Properties

Label 1680.2.t.f
Level $1680$
Weight $2$
Character orbit 1680.t
Analytic conductor $13.415$
Analytic rank $0$
Dimension $2$
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(1009,1680)]
 
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, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1680.1009");
 
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.t (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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1009.1
1.00000i
1.00000i
0 1.00000i 0 1.00000 2.00000i 0 1.00000i 0 −1.00000 0
1009.2 0 1.00000i 0 1.00000 + 2.00000i 0 1.00000i 0 −1.00000 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
5.b 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.t.f 2
3.b odd 2 1 5040.2.t.e 2
4.b odd 2 1 105.2.d.a 2
5.b even 2 1 inner 1680.2.t.f 2
5.c odd 4 1 8400.2.a.bj 1
5.c odd 4 1 8400.2.a.ch 1
12.b even 2 1 315.2.d.c 2
15.d odd 2 1 5040.2.t.e 2
20.d odd 2 1 105.2.d.a 2
20.e even 4 1 525.2.a.b 1
20.e even 4 1 525.2.a.c 1
28.d even 2 1 735.2.d.a 2
28.f even 6 2 735.2.q.b 4
28.g odd 6 2 735.2.q.a 4
60.h even 2 1 315.2.d.c 2
60.l odd 4 1 1575.2.a.e 1
60.l odd 4 1 1575.2.a.i 1
84.h odd 2 1 2205.2.d.f 2
140.c even 2 1 735.2.d.a 2
140.j odd 4 1 3675.2.a.d 1
140.j odd 4 1 3675.2.a.l 1
140.p odd 6 2 735.2.q.a 4
140.s even 6 2 735.2.q.b 4
420.o odd 2 1 2205.2.d.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.a 2 4.b odd 2 1
105.2.d.a 2 20.d odd 2 1
315.2.d.c 2 12.b even 2 1
315.2.d.c 2 60.h even 2 1
525.2.a.b 1 20.e even 4 1
525.2.a.c 1 20.e even 4 1
735.2.d.a 2 28.d even 2 1
735.2.d.a 2 140.c even 2 1
735.2.q.a 4 28.g odd 6 2
735.2.q.a 4 140.p odd 6 2
735.2.q.b 4 28.f even 6 2
735.2.q.b 4 140.s even 6 2
1575.2.a.e 1 60.l odd 4 1
1575.2.a.i 1 60.l odd 4 1
1680.2.t.f 2 1.a even 1 1 trivial
1680.2.t.f 2 5.b even 2 1 inner
2205.2.d.f 2 84.h odd 2 1
2205.2.d.f 2 420.o odd 2 1
3675.2.a.d 1 140.j odd 4 1
3675.2.a.l 1 140.j odd 4 1
5040.2.t.e 2 3.b odd 2 1
5040.2.t.e 2 15.d odd 2 1
8400.2.a.bj 1 5.c odd 4 1
8400.2.a.ch 1 5.c odd 4 1

Hecke kernels

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

\( T_{11} - 6 \) Copy content Toggle raw display
\( T_{13}^{2} + 4 \) Copy content Toggle raw display
\( T_{19} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T - 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T + 6)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( (T - 10)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T + 8)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 256 \) Copy content Toggle raw display
$71$ \( (T + 10)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 36 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 64 \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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