Properties

Label 1002.2.a.k
Level $1002$
Weight $2$
Character orbit 1002.a
Self dual yes
Analytic conductor $8.001$
Analytic rank $0$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 2x^{6} - 20x^{5} + 2x^{4} + 87x^{3} + 46x^{2} - 48x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 2x^{6} - 20x^{5} + 2x^{4} + 87x^{3} + 46x^{2} - 48x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{6} - 21\nu^{5} - 33\nu^{4} + 41\nu^{3} + 12\nu^{2} + 162\nu - 14 ) / 104 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{6} + 23\nu^{5} - \nu^{4} - 243\nu^{3} + 128\nu^{2} + 610\nu - 158 ) / 52 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -25\nu^{6} + 105\nu^{5} + 269\nu^{4} - 621\nu^{3} - 892\nu^{2} + 542\nu + 486 ) / 104 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -17\nu^{6} + 61\nu^{5} + 237\nu^{4} - 389\nu^{3} - 800\nu^{2} + 406\nu + 162 ) / 52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -59\nu^{6} + 227\nu^{5} + 743\nu^{4} - 1399\nu^{3} - 2388\nu^{2} + 1042\nu + 290 ) / 104 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} - \beta_{4} + 3\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{6} - 5\beta_{5} - 2\beta_{4} - \beta_{3} + 2\beta_{2} + 18\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 24\beta_{6} - 28\beta_{5} - 15\beta_{4} - 4\beta_{3} + 13\beta_{2} + 83\beta _1 + 80 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 119\beta_{6} - 148\beta_{5} - 59\beta_{4} - 24\beta_{3} + 74\beta_{2} + 437\beta _1 + 342 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 623\beta_{6} - 763\beta_{5} - 328\beta_{4} - 119\beta_{3} + 401\beta_{2} + 2196\beta _1 + 1865 \) 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
5.12652
2.30288
0.561628
−0.0402333
−1.55552
−2.09849
−2.29679
1.00000 1.00000 1.00000 −4.12652 1.00000 3.19234 1.00000 1.00000 −4.12652
1.2 1.00000 1.00000 1.00000 −1.30288 1.00000 1.53952 1.00000 1.00000 −1.30288
1.3 1.00000 1.00000 1.00000 0.438372 1.00000 −2.51945 1.00000 1.00000 0.438372
1.4 1.00000 1.00000 1.00000 1.04023 1.00000 4.50614 1.00000 1.00000 1.04023
1.5 1.00000 1.00000 1.00000 2.55552 1.00000 3.69934 1.00000 1.00000 2.55552
1.6 1.00000 1.00000 1.00000 3.09849 1.00000 −2.06938 1.00000 1.00000 3.09849
1.7 1.00000 1.00000 1.00000 3.29679 1.00000 −1.34851 1.00000 1.00000 3.29679
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.k 7
3.b odd 2 1 3006.2.a.u 7
4.b odd 2 1 8016.2.a.v 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1002.2.a.k 7 1.a even 1 1 trivial
3006.2.a.u 7 3.b odd 2 1
8016.2.a.v 7 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{7} - 5T_{5}^{6} - 11T_{5}^{5} + 93T_{5}^{4} - 110T_{5}^{3} - 110T_{5}^{2} + 208T_{5} - 64 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1002))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{7} \) Copy content Toggle raw display
$3$ \( (T - 1)^{7} \) Copy content Toggle raw display
$5$ \( T^{7} - 5 T^{6} - 11 T^{5} + 93 T^{4} + \cdots - 64 \) Copy content Toggle raw display
$7$ \( T^{7} - 7 T^{6} - 5 T^{5} + 99 T^{4} + \cdots + 576 \) Copy content Toggle raw display
$11$ \( T^{7} - 48 T^{5} + 12 T^{4} + \cdots + 1152 \) Copy content Toggle raw display
$13$ \( T^{7} - 6 T^{6} - 30 T^{5} + 244 T^{4} + \cdots - 384 \) Copy content Toggle raw display
$17$ \( T^{7} - 6 T^{6} - 56 T^{5} + \cdots - 1808 \) Copy content Toggle raw display
$19$ \( T^{7} + 2 T^{6} - 116 T^{5} + \cdots - 7808 \) Copy content Toggle raw display
$23$ \( T^{7} - 92 T^{5} - 140 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$29$ \( T^{7} + 4 T^{6} - 160 T^{5} + \cdots - 67328 \) Copy content Toggle raw display
$31$ \( T^{7} - 7 T^{6} - 69 T^{5} + \cdots - 17536 \) Copy content Toggle raw display
$37$ \( T^{7} + 3 T^{6} - 145 T^{5} + \cdots - 333456 \) Copy content Toggle raw display
$41$ \( T^{7} + 12 T^{6} - 104 T^{5} + \cdots - 171936 \) Copy content Toggle raw display
$43$ \( T^{7} + 2 T^{6} - 134 T^{5} + \cdots + 125792 \) Copy content Toggle raw display
$47$ \( T^{7} + 11 T^{6} - 137 T^{5} + \cdots + 11184 \) Copy content Toggle raw display
$53$ \( T^{7} - T^{6} - 101 T^{5} + 87 T^{4} + \cdots + 5072 \) Copy content Toggle raw display
$59$ \( T^{7} + 19 T^{6} - 49 T^{5} + \cdots - 189800 \) Copy content Toggle raw display
$61$ \( T^{7} - 12 T^{6} - 132 T^{5} + \cdots - 2048 \) Copy content Toggle raw display
$67$ \( T^{7} + 17 T^{6} - 371 T^{5} + \cdots - 17569912 \) Copy content Toggle raw display
$71$ \( T^{7} + 20 T^{6} + 32 T^{5} + \cdots - 2048 \) Copy content Toggle raw display
$73$ \( T^{7} - 10 T^{6} - 180 T^{5} + \cdots - 395712 \) Copy content Toggle raw display
$79$ \( T^{7} - 2 T^{6} - 260 T^{5} + \cdots - 502400 \) Copy content Toggle raw display
$83$ \( T^{7} + 7 T^{6} - 175 T^{5} + \cdots - 327912 \) Copy content Toggle raw display
$89$ \( T^{7} + 3 T^{6} - 481 T^{5} + \cdots + 5628816 \) Copy content Toggle raw display
$97$ \( T^{7} + 3 T^{6} - 137 T^{5} + \cdots - 416 \) Copy content Toggle raw display
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