Properties

Label 6084.2.b.f
Level 6084
Weight 2
Character orbit 6084.b
Analytic conductor 48.581
Analytic rank 0
Dimension 2
CM discriminant -3
Inner twists 4

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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: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -4 i q^{7} +O(q^{10})\) \( q -4 i q^{7} -8 i q^{19} + 5 q^{25} + 4 i q^{31} -10 i q^{37} -8 q^{43} -9 q^{49} + 14 q^{61} + 16 i q^{67} -10 i q^{73} -4 q^{79} -14 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 10q^{25} - 16q^{43} - 18q^{49} + 28q^{61} - 8q^{79} + O(q^{100}) \)

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 0 0 4.00000i 0 0 0
4393.2 0 0 0 0 0 4.00000i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
13.b even 2 1 inner
39.d odd 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.f 2
3.b odd 2 1 CM 6084.2.b.f 2
13.b even 2 1 inner 6084.2.b.f 2
13.d odd 4 1 36.2.a.a 1
13.d odd 4 1 6084.2.a.i 1
39.d odd 2 1 inner 6084.2.b.f 2
39.f even 4 1 36.2.a.a 1
39.f even 4 1 6084.2.a.i 1
52.f even 4 1 144.2.a.a 1
65.f even 4 1 900.2.d.b 2
65.g odd 4 1 900.2.a.g 1
65.k even 4 1 900.2.d.b 2
91.i even 4 1 1764.2.a.e 1
91.z odd 12 2 1764.2.k.h 2
91.bb even 12 2 1764.2.k.g 2
104.j odd 4 1 576.2.a.e 1
104.m even 4 1 576.2.a.f 1
117.y odd 12 2 324.2.e.c 2
117.z even 12 2 324.2.e.c 2
143.g even 4 1 4356.2.a.g 1
156.l odd 4 1 144.2.a.a 1
195.j odd 4 1 900.2.d.b 2
195.n even 4 1 900.2.a.g 1
195.u odd 4 1 900.2.d.b 2
208.l even 4 1 2304.2.d.a 2
208.m odd 4 1 2304.2.d.q 2
208.r odd 4 1 2304.2.d.q 2
208.s even 4 1 2304.2.d.a 2
260.l odd 4 1 3600.2.f.m 2
260.s odd 4 1 3600.2.f.m 2
260.u even 4 1 3600.2.a.e 1
273.o odd 4 1 1764.2.a.e 1
273.cb odd 12 2 1764.2.k.g 2
273.cd even 12 2 1764.2.k.h 2
312.w odd 4 1 576.2.a.f 1
312.y even 4 1 576.2.a.e 1
364.p odd 4 1 7056.2.a.bb 1
429.l odd 4 1 4356.2.a.g 1
468.bs even 12 2 1296.2.i.h 2
468.ch odd 12 2 1296.2.i.h 2
624.s odd 4 1 2304.2.d.a 2
624.u even 4 1 2304.2.d.q 2
624.bm even 4 1 2304.2.d.q 2
624.bo odd 4 1 2304.2.d.a 2
780.u even 4 1 3600.2.f.m 2
780.bb odd 4 1 3600.2.a.e 1
780.bn even 4 1 3600.2.f.m 2
1092.u even 4 1 7056.2.a.bb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.a.a 1 13.d odd 4 1
36.2.a.a 1 39.f even 4 1
144.2.a.a 1 52.f even 4 1
144.2.a.a 1 156.l odd 4 1
324.2.e.c 2 117.y odd 12 2
324.2.e.c 2 117.z even 12 2
576.2.a.e 1 104.j odd 4 1
576.2.a.e 1 312.y even 4 1
576.2.a.f 1 104.m even 4 1
576.2.a.f 1 312.w odd 4 1
900.2.a.g 1 65.g odd 4 1
900.2.a.g 1 195.n even 4 1
900.2.d.b 2 65.f even 4 1
900.2.d.b 2 65.k even 4 1
900.2.d.b 2 195.j odd 4 1
900.2.d.b 2 195.u odd 4 1
1296.2.i.h 2 468.bs even 12 2
1296.2.i.h 2 468.ch odd 12 2
1764.2.a.e 1 91.i even 4 1
1764.2.a.e 1 273.o odd 4 1
1764.2.k.g 2 91.bb even 12 2
1764.2.k.g 2 273.cb odd 12 2
1764.2.k.h 2 91.z odd 12 2
1764.2.k.h 2 273.cd even 12 2
2304.2.d.a 2 208.l even 4 1
2304.2.d.a 2 208.s even 4 1
2304.2.d.a 2 624.s odd 4 1
2304.2.d.a 2 624.bo odd 4 1
2304.2.d.q 2 208.m odd 4 1
2304.2.d.q 2 208.r odd 4 1
2304.2.d.q 2 624.u even 4 1
2304.2.d.q 2 624.bm even 4 1
3600.2.a.e 1 260.u even 4 1
3600.2.a.e 1 780.bb odd 4 1
3600.2.f.m 2 260.l odd 4 1
3600.2.f.m 2 260.s odd 4 1
3600.2.f.m 2 780.u even 4 1
3600.2.f.m 2 780.bn even 4 1
4356.2.a.g 1 143.g even 4 1
4356.2.a.g 1 429.l odd 4 1
6084.2.a.i 1 13.d odd 4 1
6084.2.a.i 1 39.f even 4 1
6084.2.b.f 2 1.a even 1 1 trivial
6084.2.b.f 2 3.b odd 2 1 CM
6084.2.b.f 2 13.b even 2 1 inner
6084.2.b.f 2 39.d odd 2 1 inner
7056.2.a.bb 1 364.p odd 4 1
7056.2.a.bb 1 1092.u even 4 1

Hecke kernels

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( ( 1 - 5 T^{2} )^{2} \)
$7$ \( 1 + 2 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ \( \)
$17$ \( ( 1 + 17 T^{2} )^{2} \)
$19$ \( 1 + 26 T^{2} + 361 T^{4} \)
$23$ \( ( 1 + 23 T^{2} )^{2} \)
$29$ \( ( 1 + 29 T^{2} )^{2} \)
$31$ \( 1 - 46 T^{2} + 961 T^{4} \)
$37$ \( 1 + 26 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 - 41 T^{2} )^{2} \)
$43$ \( ( 1 + 8 T + 43 T^{2} )^{2} \)
$47$ \( ( 1 - 47 T^{2} )^{2} \)
$53$ \( ( 1 + 53 T^{2} )^{2} \)
$59$ \( ( 1 - 59 T^{2} )^{2} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )^{2} \)
$67$ \( 1 + 122 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 71 T^{2} )^{2} \)
$73$ \( 1 - 46 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 + 4 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 - 83 T^{2} )^{2} \)
$89$ \( ( 1 - 89 T^{2} )^{2} \)
$97$ \( 1 + 2 T^{2} + 9409 T^{4} \)
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