Properties

Label 4851.2.a.bd
Level $4851$
Weight $2$
Character orbit 4851.a
Self dual yes
Analytic conductor $38.735$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4851.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.7354300205\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} - 2 q^{5} + (\beta + 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} - 2 q^{5} + (\beta + 3) q^{8} + ( - 2 \beta - 2) q^{10} - q^{11} + (2 \beta - 2) q^{13} + 3 q^{16} + ( - \beta - 3) q^{17} + ( - 3 \beta - 3) q^{19} + ( - 4 \beta - 2) q^{20} + ( - \beta - 1) q^{22} + 7 q^{23} - q^{25} + 2 q^{26} + ( - 3 \beta + 1) q^{29} - 4 \beta q^{31} + (\beta - 3) q^{32} + ( - 4 \beta - 5) q^{34} + ( - 6 \beta - 1) q^{37} + ( - 6 \beta - 9) q^{38} + ( - 2 \beta - 6) q^{40} + (2 \beta - 4) q^{41} + (3 \beta - 7) q^{43} + ( - 2 \beta - 1) q^{44} + (7 \beta + 7) q^{46} + (2 \beta + 7) q^{47} + ( - \beta - 1) q^{50} + ( - 2 \beta + 6) q^{52} + ( - 2 \beta + 10) q^{53} + 2 q^{55} + ( - 2 \beta - 5) q^{58} + ( - 4 \beta - 3) q^{59} + 4 q^{61} + ( - 4 \beta - 8) q^{62} + ( - 2 \beta - 7) q^{64} + ( - 4 \beta + 4) q^{65} + (2 \beta - 6) q^{67} + ( - 7 \beta - 7) q^{68} + ( - 2 \beta + 7) q^{71} + (4 \beta - 6) q^{73} + ( - 7 \beta - 13) q^{74} + ( - 9 \beta - 15) q^{76} + (8 \beta + 2) q^{79} - 6 q^{80} - 2 \beta q^{82} - 2 \beta q^{83} + (2 \beta + 6) q^{85} + ( - 4 \beta - 1) q^{86} + ( - \beta - 3) q^{88} - 10 \beta q^{89} + (14 \beta + 7) q^{92} + (9 \beta + 11) q^{94} + (6 \beta + 6) q^{95} + (6 \beta + 3) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 6 q^{8} - 4 q^{10} - 2 q^{11} - 4 q^{13} + 6 q^{16} - 6 q^{17} - 6 q^{19} - 4 q^{20} - 2 q^{22} + 14 q^{23} - 2 q^{25} + 4 q^{26} + 2 q^{29} - 6 q^{32} - 10 q^{34} - 2 q^{37} - 18 q^{38} - 12 q^{40} - 8 q^{41} - 14 q^{43} - 2 q^{44} + 14 q^{46} + 14 q^{47} - 2 q^{50} + 12 q^{52} + 20 q^{53} + 4 q^{55} - 10 q^{58} - 6 q^{59} + 8 q^{61} - 16 q^{62} - 14 q^{64} + 8 q^{65} - 12 q^{67} - 14 q^{68} + 14 q^{71} - 12 q^{73} - 26 q^{74} - 30 q^{76} + 4 q^{79} - 12 q^{80} + 12 q^{85} - 2 q^{86} - 6 q^{88} + 14 q^{92} + 22 q^{94} + 12 q^{95} + 6 q^{97}+O(q^{100}) \) 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
−1.41421
1.41421
−0.414214 0 −1.82843 −2.00000 0 0 1.58579 0 0.828427
1.2 2.41421 0 3.82843 −2.00000 0 0 4.41421 0 −4.82843
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(11\) \(1\)

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 4851.2.a.bd 2
3.b odd 2 1 1617.2.a.m 2
7.b odd 2 1 4851.2.a.be 2
7.d odd 6 2 693.2.i.f 4
21.c even 2 1 1617.2.a.n 2
21.g even 6 2 231.2.i.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.d 4 21.g even 6 2
693.2.i.f 4 7.d odd 6 2
1617.2.a.m 2 3.b odd 2 1
1617.2.a.n 2 21.c even 2 1
4851.2.a.bd 2 1.a even 1 1 trivial
4851.2.a.be 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4851))\):

\( T_{2}^{2} - 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{13}^{2} + 4T_{13} - 4 \) Copy content Toggle raw display
\( T_{17}^{2} + 6T_{17} + 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 7 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T - 9 \) Copy content Toggle raw display
$23$ \( (T - 7)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 2T - 17 \) Copy content Toggle raw display
$31$ \( T^{2} - 32 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T - 71 \) Copy content Toggle raw display
$41$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$43$ \( T^{2} + 14T + 31 \) Copy content Toggle raw display
$47$ \( T^{2} - 14T + 41 \) Copy content Toggle raw display
$53$ \( T^{2} - 20T + 92 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T - 23 \) Copy content Toggle raw display
$61$ \( (T - 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
$71$ \( T^{2} - 14T + 41 \) Copy content Toggle raw display
$73$ \( T^{2} + 12T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T - 124 \) Copy content Toggle raw display
$83$ \( T^{2} - 8 \) Copy content Toggle raw display
$89$ \( T^{2} - 200 \) Copy content Toggle raw display
$97$ \( T^{2} - 6T - 63 \) Copy content Toggle raw display
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