Properties

Label 1368.3.o.a
Level $1368$
Weight $3$
Character orbit 1368.o
Analytic conductor $37.275$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1368.o (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(37.2753001645\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -7 q^{5} + 11 q^{7} +O(q^{10})\) \( q -7 q^{5} + 11 q^{7} -3 q^{11} + 2 \beta q^{13} + 17 q^{17} -19 q^{19} -2 q^{23} + 24 q^{25} -7 \beta q^{29} -\beta q^{31} -77 q^{35} + 7 \beta q^{37} + 7 \beta q^{41} -21 q^{43} + 5 q^{47} + 72 q^{49} + \beta q^{53} + 21 q^{55} + 6 \beta q^{59} + 23 q^{61} -14 \beta q^{65} -7 \beta q^{67} + 16 \beta q^{71} + 39 q^{73} -33 q^{77} + 17 \beta q^{79} + 6 q^{83} -119 q^{85} + 21 \beta q^{89} + 22 \beta q^{91} + 133 q^{95} + 30 \beta q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 14q^{5} + 22q^{7} + O(q^{10}) \) \( 2q - 14q^{5} + 22q^{7} - 6q^{11} + 34q^{17} - 38q^{19} - 4q^{23} + 48q^{25} - 154q^{35} - 42q^{43} + 10q^{47} + 144q^{49} + 42q^{55} + 46q^{61} + 78q^{73} - 66q^{77} + 12q^{83} - 238q^{85} + 266q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times\).

\(n\) \(343\) \(685\) \(1009\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
721.1
1.41421i
1.41421i
0 0 0 −7.00000 0 11.0000 0 0 0
721.2 0 0 0 −7.00000 0 11.0000 0 0 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
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.3.o.a 2
3.b odd 2 1 152.3.e.a 2
4.b odd 2 1 2736.3.o.b 2
12.b even 2 1 304.3.e.f 2
19.b odd 2 1 inner 1368.3.o.a 2
24.f even 2 1 1216.3.e.c 2
24.h odd 2 1 1216.3.e.d 2
57.d even 2 1 152.3.e.a 2
76.d even 2 1 2736.3.o.b 2
228.b odd 2 1 304.3.e.f 2
456.l odd 2 1 1216.3.e.c 2
456.p even 2 1 1216.3.e.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.3.e.a 2 3.b odd 2 1
152.3.e.a 2 57.d even 2 1
304.3.e.f 2 12.b even 2 1
304.3.e.f 2 228.b odd 2 1
1216.3.e.c 2 24.f even 2 1
1216.3.e.c 2 456.l odd 2 1
1216.3.e.d 2 24.h odd 2 1
1216.3.e.d 2 456.p even 2 1
1368.3.o.a 2 1.a even 1 1 trivial
1368.3.o.a 2 19.b odd 2 1 inner
2736.3.o.b 2 4.b odd 2 1
2736.3.o.b 2 76.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 7 \) acting on \(S_{3}^{\mathrm{new}}(1368, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 7 + T )^{2} \)
$7$ \( ( -11 + T )^{2} \)
$11$ \( ( 3 + T )^{2} \)
$13$ \( 128 + T^{2} \)
$17$ \( ( -17 + T )^{2} \)
$19$ \( ( 19 + T )^{2} \)
$23$ \( ( 2 + T )^{2} \)
$29$ \( 1568 + T^{2} \)
$31$ \( 32 + T^{2} \)
$37$ \( 1568 + T^{2} \)
$41$ \( 1568 + T^{2} \)
$43$ \( ( 21 + T )^{2} \)
$47$ \( ( -5 + T )^{2} \)
$53$ \( 32 + T^{2} \)
$59$ \( 1152 + T^{2} \)
$61$ \( ( -23 + T )^{2} \)
$67$ \( 1568 + T^{2} \)
$71$ \( 8192 + T^{2} \)
$73$ \( ( -39 + T )^{2} \)
$79$ \( 9248 + T^{2} \)
$83$ \( ( -6 + T )^{2} \)
$89$ \( 14112 + T^{2} \)
$97$ \( 28800 + T^{2} \)
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