# 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 $1$

# Related objects

## 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: $$x^{2} - x - 26$$ x^2 - x - 26 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.

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1296))$$.

## Hecke characteristic polynomials

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