Properties

Label 1296.3.e.a
Level $1296$
Weight $3$
Character orbit 1296.e
Analytic conductor $35.313$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1296.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(35.3134422611\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 2 - 4 \zeta_{6} ) q^{5} -2 q^{7} +O(q^{10})\) \( q + ( 2 - 4 \zeta_{6} ) q^{5} -2 q^{7} + ( -1 + 2 \zeta_{6} ) q^{11} -4 q^{13} + ( -9 + 18 \zeta_{6} ) q^{17} -11 q^{19} + ( 16 - 32 \zeta_{6} ) q^{23} + 13 q^{25} + ( -26 + 52 \zeta_{6} ) q^{29} -32 q^{31} + ( -4 + 8 \zeta_{6} ) q^{35} -34 q^{37} + ( -7 + 14 \zeta_{6} ) q^{41} + 61 q^{43} + ( -28 + 56 \zeta_{6} ) q^{47} -45 q^{49} + 6 q^{55} + ( -29 + 58 \zeta_{6} ) q^{59} + 56 q^{61} + ( -8 + 16 \zeta_{6} ) q^{65} + 31 q^{67} + ( -18 + 36 \zeta_{6} ) q^{71} + 65 q^{73} + ( 2 - 4 \zeta_{6} ) q^{77} -38 q^{79} + ( -28 + 56 \zeta_{6} ) q^{83} + 54 q^{85} + ( -72 + 144 \zeta_{6} ) q^{89} + 8 q^{91} + ( -22 + 44 \zeta_{6} ) q^{95} -115 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{7} + O(q^{10}) \) \( 2 q - 4 q^{7} - 8 q^{13} - 22 q^{19} + 26 q^{25} - 64 q^{31} - 68 q^{37} + 122 q^{43} - 90 q^{49} + 12 q^{55} + 112 q^{61} + 62 q^{67} + 130 q^{73} - 76 q^{79} + 108 q^{85} + 16 q^{91} - 230 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 3.46410i 0 −2.00000 0 0 0
161.2 0 0 0 3.46410i 0 −2.00000 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.3.e.a 2
3.b odd 2 1 inner 1296.3.e.a 2
4.b odd 2 1 81.3.b.a 2
9.c even 3 1 144.3.q.a 2
9.c even 3 1 432.3.q.a 2
9.d odd 6 1 144.3.q.a 2
9.d odd 6 1 432.3.q.a 2
12.b even 2 1 81.3.b.a 2
36.f odd 6 1 9.3.d.a 2
36.f odd 6 1 27.3.d.a 2
36.h even 6 1 9.3.d.a 2
36.h even 6 1 27.3.d.a 2
72.j odd 6 1 576.3.q.a 2
72.j odd 6 1 1728.3.q.b 2
72.l even 6 1 576.3.q.b 2
72.l even 6 1 1728.3.q.a 2
72.n even 6 1 576.3.q.a 2
72.n even 6 1 1728.3.q.b 2
72.p odd 6 1 576.3.q.b 2
72.p odd 6 1 1728.3.q.a 2
180.n even 6 1 225.3.j.a 2
180.n even 6 1 675.3.j.a 2
180.p odd 6 1 225.3.j.a 2
180.p odd 6 1 675.3.j.a 2
180.v odd 12 2 225.3.i.a 4
180.v odd 12 2 675.3.i.a 4
180.x even 12 2 225.3.i.a 4
180.x even 12 2 675.3.i.a 4
252.n even 6 1 441.3.n.a 2
252.o even 6 1 441.3.n.b 2
252.r odd 6 1 441.3.j.b 2
252.s odd 6 1 441.3.r.a 2
252.u odd 6 1 441.3.j.a 2
252.bb even 6 1 441.3.j.a 2
252.bi even 6 1 441.3.r.a 2
252.bj even 6 1 441.3.j.b 2
252.bl odd 6 1 441.3.n.b 2
252.bn odd 6 1 441.3.n.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.3.d.a 2 36.f odd 6 1
9.3.d.a 2 36.h even 6 1
27.3.d.a 2 36.f odd 6 1
27.3.d.a 2 36.h even 6 1
81.3.b.a 2 4.b odd 2 1
81.3.b.a 2 12.b even 2 1
144.3.q.a 2 9.c even 3 1
144.3.q.a 2 9.d odd 6 1
225.3.i.a 4 180.v odd 12 2
225.3.i.a 4 180.x even 12 2
225.3.j.a 2 180.n even 6 1
225.3.j.a 2 180.p odd 6 1
432.3.q.a 2 9.c even 3 1
432.3.q.a 2 9.d odd 6 1
441.3.j.a 2 252.u odd 6 1
441.3.j.a 2 252.bb even 6 1
441.3.j.b 2 252.r odd 6 1
441.3.j.b 2 252.bj even 6 1
441.3.n.a 2 252.n even 6 1
441.3.n.a 2 252.bn odd 6 1
441.3.n.b 2 252.o even 6 1
441.3.n.b 2 252.bl odd 6 1
441.3.r.a 2 252.s odd 6 1
441.3.r.a 2 252.bi even 6 1
576.3.q.a 2 72.j odd 6 1
576.3.q.a 2 72.n even 6 1
576.3.q.b 2 72.l even 6 1
576.3.q.b 2 72.p odd 6 1
675.3.i.a 4 180.v odd 12 2
675.3.i.a 4 180.x even 12 2
675.3.j.a 2 180.n even 6 1
675.3.j.a 2 180.p odd 6 1
1296.3.e.a 2 1.a even 1 1 trivial
1296.3.e.a 2 3.b odd 2 1 inner
1728.3.q.a 2 72.l even 6 1
1728.3.q.a 2 72.p odd 6 1
1728.3.q.b 2 72.j odd 6 1
1728.3.q.b 2 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1296, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 12 + T^{2} \)
$7$ \( ( 2 + T )^{2} \)
$11$ \( 3 + T^{2} \)
$13$ \( ( 4 + T )^{2} \)
$17$ \( 243 + T^{2} \)
$19$ \( ( 11 + T )^{2} \)
$23$ \( 768 + T^{2} \)
$29$ \( 2028 + T^{2} \)
$31$ \( ( 32 + T )^{2} \)
$37$ \( ( 34 + T )^{2} \)
$41$ \( 147 + T^{2} \)
$43$ \( ( -61 + T )^{2} \)
$47$ \( 2352 + T^{2} \)
$53$ \( T^{2} \)
$59$ \( 2523 + T^{2} \)
$61$ \( ( -56 + T )^{2} \)
$67$ \( ( -31 + T )^{2} \)
$71$ \( 972 + T^{2} \)
$73$ \( ( -65 + T )^{2} \)
$79$ \( ( 38 + T )^{2} \)
$83$ \( 2352 + T^{2} \)
$89$ \( 15552 + T^{2} \)
$97$ \( ( 115 + T )^{2} \)
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