Properties

Label 1296.4.a.u
Level $1296$
Weight $4$
Character orbit 1296.a
Self dual yes
Analytic conductor $76.466$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1296.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(76.4664753674\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 9)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 8 - \beta ) q^{5} + ( -2 - 3 \beta ) q^{7} +O(q^{10})\) \( q + ( 8 - \beta ) q^{5} + ( -2 - 3 \beta ) q^{7} + ( -37 + 8 \beta ) q^{11} + ( 2 - 15 \beta ) q^{13} + ( 45 + 9 \beta ) q^{17} + ( 25 + 27 \beta ) q^{19} + ( -26 + 19 \beta ) q^{23} + ( -53 - 15 \beta ) q^{25} + ( -26 + \beta ) q^{29} + ( -20 - 3 \beta ) q^{31} + ( 8 - 19 \beta ) q^{35} + ( -52 + 54 \beta ) q^{37} + ( 17 + 98 \beta ) q^{41} + ( -47 + 6 \beta ) q^{43} + ( -154 - 91 \beta ) q^{47} + ( -267 + 21 \beta ) q^{49} + ( 108 - 162 \beta ) q^{53} + ( -360 + 93 \beta ) q^{55} + ( -467 + 136 \beta ) q^{59} + ( 272 - 105 \beta ) q^{61} + ( 136 - 107 \beta ) q^{65} + ( -461 - 66 \beta ) q^{67} + ( -612 - 144 \beta ) q^{71} + ( -349 + 243 \beta ) q^{73} + ( -118 + 71 \beta ) q^{77} + ( 556 - 309 \beta ) q^{79} + ( -460 + 107 \beta ) q^{83} + ( 288 + 18 \beta ) q^{85} + ( -234 + 72 \beta ) q^{89} + ( 356 + 69 \beta ) q^{91} + ( -16 + 164 \beta ) q^{95} + ( 317 + 102 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 15 q^{5} - 7 q^{7} + O(q^{10}) \) \( 2 q + 15 q^{5} - 7 q^{7} - 66 q^{11} - 11 q^{13} + 99 q^{17} + 77 q^{19} - 33 q^{23} - 121 q^{25} - 51 q^{29} - 43 q^{31} - 3 q^{35} - 50 q^{37} + 132 q^{41} - 88 q^{43} - 399 q^{47} - 513 q^{49} + 54 q^{53} - 627 q^{55} - 798 q^{59} + 439 q^{61} + 165 q^{65} - 988 q^{67} - 1368 q^{71} - 455 q^{73} - 165 q^{77} + 803 q^{79} - 813 q^{83} + 594 q^{85} - 396 q^{89} + 781 q^{91} + 132 q^{95} + 736 q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.37228
−2.37228
0 0 0 4.62772 0 −12.1168 0 0 0
1.2 0 0 0 10.3723 0 5.11684 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.4.a.u 2
3.b odd 2 1 1296.4.a.i 2
4.b odd 2 1 81.4.a.d 2
9.c even 3 2 144.4.i.c 4
9.d odd 6 2 432.4.i.c 4
12.b even 2 1 81.4.a.a 2
20.d odd 2 1 2025.4.a.g 2
36.f odd 6 2 9.4.c.a 4
36.h even 6 2 27.4.c.a 4
60.h even 2 1 2025.4.a.n 2
180.p odd 6 2 225.4.e.b 4
180.x even 12 4 225.4.k.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.c.a 4 36.f odd 6 2
27.4.c.a 4 36.h even 6 2
81.4.a.a 2 12.b even 2 1
81.4.a.d 2 4.b odd 2 1
144.4.i.c 4 9.c even 3 2
225.4.e.b 4 180.p odd 6 2
225.4.k.b 8 180.x even 12 4
432.4.i.c 4 9.d odd 6 2
1296.4.a.i 2 3.b odd 2 1
1296.4.a.u 2 1.a even 1 1 trivial
2025.4.a.g 2 20.d odd 2 1
2025.4.a.n 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 15 T_{5} + 48 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1296))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 48 - 15 T + T^{2} \)
$7$ \( -62 + 7 T + T^{2} \)
$11$ \( 561 + 66 T + T^{2} \)
$13$ \( -1826 + 11 T + T^{2} \)
$17$ \( 1782 - 99 T + T^{2} \)
$19$ \( -4532 - 77 T + T^{2} \)
$23$ \( -2706 + 33 T + T^{2} \)
$29$ \( 642 + 51 T + T^{2} \)
$31$ \( 388 + 43 T + T^{2} \)
$37$ \( -23432 + 50 T + T^{2} \)
$41$ \( -74877 - 132 T + T^{2} \)
$43$ \( 1639 + 88 T + T^{2} \)
$47$ \( -28518 + 399 T + T^{2} \)
$53$ \( -215784 - 54 T + T^{2} \)
$59$ \( 6609 + 798 T + T^{2} \)
$61$ \( -42776 - 439 T + T^{2} \)
$67$ \( 208099 + 988 T + T^{2} \)
$71$ \( 296784 + 1368 T + T^{2} \)
$73$ \( -435398 + 455 T + T^{2} \)
$79$ \( -626516 - 803 T + T^{2} \)
$83$ \( 70788 + 813 T + T^{2} \)
$89$ \( -3564 + 396 T + T^{2} \)
$97$ \( 49591 - 736 T + T^{2} \)
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