Properties

Label 2352.2.bl.n
Level $2352$
Weight $2$
Character orbit 2352.bl
Analytic conductor $18.781$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.bl (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \( 3 \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} ) q^{3} + \beta_{3} q^{5} + \beta_{1} q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} ) q^{3} + \beta_{3} q^{5} + \beta_{1} q^{9} + ( \beta_{2} - \beta_{3} ) q^{11} -2 \beta_{2} q^{13} + \beta_{2} q^{15} + ( \beta_{2} - \beta_{3} ) q^{17} + 2 \beta_{1} q^{19} + 3 \beta_{3} q^{23} + ( 1 + \beta_{1} ) q^{25} - q^{27} + 6 q^{29} + ( 8 + 8 \beta_{1} ) q^{31} -\beta_{3} q^{33} + 4 \beta_{1} q^{37} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{39} + 3 \beta_{2} q^{41} + 2 \beta_{2} q^{43} + ( \beta_{2} - \beta_{3} ) q^{45} + 12 \beta_{1} q^{47} -\beta_{3} q^{51} + ( 6 + 6 \beta_{1} ) q^{53} -6 q^{55} -2 q^{57} + ( 12 + 12 \beta_{1} ) q^{59} -12 \beta_{1} q^{65} + 3 \beta_{2} q^{69} -5 \beta_{2} q^{71} + ( -6 \beta_{2} + 6 \beta_{3} ) q^{73} + \beta_{1} q^{75} -4 \beta_{3} q^{79} + ( -1 - \beta_{1} ) q^{81} -6 q^{85} + ( 6 + 6 \beta_{1} ) q^{87} -5 \beta_{3} q^{89} + 8 \beta_{1} q^{93} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{95} -2 \beta_{2} q^{97} -\beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} - 2q^{9} + O(q^{10}) \) \( 4q + 2q^{3} - 2q^{9} - 4q^{19} + 2q^{25} - 4q^{27} + 24q^{29} + 16q^{31} - 8q^{37} - 24q^{47} + 12q^{53} - 24q^{55} - 8q^{57} + 24q^{59} + 24q^{65} - 2q^{75} - 2q^{81} - 24q^{85} + 12q^{87} - 16q^{93} + O(q^{100}) \)

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

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

Character values

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

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
0 0.500000 0.866025i 0 −2.12132 + 1.22474i 0 0 0 −0.500000 0.866025i 0
31.2 0 0.500000 0.866025i 0 2.12132 1.22474i 0 0 0 −0.500000 0.866025i 0
607.1 0 0.500000 + 0.866025i 0 −2.12132 1.22474i 0 0 0 −0.500000 + 0.866025i 0
607.2 0 0.500000 + 0.866025i 0 2.12132 + 1.22474i 0 0 0 −0.500000 + 0.866025i 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.c even 3 1 inner
28.d even 2 1 inner
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.2.bl.n 4
4.b odd 2 1 2352.2.bl.m 4
7.b odd 2 1 2352.2.bl.m 4
7.c even 3 1 336.2.b.b 2
7.c even 3 1 inner 2352.2.bl.n 4
7.d odd 6 1 336.2.b.c yes 2
7.d odd 6 1 2352.2.bl.m 4
21.g even 6 1 1008.2.b.d 2
21.h odd 6 1 1008.2.b.e 2
28.d even 2 1 inner 2352.2.bl.n 4
28.f even 6 1 336.2.b.b 2
28.f even 6 1 inner 2352.2.bl.n 4
28.g odd 6 1 336.2.b.c yes 2
28.g odd 6 1 2352.2.bl.m 4
56.j odd 6 1 1344.2.b.a 2
56.k odd 6 1 1344.2.b.a 2
56.m even 6 1 1344.2.b.d 2
56.p even 6 1 1344.2.b.d 2
84.j odd 6 1 1008.2.b.e 2
84.n even 6 1 1008.2.b.d 2
168.s odd 6 1 4032.2.b.f 2
168.v even 6 1 4032.2.b.d 2
168.ba even 6 1 4032.2.b.d 2
168.be odd 6 1 4032.2.b.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.b.b 2 7.c even 3 1
336.2.b.b 2 28.f even 6 1
336.2.b.c yes 2 7.d odd 6 1
336.2.b.c yes 2 28.g odd 6 1
1008.2.b.d 2 21.g even 6 1
1008.2.b.d 2 84.n even 6 1
1008.2.b.e 2 21.h odd 6 1
1008.2.b.e 2 84.j odd 6 1
1344.2.b.a 2 56.j odd 6 1
1344.2.b.a 2 56.k odd 6 1
1344.2.b.d 2 56.m even 6 1
1344.2.b.d 2 56.p even 6 1
2352.2.bl.m 4 4.b odd 2 1
2352.2.bl.m 4 7.b odd 2 1
2352.2.bl.m 4 7.d odd 6 1
2352.2.bl.m 4 28.g odd 6 1
2352.2.bl.n 4 1.a even 1 1 trivial
2352.2.bl.n 4 7.c even 3 1 inner
2352.2.bl.n 4 28.d even 2 1 inner
2352.2.bl.n 4 28.f even 6 1 inner
4032.2.b.d 2 168.v even 6 1
4032.2.b.d 2 168.ba even 6 1
4032.2.b.f 2 168.s odd 6 1
4032.2.b.f 2 168.be odd 6 1

Hecke kernels

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

\( T_{5}^{4} - 6 T_{5}^{2} + 36 \)
\( T_{11}^{4} - 6 T_{11}^{2} + 36 \)
\( T_{17}^{4} - 6 T_{17}^{2} + 36 \)
\( T_{19}^{2} + 2 T_{19} + 4 \)
\( T_{31}^{2} - 8 T_{31} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 1 - T + T^{2} )^{2} \)
$5$ \( 36 - 6 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 36 - 6 T^{2} + T^{4} \)
$13$ \( ( 24 + T^{2} )^{2} \)
$17$ \( 36 - 6 T^{2} + T^{4} \)
$19$ \( ( 4 + 2 T + T^{2} )^{2} \)
$23$ \( 2916 - 54 T^{2} + T^{4} \)
$29$ \( ( -6 + T )^{4} \)
$31$ \( ( 64 - 8 T + T^{2} )^{2} \)
$37$ \( ( 16 + 4 T + T^{2} )^{2} \)
$41$ \( ( 54 + T^{2} )^{2} \)
$43$ \( ( 24 + T^{2} )^{2} \)
$47$ \( ( 144 + 12 T + T^{2} )^{2} \)
$53$ \( ( 36 - 6 T + T^{2} )^{2} \)
$59$ \( ( 144 - 12 T + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( ( 150 + T^{2} )^{2} \)
$73$ \( 46656 - 216 T^{2} + T^{4} \)
$79$ \( 9216 - 96 T^{2} + T^{4} \)
$83$ \( T^{4} \)
$89$ \( 22500 - 150 T^{2} + T^{4} \)
$97$ \( ( 24 + T^{2} )^{2} \)
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