↑

Properties

Label 1710.2.l.m
Level $1710$
Weight $2$
Character orbit 1710.l
Analytic conductor $13.654$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1710,2,Mod(1261,1710)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1710, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("1710.1261");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1710.l (of order \(3\), degree \(2\), 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: \(13.6544187456\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_{2} - 1) q^{4} + \beta_{2} q^{5} + ( - \beta_{3} + 2) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_{2} - 1) q^{4} + \beta_{2} q^{5} + ( - \beta_{3} + 2) q^{7} + q^{8} + ( - \beta_{2} + 1) q^{10} - q^{11} + ( - 2 \beta_{2} + 2) q^{13} + ( - 2 \beta_{2} - \beta_1) q^{14} - \beta_{2} q^{16} + (2 \beta_{2} - 2 \beta_1) q^{17} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 - 3) q^{19} - q^{20} + \beta_{2} q^{22} + ( - 3 \beta_{3} - 3 \beta_1) q^{23} + (\beta_{2} - 1) q^{25} - 2 q^{26} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{28} + ( - 2 \beta_{2} + 2) q^{29} + (4 \beta_{3} + 4) q^{31} + (\beta_{2} - 1) q^{32} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{34} + (2 \beta_{2} + \beta_1) q^{35} - 3 \beta_{3} q^{37} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{38} + \beta_{2} q^{40} + ( - 3 \beta_{2} + 2 \beta_1) q^{41} + 2 \beta_1 q^{43} + ( - \beta_{2} + 1) q^{44} + 3 \beta_{3} q^{46} + ( - 2 \beta_{3} - 6 \beta_{2} + \cdots + 6) q^{47}+ \cdots + ( - \beta_{2} - 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{4} + 2 q^{5} + 10 q^{7} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 2 q^{4} + 2 q^{5} + 10 q^{7} + 4 q^{8} + 2 q^{10} - 4 q^{11} + 4 q^{13} - 5 q^{14} - 2 q^{16} + 2 q^{17} - 10 q^{19} - 4 q^{20} + 2 q^{22} + 3 q^{23} - 2 q^{25} - 8 q^{26} - 5 q^{28} + 4 q^{29} + 8 q^{31} - 2 q^{32} + 2 q^{34} + 5 q^{35} + 6 q^{37} + 5 q^{38} + 2 q^{40} - 4 q^{41} + 2 q^{43} + 2 q^{44} - 6 q^{46} + 14 q^{47} + 14 q^{49} + 4 q^{50} + 4 q^{52} - 11 q^{53} - 2 q^{55} + 10 q^{56} - 8 q^{58} + 25 q^{59} - 2 q^{61} - 4 q^{62} + 4 q^{64} + 8 q^{65} - 23 q^{67} - 4 q^{68} + 5 q^{70} + 16 q^{71} - 9 q^{73} - 3 q^{74} + 5 q^{76} - 10 q^{77} + 14 q^{79} + 2 q^{80} - 4 q^{82} + 2 q^{83} - 2 q^{85} + 2 q^{86} - 4 q^{88} + 7 q^{89} + 10 q^{91} + 3 q^{92} - 28 q^{94} - 5 q^{95} + 9 q^{97} - 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4 ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4\beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} - 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1710\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1261.1
−0.780776 − 1.35234i
1.28078 + 2.21837i
−0.780776 + 1.35234i
1.28078 − 2.21837i
−0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 0.438447 1.00000 0 0.500000 − 0.866025i
1261.2 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 4.56155 1.00000 0 0.500000 − 0.866025i
1531.1 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.500000 − 0.866025i 0 0.438447 1.00000 0 0.500000 + 0.866025i
1531.2 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.500000 − 0.866025i 0 4.56155 1.00000 0 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1710.2.l.m 4
3.b odd 2 1 190.2.e.c ✓ 4
12.b even 2 1 1520.2.q.h 4
15.d odd 2 1 950.2.e.h 4
15.e even 4 2 950.2.j.f 8
19.c even 3 1 inner 1710.2.l.m 4
57.f even 6 1 3610.2.a.u 2
57.h odd 6 1 190.2.e.c ✓ 4
57.h odd 6 1 3610.2.a.k 2
228.m even 6 1 1520.2.q.h 4
285.n odd 6 1 950.2.e.h 4
285.v even 12 2 950.2.j.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.e.c ✓ 4 3.b odd 2 1
190.2.e.c ✓ 4 57.h odd 6 1
950.2.e.h 4 15.d odd 2 1
950.2.e.h 4 285.n odd 6 1
950.2.j.f 8 15.e even 4 2
950.2.j.f 8 285.v even 12 2
1520.2.q.h 4 12.b even 2 1
1520.2.q.h 4 228.m even 6 1
1710.2.l.m 4 1.a even 1 1 trivial
1710.2.l.m 4 19.c even 3 1 inner
3610.2.a.k 2 57.h odd 6 1
3610.2.a.u 2 57.f even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1710, [\chi])\):

\( T_{7}^{2} - 5T_{7} + 2 \) Copy content Toggle raw display
\( T_{11} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 5 T + 2)^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( (T^{2} + 5 T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 3 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$29$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 3 T - 36)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 4 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$43$ \( T^{4} - 2 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( T^{4} - 14 T^{3} + \cdots + 1024 \) Copy content Toggle raw display
$53$ \( T^{4} + 11 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$59$ \( T^{4} - 25 T^{3} + \cdots + 23104 \) Copy content Toggle raw display
$61$ \( T^{4} + 2 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$67$ \( T^{4} + 23 T^{3} + \cdots + 16384 \) Copy content Toggle raw display
$71$ \( T^{4} - 16 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$73$ \( T^{4} + 9 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$79$ \( T^{4} - 14 T^{3} + \cdots + 1024 \) Copy content Toggle raw display
$83$ \( (T^{2} - T - 106)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 7 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$97$ \( T^{4} - 9 T^{3} + \cdots + 324 \) Copy content Toggle raw display
show more
show less