Properties

Label 69.4.c.b
Level $69$
Weight $4$
Character orbit 69.c
Analytic conductor $4.071$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-23})\)
Defining polynomial: \(x^{4} - x^{3} - 5 x^{2} - 6 x + 36\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{2} - \beta_{3} ) q^{2} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{3} + ( -4 + 6 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{4} + ( 4 + \beta_{1} + 5 \beta_{2} + 6 \beta_{3} ) q^{6} + ( 9 + 8 \beta_{2} + 10 \beta_{3} ) q^{8} + ( 10 - 8 \beta_{1} + 5 \beta_{2} - 8 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{2} - \beta_{3} ) q^{2} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{3} + ( -4 + 6 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{4} + ( 4 + \beta_{1} + 5 \beta_{2} + 6 \beta_{3} ) q^{6} + ( 9 + 8 \beta_{2} + 10 \beta_{3} ) q^{8} + ( 10 - 8 \beta_{1} + 5 \beta_{2} - 8 \beta_{3} ) q^{9} + ( 57 - 15 \beta_{1} + 6 \beta_{2} - 12 \beta_{3} ) q^{12} + ( -35 + 24 \beta_{1} + 4 \beta_{2} + 12 \beta_{3} ) q^{13} + ( 75 - 6 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{16} + ( -7 - 7 \beta_{1} - 26 \beta_{2} - 31 \beta_{3} ) q^{18} + ( 23 + 46 \beta_{3} ) q^{23} + ( -25 - 4 \beta_{1} - 47 \beta_{2} - 48 \beta_{3} ) q^{24} -125 q^{25} + ( 87 + 83 \beta_{2} + 91 \beta_{3} ) q^{26} + ( -141 - 22 \beta_{3} ) q^{27} + ( -61 - 94 \beta_{2} - 28 \beta_{3} ) q^{29} + ( -173 - 12 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{31} + ( -16 - 23 \beta_{2} - 9 \beta_{3} ) q^{32} + ( -267 + 93 \beta_{1} + 33 \beta_{2} - 21 \beta_{3} ) q^{36} + ( 247 - 98 \beta_{1} + 5 \beta_{2} - 48 \beta_{3} ) q^{39} + ( -103 - 142 \beta_{2} - 64 \beta_{3} ) q^{41} + ( 253 - 138 \beta_{1} - 23 \beta_{2} - 69 \beta_{3} ) q^{46} + ( 59 - 16 \beta_{2} + 134 \beta_{3} ) q^{47} + ( -128 + 157 \beta_{1} + 65 \beta_{2} + 12 \beta_{3} ) q^{48} + 343 q^{49} + ( 125 + 125 \beta_{2} + 125 \beta_{3} ) q^{50} + ( 760 - 330 \beta_{1} - 55 \beta_{2} - 165 \beta_{3} ) q^{52} + ( 9 + 66 \beta_{1} + 141 \beta_{2} + 163 \beta_{3} ) q^{54} + ( -765 + 366 \beta_{1} + 61 \beta_{2} + 183 \beta_{3} ) q^{58} + ( -170 - 340 \beta_{3} ) q^{59} + ( 147 + 149 \beta_{2} + 145 \beta_{3} ) q^{62} + ( 401 + 48 \beta_{1} + 8 \beta_{2} + 24 \beta_{3} ) q^{64} + ( 161 + 92 \beta_{1} - 161 \beta_{2} ) q^{69} + ( 173 + 392 \beta_{2} - 46 \beta_{3} ) q^{71} + ( 130 + 94 \beta_{1} + 245 \beta_{2} + 286 \beta_{3} ) q^{72} + ( -605 + 96 \beta_{1} + 16 \beta_{2} + 48 \beta_{3} ) q^{73} + ( 125 - 250 \beta_{1} - 125 \beta_{2} ) q^{75} + ( -304 - 67 \beta_{1} - 443 \beta_{2} - 498 \beta_{3} ) q^{78} + ( 53 - 304 \beta_{1} - 53 \beta_{2} ) q^{81} + ( -1275 + 618 \beta_{1} + 103 \beta_{2} + 309 \beta_{3} ) q^{82} + ( 607 + 226 \beta_{1} + 239 \beta_{2} + 564 \beta_{3} ) q^{87} + ( -368 - 529 \beta_{2} - 207 \beta_{3} ) q^{92} + ( 67 - 332 \beta_{1} - 193 \beta_{2} + 24 \beta_{3} ) q^{93} + ( 633 - 354 \beta_{1} - 59 \beta_{2} - 177 \beta_{3} ) q^{94} + ( 141 + 51 \beta_{1} + 66 \beta_{2} + 138 \beta_{3} ) q^{96} + ( -343 - 343 \beta_{2} - 343 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} - 18q^{4} - 5q^{6} + 38q^{9} + O(q^{10}) \) \( 4q - 4q^{3} - 18q^{4} - 5q^{6} + 38q^{9} + 225q^{12} - 148q^{13} + 302q^{16} + 79q^{18} + 86q^{24} - 500q^{25} - 520q^{27} - 688q^{31} - 999q^{36} + 976q^{39} + 1058q^{46} - 509q^{48} + 1372q^{49} + 3150q^{52} - 506q^{54} - 3182q^{58} + 1588q^{64} + 1058q^{69} - 448q^{72} - 2452q^{73} + 500q^{75} + 599q^{78} + 14q^{81} - 5306q^{82} + 1048q^{87} + 274q^{93} + 2650q^{94} + 207q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 5 x^{2} - 6 x + 36\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 5 \nu^{2} - 5 \nu - 26 \)\()/10\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 6 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 2 \beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(-5 \beta_{3} + 6\)

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
−1.82666 1.63197i
2.32666 + 0.765945i
2.32666 0.765945i
−1.82666 + 1.63197i
4.99599i −5.15331 0.665865i −16.9599 0 −3.32666 + 25.7459i 0 44.7638i 26.1132 + 6.86282i 0
68.2 0.200160i 3.15331 + 4.12997i 7.95994 0 0.826656 0.631168i 0 3.19455i −7.11325 + 26.0461i 0
68.3 0.200160i 3.15331 4.12997i 7.95994 0 0.826656 + 0.631168i 0 3.19455i −7.11325 26.0461i 0
68.4 4.99599i −5.15331 + 0.665865i −16.9599 0 −3.32666 25.7459i 0 44.7638i 26.1132 6.86282i 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.b 4
3.b odd 2 1 inner 69.4.c.b 4
23.b odd 2 1 CM 69.4.c.b 4
69.c even 2 1 inner 69.4.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.4.c.b 4 1.a even 1 1 trivial
69.4.c.b 4 3.b odd 2 1 inner
69.4.c.b 4 23.b odd 2 1 CM
69.4.c.b 4 69.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 25 T_{2}^{2} + 1 \) acting on \(S_{4}^{\mathrm{new}}(69, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 25 T^{2} + T^{4} \)
$3$ \( 729 + 108 T - 11 T^{2} + 4 T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( ( -1115 + 74 T + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( ( 12167 + T^{2} )^{2} \)
$29$ \( 3039868225 + 128302 T^{2} + T^{4} \)
$31$ \( ( 28963 + 344 T + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( 12668628025 + 319318 T^{2} + T^{4} \)
$43$ \( T^{4} \)
$47$ \( 10306107361 + 209950 T^{2} + T^{4} \)
$53$ \( T^{4} \)
$59$ \( ( 664700 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( 1050758254225 + 2098798 T^{2} + T^{4} \)
$73$ \( ( 336025 + 1226 T + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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