Properties

Label 1672.2.a.j
Level $1672$
Weight $2$
Character orbit 1672.a
Self dual yes
Analytic conductor $13.351$
Analytic rank $0$
Dimension $6$
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] = [1672,2,Mod(1,1672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1672, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1672.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1672 = 2^{3} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1672.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: \(13.3509872180\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.576096652.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 8x^{4} + 11x^{3} + 16x^{2} - 6x - 4 \) 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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + \beta_{4} q^{5} + \beta_{3} q^{7} + (\beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + \beta_{4} q^{5} + \beta_{3} q^{7} + (\beta_{2} - \beta_1 + 1) q^{9} + q^{11} + \beta_{3} q^{13} + (\beta_{4} + \beta_{2} + 1) q^{15} + ( - \beta_{4} - \beta_1 + 1) q^{17} + q^{19} + ( - \beta_{5} - \beta_{2} + 1) q^{21} + (\beta_{4} - \beta_{3} - \beta_{2} + 1) q^{23} + (\beta_{5} - \beta_{2} - \beta_1 + 1) q^{25} + (\beta_{4} - \beta_{3} + 2 \beta_{2} + 2) q^{27} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{29} + (\beta_{4} - 2 \beta_{3} - \beta_{2} + 1) q^{31} + ( - \beta_1 + 1) q^{33} + ( - \beta_{2} + 1) q^{35} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{37} + ( - \beta_{5} - \beta_{2} + 1) q^{39} + ( - \beta_{4} + \beta_{3} + \beta_1 + 1) q^{41} + (\beta_{5} - \beta_{3} + \beta_{2} + 2 \beta_1 + 3) q^{43} + ( - \beta_{4} - \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 3) q^{45} + (2 \beta_{3} + 2 \beta_1 + 2) q^{47} + ( - 2 \beta_{5} - \beta_{3} - \beta_{2}) q^{49} + ( - \beta_{4} - \beta_1 + 3) q^{51} + ( - \beta_{5} + 2 \beta_1 + 2) q^{53} + \beta_{4} q^{55} + ( - \beta_1 + 1) q^{57} + (\beta_{5} - 2 \beta_{4} + \beta_{3} + 3 \beta_{2} - \beta_1 + 4) q^{59} + (\beta_{5} - \beta_{4} + 2 \beta_{2} + \beta_1 - 1) q^{61} + ( - \beta_{5} - \beta_{4} - \beta_{2} + \beta_1) q^{63} + ( - \beta_{2} + 1) q^{65} + ( - \beta_{5} - \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 3) q^{67} + (\beta_{5} + \beta_{3} + \beta_{2} + \beta_1) q^{69} + (\beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 3 \beta_1 - 1) q^{71} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{73} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1 + 3) q^{75} + \beta_{3} q^{77} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + 3 \beta_1 - 1) q^{79} + (\beta_{5} + 3 \beta_{4} - 2 \beta_{3} + \beta_{2} - 3 \beta_1 + 1) q^{81} + ( - 3 \beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_1 + 2) q^{83} + ( - \beta_{5} + \beta_{4} + 2 \beta_{2} + \beta_1 - 5) q^{85} + ( - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{87} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} - 5 \beta_1 + 4) q^{89} + ( - 2 \beta_{5} - \beta_{3} - \beta_{2} + 7) q^{91} + (2 \beta_{5} + \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{93} + \beta_{4} q^{95} + (2 \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{97} + (\beta_{2} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} + q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{3} + q^{5} + 4 q^{9} + 6 q^{11} + 7 q^{15} + 3 q^{17} + 6 q^{19} + 7 q^{21} + 7 q^{23} + 3 q^{25} + 13 q^{27} - 4 q^{29} + 7 q^{31} + 4 q^{33} + 6 q^{35} - 2 q^{37} + 7 q^{39} + 7 q^{41} + 21 q^{43} + 11 q^{45} + 16 q^{47} + 2 q^{49} + 15 q^{51} + 17 q^{53} + q^{55} + 4 q^{57} + 19 q^{59} - 6 q^{61} + 2 q^{63} + 6 q^{65} + 14 q^{67} + q^{69} + q^{71} - 5 q^{73} + 18 q^{75} + 2 q^{81} + 11 q^{83} - 26 q^{85} + 14 q^{87} + 12 q^{89} + 44 q^{91} - 6 q^{93} + q^{95} + 14 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 8x^{4} + 11x^{3} + 16x^{2} - 6x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 6\nu^{3} + 9\nu^{2} + 4\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 8\nu^{3} + 11\nu^{2} + 14\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} + 4\nu^{3} - 23\nu^{2} + 2\nu + 14 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + \beta_{3} + \beta_{2} + 6\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} - \beta_{4} + 2\beta_{3} + 8\beta_{2} + 10\beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} - 8\beta_{4} + 12\beta_{3} + 13\beta_{2} + 43\beta _1 + 21 \) 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
2.89799
2.20649
0.634105
−0.393583
−1.13326
−2.21174
0 −1.89799 0 −1.20786 0 1.24216 0 0.602364 0
1.2 0 −1.20649 0 −0.300070 0 −4.45862 0 −1.54439 0
1.3 0 0.365895 0 3.51994 0 1.20230 0 −2.86612 0
1.4 0 1.39358 0 −3.68793 0 −0.935893 0 −1.05793 0
1.5 0 2.13326 0 0.368439 0 4.29504 0 1.55080 0
1.6 0 3.21174 0 2.30747 0 −1.34499 0 7.31527 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(-1\)
\(19\) \(-1\)

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 1672.2.a.j 6
4.b odd 2 1 3344.2.a.v 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1672.2.a.j 6 1.a even 1 1 trivial
3344.2.a.v 6 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1672))\):

\( T_{3}^{6} - 4T_{3}^{5} - 3T_{3}^{4} + 21T_{3}^{3} - 4T_{3}^{2} - 23T_{3} + 8 \) Copy content Toggle raw display
\( T_{5}^{6} - T_{5}^{5} - 16T_{5}^{4} + 15T_{5}^{3} + 37T_{5}^{2} - 4T_{5} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 4 T^{5} - 3 T^{4} + 21 T^{3} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( T^{6} - T^{5} - 16 T^{4} + 15 T^{3} + \cdots - 4 \) Copy content Toggle raw display
$7$ \( T^{6} - 22 T^{4} + 3 T^{3} + 56 T^{2} + \cdots - 36 \) Copy content Toggle raw display
$11$ \( (T - 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 22 T^{4} + 3 T^{3} + 56 T^{2} + \cdots - 36 \) Copy content Toggle raw display
$17$ \( T^{6} - 3 T^{5} - 16 T^{4} + 22 T^{3} + \cdots - 16 \) Copy content Toggle raw display
$19$ \( (T - 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 7 T^{5} - 23 T^{4} + 151 T^{3} + \cdots + 128 \) Copy content Toggle raw display
$29$ \( T^{6} + 4 T^{5} - 118 T^{4} + \cdots + 7444 \) Copy content Toggle raw display
$31$ \( T^{6} - 7 T^{5} - 78 T^{4} + \cdots + 5684 \) Copy content Toggle raw display
$37$ \( T^{6} + 2 T^{5} - 35 T^{4} + 6 T^{3} + \cdots - 128 \) Copy content Toggle raw display
$41$ \( T^{6} - 7 T^{5} - 21 T^{4} + 184 T^{3} + \cdots - 256 \) Copy content Toggle raw display
$43$ \( T^{6} - 21 T^{5} + 29 T^{4} + \cdots - 99328 \) Copy content Toggle raw display
$47$ \( T^{6} - 16 T^{5} + 8 T^{4} + \cdots - 4096 \) Copy content Toggle raw display
$53$ \( T^{6} - 17 T^{5} + 26 T^{4} + \cdots - 3584 \) Copy content Toggle raw display
$59$ \( T^{6} - 19 T^{5} - 95 T^{4} + \cdots - 56448 \) Copy content Toggle raw display
$61$ \( T^{6} + 6 T^{5} - 122 T^{4} + \cdots - 17792 \) Copy content Toggle raw display
$67$ \( T^{6} - 14 T^{5} - 167 T^{4} + \cdots + 105344 \) Copy content Toggle raw display
$71$ \( T^{6} - T^{5} - 204 T^{4} + \cdots + 76972 \) Copy content Toggle raw display
$73$ \( T^{6} + 5 T^{5} - 172 T^{4} + \cdots + 3136 \) Copy content Toggle raw display
$79$ \( T^{6} - 166 T^{4} - 444 T^{3} + \cdots + 48896 \) Copy content Toggle raw display
$83$ \( T^{6} - 11 T^{5} - 239 T^{4} + \cdots - 605504 \) Copy content Toggle raw display
$89$ \( T^{6} - 12 T^{5} - 215 T^{4} + \cdots - 127744 \) Copy content Toggle raw display
$97$ \( T^{6} - 14 T^{5} - 247 T^{4} + \cdots + 12544 \) Copy content Toggle raw display
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