Properties

Label 1984.1.s.a
Level $1984$
Weight $1$
Character orbit 1984.s
Analytic conductor $0.990$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1984 = 2^{6} \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1984.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.990144985064\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 124)
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.15376.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{3} - \zeta_{12}^{4} q^{5} + \zeta_{12}^{5} q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{5} q^{3} - \zeta_{12}^{4} q^{5} + \zeta_{12}^{5} q^{7} + \zeta_{12} q^{11} + \zeta_{12}^{4} q^{13} + \zeta_{12}^{3} q^{15} - \zeta_{12}^{2} q^{17} + \zeta_{12}^{5} q^{19} - \zeta_{12}^{4} q^{21} - \zeta_{12}^{3} q^{27} + \zeta_{12}^{3} q^{31} - q^{33} + \zeta_{12}^{3} q^{35} + \zeta_{12}^{2} q^{37} - \zeta_{12}^{3} q^{39} + \zeta_{12}^{4} q^{41} + \zeta_{12}^{5} q^{43} + \zeta_{12} q^{51} + \zeta_{12}^{4} q^{53} - \zeta_{12}^{5} q^{55} - \zeta_{12}^{4} q^{57} - \zeta_{12}^{5} q^{59} + \zeta_{12}^{2} q^{65} + \zeta_{12} q^{67} + \zeta_{12} q^{71} + \zeta_{12}^{4} q^{73} - q^{77} + \zeta_{12}^{5} q^{79} + \zeta_{12}^{2} q^{81} - \zeta_{12} q^{83} - q^{85} - \zeta_{12}^{3} q^{91} - \zeta_{12}^{2} q^{93} + \zeta_{12}^{3} q^{95} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} - 2 q^{13} - 2 q^{17} + 2 q^{21} - 4 q^{33} + 2 q^{37} - 2 q^{41} - 2 q^{53} + 2 q^{57} + 2 q^{65} - 2 q^{73} - 4 q^{77} + 2 q^{81} - 4 q^{85} - 2 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1984\mathbb{Z}\right)^\times\).

\(n\) \(63\) \(65\) \(1861\)
\(\chi(n)\) \(-1\) \(\zeta_{12}^{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 −0.866025 0.500000i 0 0 0
191.2 0 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0.866025 + 0.500000i 0 0 0
831.1 0 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 −0.866025 + 0.500000i 0 0 0
831.2 0 0.866025 0.500000i 0 0.500000 0.866025i 0 0.866025 0.500000i 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
4.b odd 2 1 inner
31.c even 3 1 inner
124.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1984.1.s.a 4
4.b odd 2 1 inner 1984.1.s.a 4
8.b even 2 1 124.1.i.a 4
8.d odd 2 1 124.1.i.a 4
24.f even 2 1 1116.1.x.a 4
24.h odd 2 1 1116.1.x.a 4
31.c even 3 1 inner 1984.1.s.a 4
40.e odd 2 1 3100.1.z.a 4
40.f even 2 1 3100.1.z.a 4
40.i odd 4 1 3100.1.t.a 4
40.i odd 4 1 3100.1.t.b 4
40.k even 4 1 3100.1.t.a 4
40.k even 4 1 3100.1.t.b 4
124.i odd 6 1 inner 1984.1.s.a 4
248.b even 2 1 3844.1.i.d 4
248.g odd 2 1 3844.1.i.d 4
248.l odd 6 1 3844.1.b.c 2
248.l odd 6 1 3844.1.i.d 4
248.m odd 6 1 124.1.i.a 4
248.m odd 6 1 3844.1.b.d 2
248.p even 6 1 124.1.i.a 4
248.p even 6 1 3844.1.b.d 2
248.q even 6 1 3844.1.b.c 2
248.q even 6 1 3844.1.i.d 4
248.r odd 10 4 3844.1.n.f 16
248.s odd 10 4 3844.1.n.e 16
248.u even 10 4 3844.1.n.e 16
248.v even 10 4 3844.1.n.f 16
248.bb even 30 4 3844.1.l.c 8
248.bb even 30 4 3844.1.n.f 16
248.bc even 30 4 3844.1.l.d 8
248.bc even 30 4 3844.1.n.e 16
248.be odd 30 4 3844.1.l.d 8
248.be odd 30 4 3844.1.n.e 16
248.bf odd 30 4 3844.1.l.c 8
248.bf odd 30 4 3844.1.n.f 16
744.s odd 6 1 1116.1.x.a 4
744.y even 6 1 1116.1.x.a 4
1240.bg even 6 1 3100.1.z.a 4
1240.bi odd 6 1 3100.1.z.a 4
1240.ch even 12 1 3100.1.t.a 4
1240.ch even 12 1 3100.1.t.b 4
1240.cj odd 12 1 3100.1.t.a 4
1240.cj odd 12 1 3100.1.t.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.1.i.a 4 8.b even 2 1
124.1.i.a 4 8.d odd 2 1
124.1.i.a 4 248.m odd 6 1
124.1.i.a 4 248.p even 6 1
1116.1.x.a 4 24.f even 2 1
1116.1.x.a 4 24.h odd 2 1
1116.1.x.a 4 744.s odd 6 1
1116.1.x.a 4 744.y even 6 1
1984.1.s.a 4 1.a even 1 1 trivial
1984.1.s.a 4 4.b odd 2 1 inner
1984.1.s.a 4 31.c even 3 1 inner
1984.1.s.a 4 124.i odd 6 1 inner
3100.1.t.a 4 40.i odd 4 1
3100.1.t.a 4 40.k even 4 1
3100.1.t.a 4 1240.ch even 12 1
3100.1.t.a 4 1240.cj odd 12 1
3100.1.t.b 4 40.i odd 4 1
3100.1.t.b 4 40.k even 4 1
3100.1.t.b 4 1240.ch even 12 1
3100.1.t.b 4 1240.cj odd 12 1
3100.1.z.a 4 40.e odd 2 1
3100.1.z.a 4 40.f even 2 1
3100.1.z.a 4 1240.bg even 6 1
3100.1.z.a 4 1240.bi odd 6 1
3844.1.b.c 2 248.l odd 6 1
3844.1.b.c 2 248.q even 6 1
3844.1.b.d 2 248.m odd 6 1
3844.1.b.d 2 248.p even 6 1
3844.1.i.d 4 248.b even 2 1
3844.1.i.d 4 248.g odd 2 1
3844.1.i.d 4 248.l odd 6 1
3844.1.i.d 4 248.q even 6 1
3844.1.l.c 8 248.bb even 30 4
3844.1.l.c 8 248.bf odd 30 4
3844.1.l.d 8 248.bc even 30 4
3844.1.l.d 8 248.be odd 30 4
3844.1.n.e 16 248.s odd 10 4
3844.1.n.e 16 248.u even 10 4
3844.1.n.e 16 248.bc even 30 4
3844.1.n.e 16 248.be odd 30 4
3844.1.n.f 16 248.r odd 10 4
3844.1.n.f 16 248.v even 10 4
3844.1.n.f 16 248.bb even 30 4
3844.1.n.f 16 248.bf odd 30 4

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1984, [\chi])\).

Hecke characteristic polynomials

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