Properties

Label 1216.1.p.a
Level $1216$
Weight $1$
Character orbit 1216.p
Analytic conductor $0.607$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -8
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1216.p (of order \(6\), degree \(2\), 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: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.1267762688.2

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{3} + ( - \zeta_{12}^{4} - \zeta_{12}^{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{3} + ( - \zeta_{12}^{4} - \zeta_{12}^{2} - 1) q^{9} - \zeta_{12}^{3} q^{11} + \zeta_{12}^{4} q^{17} - \zeta_{12}^{5} q^{19} - \zeta_{12}^{2} q^{25} + (\zeta_{12}^{5} + \zeta_{12}) q^{27} + ( - \zeta_{12}^{2} - 1) q^{33} + (\zeta_{12}^{2} + 1) q^{41} + \zeta_{12} q^{43} - q^{49} + (2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{51} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{57} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{59} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{67} + \zeta_{12}^{4} q^{73} + (\zeta_{12}^{5} - \zeta_{12}) q^{75} + (\zeta_{12}^{4} + 1) q^{81} - \zeta_{12}^{3} q^{83} + (\zeta_{12}^{2} + 1) q^{97} + (\zeta_{12}^{5} + \zeta_{12}^{3} - \zeta_{12}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} - 4 q^{17} - 2 q^{25} - 6 q^{33} + 6 q^{41} - 4 q^{49} - 2 q^{73} - 2 q^{81} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
673.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 −0.866025 + 1.50000i 0 0 0 0 0 −1.00000 1.73205i 0
673.2 0 0.866025 1.50000i 0 0 0 0 0 −1.00000 1.73205i 0
1057.1 0 −0.866025 1.50000i 0 0 0 0 0 −1.00000 + 1.73205i 0
1057.2 0 0.866025 + 1.50000i 0 0 0 0 0 −1.00000 + 1.73205i 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
8.b even 2 1 inner
19.d odd 6 1 inner
76.f even 6 1 inner
152.l odd 6 1 inner
152.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.1.p.a 4
4.b odd 2 1 inner 1216.1.p.a 4
8.b even 2 1 inner 1216.1.p.a 4
8.d odd 2 1 CM 1216.1.p.a 4
19.d odd 6 1 inner 1216.1.p.a 4
76.f even 6 1 inner 1216.1.p.a 4
152.l odd 6 1 inner 1216.1.p.a 4
152.o even 6 1 inner 1216.1.p.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.1.p.a 4 1.a even 1 1 trivial
1216.1.p.a 4 4.b odd 2 1 inner
1216.1.p.a 4 8.b even 2 1 inner
1216.1.p.a 4 8.d odd 2 1 CM
1216.1.p.a 4 19.d odd 6 1 inner
1216.1.p.a 4 76.f even 6 1 inner
1216.1.p.a 4 152.l odd 6 1 inner
1216.1.p.a 4 152.o even 6 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
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