Properties

Label 4851.2.a.y
Level 4851
Weight 2
Character orbit 4851.a
Self dual yes
Analytic conductor 38.735
Analytic rank 0
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 77)
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.

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

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
−2.23607 0 3.00000 −2.00000 0 0 −2.23607 0 4.47214
1.2 2.23607 0 3.00000 −2.00000 0 0 2.23607 0 −4.47214
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.y 2
3.b odd 2 1 539.2.a.f 2
7.b odd 2 1 693.2.a.h 2
12.b even 2 1 8624.2.a.ce 2
21.c even 2 1 77.2.a.d 2
21.g even 6 2 539.2.e.i 4
21.h odd 6 2 539.2.e.j 4
33.d even 2 1 5929.2.a.m 2
77.b even 2 1 7623.2.a.bl 2
84.h odd 2 1 1232.2.a.m 2
105.g even 2 1 1925.2.a.r 2
105.k odd 4 2 1925.2.b.h 4
168.e odd 2 1 4928.2.a.bv 2
168.i even 2 1 4928.2.a.bm 2
231.h odd 2 1 847.2.a.f 2
231.r odd 10 2 847.2.f.b 4
231.r odd 10 2 847.2.f.m 4
231.u even 10 2 847.2.f.a 4
231.u even 10 2 847.2.f.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.d 2 21.c even 2 1
539.2.a.f 2 3.b odd 2 1
539.2.e.i 4 21.g even 6 2
539.2.e.j 4 21.h odd 6 2
693.2.a.h 2 7.b odd 2 1
847.2.a.f 2 231.h odd 2 1
847.2.f.a 4 231.u even 10 2
847.2.f.b 4 231.r odd 10 2
847.2.f.m 4 231.r odd 10 2
847.2.f.n 4 231.u even 10 2
1232.2.a.m 2 84.h odd 2 1
1925.2.a.r 2 105.g even 2 1
1925.2.b.h 4 105.k odd 4 2
4851.2.a.y 2 1.a even 1 1 trivial
4928.2.a.bm 2 168.i even 2 1
4928.2.a.bv 2 168.e odd 2 1
5929.2.a.m 2 33.d even 2 1
7623.2.a.bl 2 77.b even 2 1
8624.2.a.ce 2 12.b even 2 1

Atkin-Lehner signs

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

Hecke kernels

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + 4 T^{4} \)
$3$ \( \)
$5$ \( ( 1 + 2 T + 5 T^{2} )^{2} \)
$7$ \( \)
$11$ \( ( 1 - T )^{2} \)
$13$ \( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4} \)
$17$ \( 1 + 2 T + 30 T^{2} + 34 T^{3} + 289 T^{4} \)
$19$ \( 1 + 4 T + 22 T^{2} + 76 T^{3} + 361 T^{4} \)
$23$ \( 1 - 4 T + 30 T^{2} - 92 T^{3} + 529 T^{4} \)
$29$ \( 1 + 8 T + 54 T^{2} + 232 T^{3} + 841 T^{4} \)
$31$ \( 1 - 10 T + 82 T^{2} - 310 T^{3} + 961 T^{4} \)
$37$ \( 1 + 8 T + 70 T^{2} + 296 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 18 T + 158 T^{2} + 738 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 - 8 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 10 T + 114 T^{2} - 470 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 8 T + 102 T^{2} + 424 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 2 T + 114 T^{2} - 118 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 10 T + 142 T^{2} - 610 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 20 T + 214 T^{2} - 1340 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 12 T + 158 T^{2} - 852 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 6 T + 150 T^{2} - 438 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 78 T^{2} + 6241 T^{4} \)
$83$ \( 1 - 4 T - 10 T^{2} - 332 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 - 2 T + 89 T^{2} )^{2} \)
$97$ \( 1 + 8 T + 30 T^{2} + 776 T^{3} + 9409 T^{4} \)
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