Properties

Label 2352.2.h.h
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: \( 2 \)
Twist minimal: no (minimal twist has level 336)
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.

\(f(q)\) \(=\) \( q + ( -\zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{5} + ( 1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{5} + ( 1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{11} -2 q^{13} + ( -\zeta_{8} - 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{15} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{17} -4 \zeta_{8}^{2} q^{19} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{23} + 3 q^{25} + ( \zeta_{8} + 5 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{27} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{29} -6 \zeta_{8}^{2} q^{31} + ( -2 + \zeta_{8} + \zeta_{8}^{3} ) q^{33} -4 q^{37} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{39} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{41} + 4 \zeta_{8}^{2} q^{43} + ( 4 + \zeta_{8} + \zeta_{8}^{3} ) q^{45} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{47} + ( -5 \zeta_{8} - 10 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{51} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{53} + 2 \zeta_{8}^{2} q^{55} + ( 4 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{57} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{59} + 6 q^{61} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{65} + 12 \zeta_{8}^{2} q^{67} + ( 10 - 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{69} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{71} + 2 q^{73} + ( -3 \zeta_{8} + 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{75} -12 \zeta_{8}^{2} q^{79} + ( -7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{81} + ( 10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{83} -10 q^{85} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{87} + ( -7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{89} + ( 6 + 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{93} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{95} -14 q^{97} + ( \zeta_{8} - 4 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{9} - 8q^{13} + 12q^{25} - 8q^{33} - 16q^{37} + 16q^{45} + 16q^{57} + 24q^{61} + 40q^{69} + 8q^{73} - 28q^{81} - 40q^{85} + 24q^{93} - 56q^{97} + O(q^{100}) \)

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 −1.41421 1.00000i 0 1.41421i 0 0 0 1.00000 + 2.82843i 0
2255.2 0 −1.41421 + 1.00000i 0 1.41421i 0 0 0 1.00000 2.82843i 0
2255.3 0 1.41421 1.00000i 0 1.41421i 0 0 0 1.00000 2.82843i 0
2255.4 0 1.41421 + 1.00000i 0 1.41421i 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
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b 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.h 4
3.b odd 2 1 inner 2352.2.h.h 4
4.b odd 2 1 inner 2352.2.h.h 4
7.b odd 2 1 336.2.h.a 4
12.b even 2 1 inner 2352.2.h.h 4
21.c even 2 1 336.2.h.a 4
28.d even 2 1 336.2.h.a 4
56.e even 2 1 1344.2.h.c 4
56.h odd 2 1 1344.2.h.c 4
84.h odd 2 1 336.2.h.a 4
168.e odd 2 1 1344.2.h.c 4
168.i even 2 1 1344.2.h.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.h.a 4 7.b odd 2 1
336.2.h.a 4 21.c even 2 1
336.2.h.a 4 28.d even 2 1
336.2.h.a 4 84.h odd 2 1
1344.2.h.c 4 56.e even 2 1
1344.2.h.c 4 56.h odd 2 1
1344.2.h.c 4 168.e odd 2 1
1344.2.h.c 4 168.i even 2 1
2352.2.h.h 4 1.a even 1 1 trivial
2352.2.h.h 4 3.b odd 2 1 inner
2352.2.h.h 4 4.b odd 2 1 inner
2352.2.h.h 4 12.b 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 \(S_{2}^{\mathrm{new}}(2352, [\chi])\):

\( T_{5}^{2} + 2 \)
\( T_{11}^{2} - 2 \)
\( T_{13} + 2 \)
\( T_{47}^{2} - 72 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 - 2 T^{2} + T^{4} \)
$5$ \( ( 2 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( ( -2 + T^{2} )^{2} \)
$13$ \( ( 2 + T )^{4} \)
$17$ \( ( 50 + T^{2} )^{2} \)
$19$ \( ( 16 + T^{2} )^{2} \)
$23$ \( ( -50 + T^{2} )^{2} \)
$29$ \( ( 8 + T^{2} )^{2} \)
$31$ \( ( 36 + T^{2} )^{2} \)
$37$ \( ( 4 + T )^{4} \)
$41$ \( ( 2 + T^{2} )^{2} \)
$43$ \( ( 16 + T^{2} )^{2} \)
$47$ \( ( -72 + T^{2} )^{2} \)
$53$ \( ( 128 + T^{2} )^{2} \)
$59$ \( ( -128 + T^{2} )^{2} \)
$61$ \( ( -6 + T )^{4} \)
$67$ \( ( 144 + T^{2} )^{2} \)
$71$ \( ( -50 + T^{2} )^{2} \)
$73$ \( ( -2 + T )^{4} \)
$79$ \( ( 144 + T^{2} )^{2} \)
$83$ \( ( -200 + T^{2} )^{2} \)
$89$ \( ( 98 + T^{2} )^{2} \)
$97$ \( ( 14 + T )^{4} \)
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