Properties

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

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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: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{2} ) q^{5} + ( 2 \beta_{1} + \beta_{3} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{2} ) q^{5} + ( 2 \beta_{1} + \beta_{3} ) q^{7} -3 q^{9} + ( 2 \beta_{1} - \beta_{3} ) q^{11} + ( -3 - \beta_{2} ) q^{17} + ( -\beta_{1} - 2 \beta_{3} ) q^{19} + ( -2 \beta_{1} - 4 \beta_{3} ) q^{23} + ( 10 - \beta_{2} ) q^{25} + ( -8 \beta_{1} - 5 \beta_{3} ) q^{35} + ( 2 \beta_{1} + 3 \beta_{3} ) q^{43} + ( 3 - 3 \beta_{2} ) q^{45} + ( -6 \beta_{1} + \beta_{3} ) q^{47} + ( -6 + 3 \beta_{2} ) q^{49} -7 \beta_{3} q^{55} + ( 7 + \beta_{2} ) q^{61} + ( -6 \beta_{1} - 3 \beta_{3} ) q^{63} + ( -7 + 3 \beta_{2} ) q^{73} + ( -7 + \beta_{2} ) q^{77} + 9 q^{81} + ( -2 \beta_{1} - 4 \beta_{3} ) q^{83} + ( -11 - 3 \beta_{2} ) q^{85} + ( 10 \beta_{1} + \beta_{3} ) q^{95} + ( -6 \beta_{1} + 3 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} - 12q^{9} + O(q^{10}) \) \( 4q - 2q^{5} - 12q^{9} - 14q^{17} + 38q^{25} + 6q^{45} - 18q^{49} + 30q^{61} - 22q^{73} - 26q^{77} + 36q^{81} - 50q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 4 \nu^{2} - 4 \nu - 15 \)\()/10\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 9 \nu + 5 \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{3} - 2 \nu^{2} + 2 \nu + 25 \)\()/10\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + \beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + \beta_{2} + 5 \beta_{1} + 4\)\()/2\)
\(\nu^{3}\)\(=\)\(-4 \beta_{3} - 2 \beta_{1} + 7\)

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
−1.63746 + 1.52274i
−1.63746 1.52274i
2.13746 0.656712i
2.13746 + 0.656712i
0 0 0 −4.27492 0 4.77753i 0 −3.00000 0
1215.2 0 0 0 −4.27492 0 4.77753i 0 −3.00000 0
1215.3 0 0 0 3.27492 0 0.418627i 0 −3.00000 0
1215.4 0 0 0 3.27492 0 0.418627i 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.b 4
4.b odd 2 1 inner 1216.2.h.b 4
8.b even 2 1 304.2.h.c 4
8.d odd 2 1 304.2.h.c 4
19.b odd 2 1 CM 1216.2.h.b 4
24.f even 2 1 2736.2.k.j 4
24.h odd 2 1 2736.2.k.j 4
76.d even 2 1 inner 1216.2.h.b 4
152.b even 2 1 304.2.h.c 4
152.g odd 2 1 304.2.h.c 4
456.l odd 2 1 2736.2.k.j 4
456.p even 2 1 2736.2.k.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.h.c 4 8.b even 2 1
304.2.h.c 4 8.d odd 2 1
304.2.h.c 4 152.b even 2 1
304.2.h.c 4 152.g odd 2 1
1216.2.h.b 4 1.a even 1 1 trivial
1216.2.h.b 4 4.b odd 2 1 inner
1216.2.h.b 4 19.b odd 2 1 CM
1216.2.h.b 4 76.d even 2 1 inner
2736.2.k.j 4 24.f even 2 1
2736.2.k.j 4 24.h odd 2 1
2736.2.k.j 4 456.l odd 2 1
2736.2.k.j 4 456.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 \(S_{2}^{\mathrm{new}}(1216, [\chi])\):

\( T_{3} \)
\( T_{5}^{2} + T_{5} - 14 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -14 + T + T^{2} )^{2} \)
$7$ \( 4 + 23 T^{2} + T^{4} \)
$11$ \( 196 + 47 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( -2 + 7 T + T^{2} )^{2} \)
$19$ \( ( 19 + T^{2} )^{2} \)
$23$ \( ( 76 + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( 1764 + 87 T^{2} + T^{4} \)
$47$ \( 14884 + 263 T^{2} + T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 42 - 15 T + T^{2} )^{2} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( -98 + 11 T + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( ( 76 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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