Properties

Label 69.2.a.b
Level $69$
Weight $2$
Character orbit 69.a
Self dual yes
Analytic conductor $0.551$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 69.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.550967773947\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q -\beta q^{2} - q^{3} + 3 q^{4} + ( -1 + \beta ) q^{5} + \beta q^{6} + ( 1 + \beta ) q^{7} -\beta q^{8} + q^{9} +O(q^{10})\) \( q -\beta q^{2} - q^{3} + 3 q^{4} + ( -1 + \beta ) q^{5} + \beta q^{6} + ( 1 + \beta ) q^{7} -\beta q^{8} + q^{9} + ( -5 + \beta ) q^{10} + 4 q^{11} -3 q^{12} -2 \beta q^{13} + ( -5 - \beta ) q^{14} + ( 1 - \beta ) q^{15} - q^{16} + ( -5 + \beta ) q^{17} -\beta q^{18} + ( 5 + \beta ) q^{19} + ( -3 + 3 \beta ) q^{20} + ( -1 - \beta ) q^{21} -4 \beta q^{22} + q^{23} + \beta q^{24} + ( 1 - 2 \beta ) q^{25} + 10 q^{26} - q^{27} + ( 3 + 3 \beta ) q^{28} + 2 \beta q^{29} + ( 5 - \beta ) q^{30} + ( -2 - 2 \beta ) q^{31} + 3 \beta q^{32} -4 q^{33} + ( -5 + 5 \beta ) q^{34} + 4 q^{35} + 3 q^{36} + 2 \beta q^{37} + ( -5 - 5 \beta ) q^{38} + 2 \beta q^{39} + ( -5 + \beta ) q^{40} + ( -2 - 4 \beta ) q^{41} + ( 5 + \beta ) q^{42} + ( 1 - 3 \beta ) q^{43} + 12 q^{44} + ( -1 + \beta ) q^{45} -\beta q^{46} -4 q^{47} + q^{48} + ( -1 + 2 \beta ) q^{49} + ( 10 - \beta ) q^{50} + ( 5 - \beta ) q^{51} -6 \beta q^{52} + ( -3 - \beta ) q^{53} + \beta q^{54} + ( -4 + 4 \beta ) q^{55} + ( -5 - \beta ) q^{56} + ( -5 - \beta ) q^{57} -10 q^{58} + ( 4 - 4 \beta ) q^{59} + ( 3 - 3 \beta ) q^{60} + 2 \beta q^{61} + ( 10 + 2 \beta ) q^{62} + ( 1 + \beta ) q^{63} -13 q^{64} + ( -10 + 2 \beta ) q^{65} + 4 \beta q^{66} + ( 3 - \beta ) q^{67} + ( -15 + 3 \beta ) q^{68} - q^{69} -4 \beta q^{70} -8 q^{71} -\beta q^{72} + ( -2 + 4 \beta ) q^{73} -10 q^{74} + ( -1 + 2 \beta ) q^{75} + ( 15 + 3 \beta ) q^{76} + ( 4 + 4 \beta ) q^{77} -10 q^{78} + ( 3 + 3 \beta ) q^{79} + ( 1 - \beta ) q^{80} + q^{81} + ( 20 + 2 \beta ) q^{82} + 4 q^{83} + ( -3 - 3 \beta ) q^{84} + ( 10 - 6 \beta ) q^{85} + ( 15 - \beta ) q^{86} -2 \beta q^{87} -4 \beta q^{88} + ( 1 - \beta ) q^{89} + ( -5 + \beta ) q^{90} + ( -10 - 2 \beta ) q^{91} + 3 q^{92} + ( 2 + 2 \beta ) q^{93} + 4 \beta q^{94} + 4 \beta q^{95} -3 \beta q^{96} + ( 4 + 2 \beta ) q^{97} + ( -10 + \beta ) q^{98} + 4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 6q^{4} - 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 6q^{4} - 2q^{5} + 2q^{7} + 2q^{9} - 10q^{10} + 8q^{11} - 6q^{12} - 10q^{14} + 2q^{15} - 2q^{16} - 10q^{17} + 10q^{19} - 6q^{20} - 2q^{21} + 2q^{23} + 2q^{25} + 20q^{26} - 2q^{27} + 6q^{28} + 10q^{30} - 4q^{31} - 8q^{33} - 10q^{34} + 8q^{35} + 6q^{36} - 10q^{38} - 10q^{40} - 4q^{41} + 10q^{42} + 2q^{43} + 24q^{44} - 2q^{45} - 8q^{47} + 2q^{48} - 2q^{49} + 20q^{50} + 10q^{51} - 6q^{53} - 8q^{55} - 10q^{56} - 10q^{57} - 20q^{58} + 8q^{59} + 6q^{60} + 20q^{62} + 2q^{63} - 26q^{64} - 20q^{65} + 6q^{67} - 30q^{68} - 2q^{69} - 16q^{71} - 4q^{73} - 20q^{74} - 2q^{75} + 30q^{76} + 8q^{77} - 20q^{78} + 6q^{79} + 2q^{80} + 2q^{81} + 40q^{82} + 8q^{83} - 6q^{84} + 20q^{85} + 30q^{86} + 2q^{89} - 10q^{90} - 20q^{91} + 6q^{92} + 4q^{93} + 8q^{97} - 20q^{98} + 8q^{99} + O(q^{100}) \)

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
−2.23607 −1.00000 3.00000 1.23607 2.23607 3.23607 −2.23607 1.00000 −2.76393
1.2 2.23607 −1.00000 3.00000 −3.23607 −2.23607 −1.23607 2.23607 1.00000 −7.23607
\(n\): 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 69.2.a.b 2
3.b odd 2 1 207.2.a.c 2
4.b odd 2 1 1104.2.a.m 2
5.b even 2 1 1725.2.a.ba 2
5.c odd 4 2 1725.2.b.o 4
7.b odd 2 1 3381.2.a.t 2
8.b even 2 1 4416.2.a.bm 2
8.d odd 2 1 4416.2.a.bg 2
11.b odd 2 1 8349.2.a.i 2
12.b even 2 1 3312.2.a.bb 2
15.d odd 2 1 5175.2.a.bk 2
23.b odd 2 1 1587.2.a.i 2
69.c even 2 1 4761.2.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.2.a.b 2 1.a even 1 1 trivial
207.2.a.c 2 3.b odd 2 1
1104.2.a.m 2 4.b odd 2 1
1587.2.a.i 2 23.b odd 2 1
1725.2.a.ba 2 5.b even 2 1
1725.2.b.o 4 5.c odd 4 2
3312.2.a.bb 2 12.b even 2 1
3381.2.a.t 2 7.b odd 2 1
4416.2.a.bg 2 8.d odd 2 1
4416.2.a.bm 2 8.b even 2 1
4761.2.a.v 2 69.c even 2 1
5175.2.a.bk 2 15.d odd 2 1
8349.2.a.i 2 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 5 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(69))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -5 + T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( -4 + 2 T + T^{2} \)
$7$ \( -4 - 2 T + T^{2} \)
$11$ \( ( -4 + T )^{2} \)
$13$ \( -20 + T^{2} \)
$17$ \( 20 + 10 T + T^{2} \)
$19$ \( 20 - 10 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( -20 + T^{2} \)
$31$ \( -16 + 4 T + T^{2} \)
$37$ \( -20 + T^{2} \)
$41$ \( -76 + 4 T + T^{2} \)
$43$ \( -44 - 2 T + T^{2} \)
$47$ \( ( 4 + T )^{2} \)
$53$ \( 4 + 6 T + T^{2} \)
$59$ \( -64 - 8 T + T^{2} \)
$61$ \( -20 + T^{2} \)
$67$ \( 4 - 6 T + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( -76 + 4 T + T^{2} \)
$79$ \( -36 - 6 T + T^{2} \)
$83$ \( ( -4 + T )^{2} \)
$89$ \( -4 - 2 T + T^{2} \)
$97$ \( -4 - 8 T + T^{2} \)
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