Properties

Label 8008.2.a.x
Level $8008$
Weight $2$
Character orbit 8008.a
Self dual yes
Analytic conductor $63.944$
Analytic rank $0$
Dimension $12$
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] = [8008,2,Mod(1,8008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("8008.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8008.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: \(63.9442019386\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 17 x^{10} + 79 x^{9} + 80 x^{8} - 536 x^{7} - 4 x^{6} + 1484 x^{5} - 682 x^{4} - 1431 x^{3} + 1069 x^{2} - 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{4} q^{5} + q^{7} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{4} q^{5} + q^{7} + (\beta_{2} + 1) q^{9} + q^{11} + q^{13} + ( - \beta_{9} + \beta_{8} + \beta_{4} - \beta_{2} + 2 \beta_1 - 2) q^{15} + ( - \beta_{7} + 1) q^{17} - \beta_{11} q^{19} + \beta_1 q^{21} + ( - \beta_{11} + \beta_{10} + \beta_{2} + 1) q^{23} + (\beta_{6} + \beta_{4} + 1) q^{25} + ( - \beta_{9} + \beta_{8} + \beta_{3} + 2 \beta_1 - 1) q^{27} + ( - \beta_{10} + 1) q^{29} + (\beta_{9} - \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 + 2) q^{31} + \beta_1 q^{33} + \beta_{4} q^{35} + ( - \beta_{10} + \beta_{5} + 1) q^{37} + \beta_1 q^{39} + ( - \beta_{8} - \beta_{3} - \beta_1 + 3) q^{41} + (\beta_{10} - \beta_{9} + \beta_{7} + \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{43} + (\beta_{10} - \beta_{8} + \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{45} + (\beta_{7} + \beta_{5} + \beta_{3} + \beta_1) q^{47} + q^{49} + (\beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{51} + (\beta_{11} - \beta_{5} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{53} + \beta_{4} q^{55} + ( - \beta_{11} + \beta_{9} + \beta_{5} + \beta_{2} - \beta_1) q^{57} + (\beta_{11} + \beta_{9} - \beta_{4} - \beta_{2} + 2) q^{59} + (\beta_{11} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{61} + (\beta_{2} + 1) q^{63} + \beta_{4} q^{65} + ( - \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{6} + \beta_{5} + \beta_{2} - 2 \beta_1 + 1) q^{67} + (\beta_{8} - \beta_{6} + \beta_{4} + 2 \beta_1 - 2) q^{69} + (\beta_{9} - \beta_{8} - \beta_{4} - \beta_{3} + \beta_{2} - 3 \beta_1 + 3) q^{71} + (\beta_{11} + \beta_{9} - \beta_{5} - \beta_{4} + \beta_{2} - 2 \beta_1 + 4) q^{73} + (\beta_{11} - 2 \beta_{10} - 2 \beta_{9} + \beta_{8} + \beta_{7} + \beta_{4} - 3 \beta_{2} + 3 \beta_1 - 3) q^{75} + q^{77} + (\beta_{11} + \beta_{10} - \beta_{9} - \beta_{2} + 2 \beta_1 + 1) q^{79} + (\beta_{10} + \beta_{9} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{81} + (\beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{83} + (\beta_{11} + \beta_{9} - \beta_{8} - \beta_{6} + \beta_{4} + \beta_{2} - \beta_1 + 2) q^{85} + ( - \beta_{11} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_1 + 1) q^{87} + (\beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{4} - \beta_1 + 2) q^{89} + q^{91} + (\beta_{11} + \beta_{9} - 2 \beta_{7} - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} + \cdots + 3) q^{93}+ \cdots + (\beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9} + 12 q^{11} + 12 q^{13} - 3 q^{15} + 16 q^{17} - 2 q^{19} + 4 q^{21} + 9 q^{23} + 14 q^{25} + 7 q^{27} + 15 q^{29} + 10 q^{31} + 4 q^{33} + 6 q^{35} + 18 q^{37} + 4 q^{39} + 24 q^{41} + 15 q^{45} + 5 q^{47} + 12 q^{49} + 4 q^{51} + 15 q^{53} + 6 q^{55} - 4 q^{57} + 15 q^{59} + 17 q^{61} + 14 q^{63} + 6 q^{65} - 7 q^{67} + 9 q^{71} + 32 q^{73} - 8 q^{75} + 12 q^{77} + 20 q^{79} - 4 q^{81} - 5 q^{83} + 25 q^{85} + 19 q^{87} + 16 q^{89} + 12 q^{91} + 21 q^{93} + 8 q^{95} + 10 q^{97} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 17 x^{10} + 79 x^{9} + 80 x^{8} - 536 x^{7} - 4 x^{6} + 1484 x^{5} - 682 x^{4} - 1431 x^{3} + 1069 x^{2} - 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 77 \nu^{11} + 116 \nu^{10} - 2017 \nu^{9} - 1221 \nu^{8} + 18608 \nu^{7} - 2976 \nu^{6} - 71120 \nu^{5} + 55928 \nu^{4} + 100022 \nu^{3} - 130123 \nu^{2} - 30863 \nu + 42444 ) / 6100 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 113 \nu^{11} - 416 \nu^{10} - 2453 \nu^{9} + 8491 \nu^{8} + 18752 \nu^{7} - 60424 \nu^{6} - 57900 \nu^{5} + 180912 \nu^{4} + 47918 \nu^{3} - 207707 \nu^{2} + 37113 \nu + 31376 ) / 6100 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 46 \nu^{11} - 2 \nu^{10} - 896 \nu^{9} - 468 \nu^{8} + 6284 \nu^{7} + 8097 \nu^{6} - 22080 \nu^{5} - 40751 \nu^{4} + 48306 \nu^{3} + 60651 \nu^{2} - 52764 \nu - 248 ) / 1525 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 163 \nu^{11} - 816 \nu^{10} - 1703 \nu^{9} + 15541 \nu^{8} - 5448 \nu^{7} - 100224 \nu^{6} + 95000 \nu^{5} + 259612 \nu^{4} - 246082 \nu^{3} - 231757 \nu^{2} + 144063 \nu + 18376 ) / 6100 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 19 \nu^{11} + 91 \nu^{10} + 386 \nu^{9} - 1764 \nu^{8} - 2821 \nu^{7} + 11464 \nu^{6} + 9303 \nu^{5} - 29113 \nu^{4} - 13574 \nu^{3} + 25304 \nu^{2} + 6817 \nu - 3234 ) / 610 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 124 \nu^{11} - 443 \nu^{10} - 2044 \nu^{9} + 7968 \nu^{8} + 9646 \nu^{7} - 46427 \nu^{6} - 7975 \nu^{5} + 100626 \nu^{4} - 29511 \nu^{3} - 68611 \nu^{2} + 32399 \nu + 4848 ) / 3050 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 65 \nu^{11} - 154 \nu^{10} - 1221 \nu^{9} + 2943 \nu^{8} + 7580 \nu^{7} - 19166 \nu^{6} - 17414 \nu^{5} + 51436 \nu^{4} + 6980 \nu^{3} - 53469 \nu^{2} + 16547 \nu + 9208 ) / 1220 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 103 \nu^{11} + 214 \nu^{10} + 2115 \nu^{9} - 4397 \nu^{8} - 15052 \nu^{7} + 31846 \nu^{6} + 43218 \nu^{5} - 97584 \nu^{4} - 36568 \nu^{3} + 108835 \nu^{2} - 19017 \nu - 6892 ) / 1220 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 679 \nu^{11} + 1528 \nu^{10} + 13849 \nu^{9} - 30103 \nu^{8} - 97266 \nu^{7} + 203642 \nu^{6} + 269800 \nu^{5} - 562446 \nu^{4} - 193544 \nu^{3} + 549981 \nu^{2} + \cdots - 22808 ) / 6100 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + \beta_{8} + \beta_{3} + 8\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} + \beta_{9} + \beta_{4} + \beta_{3} + 10\beta_{2} + \beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -10\beta_{9} + 12\beta_{8} - \beta_{6} - \beta_{5} - 2\beta_{4} + 11\beta_{3} + 65\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{11} + 15 \beta_{10} + 15 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - \beta_{5} + 7 \beta_{4} + 13 \beta_{3} + 93 \beta_{2} + 11 \beta _1 + 202 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3 \beta_{11} - 3 \beta_{10} - 84 \beta_{9} + 118 \beta_{8} - 3 \beta_{7} - 17 \beta_{6} - 16 \beta_{5} - 37 \beta_{4} + 102 \beta_{3} + \beta_{2} + 539 \beta _1 - 42 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 32 \beta_{11} + 165 \beta_{10} + 168 \beta_{9} - 33 \beta_{8} - 36 \beta_{7} - 19 \beta_{5} + 26 \beta_{4} + 137 \beta_{3} + 843 \beta_{2} + 99 \beta _1 + 1688 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 53 \beta_{11} - 50 \beta_{10} - 681 \beta_{9} + 1094 \beta_{8} - 55 \beta_{7} - 201 \beta_{6} - 188 \beta_{5} - 471 \beta_{4} + 914 \beta_{3} + 27 \beta_{2} + 4549 \beta _1 - 229 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 368 \beta_{11} + 1629 \beta_{10} + 1694 \beta_{9} - 381 \beta_{8} - 444 \beta_{7} - 5 \beta_{6} - 246 \beta_{5} - 81 \beta_{4} + 1349 \beta_{3} + 7560 \beta_{2} + 865 \beta _1 + 14538 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 633 \beta_{11} - 569 \beta_{10} - 5511 \beta_{9} + 9899 \beta_{8} - 695 \beta_{7} - 2073 \beta_{6} - 1951 \beta_{5} - 5165 \beta_{4} + 8149 \beta_{3} + 465 \beta_{2} + 38915 \beta _1 - 1019 \) 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
−2.95543
−2.33021
−1.78254
−1.54910
−0.219004
0.336827
0.818045
1.39079
1.75763
2.54627
2.98199
3.00474
0 −2.95543 0 3.71005 0 1.00000 0 5.73458 0
1.2 0 −2.33021 0 −0.450718 0 1.00000 0 2.42987 0
1.3 0 −1.78254 0 0.474905 0 1.00000 0 0.177459 0
1.4 0 −1.54910 0 −3.31877 0 1.00000 0 −0.600281 0
1.5 0 −0.219004 0 2.16734 0 1.00000 0 −2.95204 0
1.6 0 0.336827 0 3.95783 0 1.00000 0 −2.88655 0
1.7 0 0.818045 0 −0.570018 0 1.00000 0 −2.33080 0
1.8 0 1.39079 0 0.208204 0 1.00000 0 −1.06569 0
1.9 0 1.75763 0 −3.24204 0 1.00000 0 0.0892594 0
1.10 0 2.54627 0 3.33743 0 1.00000 0 3.48349 0
1.11 0 2.98199 0 1.64834 0 1.00000 0 5.89225 0
1.12 0 3.00474 0 −1.92256 0 1.00000 0 6.02845 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(11\) \(-1\)
\(13\) \(-1\)

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 8008.2.a.x 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8008.2.a.x 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\( T_{3}^{12} - 4 T_{3}^{11} - 17 T_{3}^{10} + 79 T_{3}^{9} + 80 T_{3}^{8} - 536 T_{3}^{7} - 4 T_{3}^{6} + 1484 T_{3}^{5} - 682 T_{3}^{4} - 1431 T_{3}^{3} + 1069 T_{3}^{2} - 64 \) Copy content Toggle raw display
\( T_{5}^{12} - 6 T_{5}^{11} - 19 T_{5}^{10} + 157 T_{5}^{9} + 38 T_{5}^{8} - 1310 T_{5}^{7} + 816 T_{5}^{6} + 3604 T_{5}^{5} - 3024 T_{5}^{4} - 2287 T_{5}^{3} + 1063 T_{5}^{2} + 340 T_{5} - 92 \) Copy content Toggle raw display
\( T_{17}^{12} - 16 T_{17}^{11} + 7 T_{17}^{10} + 1031 T_{17}^{9} - 4150 T_{17}^{8} - 13894 T_{17}^{7} + 89876 T_{17}^{6} - 12522 T_{17}^{5} - 323954 T_{17}^{4} - 38441 T_{17}^{3} + 175719 T_{17}^{2} + 32928 T_{17} - 3460 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 4 T^{11} - 17 T^{10} + 79 T^{9} + \cdots - 64 \) Copy content Toggle raw display
$5$ \( T^{12} - 6 T^{11} - 19 T^{10} + 157 T^{9} + \cdots - 92 \) Copy content Toggle raw display
$7$ \( (T - 1)^{12} \) Copy content Toggle raw display
$11$ \( (T - 1)^{12} \) Copy content Toggle raw display
$13$ \( (T - 1)^{12} \) Copy content Toggle raw display
$17$ \( T^{12} - 16 T^{11} + 7 T^{10} + \cdots - 3460 \) Copy content Toggle raw display
$19$ \( T^{12} + 2 T^{11} - 109 T^{10} + \cdots - 51984 \) Copy content Toggle raw display
$23$ \( T^{12} - 9 T^{11} - 92 T^{10} + \cdots + 2293760 \) Copy content Toggle raw display
$29$ \( T^{12} - 15 T^{11} - 6 T^{10} + \cdots + 877312 \) Copy content Toggle raw display
$31$ \( T^{12} - 10 T^{11} - 144 T^{10} + \cdots + 179200 \) Copy content Toggle raw display
$37$ \( T^{12} - 18 T^{11} - 44 T^{10} + \cdots + 32083712 \) Copy content Toggle raw display
$41$ \( T^{12} - 24 T^{11} + 100 T^{10} + \cdots + 49408 \) Copy content Toggle raw display
$43$ \( T^{12} - 200 T^{10} + \cdots + 12467456 \) Copy content Toggle raw display
$47$ \( T^{12} - 5 T^{11} - 218 T^{10} + \cdots - 1289216 \) Copy content Toggle raw display
$53$ \( T^{12} - 15 T^{11} + \cdots - 10360994036 \) Copy content Toggle raw display
$59$ \( T^{12} - 15 T^{11} - 211 T^{10} + \cdots - 4305920 \) Copy content Toggle raw display
$61$ \( T^{12} - 17 T^{11} - 98 T^{10} + \cdots - 260820 \) Copy content Toggle raw display
$67$ \( T^{12} + 7 T^{11} + \cdots - 309893103568 \) Copy content Toggle raw display
$71$ \( T^{12} - 9 T^{11} - 166 T^{10} + \cdots - 5328640 \) Copy content Toggle raw display
$73$ \( T^{12} - 32 T^{11} + 90 T^{10} + \cdots - 31928576 \) Copy content Toggle raw display
$79$ \( T^{12} - 20 T^{11} + \cdots + 61219583440 \) Copy content Toggle raw display
$83$ \( T^{12} + 5 T^{11} - 392 T^{10} + \cdots + 30750608 \) Copy content Toggle raw display
$89$ \( T^{12} - 16 T^{11} - 237 T^{10} + \cdots - 322708 \) Copy content Toggle raw display
$97$ \( T^{12} - 10 T^{11} - 420 T^{10} + \cdots - 66277120 \) Copy content Toggle raw display
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