Properties

Label 4008.2.a.h
Level $4008$
Weight $2$
Character orbit 4008.a
Self dual yes
Analytic conductor $32.004$
Analytic rank $1$
Dimension $8$
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: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 23x^{6} - 3x^{5} + 163x^{4} + 13x^{3} - 418x^{2} + 4x + 269 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + \beta_1 q^{5} + \beta_{3} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + \beta_1 q^{5} + \beta_{3} q^{7} + q^{9} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2}) q^{11} + ( - \beta_{5} - \beta_{2} - 1) q^{13} - \beta_1 q^{15} + ( - \beta_{6} - \beta_{4} - \beta_{3} - 1) q^{17} + (\beta_{7} - \beta_{4} - 2 \beta_1) q^{19} - \beta_{3} q^{21} + ( - \beta_{3} + 1) q^{23} + (\beta_{7} - \beta_{6}) q^{25} - q^{27} + (\beta_{7} - \beta_{6} + \beta_{4} - \beta_{2} + \beta_1 + 1) q^{29} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} - \beta_1 - 3) q^{31} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{2}) q^{33} + ( - \beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{35} + (\beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{2} - \beta_1 + 1) q^{37} + (\beta_{5} + \beta_{2} + 1) q^{39} + ( - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{2} - 1) q^{41} + ( - \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} - 2 \beta_1) q^{43} + \beta_1 q^{45} + (\beta_{7} - 2 \beta_{6} - \beta_{4} - 2 \beta_{2} - 5) q^{47} + (2 \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{49} + (\beta_{6} + \beta_{4} + \beta_{3} + 1) q^{51} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_1 + 2) q^{53} + ( - \beta_{7} + \beta_{6} + 3 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{55} + ( - \beta_{7} + \beta_{4} + 2 \beta_1) q^{57} + (3 \beta_{7} - \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + \beta_{2} - 2) q^{59} + ( - \beta_{5} + 2 \beta_{4} - \beta_1 - 1) q^{61} + \beta_{3} q^{63} + ( - 3 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} - 4 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{65} + ( - \beta_{7} - \beta_{6} - \beta_{4} - \beta_1 + 3) q^{67} + (\beta_{3} - 1) q^{69} + ( - \beta_{7} + 2 \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 3) q^{71} + ( - \beta_{7} + \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_1 - 5) q^{73} + ( - \beta_{7} + \beta_{6}) q^{75} + (2 \beta_{6} - 2 \beta_{4} + \beta_{2} + \beta_1 + 4) q^{77} + ( - \beta_{7} + 2 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{79} + q^{81} + ( - 2 \beta_{6} - 2 \beta_{4} + \beta_{2} - 4) q^{83} + (3 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - \beta_{2} + 2) q^{85} + ( - \beta_{7} + \beta_{6} - \beta_{4} + \beta_{2} - \beta_1 - 1) q^{87} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 + 2) q^{89} + (\beta_{7} - 3 \beta_{6} + 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 3) q^{91} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + \beta_1 + 3) q^{93} + (\beta_{7} - \beta_{5} - 2 \beta_{4} - \beta_{2} - 8) q^{95} + (\beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{2} - \beta_1 + 1) q^{97} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{3} - q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{3} - q^{7} + 8 q^{9} - 3 q^{11} - 8 q^{13} - 7 q^{17} + q^{21} + 9 q^{23} + 6 q^{25} - 8 q^{27} + 17 q^{29} - 23 q^{31} + 3 q^{33} - 15 q^{35} + 8 q^{37} + 8 q^{39} - 8 q^{41} - 2 q^{43} - 34 q^{47} + 5 q^{49} + 7 q^{51} + 12 q^{53} - 7 q^{55} - 16 q^{59} - 2 q^{61} - q^{63} - 14 q^{65} + 21 q^{67} - 9 q^{69} - 29 q^{71} - 38 q^{73} - 6 q^{75} + 20 q^{77} - 12 q^{79} + 8 q^{81} - 32 q^{83} + 23 q^{85} - 17 q^{87} + 11 q^{89} - 5 q^{91} + 23 q^{93} - 67 q^{95} + 8 q^{97} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 23x^{6} - 3x^{5} + 163x^{4} + 13x^{3} - 418x^{2} + 4x + 269 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 185\nu^{7} - 1337\nu^{6} - 3411\nu^{5} + 24599\nu^{4} + 18970\nu^{3} - 110475\nu^{2} - 18219\nu + 87357 ) / 8453 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -211\nu^{7} - 714\nu^{6} + 3799\nu^{5} + 13889\nu^{4} - 10670\nu^{3} - 61564\nu^{2} - 16368\nu + 64354 ) / 8453 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -253\nu^{7} - 776\nu^{6} + 5076\nu^{5} + 14290\nu^{4} - 24572\nu^{3} - 48820\nu^{2} + 35699\nu + 9019 ) / 8453 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -312\nu^{7} + 747\nu^{6} + 4656\nu^{5} - 11512\nu^{4} - 10289\nu^{3} + 48782\nu^{2} - 26206\nu - 39128 ) / 8453 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 545\nu^{7} + 402\nu^{6} - 10734\nu^{5} - 10235\nu^{4} + 53600\nu^{3} + 47164\nu^{2} - 66009\nu - 38506 ) / 8453 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 545\nu^{7} + 402\nu^{6} - 10734\nu^{5} - 10235\nu^{4} + 53600\nu^{3} + 55617\nu^{2} - 66009\nu - 80771 ) / 8453 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{7} - 3\beta_{6} - \beta_{5} - 2\beta_{4} + 3\beta_{3} - \beta_{2} + 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 14\beta_{7} - 17\beta_{6} - 5\beta_{4} + 2\beta_{2} + 2\beta _1 + 41 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 54\beta_{7} - 59\beta_{6} - 22\beta_{5} - 36\beta_{4} + 47\beta_{3} - 18\beta_{2} + 97\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 191\beta_{7} - 247\beta_{6} - 4\beta_{5} - 102\beta_{4} + 9\beta_{3} + 29\beta_{2} + 61\beta _1 + 422 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 804\beta_{7} - 902\beta_{6} - 332\beta_{5} - 531\beta_{4} + 624\beta_{3} - 240\beta_{2} + 1139\beta _1 + 333 \) 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
−3.24252
−2.57145
−2.12535
−0.938106
1.02139
1.94540
2.18690
3.72373
0 −1.00000 0 −3.24252 0 −3.21024 0 1.00000 0
1.2 0 −1.00000 0 −2.57145 0 1.34479 0 1.00000 0
1.3 0 −1.00000 0 −2.12535 0 2.08900 0 1.00000 0
1.4 0 −1.00000 0 −0.938106 0 4.96672 0 1.00000 0
1.5 0 −1.00000 0 1.02139 0 −1.14465 0 1.00000 0
1.6 0 −1.00000 0 1.94540 0 −4.16489 0 1.00000 0
1.7 0 −1.00000 0 2.18690 0 0.195760 0 1.00000 0
1.8 0 −1.00000 0 3.72373 0 −1.07650 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.h 8
4.b odd 2 1 8016.2.a.y 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.h 8 1.a even 1 1 trivial
8016.2.a.y 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4008))\):

\( T_{5}^{8} - 23T_{5}^{6} - 3T_{5}^{5} + 163T_{5}^{4} + 13T_{5}^{3} - 418T_{5}^{2} + 4T_{5} + 269 \) Copy content Toggle raw display
\( T_{7}^{8} + T_{7}^{7} - 30T_{7}^{6} - 39T_{7}^{5} + 181T_{7}^{4} + 166T_{7}^{3} - 253T_{7}^{2} - 188T_{7} + 45 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 23 T^{6} - 3 T^{5} + 163 T^{4} + \cdots + 269 \) Copy content Toggle raw display
$7$ \( T^{8} + T^{7} - 30 T^{6} - 39 T^{5} + \cdots + 45 \) Copy content Toggle raw display
$11$ \( T^{8} + 3 T^{7} - 47 T^{6} + \cdots - 12892 \) Copy content Toggle raw display
$13$ \( T^{8} + 8 T^{7} - 47 T^{6} + \cdots + 56452 \) Copy content Toggle raw display
$17$ \( T^{8} + 7 T^{7} - 38 T^{6} + \cdots - 2140 \) Copy content Toggle raw display
$19$ \( T^{8} - 93 T^{6} + 162 T^{5} + \cdots - 21188 \) Copy content Toggle raw display
$23$ \( T^{8} - 9 T^{7} + 5 T^{6} + 142 T^{5} + \cdots - 116 \) Copy content Toggle raw display
$29$ \( T^{8} - 17 T^{7} - 9 T^{6} + \cdots + 174684 \) Copy content Toggle raw display
$31$ \( T^{8} + 23 T^{7} + 43 T^{6} + \cdots - 1841393 \) Copy content Toggle raw display
$37$ \( T^{8} - 8 T^{7} - 73 T^{6} + \cdots + 12471 \) Copy content Toggle raw display
$41$ \( T^{8} + 8 T^{7} - 152 T^{6} + \cdots + 809352 \) Copy content Toggle raw display
$43$ \( T^{8} + 2 T^{7} - 182 T^{6} + \cdots + 315668 \) Copy content Toggle raw display
$47$ \( T^{8} + 34 T^{7} + 295 T^{6} + \cdots + 3024711 \) Copy content Toggle raw display
$53$ \( T^{8} - 12 T^{7} - 137 T^{6} + \cdots - 695153 \) Copy content Toggle raw display
$59$ \( T^{8} + 16 T^{7} + \cdots - 254184949 \) Copy content Toggle raw display
$61$ \( T^{8} + 2 T^{7} - 271 T^{6} + \cdots - 444096 \) Copy content Toggle raw display
$67$ \( T^{8} - 21 T^{7} + 38 T^{6} + \cdots + 10321 \) Copy content Toggle raw display
$71$ \( T^{8} + 29 T^{7} + 130 T^{6} + \cdots + 61884 \) Copy content Toggle raw display
$73$ \( T^{8} + 38 T^{7} + 525 T^{6} + \cdots - 42020 \) Copy content Toggle raw display
$79$ \( T^{8} + 12 T^{7} - 312 T^{6} + \cdots + 2844832 \) Copy content Toggle raw display
$83$ \( T^{8} + 32 T^{7} + 225 T^{6} + \cdots - 893029 \) Copy content Toggle raw display
$89$ \( T^{8} - 11 T^{7} - 190 T^{6} + \cdots - 1037805 \) Copy content Toggle raw display
$97$ \( T^{8} - 8 T^{7} - 222 T^{6} + \cdots - 1227013 \) Copy content Toggle raw display
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