Properties

Label 1368.2.e.d
Level $1368$
Weight $2$
Character orbit 1368.e
Analytic conductor $10.924$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

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: \(8\)
Coefficient field: 8.0.207360000.1
Defining polynomial: \(x^{8} + 6 x^{6} + 32 x^{4} + 24 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{5} ) q^{2} + ( -1 - \beta_{4} ) q^{4} + 2 \beta_{4} q^{7} + 2 \beta_{5} q^{8} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{5} ) q^{2} + ( -1 - \beta_{4} ) q^{4} + 2 \beta_{4} q^{7} + 2 \beta_{5} q^{8} -\beta_{7} q^{11} + 2 \beta_{3} q^{13} + ( -2 \beta_{1} - 2 \beta_{5} ) q^{14} + ( -2 + 2 \beta_{4} ) q^{16} + 2 \beta_{7} q^{17} + ( -2 - \beta_{6} ) q^{19} + ( -\beta_{3} + \beta_{6} ) q^{22} + ( 2 \beta_{2} + \beta_{7} ) q^{23} + 5 q^{25} + ( 2 \beta_{2} + 2 \beta_{7} ) q^{26} + ( 6 - 2 \beta_{4} ) q^{28} -\beta_{5} q^{29} + 4 \beta_{3} q^{31} -4 \beta_{1} q^{32} + ( 2 \beta_{3} - 2 \beta_{6} ) q^{34} + 2 \beta_{3} q^{37} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{5} - \beta_{7} ) q^{38} + ( -2 \beta_{1} + \beta_{5} ) q^{41} -2 \beta_{2} q^{44} + ( -3 \beta_{3} - \beta_{6} ) q^{46} + ( 2 \beta_{2} + \beta_{7} ) q^{47} -5 q^{49} + ( 5 \beta_{1} - 5 \beta_{5} ) q^{50} + ( -2 \beta_{3} - 2 \beta_{6} ) q^{52} -3 \beta_{5} q^{53} + ( 8 \beta_{1} - 4 \beta_{5} ) q^{56} + ( 1 - \beta_{4} ) q^{58} + ( 4 \beta_{1} - 2 \beta_{5} ) q^{59} + 6 \beta_{4} q^{61} + ( 4 \beta_{2} + 4 \beta_{7} ) q^{62} + 8 q^{64} + 2 \beta_{6} q^{67} + 4 \beta_{2} q^{68} -8 \beta_{5} q^{71} -12 q^{73} + ( 2 \beta_{2} + 2 \beta_{7} ) q^{74} + ( 2 - 3 \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{76} + ( 4 \beta_{2} + 2 \beta_{7} ) q^{77} + 2 \beta_{3} q^{79} + ( 3 + \beta_{4} ) q^{82} -5 \beta_{7} q^{83} + 4 \beta_{3} q^{88} + ( -14 \beta_{1} + 7 \beta_{5} ) q^{89} + 4 \beta_{6} q^{91} + ( -2 \beta_{2} - 4 \beta_{7} ) q^{92} + ( -3 \beta_{3} - \beta_{6} ) q^{94} + 4 \beta_{6} q^{97} + ( -5 \beta_{1} + 5 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} + O(q^{10}) \) \( 8 q - 8 q^{4} - 16 q^{16} - 16 q^{19} + 40 q^{25} + 48 q^{28} - 40 q^{49} + 8 q^{58} + 64 q^{64} - 96 q^{73} + 16 q^{76} + 24 q^{82} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 6 x^{6} + 32 x^{4} + 24 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} - 104 \nu \)\()/64\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + 232 \nu \)\()/64\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} - 72 \)\()/32\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{6} + 16 \nu^{4} + 96 \nu^{2} + 40 \)\()/32\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{7} + 16 \nu^{5} + 80 \nu^{3} + 8 \nu \)\()/32\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} - 6 \nu^{4} - 28 \nu^{2} - 12 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{7} + 16 \nu^{5} + 88 \nu^{3} + 8 \nu \)\()/16\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} + 3 \beta_{4} - \beta_{3} - 3\)\()/2\)
\(\nu^{3}\)\(=\)\(2 \beta_{7} - 4 \beta_{5}\)
\(\nu^{4}\)\(=\)\(-3 \beta_{6} - 7 \beta_{4} - 3 \beta_{3} - 7\)
\(\nu^{5}\)\(=\)\(-10 \beta_{7} + 22 \beta_{5} - 10 \beta_{2} - 22 \beta_{1}\)
\(\nu^{6}\)\(=\)\(32 \beta_{3} + 72\)
\(\nu^{7}\)\(=\)\(52 \beta_{2} + 116 \beta_{1}\)

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
−0.437016 + 0.756934i
1.14412 1.98168i
−0.437016 0.756934i
1.14412 + 1.98168i
−1.14412 1.98168i
0.437016 + 0.756934i
−1.14412 + 1.98168i
0.437016 0.756934i
−0.707107 1.22474i 0 −1.00000 + 1.73205i 0 0 3.46410i 2.82843 0 0
379.2 −0.707107 1.22474i 0 −1.00000 + 1.73205i 0 0 3.46410i 2.82843 0 0
379.3 −0.707107 + 1.22474i 0 −1.00000 1.73205i 0 0 3.46410i 2.82843 0 0
379.4 −0.707107 + 1.22474i 0 −1.00000 1.73205i 0 0 3.46410i 2.82843 0 0
379.5 0.707107 1.22474i 0 −1.00000 1.73205i 0 0 3.46410i −2.82843 0 0
379.6 0.707107 1.22474i 0 −1.00000 1.73205i 0 0 3.46410i −2.82843 0 0
379.7 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 0 0 3.46410i −2.82843 0 0
379.8 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 0 0 3.46410i −2.82843 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 379.8
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{5} \)
\( T_{7}^{2} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + 2 T^{2} + T^{4} )^{2} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 12 + T^{2} )^{4} \)
$11$ \( ( -10 + T^{2} )^{4} \)
$13$ \( ( -20 + T^{2} )^{4} \)
$17$ \( ( -40 + T^{2} )^{4} \)
$19$ \( ( 19 + 4 T + T^{2} )^{4} \)
$23$ \( ( 30 + T^{2} )^{4} \)
$29$ \( ( -2 + T^{2} )^{4} \)
$31$ \( ( -80 + T^{2} )^{4} \)
$37$ \( ( -20 + T^{2} )^{4} \)
$41$ \( ( 6 + T^{2} )^{4} \)
$43$ \( T^{8} \)
$47$ \( ( 30 + T^{2} )^{4} \)
$53$ \( ( -18 + T^{2} )^{4} \)
$59$ \( ( 24 + T^{2} )^{4} \)
$61$ \( ( 108 + T^{2} )^{4} \)
$67$ \( ( 60 + T^{2} )^{4} \)
$71$ \( ( -128 + T^{2} )^{4} \)
$73$ \( ( 12 + T )^{8} \)
$79$ \( ( -20 + T^{2} )^{4} \)
$83$ \( ( -250 + T^{2} )^{4} \)
$89$ \( ( 294 + T^{2} )^{4} \)
$97$ \( ( 240 + T^{2} )^{4} \)
show more
show less