Properties

Label 2842.2.a.ba
Level $2842$
Weight $2$
Character orbit 2842.a
Self dual yes
Analytic conductor $22.693$
Analytic rank $1$
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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.52756992.1
Defining polynomial: \( x^{6} - 2x^{5} - 13x^{4} + 18x^{3} + 22x^{2} - 20x - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q - q^{2} + \beta_{2} q^{3} + q^{4} - \beta_1 q^{5} - \beta_{2} q^{6} - q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_{2} q^{3} + q^{4} - \beta_1 q^{5} - \beta_{2} q^{6} - q^{8} - q^{9} + \beta_1 q^{10} + (\beta_{4} - 1) q^{11} + \beta_{2} q^{12} + \beta_1 q^{13} + ( - \beta_{5} - \beta_{4}) q^{15} + q^{16} + ( - \beta_{3} - 2 \beta_{2}) q^{17} + q^{18} + (\beta_{3} + \beta_1) q^{19} - \beta_1 q^{20} + ( - \beta_{4} + 1) q^{22} + (\beta_{5} - 5) q^{23} - \beta_{2} q^{24} + (2 \beta_{5} - \beta_{4} + 8) q^{25} - \beta_1 q^{26} - 4 \beta_{2} q^{27} - q^{29} + (\beta_{5} + \beta_{4}) q^{30} + (2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{31} - q^{32} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{33} + (\beta_{3} + 2 \beta_{2}) q^{34} - q^{36} + ( - \beta_{5} - 2 \beta_{4} - 3) q^{37} + ( - \beta_{3} - \beta_1) q^{38} + (\beta_{5} + \beta_{4}) q^{39} + \beta_1 q^{40} + ( - 3 \beta_{2} + \beta_1) q^{41} + ( - 2 \beta_{5} - 4) q^{43} + (\beta_{4} - 1) q^{44} + \beta_1 q^{45} + ( - \beta_{5} + 5) q^{46} + ( - 2 \beta_{3} + \beta_1) q^{47} + \beta_{2} q^{48} + ( - 2 \beta_{5} + \beta_{4} - 8) q^{50} + ( - \beta_{5} + \beta_{4} - 4) q^{51} + \beta_1 q^{52} + ( - 2 \beta_{5} + 4) q^{53} + 4 \beta_{2} q^{54} + ( - 2 \beta_{3} - 8 \beta_{2} + 2 \beta_1) q^{55} + 2 \beta_{5} q^{57} + q^{58} + \beta_{3} q^{59} + ( - \beta_{5} - \beta_{4}) q^{60} + (\beta_{3} + 3 \beta_{2} + \beta_1) q^{61} + ( - 2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{62} + q^{64} + ( - 2 \beta_{5} + \beta_{4} - 13) q^{65} + (\beta_{3} + \beta_{2} - \beta_1) q^{66} - 4 q^{67} + ( - \beta_{3} - 2 \beta_{2}) q^{68} + (\beta_{3} - 5 \beta_{2} + \beta_1) q^{69} + (3 \beta_{5} - 3) q^{71} + q^{72} + ( - 2 \beta_{3} + \beta_{2} - \beta_1) q^{73} + (\beta_{5} + 2 \beta_{4} + 3) q^{74} + (3 \beta_{3} + 8 \beta_{2} + \beta_1) q^{75} + (\beta_{3} + \beta_1) q^{76} + ( - \beta_{5} - \beta_{4}) q^{78} + ( - \beta_{5} + \beta_{4} - 2) q^{79} - \beta_1 q^{80} - 5 q^{81} + (3 \beta_{2} - \beta_1) q^{82} + (2 \beta_{3} + \beta_{2} - \beta_1) q^{83} + (3 \beta_{5} + 4 \beta_{4} - 3) q^{85} + (2 \beta_{5} + 4) q^{86} - \beta_{2} q^{87} + ( - \beta_{4} + 1) q^{88} + ( - \beta_{2} - 3 \beta_1) q^{89} - \beta_1 q^{90} + (\beta_{5} - 5) q^{92} + (3 \beta_{5} - \beta_{4} + 4) q^{93} + (2 \beta_{3} - \beta_1) q^{94} + ( - 3 \beta_{5} - \beta_{4} - 10) q^{95} - \beta_{2} q^{96} + ( - \beta_{3} - 6 \beta_{2}) q^{97} + ( - \beta_{4} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 6 q^{4} - 6 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} + 6 q^{4} - 6 q^{8} - 6 q^{9} - 4 q^{11} + 6 q^{16} + 6 q^{18} + 4 q^{22} - 32 q^{23} + 42 q^{25} - 6 q^{29} - 6 q^{32} - 6 q^{36} - 20 q^{37} - 20 q^{43} - 4 q^{44} + 32 q^{46} - 42 q^{50} - 20 q^{51} + 28 q^{53} - 4 q^{57} + 6 q^{58} + 6 q^{64} - 72 q^{65} - 24 q^{67} - 24 q^{71} + 6 q^{72} + 20 q^{74} - 8 q^{79} - 30 q^{81} - 16 q^{85} + 20 q^{86} + 4 q^{88} - 32 q^{92} + 16 q^{93} - 56 q^{95} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 13x^{4} + 18x^{3} + 22x^{2} - 20x - 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{4} - 5\nu^{3} - 12\nu^{2} + 47\nu + 16 ) / 11 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{5} - 3\nu^{4} - 26\nu^{3} + 21\nu^{2} + 31\nu - 9 ) / 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{4} + \nu^{3} - 35\nu^{2} - 16\nu + 65 ) / 11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6\nu^{4} - 8\nu^{3} - 72\nu^{2} + 40\nu + 63 ) / 11 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{5} - 6\nu^{4} - 52\nu^{3} + 42\nu^{2} + 84\nu - 29 ) / 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - 2\beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} - 2\beta_{3} - 2\beta_{2} - 2\beta _1 + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{5} + \beta_{4} - 22\beta_{2} - 6\beta _1 + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 20\beta_{5} + 17\beta_{4} - 24\beta_{3} - 40\beta_{2} - 32\beta _1 + 111 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 147\beta_{5} + 28\beta_{4} - 15\beta_{3} - 283\beta_{2} - 105\beta _1 + 237 ) / 2 \) 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
1.65334
−0.285415
3.87472
1.04629
−3.11384
−1.17509
−1.00000 −1.41421 1.00000 −4.16178 1.41421 0 −1.00000 −1.00000 4.16178
1.2 −1.00000 −1.41421 1.00000 −0.157351 1.41421 0 −1.00000 −1.00000 0.157351
1.3 −1.00000 −1.41421 1.00000 4.31913 1.41421 0 −1.00000 −1.00000 −4.31913
1.4 −1.00000 1.41421 1.00000 −4.31913 −1.41421 0 −1.00000 −1.00000 4.31913
1.5 −1.00000 1.41421 1.00000 0.157351 −1.41421 0 −1.00000 −1.00000 −0.157351
1.6 −1.00000 1.41421 1.00000 4.16178 −1.41421 0 −1.00000 −1.00000 −4.16178
\(n\): 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.ba 6
7.b odd 2 1 inner 2842.2.a.ba 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2842.2.a.ba 6 1.a even 1 1 trivial
2842.2.a.ba 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2842))\):

\( T_{3}^{2} - 2 \) Copy content Toggle raw display
\( T_{5}^{6} - 36T_{5}^{4} + 324T_{5}^{2} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} - 2)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} - 36 T^{4} + 324 T^{2} - 8 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( (T^{3} + 2 T^{2} - 24 T + 24)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - 36 T^{4} + 324 T^{2} - 8 \) Copy content Toggle raw display
$17$ \( T^{6} - 66 T^{4} + 1224 T^{2} + \cdots - 6728 \) Copy content Toggle raw display
$19$ \( T^{6} - 70 T^{4} + 1228 T^{2} + \cdots - 648 \) Copy content Toggle raw display
$23$ \( (T^{3} + 16 T^{2} + 68 T + 56)^{2} \) Copy content Toggle raw display
$29$ \( (T + 1)^{6} \) Copy content Toggle raw display
$31$ \( T^{6} - 212 T^{4} + 13764 T^{2} + \cdots - 250632 \) Copy content Toggle raw display
$37$ \( (T^{3} + 10 T^{2} - 72 T - 648)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 90 T^{4} + 1368 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$43$ \( (T^{3} + 10 T^{2} - 36 T - 168)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} - 268 T^{4} + 18292 T^{2} + \cdots - 60552 \) Copy content Toggle raw display
$53$ \( (T^{3} - 14 T^{2} - 4 T + 248)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 50 T^{4} + 504 T^{2} + \cdots - 648 \) Copy content Toggle raw display
$61$ \( T^{6} - 112 T^{4} + 1600 T^{2} + \cdots - 4608 \) Copy content Toggle raw display
$67$ \( (T + 4)^{6} \) Copy content Toggle raw display
$71$ \( (T^{3} + 12 T^{2} - 108 T - 648)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 218 T^{4} + 5736 T^{2} + \cdots - 8712 \) Copy content Toggle raw display
$79$ \( (T^{3} + 4 T^{2} - 44 T - 168)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} - 266 T^{4} + 17256 T^{2} + \cdots - 3528 \) Copy content Toggle raw display
$89$ \( T^{6} - 330 T^{4} + 25608 T^{2} + \cdots - 22472 \) Copy content Toggle raw display
$97$ \( T^{6} - 242 T^{4} + 14760 T^{2} + \cdots - 78408 \) Copy content Toggle raw display
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