# Properties

 Label 1088.4.a.x Level $1088$ Weight $4$ Character orbit 1088.a Self dual yes Analytic conductor $64.194$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1088 = 2^{6} \cdot 17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1088.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.1940780862$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.2636.1 Defining polynomial: $$x^{3} - 14x - 4$$ x^3 - 14*x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 17) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - \beta_1 + 1) q^{3} + ( - \beta_{2} + 3) q^{5} + (2 \beta_{2} + \beta_1 - 8) q^{7} + ( - \beta_{2} - 8 \beta_1 + 20) q^{9}+O(q^{10})$$ q + (b2 - b1 + 1) * q^3 + (-b2 + 3) * q^5 + (2*b2 + b1 - 8) * q^7 + (-b2 - 8*b1 + 20) * q^9 $$q + (\beta_{2} - \beta_1 + 1) q^{3} + ( - \beta_{2} + 3) q^{5} + (2 \beta_{2} + \beta_1 - 8) q^{7} + ( - \beta_{2} - 8 \beta_1 + 20) q^{9} + ( - \beta_{2} + 11 \beta_1 - 9) q^{11} + ( - 3 \beta_{2} - 8 \beta_1 - 9) q^{13} + (6 \beta_{2} - 2 \beta_1 - 38) q^{15} - 17 q^{17} + ( - 4 \beta_{2} + 22 \beta_1 + 28) q^{19} + ( - 15 \beta_{2} + 12 \beta_1 + 69) q^{21} + (2 \beta_{2} + 39 \beta_1 - 48) q^{23} + ( - 10 \beta_{2} + 4 \beta_1 - 71) q^{25} + (4 \beta_{2} - 40 \beta_1 - 8) q^{27} + ( - 15 \beta_{2} + 16 \beta_1 + 157) q^{29} + ( - 8 \beta_{2} - 39 \beta_1 - 74) q^{31} + ( - 17 \beta_{2} + 76 \beta_1 - 105) q^{33} + (22 \beta_{2} - 10 \beta_1 - 118) q^{35} + (25 \beta_{2} + 28 \beta_1 - 127) q^{37} + (8 \beta_{2} - 36 \beta_1 - 92) q^{39} + ( - 30 \beta_{2} - 52 \beta_1 - 88) q^{41} + (28 \beta_{2} + 2 \beta_1 + 176) q^{43} + ( - 27 \beta_{2} + 20 \beta_1 + 137) q^{45} + ( - 22 \beta_{2} + 48 \beta_1 - 206) q^{47} + ( - 47 \beta_{2} + 20 \beta_1 - 74) q^{49} + ( - 17 \beta_{2} + 17 \beta_1 - 17) q^{51} + (4 \beta_{2} - 116 \beta_1 - 102) q^{53} + (2 \beta_{2} - 18 \beta_1 - 26) q^{55} + (18 \beta_{2} + 108 \beta_1 - 246) q^{57} + ( - 130 \beta_1 + 212) q^{59} + ( - 39 \beta_{2} + 64 \beta_1 + 41) q^{61} + (48 \beta_{2} - 9 \beta_1 - 390) q^{63} + ( - 12 \beta_{2} + 28 \beta_1 + 140) q^{65} + ( - 66 \beta_{2} + 24 \beta_1 + 358) q^{67} + ( - 93 \beta_{2} + 280 \beta_1 - 161) q^{69} + ( - 36 \beta_{2} - 185 \beta_1 + 146) q^{71} + ( - 8 \beta_{2} + 16 \beta_1 + 282) q^{73} + ( - 45 \beta_{2} + 105 \beta_1 - 501) q^{75} + (9 \beta_{2} + 165) q^{77} + ( - 90 \beta_{2} + 267 \beta_1 + 228) q^{79} + (47 \beta_{2} - 20 \beta_1 - 184) q^{81} + (64 \beta_{2} + 82 \beta_1 - 820) q^{83} + (17 \beta_{2} - 51) q^{85} + (186 \beta_{2} - 46 \beta_1 - 538) q^{87} + (55 \beta_{2} - 276 \beta_1 - 75) q^{89} + (22 \beta_{2} - 64 \beta_1 - 346) q^{91} + ( - 11 \beta_{2} - 152 \beta_1 - 207) q^{93} + ( - 56 \beta_{2} - 28 \beta_1 + 176) q^{95} + (60 \beta_{2} + 140 \beta_1 - 110) q^{97} + ( - 103 \beta_{2} + 281 \beta_1 - 939) q^{99}+O(q^{100})$$ q + (b2 - b1 + 1) * q^3 + (-b2 + 3) * q^5 + (2*b2 + b1 - 8) * q^7 + (-b2 - 8*b1 + 20) * q^9 + (-b2 + 11*b1 - 9) * q^11 + (-3*b2 - 8*b1 - 9) * q^13 + (6*b2 - 2*b1 - 38) * q^15 - 17 * q^17 + (-4*b2 + 22*b1 + 28) * q^19 + (-15*b2 + 12*b1 + 69) * q^21 + (2*b2 + 39*b1 - 48) * q^23 + (-10*b2 + 4*b1 - 71) * q^25 + (4*b2 - 40*b1 - 8) * q^27 + (-15*b2 + 16*b1 + 157) * q^29 + (-8*b2 - 39*b1 - 74) * q^31 + (-17*b2 + 76*b1 - 105) * q^33 + (22*b2 - 10*b1 - 118) * q^35 + (25*b2 + 28*b1 - 127) * q^37 + (8*b2 - 36*b1 - 92) * q^39 + (-30*b2 - 52*b1 - 88) * q^41 + (28*b2 + 2*b1 + 176) * q^43 + (-27*b2 + 20*b1 + 137) * q^45 + (-22*b2 + 48*b1 - 206) * q^47 + (-47*b2 + 20*b1 - 74) * q^49 + (-17*b2 + 17*b1 - 17) * q^51 + (4*b2 - 116*b1 - 102) * q^53 + (2*b2 - 18*b1 - 26) * q^55 + (18*b2 + 108*b1 - 246) * q^57 + (-130*b1 + 212) * q^59 + (-39*b2 + 64*b1 + 41) * q^61 + (48*b2 - 9*b1 - 390) * q^63 + (-12*b2 + 28*b1 + 140) * q^65 + (-66*b2 + 24*b1 + 358) * q^67 + (-93*b2 + 280*b1 - 161) * q^69 + (-36*b2 - 185*b1 + 146) * q^71 + (-8*b2 + 16*b1 + 282) * q^73 + (-45*b2 + 105*b1 - 501) * q^75 + (9*b2 + 165) * q^77 + (-90*b2 + 267*b1 + 228) * q^79 + (47*b2 - 20*b1 - 184) * q^81 + (64*b2 + 82*b1 - 820) * q^83 + (17*b2 - 51) * q^85 + (186*b2 - 46*b1 - 538) * q^87 + (55*b2 - 276*b1 - 75) * q^89 + (22*b2 - 64*b1 - 346) * q^91 + (-11*b2 - 152*b1 - 207) * q^93 + (-56*b2 - 28*b1 + 176) * q^95 + (60*b2 + 140*b1 - 110) * q^97 + (-103*b2 + 281*b1 - 939) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 4 q^{3} + 8 q^{5} - 22 q^{7} + 59 q^{9}+O(q^{10})$$ 3 * q + 4 * q^3 + 8 * q^5 - 22 * q^7 + 59 * q^9 $$3 q + 4 q^{3} + 8 q^{5} - 22 q^{7} + 59 q^{9} - 28 q^{11} - 30 q^{13} - 108 q^{15} - 51 q^{17} + 80 q^{19} + 192 q^{21} - 142 q^{23} - 223 q^{25} - 20 q^{27} + 456 q^{29} - 230 q^{31} - 332 q^{33} - 332 q^{35} - 356 q^{37} - 268 q^{39} - 294 q^{41} + 556 q^{43} + 384 q^{45} - 640 q^{47} - 269 q^{49} - 68 q^{51} - 302 q^{53} - 76 q^{55} - 720 q^{57} + 636 q^{59} + 84 q^{61} - 1122 q^{63} + 408 q^{65} + 1008 q^{67} - 576 q^{69} + 402 q^{71} + 838 q^{73} - 1548 q^{75} + 504 q^{77} + 594 q^{79} - 505 q^{81} - 2396 q^{83} - 136 q^{85} - 1428 q^{87} - 170 q^{89} - 1016 q^{91} - 632 q^{93} + 472 q^{95} - 270 q^{97} - 2920 q^{99}+O(q^{100})$$ 3 * q + 4 * q^3 + 8 * q^5 - 22 * q^7 + 59 * q^9 - 28 * q^11 - 30 * q^13 - 108 * q^15 - 51 * q^17 + 80 * q^19 + 192 * q^21 - 142 * q^23 - 223 * q^25 - 20 * q^27 + 456 * q^29 - 230 * q^31 - 332 * q^33 - 332 * q^35 - 356 * q^37 - 268 * q^39 - 294 * q^41 + 556 * q^43 + 384 * q^45 - 640 * q^47 - 269 * q^49 - 68 * q^51 - 302 * q^53 - 76 * q^55 - 720 * q^57 + 636 * q^59 + 84 * q^61 - 1122 * q^63 + 408 * q^65 + 1008 * q^67 - 576 * q^69 + 402 * q^71 + 838 * q^73 - 1548 * q^75 + 504 * q^77 + 594 * q^79 - 505 * q^81 - 2396 * q^83 - 136 * q^85 - 1428 * q^87 - 170 * q^89 - 1016 * q^91 - 632 * q^93 + 472 * q^95 - 270 * q^97 - 2920 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 14x - 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 9$$ v^2 - 9
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 9$$ b2 + 9

## 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
 −0.287410 3.87707 −3.58966
0 −7.62999 0 11.9174 0 −26.1222 0 31.2167 0
1.2 0 3.15463 0 −3.03171 0 7.94049 0 −17.0483 0
1.3 0 8.47535 0 −0.885690 0 −3.81828 0 44.8316 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
$$2$$ $$-1$$
$$17$$ $$1$$

## 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 1088.4.a.x 3
4.b odd 2 1 1088.4.a.v 3
8.b even 2 1 272.4.a.h 3
8.d odd 2 1 17.4.a.b 3
24.f even 2 1 153.4.a.g 3
24.h odd 2 1 2448.4.a.bi 3
40.e odd 2 1 425.4.a.g 3
40.k even 4 2 425.4.b.f 6
56.e even 2 1 833.4.a.d 3
88.g even 2 1 2057.4.a.e 3
136.e odd 2 1 289.4.a.b 3
136.j odd 4 2 289.4.b.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.a.b 3 8.d odd 2 1
153.4.a.g 3 24.f even 2 1
272.4.a.h 3 8.b even 2 1
289.4.a.b 3 136.e odd 2 1
289.4.b.b 6 136.j odd 4 2
425.4.a.g 3 40.e odd 2 1
425.4.b.f 6 40.k even 4 2
833.4.a.d 3 56.e even 2 1
1088.4.a.v 3 4.b odd 2 1
1088.4.a.x 3 1.a even 1 1 trivial
2057.4.a.e 3 88.g even 2 1
2448.4.a.bi 3 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1088))$$:

 $$T_{3}^{3} - 4T_{3}^{2} - 62T_{3} + 204$$ T3^3 - 4*T3^2 - 62*T3 + 204 $$T_{5}^{3} - 8T_{5}^{2} - 44T_{5} - 32$$ T5^3 - 8*T5^2 - 44*T5 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - 4 T^{2} - 62 T + 204$$
$5$ $$T^{3} - 8 T^{2} - 44 T - 32$$
$7$ $$T^{3} + 22 T^{2} - 138 T - 792$$
$11$ $$T^{3} + 28 T^{2} - 1366 T - 4692$$
$13$ $$T^{3} + 30 T^{2} - 1472 T + 9392$$
$17$ $$(T + 17)^{3}$$
$19$ $$T^{3} - 80 T^{2} - 4632 T + 340128$$
$23$ $$T^{3} + 142 T^{2} - 15770 T - 1600544$$
$29$ $$T^{3} - 456 T^{2} + 53908 T - 1518624$$
$31$ $$T^{3} + 230 T^{2} - 11586 T + 81608$$
$37$ $$T^{3} + 356 T^{2} - 17964 T - 6176752$$
$41$ $$T^{3} + 294 T^{2} - 86564 T - 1638744$$
$43$ $$T^{3} - 556 T^{2} + 51096 T + 7270272$$
$47$ $$T^{3} + 640 T^{2} + 85328 T + 1671168$$
$53$ $$T^{3} + 302 T^{2} + \cdots - 18162072$$
$59$ $$T^{3} - 636 T^{2} + \cdots + 49419072$$
$61$ $$T^{3} - 84 T^{2} - 124412 T + 6792784$$
$67$ $$T^{3} - 1008 T^{2} + 65040 T - 765952$$
$71$ $$T^{3} - 402 T^{2} + \cdots + 274866016$$
$73$ $$T^{3} - 838 T^{2} + \cdots - 19957512$$
$79$ $$T^{3} - 594 T^{2} + \cdots + 742135824$$
$83$ $$T^{3} + 2396 T^{2} + \cdots + 142080704$$
$89$ $$T^{3} + 170 T^{2} + \cdots - 446571376$$
$97$ $$T^{3} + 270 T^{2} + \cdots - 206623000$$