Properties

Label 6084.2.b.j
Level $6084$
Weight $2$
Character orbit 6084.b
Analytic conductor $48.581$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 6084 = 2^{2} \cdot 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6084.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.5809845897\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 156)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 i q^{5} + 2 i q^{7} +O(q^{10})\) \( q + 4 i q^{5} + 2 i q^{7} -4 i q^{11} + 2 q^{17} -2 i q^{19} -11 q^{25} + 6 q^{29} -10 i q^{31} -8 q^{35} -10 i q^{37} -8 i q^{41} -4 q^{43} -4 i q^{47} + 3 q^{49} + 10 q^{53} + 16 q^{55} -8 i q^{59} -14 q^{61} + 2 i q^{67} -16 i q^{71} + 10 i q^{73} + 8 q^{77} -16 q^{79} + 8 i q^{85} -4 i q^{89} + 8 q^{95} -2 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 4q^{17} - 22q^{25} + 12q^{29} - 16q^{35} - 8q^{43} + 6q^{49} + 20q^{53} + 32q^{55} - 28q^{61} + 16q^{77} - 32q^{79} + 16q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6084\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(3043\) \(3889\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4393.1
1.00000i
1.00000i
0 0 0 4.00000i 0 2.00000i 0 0 0
4393.2 0 0 0 4.00000i 0 2.00000i 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
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6084.2.b.j 2
3.b odd 2 1 2028.2.b.a 2
13.b even 2 1 inner 6084.2.b.j 2
13.d odd 4 1 468.2.a.d 1
13.d odd 4 1 6084.2.a.b 1
39.d odd 2 1 2028.2.b.a 2
39.f even 4 1 156.2.a.a 1
39.f even 4 1 2028.2.a.c 1
39.h odd 6 2 2028.2.q.h 4
39.i odd 6 2 2028.2.q.h 4
39.k even 12 2 2028.2.i.e 2
39.k even 12 2 2028.2.i.g 2
52.f even 4 1 1872.2.a.s 1
104.j odd 4 1 7488.2.a.c 1
104.m even 4 1 7488.2.a.d 1
117.y odd 12 2 4212.2.i.b 2
117.z even 12 2 4212.2.i.l 2
156.l odd 4 1 624.2.a.e 1
156.l odd 4 1 8112.2.a.bi 1
195.j odd 4 1 3900.2.h.b 2
195.n even 4 1 3900.2.a.m 1
195.u odd 4 1 3900.2.h.b 2
273.o odd 4 1 7644.2.a.k 1
312.w odd 4 1 2496.2.a.o 1
312.y even 4 1 2496.2.a.bc 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.a.a 1 39.f even 4 1
468.2.a.d 1 13.d odd 4 1
624.2.a.e 1 156.l odd 4 1
1872.2.a.s 1 52.f even 4 1
2028.2.a.c 1 39.f even 4 1
2028.2.b.a 2 3.b odd 2 1
2028.2.b.a 2 39.d odd 2 1
2028.2.i.e 2 39.k even 12 2
2028.2.i.g 2 39.k even 12 2
2028.2.q.h 4 39.h odd 6 2
2028.2.q.h 4 39.i odd 6 2
2496.2.a.o 1 312.w odd 4 1
2496.2.a.bc 1 312.y even 4 1
3900.2.a.m 1 195.n even 4 1
3900.2.h.b 2 195.j odd 4 1
3900.2.h.b 2 195.u odd 4 1
4212.2.i.b 2 117.y odd 12 2
4212.2.i.l 2 117.z even 12 2
6084.2.a.b 1 13.d odd 4 1
6084.2.b.j 2 1.a even 1 1 trivial
6084.2.b.j 2 13.b even 2 1 inner
7488.2.a.c 1 104.j odd 4 1
7488.2.a.d 1 104.m even 4 1
7644.2.a.k 1 273.o odd 4 1
8112.2.a.bi 1 156.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6084, [\chi])\):

\( T_{5}^{2} + 16 \)
\( T_{7}^{2} + 4 \)
\( T_{11}^{2} + 16 \)
\( T_{23} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 16 + T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( 16 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( 4 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( 100 + T^{2} \)
$37$ \( 100 + T^{2} \)
$41$ \( 64 + T^{2} \)
$43$ \( ( 4 + T )^{2} \)
$47$ \( 16 + T^{2} \)
$53$ \( ( -10 + T )^{2} \)
$59$ \( 64 + T^{2} \)
$61$ \( ( 14 + T )^{2} \)
$67$ \( 4 + T^{2} \)
$71$ \( 256 + T^{2} \)
$73$ \( 100 + T^{2} \)
$79$ \( ( 16 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( 16 + T^{2} \)
$97$ \( 4 + T^{2} \)
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