Properties

Label 1344.2.b.f
Level $1344$
Weight $2$
Character orbit 1344.b
Analytic conductor $10.732$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.7318940317\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.2312.1
Defining polynomial: \(x^{4} - x^{3} - 2 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 84)
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.

\(f(q)\) \(=\) \( q + q^{3} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{5} + ( 1 - \beta_{1} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{5} + ( 1 - \beta_{1} ) q^{7} + q^{9} -\beta_{2} q^{11} -2 \beta_{2} q^{13} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{15} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{17} + ( -3 - \beta_{1} + \beta_{3} ) q^{19} + ( 1 - \beta_{1} ) q^{21} -\beta_{2} q^{23} + ( -2 - \beta_{1} + \beta_{3} ) q^{25} + q^{27} + 2 q^{29} -\beta_{2} q^{33} + ( 4 + \beta_{2} - 2 \beta_{3} ) q^{35} + ( 3 - \beta_{1} + \beta_{3} ) q^{37} -2 \beta_{2} q^{39} + ( -1 + \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{41} + ( -3 + 3 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{43} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{45} + ( -2 + 2 \beta_{1} - 2 \beta_{3} ) q^{47} + ( \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{49} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{51} + ( -4 - 2 \beta_{1} + 2 \beta_{3} ) q^{53} + ( 1 - \beta_{1} + \beta_{3} ) q^{55} + ( -3 - \beta_{1} + \beta_{3} ) q^{57} -4 q^{59} + ( -2 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{61} + ( 1 - \beta_{1} ) q^{63} + ( 2 - 2 \beta_{1} + 2 \beta_{3} ) q^{65} + ( 3 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{67} -\beta_{2} q^{69} + ( -2 + 2 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{71} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{73} + ( -2 - \beta_{1} + \beta_{3} ) q^{75} + ( 3 - \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{77} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{79} + q^{81} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{83} + ( -5 - 3 \beta_{1} + 3 \beta_{3} ) q^{85} + 2 q^{87} + ( 1 - \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{89} + ( 6 - 2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{91} + ( 6 - 6 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{95} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{97} -\beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 2q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 2q^{7} + 4q^{9} - 12q^{19} + 2q^{21} - 8q^{25} + 4q^{27} + 8q^{29} + 12q^{35} + 12q^{37} - 8q^{47} + 8q^{49} - 16q^{53} + 4q^{55} - 12q^{57} - 16q^{59} + 2q^{63} + 8q^{65} - 8q^{75} + 8q^{77} + 4q^{81} + 8q^{83} - 20q^{85} + 8q^{87} + 16q^{91} + O(q^{100}) \)

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
895.1
1.28078 0.599676i
−0.780776 1.17915i
−0.780776 + 1.17915i
1.28078 + 0.599676i
0 1.00000 0 3.33513i 0 −1.56155 + 2.13578i 0 1.00000 0
895.2 0 1.00000 0 1.69614i 0 2.56155 0.662153i 0 1.00000 0
895.3 0 1.00000 0 1.69614i 0 2.56155 + 0.662153i 0 1.00000 0
895.4 0 1.00000 0 3.33513i 0 −1.56155 2.13578i 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
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 1344.2.b.f 4
3.b odd 2 1 4032.2.b.n 4
4.b odd 2 1 1344.2.b.e 4
7.b odd 2 1 1344.2.b.e 4
8.b even 2 1 84.2.b.a 4
8.d odd 2 1 84.2.b.b yes 4
12.b even 2 1 4032.2.b.j 4
21.c even 2 1 4032.2.b.j 4
24.f even 2 1 252.2.b.d 4
24.h odd 2 1 252.2.b.e 4
28.d even 2 1 inner 1344.2.b.f 4
56.e even 2 1 84.2.b.a 4
56.h odd 2 1 84.2.b.b yes 4
56.j odd 6 2 588.2.o.a 8
56.k odd 6 2 588.2.o.a 8
56.m even 6 2 588.2.o.c 8
56.p even 6 2 588.2.o.c 8
84.h odd 2 1 4032.2.b.n 4
168.e odd 2 1 252.2.b.e 4
168.i even 2 1 252.2.b.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.b.a 4 8.b even 2 1
84.2.b.a 4 56.e even 2 1
84.2.b.b yes 4 8.d odd 2 1
84.2.b.b yes 4 56.h odd 2 1
252.2.b.d 4 24.f even 2 1
252.2.b.d 4 168.i even 2 1
252.2.b.e 4 24.h odd 2 1
252.2.b.e 4 168.e odd 2 1
588.2.o.a 8 56.j odd 6 2
588.2.o.a 8 56.k odd 6 2
588.2.o.c 8 56.m even 6 2
588.2.o.c 8 56.p even 6 2
1344.2.b.e 4 4.b odd 2 1
1344.2.b.e 4 7.b odd 2 1
1344.2.b.f 4 1.a even 1 1 trivial
1344.2.b.f 4 28.d even 2 1 inner
4032.2.b.j 4 12.b even 2 1
4032.2.b.j 4 21.c even 2 1
4032.2.b.n 4 3.b odd 2 1
4032.2.b.n 4 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1344, [\chi])\):

\( T_{5}^{4} + 14 T_{5}^{2} + 32 \)
\( T_{19}^{2} + 6 T_{19} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( 32 + 14 T^{2} + T^{4} \)
$7$ \( 49 - 14 T - 2 T^{2} - 2 T^{3} + T^{4} \)
$11$ \( 8 + 10 T^{2} + T^{4} \)
$13$ \( 128 + 40 T^{2} + T^{4} \)
$17$ \( 512 + 46 T^{2} + T^{4} \)
$19$ \( ( -8 + 6 T + T^{2} )^{2} \)
$23$ \( 8 + 10 T^{2} + T^{4} \)
$29$ \( ( -2 + T )^{4} \)
$31$ \( T^{4} \)
$37$ \( ( -8 - 6 T + T^{2} )^{2} \)
$41$ \( 128 + 62 T^{2} + T^{4} \)
$43$ \( 5408 + 148 T^{2} + T^{4} \)
$47$ \( ( -64 + 4 T + T^{2} )^{2} \)
$53$ \( ( -52 + 8 T + T^{2} )^{2} \)
$59$ \( ( 4 + T )^{4} \)
$61$ \( 2048 + 112 T^{2} + T^{4} \)
$67$ \( 512 + 124 T^{2} + T^{4} \)
$71$ \( 2312 + 170 T^{2} + T^{4} \)
$73$ \( 512 + 56 T^{2} + T^{4} \)
$79$ \( 128 + 28 T^{2} + T^{4} \)
$83$ \( ( -64 - 4 T + T^{2} )^{2} \)
$89$ \( 128 + 62 T^{2} + T^{4} \)
$97$ \( 8192 + 184 T^{2} + T^{4} \)
show more
show less