Properties

Label 544.1.n.a
Level $544$
Weight $1$
Character orbit 544.n
Analytic conductor $0.271$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 544 = 2^{5} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 544.n (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.271491366872\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 136)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.314432.1
Artin image: $C_4{\rm wrC}_2$
Artin field: Galois closure of 8.0.321978368.6

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + ( 1 + i ) q^{3} + i q^{9} +O(q^{10})\) \( q + ( 1 + i ) q^{3} + i q^{9} + ( -1 + i ) q^{11} -i q^{17} -2 i q^{19} + i q^{25} -2 q^{33} + ( -1 + i ) q^{41} -i q^{49} + ( 1 - i ) q^{51} + ( 2 - 2 i ) q^{57} + ( -1 - i ) q^{73} + ( -1 + i ) q^{75} + q^{81} + ( 1 + i ) q^{97} + ( -1 - i ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{11} - 4q^{33} - 2q^{41} + 2q^{51} + 4q^{57} - 2q^{73} - 2q^{75} + 2q^{81} + 2q^{97} - 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/544\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(511\) \(513\)
\(\chi(n)\) \(-1\) \(-1\) \(i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
1.00000i
1.00000i
0 1.00000 + 1.00000i 0 0 0 0 0 1.00000i 0
463.1 0 1.00000 1.00000i 0 0 0 0 0 1.00000i 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.c even 4 1 inner
136.j odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 544.1.n.a 2
4.b odd 2 1 136.1.j.a 2
8.b even 2 1 136.1.j.a 2
8.d odd 2 1 CM 544.1.n.a 2
12.b even 2 1 1224.1.s.a 2
17.c even 4 1 inner 544.1.n.a 2
20.d odd 2 1 3400.1.y.a 2
20.e even 4 1 3400.1.bc.a 2
20.e even 4 1 3400.1.bc.b 2
24.h odd 2 1 1224.1.s.a 2
40.f even 2 1 3400.1.y.a 2
40.i odd 4 1 3400.1.bc.a 2
40.i odd 4 1 3400.1.bc.b 2
68.d odd 2 1 2312.1.j.b 2
68.f odd 4 1 136.1.j.a 2
68.f odd 4 1 2312.1.j.b 2
68.g odd 8 2 2312.1.e.a 2
68.g odd 8 2 2312.1.f.b 2
68.i even 16 8 2312.1.p.e 8
136.h even 2 1 2312.1.j.b 2
136.i even 4 1 136.1.j.a 2
136.i even 4 1 2312.1.j.b 2
136.j odd 4 1 inner 544.1.n.a 2
136.o even 8 2 2312.1.e.a 2
136.o even 8 2 2312.1.f.b 2
136.q odd 16 8 2312.1.p.e 8
204.l even 4 1 1224.1.s.a 2
340.i even 4 1 3400.1.bc.b 2
340.n odd 4 1 3400.1.y.a 2
340.s even 4 1 3400.1.bc.a 2
408.t odd 4 1 1224.1.s.a 2
680.s odd 4 1 3400.1.bc.b 2
680.be even 4 1 3400.1.y.a 2
680.bk odd 4 1 3400.1.bc.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.1.j.a 2 4.b odd 2 1
136.1.j.a 2 8.b even 2 1
136.1.j.a 2 68.f odd 4 1
136.1.j.a 2 136.i even 4 1
544.1.n.a 2 1.a even 1 1 trivial
544.1.n.a 2 8.d odd 2 1 CM
544.1.n.a 2 17.c even 4 1 inner
544.1.n.a 2 136.j odd 4 1 inner
1224.1.s.a 2 12.b even 2 1
1224.1.s.a 2 24.h odd 2 1
1224.1.s.a 2 204.l even 4 1
1224.1.s.a 2 408.t odd 4 1
2312.1.e.a 2 68.g odd 8 2
2312.1.e.a 2 136.o even 8 2
2312.1.f.b 2 68.g odd 8 2
2312.1.f.b 2 136.o even 8 2
2312.1.j.b 2 68.d odd 2 1
2312.1.j.b 2 68.f odd 4 1
2312.1.j.b 2 136.h even 2 1
2312.1.j.b 2 136.i even 4 1
2312.1.p.e 8 68.i even 16 8
2312.1.p.e 8 136.q odd 16 8
3400.1.y.a 2 20.d odd 2 1
3400.1.y.a 2 40.f even 2 1
3400.1.y.a 2 340.n odd 4 1
3400.1.y.a 2 680.be even 4 1
3400.1.bc.a 2 20.e even 4 1
3400.1.bc.a 2 40.i odd 4 1
3400.1.bc.a 2 340.s even 4 1
3400.1.bc.a 2 680.bk odd 4 1
3400.1.bc.b 2 20.e even 4 1
3400.1.bc.b 2 40.i odd 4 1
3400.1.bc.b 2 340.i even 4 1
3400.1.bc.b 2 680.s odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(544, [\chi])\).

Hecke characteristic polynomials

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