Properties

Label 6016.2.a.h
Level $6016$
Weight $2$
Character orbit 6016.a
Self dual yes
Analytic conductor $48.038$
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] = [6016,2,Mod(1,6016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6016, 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("6016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6016 = 2^{7} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6016.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: \(48.0380018560\)
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: \( x^{8} - 2x^{7} - 10x^{6} + 20x^{5} + 19x^{4} - 34x^{3} - 10x^{2} + 6x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{2} + 1) q^{5} + (\beta_{6} + 1) q^{7} + ( - \beta_{4} + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{2} + 1) q^{5} + (\beta_{6} + 1) q^{7} + ( - \beta_{4} + \beta_{3}) q^{9} + ( - \beta_{4} + 1) q^{11} + ( - \beta_{3} + \beta_1 - 1) q^{13} + ( - \beta_{6} + \beta_{5} + \beta_{2} + \beta_1) q^{15} + ( - \beta_{7} + \beta_{5} + \beta_{4} - 2 \beta_{3} - 2) q^{17} + ( - \beta_{7} + \beta_{5} - \beta_1 - 1) q^{19} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{21} + (\beta_{7} - \beta_{5} - \beta_{4} + \beta_1) q^{23} + (2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 3) q^{25} + (\beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} + 1) q^{27} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} - \beta_1 + 3) q^{29} + ( - 2 \beta_{7} - \beta_{6} + \beta_1) q^{31} + ( - \beta_{3} + 3 \beta_1 - 1) q^{33} + (2 \beta_{7} - \beta_{5} + \beta_{4} + 3 \beta_{2} + 5) q^{35} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} - 1) q^{37} + ( - \beta_{7} - \beta_{6} - 2 \beta_{4} - \beta_{2} - 2 \beta_1 + 1) q^{39} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 3) q^{41} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4}) q^{43} + (\beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + \beta_1 + 3) q^{45} - q^{47} + (\beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} + 1) q^{49} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{4} - 4 \beta_1 + 1) q^{51} + (\beta_{6} - 2 \beta_{5} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{53} + ( - \beta_{6} - \beta_{3} + 2 \beta_{2} + 3) q^{55} + (\beta_{7} + \beta_{4} + 2 \beta_{2} + \beta_1 + 1) q^{57} + ( - \beta_{7} + 3 \beta_{6} + 2 \beta_{4} - \beta_{3} - 2 \beta_1 + 2) q^{59} + ( - 2 \beta_{7} - \beta_{5} - 2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{61} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + 3 \beta_1 + 1) q^{63} + ( - \beta_{7} - \beta_{6} - 2 \beta_{3} - 2) q^{65} + (\beta_{7} + \beta_{5} - \beta_{4} - 3 \beta_1 + 2) q^{67} + ( - \beta_{7} - \beta_{4} - \beta_{3} - 2 \beta_{2} - 2) q^{69} + (\beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 5) q^{71} + (2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{73} + ( - \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + \beta_1 + 3) q^{75} + (\beta_{6} - \beta_{5} + \beta_{4} + \beta_1 + 4) q^{77} + (\beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 3) q^{79} + ( - \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_1) q^{81} + ( - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{83} + ( - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 3) q^{85} + (2 \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 6 \beta_1 + 3) q^{87} + (2 \beta_{6} + \beta_{4} + \beta_{3} + 2 \beta_1 - 3) q^{89} + ( - \beta_{7} - \beta_{2} - \beta_1 + 1) q^{91} + (\beta_{2} + 2 \beta_1 + 5) q^{93} + (2 \beta_{7} - \beta_{6} - \beta_{4} + 2 \beta_{3} - \beta_{2} + 3 \beta_1 + 2) q^{95} + (2 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 3) q^{97} + ( - \beta_{7} - \beta_{6} - \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} + 4 q^{5} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} + 4 q^{5} + 6 q^{7} + 12 q^{11} - 2 q^{13} + 2 q^{15} - 8 q^{17} - 6 q^{19} + 6 q^{21} + 2 q^{23} + 8 q^{25} - 4 q^{27} + 24 q^{29} + 8 q^{31} + 2 q^{33} + 18 q^{35} - 2 q^{37} + 20 q^{39} - 8 q^{41} - 6 q^{43} + 20 q^{45} - 8 q^{47} - 4 q^{49} + 14 q^{51} + 4 q^{53} + 22 q^{55} - 4 q^{57} + 4 q^{59} + 20 q^{61} + 8 q^{63} - 4 q^{65} + 14 q^{67} + 2 q^{69} + 16 q^{71} - 16 q^{73} + 28 q^{75} + 26 q^{77} + 24 q^{79} - 4 q^{81} + 10 q^{83} - 4 q^{85} + 10 q^{87} - 32 q^{89} + 12 q^{91} + 40 q^{93} + 20 q^{95} + 16 q^{97} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 10x^{6} + 20x^{5} + 19x^{4} - 34x^{3} - 10x^{2} + 6x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 4\nu^{6} + 11\nu^{5} - 42\nu^{4} - 25\nu^{3} + 93\nu^{2} + 13\nu - 32 ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + \nu^{6} + 11\nu^{5} - 9\nu^{4} - 28\nu^{3} + 9\nu^{2} + 19\nu - 2 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + \nu^{6} + 11\nu^{5} - 9\nu^{4} - 28\nu^{3} + 6\nu^{2} + 19\nu + 7 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{7} + 8\nu^{6} + 13\nu^{5} - 75\nu^{4} + 31\nu^{3} + 105\nu^{2} - 73\nu - 10 ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{7} + 11\nu^{6} + 46\nu^{5} - 111\nu^{4} - 53\nu^{3} + 195\nu^{2} - 34\nu - 43 ) / 9 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} - 2\nu^{6} - 10\nu^{5} + 20\nu^{4} + 19\nu^{3} - 34\nu^{2} - 10\nu + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} + 6\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} + \beta_{5} - 7\beta_{4} + 8\beta_{3} - 2\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{7} + 8\beta_{6} + 11\beta_{4} - \beta_{3} + 11\beta_{2} + 41\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{7} - 12\beta_{6} + 11\beta_{5} - 50\beta_{4} + 59\beta_{3} + 2\beta_{2} - 26\beta _1 + 123 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 70\beta_{7} + 57\beta_{6} + 2\beta_{5} + 97\beta_{4} - 18\beta_{3} + 95\beta_{2} + 294\beta _1 + 24 \) Copy content Toggle raw display

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
−2.78948
−1.33694
−0.483326
−0.149653
0.422477
1.55261
2.12617
2.65815
0 −2.78948 0 0.547094 0 −0.876372 0 4.78119 0
1.2 0 −1.33694 0 3.87774 0 4.05948 0 −1.21259 0
1.3 0 −0.483326 0 −0.806701 0 2.98560 0 −2.76640 0
1.4 0 −0.149653 0 −2.53341 0 −2.71400 0 −2.97760 0
1.5 0 0.422477 0 −0.440372 0 −2.26916 0 −2.82151 0
1.6 0 1.55261 0 1.91860 0 0.0287238 0 −0.589408 0
1.7 0 2.12617 0 −2.49763 0 3.36022 0 1.52059 0
1.8 0 2.65815 0 3.93469 0 1.42550 0 4.06574 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(47\) \(1\)

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 6016.2.a.h yes 8
4.b odd 2 1 6016.2.a.f yes 8
8.b even 2 1 6016.2.a.e 8
8.d odd 2 1 6016.2.a.g yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6016.2.a.e 8 8.b even 2 1
6016.2.a.f yes 8 4.b odd 2 1
6016.2.a.g yes 8 8.d odd 2 1
6016.2.a.h yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6016))\):

\( T_{3}^{8} - 2T_{3}^{7} - 10T_{3}^{6} + 20T_{3}^{5} + 19T_{3}^{4} - 34T_{3}^{3} - 10T_{3}^{2} + 6T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{8} - 4T_{5}^{7} - 16T_{5}^{6} + 54T_{5}^{5} + 92T_{5}^{4} - 172T_{5}^{3} - 156T_{5}^{2} + 52T_{5} + 36 \) Copy content Toggle raw display
\( T_{13}^{8} + 2T_{13}^{7} - 38T_{13}^{6} - 86T_{13}^{5} + 122T_{13}^{4} + 180T_{13}^{3} - 216T_{13}^{2} + 40T_{13} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} - 10 T^{6} + 20 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{7} - 16 T^{6} + 54 T^{5} + \cdots + 36 \) Copy content Toggle raw display
$7$ \( T^{8} - 6 T^{7} - 8 T^{6} + 86 T^{5} + \cdots - 9 \) Copy content Toggle raw display
$11$ \( T^{8} - 12 T^{7} + 34 T^{6} + \cdots + 108 \) Copy content Toggle raw display
$13$ \( T^{8} + 2 T^{7} - 38 T^{6} - 86 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$17$ \( T^{8} + 8 T^{7} - 54 T^{6} - 532 T^{5} + \cdots + 813 \) Copy content Toggle raw display
$19$ \( T^{8} + 6 T^{7} - 24 T^{6} - 138 T^{5} + \cdots - 876 \) Copy content Toggle raw display
$23$ \( T^{8} - 2 T^{7} - 62 T^{6} + \cdots + 5756 \) Copy content Toggle raw display
$29$ \( T^{8} - 24 T^{7} + 146 T^{6} + \cdots - 235868 \) Copy content Toggle raw display
$31$ \( T^{8} - 8 T^{7} - 104 T^{6} + \cdots - 42300 \) Copy content Toggle raw display
$37$ \( T^{8} + 2 T^{7} - 178 T^{6} + \cdots - 495945 \) Copy content Toggle raw display
$41$ \( T^{8} + 8 T^{7} - 106 T^{6} + \cdots - 2428 \) Copy content Toggle raw display
$43$ \( T^{8} + 6 T^{7} - 94 T^{6} + \cdots - 241884 \) Copy content Toggle raw display
$47$ \( (T + 1)^{8} \) Copy content Toggle raw display
$53$ \( T^{8} - 4 T^{7} - 148 T^{6} + \cdots - 99081 \) Copy content Toggle raw display
$59$ \( T^{8} - 4 T^{7} - 262 T^{6} + \cdots + 1889549 \) Copy content Toggle raw display
$61$ \( T^{8} - 20 T^{7} - 96 T^{6} + \cdots + 675827 \) Copy content Toggle raw display
$67$ \( T^{8} - 14 T^{7} - 114 T^{6} + \cdots - 28900 \) Copy content Toggle raw display
$71$ \( T^{8} - 16 T^{7} - 98 T^{6} + \cdots - 20817 \) Copy content Toggle raw display
$73$ \( T^{8} + 16 T^{7} - 240 T^{6} + \cdots + 379332 \) Copy content Toggle raw display
$79$ \( T^{8} - 24 T^{7} - 62 T^{6} + \cdots + 1776807 \) Copy content Toggle raw display
$83$ \( T^{8} - 10 T^{7} - 4 T^{6} + 112 T^{5} + \cdots + 48 \) Copy content Toggle raw display
$89$ \( T^{8} + 32 T^{7} + 246 T^{6} + \cdots + 601221 \) Copy content Toggle raw display
$97$ \( T^{8} - 16 T^{7} - 2 T^{6} + \cdots - 87863 \) Copy content Toggle raw display
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