Properties

Label 6762.2.a.cb
Level $6762$
Weight $2$
Character orbit 6762.a
Self dual yes
Analytic conductor $53.995$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + ( 1 + \beta ) q^{5} - q^{6} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} + ( 1 + \beta ) q^{5} - q^{6} + q^{8} + q^{9} + ( 1 + \beta ) q^{10} + ( -3 + \beta ) q^{11} - q^{12} + 2 \beta q^{13} + ( -1 - \beta ) q^{15} + q^{16} + 4 q^{17} + q^{18} + ( 1 + 3 \beta ) q^{19} + ( 1 + \beta ) q^{20} + ( -3 + \beta ) q^{22} + q^{23} - q^{24} + ( 1 + 2 \beta ) q^{25} + 2 \beta q^{26} - q^{27} + 2 \beta q^{29} + ( -1 - \beta ) q^{30} + ( -2 - 2 \beta ) q^{31} + q^{32} + ( 3 - \beta ) q^{33} + 4 q^{34} + q^{36} + ( 9 - \beta ) q^{37} + ( 1 + 3 \beta ) q^{38} -2 \beta q^{39} + ( 1 + \beta ) q^{40} + 2 q^{41} + ( -7 - \beta ) q^{43} + ( -3 + \beta ) q^{44} + ( 1 + \beta ) q^{45} + q^{46} -4 q^{47} - q^{48} + ( 1 + 2 \beta ) q^{50} -4 q^{51} + 2 \beta q^{52} + ( 3 - \beta ) q^{53} - q^{54} + ( 2 - 2 \beta ) q^{55} + ( -1 - 3 \beta ) q^{57} + 2 \beta q^{58} -4 \beta q^{59} + ( -1 - \beta ) q^{60} + ( -3 - \beta ) q^{61} + ( -2 - 2 \beta ) q^{62} + q^{64} + ( 10 + 2 \beta ) q^{65} + ( 3 - \beta ) q^{66} + ( 3 - 3 \beta ) q^{67} + 4 q^{68} - q^{69} -4 \beta q^{71} + q^{72} -2 \beta q^{73} + ( 9 - \beta ) q^{74} + ( -1 - 2 \beta ) q^{75} + ( 1 + 3 \beta ) q^{76} -2 \beta q^{78} -2 \beta q^{79} + ( 1 + \beta ) q^{80} + q^{81} + 2 q^{82} + ( -11 + \beta ) q^{83} + ( 4 + 4 \beta ) q^{85} + ( -7 - \beta ) q^{86} -2 \beta q^{87} + ( -3 + \beta ) q^{88} + ( 6 - 2 \beta ) q^{89} + ( 1 + \beta ) q^{90} + q^{92} + ( 2 + 2 \beta ) q^{93} -4 q^{94} + ( 16 + 4 \beta ) q^{95} - q^{96} + ( 4 + 2 \beta ) q^{97} + ( -3 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{8} + 2q^{9} + 2q^{10} - 6q^{11} - 2q^{12} - 2q^{15} + 2q^{16} + 8q^{17} + 2q^{18} + 2q^{19} + 2q^{20} - 6q^{22} + 2q^{23} - 2q^{24} + 2q^{25} - 2q^{27} - 2q^{30} - 4q^{31} + 2q^{32} + 6q^{33} + 8q^{34} + 2q^{36} + 18q^{37} + 2q^{38} + 2q^{40} + 4q^{41} - 14q^{43} - 6q^{44} + 2q^{45} + 2q^{46} - 8q^{47} - 2q^{48} + 2q^{50} - 8q^{51} + 6q^{53} - 2q^{54} + 4q^{55} - 2q^{57} - 2q^{60} - 6q^{61} - 4q^{62} + 2q^{64} + 20q^{65} + 6q^{66} + 6q^{67} + 8q^{68} - 2q^{69} + 2q^{72} + 18q^{74} - 2q^{75} + 2q^{76} + 2q^{80} + 2q^{81} + 4q^{82} - 22q^{83} + 8q^{85} - 14q^{86} - 6q^{88} + 12q^{89} + 2q^{90} + 2q^{92} + 4q^{93} - 8q^{94} + 32q^{95} - 2q^{96} + 8q^{97} - 6q^{99} + O(q^{100}) \)

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
−0.618034
1.61803
1.00000 −1.00000 1.00000 −1.23607 −1.00000 0 1.00000 1.00000 −1.23607
1.2 1.00000 −1.00000 1.00000 3.23607 −1.00000 0 1.00000 1.00000 3.23607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)
\(23\) \(-1\)

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 6762.2.a.cb 2
7.b odd 2 1 138.2.a.d 2
21.c even 2 1 414.2.a.f 2
28.d even 2 1 1104.2.a.j 2
35.c odd 2 1 3450.2.a.be 2
35.f even 4 2 3450.2.d.x 4
56.e even 2 1 4416.2.a.bl 2
56.h odd 2 1 4416.2.a.bh 2
84.h odd 2 1 3312.2.a.bc 2
161.c even 2 1 3174.2.a.s 2
483.c odd 2 1 9522.2.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.a.d 2 7.b odd 2 1
414.2.a.f 2 21.c even 2 1
1104.2.a.j 2 28.d even 2 1
3174.2.a.s 2 161.c even 2 1
3312.2.a.bc 2 84.h odd 2 1
3450.2.a.be 2 35.c odd 2 1
3450.2.d.x 4 35.f even 4 2
4416.2.a.bh 2 56.h odd 2 1
4416.2.a.bl 2 56.e even 2 1
6762.2.a.cb 2 1.a even 1 1 trivial
9522.2.a.q 2 483.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6762))\):

\( T_{5}^{2} - 2 T_{5} - 4 \)
\( T_{11}^{2} + 6 T_{11} + 4 \)
\( T_{13}^{2} - 20 \)
\( T_{17} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( -4 - 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 4 + 6 T + T^{2} \)
$13$ \( -20 + T^{2} \)
$17$ \( ( -4 + T )^{2} \)
$19$ \( -44 - 2 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( -20 + T^{2} \)
$31$ \( -16 + 4 T + T^{2} \)
$37$ \( 76 - 18 T + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( 44 + 14 T + T^{2} \)
$47$ \( ( 4 + T )^{2} \)
$53$ \( 4 - 6 T + T^{2} \)
$59$ \( -80 + T^{2} \)
$61$ \( 4 + 6 T + T^{2} \)
$67$ \( -36 - 6 T + T^{2} \)
$71$ \( -80 + T^{2} \)
$73$ \( -20 + T^{2} \)
$79$ \( -20 + T^{2} \)
$83$ \( 116 + 22 T + T^{2} \)
$89$ \( 16 - 12 T + T^{2} \)
$97$ \( -4 - 8 T + T^{2} \)
show more
show less