Properties

Label 68.2.a.a
Level $68$
Weight $2$
Character orbit 68.a
Self dual yes
Analytic conductor $0.543$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 68.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.542982733745\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} - 2 \beta q^{5} + ( - \beta - 1) q^{7} + (2 \beta + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{3} - 2 \beta q^{5} + ( - \beta - 1) q^{7} + (2 \beta + 1) q^{9} + (\beta - 3) q^{11} + (2 \beta + 2) q^{13} + ( - 2 \beta - 6) q^{15} - q^{17} + ( - 2 \beta + 2) q^{19} + ( - 2 \beta - 4) q^{21} + (\beta - 3) q^{23} + 7 q^{25} + 4 q^{27} + 2 \beta q^{29} + (3 \beta - 1) q^{31} - 2 \beta q^{33} + (2 \beta + 6) q^{35} + ( - 2 \beta + 8) q^{37} + (4 \beta + 8) q^{39} - 6 q^{41} + ( - 6 \beta + 2) q^{43} + ( - 2 \beta - 12) q^{45} - 4 \beta q^{47} + (2 \beta - 3) q^{49} + ( - \beta - 1) q^{51} + (4 \beta + 6) q^{53} + (6 \beta - 6) q^{55} - 4 q^{57} + ( - 2 \beta + 6) q^{59} + (2 \beta - 4) q^{61} + ( - 3 \beta - 7) q^{63} + ( - 4 \beta - 12) q^{65} + (4 \beta + 8) q^{67} - 2 \beta q^{69} + ( - 3 \beta - 3) q^{71} + 2 q^{73} + (7 \beta + 7) q^{75} + 2 \beta q^{77} + ( - 3 \beta - 7) q^{79} + ( - 2 \beta + 1) q^{81} + (2 \beta - 6) q^{83} + 2 \beta q^{85} + (2 \beta + 6) q^{87} + ( - 2 \beta + 6) q^{89} + ( - 4 \beta - 8) q^{91} + (2 \beta + 8) q^{93} + ( - 4 \beta + 12) q^{95} + ( - 4 \beta + 2) q^{97} + ( - 5 \beta + 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} - 6 q^{11} + 4 q^{13} - 12 q^{15} - 2 q^{17} + 4 q^{19} - 8 q^{21} - 6 q^{23} + 14 q^{25} + 8 q^{27} - 2 q^{31} + 12 q^{35} + 16 q^{37} + 16 q^{39} - 12 q^{41} + 4 q^{43} - 24 q^{45} - 6 q^{49} - 2 q^{51} + 12 q^{53} - 12 q^{55} - 8 q^{57} + 12 q^{59} - 8 q^{61} - 14 q^{63} - 24 q^{65} + 16 q^{67} - 6 q^{71} + 4 q^{73} + 14 q^{75} - 14 q^{79} + 2 q^{81} - 12 q^{83} + 12 q^{87} + 12 q^{89} - 16 q^{91} + 16 q^{93} + 24 q^{95} + 4 q^{97} + 6 q^{99}+O(q^{100}) \) 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
−1.73205
1.73205
0 −0.732051 0 3.46410 0 0.732051 0 −2.46410 0
1.2 0 2.73205 0 −3.46410 0 −2.73205 0 4.46410 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(1\)

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 68.2.a.a 2
3.b odd 2 1 612.2.a.e 2
4.b odd 2 1 272.2.a.e 2
5.b even 2 1 1700.2.a.d 2
5.c odd 4 2 1700.2.e.c 4
7.b odd 2 1 3332.2.a.h 2
8.b even 2 1 1088.2.a.p 2
8.d odd 2 1 1088.2.a.t 2
11.b odd 2 1 8228.2.a.k 2
12.b even 2 1 2448.2.a.y 2
17.b even 2 1 1156.2.a.a 2
17.c even 4 2 1156.2.b.c 4
17.d even 8 4 1156.2.e.d 8
17.e odd 16 8 1156.2.h.f 16
20.d odd 2 1 6800.2.a.bh 2
24.f even 2 1 9792.2.a.cs 2
24.h odd 2 1 9792.2.a.cr 2
68.d odd 2 1 4624.2.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.2.a.a 2 1.a even 1 1 trivial
272.2.a.e 2 4.b odd 2 1
612.2.a.e 2 3.b odd 2 1
1088.2.a.p 2 8.b even 2 1
1088.2.a.t 2 8.d odd 2 1
1156.2.a.a 2 17.b even 2 1
1156.2.b.c 4 17.c even 4 2
1156.2.e.d 8 17.d even 8 4
1156.2.h.f 16 17.e odd 16 8
1700.2.a.d 2 5.b even 2 1
1700.2.e.c 4 5.c odd 4 2
2448.2.a.y 2 12.b even 2 1
3332.2.a.h 2 7.b odd 2 1
4624.2.a.x 2 68.d odd 2 1
6800.2.a.bh 2 20.d odd 2 1
8228.2.a.k 2 11.b odd 2 1
9792.2.a.cr 2 24.h odd 2 1
9792.2.a.cs 2 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(\Gamma_0(68))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$5$ \( T^{2} - 12 \) Copy content Toggle raw display
$7$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T - 8 \) Copy content Toggle raw display
$17$ \( (T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 4T - 8 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$29$ \( T^{2} - 12 \) Copy content Toggle raw display
$31$ \( T^{2} + 2T - 26 \) Copy content Toggle raw display
$37$ \( T^{2} - 16T + 52 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 104 \) Copy content Toggle raw display
$47$ \( T^{2} - 48 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T - 12 \) Copy content Toggle raw display
$59$ \( T^{2} - 12T + 24 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 16T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T - 18 \) Copy content Toggle raw display
$73$ \( (T - 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 14T + 22 \) Copy content Toggle raw display
$83$ \( T^{2} + 12T + 24 \) Copy content Toggle raw display
$89$ \( T^{2} - 12T + 24 \) Copy content Toggle raw display
$97$ \( T^{2} - 4T - 44 \) Copy content Toggle raw display
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