Properties

Label 1216.2.i.a
Level $1216$
Weight $2$
Character orbit 1216.i
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.i (of order \(3\), 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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 608)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -3 + 3 \zeta_{6} ) q^{3} + ( 2 - 2 \zeta_{6} ) q^{5} + 2 q^{7} -6 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -3 + 3 \zeta_{6} ) q^{3} + ( 2 - 2 \zeta_{6} ) q^{5} + 2 q^{7} -6 \zeta_{6} q^{9} -3 q^{11} + 2 \zeta_{6} q^{13} + 6 \zeta_{6} q^{15} + ( 6 - 6 \zeta_{6} ) q^{17} + ( 2 + 3 \zeta_{6} ) q^{19} + ( -6 + 6 \zeta_{6} ) q^{21} -4 \zeta_{6} q^{23} + \zeta_{6} q^{25} + 9 q^{27} -6 \zeta_{6} q^{29} + 8 q^{31} + ( 9 - 9 \zeta_{6} ) q^{33} + ( 4 - 4 \zeta_{6} ) q^{35} + 8 q^{37} -6 q^{39} + ( 3 - 3 \zeta_{6} ) q^{41} + ( 8 - 8 \zeta_{6} ) q^{43} -12 q^{45} + 10 \zeta_{6} q^{47} -3 q^{49} + 18 \zeta_{6} q^{51} + 2 \zeta_{6} q^{53} + ( -6 + 6 \zeta_{6} ) q^{55} + ( -15 + 6 \zeta_{6} ) q^{57} + ( 3 - 3 \zeta_{6} ) q^{59} + 14 \zeta_{6} q^{61} -12 \zeta_{6} q^{63} + 4 q^{65} -7 \zeta_{6} q^{67} + 12 q^{69} + ( -2 + 2 \zeta_{6} ) q^{71} + ( -11 + 11 \zeta_{6} ) q^{73} -3 q^{75} -6 q^{77} + ( -4 + 4 \zeta_{6} ) q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} -15 q^{83} -12 \zeta_{6} q^{85} + 18 q^{87} + 18 \zeta_{6} q^{89} + 4 \zeta_{6} q^{91} + ( -24 + 24 \zeta_{6} ) q^{93} + ( 10 - 4 \zeta_{6} ) q^{95} + ( 3 - 3 \zeta_{6} ) q^{97} + 18 \zeta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{3} + 2q^{5} + 4q^{7} - 6q^{9} + O(q^{10}) \) \( 2q - 3q^{3} + 2q^{5} + 4q^{7} - 6q^{9} - 6q^{11} + 2q^{13} + 6q^{15} + 6q^{17} + 7q^{19} - 6q^{21} - 4q^{23} + q^{25} + 18q^{27} - 6q^{29} + 16q^{31} + 9q^{33} + 4q^{35} + 16q^{37} - 12q^{39} + 3q^{41} + 8q^{43} - 24q^{45} + 10q^{47} - 6q^{49} + 18q^{51} + 2q^{53} - 6q^{55} - 24q^{57} + 3q^{59} + 14q^{61} - 12q^{63} + 8q^{65} - 7q^{67} + 24q^{69} - 2q^{71} - 11q^{73} - 6q^{75} - 12q^{77} - 4q^{79} - 9q^{81} - 30q^{83} - 12q^{85} + 36q^{87} + 18q^{89} + 4q^{91} - 24q^{93} + 16q^{95} + 3q^{97} + 18q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.50000 2.59808i 0 1.00000 + 1.73205i 0 2.00000 0 −3.00000 + 5.19615i 0
961.1 0 −1.50000 + 2.59808i 0 1.00000 1.73205i 0 2.00000 0 −3.00000 5.19615i 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.i.a 2
4.b odd 2 1 1216.2.i.j 2
8.b even 2 1 608.2.i.b yes 2
8.d odd 2 1 608.2.i.a 2
19.c even 3 1 inner 1216.2.i.a 2
76.g odd 6 1 1216.2.i.j 2
152.k odd 6 1 608.2.i.a 2
152.p even 6 1 608.2.i.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.2.i.a 2 8.d odd 2 1
608.2.i.a 2 152.k odd 6 1
608.2.i.b yes 2 8.b even 2 1
608.2.i.b yes 2 152.p even 6 1
1216.2.i.a 2 1.a even 1 1 trivial
1216.2.i.a 2 19.c even 3 1 inner
1216.2.i.j 2 4.b odd 2 1
1216.2.i.j 2 76.g odd 6 1

Hecke kernels

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

\( T_{3}^{2} + 3 T_{3} + 9 \)
\( T_{5}^{2} - 2 T_{5} + 4 \)
\( T_{7} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + 3 T + T^{2} \)
$5$ \( 4 - 2 T + T^{2} \)
$7$ \( ( -2 + T )^{2} \)
$11$ \( ( 3 + T )^{2} \)
$13$ \( 4 - 2 T + T^{2} \)
$17$ \( 36 - 6 T + T^{2} \)
$19$ \( 19 - 7 T + T^{2} \)
$23$ \( 16 + 4 T + T^{2} \)
$29$ \( 36 + 6 T + T^{2} \)
$31$ \( ( -8 + T )^{2} \)
$37$ \( ( -8 + T )^{2} \)
$41$ \( 9 - 3 T + T^{2} \)
$43$ \( 64 - 8 T + T^{2} \)
$47$ \( 100 - 10 T + T^{2} \)
$53$ \( 4 - 2 T + T^{2} \)
$59$ \( 9 - 3 T + T^{2} \)
$61$ \( 196 - 14 T + T^{2} \)
$67$ \( 49 + 7 T + T^{2} \)
$71$ \( 4 + 2 T + T^{2} \)
$73$ \( 121 + 11 T + T^{2} \)
$79$ \( 16 + 4 T + T^{2} \)
$83$ \( ( 15 + T )^{2} \)
$89$ \( 324 - 18 T + T^{2} \)
$97$ \( 9 - 3 T + T^{2} \)
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