Properties

Label 2368.2.g.j
Level $2368$
Weight $2$
Character orbit 2368.g
Analytic conductor $18.909$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(18.9085751986\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{21})\)
Defining polynomial: \( x^{4} + 11x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 1) q^{3} + \beta_1 q^{5} - 2 q^{7} + ( - \beta_{3} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 1) q^{3} + \beta_1 q^{5} - 2 q^{7} + ( - \beta_{3} + 3) q^{9} + (\beta_{3} - 2) q^{11} + \beta_{2} q^{13} + (\beta_{2} + 3 \beta_1) q^{15} + 2 \beta_{2} q^{17} + 2 \beta_1 q^{19} + (2 \beta_{3} - 2) q^{21} - \beta_{2} q^{23} + (3 \beta_{3} - 4) q^{25} + 5 q^{27} - \beta_{2} q^{29} + ( - 2 \beta_{2} + \beta_1) q^{31} + (2 \beta_{3} - 7) q^{33} - 2 \beta_1 q^{35} + (\beta_{2} - \beta_1 - 4) q^{37} + ( - 2 \beta_{2} - \beta_1) q^{39} + (\beta_{3} + 7) q^{41} + (2 \beta_{2} + 2 \beta_1) q^{43} + (\beta_{2} + 5 \beta_1) q^{45} + ( - 2 \beta_{3} - 2) q^{47} - 3 q^{49} + ( - 4 \beta_{2} - 2 \beta_1) q^{51} + (2 \beta_{3} - 4) q^{53} + ( - \beta_{2} - 4 \beta_1) q^{55} + (2 \beta_{2} + 6 \beta_1) q^{57} + 2 \beta_1 q^{59} + ( - 4 \beta_{2} - 3 \beta_1) q^{61} + (2 \beta_{3} - 6) q^{63} + 3 q^{65} + ( - 3 \beta_{3} + 2) q^{67} + (2 \beta_{2} + \beta_1) q^{69} + ( - 4 \beta_{3} + 2) q^{71} + ( - 3 \beta_{3} + 4) q^{73} + (4 \beta_{3} - 19) q^{75} + ( - 2 \beta_{3} + 4) q^{77} + (3 \beta_{2} + 4 \beta_1) q^{79} + ( - 2 \beta_{3} - 4) q^{81} + ( - 4 \beta_{3} - 4) q^{83} + 6 q^{85} + (2 \beta_{2} + \beta_1) q^{87} + (2 \beta_{2} + 2 \beta_1) q^{89} - 2 \beta_{2} q^{91} + (5 \beta_{2} + 5 \beta_1) q^{93} + (6 \beta_{3} - 18) q^{95} + ( - 4 \beta_{2} - 2 \beta_1) q^{97} + (4 \beta_{3} - 11) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 8 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 8 q^{7} + 10 q^{9} - 6 q^{11} - 4 q^{21} - 10 q^{25} + 20 q^{27} - 24 q^{33} - 16 q^{37} + 30 q^{41} - 12 q^{47} - 12 q^{49} - 12 q^{53} - 20 q^{63} + 12 q^{65} + 2 q^{67} + 10 q^{73} - 68 q^{75} + 12 q^{77} - 20 q^{81} - 24 q^{83} + 24 q^{85} - 60 q^{95} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 11x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + \nu ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{3} + 17\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{2} + 17\beta_1 ) / 3 \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
1.79129i
1.79129i
2.79129i
2.79129i
0 −1.79129 0 0.791288i 0 −2.00000 0 0.208712 0
961.2 0 −1.79129 0 0.791288i 0 −2.00000 0 0.208712 0
961.3 0 2.79129 0 3.79129i 0 −2.00000 0 4.79129 0
961.4 0 2.79129 0 3.79129i 0 −2.00000 0 4.79129 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.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2368.2.g.j 4
4.b odd 2 1 2368.2.g.h 4
8.b even 2 1 74.2.b.a 4
8.d odd 2 1 592.2.g.c 4
24.f even 2 1 5328.2.h.m 4
24.h odd 2 1 666.2.c.b 4
37.b even 2 1 inner 2368.2.g.j 4
40.f even 2 1 1850.2.d.e 4
40.i odd 4 1 1850.2.c.g 4
40.i odd 4 1 1850.2.c.h 4
148.b odd 2 1 2368.2.g.h 4
296.e even 2 1 74.2.b.a 4
296.h odd 2 1 592.2.g.c 4
296.m odd 4 1 2738.2.a.h 2
296.m odd 4 1 2738.2.a.k 2
888.c even 2 1 5328.2.h.m 4
888.i odd 2 1 666.2.c.b 4
1480.j even 2 1 1850.2.d.e 4
1480.x odd 4 1 1850.2.c.g 4
1480.x odd 4 1 1850.2.c.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.b.a 4 8.b even 2 1
74.2.b.a 4 296.e even 2 1
592.2.g.c 4 8.d odd 2 1
592.2.g.c 4 296.h odd 2 1
666.2.c.b 4 24.h odd 2 1
666.2.c.b 4 888.i odd 2 1
1850.2.c.g 4 40.i odd 4 1
1850.2.c.g 4 1480.x odd 4 1
1850.2.c.h 4 40.i odd 4 1
1850.2.c.h 4 1480.x odd 4 1
1850.2.d.e 4 40.f even 2 1
1850.2.d.e 4 1480.j even 2 1
2368.2.g.h 4 4.b odd 2 1
2368.2.g.h 4 148.b odd 2 1
2368.2.g.j 4 1.a even 1 1 trivial
2368.2.g.j 4 37.b even 2 1 inner
2738.2.a.h 2 296.m odd 4 1
2738.2.a.k 2 296.m odd 4 1
5328.2.h.m 4 24.f even 2 1
5328.2.h.m 4 888.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2368, [\chi])\):

\( T_{3}^{2} - T_{3} - 5 \) Copy content Toggle raw display
\( T_{5}^{4} + 15T_{5}^{2} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - T - 5)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 15T^{2} + 9 \) Copy content Toggle raw display
$7$ \( (T + 2)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 3 T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 15T^{2} + 9 \) Copy content Toggle raw display
$17$ \( T^{4} + 60T^{2} + 144 \) Copy content Toggle raw display
$19$ \( T^{4} + 60T^{2} + 144 \) Copy content Toggle raw display
$23$ \( T^{4} + 15T^{2} + 9 \) Copy content Toggle raw display
$29$ \( T^{4} + 15T^{2} + 9 \) Copy content Toggle raw display
$31$ \( T^{4} + 99T^{2} + 2025 \) Copy content Toggle raw display
$37$ \( (T^{2} + 8 T + 37)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 15 T + 51)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 6 T - 12)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T - 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 60T^{2} + 144 \) Copy content Toggle raw display
$61$ \( T^{4} + 231 T^{2} + 11025 \) Copy content Toggle raw display
$67$ \( (T^{2} - T - 47)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 84)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 5 T - 41)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 231 T^{2} + 11025 \) Copy content Toggle raw display
$83$ \( (T^{2} + 12 T - 48)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 204T^{2} + 3600 \) Copy content Toggle raw display
show more
show less