Properties

Label 2352.1.by.a
Level $2352$
Weight $1$
Character orbit 2352.by
Analytic conductor $1.174$
Analytic rank $0$
Dimension $6$
Projective image $D_{14}$
CM discriminant -3
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2352.by (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.17380090971\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
Defining polynomial: \(x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{14}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{14} + \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{14} q^{3} + \zeta_{14}^{5} q^{7} + \zeta_{14}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{14} q^{3} + \zeta_{14}^{5} q^{7} + \zeta_{14}^{2} q^{9} + ( -\zeta_{14}^{4} + \zeta_{14}^{6} ) q^{13} + ( -\zeta_{14}^{2} + \zeta_{14}^{5} ) q^{19} -\zeta_{14}^{6} q^{21} -\zeta_{14}^{2} q^{25} -\zeta_{14}^{3} q^{27} + ( -\zeta_{14}^{3} + \zeta_{14}^{4} ) q^{31} + ( -1 + \zeta_{14} ) q^{37} + ( 1 + \zeta_{14}^{5} ) q^{39} + ( -\zeta_{14} - \zeta_{14}^{4} ) q^{43} -\zeta_{14}^{3} q^{49} + ( \zeta_{14}^{3} - \zeta_{14}^{6} ) q^{57} + ( -1 - \zeta_{14} ) q^{61} - q^{63} + ( -\zeta_{14} - \zeta_{14}^{6} ) q^{67} + ( \zeta_{14}^{5} + \zeta_{14}^{6} ) q^{73} + \zeta_{14}^{3} q^{75} + ( \zeta_{14}^{3} + \zeta_{14}^{4} ) q^{79} + \zeta_{14}^{4} q^{81} + ( \zeta_{14}^{2} - \zeta_{14}^{4} ) q^{91} + ( \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{93} + ( -\zeta_{14}^{2} - \zeta_{14}^{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{3} + q^{7} - q^{9} + O(q^{10}) \) \( 6q - q^{3} + q^{7} - q^{9} + 2q^{19} + q^{21} + q^{25} - q^{27} - 2q^{31} - 5q^{37} + 7q^{39} - q^{49} + 2q^{57} - 7q^{61} - 6q^{63} + q^{75} - q^{81} - 2q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-\zeta_{14}^{2}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
335.1
0.900969 0.433884i
−0.623490 + 0.781831i
0.222521 0.974928i
0.222521 + 0.974928i
−0.623490 0.781831i
0.900969 + 0.433884i
0 −0.900969 + 0.433884i 0 0 0 −0.623490 0.781831i 0 0.623490 0.781831i 0
671.1 0 0.623490 0.781831i 0 0 0 0.222521 0.974928i 0 −0.222521 0.974928i 0
1007.1 0 −0.222521 + 0.974928i 0 0 0 0.900969 0.433884i 0 −0.900969 0.433884i 0
1343.1 0 −0.222521 0.974928i 0 0 0 0.900969 + 0.433884i 0 −0.900969 + 0.433884i 0
1679.1 0 0.623490 + 0.781831i 0 0 0 0.222521 + 0.974928i 0 −0.222521 + 0.974928i 0
2015.1 0 −0.900969 0.433884i 0 0 0 −0.623490 + 0.781831i 0 0.623490 + 0.781831i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2015.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
196.j even 14 1 inner
588.r odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.1.by.a 6
3.b odd 2 1 CM 2352.1.by.a 6
4.b odd 2 1 2352.1.by.b yes 6
12.b even 2 1 2352.1.by.b yes 6
49.f odd 14 1 2352.1.by.b yes 6
147.k even 14 1 2352.1.by.b yes 6
196.j even 14 1 inner 2352.1.by.a 6
588.r odd 14 1 inner 2352.1.by.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.1.by.a 6 1.a even 1 1 trivial
2352.1.by.a 6 3.b odd 2 1 CM
2352.1.by.a 6 196.j even 14 1 inner
2352.1.by.a 6 588.r odd 14 1 inner
2352.1.by.b yes 6 4.b odd 2 1
2352.1.by.b yes 6 12.b even 2 1
2352.1.by.b yes 6 49.f odd 14 1
2352.1.by.b yes 6 147.k even 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{19}^{3} - T_{19}^{2} - 2 T_{19} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2352, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
$11$ \( T^{6} \)
$13$ \( 7 + 7 T - 7 T^{3} + T^{6} \)
$17$ \( T^{6} \)
$19$ \( ( 1 - 2 T - T^{2} + T^{3} )^{2} \)
$23$ \( T^{6} \)
$29$ \( T^{6} \)
$31$ \( ( -1 - 2 T + T^{2} + T^{3} )^{2} \)
$37$ \( 1 + 3 T + 9 T^{2} + 13 T^{3} + 11 T^{4} + 5 T^{5} + T^{6} \)
$41$ \( T^{6} \)
$43$ \( 7 + 14 T + 7 T^{2} + T^{6} \)
$47$ \( T^{6} \)
$53$ \( T^{6} \)
$59$ \( T^{6} \)
$61$ \( 7 + 21 T + 35 T^{2} + 35 T^{3} + 21 T^{4} + 7 T^{5} + T^{6} \)
$67$ \( 7 + 14 T^{2} + 7 T^{4} + T^{6} \)
$71$ \( T^{6} \)
$73$ \( 7 - 14 T + 7 T^{2} + T^{6} \)
$79$ \( 7 + 14 T^{2} + 7 T^{4} + T^{6} \)
$83$ \( T^{6} \)
$89$ \( T^{6} \)
$97$ \( 7 + 14 T^{2} + 7 T^{4} + T^{6} \)
show more
show less