Properties

Label 1296.2.i.o
Level $1296$
Weight $2$
Character orbit 1296.i
Analytic conductor $10.349$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1296.i (of order \(3\), 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: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 54)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + 3 \zeta_{6} q^{5} + ( -1 + \zeta_{6} ) q^{7} +O(q^{10})\) \( q + 3 \zeta_{6} q^{5} + ( -1 + \zeta_{6} ) q^{7} + ( 3 - 3 \zeta_{6} ) q^{11} + 4 \zeta_{6} q^{13} -2 q^{19} + 6 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + ( 6 - 6 \zeta_{6} ) q^{29} + 5 \zeta_{6} q^{31} -3 q^{35} + 2 q^{37} -6 \zeta_{6} q^{41} + ( -10 + 10 \zeta_{6} ) q^{43} + ( -6 + 6 \zeta_{6} ) q^{47} + 6 \zeta_{6} q^{49} -9 q^{53} + 9 q^{55} -12 \zeta_{6} q^{59} + ( -8 + 8 \zeta_{6} ) q^{61} + ( -12 + 12 \zeta_{6} ) q^{65} + 14 \zeta_{6} q^{67} -7 q^{73} + 3 \zeta_{6} q^{77} + ( 8 - 8 \zeta_{6} ) q^{79} + ( 3 - 3 \zeta_{6} ) q^{83} + 18 q^{89} -4 q^{91} -6 \zeta_{6} q^{95} + ( 1 - \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{5} - q^{7} + O(q^{10}) \) \( 2q + 3q^{5} - q^{7} + 3q^{11} + 4q^{13} - 4q^{19} + 6q^{23} - 4q^{25} + 6q^{29} + 5q^{31} - 6q^{35} + 4q^{37} - 6q^{41} - 10q^{43} - 6q^{47} + 6q^{49} - 18q^{53} + 18q^{55} - 12q^{59} - 8q^{61} - 12q^{65} + 14q^{67} - 14q^{73} + 3q^{77} + 8q^{79} + 3q^{83} + 36q^{89} - 8q^{91} - 6q^{95} + q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 1.50000 2.59808i 0 −0.500000 0.866025i 0 0 0
865.1 0 0 0 1.50000 + 2.59808i 0 −0.500000 + 0.866025i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.2.i.o 2
3.b odd 2 1 1296.2.i.c 2
4.b odd 2 1 162.2.c.b 2
9.c even 3 1 432.2.a.b 1
9.c even 3 1 inner 1296.2.i.o 2
9.d odd 6 1 432.2.a.g 1
9.d odd 6 1 1296.2.i.c 2
12.b even 2 1 162.2.c.c 2
36.f odd 6 1 54.2.a.b yes 1
36.f odd 6 1 162.2.c.b 2
36.h even 6 1 54.2.a.a 1
36.h even 6 1 162.2.c.c 2
72.j odd 6 1 1728.2.a.d 1
72.l even 6 1 1728.2.a.c 1
72.n even 6 1 1728.2.a.z 1
72.p odd 6 1 1728.2.a.y 1
180.n even 6 1 1350.2.a.r 1
180.p odd 6 1 1350.2.a.h 1
180.v odd 12 2 1350.2.c.b 2
180.x even 12 2 1350.2.c.k 2
252.s odd 6 1 2646.2.a.a 1
252.bi even 6 1 2646.2.a.bd 1
396.k even 6 1 6534.2.a.b 1
396.o odd 6 1 6534.2.a.bc 1
468.x even 6 1 9126.2.a.u 1
468.bg odd 6 1 9126.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.a.a 1 36.h even 6 1
54.2.a.b yes 1 36.f odd 6 1
162.2.c.b 2 4.b odd 2 1
162.2.c.b 2 36.f odd 6 1
162.2.c.c 2 12.b even 2 1
162.2.c.c 2 36.h even 6 1
432.2.a.b 1 9.c even 3 1
432.2.a.g 1 9.d odd 6 1
1296.2.i.c 2 3.b odd 2 1
1296.2.i.c 2 9.d odd 6 1
1296.2.i.o 2 1.a even 1 1 trivial
1296.2.i.o 2 9.c even 3 1 inner
1350.2.a.h 1 180.p odd 6 1
1350.2.a.r 1 180.n even 6 1
1350.2.c.b 2 180.v odd 12 2
1350.2.c.k 2 180.x even 12 2
1728.2.a.c 1 72.l even 6 1
1728.2.a.d 1 72.j odd 6 1
1728.2.a.y 1 72.p odd 6 1
1728.2.a.z 1 72.n even 6 1
2646.2.a.a 1 252.s odd 6 1
2646.2.a.bd 1 252.bi even 6 1
6534.2.a.b 1 396.k even 6 1
6534.2.a.bc 1 396.o odd 6 1
9126.2.a.r 1 468.bg odd 6 1
9126.2.a.u 1 468.x even 6 1

Hecke kernels

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

\( T_{5}^{2} - 3 T_{5} + 9 \)
\( T_{7}^{2} + T_{7} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 9 - 3 T + T^{2} \)
$7$ \( 1 + T + T^{2} \)
$11$ \( 9 - 3 T + T^{2} \)
$13$ \( 16 - 4 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( 36 - 6 T + T^{2} \)
$29$ \( 36 - 6 T + T^{2} \)
$31$ \( 25 - 5 T + T^{2} \)
$37$ \( ( -2 + T )^{2} \)
$41$ \( 36 + 6 T + T^{2} \)
$43$ \( 100 + 10 T + T^{2} \)
$47$ \( 36 + 6 T + T^{2} \)
$53$ \( ( 9 + T )^{2} \)
$59$ \( 144 + 12 T + T^{2} \)
$61$ \( 64 + 8 T + T^{2} \)
$67$ \( 196 - 14 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 7 + T )^{2} \)
$79$ \( 64 - 8 T + T^{2} \)
$83$ \( 9 - 3 T + T^{2} \)
$89$ \( ( -18 + T )^{2} \)
$97$ \( 1 - T + T^{2} \)
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