↑

Properties

Label 4002.2.a.bj
Level $4002$
Weight $2$
Character orbit 4002.a
Self dual yes
Analytic conductor $31.956$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4002,2,Mod(1,4002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4002, 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("4002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4002.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: \(31.9561308889\)
Analytic rank: \(0\)
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: \( x^{8} - x^{7} - 26x^{6} + 4x^{5} + 209x^{4} + 113x^{3} - 436x^{2} - 360x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + \beta_1 q^{5} - q^{6} - \beta_{4} q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} + \beta_1 q^{5} - q^{6} - \beta_{4} q^{7} + q^{8} + q^{9} + \beta_1 q^{10} + ( - \beta_{5} - \beta_{2}) q^{11} - q^{12} + (\beta_{7} + 1) q^{13} - \beta_{4} q^{14} - \beta_1 q^{15} + q^{16} + ( - \beta_{7} - \beta_{4} + \beta_{2} + \cdots + 1) q^{17}+ \cdots + ( - \beta_{5} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9} + q^{10} + 3 q^{11} - 8 q^{12} + 9 q^{13} - q^{15} + 8 q^{16} + 8 q^{17} + 8 q^{18} + q^{20} + 3 q^{22} + 8 q^{23} - 8 q^{24} + 13 q^{25} + 9 q^{26} - 8 q^{27} + 8 q^{29} - q^{30} - 9 q^{31} + 8 q^{32} - 3 q^{33} + 8 q^{34} - 2 q^{35} + 8 q^{36} + 7 q^{37} - 9 q^{39} + q^{40} + 3 q^{41} + 16 q^{43} + 3 q^{44} + q^{45} + 8 q^{46} + 24 q^{47} - 8 q^{48} + 6 q^{49} + 13 q^{50} - 8 q^{51} + 9 q^{52} + 8 q^{53} - 8 q^{54} + 13 q^{55} + 8 q^{58} - 3 q^{59} - q^{60} + 31 q^{61} - 9 q^{62} + 8 q^{64} + 13 q^{65} - 3 q^{66} - 11 q^{67} + 8 q^{68} - 8 q^{69} - 2 q^{70} + 7 q^{71} + 8 q^{72} + 14 q^{73} + 7 q^{74} - 13 q^{75} + 10 q^{77} - 9 q^{78} + 12 q^{79} + q^{80} + 8 q^{81} + 3 q^{82} - 8 q^{83} + 22 q^{85} + 16 q^{86} - 8 q^{87} + 3 q^{88} - 12 q^{89} + q^{90} + 28 q^{91} + 8 q^{92} + 9 q^{93} + 24 q^{94} - 2 q^{95} - 8 q^{96} + 16 q^{97} + 6 q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 26x^{6} + 4x^{5} + 209x^{4} + 113x^{3} - 436x^{2} - 360x - 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 45\nu^{7} - 144\nu^{6} - 770\nu^{5} + 1874\nu^{4} + 3119\nu^{3} - 3940\nu^{2} + 1944\nu + 1488 ) / 1664 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 89\nu^{7} - 368\nu^{6} - 1338\nu^{5} + 5130\nu^{4} + 4819\nu^{3} - 13108\nu^{2} + 184\nu - 3824 ) / 1664 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -53\nu^{7} + 128\nu^{6} + 1138\nu^{5} - 1634\nu^{4} - 7815\nu^{3} + 2676\nu^{2} + 15016\nu + 4016 ) / 832 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9\nu^{7} - 34\nu^{6} - 154\nu^{5} + 510\nu^{4} + 759\nu^{3} - 1698\nu^{2} - 984\nu + 464 ) / 104 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -251\nu^{7} + 720\nu^{6} + 4942\nu^{5} - 10046\nu^{4} - 29193\nu^{3} + 25628\nu^{2} + 42072\nu + 5456 ) / 1664 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 311\nu^{7} - 912\nu^{6} - 6246\nu^{5} + 12822\nu^{4} + 38621\nu^{3} - 32268\nu^{2} - 61112\nu - 8464 ) / 1664 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - 2\beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} - 3\beta_{2} + 10\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{7} - 17\beta_{6} - 12\beta_{5} + 15\beta_{4} + 12\beta_{3} - 31\beta_{2} + 7\beta _1 + 82 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -27\beta_{7} - 37\beta_{6} - 7\beta_{5} - 9\beta_{4} + 26\beta_{3} - 70\beta_{2} + 119\beta _1 + 181 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -78\beta_{7} - 293\beta_{6} - 157\beta_{5} + 189\beta_{4} + 163\beta_{3} - 470\beta_{2} + 178\beta _1 + 1147 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -559\beta_{7} - 881\beta_{6} - 210\beta_{5} - 17\beta_{4} + 485\beta_{3} - 1341\beta_{2} + 1578\beta _1 + 3378 \) 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
−3.17046
−2.26152
−2.09236
−0.869754
−0.0471692
1.78927
3.49512
4.15688
1.00000 −1.00000 1.00000 −3.17046 −1.00000 −4.05963 1.00000 1.00000 −3.17046
1.2 1.00000 −1.00000 1.00000 −2.26152 −1.00000 3.32565 1.00000 1.00000 −2.26152
1.3 1.00000 −1.00000 1.00000 −2.09236 −1.00000 1.21381 1.00000 1.00000 −2.09236
1.4 1.00000 −1.00000 1.00000 −0.869754 −1.00000 3.97135 1.00000 1.00000 −0.869754
1.5 1.00000 −1.00000 1.00000 −0.0471692 −1.00000 −3.98374 1.00000 1.00000 −0.0471692
1.6 1.00000 −1.00000 1.00000 1.78927 −1.00000 0.126532 1.00000 1.00000 1.78927
1.7 1.00000 −1.00000 1.00000 3.49512 −1.00000 −1.05589 1.00000 1.00000 3.49512
1.8 1.00000 −1.00000 1.00000 4.15688 −1.00000 0.461913 1.00000 1.00000 4.15688
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(23\) \(-1\)
\(29\) \(-1\)

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 4002.2.a.bj ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.bj ✓ 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\( T_{5}^{8} - T_{5}^{7} - 26T_{5}^{6} + 4T_{5}^{5} + 209T_{5}^{4} + 113T_{5}^{3} - 436T_{5}^{2} - 360T_{5} - 16 \) Copy content Toggle raw display
\( T_{7}^{8} - 31T_{7}^{6} + 10T_{7}^{5} + 256T_{7}^{4} - 168T_{7}^{3} - 248T_{7}^{2} + 160T_{7} - 16 \) Copy content Toggle raw display
\( T_{11}^{8} - 3T_{11}^{7} - 59T_{11}^{6} + 233T_{11}^{5} + 888T_{11}^{4} - 4934T_{11}^{3} + 784T_{11}^{2} + 22144T_{11} - 25920 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( (T + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - T^{7} + \cdots - 16 \) Copy content Toggle raw display
$7$ \( T^{8} - 31 T^{6} + \cdots - 16 \) Copy content Toggle raw display
$11$ \( T^{8} - 3 T^{7} + \cdots - 25920 \) Copy content Toggle raw display
$13$ \( T^{8} - 9 T^{7} + \cdots - 3664 \) Copy content Toggle raw display
$17$ \( T^{8} - 8 T^{7} + \cdots + 1728 \) Copy content Toggle raw display
$19$ \( T^{8} - 31 T^{6} + \cdots - 16 \) Copy content Toggle raw display
$23$ \( (T - 1)^{8} \) Copy content Toggle raw display
$29$ \( (T - 1)^{8} \) Copy content Toggle raw display
$31$ \( T^{8} + 9 T^{7} + \cdots - 957664 \) Copy content Toggle raw display
$37$ \( T^{8} - 7 T^{7} + \cdots - 208520 \) Copy content Toggle raw display
$41$ \( T^{8} - 3 T^{7} + \cdots - 5000 \) Copy content Toggle raw display
$43$ \( T^{8} - 16 T^{7} + \cdots + 2837152 \) Copy content Toggle raw display
$47$ \( T^{8} - 24 T^{7} + \cdots - 1160704 \) Copy content Toggle raw display
$53$ \( T^{8} - 8 T^{7} + \cdots - 1116672 \) Copy content Toggle raw display
$59$ \( T^{8} + 3 T^{7} + \cdots + 9582624 \) Copy content Toggle raw display
$61$ \( T^{8} - 31 T^{7} + \cdots + 2726048 \) Copy content Toggle raw display
$67$ \( T^{8} + 11 T^{7} + \cdots - 7076672 \) Copy content Toggle raw display
$71$ \( T^{8} - 7 T^{7} + \cdots - 109792 \) Copy content Toggle raw display
$73$ \( T^{8} - 14 T^{7} + \cdots - 20453504 \) Copy content Toggle raw display
$79$ \( T^{8} - 12 T^{7} + \cdots - 285376 \) Copy content Toggle raw display
$83$ \( T^{8} + 8 T^{7} + \cdots + 25728 \) Copy content Toggle raw display
$89$ \( T^{8} + 12 T^{7} + \cdots + 504320 \) Copy content Toggle raw display
$97$ \( T^{8} - 16 T^{7} + \cdots - 859392 \) Copy content Toggle raw display
show more
show less