# Properties

 Label 441.4.e.w Level $441$ Weight $4$ Character orbit 441.e Analytic conductor $26.020$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 441.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$26.0198423125$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.9924270768.1 Defining polynomial: $$x^{6} - x^{5} + 25 x^{4} + 12 x^{3} + 582 x^{2} - 144 x + 36$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -8 + \beta_{1} + \beta_{2} + 8 \beta_{4} + \beta_{5} ) q^{4} + ( \beta_{1} + \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{5} + ( -10 + 9 \beta_{2} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -8 + \beta_{1} + \beta_{2} + 8 \beta_{4} + \beta_{5} ) q^{4} + ( \beta_{1} + \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{5} + ( -10 + 9 \beta_{2} + \beta_{3} ) q^{8} + ( -22 - 11 \beta_{1} - 11 \beta_{2} + 22 \beta_{4} + \beta_{5} ) q^{10} + ( 12 + \beta_{1} + \beta_{2} - 12 \beta_{4} + 3 \beta_{5} ) q^{11} + ( -19 + 5 \beta_{2} - \beta_{3} ) q^{13} + ( -19 \beta_{1} + \beta_{3} - 74 \beta_{4} - \beta_{5} ) q^{16} + ( -16 + 16 \beta_{4} + 4 \beta_{5} ) q^{17} + ( -7 \beta_{1} + \beta_{3} - 65 \beta_{4} - \beta_{5} ) q^{19} + ( 150 + 11 \beta_{2} - 3 \beta_{3} ) q^{20} + ( 2 + 13 \beta_{2} + \beta_{3} ) q^{22} + ( -24 \beta_{1} + 4 \beta_{3} + 80 \beta_{4} - 4 \beta_{5} ) q^{23} + ( -53 - 29 \beta_{1} - 29 \beta_{2} + 53 \beta_{4} + \beta_{5} ) q^{25} + ( -16 \beta_{1} + 5 \beta_{3} - 86 \beta_{4} - 5 \beta_{5} ) q^{26} + ( -26 - 25 \beta_{2} + 5 \beta_{3} ) q^{29} + ( -39 - 22 \beta_{1} - 22 \beta_{2} + 39 \beta_{4} + 2 \beta_{5} ) q^{31} + ( 218 - 29 \beta_{1} - 29 \beta_{2} - 218 \beta_{4} - 11 \beta_{5} ) q^{32} + ( 24 + 48 \beta_{2} ) q^{34} + ( -19 \beta_{1} + \beta_{3} - 81 \beta_{4} - \beta_{5} ) q^{37} + ( 106 - 80 \beta_{1} - 80 \beta_{2} - 106 \beta_{4} - 7 \beta_{5} ) q^{38} + ( 75 \beta_{1} + 3 \beta_{3} - 18 \beta_{4} - 3 \beta_{5} ) q^{40} + ( 82 + 2 \beta_{2} + 14 \beta_{3} ) q^{41} + ( 143 + 69 \beta_{2} + 3 \beta_{3} ) q^{43} + ( -11 \beta_{1} - 11 \beta_{3} - 298 \beta_{4} + 11 \beta_{5} ) q^{44} + ( 360 + 24 \beta_{1} + 24 \beta_{2} - 360 \beta_{4} - 24 \beta_{5} ) q^{46} + ( 72 \beta_{1} - 28 \beta_{3} + 46 \beta_{4} + 28 \beta_{5} ) q^{47} + ( 470 + 32 \beta_{2} - 29 \beta_{3} ) q^{50} + ( 74 - 102 \beta_{1} - 102 \beta_{2} - 74 \beta_{4} - 24 \beta_{5} ) q^{52} + ( 154 + 69 \beta_{1} + 69 \beta_{2} - 154 \beta_{4} + 11 \beta_{5} ) q^{53} + ( 350 + 19 \beta_{2} + 25 \beta_{3} ) q^{55} + ( -41 \beta_{1} - 25 \beta_{3} + 430 \beta_{4} + 25 \beta_{5} ) q^{58} + ( -358 + 69 \beta_{1} + 69 \beta_{2} + 358 \beta_{4} - 29 \beta_{5} ) q^{59} + ( -100 \beta_{1} - 20 \beta_{3} + 10 \beta_{4} + 20 \beta_{5} ) q^{61} + ( 364 + 33 \beta_{2} - 22 \beta_{3} ) q^{62} + ( -194 - 183 \beta_{2} - 21 \beta_{3} ) q^{64} + ( 50 \beta_{1} - 18 \beta_{3} - 174 \beta_{4} + 18 \beta_{5} ) q^{65} + ( 215 + 17 \beta_{1} + 17 \beta_{2} - 215 \beta_{4} + 47 \beta_{5} ) q^{67} + ( -24 \beta_{1} + 16 \beta_{3} - 640 \beta_{4} - 16 \beta_{5} ) q^{68} + ( -66 + 120 \beta_{2} + 12 \beta_{3} ) q^{71} + ( 363 + 101 \beta_{1} + 101 \beta_{2} - 363 \beta_{4} + 23 \beta_{5} ) q^{73} + ( 298 - 108 \beta_{1} - 108 \beta_{2} - 298 \beta_{4} - 19 \beta_{5} ) q^{74} + ( 718 - 186 \beta_{2} - 72 \beta_{3} ) q^{76} + ( 36 \beta_{1} + 48 \beta_{3} - 299 \beta_{4} - 48 \beta_{5} ) q^{79} + ( -18 + 121 \beta_{1} + 121 \beta_{2} + 18 \beta_{4} + 51 \beta_{5} ) q^{80} + ( -32 \beta_{1} + 2 \beta_{3} + 52 \beta_{4} - 2 \beta_{5} ) q^{82} + ( 156 - 51 \beta_{2} + 27 \beta_{3} ) q^{83} + ( 624 + 72 \beta_{2} ) q^{85} + ( 50 \beta_{1} + 69 \beta_{3} - 1086 \beta_{4} - 69 \beta_{5} ) q^{86} + ( 258 - 117 \beta_{1} - 117 \beta_{2} - 258 \beta_{4} - 3 \beta_{5} ) q^{88} + ( -170 \beta_{1} + 22 \beta_{3} - 532 \beta_{4} - 22 \beta_{5} ) q^{89} + ( 112 - 336 \beta_{2} + 56 \beta_{3} ) q^{92} + ( -984 + 342 \beta_{1} + 342 \beta_{2} + 984 \beta_{4} + 72 \beta_{5} ) q^{94} + ( -246 - 2 \beta_{1} - 2 \beta_{2} + 246 \beta_{4} + 54 \beta_{5} ) q^{95} + ( -24 - 53 \beta_{2} + 49 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + q^{2} - 25q^{4} - 11q^{5} - 78q^{8} + O(q^{10})$$ $$6q + q^{2} - 25q^{4} - 11q^{5} - 78q^{8} - 55q^{10} + 35q^{11} - 124q^{13} - 241q^{16} - 48q^{17} - 202q^{19} + 878q^{20} - 14q^{22} + 216q^{23} - 130q^{25} - 274q^{26} - 106q^{29} - 95q^{31} + 683q^{32} + 48q^{34} - 262q^{37} + 398q^{38} + 21q^{40} + 488q^{41} + 720q^{43} - 905q^{44} + 1056q^{46} + 210q^{47} + 2756q^{50} + 324q^{52} + 393q^{53} + 2062q^{55} + 1249q^{58} - 1143q^{59} - 70q^{61} + 2118q^{62} - 798q^{64} - 472q^{65} + 628q^{67} - 1944q^{68} - 636q^{71} + 988q^{73} + 1002q^{74} + 4680q^{76} - 861q^{79} - 175q^{80} + 124q^{82} + 1038q^{83} + 3600q^{85} - 3208q^{86} + 891q^{88} - 1766q^{89} + 1344q^{92} - 3294q^{94} - 736q^{95} - 38q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 25 x^{4} + 12 x^{3} + 582 x^{2} - 144 x + 36$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - 25 \nu^{4} + 625 \nu^{3} - 582 \nu^{2} + 144 \nu - 3600$$$$)/14406$$ $$\beta_{3}$$ $$=$$ $$($$$$-25 \nu^{5} + 625 \nu^{4} - 1219 \nu^{3} + 14550 \nu^{2} - 3600 \nu + 234060$$$$)/14406$$ $$\beta_{4}$$ $$=$$ $$($$$$100 \nu^{5} - 99 \nu^{4} + 2475 \nu^{3} + 1825 \nu^{2} + 57618 \nu + 150$$$$)/14406$$ $$\beta_{5}$$ $$=$$ $$($$$$-1601 \nu^{5} + 1609 \nu^{4} - 40225 \nu^{3} - 14212 \nu^{2} - 936438 \nu + 231696$$$$)/14406$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + 16 \beta_{4} + \beta_{2} + \beta_{1} - 16$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 25 \beta_{2} - 10$$ $$\nu^{4}$$ $$=$$ $$-25 \beta_{5} - 394 \beta_{4} + 25 \beta_{3} - 43 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-43 \beta_{5} - 538 \beta_{4} - 637 \beta_{2} - 637 \beta_{1} + 538$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$-\beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
226.1
 −2.27818 + 3.94593i 0.124036 − 0.214837i 2.65415 − 4.59712i −2.27818 − 3.94593i 0.124036 + 0.214837i 2.65415 + 4.59712i
−2.27818 + 3.94593i 0 −6.38024 11.0509i −8.93660 + 15.4786i 0 0 21.6905 0 −40.7184 70.5264i
226.2 0.124036 0.214837i 0 3.96923 + 6.87491i 6.21730 10.7687i 0 0 3.95388 0 −1.54234 2.67141i
226.3 2.65415 4.59712i 0 −10.0890 17.4746i −2.78070 + 4.81631i 0 0 −64.6443 0 14.7608 + 25.5664i
361.1 −2.27818 3.94593i 0 −6.38024 + 11.0509i −8.93660 15.4786i 0 0 21.6905 0 −40.7184 + 70.5264i
361.2 0.124036 + 0.214837i 0 3.96923 6.87491i 6.21730 + 10.7687i 0 0 3.95388 0 −1.54234 + 2.67141i
361.3 2.65415 + 4.59712i 0 −10.0890 + 17.4746i −2.78070 4.81631i 0 0 −64.6443 0 14.7608 25.5664i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 361.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{2}^{6} - T_{2}^{5} + 25 T_{2}^{4} + 12 T_{2}^{3} + 582 T_{2}^{2} - 144 T_{2} + 36$$ $$T_{5}^{6} + 11 T_{5}^{5} + 313 T_{5}^{4} + 360 T_{5}^{3} + 50460 T_{5}^{2} + 237312 T_{5} + 1527696$$ $$T_{13}^{3} + 62 T_{13}^{2} + 425 T_{13} - 18452$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$36 - 144 T + 582 T^{2} + 12 T^{3} + 25 T^{4} - T^{5} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$1527696 + 237312 T + 50460 T^{2} + 360 T^{3} + 313 T^{4} + 11 T^{5} + T^{6}$$
$7$ $$T^{6}$$
$11$ $$91470096 + 13083552 T + 1536684 T^{2} + 67008 T^{3} + 2593 T^{4} - 35 T^{5} + T^{6}$$
$13$ $$( -18452 + 425 T + 62 T^{2} + T^{3} )^{2}$$
$17$ $$12745506816 + 270950400 T + 11179008 T^{2} + 110592 T^{3} + 4704 T^{4} + 48 T^{5} + T^{6}$$
$19$ $$54664310416 + 2871346924 T + 103594553 T^{2} + 2013154 T^{3} + 28523 T^{4} + 202 T^{5} + T^{6}$$
$23$ $$2498119335936 - 1062125568 T + 341849088 T^{2} - 3015936 T^{3} + 47328 T^{4} - 216 T^{5} + T^{6}$$
$29$ $$( 824976 - 20472 T + 53 T^{2} + T^{3} )^{2}$$
$31$ $$139783329 + 118241823 T + 101143186 T^{2} - 926449 T^{3} + 19026 T^{4} + 95 T^{5} + T^{6}$$
$37$ $$2415919104 + 692502528 T + 185622097 T^{2} + 3593014 T^{3} + 54555 T^{4} + 262 T^{5} + T^{6}$$
$41$ $$( -300384 - 18780 T - 244 T^{2} + T^{3} )^{2}$$
$43$ $$( 18269746 - 72363 T - 360 T^{2} + T^{3} )^{2}$$
$47$ $$26205471480384 + 1261946958048 T + 59695121376 T^{2} + 62006616 T^{3} + 290616 T^{4} - 210 T^{5} + T^{6}$$
$53$ $$1100208565649664 - 2677964032512 T + 19553872752 T^{2} - 34609536 T^{3} + 235185 T^{4} - 393 T^{5} + T^{6}$$
$59$ $$10094008708475136 - 13372818322176 T + 132552677808 T^{2} + 353075760 T^{3} + 1173345 T^{4} + 1143 T^{5} + T^{6}$$
$61$ $$7162406161000000 + 28850707900000 T + 122136980000 T^{2} + 145399000 T^{3} + 345800 T^{4} + 70 T^{5} + T^{6}$$
$67$ $$783608160972004 + 8536829868926 T + 75422826113 T^{2} + 247502768 T^{3} + 699347 T^{4} - 628 T^{5} + T^{6}$$
$71$ $$( 28535976 - 330804 T + 318 T^{2} + T^{3} )^{2}$$
$73$ $$20508278645865924 - 623666998890 T + 141507598609 T^{2} - 282111496 T^{3} + 980499 T^{4} - 988 T^{5} + T^{6}$$
$79$ $$37619060662457569 + 50021533268637 T + 233509331958 T^{2} + 165859913 T^{3} + 999222 T^{4} + 861 T^{5} + T^{6}$$
$83$ $$( 47916036 - 131616 T - 519 T^{2} + T^{3} )^{2}$$
$89$ $$169118164647936 - 3614222868480 T + 100205551104 T^{2} + 516815808 T^{3} + 2840836 T^{4} + 1766 T^{5} + T^{6}$$
$97$ $$( -44776452 - 569600 T + 19 T^{2} + T^{3} )^{2}$$