Properties

Label 1296.4.a.i
Level $1296$
Weight $4$
Character orbit 1296.a
Self dual yes
Analytic conductor $76.466$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta - 7) q^{5} + (3 \beta - 5) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 7) q^{5} + (3 \beta - 5) q^{7} + (8 \beta + 29) q^{11} + (15 \beta - 13) q^{13} + (9 \beta - 54) q^{17} + ( - 27 \beta + 52) q^{19} + (19 \beta + 7) q^{23} + (15 \beta - 68) q^{25} + (\beta + 25) q^{29} + (3 \beta - 23) q^{31} + ( - 19 \beta + 11) q^{35} + ( - 54 \beta + 2) q^{37} + (98 \beta - 115) q^{41} + ( - 6 \beta - 41) q^{43} + ( - 91 \beta + 245) q^{47} + ( - 21 \beta - 246) q^{49} + ( - 162 \beta + 54) q^{53} + ( - 93 \beta - 267) q^{55} + (136 \beta + 331) q^{59} + (105 \beta + 167) q^{61} + ( - 107 \beta - 29) q^{65} + (66 \beta - 527) q^{67} + ( - 144 \beta + 756) q^{71} + ( - 243 \beta - 106) q^{73} + (71 \beta + 47) q^{77} + (309 \beta + 247) q^{79} + (107 \beta + 353) q^{83} + ( - 18 \beta + 306) q^{85} + (72 \beta + 162) q^{89} + ( - 69 \beta + 425) q^{91} + (164 \beta - 148) q^{95} + ( - 102 \beta + 419) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 15 q^{5} - 7 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 15 q^{5} - 7 q^{7} + 66 q^{11} - 11 q^{13} - 99 q^{17} + 77 q^{19} + 33 q^{23} - 121 q^{25} + 51 q^{29} - 43 q^{31} + 3 q^{35} - 50 q^{37} - 132 q^{41} - 88 q^{43} + 399 q^{47} - 513 q^{49} - 54 q^{53} - 627 q^{55} + 798 q^{59} + 439 q^{61} - 165 q^{65} - 988 q^{67} + 1368 q^{71} - 455 q^{73} + 165 q^{77} + 803 q^{79} + 813 q^{83} + 594 q^{85} + 396 q^{89} + 781 q^{91} - 132 q^{95} + 736 q^{97}+O(q^{100}) \) 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
3.37228
−2.37228
0 0 0 −10.3723 0 5.11684 0 0 0
1.2 0 0 0 −4.62772 0 −12.1168 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.i 2
3.b odd 2 1 1296.4.a.u 2
4.b odd 2 1 81.4.a.a 2
9.c even 3 2 432.4.i.c 4
9.d odd 6 2 144.4.i.c 4
12.b even 2 1 81.4.a.d 2
20.d odd 2 1 2025.4.a.n 2
36.f odd 6 2 27.4.c.a 4
36.h even 6 2 9.4.c.a 4
60.h even 2 1 2025.4.a.g 2
180.n even 6 2 225.4.e.b 4
180.v odd 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.h even 6 2
27.4.c.a 4 36.f odd 6 2
81.4.a.a 2 4.b odd 2 1
81.4.a.d 2 12.b even 2 1
144.4.i.c 4 9.d odd 6 2
225.4.e.b 4 180.n even 6 2
225.4.k.b 8 180.v odd 12 4
432.4.i.c 4 9.c even 3 2
1296.4.a.i 2 1.a even 1 1 trivial
1296.4.a.u 2 3.b odd 2 1
2025.4.a.g 2 60.h even 2 1
2025.4.a.n 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 15T_{5} + 48 \) 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
$5$ \( T^{2} + 15T + 48 \) Copy content Toggle raw display
$7$ \( T^{2} + 7T - 62 \) Copy content Toggle raw display
$11$ \( T^{2} - 66T + 561 \) Copy content Toggle raw display
$13$ \( T^{2} + 11T - 1826 \) Copy content Toggle raw display
$17$ \( T^{2} + 99T + 1782 \) Copy content Toggle raw display
$19$ \( T^{2} - 77T - 4532 \) Copy content Toggle raw display
$23$ \( T^{2} - 33T - 2706 \) Copy content Toggle raw display
$29$ \( T^{2} - 51T + 642 \) Copy content Toggle raw display
$31$ \( T^{2} + 43T + 388 \) Copy content Toggle raw display
$37$ \( T^{2} + 50T - 23432 \) Copy content Toggle raw display
$41$ \( T^{2} + 132T - 74877 \) Copy content Toggle raw display
$43$ \( T^{2} + 88T + 1639 \) Copy content Toggle raw display
$47$ \( T^{2} - 399T - 28518 \) Copy content Toggle raw display
$53$ \( T^{2} + 54T - 215784 \) Copy content Toggle raw display
$59$ \( T^{2} - 798T + 6609 \) Copy content Toggle raw display
$61$ \( T^{2} - 439T - 42776 \) Copy content Toggle raw display
$67$ \( T^{2} + 988T + 208099 \) Copy content Toggle raw display
$71$ \( T^{2} - 1368 T + 296784 \) Copy content Toggle raw display
$73$ \( T^{2} + 455T - 435398 \) Copy content Toggle raw display
$79$ \( T^{2} - 803T - 626516 \) Copy content Toggle raw display
$83$ \( T^{2} - 813T + 70788 \) Copy content Toggle raw display
$89$ \( T^{2} - 396T - 3564 \) Copy content Toggle raw display
$97$ \( T^{2} - 736T + 49591 \) Copy content Toggle raw display
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