Properties

Label 693.2.a.j
Level $693$
Weight $2$
Character orbit 693.a
Self dual yes
Analytic conductor $5.534$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.53363286007\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Defining polynomial: \(x^{2} - x - 5\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( 3 + \beta ) q^{4} -3 q^{5} + q^{7} + ( 5 + 2 \beta ) q^{8} +O(q^{10})\) \( q + \beta q^{2} + ( 3 + \beta ) q^{4} -3 q^{5} + q^{7} + ( 5 + 2 \beta ) q^{8} -3 \beta q^{10} + q^{11} + q^{13} + \beta q^{14} + ( 4 + 5 \beta ) q^{16} + ( -4 + 2 \beta ) q^{17} + ( -3 + 2 \beta ) q^{19} + ( -9 - 3 \beta ) q^{20} + \beta q^{22} + ( 2 - 2 \beta ) q^{23} + 4 q^{25} + \beta q^{26} + ( 3 + \beta ) q^{28} + ( 1 - 4 \beta ) q^{29} -2 \beta q^{31} + ( 15 + 5 \beta ) q^{32} + ( 10 - 2 \beta ) q^{34} -3 q^{35} + q^{37} + ( 10 - \beta ) q^{38} + ( -15 - 6 \beta ) q^{40} + ( 4 - 4 \beta ) q^{41} + ( -2 - 2 \beta ) q^{43} + ( 3 + \beta ) q^{44} -10 q^{46} + ( -5 - 2 \beta ) q^{47} + q^{49} + 4 \beta q^{50} + ( 3 + \beta ) q^{52} + ( 6 - 2 \beta ) q^{53} -3 q^{55} + ( 5 + 2 \beta ) q^{56} + ( -20 - 3 \beta ) q^{58} + ( -1 + 2 \beta ) q^{59} + 10 q^{61} + ( -10 - 2 \beta ) q^{62} + ( 17 + 10 \beta ) q^{64} -3 q^{65} + ( 5 - 2 \beta ) q^{67} + ( -2 + 4 \beta ) q^{68} -3 \beta q^{70} + ( -4 + 4 \beta ) q^{71} + 7 q^{73} + \beta q^{74} + ( 1 + 5 \beta ) q^{76} + q^{77} -4 \beta q^{79} + ( -12 - 15 \beta ) q^{80} -20 q^{82} + ( 8 - 2 \beta ) q^{83} + ( 12 - 6 \beta ) q^{85} + ( -10 - 4 \beta ) q^{86} + ( 5 + 2 \beta ) q^{88} + ( -2 + 4 \beta ) q^{89} + q^{91} + ( -4 - 6 \beta ) q^{92} + ( -10 - 7 \beta ) q^{94} + ( 9 - 6 \beta ) q^{95} + ( -6 - 2 \beta ) q^{97} + \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 7 q^{4} - 6 q^{5} + 2 q^{7} + 12 q^{8} + O(q^{10}) \) \( 2 q + q^{2} + 7 q^{4} - 6 q^{5} + 2 q^{7} + 12 q^{8} - 3 q^{10} + 2 q^{11} + 2 q^{13} + q^{14} + 13 q^{16} - 6 q^{17} - 4 q^{19} - 21 q^{20} + q^{22} + 2 q^{23} + 8 q^{25} + q^{26} + 7 q^{28} - 2 q^{29} - 2 q^{31} + 35 q^{32} + 18 q^{34} - 6 q^{35} + 2 q^{37} + 19 q^{38} - 36 q^{40} + 4 q^{41} - 6 q^{43} + 7 q^{44} - 20 q^{46} - 12 q^{47} + 2 q^{49} + 4 q^{50} + 7 q^{52} + 10 q^{53} - 6 q^{55} + 12 q^{56} - 43 q^{58} + 20 q^{61} - 22 q^{62} + 44 q^{64} - 6 q^{65} + 8 q^{67} - 3 q^{70} - 4 q^{71} + 14 q^{73} + q^{74} + 7 q^{76} + 2 q^{77} - 4 q^{79} - 39 q^{80} - 40 q^{82} + 14 q^{83} + 18 q^{85} - 24 q^{86} + 12 q^{88} + 2 q^{91} - 14 q^{92} - 27 q^{94} + 12 q^{95} - 14 q^{97} + q^{98} + O(q^{100}) \)

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
−1.79129
2.79129
−1.79129 0 1.20871 −3.00000 0 1.00000 1.41742 0 5.37386
1.2 2.79129 0 5.79129 −3.00000 0 1.00000 10.5826 0 −8.37386
\(n\): 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 693.2.a.j 2
3.b odd 2 1 231.2.a.b 2
7.b odd 2 1 4851.2.a.ba 2
11.b odd 2 1 7623.2.a.bf 2
12.b even 2 1 3696.2.a.bl 2
15.d odd 2 1 5775.2.a.bn 2
21.c even 2 1 1617.2.a.o 2
33.d even 2 1 2541.2.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.b 2 3.b odd 2 1
693.2.a.j 2 1.a even 1 1 trivial
1617.2.a.o 2 21.c even 2 1
2541.2.a.z 2 33.d even 2 1
3696.2.a.bl 2 12.b even 2 1
4851.2.a.ba 2 7.b odd 2 1
5775.2.a.bn 2 15.d odd 2 1
7623.2.a.bf 2 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(693))\):

\( T_{2}^{2} - T_{2} - 5 \)
\( T_{5} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -5 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 3 + T )^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( -12 + 6 T + T^{2} \)
$19$ \( -17 + 4 T + T^{2} \)
$23$ \( -20 - 2 T + T^{2} \)
$29$ \( -83 + 2 T + T^{2} \)
$31$ \( -20 + 2 T + T^{2} \)
$37$ \( ( -1 + T )^{2} \)
$41$ \( -80 - 4 T + T^{2} \)
$43$ \( -12 + 6 T + T^{2} \)
$47$ \( 15 + 12 T + T^{2} \)
$53$ \( 4 - 10 T + T^{2} \)
$59$ \( -21 + T^{2} \)
$61$ \( ( -10 + T )^{2} \)
$67$ \( -5 - 8 T + T^{2} \)
$71$ \( -80 + 4 T + T^{2} \)
$73$ \( ( -7 + T )^{2} \)
$79$ \( -80 + 4 T + T^{2} \)
$83$ \( 28 - 14 T + T^{2} \)
$89$ \( -84 + T^{2} \)
$97$ \( 28 + 14 T + T^{2} \)
show more
show less