Properties

Label 1296.2.s.b
Level $1296$
Weight $2$
Character orbit 1296.s
Analytic conductor $10.349$
Analytic rank $1$
Dimension $2$
CM discriminant -3
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1296.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.3486121020\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 - 2 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( -2 - 2 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{13} + ( 2 - 4 \zeta_{6} ) q^{19} + ( -5 + 5 \zeta_{6} ) q^{25} + ( -12 + 6 \zeta_{6} ) q^{31} -10 q^{37} + ( -6 - 6 \zeta_{6} ) q^{43} + 5 \zeta_{6} q^{49} + ( -14 + 14 \zeta_{6} ) q^{61} + ( -4 + 2 \zeta_{6} ) q^{67} + 10 q^{73} + ( -10 - 10 \zeta_{6} ) q^{79} + ( 4 - 8 \zeta_{6} ) q^{91} + ( 14 - 14 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{7} + O(q^{10}) \) \( 2 q - 6 q^{7} + 2 q^{13} - 5 q^{25} - 18 q^{31} - 20 q^{37} - 18 q^{43} + 5 q^{49} - 14 q^{61} - 6 q^{67} + 20 q^{73} - 30 q^{79} + 14 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
431.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0 0 −3.00000 1.73205i 0 0 0
863.1 0 0 0 0 0 −3.00000 + 1.73205i 0 0 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 \(\Q(\sqrt{-3}) \)
36.f odd 6 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.2.s.b 2
3.b odd 2 1 CM 1296.2.s.b 2
4.b odd 2 1 1296.2.s.e 2
9.c even 3 1 48.2.c.a 2
9.c even 3 1 1296.2.s.e 2
9.d odd 6 1 48.2.c.a 2
9.d odd 6 1 1296.2.s.e 2
12.b even 2 1 1296.2.s.e 2
36.f odd 6 1 48.2.c.a 2
36.f odd 6 1 inner 1296.2.s.b 2
36.h even 6 1 48.2.c.a 2
36.h even 6 1 inner 1296.2.s.b 2
45.h odd 6 1 1200.2.h.e 2
45.j even 6 1 1200.2.h.e 2
45.k odd 12 2 1200.2.o.i 4
45.l even 12 2 1200.2.o.i 4
63.l odd 6 1 2352.2.h.c 2
63.o even 6 1 2352.2.h.c 2
72.j odd 6 1 192.2.c.a 2
72.l even 6 1 192.2.c.a 2
72.n even 6 1 192.2.c.a 2
72.p odd 6 1 192.2.c.a 2
144.u even 12 2 768.2.f.d 4
144.v odd 12 2 768.2.f.d 4
144.w odd 12 2 768.2.f.d 4
144.x even 12 2 768.2.f.d 4
180.n even 6 1 1200.2.h.e 2
180.p odd 6 1 1200.2.h.e 2
180.v odd 12 2 1200.2.o.i 4
180.x even 12 2 1200.2.o.i 4
252.s odd 6 1 2352.2.h.c 2
252.bi even 6 1 2352.2.h.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.c.a 2 9.c even 3 1
48.2.c.a 2 9.d odd 6 1
48.2.c.a 2 36.f odd 6 1
48.2.c.a 2 36.h even 6 1
192.2.c.a 2 72.j odd 6 1
192.2.c.a 2 72.l even 6 1
192.2.c.a 2 72.n even 6 1
192.2.c.a 2 72.p odd 6 1
768.2.f.d 4 144.u even 12 2
768.2.f.d 4 144.v odd 12 2
768.2.f.d 4 144.w odd 12 2
768.2.f.d 4 144.x even 12 2
1200.2.h.e 2 45.h odd 6 1
1200.2.h.e 2 45.j even 6 1
1200.2.h.e 2 180.n even 6 1
1200.2.h.e 2 180.p odd 6 1
1200.2.o.i 4 45.k odd 12 2
1200.2.o.i 4 45.l even 12 2
1200.2.o.i 4 180.v odd 12 2
1200.2.o.i 4 180.x even 12 2
1296.2.s.b 2 1.a even 1 1 trivial
1296.2.s.b 2 3.b odd 2 1 CM
1296.2.s.b 2 36.f odd 6 1 inner
1296.2.s.b 2 36.h even 6 1 inner
1296.2.s.e 2 4.b odd 2 1
1296.2.s.e 2 9.c even 3 1
1296.2.s.e 2 9.d odd 6 1
1296.2.s.e 2 12.b even 2 1
2352.2.h.c 2 63.l odd 6 1
2352.2.h.c 2 63.o even 6 1
2352.2.h.c 2 252.s odd 6 1
2352.2.h.c 2 252.bi even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1296, [\chi])\):

\( T_{5} \)
\( T_{7}^{2} + 6 T_{7} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 12 + 6 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 4 - 2 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 12 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 108 + 18 T + T^{2} \)
$37$ \( ( 10 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( 108 + 18 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 196 + 14 T + T^{2} \)
$67$ \( 12 + 6 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -10 + T )^{2} \)
$79$ \( 300 + 30 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 196 - 14 T + T^{2} \)
show more
show less