Properties

Label 3520.1.i.b
Level $3520$
Weight $1$
Character orbit 3520.i
Self dual yes
Analytic conductor $1.757$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -11, -55, 5
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3520 = 2^{6} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3520.i (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.75670884447\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{5}, \sqrt{-11})\)
Artin image: $D_4$
Artin field: Galois closure of 4.2.17600.2

$q$-expansion

\(f(q)\) \(=\) \( q + q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + q^{9} + q^{11} + q^{25} - 2 q^{31} + q^{45} - q^{49} + q^{55} - 2 q^{59} + 2 q^{71} + q^{81} - 2 q^{89} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3520\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(1541\) \(2751\) \(2817\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1
0
0 0 0 1.00000 0 0 0 1.00000 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
5.b even 2 1 RM by \(\Q(\sqrt{5}) \)
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
55.d odd 2 1 CM by \(\Q(\sqrt{-55}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3520.1.i.b 1
4.b odd 2 1 3520.1.i.a 1
5.b even 2 1 RM 3520.1.i.b 1
8.b even 2 1 55.1.d.a 1
8.d odd 2 1 880.1.i.a 1
11.b odd 2 1 CM 3520.1.i.b 1
20.d odd 2 1 3520.1.i.a 1
24.h odd 2 1 495.1.h.a 1
40.e odd 2 1 880.1.i.a 1
40.f even 2 1 55.1.d.a 1
40.i odd 4 2 275.1.c.a 1
44.c even 2 1 3520.1.i.a 1
55.d odd 2 1 CM 3520.1.i.b 1
56.h odd 2 1 2695.1.g.c 1
56.j odd 6 2 2695.1.q.b 2
56.p even 6 2 2695.1.q.c 2
88.b odd 2 1 55.1.d.a 1
88.g even 2 1 880.1.i.a 1
88.o even 10 4 605.1.h.a 4
88.p odd 10 4 605.1.h.a 4
120.i odd 2 1 495.1.h.a 1
120.w even 4 2 2475.1.b.a 1
220.g even 2 1 3520.1.i.a 1
264.m even 2 1 495.1.h.a 1
280.c odd 2 1 2695.1.g.c 1
280.bf even 6 2 2695.1.q.c 2
280.bk odd 6 2 2695.1.q.b 2
440.c even 2 1 880.1.i.a 1
440.o odd 2 1 55.1.d.a 1
440.t even 4 2 275.1.c.a 1
440.ba odd 10 4 605.1.h.a 4
440.bd even 10 4 605.1.h.a 4
440.bp odd 20 8 3025.1.x.a 4
440.bu even 20 8 3025.1.x.a 4
616.o even 2 1 2695.1.g.c 1
616.s even 6 2 2695.1.q.b 2
616.bg odd 6 2 2695.1.q.c 2
1320.u even 2 1 495.1.h.a 1
1320.bn odd 4 2 2475.1.b.a 1
3080.k even 2 1 2695.1.g.c 1
3080.bz odd 6 2 2695.1.q.c 2
3080.dc even 6 2 2695.1.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.1.d.a 1 8.b even 2 1
55.1.d.a 1 40.f even 2 1
55.1.d.a 1 88.b odd 2 1
55.1.d.a 1 440.o odd 2 1
275.1.c.a 1 40.i odd 4 2
275.1.c.a 1 440.t even 4 2
495.1.h.a 1 24.h odd 2 1
495.1.h.a 1 120.i odd 2 1
495.1.h.a 1 264.m even 2 1
495.1.h.a 1 1320.u even 2 1
605.1.h.a 4 88.o even 10 4
605.1.h.a 4 88.p odd 10 4
605.1.h.a 4 440.ba odd 10 4
605.1.h.a 4 440.bd even 10 4
880.1.i.a 1 8.d odd 2 1
880.1.i.a 1 40.e odd 2 1
880.1.i.a 1 88.g even 2 1
880.1.i.a 1 440.c even 2 1
2475.1.b.a 1 120.w even 4 2
2475.1.b.a 1 1320.bn odd 4 2
2695.1.g.c 1 56.h odd 2 1
2695.1.g.c 1 280.c odd 2 1
2695.1.g.c 1 616.o even 2 1
2695.1.g.c 1 3080.k even 2 1
2695.1.q.b 2 56.j odd 6 2
2695.1.q.b 2 280.bk odd 6 2
2695.1.q.b 2 616.s even 6 2
2695.1.q.b 2 3080.dc even 6 2
2695.1.q.c 2 56.p even 6 2
2695.1.q.c 2 280.bf even 6 2
2695.1.q.c 2 616.bg odd 6 2
2695.1.q.c 2 3080.bz odd 6 2
3025.1.x.a 4 440.bp odd 20 8
3025.1.x.a 4 440.bu even 20 8
3520.1.i.a 1 4.b odd 2 1
3520.1.i.a 1 20.d odd 2 1
3520.1.i.a 1 44.c even 2 1
3520.1.i.a 1 220.g even 2 1
3520.1.i.b 1 1.a even 1 1 trivial
3520.1.i.b 1 5.b even 2 1 RM
3520.1.i.b 1 11.b odd 2 1 CM
3520.1.i.b 1 55.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3520, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{31} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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