Properties

Label 4008.2.a.m
Level $4008$
Weight $2$
Character orbit 4008.a
Self dual yes
Analytic conductor $32.004$
Analytic rank $0$
Dimension $13$
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] = [4008,2,Mod(1,4008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4008.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4008.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: \(32.0040411301\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 2 x^{12} - 49 x^{11} + 99 x^{10} + 901 x^{9} - 1879 x^{8} - 7582 x^{7} + 16968 x^{6} + 26911 x^{5} - 72240 x^{4} - 14532 x^{3} + 112850 x^{2} - 72184 x + 12144 \) 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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_1 q^{5} - \beta_{9} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta_1 q^{5} - \beta_{9} q^{7} + q^{9} + (\beta_{5} + 1) q^{11} + (\beta_{3} + 1) q^{13} + \beta_1 q^{15} + ( - \beta_{2} + 1) q^{17} + (\beta_{10} - \beta_{9} + 1) q^{19} - \beta_{9} q^{21} + (\beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} + \cdots + 2) q^{23}+ \cdots + (\beta_{5} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9} + 11 q^{11} + 12 q^{13} + 2 q^{15} + 15 q^{17} + 14 q^{19} + q^{21} + 9 q^{23} + 37 q^{25} + 13 q^{27} - 3 q^{29} - 17 q^{31} + 11 q^{33} + 15 q^{35} + 16 q^{37} + 12 q^{39} + 12 q^{41} + 20 q^{43} + 2 q^{45} - 6 q^{47} + 26 q^{49} + 15 q^{51} - 12 q^{53} + 7 q^{55} + 14 q^{57} + 14 q^{59} + 24 q^{61} + q^{63} + 8 q^{65} + 3 q^{67} + 9 q^{69} + 17 q^{71} + 34 q^{73} + 37 q^{75} + 30 q^{77} + 10 q^{79} + 13 q^{81} + 44 q^{83} + 25 q^{85} - 3 q^{87} + 25 q^{89} + 29 q^{91} - 17 q^{93} - 15 q^{95} + 38 q^{97} + 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 2 x^{12} - 49 x^{11} + 99 x^{10} + 901 x^{9} - 1879 x^{8} - 7582 x^{7} + 16968 x^{6} + 26911 x^{5} - 72240 x^{4} - 14532 x^{3} + 112850 x^{2} - 72184 x + 12144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6207348777 \nu^{12} - 12640530894 \nu^{11} + 576861363065 \nu^{10} + 295838273673 \nu^{9} - 16627326024925 \nu^{8} + \cdots + 12\!\cdots\!20 ) / 111451400959900 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 126516435801 \nu^{12} - 351643713690 \nu^{11} - 4532098501229 \nu^{10} + 14015645726951 \nu^{9} + 48247617829913 \nu^{8} + \cdots + 36\!\cdots\!88 ) / 557257004799500 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8155241826 \nu^{12} - 24439858116 \nu^{11} - 363301449706 \nu^{10} + 1102737562371 \nu^{9} + 5918945839452 \nu^{8} + \cdots - 261781210799648 ) / 27862850239975 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 42253210649 \nu^{12} - 42904580362 \nu^{11} - 2192557864275 \nu^{10} + 1857640717639 \nu^{9} + 43420560752365 \nu^{8} + \cdots - 9603799562310 ) / 55725700479950 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 42253210649 \nu^{12} - 42904580362 \nu^{11} - 2192557864275 \nu^{10} + 1857640717639 \nu^{9} + 43420560752365 \nu^{8} + \cdots - 455409403401910 ) / 55725700479950 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 138385219703 \nu^{12} - 105435570025 \nu^{11} - 6791649579897 \nu^{10} + 5771355044103 \nu^{9} + 127066606088584 \nu^{8} + \cdots - 29\!\cdots\!16 ) / 139314251199875 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 111023509273 \nu^{12} - 148746957474 \nu^{11} - 5533191476745 \nu^{10} + 7117312024823 \nu^{9} + 104000386651945 \nu^{8} + \cdots - 629228580752360 ) / 111451400959900 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 323942240303 \nu^{12} + 655210507370 \nu^{11} + 16238231918487 \nu^{10} - 30174775561853 \nu^{9} + \cdots + 68\!\cdots\!36 ) / 278628502399750 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 208422301596 \nu^{12} + 101272023955 \nu^{11} - 9810853937919 \nu^{10} - 3015289235129 \nu^{9} + 173349897433318 \nu^{8} + \cdots - 13\!\cdots\!57 ) / 139314251199875 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 889692540773 \nu^{12} - 1710596413190 \nu^{11} - 43690158347557 \nu^{10} + 79678278406623 \nu^{9} + 807810238699029 \nu^{8} + \cdots - 20\!\cdots\!96 ) / 557257004799500 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 224267911141 \nu^{12} + 63390661414 \nu^{11} - 11028950583521 \nu^{10} - 1496522255749 \nu^{9} + 205661421513157 \nu^{8} + \cdots - 22\!\cdots\!28 ) / 111451400959900 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} + 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{12} - \beta_{11} + \beta_{10} + \beta_{8} + 2\beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{3} + 11\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 5 \beta_{12} - \beta_{11} + 5 \beta_{10} - \beta_{9} + 2 \beta_{8} + 19 \beta_{6} - 17 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 2 \beta _1 + 91 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 30 \beta_{12} - 18 \beta_{11} + 29 \beta_{10} + 7 \beta_{9} + 20 \beta_{8} + 3 \beta_{7} + 46 \beta_{6} - 16 \beta_{5} + 43 \beta_{4} - 21 \beta_{3} - 7 \beta_{2} + 143 \beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 137 \beta_{12} - 28 \beta_{11} + 144 \beta_{10} - 42 \beta_{9} + 41 \beta_{8} - 12 \beta_{7} + 335 \beta_{6} - 280 \beta_{5} - 100 \beta_{4} + 40 \beta_{3} - 102 \beta_{2} + 49 \beta _1 + 1184 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 665 \beta_{12} - 281 \beta_{11} + 632 \beta_{10} + 213 \beta_{9} + 350 \beta_{8} + 97 \beta_{7} + 905 \beta_{6} - 247 \beta_{5} + 753 \beta_{4} - 381 \beta_{3} - 199 \beta_{2} + 2061 \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2931 \beta_{12} - 676 \beta_{11} + 3142 \beta_{10} - 1097 \beta_{9} + 716 \beta_{8} - 320 \beta_{7} + 5857 \beta_{6} - 4654 \beta_{5} - 1956 \beta_{4} + 668 \beta_{3} - 2055 \beta_{2} + 1017 \beta _1 + 16711 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 13266 \beta_{12} - 4390 \beta_{11} + 12482 \beta_{10} + 4644 \beta_{9} + 6123 \beta_{8} + 2314 \beta_{7} + 16957 \beta_{6} - 4186 \beta_{5} + 12395 \beta_{4} - 6687 \beta_{3} - 4266 \beta_{2} + \cdots + 1836 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 57319 \beta_{12} - 14661 \beta_{11} + 61925 \beta_{10} - 23592 \beta_{9} + 12118 \beta_{8} - 6102 \beta_{7} + 102280 \beta_{6} - 78127 \beta_{5} - 34966 \beta_{4} + 10790 \beta_{3} + \cdots + 249558 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 251329 \beta_{12} - 70391 \beta_{11} + 235453 \beta_{10} + 89347 \beta_{9} + 107880 \beta_{8} + 48175 \beta_{7} + 310864 \beta_{6} - 76539 \beta_{5} + 199760 \beta_{4} - 116067 \beta_{3} + \cdots + 60252 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1073038 \beta_{12} - 294937 \beta_{11} + 1162932 \beta_{10} - 459487 \beta_{9} + 205019 \beta_{8} - 103543 \beta_{7} + 1786032 \beta_{6} - 1320651 \beta_{5} - 597828 \beta_{4} + \cdots + 3880740 \) 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
−4.02595
−3.60065
−3.05600
−2.71099
−2.04519
0.284022
0.649069
1.25667
1.68837
2.79706
3.18209
3.38131
4.20017
0 1.00000 0 −4.02595 0 0.910115 0 1.00000 0
1.2 0 1.00000 0 −3.60065 0 −4.85776 0 1.00000 0
1.3 0 1.00000 0 −3.05600 0 2.59989 0 1.00000 0
1.4 0 1.00000 0 −2.71099 0 −0.775439 0 1.00000 0
1.5 0 1.00000 0 −2.04519 0 1.81275 0 1.00000 0
1.6 0 1.00000 0 0.284022 0 −3.90646 0 1.00000 0
1.7 0 1.00000 0 0.649069 0 4.19646 0 1.00000 0
1.8 0 1.00000 0 1.25667 0 −4.55521 0 1.00000 0
1.9 0 1.00000 0 1.68837 0 0.722089 0 1.00000 0
1.10 0 1.00000 0 2.79706 0 4.56518 0 1.00000 0
1.11 0 1.00000 0 3.18209 0 −1.39408 0 1.00000 0
1.12 0 1.00000 0 3.38131 0 2.18014 0 1.00000 0
1.13 0 1.00000 0 4.20017 0 −0.497670 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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 4008.2.a.m 13
4.b odd 2 1 8016.2.a.bg 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.m 13 1.a even 1 1 trivial
8016.2.a.bg 13 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4008))\):

\( T_{5}^{13} - 2 T_{5}^{12} - 49 T_{5}^{11} + 99 T_{5}^{10} + 901 T_{5}^{9} - 1879 T_{5}^{8} - 7582 T_{5}^{7} + 16968 T_{5}^{6} + 26911 T_{5}^{5} - 72240 T_{5}^{4} - 14532 T_{5}^{3} + 112850 T_{5}^{2} - 72184 T_{5} + 12144 \) Copy content Toggle raw display
\( T_{7}^{13} - T_{7}^{12} - 58 T_{7}^{11} + 81 T_{7}^{10} + 1147 T_{7}^{9} - 2082 T_{7}^{8} - 8413 T_{7}^{7} + 19342 T_{7}^{6} + 13575 T_{7}^{5} - 44740 T_{7}^{4} - 1584 T_{7}^{3} + 30480 T_{7}^{2} - 2176 T_{7} - 6016 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} \) Copy content Toggle raw display
$3$ \( (T - 1)^{13} \) Copy content Toggle raw display
$5$ \( T^{13} - 2 T^{12} - 49 T^{11} + \cdots + 12144 \) Copy content Toggle raw display
$7$ \( T^{13} - T^{12} - 58 T^{11} + 81 T^{10} + \cdots - 6016 \) Copy content Toggle raw display
$11$ \( T^{13} - 11 T^{12} - 29 T^{11} + \cdots + 123008 \) Copy content Toggle raw display
$13$ \( T^{13} - 12 T^{12} - 61 T^{11} + \cdots + 1786432 \) Copy content Toggle raw display
$17$ \( T^{13} - 15 T^{12} - 54 T^{11} + \cdots - 23680624 \) Copy content Toggle raw display
$19$ \( T^{13} - 14 T^{12} - 25 T^{11} + \cdots + 180736 \) Copy content Toggle raw display
$23$ \( T^{13} - 9 T^{12} - 209 T^{11} + \cdots + 368197632 \) Copy content Toggle raw display
$29$ \( T^{13} + 3 T^{12} - 151 T^{11} + \cdots - 134409216 \) Copy content Toggle raw display
$31$ \( T^{13} + 17 T^{12} - 49 T^{11} + \cdots + 5259264 \) Copy content Toggle raw display
$37$ \( T^{13} - 16 T^{12} - 97 T^{11} + \cdots + 108176 \) Copy content Toggle raw display
$41$ \( T^{13} - 12 T^{12} + \cdots - 262115744 \) Copy content Toggle raw display
$43$ \( T^{13} - 20 T^{12} + \cdots - 1580774304 \) Copy content Toggle raw display
$47$ \( T^{13} + 6 T^{12} - 351 T^{11} + \cdots + 232066544 \) Copy content Toggle raw display
$53$ \( T^{13} + 12 T^{12} - 143 T^{11} + \cdots + 56844352 \) Copy content Toggle raw display
$59$ \( T^{13} - 14 T^{12} - 187 T^{11} + \cdots - 6069184 \) Copy content Toggle raw display
$61$ \( T^{13} - 24 T^{12} - 87 T^{11} + \cdots - 2717696 \) Copy content Toggle raw display
$67$ \( T^{13} - 3 T^{12} + \cdots - 7323689064 \) Copy content Toggle raw display
$71$ \( T^{13} - 17 T^{12} + \cdots + 618043392 \) Copy content Toggle raw display
$73$ \( T^{13} - 34 T^{12} + \cdots + 3166561152 \) Copy content Toggle raw display
$79$ \( T^{13} - 10 T^{12} + \cdots - 341278848 \) Copy content Toggle raw display
$83$ \( T^{13} - 44 T^{12} + \cdots + 43837825568 \) Copy content Toggle raw display
$89$ \( T^{13} - 25 T^{12} + \cdots + 259651296 \) Copy content Toggle raw display
$97$ \( T^{13} - 38 T^{12} + \cdots - 428877315008 \) Copy content Toggle raw display
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