# Properties

 Label 441.4.e.y Level $441$ Weight $4$ Character orbit 441.e Analytic conductor $26.020$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.5922408960000.19 Defining polynomial: $$x^{8} - 4 x^{7} - 54 x^{6} + 176 x^{5} + 1307 x^{4} - 2912 x^{3} - 15314 x^{2} + 16800 x + 86044$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$7^{2}$$ Twist minimal: no (minimal twist has level 49) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{2} + \beta_{6} ) q^{2} + ( -8 - 8 \beta_{1} + \beta_{6} ) q^{4} + ( \beta_{4} + \beta_{7} ) q^{5} + ( -17 + \beta_{2} ) q^{8} +O(q^{10})$$ $$q + ( 1 - \beta_{2} + \beta_{6} ) q^{2} + ( -8 - 8 \beta_{1} + \beta_{6} ) q^{4} + ( \beta_{4} + \beta_{7} ) q^{5} + ( -17 + \beta_{2} ) q^{8} + ( 2 \beta_{3} + 4 \beta_{4} ) q^{10} + ( 28 + 28 \beta_{1} + 6 \beta_{6} ) q^{11} + ( 6 \beta_{3} - 6 \beta_{5} - \beta_{7} ) q^{13} + ( -9 - 48 \beta_{1} + 9 \beta_{2} - 9 \beta_{6} ) q^{16} + ( -\beta_{3} + 9 \beta_{4} ) q^{17} + ( -4 \beta_{4} - 11 \beta_{5} - 4 \beta_{7} ) q^{19} + ( 2 \beta_{3} - 2 \beta_{5} - 12 \beta_{7} ) q^{20} + ( -74 - 22 \beta_{2} ) q^{22} + ( 8 - 84 \beta_{1} - 8 \beta_{2} + 8 \beta_{6} ) q^{23} + ( 43 + 43 \beta_{1} + 22 \beta_{6} ) q^{25} + ( -16 \beta_{4} + 16 \beta_{5} - 16 \beta_{7} ) q^{26} + ( -58 - 14 \beta_{2} ) q^{29} + ( 38 \beta_{3} - 4 \beta_{4} ) q^{31} + ( 16 + 16 \beta_{1} + 47 \beta_{6} ) q^{32} + ( 21 \beta_{3} - 21 \beta_{5} - 34 \beta_{7} ) q^{34} + ( 6 + 56 \beta_{1} - 6 \beta_{2} + 6 \beta_{6} ) q^{37} + ( 25 \beta_{3} - 38 \beta_{4} ) q^{38} + ( -20 \beta_{4} - 2 \beta_{5} - 20 \beta_{7} ) q^{40} + ( 31 \beta_{3} - 31 \beta_{5} + 3 \beta_{7} ) q^{41} + ( 170 - 70 \beta_{2} ) q^{43} + ( -26 - 128 \beta_{1} + 26 \beta_{2} - 26 \beta_{6} ) q^{44} + ( -128 - 128 \beta_{1} + 92 \beta_{6} ) q^{46} + ( -32 \beta_{4} + 26 \beta_{5} - 32 \beta_{7} ) q^{47} + ( -331 - 21 \beta_{2} ) q^{50} + ( -32 \beta_{3} - 24 \beta_{4} ) q^{52} + ( -14 - 14 \beta_{1} - 36 \beta_{6} ) q^{53} + ( 12 \beta_{3} - 12 \beta_{5} + 4 \beta_{7} ) q^{55} + ( -58 - 224 \beta_{1} + 58 \beta_{2} - 58 \beta_{6} ) q^{58} + ( 49 \beta_{3} - 20 \beta_{4} ) q^{59} + ( -27 \beta_{4} + 68 \beta_{5} - 27 \beta_{7} ) q^{61} + ( -122 \beta_{3} + 122 \beta_{5} - 60 \beta_{7} ) q^{62} + ( -471 + 103 \beta_{2} ) q^{64} + ( -70 + 154 \beta_{1} + 70 \beta_{2} - 70 \beta_{6} ) q^{65} + ( 448 + 448 \beta_{1} - 76 \beta_{6} ) q^{67} + ( -106 \beta_{4} - 13 \beta_{5} - 106 \beta_{7} ) q^{68} + ( -548 - 28 \beta_{2} ) q^{71} + ( 37 \beta_{3} + 25 \beta_{4} ) q^{73} + ( -96 - 96 \beta_{1} - 50 \beta_{6} ) q^{74} + ( -63 \beta_{3} + 63 \beta_{5} + 70 \beta_{7} ) q^{76} + ( 188 - 168 \beta_{1} - 188 \beta_{2} + 188 \beta_{6} ) q^{79} + ( -18 \beta_{3} + 12 \beta_{4} ) q^{80} + ( -50 \beta_{4} + 99 \beta_{5} - 50 \beta_{7} ) q^{82} + ( -\beta_{3} + \beta_{5} - 8 \beta_{7} ) q^{83} + ( -916 + 190 \beta_{2} ) q^{85} + ( 170 - 1120 \beta_{1} - 170 \beta_{2} + 170 \beta_{6} ) q^{86} + ( -352 - 352 \beta_{1} - 74 \beta_{6} ) q^{88} + ( 75 \beta_{4} + 157 \beta_{5} + 75 \beta_{7} ) q^{89} + ( -956 + 156 \beta_{2} ) q^{92} + ( -142 \beta_{3} - 76 \beta_{4} ) q^{94} + ( 460 + 460 \beta_{1} - 176 \beta_{6} ) q^{95} + ( -189 \beta_{3} + 189 \beta_{5} - 91 \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{2} - 34q^{4} - 132q^{8} + O(q^{10})$$ $$8q + 2q^{2} - 34q^{4} - 132q^{8} + 100q^{11} + 174q^{16} - 680q^{22} + 352q^{23} + 128q^{25} - 520q^{29} - 30q^{32} - 212q^{37} + 1080q^{43} + 460q^{44} - 696q^{46} - 2732q^{50} + 16q^{53} + 780q^{58} - 3356q^{64} - 756q^{65} + 1944q^{67} - 4496q^{71} - 284q^{74} + 1048q^{79} - 6568q^{85} + 4820q^{86} - 1260q^{88} - 7024q^{92} + 2192q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} - 54 x^{6} + 176 x^{5} + 1307 x^{4} - 2912 x^{3} - 15314 x^{2} + 16800 x + 86044$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{6} + 6 \nu^{5} + 149 \nu^{4} - 308 \nu^{3} - 3293 \nu^{2} + 3448 \nu + 22372$$$$)/8946$$ $$\beta_{2}$$ $$=$$ $$($$$$188 \nu^{7} - 658 \nu^{6} - 11168 \nu^{5} + 29565 \nu^{4} + 311680 \nu^{3} - 497414 \nu^{2} - 5795360 \nu + 4518059$$$$)/3072951$$ $$\beta_{3}$$ $$=$$ $$($$$$-94 \nu^{7} + 558 \nu^{6} + 4897 \nu^{5} - 31843 \nu^{4} - 120574 \nu^{3} + 1137914 \nu^{2} + 454250 \nu - 11478684$$$$)/1024317$$ $$\beta_{4}$$ $$=$$ $$($$$$1034 \nu^{7} - 2932 \nu^{6} - 63485 \nu^{5} + 111426 \nu^{4} + 1820038 \nu^{3} - 68156 \nu^{2} - 17694113 \nu - 24320296$$$$)/3072951$$ $$\beta_{5}$$ $$=$$ $$($$$$-685 \nu^{7} + 10069 \nu^{6} + 3667 \nu^{5} - 388145 \nu^{4} + 210780 \nu^{3} + 6478590 \nu^{2} - 3760032 \nu - 36898988$$$$)/2048634$$ $$\beta_{6}$$ $$=$$ $$($$$$2744 \nu^{7} - 8917 \nu^{6} - 155726 \nu^{5} + 356991 \nu^{4} + 3179236 \nu^{3} - 3891986 \nu^{2} - 26106296 \nu + 2554244$$$$)/6145902$$ $$\beta_{7}$$ $$=$$ $$($$$$5467 \nu^{7} + 2506 \nu^{6} - 235570 \nu^{5} - 369522 \nu^{4} + 3410879 \nu^{3} + 8909612 \nu^{2} - 8027680 \nu - 44176664$$$$)/6145902$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} - \beta_{3} - 7 \beta_{2} + 7$$$$)/7$$ $$\nu^{2}$$ $$=$$ $$($$$$4 \beta_{4} + 10 \beta_{3} - 7 \beta_{2} + 14 \beta_{1} + 119$$$$)/7$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{7} - 42 \beta_{6} - 2 \beta_{5} + 54 \beta_{4} - 31 \beta_{3} - 77 \beta_{2} + 189$$$$)/7$$ $$\nu^{4}$$ $$=$$ $$($$$$16 \beta_{7} - 84 \beta_{6} + 40 \beta_{5} + 200 \beta_{4} + 236 \beta_{3} - 147 \beta_{2} + 1316 \beta_{1} + 2023$$$$)/7$$ $$\nu^{5}$$ $$=$$ $$($$$$356 \beta_{7} - 2240 \beta_{6} - 216 \beta_{5} + 1686 \beta_{4} - 315 \beta_{3} + 245 \beta_{2} + 2240 \beta_{1} + 3451$$$$)/7$$ $$\nu^{6}$$ $$=$$ $$($$$$1952 \beta_{7} - 6510 \beta_{6} + 2640 \beta_{5} + 6780 \beta_{4} + 3222 \beta_{3} + 1099 \beta_{2} + 50400 \beta_{1} + 26397$$$$)/7$$ $$\nu^{7}$$ $$=$$ $$($$$$22148 \beta_{7} - 73010 \beta_{6} - 6566 \beta_{5} + 44318 \beta_{4} + 2477 \beta_{3} + 49287 \beta_{2} + 139552 \beta_{1} + 28329$$$$)/7$$

## Character values

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

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

## 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
 −2.82402 + 1.22474i −4.23824 − 1.22474i 5.23824 + 1.22474i 3.82402 − 1.22474i −2.82402 − 1.22474i −4.23824 + 1.22474i 5.23824 − 1.22474i 3.82402 + 1.22474i
−1.76556 + 3.05805i 0 −2.23444 3.87016i −1.03865 + 1.79899i 0 0 −12.4689 0 −3.66760 6.35247i
226.2 −1.76556 + 3.05805i 0 −2.23444 3.87016i 1.03865 1.79899i 0 0 −12.4689 0 3.66760 + 6.35247i
226.3 2.26556 3.92407i 0 −6.26556 10.8523i −6.73953 + 11.6732i 0 0 −20.5311 0 30.5377 + 52.8928i
226.4 2.26556 3.92407i 0 −6.26556 10.8523i 6.73953 11.6732i 0 0 −20.5311 0 −30.5377 52.8928i
361.1 −1.76556 3.05805i 0 −2.23444 + 3.87016i −1.03865 1.79899i 0 0 −12.4689 0 −3.66760 + 6.35247i
361.2 −1.76556 3.05805i 0 −2.23444 + 3.87016i 1.03865 + 1.79899i 0 0 −12.4689 0 3.66760 6.35247i
361.3 2.26556 + 3.92407i 0 −6.26556 + 10.8523i −6.73953 11.6732i 0 0 −20.5311 0 30.5377 52.8928i
361.4 2.26556 + 3.92407i 0 −6.26556 + 10.8523i 6.73953 + 11.6732i 0 0 −20.5311 0 −30.5377 + 52.8928i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 361.4 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.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.e.y 8
3.b odd 2 1 49.4.c.e 8
7.b odd 2 1 inner 441.4.e.y 8
7.c even 3 1 441.4.a.u 4
7.c even 3 1 inner 441.4.e.y 8
7.d odd 6 1 441.4.a.u 4
7.d odd 6 1 inner 441.4.e.y 8
21.c even 2 1 49.4.c.e 8
21.g even 6 1 49.4.a.e 4
21.g even 6 1 49.4.c.e 8
21.h odd 6 1 49.4.a.e 4
21.h odd 6 1 49.4.c.e 8
84.j odd 6 1 784.4.a.bf 4
84.n even 6 1 784.4.a.bf 4
105.o odd 6 1 1225.4.a.bb 4
105.p even 6 1 1225.4.a.bb 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.a.e 4 21.g even 6 1
49.4.a.e 4 21.h odd 6 1
49.4.c.e 8 3.b odd 2 1
49.4.c.e 8 21.c even 2 1
49.4.c.e 8 21.g even 6 1
49.4.c.e 8 21.h odd 6 1
441.4.a.u 4 7.c even 3 1
441.4.a.u 4 7.d odd 6 1
441.4.e.y 8 1.a even 1 1 trivial
441.4.e.y 8 7.b odd 2 1 inner
441.4.e.y 8 7.c even 3 1 inner
441.4.e.y 8 7.d odd 6 1 inner
784.4.a.bf 4 84.j odd 6 1
784.4.a.bf 4 84.n even 6 1
1225.4.a.bb 4 105.o odd 6 1
1225.4.a.bb 4 105.p even 6 1

## Hecke kernels

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

 $$T_{2}^{4} - T_{2}^{3} + 17 T_{2}^{2} + 16 T_{2} + 256$$ $$T_{5}^{8} + 186 T_{5}^{6} + 33812 T_{5}^{4} + 145824 T_{5}^{2} + 614656$$ $$T_{13}^{4} - 3234 T_{13}^{2} + 2458624$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 256 + 16 T + 17 T^{2} - T^{3} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$614656 + 145824 T^{2} + 33812 T^{4} + 186 T^{6} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$( 1600 - 2000 T + 2460 T^{2} - 50 T^{3} + T^{4} )^{2}$$
$13$ $$( 2458624 - 3234 T^{2} + T^{4} )^{2}$$
$17$ $$95001747709456 + 141953618576 T^{2} + 202363212 T^{4} + 14564 T^{6} + T^{8}$$
$19$ $$2765866292982016 + 775514317984 T^{2} + 164853012 T^{4} + 14746 T^{6} + T^{8}$$
$23$ $$( 44943616 - 1179904 T + 24272 T^{2} - 176 T^{3} + T^{4} )^{2}$$
$29$ $$( 1040 + 130 T + T^{2} )^{4}$$
$31$ $$396154108207169536 + 63006232805376 T^{2} + 9391403072 T^{4} + 100104 T^{6} + T^{8}$$
$37$ $$( 4946176 + 235744 T + 9012 T^{2} + 106 T^{3} + T^{4} )^{2}$$
$41$ $$( 307721764 - 66836 T^{2} + T^{4} )^{2}$$
$43$ $$( -61400 - 270 T + T^{2} )^{4}$$
$47$ $$69422863899074560000 + 1560090870016000 T^{2} + 26726779200 T^{4} + 187240 T^{6} + T^{8}$$
$53$ $$( 442849936 + 168352 T + 21108 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$59$ $$2563073382676500736 + 303148207106976 T^{2} + 34253977172 T^{4} + 189354 T^{6} + T^{8}$$
$61$ $$25223230345416683776 + 1809354357370784 T^{2} + 124769317332 T^{4} + 360266 T^{6} + T^{8}$$
$67$ $$( 20259536896 - 138350592 T + 802448 T^{2} - 972 T^{3} + T^{4} )^{2}$$
$71$ $$( 303104 + 1124 T + T^{2} )^{4}$$
$73$ $$15\!\cdots\!76$$$$+ 3439588972084944 T^{2} + 64165647212 T^{4} + 276756 T^{6} + T^{8}$$
$79$ $$( 255728444416 + 264984704 T + 780272 T^{2} - 524 T^{3} + T^{4} )^{2}$$
$83$ $$( 6492304 - 11466 T^{2} + T^{4} )^{2}$$
$89$ $$80\!\cdots\!56$$$$+ 10308817483656026384 T^{2} + 10287783492492 T^{4} + 3623876 T^{6} + T^{8}$$
$97$ $$( 841222821124 - 3082884 T^{2} + T^{4} )^{2}$$