Properties

Label 3042.2.b.r
Level $3042$
Weight $2$
Character orbit 3042.b
Analytic conductor $24.290$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1014)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} - q^{4} + ( - \beta_{5} + \beta_{3} + \beta_1) q^{5} + ( - \beta_{3} + 2 \beta_1) q^{7} - \beta_{5} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} - q^{4} + ( - \beta_{5} + \beta_{3} + \beta_1) q^{5} + ( - \beta_{3} + 2 \beta_1) q^{7} - \beta_{5} q^{8} + ( - 2 \beta_{4} - \beta_{2} + 2) q^{10} + (2 \beta_{3} + \beta_1) q^{11} + ( - \beta_{4} + \beta_{2} - 1) q^{14} + q^{16} + ( - 2 \beta_{4} + 2 \beta_{2} + 4) q^{17} + ( - 4 \beta_{5} - 4 \beta_{3} + 4 \beta_1) q^{19} + (\beta_{5} - \beta_{3} - \beta_1) q^{20} + ( - 3 \beta_{4} - 2 \beta_{2} + 2) q^{22} + ( - 2 \beta_{2} + 6) q^{23} + (7 \beta_{4} + 3 \beta_{2} - 4) q^{25} + (\beta_{3} - 2 \beta_1) q^{28} + ( - 5 \beta_{4} - 2 \beta_{2} - 2) q^{29} + ( - 4 \beta_{3} + 5 \beta_1) q^{31} + \beta_{5} q^{32} + (6 \beta_{5} + 2 \beta_{3} - 4 \beta_1) q^{34} + (\beta_{4} + \beta_{2} - 2) q^{35} + ( - 6 \beta_{5} - 4 \beta_{3} + 2 \beta_1) q^{37} + 4 \beta_{2} q^{38} + (2 \beta_{4} + \beta_{2} - 2) q^{40} + ( - 4 \beta_{5} - 2 \beta_1) q^{41} + ( - 4 \beta_{4} - 6 \beta_{2} + 6) q^{43} + ( - 2 \beta_{3} - \beta_1) q^{44} + (4 \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{46} + 4 \beta_{3} q^{47} + ( - 3 \beta_{4} + 4 \beta_{2} + 1) q^{49} + ( - \beta_{5} + 3 \beta_{3} + 4 \beta_1) q^{50} + (6 \beta_{4} + 9 \beta_{2} - 10) q^{53} + (8 \beta_{4} + 3 \beta_{2} - 11) q^{55} + (\beta_{4} - \beta_{2} + 1) q^{56} + ( - 4 \beta_{5} - 2 \beta_{3} - 3 \beta_1) q^{58} + (2 \beta_{5} - \beta_{3} + 2 \beta_1) q^{59} + ( - 2 \beta_{4} - 2 \beta_{2} - 2) q^{61} + ( - \beta_{4} + 4 \beta_{2} - 4) q^{62} - q^{64} + (8 \beta_{5} + 8 \beta_{3} - 10 \beta_1) q^{67} + (2 \beta_{4} - 2 \beta_{2} - 4) q^{68} + ( - \beta_{5} + \beta_{3}) q^{70} + ( - 2 \beta_{5} - 6 \beta_{3} + 6 \beta_1) q^{71} + (\beta_{5} - 3 \beta_{3} - \beta_1) q^{73} + (2 \beta_{4} + 4 \beta_{2} + 2) q^{74} + (4 \beta_{5} + 4 \beta_{3} - 4 \beta_1) q^{76} + (\beta_{4} + 2 \beta_{2} - 3) q^{77} + (5 \beta_{4} + 11 \beta_{2} - 7) q^{79} + ( - \beta_{5} + \beta_{3} + \beta_1) q^{80} + (2 \beta_{4} + 4) q^{82} + (2 \beta_{5} + 11 \beta_{3} - 2 \beta_1) q^{83} + ( - 8 \beta_{5} + 8 \beta_{3} + 6 \beta_1) q^{85} + ( - 6 \beta_{3} + 2 \beta_1) q^{86} + (3 \beta_{4} + 2 \beta_{2} - 2) q^{88} + (2 \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{89} + (2 \beta_{2} - 6) q^{92} + ( - 4 \beta_{4} - 4 \beta_{2} + 4) q^{94} + (4 \beta_{4} + 4 \beta_{2} - 4) q^{95} + ( - 3 \beta_{5} + \beta_{3} + 3 \beta_1) q^{97} + (5 \beta_{5} + 4 \beta_{3} - 7 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} + 6 q^{10} - 6 q^{14} + 6 q^{16} + 24 q^{17} + 2 q^{22} + 32 q^{23} - 4 q^{25} - 26 q^{29} - 8 q^{35} + 8 q^{38} - 6 q^{40} + 16 q^{43} + 8 q^{49} - 30 q^{53} - 44 q^{55} + 6 q^{56} - 20 q^{61} - 18 q^{62} - 6 q^{64} - 24 q^{68} + 24 q^{74} - 12 q^{77} - 10 q^{79} + 28 q^{82} - 2 q^{88} - 32 q^{92} + 8 q^{94} - 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 5x^{4} + 6x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 3\nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 4\nu^{3} + 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 3\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{3} + 9\beta_1 \) 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
0.445042i
1.80194i
1.24698i
1.24698i
1.80194i
0.445042i
1.00000i 0 −1.00000 0.692021i 0 0.356896i 1.00000i 0 −0.692021
1351.2 1.00000i 0 −1.00000 0.356896i 0 4.04892i 1.00000i 0 −0.356896
1351.3 1.00000i 0 −1.00000 4.04892i 0 0.692021i 1.00000i 0 4.04892
1351.4 1.00000i 0 −1.00000 4.04892i 0 0.692021i 1.00000i 0 4.04892
1351.5 1.00000i 0 −1.00000 0.356896i 0 4.04892i 1.00000i 0 −0.356896
1351.6 1.00000i 0 −1.00000 0.692021i 0 0.356896i 1.00000i 0 −0.692021
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1351.6
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.r 6
3.b odd 2 1 1014.2.b.g 6
13.b even 2 1 inner 3042.2.b.r 6
13.d odd 4 1 3042.2.a.bd 3
13.d odd 4 1 3042.2.a.be 3
39.d odd 2 1 1014.2.b.g 6
39.f even 4 1 1014.2.a.m 3
39.f even 4 1 1014.2.a.o yes 3
39.h odd 6 2 1014.2.i.g 12
39.i odd 6 2 1014.2.i.g 12
39.k even 12 2 1014.2.e.k 6
39.k even 12 2 1014.2.e.m 6
156.l odd 4 1 8112.2.a.bz 3
156.l odd 4 1 8112.2.a.ce 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1014.2.a.m 3 39.f even 4 1
1014.2.a.o yes 3 39.f even 4 1
1014.2.b.g 6 3.b odd 2 1
1014.2.b.g 6 39.d odd 2 1
1014.2.e.k 6 39.k even 12 2
1014.2.e.m 6 39.k even 12 2
1014.2.i.g 12 39.h odd 6 2
1014.2.i.g 12 39.i odd 6 2
3042.2.a.bd 3 13.d odd 4 1
3042.2.a.be 3 13.d odd 4 1
3042.2.b.r 6 1.a even 1 1 trivial
3042.2.b.r 6 13.b even 2 1 inner
8112.2.a.bz 3 156.l odd 4 1
8112.2.a.ce 3 156.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 \(S_{2}^{\mathrm{new}}(3042, [\chi])\):

\( T_{5}^{6} + 17T_{5}^{4} + 10T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{6} + 17T_{7}^{4} + 10T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{17}^{3} - 12T_{17}^{2} + 20T_{17} + 104 \) Copy content Toggle raw display
\( T_{23}^{3} - 16T_{23}^{2} + 76T_{23} - 104 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 17 T^{4} + 10 T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{6} + 17 T^{4} + 10 T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{6} + 33 T^{4} + 230 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( (T^{3} - 12 T^{2} + 20 T + 104)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 80 T^{4} + 1536 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$23$ \( (T^{3} - 16 T^{2} + 76 T - 104)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 13 T^{2} + 12 T - 223)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 125 T^{4} + 1006 T^{2} + \cdots + 841 \) Copy content Toggle raw display
$37$ \( T^{6} + 104 T^{4} + 208 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$41$ \( T^{6} + 84 T^{4} + 1568 T^{2} + \cdots + 3136 \) Copy content Toggle raw display
$43$ \( (T^{3} - 8 T^{2} - 44 T + 344)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 80 T^{4} + 1536 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( (T^{3} + 15 T^{2} - 72 T - 1247)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 41 T^{4} + 166 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$61$ \( (T^{3} + 10 T^{2} + 24 T + 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 404 T^{4} + 47200 T^{2} + \cdots + 1236544 \) Copy content Toggle raw display
$71$ \( T^{6} + 180 T^{4} + 6432 T^{2} + \cdots + 10816 \) Copy content Toggle raw display
$73$ \( T^{6} + 69 T^{4} + 614 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$79$ \( (T^{3} + 5 T^{2} - 204 T - 1469)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 497 T^{4} + 70854 T^{2} + \cdots + 2181529 \) Copy content Toggle raw display
$89$ \( T^{6} + 52 T^{4} + 416 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$97$ \( T^{6} + 77 T^{4} + 294 T^{2} + \cdots + 49 \) Copy content Toggle raw display
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