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Properties

Label 4005.2.a.r
Level $4005$
Weight $2$
Character orbit 4005.a
Self dual yes
Analytic conductor $31.980$
Analytic rank $0$
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: \(0\)
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: \( x^{10} - 4x^{9} - 8x^{8} + 35x^{7} + 29x^{6} - 103x^{5} - 57x^{4} + 106x^{3} + 29x^{2} - 39x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} - q^{5} + (\beta_{7} + \beta_{3} + \beta_{2} + 1) q^{7} + (\beta_{3} + \beta_1 - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} - q^{5} + (\beta_{7} + \beta_{3} + \beta_{2} + 1) q^{7} + (\beta_{3} + \beta_1 - 2) q^{8} + ( - \beta_1 + 1) q^{10} + ( - \beta_{9} + \beta_{5} + \beta_{2} + 1) q^{11} + ( - \beta_{9} + \beta_{7} + \cdots + 3 \beta_1) q^{13}+ \cdots + ( - 2 \beta_{9} + 3 \beta_{8} + \cdots + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{2} + 14 q^{4} - 10 q^{5} + 7 q^{7} - 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 6 q^{2} + 14 q^{4} - 10 q^{5} + 7 q^{7} - 15 q^{8} + 6 q^{10} + 10 q^{11} + 7 q^{13} + 7 q^{14} + 22 q^{16} - 11 q^{17} + 10 q^{19} - 14 q^{20} + 8 q^{22} - 6 q^{23} + 10 q^{25} + 14 q^{26} + 16 q^{28} + 3 q^{29} + 12 q^{31} - 21 q^{32} - 6 q^{34} - 7 q^{35} - 19 q^{37} - 6 q^{38} + 15 q^{40} - 13 q^{41} + 9 q^{43} + 26 q^{44} + 8 q^{46} - 21 q^{47} + 53 q^{49} - 6 q^{50} - 43 q^{52} - 7 q^{53} - 10 q^{55} + 53 q^{56} - 42 q^{58} + 19 q^{59} + 4 q^{61} + 28 q^{62} + 5 q^{64} - 7 q^{65} - 6 q^{67} - 2 q^{68} - 7 q^{70} + 6 q^{71} + 6 q^{73} - 2 q^{76} + 40 q^{77} + 25 q^{79} - 22 q^{80} + q^{82} + 22 q^{83} + 11 q^{85} + 2 q^{86} + 20 q^{88} - 10 q^{89} - 10 q^{91} - 10 q^{92} + 25 q^{94} - 10 q^{95} + 40 q^{97} + 56 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4x^{9} - 8x^{8} + 35x^{7} + 29x^{6} - 103x^{5} - 57x^{4} + 106x^{3} + 29x^{2} - 39x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu^{2} - 2\nu + 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{9} - 5\nu^{8} - 2\nu^{7} + 32\nu^{6} - 6\nu^{5} - 62\nu^{4} + 6\nu^{3} + 23\nu^{2} - 5\nu + 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{8} - 5\nu^{7} - 2\nu^{6} + 32\nu^{5} - 5\nu^{4} - 66\nu^{3} + 4\nu^{2} + 36\nu - 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{9} + 4\nu^{8} + 7\nu^{7} - 30\nu^{6} - 27\nu^{5} + 71\nu^{4} + 62\nu^{3} - 41\nu^{2} - 31\nu + 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{9} - 3\nu^{8} - 12\nu^{7} + 28\nu^{6} + 58\nu^{5} - 73\nu^{4} - 122\nu^{3} + 32\nu^{2} + 55\nu - 3 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( \nu^{9} - 2\nu^{8} - 16\nu^{7} + 21\nu^{6} + 90\nu^{5} - 54\nu^{4} - 199\nu^{3} + 6\nu^{2} + 99\nu - 3 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( 2\nu^{9} - 8\nu^{8} - 14\nu^{7} + 61\nu^{6} + 49\nu^{5} - 142\nu^{4} - 101\nu^{3} + 72\nu^{2} + 39\nu - 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_{2} + 5\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} + 2\beta_{5} + \beta_{4} + 4\beta_{3} + 13\beta_{2} + 9\beta _1 + 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{7} - \beta_{6} + 7\beta_{5} + 3\beta_{4} + 18\beta_{3} + 44\beta_{2} + 32\beta _1 + 52 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{9} - 20\beta_{7} - 3\beta_{6} + 35\beta_{5} + 15\beta_{4} + 67\beta_{3} + 161\beta_{2} + 78\beta _1 + 189 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 5 \beta_{9} + \beta_{8} - 77 \beta_{7} - 15 \beta_{6} + 126 \beta_{5} + 51 \beta_{4} + 250 \beta_{3} + \cdots + 596 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 27 \beta_{9} + 5 \beta_{8} - 302 \beta_{7} - 49 \beta_{6} + 487 \beta_{5} + 194 \beta_{4} + 894 \beta_{3} + \cdots + 2049 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 113 \beta_{9} + 27 \beta_{8} - 1110 \beta_{7} - 185 \beta_{6} + 1733 \beta_{5} + 673 \beta_{4} + \cdots + 6846 \) 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
−1.71694
−1.53398
−1.32759
−1.05100
0.0538492
0.704585
0.770724
2.21375
2.43743
3.44917
−2.71694 0 5.38176 −1.00000 0 0.704196 −9.18805 0 2.71694
1.2 −2.53398 0 4.42108 −1.00000 0 −3.60443 −6.13497 0 2.53398
1.3 −2.32759 0 3.41766 −1.00000 0 −0.231668 −3.29972 0 2.32759
1.4 −2.05100 0 2.20661 −1.00000 0 4.21605 −0.423758 0 2.05100
1.5 −0.946151 0 −1.10480 −1.00000 0 2.86767 2.93761 0 0.946151
1.6 −0.295415 0 −1.91273 −1.00000 0 3.54012 1.15588 0 0.295415
1.7 −0.229276 0 −1.94743 −1.00000 0 −3.83075 0.905050 0 0.229276
1.8 1.21375 0 −0.526821 −1.00000 0 −4.72892 −3.06692 0 −1.21375
1.9 1.43743 0 0.0662179 −1.00000 0 3.21347 −2.77969 0 −1.43743
1.10 2.44917 0 3.99846 −1.00000 0 4.85425 4.89457 0 −2.44917
\(n\): 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.r 10
3.b odd 2 1 1335.2.a.k ✓ 10
15.d odd 2 1 6675.2.a.y 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1335.2.a.k ✓ 10 3.b odd 2 1
4005.2.a.r 10 1.a even 1 1 trivial
6675.2.a.y 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):

\( T_{2}^{10} + 6T_{2}^{9} + T_{2}^{8} - 53T_{2}^{7} - 76T_{2}^{6} + 106T_{2}^{5} + 234T_{2}^{4} - 11T_{2}^{3} - 178T_{2}^{2} - 77T_{2} - 9 \) Copy content Toggle raw display
\( T_{7}^{10} - 7 T_{7}^{9} - 37 T_{7}^{8} + 346 T_{7}^{7} + 154 T_{7}^{6} - 5463 T_{7}^{5} + 6423 T_{7}^{4} + \cdots + 7112 \) Copy content Toggle raw display
\( T_{11}^{10} - 10 T_{11}^{9} - 36 T_{11}^{8} + 540 T_{11}^{7} - 4 T_{11}^{6} - 8264 T_{11}^{5} + \cdots + 3328 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 6 T^{9} + \cdots - 9 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T + 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - 7 T^{9} + \cdots + 7112 \) Copy content Toggle raw display
$11$ \( T^{10} - 10 T^{9} + \cdots + 3328 \) Copy content Toggle raw display
$13$ \( T^{10} - 7 T^{9} + \cdots + 95434 \) Copy content Toggle raw display
$17$ \( T^{10} + 11 T^{9} + \cdots - 22662 \) Copy content Toggle raw display
$19$ \( T^{10} - 10 T^{9} + \cdots + 200448 \) Copy content Toggle raw display
$23$ \( T^{10} + 6 T^{9} + \cdots + 134656 \) Copy content Toggle raw display
$29$ \( T^{10} - 3 T^{9} + \cdots + 113902 \) Copy content Toggle raw display
$31$ \( T^{10} - 12 T^{9} + \cdots + 131072 \) Copy content Toggle raw display
$37$ \( T^{10} + 19 T^{9} + \cdots + 8014 \) Copy content Toggle raw display
$41$ \( T^{10} + 13 T^{9} + \cdots - 1042058 \) Copy content Toggle raw display
$43$ \( T^{10} - 9 T^{9} + \cdots + 8694092 \) Copy content Toggle raw display
$47$ \( T^{10} + 21 T^{9} + \cdots - 1206968 \) Copy content Toggle raw display
$53$ \( T^{10} + 7 T^{9} + \cdots - 27969554 \) Copy content Toggle raw display
$59$ \( T^{10} - 19 T^{9} + \cdots - 297948 \) Copy content Toggle raw display
$61$ \( T^{10} - 4 T^{9} + \cdots + 547456 \) Copy content Toggle raw display
$67$ \( T^{10} + 6 T^{9} + \cdots - 258816 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 175440384 \) Copy content Toggle raw display
$73$ \( T^{10} - 6 T^{9} + \cdots + 697728 \) Copy content Toggle raw display
$79$ \( T^{10} - 25 T^{9} + \cdots - 2416 \) Copy content Toggle raw display
$83$ \( T^{10} - 22 T^{9} + \cdots + 511744 \) Copy content Toggle raw display
$89$ \( (T + 1)^{10} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 10800022528 \) Copy content Toggle raw display
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