# Properties

 Label 153.4.a.g Level $153$ Weight $4$ Character orbit 153.a Self dual yes Analytic conductor $9.027$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$153 = 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 153.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.02729223088$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.2636.1 Defining polynomial: $$x^{3} - 14x - 4$$ x^3 - 14*x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - \beta_1) q^{2} + ( - \beta_{2} - 3 \beta_1 + 8) q^{4} + ( - 2 \beta_{2} + 2) q^{5} + ( - 4 \beta_{2} - \beta_1 + 6) q^{7} + (9 \beta_{2} - 5 \beta_1 + 16) q^{8}+O(q^{10})$$ q + (b2 - b1) * q^2 + (-b2 - 3*b1 + 8) * q^4 + (-2*b2 + 2) * q^5 + (-4*b2 - b1 + 6) * q^7 + (9*b2 - 5*b1 + 16) * q^8 $$q + (\beta_{2} - \beta_1) q^{2} + ( - \beta_{2} - 3 \beta_1 + 8) q^{4} + ( - 2 \beta_{2} + 2) q^{5} + ( - 4 \beta_{2} - \beta_1 + 6) q^{7} + (9 \beta_{2} - 5 \beta_1 + 16) q^{8} + (8 \beta_{2} - 16) q^{10} + (2 \beta_{2} - 11 \beta_1 + 10) q^{11} + (6 \beta_{2} + 8 \beta_1 + 12) q^{13} + (20 \beta_{2} - 4 \beta_1 - 24) q^{14} + (7 \beta_{2} - 11 \beta_1 + 48) q^{16} + 17 q^{17} + ( - 8 \beta_{2} + 22 \beta_1 + 24) q^{19} + ( - 24 \beta_{2} + 8 \beta_1 + 48) q^{20} + (26 \beta_{2} - 34 \beta_1 + 104) q^{22} + (4 \beta_{2} + 39 \beta_1 - 46) q^{23} + ( - 20 \beta_{2} + 4 \beta_1 - 81) q^{25} + ( - 22 \beta_{2} - 2 \beta_1 - 16) q^{26} + ( - 44 \beta_{2} + 4 \beta_1 + 144) q^{28} + ( - 30 \beta_{2} + 16 \beta_1 + 142) q^{29} + (16 \beta_{2} + 39 \beta_1 + 82) q^{31} + ( - 23 \beta_{2} - 37 \beta_1 + 16) q^{32} + (17 \beta_{2} - 17 \beta_1) q^{34} + ( - 44 \beta_{2} + 10 \beta_1 + 96) q^{35} + ( - 50 \beta_{2} - 28 \beta_1 + 102) q^{37} + (4 \beta_{2} + 28 \beta_1 - 240) q^{38} + (40 \beta_{2} - 8 \beta_1 - 128) q^{40} + (60 \beta_{2} + 52 \beta_1 + 118) q^{41} + (56 \beta_{2} + 2 \beta_1 + 204) q^{43} + (78 \beta_{2} - 110 \beta_1 + 400) q^{44} + ( - 136 \beta_{2} + 120 \beta_1 - 280) q^{46} + ( - 44 \beta_{2} + 48 \beta_1 - 228) q^{47} + ( - 94 \beta_{2} + 20 \beta_1 - 121) q^{49} + ( - 29 \beta_{2} + 109 \beta_1 - 192) q^{50} + (6 \beta_{2} - 30 \beta_1 - 256) q^{52} + (8 \beta_{2} - 116 \beta_1 - 98) q^{53} + ( - 4 \beta_{2} + 18 \beta_1 + 24) q^{55} + (108 \beta_{2} - 60 \beta_1 - 192) q^{56} + (200 \beta_{2} - 80 \beta_1 - 368) q^{58} + (130 \beta_1 - 212) q^{59} + (78 \beta_{2} - 64 \beta_1 - 2) q^{61} + ( - 44 \beta_{2} - 20 \beta_1 - 184) q^{62} + (103 \beta_{2} + 21 \beta_1 - 272) q^{64} + (24 \beta_{2} - 28 \beta_1 - 128) q^{65} + ( - 132 \beta_{2} + 24 \beta_1 + 292) q^{67} + ( - 17 \beta_{2} - 51 \beta_1 + 136) q^{68} + (208 \beta_{2} - 32 \beta_1 - 432) q^{70} + ( - 72 \beta_{2} - 185 \beta_1 + 110) q^{71} + ( - 16 \beta_{2} + 16 \beta_1 + 274) q^{73} + (308 \beta_{2} - 108 \beta_1 - 176) q^{74} + ( - 244 \beta_{2} + 116 \beta_1 - 384) q^{76} + (18 \beta_{2} + 174) q^{77} + (180 \beta_{2} - 267 \beta_1 - 138) q^{79} + ( - 40 \beta_{2} + 8 \beta_1) q^{80} + ( - 166 \beta_{2} - 74 \beta_1 + 64) q^{82} + ( - 128 \beta_{2} - 82 \beta_1 + 756) q^{83} + ( - 34 \beta_{2} + 34) q^{85} + (32 \beta_{2} - 256 \beta_1 + 432) q^{86} + (178 \beta_{2} - 426 \beta_1 + 672) q^{88} + ( - 110 \beta_{2} + 276 \beta_1 + 20) q^{89} + (44 \beta_{2} - 64 \beta_1 - 324) q^{91} + ( - 144 \beta_{2} + 344 \beta_1 - 1680) q^{92} + ( - 192 \beta_{2} + 368 \beta_1 - 736) q^{94} + ( - 112 \beta_{2} - 28 \beta_1 + 120) q^{95} + (120 \beta_{2} + 140 \beta_1 - 50) q^{97} + (121 \beta_{2} + 255 \beta_1 - 912) q^{98}+O(q^{100})$$ q + (b2 - b1) * q^2 + (-b2 - 3*b1 + 8) * q^4 + (-2*b2 + 2) * q^5 + (-4*b2 - b1 + 6) * q^7 + (9*b2 - 5*b1 + 16) * q^8 + (8*b2 - 16) * q^10 + (2*b2 - 11*b1 + 10) * q^11 + (6*b2 + 8*b1 + 12) * q^13 + (20*b2 - 4*b1 - 24) * q^14 + (7*b2 - 11*b1 + 48) * q^16 + 17 * q^17 + (-8*b2 + 22*b1 + 24) * q^19 + (-24*b2 + 8*b1 + 48) * q^20 + (26*b2 - 34*b1 + 104) * q^22 + (4*b2 + 39*b1 - 46) * q^23 + (-20*b2 + 4*b1 - 81) * q^25 + (-22*b2 - 2*b1 - 16) * q^26 + (-44*b2 + 4*b1 + 144) * q^28 + (-30*b2 + 16*b1 + 142) * q^29 + (16*b2 + 39*b1 + 82) * q^31 + (-23*b2 - 37*b1 + 16) * q^32 + (17*b2 - 17*b1) * q^34 + (-44*b2 + 10*b1 + 96) * q^35 + (-50*b2 - 28*b1 + 102) * q^37 + (4*b2 + 28*b1 - 240) * q^38 + (40*b2 - 8*b1 - 128) * q^40 + (60*b2 + 52*b1 + 118) * q^41 + (56*b2 + 2*b1 + 204) * q^43 + (78*b2 - 110*b1 + 400) * q^44 + (-136*b2 + 120*b1 - 280) * q^46 + (-44*b2 + 48*b1 - 228) * q^47 + (-94*b2 + 20*b1 - 121) * q^49 + (-29*b2 + 109*b1 - 192) * q^50 + (6*b2 - 30*b1 - 256) * q^52 + (8*b2 - 116*b1 - 98) * q^53 + (-4*b2 + 18*b1 + 24) * q^55 + (108*b2 - 60*b1 - 192) * q^56 + (200*b2 - 80*b1 - 368) * q^58 + (130*b1 - 212) * q^59 + (78*b2 - 64*b1 - 2) * q^61 + (-44*b2 - 20*b1 - 184) * q^62 + (103*b2 + 21*b1 - 272) * q^64 + (24*b2 - 28*b1 - 128) * q^65 + (-132*b2 + 24*b1 + 292) * q^67 + (-17*b2 - 51*b1 + 136) * q^68 + (208*b2 - 32*b1 - 432) * q^70 + (-72*b2 - 185*b1 + 110) * q^71 + (-16*b2 + 16*b1 + 274) * q^73 + (308*b2 - 108*b1 - 176) * q^74 + (-244*b2 + 116*b1 - 384) * q^76 + (18*b2 + 174) * q^77 + (180*b2 - 267*b1 - 138) * q^79 + (-40*b2 + 8*b1) * q^80 + (-166*b2 - 74*b1 + 64) * q^82 + (-128*b2 - 82*b1 + 756) * q^83 + (-34*b2 + 34) * q^85 + (32*b2 - 256*b1 + 432) * q^86 + (178*b2 - 426*b1 + 672) * q^88 + (-110*b2 + 276*b1 + 20) * q^89 + (44*b2 - 64*b1 - 324) * q^91 + (-144*b2 + 344*b1 - 1680) * q^92 + (-192*b2 + 368*b1 - 736) * q^94 + (-112*b2 - 28*b1 + 120) * q^95 + (120*b2 + 140*b1 - 50) * q^97 + (121*b2 + 255*b1 - 912) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 25 q^{4} + 8 q^{5} + 22 q^{7} + 39 q^{8}+O(q^{10})$$ 3 * q - q^2 + 25 * q^4 + 8 * q^5 + 22 * q^7 + 39 * q^8 $$3 q - q^{2} + 25 q^{4} + 8 q^{5} + 22 q^{7} + 39 q^{8} - 56 q^{10} + 28 q^{11} + 30 q^{13} - 92 q^{14} + 137 q^{16} + 51 q^{17} + 80 q^{19} + 168 q^{20} + 286 q^{22} - 142 q^{23} - 223 q^{25} - 26 q^{26} + 476 q^{28} + 456 q^{29} + 230 q^{31} + 71 q^{32} - 17 q^{34} + 332 q^{35} + 356 q^{37} - 724 q^{38} - 424 q^{40} + 294 q^{41} + 556 q^{43} + 1122 q^{44} - 704 q^{46} - 640 q^{47} - 269 q^{49} - 547 q^{50} - 774 q^{52} - 302 q^{53} + 76 q^{55} - 684 q^{56} - 1304 q^{58} - 636 q^{59} - 84 q^{61} - 508 q^{62} - 919 q^{64} - 408 q^{65} + 1008 q^{67} + 425 q^{68} - 1504 q^{70} + 402 q^{71} + 838 q^{73} - 836 q^{74} - 908 q^{76} + 504 q^{77} - 594 q^{79} + 40 q^{80} + 358 q^{82} + 2396 q^{83} + 136 q^{85} + 1264 q^{86} + 1838 q^{88} + 170 q^{89} - 1016 q^{91} - 4896 q^{92} - 2016 q^{94} + 472 q^{95} - 270 q^{97} - 2857 q^{98}+O(q^{100})$$ 3 * q - q^2 + 25 * q^4 + 8 * q^5 + 22 * q^7 + 39 * q^8 - 56 * q^10 + 28 * q^11 + 30 * q^13 - 92 * q^14 + 137 * q^16 + 51 * q^17 + 80 * q^19 + 168 * q^20 + 286 * q^22 - 142 * q^23 - 223 * q^25 - 26 * q^26 + 476 * q^28 + 456 * q^29 + 230 * q^31 + 71 * q^32 - 17 * q^34 + 332 * q^35 + 356 * q^37 - 724 * q^38 - 424 * q^40 + 294 * q^41 + 556 * q^43 + 1122 * q^44 - 704 * q^46 - 640 * q^47 - 269 * q^49 - 547 * q^50 - 774 * q^52 - 302 * q^53 + 76 * q^55 - 684 * q^56 - 1304 * q^58 - 636 * q^59 - 84 * q^61 - 508 * q^62 - 919 * q^64 - 408 * q^65 + 1008 * q^67 + 425 * q^68 - 1504 * q^70 + 402 * q^71 + 838 * q^73 - 836 * q^74 - 908 * q^76 + 504 * q^77 - 594 * q^79 + 40 * q^80 + 358 * q^82 + 2396 * q^83 + 136 * q^85 + 1264 * q^86 + 1838 * q^88 + 170 * q^89 - 1016 * q^91 - 4896 * q^92 - 2016 * q^94 + 472 * q^95 - 270 * q^97 - 2857 * q^98

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} - 10 ) / 2$$ (v^2 - 10) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2} + 10$$ 2*b2 + 10

## 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
−4.67129 0 13.8209 11.9174 0 26.1222 −27.1912 0 −55.6696
1.2 −1.36122 0 −6.14708 −3.03171 0 −7.94049 19.2573 0 4.12682
1.3 5.03251 0 17.3261 −0.885690 0 3.81828 46.9339 0 −4.45724
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$17$$ $$-1$$

## 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 153.4.a.g 3
3.b odd 2 1 17.4.a.b 3
4.b odd 2 1 2448.4.a.bi 3
12.b even 2 1 272.4.a.h 3
15.d odd 2 1 425.4.a.g 3
15.e even 4 2 425.4.b.f 6
21.c even 2 1 833.4.a.d 3
24.f even 2 1 1088.4.a.x 3
24.h odd 2 1 1088.4.a.v 3
33.d even 2 1 2057.4.a.e 3
51.c odd 2 1 289.4.a.b 3
51.f 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 3.b odd 2 1
153.4.a.g 3 1.a even 1 1 trivial
272.4.a.h 3 12.b even 2 1
289.4.a.b 3 51.c odd 2 1
289.4.b.b 6 51.f odd 4 2
425.4.a.g 3 15.d odd 2 1
425.4.b.f 6 15.e even 4 2
833.4.a.d 3 21.c even 2 1
1088.4.a.v 3 24.h odd 2 1
1088.4.a.x 3 24.f even 2 1
2057.4.a.e 3 33.d even 2 1
2448.4.a.bi 3 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(153))$$:

 $$T_{2}^{3} + T_{2}^{2} - 24T_{2} - 32$$ T2^3 + T2^2 - 24*T2 - 32 $$T_{5}^{3} - 8T_{5}^{2} - 44T_{5} - 32$$ T5^3 - 8*T5^2 - 44*T5 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + T^{2} - 24 T - 32$$
$3$ $$T^{3}$$
$5$ $$T^{3} - 8 T^{2} - 44 T - 32$$
$7$ $$T^{3} - 22 T^{2} - 138 T + 792$$
$11$ $$T^{3} - 28 T^{2} - 1366 T + 4692$$
$13$ $$T^{3} - 30 T^{2} - 1472 T - 9392$$
$17$ $$(T - 17)^{3}$$
$19$ $$T^{3} - 80 T^{2} - 4632 T + 340128$$
$23$ $$T^{3} + 142 T^{2} - 15770 T - 1600544$$
$29$ $$T^{3} - 456 T^{2} + 53908 T - 1518624$$
$31$ $$T^{3} - 230 T^{2} - 11586 T - 81608$$
$37$ $$T^{3} - 356 T^{2} - 17964 T + 6176752$$
$41$ $$T^{3} - 294 T^{2} - 86564 T + 1638744$$
$43$ $$T^{3} - 556 T^{2} + 51096 T + 7270272$$
$47$ $$T^{3} + 640 T^{2} + 85328 T + 1671168$$
$53$ $$T^{3} + 302 T^{2} + \cdots - 18162072$$
$59$ $$T^{3} + 636 T^{2} + \cdots - 49419072$$
$61$ $$T^{3} + 84 T^{2} - 124412 T - 6792784$$
$67$ $$T^{3} - 1008 T^{2} + 65040 T - 765952$$
$71$ $$T^{3} - 402 T^{2} + \cdots + 274866016$$
$73$ $$T^{3} - 838 T^{2} + \cdots - 19957512$$
$79$ $$T^{3} + 594 T^{2} + \cdots - 742135824$$
$83$ $$T^{3} - 2396 T^{2} + \cdots - 142080704$$
$89$ $$T^{3} - 170 T^{2} + \cdots + 446571376$$
$97$ $$T^{3} + 270 T^{2} + \cdots - 206623000$$