Properties

Label 1368.2.e.b
Level $1368$
Weight $2$
Character orbit 1368.e
Analytic conductor $10.924$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(10.9235349965\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial: \(x^{4} - x^{3} + 2 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 - \beta_{3} ) q^{4} + ( -1 + 2 \beta_{3} ) q^{7} + ( \beta_{1} - 2 \beta_{2} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 - \beta_{3} ) q^{4} + ( -1 + 2 \beta_{3} ) q^{7} + ( \beta_{1} - 2 \beta_{2} ) q^{8} + 4 q^{11} + ( -\beta_{1} - \beta_{2} ) q^{13} + ( -\beta_{1} + 4 \beta_{2} ) q^{14} + ( -3 - \beta_{3} ) q^{16} + q^{17} + ( 4 + \beta_{1} - \beta_{2} ) q^{19} + 4 \beta_{1} q^{22} + ( 1 - 2 \beta_{3} ) q^{23} + 5 q^{25} + ( -3 + \beta_{3} ) q^{26} + ( 7 + \beta_{3} ) q^{28} + ( \beta_{1} + \beta_{2} ) q^{29} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{31} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{32} + \beta_{1} q^{34} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{37} + ( -1 + 4 \beta_{1} - \beta_{3} ) q^{38} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{41} + ( 4 - 4 \beta_{3} ) q^{44} + ( \beta_{1} - 4 \beta_{2} ) q^{46} + ( -2 + 4 \beta_{3} ) q^{47} -8 q^{49} + 5 \beta_{1} q^{50} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{52} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{53} + ( 7 \beta_{1} + 2 \beta_{2} ) q^{56} + ( 3 - \beta_{3} ) q^{58} + ( 5 \beta_{1} - 5 \beta_{2} ) q^{59} + ( 12 - 4 \beta_{3} ) q^{62} + ( -7 + 3 \beta_{3} ) q^{64} + ( 7 \beta_{1} - 7 \beta_{2} ) q^{67} + ( 1 - \beta_{3} ) q^{68} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{71} + 3 q^{73} + ( 6 - 2 \beta_{3} ) q^{74} + ( 4 - \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{76} + ( -4 + 8 \beta_{3} ) q^{77} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{79} + ( 4 + 4 \beta_{3} ) q^{82} + 8 q^{83} + ( 4 \beta_{1} - 8 \beta_{2} ) q^{88} + ( 8 \beta_{1} - 8 \beta_{2} ) q^{89} + ( 5 \beta_{1} - 5 \beta_{2} ) q^{91} + ( -7 - \beta_{3} ) q^{92} + ( -2 \beta_{1} + 8 \beta_{2} ) q^{94} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{97} -8 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + O(q^{10}) \) \( 4 q + 2 q^{4} + 16 q^{11} - 14 q^{16} + 4 q^{17} + 16 q^{19} + 20 q^{25} - 10 q^{26} + 30 q^{28} - 6 q^{38} + 8 q^{44} - 32 q^{49} + 10 q^{58} + 40 q^{62} - 22 q^{64} + 2 q^{68} + 12 q^{73} + 20 q^{74} + 8 q^{76} + 24 q^{82} + 32 q^{83} - 30 q^{92} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 2 x^{2} + x + 1\):

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1
0.809017 1.40126i
0.809017 + 1.40126i
−0.309017 + 0.535233i
−0.309017 0.535233i
−1.11803 0.866025i 0 0.500000 + 1.93649i 0 0 3.87298i 1.11803 2.59808i 0 0
379.2 −1.11803 + 0.866025i 0 0.500000 1.93649i 0 0 3.87298i 1.11803 + 2.59808i 0 0
379.3 1.11803 0.866025i 0 0.500000 1.93649i 0 0 3.87298i −1.11803 2.59808i 0 0
379.4 1.11803 + 0.866025i 0 0.500000 + 1.93649i 0 0 3.87298i −1.11803 + 2.59808i 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
8.d odd 2 1 inner
19.b odd 2 1 inner
152.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.2.e.b 4
3.b odd 2 1 152.2.b.b 4
4.b odd 2 1 5472.2.e.b 4
8.b even 2 1 5472.2.e.b 4
8.d odd 2 1 inner 1368.2.e.b 4
12.b even 2 1 608.2.b.b 4
19.b odd 2 1 inner 1368.2.e.b 4
24.f even 2 1 152.2.b.b 4
24.h odd 2 1 608.2.b.b 4
57.d even 2 1 152.2.b.b 4
76.d even 2 1 5472.2.e.b 4
152.b even 2 1 inner 1368.2.e.b 4
152.g odd 2 1 5472.2.e.b 4
228.b odd 2 1 608.2.b.b 4
456.l odd 2 1 152.2.b.b 4
456.p even 2 1 608.2.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.b.b 4 3.b odd 2 1
152.2.b.b 4 24.f even 2 1
152.2.b.b 4 57.d even 2 1
152.2.b.b 4 456.l odd 2 1
608.2.b.b 4 12.b even 2 1
608.2.b.b 4 24.h odd 2 1
608.2.b.b 4 228.b odd 2 1
608.2.b.b 4 456.p even 2 1
1368.2.e.b 4 1.a even 1 1 trivial
1368.2.e.b 4 8.d odd 2 1 inner
1368.2.e.b 4 19.b odd 2 1 inner
1368.2.e.b 4 152.b even 2 1 inner
5472.2.e.b 4 4.b odd 2 1
5472.2.e.b 4 8.b even 2 1
5472.2.e.b 4 76.d even 2 1
5472.2.e.b 4 152.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1368, [\chi])\):

\( T_{5} \)
\( T_{7}^{2} + 15 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 15 + T^{2} )^{2} \)
$11$ \( ( -4 + T )^{4} \)
$13$ \( ( -5 + T^{2} )^{2} \)
$17$ \( ( -1 + T )^{4} \)
$19$ \( ( 19 - 8 T + T^{2} )^{2} \)
$23$ \( ( 15 + T^{2} )^{2} \)
$29$ \( ( -5 + T^{2} )^{2} \)
$31$ \( ( -80 + T^{2} )^{2} \)
$37$ \( ( -20 + T^{2} )^{2} \)
$41$ \( ( 48 + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( ( 60 + T^{2} )^{2} \)
$53$ \( ( -45 + T^{2} )^{2} \)
$59$ \( ( 75 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( ( 147 + T^{2} )^{2} \)
$71$ \( ( -80 + T^{2} )^{2} \)
$73$ \( ( -3 + T )^{4} \)
$79$ \( ( -80 + T^{2} )^{2} \)
$83$ \( ( -8 + T )^{4} \)
$89$ \( ( 192 + T^{2} )^{2} \)
$97$ \( ( 48 + T^{2} )^{2} \)
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