Properties

Label 1088.1.p.a
Level $1088$
Weight $1$
Character orbit 1088.p
Analytic conductor $0.543$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1088 = 2^{6} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1088.p (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.542982733745\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 68)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.19652.1
Artin image: $C_4{\rm wrC}_2$
Artin field: Galois closure of 8.0.321978368.5

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( 1 - i ) q^{5} -i q^{9} +O(q^{10})\) \( q + ( 1 - i ) q^{5} -i q^{9} - q^{17} -i q^{25} + ( -1 + i ) q^{29} + ( 1 - i ) q^{37} + ( 1 + i ) q^{41} + ( -1 - i ) q^{45} + i q^{49} + ( 1 + i ) q^{61} + ( 1 - i ) q^{73} - q^{81} + ( -1 + i ) q^{85} + ( -1 + i ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} + O(q^{10}) \) \( 2q + 2q^{5} - 2q^{17} - 2q^{29} + 2q^{37} + 2q^{41} - 2q^{45} + 2q^{61} + 2q^{73} - 2q^{81} - 2q^{85} - 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1088\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
1.00000i
1.00000i
0 0 0 1.00000 1.00000i 0 0 0 1.00000i 0
319.1 0 0 0 1.00000 + 1.00000i 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
17.c even 4 1 inner
68.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1088.1.p.a 2
4.b odd 2 1 CM 1088.1.p.a 2
8.b even 2 1 68.1.f.a 2
8.d odd 2 1 68.1.f.a 2
17.c even 4 1 inner 1088.1.p.a 2
24.f even 2 1 612.1.l.a 2
24.h odd 2 1 612.1.l.a 2
40.e odd 2 1 1700.1.p.a 2
40.f even 2 1 1700.1.p.a 2
40.i odd 4 1 1700.1.n.a 2
40.i odd 4 1 1700.1.n.b 2
40.k even 4 1 1700.1.n.a 2
40.k even 4 1 1700.1.n.b 2
56.e even 2 1 3332.1.m.b 2
56.h odd 2 1 3332.1.m.b 2
56.j odd 6 2 3332.1.bc.b 4
56.k odd 6 2 3332.1.bc.c 4
56.m even 6 2 3332.1.bc.b 4
56.p even 6 2 3332.1.bc.c 4
68.f odd 4 1 inner 1088.1.p.a 2
136.e odd 2 1 1156.1.f.b 2
136.h even 2 1 1156.1.f.b 2
136.i even 4 1 68.1.f.a 2
136.i even 4 1 1156.1.f.b 2
136.j odd 4 1 68.1.f.a 2
136.j odd 4 1 1156.1.f.b 2
136.o even 8 2 1156.1.c.b 2
136.o even 8 2 1156.1.d.a 2
136.p odd 8 2 1156.1.c.b 2
136.p odd 8 2 1156.1.d.a 2
136.q odd 16 8 1156.1.g.b 8
136.s even 16 8 1156.1.g.b 8
408.q even 4 1 612.1.l.a 2
408.t odd 4 1 612.1.l.a 2
680.s odd 4 1 1700.1.n.a 2
680.t even 4 1 1700.1.n.b 2
680.bc odd 4 1 1700.1.p.a 2
680.be even 4 1 1700.1.p.a 2
680.bk odd 4 1 1700.1.n.b 2
680.bl even 4 1 1700.1.n.a 2
952.v odd 4 1 3332.1.m.b 2
952.x even 4 1 3332.1.m.b 2
952.bw even 12 2 3332.1.bc.c 4
952.by odd 12 2 3332.1.bc.c 4
952.cb odd 12 2 3332.1.bc.b 4
952.cd even 12 2 3332.1.bc.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.1.f.a 2 8.b even 2 1
68.1.f.a 2 8.d odd 2 1
68.1.f.a 2 136.i even 4 1
68.1.f.a 2 136.j odd 4 1
612.1.l.a 2 24.f even 2 1
612.1.l.a 2 24.h odd 2 1
612.1.l.a 2 408.q even 4 1
612.1.l.a 2 408.t odd 4 1
1088.1.p.a 2 1.a even 1 1 trivial
1088.1.p.a 2 4.b odd 2 1 CM
1088.1.p.a 2 17.c even 4 1 inner
1088.1.p.a 2 68.f odd 4 1 inner
1156.1.c.b 2 136.o even 8 2
1156.1.c.b 2 136.p odd 8 2
1156.1.d.a 2 136.o even 8 2
1156.1.d.a 2 136.p odd 8 2
1156.1.f.b 2 136.e odd 2 1
1156.1.f.b 2 136.h even 2 1
1156.1.f.b 2 136.i even 4 1
1156.1.f.b 2 136.j odd 4 1
1156.1.g.b 8 136.q odd 16 8
1156.1.g.b 8 136.s even 16 8
1700.1.n.a 2 40.i odd 4 1
1700.1.n.a 2 40.k even 4 1
1700.1.n.a 2 680.s odd 4 1
1700.1.n.a 2 680.bl even 4 1
1700.1.n.b 2 40.i odd 4 1
1700.1.n.b 2 40.k even 4 1
1700.1.n.b 2 680.t even 4 1
1700.1.n.b 2 680.bk odd 4 1
1700.1.p.a 2 40.e odd 2 1
1700.1.p.a 2 40.f even 2 1
1700.1.p.a 2 680.bc odd 4 1
1700.1.p.a 2 680.be even 4 1
3332.1.m.b 2 56.e even 2 1
3332.1.m.b 2 56.h odd 2 1
3332.1.m.b 2 952.v odd 4 1
3332.1.m.b 2 952.x even 4 1
3332.1.bc.b 4 56.j odd 6 2
3332.1.bc.b 4 56.m even 6 2
3332.1.bc.b 4 952.cb odd 12 2
3332.1.bc.b 4 952.cd even 12 2
3332.1.bc.c 4 56.k odd 6 2
3332.1.bc.c 4 56.p even 6 2
3332.1.bc.c 4 952.bw even 12 2
3332.1.bc.c 4 952.by odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1088, [\chi])\).

Hecke characteristic polynomials

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