Properties

Label 8008.2.a.y
Level $8008$
Weight $2$
Character orbit 8008.a
Self dual yes
Analytic conductor $63.944$
Analytic rank $1$
Dimension $14$
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: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 3 x^{13} - 27 x^{12} + 78 x^{11} + 273 x^{10} - 750 x^{9} - 1306 x^{8} + 3378 x^{7} + 2996 x^{6} - 7275 x^{5} - 2804 x^{4} + 6417 x^{3} + 538 x^{2} - 1032 x - 128 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{10} q^{5} + q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{10} q^{5} + q^{7} + (\beta_{2} + 2) q^{9} - q^{11} - q^{13} + (\beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} + \beta_{7} - \beta_{2} + 2 \beta_1 - 3) q^{15} - \beta_{8} q^{17} + ( - \beta_{6} - 1) q^{19} - \beta_1 q^{21} + (\beta_{12} + \beta_{9}) q^{23} + ( - \beta_{10} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_1 + 1) q^{25} + ( - \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} - \beta_{8} - \beta_{7} - \beta_{5} - \beta_{4} - 3 \beta_1) q^{27} + ( - \beta_{13} - \beta_{11} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{29} + ( - \beta_{10} + \beta_{8} + \beta_{4} - \beta_{2} + \beta_1 - 2) q^{31} + \beta_1 q^{33} + \beta_{10} q^{35} + ( - \beta_{13} - \beta_{7} + \beta_{2} + 1) q^{37} + \beta_1 q^{39} + ( - \beta_{11} + \beta_{9} + \beta_{8} + \beta_{6} + \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{41} + (\beta_{9} + \beta_{8} + \beta_{6} + \beta_{5} - 2 \beta_{2} + 2 \beta_1 - 2) q^{43} + (\beta_{12} - \beta_{11} + 3 \beta_{10} + 2 \beta_{9} + \beta_{8} + 2 \beta_{5} + \beta_{3} - 2 \beta_{2} + \cdots - 2) q^{45}+ \cdots + ( - \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9} - 14 q^{11} - 14 q^{13} - 6 q^{15} - 6 q^{17} - 13 q^{19} - 3 q^{21} - 9 q^{23} + 22 q^{25} - 18 q^{27} + 2 q^{29} - 2 q^{31} + 3 q^{33} - 6 q^{35} - q^{37} + 3 q^{39} - 16 q^{41} - 15 q^{43} - 44 q^{45} - 8 q^{47} + 14 q^{49} - 14 q^{51} - 6 q^{53} + 6 q^{55} - 10 q^{57} - 36 q^{59} - 19 q^{61} + 21 q^{63} + 6 q^{65} - 34 q^{67} - q^{69} - 10 q^{71} + 9 q^{73} - 44 q^{75} - 14 q^{77} - q^{79} + 42 q^{81} - 56 q^{83} + 21 q^{85} - 5 q^{87} - 14 q^{89} - 14 q^{91} - 20 q^{93} + q^{95} - 14 q^{97} - 21 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 3 x^{13} - 27 x^{12} + 78 x^{11} + 273 x^{10} - 750 x^{9} - 1306 x^{8} + 3378 x^{7} + 2996 x^{6} - 7275 x^{5} - 2804 x^{4} + 6417 x^{3} + 538 x^{2} - 1032 x - 128 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11074521 \nu^{13} + 106614016 \nu^{12} + 202633696 \nu^{11} - 2843591948 \nu^{10} - 1105297503 \nu^{9} + 27555851639 \nu^{8} + \cdots - 16683138390 ) / 15572760974 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 58724453 \nu^{13} + 415732687 \nu^{12} + 429387489 \nu^{11} - 9589273014 \nu^{10} + 12008354815 \nu^{9} + 76649830082 \nu^{8} + \cdots + 264264384876 ) / 31145521948 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 84573421 \nu^{13} + 56007919 \nu^{12} + 1990802275 \nu^{11} + 1134603604 \nu^{10} - 16590727105 \nu^{9} - 46010891430 \nu^{8} + \cdots - 403244409496 ) / 31145521948 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 86197394 \nu^{13} - 519312690 \nu^{12} + 4042783649 \nu^{11} + 14135363599 \nu^{10} - 62980268748 \nu^{9} - 143598407974 \nu^{8} + \cdots - 148105342534 ) / 15572760974 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 100221529 \nu^{13} + 20088978 \nu^{12} + 3433412919 \nu^{11} - 872022317 \nu^{10} - 43636307733 \nu^{9} + 14040705209 \nu^{8} + \cdots + 51792914190 ) / 15572760974 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 204529955 \nu^{13} + 624487815 \nu^{12} + 5370665871 \nu^{11} - 15034911330 \nu^{10} - 53506698243 \nu^{9} + 127831458804 \nu^{8} + \cdots - 12295329000 ) / 31145521948 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 217441493 \nu^{13} + 681733073 \nu^{12} + 6654223007 \nu^{11} - 21109853824 \nu^{10} - 75443571221 \nu^{9} + 245864340184 \nu^{8} + \cdots + 306825120252 ) / 31145521948 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 208926672 \nu^{13} - 491561333 \nu^{12} - 5735162063 \nu^{11} + 12048495611 \nu^{10} + 58834255396 \nu^{9} - 105792814085 \nu^{8} + \cdots - 9943373676 ) / 15572760974 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 488425337 \nu^{13} - 1340879233 \nu^{12} - 12295355455 \nu^{11} + 31978291166 \nu^{10} + 110106082525 \nu^{9} - 271168142180 \nu^{8} + \cdots - 141640358616 ) / 31145521948 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 282478821 \nu^{13} + 525449102 \nu^{12} + 7980519403 \nu^{11} - 11978040794 \nu^{10} - 86063793263 \nu^{9} + 93824334855 \nu^{8} + \cdots + 816058514 ) / 15572760974 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 461482718 \nu^{13} + 1119246863 \nu^{12} + 12534518457 \nu^{11} - 27398869301 \nu^{10} - 127270246922 \nu^{9} + \cdots + 105424374118 ) / 15572760974 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} + \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{13} + \beta_{12} + \beta_{11} - 2\beta_{10} - \beta_{5} + 2\beta_{4} - \beta_{3} + 12\beta_{2} + 39 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14 \beta_{13} - 15 \beta_{12} + 16 \beta_{11} + 10 \beta_{10} - 2 \beta_{9} + 13 \beta_{8} + 17 \beta_{7} - \beta_{6} + 14 \beta_{5} + 15 \beta_{4} + \beta_{3} + 90 \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 15 \beta_{13} + 13 \beta_{12} + 18 \beta_{11} - 38 \beta_{10} - 3 \beta_{9} - \beta_{8} + 3 \beta_{7} - 2 \beta_{6} - 15 \beta_{5} + 29 \beta_{4} - 19 \beta_{3} + 131 \beta_{2} + 4 \beta _1 + 353 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 167 \beta_{13} - 186 \beta_{12} + 205 \beta_{11} + 77 \beta_{10} - 43 \beta_{9} + 139 \beta_{8} + 215 \beta_{7} - 18 \beta_{6} + 156 \beta_{5} + 181 \beta_{4} + 16 \beta_{3} + 6 \beta_{2} + 932 \beta _1 - 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 160 \beta_{13} + 116 \beta_{12} + 257 \beta_{11} - 531 \beta_{10} - 80 \beta_{9} - 17 \beta_{8} + 76 \beta_{7} - 42 \beta_{6} - 187 \beta_{5} + 345 \beta_{4} - 261 \beta_{3} + 1403 \beta_{2} + 106 \beta _1 + 3423 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1906 \beta_{13} - 2198 \beta_{12} + 2435 \beta_{11} + 482 \beta_{10} - 667 \beta_{9} + 1449 \beta_{8} + 2498 \beta_{7} - 238 \beta_{6} + 1627 \beta_{5} + 2054 \beta_{4} + 185 \beta_{3} + 160 \beta_{2} + 9828 \beta _1 - 200 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1452 \beta_{13} + 725 \beta_{12} + 3377 \beta_{11} - 6685 \beta_{10} - 1435 \beta_{9} - 153 \beta_{8} + 1309 \beta_{7} - 611 \beta_{6} - 2214 \beta_{5} + 3921 \beta_{4} - 3166 \beta_{3} + 14968 \beta_{2} + \cdots + 34307 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 21440 \beta_{13} - 25547 \beta_{12} + 28043 \beta_{11} + 1700 \beta_{10} - 9180 \beta_{9} + 15259 \beta_{8} + 28129 \beta_{7} - 2822 \beta_{6} + 16505 \beta_{5} + 22781 \beta_{4} + 1878 \beta_{3} + \cdots - 2231 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 11353 \beta_{13} + 597 \beta_{12} + 42620 \beta_{11} - 80507 \beta_{10} - 21831 \beta_{9} - 402 \beta_{8} + 19277 \beta_{7} - 7711 \beta_{6} - 25713 \beta_{5} + 44080 \beta_{4} - 36193 \beta_{3} + \cdots + 349673 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 240060 \beta_{13} - 294913 \beta_{12} + 318651 \beta_{11} - 16807 \beta_{10} - 119388 \beta_{9} + 163166 \beta_{8} + 312802 \beta_{7} - 31947 \beta_{6} + 164782 \beta_{5} + 250307 \beta_{4} + \cdots - 23048 \) 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
3.38813
3.22236
2.52493
1.97367
1.53687
1.36889
0.528665
−0.129238
−0.374242
−1.44419
−1.85932
−2.05490
−2.46105
−3.22058
0 −3.38813 0 −4.16385 0 1.00000 0 8.47945 0
1.2 0 −3.22236 0 1.94503 0 1.00000 0 7.38359 0
1.3 0 −2.52493 0 3.43935 0 1.00000 0 3.37528 0
1.4 0 −1.97367 0 −1.63177 0 1.00000 0 0.895375 0
1.5 0 −1.53687 0 −0.216819 0 1.00000 0 −0.638026 0
1.6 0 −1.36889 0 −3.92349 0 1.00000 0 −1.12615 0
1.7 0 −0.528665 0 0.933973 0 1.00000 0 −2.72051 0
1.8 0 0.129238 0 −0.197815 0 1.00000 0 −2.98330 0
1.9 0 0.374242 0 3.66821 0 1.00000 0 −2.85994 0
1.10 0 1.44419 0 −3.20437 0 1.00000 0 −0.914318 0
1.11 0 1.85932 0 1.13919 0 1.00000 0 0.457073 0
1.12 0 2.05490 0 0.929320 0 1.00000 0 1.22261 0
1.13 0 2.46105 0 −1.12857 0 1.00000 0 3.05676 0
1.14 0 3.22058 0 −3.58838 0 1.00000 0 7.37212 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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.y 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8008.2.a.y 14 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\( T_{3}^{14} + 3 T_{3}^{13} - 27 T_{3}^{12} - 78 T_{3}^{11} + 273 T_{3}^{10} + 750 T_{3}^{9} - 1306 T_{3}^{8} - 3378 T_{3}^{7} + 2996 T_{3}^{6} + 7275 T_{3}^{5} - 2804 T_{3}^{4} - 6417 T_{3}^{3} + 538 T_{3}^{2} + 1032 T_{3} - 128 \) Copy content Toggle raw display
\( T_{5}^{14} + 6 T_{5}^{13} - 28 T_{5}^{12} - 201 T_{5}^{11} + 229 T_{5}^{10} + 2321 T_{5}^{9} - 472 T_{5}^{8} - 10984 T_{5}^{7} + 500 T_{5}^{6} + 21765 T_{5}^{5} - 2561 T_{5}^{4} - 16489 T_{5}^{3} + 2635 T_{5}^{2} + 2934 T_{5} + 360 \) Copy content Toggle raw display
\( T_{17}^{14} + 6 T_{17}^{13} - 113 T_{17}^{12} - 811 T_{17}^{11} + 3506 T_{17}^{10} + 34742 T_{17}^{9} - 8372 T_{17}^{8} - 532476 T_{17}^{7} - 699242 T_{17}^{6} + 2366025 T_{17}^{5} + 3832443 T_{17}^{4} - 4563644 T_{17}^{3} + \cdots - 1063168 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + 3 T^{13} - 27 T^{12} - 78 T^{11} + \cdots - 128 \) Copy content Toggle raw display
$5$ \( T^{14} + 6 T^{13} - 28 T^{12} - 201 T^{11} + \cdots + 360 \) Copy content Toggle raw display
$7$ \( (T - 1)^{14} \) Copy content Toggle raw display
$11$ \( (T + 1)^{14} \) Copy content Toggle raw display
$13$ \( (T + 1)^{14} \) Copy content Toggle raw display
$17$ \( T^{14} + 6 T^{13} - 113 T^{12} + \cdots - 1063168 \) Copy content Toggle raw display
$19$ \( T^{14} + 13 T^{13} + \cdots - 139618496 \) Copy content Toggle raw display
$23$ \( T^{14} + 9 T^{13} - 145 T^{12} + \cdots - 329832448 \) Copy content Toggle raw display
$29$ \( T^{14} - 2 T^{13} - 275 T^{12} + \cdots - 159660032 \) Copy content Toggle raw display
$31$ \( T^{14} + 2 T^{13} - 241 T^{12} + \cdots - 73882112 \) Copy content Toggle raw display
$37$ \( T^{14} + T^{13} - 278 T^{12} + \cdots - 94912000 \) Copy content Toggle raw display
$41$ \( T^{14} + 16 T^{13} + \cdots + 2750476288 \) Copy content Toggle raw display
$43$ \( T^{14} + 15 T^{13} + \cdots - 1039623840 \) Copy content Toggle raw display
$47$ \( T^{14} + 8 T^{13} + \cdots - 29096382464 \) Copy content Toggle raw display
$53$ \( T^{14} + 6 T^{13} + \cdots - 4953257752 \) Copy content Toggle raw display
$59$ \( T^{14} + 36 T^{13} + \cdots + 2664831254528 \) Copy content Toggle raw display
$61$ \( T^{14} + 19 T^{13} - 42 T^{12} + \cdots - 8909920 \) Copy content Toggle raw display
$67$ \( T^{14} + 34 T^{13} + \cdots + 69751030496 \) Copy content Toggle raw display
$71$ \( T^{14} + 10 T^{13} + \cdots - 50501104640 \) Copy content Toggle raw display
$73$ \( T^{14} - 9 T^{13} + \cdots + 782936957952 \) Copy content Toggle raw display
$79$ \( T^{14} + T^{13} - 720 T^{12} + \cdots + 20171040768 \) Copy content Toggle raw display
$83$ \( T^{14} + 56 T^{13} + \cdots - 40240582000 \) Copy content Toggle raw display
$89$ \( T^{14} + 14 T^{13} + \cdots - 118735614796 \) Copy content Toggle raw display
$97$ \( T^{14} + 14 T^{13} + \cdots - 901202688 \) Copy content Toggle raw display
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