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Properties

Label 1710.2.l.q
Level $1710$
Weight $2$
Character orbit 1710.l
Analytic conductor $13.654$
Analytic rank $0$
Dimension $6$
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] = [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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.29654208.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 14x^{4} + 49x^{2} + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 570)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} - \beta_1 q^{4} + ( - \beta_1 + 1) q^{5} - \beta_{3} q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} - \beta_1 q^{4} + ( - \beta_1 + 1) q^{5} - \beta_{3} q^{7} - q^{8} - \beta_1 q^{10} + ( - \beta_{2} - 1) q^{11} + ( - \beta_{4} - \beta_{2}) q^{13} + \beta_{5} q^{14} + (\beta_1 - 1) q^{16} + (\beta_{5} + 2 \beta_{4} + \beta_1 - 1) q^{17} + ( - \beta_{5} - \beta_{4} - \beta_{2}) q^{19} - q^{20} + (\beta_{4} + \beta_1 - 1) q^{22} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{23} - \beta_1 q^{25} - \beta_{2} q^{26} + (\beta_{5} + \beta_{3}) q^{28} + (\beta_{5} - 2 \beta_{4} + \cdots + 3 \beta_1) q^{29}+ \cdots + ( - 2 \beta_{5} - 2 \beta_{4} + \cdots + 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} - 3 q^{4} + 3 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{2} - 3 q^{4} + 3 q^{5} + 2 q^{7} - 6 q^{8} - 3 q^{10} - 8 q^{11} - q^{13} + q^{14} - 3 q^{16} - 4 q^{17} - 2 q^{19} - 6 q^{20} - 4 q^{22} - 4 q^{23} - 3 q^{25} - 2 q^{26} - q^{28} + 6 q^{29} + 2 q^{31} + 3 q^{32} + 4 q^{34} + q^{35} - 6 q^{37} - q^{38} - 3 q^{40} + 8 q^{41} - 11 q^{43} + 4 q^{44} - 8 q^{46} + 36 q^{49} - 6 q^{50} - q^{52} + 18 q^{53} - 4 q^{55} - 2 q^{56} + 12 q^{58} + 18 q^{59} + 3 q^{61} + q^{62} + 6 q^{64} - 2 q^{65} - 15 q^{67} + 8 q^{68} - q^{70} - 4 q^{71} + 21 q^{73} - 3 q^{74} + q^{76} - 32 q^{77} + 7 q^{79} + 3 q^{80} - 8 q^{82} - 4 q^{83} + 4 q^{85} + 11 q^{86} + 8 q^{88} - 18 q^{89} - 15 q^{91} - 4 q^{92} - q^{95} - 38 q^{97} + 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 14x^{4} + 49x^{2} + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 7\nu + 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} - 9\nu^{2} - 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 12\nu^{3} - 2\nu^{2} + 35\nu - 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 12\nu^{3} + 9\nu^{2} - 29\nu + 10 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -14\beta_{5} - 14\beta_{4} - 7\beta_{3} - 7\beta_{2} + 12\beta _1 - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{3} - 9\beta_{2} + 35 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 98\beta_{5} + 110\beta_{4} + 49\beta_{3} + 55\beta_{2} - 144\beta _1 + 72 ) / 3 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1261.1
− 2.35084i
2.86514i
− 0.514306i
2.35084i
− 2.86514i
0.514306i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −4.59821 −1.00000 0 −0.500000 + 0.866025i
1261.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 1.75353 −1.00000 0 −0.500000 + 0.866025i
1261.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 3.84469 −1.00000 0 −0.500000 + 0.866025i
1531.1 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.500000 − 0.866025i 0 −4.59821 −1.00000 0 −0.500000 − 0.866025i
1531.2 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.500000 − 0.866025i 0 1.75353 −1.00000 0 −0.500000 − 0.866025i
1531.3 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.500000 − 0.866025i 0 3.84469 −1.00000 0 −0.500000 − 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1261.3
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.q 6
3.b odd 2 1 570.2.i.j ✓ 6
19.c even 3 1 inner 1710.2.l.q 6
57.h odd 6 1 570.2.i.j ✓ 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.j ✓ 6 3.b odd 2 1
570.2.i.j ✓ 6 57.h odd 6 1
1710.2.l.q 6 1.a even 1 1 trivial
1710.2.l.q 6 19.c even 3 1 inner

Hecke kernels

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

\( T_{7}^{3} - T_{7}^{2} - 19T_{7} + 31 \) Copy content Toggle raw display
\( T_{11}^{3} + 4T_{11}^{2} - 11T_{11} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$7$ \( (T^{3} - T^{2} - 19 T + 31)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} + 4 T^{2} - 11 T - 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + T^{5} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{6} + 4 T^{5} + \cdots + 46656 \) Copy content Toggle raw display
$19$ \( T^{6} + 2 T^{5} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} + 4 T^{5} + \cdots + 36 \) Copy content Toggle raw display
$29$ \( T^{6} - 6 T^{5} + \cdots + 467856 \) Copy content Toggle raw display
$31$ \( (T^{3} - T^{2} - 16 T - 8)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1)^{6} \) Copy content Toggle raw display
$41$ \( T^{6} - 8 T^{5} + \cdots + 1077444 \) Copy content Toggle raw display
$43$ \( T^{6} + 11 T^{5} + \cdots + 1296 \) Copy content Toggle raw display
$47$ \( T^{6} + 84 T^{4} + \cdots + 20736 \) Copy content Toggle raw display
$53$ \( T^{6} - 18 T^{5} + \cdots + 11664 \) Copy content Toggle raw display
$59$ \( (T^{2} - 6 T + 36)^{3} \) Copy content Toggle raw display
$61$ \( T^{6} - 3 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$67$ \( T^{6} + 15 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$71$ \( T^{6} + 4 T^{5} + \cdots + 46656 \) Copy content Toggle raw display
$73$ \( T^{6} - 21 T^{5} + \cdots + 45796 \) Copy content Toggle raw display
$79$ \( T^{6} - 7 T^{5} + \cdots + 144 \) Copy content Toggle raw display
$83$ \( (T^{3} + 2 T^{2} + \cdots - 1032)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 18 T^{5} + \cdots + 11664 \) Copy content Toggle raw display
$97$ \( T^{6} + 38 T^{5} + \cdots + 3283344 \) Copy content Toggle raw display
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