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Properties

Label 1680.2.bg.t
Level $1680$
Weight $2$
Character orbit 1680.bg
Analytic conductor $13.415$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1680,2,Mod(961,1680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1680, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1680.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.bg (of order \(3\), degree \(2\), not minimal)

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: no
Analytic conductor: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{3} - \beta_1 q^{5} + (\beta_{3} - \beta_1) q^{7} + \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{3} - \beta_1 q^{5} + (\beta_{3} - \beta_1) q^{7} + \beta_1 q^{9} + (\beta_{3} + \beta_{2}) q^{11} + ( - 2 \beta_{3} + \beta_{2} - 1) q^{13} + q^{15} + (\beta_{3} + \beta_{2}) q^{17} + 7 \beta_1 q^{19} + (\beta_{2} + 1) q^{21} + (\beta_{3} - 2 \beta_{2}) q^{23} + ( - \beta_1 - 1) q^{25} - q^{27} + ( - 2 \beta_{3} + \beta_{2} - 6) q^{29} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{31} + ( - \beta_{3} + 2 \beta_{2}) q^{33} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{35} + (\beta_{3} - 2 \beta_{2} - \beta_1) q^{37} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{39} + (2 \beta_{3} - \beta_{2}) q^{41} + (2 \beta_{3} - \beta_{2} + 1) q^{43} + (\beta_1 + 1) q^{45} + 6 \beta_1 q^{47} + (2 \beta_{3} - 2 \beta_{2} + 5 \beta_1 + 5) q^{49} + ( - \beta_{3} + 2 \beta_{2}) q^{51} + (2 \beta_{3} + 2 \beta_{2}) q^{53} + (2 \beta_{3} - \beta_{2}) q^{55} - 7 q^{57} + (\beta_{3} + \beta_{2} - 6 \beta_1 - 6) q^{59} + (2 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{61} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{63} + ( - \beta_{3} + 2 \beta_{2} + \beta_1) q^{65} + (\beta_{3} + \beta_{2} - \beta_1 - 1) q^{67} + (2 \beta_{3} - \beta_{2}) q^{69} + ( - 6 \beta_{3} + 3 \beta_{2}) q^{71} + (\beta_{3} + \beta_{2} - 5 \beta_1 - 5) q^{73} - \beta_1 q^{75} + (2 \beta_{3} - \beta_{2} + 12 \beta_1 + 6) q^{77} - 11 \beta_1 q^{79} + ( - \beta_1 - 1) q^{81} + ( - 2 \beta_{3} + \beta_{2} + 6) q^{83} + (2 \beta_{3} - \beta_{2}) q^{85} + ( - \beta_{3} - \beta_{2} - 6 \beta_1 - 6) q^{87} + ( - \beta_{3} + 2 \beta_{2} - 6 \beta_1) q^{89} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots - 12) q^{91}+ \cdots + ( - 2 \beta_{3} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 2 q^{5} + 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 2 q^{5} + 2 q^{7} - 2 q^{9} - 4 q^{13} + 4 q^{15} - 14 q^{19} + 4 q^{21} - 2 q^{25} - 4 q^{27} - 24 q^{29} - 2 q^{31} - 2 q^{35} + 2 q^{37} - 2 q^{39} + 4 q^{43} + 2 q^{45} - 12 q^{47} + 10 q^{49} - 28 q^{57} - 12 q^{59} + 8 q^{61} + 2 q^{63} - 2 q^{65} - 2 q^{67} - 10 q^{73} + 2 q^{75} + 22 q^{79} - 2 q^{81} + 24 q^{83} - 12 q^{87} + 12 q^{89} - 38 q^{91} + 2 q^{93} + 14 q^{95} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(\beta_{1}\) \(1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
−0.707107 − 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 − 1.22474i
0 0.500000 + 0.866025i 0 0.500000 − 0.866025i 0 −1.62132 − 2.09077i 0 −0.500000 + 0.866025i 0
961.2 0 0.500000 + 0.866025i 0 0.500000 − 0.866025i 0 2.62132 + 0.358719i 0 −0.500000 + 0.866025i 0
1201.1 0 0.500000 − 0.866025i 0 0.500000 + 0.866025i 0 −1.62132 + 2.09077i 0 −0.500000 − 0.866025i 0
1201.2 0 0.500000 − 0.866025i 0 0.500000 + 0.866025i 0 2.62132 − 0.358719i 0 −0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.2.bg.t 4
4.b odd 2 1 420.2.q.d ✓ 4
7.c even 3 1 inner 1680.2.bg.t 4
12.b even 2 1 1260.2.s.e 4
20.d odd 2 1 2100.2.q.k 4
20.e even 4 2 2100.2.bc.f 8
28.d even 2 1 2940.2.q.q 4
28.f even 6 1 2940.2.a.p 2
28.f even 6 1 2940.2.q.q 4
28.g odd 6 1 420.2.q.d ✓ 4
28.g odd 6 1 2940.2.a.r 2
84.j odd 6 1 8820.2.a.bf 2
84.n even 6 1 1260.2.s.e 4
84.n even 6 1 8820.2.a.bk 2
140.p odd 6 1 2100.2.q.k 4
140.w even 12 2 2100.2.bc.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.d ✓ 4 4.b odd 2 1
420.2.q.d ✓ 4 28.g odd 6 1
1260.2.s.e 4 12.b even 2 1
1260.2.s.e 4 84.n even 6 1
1680.2.bg.t 4 1.a even 1 1 trivial
1680.2.bg.t 4 7.c even 3 1 inner
2100.2.q.k 4 20.d odd 2 1
2100.2.q.k 4 140.p odd 6 1
2100.2.bc.f 8 20.e even 4 2
2100.2.bc.f 8 140.w even 12 2
2940.2.a.p 2 28.f even 6 1
2940.2.a.r 2 28.g odd 6 1
2940.2.q.q 4 28.d even 2 1
2940.2.q.q 4 28.f even 6 1
8820.2.a.bf 2 84.j odd 6 1
8820.2.a.bk 2 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1680, [\chi])\):

\( T_{11}^{4} + 18T_{11}^{2} + 324 \) Copy content Toggle raw display
\( T_{13}^{2} + 2T_{13} - 17 \) Copy content Toggle raw display
\( T_{17}^{4} + 18T_{17}^{2} + 324 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$11$ \( T^{4} + 18T^{2} + 324 \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T - 17)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 18T^{2} + 324 \) Copy content Toggle raw display
$19$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 18T^{2} + 324 \) Copy content Toggle raw display
$29$ \( (T^{2} + 12 T + 18)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2 T^{3} + \cdots + 5041 \) Copy content Toggle raw display
$37$ \( T^{4} - 2 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$41$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T - 17)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 72T^{2} + 5184 \) Copy content Toggle raw display
$59$ \( T^{4} + 12 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$61$ \( T^{4} - 8 T^{3} + \cdots + 3136 \) Copy content Toggle raw display
$67$ \( T^{4} + 2 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$71$ \( (T^{2} - 162)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 10 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$79$ \( (T^{2} - 11 T + 121)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 12 T + 18)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 12 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$97$ \( (T^{2} - 16 T - 8)^{2} \) Copy content Toggle raw display
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