Properties

Label 1116.1.x.a
Level $1116$
Weight $1$
Character orbit 1116.x
Analytic conductor $0.557$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1116 = 2^{2} \cdot 3^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1116.x (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.556956554098\)
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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q - \zeta_{12}^{3} q^{2} - q^{4} + \zeta_{12}^{2} q^{5} - \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12}^{3} q^{2} - q^{4} + \zeta_{12}^{2} q^{5} - \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} - \zeta_{12}^{5} q^{10} - \zeta_{12}^{5} q^{11} + \zeta_{12}^{2} q^{13} + \zeta_{12}^{4} q^{14} + q^{16} - \zeta_{12}^{4} q^{17} + \zeta_{12} q^{19} - \zeta_{12}^{2} q^{20} - \zeta_{12}^{2} q^{22} - \zeta_{12}^{5} q^{26} + \zeta_{12} q^{28} - \zeta_{12}^{3} q^{31} - \zeta_{12}^{3} q^{32} - \zeta_{12} q^{34} - \zeta_{12}^{3} q^{35} + \zeta_{12}^{4} q^{37} - \zeta_{12}^{4} q^{38} + \zeta_{12}^{5} q^{40} + \zeta_{12}^{2} q^{41} + \zeta_{12} q^{43} + \zeta_{12}^{5} q^{44} - \zeta_{12}^{2} q^{52} - \zeta_{12}^{2} q^{53} + \zeta_{12} q^{55} - \zeta_{12}^{4} q^{56} + \zeta_{12} q^{59} - q^{62} - q^{64} + \zeta_{12}^{4} q^{65} + \zeta_{12}^{5} q^{67} + \zeta_{12}^{4} q^{68} - q^{70} + \zeta_{12}^{5} q^{71} - \zeta_{12}^{2} q^{73} + \zeta_{12} q^{74} - \zeta_{12} q^{76} - q^{77} - \zeta_{12} q^{79} + \zeta_{12}^{2} q^{80} - \zeta_{12}^{5} q^{82} + \zeta_{12}^{5} q^{83} + q^{85} - \zeta_{12}^{4} q^{86} + \zeta_{12}^{2} q^{88} - \zeta_{12}^{3} q^{91} + \zeta_{12}^{3} q^{95} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 2 q^{5} + 2 q^{13} - 2 q^{14} + 4 q^{16} + 2 q^{17} - 2 q^{20} - 2 q^{22} - 2 q^{37} + 2 q^{38} + 2 q^{41} - 2 q^{52} - 2 q^{53} + 2 q^{56} - 4 q^{62} - 4 q^{64} - 2 q^{65} - 2 q^{68} - 4 q^{70} - 2 q^{73} - 4 q^{77} + 2 q^{80} + 4 q^{85} + 2 q^{86} + 2 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1116\mathbb{Z}\right)^\times\).

\(n\) \(497\) \(559\) \(685\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
811.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
1.00000i 0 −1.00000 0.500000 + 0.866025i 0 −0.866025 0.500000i 1.00000i 0 0.866025 0.500000i
811.2 1.00000i 0 −1.00000 0.500000 + 0.866025i 0 0.866025 + 0.500000i 1.00000i 0 −0.866025 + 0.500000i
955.1 1.00000i 0 −1.00000 0.500000 0.866025i 0 0.866025 0.500000i 1.00000i 0 −0.866025 0.500000i
955.2 1.00000i 0 −1.00000 0.500000 0.866025i 0 −0.866025 + 0.500000i 1.00000i 0 0.866025 + 0.500000i
\(n\): 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 1116.1.x.a 4
3.b odd 2 1 124.1.i.a 4
4.b odd 2 1 inner 1116.1.x.a 4
12.b even 2 1 124.1.i.a 4
15.d odd 2 1 3100.1.z.a 4
15.e even 4 1 3100.1.t.a 4
15.e even 4 1 3100.1.t.b 4
24.f even 2 1 1984.1.s.a 4
24.h odd 2 1 1984.1.s.a 4
31.c even 3 1 inner 1116.1.x.a 4
60.h even 2 1 3100.1.z.a 4
60.l odd 4 1 3100.1.t.a 4
60.l odd 4 1 3100.1.t.b 4
93.c even 2 1 3844.1.i.d 4
93.g even 6 1 3844.1.b.c 2
93.g even 6 1 3844.1.i.d 4
93.h odd 6 1 124.1.i.a 4
93.h odd 6 1 3844.1.b.d 2
93.k even 10 4 3844.1.n.f 16
93.l odd 10 4 3844.1.n.e 16
93.o odd 30 4 3844.1.l.d 8
93.o odd 30 4 3844.1.n.e 16
93.p even 30 4 3844.1.l.c 8
93.p even 30 4 3844.1.n.f 16
124.i odd 6 1 inner 1116.1.x.a 4
372.b odd 2 1 3844.1.i.d 4
372.p even 6 1 124.1.i.a 4
372.p even 6 1 3844.1.b.d 2
372.q odd 6 1 3844.1.b.c 2
372.q odd 6 1 3844.1.i.d 4
372.t even 10 4 3844.1.n.e 16
372.u odd 10 4 3844.1.n.f 16
372.bc odd 30 4 3844.1.l.c 8
372.bc odd 30 4 3844.1.n.f 16
372.bd even 30 4 3844.1.l.d 8
372.bd even 30 4 3844.1.n.e 16
465.u odd 6 1 3100.1.z.a 4
465.be even 12 1 3100.1.t.a 4
465.be even 12 1 3100.1.t.b 4
744.s odd 6 1 1984.1.s.a 4
744.y even 6 1 1984.1.s.a 4
1860.bc even 6 1 3100.1.z.a 4
1860.cf odd 12 1 3100.1.t.a 4
1860.cf 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 3.b odd 2 1
124.1.i.a 4 12.b even 2 1
124.1.i.a 4 93.h odd 6 1
124.1.i.a 4 372.p even 6 1
1116.1.x.a 4 1.a even 1 1 trivial
1116.1.x.a 4 4.b odd 2 1 inner
1116.1.x.a 4 31.c even 3 1 inner
1116.1.x.a 4 124.i odd 6 1 inner
1984.1.s.a 4 24.f even 2 1
1984.1.s.a 4 24.h odd 2 1
1984.1.s.a 4 744.s odd 6 1
1984.1.s.a 4 744.y even 6 1
3100.1.t.a 4 15.e even 4 1
3100.1.t.a 4 60.l odd 4 1
3100.1.t.a 4 465.be even 12 1
3100.1.t.a 4 1860.cf odd 12 1
3100.1.t.b 4 15.e even 4 1
3100.1.t.b 4 60.l odd 4 1
3100.1.t.b 4 465.be even 12 1
3100.1.t.b 4 1860.cf odd 12 1
3100.1.z.a 4 15.d odd 2 1
3100.1.z.a 4 60.h even 2 1
3100.1.z.a 4 465.u odd 6 1
3100.1.z.a 4 1860.bc even 6 1
3844.1.b.c 2 93.g even 6 1
3844.1.b.c 2 372.q odd 6 1
3844.1.b.d 2 93.h odd 6 1
3844.1.b.d 2 372.p even 6 1
3844.1.i.d 4 93.c even 2 1
3844.1.i.d 4 93.g even 6 1
3844.1.i.d 4 372.b odd 2 1
3844.1.i.d 4 372.q odd 6 1
3844.1.l.c 8 93.p even 30 4
3844.1.l.c 8 372.bc odd 30 4
3844.1.l.d 8 93.o odd 30 4
3844.1.l.d 8 372.bd even 30 4
3844.1.n.e 16 93.l odd 10 4
3844.1.n.e 16 93.o odd 30 4
3844.1.n.e 16 372.t even 10 4
3844.1.n.e 16 372.bd even 30 4
3844.1.n.f 16 93.k even 10 4
3844.1.n.f 16 93.p even 30 4
3844.1.n.f 16 372.u odd 10 4
3844.1.n.f 16 372.bc odd 30 4

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1116, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) 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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