Properties

Label 1216.2.c.i
Level $1216$
Weight $2$
Character orbit 1216.c
Analytic conductor $9.710$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.2702336256.1
Defining polynomial: \(x^{8} + 9 x^{6} + 56 x^{4} + 225 x^{2} + 625\)
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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{5} + \beta_{5} q^{7} + 3 q^{9} +O(q^{10})\) \( q -\beta_{3} q^{5} + \beta_{5} q^{7} + 3 q^{9} + ( -\beta_{2} - 2 \beta_{4} ) q^{11} + ( -\beta_{1} + \beta_{3} ) q^{13} -\beta_{6} q^{17} -\beta_{4} q^{19} + ( \beta_{5} - \beta_{7} ) q^{23} + ( -1 + \beta_{6} ) q^{25} + ( -\beta_{1} + \beta_{3} ) q^{29} -2 \beta_{7} q^{31} + ( \beta_{2} - 2 \beta_{4} ) q^{35} + ( -\beta_{1} - 3 \beta_{3} ) q^{37} + ( -2 - 2 \beta_{6} ) q^{41} + ( \beta_{2} - 6 \beta_{4} ) q^{43} -3 \beta_{3} q^{45} -\beta_{5} q^{47} + ( 3 + 3 \beta_{6} ) q^{49} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{53} + ( 2 \beta_{5} - 5 \beta_{7} ) q^{55} + 2 \beta_{2} q^{59} -\beta_{3} q^{61} + 3 \beta_{5} q^{63} + 8 q^{65} + 2 \beta_{2} q^{67} + ( 2 \beta_{5} + 4 \beta_{7} ) q^{71} + \beta_{6} q^{73} + ( -2 \beta_{1} + \beta_{3} ) q^{77} + ( 2 \beta_{5} - 4 \beta_{7} ) q^{79} + 9 q^{81} + ( -4 \beta_{2} - 4 \beta_{4} ) q^{83} + ( -2 \beta_{1} - 3 \beta_{3} ) q^{85} -10 q^{89} + ( -4 \beta_{2} + 12 \beta_{4} ) q^{91} -\beta_{7} q^{95} + ( 10 - 2 \beta_{6} ) q^{97} + ( -3 \beta_{2} - 6 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 24q^{9} + O(q^{10}) \) \( 8q + 24q^{9} - 4q^{17} - 4q^{25} - 24q^{41} + 36q^{49} + 64q^{65} + 4q^{73} + 72q^{81} - 80q^{89} + 72q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 9 x^{6} + 56 x^{4} + 225 x^{2} + 625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 84 \nu^{4} + 756 \nu^{2} + 2325 \)\()/700\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} - 14 \nu^{5} - 126 \nu^{3} - 855 \nu \)\()/350\)
\(\beta_{3}\)\(=\)\((\)\( 17 \nu^{6} + 28 \nu^{4} + 252 \nu^{2} + 325 \)\()/700\)
\(\beta_{4}\)\(=\)\((\)\( -9 \nu^{7} - 56 \nu^{5} - 154 \nu^{3} - 625 \nu \)\()/1750\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{7} + 52 \nu^{5} + 268 \nu^{3} + 575 \nu \)\()/500\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} - 9 \nu^{4} - 31 \nu^{2} - 100 \)\()/25\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} + 4 \nu^{5} + 36 \nu^{3} + 45 \nu \)\()/100\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} - \beta_{5} - 2 \beta_{4} - 2 \beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{6} + 3 \beta_{3} + 5 \beta_{1} - 10\)\()/4\)
\(\nu^{3}\)\(=\)\(3 \beta_{7} + \beta_{5} + 7 \beta_{4}\)
\(\nu^{4}\)\(=\)\((\)\(-18 \beta_{6} - 29 \beta_{3} - 11 \beta_{1} - 22\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-67 \beta_{7} + 45 \beta_{5} - 90 \beta_{4} + 22 \beta_{2}\)\()/4\)
\(\nu^{6}\)\(=\)\(42 \beta_{3} - 14 \beta_{1} + 27\)
\(\nu^{7}\)\(=\)\((\)\(281 \beta_{7} - 279 \beta_{5} - 558 \beta_{4} + 2 \beta_{2}\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
609.1
−0.656712 2.13746i
0.656712 + 2.13746i
1.52274 1.63746i
−1.52274 + 1.63746i
1.52274 + 1.63746i
−1.52274 1.63746i
−0.656712 + 2.13746i
0.656712 2.13746i
0 0 0 3.04547i 0 −0.418627 0 3.00000 0
609.2 0 0 0 3.04547i 0 0.418627 0 3.00000 0
609.3 0 0 0 1.31342i 0 −4.77753 0 3.00000 0
609.4 0 0 0 1.31342i 0 4.77753 0 3.00000 0
609.5 0 0 0 1.31342i 0 −4.77753 0 3.00000 0
609.6 0 0 0 1.31342i 0 4.77753 0 3.00000 0
609.7 0 0 0 3.04547i 0 −0.418627 0 3.00000 0
609.8 0 0 0 3.04547i 0 0.418627 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 609.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.c.i 8
4.b odd 2 1 inner 1216.2.c.i 8
8.b even 2 1 inner 1216.2.c.i 8
8.d odd 2 1 inner 1216.2.c.i 8
16.e even 4 1 4864.2.a.bj 4
16.e even 4 1 4864.2.a.bk 4
16.f odd 4 1 4864.2.a.bj 4
16.f odd 4 1 4864.2.a.bk 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.c.i 8 1.a even 1 1 trivial
1216.2.c.i 8 4.b odd 2 1 inner
1216.2.c.i 8 8.b even 2 1 inner
1216.2.c.i 8 8.d odd 2 1 inner
4864.2.a.bj 4 16.e even 4 1
4864.2.a.bj 4 16.f odd 4 1
4864.2.a.bk 4 16.e even 4 1
4864.2.a.bk 4 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1216, [\chi])\):

\( T_{3} \)
\( T_{5}^{4} + 11 T_{5}^{2} + 16 \)
\( T_{7}^{4} - 23 T_{7}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 16 + 11 T^{2} + T^{4} )^{2} \)
$7$ \( ( 4 - 23 T^{2} + T^{4} )^{2} \)
$11$ \( ( 144 + 33 T^{2} + T^{4} )^{2} \)
$13$ \( ( 256 + 44 T^{2} + T^{4} )^{2} \)
$17$ \( ( -14 + T + T^{2} )^{4} \)
$19$ \( ( 1 + T^{2} )^{4} \)
$23$ \( ( -12 + T^{2} )^{4} \)
$29$ \( ( 256 + 44 T^{2} + T^{4} )^{2} \)
$31$ \( ( 256 - 44 T^{2} + T^{4} )^{2} \)
$37$ \( ( 64 + 92 T^{2} + T^{4} )^{2} \)
$41$ \( ( -48 + 6 T + T^{2} )^{4} \)
$43$ \( ( 784 + 113 T^{2} + T^{4} )^{2} \)
$47$ \( ( 4 - 23 T^{2} + T^{4} )^{2} \)
$53$ \( ( 108 + T^{2} )^{4} \)
$59$ \( ( 3136 + 116 T^{2} + T^{4} )^{2} \)
$61$ \( ( 16 + 11 T^{2} + T^{4} )^{2} \)
$67$ \( ( 3136 + 116 T^{2} + T^{4} )^{2} \)
$71$ \( ( 28224 - 348 T^{2} + T^{4} )^{2} \)
$73$ \( ( -14 - T + T^{2} )^{4} \)
$79$ \( ( 3136 - 188 T^{2} + T^{4} )^{2} \)
$83$ \( ( 50176 + 464 T^{2} + T^{4} )^{2} \)
$89$ \( ( 10 + T )^{8} \)
$97$ \( ( 24 - 18 T + T^{2} )^{4} \)
show more
show less