Properties

Label 1296.2.s.e
Level $1296$
Weight $2$
Character orbit 1296.s
Analytic conductor $10.349$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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.e 2
3.b odd 2 1 CM 1296.2.s.e 2
4.b odd 2 1 1296.2.s.b 2
9.c even 3 1 48.2.c.a 2
9.c even 3 1 1296.2.s.b 2
9.d odd 6 1 48.2.c.a 2
9.d odd 6 1 1296.2.s.b 2
12.b even 2 1 1296.2.s.b 2
36.f odd 6 1 48.2.c.a 2
36.f odd 6 1 inner 1296.2.s.e 2
36.h even 6 1 48.2.c.a 2
36.h even 6 1 inner 1296.2.s.e 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 4.b odd 2 1
1296.2.s.b 2 9.c even 3 1
1296.2.s.b 2 9.d odd 6 1
1296.2.s.b 2 12.b even 2 1
1296.2.s.e 2 1.a even 1 1 trivial
1296.2.s.e 2 3.b odd 2 1 CM
1296.2.s.e 2 36.f odd 6 1 inner
1296.2.s.e 2 36.h even 6 1 inner
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 \(S_{2}^{\mathrm{new}}(1296, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{2} - 6T_{7} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

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