# Properties

 Label 441.4.a.s Level $441$ Weight $4$ Character orbit 441.a Self dual yes Analytic conductor $26.020$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 441.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.0198423125$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.57516.1 Defining polynomial: $$x^{3} - x^{2} - 24 x + 6$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 8 + \beta_{1} + \beta_{2} ) q^{4} + ( -4 + \beta_{1} - \beta_{2} ) q^{5} + ( -10 - 9 \beta_{1} - \beta_{2} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 8 + \beta_{1} + \beta_{2} ) q^{4} + ( -4 + \beta_{1} - \beta_{2} ) q^{5} + ( -10 - 9 \beta_{1} - \beta_{2} ) q^{8} + ( -22 + 11 \beta_{1} - \beta_{2} ) q^{10} + ( -12 + \beta_{1} + 3 \beta_{2} ) q^{11} + ( 19 + 5 \beta_{1} - \beta_{2} ) q^{13} + ( 74 + 19 \beta_{1} + \beta_{2} ) q^{16} + ( -16 - 4 \beta_{2} ) q^{17} + ( -65 - 7 \beta_{1} - \beta_{2} ) q^{19} + ( -150 + 11 \beta_{1} - 3 \beta_{2} ) q^{20} + ( 2 - 13 \beta_{1} - \beta_{2} ) q^{22} + ( -80 + 24 \beta_{1} + 4 \beta_{2} ) q^{23} + ( 53 - 29 \beta_{1} + \beta_{2} ) q^{25} + ( -86 - 16 \beta_{1} - 5 \beta_{2} ) q^{26} + ( -26 + 25 \beta_{1} - 5 \beta_{2} ) q^{29} + ( -39 + 22 \beta_{1} - 2 \beta_{2} ) q^{31} + ( -218 - 29 \beta_{1} - 11 \beta_{2} ) q^{32} + ( -24 + 48 \beta_{1} ) q^{34} + ( 81 + 19 \beta_{1} + \beta_{2} ) q^{37} + ( 106 + 80 \beta_{1} + 7 \beta_{2} ) q^{38} + ( -18 + 75 \beta_{1} - 3 \beta_{2} ) q^{40} + ( -82 + 2 \beta_{1} + 14 \beta_{2} ) q^{41} + ( 143 - 69 \beta_{1} - 3 \beta_{2} ) q^{43} + ( 298 + 11 \beta_{1} - 11 \beta_{2} ) q^{44} + ( -360 + 24 \beta_{1} - 24 \beta_{2} ) q^{46} + ( 46 + 72 \beta_{1} + 28 \beta_{2} ) q^{47} + ( 470 - 32 \beta_{1} + 29 \beta_{2} ) q^{50} + ( 74 + 102 \beta_{1} + 24 \beta_{2} ) q^{52} + ( -154 + 69 \beta_{1} + 11 \beta_{2} ) q^{53} + ( -350 + 19 \beta_{1} + 25 \beta_{2} ) q^{55} + ( -430 + 41 \beta_{1} - 25 \beta_{2} ) q^{58} + ( -358 - 69 \beta_{1} + 29 \beta_{2} ) q^{59} + ( 10 - 100 \beta_{1} + 20 \beta_{2} ) q^{61} + ( -364 + 33 \beta_{1} - 22 \beta_{2} ) q^{62} + ( -194 + 183 \beta_{1} + 21 \beta_{2} ) q^{64} + ( 174 - 50 \beta_{1} - 18 \beta_{2} ) q^{65} + ( -215 + 17 \beta_{1} + 47 \beta_{2} ) q^{67} + ( -640 - 24 \beta_{1} - 16 \beta_{2} ) q^{68} + ( -66 - 120 \beta_{1} - 12 \beta_{2} ) q^{71} + ( 363 - 101 \beta_{1} - 23 \beta_{2} ) q^{73} + ( -298 - 108 \beta_{1} - 19 \beta_{2} ) q^{74} + ( -718 - 186 \beta_{1} - 72 \beta_{2} ) q^{76} + ( 299 - 36 \beta_{1} + 48 \beta_{2} ) q^{79} + ( -18 - 121 \beta_{1} - 51 \beta_{2} ) q^{80} + ( 52 - 32 \beta_{1} - 2 \beta_{2} ) q^{82} + ( -156 - 51 \beta_{1} + 27 \beta_{2} ) q^{83} + ( 624 - 72 \beta_{1} ) q^{85} + ( 1086 - 50 \beta_{1} + 69 \beta_{2} ) q^{86} + ( -258 - 117 \beta_{1} - 3 \beta_{2} ) q^{88} + ( -532 - 170 \beta_{1} - 22 \beta_{2} ) q^{89} + ( 112 + 336 \beta_{1} - 56 \beta_{2} ) q^{92} + ( -984 - 342 \beta_{1} - 72 \beta_{2} ) q^{94} + ( 246 - 2 \beta_{1} + 54 \beta_{2} ) q^{95} + ( 24 - 53 \beta_{1} + 49 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - q^{2} + 25q^{4} - 11q^{5} - 39q^{8} + O(q^{10})$$ $$3q - q^{2} + 25q^{4} - 11q^{5} - 39q^{8} - 55q^{10} - 35q^{11} + 62q^{13} + 241q^{16} - 48q^{17} - 202q^{19} - 439q^{20} - 7q^{22} - 216q^{23} + 130q^{25} - 274q^{26} - 53q^{29} - 95q^{31} - 683q^{32} - 24q^{34} + 262q^{37} + 398q^{38} + 21q^{40} - 244q^{41} + 360q^{43} + 905q^{44} - 1056q^{46} + 210q^{47} + 1378q^{50} + 324q^{52} - 393q^{53} - 1031q^{55} - 1249q^{58} - 1143q^{59} - 70q^{61} - 1059q^{62} - 399q^{64} + 472q^{65} - 628q^{67} - 1944q^{68} - 318q^{71} + 988q^{73} - 1002q^{74} - 2340q^{76} + 861q^{79} - 175q^{80} + 124q^{82} - 519q^{83} + 1800q^{85} + 3208q^{86} - 891q^{88} - 1766q^{89} + 672q^{92} - 3294q^{94} + 736q^{95} + 19q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 24 x + 6$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 16$$

## 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
 5.30829 0.248072 −4.55637
−5.30829 0 20.1780 −5.56140 0 0 −64.6443 0 29.5215
1.2 −0.248072 0 −7.93846 12.4346 0 0 3.95388 0 −3.08468
1.3 4.55637 0 12.7605 −17.8732 0 0 21.6905 0 −81.4369
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$1$$

## 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 441.4.a.s 3
3.b odd 2 1 147.4.a.l 3
7.b odd 2 1 441.4.a.t 3
7.c even 3 2 63.4.e.c 6
7.d odd 6 2 441.4.e.w 6
12.b even 2 1 2352.4.a.ci 3
21.c even 2 1 147.4.a.m 3
21.g even 6 2 147.4.e.n 6
21.h odd 6 2 21.4.e.b 6
84.h odd 2 1 2352.4.a.cg 3
84.n even 6 2 336.4.q.k 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.b 6 21.h odd 6 2
63.4.e.c 6 7.c even 3 2
147.4.a.l 3 3.b odd 2 1
147.4.a.m 3 21.c even 2 1
147.4.e.n 6 21.g even 6 2
336.4.q.k 6 84.n even 6 2
441.4.a.s 3 1.a even 1 1 trivial
441.4.a.t 3 7.b odd 2 1
441.4.e.w 6 7.d odd 6 2
2352.4.a.cg 3 84.h odd 2 1
2352.4.a.ci 3 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2}^{3} + T_{2}^{2} - 24 T_{2} - 6$$ $$T_{5}^{3} + 11 T_{5}^{2} - 192 T_{5} - 1236$$ $$T_{13}^{3} - 62 T_{13}^{2} + 425 T_{13} + 18452$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-6 - 24 T + T^{2} + T^{3}$$
$3$ $$T^{3}$$
$5$ $$-1236 - 192 T + 11 T^{2} + T^{3}$$
$7$ $$T^{3}$$
$11$ $$9564 - 1368 T + 35 T^{2} + T^{3}$$
$13$ $$18452 + 425 T - 62 T^{2} + T^{3}$$
$17$ $$-112896 - 2400 T + 48 T^{2} + T^{3}$$
$19$ $$233804 + 12281 T + 202 T^{2} + T^{3}$$
$23$ $$-1580544 - 672 T + 216 T^{2} + T^{3}$$
$29$ $$824976 - 20472 T + 53 T^{2} + T^{3}$$
$31$ $$-11823 - 10001 T + 95 T^{2} + T^{3}$$
$37$ $$-49152 + 14089 T - 262 T^{2} + T^{3}$$
$41$ $$300384 - 18780 T + 244 T^{2} + T^{3}$$
$43$ $$18269746 - 72363 T - 360 T^{2} + T^{3}$$
$47$ $$-5119128 - 246516 T - 210 T^{2} + T^{3}$$
$53$ $$-33169392 - 80736 T + 393 T^{2} + T^{3}$$
$59$ $$-100468944 + 133104 T + 1143 T^{2} + T^{3}$$
$61$ $$-84631000 - 340900 T + 70 T^{2} + T^{3}$$
$67$ $$27993002 - 304963 T + 628 T^{2} + T^{3}$$
$71$ $$28535976 - 330804 T + 318 T^{2} + T^{3}$$
$73$ $$143207118 - 4355 T - 988 T^{2} + T^{3}$$
$79$ $$193956337 - 257901 T - 861 T^{2} + T^{3}$$
$83$ $$-47916036 - 131616 T + 519 T^{2} + T^{3}$$
$89$ $$-13004544 + 277920 T + 1766 T^{2} + T^{3}$$
$97$ $$44776452 - 569600 T - 19 T^{2} + T^{3}$$