Properties

Label 1216.1.e.b
Level $1216$
Weight $1$
Character orbit 1216.e
Self dual yes
Analytic conductor $0.607$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -19
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.606863055362\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 76)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.76.1
Artin image $D_6$
Artin field Galois closure of 6.0.739328.2

$q$-expansion

\(f(q)\) \(=\) \( q + q^{5} + q^{7} + q^{9} + O(q^{10}) \) \( q + q^{5} + q^{7} + q^{9} - q^{11} - q^{17} + q^{19} - 2q^{23} + q^{35} - q^{43} + q^{45} + q^{47} - q^{55} + q^{61} + q^{63} - q^{73} - q^{77} + q^{81} + 2q^{83} - q^{85} + q^{95} - q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1025.1
0
0 0 0 1.00000 0 1.00000 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.1.e.b 1
4.b odd 2 1 1216.1.e.a 1
8.b even 2 1 304.1.e.a 1
8.d odd 2 1 76.1.c.a 1
19.b odd 2 1 CM 1216.1.e.b 1
24.f even 2 1 684.1.h.a 1
24.h odd 2 1 2736.1.o.b 1
40.e odd 2 1 1900.1.e.a 1
40.k even 4 2 1900.1.g.a 2
56.e even 2 1 3724.1.e.c 1
56.k odd 6 2 3724.1.bc.c 2
56.m even 6 2 3724.1.bc.b 2
76.d even 2 1 1216.1.e.a 1
152.b even 2 1 76.1.c.a 1
152.g odd 2 1 304.1.e.a 1
152.k odd 6 2 1444.1.h.a 2
152.o even 6 2 1444.1.h.a 2
152.u odd 18 6 1444.1.j.a 6
152.v even 18 6 1444.1.j.a 6
456.l odd 2 1 684.1.h.a 1
456.p even 2 1 2736.1.o.b 1
760.p even 2 1 1900.1.e.a 1
760.y odd 4 2 1900.1.g.a 2
1064.p odd 2 1 3724.1.e.c 1
1064.bi odd 6 2 3724.1.bc.b 2
1064.cm even 6 2 3724.1.bc.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.1.c.a 1 8.d odd 2 1
76.1.c.a 1 152.b even 2 1
304.1.e.a 1 8.b even 2 1
304.1.e.a 1 152.g odd 2 1
684.1.h.a 1 24.f even 2 1
684.1.h.a 1 456.l odd 2 1
1216.1.e.a 1 4.b odd 2 1
1216.1.e.a 1 76.d even 2 1
1216.1.e.b 1 1.a even 1 1 trivial
1216.1.e.b 1 19.b odd 2 1 CM
1444.1.h.a 2 152.k odd 6 2
1444.1.h.a 2 152.o even 6 2
1444.1.j.a 6 152.u odd 18 6
1444.1.j.a 6 152.v even 18 6
1900.1.e.a 1 40.e odd 2 1
1900.1.e.a 1 760.p even 2 1
1900.1.g.a 2 40.k even 4 2
1900.1.g.a 2 760.y odd 4 2
2736.1.o.b 1 24.h odd 2 1
2736.1.o.b 1 456.p even 2 1
3724.1.e.c 1 56.e even 2 1
3724.1.e.c 1 1064.p odd 2 1
3724.1.bc.b 2 56.m even 6 2
3724.1.bc.b 2 1064.bi odd 6 2
3724.1.bc.c 2 56.k odd 6 2
3724.1.bc.c 2 1064.cm even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 1 \) acting on \(S_{1}^{\mathrm{new}}(1216, [\chi])\).

Hecke characteristic polynomials

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