Properties

Label 3042.2.b.l
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: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{4} - \beta_{2} q^{5} + (\beta_{2} - 3 \beta_1) q^{7} - \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{4} - \beta_{2} q^{5} + (\beta_{2} - 3 \beta_1) q^{7} - \beta_1 q^{8} + \beta_{3} q^{10} + ( - \beta_{2} + 3 \beta_1) q^{11} + ( - \beta_{3} + 3) q^{14} + q^{16} + 3 \beta_{3} q^{17} + (\beta_{2} + 3 \beta_1) q^{19} + \beta_{2} q^{20} + (\beta_{3} - 3) q^{22} + ( - 3 \beta_{3} - 3) q^{23} + 2 q^{25} + ( - \beta_{2} + 3 \beta_1) q^{28} + 3 q^{29} + (2 \beta_{2} + 6 \beta_1) q^{31} + \beta_1 q^{32} + 3 \beta_{2} q^{34} + ( - 3 \beta_{3} + 3) q^{35} - 3 \beta_1 q^{37} + ( - \beta_{3} - 3) q^{38} - \beta_{3} q^{40} + ( - 2 \beta_{2} - 3 \beta_1) q^{41} + (3 \beta_{3} - 1) q^{43} + (\beta_{2} - 3 \beta_1) q^{44} + ( - 3 \beta_{2} - 3 \beta_1) q^{46} + ( - \beta_{2} - 3 \beta_1) q^{47} + (6 \beta_{3} - 5) q^{49} + 2 \beta_1 q^{50} - 3 q^{53} + (3 \beta_{3} - 3) q^{55} + (\beta_{3} - 3) q^{56} + 3 \beta_1 q^{58} - 8 \beta_{2} q^{59} + (3 \beta_{3} + 10) q^{61} + ( - 2 \beta_{3} - 6) q^{62} - q^{64} + ( - \beta_{2} + 9 \beta_1) q^{67} - 3 \beta_{3} q^{68} + ( - 3 \beta_{2} + 3 \beta_1) q^{70} + (3 \beta_{2} - 3 \beta_1) q^{71} - 7 \beta_{2} q^{73} + 3 q^{74} + ( - \beta_{2} - 3 \beta_1) q^{76} + ( - 6 \beta_{3} + 12) q^{77} + (6 \beta_{3} - 2) q^{79} - \beta_{2} q^{80} + (2 \beta_{3} + 3) q^{82} + ( - 5 \beta_{2} + 3 \beta_1) q^{83} - 9 \beta_1 q^{85} + (3 \beta_{2} - \beta_1) q^{86} + ( - \beta_{3} + 3) q^{88} + (2 \beta_{2} + 6 \beta_1) q^{89} + (3 \beta_{3} + 3) q^{92} + (\beta_{3} + 3) q^{94} + (3 \beta_{3} + 3) q^{95} + 6 \beta_1 q^{97} + (6 \beta_{2} - 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 12 q^{14} + 4 q^{16} - 12 q^{22} - 12 q^{23} + 8 q^{25} + 12 q^{29} + 12 q^{35} - 12 q^{38} - 4 q^{43} - 20 q^{49} - 12 q^{53} - 12 q^{55} - 12 q^{56} + 40 q^{61} - 24 q^{62} - 4 q^{64} + 12 q^{74} + 48 q^{77} - 8 q^{79} + 12 q^{82} + 12 q^{88} + 12 q^{92} + 12 q^{94} + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

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 1.73205i 0 4.73205i 1.00000i 0 −1.73205
1351.2 1.00000i 0 −1.00000 1.73205i 0 1.26795i 1.00000i 0 1.73205
1351.3 1.00000i 0 −1.00000 1.73205i 0 1.26795i 1.00000i 0 1.73205
1351.4 1.00000i 0 −1.00000 1.73205i 0 4.73205i 1.00000i 0 −1.73205
\(n\): 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.l 4
3.b odd 2 1 1014.2.b.d 4
13.b even 2 1 inner 3042.2.b.l 4
13.c even 3 1 234.2.l.a 4
13.d odd 4 1 3042.2.a.s 2
13.d odd 4 1 3042.2.a.v 2
13.e even 6 1 234.2.l.a 4
39.d odd 2 1 1014.2.b.d 4
39.f even 4 1 1014.2.a.h 2
39.f even 4 1 1014.2.a.j 2
39.h odd 6 1 78.2.i.b 4
39.h odd 6 1 1014.2.i.f 4
39.i odd 6 1 78.2.i.b 4
39.i odd 6 1 1014.2.i.f 4
39.k even 12 2 1014.2.e.h 4
39.k even 12 2 1014.2.e.j 4
52.i odd 6 1 1872.2.by.k 4
52.j odd 6 1 1872.2.by.k 4
156.l odd 4 1 8112.2.a.bq 2
156.l odd 4 1 8112.2.a.bx 2
156.p even 6 1 624.2.bv.d 4
156.r even 6 1 624.2.bv.d 4
195.x odd 6 1 1950.2.bc.c 4
195.y odd 6 1 1950.2.bc.c 4
195.bf even 12 1 1950.2.y.a 4
195.bf even 12 1 1950.2.y.h 4
195.bl even 12 1 1950.2.y.a 4
195.bl even 12 1 1950.2.y.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.i.b 4 39.h odd 6 1
78.2.i.b 4 39.i odd 6 1
234.2.l.a 4 13.c even 3 1
234.2.l.a 4 13.e even 6 1
624.2.bv.d 4 156.p even 6 1
624.2.bv.d 4 156.r even 6 1
1014.2.a.h 2 39.f even 4 1
1014.2.a.j 2 39.f even 4 1
1014.2.b.d 4 3.b odd 2 1
1014.2.b.d 4 39.d odd 2 1
1014.2.e.h 4 39.k even 12 2
1014.2.e.j 4 39.k even 12 2
1014.2.i.f 4 39.h odd 6 1
1014.2.i.f 4 39.i odd 6 1
1872.2.by.k 4 52.i odd 6 1
1872.2.by.k 4 52.j odd 6 1
1950.2.y.a 4 195.bf even 12 1
1950.2.y.a 4 195.bl even 12 1
1950.2.y.h 4 195.bf even 12 1
1950.2.y.h 4 195.bl even 12 1
1950.2.bc.c 4 195.x odd 6 1
1950.2.bc.c 4 195.y odd 6 1
3042.2.a.s 2 13.d odd 4 1
3042.2.a.v 2 13.d odd 4 1
3042.2.b.l 4 1.a even 1 1 trivial
3042.2.b.l 4 13.b even 2 1 inner
8112.2.a.bq 2 156.l odd 4 1
8112.2.a.bx 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 \(S_{2}^{\mathrm{new}}(3042, [\chi])\):

\( T_{5}^{2} + 3 \) Copy content Toggle raw display
\( T_{7}^{4} + 24T_{7}^{2} + 36 \) Copy content Toggle raw display
\( T_{17}^{2} - 27 \) Copy content Toggle raw display
\( T_{23}^{2} + 6T_{23} - 18 \) Copy content Toggle raw display

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
$5$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$11$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$23$ \( (T^{2} + 6 T - 18)^{2} \) Copy content Toggle raw display
$29$ \( (T - 3)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$37$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 42T^{2} + 9 \) Copy content Toggle raw display
$43$ \( (T^{2} + 2 T - 26)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$53$ \( (T + 3)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 192)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 20 T + 73)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 168T^{2} + 6084 \) Copy content Toggle raw display
$71$ \( T^{4} + 72T^{2} + 324 \) Copy content Toggle raw display
$73$ \( (T^{2} + 147)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T - 104)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 168T^{2} + 4356 \) Copy content Toggle raw display
$89$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$97$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
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