Properties

Label 65.2.a.b
Level $65$
Weight $2$
Character orbit 65.a
Self dual yes
Analytic conductor $0.519$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 65.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.519027613138\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + \beta q^{3} + ( - 2 \beta + 1) q^{4} + q^{5} + ( - \beta + 2) q^{6} + ( - 2 \beta + 2) q^{7} + (\beta - 3) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} + \beta q^{3} + ( - 2 \beta + 1) q^{4} + q^{5} + ( - \beta + 2) q^{6} + ( - 2 \beta + 2) q^{7} + (\beta - 3) q^{8} - q^{9} + (\beta - 1) q^{10} + ( - \beta + 2) q^{11} + (\beta - 4) q^{12} - q^{13} + (4 \beta - 6) q^{14} + \beta q^{15} + 3 q^{16} + ( - 2 \beta - 2) q^{17} + ( - \beta + 1) q^{18} + (\beta + 2) q^{19} + ( - 2 \beta + 1) q^{20} + (2 \beta - 4) q^{21} + (3 \beta - 4) q^{22} - \beta q^{23} + ( - 3 \beta + 2) q^{24} + q^{25} + ( - \beta + 1) q^{26} - 4 \beta q^{27} + ( - 6 \beta + 10) q^{28} + 4 \beta q^{29} + ( - \beta + 2) q^{30} + (3 \beta + 6) q^{31} + (\beta + 3) q^{32} + (2 \beta - 2) q^{33} - 2 q^{34} + ( - 2 \beta + 2) q^{35} + (2 \beta - 1) q^{36} + 6 \beta q^{37} + \beta q^{38} - \beta q^{39} + (\beta - 3) q^{40} + ( - 2 \beta - 6) q^{41} + ( - 6 \beta + 8) q^{42} + (5 \beta - 4) q^{43} + ( - 5 \beta + 6) q^{44} - q^{45} + (\beta - 2) q^{46} + (2 \beta - 2) q^{47} + 3 \beta q^{48} + ( - 8 \beta + 5) q^{49} + (\beta - 1) q^{50} + ( - 2 \beta - 4) q^{51} + (2 \beta - 1) q^{52} + ( - 6 \beta - 6) q^{53} + (4 \beta - 8) q^{54} + ( - \beta + 2) q^{55} + (8 \beta - 10) q^{56} + (2 \beta + 2) q^{57} + ( - 4 \beta + 8) q^{58} + (3 \beta + 6) q^{59} + (\beta - 4) q^{60} - 8 q^{61} + 3 \beta q^{62} + (2 \beta - 2) q^{63} + (2 \beta - 7) q^{64} - q^{65} + ( - 4 \beta + 6) q^{66} - 2 q^{67} + (2 \beta + 6) q^{68} - 2 q^{69} + (4 \beta - 6) q^{70} + ( - 7 \beta + 2) q^{71} + ( - \beta + 3) q^{72} - 6 \beta q^{73} + ( - 6 \beta + 12) q^{74} + \beta q^{75} + ( - 3 \beta - 2) q^{76} + ( - 6 \beta + 8) q^{77} + (\beta - 2) q^{78} + 6 \beta q^{79} + 3 q^{80} - 5 q^{81} + ( - 4 \beta + 2) q^{82} + ( - 2 \beta - 6) q^{83} + (10 \beta - 12) q^{84} + ( - 2 \beta - 2) q^{85} + ( - 9 \beta + 14) q^{86} + 8 q^{87} + (5 \beta - 8) q^{88} + 6 q^{89} + ( - \beta + 1) q^{90} + (2 \beta - 2) q^{91} + ( - \beta + 4) q^{92} + (6 \beta + 6) q^{93} + ( - 4 \beta + 6) q^{94} + (\beta + 2) q^{95} + (3 \beta + 2) q^{96} + (4 \beta - 2) q^{97} + (13 \beta - 21) q^{98} + (\beta - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{6} + 4 q^{7} - 6 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{6} + 4 q^{7} - 6 q^{8} - 2 q^{9} - 2 q^{10} + 4 q^{11} - 8 q^{12} - 2 q^{13} - 12 q^{14} + 6 q^{16} - 4 q^{17} + 2 q^{18} + 4 q^{19} + 2 q^{20} - 8 q^{21} - 8 q^{22} + 4 q^{24} + 2 q^{25} + 2 q^{26} + 20 q^{28} + 4 q^{30} + 12 q^{31} + 6 q^{32} - 4 q^{33} - 4 q^{34} + 4 q^{35} - 2 q^{36} - 6 q^{40} - 12 q^{41} + 16 q^{42} - 8 q^{43} + 12 q^{44} - 2 q^{45} - 4 q^{46} - 4 q^{47} + 10 q^{49} - 2 q^{50} - 8 q^{51} - 2 q^{52} - 12 q^{53} - 16 q^{54} + 4 q^{55} - 20 q^{56} + 4 q^{57} + 16 q^{58} + 12 q^{59} - 8 q^{60} - 16 q^{61} - 4 q^{63} - 14 q^{64} - 2 q^{65} + 12 q^{66} - 4 q^{67} + 12 q^{68} - 4 q^{69} - 12 q^{70} + 4 q^{71} + 6 q^{72} + 24 q^{74} - 4 q^{76} + 16 q^{77} - 4 q^{78} + 6 q^{80} - 10 q^{81} + 4 q^{82} - 12 q^{83} - 24 q^{84} - 4 q^{85} + 28 q^{86} + 16 q^{87} - 16 q^{88} + 12 q^{89} + 2 q^{90} - 4 q^{91} + 8 q^{92} + 12 q^{93} + 12 q^{94} + 4 q^{95} + 4 q^{96} - 4 q^{97} - 42 q^{98} - 4 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
−2.41421 −1.41421 3.82843 1.00000 3.41421 4.82843 −4.41421 −1.00000 −2.41421
1.2 0.414214 1.41421 −1.82843 1.00000 0.585786 −0.828427 −1.58579 −1.00000 0.414214
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(13\) \(1\)

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 65.2.a.b 2
3.b odd 2 1 585.2.a.m 2
4.b odd 2 1 1040.2.a.j 2
5.b even 2 1 325.2.a.i 2
5.c odd 4 2 325.2.b.f 4
7.b odd 2 1 3185.2.a.j 2
8.b even 2 1 4160.2.a.bf 2
8.d odd 2 1 4160.2.a.z 2
11.b odd 2 1 7865.2.a.j 2
12.b even 2 1 9360.2.a.cd 2
13.b even 2 1 845.2.a.g 2
13.c even 3 2 845.2.e.h 4
13.d odd 4 2 845.2.c.b 4
13.e even 6 2 845.2.e.c 4
13.f odd 12 4 845.2.m.f 8
15.d odd 2 1 2925.2.a.u 2
15.e even 4 2 2925.2.c.r 4
20.d odd 2 1 5200.2.a.bu 2
39.d odd 2 1 7605.2.a.x 2
65.d even 2 1 4225.2.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.b 2 1.a even 1 1 trivial
325.2.a.i 2 5.b even 2 1
325.2.b.f 4 5.c odd 4 2
585.2.a.m 2 3.b odd 2 1
845.2.a.g 2 13.b even 2 1
845.2.c.b 4 13.d odd 4 2
845.2.e.c 4 13.e even 6 2
845.2.e.h 4 13.c even 3 2
845.2.m.f 8 13.f odd 12 4
1040.2.a.j 2 4.b odd 2 1
2925.2.a.u 2 15.d odd 2 1
2925.2.c.r 4 15.e even 4 2
3185.2.a.j 2 7.b odd 2 1
4160.2.a.z 2 8.d odd 2 1
4160.2.a.bf 2 8.b even 2 1
4225.2.a.r 2 65.d even 2 1
5200.2.a.bu 2 20.d odd 2 1
7605.2.a.x 2 39.d odd 2 1
7865.2.a.j 2 11.b odd 2 1
9360.2.a.cd 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(65))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 2 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$11$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$23$ \( T^{2} - 2 \) Copy content Toggle raw display
$29$ \( T^{2} - 32 \) Copy content Toggle raw display
$31$ \( T^{2} - 12T + 18 \) Copy content Toggle raw display
$37$ \( T^{2} - 72 \) Copy content Toggle raw display
$41$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
$43$ \( T^{2} + 8T - 34 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 12T - 36 \) Copy content Toggle raw display
$59$ \( T^{2} - 12T + 18 \) Copy content Toggle raw display
$61$ \( (T + 8)^{2} \) Copy content Toggle raw display
$67$ \( (T + 2)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 94 \) Copy content Toggle raw display
$73$ \( T^{2} - 72 \) Copy content Toggle raw display
$79$ \( T^{2} - 72 \) Copy content Toggle raw display
$83$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
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