Properties

Label 1296.2.i.e
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: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
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.

\(f(q)\) \(=\) \( q -2 \zeta_{6} q^{5} +O(q^{10})\) \( q -2 \zeta_{6} q^{5} + ( -4 + 4 \zeta_{6} ) q^{11} + 2 \zeta_{6} q^{13} -2 q^{17} + 4 q^{19} + 8 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} + ( 6 - 6 \zeta_{6} ) q^{29} + 8 \zeta_{6} q^{31} + 6 q^{37} -6 \zeta_{6} q^{41} + ( 4 - 4 \zeta_{6} ) q^{43} + 7 \zeta_{6} q^{49} + 2 q^{53} + 8 q^{55} -4 \zeta_{6} q^{59} + ( 2 - 2 \zeta_{6} ) q^{61} + ( 4 - 4 \zeta_{6} ) q^{65} -4 \zeta_{6} q^{67} + 8 q^{71} + 10 q^{73} + ( -8 + 8 \zeta_{6} ) q^{79} + ( 4 - 4 \zeta_{6} ) q^{83} + 4 \zeta_{6} q^{85} + 6 q^{89} -8 \zeta_{6} q^{95} + ( -2 + 2 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + O(q^{10}) \) \( 2q - 2q^{5} - 4q^{11} + 2q^{13} - 4q^{17} + 8q^{19} + 8q^{23} + q^{25} + 6q^{29} + 8q^{31} + 12q^{37} - 6q^{41} + 4q^{43} + 7q^{49} + 4q^{53} + 16q^{55} - 4q^{59} + 2q^{61} + 4q^{65} - 4q^{67} + 16q^{71} + 20q^{73} - 8q^{79} + 4q^{83} + 4q^{85} + 12q^{89} - 8q^{95} - 2q^{97} + O(q^{100}) \)

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 −1.00000 + 1.73205i 0 0 0 0 0
865.1 0 0 0 −1.00000 1.73205i 0 0 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.e 2
3.b odd 2 1 1296.2.i.m 2
4.b odd 2 1 648.2.i.b 2
9.c even 3 1 144.2.a.b 1
9.c even 3 1 inner 1296.2.i.e 2
9.d odd 6 1 48.2.a.a 1
9.d odd 6 1 1296.2.i.m 2
12.b even 2 1 648.2.i.g 2
36.f odd 6 1 72.2.a.a 1
36.f odd 6 1 648.2.i.b 2
36.h even 6 1 24.2.a.a 1
36.h even 6 1 648.2.i.g 2
45.h odd 6 1 1200.2.a.d 1
45.j even 6 1 3600.2.a.v 1
45.k odd 12 2 3600.2.f.r 2
45.l even 12 2 1200.2.f.b 2
63.i even 6 1 2352.2.q.r 2
63.j odd 6 1 2352.2.q.l 2
63.l odd 6 1 7056.2.a.q 1
63.n odd 6 1 2352.2.q.l 2
63.o even 6 1 2352.2.a.i 1
63.s even 6 1 2352.2.q.r 2
72.j odd 6 1 192.2.a.b 1
72.l even 6 1 192.2.a.d 1
72.n even 6 1 576.2.a.b 1
72.p odd 6 1 576.2.a.d 1
99.g even 6 1 5808.2.a.s 1
117.n odd 6 1 8112.2.a.be 1
144.u even 12 2 768.2.d.e 2
144.v odd 12 2 2304.2.d.i 2
144.w odd 12 2 768.2.d.d 2
144.x even 12 2 2304.2.d.k 2
180.n even 6 1 600.2.a.h 1
180.p odd 6 1 1800.2.a.m 1
180.v odd 12 2 600.2.f.e 2
180.x even 12 2 1800.2.f.c 2
252.n even 6 1 3528.2.s.y 2
252.o even 6 1 1176.2.q.i 2
252.r odd 6 1 1176.2.q.a 2
252.s odd 6 1 1176.2.a.i 1
252.u odd 6 1 3528.2.s.j 2
252.bb even 6 1 1176.2.q.i 2
252.bi even 6 1 3528.2.a.d 1
252.bj even 6 1 3528.2.s.y 2
252.bl odd 6 1 3528.2.s.j 2
252.bn odd 6 1 1176.2.q.a 2
360.bd even 6 1 4800.2.a.q 1
360.bh odd 6 1 4800.2.a.cc 1
360.br even 12 2 4800.2.f.bg 2
360.bt odd 12 2 4800.2.f.d 2
396.k even 6 1 8712.2.a.u 1
396.o odd 6 1 2904.2.a.c 1
468.x even 6 1 4056.2.a.i 1
468.ch odd 12 2 4056.2.c.e 2
504.cc even 6 1 9408.2.a.cc 1
504.co odd 6 1 9408.2.a.h 1
612.n even 6 1 6936.2.a.p 1
684.bh odd 6 1 8664.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 36.h even 6 1
48.2.a.a 1 9.d odd 6 1
72.2.a.a 1 36.f odd 6 1
144.2.a.b 1 9.c even 3 1
192.2.a.b 1 72.j odd 6 1
192.2.a.d 1 72.l even 6 1
576.2.a.b 1 72.n even 6 1
576.2.a.d 1 72.p odd 6 1
600.2.a.h 1 180.n even 6 1
600.2.f.e 2 180.v odd 12 2
648.2.i.b 2 4.b odd 2 1
648.2.i.b 2 36.f odd 6 1
648.2.i.g 2 12.b even 2 1
648.2.i.g 2 36.h even 6 1
768.2.d.d 2 144.w odd 12 2
768.2.d.e 2 144.u even 12 2
1176.2.a.i 1 252.s odd 6 1
1176.2.q.a 2 252.r odd 6 1
1176.2.q.a 2 252.bn odd 6 1
1176.2.q.i 2 252.o even 6 1
1176.2.q.i 2 252.bb even 6 1
1200.2.a.d 1 45.h odd 6 1
1200.2.f.b 2 45.l even 12 2
1296.2.i.e 2 1.a even 1 1 trivial
1296.2.i.e 2 9.c even 3 1 inner
1296.2.i.m 2 3.b odd 2 1
1296.2.i.m 2 9.d odd 6 1
1800.2.a.m 1 180.p odd 6 1
1800.2.f.c 2 180.x even 12 2
2304.2.d.i 2 144.v odd 12 2
2304.2.d.k 2 144.x even 12 2
2352.2.a.i 1 63.o even 6 1
2352.2.q.l 2 63.j odd 6 1
2352.2.q.l 2 63.n odd 6 1
2352.2.q.r 2 63.i even 6 1
2352.2.q.r 2 63.s even 6 1
2904.2.a.c 1 396.o odd 6 1
3528.2.a.d 1 252.bi even 6 1
3528.2.s.j 2 252.u odd 6 1
3528.2.s.j 2 252.bl odd 6 1
3528.2.s.y 2 252.n even 6 1
3528.2.s.y 2 252.bj even 6 1
3600.2.a.v 1 45.j even 6 1
3600.2.f.r 2 45.k odd 12 2
4056.2.a.i 1 468.x even 6 1
4056.2.c.e 2 468.ch odd 12 2
4800.2.a.q 1 360.bd even 6 1
4800.2.a.cc 1 360.bh odd 6 1
4800.2.f.d 2 360.bt odd 12 2
4800.2.f.bg 2 360.br even 12 2
5808.2.a.s 1 99.g even 6 1
6936.2.a.p 1 612.n even 6 1
7056.2.a.q 1 63.l odd 6 1
8112.2.a.be 1 117.n odd 6 1
8664.2.a.j 1 684.bh odd 6 1
8712.2.a.u 1 396.k even 6 1
9408.2.a.h 1 504.co odd 6 1
9408.2.a.cc 1 504.cc even 6 1

Hecke kernels

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

\( T_{5}^{2} + 2 T_{5} + 4 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \)
$7$ \( 1 - 7 T^{2} + 49 T^{4} \)
$11$ \( 1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 7 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \)
$17$ \( ( 1 + 2 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 - 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4} \)
$29$ \( 1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4} \)
$31$ \( 1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - 6 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 - 2 T + 53 T^{2} )^{2} \)
$59$ \( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 8 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 10 T + 73 T^{2} )^{2} \)
$79$ \( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 4 T - 67 T^{2} - 332 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 - 6 T + 89 T^{2} )^{2} \)
$97$ \( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \)
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