Properties

Label 2368.1.k.b
Level 2368
Weight 1
Character orbit 2368.k
Analytic conductor 1.182
Analytic rank 0
Dimension 2
Projective image \(S_{4}\)
CM/RM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 2368 = 2^{6} \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2368.k (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.18178594991\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 148)
Projective image \(S_{4}\)
Projective field Galois closure of 4.0.202612.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -i q^{3} + q^{7} +O(q^{10})\) \( q -i q^{3} + q^{7} + i q^{11} + ( -1 + i ) q^{17} + ( 1 - i ) q^{19} -i q^{21} + ( 1 - i ) q^{23} + i q^{25} -i q^{27} + ( 1 + i ) q^{29} + q^{33} + i q^{37} -i q^{41} - q^{47} + ( 1 + i ) q^{51} - q^{53} + ( -1 - i ) q^{57} + ( -1 - i ) q^{69} - q^{71} -i q^{73} + q^{75} + i q^{77} + ( 1 - i ) q^{79} - q^{81} - q^{83} + ( 1 - i ) q^{87} + ( -1 - i ) q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{7} + O(q^{10}) \) \( 2q + 2q^{7} - 2q^{17} + 2q^{19} + 2q^{23} + 2q^{29} + 2q^{33} - 2q^{47} + 2q^{51} - 2q^{53} - 2q^{57} - 2q^{69} - 2q^{71} + 2q^{75} + 2q^{79} - 2q^{81} - 2q^{83} + 2q^{87} - 2q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2368\mathbb{Z}\right)^\times\).

\(n\) \(705\) \(1407\) \(1925\)
\(\chi(n)\) \(i\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1153.1
1.00000i
1.00000i
0 1.00000i 0 0 0 1.00000 0 0 0
2177.1 0 1.00000i 0 0 0 1.00000 0 0 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
37.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2368.1.k.b 2
4.b odd 2 1 2368.1.k.a 2
8.b even 2 1 592.1.k.b 2
8.d odd 2 1 148.1.f.a 2
24.f even 2 1 1332.1.o.a 2
37.d odd 4 1 inner 2368.1.k.b 2
40.e odd 2 1 3700.1.j.c 2
40.k even 4 1 3700.1.t.a 2
40.k even 4 1 3700.1.t.b 2
148.g even 4 1 2368.1.k.a 2
296.j even 4 1 148.1.f.a 2
296.m odd 4 1 592.1.k.b 2
888.u odd 4 1 1332.1.o.a 2
1480.v odd 4 1 3700.1.t.b 2
1480.bc odd 4 1 3700.1.t.a 2
1480.bn even 4 1 3700.1.j.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
148.1.f.a 2 8.d odd 2 1
148.1.f.a 2 296.j even 4 1
592.1.k.b 2 8.b even 2 1
592.1.k.b 2 296.m odd 4 1
1332.1.o.a 2 24.f even 2 1
1332.1.o.a 2 888.u odd 4 1
2368.1.k.a 2 4.b odd 2 1
2368.1.k.a 2 148.g even 4 1
2368.1.k.b 2 1.a even 1 1 trivial
2368.1.k.b 2 37.d odd 4 1 inner
3700.1.j.c 2 40.e odd 2 1
3700.1.j.c 2 1480.bn even 4 1
3700.1.t.a 2 40.k even 4 1
3700.1.t.a 2 1480.bc odd 4 1
3700.1.t.b 2 40.k even 4 1
3700.1.t.b 2 1480.v odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2368, [\chi])\):

\( T_{3}^{2} + 1 \)
\( T_{5} \)
\( T_{7} - 1 \)

Hecke characteristic polynomials

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