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Properties

Label 1680.2.f.g
Level $1680$
Weight $2$
Character orbit 1680.f
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(881,1680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1680.881");
 
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.f (of order \(2\), degree \(1\), 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: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

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

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + q^{5} + (\beta_{2} + 2) q^{7} + ( - \beta_{3} + \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + q^{5} + (\beta_{2} + 2) q^{7} + ( - \beta_{3} + \beta_{2} + 1) q^{9} + ( - \beta_{3} - \beta_1) q^{11} + ( - 3 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{13} - \beta_1 q^{15} + (\beta_{3} - \beta_1 + 2) q^{17} - 2 \beta_{2} q^{19} + ( - 2 \beta_{3} - \beta_{2} - 3 \beta_1 - 1) q^{21} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{23} + q^{25} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 4) q^{27} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{29} + 2 \beta_{2} q^{31} + ( - \beta_{3} + \beta_{2} - 2) q^{33} + (\beta_{2} + 2) q^{35} + ( - 2 \beta_{3} + 2 \beta_1 - 2) q^{37} + (\beta_{3} + 5 \beta_{2} + 2 \beta_1 - 4) q^{39} - 6 q^{41} + ( - 2 \beta_{3} + 2 \beta_1) q^{43} + ( - \beta_{3} + \beta_{2} + 1) q^{45} + ( - \beta_{3} + \beta_1 + 4) q^{47} + (4 \beta_{2} + 1) q^{49} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{51} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{53} + ( - \beta_{3} - \beta_1) q^{55} + (4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{57} + (2 \beta_{3} - 2 \beta_1 + 4) q^{59} - 4 \beta_{2} q^{61} + ( - \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 2) q^{63} + ( - 3 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{65} + ( - 2 \beta_{3} + 2 \beta_1) q^{67} + (6 \beta_{3} + 2 \beta_1 + 6) q^{69} + ( - 4 \beta_{3} - 6 \beta_{2} - 4 \beta_1) q^{71} - 4 \beta_{2} q^{73} - \beta_1 q^{75} + ( - 3 \beta_{3} - \beta_1 - 2) q^{77} + (\beta_{3} - \beta_1) q^{79} + (3 \beta_{2} + 5 \beta_1 - 3) q^{81} + (4 \beta_{3} - 4 \beta_1 + 8) q^{83} + (\beta_{3} - \beta_1 + 2) q^{85} + ( - 3 \beta_{3} - 3 \beta_{2} - 2 \beta_1) q^{87} + (2 \beta_{3} - 2 \beta_1 + 10) q^{89} + ( - 9 \beta_{3} - 4 \beta_{2} - 3 \beta_1) q^{91} + ( - 4 \beta_{3} - 2 \beta_{2} + \cdots - 2) q^{93}+ \cdots + ( - 2 \beta_{3} - \beta_{2} + \beta_1 - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} + 4 q^{5} + 8 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} + 4 q^{5} + 8 q^{7} + 5 q^{9} - q^{15} + 6 q^{17} - 5 q^{21} + 4 q^{25} - 16 q^{27} - 7 q^{33} + 8 q^{35} - 4 q^{37} - 15 q^{39} - 24 q^{41} + 4 q^{43} + 5 q^{45} + 18 q^{47} + 4 q^{49} + 15 q^{51} + 6 q^{57} + 12 q^{59} - 5 q^{63} + 4 q^{67} + 20 q^{69} - q^{75} - 6 q^{77} - 2 q^{79} - 7 q^{81} + 24 q^{83} + 6 q^{85} + q^{87} + 36 q^{89} + 6 q^{91} - 6 q^{93} - 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - \nu^{2} - 2\nu - 3 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + \beta_{2} + 2\beta _1 + 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
1.68614 + 0.396143i
1.68614 − 0.396143i
−1.18614 + 1.26217i
−1.18614 − 1.26217i
0 −1.68614 − 0.396143i 0 1.00000 0 2.00000 + 1.73205i 0 2.68614 + 1.33591i 0
881.2 0 −1.68614 + 0.396143i 0 1.00000 0 2.00000 − 1.73205i 0 2.68614 − 1.33591i 0
881.3 0 1.18614 − 1.26217i 0 1.00000 0 2.00000 − 1.73205i 0 −0.186141 − 2.99422i 0
881.4 0 1.18614 + 1.26217i 0 1.00000 0 2.00000 + 1.73205i 0 −0.186141 + 2.99422i 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
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.2.f.g 4
3.b odd 2 1 1680.2.f.h 4
4.b odd 2 1 105.2.b.d yes 4
7.b odd 2 1 1680.2.f.h 4
12.b even 2 1 105.2.b.c ✓ 4
20.d odd 2 1 525.2.b.e 4
20.e even 4 2 525.2.g.d 8
21.c even 2 1 inner 1680.2.f.g 4
28.d even 2 1 105.2.b.c ✓ 4
28.f even 6 1 735.2.s.h 4
28.f even 6 1 735.2.s.i 4
28.g odd 6 1 735.2.s.g 4
28.g odd 6 1 735.2.s.j 4
60.h even 2 1 525.2.b.g 4
60.l odd 4 2 525.2.g.e 8
84.h odd 2 1 105.2.b.d yes 4
84.j odd 6 1 735.2.s.g 4
84.j odd 6 1 735.2.s.j 4
84.n even 6 1 735.2.s.h 4
84.n even 6 1 735.2.s.i 4
140.c even 2 1 525.2.b.g 4
140.j odd 4 2 525.2.g.e 8
420.o odd 2 1 525.2.b.e 4
420.w even 4 2 525.2.g.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.b.c ✓ 4 12.b even 2 1
105.2.b.c ✓ 4 28.d even 2 1
105.2.b.d yes 4 4.b odd 2 1
105.2.b.d yes 4 84.h odd 2 1
525.2.b.e 4 20.d odd 2 1
525.2.b.e 4 420.o odd 2 1
525.2.b.g 4 60.h even 2 1
525.2.b.g 4 140.c even 2 1
525.2.g.d 8 20.e even 4 2
525.2.g.d 8 420.w even 4 2
525.2.g.e 8 60.l odd 4 2
525.2.g.e 8 140.j odd 4 2
735.2.s.g 4 28.g odd 6 1
735.2.s.g 4 84.j odd 6 1
735.2.s.h 4 28.f even 6 1
735.2.s.h 4 84.n even 6 1
735.2.s.i 4 28.f even 6 1
735.2.s.i 4 84.n even 6 1
735.2.s.j 4 28.g odd 6 1
735.2.s.j 4 84.j odd 6 1
1680.2.f.g 4 1.a even 1 1 trivial
1680.2.f.g 4 21.c even 2 1 inner
1680.2.f.h 4 3.b odd 2 1
1680.2.f.h 4 7.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 \(S_{2}^{\mathrm{new}}(1680, [\chi])\):

\( T_{11}^{4} + 7T_{11}^{2} + 4 \) Copy content Toggle raw display
\( T_{17}^{2} - 3T_{17} - 6 \) Copy content Toggle raw display
\( T_{41} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + T^{3} - 2 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 4 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 7T^{2} + 4 \) Copy content Toggle raw display
$13$ \( T^{4} + 51T^{2} + 576 \) Copy content Toggle raw display
$17$ \( (T^{2} - 3 T - 6)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 76T^{2} + 256 \) Copy content Toggle raw display
$29$ \( T^{4} + 19T^{2} + 16 \) Copy content Toggle raw display
$31$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2 T - 32)^{2} \) Copy content Toggle raw display
$41$ \( (T + 6)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T - 32)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 9 T + 12)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 76T^{2} + 256 \) Copy content Toggle raw display
$59$ \( (T^{2} - 6 T - 24)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 2 T - 32)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 184T^{2} + 16 \) Copy content Toggle raw display
$73$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + T - 8)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 12 T - 96)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 18 T + 48)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 123T^{2} + 144 \) Copy content Toggle raw display
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