Properties

Label 2352.2.h.k
Level $2352$
Weight $2$
Character orbit 2352.h
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.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{8} - \zeta_{8}^{3} ) q^{3} -2 \zeta_{8}^{2} q^{5} + ( -1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{8} - \zeta_{8}^{3} ) q^{3} -2 \zeta_{8}^{2} q^{5} + ( -1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{11} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{13} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{15} + 2 \zeta_{8}^{2} q^{17} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{23} + q^{25} + ( -5 - \zeta_{8} - \zeta_{8}^{3} ) q^{27} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{29} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{31} + ( 3 \zeta_{8} - 6 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{33} + 6 q^{37} + ( -\zeta_{8} + 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{39} + 10 \zeta_{8}^{2} q^{41} -12 \zeta_{8}^{2} q^{43} + ( -4 \zeta_{8} + 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{45} -6 q^{47} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{51} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{53} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{55} -12 q^{59} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{61} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{65} + 12 \zeta_{8}^{2} q^{67} + ( 3 \zeta_{8} - 6 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{69} + ( -9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{71} + ( 7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{73} + ( 1 - \zeta_{8} - \zeta_{8}^{3} ) q^{75} + 12 \zeta_{8}^{2} q^{79} + ( -7 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{81} -6 q^{83} + 4 q^{85} + ( -12 - 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{87} + 2 \zeta_{8}^{2} q^{89} + ( -12 - 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{93} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{97} + ( -3 \zeta_{8} - 12 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 4q^{9} + 4q^{25} - 20q^{27} + 24q^{37} - 24q^{47} - 48q^{59} + 4q^{75} - 28q^{81} - 24q^{83} + 16q^{85} - 48q^{87} - 48q^{93} + O(q^{100}) \)

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
0.707107 + 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
0 1.00000 1.41421i 0 2.00000i 0 0 0 −1.00000 2.82843i 0
2255.2 0 1.00000 1.41421i 0 2.00000i 0 0 0 −1.00000 2.82843i 0
2255.3 0 1.00000 + 1.41421i 0 2.00000i 0 0 0 −1.00000 + 2.82843i 0
2255.4 0 1.00000 + 1.41421i 0 2.00000i 0 0 0 −1.00000 + 2.82843i 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
12.b even 2 1 inner
21.c even 2 1 inner
28.d 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.k yes 4
3.b odd 2 1 2352.2.h.f 4
4.b odd 2 1 2352.2.h.f 4
7.b odd 2 1 2352.2.h.f 4
12.b even 2 1 inner 2352.2.h.k yes 4
21.c even 2 1 inner 2352.2.h.k yes 4
28.d even 2 1 inner 2352.2.h.k yes 4
84.h odd 2 1 2352.2.h.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.2.h.f 4 3.b odd 2 1
2352.2.h.f 4 4.b odd 2 1
2352.2.h.f 4 7.b odd 2 1
2352.2.h.f 4 84.h odd 2 1
2352.2.h.k yes 4 1.a even 1 1 trivial
2352.2.h.k yes 4 12.b even 2 1 inner
2352.2.h.k yes 4 21.c even 2 1 inner
2352.2.h.k yes 4 28.d even 2 1 inner

Hecke kernels

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

\( T_{5}^{2} + 4 \)
\( T_{11}^{2} - 18 \)
\( T_{13}^{2} - 2 \)
\( T_{47} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 3 - 2 T + T^{2} )^{2} \)
$5$ \( ( 4 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( ( -18 + T^{2} )^{2} \)
$13$ \( ( -2 + T^{2} )^{2} \)
$17$ \( ( 4 + T^{2} )^{2} \)
$19$ \( T^{4} \)
$23$ \( ( -18 + T^{2} )^{2} \)
$29$ \( ( 72 + T^{2} )^{2} \)
$31$ \( ( 72 + T^{2} )^{2} \)
$37$ \( ( -6 + T )^{4} \)
$41$ \( ( 100 + T^{2} )^{2} \)
$43$ \( ( 144 + T^{2} )^{2} \)
$47$ \( ( 6 + T )^{4} \)
$53$ \( ( 32 + T^{2} )^{2} \)
$59$ \( ( 12 + T )^{4} \)
$61$ \( ( -2 + T^{2} )^{2} \)
$67$ \( ( 144 + T^{2} )^{2} \)
$71$ \( ( -162 + T^{2} )^{2} \)
$73$ \( ( -98 + T^{2} )^{2} \)
$79$ \( ( 144 + T^{2} )^{2} \)
$83$ \( ( 6 + T )^{4} \)
$89$ \( ( 4 + T^{2} )^{2} \)
$97$ \( ( -50 + T^{2} )^{2} \)
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