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Properties

Label 1680.2.f.j
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: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 420)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

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

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} + \beta_{2} - 2\beta_1 ) / 2 \) 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
− 1.61803i
1.61803i
− 0.618034i
0.618034i
0 −0.618034 − 1.61803i 0 1.00000 0 2.61803 + 0.381966i 0 −2.23607 + 2.00000i 0
881.2 0 −0.618034 + 1.61803i 0 1.00000 0 2.61803 − 0.381966i 0 −2.23607 − 2.00000i 0
881.3 0 1.61803 − 0.618034i 0 1.00000 0 0.381966 − 2.61803i 0 2.23607 − 2.00000i 0
881.4 0 1.61803 + 0.618034i 0 1.00000 0 0.381966 + 2.61803i 0 2.23607 + 2.00000i 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.j 4
3.b odd 2 1 1680.2.f.f 4
4.b odd 2 1 420.2.d.c ✓ 4
7.b odd 2 1 1680.2.f.f 4
12.b even 2 1 420.2.d.d yes 4
20.d odd 2 1 2100.2.d.h 4
20.e even 4 1 2100.2.f.a 4
20.e even 4 1 2100.2.f.g 4
21.c even 2 1 inner 1680.2.f.j 4
28.d even 2 1 420.2.d.d yes 4
60.h even 2 1 2100.2.d.g 4
60.l odd 4 1 2100.2.f.b 4
60.l odd 4 1 2100.2.f.h 4
84.h odd 2 1 420.2.d.c ✓ 4
140.c even 2 1 2100.2.d.g 4
140.j odd 4 1 2100.2.f.b 4
140.j odd 4 1 2100.2.f.h 4
420.o odd 2 1 2100.2.d.h 4
420.w even 4 1 2100.2.f.a 4
420.w even 4 1 2100.2.f.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.d.c ✓ 4 4.b odd 2 1
420.2.d.c ✓ 4 84.h odd 2 1
420.2.d.d yes 4 12.b even 2 1
420.2.d.d yes 4 28.d even 2 1
1680.2.f.f 4 3.b odd 2 1
1680.2.f.f 4 7.b odd 2 1
1680.2.f.j 4 1.a even 1 1 trivial
1680.2.f.j 4 21.c even 2 1 inner
2100.2.d.g 4 60.h even 2 1
2100.2.d.g 4 140.c even 2 1
2100.2.d.h 4 20.d odd 2 1
2100.2.d.h 4 420.o odd 2 1
2100.2.f.a 4 20.e even 4 1
2100.2.f.a 4 420.w even 4 1
2100.2.f.b 4 60.l odd 4 1
2100.2.f.b 4 140.j odd 4 1
2100.2.f.g 4 20.e even 4 1
2100.2.f.g 4 420.w even 4 1
2100.2.f.h 4 60.l odd 4 1
2100.2.f.h 4 140.j odd 4 1

Hecke kernels

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

\( T_{11}^{4} + 28T_{11}^{2} + 16 \) Copy content Toggle raw display
\( T_{17}^{2} - 20 \) Copy content Toggle raw display
\( T_{41}^{2} - 16T_{41} + 44 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 6 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$11$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$17$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 60T^{2} + 400 \) Copy content Toggle raw display
$23$ \( T^{4} + 92T^{2} + 1936 \) Copy content Toggle raw display
$29$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$37$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 16 T + 44)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 10 T + 20)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 2 T - 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 108T^{2} + 1936 \) Copy content Toggle raw display
$59$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 192T^{2} + 4096 \) Copy content Toggle raw display
$67$ \( (T^{2} + 6 T - 36)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 252 T^{2} + 15376 \) Copy content Toggle raw display
$73$ \( T^{4} + 172T^{2} + 5776 \) Copy content Toggle raw display
$79$ \( (T^{2} + 12 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 10 T + 20)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 16 T + 44)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 268 T^{2} + 13456 \) Copy content Toggle raw display
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