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Properties

Label 4014.2.a.x
Level $4014$
Weight $2$
Character orbit 4014.a
Self dual yes
Analytic conductor $32.052$
Analytic rank $0$
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] = [4014,2,Mod(1,4014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4014, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4014.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4014.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: \(32.0519513713\)
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: \( x^{8} - 2x^{7} - 14x^{6} + 28x^{5} + 43x^{4} - 90x^{3} - 23x^{2} + 82x - 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( - \beta_{6} + \beta_{5} + 1) q^{5} + ( - \beta_{7} - 1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + ( - \beta_{6} + \beta_{5} + 1) q^{5} + ( - \beta_{7} - 1) q^{7} - q^{8} + (\beta_{6} - \beta_{5} - 1) q^{10} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 1) q^{11}+ \cdots + ( - \beta_{6} + \beta_{5} + 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} + 6 q^{5} - 6 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 8 q^{4} + 6 q^{5} - 6 q^{7} - 8 q^{8} - 6 q^{10} + 11 q^{11} - q^{13} + 6 q^{14} + 8 q^{16} + 16 q^{17} - q^{19} + 6 q^{20} - 11 q^{22} + 14 q^{23} + 10 q^{25} + q^{26} - 6 q^{28} + 21 q^{29} - 6 q^{31} - 8 q^{32} - 16 q^{34} + 8 q^{35} - 14 q^{37} + q^{38} - 6 q^{40} + 16 q^{41} - 29 q^{43} + 11 q^{44} - 14 q^{46} + 9 q^{47} - 2 q^{49} - 10 q^{50} - q^{52} + 11 q^{53} - 22 q^{55} + 6 q^{56} - 21 q^{58} + 21 q^{59} + 3 q^{61} + 6 q^{62} + 8 q^{64} + 24 q^{65} - 20 q^{67} + 16 q^{68} - 8 q^{70} + 32 q^{71} + 13 q^{73} + 14 q^{74} - q^{76} + 4 q^{77} + 21 q^{79} + 6 q^{80} - 16 q^{82} + 28 q^{83} - 14 q^{85} + 29 q^{86} - 11 q^{88} + 54 q^{89} - 36 q^{91} + 14 q^{92} - 9 q^{94} + 30 q^{95} + 10 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 14x^{6} + 28x^{5} + 43x^{4} - 90x^{3} - 23x^{2} + 82x - 28 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13\nu^{7} - 22\nu^{6} - 144\nu^{5} + 260\nu^{4} + 57\nu^{3} - 466\nu^{2} + 841\nu + 116 ) / 194 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -8\nu^{7} + 21\nu^{6} + 111\nu^{5} - 257\nu^{4} - 341\nu^{3} + 533\nu^{2} + 251\nu - 243 ) / 97 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 39\nu^{7} - 66\nu^{6} - 626\nu^{5} + 974\nu^{4} + 2693\nu^{3} - 3532\nu^{2} - 3297\nu + 3258 ) / 194 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 85\nu^{7} - 114\nu^{6} - 1240\nu^{5} + 1506\nu^{4} + 4387\nu^{3} - 4196\nu^{2} - 4425\nu + 3370 ) / 194 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 89\nu^{7} - 76\nu^{6} - 1344\nu^{5} + 1004\nu^{4} + 5091\nu^{3} - 2862\nu^{2} - 5181\nu + 2764 ) / 194 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 49\nu^{7} - 68\nu^{6} - 692\nu^{5} + 883\nu^{4} + 2222\nu^{3} - 2234\nu^{2} - 1889\nu + 1355 ) / 97 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{5} - \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + 2\beta_{5} - \beta_{4} + \beta_{3} - 2\beta_{2} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 11\beta_{7} - \beta_{6} - 10\beta_{5} + \beta_{4} + 3\beta_{3} - 10\beta_{2} + 12\beta _1 + 31 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -14\beta_{6} + 27\beta_{5} - 13\beta_{4} + 16\beta_{3} - 22\beta_{2} + 75\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 110\beta_{7} - 12\beta_{6} - 96\beta_{5} + 11\beta_{4} + 45\beta_{3} - 97\beta_{2} + 123\beta _1 + 288 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{7} - 151\beta_{6} + 292\beta_{5} - 141\beta_{4} + 189\beta_{3} - 220\beta_{2} + 735\beta _1 + 24 \) 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
1.44360
−3.14687
3.17253
−1.34121
2.16537
0.769489
−1.60467
0.541762
−1.00000 0 1.00000 −2.94121 0 −2.83796 −1.00000 0 2.94121
1.2 −1.00000 0 1.00000 −2.03557 0 −4.04095 −1.00000 0 2.03557
1.3 −1.00000 0 1.00000 −0.585733 0 −1.49689 −1.00000 0 0.585733
1.4 −1.00000 0 1.00000 0.350247 0 3.21437 −1.00000 0 −0.350247
1.5 −1.00000 0 1.00000 1.09191 0 1.60541 −1.00000 0 −1.09191
1.6 −1.00000 0 1.00000 2.40529 0 2.01389 −1.00000 0 −2.40529
1.7 −1.00000 0 1.00000 3.83272 0 −2.71599 −1.00000 0 −3.83272
1.8 −1.00000 0 1.00000 3.88234 0 −1.74188 −1.00000 0 −3.88234
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(223\) \(-1\)

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 4014.2.a.x ✓ 8
3.b odd 2 1 4014.2.a.y yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4014.2.a.x ✓ 8 1.a even 1 1 trivial
4014.2.a.y yes 8 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4014))\):

\( T_{5}^{8} - 6T_{5}^{7} - 7T_{5}^{6} + 82T_{5}^{5} - 34T_{5}^{4} - 268T_{5}^{3} + 189T_{5}^{2} + 104T_{5} - 48 \) Copy content Toggle raw display
\( T_{7}^{8} + 6T_{7}^{7} - 9T_{7}^{6} - 102T_{7}^{5} - 56T_{7}^{4} + 478T_{7}^{3} + 515T_{7}^{2} - 658T_{7} - 844 \) Copy content Toggle raw display
\( T_{11}^{8} - 11T_{11}^{7} + 13T_{11}^{6} + 200T_{11}^{5} - 487T_{11}^{4} - 1017T_{11}^{3} + 2664T_{11}^{2} + 1227T_{11} - 981 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 6 T^{7} + \cdots - 48 \) Copy content Toggle raw display
$7$ \( T^{8} + 6 T^{7} + \cdots - 844 \) Copy content Toggle raw display
$11$ \( T^{8} - 11 T^{7} + \cdots - 981 \) Copy content Toggle raw display
$13$ \( T^{8} + T^{7} + \cdots + 6621 \) Copy content Toggle raw display
$17$ \( T^{8} - 16 T^{7} + \cdots + 156 \) Copy content Toggle raw display
$19$ \( T^{8} + T^{7} + \cdots + 105249 \) Copy content Toggle raw display
$23$ \( T^{8} - 14 T^{7} + \cdots - 27648 \) Copy content Toggle raw display
$29$ \( T^{8} - 21 T^{7} + \cdots + 312249 \) Copy content Toggle raw display
$31$ \( T^{8} + 6 T^{7} + \cdots + 1834244 \) Copy content Toggle raw display
$37$ \( T^{8} + 14 T^{7} + \cdots - 93188 \) Copy content Toggle raw display
$41$ \( T^{8} - 16 T^{7} + \cdots - 144 \) Copy content Toggle raw display
$43$ \( T^{8} + 29 T^{7} + \cdots + 207987 \) Copy content Toggle raw display
$47$ \( T^{8} - 9 T^{7} + \cdots - 1613487 \) Copy content Toggle raw display
$53$ \( T^{8} - 11 T^{7} + \cdots + 396921 \) Copy content Toggle raw display
$59$ \( T^{8} - 21 T^{7} + \cdots + 124461 \) Copy content Toggle raw display
$61$ \( T^{8} - 3 T^{7} + \cdots + 380517 \) Copy content Toggle raw display
$67$ \( T^{8} + 20 T^{7} + \cdots + 5401292 \) Copy content Toggle raw display
$71$ \( T^{8} - 32 T^{7} + \cdots - 24100236 \) Copy content Toggle raw display
$73$ \( T^{8} - 13 T^{7} + \cdots + 401875 \) Copy content Toggle raw display
$79$ \( T^{8} - 21 T^{7} + \cdots + 3312389 \) Copy content Toggle raw display
$83$ \( T^{8} - 28 T^{7} + \cdots + 6789312 \) Copy content Toggle raw display
$89$ \( T^{8} - 54 T^{7} + \cdots - 330048 \) Copy content Toggle raw display
$97$ \( T^{8} - 10 T^{7} + \cdots - 492 \) Copy content Toggle raw display
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