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Properties

Label 1710.2.l.r
Level $1710$
Weight $2$
Character orbit 1710.l
Analytic conductor $13.654$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.4678560000.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 9x^{6} + 62x^{4} + 171x^{2} + 361 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_1 - 1) q^{4} + \beta_1 q^{5} + (\beta_{3} + 1) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_1 - 1) q^{4} + \beta_1 q^{5} + (\beta_{3} + 1) q^{7} + q^{8} + ( - \beta_1 + 1) q^{10} + ( - \beta_{6} + \beta_{4}) q^{11} + ( - \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{13}+ \cdots + (\beta_{6} + \beta_{5} - 2 \beta_{2} - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 4 q^{4} + 4 q^{5} + 12 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} - 4 q^{4} + 4 q^{5} + 12 q^{7} + 8 q^{8} + 4 q^{10} - 4 q^{11} - 6 q^{14} - 4 q^{16} + 8 q^{17} - 8 q^{20} + 2 q^{22} - 4 q^{25} - 6 q^{28} + 4 q^{29} - 32 q^{31} - 4 q^{32} + 8 q^{34} + 6 q^{35} - 4 q^{37} + 4 q^{40} - 10 q^{41} - 8 q^{43} + 2 q^{44} - 4 q^{47} + 8 q^{49} + 8 q^{50} - 8 q^{53} - 2 q^{55} + 12 q^{56} - 8 q^{58} - 20 q^{59} - 12 q^{61} + 16 q^{62} + 8 q^{64} + 12 q^{67} - 16 q^{68} + 6 q^{70} - 8 q^{71} - 4 q^{73} + 2 q^{74} + 32 q^{77} + 16 q^{79} + 4 q^{80} - 10 q^{82} + 8 q^{83} - 8 q^{85} - 8 q^{86} - 4 q^{88} - 2 q^{89} + 8 q^{91} + 8 q^{94} - 16 q^{97} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 9x^{6} + 62x^{4} + 171x^{2} + 361 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 9\nu^{6} + 62\nu^{4} + 558\nu^{2} + 1539 ) / 1178 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13\nu^{6} + 155\nu^{4} + 806\nu^{2} - 589\nu + 2223 ) / 589 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9\nu^{7} - 19\nu^{6} + 62\nu^{5} + 558\nu^{3} + 361\nu + 2641 ) / 1178 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 26\nu^{7} + 7\nu^{6} + 310\nu^{5} + 310\nu^{4} + 1612\nu^{3} + 1612\nu^{2} + 4446\nu + 5909 ) / 1178 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -19\nu^{7} - 55\nu^{6} - 248\nu^{4} - 1054\nu^{2} + 2641\nu + 1197 ) / 1178 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -19\nu^{7} + 64\nu^{6} + 310\nu^{4} + 1612\nu^{2} + 2641\nu + 342 ) / 1178 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -45\nu^{7} + 62\nu^{6} - 310\nu^{5} + 558\nu^{4} - 1612\nu^{3} + 2666\nu^{2} - 1805\nu + 4712 ) / 1178 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{7} - \beta_{6} - \beta_{5} + 2\beta_{4} - 6\beta_{2} + 3\beta _1 + 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - \beta_{6} + 2\beta_{5} - \beta_{4} + 27\beta _1 - 28 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + 5\beta_{6} - \beta_{5} - 4\beta_{4} + 15\beta_{3} - 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6\beta_{7} - 3\beta_{6} - 3\beta_{5} + 6\beta_{4} - 43\beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -53\beta_{7} + \beta_{6} + 52\beta_{5} + \beta_{4} - 156\beta_{3} + 156\beta_{2} - 105\beta _1 + 52 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -31\beta_{7} + 62\beta_{6} - 31\beta_{5} - 31\beta_{4} + 293 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 278\beta_{7} - 325\beta_{6} - 325\beta_{5} + 278\beta_{4} - 834\beta_{2} + 603\beta _1 + 278 ) / 6 \) Copy content Toggle raw display

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

Character values

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

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

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
1.18512 + 2.05269i
0.919506 + 1.59263i
−1.18512 − 2.05269i
−0.919506 − 1.59263i
1.18512 − 2.05269i
0.919506 − 1.59263i
−1.18512 + 2.05269i
−0.919506 + 1.59263i
−0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −1.98827 1.00000 0 0.500000 − 0.866025i
1261.2 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 0.779022 1.00000 0 0.500000 − 0.866025i
1261.3 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 2.75221 1.00000 0 0.500000 − 0.866025i
1261.4 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 4.45705 1.00000 0 0.500000 − 0.866025i
1531.1 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.500000 − 0.866025i 0 −1.98827 1.00000 0 0.500000 + 0.866025i
1531.2 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.500000 − 0.866025i 0 0.779022 1.00000 0 0.500000 + 0.866025i
1531.3 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.500000 − 0.866025i 0 2.75221 1.00000 0 0.500000 + 0.866025i
1531.4 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.500000 − 0.866025i 0 4.45705 1.00000 0 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1261.4
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.r ✓ 8
3.b odd 2 1 1710.2.l.s yes 8
19.c even 3 1 inner 1710.2.l.r ✓ 8
57.h odd 6 1 1710.2.l.s yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1710.2.l.r ✓ 8 1.a even 1 1 trivial
1710.2.l.r ✓ 8 19.c even 3 1 inner
1710.2.l.s yes 8 3.b odd 2 1
1710.2.l.s yes 8 57.h odd 6 1

Hecke kernels

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

\( T_{7}^{4} - 6T_{7}^{3} + 2T_{7}^{2} + 26T_{7} - 19 \) Copy content Toggle raw display
\( T_{11}^{4} + 2T_{11}^{3} - 32T_{11}^{2} - 78T_{11} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 6 T^{3} + 2 T^{2} + \cdots - 19)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} - 32 T^{2} + \cdots - 9)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 44 T^{6} + \cdots + 92416 \) Copy content Toggle raw display
$17$ \( T^{8} - 8 T^{7} + \cdots + 156816 \) Copy content Toggle raw display
$19$ \( T^{8} - 28 T^{6} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( (T^{4} + 45 T^{2} + 2025)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 4 T^{7} + \cdots + 156816 \) Copy content Toggle raw display
$31$ \( (T^{4} + 16 T^{3} + \cdots - 3904)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2 T^{3} - 32 T^{2} + \cdots - 9)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 10 T^{7} + \cdots + 5517801 \) Copy content Toggle raw display
$43$ \( T^{8} + 8 T^{7} + \cdots + 3748096 \) Copy content Toggle raw display
$47$ \( T^{8} + 4 T^{7} + \cdots + 17438976 \) Copy content Toggle raw display
$53$ \( T^{8} + 8 T^{7} + \cdots + 29241 \) Copy content Toggle raw display
$59$ \( T^{8} + 20 T^{7} + \cdots + 17438976 \) Copy content Toggle raw display
$61$ \( T^{8} + 12 T^{7} + \cdots + 59536 \) Copy content Toggle raw display
$67$ \( T^{8} - 12 T^{7} + \cdots + 234256 \) Copy content Toggle raw display
$71$ \( T^{8} + 8 T^{7} + \cdots + 156816 \) Copy content Toggle raw display
$73$ \( T^{8} + 4 T^{7} + \cdots + 4822416 \) Copy content Toggle raw display
$79$ \( T^{8} - 16 T^{7} + \cdots + 144384256 \) Copy content Toggle raw display
$83$ \( (T^{4} - 4 T^{3} + \cdots + 144)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 2 T^{7} + \cdots + 17214201 \) Copy content Toggle raw display
$97$ \( T^{8} + 16 T^{7} + \cdots + 51322896 \) Copy content Toggle raw display
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