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Properties

Label 1680.2.bg.s
Level $1680$
Weight $2$
Character orbit 1680.bg
Analytic conductor $13.415$
Analytic rank $0$
Dimension $4$
CM no
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: \(\Q(\sqrt{-3}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 840)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{3} - \beta_{2} q^{5} + (\beta_{3} + \beta_1) q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{3} - \beta_{2} q^{5} + (\beta_{3} + \beta_1) q^{7} + \beta_{2} q^{9} + (\beta_{2} + \beta_1 + 1) q^{11} + (\beta_{3} - 2) q^{13} + q^{15} + (3 \beta_{2} - \beta_1 + 3) q^{17} - \beta_{2} q^{19} + \beta_{3} q^{21} + (\beta_{3} + \beta_{2} + \beta_1) q^{23} + ( - \beta_{2} - 1) q^{25} - q^{27} + (\beta_{3} + 1) q^{29} + ( - 5 \beta_{2} - 5) q^{31} + (\beta_{3} + \beta_{2} + \beta_1) q^{33} + \beta_1 q^{35} + (\beta_{3} + \beta_1) q^{37} + ( - 2 \beta_{2} - \beta_1 - 2) q^{39} + (3 \beta_{3} + 1) q^{41} + ( - \beta_{3} + 2) q^{43} + (\beta_{2} + 1) q^{45} + (4 \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{47} + ( - 7 \beta_{2} - 7) q^{49} + ( - \beta_{3} + 3 \beta_{2} - \beta_1) q^{51} + ( - 6 \beta_{2} - 2 \beta_1 - 6) q^{53} + ( - \beta_{3} + 1) q^{55} + q^{57} + ( - 5 \beta_{2} + \beta_1 - 5) q^{59} + (4 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{61} - \beta_1 q^{63} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{65} + (2 \beta_{2} + 3 \beta_1 + 2) q^{67} + (\beta_{3} - 1) q^{69} + ( - \beta_{3} + 9) q^{71} + (10 \beta_{2} - \beta_1 + 10) q^{73} - \beta_{2} q^{75} + (\beta_{3} - 7) q^{77} + 13 \beta_{2} q^{79} + ( - \beta_{2} - 1) q^{81} + (3 \beta_{3} - 5) q^{83} + (\beta_{3} + 3) q^{85} + (\beta_{2} - \beta_1 + 1) q^{87} + ( - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{89} + ( - 2 \beta_{3} - 7 \beta_{2} - 2 \beta_1) q^{91} - 5 \beta_{2} q^{93} + ( - \beta_{2} - 1) q^{95} + ( - 4 \beta_{3} - 4) q^{97} + (\beta_{3} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 2 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 2 q^{5} - 2 q^{9} + 2 q^{11} - 8 q^{13} + 4 q^{15} + 6 q^{17} + 2 q^{19} - 2 q^{23} - 2 q^{25} - 4 q^{27} + 4 q^{29} - 10 q^{31} - 2 q^{33} - 4 q^{39} + 4 q^{41} + 8 q^{43} + 2 q^{45} - 4 q^{47} - 14 q^{49} - 6 q^{51} - 12 q^{53} + 4 q^{55} + 4 q^{57} - 10 q^{59} + 8 q^{61} - 4 q^{65} + 4 q^{67} - 4 q^{69} + 36 q^{71} + 20 q^{73} + 2 q^{75} - 28 q^{77} - 26 q^{79} - 2 q^{81} - 20 q^{83} + 12 q^{85} + 2 q^{87} - 6 q^{89} + 14 q^{91} + 10 q^{93} - 2 q^{95} - 16 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 7 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{3} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(1\) \(1\) \(1\)

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
1.32288 + 2.29129i
−1.32288 − 2.29129i
1.32288 − 2.29129i
−1.32288 + 2.29129i
0 0.500000 + 0.866025i 0 0.500000 − 0.866025i 0 −1.32288 + 2.29129i 0 −0.500000 + 0.866025i 0
961.2 0 0.500000 + 0.866025i 0 0.500000 − 0.866025i 0 1.32288 − 2.29129i 0 −0.500000 + 0.866025i 0
1201.1 0 0.500000 − 0.866025i 0 0.500000 + 0.866025i 0 −1.32288 − 2.29129i 0 −0.500000 − 0.866025i 0
1201.2 0 0.500000 − 0.866025i 0 0.500000 + 0.866025i 0 1.32288 + 2.29129i 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.s 4
4.b odd 2 1 840.2.bg.h ✓ 4
7.c even 3 1 inner 1680.2.bg.s 4
12.b even 2 1 2520.2.bi.j 4
28.f even 6 1 5880.2.a.bo 2
28.g odd 6 1 840.2.bg.h ✓ 4
28.g odd 6 1 5880.2.a.bq 2
84.n even 6 1 2520.2.bi.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.bg.h ✓ 4 4.b odd 2 1
840.2.bg.h ✓ 4 28.g odd 6 1
1680.2.bg.s 4 1.a even 1 1 trivial
1680.2.bg.s 4 7.c even 3 1 inner
2520.2.bi.j 4 12.b even 2 1
2520.2.bi.j 4 84.n even 6 1
5880.2.a.bo 2 28.f even 6 1
5880.2.a.bq 2 28.g odd 6 1

Hecke kernels

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

\( T_{11}^{4} - 2T_{11}^{3} + 10T_{11}^{2} + 12T_{11} + 36 \) Copy content Toggle raw display
\( T_{13}^{2} + 4T_{13} - 3 \) Copy content Toggle raw display
\( T_{17}^{4} - 6T_{17}^{3} + 34T_{17}^{2} - 12T_{17} + 4 \) 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
$7$ \( T^{4} + 7T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T - 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 2 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$29$ \( (T^{2} - 2 T - 6)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 7T^{2} + 49 \) Copy content Toggle raw display
$41$ \( (T^{2} - 2 T - 62)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 4 T - 3)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 4 T^{3} + \cdots + 11664 \) Copy content Toggle raw display
$53$ \( T^{4} + 12 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$59$ \( T^{4} + 10 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$61$ \( T^{4} - 8 T^{3} + \cdots + 9216 \) Copy content Toggle raw display
$67$ \( T^{4} - 4 T^{3} + \cdots + 3481 \) Copy content Toggle raw display
$71$ \( (T^{2} - 18 T + 74)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 20 T^{3} + \cdots + 8649 \) Copy content Toggle raw display
$79$ \( (T^{2} + 13 T + 169)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 10 T - 38)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 6 T^{3} + \cdots + 2916 \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T - 96)^{2} \) Copy content Toggle raw display
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