Properties

Label 1380.2.f.a
Level $1380$
Weight $2$
Character orbit 1380.f
Analytic conductor $11.019$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1380.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.0193554789\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \(x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{3} q^{3} + ( -\beta_{1} + \beta_{4} ) q^{5} -\beta_{3} q^{7} - q^{9} +O(q^{10})\) \( q -\beta_{3} q^{3} + ( -\beta_{1} + \beta_{4} ) q^{5} -\beta_{3} q^{7} - q^{9} -\beta_{1} q^{11} + ( \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{13} + ( -\beta_{2} - \beta_{5} ) q^{15} + ( -\beta_{4} + 2 \beta_{5} ) q^{17} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{19} - q^{21} -\beta_{3} q^{23} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{25} + \beta_{3} q^{27} + ( 2 - 3 \beta_{2} ) q^{29} + ( -4 - \beta_{1} + \beta_{2} ) q^{31} -\beta_{5} q^{33} + ( -\beta_{2} - \beta_{5} ) q^{35} + ( -3 \beta_{4} + \beta_{5} ) q^{37} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{39} + ( -3 - \beta_{1} - 4 \beta_{2} ) q^{41} + ( \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{43} + ( \beta_{1} - \beta_{4} ) q^{45} + ( -4 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{47} + 6 q^{49} + ( -2 \beta_{1} + \beta_{2} ) q^{51} + ( -\beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{53} + ( 3 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{55} + ( -\beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{57} + ( -2 - \beta_{2} ) q^{59} + ( -7 + 4 \beta_{1} - \beta_{2} ) q^{61} + \beta_{3} q^{63} + ( \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{65} + ( 2 \beta_{3} - 5 \beta_{4} - \beta_{5} ) q^{67} - q^{69} + ( -1 - 3 \beta_{1} + 4 \beta_{2} ) q^{71} + ( -7 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{73} + ( 2 + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{75} -\beta_{5} q^{77} + ( 2 - 6 \beta_{1} + 2 \beta_{2} ) q^{79} + q^{81} + ( 8 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{83} + ( \beta_{1} - 3 \beta_{2} + 5 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{85} + ( -2 \beta_{3} - 3 \beta_{4} ) q^{87} + ( -2 - 4 \beta_{2} ) q^{89} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{91} + ( 4 \beta_{3} + \beta_{4} - \beta_{5} ) q^{93} + ( 5 - 3 \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{95} + ( -3 \beta_{3} + 7 \beta_{4} - 5 \beta_{5} ) q^{97} + \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{9} + O(q^{10}) \) \( 6q - 6q^{9} + 2q^{15} + 4q^{19} - 6q^{21} + 2q^{25} + 18q^{29} - 26q^{31} + 2q^{35} + 4q^{39} - 10q^{41} + 36q^{49} - 2q^{51} + 16q^{55} - 10q^{59} - 40q^{61} + 4q^{65} - 6q^{69} - 14q^{71} + 8q^{75} + 8q^{79} + 6q^{81} + 6q^{85} - 4q^{89} + 4q^{91} + 28q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{5} + 8 \nu^{4} - 4 \nu^{3} - \nu^{2} + 2 \nu + 38 \)\()/23\)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{5} + 17 \nu^{4} - 20 \nu^{3} - 5 \nu^{2} + 10 \nu + 29 \)\()/23\)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{5} - 10 \nu^{4} + 5 \nu^{3} + 30 \nu^{2} + 32 \nu - 13 \)\()/23\)
\(\beta_{4}\)\(=\)\((\)\( -11 \nu^{5} + 19 \nu^{4} - 21 \nu^{3} - 11 \nu^{2} - 70 \nu + 27 \)\()/23\)
\(\beta_{5}\)\(=\)\((\)\( -14 \nu^{5} + 20 \nu^{4} - 10 \nu^{3} - 37 \nu^{2} - 64 \nu + 26 \)\()/23\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{5} + 2 \beta_{3}\)
\(\nu^{3}\)\(=\)\(2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_{1} - 2\)
\(\nu^{4}\)\(=\)\(-\beta_{2} + 5 \beta_{1} - 7\)
\(\nu^{5}\)\(=\)\(-8 \beta_{5} + 3 \beta_{4} - 9 \beta_{3} - 3 \beta_{2} + 8 \beta_{1} - 9\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(461\) \(691\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
829.1
0.403032 + 0.403032i
−0.854638 0.854638i
1.45161 + 1.45161i
0.403032 0.403032i
−0.854638 + 0.854638i
1.45161 1.45161i
0 1.00000i 0 −1.67513 1.48119i 0 1.00000i 0 −1.00000 0
829.2 0 1.00000i 0 −0.539189 + 2.17009i 0 1.00000i 0 −1.00000 0
829.3 0 1.00000i 0 2.21432 + 0.311108i 0 1.00000i 0 −1.00000 0
829.4 0 1.00000i 0 −1.67513 + 1.48119i 0 1.00000i 0 −1.00000 0
829.5 0 1.00000i 0 −0.539189 2.17009i 0 1.00000i 0 −1.00000 0
829.6 0 1.00000i 0 2.21432 0.311108i 0 1.00000i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 829.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1380.2.f.a 6
3.b odd 2 1 4140.2.f.a 6
5.b even 2 1 inner 1380.2.f.a 6
5.c odd 4 1 6900.2.a.y 3
5.c odd 4 1 6900.2.a.z 3
15.d odd 2 1 4140.2.f.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.f.a 6 1.a even 1 1 trivial
1380.2.f.a 6 5.b even 2 1 inner
4140.2.f.a 6 3.b odd 2 1
4140.2.f.a 6 15.d odd 2 1
6900.2.a.y 3 5.c odd 4 1
6900.2.a.z 3 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1380, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( ( 1 + T^{2} )^{3} \)
$5$ \( 125 - 5 T^{2} - 16 T^{3} - T^{4} + T^{6} \)
$7$ \( ( 1 + T^{2} )^{3} \)
$11$ \( ( -2 - 4 T + T^{3} )^{2} \)
$13$ \( 100 + 156 T^{2} + 32 T^{4} + T^{6} \)
$17$ \( 625 + 275 T^{2} + 31 T^{4} + T^{6} \)
$19$ \( ( -10 - 14 T - 2 T^{2} + T^{3} )^{2} \)
$23$ \( ( 1 + T^{2} )^{3} \)
$29$ \( ( 61 - 3 T - 9 T^{2} + T^{3} )^{2} \)
$31$ \( ( 59 + 51 T + 13 T^{2} + T^{3} )^{2} \)
$37$ \( 625 + 475 T^{2} + 59 T^{4} + T^{6} \)
$41$ \( ( 25 - 57 T + 5 T^{2} + T^{3} )^{2} \)
$43$ \( 5776 + 1600 T^{2} + 80 T^{4} + T^{6} \)
$47$ \( 3364 + 1080 T^{2} + 92 T^{4} + T^{6} \)
$53$ \( 4489 + 839 T^{2} + 51 T^{4} + T^{6} \)
$59$ \( ( -1 + 5 T + 5 T^{2} + T^{3} )^{2} \)
$61$ \( ( 74 + 74 T + 20 T^{2} + T^{3} )^{2} \)
$67$ \( 26569 + 9083 T^{2} + 195 T^{4} + T^{6} \)
$71$ \( ( -5 - 49 T + 7 T^{2} + T^{3} )^{2} \)
$73$ \( 42436 + 9980 T^{2} + 208 T^{4} + T^{6} \)
$79$ \( ( -416 - 128 T - 4 T^{2} + T^{3} )^{2} \)
$83$ \( 1849 + 10963 T^{2} + 215 T^{4} + T^{6} \)
$89$ \( ( -40 - 52 T + 2 T^{2} + T^{3} )^{2} \)
$97$ \( 781456 + 40400 T^{2} + 388 T^{4} + T^{6} \)
show more
show less