Properties

Label 1216.2.c.c
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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + 3 i q^{3} + 4 i q^{5} + q^{7} -6 q^{9} +O(q^{10})\) \( q + 3 i q^{3} + 4 i q^{5} + q^{7} -6 q^{9} -5 i q^{13} -12 q^{15} -5 q^{17} + i q^{19} + 3 i q^{21} -3 q^{23} -11 q^{25} -9 i q^{27} + 7 i q^{29} + 10 q^{31} + 4 i q^{35} + 2 i q^{37} + 15 q^{39} -6 q^{41} -4 i q^{43} -24 i q^{45} + 8 q^{47} -6 q^{49} -15 i q^{51} + 9 i q^{53} -3 q^{57} + i q^{59} -2 i q^{61} -6 q^{63} + 20 q^{65} + 7 i q^{67} -9 i q^{69} -12 q^{71} + 11 q^{73} -33 i q^{75} + 16 q^{79} + 9 q^{81} + 14 i q^{83} -20 i q^{85} -21 q^{87} -4 q^{89} -5 i q^{91} + 30 i q^{93} -4 q^{95} -12 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{7} - 12q^{9} + O(q^{10}) \) \( 2q + 2q^{7} - 12q^{9} - 24q^{15} - 10q^{17} - 6q^{23} - 22q^{25} + 20q^{31} + 30q^{39} - 12q^{41} + 16q^{47} - 12q^{49} - 6q^{57} - 12q^{63} + 40q^{65} - 24q^{71} + 22q^{73} + 32q^{79} + 18q^{81} - 42q^{87} - 8q^{89} - 8q^{95} - 24q^{97} + O(q^{100}) \)

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 3.00000i 0 4.00000i 0 1.00000 0 −6.00000 0
609.2 0 3.00000i 0 4.00000i 0 1.00000 0 −6.00000 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.c yes 2
4.b odd 2 1 1216.2.c.b 2
8.b even 2 1 inner 1216.2.c.c yes 2
8.d odd 2 1 1216.2.c.b 2
16.e even 4 1 4864.2.a.b 1
16.e even 4 1 4864.2.a.o 1
16.f odd 4 1 4864.2.a.a 1
16.f odd 4 1 4864.2.a.p 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.c.b 2 4.b odd 2 1
1216.2.c.b 2 8.d odd 2 1
1216.2.c.c yes 2 1.a even 1 1 trivial
1216.2.c.c yes 2 8.b even 2 1 inner
4864.2.a.a 1 16.f odd 4 1
4864.2.a.b 1 16.e even 4 1
4864.2.a.o 1 16.e even 4 1
4864.2.a.p 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 \(S_{2}^{\mathrm{new}}(1216, [\chi])\):

\( T_{3}^{2} + 9 \)
\( T_{5}^{2} + 16 \)
\( T_{7} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( 16 + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( 25 + T^{2} \)
$17$ \( ( 5 + T )^{2} \)
$19$ \( 1 + T^{2} \)
$23$ \( ( 3 + T )^{2} \)
$29$ \( 49 + T^{2} \)
$31$ \( ( -10 + T )^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( ( -8 + T )^{2} \)
$53$ \( 81 + T^{2} \)
$59$ \( 1 + T^{2} \)
$61$ \( 4 + T^{2} \)
$67$ \( 49 + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( ( -11 + T )^{2} \)
$79$ \( ( -16 + T )^{2} \)
$83$ \( 196 + T^{2} \)
$89$ \( ( 4 + T )^{2} \)
$97$ \( ( 12 + T )^{2} \)
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