Properties

Label 1216.2.h.a
Level $1216$
Weight $2$
Character orbit 1216.h
Analytic conductor $9.710$
Analytic rank $0$
Dimension $2$
CM discriminant -19
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-19}) \)
Defining polynomial: \(x^{2} - x + 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 304)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} + \beta q^{7} -3 q^{9} +O(q^{10})\) \( q + q^{5} + \beta q^{7} -3 q^{9} -\beta q^{11} + 7 q^{17} + \beta q^{19} + 2 \beta q^{23} -4 q^{25} + \beta q^{35} + 3 \beta q^{43} -3 q^{45} + \beta q^{47} -12 q^{49} -\beta q^{55} -15 q^{61} -3 \beta q^{63} + 11 q^{73} + 19 q^{77} + 9 q^{81} + 2 \beta q^{83} + 7 q^{85} + \beta q^{95} + 3 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} - 6q^{9} + O(q^{10}) \) \( 2q + 2q^{5} - 6q^{9} + 14q^{17} - 8q^{25} - 6q^{45} - 24q^{49} - 30q^{61} + 22q^{73} + 38q^{77} + 18q^{81} + 14q^{85} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1215.1
0.500000 2.17945i
0.500000 + 2.17945i
0 0 0 1.00000 0 4.35890i 0 −3.00000 0
1215.2 0 0 0 1.00000 0 4.35890i 0 −3.00000 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 \(\Q(\sqrt{-19}) \)
4.b odd 2 1 inner
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.h.a 2
4.b odd 2 1 inner 1216.2.h.a 2
8.b even 2 1 304.2.h.a 2
8.d odd 2 1 304.2.h.a 2
19.b odd 2 1 CM 1216.2.h.a 2
24.f even 2 1 2736.2.k.g 2
24.h odd 2 1 2736.2.k.g 2
76.d even 2 1 inner 1216.2.h.a 2
152.b even 2 1 304.2.h.a 2
152.g odd 2 1 304.2.h.a 2
456.l odd 2 1 2736.2.k.g 2
456.p even 2 1 2736.2.k.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.h.a 2 8.b even 2 1
304.2.h.a 2 8.d odd 2 1
304.2.h.a 2 152.b even 2 1
304.2.h.a 2 152.g odd 2 1
1216.2.h.a 2 1.a even 1 1 trivial
1216.2.h.a 2 4.b odd 2 1 inner
1216.2.h.a 2 19.b odd 2 1 CM
1216.2.h.a 2 76.d even 2 1 inner
2736.2.k.g 2 24.f even 2 1
2736.2.k.g 2 24.h odd 2 1
2736.2.k.g 2 456.l odd 2 1
2736.2.k.g 2 456.p even 2 1

Hecke kernels

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

\( T_{3} \)
\( T_{5} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( 19 + T^{2} \)
$11$ \( 19 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( -7 + T )^{2} \)
$19$ \( 19 + T^{2} \)
$23$ \( 76 + T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( 171 + T^{2} \)
$47$ \( 19 + T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 15 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -11 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( 76 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
show more
show less