Properties

Label 1224.1.m.a
Level 1224
Weight 1
Character orbit 1224.m
Analytic conductor 0.611
Analytic rank 0
Dimension 2
Projective image \(S_{4}\)
CM/RM no
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.610855575463\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(S_{4}\)
Projective field Galois closure of 4.2.7344.1
Artin image $\GL(2,3)$
Artin field Galois closure of 8.2.11002604544.1

$q$-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.

\(f(q)\) \(=\) \( q - q^{5} -\beta q^{7} +O(q^{10})\) \( q - q^{5} -\beta q^{7} - q^{11} - q^{13} - q^{17} - q^{19} + q^{23} -\beta q^{31} + \beta q^{35} + \beta q^{37} - q^{41} + q^{43} -\beta q^{47} - q^{49} -\beta q^{53} + q^{55} -\beta q^{61} + q^{65} + \beta q^{73} + \beta q^{77} -\beta q^{83} + q^{85} + \beta q^{89} + \beta q^{91} + q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + O(q^{10}) \) \( 2q - 2q^{5} - 2q^{11} - 2q^{13} - 2q^{17} - 2q^{19} + 2q^{23} - 2q^{41} + 2q^{43} - 2q^{49} + 2q^{55} + 2q^{65} + 2q^{85} + 2q^{95} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1
1.41421i
1.41421i
0 0 0 −1.00000 0 1.41421i 0 0 0
305.2 0 0 0 −1.00000 0 1.41421i 0 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
51.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.1.m.a 2
3.b odd 2 1 1224.1.m.b yes 2
4.b odd 2 1 2448.1.m.a 2
12.b even 2 1 2448.1.m.b 2
17.b even 2 1 1224.1.m.b yes 2
51.c odd 2 1 inner 1224.1.m.a 2
68.d odd 2 1 2448.1.m.b 2
204.h even 2 1 2448.1.m.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1224.1.m.a 2 1.a even 1 1 trivial
1224.1.m.a 2 51.c odd 2 1 inner
1224.1.m.b yes 2 3.b odd 2 1
1224.1.m.b yes 2 17.b even 2 1
2448.1.m.a 2 4.b odd 2 1
2448.1.m.a 2 204.h even 2 1
2448.1.m.b 2 12.b even 2 1
2448.1.m.b 2 68.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1224, [\chi])\).

Hecke characteristic polynomials

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