Properties

Label 1216.2.c.d
Level $1216$
Weight $2$
Character orbit 1216.c
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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

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

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{3} + 3 q^{7} + 2 q^{9} +O(q^{10})\) \( q + i q^{3} + 3 q^{7} + 2 q^{9} + 3 i q^{13} + 3 q^{17} -i q^{19} + 3 i q^{21} -9 q^{23} + 5 q^{25} + 5 i q^{27} -9 i q^{29} + 6 q^{31} -6 i q^{37} -3 q^{39} + 6 q^{41} + 8 i q^{43} + 2 q^{49} + 3 i q^{51} + 9 i q^{53} + q^{57} + 3 i q^{59} + 6 i q^{61} + 6 q^{63} + 5 i q^{67} -9 i q^{69} + 11 q^{73} + 5 i q^{75} -12 q^{79} + q^{81} + 6 i q^{83} + 9 q^{87} + 9 i q^{91} + 6 i q^{93} -8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{7} + 4q^{9} + O(q^{10}) \) \( 2q + 6q^{7} + 4q^{9} + 6q^{17} - 18q^{23} + 10q^{25} + 12q^{31} - 6q^{39} + 12q^{41} + 4q^{49} + 2q^{57} + 12q^{63} + 22q^{73} - 24q^{79} + 2q^{81} + 18q^{87} - 16q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
609.1
1.00000i
1.00000i
0 1.00000i 0 0 0 3.00000 0 2.00000 0
609.2 0 1.00000i 0 0 0 3.00000 0 2.00000 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
8.b 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.c.d yes 2
4.b odd 2 1 1216.2.c.a 2
8.b even 2 1 inner 1216.2.c.d yes 2
8.d odd 2 1 1216.2.c.a 2
16.e even 4 1 4864.2.a.d 1
16.e even 4 1 4864.2.a.l 1
16.f odd 4 1 4864.2.a.e 1
16.f odd 4 1 4864.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.c.a 2 4.b odd 2 1
1216.2.c.a 2 8.d odd 2 1
1216.2.c.d yes 2 1.a even 1 1 trivial
1216.2.c.d yes 2 8.b even 2 1 inner
4864.2.a.d 1 16.e even 4 1
4864.2.a.e 1 16.f odd 4 1
4864.2.a.l 1 16.e even 4 1
4864.2.a.m 1 16.f odd 4 1

Hecke kernels

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

\( T_{3}^{2} + 1 \)
\( T_{5} \)
\( T_{7} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -3 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( 9 + T^{2} \)
$17$ \( ( -3 + T )^{2} \)
$19$ \( 1 + T^{2} \)
$23$ \( ( 9 + T )^{2} \)
$29$ \( 81 + T^{2} \)
$31$ \( ( -6 + T )^{2} \)
$37$ \( 36 + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 81 + T^{2} \)
$59$ \( 9 + T^{2} \)
$61$ \( 36 + T^{2} \)
$67$ \( 25 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -11 + T )^{2} \)
$79$ \( ( 12 + T )^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( 8 + T )^{2} \)
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