Properties

Label 3.13.b.a
Level $3$
Weight $13$
Character orbit 3.b
Self dual yes
Analytic conductor $2.742$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 3.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(2.74198145183\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 729 q^{3} + 4096 q^{4} - 153502 q^{7} + 531441 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 729 q^{3} + 4096 q^{4} - 153502 q^{7} + 531441 q^{9} + 2985984 q^{12} - 9397582 q^{13} + 16777216 q^{16} + 17886962 q^{19} - 111902958 q^{21} + 244140625 q^{25} + 387420489 q^{27} - 628744192 q^{28} - 530187838 q^{31} + 2176782336 q^{36} + 2826257618 q^{37} - 6850837278 q^{39} - 235885102 q^{43} + 12230590464 q^{48} + 9721576803 q^{49} - 38492495872 q^{52} + 13039595298 q^{57} + 74063873522 q^{61} - 81577256382 q^{63} + 68719476736 q^{64} - 151031344462 q^{67} + 104459767778 q^{73} + 177978515625 q^{75} + 73264996352 q^{76} - 444304748158 q^{79} + 282429536481 q^{81} - 458354515968 q^{84} + 1442547632164 q^{91} - 386506933902 q^{93} - 1662757858942 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0
0 729.000 4096.00 0 0 −153502. 0 531441. 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.13.b.a 1
3.b odd 2 1 CM 3.13.b.a 1
4.b odd 2 1 48.13.e.a 1
5.b even 2 1 75.13.c.a 1
5.c odd 4 2 75.13.d.a 2
8.b even 2 1 192.13.e.a 1
8.d odd 2 1 192.13.e.b 1
9.c even 3 2 81.13.d.a 2
9.d odd 6 2 81.13.d.a 2
12.b even 2 1 48.13.e.a 1
15.d odd 2 1 75.13.c.a 1
15.e even 4 2 75.13.d.a 2
24.f even 2 1 192.13.e.b 1
24.h odd 2 1 192.13.e.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.13.b.a 1 1.a even 1 1 trivial
3.13.b.a 1 3.b odd 2 1 CM
48.13.e.a 1 4.b odd 2 1
48.13.e.a 1 12.b even 2 1
75.13.c.a 1 5.b even 2 1
75.13.c.a 1 15.d odd 2 1
75.13.d.a 2 5.c odd 4 2
75.13.d.a 2 15.e even 4 2
81.13.d.a 2 9.c even 3 2
81.13.d.a 2 9.d odd 6 2
192.13.e.a 1 8.b even 2 1
192.13.e.a 1 24.h odd 2 1
192.13.e.b 1 8.d odd 2 1
192.13.e.b 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{13}^{\mathrm{new}}(3, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 729 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 153502 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 9397582 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 17886962 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 530187838 \) Copy content Toggle raw display
$37$ \( T - 2826257618 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 235885102 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 74063873522 \) Copy content Toggle raw display
$67$ \( T + 151031344462 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 104459767778 \) Copy content Toggle raw display
$79$ \( T + 444304748158 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T + 1662757858942 \) Copy content Toggle raw display
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