Properties

Label 4761.2.a.w
Level $4761$
Weight $2$
Character orbit 4761.a
Self dual yes
Analytic conductor $38.017$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4761 = 3^{2} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4761.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.0167764023\)
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 23)
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.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( -1 + \beta ) q^{4} -2 \beta q^{5} + ( -2 + 2 \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} +O(q^{10})\) \( q + \beta q^{2} + ( -1 + \beta ) q^{4} -2 \beta q^{5} + ( -2 + 2 \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} + ( -2 - 2 \beta ) q^{10} + ( -4 + 2 \beta ) q^{11} + 3 q^{13} + 2 q^{14} -3 \beta q^{16} + ( 2 + 2 \beta ) q^{17} + 2 q^{19} -2 q^{20} + ( 2 - 2 \beta ) q^{22} + ( -1 + 4 \beta ) q^{25} + 3 \beta q^{26} + ( 4 - 2 \beta ) q^{28} + 3 q^{29} + ( 3 - 6 \beta ) q^{31} + ( -5 + \beta ) q^{32} + ( 2 + 4 \beta ) q^{34} -4 q^{35} -2 \beta q^{37} + 2 \beta q^{38} + ( 4 + 2 \beta ) q^{40} + ( 1 - 4 \beta ) q^{41} + ( 6 - 4 \beta ) q^{44} + ( 1 - 2 \beta ) q^{47} + ( 1 - 4 \beta ) q^{49} + ( 4 + 3 \beta ) q^{50} + ( -3 + 3 \beta ) q^{52} + ( -2 - 4 \beta ) q^{53} + ( -4 + 4 \beta ) q^{55} + ( -6 + 2 \beta ) q^{56} + 3 \beta q^{58} + ( -4 + 4 \beta ) q^{59} + ( 2 - 8 \beta ) q^{61} + ( -6 - 3 \beta ) q^{62} + ( 1 + 2 \beta ) q^{64} -6 \beta q^{65} + ( 4 + 2 \beta ) q^{67} + 2 \beta q^{68} -4 \beta q^{70} + ( -11 + 2 \beta ) q^{71} + ( 9 + 4 \beta ) q^{73} + ( -2 - 2 \beta ) q^{74} + ( -2 + 2 \beta ) q^{76} + ( 12 - 8 \beta ) q^{77} + ( 6 - 8 \beta ) q^{79} + ( 6 + 6 \beta ) q^{80} + ( -4 - 3 \beta ) q^{82} + ( -10 - 2 \beta ) q^{83} + ( -4 - 8 \beta ) q^{85} + ( -8 + 6 \beta ) q^{88} + ( -8 + 4 \beta ) q^{89} + ( -6 + 6 \beta ) q^{91} + ( -2 - \beta ) q^{94} -4 \beta q^{95} + ( -14 + 6 \beta ) q^{97} + ( -4 - 3 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} - 2q^{5} - 2q^{7} + O(q^{10}) \) \( 2q + q^{2} - q^{4} - 2q^{5} - 2q^{7} - 6q^{10} - 6q^{11} + 6q^{13} + 4q^{14} - 3q^{16} + 6q^{17} + 4q^{19} - 4q^{20} + 2q^{22} + 2q^{25} + 3q^{26} + 6q^{28} + 6q^{29} - 9q^{32} + 8q^{34} - 8q^{35} - 2q^{37} + 2q^{38} + 10q^{40} - 2q^{41} + 8q^{44} - 2q^{49} + 11q^{50} - 3q^{52} - 8q^{53} - 4q^{55} - 10q^{56} + 3q^{58} - 4q^{59} - 4q^{61} - 15q^{62} + 4q^{64} - 6q^{65} + 10q^{67} + 2q^{68} - 4q^{70} - 20q^{71} + 22q^{73} - 6q^{74} - 2q^{76} + 16q^{77} + 4q^{79} + 18q^{80} - 11q^{82} - 22q^{83} - 16q^{85} - 10q^{88} - 12q^{89} - 6q^{91} - 5q^{94} - 4q^{95} - 22q^{97} - 11q^{98} + O(q^{100}) \)

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
−0.618034
1.61803
−0.618034 0 −1.61803 1.23607 0 −3.23607 2.23607 0 −0.763932
1.2 1.61803 0 0.618034 −3.23607 0 1.23607 −2.23607 0 −5.23607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(23\) \(-1\)

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 4761.2.a.w 2
3.b odd 2 1 529.2.a.a 2
12.b even 2 1 8464.2.a.bb 2
23.b odd 2 1 207.2.a.d 2
69.c even 2 1 23.2.a.a 2
69.g even 22 10 529.2.c.o 20
69.h odd 22 10 529.2.c.n 20
92.b even 2 1 3312.2.a.ba 2
115.c odd 2 1 5175.2.a.be 2
276.h odd 2 1 368.2.a.h 2
345.h even 2 1 575.2.a.f 2
345.l odd 4 2 575.2.b.d 4
483.c odd 2 1 1127.2.a.c 2
552.b even 2 1 1472.2.a.t 2
552.h odd 2 1 1472.2.a.s 2
759.h odd 2 1 2783.2.a.c 2
897.g even 2 1 3887.2.a.i 2
1173.b even 2 1 6647.2.a.b 2
1311.h odd 2 1 8303.2.a.e 2
1380.b odd 2 1 9200.2.a.bt 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.2.a.a 2 69.c even 2 1
207.2.a.d 2 23.b odd 2 1
368.2.a.h 2 276.h odd 2 1
529.2.a.a 2 3.b odd 2 1
529.2.c.n 20 69.h odd 22 10
529.2.c.o 20 69.g even 22 10
575.2.a.f 2 345.h even 2 1
575.2.b.d 4 345.l odd 4 2
1127.2.a.c 2 483.c odd 2 1
1472.2.a.s 2 552.h odd 2 1
1472.2.a.t 2 552.b even 2 1
2783.2.a.c 2 759.h odd 2 1
3312.2.a.ba 2 92.b even 2 1
3887.2.a.i 2 897.g even 2 1
4761.2.a.w 2 1.a even 1 1 trivial
5175.2.a.be 2 115.c odd 2 1
6647.2.a.b 2 1173.b even 2 1
8303.2.a.e 2 1311.h odd 2 1
8464.2.a.bb 2 12.b even 2 1
9200.2.a.bt 2 1380.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4761))\):

\( T_{2}^{2} - T_{2} - 1 \)
\( T_{5}^{2} + 2 T_{5} - 4 \)
\( T_{7}^{2} + 2 T_{7} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( -4 + 2 T + T^{2} \)
$7$ \( -4 + 2 T + T^{2} \)
$11$ \( 4 + 6 T + T^{2} \)
$13$ \( ( -3 + T )^{2} \)
$17$ \( 4 - 6 T + T^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -3 + T )^{2} \)
$31$ \( -45 + T^{2} \)
$37$ \( -4 + 2 T + T^{2} \)
$41$ \( -19 + 2 T + T^{2} \)
$43$ \( T^{2} \)
$47$ \( -5 + T^{2} \)
$53$ \( -4 + 8 T + T^{2} \)
$59$ \( -16 + 4 T + T^{2} \)
$61$ \( -76 + 4 T + T^{2} \)
$67$ \( 20 - 10 T + T^{2} \)
$71$ \( 95 + 20 T + T^{2} \)
$73$ \( 101 - 22 T + T^{2} \)
$79$ \( -76 - 4 T + T^{2} \)
$83$ \( 116 + 22 T + T^{2} \)
$89$ \( 16 + 12 T + T^{2} \)
$97$ \( 76 + 22 T + T^{2} \)
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