Properties

Label 69.4.c.a
Level $69$
Weight $4$
Character orbit 69.c
Analytic conductor $4.071$
Analytic rank $0$
Dimension $2$
CM discriminant -23
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 69.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.07113179040\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-23}) \)
Defining polynomial: \(x^{2} - x + 6\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-23}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} + ( 2 - \beta ) q^{3} -15 q^{4} + ( -23 - 2 \beta ) q^{6} + 7 \beta q^{8} + ( -19 - 4 \beta ) q^{9} +O(q^{10})\) \( q -\beta q^{2} + ( 2 - \beta ) q^{3} -15 q^{4} + ( -23 - 2 \beta ) q^{6} + 7 \beta q^{8} + ( -19 - 4 \beta ) q^{9} + ( -30 + 15 \beta ) q^{12} + 74 q^{13} + 41 q^{16} + ( -92 + 19 \beta ) q^{18} -23 \beta q^{23} + ( 161 + 14 \beta ) q^{24} -125 q^{25} -74 \beta q^{26} + ( -130 + 11 \beta ) q^{27} -28 \beta q^{29} + 344 q^{31} + 15 \beta q^{32} + ( 285 + 60 \beta ) q^{36} + ( 148 - 74 \beta ) q^{39} -64 \beta q^{41} -529 q^{46} + 134 \beta q^{47} + ( 82 - 41 \beta ) q^{48} + 343 q^{49} + 125 \beta q^{50} -1110 q^{52} + ( 253 + 130 \beta ) q^{54} -644 q^{58} + 170 \beta q^{59} -344 \beta q^{62} + 673 q^{64} + ( -529 - 46 \beta ) q^{69} -46 \beta q^{71} + ( 644 - 133 \beta ) q^{72} + 1226 q^{73} + ( -250 + 125 \beta ) q^{75} + ( -1702 - 148 \beta ) q^{78} + ( -7 + 152 \beta ) q^{81} -1472 q^{82} + ( -644 - 56 \beta ) q^{87} + 345 \beta q^{92} + ( 688 - 344 \beta ) q^{93} + 3082 q^{94} + ( 345 + 30 \beta ) q^{96} -343 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{3} - 30q^{4} - 46q^{6} - 38q^{9} + O(q^{10}) \) \( 2q + 4q^{3} - 30q^{4} - 46q^{6} - 38q^{9} - 60q^{12} + 148q^{13} + 82q^{16} - 184q^{18} + 322q^{24} - 250q^{25} - 260q^{27} + 688q^{31} + 570q^{36} + 296q^{39} - 1058q^{46} + 164q^{48} + 686q^{49} - 2220q^{52} + 506q^{54} - 1288q^{58} + 1346q^{64} - 1058q^{69} + 1288q^{72} + 2452q^{73} - 500q^{75} - 3404q^{78} - 14q^{81} - 2944q^{82} - 1288q^{87} + 1376q^{93} + 6164q^{94} + 690q^{96} + O(q^{100}) \)

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
0.500000 + 2.39792i
0.500000 2.39792i
4.79583i 2.00000 4.79583i −15.0000 0 −23.0000 9.59166i 0 33.5708i −19.0000 19.1833i 0
68.2 4.79583i 2.00000 + 4.79583i −15.0000 0 −23.0000 + 9.59166i 0 33.5708i −19.0000 + 19.1833i 0
\(n\): e.g. 2-40 or 990-1000
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 69.4.c.a 2
3.b odd 2 1 inner 69.4.c.a 2
23.b odd 2 1 CM 69.4.c.a 2
69.c even 2 1 inner 69.4.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.4.c.a 2 1.a even 1 1 trivial
69.4.c.a 2 3.b odd 2 1 inner
69.4.c.a 2 23.b odd 2 1 CM
69.4.c.a 2 69.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 23 \) acting on \(S_{4}^{\mathrm{new}}(69, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 23 + T^{2} \)
$3$ \( 27 - 4 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( -74 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( 12167 + T^{2} \)
$29$ \( 18032 + T^{2} \)
$31$ \( ( -344 + T )^{2} \)
$37$ \( T^{2} \)
$41$ \( 94208 + T^{2} \)
$43$ \( T^{2} \)
$47$ \( 412988 + T^{2} \)
$53$ \( T^{2} \)
$59$ \( 664700 + T^{2} \)
$61$ \( T^{2} \)
$67$ \( T^{2} \)
$71$ \( 48668 + T^{2} \)
$73$ \( ( -1226 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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