Properties

Label 2352.3.f.e
Level $2352$
Weight $3$
Character orbit 2352.f
Analytic conductor $64.087$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2352.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(64.0873581775\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: no (minimal twist has level 42)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( -\beta_{1} + 2 \beta_{2} ) q^{5} -3 q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} + ( -\beta_{1} + 2 \beta_{2} ) q^{5} -3 q^{9} -6 q^{11} + ( -4 \beta_{1} - \beta_{2} ) q^{13} + ( -6 + \beta_{3} ) q^{15} + ( -\beta_{1} + 8 \beta_{2} ) q^{17} + ( -\beta_{1} + 7 \beta_{2} ) q^{19} + ( 12 + 3 \beta_{3} ) q^{23} + ( -11 + 4 \beta_{3} ) q^{25} -3 \beta_{2} q^{27} -4 \beta_{3} q^{29} + ( 3 \beta_{1} + 17 \beta_{2} ) q^{31} -6 \beta_{2} q^{33} + ( -11 - 2 \beta_{3} ) q^{37} + ( 3 + 4 \beta_{3} ) q^{39} + ( -2 \beta_{1} - 26 \beta_{2} ) q^{41} + ( -7 + \beta_{3} ) q^{43} + ( 3 \beta_{1} - 6 \beta_{2} ) q^{45} + ( \beta_{1} + 22 \beta_{2} ) q^{47} + ( -24 + \beta_{3} ) q^{51} + ( -60 - 3 \beta_{3} ) q^{53} + ( 6 \beta_{1} - 12 \beta_{2} ) q^{55} + ( -21 + \beta_{3} ) q^{57} + ( -7 \beta_{1} - 4 \beta_{2} ) q^{59} + ( 4 \beta_{1} - 12 \beta_{2} ) q^{61} + ( -90 + 7 \beta_{3} ) q^{65} + ( 55 - 7 \beta_{3} ) q^{67} + ( 9 \beta_{1} + 12 \beta_{2} ) q^{69} + ( -78 - 7 \beta_{3} ) q^{71} + ( -20 \beta_{1} + 11 \beta_{2} ) q^{73} + ( 12 \beta_{1} - 11 \beta_{2} ) q^{75} + ( -5 - 11 \beta_{3} ) q^{79} + 9 q^{81} + ( -19 \beta_{1} - 10 \beta_{2} ) q^{83} + ( -72 + 10 \beta_{3} ) q^{85} -12 \beta_{1} q^{87} -12 \beta_{2} q^{89} + ( -51 - 3 \beta_{3} ) q^{93} + ( -66 + 9 \beta_{3} ) q^{95} + ( 2 \beta_{1} - 12 \beta_{2} ) q^{97} + 18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{9} + O(q^{10}) \) \( 4q - 12q^{9} - 24q^{11} - 24q^{15} + 48q^{23} - 44q^{25} - 44q^{37} + 12q^{39} - 28q^{43} - 96q^{51} - 240q^{53} - 84q^{57} - 360q^{65} + 220q^{67} - 312q^{71} - 20q^{79} + 36q^{81} - 288q^{85} - 204q^{93} - 264q^{95} + 72q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 4 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\( -3 \nu^{3} \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 3 \beta_{1}\)\()/12\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{3}\)\()/3\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−0.707107 + 1.22474i
0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 1.22474i
0 1.73205i 0 8.36308i 0 0 0 −3.00000 0
97.2 0 1.73205i 0 1.43488i 0 0 0 −3.00000 0
97.3 0 1.73205i 0 1.43488i 0 0 0 −3.00000 0
97.4 0 1.73205i 0 8.36308i 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
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.3.f.e 4
4.b odd 2 1 294.3.c.a 4
7.b odd 2 1 inner 2352.3.f.e 4
7.c even 3 1 336.3.bh.e 4
7.d odd 6 1 336.3.bh.e 4
12.b even 2 1 882.3.c.b 4
21.g even 6 1 1008.3.cg.h 4
21.h odd 6 1 1008.3.cg.h 4
28.d even 2 1 294.3.c.a 4
28.f even 6 1 42.3.g.a 4
28.f even 6 1 294.3.g.a 4
28.g odd 6 1 42.3.g.a 4
28.g odd 6 1 294.3.g.a 4
84.h odd 2 1 882.3.c.b 4
84.j odd 6 1 126.3.n.a 4
84.j odd 6 1 882.3.n.e 4
84.n even 6 1 126.3.n.a 4
84.n even 6 1 882.3.n.e 4
140.p odd 6 1 1050.3.p.a 4
140.s even 6 1 1050.3.p.a 4
140.w even 12 2 1050.3.q.a 8
140.x odd 12 2 1050.3.q.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.g.a 4 28.f even 6 1
42.3.g.a 4 28.g odd 6 1
126.3.n.a 4 84.j odd 6 1
126.3.n.a 4 84.n even 6 1
294.3.c.a 4 4.b odd 2 1
294.3.c.a 4 28.d even 2 1
294.3.g.a 4 28.f even 6 1
294.3.g.a 4 28.g odd 6 1
336.3.bh.e 4 7.c even 3 1
336.3.bh.e 4 7.d odd 6 1
882.3.c.b 4 12.b even 2 1
882.3.c.b 4 84.h odd 2 1
882.3.n.e 4 84.j odd 6 1
882.3.n.e 4 84.n even 6 1
1008.3.cg.h 4 21.g even 6 1
1008.3.cg.h 4 21.h odd 6 1
1050.3.p.a 4 140.p odd 6 1
1050.3.p.a 4 140.s even 6 1
1050.3.q.a 8 140.w even 12 2
1050.3.q.a 8 140.x odd 12 2
2352.3.f.e 4 1.a even 1 1 trivial
2352.3.f.e 4 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2352, [\chi])\):

\( T_{5}^{4} + 72 T_{5}^{2} + 144 \)
\( T_{11} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 3 + T^{2} )^{2} \)
$5$ \( 144 + 72 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 6 + T )^{4} \)
$13$ \( 145161 + 774 T^{2} + T^{4} \)
$17$ \( 28224 + 432 T^{2} + T^{4} \)
$19$ \( 15129 + 342 T^{2} + T^{4} \)
$23$ \( ( -504 - 24 T + T^{2} )^{2} \)
$29$ \( ( -1152 + T^{2} )^{2} \)
$31$ \( 423801 + 2166 T^{2} + T^{4} \)
$37$ \( ( -167 + 22 T + T^{2} )^{2} \)
$41$ \( 3732624 + 4248 T^{2} + T^{4} \)
$43$ \( ( -23 + 14 T + T^{2} )^{2} \)
$47$ \( 2039184 + 2952 T^{2} + T^{4} \)
$53$ \( ( 2952 + 120 T + T^{2} )^{2} \)
$59$ \( 1272384 + 2448 T^{2} + T^{4} \)
$61$ \( 2304 + 1632 T^{2} + T^{4} \)
$67$ \( ( -503 - 110 T + T^{2} )^{2} \)
$71$ \( ( 2556 + 156 T + T^{2} )^{2} \)
$73$ \( 85322169 + 19926 T^{2} + T^{4} \)
$79$ \( ( -8687 + 10 T + T^{2} )^{2} \)
$83$ \( 69956496 + 17928 T^{2} + T^{4} \)
$89$ \( ( 432 + T^{2} )^{2} \)
$97$ \( 112896 + 1056 T^{2} + T^{4} \)
show more
show less