Properties

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

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.61803
−0.618034
−1.61803 0 0.618034 −1.00000 0 1.00000 2.23607 0 1.61803
1.2 0.618034 0 −1.61803 −1.00000 0 1.00000 −2.23607 0 −0.618034
\(n\): 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 693.2.a.f 2
3.b odd 2 1 231.2.a.c 2
7.b odd 2 1 4851.2.a.w 2
11.b odd 2 1 7623.2.a.bm 2
12.b even 2 1 3696.2.a.be 2
15.d odd 2 1 5775.2.a.be 2
21.c even 2 1 1617.2.a.p 2
33.d even 2 1 2541.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.c 2 3.b odd 2 1
693.2.a.f 2 1.a even 1 1 trivial
1617.2.a.p 2 21.c even 2 1
2541.2.a.t 2 33.d even 2 1
3696.2.a.be 2 12.b even 2 1
4851.2.a.w 2 7.b odd 2 1
5775.2.a.be 2 15.d odd 2 1
7623.2.a.bm 2 11.b odd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(693))\):

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4} \)
$3$ \( \)
$5$ \( ( 1 + T + 5 T^{2} )^{2} \)
$7$ \( ( 1 - T )^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( 1 + 2 T + 7 T^{2} + 26 T^{3} + 169 T^{4} \)
$17$ \( 1 + 6 T + 38 T^{2} + 102 T^{3} + 289 T^{4} \)
$19$ \( 1 - 7 T^{2} + 361 T^{4} \)
$23$ \( 1 - 2 T + 2 T^{2} - 46 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 5 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 6 T + 66 T^{2} + 186 T^{3} + 961 T^{4} \)
$37$ \( ( 1 + 7 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 4 T + 66 T^{2} + 164 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 2 T + 42 T^{2} + 86 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 4 T + 93 T^{2} - 188 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 2 T - 18 T^{2} - 106 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 7 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 - 2 T + 61 T^{2} )^{2} \)
$67$ \( 1 + 24 T + 273 T^{2} + 1608 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 4 T + 126 T^{2} + 284 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 18 T + 207 T^{2} - 1314 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 20 T + 238 T^{2} + 1580 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 18 T + 242 T^{2} + 1494 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 158 T^{2} + 7921 T^{4} \)
$97$ \( 1 - 6 T + 158 T^{2} - 582 T^{3} + 9409 T^{4} \)
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