Properties

Label 1224.1.n.a
Level $1224$
Weight $1$
Character orbit 1224.n
Self dual yes
Analytic conductor $0.611$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -8, -136, 17
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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 136)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-2}, \sqrt{17})\)
Artin image $D_4$
Artin field Galois closure of 4.0.9792.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} + q^{8} + q^{16} + q^{17} - 2q^{19} - q^{25} + q^{32} + q^{34} - 2q^{38} + 2q^{43} - q^{49} - q^{50} - 2q^{59} + q^{64} - 2q^{67} + q^{68} - 2q^{76} - 2q^{83} + 2q^{86} + 2q^{89} - q^{98} + O(q^{100}) \)

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
0
1.00000 0 1.00000 0 0 0 1.00000 0 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.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
17.b even 2 1 RM by \(\Q(\sqrt{17}) \)
136.e odd 2 1 CM by \(\Q(\sqrt{-34}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.1.n.a 1
3.b odd 2 1 136.1.e.a 1
8.d odd 2 1 CM 1224.1.n.a 1
12.b even 2 1 544.1.e.a 1
15.d odd 2 1 3400.1.g.a 1
15.e even 4 2 3400.1.k.a 2
17.b even 2 1 RM 1224.1.n.a 1
24.f even 2 1 136.1.e.a 1
24.h odd 2 1 544.1.e.a 1
51.c odd 2 1 136.1.e.a 1
51.f odd 4 2 2312.1.f.a 1
51.g odd 8 4 2312.1.j.a 2
51.i even 16 8 2312.1.p.c 4
120.m even 2 1 3400.1.g.a 1
120.q odd 4 2 3400.1.k.a 2
136.e odd 2 1 CM 1224.1.n.a 1
204.h even 2 1 544.1.e.a 1
255.h odd 2 1 3400.1.g.a 1
255.o even 4 2 3400.1.k.a 2
408.b odd 2 1 544.1.e.a 1
408.h even 2 1 136.1.e.a 1
408.q even 4 2 2312.1.f.a 1
408.bd even 8 4 2312.1.j.a 2
408.bg odd 16 8 2312.1.p.c 4
2040.p even 2 1 3400.1.g.a 1
2040.cp odd 4 2 3400.1.k.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.1.e.a 1 3.b odd 2 1
136.1.e.a 1 24.f even 2 1
136.1.e.a 1 51.c odd 2 1
136.1.e.a 1 408.h even 2 1
544.1.e.a 1 12.b even 2 1
544.1.e.a 1 24.h odd 2 1
544.1.e.a 1 204.h even 2 1
544.1.e.a 1 408.b odd 2 1
1224.1.n.a 1 1.a even 1 1 trivial
1224.1.n.a 1 8.d odd 2 1 CM
1224.1.n.a 1 17.b even 2 1 RM
1224.1.n.a 1 136.e odd 2 1 CM
2312.1.f.a 1 51.f odd 4 2
2312.1.f.a 1 408.q even 4 2
2312.1.j.a 2 51.g odd 8 4
2312.1.j.a 2 408.bd even 8 4
2312.1.p.c 4 51.i even 16 8
2312.1.p.c 4 408.bg odd 16 8
3400.1.g.a 1 15.d odd 2 1
3400.1.g.a 1 120.m even 2 1
3400.1.g.a 1 255.h odd 2 1
3400.1.g.a 1 2040.p even 2 1
3400.1.k.a 2 15.e even 4 2
3400.1.k.a 2 120.q odd 4 2
3400.1.k.a 2 255.o even 4 2
3400.1.k.a 2 2040.cp odd 4 2

Hecke kernels

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

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

Hecke characteristic polynomials

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