# Properties

 Label 1078.2.c.b Level $1078$ Weight $2$ Character orbit 1078.c 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.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

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

 $$f(q)$$ $$=$$ $$q + \beta_{9} q^{2} + \beta_{14} q^{3} - q^{4} + (\beta_{11} - \beta_{8}) q^{5} - \beta_{13} q^{6} - \beta_{9} q^{8} + ( - \beta_{2} - 2) q^{9}+O(q^{10})$$ q + b9 * q^2 + b14 * q^3 - q^4 + (b11 - b8) * q^5 - b13 * q^6 - b9 * q^8 + (-b2 - 2) * q^9 $$q + \beta_{9} q^{2} + \beta_{14} q^{3} - q^{4} + (\beta_{11} - \beta_{8}) q^{5} - \beta_{13} q^{6} - \beta_{9} q^{8} + ( - \beta_{2} - 2) q^{9} + (\beta_{10} + \beta_{6}) q^{10} + ( - \beta_{12} + \beta_{9} - \beta_{3} - 1) q^{11} - \beta_{14} q^{12} + ( - \beta_{10} - \beta_{4}) q^{13} + ( - 3 \beta_1 - 2) q^{15} + q^{16} + ( - \beta_{13} - 2 \beta_{10} - \beta_{6} + 2 \beta_{4}) q^{17} + ( - \beta_{15} - 2 \beta_{9}) q^{18} + ( - \beta_{13} + \beta_{10} + 2 \beta_{6} - \beta_{4}) q^{19} + ( - \beta_{11} + \beta_{8}) q^{20} + ( - \beta_{9} + \beta_{7} + \beta_1) q^{22} + ( - \beta_{3} + \beta_{2} - 2) q^{23} + \beta_{13} q^{24} + ( - \beta_{2} - 2 \beta_1 - 1) q^{25} + (\beta_{11} + \beta_{5}) q^{26} + ( - \beta_{14} + 4 \beta_{11} - 2 \beta_{8} - 3 \beta_{5}) q^{27} + (\beta_{12} - 2 \beta_{9} + \beta_{7}) q^{29} + ( - 3 \beta_{12} + \beta_{9}) q^{30} + ( - \beta_{14} + 2 \beta_{11} - 3 \beta_{8} - 2 \beta_{5}) q^{31} + \beta_{9} q^{32} + ( - \beta_{14} - \beta_{13} + \beta_{11} - 3 \beta_{10} + \beta_{8} - 2 \beta_{6} - \beta_{5}) q^{33} + ( - \beta_{14} + 2 \beta_{11} - \beta_{8} - 2 \beta_{5}) q^{34} + (\beta_{2} + 2) q^{36} + ( - \beta_{3} + 2) q^{37} + ( - \beta_{14} - \beta_{11} + 2 \beta_{8} + \beta_{5}) q^{38} + (\beta_{15} + 4 \beta_{12} - \beta_{9} + \beta_{7}) q^{39} + ( - \beta_{10} - \beta_{6}) q^{40} + ( - 3 \beta_{13} + \beta_{10} - 2 \beta_{4}) q^{41} + (\beta_{15} + 3 \beta_{12} + \beta_{9} + 3 \beta_{7}) q^{43} + (\beta_{12} - \beta_{9} + \beta_{3} + 1) q^{44} + (\beta_{14} - 6 \beta_{11} + 3 \beta_{8}) q^{45} + (\beta_{15} - 2 \beta_{9} + \beta_{7}) q^{46} - 2 \beta_{5} q^{47} + \beta_{14} q^{48} + ( - \beta_{15} - 2 \beta_{12} + \beta_{9}) q^{50} + ( - 3 \beta_{15} + \beta_{12} - 7 \beta_{9} + \beta_{7}) q^{51} + (\beta_{10} + \beta_{4}) q^{52} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 4) q^{53} + (\beta_{13} + 4 \beta_{10} + 2 \beta_{6} - 3 \beta_{4}) q^{54} + ( - \beta_{14} + 2 \beta_{13} - 3 \beta_{11} + 2 \beta_{8} + \beta_{5} - \beta_{4}) q^{55} + ( - 2 \beta_{12} - 4 \beta_{9} + \beta_{7}) q^{57} + (\beta_{3} - \beta_1 + 1) q^{58} + (3 \beta_{14} + 2 \beta_{11} + \beta_{5}) q^{59} + (3 \beta_1 + 2) q^{60} + ( - \beta_{13} - \beta_{10} - 3 \beta_{6} - 2 \beta_{4}) q^{61} + (\beta_{13} + 2 \beta_{10} + 3 \beta_{6} - 2 \beta_{4}) q^{62} - q^{64} + (2 \beta_{15} + 4 \beta_{12} + 2 \beta_{9} + \beta_{7}) q^{65} + ( - \beta_{14} + \beta_{13} + 3 \beta_{11} + \beta_{10} - 2 \beta_{8} - \beta_{6} - \beta_{4}) q^{66} + (2 \beta_{3} + \beta_1 - 1) q^{67} + (\beta_{13} + 2 \beta_{10} + \beta_{6} - 2 \beta_{4}) q^{68} + ( - 3 \beta_{11} + 3 \beta_{8} + 2 \beta_{5}) q^{69} + ( - 2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{71} + (\beta_{15} + 2 \beta_{9}) q^{72} + ( - 5 \beta_{10} - \beta_{6} + 2 \beta_{4}) q^{73} + (2 \beta_{9} + \beta_{7}) q^{74} + ( - \beta_{14} - 2 \beta_{11} + 2 \beta_{8} - 3 \beta_{5}) q^{75} + (\beta_{13} - \beta_{10} - 2 \beta_{6} + \beta_{4}) q^{76} + (\beta_{3} - \beta_{2} - 4 \beta_1 - 3) q^{78} + (3 \beta_{15} + 2 \beta_{12} - 3 \beta_{9} + \beta_{7}) q^{79} + (\beta_{11} - \beta_{8}) q^{80} + (2 \beta_{3} + \beta_{2} - 4 \beta_1 - 1) q^{81} + ( - 3 \beta_{14} - \beta_{11} + 2 \beta_{5}) q^{82} + ( - 3 \beta_{13} + 2 \beta_{10} - \beta_{6} - 2 \beta_{4}) q^{83} + ( - 6 \beta_{12} + 4 \beta_{9} - 2 \beta_{7}) q^{85} + (3 \beta_{3} - \beta_{2} - 3 \beta_1 - 4) q^{86} + (2 \beta_{13} + 4 \beta_{10} + \beta_{6} - \beta_{4}) q^{87} + (\beta_{9} - \beta_{7} - \beta_1) q^{88} + (3 \beta_{14} - 4 \beta_{11} - \beta_{8} + \beta_{5}) q^{89} + ( - \beta_{13} - 6 \beta_{10} - 3 \beta_{6}) q^{90} + (\beta_{3} - \beta_{2} + 2) q^{92} + ( - \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 4) q^{93} - 2 \beta_{4} q^{94} + ( - 3 \beta_{12} - 5 \beta_{9} + \beta_{7}) q^{95} - \beta_{13} q^{96} + ( - 2 \beta_{14} + 3 \beta_{11} - 2 \beta_{8} + 4 \beta_{5}) q^{97} + ( - \beta_{15} + 5 \beta_{12} - 5 \beta_{9} + \beta_{7} - \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{99}+O(q^{100})$$ q + b9 * q^2 + b14 * q^3 - q^4 + (b11 - b8) * q^5 - b13 * q^6 - b9 * q^8 + (-b2 - 2) * q^9 + (b10 + b6) * q^10 + (-b12 + b9 - b3 - 1) * q^11 - b14 * q^12 + (-b10 - b4) * q^13 + (-3*b1 - 2) * q^15 + q^16 + (-b13 - 2*b10 - b6 + 2*b4) * q^17 + (-b15 - 2*b9) * q^18 + (-b13 + b10 + 2*b6 - b4) * q^19 + (-b11 + b8) * q^20 + (-b9 + b7 + b1) * q^22 + (-b3 + b2 - 2) * q^23 + b13 * q^24 + (-b2 - 2*b1 - 1) * q^25 + (b11 + b5) * q^26 + (-b14 + 4*b11 - 2*b8 - 3*b5) * q^27 + (b12 - 2*b9 + b7) * q^29 + (-3*b12 + b9) * q^30 + (-b14 + 2*b11 - 3*b8 - 2*b5) * q^31 + b9 * q^32 + (-b14 - b13 + b11 - 3*b10 + b8 - 2*b6 - b5) * q^33 + (-b14 + 2*b11 - b8 - 2*b5) * q^34 + (b2 + 2) * q^36 + (-b3 + 2) * q^37 + (-b14 - b11 + 2*b8 + b5) * q^38 + (b15 + 4*b12 - b9 + b7) * q^39 + (-b10 - b6) * q^40 + (-3*b13 + b10 - 2*b4) * q^41 + (b15 + 3*b12 + b9 + 3*b7) * q^43 + (b12 - b9 + b3 + 1) * q^44 + (b14 - 6*b11 + 3*b8) * q^45 + (b15 - 2*b9 + b7) * q^46 - 2*b5 * q^47 + b14 * q^48 + (-b15 - 2*b12 + b9) * q^50 + (-3*b15 + b12 - 7*b9 + b7) * q^51 + (b10 + b4) * q^52 + (-2*b3 + b2 + b1 + 4) * q^53 + (b13 + 4*b10 + 2*b6 - 3*b4) * q^54 + (-b14 + 2*b13 - 3*b11 + 2*b8 + b5 - b4) * q^55 + (-2*b12 - 4*b9 + b7) * q^57 + (b3 - b1 + 1) * q^58 + (3*b14 + 2*b11 + b5) * q^59 + (3*b1 + 2) * q^60 + (-b13 - b10 - 3*b6 - 2*b4) * q^61 + (b13 + 2*b10 + 3*b6 - 2*b4) * q^62 - q^64 + (2*b15 + 4*b12 + 2*b9 + b7) * q^65 + (-b14 + b13 + 3*b11 + b10 - 2*b8 - b6 - b4) * q^66 + (2*b3 + b1 - 1) * q^67 + (b13 + 2*b10 + b6 - 2*b4) * q^68 + (-3*b11 + 3*b8 + 2*b5) * q^69 + (-2*b3 - 2*b2 - b1) * q^71 + (b15 + 2*b9) * q^72 + (-5*b10 - b6 + 2*b4) * q^73 + (2*b9 + b7) * q^74 + (-b14 - 2*b11 + 2*b8 - 3*b5) * q^75 + (b13 - b10 - 2*b6 + b4) * q^76 + (b3 - b2 - 4*b1 - 3) * q^78 + (3*b15 + 2*b12 - 3*b9 + b7) * q^79 + (b11 - b8) * q^80 + (2*b3 + b2 - 4*b1 - 1) * q^81 + (-3*b14 - b11 + 2*b5) * q^82 + (-3*b13 + 2*b10 - b6 - 2*b4) * q^83 + (-6*b12 + 4*b9 - 2*b7) * q^85 + (3*b3 - b2 - 3*b1 - 4) * q^86 + (2*b13 + 4*b10 + b6 - b4) * q^87 + (b9 - b7 - b1) * q^88 + (3*b14 - 4*b11 - b8 + b5) * q^89 + (-b13 - 6*b10 - 3*b6) * q^90 + (b3 - b2 + 2) * q^92 + (-b3 + 3*b2 - 3*b1 + 4) * q^93 - 2*b4 * q^94 + (-3*b12 - 5*b9 + b7) * q^95 - b13 * q^96 + (-2*b14 + 3*b11 - 2*b8 + 4*b5) * q^97 + (-b15 + 5*b12 - 5*b9 + b7 - b3 + 2*b2 + b1 + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 16 q^{4} - 32 q^{9}+O(q^{10})$$ 16 * q - 16 * q^4 - 32 * q^9 $$16 q - 16 q^{4} - 32 q^{9} - 16 q^{11} - 8 q^{15} + 16 q^{16} - 8 q^{22} - 32 q^{23} + 32 q^{36} + 32 q^{37} + 16 q^{44} + 56 q^{53} + 24 q^{58} + 8 q^{60} - 16 q^{64} - 24 q^{67} + 8 q^{71} - 16 q^{78} + 16 q^{81} - 40 q^{86} + 8 q^{88} + 32 q^{92} + 88 q^{93} + 56 q^{99}+O(q^{100})$$ 16 * q - 16 * q^4 - 32 * q^9 - 16 * q^11 - 8 * q^15 + 16 * q^16 - 8 * q^22 - 32 * q^23 + 32 * q^36 + 32 * q^37 + 16 * q^44 + 56 * q^53 + 24 * q^58 + 8 * q^60 - 16 * q^64 - 24 * q^67 + 8 * q^71 - 16 * q^78 + 16 * q^81 - 40 * q^86 + 8 * q^88 + 32 * q^92 + 88 * q^93 + 56 * q^99

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

 $$\beta_{1}$$ $$=$$ $$( - 73568941 \nu^{15} - 2828429898 \nu^{14} + 10706428236 \nu^{13} - 20127265496 \nu^{12} + 35741021050 \nu^{11} + \cdots - 10982429182739 ) / 3707507912227$$ (-73568941*v^15 - 2828429898*v^14 + 10706428236*v^13 - 20127265496*v^12 + 35741021050*v^11 - 17927851065*v^10 + 58772639836*v^9 - 308269732922*v^8 + 50382616322*v^7 - 35812385940*v^6 - 245517720058*v^5 + 1525793905272*v^4 + 747026251899*v^3 + 178512937629*v^2 + 28424647318*v - 10982429182739) / 3707507912227 $$\beta_{2}$$ $$=$$ $$( - 973484628 \nu^{15} + 16798064640 \nu^{14} - 57412057192 \nu^{13} + 106920971791 \nu^{12} - 159815035576 \nu^{11} + \cdots + 17707406591301 ) / 3707507912227$$ (-973484628*v^15 + 16798064640*v^14 - 57412057192*v^13 + 106920971791*v^12 - 159815035576*v^11 + 58809142080*v^10 - 208743822856*v^9 + 1110018585936*v^8 - 324206168264*v^7 + 209105071620*v^6 + 1201975196104*v^5 - 5438378020432*v^4 - 2697240781348*v^3 - 647205045504*v^2 - 103443810800*v + 17707406591301) / 3707507912227 $$\beta_{3}$$ $$=$$ $$( - 4604477275 \nu^{15} + 26065356324 \nu^{14} - 75543186530 \nu^{13} + 146897647868 \nu^{12} - 132924225826 \nu^{11} + \cdots + 1817177370827 ) / 3707507912227$$ (-4604477275*v^15 + 26065356324*v^14 - 75543186530*v^13 + 146897647868*v^12 - 132924225826*v^11 - 13305240150*v^10 - 16121748394*v^9 + 141233089912*v^8 - 557871539990*v^7 + 316601789265*v^6 + 1303176243718*v^5 - 490404558308*v^4 - 369763065215*v^3 - 98321373186*v^2 - 17113540344*v + 1817177370827) / 3707507912227 $$\beta_{4}$$ $$=$$ $$( - 275511061668 \nu^{15} + 1657750582996 \nu^{14} - 4984224043414 \nu^{13} + 9981450357924 \nu^{12} - 9469500994272 \nu^{11} + \cdots - 1555184012514 ) / 3707507912227$$ (-275511061668*v^15 + 1657750582996*v^14 - 4984224043414*v^13 + 9981450357924*v^12 - 9469500994272*v^11 - 4940688808597*v^10 + 20095109174076*v^9 - 36700380318006*v^8 + 25948536491036*v^7 + 28399264833487*v^6 - 41476514830836*v^5 + 119515269818808*v^4 - 137980607533908*v^3 - 79280602131237*v^2 - 12412885174910*v - 1555184012514) / 3707507912227 $$\beta_{5}$$ $$=$$ $$( - 464283294072 \nu^{15} + 2897231896860 \nu^{14} - 9050241140038 \nu^{13} + 18865423115472 \nu^{12} - 20243277302808 \nu^{11} + \cdots - 1395125725490 ) / 3707507912227$$ (-464283294072*v^15 + 2897231896860*v^14 - 9050241140038*v^13 + 18865423115472*v^12 - 20243277302808*v^11 - 3659327351235*v^10 + 34524757354698*v^9 - 69586426065626*v^8 + 59597927208868*v^7 + 33784753952673*v^6 - 76446349441200*v^5 + 218895832838984*v^4 - 286618252654908*v^3 - 66966547472787*v^2 - 10425321369434*v - 1395125725490) / 3707507912227 $$\beta_{6}$$ $$=$$ $$( 476129408728 \nu^{15} - 2861932186210 \nu^{14} + 8584036352598 \nu^{13} - 17121557060007 \nu^{12} + 16021649705480 \nu^{11} + \cdots + 2687120456379 ) / 3707507912227$$ (476129408728*v^15 - 2861932186210*v^14 + 8584036352598*v^13 - 17121557060007*v^12 + 16021649705480*v^11 + 9119986400225*v^10 - 35176977398796*v^9 + 63105295993728*v^8 - 43746959330988*v^7 - 50114785825646*v^6 + 71179229667320*v^5 - 205114133957706*v^4 + 238865093218054*v^3 + 137040061864945*v^2 + 21456435065178*v + 2687120456379) / 3707507912227 $$\beta_{7}$$ $$=$$ $$( 770100281810 \nu^{15} - 4733263012640 \nu^{14} + 14562049553051 \nu^{13} - 29920520825502 \nu^{12} + 30799143245438 \nu^{11} + \cdots + 4887214330899 ) / 3707507912227$$ (770100281810*v^15 - 4733263012640*v^14 + 14562049553051*v^13 - 29920520825502*v^12 + 30799143245438*v^11 + 8829439453946*v^10 - 56098808754161*v^9 + 110050939842954*v^8 - 89459573480504*v^7 - 62707181505525*v^6 + 117223638772823*v^5 - 351034936647114*v^4 + 441822991616242*v^3 + 152138648268578*v^2 + 39169305360240*v + 4887214330899) / 3707507912227 $$\beta_{8}$$ $$=$$ $$( - 805007695745 \nu^{15} + 5029032860754 \nu^{14} - 15724330059984 \nu^{13} + 32793740003767 \nu^{12} - 35212340782612 \nu^{11} + \cdots - 2408034273855 ) / 3707507912227$$ (-805007695745*v^15 + 5029032860754*v^14 - 15724330059984*v^13 + 32793740003767*v^12 - 35212340782612*v^11 - 6414480346779*v^10 + 60443911042560*v^9 - 121464878425672*v^8 + 103864347546612*v^7 + 58759557056628*v^6 - 132943428475514*v^5 + 378138116252806*v^4 - 494704696217447*v^3 - 115575276518547*v^2 - 17991168996474*v - 2408034273855) / 3707507912227 $$\beta_{9}$$ $$=$$ $$( - 1616321538 \nu^{15} + 9938916556 \nu^{14} - 30594542475 \nu^{13} + 62871582510 \nu^{12} - 64714538406 \nu^{11} - 18641341380 \nu^{10} + \cdots - 10233974547 ) / 3810388399$$ (-1616321538*v^15 + 9938916556*v^14 - 30594542475*v^13 + 62871582510*v^12 - 64714538406*v^11 - 18641341380*v^10 + 118271636061*v^9 - 231117756606*v^8 + 186162436800*v^7 + 134499720676*v^6 - 250255719435*v^5 + 736991238906*v^4 - 924410484114*v^3 - 318408624800*v^2 - 82003950372*v - 10233974547) / 3810388399 $$\beta_{10}$$ $$=$$ $$( 854458 \nu^{15} - 5149288 \nu^{14} + 15489103 \nu^{13} - 31002870 \nu^{12} + 29353426 \nu^{11} + 15677868 \nu^{10} - 63062803 \nu^{9} + 114195864 \nu^{8} + \cdots + 4877403 ) / 1828267$$ (854458*v^15 - 5149288*v^14 + 15489103*v^13 - 31002870*v^12 + 29353426*v^11 + 15677868*v^10 - 63062803*v^9 + 114195864*v^8 - 80446252*v^7 - 89178588*v^6 + 128785525*v^5 - 370469382*v^4 + 434157674*v^3 + 248804552*v^2 + 38956914*v + 4877403) / 1828267 $$\beta_{11}$$ $$=$$ $$( 1570148876 \nu^{15} - 9814610316 \nu^{14} + 30686832232 \nu^{13} - 63994473524 \nu^{12} + 68737309508 \nu^{11} + 12449599248 \nu^{10} + \cdots + 4714210963 ) / 1973128213$$ (1570148876*v^15 - 9814610316*v^14 + 30686832232*v^13 - 63994473524*v^12 + 68737309508*v^11 + 12449599248*v^10 - 117627630280*v^9 + 236328234138*v^8 - 202589468876*v^7 - 114627530376*v^6 + 259195647184*v^5 - 741062624316*v^4 + 968316440200*v^3 + 226232221584*v^2 + 35218281816*v + 4714210963) / 1973128213 $$\beta_{12}$$ $$=$$ $$( 3087253540240 \nu^{15} - 18982048227910 \nu^{14} + 58429841322972 \nu^{13} - 120068214463500 \nu^{12} + \cdots + 19545417290592 ) / 3707507912227$$ (3087253540240*v^15 - 18982048227910*v^14 + 58429841322972*v^13 - 120068214463500*v^12 + 123576350616332*v^11 + 35598713867733*v^10 - 225839628061730*v^9 + 441382360047234*v^8 - 355634599263140*v^7 - 256040217912758*v^6 + 477587659531454*v^5 - 1407573904107408*v^4 + 1765495989661296*v^3 + 608117198238559*v^2 + 156616076158474*v + 19545417290592) / 3707507912227 $$\beta_{13}$$ $$=$$ $$( 3131859052620 \nu^{15} - 18867452624394 \nu^{14} + 56728752984694 \nu^{13} - 113478926578056 \nu^{12} + \cdots + 17856504555702 ) / 3707507912227$$ (3131859052620*v^15 - 18867452624394*v^14 + 56728752984694*v^13 - 113478926578056*v^12 + 107222867624596*v^11 + 57967676009072*v^10 - 231355635133834*v^9 + 418078071142320*v^8 - 293684812663328*v^7 - 327591625067831*v^6 + 471436953125158*v^5 - 1356409325056608*v^4 + 1589452776259132*v^3 + 910890443850692*v^2 + 142623474998120*v + 17856504555702) / 3707507912227 $$\beta_{14}$$ $$=$$ $$( - 5329561831437 \nu^{15} + 33315128327418 \nu^{14} - 104171995568578 \nu^{13} + 217250160715424 \nu^{12} + \cdots - 15992012287118 ) / 3707507912227$$ (-5329561831437*v^15 + 33315128327418*v^14 - 104171995568578*v^13 + 217250160715424*v^12 - 233356529166572*v^11 - 42306147026280*v^10 + 399610538807086*v^9 - 802749774453640*v^8 + 687914400857440*v^7 + 389163059670207*v^6 - 880025451774682*v^5 + 2513669318791624*v^4 - 3284887445773991*v^3 - 767456463763692*v^2 - 119471231971904*v - 15992012287118) / 3707507912227 $$\beta_{15}$$ $$=$$ $$( - 7516121976528 \nu^{15} + 46209561787616 \nu^{14} - 142236306189628 \nu^{13} + 292274046112479 \nu^{12} + \cdots - 47582417802291 ) / 3707507912227$$ (-7516121976528*v^15 + 46209561787616*v^14 - 142236306189628*v^13 + 292274046112479*v^12 - 300793116749828*v^11 - 86647872654676*v^10 + 549671926014968*v^9 - 1074450180412176*v^8 + 866011308655208*v^7 + 622075435017104*v^6 - 1161717303443108*v^5 + 3426608236011048*v^4 - 4298074501231946*v^3 - 1480448402799748*v^2 - 381276303387704*v - 47582417802291) / 3707507912227
 $$\nu$$ $$=$$ $$( \beta_{12} + \beta_{9} - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + \beta _1 + 2 ) / 4$$ (b12 + b9 - b7 + b5 + b4 + b3 + b1 + 2) / 4 $$\nu^{2}$$ $$=$$ $$( -\beta_{13} + \beta_{12} + 2\beta_{10} + 2\beta_{9} + \beta_{4} ) / 2$$ (-b13 + b12 + 2*b10 + 2*b9 + b4) / 2 $$\nu^{3}$$ $$=$$ $$( -\beta_{14} - \beta_{13} - 2\beta_{11} + 2\beta_{10} + \beta_{8} + \beta_{6} - 3\beta_{5} + 3\beta_{4} ) / 2$$ (-b14 - b13 - 2*b11 + 2*b10 + b8 + b6 - 3*b5 + 3*b4) / 2 $$\nu^{4}$$ $$=$$ $$( -5\beta_{14} - 9\beta_{11} + 3\beta_{8} - 5\beta_{5} + \beta_{2} + 6\beta _1 + 13 ) / 2$$ (-5*b14 - 9*b11 + 3*b8 - 5*b5 + b2 + 6*b1 + 13) / 2 $$\nu^{5}$$ $$=$$ $$( 8 \beta_{15} - 8 \beta_{14} + 8 \beta_{13} + 35 \beta_{12} - 14 \beta_{11} - 14 \beta_{10} + 29 \beta_{9} + 8 \beta_{8} - 3 \beta_{7} - 8 \beta_{6} - 11 \beta_{5} - 11 \beta_{4} + 3 \beta_{3} + 8 \beta_{2} + 35 \beta _1 + 64 ) / 4$$ (8*b15 - 8*b14 + 8*b13 + 35*b12 - 14*b11 - 14*b10 + 29*b9 + 8*b8 - 3*b7 - 8*b6 - 11*b5 - 11*b4 + 3*b3 + 8*b2 + 35*b1 + 64) / 4 $$\nu^{6}$$ $$=$$ $$8\beta_{15} + 34\beta_{12} + 29\beta_{9} + \beta_{7}$$ 8*b15 + 34*b12 + 29*b9 + b7 $$\nu^{7}$$ $$=$$ $$( 51 \beta_{15} - 49 \beta_{14} - 49 \beta_{13} + 196 \beta_{12} - 82 \beta_{11} + 82 \beta_{10} + 142 \beta_{9} + 51 \beta_{8} + 2 \beta_{7} + 51 \beta_{6} - 47 \beta_{5} + 47 \beta_{4} + 2 \beta_{3} - 51 \beta_{2} - 196 \beta _1 - 338 ) / 4$$ (51*b15 - 49*b14 - 49*b13 + 196*b12 - 82*b11 + 82*b10 + 142*b9 + 51*b8 + 2*b7 + 51*b6 - 47*b5 + 47*b4 + 2*b3 - 51*b2 - 196*b1 - 338) / 4 $$\nu^{8}$$ $$=$$ $$( -148\beta_{14} - 245\beta_{11} + 150\beta_{8} - 118\beta_{5} + 10\beta_{3} - 50\beta_{2} - 188\beta _1 - 323 ) / 2$$ (-148*b14 - 245*b11 + 150*b8 - 118*b5 + 10*b3 - 50*b2 - 188*b1 - 323) / 2 $$\nu^{9}$$ $$=$$ $$( - 278 \beta_{14} + 278 \beta_{13} - 456 \beta_{11} - 456 \beta_{10} + 298 \beta_{8} - 298 \beta_{6} - 223 \beta_{5} - 223 \beta_{4} ) / 2$$ (-278*b14 + 278*b13 - 456*b11 - 456*b10 + 298*b8 - 298*b6 - 223*b5 - 223*b4) / 2 $$\nu^{10}$$ $$=$$ $$( 288 \beta_{15} + 809 \beta_{13} + 1027 \beta_{12} - 1320 \beta_{10} + 674 \beta_{9} + 70 \beta_{7} - 864 \beta_{6} - 599 \beta_{4} ) / 2$$ (288*b15 + 809*b13 + 1027*b12 - 1320*b10 + 674*b9 + 70*b7 - 864*b6 - 599*b4) / 2 $$\nu^{11}$$ $$=$$ $$( 1673 \beta_{15} + 1533 \beta_{14} + 1533 \beta_{13} + 5864 \beta_{12} + 2490 \beta_{11} - 2490 \beta_{10} + 3716 \beta_{9} - 1673 \beta_{8} + 408 \beta_{7} - 1673 \beta_{6} + 1125 \beta_{5} - 1125 \beta_{4} + \cdots - 9580 ) / 4$$ (1673*b15 + 1533*b14 + 1533*b13 + 5864*b12 + 2490*b11 - 2490*b10 + 3716*b9 - 1673*b8 + 408*b7 - 1673*b6 + 1125*b5 - 1125*b4 + 408*b3 - 1673*b2 - 5864*b1 - 9580) / 4 $$\nu^{12}$$ $$=$$ $$428\beta_{3} - 1603\beta_{2} - 5576\beta _1 - 9071$$ 428*b3 - 1603*b2 - 5576*b1 - 9071 $$\nu^{13}$$ $$=$$ $$( - 9210 \beta_{15} - 8354 \beta_{14} + 8354 \beta_{13} - 31783 \beta_{12} - 13502 \beta_{11} - 13502 \beta_{10} - 19597 \beta_{9} + 9210 \beta_{8} - 2489 \beta_{7} - 9210 \beta_{6} - 5865 \beta_{5} + \cdots - 51380 ) / 4$$ (-9210*b15 - 8354*b14 + 8354*b13 - 31783*b12 - 13502*b11 - 13502*b10 - 19597*b9 + 9210*b8 - 2489*b7 - 9210*b6 - 5865*b5 - 5865*b4 + 2489*b3 - 9210*b2 - 31783*b1 - 51380) / 4 $$\nu^{14}$$ $$=$$ $$( - 8782 \beta_{15} + 23857 \beta_{13} - 30180 \beta_{12} - 38504 \beta_{10} - 18481 \beta_{9} - 2459 \beta_{7} - 26346 \beta_{6} - 16480 \beta_{4} ) / 2$$ (-8782*b15 + 23857*b13 - 30180*b12 - 38504*b10 - 18481*b9 - 2459*b7 - 26346*b6 - 16480*b4) / 2 $$\nu^{15}$$ $$=$$ $$( 45285 \beta_{14} + 45285 \beta_{13} + 73006 \beta_{11} - 73006 \beta_{10} - 50203 \beta_{8} - 50203 \beta_{6} + 31097 \beta_{5} - 31097 \beta_{4} ) / 2$$ (45285*b14 + 45285*b13 + 73006*b11 - 73006*b10 - 50203*b8 - 50203*b6 + 31097*b5 - 31097*b4) / 2

## Character values

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

 $$n$$ $$199$$ $$981$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1077.1
 −0.186243 + 0.0499037i 0.601150 − 2.24352i −0.347596 + 1.29724i 1.60599 − 0.430324i 0.430324 − 1.60599i −1.29724 + 0.347596i 2.24352 − 0.601150i −0.0499037 + 0.186243i −0.0499037 − 0.186243i 2.24352 + 0.601150i −1.29724 − 0.347596i 0.430324 + 1.60599i 1.60599 + 0.430324i −0.347596 − 1.29724i 0.601150 + 2.24352i −0.186243 − 0.0499037i
1.00000i 3.12703i −1.00000 2.20288i −3.12703 0 1.00000i −6.77832 2.20288
1077.2 1.00000i 2.59518i −1.00000 3.53900i −2.59518 0 1.00000i −3.73495 −3.53900
1077.3 1.00000i 1.55924i −1.00000 1.01909i −1.55924 0 1.00000i 0.568783 1.01909
1077.4 1.00000i 1.02738i −1.00000 1.25868i −1.02738 0 1.00000i 1.94448 −1.25868
1077.5 1.00000i 1.02738i −1.00000 1.25868i 1.02738 0 1.00000i 1.94448 1.25868
1077.6 1.00000i 1.55924i −1.00000 1.01909i 1.55924 0 1.00000i 0.568783 −1.01909
1077.7 1.00000i 2.59518i −1.00000 3.53900i 2.59518 0 1.00000i −3.73495 3.53900
1077.8 1.00000i 3.12703i −1.00000 2.20288i 3.12703 0 1.00000i −6.77832 −2.20288
1077.9 1.00000i 3.12703i −1.00000 2.20288i 3.12703 0 1.00000i −6.77832 −2.20288
1077.10 1.00000i 2.59518i −1.00000 3.53900i 2.59518 0 1.00000i −3.73495 3.53900
1077.11 1.00000i 1.55924i −1.00000 1.01909i 1.55924 0 1.00000i 0.568783 −1.01909
1077.12 1.00000i 1.02738i −1.00000 1.25868i 1.02738 0 1.00000i 1.94448 1.25868
1077.13 1.00000i 1.02738i −1.00000 1.25868i −1.02738 0 1.00000i 1.94448 −1.25868
1077.14 1.00000i 1.55924i −1.00000 1.01909i −1.55924 0 1.00000i 0.568783 1.01909
1077.15 1.00000i 2.59518i −1.00000 3.53900i −2.59518 0 1.00000i −3.73495 −3.53900
1077.16 1.00000i 3.12703i −1.00000 2.20288i −3.12703 0 1.00000i −6.77832 2.20288
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1077.16 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
11.b odd 2 1 inner
77.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.c.b 16
7.b odd 2 1 inner 1078.2.c.b 16
7.c even 3 1 154.2.i.a 16
7.c even 3 1 1078.2.i.c 16
7.d odd 6 1 154.2.i.a 16
7.d odd 6 1 1078.2.i.c 16
11.b odd 2 1 inner 1078.2.c.b 16
21.g even 6 1 1386.2.bk.c 16
21.h odd 6 1 1386.2.bk.c 16
28.f even 6 1 1232.2.bn.b 16
28.g odd 6 1 1232.2.bn.b 16
77.b even 2 1 inner 1078.2.c.b 16
77.h odd 6 1 154.2.i.a 16
77.h odd 6 1 1078.2.i.c 16
77.i even 6 1 154.2.i.a 16
77.i even 6 1 1078.2.i.c 16
231.k odd 6 1 1386.2.bk.c 16
231.l even 6 1 1386.2.bk.c 16
308.m odd 6 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.c even 3 1
154.2.i.a 16 7.d odd 6 1
154.2.i.a 16 77.h odd 6 1
154.2.i.a 16 77.i even 6 1
1078.2.c.b 16 1.a even 1 1 trivial
1078.2.c.b 16 7.b odd 2 1 inner
1078.2.c.b 16 11.b odd 2 1 inner
1078.2.c.b 16 77.b even 2 1 inner
1078.2.i.c 16 7.c even 3 1
1078.2.i.c 16 7.d odd 6 1
1078.2.i.c 16 77.h odd 6 1
1078.2.i.c 16 77.i even 6 1
1232.2.bn.b 16 28.f even 6 1
1232.2.bn.b 16 28.g odd 6 1
1232.2.bn.b 16 308.m odd 6 1
1232.2.bn.b 16 308.n even 6 1
1386.2.bk.c 16 21.g even 6 1
1386.2.bk.c 16 21.h odd 6 1
1386.2.bk.c 16 231.k odd 6 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 $$T_{3}^{8} + 20T_{3}^{6} + 126T_{3}^{4} + 272T_{3}^{2} + 169$$ acting on $$S_{2}^{\mathrm{new}}(1078, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{8}$$
$3$ $$(T^{8} + 20 T^{6} + 126 T^{4} + 272 T^{2} + \cdots + 169)^{2}$$
$5$ $$(T^{8} + 20 T^{6} + 108 T^{4} + 188 T^{2} + \cdots + 100)^{2}$$
$7$ $$T^{16}$$
$11$ $$(T^{8} + 8 T^{7} + 20 T^{6} - 40 T^{5} + \cdots + 14641)^{2}$$
$13$ $$(T^{8} - 44 T^{6} + 510 T^{4} - 908 T^{2} + \cdots + 25)^{2}$$
$17$ $$(T^{8} - 92 T^{6} + 2352 T^{4} + \cdots + 78400)^{2}$$
$19$ $$(T^{8} - 80 T^{6} + 1272 T^{4} - 164 T^{2} + \cdots + 4)^{2}$$
$23$ $$(T^{4} + 8 T^{3} - 6 T^{2} - 34 T - 14)^{4}$$
$29$ $$(T^{8} + 36 T^{6} + 366 T^{4} + 900 T^{2} + \cdots + 9)^{2}$$
$31$ $$(T^{8} + 164 T^{6} + 7200 T^{4} + \cdots + 3136)^{2}$$
$37$ $$(T^{4} - 8 T^{3} + 12 T^{2} - 2 T - 2)^{4}$$
$41$ $$(T^{8} - 200 T^{6} + 12588 T^{4} + \cdots + 1674436)^{2}$$
$43$ $$(T^{8} + 244 T^{6} + 19536 T^{4} + \cdots + 1201216)^{2}$$
$47$ $$(T^{8} + 80 T^{6} + 1920 T^{4} + \cdots + 4096)^{2}$$
$53$ $$(T^{4} - 14 T^{3} + 24 T^{2} + 322 T - 1082)^{4}$$
$59$ $$(T^{8} + 296 T^{6} + 19974 T^{4} + \cdots + 2356225)^{2}$$
$61$ $$(T^{8} - 300 T^{6} + 26838 T^{4} + \cdots + 1306449)^{2}$$
$67$ $$(T^{4} + 6 T^{3} - 54 T^{2} - 48 T + 417)^{4}$$
$71$ $$(T^{4} - 2 T^{3} - 174 T^{2} - 338 T + 3262)^{4}$$
$73$ $$(T^{8} - 204 T^{6} + 11868 T^{4} + \cdots + 125316)^{2}$$
$79$ $$(T^{8} + 400 T^{6} + 46182 T^{4} + \cdots + 33953929)^{2}$$
$83$ $$(T^{8} - 260 T^{6} + 9888 T^{4} + \cdots + 107584)^{2}$$
$89$ $$(T^{8} + 456 T^{6} + 62088 T^{4} + \cdots + 49617936)^{2}$$
$97$ $$(T^{8} + 644 T^{6} + 145902 T^{4} + \cdots + 496532089)^{2}$$