Properties

Label 4005.2.a.p
Level $4005$
Weight $2$
Character orbit 4005.a
Self dual yes
Analytic conductor $31.980$
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] = [4005,2,Mod(1,4005)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4005, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4005.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4005.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: \(31.9800860095\)
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: \( x^{8} - x^{7} - 11x^{6} + 9x^{5} + 34x^{4} - 19x^{3} - 27x^{2} + 11x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 445)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_1 + 1) q^{4} + q^{5} + ( - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{7} + ( - \beta_{2} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_1 + 1) q^{4} + q^{5} + ( - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{7} + ( - \beta_{2} - \beta_1) q^{8} - \beta_1 q^{10} + (\beta_{2} - 2) q^{11} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \beta_{3} + \beta_1 - 2) q^{13} + (\beta_{7} + \beta_{6} + \beta_{2} + 2 \beta_1 - 2) q^{14} + (\beta_{7} - 2 \beta_{4} - \beta_{3} - \beta_1 + 1) q^{16} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 2) q^{17} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - 2 \beta_1 + 3) q^{19} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_1 + 1) q^{20} + ( - \beta_{7} - \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 4 \beta_1) q^{22} + (2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_1 + 1) q^{23} + q^{25} + (3 \beta_{7} + 2 \beta_{6} + \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - \beta_1 + 1) q^{26} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} + 4 \beta_{3} + 5 \beta_1 - 5) q^{28} + (\beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_1 - 2) q^{29} + (2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2}) q^{31} + ( - \beta_{7} - 2 \beta_{6} - \beta_{5} - 1) q^{32} + (\beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + 2 \beta_1 - 2) q^{34} + ( - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{35} + (\beta_{7} + 2 \beta_{6} + 2 \beta_{5} - \beta_{3} - \beta_{2} - \beta_1) q^{37} + ( - 2 \beta_{7} - 2 \beta_{6} + \beta_{3} - 2 \beta_1 + 3) q^{38} + ( - \beta_{2} - \beta_1) q^{40} + (\beta_{7} - 2 \beta_{6} - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{41} + (\beta_{7} - 4 \beta_{4} - 2 \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{43} + (\beta_{7} + 2 \beta_{6} - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 5 \beta_1 - 1) q^{44} + ( - 2 \beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{46} + (2 \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{47} + (\beta_{7} + \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - \beta_{2} - 4 \beta_1 + 1) q^{49} - \beta_1 q^{50} + ( - 3 \beta_{7} - 4 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + \cdots - 3) q^{52}+ \cdots + ( - 2 \beta_{6} + 2 \beta_{5} - 5 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} - 8 \beta_1 + 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} + 7 q^{4} + 8 q^{5} - 6 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{2} + 7 q^{4} + 8 q^{5} - 6 q^{7} - 3 q^{8} - q^{10} - 14 q^{11} - 7 q^{13} - 15 q^{14} + 9 q^{16} - 17 q^{17} + 17 q^{19} + 7 q^{20} + 2 q^{22} + q^{23} + 8 q^{25} - 3 q^{26} - 29 q^{28} - 10 q^{29} + q^{31} - 2 q^{32} - 16 q^{34} - 6 q^{35} - 11 q^{37} + 30 q^{38} - 3 q^{40} - 15 q^{41} - 5 q^{43} - 7 q^{44} - 12 q^{46} - 12 q^{47} + 4 q^{49} - q^{50} - 14 q^{52} + q^{53} - 14 q^{55} - 3 q^{56} - 37 q^{58} - 26 q^{59} + 13 q^{61} - 22 q^{62} - 15 q^{64} - 7 q^{65} - 25 q^{67} - 23 q^{68} - 15 q^{70} - 28 q^{71} - 17 q^{73} + 5 q^{74} + 8 q^{76} - 7 q^{79} + 9 q^{80} + 5 q^{82} - 44 q^{83} - 17 q^{85} + 13 q^{86} - 66 q^{88} - 8 q^{89} + 27 q^{91} - 15 q^{92} - 27 q^{94} + 17 q^{95} + q^{97} + 34 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 11x^{6} + 9x^{5} + 34x^{4} - 19x^{3} - 27x^{2} + 11x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{7} + \nu^{6} + 23\nu^{5} - 8\nu^{4} - 76\nu^{3} + 12\nu^{2} + 65\nu - 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{7} + \nu^{6} + 11\nu^{5} - 9\nu^{4} - 34\nu^{3} + 18\nu^{2} + 27\nu - 7 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{7} + 3\nu^{6} + 45\nu^{5} - 26\nu^{4} - 144\nu^{3} + 50\nu^{2} + 121\nu - 27 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{7} - 4\nu^{6} - 55\nu^{5} + 34\nu^{4} + 170\nu^{3} - 61\nu^{2} - 136\nu + 27 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -6\nu^{7} + 5\nu^{6} + 67\nu^{5} - 42\nu^{4} - 212\nu^{3} + 72\nu^{2} + 175\nu - 29 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} - \beta_{3} - \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + 6\beta_{5} - 8\beta_{4} - 7\beta_{3} - 7\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + 2\beta_{6} + \beta_{5} + 8\beta_{2} + 28\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11\beta_{7} + 2\beta_{6} + 37\beta_{5} - 54\beta_{4} - 48\beta_{3} - 47\beta _1 + 86 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13\beta_{7} + 24\beta_{6} + 12\beta_{5} - \beta_{4} - 3\beta_{3} + 54\beta_{2} + 163\beta _1 + 9 \) 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
2.51468
2.23321
1.16343
0.217002
0.151894
−1.11667
−1.66289
−2.50065
−2.51468 0 4.32363 1.00000 0 −2.83060 −5.84320 0 −2.51468
1.2 −2.23321 0 2.98721 1.00000 0 1.87039 −2.20463 0 −2.23321
1.3 −1.16343 0 −0.646427 1.00000 0 2.23155 3.07894 0 −1.16343
1.4 −0.217002 0 −1.95291 1.00000 0 −2.89958 0.857790 0 −0.217002
1.5 −0.151894 0 −1.97693 1.00000 0 2.88745 0.604071 0 −0.151894
1.6 1.11667 0 −0.753041 1.00000 0 −1.16728 −3.07425 0 1.11667
1.7 1.66289 0 0.765209 1.00000 0 −1.19579 −2.05332 0 1.66289
1.8 2.50065 0 4.25326 1.00000 0 −4.89614 5.63461 0 2.50065
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(89\) \(1\)

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 4005.2.a.p 8
3.b odd 2 1 445.2.a.g 8
12.b even 2 1 7120.2.a.bk 8
15.d odd 2 1 2225.2.a.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
445.2.a.g 8 3.b odd 2 1
2225.2.a.l 8 15.d odd 2 1
4005.2.a.p 8 1.a even 1 1 trivial
7120.2.a.bk 8 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):

\( T_{2}^{8} + T_{2}^{7} - 11T_{2}^{6} - 9T_{2}^{5} + 34T_{2}^{4} + 19T_{2}^{3} - 27T_{2}^{2} - 11T_{2} - 1 \) Copy content Toggle raw display
\( T_{7}^{8} + 6T_{7}^{7} - 12T_{7}^{6} - 102T_{7}^{5} + 11T_{7}^{4} + 536T_{7}^{3} + 244T_{7}^{2} - 856T_{7} - 676 \) Copy content Toggle raw display
\( T_{11}^{8} + 14T_{11}^{7} + 50T_{11}^{6} - 75T_{11}^{5} - 599T_{11}^{4} - 112T_{11}^{3} + 2192T_{11}^{2} + 704T_{11} - 2752 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{7} - 11 T^{6} - 9 T^{5} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 6 T^{7} - 12 T^{6} - 102 T^{5} + \cdots - 676 \) Copy content Toggle raw display
$11$ \( T^{8} + 14 T^{7} + 50 T^{6} + \cdots - 2752 \) Copy content Toggle raw display
$13$ \( T^{8} + 7 T^{7} - 40 T^{6} - 270 T^{5} + \cdots - 64 \) Copy content Toggle raw display
$17$ \( T^{8} + 17 T^{7} + 65 T^{6} + \cdots - 2176 \) Copy content Toggle raw display
$19$ \( T^{8} - 17 T^{7} + 80 T^{6} + \cdots + 7628 \) Copy content Toggle raw display
$23$ \( T^{8} - T^{7} - 67 T^{6} - 29 T^{5} + \cdots + 1004 \) Copy content Toggle raw display
$29$ \( T^{8} + 10 T^{7} - 98 T^{6} + \cdots - 14864 \) Copy content Toggle raw display
$31$ \( T^{8} - T^{7} - 136 T^{6} + \cdots - 59996 \) Copy content Toggle raw display
$37$ \( T^{8} + 11 T^{7} - 25 T^{6} + \cdots - 256 \) Copy content Toggle raw display
$41$ \( T^{8} + 15 T^{7} - 5 T^{6} + \cdots + 11344 \) Copy content Toggle raw display
$43$ \( T^{8} + 5 T^{7} - 196 T^{6} + \cdots - 644492 \) Copy content Toggle raw display
$47$ \( T^{8} + 12 T^{7} - 80 T^{6} + \cdots + 249712 \) Copy content Toggle raw display
$53$ \( T^{8} - T^{7} - 286 T^{6} + \cdots + 1486976 \) Copy content Toggle raw display
$59$ \( T^{8} + 26 T^{7} + 196 T^{6} + \cdots - 77428 \) Copy content Toggle raw display
$61$ \( T^{8} - 13 T^{7} - 251 T^{6} + \cdots + 16005968 \) Copy content Toggle raw display
$67$ \( T^{8} + 25 T^{7} - 187 T^{6} + \cdots - 58944976 \) Copy content Toggle raw display
$71$ \( T^{8} + 28 T^{7} + 286 T^{6} + \cdots - 11968 \) Copy content Toggle raw display
$73$ \( T^{8} + 17 T^{7} - 139 T^{6} + \cdots + 548656 \) Copy content Toggle raw display
$79$ \( T^{8} + 7 T^{7} - 271 T^{6} + \cdots + 226304 \) Copy content Toggle raw display
$83$ \( T^{8} + 44 T^{7} + 619 T^{6} + \cdots - 6725668 \) Copy content Toggle raw display
$89$ \( (T + 1)^{8} \) Copy content Toggle raw display
$97$ \( T^{8} - T^{7} - 356 T^{6} + \cdots + 1330048 \) Copy content Toggle raw display
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