Properties

Label 4851.2.a.w
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: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} + ( -1 + \beta ) q^{4} + q^{5} + ( -1 + 2 \beta ) q^{8} +O(q^{10})\) \( q -\beta q^{2} + ( -1 + \beta ) q^{4} + q^{5} + ( -1 + 2 \beta ) q^{8} -\beta q^{10} - q^{11} + ( -1 + 4 \beta ) q^{13} -3 \beta q^{16} + ( 4 - 2 \beta ) q^{17} + ( 3 - 6 \beta ) q^{19} + ( -1 + \beta ) q^{20} + \beta q^{22} + ( -2 + 6 \beta ) q^{23} -4 q^{25} + ( -4 - 3 \beta ) q^{26} -5 q^{29} + ( 4 - 2 \beta ) q^{31} + ( 5 - \beta ) q^{32} + ( 2 - 2 \beta ) q^{34} -7 q^{37} + ( 6 + 3 \beta ) q^{38} + ( -1 + 2 \beta ) q^{40} + 4 \beta q^{41} + ( 2 - 6 \beta ) q^{43} + ( 1 - \beta ) q^{44} + ( -6 - 4 \beta ) q^{46} + ( -1 - 2 \beta ) q^{47} + 4 \beta q^{50} + ( 5 - \beta ) q^{52} + ( 6 - 10 \beta ) q^{53} - q^{55} + 5 \beta q^{58} + ( -5 + 10 \beta ) q^{59} -2 q^{61} + ( 2 - 2 \beta ) q^{62} + ( 1 + 2 \beta ) q^{64} + ( -1 + 4 \beta ) q^{65} + ( -11 - 2 \beta ) q^{67} + ( -6 + 4 \beta ) q^{68} -4 \beta q^{71} + ( -7 - 4 \beta ) q^{73} + 7 \beta q^{74} + ( -9 + 3 \beta ) q^{76} + ( -12 + 4 \beta ) q^{79} -3 \beta q^{80} + ( -4 - 4 \beta ) q^{82} + ( 8 + 2 \beta ) q^{83} + ( 4 - 2 \beta ) q^{85} + ( 6 + 4 \beta ) q^{86} + ( 1 - 2 \beta ) q^{88} + ( -2 + 4 \beta ) q^{89} + ( 8 - 2 \beta ) q^{92} + ( 2 + 3 \beta ) q^{94} + ( 3 - 6 \beta ) q^{95} + ( -6 + 6 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} + 2 q^{5} + O(q^{10}) \) \( 2 q - q^{2} - q^{4} + 2 q^{5} - q^{10} - 2 q^{11} + 2 q^{13} - 3 q^{16} + 6 q^{17} - q^{20} + q^{22} + 2 q^{23} - 8 q^{25} - 11 q^{26} - 10 q^{29} + 6 q^{31} + 9 q^{32} + 2 q^{34} - 14 q^{37} + 15 q^{38} + 4 q^{41} - 2 q^{43} + q^{44} - 16 q^{46} - 4 q^{47} + 4 q^{50} + 9 q^{52} + 2 q^{53} - 2 q^{55} + 5 q^{58} - 4 q^{61} + 2 q^{62} + 4 q^{64} + 2 q^{65} - 24 q^{67} - 8 q^{68} - 4 q^{71} - 18 q^{73} + 7 q^{74} - 15 q^{76} - 20 q^{79} - 3 q^{80} - 12 q^{82} + 18 q^{83} + 6 q^{85} + 16 q^{86} + 14 q^{92} + 7 q^{94} - 6 q^{97} + O(q^{100}) \)

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
1.61803
−0.618034
−1.61803 0 0.618034 1.00000 0 0 2.23607 0 −1.61803
1.2 0.618034 0 −1.61803 1.00000 0 0 −2.23607 0 0.618034
\(n\): 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.w 2
3.b odd 2 1 1617.2.a.p 2
7.b odd 2 1 693.2.a.f 2
21.c even 2 1 231.2.a.c 2
77.b even 2 1 7623.2.a.bm 2
84.h odd 2 1 3696.2.a.be 2
105.g even 2 1 5775.2.a.be 2
231.h odd 2 1 2541.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.c 2 21.c even 2 1
693.2.a.f 2 7.b odd 2 1
1617.2.a.p 2 3.b odd 2 1
2541.2.a.t 2 231.h odd 2 1
3696.2.a.be 2 84.h odd 2 1
4851.2.a.w 2 1.a even 1 1 trivial
5775.2.a.be 2 105.g even 2 1
7623.2.a.bm 2 77.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4851))\):

\( T_{2}^{2} + T_{2} - 1 \)
\( T_{5} - 1 \)
\( T_{13}^{2} - 2 T_{13} - 19 \)
\( T_{17}^{2} - 6 T_{17} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( -19 - 2 T + T^{2} \)
$17$ \( 4 - 6 T + T^{2} \)
$19$ \( -45 + T^{2} \)
$23$ \( -44 - 2 T + T^{2} \)
$29$ \( ( 5 + T )^{2} \)
$31$ \( 4 - 6 T + T^{2} \)
$37$ \( ( 7 + T )^{2} \)
$41$ \( -16 - 4 T + T^{2} \)
$43$ \( -44 + 2 T + T^{2} \)
$47$ \( -1 + 4 T + T^{2} \)
$53$ \( -124 - 2 T + T^{2} \)
$59$ \( -125 + T^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( 139 + 24 T + T^{2} \)
$71$ \( -16 + 4 T + T^{2} \)
$73$ \( 61 + 18 T + T^{2} \)
$79$ \( 80 + 20 T + T^{2} \)
$83$ \( 76 - 18 T + T^{2} \)
$89$ \( -20 + T^{2} \)
$97$ \( -36 + 6 T + T^{2} \)
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