Properties

Label 1296.2.s.h
Level $1296$
Weight $2$
Character orbit 1296.s
Analytic conductor $10.349$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $8$

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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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + (4 \beta_{2} - 4) q^{13} + \beta_{3} q^{17} + 13 \beta_{2} q^{25} + ( - \beta_{3} + \beta_1) q^{29} + 2 q^{37} - 3 \beta_1 q^{41} + (7 \beta_{2} - 7) q^{49} - 3 \beta_{3} q^{53} + 10 \beta_{2} q^{61} + (4 \beta_{3} - 4 \beta_1) q^{65} + 16 q^{73} + (18 \beta_{2} - 18) q^{85} - \beta_{3} q^{89} + 8 \beta_{2} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{13} + 26 q^{25} + 8 q^{37} - 14 q^{49} + 20 q^{61} + 64 q^{73} - 36 q^{85} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{3} ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(1 - \beta_{2}\)

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
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
0 0 0 −3.67423 + 2.12132i 0 0 0 0 0
431.2 0 0 0 3.67423 2.12132i 0 0 0 0 0
863.1 0 0 0 −3.67423 2.12132i 0 0 0 0 0
863.2 0 0 0 3.67423 + 2.12132i 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
12.b even 2 1 inner
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.h 4
3.b odd 2 1 inner 1296.2.s.h 4
4.b odd 2 1 CM 1296.2.s.h 4
9.c even 3 1 144.2.c.a 2
9.c even 3 1 inner 1296.2.s.h 4
9.d odd 6 1 144.2.c.a 2
9.d odd 6 1 inner 1296.2.s.h 4
12.b even 2 1 inner 1296.2.s.h 4
36.f odd 6 1 144.2.c.a 2
36.f odd 6 1 inner 1296.2.s.h 4
36.h even 6 1 144.2.c.a 2
36.h even 6 1 inner 1296.2.s.h 4
45.h odd 6 1 3600.2.h.b 2
45.j even 6 1 3600.2.h.b 2
45.k odd 12 2 3600.2.o.a 4
45.l even 12 2 3600.2.o.a 4
63.l odd 6 1 7056.2.h.b 2
63.o even 6 1 7056.2.h.b 2
72.j odd 6 1 576.2.c.a 2
72.l even 6 1 576.2.c.a 2
72.n even 6 1 576.2.c.a 2
72.p odd 6 1 576.2.c.a 2
144.u even 12 2 2304.2.f.f 4
144.v odd 12 2 2304.2.f.f 4
144.w odd 12 2 2304.2.f.f 4
144.x even 12 2 2304.2.f.f 4
180.n even 6 1 3600.2.h.b 2
180.p odd 6 1 3600.2.h.b 2
180.v odd 12 2 3600.2.o.a 4
180.x even 12 2 3600.2.o.a 4
252.s odd 6 1 7056.2.h.b 2
252.bi even 6 1 7056.2.h.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.c.a 2 9.c even 3 1
144.2.c.a 2 9.d odd 6 1
144.2.c.a 2 36.f odd 6 1
144.2.c.a 2 36.h even 6 1
576.2.c.a 2 72.j odd 6 1
576.2.c.a 2 72.l even 6 1
576.2.c.a 2 72.n even 6 1
576.2.c.a 2 72.p odd 6 1
1296.2.s.h 4 1.a even 1 1 trivial
1296.2.s.h 4 3.b odd 2 1 inner
1296.2.s.h 4 4.b odd 2 1 CM
1296.2.s.h 4 9.c even 3 1 inner
1296.2.s.h 4 9.d odd 6 1 inner
1296.2.s.h 4 12.b even 2 1 inner
1296.2.s.h 4 36.f odd 6 1 inner
1296.2.s.h 4 36.h even 6 1 inner
2304.2.f.f 4 144.u even 12 2
2304.2.f.f 4 144.v odd 12 2
2304.2.f.f 4 144.w odd 12 2
2304.2.f.f 4 144.x even 12 2
3600.2.h.b 2 45.h odd 6 1
3600.2.h.b 2 45.j even 6 1
3600.2.h.b 2 180.n even 6 1
3600.2.h.b 2 180.p odd 6 1
3600.2.o.a 4 45.k odd 12 2
3600.2.o.a 4 45.l even 12 2
3600.2.o.a 4 180.v odd 12 2
3600.2.o.a 4 180.x even 12 2
7056.2.h.b 2 63.l odd 6 1
7056.2.h.b 2 63.o even 6 1
7056.2.h.b 2 252.s odd 6 1
7056.2.h.b 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 \(S_{2}^{\mathrm{new}}(1296, [\chi])\):

\( T_{5}^{4} - 18T_{5}^{2} + 324 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 18T^{2} + 324 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 18T^{2} + 324 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T - 2)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 162 T^{2} + 26244 \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 162)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T - 16)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
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