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Properties

Label 4005.2.a.s
Level $4005$
Weight $2$
Character orbit 4005.a
Self dual yes
Analytic conductor $31.980$
Analytic rank $1$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} - 16x^{8} + 15x^{7} + 85x^{6} - 75x^{5} - 163x^{4} + 138x^{3} + 78x^{2} - 67x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1335)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - q^{5} + \beta_{6} q^{7} + ( - \beta_{5} + \beta_{4} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - q^{5} + \beta_{6} q^{7} + ( - \beta_{5} + \beta_{4} - \beta_1) q^{8} + \beta_1 q^{10} + ( - \beta_{9} - \beta_{4} - 1) q^{11} + ( - \beta_{9} + \beta_{7} + \cdots + \beta_1) q^{13}+ \cdots + ( - \beta_{9} + 2 \beta_{7} + \beta_{6} + \cdots + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7} + q^{10} - 10 q^{11} + 5 q^{13} - 13 q^{14} + 19 q^{16} - 9 q^{17} - 6 q^{19} - 13 q^{20} + 3 q^{23} + 10 q^{25} - 14 q^{26} - 18 q^{28} - 38 q^{29} + 2 q^{31} - 16 q^{32} - 8 q^{34} + q^{35} + 9 q^{37} - 20 q^{38} - 36 q^{41} - 7 q^{43} - 16 q^{44} + 2 q^{46} + 23 q^{47} + 25 q^{49} - q^{50} + 13 q^{52} - 27 q^{53} + 10 q^{55} - 41 q^{56} - 32 q^{58} - 20 q^{59} + 30 q^{61} + 2 q^{62} - 2 q^{64} - 5 q^{65} - 5 q^{67} + 10 q^{68} + 13 q^{70} - 24 q^{71} - 19 q^{73} - 42 q^{74} - 30 q^{76} - 18 q^{77} + 12 q^{79} - 19 q^{80} + 29 q^{82} - 3 q^{83} + 9 q^{85} - 38 q^{86} - 16 q^{88} + 10 q^{89} - 6 q^{91} - 32 q^{92} + 17 q^{94} + 6 q^{95} + 3 q^{97} + 37 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 16x^{8} + 15x^{7} + 85x^{6} - 75x^{5} - 163x^{4} + 138x^{3} + 78x^{2} - 67x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{9} - 8\nu^{8} - 106\nu^{7} + 77\nu^{6} + 733\nu^{5} - 168\nu^{4} - 1823\nu^{3} + 59\nu^{2} + 1140\nu - 95 ) / 77 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -9\nu^{9} - \nu^{8} + 160\nu^{7} - 919\nu^{5} + 56\nu^{4} + 1803\nu^{3} - 137\nu^{2} - 743\nu + 94 ) / 77 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{9} - \nu^{8} + 160\nu^{7} - 919\nu^{5} + 56\nu^{4} + 1880\nu^{3} - 137\nu^{2} - 1128\nu + 94 ) / 77 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13 \nu^{9} + 10 \nu^{8} - 214 \nu^{7} - 154 \nu^{6} + 1182 \nu^{5} + 749 \nu^{4} - 2399 \nu^{3} + \cdots + 292 ) / 77 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{9} - 2\nu^{8} - 65\nu^{7} + 22\nu^{6} + 351\nu^{5} - 75\nu^{4} - 684\nu^{3} + 111\nu^{2} + 340\nu - 65 ) / 11 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4\nu^{9} - 2\nu^{8} - 65\nu^{7} + 22\nu^{6} + 351\nu^{5} - 64\nu^{4} - 684\nu^{3} + 34\nu^{2} + 340\nu + 1 ) / 11 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 32 \nu^{9} - 5 \nu^{8} - 509 \nu^{7} + 77 \nu^{6} + 2720 \nu^{5} - 413 \nu^{4} - 5384 \nu^{3} + \cdots - 531 ) / 77 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{4} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - \beta_{7} + 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} - \beta_{7} + 8\beta_{5} - 7\beta_{4} + \beta_{3} + 29\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{9} + 9\beta_{8} - 10\beta_{7} - \beta_{6} - \beta_{4} + 45\beta_{2} + \beta _1 + 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11\beta_{9} + \beta_{8} - 11\beta_{7} + 56\beta_{5} - 43\beta_{4} + 9\beta_{3} + 2\beta_{2} + 177\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13\beta_{9} + 63\beta_{8} - 78\beta_{7} - 9\beta_{6} - 14\beta_{4} - \beta_{3} + 287\beta_{2} + 14\beta _1 + 534 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 92 \beta_{9} + 17 \beta_{8} - 91 \beta_{7} + \beta_{6} + 379 \beta_{5} - 257 \beta_{4} + 58 \beta_{3} + \cdots + 110 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
2.58455
2.52467
1.69046
0.970031
0.644507
0.0651711
−0.875863
−1.67217
−2.41366
−2.51771
−2.58455 0 4.67989 −1.00000 0 1.54056 −6.92630 0 2.58455
1.2 −2.52467 0 4.37398 −1.00000 0 −0.151868 −5.99353 0 2.52467
1.3 −1.69046 0 0.857665 −1.00000 0 0.437617 1.93108 0 1.69046
1.4 −0.970031 0 −1.05904 −1.00000 0 −4.84165 2.96736 0 0.970031
1.5 −0.644507 0 −1.58461 −1.00000 0 2.88508 2.31031 0 0.644507
1.6 −0.0651711 0 −1.99575 −1.00000 0 4.79409 0.260408 0 0.0651711
1.7 0.875863 0 −1.23286 −1.00000 0 −3.38461 −2.83155 0 −0.875863
1.8 1.67217 0 0.796137 −1.00000 0 3.27977 −2.01306 0 −1.67217
1.9 2.41366 0 3.82573 −1.00000 0 −2.78117 4.40669 0 −2.41366
1.10 2.51771 0 4.33887 −1.00000 0 −2.77781 5.88859 0 −2.51771
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.s 10
3.b odd 2 1 1335.2.a.j ✓ 10
15.d odd 2 1 6675.2.a.z 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1335.2.a.j ✓ 10 3.b odd 2 1
4005.2.a.s 10 1.a even 1 1 trivial
6675.2.a.z 10 15.d odd 2 1

Hecke kernels

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

\( T_{2}^{10} + T_{2}^{9} - 16T_{2}^{8} - 15T_{2}^{7} + 85T_{2}^{6} + 75T_{2}^{5} - 163T_{2}^{4} - 138T_{2}^{3} + 78T_{2}^{2} + 67T_{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{10} + T_{7}^{9} - 47 T_{7}^{8} - 40 T_{7}^{7} + 732 T_{7}^{6} + 459 T_{7}^{5} - 4589 T_{7}^{4} + \cdots - 588 \) Copy content Toggle raw display
\( T_{11}^{10} + 10 T_{11}^{9} - 12 T_{11}^{8} - 328 T_{11}^{7} - 164 T_{11}^{6} + 4080 T_{11}^{5} + \cdots + 36608 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + T^{9} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T + 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + T^{9} + \cdots - 588 \) Copy content Toggle raw display
$11$ \( T^{10} + 10 T^{9} + \cdots + 36608 \) Copy content Toggle raw display
$13$ \( T^{10} - 5 T^{9} + \cdots + 28814 \) Copy content Toggle raw display
$17$ \( T^{10} + 9 T^{9} + \cdots + 2272864 \) Copy content Toggle raw display
$19$ \( T^{10} + 6 T^{9} + \cdots - 128 \) Copy content Toggle raw display
$23$ \( T^{10} - 3 T^{9} + \cdots - 724416 \) Copy content Toggle raw display
$29$ \( T^{10} + 38 T^{9} + \cdots + 1064987 \) Copy content Toggle raw display
$31$ \( T^{10} - 2 T^{9} + \cdots + 2688 \) Copy content Toggle raw display
$37$ \( T^{10} - 9 T^{9} + \cdots - 26502742 \) Copy content Toggle raw display
$41$ \( T^{10} + 36 T^{9} + \cdots - 3961769 \) Copy content Toggle raw display
$43$ \( T^{10} + 7 T^{9} + \cdots - 2199092 \) Copy content Toggle raw display
$47$ \( T^{10} - 23 T^{9} + \cdots + 105964 \) Copy content Toggle raw display
$53$ \( T^{10} + 27 T^{9} + \cdots - 394702 \) Copy content Toggle raw display
$59$ \( T^{10} + 20 T^{9} + \cdots - 83126231 \) Copy content Toggle raw display
$61$ \( T^{10} - 30 T^{9} + \cdots + 1247488 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots - 5370039744 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 449773568 \) Copy content Toggle raw display
$73$ \( T^{10} + 19 T^{9} + \cdots - 481088 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 519934779 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 667123776 \) Copy content Toggle raw display
$89$ \( (T - 1)^{10} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 688855552 \) Copy content Toggle raw display
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