{2} + \cdots - 226 \) Copy content Toggle raw display
$41$ \( T^{5} - 4 T^{4} - 110 T^{3} + \cdots + 4176 \) Copy content Toggle raw display
$43$ \( T^{5} - 18 T^{4} - 24 T^{3} + \cdots + 17424 \) Copy content Toggle raw display
$47$ \( T^{5} + 7 T^{4} - 61 T^{3} - 243 T^{2} + \cdots - 156 \) Copy content Toggle raw display
$53$ \( T^{5} - 3 T^{4} - 209 T^{3} + \cdots - 43392 \) Copy content Toggle raw display
$59$ \( T^{5} - 13 T^{4} - 127 T^{3} + \cdots - 3306 \) Copy content Toggle raw display
$61$ \( T^{5} - 16 T^{4} - 28 T^{3} + \cdots - 14848 \) Copy content Toggle raw display
$67$ \( T^{5} - 9 T^{4} - 155 T^{3} + \cdots - 36476 \) Copy content Toggle raw display
$71$ \( T^{5} + 10 T^{4} - 184 T^{3} + \cdots + 3456 \) Copy content Toggle raw display
$73$ \( T^{5} - 8 T^{4} - 104 T^{3} + \cdots - 2768 \) Copy content Toggle raw display
$79$ \( T^{5} - 6 T^{4} - 102 T^{3} + \cdots - 6336 \) Copy content Toggle raw display
$83$ \( T^{5} - T^{4} - 281 T^{3} + \cdots + 10266 \) Copy content Toggle raw display
$89$ \( T^{5} + 5 T^{4} - 205 T^{3} + \cdots + 50268 \) Copy content Toggle raw display
$97$ \( T^{5} + 7 T^{4} - 253 T^{3} + \cdots + 27848 \) Copy content Toggle raw display
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