Properties

Label 1216.1.g.a
Level $1216$
Weight $1$
Character orbit 1216.g
Analytic conductor $0.607$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -8, -19, 152
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.606863055362\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-2}, \sqrt{-19})\)
Artin image: $D_4:C_2$
Artin field: Galois closure of 8.0.378535936.1

$q$-expansion

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

\(f(q)\) \(=\) \( q - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{9} - i q^{11} + q^{17} - i q^{19} + q^{25} + i q^{43} - q^{49} - q^{73} + q^{81} - i q^{83} + 2 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} + 4 q^{17} + 2 q^{25} - 2 q^{49} - 4 q^{73} + 2 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
417.1
1.00000i
1.00000i
0 0 0 0 0 0 0 −1.00000 0
417.2 0 0 0 0 0 0 0 −1.00000 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 \(\Q(\sqrt{-2}) \)
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
152.b even 2 1 RM by \(\Q(\sqrt{38}) \)
4.b odd 2 1 inner
8.b even 2 1 inner
76.d even 2 1 inner
152.g odd 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{1}^{\mathrm{new}}(1216, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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