Properties

Label 2352.2.h.a
Level $2352$
Weight $2$
Character orbit 2352.h
Analytic conductor $18.781$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + 2 \zeta_{6} ) q^{3} -3 q^{9} +O(q^{10})\) \( q + ( -1 + 2 \zeta_{6} ) q^{3} -3 q^{9} -7 q^{13} + ( -3 + 6 \zeta_{6} ) q^{19} + 5 q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 5 - 10 \zeta_{6} ) q^{31} - q^{37} + ( 7 - 14 \zeta_{6} ) q^{39} + ( 7 - 14 \zeta_{6} ) q^{43} -9 q^{57} -14 q^{61} + ( 7 - 14 \zeta_{6} ) q^{67} -7 q^{73} + ( -5 + 10 \zeta_{6} ) q^{75} + ( 7 - 14 \zeta_{6} ) q^{79} + 9 q^{81} + 15 q^{93} + 14 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{9} + O(q^{10}) \) \( 2 q - 6 q^{9} - 14 q^{13} + 10 q^{25} - 2 q^{37} - 18 q^{57} - 28 q^{61} - 14 q^{73} + 18 q^{81} + 30 q^{93} + 28 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2255.1
0.500000 0.866025i
0.500000 + 0.866025i
0 1.73205i 0 0 0 0 0 −3.00000 0
2255.2 0 1.73205i 0 0 0 0 0 −3.00000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.2.h.a 2
3.b odd 2 1 CM 2352.2.h.a 2
4.b odd 2 1 inner 2352.2.h.a 2
7.b odd 2 1 2352.2.h.e 2
7.d odd 6 1 336.2.bj.b 2
7.d odd 6 1 336.2.bj.c yes 2
12.b even 2 1 inner 2352.2.h.a 2
21.c even 2 1 2352.2.h.e 2
21.g even 6 1 336.2.bj.b 2
21.g even 6 1 336.2.bj.c yes 2
28.d even 2 1 2352.2.h.e 2
28.f even 6 1 336.2.bj.b 2
28.f even 6 1 336.2.bj.c yes 2
84.h odd 2 1 2352.2.h.e 2
84.j odd 6 1 336.2.bj.b 2
84.j odd 6 1 336.2.bj.c yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bj.b 2 7.d odd 6 1
336.2.bj.b 2 21.g even 6 1
336.2.bj.b 2 28.f even 6 1
336.2.bj.b 2 84.j odd 6 1
336.2.bj.c yes 2 7.d odd 6 1
336.2.bj.c yes 2 21.g even 6 1
336.2.bj.c yes 2 28.f even 6 1
336.2.bj.c yes 2 84.j odd 6 1
2352.2.h.a 2 1.a even 1 1 trivial
2352.2.h.a 2 3.b odd 2 1 CM
2352.2.h.a 2 4.b odd 2 1 inner
2352.2.h.a 2 12.b even 2 1 inner
2352.2.h.e 2 7.b odd 2 1
2352.2.h.e 2 21.c even 2 1
2352.2.h.e 2 28.d even 2 1
2352.2.h.e 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2352, [\chi])\):

\( T_{5} \)
\( T_{11} \)
\( T_{13} + 7 \)
\( T_{47} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( 7 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 27 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 75 + T^{2} \)
$37$ \( ( 1 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( 147 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 14 + T )^{2} \)
$67$ \( 147 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 7 + T )^{2} \)
$79$ \( 147 + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( -14 + T )^{2} \)
show more
show less