Properties

Label 1224.1.n.b
Level $1224$
Weight $1$
Character orbit 1224.n
Self dual yes
Analytic conductor $0.611$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -136
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1224.n (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.610855575463\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{4}\)
Projective field Galois closure of 4.0.9792.1
Artin image $D_8$
Artin field Galois closure of 8.0.14670139392.2

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} -\beta q^{5} -\beta q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} -\beta q^{5} -\beta q^{7} - q^{8} + \beta q^{10} + \beta q^{14} + q^{16} + q^{17} -\beta q^{20} + \beta q^{23} + q^{25} -\beta q^{28} + \beta q^{29} + \beta q^{31} - q^{32} - q^{34} + 2 q^{35} + \beta q^{37} + \beta q^{40} -2 q^{43} -\beta q^{46} + q^{49} - q^{50} + \beta q^{56} -\beta q^{58} -\beta q^{61} -\beta q^{62} + q^{64} + q^{68} -2 q^{70} -\beta q^{71} -\beta q^{74} + \beta q^{79} -\beta q^{80} -\beta q^{85} + 2 q^{86} + 2 q^{89} + \beta q^{92} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{8} + 2q^{16} + 2q^{17} + 2q^{25} - 2q^{32} - 2q^{34} + 4q^{35} - 4q^{43} + 2q^{49} - 2q^{50} + 2q^{64} + 2q^{68} - 4q^{70} + 4q^{86} + 4q^{89} - 2q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times\).

\(n\) \(137\) \(613\) \(649\) \(919\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
883.1
1.41421
−1.41421
−1.00000 0 1.00000 −1.41421 0 −1.41421 −1.00000 0 1.41421
883.2 −1.00000 0 1.00000 1.41421 0 1.41421 −1.00000 0 −1.41421
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
136.e odd 2 1 CM by \(\Q(\sqrt{-34}) \)
8.d odd 2 1 inner
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.1.n.b 2
3.b odd 2 1 1224.1.n.c yes 2
8.d odd 2 1 inner 1224.1.n.b 2
17.b even 2 1 inner 1224.1.n.b 2
24.f even 2 1 1224.1.n.c yes 2
51.c odd 2 1 1224.1.n.c yes 2
136.e odd 2 1 CM 1224.1.n.b 2
408.h even 2 1 1224.1.n.c yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1224.1.n.b 2 1.a even 1 1 trivial
1224.1.n.b 2 8.d odd 2 1 inner
1224.1.n.b 2 17.b even 2 1 inner
1224.1.n.b 2 136.e odd 2 1 CM
1224.1.n.c yes 2 3.b odd 2 1
1224.1.n.c yes 2 24.f even 2 1
1224.1.n.c yes 2 51.c odd 2 1
1224.1.n.c yes 2 408.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1224, [\chi])\):

\( T_{5}^{2} - 2 \)
\( T_{89} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ 1
$5$ \( 1 + T^{4} \)
$7$ \( 1 + T^{4} \)
$11$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$13$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$17$ \( ( 1 - T )^{2} \)
$19$ \( ( 1 + T^{2} )^{2} \)
$23$ \( 1 + T^{4} \)
$29$ \( 1 + T^{4} \)
$31$ \( 1 + T^{4} \)
$37$ \( 1 + T^{4} \)
$41$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$43$ \( ( 1 + T )^{4} \)
$47$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$53$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$59$ \( ( 1 + T^{2} )^{2} \)
$61$ \( 1 + T^{4} \)
$67$ \( ( 1 + T^{2} )^{2} \)
$71$ \( 1 + T^{4} \)
$73$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$79$ \( 1 + T^{4} \)
$83$ \( ( 1 + T^{2} )^{2} \)
$89$ \( ( 1 - T )^{4} \)
$97$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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