Properties

Label 138.2.a.d
Level $138$
Weight $2$
Character orbit 138.a
Self dual yes
Analytic conductor $1.102$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 138.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.10193554789\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

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

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 −3.23607 1.00000 4.47214 1.00000 1.00000 −3.23607
1.2 1.00000 1.00000 1.00000 1.23607 1.00000 −4.47214 1.00000 1.00000 1.23607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 138.2.a.d 2
3.b odd 2 1 414.2.a.f 2
4.b odd 2 1 1104.2.a.j 2
5.b even 2 1 3450.2.a.be 2
5.c odd 4 2 3450.2.d.x 4
7.b odd 2 1 6762.2.a.cb 2
8.b even 2 1 4416.2.a.bh 2
8.d odd 2 1 4416.2.a.bl 2
12.b even 2 1 3312.2.a.bc 2
23.b odd 2 1 3174.2.a.s 2
69.c even 2 1 9522.2.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.a.d 2 1.a even 1 1 trivial
414.2.a.f 2 3.b odd 2 1
1104.2.a.j 2 4.b odd 2 1
3174.2.a.s 2 23.b odd 2 1
3312.2.a.bc 2 12.b even 2 1
3450.2.a.be 2 5.b even 2 1
3450.2.d.x 4 5.c odd 4 2
4416.2.a.bh 2 8.b even 2 1
4416.2.a.bl 2 8.d odd 2 1
6762.2.a.cb 2 7.b odd 2 1
9522.2.a.q 2 69.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 2T_{5} - 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(138))\). Copy content Toggle raw display

Hecke characteristic polynomials

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