Properties

Label 3042.2.b.i
Level $3042$
Weight $2$
Character orbit 3042.b
Analytic conductor $24.290$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3042.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.2904922949\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12}^{3} q^{2} - q^{4} + ( -1 + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{5} + ( 1 - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{7} -\zeta_{12}^{3} q^{8} +O(q^{10})\) \( q + \zeta_{12}^{3} q^{2} - q^{4} + ( -1 + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{5} + ( 1 - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{7} -\zeta_{12}^{3} q^{8} + ( -2 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{10} + ( -1 + 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{11} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{14} + q^{16} + ( -4 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{17} + ( 1 - 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{19} + ( 1 - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{20} + ( 3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{22} + ( -1 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{23} + ( -2 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{25} + ( -1 + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{28} + ( 1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{29} + ( 2 - 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{31} + \zeta_{12}^{3} q^{32} + ( 1 - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{34} + ( 5 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{35} + ( 2 - 4 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{37} + ( 3 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{38} + ( 2 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{40} + ( 6 - 12 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{41} + ( 1 + 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{43} + ( 1 - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{44} + ( -3 + 6 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{46} + ( 3 - 6 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{47} + ( 3 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{49} + ( 4 - 8 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{50} + ( 3 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{53} + ( 3 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{55} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{56} + ( -2 + 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{58} -8 \zeta_{12}^{3} q^{59} + ( -4 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{61} + ( -2 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{62} - q^{64} + ( 7 - 14 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{67} + ( 4 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{68} + ( -3 + 6 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{70} + ( -1 + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{71} + ( -1 + 2 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{73} + ( -7 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{74} + ( -1 + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{76} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{77} + ( -6 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{79} + ( -1 + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{80} + ( -1 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{82} + ( 3 - 6 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{83} + ( 6 - 12 \zeta_{12}^{2} - 11 \zeta_{12}^{3} ) q^{85} + ( -5 + 10 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{86} + ( -3 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{88} + ( -2 + 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{89} + ( 1 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{92} + ( -3 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{94} + ( 9 + 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{95} + 6 \zeta_{12}^{3} q^{97} + ( 2 - 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + O(q^{10}) \) \( 4q - 4q^{4} - 8q^{10} + 4q^{14} + 4q^{16} - 16q^{17} + 12q^{22} - 4q^{23} - 8q^{25} + 4q^{29} + 20q^{35} + 12q^{38} + 8q^{40} + 4q^{43} + 12q^{49} + 12q^{53} + 12q^{55} - 4q^{56} - 16q^{61} - 8q^{62} - 4q^{64} + 16q^{68} - 28q^{74} - 24q^{79} - 4q^{82} - 12q^{88} + 4q^{92} - 12q^{94} + 36q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1351.1
0.866025 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
1.00000i 0 −1.00000 3.73205i 0 2.73205i 1.00000i 0 −3.73205
1351.2 1.00000i 0 −1.00000 0.267949i 0 0.732051i 1.00000i 0 −0.267949
1351.3 1.00000i 0 −1.00000 0.267949i 0 0.732051i 1.00000i 0 −0.267949
1351.4 1.00000i 0 −1.00000 3.73205i 0 2.73205i 1.00000i 0 −3.73205
\(n\): 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 3042.2.b.i 4
3.b odd 2 1 1014.2.b.e 4
13.b even 2 1 inner 3042.2.b.i 4
13.c even 3 1 234.2.l.c 4
13.d odd 4 1 3042.2.a.p 2
13.d odd 4 1 3042.2.a.y 2
13.e even 6 1 234.2.l.c 4
39.d odd 2 1 1014.2.b.e 4
39.f even 4 1 1014.2.a.i 2
39.f even 4 1 1014.2.a.k 2
39.h odd 6 1 78.2.i.a 4
39.h odd 6 1 1014.2.i.a 4
39.i odd 6 1 78.2.i.a 4
39.i odd 6 1 1014.2.i.a 4
39.k even 12 2 1014.2.e.g 4
39.k even 12 2 1014.2.e.i 4
52.i odd 6 1 1872.2.by.h 4
52.j odd 6 1 1872.2.by.h 4
156.l odd 4 1 8112.2.a.bj 2
156.l odd 4 1 8112.2.a.bp 2
156.p even 6 1 624.2.bv.e 4
156.r even 6 1 624.2.bv.e 4
195.x odd 6 1 1950.2.bc.d 4
195.y odd 6 1 1950.2.bc.d 4
195.bf even 12 1 1950.2.y.b 4
195.bf even 12 1 1950.2.y.g 4
195.bl even 12 1 1950.2.y.b 4
195.bl even 12 1 1950.2.y.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.i.a 4 39.h odd 6 1
78.2.i.a 4 39.i odd 6 1
234.2.l.c 4 13.c even 3 1
234.2.l.c 4 13.e even 6 1
624.2.bv.e 4 156.p even 6 1
624.2.bv.e 4 156.r even 6 1
1014.2.a.i 2 39.f even 4 1
1014.2.a.k 2 39.f even 4 1
1014.2.b.e 4 3.b odd 2 1
1014.2.b.e 4 39.d odd 2 1
1014.2.e.g 4 39.k even 12 2
1014.2.e.i 4 39.k even 12 2
1014.2.i.a 4 39.h odd 6 1
1014.2.i.a 4 39.i odd 6 1
1872.2.by.h 4 52.i odd 6 1
1872.2.by.h 4 52.j odd 6 1
1950.2.y.b 4 195.bf even 12 1
1950.2.y.b 4 195.bl even 12 1
1950.2.y.g 4 195.bf even 12 1
1950.2.y.g 4 195.bl even 12 1
1950.2.bc.d 4 195.x odd 6 1
1950.2.bc.d 4 195.y odd 6 1
3042.2.a.p 2 13.d odd 4 1
3042.2.a.y 2 13.d odd 4 1
3042.2.b.i 4 1.a even 1 1 trivial
3042.2.b.i 4 13.b even 2 1 inner
8112.2.a.bj 2 156.l odd 4 1
8112.2.a.bp 2 156.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 \(S_{2}^{\mathrm{new}}(3042, [\chi])\):

\( T_{5}^{4} + 14 T_{5}^{2} + 1 \)
\( T_{7}^{4} + 8 T_{7}^{2} + 4 \)
\( T_{17}^{2} + 8 T_{17} + 13 \)
\( T_{23}^{2} + 2 T_{23} - 26 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 1 + 14 T^{2} + T^{4} \)
$7$ \( 4 + 8 T^{2} + T^{4} \)
$11$ \( 36 + 24 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( 13 + 8 T + T^{2} )^{2} \)
$19$ \( 36 + 24 T^{2} + T^{4} \)
$23$ \( ( -26 + 2 T + T^{2} )^{2} \)
$29$ \( ( -11 - 2 T + T^{2} )^{2} \)
$31$ \( 64 + 32 T^{2} + T^{4} \)
$37$ \( 1369 + 122 T^{2} + T^{4} \)
$41$ \( 11449 + 218 T^{2} + T^{4} \)
$43$ \( ( -74 - 2 T + T^{2} )^{2} \)
$47$ \( 324 + 72 T^{2} + T^{4} \)
$53$ \( ( -3 - 6 T + T^{2} )^{2} \)
$59$ \( ( 64 + T^{2} )^{2} \)
$61$ \( ( -11 + 8 T + T^{2} )^{2} \)
$67$ \( 21316 + 296 T^{2} + T^{4} \)
$71$ \( 36 + 24 T^{2} + T^{4} \)
$73$ \( 3721 + 134 T^{2} + T^{4} \)
$79$ \( ( 24 + 12 T + T^{2} )^{2} \)
$83$ \( 4 + 104 T^{2} + T^{4} \)
$89$ \( 576 + 96 T^{2} + T^{4} \)
$97$ \( ( 36 + T^{2} )^{2} \)
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