# Properties

 Label 1078.2.i.c Level $1078$ Weight $2$ Character orbit 1078.i Analytic conductor $8.608$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1078 = 2 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1078.i (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.60787333789$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} - 102 x^{7} + 144 x^{6} - 432 x^{5} + 502 x^{4} + 288 x^{3} + 72 x^{2} + 12 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 154) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( \beta_{3} - \beta_{4} - \beta_{13} ) q^{3} + \beta_{10} q^{4} + ( \beta_{8} + \beta_{10} ) q^{5} + ( \beta_{6} - \beta_{14} ) q^{6} -\beta_{11} q^{8} + ( 3 + \beta_{3} + 2 \beta_{7} - \beta_{8} - 4 \beta_{10} - \beta_{13} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( \beta_{3} - \beta_{4} - \beta_{13} ) q^{3} + \beta_{10} q^{4} + ( \beta_{8} + \beta_{10} ) q^{5} + ( \beta_{6} - \beta_{14} ) q^{6} -\beta_{11} q^{8} + ( 3 + \beta_{3} + 2 \beta_{7} - \beta_{8} - 4 \beta_{10} - \beta_{13} ) q^{9} + ( -\beta_{5} - \beta_{11} ) q^{10} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{11} + \beta_{13} + \beta_{14} ) q^{11} + ( -\beta_{4} - \beta_{12} ) q^{12} + ( 3 \beta_{1} - \beta_{2} - \beta_{6} + \beta_{11} + \beta_{14} + \beta_{15} ) q^{13} + ( 1 - 3 \beta_{4} - 3 \beta_{12} - 3 \beta_{13} ) q^{15} + ( -1 + \beta_{10} ) q^{16} + ( 2 \beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} + 2 \beta_{11} + 2 \beta_{14} + 3 \beta_{15} ) q^{17} + ( 3 \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} - 2 \beta_{9} + 4 \beta_{11} - \beta_{14} ) q^{18} + ( 2 \beta_{5} + \beta_{6} - 2 \beta_{9} + 2 \beta_{11} + \beta_{14} - \beta_{15} ) q^{19} + ( -1 - \beta_{7} + \beta_{8} + 2 \beta_{10} ) q^{20} + ( -1 + \beta_{4} + \beta_{6} + \beta_{12} + \beta_{13} + \beta_{14} ) q^{22} + ( -\beta_{3} - \beta_{4} - 2 \beta_{7} + \beta_{8} + \beta_{10} + \beta_{12} + \beta_{13} ) q^{23} + ( -\beta_{1} - \beta_{2} + \beta_{6} - \beta_{15} ) q^{24} + ( -1 + \beta_{3} + \beta_{4} - \beta_{7} + 2 \beta_{8} + \beta_{10} + \beta_{12} ) q^{25} + ( 1 + \beta_{3} + \beta_{4} + \beta_{10} + \beta_{12} + \beta_{13} ) q^{26} + ( 4 + 4 \beta_{3} + 3 \beta_{4} + 2 \beta_{7} - 2 \beta_{8} - 8 \beta_{10} + \beta_{12} - \beta_{13} ) q^{27} + ( -\beta_{1} - \beta_{2} + \beta_{6} - \beta_{11} + \beta_{14} - \beta_{15} ) q^{29} + ( -2 \beta_{1} + 3 \beta_{6} - 3 \beta_{14} - 3 \beta_{15} ) q^{30} + ( -1 - \beta_{3} + \beta_{4} - 3 \beta_{7} + 2 \beta_{10} + 2 \beta_{12} + 3 \beta_{13} ) q^{31} + ( -\beta_{1} - \beta_{11} ) q^{32} + ( 2 - 4 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{8} + 2 \beta_{9} + \beta_{10} - 6 \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{33} + ( 2 + 3 \beta_{3} + 2 \beta_{4} + \beta_{7} - \beta_{8} - 4 \beta_{10} + \beta_{12} - \beta_{13} ) q^{34} + ( 2 - \beta_{4} + \beta_{7} + \beta_{8} - \beta_{12} - \beta_{13} ) q^{36} + ( -3 - \beta_{4} + 3 \beta_{10} + \beta_{12} ) q^{37} + ( -\beta_{3} + \beta_{4} + 2 \beta_{7} - \beta_{10} - \beta_{12} ) q^{38} + ( 2 \beta_{2} - \beta_{5} + 2 \beta_{6} + 2 \beta_{9} - 2 \beta_{11} - 3 \beta_{14} + \beta_{15} ) q^{39} + ( -\beta_{1} - \beta_{5} + \beta_{9} - 2 \beta_{11} ) q^{40} + ( -2 \beta_{2} + \beta_{6} - \beta_{11} - \beta_{14} + 2 \beta_{15} ) q^{41} + ( -\beta_{1} - 2 \beta_{2} + 2 \beta_{5} + 3 \beta_{6} - \beta_{9} + 6 \beta_{11} + 3 \beta_{14} - 2 \beta_{15} ) q^{43} + ( \beta_{4} - \beta_{6} - \beta_{12} + \beta_{14} + \beta_{15} ) q^{44} + ( 9 + \beta_{3} - \beta_{4} + 3 \beta_{7} - 6 \beta_{10} - \beta_{13} ) q^{45} + ( \beta_{1} - 2 \beta_{2} - \beta_{5} - 2 \beta_{6} + 2 \beta_{9} - \beta_{11} + \beta_{14} + \beta_{15} ) q^{46} + ( -2 \beta_{3} - 2 \beta_{4} - 2 \beta_{12} - 2 \beta_{13} ) q^{47} + ( -\beta_{3} - \beta_{12} + \beta_{13} ) q^{48} + ( \beta_{2} - 2 \beta_{5} + \beta_{9} - \beta_{11} + \beta_{15} ) q^{50} + ( -3 \beta_{1} + \beta_{2} + 3 \beta_{5} - 4 \beta_{6} + 3 \beta_{9} + 3 \beta_{14} + 3 \beta_{15} ) q^{51} + ( 2 \beta_{1} - \beta_{11} + \beta_{14} + \beta_{15} ) q^{52} + ( 1 + 2 \beta_{3} + 2 \beta_{4} + \beta_{7} - 2 \beta_{8} - 7 \beta_{10} + 2 \beta_{13} ) q^{53} + ( 5 \beta_{1} + 4 \beta_{2} + 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{9} + 8 \beta_{11} - \beta_{14} + \beta_{15} ) q^{54} + ( -3 + \beta_{1} - \beta_{2} - \beta_{4} - 3 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 6 \beta_{10} + \beta_{12} - \beta_{13} + 3 \beta_{14} + \beta_{15} ) q^{55} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{6} - 3 \beta_{11} + \beta_{14} + 2 \beta_{15} ) q^{57} + ( -1 - \beta_{3} + \beta_{4} + \beta_{10} - \beta_{12} + \beta_{13} ) q^{58} + ( -4 + 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{10} - \beta_{12} - 4 \beta_{13} ) q^{59} + ( -3 \beta_{3} - 3 \beta_{4} + \beta_{10} - 3 \beta_{12} ) q^{60} + ( -2 \beta_{1} + \beta_{2} - 3 \beta_{5} + 2 \beta_{6} + 3 \beta_{9} - 2 \beta_{11} + \beta_{14} - \beta_{15} ) q^{61} + ( \beta_{1} - 2 \beta_{2} - 3 \beta_{6} + 3 \beta_{9} - 2 \beta_{11} + 3 \beta_{14} + 2 \beta_{15} ) q^{62} - q^{64} + ( 4 \beta_{1} + \beta_{2} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{9} + \beta_{14} + \beta_{15} ) q^{65} + ( -4 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} + 3 \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{66} + ( -3 \beta_{3} - 3 \beta_{4} + 4 \beta_{10} - \beta_{12} - 2 \beta_{13} ) q^{67} + ( 3 \beta_{1} + 3 \beta_{2} + \beta_{5} + 2 \beta_{6} - \beta_{9} + 4 \beta_{11} - \beta_{14} + \beta_{15} ) q^{68} + ( -3 - 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{7} + 3 \beta_{8} + 6 \beta_{10} ) q^{69} + ( 3 - 2 \beta_{3} + \beta_{4} - 2 \beta_{7} - 2 \beta_{8} - \beta_{12} - \beta_{13} ) q^{71} + ( \beta_{1} - \beta_{5} + \beta_{6} - \beta_{9} - \beta_{14} - \beta_{15} ) q^{72} + ( -2 \beta_{1} + \beta_{5} + 5 \beta_{11} + 2 \beta_{14} + 2 \beta_{15} ) q^{73} + ( -2 \beta_{1} - \beta_{2} - \beta_{6} - 3 \beta_{11} + \beta_{15} ) q^{74} + ( -3 \beta_{3} - 4 \beta_{4} - 2 \beta_{8} - 2 \beta_{10} - 4 \beta_{12} - 3 \beta_{13} ) q^{75} + ( -\beta_{1} + \beta_{2} - 2 \beta_{9} + \beta_{11} - \beta_{15} ) q^{76} + ( \beta_{3} - 3 \beta_{4} - \beta_{7} - \beta_{8} - 2 \beta_{12} - 2 \beta_{13} ) q^{78} + ( -4 \beta_{1} + \beta_{2} - 3 \beta_{5} + \beta_{6} - 3 \beta_{9} - 2 \beta_{14} - 2 \beta_{15} ) q^{79} + ( -1 - \beta_{7} + \beta_{10} ) q^{80} + ( 1 + 3 \beta_{3} + 3 \beta_{4} + \beta_{7} - 2 \beta_{8} - 3 \beta_{10} + 5 \beta_{12} - 2 \beta_{13} ) q^{81} + ( -1 + 2 \beta_{3} - \beta_{4} - \beta_{10} - \beta_{12} + 2 \beta_{13} ) q^{82} + ( -3 \beta_{1} - 2 \beta_{2} + \beta_{6} - \beta_{9} - 2 \beta_{11} - \beta_{14} + 2 \beta_{15} ) q^{83} + ( 6 \beta_{1} + 6 \beta_{2} - 2 \beta_{6} + 2 \beta_{11} - 2 \beta_{14} + 6 \beta_{15} ) q^{85} + ( 5 - 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{7} - \beta_{8} - 6 \beta_{10} - 3 \beta_{12} + 2 \beta_{13} ) q^{86} + ( -2 \beta_{2} - \beta_{5} + 2 \beta_{6} - 4 \beta_{11} - \beta_{14} - 3 \beta_{15} ) q^{87} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{10} + \beta_{12} - \beta_{15} ) q^{88} + ( -5 + \beta_{3} + 4 \beta_{4} + \beta_{8} - 4 \beta_{10} + 4 \beta_{12} + \beta_{13} ) q^{89} + ( 9 \beta_{1} + \beta_{6} - 3 \beta_{9} + 6 \beta_{11} - \beta_{14} ) q^{90} + ( 1 + \beta_{3} + \beta_{4} - \beta_{7} - \beta_{8} + 2 \beta_{12} + 2 \beta_{13} ) q^{92} + ( -11 - 6 \beta_{3} - \beta_{4} - 6 \beta_{7} + 3 \beta_{8} + 14 \beta_{10} + \beta_{12} + 6 \beta_{13} ) q^{93} + ( -2 \beta_{1} - 2 \beta_{14} - 2 \beta_{15} ) q^{94} + ( 5 \beta_{1} - 4 \beta_{2} - 4 \beta_{6} + 4 \beta_{11} + 3 \beta_{14} + \beta_{15} ) q^{95} + ( -\beta_{1} - \beta_{2} + \beta_{14} - \beta_{15} ) q^{96} + ( 3 - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{7} - 2 \beta_{8} - 6 \beta_{10} + 2 \beta_{12} - 2 \beta_{13} ) q^{97} + ( 4 - 7 \beta_{1} - 6 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} - 6 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} + \beta_{14} - 6 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 8 q^{4} + 12 q^{5} + 16 q^{9} + O(q^{10})$$ $$16 q + 8 q^{4} + 12 q^{5} + 16 q^{9} + 8 q^{11} - 8 q^{15} - 8 q^{16} - 8 q^{22} + 16 q^{23} + 36 q^{26} + 12 q^{31} + 24 q^{33} + 32 q^{36} - 16 q^{37} - 12 q^{38} - 8 q^{44} + 108 q^{45} - 24 q^{47} - 28 q^{53} - 12 q^{58} - 60 q^{59} - 4 q^{60} - 16 q^{64} - 48 q^{66} + 12 q^{67} + 8 q^{71} - 60 q^{75} - 16 q^{78} - 12 q^{80} - 8 q^{81} + 20 q^{86} - 4 q^{88} - 96 q^{89} + 32 q^{92} - 44 q^{93} + 56 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} - 102 x^{7} + 144 x^{6} - 432 x^{5} + 502 x^{4} + 288 x^{3} + 72 x^{2} + 12 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-80029143512 \nu^{15} + 385788744870 \nu^{14} - 820783926284 \nu^{13} + 848040618120 \nu^{12} + 1720995499366 \nu^{11} - 6827412737484 \nu^{10} + 6402779061545 \nu^{9} - 3349022853273 \nu^{8} - 9000312527194 \nu^{7} + 24987667997140 \nu^{6} - 8831452106135 \nu^{5} + 17087672692002 \nu^{4} + 9515651467064 \nu^{3} - 97367215891956 \nu^{2} + 394928996361 \nu + 33432180594$$$$)/ 3707507912227$$ $$\beta_{2}$$ $$=$$ $$($$$$-115452774644 \nu^{15} + 886173093256 \nu^{14} - 3342656190846 \nu^{13} + 8285369121684 \nu^{12} - 12911491993004 \nu^{11} + 8714136666371 \nu^{10} + 7289551050986 \nu^{9} - 30002334611460 \nu^{8} + 43949161545424 \nu^{7} - 21576071160793 \nu^{6} - 23813610618566 \nu^{5} + 85339924434804 \nu^{4} - 157011910468036 \nu^{3} + 111746321740448 \nu^{2} - 13202743167632 \nu - 1622048373702$$$$)/ 3707507912227$$ $$\beta_{3}$$ $$=$$ $$($$$$229876192869 \nu^{15} - 1434169055319 \nu^{14} + 4481995762636 \nu^{13} - 9349199101054 \nu^{12} + 10037306027966 \nu^{11} + 1831974981075 \nu^{10} - 17299825871464 \nu^{9} + 35017964444230 \nu^{8} - 30103090682590 \nu^{7} - 16716169888734 \nu^{6} + 38997521702488 \nu^{5} - 110456015651282 \nu^{4} + 142750731668897 \nu^{3} + 33344856580986 \nu^{2} + 5189891590886 \nu + 7097366139528$$$$)/ 3707507912227$$ $$\beta_{4}$$ $$=$$ $$($$$$234407101203 \nu^{15} - 1463062841541 \nu^{14} + 4568245377402 \nu^{13} - 9516224014418 \nu^{12} + 10205971274842 \nu^{11} + 1827352370160 \nu^{10} - 17224931483234 \nu^{9} + 34568461621396 \nu^{8} - 29494836526278 \nu^{7} - 17068584063939 \nu^{6} + 37448827738712 \nu^{5} - 108439817187702 \nu^{4} + 143867520986011 \nu^{3} + 33621690891801 \nu^{2} + 5235429778548 \nu - 5702240414038$$$$)/ 3707507912227$$ $$\beta_{5}$$ $$=$$ $$($$$$-323659513794 \nu^{15} + 2126436132890 \nu^{14} - 7021940502570 \nu^{13} + 15520046894136 \nu^{12} - 19351307844336 \nu^{11} + 4000851706865 \nu^{10} + 22846579328323 \nu^{9} - 56754662154696 \nu^{8} + 61179136551334 \nu^{7} + 2104319424462 \nu^{6} - 54144940005777 \nu^{5} + 172779950825427 \nu^{4} - 256920959600440 \nu^{3} + 39533236474234 \nu^{2} - 22206981366431 \nu - 2750714974023$$$$)/ 3707507912227$$ $$\beta_{6}$$ $$=$$ $$($$$$-394538957168 \nu^{15} + 2231811901383 \nu^{14} - 6199106801734 \nu^{13} + 11192177985162 \nu^{12} - 6728239255060 \nu^{11} - 15500105696193 \nu^{10} + 30089262246160 \nu^{9} - 41574777060864 \nu^{8} + 11544594088212 \nu^{7} + 67008107672279 \nu^{6} - 55357249671040 \nu^{5} + 144140956476978 \nu^{4} - 124089191799984 \nu^{3} - 224470101825857 \nu^{2} - 11832162962962 \nu - 1505816495286$$$$)/ 3707507912227$$ $$\beta_{7}$$ $$=$$ $$($$$$401980321088 \nu^{15} - 2507531613006 \nu^{14} + 7838812215514 \nu^{13} - 16353473148736 \nu^{12} + 17544133384043 \nu^{11} + 3227680818897 \nu^{10} - 30296941112790 \nu^{9} + 61133313639343 \nu^{8} - 52069085549277 \nu^{7} - 29293132185474 \nu^{6} + 66949942975780 \nu^{5} - 191025350183983 \nu^{4} + 246377240843999 \nu^{3} + 57553292205336 \nu^{2} + 8958074916496 \nu + 6420259797322$$$$)/ 3707507912227$$ $$\beta_{8}$$ $$=$$ $$($$$$-403027374657 \nu^{15} + 2521501247748 \nu^{14} - 7885517844470 \nu^{13} + 16440266855031 \nu^{12} - 17668207398569 \nu^{11} - 3186799527882 \nu^{10} + 30146969929770 \nu^{9} - 60331564786329 \nu^{8} + 51795261997335 \nu^{7} + 29466424871154 \nu^{6} - 65993485499734 \nu^{5} + 187112766068823 \nu^{4} - 248327455373448 \nu^{3} - 58021984313211 \nu^{2} - 9033094079978 \nu + 4012225523467$$$$)/ 3707507912227$$ $$\beta_{9}$$ $$=$$ $$($$$$556158552240 \nu^{15} - 3247720931080 \nu^{14} + 9404820278882 \nu^{13} - 17969597678127 \nu^{12} + 14300654206114 \nu^{11} + 15947399137709 \nu^{10} - 41579756460341 \nu^{9} + 66454318847001 \nu^{8} - 34746646803794 \nu^{7} - 75102453822786 \nu^{6} + 80010681773455 \nu^{5} - 222201806649708 \nu^{4} + 229349441750990 \nu^{3} + 234407277756901 \nu^{2} + 21061506068817 \nu + 2653688275785$$$$)/ 3707507912227$$ $$\beta_{10}$$ $$=$$ $$($$$$-785074438 \nu^{15} + 4907305158 \nu^{14} - 15343416116 \nu^{13} + 31997236762 \nu^{12} - 34368654754 \nu^{11} - 6224799624 \nu^{10} + 58813815140 \nu^{9} - 118164117069 \nu^{8} + 101294734438 \nu^{7} + 57313765188 \nu^{6} - 129597823592 \nu^{5} + 370531312158 \nu^{4} - 484158220100 \nu^{3} - 113116110792 \nu^{2} - 17609140908 \nu - 1370541375$$$$)/ 1973128213$$ $$\beta_{11}$$ $$=$$ $$($$$$-1616321538 \nu^{15} + 9938916556 \nu^{14} - 30594542475 \nu^{13} + 62871582510 \nu^{12} - 64714538406 \nu^{11} - 18641341380 \nu^{10} + 118271636061 \nu^{9} - 231117756606 \nu^{8} + 186162436800 \nu^{7} + 134499720676 \nu^{6} - 250255719435 \nu^{5} + 736991238906 \nu^{4} - 924410484114 \nu^{3} - 318408624800 \nu^{2} - 82003950372 \nu - 10233974547$$$$)/ 3810388399$$ $$\beta_{12}$$ $$=$$ $$($$$$2432602484212 \nu^{15} - 15210362430228 \nu^{14} + 47566230428388 \nu^{13} - 99202432432724 \nu^{12} + 106574496442407 \nu^{11} + 19314445911990 \nu^{10} - 182513504406276 \nu^{9} + 366427539327546 \nu^{8} - 314133045516125 \nu^{7} - 177707059051737 \nu^{6} + 401666792306712 \nu^{5} - 1146623846023684 \nu^{4} + 1499508109685491 \nu^{3} + 350334214614267 \nu^{2} + 54537167624894 \nu + 3660982645558$$$$)/ 3707507912227$$ $$\beta_{13}$$ $$=$$ $$($$$$-2667083154356 \nu^{15} + 16670596841871 \nu^{14} - 52123769377554 \nu^{13} + 108698529181646 \nu^{12} - 116744726696199 \nu^{11} - 21159726133215 \nu^{10} + 199797208529346 \nu^{9} - 401304270681864 \nu^{8} + 343678264658725 \nu^{7} + 194739830729736 \nu^{6} - 439361137765482 \nu^{5} + 1256589457116658 \nu^{4} - 1642628604419603 \nu^{3} - 383777392568439 \nu^{2} - 59744172756124 \nu - 5233663502032$$$$)/ 3707507912227$$ $$\beta_{14}$$ $$=$$ $$($$$$2737320095452 \nu^{15} - 16635640723011 \nu^{14} + 50529646182960 \nu^{13} - 102286748592894 \nu^{12} + 100494628369536 \nu^{11} + 42467570312879 \nu^{10} - 201266372887674 \nu^{9} + 376503294081456 \nu^{8} - 282140218575116 \nu^{7} - 260583517395552 \nu^{6} + 416079703454118 \nu^{5} - 1212268368579630 \nu^{4} + 1465363584459148 \nu^{3} + 686420342024835 \nu^{2} + 130791312035158 \nu + 16350688060416$$$$)/ 3707507912227$$ $$\beta_{15}$$ $$=$$ $$($$$$-2891771622084 \nu^{15} + 17710086389784 \nu^{14} - 54266401205842 \nu^{13} + 110934804723696 \nu^{12} - 112385854122694 \nu^{11} - 37485437796620 \nu^{10} + 212147297949199 \nu^{9} - 408031002582501 \nu^{8} + 320685750244910 \nu^{7} + 252628621076411 \nu^{6} - 444942596806753 \nu^{5} + 1305146306980602 \nu^{4} - 1617999730660324 \nu^{3} - 622496304087051 \nu^{2} - 143808261987203 \nu - 17956801097484$$$$)/ 3707507912227$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$-\beta_{15} - \beta_{14} + \beta_{6} - 3 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{15} - 4 \beta_{14} - \beta_{13} + \beta_{12} - 2 \beta_{11} + 4 \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + 4 \beta_{6} + 3 \beta_{4} + 4 \beta_{3} + 3 \beta_{2} - 6 \beta_{1} - 2$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$5 \beta_{12} + 9 \beta_{10} + 2 \beta_{8} - \beta_{7} + 5 \beta_{4} + 5 \beta_{3} - 1$$ $$\nu^{5}$$ $$=$$ $$($$$$-8 \beta_{15} + 8 \beta_{14} + 11 \beta_{13} + 19 \beta_{12} + 28 \beta_{11} + 14 \beta_{10} - 8 \beta_{9} + 8 \beta_{8} - 11 \beta_{6} + 8 \beta_{5} + 19 \beta_{4} + 11 \beta_{3} - 19 \beta_{2} + 6 \beta_{1} + 6$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-26 \beta_{15} + \beta_{14} + 46 \beta_{11} - 8 \beta_{9} + \beta_{6} + 16 \beta_{5} - 26 \beta_{2} - 18 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$($$$$-96 \beta_{15} - 47 \beta_{14} - 96 \beta_{13} - 47 \beta_{12} + 82 \beta_{11} + 82 \beta_{10} - 51 \beta_{7} + 49 \beta_{6} + 51 \beta_{5} - 49 \beta_{4} + 49 \beta_{3} - 49 \beta_{2} - 127 \beta_{1} - 113$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$-138 \beta_{13} + 10 \beta_{12} + 245 \beta_{10} + 50 \beta_{8} - 100 \beta_{7} - 10 \beta_{4} + 138 \beta_{3} - 195$$ $$\nu^{9}$$ $$=$$ $$($$$$223 \beta_{15} + 501 \beta_{14} - 278 \beta_{13} + 278 \beta_{12} + 456 \beta_{11} + 912 \beta_{10} - 298 \beta_{9} + 298 \beta_{8} - 298 \beta_{7} - 501 \beta_{6} + 223 \beta_{4} + 501 \beta_{3} - 223 \beta_{2} + 837 \beta_{1} - 456$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$-70 \beta_{15} + 739 \beta_{14} + 1320 \beta_{11} - 576 \beta_{9} - 669 \beta_{6} + 288 \beta_{5} - 669 \beta_{2} + 962 \beta_{1}$$ $$\nu^{11}$$ $$=$$ $$($$$$-1533 \beta_{15} + 1533 \beta_{14} - 1125 \beta_{13} - 2658 \beta_{12} + 4980 \beta_{11} - 2490 \beta_{10} - 1673 \beta_{9} - 1673 \beta_{8} - 1125 \beta_{6} + 1673 \beta_{5} - 2658 \beta_{4} - 1125 \beta_{3} - 2658 \beta_{2} + 957 \beta_{1} - 817$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$-3545 \beta_{13} - 3545 \beta_{12} - 1603 \beta_{8} - 1603 \beta_{7} - 3973 \beta_{4} + 428 \beta_{3} - 3923$$ $$\nu^{13}$$ $$=$$ $$($$$$14219 \beta_{15} + 5865 \beta_{14} - 14219 \beta_{13} - 5865 \beta_{12} - 13502 \beta_{11} + 13502 \beta_{10} - 9210 \beta_{7} - 8354 \beta_{6} - 9210 \beta_{5} - 8354 \beta_{4} + 8354 \beta_{3} + 8354 \beta_{2} + 18511 \beta_{1} - 17794$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$18939 \beta_{15} + 18939 \beta_{14} - 8782 \beta_{9} - 21398 \beta_{6} - 8782 \beta_{5} + 2459 \beta_{2} + 39879 \beta_{1}$$ $$\nu^{15}$$ $$=$$ $$($$$$31097 \beta_{15} + 76382 \beta_{14} + 45285 \beta_{13} - 45285 \beta_{12} + 73006 \beta_{11} - 146012 \beta_{10} - 50203 \beta_{9} - 50203 \beta_{8} + 50203 \beta_{7} - 76382 \beta_{6} - 31097 \beta_{4} - 76382 \beta_{3} - 31097 \beta_{2} + 126906 \beta_{1} + 73006$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$981$$ $$\chi(n)$$ $$1 - \beta_{10}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
901.1
 2.24352 + 0.601150i −1.29724 − 0.347596i 1.60599 + 0.430324i −0.186243 − 0.0499037i 0.601150 − 2.24352i −0.347596 + 1.29724i 0.430324 − 1.60599i −0.0499037 + 0.186243i 2.24352 − 0.601150i −1.29724 + 0.347596i 1.60599 − 0.430324i −0.186243 + 0.0499037i 0.601150 + 2.24352i −0.347596 − 1.29724i 0.430324 + 1.60599i −0.0499037 − 0.186243i
−0.866025 0.500000i −2.24749 + 1.29759i 0.500000 + 0.866025i 3.06486 + 1.76950i 2.59518 0 1.00000i 1.86747 3.23456i −1.76950 3.06486i
901.2 −0.866025 0.500000i −1.35034 + 0.779618i 0.500000 + 0.866025i −0.882559 0.509546i 1.55924 0 1.00000i −0.284392 + 0.492581i 0.509546 + 0.882559i
901.3 −0.866025 0.500000i 0.889740 0.513691i 0.500000 + 0.866025i −1.09005 0.629341i −1.02738 0 1.00000i −0.972242 + 1.68397i 0.629341 + 1.09005i
901.4 −0.866025 0.500000i 2.70809 1.56352i 0.500000 + 0.866025i 1.90775 + 1.10144i −3.12703 0 1.00000i 3.38916 5.87020i −1.10144 1.90775i
901.5 0.866025 + 0.500000i −2.24749 + 1.29759i 0.500000 + 0.866025i 3.06486 + 1.76950i −2.59518 0 1.00000i 1.86747 3.23456i 1.76950 + 3.06486i
901.6 0.866025 + 0.500000i −1.35034 + 0.779618i 0.500000 + 0.866025i −0.882559 0.509546i −1.55924 0 1.00000i −0.284392 + 0.492581i −0.509546 0.882559i
901.7 0.866025 + 0.500000i 0.889740 0.513691i 0.500000 + 0.866025i −1.09005 0.629341i 1.02738 0 1.00000i −0.972242 + 1.68397i −0.629341 1.09005i
901.8 0.866025 + 0.500000i 2.70809 1.56352i 0.500000 + 0.866025i 1.90775 + 1.10144i 3.12703 0 1.00000i 3.38916 5.87020i 1.10144 + 1.90775i
1011.1 −0.866025 + 0.500000i −2.24749 1.29759i 0.500000 0.866025i 3.06486 1.76950i 2.59518 0 1.00000i 1.86747 + 3.23456i −1.76950 + 3.06486i
1011.2 −0.866025 + 0.500000i −1.35034 0.779618i 0.500000 0.866025i −0.882559 + 0.509546i 1.55924 0 1.00000i −0.284392 0.492581i 0.509546 0.882559i
1011.3 −0.866025 + 0.500000i 0.889740 + 0.513691i 0.500000 0.866025i −1.09005 + 0.629341i −1.02738 0 1.00000i −0.972242 1.68397i 0.629341 1.09005i
1011.4 −0.866025 + 0.500000i 2.70809 + 1.56352i 0.500000 0.866025i 1.90775 1.10144i −3.12703 0 1.00000i 3.38916 + 5.87020i −1.10144 + 1.90775i
1011.5 0.866025 0.500000i −2.24749 1.29759i 0.500000 0.866025i 3.06486 1.76950i −2.59518 0 1.00000i 1.86747 + 3.23456i 1.76950 3.06486i
1011.6 0.866025 0.500000i −1.35034 0.779618i 0.500000 0.866025i −0.882559 + 0.509546i −1.55924 0 1.00000i −0.284392 0.492581i −0.509546 + 0.882559i
1011.7 0.866025 0.500000i 0.889740 + 0.513691i 0.500000 0.866025i −1.09005 + 0.629341i 1.02738 0 1.00000i −0.972242 1.68397i −0.629341 + 1.09005i
1011.8 0.866025 0.500000i 2.70809 + 1.56352i 0.500000 0.866025i 1.90775 1.10144i 3.12703 0 1.00000i 3.38916 + 5.87020i 1.10144 1.90775i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1011.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
11.b odd 2 1 inner
77.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.i.c 16
7.b odd 2 1 154.2.i.a 16
7.c even 3 1 154.2.i.a 16
7.c even 3 1 1078.2.c.b 16
7.d odd 6 1 1078.2.c.b 16
7.d odd 6 1 inner 1078.2.i.c 16
11.b odd 2 1 inner 1078.2.i.c 16
21.c even 2 1 1386.2.bk.c 16
21.h odd 6 1 1386.2.bk.c 16
28.d even 2 1 1232.2.bn.b 16
28.g odd 6 1 1232.2.bn.b 16
77.b even 2 1 154.2.i.a 16
77.h odd 6 1 154.2.i.a 16
77.h odd 6 1 1078.2.c.b 16
77.i even 6 1 1078.2.c.b 16
77.i even 6 1 inner 1078.2.i.c 16
231.h odd 2 1 1386.2.bk.c 16
231.l even 6 1 1386.2.bk.c 16
308.g odd 2 1 1232.2.bn.b 16
308.n even 6 1 1232.2.bn.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.i.a 16 7.b odd 2 1
154.2.i.a 16 7.c even 3 1
154.2.i.a 16 77.b even 2 1
154.2.i.a 16 77.h odd 6 1
1078.2.c.b 16 7.c even 3 1
1078.2.c.b 16 7.d odd 6 1
1078.2.c.b 16 77.h odd 6 1
1078.2.c.b 16 77.i even 6 1
1078.2.i.c 16 1.a even 1 1 trivial
1078.2.i.c 16 7.d odd 6 1 inner
1078.2.i.c 16 11.b odd 2 1 inner
1078.2.i.c 16 77.i even 6 1 inner
1232.2.bn.b 16 28.d even 2 1
1232.2.bn.b 16 28.g odd 6 1
1232.2.bn.b 16 308.g odd 2 1
1232.2.bn.b 16 308.n even 6 1
1386.2.bk.c 16 21.c even 2 1
1386.2.bk.c 16 21.h odd 6 1
1386.2.bk.c 16 231.h odd 2 1
1386.2.bk.c 16 231.l even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 10 T_{3}^{6} + 87 T_{3}^{4} + 60 T_{3}^{3} - 118 T_{3}^{2} - 78 T_{3} + 169$$ acting on $$S_{2}^{\mathrm{new}}(1078, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{4}$$
$3$ $$( 169 - 78 T - 118 T^{2} + 60 T^{3} + 87 T^{4} - 10 T^{6} + T^{8} )^{2}$$
$5$ $$( 100 + 180 T + 68 T^{2} - 72 T^{3} - 30 T^{4} + 24 T^{5} + 8 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$7$ $$T^{16}$$
$11$ $$214358881 - 155897368 T + 77948684 T^{2} - 38652240 T^{3} + 15841562 T^{4} - 6058712 T^{5} + 2081200 T^{6} - 682968 T^{7} + 220771 T^{8} - 62088 T^{9} + 17200 T^{10} - 4552 T^{11} + 1082 T^{12} - 240 T^{13} + 44 T^{14} - 8 T^{15} + T^{16}$$
$13$ $$( 25 - 908 T^{2} + 510 T^{4} - 44 T^{6} + T^{8} )^{2}$$
$17$ $$6146560000 + 1816371200 T^{2} + 352359424 T^{4} + 40065536 T^{6} + 3322048 T^{8} + 170048 T^{10} + 6112 T^{12} + 92 T^{14} + T^{16}$$
$19$ $$16 + 656 T^{2} + 21808 T^{4} + 207968 T^{6} + 1604860 T^{8} + 101432 T^{10} + 5128 T^{12} + 80 T^{14} + T^{16}$$
$23$ $$( 196 - 476 T + 1072 T^{2} - 428 T^{3} + 322 T^{4} - 20 T^{5} + 70 T^{6} - 8 T^{7} + T^{8} )^{2}$$
$29$ $$( 9 + 900 T^{2} + 366 T^{4} + 36 T^{6} + T^{8} )^{2}$$
$31$ $$( 3136 - 21504 T + 53408 T^{2} - 29184 T^{3} + 5064 T^{4} + 456 T^{5} - 64 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$37$ $$( 4 - 4 T + 28 T^{2} + 56 T^{3} + 130 T^{4} + 92 T^{5} + 52 T^{6} + 8 T^{7} + T^{8} )^{2}$$
$41$ $$( 1674436 - 287612 T^{2} + 12588 T^{4} - 200 T^{6} + T^{8} )^{2}$$
$43$ $$( 1201216 + 527488 T^{2} + 19536 T^{4} + 244 T^{6} + T^{8} )^{2}$$
$47$ $$( 4096 - 12288 T + 11264 T^{2} + 3072 T^{3} - 576 T^{4} - 192 T^{5} + 32 T^{6} + 12 T^{7} + T^{8} )^{2}$$
$53$ $$( 1170724 + 348404 T + 129652 T^{2} + 22568 T^{3} + 6166 T^{4} + 980 T^{5} + 172 T^{6} + 14 T^{7} + T^{8} )^{2}$$
$59$ $$( 2356225 + 1768320 T + 439298 T^{2} - 2304 T^{3} - 9981 T^{4} + 60 T^{5} + 302 T^{6} + 30 T^{7} + T^{8} )^{2}$$
$61$ $$1706808989601 + 826402153644 T^{2} + 365064614874 T^{4} + 16192668528 T^{6} + 529204995 T^{8} + 6786288 T^{10} + 63162 T^{12} + 300 T^{14} + T^{16}$$
$67$ $$( 173889 + 20016 T + 24822 T^{2} + 2412 T^{3} + 2787 T^{4} + 228 T^{5} + 90 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$71$ $$( 3262 - 338 T - 174 T^{2} - 2 T^{3} + T^{4} )^{4}$$
$73$ $$15704099856 + 18167311152 T^{2} + 19529630496 T^{4} + 1669398768 T^{6} + 111149820 T^{8} + 2131128 T^{10} + 29748 T^{12} + 204 T^{14} + T^{16}$$
$79$ $$1152869294537041 - 71748319001332 T^{2} + 2897165070586 T^{4} - 70424410456 T^{6} + 1253579995 T^{8} - 14246584 T^{10} + 113818 T^{12} - 400 T^{14} + T^{16}$$
$83$ $$( 107584 - 77504 T^{2} + 9888 T^{4} - 260 T^{6} + T^{8} )^{2}$$
$89$ $$( 49617936 + 11411280 T - 224064 T^{2} - 252720 T^{3} + 5460 T^{4} + 7488 T^{5} + 924 T^{6} + 48 T^{7} + T^{8} )^{2}$$
$97$ $$( 496532089 + 14100500 T^{2} + 145902 T^{4} + 644 T^{6} + T^{8} )^{2}$$