Properties

Label 1216.2.n.a
Level $1216$
Weight $2$
Character orbit 1216.n
Analytic conductor $9.710$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 304)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{3} + ( 3 - 3 \zeta_{6} ) q^{5} + ( -2 + 4 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{3} + ( 3 - 3 \zeta_{6} ) q^{5} + ( -2 + 4 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} + ( -2 + 4 \zeta_{6} ) q^{11} + ( 6 - 3 \zeta_{6} ) q^{13} + 3 \zeta_{6} q^{15} + ( -3 + 3 \zeta_{6} ) q^{17} + ( -3 - 2 \zeta_{6} ) q^{19} + ( -2 - 2 \zeta_{6} ) q^{21} + ( 6 - 3 \zeta_{6} ) q^{23} -4 \zeta_{6} q^{25} -5 q^{27} + ( -10 + 5 \zeta_{6} ) q^{29} -4 q^{31} + ( -2 - 2 \zeta_{6} ) q^{33} + ( 6 + 6 \zeta_{6} ) q^{35} + ( -3 + 6 \zeta_{6} ) q^{39} + ( 5 + 5 \zeta_{6} ) q^{41} + ( 7 + 7 \zeta_{6} ) q^{43} + 6 q^{45} + ( -2 + \zeta_{6} ) q^{47} -5 q^{49} -3 \zeta_{6} q^{51} + ( -2 + \zeta_{6} ) q^{53} + ( 6 + 6 \zeta_{6} ) q^{55} + ( 5 - 3 \zeta_{6} ) q^{57} + ( 3 - 3 \zeta_{6} ) q^{59} + 7 \zeta_{6} q^{61} + ( -8 + 4 \zeta_{6} ) q^{63} + ( 9 - 18 \zeta_{6} ) q^{65} + 5 \zeta_{6} q^{67} + ( -3 + 6 \zeta_{6} ) q^{69} + ( 9 - 9 \zeta_{6} ) q^{71} + ( -7 + 7 \zeta_{6} ) q^{73} + 4 q^{75} -12 q^{77} + ( -7 + 7 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} + ( 2 - 4 \zeta_{6} ) q^{83} + 9 \zeta_{6} q^{85} + ( 5 - 10 \zeta_{6} ) q^{87} + ( 10 - 5 \zeta_{6} ) q^{89} + 18 \zeta_{6} q^{91} + ( 4 - 4 \zeta_{6} ) q^{93} + ( -15 + 9 \zeta_{6} ) q^{95} + ( 5 + 5 \zeta_{6} ) q^{97} + ( -8 + 4 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + 3q^{5} + 2q^{9} + O(q^{10}) \) \( 2q - q^{3} + 3q^{5} + 2q^{9} + 9q^{13} + 3q^{15} - 3q^{17} - 8q^{19} - 6q^{21} + 9q^{23} - 4q^{25} - 10q^{27} - 15q^{29} - 8q^{31} - 6q^{33} + 18q^{35} + 15q^{41} + 21q^{43} + 12q^{45} - 3q^{47} - 10q^{49} - 3q^{51} - 3q^{53} + 18q^{55} + 7q^{57} + 3q^{59} + 7q^{61} - 12q^{63} + 5q^{67} + 9q^{71} - 7q^{73} + 8q^{75} - 24q^{77} - 7q^{79} - q^{81} + 9q^{85} + 15q^{89} + 18q^{91} + 4q^{93} - 21q^{95} + 15q^{97} - 12q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(-1\) \(\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
255.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −0.500000 + 0.866025i 0 1.50000 2.59808i 0 3.46410i 0 1.00000 + 1.73205i 0
639.1 0 −0.500000 0.866025i 0 1.50000 + 2.59808i 0 3.46410i 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
76.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.n.a 2
4.b odd 2 1 1216.2.n.b 2
8.b even 2 1 304.2.n.b yes 2
8.d odd 2 1 304.2.n.a 2
19.d odd 6 1 1216.2.n.b 2
24.f even 2 1 2736.2.bm.g 2
24.h odd 2 1 2736.2.bm.h 2
76.f even 6 1 inner 1216.2.n.a 2
152.l odd 6 1 304.2.n.a 2
152.o even 6 1 304.2.n.b yes 2
456.s odd 6 1 2736.2.bm.h 2
456.v even 6 1 2736.2.bm.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.n.a 2 8.d odd 2 1
304.2.n.a 2 152.l odd 6 1
304.2.n.b yes 2 8.b even 2 1
304.2.n.b yes 2 152.o even 6 1
1216.2.n.a 2 1.a even 1 1 trivial
1216.2.n.a 2 76.f even 6 1 inner
1216.2.n.b 2 4.b odd 2 1
1216.2.n.b 2 19.d odd 6 1
2736.2.bm.g 2 24.f even 2 1
2736.2.bm.g 2 456.v even 6 1
2736.2.bm.h 2 24.h odd 2 1
2736.2.bm.h 2 456.s odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1216, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( 9 - 3 T + T^{2} \)
$7$ \( 12 + T^{2} \)
$11$ \( 12 + T^{2} \)
$13$ \( 27 - 9 T + T^{2} \)
$17$ \( 9 + 3 T + T^{2} \)
$19$ \( 19 + 8 T + T^{2} \)
$23$ \( 27 - 9 T + T^{2} \)
$29$ \( 75 + 15 T + T^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( T^{2} \)
$41$ \( 75 - 15 T + T^{2} \)
$43$ \( 147 - 21 T + T^{2} \)
$47$ \( 3 + 3 T + T^{2} \)
$53$ \( 3 + 3 T + T^{2} \)
$59$ \( 9 - 3 T + T^{2} \)
$61$ \( 49 - 7 T + T^{2} \)
$67$ \( 25 - 5 T + T^{2} \)
$71$ \( 81 - 9 T + T^{2} \)
$73$ \( 49 + 7 T + T^{2} \)
$79$ \( 49 + 7 T + T^{2} \)
$83$ \( 12 + T^{2} \)
$89$ \( 75 - 15 T + T^{2} \)
$97$ \( 75 - 15 T + T^{2} \)
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