Properties

Label 1296.4.a.l
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: \(\Q(\sqrt{105}) \)
Defining polynomial: \( x^{2} - x - 26 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 18)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(-1 + 3\sqrt{105})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 5) q^{5} + (\beta - 9) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 5) q^{5} + (\beta - 9) q^{7} + ( - 2 \beta + 11) q^{11} + (\beta + 31) q^{13} + (\beta + 2) q^{17} + (5 \beta - 64) q^{19} + ( - \beta - 35) q^{23} + (9 \beta + 136) q^{25} + ( - 7 \beta + 115) q^{29} + ( - 3 \beta - 107) q^{31} + (5 \beta - 191) q^{35} + (14 \beta + 138) q^{37} + (2 \beta + 235) q^{41} + (24 \beta + 55) q^{43} + ( - 15 \beta - 249) q^{47} + ( - 19 \beta - 26) q^{49} + (14 \beta + 82) q^{53} + ( - 3 \beta + 417) q^{55} + (2 \beta - 83) q^{59} + (15 \beta - 517) q^{61} + ( - 35 \beta - 391) q^{65} + ( - 12 \beta + 577) q^{67} + (32 \beta + 172) q^{71} + ( - 21 \beta - 166) q^{73} + (31 \beta - 571) q^{77} + ( - 13 \beta - 181) q^{79} + ( - 21 \beta - 621) q^{83} + ( - 6 \beta - 246) q^{85} + (40 \beta - 226) q^{89} + (21 \beta - 43) q^{91} + (44 \beta - 860) q^{95} + (22 \beta - 53) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9 q^{5} - 19 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 9 q^{5} - 19 q^{7} + 24 q^{11} + 61 q^{13} + 3 q^{17} - 133 q^{19} - 69 q^{23} + 263 q^{25} + 237 q^{29} - 211 q^{31} - 387 q^{35} + 262 q^{37} + 468 q^{41} + 86 q^{43} - 483 q^{47} - 33 q^{49} + 150 q^{53} + 837 q^{55} - 168 q^{59} - 1049 q^{61} - 747 q^{65} + 1166 q^{67} + 312 q^{71} - 311 q^{73} - 1173 q^{77} - 349 q^{79} - 1221 q^{83} - 486 q^{85} - 492 q^{89} - 107 q^{91} - 1764 q^{95} - 128 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
5.62348
−4.62348
0 0 0 −19.8704 0 5.87043 0 0 0
1.2 0 0 0 10.8704 0 −24.8704 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.l 2
3.b odd 2 1 1296.4.a.r 2
4.b odd 2 1 162.4.a.f 2
9.c even 3 2 144.4.i.b 4
9.d odd 6 2 432.4.i.b 4
12.b even 2 1 162.4.a.g 2
36.f odd 6 2 18.4.c.b 4
36.h even 6 2 54.4.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.4.c.b 4 36.f odd 6 2
54.4.c.b 4 36.h even 6 2
144.4.i.b 4 9.c even 3 2
162.4.a.f 2 4.b odd 2 1
162.4.a.g 2 12.b even 2 1
432.4.i.b 4 9.d odd 6 2
1296.4.a.l 2 1.a even 1 1 trivial
1296.4.a.r 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 9T_{5} - 216 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1296))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 9T - 216 \) Copy content Toggle raw display
$7$ \( T^{2} + 19T - 146 \) Copy content Toggle raw display
$11$ \( T^{2} - 24T - 801 \) Copy content Toggle raw display
$13$ \( T^{2} - 61T + 694 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T - 234 \) Copy content Toggle raw display
$19$ \( T^{2} + 133T - 1484 \) Copy content Toggle raw display
$23$ \( T^{2} + 69T + 954 \) Copy content Toggle raw display
$29$ \( T^{2} - 237T + 2466 \) Copy content Toggle raw display
$31$ \( T^{2} + 211T + 9004 \) Copy content Toggle raw display
$37$ \( T^{2} - 262T - 29144 \) Copy content Toggle raw display
$41$ \( T^{2} - 468T + 53811 \) Copy content Toggle raw display
$43$ \( T^{2} - 86T - 134231 \) Copy content Toggle raw display
$47$ \( T^{2} + 483T + 5166 \) Copy content Toggle raw display
$53$ \( T^{2} - 150T - 40680 \) Copy content Toggle raw display
$59$ \( T^{2} + 168T + 6111 \) Copy content Toggle raw display
$61$ \( T^{2} + 1049 T + 221944 \) Copy content Toggle raw display
$67$ \( T^{2} - 1166 T + 305869 \) Copy content Toggle raw display
$71$ \( T^{2} - 312T - 217584 \) Copy content Toggle raw display
$73$ \( T^{2} + 311T - 80006 \) Copy content Toggle raw display
$79$ \( T^{2} + 349T - 9476 \) Copy content Toggle raw display
$83$ \( T^{2} + 1221 T + 268524 \) Copy content Toggle raw display
$89$ \( T^{2} + 492T - 317484 \) Copy content Toggle raw display
$97$ \( T^{2} + 128T - 110249 \) Copy content Toggle raw display
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