Properties

Label 69.2.c.a
Level $69$
Weight $2$
Character orbit 69.c
Analytic conductor $0.551$
Analytic rank $0$
Dimension $6$
CM discriminant -23
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 69.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.550967773947\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.8869743.1
Defining polynomial: \(x^{6} - 3 x^{3} + 8\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} -\beta_{1} q^{3} + ( -2 + \beta_{1} + \beta_{3} ) q^{4} + ( 1 + \beta_{2} + \beta_{4} ) q^{6} + ( -1 - 2 \beta_{2} - \beta_{4} + \beta_{5} ) q^{8} + ( -\beta_{2} - \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} -\beta_{1} q^{3} + ( -2 + \beta_{1} + \beta_{3} ) q^{4} + ( 1 + \beta_{2} + \beta_{4} ) q^{6} + ( -1 - 2 \beta_{2} - \beta_{4} + \beta_{5} ) q^{8} + ( -\beta_{2} - \beta_{3} ) q^{9} + ( -2 + 2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{12} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{13} + ( 4 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{16} + ( 4 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} ) q^{18} + ( 1 + \beta_{4} - \beta_{5} ) q^{23} + ( -2 + \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{24} -5 q^{25} + ( -1 - 3 \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{26} + ( 1 - \beta_{4} + \beta_{5} ) q^{27} + ( 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{29} + ( \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{31} + ( 2 + 3 \beta_{1} + 6 \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{32} + ( -5 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{5} ) q^{36} + ( -5 + 3 \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{39} + ( -3 \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{41} + ( 4 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{46} + ( -3 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{47} + ( 11 - 4 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + \beta_{5} ) q^{48} + 7 q^{49} -5 \beta_{2} q^{50} + ( -1 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{52} + ( -4 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{54} + ( 5 + \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{58} + ( 2 + 2 \beta_{4} - 2 \beta_{5} ) q^{59} + ( -1 + 3 \beta_{1} - 4 \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{62} + ( -15 + 8 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{64} + ( -\beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{69} + ( 3 \beta_{1} + 5 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{71} + ( -8 + 2 \beta_{1} - 7 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{72} + ( -5 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{73} + 5 \beta_{1} q^{75} + ( -8 + \beta_{1} - 4 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{78} + ( -\beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{81} + ( 11 - 5 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{82} + ( 7 + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{87} + ( -2 - 3 \beta_{1} - 6 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{92} + ( 1 - 3 \beta_{2} + \beta_{3} - 4 \beta_{5} ) q^{93} + ( -13 + \beta_{1} + 2 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{94} + ( 14 - 2 \beta_{1} + 12 \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{96} + 7 \beta_{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 12q^{4} + 3q^{6} + O(q^{10}) \) \( 6q - 12q^{4} + 3q^{6} - 15q^{12} + 24q^{16} + 21q^{18} - 6q^{24} - 30q^{25} + 12q^{27} - 33q^{36} - 24q^{39} + 69q^{48} + 42q^{49} - 6q^{52} + 30q^{58} - 90q^{64} - 42q^{72} - 51q^{78} + 66q^{82} + 48q^{87} - 6q^{93} - 78q^{94} + 69q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{3} + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + \nu^{2} \)\()/4\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 3 \nu^{2} + 4 \nu \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} - 2 \nu^{4} + 3 \nu^{2} + 6 \nu \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} + 2 \nu^{4} - 4 \nu^{3} + 3 \nu^{2} - 2 \nu + 4 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{5} + 2 \nu^{4} + 4 \nu^{3} + 3 \nu^{2} - 2 \nu - 4 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{4} + 2 \beta_{3} + 4 \beta_{2}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 6 \beta_{1}\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{5} - \beta_{4} + 2\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(5 \beta_{5} + 5 \beta_{4} - 2 \beta_{3} + 8 \beta_{2}\)\()/6\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{5} - \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 18 \beta_{1}\)\()/6\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(47\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
−0.261988 1.38973i
−1.07255 0.921756i
1.33454 0.467979i
1.33454 + 0.467979i
−1.07255 + 0.921756i
−0.261988 + 1.38973i
2.77947i 1.60074 + 0.661546i −5.72545 0 1.83875 4.44920i 0 10.3548i 2.12471 + 2.11792i 0
68.2 1.84351i −1.37328 1.05550i −1.39853 0 −1.94584 + 2.53166i 0 1.10881i 0.771819 + 2.89902i 0
68.3 0.935958i −0.227452 + 1.71705i 1.12398 0 1.60709 + 0.212885i 0 2.92392i −2.89653 0.781094i 0
68.4 0.935958i −0.227452 1.71705i 1.12398 0 1.60709 0.212885i 0 2.92392i −2.89653 + 0.781094i 0
68.5 1.84351i −1.37328 + 1.05550i −1.39853 0 −1.94584 2.53166i 0 1.10881i 0.771819 2.89902i 0
68.6 2.77947i 1.60074 0.661546i −5.72545 0 1.83875 + 4.44920i 0 10.3548i 2.12471 2.11792i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 CM by \(\Q(\sqrt{-23}) \)
3.b odd 2 1 inner
69.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.2.c.a 6
3.b odd 2 1 inner 69.2.c.a 6
4.b odd 2 1 1104.2.m.a 6
12.b even 2 1 1104.2.m.a 6
23.b odd 2 1 CM 69.2.c.a 6
69.c even 2 1 inner 69.2.c.a 6
92.b even 2 1 1104.2.m.a 6
276.h odd 2 1 1104.2.m.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.2.c.a 6 1.a even 1 1 trivial
69.2.c.a 6 3.b odd 2 1 inner
69.2.c.a 6 23.b odd 2 1 CM
69.2.c.a 6 69.c even 2 1 inner
1104.2.m.a 6 4.b odd 2 1
1104.2.m.a 6 12.b even 2 1
1104.2.m.a 6 92.b even 2 1
1104.2.m.a 6 276.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(69, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 23 + 36 T^{2} + 12 T^{4} + T^{6} \)
$3$ \( 27 - 4 T^{3} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( T^{6} \)
$11$ \( T^{6} \)
$13$ \( ( -74 - 39 T + T^{3} )^{2} \)
$17$ \( T^{6} \)
$19$ \( T^{6} \)
$23$ \( ( 23 + T^{2} )^{3} \)
$29$ \( 18032 + 7569 T^{2} + 174 T^{4} + T^{6} \)
$31$ \( ( -344 - 93 T + T^{3} )^{2} \)
$37$ \( T^{6} \)
$41$ \( 94208 + 15129 T^{2} + 246 T^{4} + T^{6} \)
$43$ \( T^{6} \)
$47$ \( 412988 + 19881 T^{2} + 282 T^{4} + T^{6} \)
$53$ \( T^{6} \)
$59$ \( ( 92 + T^{2} )^{3} \)
$61$ \( T^{6} \)
$67$ \( T^{6} \)
$71$ \( 48668 + 45369 T^{2} + 426 T^{4} + T^{6} \)
$73$ \( ( -1226 - 219 T + T^{3} )^{2} \)
$79$ \( T^{6} \)
$83$ \( T^{6} \)
$89$ \( T^{6} \)
$97$ \( T^{6} \)
show more
show less