# Properties

 Label 441.4.p.c Level $441$ Weight $4$ Character orbit 441.p Analytic conductor $26.020$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 441.p (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$26.0198423125$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 48 x^{14} + 1647 x^{12} - 27620 x^{10} + 336765 x^{8} - 1200006 x^{6} + 3242464 x^{4} - 1762200 x^{2} + 810000$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{4}\cdot 3^{8}$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 4 + \beta_{2} - 4 \beta_{3} + \beta_{9} ) q^{4} + ( -\beta_{12} + \beta_{13} ) q^{5} + ( -4 \beta_{6} + \beta_{11} + \beta_{12} - 2 \beta_{13} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 4 + \beta_{2} - 4 \beta_{3} + \beta_{9} ) q^{4} + ( -\beta_{12} + \beta_{13} ) q^{5} + ( -4 \beta_{6} + \beta_{11} + \beta_{12} - 2 \beta_{13} ) q^{8} + ( 3 + \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} ) q^{10} + ( 2 \beta_{1} + 2 \beta_{6} + 3 \beta_{8} + 3 \beta_{11} - \beta_{15} ) q^{11} + ( 8 + \beta_{2} - 14 \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} ) q^{13} + ( -1 - 23 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} - \beta_{9} + \beta_{10} ) q^{16} + ( 2 \beta_{1} - 2 \beta_{6} + \beta_{8} - \beta_{11} + 3 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{17} + ( 48 + 2 \beta_{2} - 27 \beta_{3} - 6 \beta_{4} - 3 \beta_{7} + \beta_{9} ) q^{19} + ( 24 \beta_{1} + 12 \beta_{6} + 12 \beta_{8} + 6 \beta_{11} - \beta_{14} - \beta_{15} ) q^{20} + ( 28 - \beta_{2} - 12 \beta_{4} + 2 \beta_{5} + 2 \beta_{10} ) q^{22} + ( -14 \beta_{1} + 17 \beta_{8} + 3 \beta_{12} + 3 \beta_{13} + \beta_{14} ) q^{23} + ( -1 + 3 \beta_{2} + 6 \beta_{3} + 4 \beta_{4} - \beta_{5} + 4 \beta_{7} + 3 \beta_{9} + 2 \beta_{10} ) q^{25} + ( -23 \beta_{1} - 46 \beta_{6} + 20 \beta_{8} + 40 \beta_{11} + 4 \beta_{12} - 4 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{26} + ( -38 \beta_{6} + 2 \beta_{11} - 4 \beta_{12} + 8 \beta_{13} - \beta_{14} + \beta_{15} ) q^{29} + ( -50 + 2 \beta_{2} - 44 \beta_{3} + \beta_{4} - 7 \beta_{5} - \beta_{7} - 2 \beta_{9} ) q^{31} + ( 12 \beta_{1} + 12 \beta_{6} - 19 \beta_{8} - 19 \beta_{11} - 10 \beta_{12} + 5 \beta_{13} + 2 \beta_{15} ) q^{32} + ( 22 + 2 \beta_{2} - 22 \beta_{3} + 11 \beta_{4} + 9 \beta_{5} + 22 \beta_{7} + 4 \beta_{9} - 9 \beta_{10} ) q^{34} + ( 4 - 155 \beta_{3} + 8 \beta_{5} + \beta_{7} - \beta_{9} - 4 \beta_{10} ) q^{37} + ( 29 \beta_{1} - 29 \beta_{6} + 32 \beta_{8} - 32 \beta_{11} - 12 \beta_{13} - 6 \beta_{14} + 3 \beta_{15} ) q^{38} + ( 260 + 2 \beta_{2} - 138 \beta_{3} - 14 \beta_{4} - 7 \beta_{7} + \beta_{9} - 2 \beta_{10} ) q^{40} + ( 68 \beta_{1} + 34 \beta_{6} + 20 \beta_{8} + 10 \beta_{11} - 5 \beta_{12} + 3 \beta_{14} + 3 \beta_{15} ) q^{41} + ( 14 - \beta_{2} - 3 \beta_{4} - 10 \beta_{5} - 10 \beta_{10} ) q^{43} + ( 6 \beta_{1} + 85 \beta_{8} - 15 \beta_{12} - 15 \beta_{13} - 4 \beta_{14} ) q^{44} + ( -171 - 20 \beta_{2} + 167 \beta_{3} + \beta_{4} + 5 \beta_{5} + \beta_{7} - 20 \beta_{9} - 10 \beta_{10} ) q^{46} + ( 8 \beta_{1} + 16 \beta_{6} + 25 \beta_{8} + 50 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - 4 \beta_{14} + 8 \beta_{15} ) q^{47} + ( -35 \beta_{6} + 59 \beta_{11} + \beta_{12} - 2 \beta_{13} + 4 \beta_{14} - 4 \beta_{15} ) q^{50} + ( -185 - 22 \beta_{2} - 186 \beta_{3} - 17 \beta_{4} + 18 \beta_{5} + 17 \beta_{7} + 22 \beta_{9} ) q^{52} + ( 26 \beta_{1} + 26 \beta_{6} - 14 \beta_{8} - 14 \beta_{11} + 40 \beta_{12} - 20 \beta_{13} + 3 \beta_{15} ) q^{53} + ( -3 - 23 \beta_{2} + 20 \beta_{3} + 7 \beta_{4} - 4 \beta_{5} + 14 \beta_{7} - 46 \beta_{9} + 4 \beta_{10} ) q^{55} + ( 2 - 422 \beta_{3} + 4 \beta_{5} + 5 \beta_{7} + 23 \beta_{9} - 2 \beta_{10} ) q^{58} + ( -56 \beta_{1} + 56 \beta_{6} + 63 \beta_{8} - 63 \beta_{11} + 17 \beta_{13} ) q^{59} + ( 139 - 48 \beta_{2} - 61 \beta_{3} + 6 \beta_{4} + 3 \beta_{7} - 24 \beta_{9} + 11 \beta_{10} ) q^{61} + ( -94 \beta_{1} - 47 \beta_{6} + 30 \beta_{8} + 15 \beta_{11} + 11 \beta_{12} + \beta_{14} + \beta_{15} ) q^{62} + ( 347 + 25 \beta_{2} + 6 \beta_{4} + 9 \beta_{5} + 9 \beta_{10} ) q^{64} + ( 36 \beta_{1} + 109 \beta_{8} + 18 \beta_{12} + 18 \beta_{13} + 4 \beta_{14} ) q^{65} + ( 43 + 3 \beta_{2} - 24 \beta_{3} + 14 \beta_{4} - 5 \beta_{5} + 14 \beta_{7} + 3 \beta_{9} + 10 \beta_{10} ) q^{67} + ( -6 \beta_{1} - 12 \beta_{6} + 95 \beta_{8} + 190 \beta_{11} - 21 \beta_{12} + 21 \beta_{13} + 3 \beta_{14} - 6 \beta_{15} ) q^{68} + ( 52 \beta_{6} - \beta_{11} + 12 \beta_{12} - 24 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{71} + ( -163 + 25 \beta_{2} - 176 \beta_{3} + 28 \beta_{4} - 15 \beta_{5} - 28 \beta_{7} - 25 \beta_{9} ) q^{73} + ( -113 \beta_{1} - 113 \beta_{6} - 38 \beta_{8} - 38 \beta_{11} - 20 \beta_{12} + 10 \beta_{13} - \beta_{15} ) q^{74} + ( 203 + 24 \beta_{2} - 500 \beta_{3} - 47 \beta_{4} - 30 \beta_{5} - 94 \beta_{7} + 48 \beta_{9} + 30 \beta_{10} ) q^{76} + ( -25 - 252 \beta_{3} - 50 \beta_{5} - 35 \beta_{7} - 48 \beta_{9} + 25 \beta_{10} ) q^{79} + ( 30 \beta_{1} - 30 \beta_{6} + 36 \beta_{8} - 36 \beta_{11} - 20 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{80} + ( 861 + 96 \beta_{2} - 425 \beta_{3} + 24 \beta_{4} + 12 \beta_{7} + 48 \beta_{9} - 13 \beta_{10} ) q^{82} + ( -20 \beta_{1} - 10 \beta_{6} + 158 \beta_{8} + 79 \beta_{11} + 26 \beta_{12} - 5 \beta_{14} - 5 \beta_{15} ) q^{83} + ( -272 - 48 \beta_{2} + 9 \beta_{4} + 23 \beta_{5} + 23 \beta_{10} ) q^{85} + ( -67 \beta_{1} + 154 \beta_{8} + 18 \beta_{12} + 18 \beta_{13} - 3 \beta_{14} ) q^{86} + ( 50 + 47 \beta_{2} - 113 \beta_{3} - 70 \beta_{4} - 7 \beta_{5} - 70 \beta_{7} + 47 \beta_{9} + 14 \beta_{10} ) q^{88} + ( 52 \beta_{1} + 104 \beta_{6} + 54 \beta_{8} + 108 \beta_{11} + 72 \beta_{12} - 72 \beta_{13} + 2 \beta_{14} - 4 \beta_{15} ) q^{89} + ( 258 \beta_{6} + 67 \beta_{11} - 7 \beta_{12} + 14 \beta_{13} - 7 \beta_{14} + 7 \beta_{15} ) q^{92} + ( 115 + 22 \beta_{2} + 128 \beta_{3} + 9 \beta_{4} - 22 \beta_{5} - 9 \beta_{7} - 22 \beta_{9} ) q^{94} + ( 144 \beta_{1} + 144 \beta_{6} + 9 \beta_{8} + 9 \beta_{11} - 88 \beta_{12} + 44 \beta_{13} - 12 \beta_{15} ) q^{95} + ( 401 + 47 \beta_{2} - 794 \beta_{3} + 4 \beta_{4} + 29 \beta_{5} + 8 \beta_{7} + 94 \beta_{9} - 29 \beta_{10} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 32q^{4} + O(q^{10})$$ $$16q + 32q^{4} + 72q^{10} - 188q^{16} + 612q^{19} + 528q^{22} - 20q^{25} - 1128q^{31} - 1196q^{37} + 3204q^{40} + 328q^{43} - 1392q^{46} - 4452q^{52} - 3372q^{58} + 1632q^{61} + 5432q^{64} + 308q^{67} - 4068q^{73} - 2176q^{79} + 10188q^{82} - 4608q^{85} + 708q^{88} + 2916q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 48 x^{14} + 1647 x^{12} - 27620 x^{10} + 336765 x^{8} - 1200006 x^{6} + 3242464 x^{4} - 1762200 x^{2} + 810000$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-10313805 \nu^{14} + 461882826 \nu^{12} - 15525573175 \nu^{10} + 234921068610 \nu^{8} - 2717580554397 \nu^{6} + 2415597870280 \nu^{4} - 1315137294000 \nu^{2} - 278327304980712$$$$)/ 24355954733376$$ $$\beta_{3}$$ $$=$$ $$($$$$-2582250337 \nu^{14} + 122400945426 \nu^{12} - 4183683881139 \nu^{10} + 68992918331690 \nu^{8} - 834373374448305 \nu^{6} + 2691078814742472 \nu^{4} - 8010514076168368 \nu^{2} + 4353170949761400$$$$)/ 3653393210006400$$ $$\beta_{4}$$ $$=$$ $$($$$$131851797 \nu^{14} - 6000144674 \nu^{12} + 198479098895 \nu^{10} - 3003233534994 \nu^{8} + 33064412041565 \nu^{6} - 30881029845512 \nu^{4} + 16812729687600 \nu^{2} + 301359467694888$$$$)/ 73067864200128$$ $$\beta_{5}$$ $$=$$ $$($$$$-20174289101 \nu^{14} + 1199936429098 \nu^{12} - 44140804007447 \nu^{10} + 925727478351170 \nu^{8} - 12720550693019965 \nu^{6} + 92682920898111656 \nu^{4} - 226162470539390064 \nu^{2} + 211548604260850200$$$$)/ 10960179630019200$$ $$\beta_{6}$$ $$=$$ $$($$$$-2582250337 \nu^{15} + 122400945426 \nu^{13} - 4183683881139 \nu^{11} + 68992918331690 \nu^{9} - 834373374448305 \nu^{7} + 2691078814742472 \nu^{5} - 8010514076168368 \nu^{3} + 699777739755000 \nu$$$$)/ 3653393210006400$$ $$\beta_{7}$$ $$=$$ $$($$$$6165622857 \nu^{14} - 304531357386 \nu^{12} + 10574100606979 \nu^{10} - 183212343919090 \nu^{8} + 2271828840408105 \nu^{6} - 9221274840461992 \nu^{4} + 22001675288833648 \nu^{2} - 11959302579305400$$$$)/ 1826696605003200$$ $$\beta_{8}$$ $$=$$ $$($$$$-937699527 \nu^{15} + 42257710318 \nu^{13} - 1411537509445 \nu^{11} + 21358303256454 \nu^{9} - 240064275625255 \nu^{7} + 219618751797592 \nu^{5} - 119568250371600 \nu^{3} - 4112177675462712 \nu$$$$)/ 730678642001280$$ $$\beta_{9}$$ $$=$$ $$($$$$-4906655549 \nu^{14} + 233254820202 \nu^{12} - 7979228432903 \nu^{10} + 132112809948130 \nu^{8} - 1600807235036685 \nu^{6} + 5321767682727944 \nu^{4} - 15379250851652336 \nu^{2} + 8357738104027800$$$$)/ 608898868334400$$ $$\beta_{10}$$ $$=$$ $$($$$$151546793651 \nu^{14} - 7122898433398 \nu^{12} + 241898414566697 \nu^{10} - 3918044005070270 \nu^{8} + 46284369537850915 \nu^{6} - 123451695584498456 \nu^{4} + 242914084530530064 \nu^{2} + 56141855905743000$$$$)/ 10960179630019200$$ $$\beta_{11}$$ $$=$$ $$($$$$253797 \nu^{15} - 12085406 \nu^{13} + 413081109 \nu^{11} - 6837155440 \nu^{9} + 82382868455 \nu^{7} - 265706935032 \nu^{5} + 646432035208 \nu^{3} - 69093405000 \nu$$$$)/ 49371246000$$ $$\beta_{12}$$ $$=$$ $$($$$$-144886104737 \nu^{15} + 6952944162226 \nu^{13} - 238669929280139 \nu^{11} + 4004343941493890 \nu^{9} - 48919377548687305 \nu^{7} + 176406493692653672 \nu^{5} - 507004002338951568 \nu^{3} + 505456539561487800 \nu$$$$)/ 27400449075048000$$ $$\beta_{13}$$ $$=$$ $$($$$$383305735249 \nu^{15} - 18114452956802 \nu^{13} + 618137711412403 \nu^{11} - 10139133031594330 \nu^{9} + 121958132016661985 \nu^{7} - 374719488978256744 \nu^{5} + 1025934682367273136 \nu^{3} + 362143867944097800 \nu$$$$)/ 54800898150096000$$ $$\beta_{14}$$ $$=$$ $$($$$$-1280411313 \nu^{15} + 57618445232 \nu^{13} - 1927428289955 \nu^{11} + 29164366973226 \nu^{9} - 328956825464915 \nu^{7} + 299885332403048 \nu^{5} - 163268228300400 \nu^{3} - 7657308044504568 \nu$$$$)/ 91334830250160$$ $$\beta_{15}$$ $$=$$ $$($$$$530090206069 \nu^{15} - 25526553824562 \nu^{13} + 876699628157143 \nu^{11} - 14764864792789530 \nu^{9} + 180388670933343285 \nu^{7} - 657910940475690664 \nu^{5} + 1738056090103156016 \nu^{3} - 944609526343891800 \nu$$$$)/ 9133483025016000$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} - 12 \beta_{3} + \beta_{2} + 12$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{13} + \beta_{12} + \beta_{11} - 20 \beta_{6}$$ $$\nu^{4}$$ $$=$$ $$\beta_{10} + 23 \beta_{9} - 2 \beta_{7} - 2 \beta_{5} - 247 \beta_{3} - 1$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{15} - 27 \beta_{13} + 54 \beta_{12} + 13 \beta_{11} + 13 \beta_{8} - 436 \beta_{6} - 436 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-31 \beta_{10} - 31 \beta_{5} + 86 \beta_{4} - 511 \beta_{2} - 5437$$ $$\nu^{7}$$ $$=$$ $$86 \beta_{14} + 659 \beta_{13} + 659 \beta_{12} - 63 \beta_{8} - 9656 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-1662 \beta_{10} - 11375 \beta_{9} + 2814 \beta_{7} + 831 \beta_{5} + 2814 \beta_{4} + 122445 \beta_{3} - 11375 \beta_{2} - 120462$$ $$\nu^{9}$$ $$=$$ $$-2814 \beta_{15} + 2814 \beta_{14} + 31702 \beta_{13} - 15851 \beta_{12} + 9607 \beta_{11} + 215296 \beta_{6}$$ $$\nu^{10}$$ $$=$$ $$-21479 \beta_{10} - 254407 \beta_{9} + 82486 \beta_{7} + 42958 \beta_{5} + 2762309 \beta_{3} + 21479$$ $$\nu^{11}$$ $$=$$ $$-82486 \beta_{15} + 379851 \beta_{13} - 759702 \beta_{12} + 395191 \beta_{11} + 395191 \beta_{8} + 4825904 \beta_{6} + 4825904 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$544823 \beta_{10} + 544823 \beta_{5} - 2277118 \beta_{4} + 5717999 \beta_{2} + 61427333$$ $$\nu^{13}$$ $$=$$ $$-2277118 \beta_{14} - 9084763 \beta_{13} - 9084763 \beta_{12} + 12792423 \beta_{8} + 108707968 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$27277998 \beta_{10} + 129130903 \beta_{9} - 60540774 \beta_{7} - 13638999 \beta_{5} - 60540774 \beta_{4} - 1425506037 \beta_{3} + 129130903 \beta_{2} + 1378604262$$ $$\nu^{15}$$ $$=$$ $$60540774 \beta_{15} - 60540774 \beta_{14} - 433899350 \beta_{13} + 216949675 \beta_{12} - 373149623 \beta_{11} - 2460222704 \beta_{6}$$

## Character values

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

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$\beta_{3}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
80.1
 −4.21355 + 2.43270i −3.91663 + 2.26127i −1.57646 + 0.910170i −0.648633 + 0.374489i 0.648633 − 0.374489i 1.57646 − 0.910170i 3.91663 − 2.26127i 4.21355 − 2.43270i −4.21355 − 2.43270i −3.91663 − 2.26127i −1.57646 − 0.910170i −0.648633 − 0.374489i 0.648633 + 0.374489i 1.57646 + 0.910170i 3.91663 + 2.26127i 4.21355 + 2.43270i
−4.21355 + 2.43270i 0 7.83601 13.5724i −6.38217 11.0542i 0 0 37.3274i 0 53.7832 + 31.0517i
80.2 −3.91663 + 2.26127i 0 6.22668 10.7849i 0.632851 + 1.09613i 0 0 20.1405i 0 −4.95730 2.86210i
80.3 −1.57646 + 0.910170i 0 −2.34318 + 4.05851i 7.54372 + 13.0661i 0 0 23.0935i 0 −23.7848 13.7321i
80.4 −0.648633 + 0.374489i 0 −3.71952 + 6.44239i 5.42768 + 9.40102i 0 0 11.5635i 0 −7.04115 4.06521i
80.5 0.648633 0.374489i 0 −3.71952 + 6.44239i −5.42768 9.40102i 0 0 11.5635i 0 −7.04115 4.06521i
80.6 1.57646 0.910170i 0 −2.34318 + 4.05851i −7.54372 13.0661i 0 0 23.0935i 0 −23.7848 13.7321i
80.7 3.91663 2.26127i 0 6.22668 10.7849i −0.632851 1.09613i 0 0 20.1405i 0 −4.95730 2.86210i
80.8 4.21355 2.43270i 0 7.83601 13.5724i 6.38217 + 11.0542i 0 0 37.3274i 0 53.7832 + 31.0517i
215.1 −4.21355 2.43270i 0 7.83601 + 13.5724i −6.38217 + 11.0542i 0 0 37.3274i 0 53.7832 31.0517i
215.2 −3.91663 2.26127i 0 6.22668 + 10.7849i 0.632851 1.09613i 0 0 20.1405i 0 −4.95730 + 2.86210i
215.3 −1.57646 0.910170i 0 −2.34318 4.05851i 7.54372 13.0661i 0 0 23.0935i 0 −23.7848 + 13.7321i
215.4 −0.648633 0.374489i 0 −3.71952 6.44239i 5.42768 9.40102i 0 0 11.5635i 0 −7.04115 + 4.06521i
215.5 0.648633 + 0.374489i 0 −3.71952 6.44239i −5.42768 + 9.40102i 0 0 11.5635i 0 −7.04115 + 4.06521i
215.6 1.57646 + 0.910170i 0 −2.34318 4.05851i −7.54372 + 13.0661i 0 0 23.0935i 0 −23.7848 + 13.7321i
215.7 3.91663 + 2.26127i 0 6.22668 + 10.7849i −0.632851 + 1.09613i 0 0 20.1405i 0 −4.95730 + 2.86210i
215.8 4.21355 + 2.43270i 0 7.83601 + 13.5724i 6.38217 11.0542i 0 0 37.3274i 0 53.7832 31.0517i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 215.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.p.c 16
3.b odd 2 1 inner 441.4.p.c 16
7.b odd 2 1 63.4.p.a 16
7.c even 3 1 63.4.p.a 16
7.c even 3 1 441.4.c.a 16
7.d odd 6 1 441.4.c.a 16
7.d odd 6 1 inner 441.4.p.c 16
21.c even 2 1 63.4.p.a 16
21.g even 6 1 441.4.c.a 16
21.g even 6 1 inner 441.4.p.c 16
21.h odd 6 1 63.4.p.a 16
21.h odd 6 1 441.4.c.a 16
28.d even 2 1 1008.4.bt.a 16
28.g odd 6 1 1008.4.bt.a 16
84.h odd 2 1 1008.4.bt.a 16
84.n even 6 1 1008.4.bt.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.p.a 16 7.b odd 2 1
63.4.p.a 16 7.c even 3 1
63.4.p.a 16 21.c even 2 1
63.4.p.a 16 21.h odd 6 1
441.4.c.a 16 7.c even 3 1
441.4.c.a 16 7.d odd 6 1
441.4.c.a 16 21.g even 6 1
441.4.c.a 16 21.h odd 6 1
441.4.p.c 16 1.a even 1 1 trivial
441.4.p.c 16 3.b odd 2 1 inner
441.4.p.c 16 7.d odd 6 1 inner
441.4.p.c 16 21.g even 6 1 inner
1008.4.bt.a 16 28.d even 2 1
1008.4.bt.a 16 28.g odd 6 1
1008.4.bt.a 16 84.h odd 2 1
1008.4.bt.a 16 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} - \cdots$$ acting on $$S_{4}^{\mathrm{new}}(441, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$810000 - 1762200 T^{2} + 3242464 T^{4} - 1200006 T^{6} + 336765 T^{8} - 27620 T^{10} + 1647 T^{12} - 48 T^{14} + T^{16}$$
$3$ $$T^{16}$$
$5$ $$49018425731856 + 31530370595472 T^{2} + 19693854748764 T^{4} + 370814223780 T^{6} + 4739623389 T^{8} + 33794766 T^{10} + 176175 T^{12} + 510 T^{14} + T^{16}$$
$7$ $$T^{16}$$
$11$ $$17\!\cdots\!16$$$$-$$$$20\!\cdots\!60$$$$T^{2} +$$$$21\!\cdots\!20$$$$T^{4} - 261274857054043068 T^{6} + 219830806138269 T^{8} - 105678094310 T^{10} + 37127511 T^{12} - 7446 T^{14} + T^{16}$$
$13$ $$( 2752420357764 + 38728522980 T^{2} + 47288277 T^{4} + 13074 T^{6} + T^{8} )^{2}$$
$17$ $$88\!\cdots\!56$$$$+$$$$64\!\cdots\!28$$$$T^{2} +$$$$29\!\cdots\!52$$$$T^{4} + 85531884162965662464 T^{6} + 18421656867340560 T^{8} + 2793093124176 T^{10} + 313601004 T^{12} + 22356 T^{14} + T^{16}$$
$19$ $$( 567106596 - 20223801360 T + 240424077474 T^{2} - 756672840 T^{3} - 85804785 T^{4} + 272646 T^{5} + 30321 T^{6} - 306 T^{7} + T^{8} )^{2}$$
$23$ $$10\!\cdots\!76$$$$-$$$$42\!\cdots\!28$$$$T^{2} +$$$$11\!\cdots\!40$$$$T^{4} -$$$$18\!\cdots\!88$$$$T^{6} + 225387671576342544 T^{8} - 18841701469712 T^{10} + 1152260940 T^{12} - 42372 T^{14} + T^{16}$$
$29$ $$( 253130837240241216 + 47051469656336 T^{2} + 3179040801 T^{4} + 93078 T^{6} + T^{8} )^{2}$$
$31$ $$( 20564942686042041 + 464084387243064 T + 882141790320 T^{2} - 58872659328 T^{3} - 134048907 T^{4} + 10260288 T^{5} + 124224 T^{6} + 564 T^{7} + T^{8} )^{2}$$
$37$ $$( 36871673487280324 + 1475505352165768 T + 38424543760414 T^{2} + 595549862012 T^{3} + 6745700747 T^{4} + 48851570 T^{5} + 250213 T^{6} + 598 T^{7} + T^{8} )^{2}$$
$41$ $$( 32324337336074547456 - 2566451699751744 T^{2} + 53591867172 T^{4} - 402972 T^{6} + T^{8} )^{2}$$
$43$ $$( 144774182 - 27889316 T - 195789 T^{2} - 82 T^{3} + T^{4} )^{4}$$
$47$ $$16\!\cdots\!36$$$$+$$$$61\!\cdots\!20$$$$T^{2} +$$$$19\!\cdots\!40$$$$T^{4} +$$$$13\!\cdots\!36$$$$T^{6} +$$$$68\!\cdots\!04$$$$T^{8} + 13314700787201280 T^{10} + 192810529224 T^{12} + 472272 T^{14} + T^{16}$$
$53$ $$24\!\cdots\!16$$$$-$$$$74\!\cdots\!80$$$$T^{2} +$$$$15\!\cdots\!04$$$$T^{4} -$$$$17\!\cdots\!44$$$$T^{6} +$$$$15\!\cdots\!57$$$$T^{8} - 79972866291229718 T^{10} + 306776980515 T^{12} - 684342 T^{14} + T^{16}$$
$59$ $$20\!\cdots\!36$$$$+$$$$66\!\cdots\!48$$$$T^{2} +$$$$15\!\cdots\!72$$$$T^{4} +$$$$16\!\cdots\!36$$$$T^{6} +$$$$11\!\cdots\!25$$$$T^{8} + 430452358387636086 T^{10} + 1125567820107 T^{12} + 1243590 T^{14} + T^{16}$$
$61$ $$( 345703894859082816 + 41677416905608512 T + 1502930502375120 T^{2} - 20726360884656 T^{3} + 65628156924 T^{4} + 238596768 T^{5} - 70446 T^{6} - 816 T^{7} + T^{8} )^{2}$$
$67$ $$( 20749646356594032784 + 38600086175507024 T + 846328311384796 T^{2} - 37829515276 T^{3} + 25660342357 T^{4} + 9236990 T^{5} + 193747 T^{6} - 154 T^{7} + T^{8} )^{2}$$
$71$ $$( 2630939135312864016 + 579610290699200 T^{2} + 28173681528 T^{4} + 317232 T^{6} + T^{8} )^{2}$$
$73$ $$($$$$91\!\cdots\!00$$$$+$$$$52\!\cdots\!00$$$$T + 1158009552606105108 T^{2} + 844955676736164 T^{3} - 643525312275 T^{4} - 986638482 T^{5} + 893979 T^{6} + 2034 T^{7} + T^{8} )^{2}$$
$79$ $$($$$$84\!\cdots\!49$$$$- 24278780400738095116 T + 117506333264689432 T^{2} - 62832634078240 T^{3} + 465213348493 T^{4} - 36829816 T^{5} + 1703128 T^{6} + 1088 T^{7} + T^{8} )^{2}$$
$83$ $$($$$$75\!\cdots\!64$$$$- 194675614013208132 T^{2} + 1143650214837 T^{4} - 1941810 T^{6} + T^{8} )^{2}$$
$89$ $$47\!\cdots\!16$$$$+$$$$12\!\cdots\!12$$$$T^{2} +$$$$25\!\cdots\!88$$$$T^{4} +$$$$21\!\cdots\!68$$$$T^{6} +$$$$14\!\cdots\!08$$$$T^{8} + 3681748657198486656 T^{10} + 8361242100720 T^{12} + 3090888 T^{14} + T^{16}$$
$97$ $$($$$$64\!\cdots\!56$$$$+ 3466733814045027288 T^{2} + 6247433928969 T^{4} + 4322814 T^{6} + T^{8} )^{2}$$