# Properties

 Label 4410.2.d.b Level $4410$ Weight $2$ Character orbit 4410.d Analytic conductor $35.214$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4410.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$35.2140272914$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 6 x^{15} + 21 x^{14} - 54 x^{13} + 113 x^{12} - 168 x^{11} + 186 x^{10} - 84 x^{9} - 6 x^{8} - 420 x^{7} + 4650 x^{6} - 21000 x^{5} + 70625 x^{4} - 168750 x^{3} + 328125 x^{2} - 468750 x + 390625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 630) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + \beta_{10} q^{5} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} + \beta_{10} q^{5} + q^{8} + \beta_{10} q^{10} + ( -\beta_{3} + \beta_{8} ) q^{11} + ( -\beta_{2} + \beta_{5} + \beta_{6} + \beta_{10} - \beta_{14} ) q^{13} + q^{16} + ( -\beta_{2} + \beta_{5} - \beta_{10} + \beta_{14} ) q^{17} + ( \beta_{1} + \beta_{9} ) q^{19} + \beta_{10} q^{20} + ( -\beta_{3} + \beta_{8} ) q^{22} + ( \beta_{3} + \beta_{7} - \beta_{8} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} ) q^{23} + ( 1 + \beta_{3} + \beta_{12} + \beta_{15} ) q^{25} + ( -\beta_{2} + \beta_{5} + \beta_{6} + \beta_{10} - \beta_{14} ) q^{26} + ( -\beta_{11} + \beta_{13} ) q^{29} + ( -2 \beta_{2} + \beta_{5} - \beta_{10} + 2 \beta_{14} ) q^{31} + q^{32} + ( -\beta_{2} + \beta_{5} - \beta_{10} + \beta_{14} ) q^{34} + ( -3 \beta_{3} + \beta_{15} ) q^{37} + ( \beta_{1} + \beta_{9} ) q^{38} + \beta_{10} q^{40} + ( -\beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{10} - \beta_{14} ) q^{41} + ( \beta_{3} - \beta_{8} - \beta_{11} + \beta_{13} - \beta_{15} ) q^{43} + ( -\beta_{3} + \beta_{8} ) q^{44} + ( \beta_{3} + \beta_{7} - \beta_{8} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} ) q^{46} + ( -\beta_{5} + 3 \beta_{9} + \beta_{10} ) q^{47} + ( 1 + \beta_{3} + \beta_{12} + \beta_{15} ) q^{50} + ( -\beta_{2} + \beta_{5} + \beta_{6} + \beta_{10} - \beta_{14} ) q^{52} + ( -1 - \beta_{3} + \beta_{8} + \beta_{11} + 2 \beta_{12} + \beta_{13} + \beta_{15} ) q^{53} + ( -\beta_{1} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} ) q^{55} + ( -\beta_{11} + \beta_{13} ) q^{58} + ( \beta_{4} + \beta_{5} + \beta_{10} ) q^{59} + ( -2 \beta_{1} + \beta_{2} + \beta_{5} - \beta_{10} - \beta_{14} ) q^{61} + ( -2 \beta_{2} + \beta_{5} - \beta_{10} + 2 \beta_{14} ) q^{62} + q^{64} + ( 3 + 2 \beta_{3} + \beta_{7} + \beta_{8} + 2 \beta_{12} + \beta_{13} - \beta_{15} ) q^{65} + 2 \beta_{8} q^{67} + ( -\beta_{2} + \beta_{5} - \beta_{10} + \beta_{14} ) q^{68} + ( -3 \beta_{3} + \beta_{8} - \beta_{11} + \beta_{13} - 3 \beta_{15} ) q^{71} + ( -2 \beta_{2} + 2 \beta_{5} + 2 \beta_{10} - 2 \beta_{14} ) q^{73} + ( -3 \beta_{3} + \beta_{15} ) q^{74} + ( \beta_{1} + \beta_{9} ) q^{76} + ( -2 \beta_{7} + \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{79} + \beta_{10} q^{80} + ( -\beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{10} - \beta_{14} ) q^{82} + ( -\beta_{2} - 2 \beta_{5} + 6 \beta_{9} + 2 \beta_{10} + \beta_{14} ) q^{83} + ( 3 + 2 \beta_{3} + \beta_{7} - \beta_{11} - 2 \beta_{12} - \beta_{15} ) q^{85} + ( \beta_{3} - \beta_{8} - \beta_{11} + \beta_{13} - \beta_{15} ) q^{86} + ( -\beta_{3} + \beta_{8} ) q^{88} + ( \beta_{2} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{10} + \beta_{14} ) q^{89} + ( \beta_{3} + \beta_{7} - \beta_{8} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} ) q^{92} + ( -\beta_{5} + 3 \beta_{9} + \beta_{10} ) q^{94} + ( 2 + \beta_{3} + \beta_{7} - 3 \beta_{8} + \beta_{11} + 2 \beta_{13} - 2 \beta_{15} ) q^{95} + ( \beta_{2} + 2 \beta_{5} + 2 \beta_{10} + \beta_{14} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 16q^{2} + 16q^{4} + 16q^{8} + O(q^{10})$$ $$16q + 16q^{2} + 16q^{4} + 16q^{8} + 16q^{16} + 16q^{23} + 12q^{25} + 16q^{32} + 16q^{46} + 12q^{50} - 32q^{53} + 16q^{64} + 40q^{65} - 8q^{79} + 64q^{85} + 16q^{92} + 24q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 6 x^{15} + 21 x^{14} - 54 x^{13} + 113 x^{12} - 168 x^{11} + 186 x^{10} - 84 x^{9} - 6 x^{8} - 420 x^{7} + 4650 x^{6} - 21000 x^{5} + 70625 x^{4} - 168750 x^{3} + 328125 x^{2} - 468750 x + 390625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$189 \nu^{15} - 7678 \nu^{14} + 25998 \nu^{13} - 49670 \nu^{12} + 103248 \nu^{11} - 101484 \nu^{10} - 27834 \nu^{9} + 259320 \nu^{8} + 415452 \nu^{7} - 870576 \nu^{6} + 8780040 \nu^{5} - 32915700 \nu^{4} + 95015625 \nu^{3} - 114151250 \nu^{2} + 262743750 \nu - 198906250$$$$)/ 113625000$$ $$\beta_{2}$$ $$=$$ $$($$$$1553 \nu^{15} - 5487 \nu^{14} + 37592 \nu^{13} - 86901 \nu^{12} + 157080 \nu^{11} - 129186 \nu^{10} + 488220 \nu^{9} + 790644 \nu^{8} + 88968 \nu^{7} + 199494 \nu^{6} + 6932940 \nu^{5} - 48097950 \nu^{4} + 103996375 \nu^{3} - 263960625 \nu^{2} + 510212500 \nu - 346640625$$$$)/ 340875000$$ $$\beta_{3}$$ $$=$$ $$($$$$-3503 \nu^{15} + 4278 \nu^{14} - 26998 \nu^{13} + 46872 \nu^{12} - 19254 \nu^{11} - 120366 \nu^{10} - 29238 \nu^{9} - 736638 \nu^{8} - 12822 \nu^{7} - 2041800 \nu^{6} - 15270150 \nu^{5} + 14222250 \nu^{4} - 64549375 \nu^{3} + 33712500 \nu^{2} - 69500000 \nu - 491250000$$$$)/ 568125000$$ $$\beta_{4}$$ $$=$$ $$($$$$-47 \nu^{15} - 6 \nu^{14} - 104 \nu^{13} - 90 \nu^{12} - 354 \nu^{11} - 918 \nu^{10} - 5118 \nu^{9} - 5610 \nu^{8} - 13746 \nu^{7} - 7452 \nu^{6} - 79170 \nu^{5} + 164850 \nu^{4} - 382375 \nu^{3} - 1177500 \nu^{2} - 606250 \nu - 3281250$$$$)/3375000$$ $$\beta_{5}$$ $$=$$ $$($$$$-28283 \nu^{15} + 58968 \nu^{14} - 195713 \nu^{13} + 336102 \nu^{12} - 463584 \nu^{11} - 162846 \nu^{10} + 1249302 \nu^{9} + 527442 \nu^{8} - 6240882 \nu^{7} + 11659590 \nu^{6} - 79847700 \nu^{5} + 266151750 \nu^{4} - 557861875 \nu^{3} + 1084443750 \nu^{2} - 1492703125 \nu + 81562500$$$$)/ 1704375000$$ $$\beta_{6}$$ $$=$$ $$($$$$-221 \nu^{15} + 1326 \nu^{14} - 2816 \nu^{13} + 5859 \nu^{12} - 9648 \nu^{11} + 12828 \nu^{10} - 7506 \nu^{9} + 28464 \nu^{8} - 28224 \nu^{7} + 246270 \nu^{6} - 1110600 \nu^{5} + 3376500 \nu^{4} - 7088125 \nu^{3} + 16706250 \nu^{2} - 22375000 \nu + 31171875$$$$)/11250000$$ $$\beta_{7}$$ $$=$$ $$($$$$-72631 \nu^{15} + 152446 \nu^{14} - 816586 \nu^{13} + 1097059 \nu^{12} - 2564568 \nu^{11} + 1295088 \nu^{10} - 3946746 \nu^{9} - 7849236 \nu^{8} - 10144404 \nu^{7} + 81074310 \nu^{6} - 315330600 \nu^{5} + 552840000 \nu^{4} - 2250321875 \nu^{3} + 3728618750 \nu^{2} - 7280656250 \nu + 4389921875$$$$)/ 3408750000$$ $$\beta_{8}$$ $$=$$ $$($$$$28999 \nu^{15} - 180524 \nu^{14} + 538634 \nu^{13} - 1227551 \nu^{12} + 2136732 \nu^{11} - 2848872 \nu^{10} + 1836654 \nu^{9} - 1678296 \nu^{8} - 5331624 \nu^{7} - 9287550 \nu^{6} + 74294100 \nu^{5} - 578292000 \nu^{4} + 1666289375 \nu^{3} - 3710312500 \nu^{2} + 6112843750 \nu - 7956484375$$$$)/ 1136250000$$ $$\beta_{9}$$ $$=$$ $$($$$$-22146 \nu^{15} + 79646 \nu^{14} - 238236 \nu^{13} + 546479 \nu^{12} - 982878 \nu^{11} + 1301988 \nu^{10} - 369666 \nu^{9} - 1282116 \nu^{8} - 43854 \nu^{7} + 10333650 \nu^{6} - 65558250 \nu^{5} + 287925000 \nu^{4} - 737662500 \nu^{3} + 1557531250 \nu^{2} - 2635218750 \nu + 3061796875$$$$)/ 852187500$$ $$\beta_{10}$$ $$=$$ $$($$$$10646 \nu^{15} - 45366 \nu^{14} + 129881 \nu^{13} - 303924 \nu^{12} + 483708 \nu^{11} - 749898 \nu^{10} + 628476 \nu^{9} + 35346 \nu^{8} - 206466 \nu^{7} - 7484130 \nu^{6} + 35428200 \nu^{5} - 165279750 \nu^{4} + 435921250 \nu^{3} - 926287500 \nu^{2} + 1293390625 \nu - 1730156250$$$$)/ 340875000$$ $$\beta_{11}$$ $$=$$ $$($$$$63739 \nu^{15} - 239664 \nu^{14} + 730399 \nu^{13} - 1660236 \nu^{12} + 2476302 \nu^{11} - 3388392 \nu^{10} + 3230094 \nu^{9} + 2386644 \nu^{8} - 993864 \nu^{7} - 31295250 \nu^{6} + 222951450 \nu^{5} - 869065500 \nu^{4} + 2373254375 \nu^{3} - 4732575000 \nu^{2} + 10084203125 \nu - 8881406250$$$$)/ 1704375000$$ $$\beta_{12}$$ $$=$$ $$($$$$134847 \nu^{15} - 489382 \nu^{14} + 1702962 \nu^{13} - 3465913 \nu^{12} + 8035536 \nu^{11} - 7941696 \nu^{10} + 8198442 \nu^{9} + 2114052 \nu^{8} - 9342132 \nu^{7} - 46368690 \nu^{6} + 612694800 \nu^{5} - 1823016000 \nu^{4} + 5620186875 \nu^{3} - 12008093750 \nu^{2} + 19754156250 \nu - 22617265625$$$$)/ 3408750000$$ $$\beta_{13}$$ $$=$$ $$($$$$-82424 \nu^{15} + 333594 \nu^{14} - 1192979 \nu^{13} + 2832846 \nu^{12} - 5451762 \nu^{11} + 6806232 \nu^{10} - 6953964 \nu^{9} + 1982616 \nu^{8} - 6017556 \nu^{7} + 23859630 \nu^{6} - 328546950 \nu^{5} + 1225090500 \nu^{4} - 3833020000 \nu^{3} + 8548481250 \nu^{2} - 16017953125 \nu + 17355937500$$$$)/ 1704375000$$ $$\beta_{14}$$ $$=$$ $$($$$$113839 \nu^{15} - 393129 \nu^{14} + 1157764 \nu^{13} - 2573376 \nu^{12} + 4791012 \nu^{11} - 4615362 \nu^{10} + 5948364 \nu^{9} + 6684504 \nu^{8} + 5982396 \nu^{7} - 45729360 \nu^{6} + 347872800 \nu^{5} - 1325565750 \nu^{4} + 4053550625 \nu^{3} - 7391446875 \nu^{2} + 13369062500 \nu - 12730312500$$$$)/ 1704375000$$ $$\beta_{15}$$ $$=$$ $$($$$$-298929 \nu^{15} + 1400564 \nu^{14} - 4250574 \nu^{13} + 10377581 \nu^{12} - 18306312 \nu^{11} + 22180692 \nu^{10} - 17319114 \nu^{9} + 3731676 \nu^{8} + 39547164 \nu^{7} + 199376490 \nu^{6} - 1103936400 \nu^{5} + 4897207500 \nu^{4} - 13935140625 \nu^{3} + 30395350000 \nu^{2} - 51938718750 \nu + 63741953125$$$$)/ 3408750000$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{11} - \beta_{10} + \beta_{3} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{15} - \beta_{14} - \beta_{12} - \beta_{10} - \beta_{9} + \beta_{5} - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_{1} - 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{14} + 2 \beta_{10} + 3 \beta_{9} + 2 \beta_{6} - 3 \beta_{2} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{15} + 6 \beta_{14} - 4 \beta_{13} - 3 \beta_{12} - 2 \beta_{11} - 3 \beta_{10} + \beta_{9} + \beta_{8} - 6 \beta_{7} + 9 \beta_{5} + 9 \beta_{4} - 10 \beta_{3} - 9 \beta_{2} - 3 \beta_{1} - 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-6 \beta_{14} - 3 \beta_{13} + 12 \beta_{12} + 8 \beta_{11} + 4 \beta_{10} + 6 \beta_{9} - 18 \beta_{8} + 6 \beta_{7} - 8 \beta_{6} + 7 \beta_{5} + 6 \beta_{4} - 25 \beta_{3} - 4 \beta_{2} + 4 \beta_{1} - 19$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$14 \beta_{15} - 6 \beta_{13} + 16 \beta_{12} + 6 \beta_{11} + 14 \beta_{8} + 24 \beta_{7} - 38 \beta_{3} - 29$$ $$\nu^{7}$$ $$=$$ $$($$$$66 \beta_{15} + 76 \beta_{14} - 104 \beta_{13} - 84 \beta_{12} - 51 \beta_{11} + 23 \beta_{10} + 6 \beta_{9} - 24 \beta_{8} - 6 \beta_{7} - 40 \beta_{6} - 76 \beta_{5} + 66 \beta_{4} + 95 \beta_{3} + 54 \beta_{2} + 28 \beta_{1} - 131$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$97 \beta_{15} + 171 \beta_{14} + 82 \beta_{13} - 3 \beta_{12} + 20 \beta_{11} - 63 \beta_{10} - 349 \beta_{9} - 58 \beta_{8} - 42 \beta_{7} + 249 \beta_{5} + 261 \beta_{4} - 53 \beta_{3} + 294 \beta_{2} + 3 \beta_{1} - 205$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$45 \beta_{14} + 132 \beta_{10} + 345 \beta_{9} - 110 \beta_{6} + 190 \beta_{5} - 144 \beta_{4} + 209 \beta_{2} - 77 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$($$$$-540 \beta_{15} + 766 \beta_{14} + 408 \beta_{13} - 121 \beta_{12} - 444 \beta_{11} - 1211 \beta_{10} + 1813 \beta_{9} + 23 \beta_{8} + 108 \beta_{7} - 288 \beta_{6} - 1345 \beta_{5} + 321 \beta_{4} - 70 \beta_{3} + 1211 \beta_{2} - 121 \beta_{1} - 1957$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-1008 \beta_{15} + 564 \beta_{14} + 209 \beta_{13} + 576 \beta_{12} - 2736 \beta_{11} - 2928 \beta_{10} - 2052 \beta_{9} - 396 \beta_{8} + 252 \beta_{7} - 576 \beta_{6} - 17 \beta_{5} - 1260 \beta_{4} + 2455 \beta_{3} + 624 \beta_{2} + 192 \beta_{1} + 3377$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$2124 \beta_{15} + 1452 \beta_{13} + 3456 \beta_{12} + 2004 \beta_{11} + 2412 \beta_{8} - 864 \beta_{7} + 5700 \beta_{3} + 1801$$ $$\nu^{13}$$ $$=$$ $$($$$$3708 \beta_{15} - 12624 \beta_{14} - 144 \beta_{13} - 288 \beta_{12} - 35 \beta_{11} - 61 \beta_{10} - 7668 \beta_{9} + 17136 \beta_{8} - 14220 \beta_{7} - 2304 \beta_{6} - 48 \beta_{5} - 7524 \beta_{4} - 12743 \beta_{3} + 1500 \beta_{2} + 96 \beta_{1} - 8963$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$-11377 \beta_{15} - 28693 \beta_{14} - 9636 \beta_{13} - 3169 \beta_{12} - 8472 \beta_{11} + 22247 \beta_{10} - 1477 \beta_{9} - 10116 \beta_{8} - 9612 \beta_{7} + 27936 \beta_{6} - 25739 \beta_{5} - 21423 \beta_{4} - 7621 \beta_{3} - 3026 \beta_{2} + 3169 \beta_{1} - 9973$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$-59 \beta_{14} + 1598 \beta_{10} - 34197 \beta_{9} + 14042 \beta_{6} - 12180 \beta_{5} - 21744 \beta_{4} - 26799 \beta_{2} - 25031 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4410\mathbb{Z}\right)^\times$$.

 $$n$$ $$1081$$ $$2647$$ $$3431$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4409.1
 0.948234 + 2.02506i 0.948234 − 2.02506i 1.98669 − 1.02619i 1.98669 + 1.02619i −0.442358 + 2.19188i −0.442358 − 2.19188i 2.11423 + 0.728019i 2.11423 − 0.728019i 1.68760 + 1.46697i 1.68760 − 1.46697i −2.11940 − 0.712845i −2.11940 + 0.712845i 0.104634 + 2.23362i 0.104634 − 2.23362i −1.27963 − 1.83372i −1.27963 + 1.83372i
1.00000 0 1.00000 −2.22787 0.191334i 0 0 1.00000 0 −2.22787 0.191334i
4409.2 1.00000 0 1.00000 −2.22787 + 0.191334i 0 0 1.00000 0 −2.22787 + 0.191334i
4409.3 1.00000 0 1.00000 −1.88205 1.20743i 0 0 1.00000 0 −1.88205 1.20743i
4409.4 1.00000 0 1.00000 −1.88205 + 1.20743i 0 0 1.00000 0 −1.88205 + 1.20743i
4409.5 1.00000 0 1.00000 −1.67704 1.47903i 0 0 1.00000 0 −1.67704 1.47903i
4409.6 1.00000 0 1.00000 −1.67704 + 1.47903i 0 0 1.00000 0 −1.67704 + 1.47903i
4409.7 1.00000 0 1.00000 −0.426635 2.19499i 0 0 1.00000 0 −0.426635 2.19499i
4409.8 1.00000 0 1.00000 −0.426635 + 2.19499i 0 0 1.00000 0 −0.426635 + 2.19499i
4409.9 1.00000 0 1.00000 0.426635 2.19499i 0 0 1.00000 0 0.426635 2.19499i
4409.10 1.00000 0 1.00000 0.426635 + 2.19499i 0 0 1.00000 0 0.426635 + 2.19499i
4409.11 1.00000 0 1.00000 1.67704 1.47903i 0 0 1.00000 0 1.67704 1.47903i
4409.12 1.00000 0 1.00000 1.67704 + 1.47903i 0 0 1.00000 0 1.67704 + 1.47903i
4409.13 1.00000 0 1.00000 1.88205 1.20743i 0 0 1.00000 0 1.88205 1.20743i
4409.14 1.00000 0 1.00000 1.88205 + 1.20743i 0 0 1.00000 0 1.88205 + 1.20743i
4409.15 1.00000 0 1.00000 2.22787 0.191334i 0 0 1.00000 0 2.22787 0.191334i
4409.16 1.00000 0 1.00000 2.22787 + 0.191334i 0 0 1.00000 0 2.22787 + 0.191334i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4409.16 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
15.d odd 2 1 inner
105.g even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4410.2.d.b 16
3.b odd 2 1 4410.2.d.a 16
5.b even 2 1 4410.2.d.a 16
7.b odd 2 1 inner 4410.2.d.b 16
7.c even 3 1 630.2.bo.a 16
7.d odd 6 1 630.2.bo.a 16
15.d odd 2 1 inner 4410.2.d.b 16
21.c even 2 1 4410.2.d.a 16
21.g even 6 1 630.2.bo.b yes 16
21.h odd 6 1 630.2.bo.b yes 16
35.c odd 2 1 4410.2.d.a 16
35.i odd 6 1 630.2.bo.b yes 16
35.j even 6 1 630.2.bo.b yes 16
35.k even 12 2 3150.2.bf.f 32
35.l odd 12 2 3150.2.bf.f 32
105.g even 2 1 inner 4410.2.d.b 16
105.o odd 6 1 630.2.bo.a 16
105.p even 6 1 630.2.bo.a 16
105.w odd 12 2 3150.2.bf.f 32
105.x even 12 2 3150.2.bf.f 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.bo.a 16 7.c even 3 1
630.2.bo.a 16 7.d odd 6 1
630.2.bo.a 16 105.o odd 6 1
630.2.bo.a 16 105.p even 6 1
630.2.bo.b yes 16 21.g even 6 1
630.2.bo.b yes 16 21.h odd 6 1
630.2.bo.b yes 16 35.i odd 6 1
630.2.bo.b yes 16 35.j even 6 1
3150.2.bf.f 32 35.k even 12 2
3150.2.bf.f 32 35.l odd 12 2
3150.2.bf.f 32 105.w odd 12 2
3150.2.bf.f 32 105.x even 12 2
4410.2.d.a 16 3.b odd 2 1
4410.2.d.a 16 5.b even 2 1
4410.2.d.a 16 21.c even 2 1
4410.2.d.a 16 35.c odd 2 1
4410.2.d.b 16 1.a even 1 1 trivial
4410.2.d.b 16 7.b odd 2 1 inner
4410.2.d.b 16 15.d odd 2 1 inner
4410.2.d.b 16 105.g even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4410, [\chi])$$:

 $$T_{11}^{8} + 50 T_{11}^{6} + 426 T_{11}^{4} + 1154 T_{11}^{2} + 961$$ $$T_{23}^{4} - 4 T_{23}^{3} - 53 T_{23}^{2} + 264 T_{23} - 294$$