Properties

Label 2850.2.a.bn
Level $2850$
Weight $2$
Character orbit 2850.a
Self dual yes
Analytic conductor $22.757$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.7573645761\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 570)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + q^{6} + \beta_{2} q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} + q^{6} + \beta_{2} q^{7} + q^{8} + q^{9} + (\beta_{2} + \beta_1 - 1) q^{11} + q^{12} + ( - \beta_{2} + 2) q^{13} + \beta_{2} q^{14} + q^{16} + ( - \beta_1 + 3) q^{17} + q^{18} - q^{19} + \beta_{2} q^{21} + (\beta_{2} + \beta_1 - 1) q^{22} + ( - \beta_{2} + 2) q^{23} + q^{24} + ( - \beta_{2} + 2) q^{26} + q^{27} + \beta_{2} q^{28} + ( - \beta_{2} - \beta_1 + 3) q^{29} + ( - \beta_{2} + \beta_1 + 5) q^{31} + q^{32} + (\beta_{2} + \beta_1 - 1) q^{33} + ( - \beta_1 + 3) q^{34} + q^{36} + (\beta_{2} - 2 \beta_1) q^{37} - q^{38} + ( - \beta_{2} + 2) q^{39} - \beta_{2} q^{41} + \beta_{2} q^{42} + ( - \beta_{2} + \beta_1 - 3) q^{43} + (\beta_{2} + \beta_1 - 1) q^{44} + ( - \beta_{2} + 2) q^{46} + (\beta_{2} + 2 \beta_1) q^{47} + q^{48} + ( - 2 \beta_{2} - 2 \beta_1 + 3) q^{49} + ( - \beta_1 + 3) q^{51} + ( - \beta_{2} + 2) q^{52} + ( - 2 \beta_1 + 4) q^{53} + q^{54} + \beta_{2} q^{56} - q^{57} + ( - \beta_{2} - \beta_1 + 3) q^{58} + (\beta_1 + 5) q^{59} + 2 q^{61} + ( - \beta_{2} + \beta_1 + 5) q^{62} + \beta_{2} q^{63} + q^{64} + (\beta_{2} + \beta_1 - 1) q^{66} + (2 \beta_1 - 2) q^{67} + ( - \beta_1 + 3) q^{68} + ( - \beta_{2} + 2) q^{69} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{71} + q^{72} + (2 \beta_{2} + 2 \beta_1 - 2) q^{73} + (\beta_{2} - 2 \beta_1) q^{74} - q^{76} + ( - 4 \beta_{2} + 8) q^{77} + ( - \beta_{2} + 2) q^{78} + (\beta_{2} - \beta_1 + 7) q^{79} + q^{81} - \beta_{2} q^{82} + ( - 3 \beta_1 - 1) q^{83} + \beta_{2} q^{84} + ( - \beta_{2} + \beta_1 - 3) q^{86} + ( - \beta_{2} - \beta_1 + 3) q^{87} + (\beta_{2} + \beta_1 - 1) q^{88} + (\beta_{2} + 2 \beta_1 + 2) q^{89} + (4 \beta_{2} + 2 \beta_1 - 10) q^{91} + ( - \beta_{2} + 2) q^{92} + ( - \beta_{2} + \beta_1 + 5) q^{93} + (\beta_{2} + 2 \beta_1) q^{94} + q^{96} + ( - 3 \beta_{2} - 3 \beta_1 - 3) q^{97} + ( - 2 \beta_{2} - 2 \beta_1 + 3) q^{98} + (\beta_{2} + \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} + 3 q^{8} + 3 q^{9} - 4 q^{11} + 3 q^{12} + 6 q^{13} + 3 q^{16} + 10 q^{17} + 3 q^{18} - 3 q^{19} - 4 q^{22} + 6 q^{23} + 3 q^{24} + 6 q^{26} + 3 q^{27} + 10 q^{29} + 14 q^{31} + 3 q^{32} - 4 q^{33} + 10 q^{34} + 3 q^{36} + 2 q^{37} - 3 q^{38} + 6 q^{39} - 10 q^{43} - 4 q^{44} + 6 q^{46} - 2 q^{47} + 3 q^{48} + 11 q^{49} + 10 q^{51} + 6 q^{52} + 14 q^{53} + 3 q^{54} - 3 q^{57} + 10 q^{58} + 14 q^{59} + 6 q^{61} + 14 q^{62} + 3 q^{64} - 4 q^{66} - 8 q^{67} + 10 q^{68} + 6 q^{69} - 4 q^{71} + 3 q^{72} - 8 q^{73} + 2 q^{74} - 3 q^{76} + 24 q^{77} + 6 q^{78} + 22 q^{79} + 3 q^{81} - 10 q^{86} + 10 q^{87} - 4 q^{88} + 4 q^{89} - 32 q^{91} + 6 q^{92} + 14 q^{93} - 2 q^{94} + 3 q^{96} - 6 q^{97} + 11 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 + 5 ) / 2 \) 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
0.311108
2.17009
−1.48119
1.00000 1.00000 1.00000 0 1.00000 −4.42864 1.00000 1.00000 0
1.2 1.00000 1.00000 1.00000 0 1.00000 1.07838 1.00000 1.00000 0
1.3 1.00000 1.00000 1.00000 0 1.00000 3.35026 1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(19\) \(1\)

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 2850.2.a.bn 3
3.b odd 2 1 8550.2.a.cf 3
5.b even 2 1 2850.2.a.bk 3
5.c odd 4 2 570.2.d.d 6
15.d odd 2 1 8550.2.a.cr 3
15.e even 4 2 1710.2.d.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.d.d 6 5.c odd 4 2
1710.2.d.e 6 15.e even 4 2
2850.2.a.bk 3 5.b even 2 1
2850.2.a.bn 3 1.a even 1 1 trivial
8550.2.a.cf 3 3.b odd 2 1
8550.2.a.cr 3 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2850))\):

\( T_{7}^{3} - 16T_{7} + 16 \) Copy content Toggle raw display
\( T_{11}^{3} + 4T_{11}^{2} - 16T_{11} - 32 \) Copy content Toggle raw display
\( T_{13}^{3} - 6T_{13}^{2} - 4T_{13} + 8 \) Copy content Toggle raw display
\( T_{23}^{3} - 6T_{23}^{2} - 4T_{23} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 16T + 16 \) Copy content Toggle raw display
$11$ \( T^{3} + 4 T^{2} - 16 T - 32 \) Copy content Toggle raw display
$13$ \( T^{3} - 6 T^{2} - 4 T + 8 \) Copy content Toggle raw display
$17$ \( T^{3} - 10 T^{2} + 20 T + 8 \) Copy content Toggle raw display
$19$ \( (T + 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 6 T^{2} - 4 T + 8 \) Copy content Toggle raw display
$29$ \( T^{3} - 10 T^{2} + 12 T + 40 \) Copy content Toggle raw display
$31$ \( T^{3} - 14 T^{2} + 28 T + 152 \) Copy content Toggle raw display
$37$ \( T^{3} - 2 T^{2} - 84 T - 232 \) Copy content Toggle raw display
$41$ \( T^{3} - 16T - 16 \) Copy content Toggle raw display
$43$ \( T^{3} + 10 T^{2} - 4 T - 8 \) Copy content Toggle raw display
$47$ \( T^{3} + 2 T^{2} - 52 T - 184 \) Copy content Toggle raw display
$53$ \( T^{3} - 14 T^{2} + 12 T + 152 \) Copy content Toggle raw display
$59$ \( T^{3} - 14 T^{2} + 52 T - 40 \) Copy content Toggle raw display
$61$ \( (T - 2)^{3} \) Copy content Toggle raw display
$67$ \( T^{3} + 8 T^{2} - 32 T - 128 \) Copy content Toggle raw display
$71$ \( T^{3} + 4 T^{2} - 80 T - 64 \) Copy content Toggle raw display
$73$ \( T^{3} + 8 T^{2} - 64 T - 256 \) Copy content Toggle raw display
$79$ \( T^{3} - 22 T^{2} + 124 T - 200 \) Copy content Toggle raw display
$83$ \( T^{3} - 120T + 16 \) Copy content Toggle raw display
$89$ \( T^{3} - 4 T^{2} - 48 T - 80 \) Copy content Toggle raw display
$97$ \( T^{3} + 6 T^{2} - 180 T - 216 \) Copy content Toggle raw display
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