Properties

Label 2842.2.a.bb
Level $2842$
Weight $2$
Character orbit 2842.a
Self dual yes
Analytic conductor $22.693$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2842 = 2 \cdot 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2842.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.6934842544\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.401917952.1
Defining polynomial: \( x^{6} - 14x^{4} + 49x^{2} - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} + ( - \beta_{2} + \beta_1) q^{5} + \beta_1 q^{6} + q^{8} + ( - \beta_{4} + \beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} + ( - \beta_{2} + \beta_1) q^{5} + \beta_1 q^{6} + q^{8} + ( - \beta_{4} + \beta_{3} + 2) q^{9} + ( - \beta_{2} + \beta_1) q^{10} + (\beta_{4} + 1) q^{11} + \beta_1 q^{12} + ( - \beta_{5} - 2 \beta_{2}) q^{13} + ( - \beta_{4} + 5) q^{15} + q^{16} + (\beta_{5} - \beta_1) q^{17} + ( - \beta_{4} + \beta_{3} + 2) q^{18} + ( - \beta_{2} + 2 \beta_1) q^{19} + ( - \beta_{2} + \beta_1) q^{20} + (\beta_{4} + 1) q^{22} + (2 \beta_{4} - 2) q^{23} + \beta_1 q^{24} + ( - \beta_{4} - \beta_{3} + 2) q^{25} + ( - \beta_{5} - 2 \beta_{2}) q^{26} + (3 \beta_{2} + \beta_1) q^{27} - q^{29} + ( - \beta_{4} + 5) q^{30} + (3 \beta_{2} + \beta_1) q^{31} + q^{32} + (\beta_{5} + 2 \beta_{2}) q^{33} + (\beta_{5} - \beta_1) q^{34} + ( - \beta_{4} + \beta_{3} + 2) q^{36} - 2 q^{37} + ( - \beta_{2} + 2 \beta_1) q^{38} + ( - \beta_{4} - 3 \beta_{3} - 1) q^{39} + ( - \beta_{2} + \beta_1) q^{40} + ( - \beta_{5} - 4 \beta_{2} + \beta_1) q^{41} + (3 \beta_{4} + \beta_{3} + 1) q^{43} + (\beta_{4} + 1) q^{44} + ( - \beta_{5} + \beta_{2} + 3 \beta_1) q^{45} + (2 \beta_{4} - 2) q^{46} + (3 \beta_{2} + \beta_1) q^{47} + \beta_1 q^{48} + ( - \beta_{4} - \beta_{3} + 2) q^{50} + (2 \beta_{4} - 4) q^{51} + ( - \beta_{5} - 2 \beta_{2}) q^{52} + ( - \beta_{4} + 3) q^{53} + (3 \beta_{2} + \beta_1) q^{54} + (2 \beta_{5} + \beta_{2} - \beta_1) q^{55} + ( - 2 \beta_{4} + \beta_{3} + 10) q^{57} - q^{58} + ( - \beta_{5} + 2 \beta_{2} - 3 \beta_1) q^{59} + ( - \beta_{4} + 5) q^{60} + (2 \beta_{5} - 2 \beta_{2} - 4 \beta_1) q^{61} + (3 \beta_{2} + \beta_1) q^{62} + q^{64} + ( - 3 \beta_{4} - 2 \beta_{3} + 3) q^{65} + (\beta_{5} + 2 \beta_{2}) q^{66} + (2 \beta_{4} - 2 \beta_{3} - 2) q^{67} + (\beta_{5} - \beta_1) q^{68} + (2 \beta_{5} + 4 \beta_{2} - 4 \beta_1) q^{69} + 8 q^{71} + ( - \beta_{4} + \beta_{3} + 2) q^{72} + ( - \beta_{5} + 2 \beta_{2} - 3 \beta_1) q^{73} - 2 q^{74} + ( - 2 \beta_{5} - 7 \beta_{2} + 2 \beta_1) q^{75} + ( - \beta_{2} + 2 \beta_1) q^{76} + ( - \beta_{4} - 3 \beta_{3} - 1) q^{78} + ( - \beta_{4} + \beta_{3} + 11) q^{79} + ( - \beta_{2} + \beta_1) q^{80} + (2 \beta_{4} + \beta_{3} - 1) q^{81} + ( - \beta_{5} - 4 \beta_{2} + \beta_1) q^{82} + (\beta_{5} + 4 \beta_{2} - \beta_1) q^{83} + (4 \beta_{4} - 4) q^{85} + (3 \beta_{4} + \beta_{3} + 1) q^{86} - \beta_1 q^{87} + (\beta_{4} + 1) q^{88} + ( - \beta_{5} - 4 \beta_{2} - \beta_1) q^{89} + ( - \beta_{5} + \beta_{2} + 3 \beta_1) q^{90} + (2 \beta_{4} - 2) q^{92} + ( - \beta_{4} + 4 \beta_{3} + 5) q^{93} + (3 \beta_{2} + \beta_1) q^{94} + ( - 2 \beta_{4} - \beta_{3} + 12) q^{95} + \beta_1 q^{96} + ( - \beta_{5} - 6 \beta_{2} - 3 \beta_1) q^{97} + ( - 2 \beta_{4} + 3 \beta_{3} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 10 q^{9} + 8 q^{11} + 28 q^{15} + 6 q^{16} + 10 q^{18} + 8 q^{22} - 8 q^{23} + 10 q^{25} - 6 q^{29} + 28 q^{30} + 6 q^{32} + 10 q^{36} - 12 q^{37} - 8 q^{39} + 12 q^{43} + 8 q^{44} - 8 q^{46} + 10 q^{50} - 20 q^{51} + 16 q^{53} + 56 q^{57} - 6 q^{58} + 28 q^{60} + 6 q^{64} + 12 q^{65} - 8 q^{67} + 48 q^{71} + 10 q^{72} - 12 q^{74} - 8 q^{78} + 64 q^{79} - 2 q^{81} - 16 q^{85} + 12 q^{86} + 8 q^{88} - 8 q^{92} + 28 q^{93} + 68 q^{95} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 14x^{4} + 49x^{2} - 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 7\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - 7\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - 10\nu^{2} + 15 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - 12\nu^{3} + 32\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{4} + 10\beta_{3} + 35 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{5} + 36\beta_{2} + 52\beta_1 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.90839
−2.26409
−0.644301
0.644301
2.26409
2.90839
1.00000 −2.90839 1.00000 −1.49418 −2.90839 0 1.00000 5.45876 −1.49418
1.2 1.00000 −2.26409 1.00000 −3.67831 −2.26409 0 1.00000 2.12612 −3.67831
1.3 1.00000 −0.644301 1.00000 −2.05851 −0.644301 0 1.00000 −2.58488 −2.05851
1.4 1.00000 0.644301 1.00000 2.05851 0.644301 0 1.00000 −2.58488 2.05851
1.5 1.00000 2.26409 1.00000 3.67831 2.26409 0 1.00000 2.12612 3.67831
1.6 1.00000 2.90839 1.00000 1.49418 2.90839 0 1.00000 5.45876 1.49418
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(29\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2842.2.a.bb 6
7.b odd 2 1 inner 2842.2.a.bb 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2842.2.a.bb 6 1.a even 1 1 trivial
2842.2.a.bb 6 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2842))\):

\( T_{3}^{6} - 14T_{3}^{4} + 49T_{3}^{2} - 18 \) Copy content Toggle raw display
\( T_{5}^{6} - 20T_{5}^{4} + 97T_{5}^{2} - 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 14 T^{4} + 49 T^{2} - 18 \) Copy content Toggle raw display
$5$ \( T^{6} - 20 T^{4} + 97 T^{2} + \cdots - 128 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( (T^{3} - 4 T^{2} - 7 T + 18)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - 60 T^{4} + 1105 T^{2} + \cdots - 5832 \) Copy content Toggle raw display
$17$ \( T^{6} - 50 T^{4} + 640 T^{2} + \cdots - 512 \) Copy content Toggle raw display
$19$ \( T^{6} - 62 T^{4} + 1084 T^{2} + \cdots - 5000 \) Copy content Toggle raw display
$23$ \( (T^{3} + 4 T^{2} - 44 T - 32)^{2} \) Copy content Toggle raw display
$29$ \( (T + 1)^{6} \) Copy content Toggle raw display
$31$ \( T^{6} - 68 T^{4} + 913 T^{2} + \cdots - 2592 \) Copy content Toggle raw display
$37$ \( (T + 2)^{6} \) Copy content Toggle raw display
$41$ \( T^{6} - 130 T^{4} + 2432 T^{2} + \cdots - 512 \) Copy content Toggle raw display
$43$ \( (T^{3} - 6 T^{2} - 143 T + 958)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} - 68 T^{4} + 913 T^{2} + \cdots - 2592 \) Copy content Toggle raw display
$53$ \( (T^{3} - 8 T^{2} + 9 T + 10)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 226 T^{4} + 15968 T^{2} + \cdots - 359552 \) Copy content Toggle raw display
$61$ \( T^{6} - 376 T^{4} + 46864 T^{2} + \cdots - 1936512 \) Copy content Toggle raw display
$67$ \( (T^{3} + 4 T^{2} - 60 T - 144)^{2} \) Copy content Toggle raw display
$71$ \( (T - 8)^{6} \) Copy content Toggle raw display
$73$ \( T^{6} - 226 T^{4} + 15968 T^{2} + \cdots - 359552 \) Copy content Toggle raw display
$79$ \( (T^{3} - 32 T^{2} + 325 T - 1032)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} - 130 T^{4} + 2432 T^{2} + \cdots - 512 \) Copy content Toggle raw display
$89$ \( T^{6} - 146 T^{4} + 3888 T^{2} + \cdots - 2592 \) Copy content Toggle raw display
$97$ \( T^{6} - 386 T^{4} + 6496 T^{2} + \cdots - 3200 \) Copy content Toggle raw display
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