Properties

Label 1216.2.t.a
Level $1216$
Weight $2$
Character orbit 1216.t
Analytic conductor $9.710$
Analytic rank $0$
Dimension $8$
CM discriminant -8
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{3} + ( 2 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{9} +O(q^{10})\) \( q + ( -2 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{3} + ( 2 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{9} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 3 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{11} + ( 6 - 6 \zeta_{24}^{4} ) q^{17} + ( -3 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{19} -5 \zeta_{24}^{4} q^{25} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 13 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{27} + ( 3 + 4 \zeta_{24} - 2 \zeta_{24}^{3} - 3 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{33} + ( -3 + 8 \zeta_{24} - 4 \zeta_{24}^{3} + 3 \zeta_{24}^{4} - 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{41} -10 \zeta_{24}^{2} q^{43} -7 q^{49} + ( -6 \zeta_{24} + 6 \zeta_{24}^{2} - 12 \zeta_{24}^{3} + 12 \zeta_{24}^{5} - 6 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{51} + ( -5 + 2 \zeta_{24} - 4 \zeta_{24}^{3} + 12 \zeta_{24}^{4} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{57} + ( -10 \zeta_{24} - 3 \zeta_{24}^{2} - 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} - 5 \zeta_{24}^{7} ) q^{59} + ( -3 \zeta_{24} + 7 \zeta_{24}^{2} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 7 \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{67} + ( 1 - 12 \zeta_{24} + 6 \zeta_{24}^{3} - \zeta_{24}^{4} + 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{73} + ( 5 \zeta_{24} - 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} - 5 \zeta_{24}^{6} + 10 \zeta_{24}^{7} ) q^{75} + ( -19 + 20 \zeta_{24} - 10 \zeta_{24}^{3} + 19 \zeta_{24}^{4} - 10 \zeta_{24}^{5} - 10 \zeta_{24}^{7} ) q^{81} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 9 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{83} -18 \zeta_{24}^{4} q^{89} + ( 5 + 12 \zeta_{24} - 6 \zeta_{24}^{3} - 5 \zeta_{24}^{4} - 6 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{97} + ( 2 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 16q^{9} + O(q^{10}) \) \( 8q + 16q^{9} + 24q^{17} - 20q^{25} + 12q^{33} - 12q^{41} - 56q^{49} + 8q^{57} + 4q^{73} - 76q^{81} - 72q^{89} + 20q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(-\zeta_{24}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
353.1
−0.258819 + 0.965926i
0.965926 + 0.258819i
0.258819 0.965926i
−0.965926 0.258819i
−0.258819 0.965926i
0.965926 0.258819i
0.258819 + 0.965926i
−0.965926 + 0.258819i
0 −2.98735 1.72474i 0 0 0 0 0 4.44949 + 7.70674i 0
353.2 0 −1.25529 0.724745i 0 0 0 0 0 −0.449490 0.778539i 0
353.3 0 1.25529 + 0.724745i 0 0 0 0 0 −0.449490 0.778539i 0
353.4 0 2.98735 + 1.72474i 0 0 0 0 0 4.44949 + 7.70674i 0
1185.1 0 −2.98735 + 1.72474i 0 0 0 0 0 4.44949 7.70674i 0
1185.2 0 −1.25529 + 0.724745i 0 0 0 0 0 −0.449490 + 0.778539i 0
1185.3 0 1.25529 0.724745i 0 0 0 0 0 −0.449490 + 0.778539i 0
1185.4 0 2.98735 1.72474i 0 0 0 0 0 4.44949 7.70674i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1185.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
8.b even 2 1 inner
19.c even 3 1 inner
76.g odd 6 1 inner
152.k odd 6 1 inner
152.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.t.a 8
4.b odd 2 1 inner 1216.2.t.a 8
8.b even 2 1 inner 1216.2.t.a 8
8.d odd 2 1 CM 1216.2.t.a 8
19.c even 3 1 inner 1216.2.t.a 8
76.g odd 6 1 inner 1216.2.t.a 8
152.k odd 6 1 inner 1216.2.t.a 8
152.p even 6 1 inner 1216.2.t.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.t.a 8 1.a even 1 1 trivial
1216.2.t.a 8 4.b odd 2 1 inner
1216.2.t.a 8 8.b even 2 1 inner
1216.2.t.a 8 8.d odd 2 1 CM
1216.2.t.a 8 19.c even 3 1 inner
1216.2.t.a 8 76.g odd 6 1 inner
1216.2.t.a 8 152.k odd 6 1 inner
1216.2.t.a 8 152.p even 6 1 inner

Hecke kernels

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

\( T_{3}^{8} - 14 T_{3}^{6} + 171 T_{3}^{4} - 350 T_{3}^{2} + 625 \)
\( T_{5} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 625 - 350 T^{2} + 171 T^{4} - 14 T^{6} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( T^{8} \)
$11$ \( ( 9 + 30 T^{2} + T^{4} )^{2} \)
$13$ \( T^{8} \)
$17$ \( ( 36 - 6 T + T^{2} )^{4} \)
$19$ \( 130321 + 12274 T^{2} + 795 T^{4} + 34 T^{6} + T^{8} \)
$23$ \( T^{8} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( ( 7569 - 522 T + 123 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$43$ \( ( 10000 - 100 T^{2} + T^{4} )^{2} \)
$47$ \( T^{8} \)
$53$ \( T^{8} \)
$59$ \( 395254161 - 6322158 T^{2} + 81243 T^{4} - 318 T^{6} + T^{8} \)
$61$ \( T^{8} \)
$67$ \( 625 - 5150 T^{2} + 42411 T^{4} - 206 T^{6} + T^{8} \)
$71$ \( T^{8} \)
$73$ \( ( 46225 + 430 T + 219 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$79$ \( T^{8} \)
$83$ \( ( 5625 + 174 T^{2} + T^{4} )^{2} \)
$89$ \( ( 324 + 18 T + T^{2} )^{4} \)
$97$ \( ( 36481 + 1910 T + 291 T^{2} - 10 T^{3} + T^{4} )^{2} \)
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