↑

Properties

Label 4014.2.a.v
Level $4014$
Weight $2$
Character orbit 4014.a
Self dual yes
Analytic conductor $32.052$
Analytic rank $0$
Dimension $7$
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] = [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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 18x^{5} - 8x^{4} + 51x^{3} + 47x^{2} - 2x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1338)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( - \beta_{4} - 1) q^{5} - \beta_{3} q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + ( - \beta_{4} - 1) q^{5} - \beta_{3} q^{7} - q^{8} + (\beta_{4} + 1) q^{10} + ( - \beta_{6} + \beta_{2}) q^{11} + ( - \beta_{5} + \beta_{2} + 1) q^{13} + \beta_{3} q^{14} + q^{16} + ( - \beta_{5} - \beta_{3} - 3) q^{17} + ( - \beta_{6} + \beta_1) q^{19} + ( - \beta_{4} - 1) q^{20} + (\beta_{6} - \beta_{2}) q^{22} + (\beta_{6} - \beta_{5} + \beta_{2} + \cdots - 1) q^{23}+ \cdots + ( - \beta_{6} + 2 \beta_{4} + 3 \beta_{2} + \cdots - 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 7 q^{2} + 7 q^{4} - 6 q^{5} + 3 q^{7} - 7 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 7 q^{2} + 7 q^{4} - 6 q^{5} + 3 q^{7} - 7 q^{8} + 6 q^{10} + q^{11} + 8 q^{13} - 3 q^{14} + 7 q^{16} - 16 q^{17} + 2 q^{19} - 6 q^{20} - q^{22} - 8 q^{23} + 19 q^{25} - 8 q^{26} + 3 q^{28} - 4 q^{29} + 11 q^{31} - 7 q^{32} + 16 q^{34} + 17 q^{37} - 2 q^{38} + 6 q^{40} - 18 q^{41} - q^{43} + q^{44} + 8 q^{46} - 5 q^{47} + 24 q^{49} - 19 q^{50} + 8 q^{52} - 6 q^{53} + 21 q^{55} - 3 q^{56} + 4 q^{58} + 7 q^{59} + 24 q^{61} - 11 q^{62} + 7 q^{64} - 11 q^{65} - 4 q^{67} - 16 q^{68} + 7 q^{71} + 28 q^{73} - 17 q^{74} + 2 q^{76} + 2 q^{77} - 6 q^{80} + 18 q^{82} + 2 q^{83} + 4 q^{85} + q^{86} - q^{88} - 5 q^{89} + 2 q^{91} - 8 q^{92} + 5 q^{94} + 14 q^{95} + 23 q^{97} - 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 18x^{5} - 8x^{4} + 51x^{3} + 47x^{2} - 2x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{6} - \nu^{5} - 17\nu^{4} + 9\nu^{3} + 42\nu^{2} + 5\nu - 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -9\nu^{6} + 7\nu^{5} + 156\nu^{4} - 49\nu^{3} - 413\nu^{2} - 104\nu + 88 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -21\nu^{6} + 16\nu^{5} + 366\nu^{4} - 112\nu^{3} - 989\nu^{2} - 221\nu + 226 ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\nu^{6} - 9\nu^{5} - 191\nu^{4} + 68\nu^{3} + 512\nu^{2} + 104\nu - 123 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -14\nu^{6} + 10\nu^{5} + 245\nu^{4} - 63\nu^{3} - 670\nu^{2} - 181\nu + 148 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} - \beta_{3} - 2\beta_{2} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{6} - \beta_{5} - 4\beta_{4} - \beta_{2} + 13\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 17\beta_{6} + 11\beta_{5} - 4\beta_{4} - 18\beta_{3} - 29\beta_{2} + 18\beta _1 + 59 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 40\beta_{6} - 17\beta_{5} - 70\beta_{4} - 11\beta_{3} - 29\beta_{2} + 188\beta _1 + 57 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 269\beta_{6} + 137\beta_{5} - 102\beta_{4} - 275\beta_{3} - 428\beta_{2} + 330\beta _1 + 831 \) 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
2.01737
−3.57857
0.369925
−0.718922
4.01465
−1.31546
−0.788998
−1.00000 0 1.00000 −3.79108 0 −2.04653 −1.00000 0 3.79108
1.2 −1.00000 0 1.00000 −3.27125 0 3.11004 −1.00000 0 3.27125
1.3 −1.00000 0 1.00000 −2.69148 0 2.17419 −1.00000 0 2.69148
1.4 −1.00000 0 1.00000 −2.18886 0 −2.20076 −1.00000 0 2.18886
1.5 −1.00000 0 1.00000 0.603045 0 4.55050 −1.00000 0 −0.603045
1.6 −1.00000 0 1.00000 1.60399 0 −4.86544 −1.00000 0 −1.60399
1.7 −1.00000 0 1.00000 3.73564 0 2.27800 −1.00000 0 −3.73564
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.v 7
3.b odd 2 1 1338.2.a.j ✓ 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1338.2.a.j ✓ 7 3.b odd 2 1
4014.2.a.v 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4014))\):

\( T_{5}^{7} + 6T_{5}^{6} - 9T_{5}^{5} - 105T_{5}^{4} - 91T_{5}^{3} + 316T_{5}^{2} + 304T_{5} - 264 \) Copy content Toggle raw display
\( T_{7}^{7} - 3T_{7}^{6} - 32T_{7}^{5} + 101T_{7}^{4} + 224T_{7}^{3} - 736T_{7}^{2} - 448T_{7} + 1536 \) Copy content Toggle raw display
\( T_{11}^{7} - T_{11}^{6} - 48T_{11}^{5} + 8T_{11}^{4} + 624T_{11}^{3} + 305T_{11}^{2} - 2384T_{11} - 2512 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{7} \) Copy content Toggle raw display
$3$ \( T^{7} \) Copy content Toggle raw display
$5$ \( T^{7} + 6 T^{6} + \cdots - 264 \) Copy content Toggle raw display
$7$ \( T^{7} - 3 T^{6} + \cdots + 1536 \) Copy content Toggle raw display
$11$ \( T^{7} - T^{6} + \cdots - 2512 \) Copy content Toggle raw display
$13$ \( T^{7} - 8 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{7} + 16 T^{6} + \cdots + 6336 \) Copy content Toggle raw display
$19$ \( T^{7} - 2 T^{6} + \cdots - 6064 \) Copy content Toggle raw display
$23$ \( T^{7} + 8 T^{6} + \cdots - 18112 \) Copy content Toggle raw display
$29$ \( T^{7} + 4 T^{6} + \cdots - 1008 \) Copy content Toggle raw display
$31$ \( T^{7} - 11 T^{6} + \cdots - 15488 \) Copy content Toggle raw display
$37$ \( T^{7} - 17 T^{6} + \cdots - 124128 \) Copy content Toggle raw display
$41$ \( T^{7} + 18 T^{6} + \cdots + 412192 \) Copy content Toggle raw display
$43$ \( T^{7} + T^{6} + \cdots + 537552 \) Copy content Toggle raw display
$47$ \( T^{7} + 5 T^{6} + \cdots - 1909248 \) Copy content Toggle raw display
$53$ \( T^{7} + 6 T^{6} + \cdots - 97956 \) Copy content Toggle raw display
$59$ \( T^{7} - 7 T^{6} + \cdots - 59056 \) Copy content Toggle raw display
$61$ \( T^{7} - 24 T^{6} + \cdots + 5464 \) Copy content Toggle raw display
$67$ \( T^{7} + 4 T^{6} + \cdots - 13872 \) Copy content Toggle raw display
$71$ \( T^{7} - 7 T^{6} + \cdots - 512 \) Copy content Toggle raw display
$73$ \( T^{7} - 28 T^{6} + \cdots - 8586 \) Copy content Toggle raw display
$79$ \( T^{7} - 275 T^{5} + \cdots + 1214688 \) Copy content Toggle raw display
$83$ \( T^{7} - 2 T^{6} + \cdots + 2585664 \) Copy content Toggle raw display
$89$ \( T^{7} + 5 T^{6} + \cdots + 119008 \) Copy content Toggle raw display
$97$ \( T^{7} - 23 T^{6} + \cdots + 1877728 \) Copy content Toggle raw display
show more
show less