Properties

Label 4368.2.h.f
Level $4368$
Weight $2$
Character orbit 4368.h
Analytic conductor $34.879$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4368.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(34.8786556029\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + 2 i q^{5} -i q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + 2 i q^{5} -i q^{7} + q^{9} + ( 2 + 3 i ) q^{13} -2 i q^{15} -2 q^{17} -4 i q^{19} + i q^{21} + 6 q^{23} + q^{25} - q^{27} + 2 q^{35} -2 i q^{37} + ( -2 - 3 i ) q^{39} -4 q^{43} + 2 i q^{45} -8 i q^{47} - q^{49} + 2 q^{51} + 4 q^{53} + 4 i q^{57} + 6 i q^{59} + 12 q^{61} -i q^{63} + ( -6 + 4 i ) q^{65} + 2 i q^{67} -6 q^{69} -14 i q^{73} - q^{75} + q^{81} + 14 i q^{83} -4 i q^{85} + 4 i q^{89} + ( 3 - 2 i ) q^{91} + 8 q^{95} -2 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{9} + 4q^{13} - 4q^{17} + 12q^{23} + 2q^{25} - 2q^{27} + 4q^{35} - 4q^{39} - 8q^{43} - 2q^{49} + 4q^{51} + 8q^{53} + 24q^{61} - 12q^{65} - 12q^{69} - 2q^{75} + 2q^{81} + 6q^{91} + 16q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times\).

\(n\) \(1093\) \(1249\) \(1457\) \(2017\) \(3823\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.00000i
1.00000i
0 −1.00000 0 2.00000i 0 1.00000i 0 1.00000 0
337.2 0 −1.00000 0 2.00000i 0 1.00000i 0 1.00000 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
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4368.2.h.f 2
4.b odd 2 1 546.2.c.b 2
12.b even 2 1 1638.2.c.b 2
13.b even 2 1 inner 4368.2.h.f 2
28.d even 2 1 3822.2.c.c 2
52.b odd 2 1 546.2.c.b 2
52.f even 4 1 7098.2.a.k 1
52.f even 4 1 7098.2.a.bc 1
156.h even 2 1 1638.2.c.b 2
364.h even 2 1 3822.2.c.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.c.b 2 4.b odd 2 1
546.2.c.b 2 52.b odd 2 1
1638.2.c.b 2 12.b even 2 1
1638.2.c.b 2 156.h even 2 1
3822.2.c.c 2 28.d even 2 1
3822.2.c.c 2 364.h even 2 1
4368.2.h.f 2 1.a even 1 1 trivial
4368.2.h.f 2 13.b even 2 1 inner
7098.2.a.k 1 52.f even 4 1
7098.2.a.bc 1 52.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4368, [\chi])\):

\( T_{5}^{2} + 4 \)
\( T_{11} \)
\( T_{17} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( 4 + T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 13 - 4 T + T^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( 16 + T^{2} \)
$23$ \( ( -6 + T )^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 4 + T )^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( ( -4 + T )^{2} \)
$59$ \( 36 + T^{2} \)
$61$ \( ( -12 + T )^{2} \)
$67$ \( 4 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 196 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( 196 + T^{2} \)
$89$ \( 16 + T^{2} \)
$97$ \( 4 + T^{2} \)
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