Properties

Label 2842.2.a.w
Level $2842$
Weight $2$
Character orbit 2842.a
Self dual yes
Analytic conductor $22.693$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(5\)
Coefficient field: 5.5.345065.1
Defining polynomial: \( x^{5} - 2x^{4} - 7x^{3} + 8x^{2} + 14x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 406)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + (\beta_1 - 1) q^{3} + q^{4} + ( - \beta_{4} - \beta_{2} - 2) q^{5} + (\beta_1 - 1) q^{6} + q^{8} + (\beta_{4} + \beta_{3} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + (\beta_1 - 1) q^{3} + q^{4} + ( - \beta_{4} - \beta_{2} - 2) q^{5} + (\beta_1 - 1) q^{6} + q^{8} + (\beta_{4} + \beta_{3} - \beta_1 + 2) q^{9} + ( - \beta_{4} - \beta_{2} - 2) q^{10} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{11} + (\beta_1 - 1) q^{12} + (\beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{13} + (\beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{15} + q^{16} + ( - 2 \beta_{4} - \beta_{3} - \beta_1 - 4) q^{17} + (\beta_{4} + \beta_{3} - \beta_1 + 2) q^{18} + (2 \beta_{4} + \beta_1) q^{19} + ( - \beta_{4} - \beta_{2} - 2) q^{20} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{22} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{23} + (\beta_1 - 1) q^{24} + (4 \beta_{4} + 3 \beta_{2} + 3) q^{25} + (\beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{26} + ( - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{27} + q^{29} + (\beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{30} + (3 \beta_{4} + 3 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{31} + q^{32} + (2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 1) q^{33} + ( - 2 \beta_{4} - \beta_{3} - \beta_1 - 4) q^{34} + (\beta_{4} + \beta_{3} - \beta_1 + 2) q^{36} + ( - \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{37} + (2 \beta_{4} + \beta_1) q^{38} + ( - 4 \beta_{4} + \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 5) q^{39} + ( - \beta_{4} - \beta_{2} - 2) q^{40} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{41} + ( - \beta_{4} - \beta_{3} + \beta_1 - 1) q^{43} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{44} + ( - 3 \beta_{4} + \beta_{3} - 2 \beta_{2} - 4) q^{45} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{46} + (2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{47} + (\beta_1 - 1) q^{48} + (4 \beta_{4} + 3 \beta_{2} + 3) q^{50} + (\beta_{4} - \beta_{3} - 4 \beta_1 + 1) q^{51} + (\beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{52} + (2 \beta_{4} - 4 \beta_{3} + 3 \beta_1 - 1) q^{53} + ( - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{54} + ( - \beta_{3} - 2 \beta_1 - 2) q^{55} + ( - 3 \beta_{4} + \beta_{3} - 2 \beta_{2}) q^{57} + q^{58} + (\beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{59} + (\beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{60} + ( - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 3) q^{61} + (3 \beta_{4} + 3 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{62} + q^{64} + ( - 2 \beta_{4} + 4 \beta_{3} - \beta_{2} + 3 \beta_1 - 2) q^{65} + (2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 1) q^{66} + (\beta_{4} + 3 \beta_{3} + 2 \beta_1 - 2) q^{67} + ( - 2 \beta_{4} - \beta_{3} - \beta_1 - 4) q^{68} + ( - 3 \beta_{3} - \beta_1 - 4) q^{69} + ( - 4 \beta_{4} - 2 \beta_{2} + \beta_1 - 4) q^{71} + (\beta_{4} + \beta_{3} - \beta_1 + 2) q^{72} + ( - 2 \beta_{4} - 5 \beta_{3} + \beta_{2} + 3 \beta_1 - 5) q^{73} + ( - \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{74} + ( - 5 \beta_{4} + 6 \beta_{3} - 4 \beta_{2} - 2) q^{75} + (2 \beta_{4} + \beta_1) q^{76} + ( - 4 \beta_{4} + \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 5) q^{78} + (5 \beta_{4} + 3 \beta_{3} - \beta_{2} - 2 \beta_1 + 6) q^{79} + ( - \beta_{4} - \beta_{2} - 2) q^{80} + ( - 3 \beta_{4} - 2 \beta_{3} - \beta_{2} - 6) q^{81} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{82} + (\beta_{4} + 6 \beta_{3} - 4 \beta_{2} - \beta_1 + 2) q^{83} + (7 \beta_{4} + 2 \beta_{3} + 5 \beta_{2} + \beta_1 + 12) q^{85} + ( - \beta_{4} - \beta_{3} + \beta_1 - 1) q^{86} + (\beta_1 - 1) q^{87} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{88} + (3 \beta_{4} + 6 \beta_{2} - \beta_1 + 1) q^{89} + ( - 3 \beta_{4} + \beta_{3} - 2 \beta_{2} - 4) q^{90} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{92} + (\beta_{3} + 3 \beta_{2} + \beta_1) q^{93} + (2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{94} + ( - 4 \beta_{4} - 2 \beta_{2} + \beta_1 - 5) q^{95} + (\beta_1 - 1) q^{96} + (3 \beta_{4} + 3 \beta_{3} + 5 \beta_1 - 2) q^{97} + (\beta_{3} - 3 \beta_{2} + 2 \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{2} - 3 q^{3} + 5 q^{4} - 7 q^{5} - 3 q^{6} + 5 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{2} - 3 q^{3} + 5 q^{4} - 7 q^{5} - 3 q^{6} + 5 q^{8} + 4 q^{9} - 7 q^{10} - 3 q^{12} - 8 q^{13} + 4 q^{15} + 5 q^{16} - 16 q^{17} + 4 q^{18} - 2 q^{19} - 7 q^{20} - 5 q^{23} - 3 q^{24} + 4 q^{25} - 8 q^{26} - 9 q^{27} + 5 q^{29} + 4 q^{30} - 5 q^{31} + 5 q^{32} - 3 q^{33} - 16 q^{34} + 4 q^{36} - 2 q^{38} - 20 q^{39} - 7 q^{40} - 17 q^{41} + q^{43} - 14 q^{45} - 5 q^{46} + 4 q^{47} - 3 q^{48} + 4 q^{50} - 3 q^{51} - 8 q^{52} + 5 q^{53} - 9 q^{54} - 12 q^{55} + 6 q^{57} + 5 q^{58} - 17 q^{59} + 4 q^{60} - 13 q^{61} - 5 q^{62} + 5 q^{64} - 7 q^{65} - 3 q^{66} - 14 q^{67} - 16 q^{68} - 16 q^{69} - 8 q^{71} + 4 q^{72} - 6 q^{73} - 8 q^{75} - 2 q^{76} - 20 q^{78} + 11 q^{79} - 7 q^{80} - 19 q^{81} - 17 q^{82} - 2 q^{83} + 39 q^{85} + q^{86} - 3 q^{87} - 9 q^{89} - 14 q^{90} - 5 q^{92} - 3 q^{93} + 4 q^{94} - 13 q^{95} - 3 q^{96} - 12 q^{97} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 7x^{3} + 8x^{2} + 14x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 2\nu^{2} - 3\nu + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 3\nu^{3} - 2\nu^{2} + 8\nu - 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{4} + 3\nu^{3} + 3\nu^{2} - 9\nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{4} + 2\beta_{3} + \beta_{2} + 5\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{4} + 9\beta_{3} + 3\beta_{2} + 9\beta _1 + 25 \) 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
−1.66755
−1.52092
0.196248
2.04534
2.94688
1.00000 −2.66755 1.00000 0.489139 −2.66755 0 1.00000 4.11583 0.489139
1.2 1.00000 −2.52092 1.00000 −4.14053 −2.52092 0 1.00000 3.35505 −4.14053
1.3 1.00000 −0.803752 1.00000 −0.712284 −0.803752 0 1.00000 −2.35398 −0.712284
1.4 1.00000 1.04534 1.00000 0.635591 1.04534 0 1.00000 −1.90726 0.635591
1.5 1.00000 1.94688 1.00000 −3.27192 1.94688 0 1.00000 0.790361 −3.27192
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2842.2.a.w 5
7.b odd 2 1 2842.2.a.y 5
7.c even 3 2 406.2.e.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
406.2.e.b 10 7.c even 3 2
2842.2.a.w 5 1.a even 1 1 trivial
2842.2.a.y 5 7.b odd 2 1

Hecke kernels

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

\( T_{3}^{5} + 3T_{3}^{4} - 5T_{3}^{3} - 15T_{3}^{2} + 6T_{3} + 11 \) Copy content Toggle raw display
\( T_{5}^{5} + 7T_{5}^{4} + 10T_{5}^{3} - 9T_{5}^{2} - 5T_{5} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{5} + 3 T^{4} - 5 T^{3} - 15 T^{2} + \cdots + 11 \) Copy content Toggle raw display
$5$ \( T^{5} + 7 T^{4} + 10 T^{3} - 9 T^{2} + \cdots + 3 \) Copy content Toggle raw display
$7$ \( T^{5} \) Copy content Toggle raw display
$11$ \( T^{5} - 29 T^{3} + 60 T^{2} + 18 T - 61 \) Copy content Toggle raw display
$13$ \( T^{5} + 8 T^{4} - 10 T^{3} - 229 T^{2} + \cdots - 539 \) Copy content Toggle raw display
$17$ \( T^{5} + 16 T^{4} + 67 T^{3} + \cdots - 1531 \) Copy content Toggle raw display
$19$ \( T^{5} + 2 T^{4} - 37 T^{3} + 18 T^{2} + \cdots - 121 \) Copy content Toggle raw display
$23$ \( T^{5} + 5 T^{4} - 29 T^{3} - 27 T^{2} + \cdots - 49 \) Copy content Toggle raw display
$29$ \( (T - 1)^{5} \) Copy content Toggle raw display
$31$ \( T^{5} + 5 T^{4} - 88 T^{3} + \cdots - 3265 \) Copy content Toggle raw display
$37$ \( T^{5} - 21 T^{3} + 26 T^{2} - 4 T - 3 \) Copy content Toggle raw display
$41$ \( T^{5} + 17 T^{4} + 68 T^{3} + \cdots - 4137 \) Copy content Toggle raw display
$43$ \( T^{5} - T^{4} - 17 T^{3} + 11 T^{2} + \cdots - 15 \) Copy content Toggle raw display
$47$ \( T^{5} - 4 T^{4} - 103 T^{3} + \cdots - 1737 \) Copy content Toggle raw display
$53$ \( T^{5} - 5 T^{4} - 235 T^{3} + \cdots + 50625 \) Copy content Toggle raw display
$59$ \( T^{5} + 17 T^{4} + 47 T^{3} + \cdots + 2205 \) Copy content Toggle raw display
$61$ \( T^{5} + 13 T^{4} - 10 T^{3} - 130 T^{2} + \cdots + 21 \) Copy content Toggle raw display
$67$ \( T^{5} + 14 T^{4} - 64 T^{3} + \cdots - 11891 \) Copy content Toggle raw display
$71$ \( T^{5} + 8 T^{4} - 107 T^{3} + 76 T^{2} + \cdots + 35 \) Copy content Toggle raw display
$73$ \( T^{5} + 6 T^{4} - 183 T^{3} + \cdots + 6729 \) Copy content Toggle raw display
$79$ \( T^{5} - 11 T^{4} - 257 T^{3} + \cdots - 229 \) Copy content Toggle raw display
$83$ \( T^{5} + 2 T^{4} - 293 T^{3} + \cdots + 11319 \) Copy content Toggle raw display
$89$ \( T^{5} + 9 T^{4} - 257 T^{3} + \cdots - 52133 \) Copy content Toggle raw display
$97$ \( T^{5} + 12 T^{4} - 275 T^{3} + \cdots + 58093 \) Copy content Toggle raw display
show more
show less