Properties

Label 4928.2.a.bm
Level 4928
Weight 2
Character orbit 4928.a
Self dual yes
Analytic conductor 39.350
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4928 = 2^{6} \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4928.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.3502781161\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q -2 \beta q^{3} + 2 q^{5} + q^{7} + ( 1 + 4 \beta ) q^{9} +O(q^{10})\) \( q -2 \beta q^{3} + 2 q^{5} + q^{7} + ( 1 + 4 \beta ) q^{9} + q^{11} + ( -2 + 2 \beta ) q^{13} -4 \beta q^{15} + ( -2 + 2 \beta ) q^{17} + ( -4 + 4 \beta ) q^{19} -2 \beta q^{21} -4 \beta q^{23} - q^{25} + ( -8 - 4 \beta ) q^{27} + ( -6 + 4 \beta ) q^{29} + ( -4 - 2 \beta ) q^{31} -2 \beta q^{33} + 2 q^{35} + ( 6 - 4 \beta ) q^{37} -4 q^{39} + ( -10 + 2 \beta ) q^{41} -8 q^{43} + ( 2 + 8 \beta ) q^{45} + ( 4 + 2 \beta ) q^{47} + q^{49} -4 q^{51} + ( -2 - 4 \beta ) q^{53} + 2 q^{55} -8 q^{57} -2 \beta q^{59} + ( 6 - 2 \beta ) q^{61} + ( 1 + 4 \beta ) q^{63} + ( -4 + 4 \beta ) q^{65} + ( -12 + 4 \beta ) q^{67} + ( 8 + 8 \beta ) q^{69} + ( -8 + 4 \beta ) q^{71} + ( -2 - 2 \beta ) q^{73} + 2 \beta q^{75} + q^{77} + ( -4 + 8 \beta ) q^{79} + ( 5 + 12 \beta ) q^{81} + ( 4 - 12 \beta ) q^{83} + ( -4 + 4 \beta ) q^{85} + ( -8 + 4 \beta ) q^{87} + 2 q^{89} + ( -2 + 2 \beta ) q^{91} + ( 4 + 12 \beta ) q^{93} + ( -8 + 8 \beta ) q^{95} + ( 10 - 12 \beta ) q^{97} + ( 1 + 4 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 4q^{5} + 2q^{7} + 6q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 4q^{5} + 2q^{7} + 6q^{9} + 2q^{11} - 2q^{13} - 4q^{15} - 2q^{17} - 4q^{19} - 2q^{21} - 4q^{23} - 2q^{25} - 20q^{27} - 8q^{29} - 10q^{31} - 2q^{33} + 4q^{35} + 8q^{37} - 8q^{39} - 18q^{41} - 16q^{43} + 12q^{45} + 10q^{47} + 2q^{49} - 8q^{51} - 8q^{53} + 4q^{55} - 16q^{57} - 2q^{59} + 10q^{61} + 6q^{63} - 4q^{65} - 20q^{67} + 24q^{69} - 12q^{71} - 6q^{73} + 2q^{75} + 2q^{77} + 22q^{81} - 4q^{83} - 4q^{85} - 12q^{87} + 4q^{89} - 2q^{91} + 20q^{93} - 8q^{95} + 8q^{97} + 6q^{99} + O(q^{100}) \)

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
0 −3.23607 0 2.00000 0 1.00000 0 7.47214 0
1.2 0 1.23607 0 2.00000 0 1.00000 0 −1.47214 0
\(n\): 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 4928.2.a.bm 2
4.b odd 2 1 4928.2.a.bv 2
8.b even 2 1 77.2.a.d 2
8.d odd 2 1 1232.2.a.m 2
24.h odd 2 1 693.2.a.h 2
40.f even 2 1 1925.2.a.r 2
40.i odd 4 2 1925.2.b.h 4
56.e even 2 1 8624.2.a.ce 2
56.h odd 2 1 539.2.a.f 2
56.j odd 6 2 539.2.e.j 4
56.p even 6 2 539.2.e.i 4
88.b odd 2 1 847.2.a.f 2
88.o even 10 2 847.2.f.a 4
88.o even 10 2 847.2.f.n 4
88.p odd 10 2 847.2.f.b 4
88.p odd 10 2 847.2.f.m 4
168.i even 2 1 4851.2.a.y 2
264.m even 2 1 7623.2.a.bl 2
616.o even 2 1 5929.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.d 2 8.b even 2 1
539.2.a.f 2 56.h odd 2 1
539.2.e.i 4 56.p even 6 2
539.2.e.j 4 56.j odd 6 2
693.2.a.h 2 24.h odd 2 1
847.2.a.f 2 88.b odd 2 1
847.2.f.a 4 88.o even 10 2
847.2.f.b 4 88.p odd 10 2
847.2.f.m 4 88.p odd 10 2
847.2.f.n 4 88.o even 10 2
1232.2.a.m 2 8.d odd 2 1
1925.2.a.r 2 40.f even 2 1
1925.2.b.h 4 40.i odd 4 2
4851.2.a.y 2 168.i even 2 1
4928.2.a.bm 2 1.a even 1 1 trivial
4928.2.a.bv 2 4.b odd 2 1
5929.2.a.m 2 616.o even 2 1
7623.2.a.bl 2 264.m even 2 1
8624.2.a.ce 2 56.e even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(11\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4928))\):

\( T_{3}^{2} + 2 T_{3} - 4 \)
\( T_{5} - 2 \)
\( T_{13}^{2} + 2 T_{13} - 4 \)
\( T_{17}^{2} + 2 T_{17} - 4 \)
\( T_{19}^{2} + 4 T_{19} - 16 \)
\( T_{23}^{2} + 4 T_{23} - 16 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \)
$5$ \( ( 1 - 2 T + 5 T^{2} )^{2} \)
$7$ \( ( 1 - T )^{2} \)
$11$ \( ( 1 - T )^{2} \)
$13$ \( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4} \)
$17$ \( 1 + 2 T + 30 T^{2} + 34 T^{3} + 289 T^{4} \)
$19$ \( 1 + 4 T + 22 T^{2} + 76 T^{3} + 361 T^{4} \)
$23$ \( 1 + 4 T + 30 T^{2} + 92 T^{3} + 529 T^{4} \)
$29$ \( 1 + 8 T + 54 T^{2} + 232 T^{3} + 841 T^{4} \)
$31$ \( 1 + 10 T + 82 T^{2} + 310 T^{3} + 961 T^{4} \)
$37$ \( 1 - 8 T + 70 T^{2} - 296 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 18 T + 158 T^{2} + 738 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 + 8 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 10 T + 114 T^{2} - 470 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 8 T + 102 T^{2} + 424 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 2 T + 114 T^{2} + 118 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 10 T + 142 T^{2} - 610 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 20 T + 214 T^{2} + 1340 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 12 T + 158 T^{2} + 852 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 6 T + 150 T^{2} + 438 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 78 T^{2} + 6241 T^{4} \)
$83$ \( 1 + 4 T - 10 T^{2} + 332 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 - 2 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 8 T + 30 T^{2} - 776 T^{3} + 9409 T^{4} \)
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