Properties

Label 1216.1.q.a
Level $1216$
Weight $1$
Character orbit 1216.q
Analytic conductor $0.607$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1216.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.606863055362\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 608)
Projective image \(A_{4}\)
Projective field Galois closure of 4.0.23104.1
Artin image $\SL(2,3):C_2$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{3} + \zeta_{12}^{4} q^{5} +O(q^{10})\) \( q + \zeta_{12} q^{3} + \zeta_{12}^{4} q^{5} + \zeta_{12}^{2} q^{13} + \zeta_{12}^{5} q^{15} + \zeta_{12}^{4} q^{17} + \zeta_{12}^{3} q^{19} -\zeta_{12}^{5} q^{23} -\zeta_{12}^{3} q^{27} + \zeta_{12}^{2} q^{29} -2 \zeta_{12}^{3} q^{31} + \zeta_{12}^{3} q^{39} + \zeta_{12}^{4} q^{41} -\zeta_{12} q^{43} + \zeta_{12}^{5} q^{47} + q^{49} + \zeta_{12}^{5} q^{51} -\zeta_{12}^{2} q^{53} + \zeta_{12}^{4} q^{57} -\zeta_{12} q^{59} -\zeta_{12}^{2} q^{61} - q^{65} -\zeta_{12}^{5} q^{67} + q^{69} + \zeta_{12} q^{71} -\zeta_{12}^{4} q^{73} + \zeta_{12} q^{79} -\zeta_{12}^{4} q^{81} -2 \zeta_{12}^{3} q^{83} -\zeta_{12}^{2} q^{85} + \zeta_{12}^{3} q^{87} -\zeta_{12}^{2} q^{89} -2 \zeta_{12}^{4} q^{93} -\zeta_{12} q^{95} + \zeta_{12}^{4} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} + O(q^{10}) \) \( 4q - 2q^{5} + 2q^{13} - 2q^{17} + 2q^{29} - 2q^{41} + 4q^{49} - 2q^{53} - 2q^{57} - 2q^{61} - 4q^{65} + 4q^{69} + 2q^{73} + 2q^{81} - 2q^{85} - 2q^{89} + 4q^{93} - 2q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
767.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0 −0.866025 + 0.500000i 0 −0.500000 0.866025i 0 0 0 0 0
767.2 0 0.866025 0.500000i 0 −0.500000 0.866025i 0 0 0 0 0
1151.1 0 −0.866025 0.500000i 0 −0.500000 + 0.866025i 0 0 0 0 0
1151.2 0 0.866025 + 0.500000i 0 −0.500000 + 0.866025i 0 0 0 0 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
4.b odd 2 1 inner
19.c even 3 1 inner
76.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.1.q.a 4
4.b odd 2 1 inner 1216.1.q.a 4
8.b even 2 1 608.1.q.a 4
8.d odd 2 1 608.1.q.a 4
19.c even 3 1 inner 1216.1.q.a 4
76.g odd 6 1 inner 1216.1.q.a 4
152.k odd 6 1 608.1.q.a 4
152.p even 6 1 608.1.q.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.1.q.a 4 8.b even 2 1
608.1.q.a 4 8.d odd 2 1
608.1.q.a 4 152.k odd 6 1
608.1.q.a 4 152.p even 6 1
1216.1.q.a 4 1.a even 1 1 trivial
1216.1.q.a 4 4.b odd 2 1 inner
1216.1.q.a 4 19.c even 3 1 inner
1216.1.q.a 4 76.g odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1216, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( ( 1 + T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( ( 1 - T + T^{2} )^{2} \)
$17$ \( ( 1 + T + T^{2} )^{2} \)
$19$ \( ( 1 + T^{2} )^{2} \)
$23$ \( 1 - T^{2} + T^{4} \)
$29$ \( ( 1 - T + T^{2} )^{2} \)
$31$ \( ( 4 + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( ( 1 + T + T^{2} )^{2} \)
$43$ \( 1 - T^{2} + T^{4} \)
$47$ \( 1 - T^{2} + T^{4} \)
$53$ \( ( 1 + T + T^{2} )^{2} \)
$59$ \( 1 - T^{2} + T^{4} \)
$61$ \( ( 1 + T + T^{2} )^{2} \)
$67$ \( 1 - T^{2} + T^{4} \)
$71$ \( 1 - T^{2} + T^{4} \)
$73$ \( ( 1 - T + T^{2} )^{2} \)
$79$ \( 1 - T^{2} + T^{4} \)
$83$ \( ( 4 + T^{2} )^{2} \)
$89$ \( ( 1 + T + T^{2} )^{2} \)
$97$ \( ( 1 + T + T^{2} )^{2} \)
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