# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1078,2,Mod(1077,1078)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1078, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1078.1077");

S:= CuspForms(chi, 2);

N := Newforms(S);

 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

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$8.60787333789$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 32x^{14} + 512x^{12} - 2272x^{10} - 1087x^{8} + 72448x^{6} + 819200x^{4} + 1310720x^{2} + 1048576$$ x^16 - 32*x^14 + 512*x^12 - 2272*x^10 - 1087*x^8 + 72448*x^6 + 819200*x^4 + 1310720*x^2 + 1048576 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 32x^{14} + 512x^{12} - 2272x^{10} - 1087x^{8} + 72448x^{6} + 819200x^{4} + 1310720x^{2} + 1048576$$ :

 $$\beta_{1}$$ $$=$$ $$( - 32173 \nu^{14} + 668960 \nu^{12} - 6919680 \nu^{10} - 46669984 \nu^{8} - 133787245 \nu^{6} + 474538112 \nu^{4} + 1011889152 \nu^{2} + \cdots - 481919500288 ) / 341451177984$$ (-32173*v^14 + 668960*v^12 - 6919680*v^10 - 46669984*v^8 - 133787245*v^6 + 474538112*v^4 + 1011889152*v^2 - 481919500288) / 341451177984 $$\beta_{2}$$ $$=$$ $$( 21649515 \nu^{14} - 500615872 \nu^{12} + 4404178432 \nu^{10} + 73242356320 \nu^{8} - 905783138901 \nu^{6} + \cdots + 115772308922368 ) / 28937987334144$$ (21649515*v^14 - 500615872*v^12 + 4404178432*v^10 + 73242356320*v^8 - 905783138901*v^6 + 4403763700640*v^4 + 35129391631360*v^2 + 115772308922368) / 28937987334144 $$\beta_{3}$$ $$=$$ $$( 240905 \nu^{14} - 7913160 \nu^{12} + 129966848 \nu^{10} - 662732768 \nu^{8} + 491616713 \nu^{6} + 13417758552 \nu^{4} + \cdots + 167526400000 ) / 149937758208$$ (240905*v^14 - 7913160*v^12 + 129966848*v^10 - 662732768*v^8 + 491616713*v^6 + 13417758552*v^4 + 194978259968*v^2 + 167526400000) / 149937758208 $$\beta_{4}$$ $$=$$ $$( 42729 \nu^{14} - 1400096 \nu^{12} + 23098880 \nu^{10} - 119264224 \nu^{8} + 119589033 \nu^{6} + 2634257152 \nu^{4} + 35772670976 \nu^{2} + \cdots + 30704009216 ) / 19295207424$$ (42729*v^14 - 1400096*v^12 + 23098880*v^10 - 119264224*v^8 + 119589033*v^6 + 2634257152*v^4 + 35772670976*v^2 + 30704009216) / 19295207424 $$\beta_{5}$$ $$=$$ $$( 116266153 \nu^{15} - 3358638720 \nu^{13} + 46096726528 \nu^{11} - 28654574560 \nu^{9} - 1703395919255 \nu^{7} + \cdots + 298669982056448 \nu ) / 308671864897536$$ (116266153*v^15 - 3358638720*v^13 + 46096726528*v^11 - 28654574560*v^9 - 1703395919255*v^7 + 10664594701728*v^5 + 114707998022656*v^3 + 298669982056448*v) / 308671864897536 $$\beta_{6}$$ $$=$$ $$( - 26548369 \nu^{14} + 933827904 \nu^{12} - 16598608384 \nu^{10} + 111831131872 \nu^{8} - 267579354193 \nu^{6} + \cdots + 8620523487232 ) / 9645995778048$$ (-26548369*v^14 + 933827904*v^12 - 16598608384*v^10 + 111831131872*v^8 - 267579354193*v^6 - 2009136336864*v^4 - 11825933968384*v^2 + 8620523487232) / 9645995778048 $$\beta_{7}$$ $$=$$ $$( 1040915 \nu^{14} - 33066624 \nu^{12} + 519454208 \nu^{10} - 2062231712 \nu^{8} - 5141852845 \nu^{6} + 101561457888 \nu^{4} + \cdots + 849802264576 ) / 256088383488$$ (1040915*v^14 - 33066624*v^12 + 519454208*v^10 - 2062231712*v^8 - 5141852845*v^6 + 101561457888*v^4 + 743042069504*v^2 + 849802264576) / 256088383488 $$\beta_{8}$$ $$=$$ $$( - 135220701 \nu^{14} + 4561306816 \nu^{12} - 76278327808 \nu^{10} + 410096159072 \nu^{8} - 107080965789 \nu^{6} + \cdots - 49543323222016 ) / 28937987334144$$ (-135220701*v^14 + 4561306816*v^12 - 76278327808*v^10 + 410096159072*v^8 - 107080965789*v^6 - 12255806975456*v^4 - 84320719473664*v^2 - 49543323222016) / 28937987334144 $$\beta_{9}$$ $$=$$ $$( - 263077499 \nu^{15} + 7568932160 \nu^{13} - 104813186560 \nu^{11} + 66556609440 \nu^{9} + 3864561461317 \nu^{7} + \cdots - 723250826477568 \nu ) / 308671864897536$$ (-263077499*v^15 + 7568932160*v^13 - 104813186560*v^11 + 66556609440*v^9 + 3864561461317*v^7 - 27621977981728*v^5 - 279805449960448*v^3 - 723250826477568*v) / 308671864897536 $$\beta_{10}$$ $$=$$ $$( 15339039 \nu^{15} - 510774904 \nu^{13} + 8514025600 \nu^{11} - 45705450272 \nu^{9} + 42003795807 \nu^{7} + 997736067176 \nu^{5} + \cdots - 13167578624000 \nu ) / 14468993667072$$ (15339039*v^15 - 510774904*v^13 + 8514025600*v^11 - 45705450272*v^9 + 42003795807*v^7 + 997736067176*v^5 + 13433955225472*v^3 - 13167578624000*v) / 14468993667072 $$\beta_{11}$$ $$=$$ $$( 133618959 \nu^{15} - 4312976672 \nu^{13} + 70457976320 \nu^{11} - 349822477600 \nu^{9} + 381384242511 \nu^{7} + \cdots + 289533980278784 \nu ) / 115751949336576$$ (133618959*v^15 - 4312976672*v^13 + 70457976320*v^11 - 349822477600*v^9 + 381384242511*v^7 + 7821020117440*v^5 + 107128611381248*v^3 + 289533980278784*v) / 115751949336576 $$\beta_{12}$$ $$=$$ $$( - 10897203 \nu^{15} + 359824640 \nu^{13} - 5954700800 \nu^{11} + 30984053920 \nu^{9} - 17555552883 \nu^{7} + \cdots - 4233948004352 \nu ) / 8194828271616$$ (-10897203*v^15 + 359824640*v^13 - 5954700800*v^11 + 30984053920*v^9 - 17555552883*v^7 - 851122418272*v^5 - 7459200822272*v^3 - 4233948004352*v) / 8194828271616 $$\beta_{13}$$ $$=$$ $$( 77172743 \nu^{15} - 2548789688 \nu^{13} + 42111652864 \nu^{11} - 219318324768 \nu^{9} + 178889617223 \nu^{7} + \cdots + 34935425286144 \nu ) / 28937987334144$$ (77172743*v^15 - 2548789688*v^13 + 42111652864*v^11 - 219318324768*v^9 + 178889617223*v^7 + 4585099534888*v^5 + 64498714624768*v^3 + 34935425286144*v) / 28937987334144 $$\beta_{14}$$ $$=$$ $$( - 319597619 \nu^{15} + 10421936192 \nu^{13} - 170792382976 \nu^{11} + 861452174496 \nu^{9} - 760912344947 \nu^{7} + \cdots - 303112411742208 \nu ) / 115751949336576$$ (-319597619*v^15 + 10421936192*v^13 - 170792382976*v^11 + 861452174496*v^9 - 760912344947*v^7 - 18179529719584*v^5 - 257651828076544*v^3 - 303112411742208*v) / 115751949336576 $$\beta_{15}$$ $$=$$ $$( - 25933907 \nu^{15} + 863194304 \nu^{13} - 14336292352 \nu^{11} + 75544371360 \nu^{9} - 37801257875 \nu^{7} + \cdots - 10214744358912 \nu ) / 8194828271616$$ (-25933907*v^15 + 863194304*v^13 - 14336292352*v^11 + 75544371360*v^9 - 37801257875*v^7 - 2043398985376*v^5 - 17995089961984*v^3 - 10214744358912*v) / 8194828271616
 $$\nu$$ $$=$$ $$( \beta_{14} + \beta_{13} + \beta_{11} - \beta_{10} ) / 2$$ (b14 + b13 + b11 - b10) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{8} + \beta_{7} + \beta_{6} + 8\beta_{4} - 8\beta_{3} - \beta_{2} + 8\beta _1 + 9 ) / 2$$ (b8 + b7 + b6 + 8*b4 - 8*b3 - b2 + 8*b1 + 9) / 2 $$\nu^{3}$$ $$=$$ $$( 4\beta_{15} + 17\beta_{14} + 9\beta_{13} - 4\beta_{12} + 17\beta_{11} + 9\beta_{10} + 4\beta_{9} + 12\beta_{5} ) / 2$$ (4*b15 + 17*b14 + 9*b13 - 4*b12 + 17*b11 + 9*b10 + 4*b9 + 12*b5) / 2 $$\nu^{4}$$ $$=$$ $$( -17\beta_{8} + 17\beta_{7} + 33\beta_{6} + 136\beta_{4} - 208\beta_{3} - 33\beta_{2} + 33 ) / 2$$ (-17*b8 + 17*b7 + 33*b6 + 136*b4 - 208*b3 - 33*b2 + 33) / 2 $$\nu^{5}$$ $$=$$ $$( - 68 \beta_{15} + 121 \beta_{14} - 121 \beta_{13} + 132 \beta_{12} + 273 \beta_{11} + 273 \beta_{10} + 132 \beta_{9} + 332 \beta_{5} ) / 2$$ (-68*b15 + 121*b14 - 121*b13 + 132*b12 + 273*b11 + 273*b10 + 132*b9 + 332*b5) / 2 $$\nu^{6}$$ $$=$$ $$( -737\beta_{8} - 321\beta_{7} + 737\beta_{6} + 1576\beta_{4} - 2184\beta_{3} - 321\beta_{2} - 1576\beta _1 - 1863 ) / 2$$ (-737*b8 - 321*b7 + 737*b6 + 1576*b4 - 2184*b3 - 321*b2 - 1576*b1 - 1863) / 2 $$\nu^{7}$$ $$=$$ $$( - 2948 \beta_{15} - 1897 \beta_{14} - 4497 \beta_{13} + 7180 \beta_{12} + 1897 \beta_{11} + 4497 \beta_{10} + 1284 \beta_{9} + 2948 \beta_{5} ) / 2$$ (-2948*b15 - 1897*b14 - 4497*b13 + 7180*b12 + 1897*b11 + 4497*b10 + 1284*b9 + 2948*b5) / 2 $$\nu^{8}$$ $$=$$ $$( -14625\beta_{8} - 14625\beta_{7} + 6129\beta_{6} + 6129\beta_{2} - 35976\beta _1 - 57281 ) / 2$$ (-14625*b8 - 14625*b7 + 6129*b6 + 6129*b2 - 35976*b1 - 57281) / 2 $$\nu^{9}$$ $$=$$ $$( - 58500 \beta_{15} - 76177 \beta_{14} - 76177 \beta_{13} + 141516 \beta_{12} - 31705 \beta_{11} + 31705 \beta_{10} - 24516 \beta_{9} - 58500 \beta_{5} ) / 2$$ (-58500*b15 - 76177*b14 - 76177*b13 + 141516*b12 - 31705*b11 + 31705*b10 - 24516*b9 - 58500*b5) / 2 $$\nu^{10}$$ $$=$$ $$( - 114721 \beta_{8} - 276193 \beta_{7} - 114721 \beta_{6} - 431528 \beta_{4} + 609416 \beta_{3} + 276193 \beta_{2} - 431528 \beta _1 - 885609 ) / 2$$ (-114721*b8 - 276193*b7 - 114721*b6 - 431528*b4 + 609416*b3 + 276193*b2 - 431528*b1 - 885609) / 2 $$\nu^{11}$$ $$=$$ $$( - 458884 \beta_{15} - 1317137 \beta_{14} - 546249 \beta_{13} + 1104772 \beta_{12} - 1317137 \beta_{11} - 546249 \beta_{10} - 1104772 \beta_{9} - 2668428 \beta_{5} ) / 2$$ (-458884*b15 - 1317137*b14 - 546249*b13 + 1104772*b12 - 1317137*b11 - 546249*b10 - 1104772*b9 - 2668428*b5) / 2 $$\nu^{12}$$ $$=$$ $$( 2109905 \beta_{8} - 2109905 \beta_{7} - 5090337 \beta_{6} - 10537096 \beta_{4} + 14907088 \beta_{3} + 5090337 \beta_{2} - 5090337 ) / 2$$ (2109905*b8 - 2109905*b7 - 5090337*b6 - 10537096*b4 + 14907088*b3 + 5090337*b2 - 5090337) / 2 $$\nu^{13}$$ $$=$$ $$( 8439620 \beta_{15} - 9563449 \beta_{14} + 9563449 \beta_{13} - 20361348 \beta_{12} - 23080977 \beta_{11} - 23080977 \beta_{10} - 20361348 \beta_{9} - 49162316 \beta_{5} ) / 2$$ (8439620*b15 - 9563449*b14 + 9563449*b13 - 20361348*b12 - 23080977*b11 - 23080977*b10 - 20361348*b9 - 49162316*b5) / 2 $$\nu^{14}$$ $$=$$ $$( 92604641 \beta_{8} + 38364417 \beta_{7} - 92604641 \beta_{6} - 130577704 \beta_{4} + 184647816 \beta_{3} + 38364417 \beta_{2} + 130577704 \beta _1 + 146283399 ) / 2$$ (92604641*b8 + 38364417*b7 - 92604641*b6 - 130577704*b4 + 184647816*b3 + 38364417*b2 + 130577704*b1 + 146283399) / 2 $$\nu^{15}$$ $$=$$ $$( 370418564 \beta_{15} + 168942121 \beta_{14} + 407830161 \beta_{13} - 894294796 \beta_{12} - 168942121 \beta_{11} - 407830161 \beta_{10} - 153457668 \beta_{9} + \cdots - 370418564 \beta_{5} ) / 2$$ (370418564*b15 + 168942121*b14 + 407830161*b13 - 894294796*b12 - 168942121*b11 - 407830161*b10 - 153457668*b9 - 370418564*b5) / 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1077.1
 2.98338 − 1.23576i −3.90726 + 1.61844i −0.807586 − 1.94969i 0.424903 + 1.02581i −0.424903 − 1.02581i 0.807586 + 1.94969i 3.90726 − 1.61844i −2.98338 + 1.23576i −2.98338 − 1.23576i 3.90726 + 1.61844i 0.807586 − 1.94969i −0.424903 + 1.02581i 0.424903 − 1.02581i −0.807586 + 1.94969i −3.90726 − 1.61844i 2.98338 + 1.23576i
1.00000i 1.84776i −1.00000 3.23688i −1.84776 0 1.00000i −0.414214 −3.23688
1077.2 1.00000i 1.84776i −1.00000 2.47151i −1.84776 0 1.00000i −0.414214 2.47151
1077.3 1.00000i 0.765367i −1.00000 2.05161i −0.765367 0 1.00000i 2.41421 −2.05161
1077.4 1.00000i 0.765367i −1.00000 3.89937i −0.765367 0 1.00000i 2.41421 3.89937
1077.5 1.00000i 0.765367i −1.00000 3.89937i 0.765367 0 1.00000i 2.41421 −3.89937
1077.6 1.00000i 0.765367i −1.00000 2.05161i 0.765367 0 1.00000i 2.41421 2.05161
1077.7 1.00000i 1.84776i −1.00000 2.47151i 1.84776 0 1.00000i −0.414214 −2.47151
1077.8 1.00000i 1.84776i −1.00000 3.23688i 1.84776 0 1.00000i −0.414214 3.23688
1077.9 1.00000i 1.84776i −1.00000 3.23688i 1.84776 0 1.00000i −0.414214 3.23688
1077.10 1.00000i 1.84776i −1.00000 2.47151i 1.84776 0 1.00000i −0.414214 −2.47151
1077.11 1.00000i 0.765367i −1.00000 2.05161i 0.765367 0 1.00000i 2.41421 2.05161
1077.12 1.00000i 0.765367i −1.00000 3.89937i 0.765367 0 1.00000i 2.41421 −3.89937
1077.13 1.00000i 0.765367i −1.00000 3.89937i −0.765367 0 1.00000i 2.41421 3.89937
1077.14 1.00000i 0.765367i −1.00000 2.05161i −0.765367 0 1.00000i 2.41421 −2.05161
1077.15 1.00000i 1.84776i −1.00000 2.47151i −1.84776 0 1.00000i −0.414214 2.47151
1077.16 1.00000i 1.84776i −1.00000 3.23688i −1.84776 0 1.00000i −0.414214 −3.23688
 $$n$$: 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.c 16
7.b odd 2 1 inner 1078.2.c.c 16
7.c even 3 2 1078.2.i.d 32
7.d odd 6 2 1078.2.i.d 32
11.b odd 2 1 inner 1078.2.c.c 16
77.b even 2 1 inner 1078.2.c.c 16
77.h odd 6 2 1078.2.i.d 32
77.i even 6 2 1078.2.i.d 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1078.2.c.c 16 1.a even 1 1 trivial
1078.2.c.c 16 7.b odd 2 1 inner
1078.2.c.c 16 11.b odd 2 1 inner
1078.2.c.c 16 77.b even 2 1 inner
1078.2.i.d 32 7.c even 3 2
1078.2.i.d 32 7.d odd 6 2
1078.2.i.d 32 77.h odd 6 2
1078.2.i.d 32 77.i even 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 4T_{3}^{2} + 2$$ acting on $$S_{2}^{\mathrm{new}}(1078, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{8}$$
$3$ $$(T^{4} + 4 T^{2} + 2)^{4}$$
$5$ $$(T^{8} + 36 T^{6} + 450 T^{4} + 2304 T^{2} + \cdots + 4096)^{2}$$
$7$ $$T^{16}$$
$11$ $$(T^{8} - 8 T^{7} + 34 T^{6} - 128 T^{5} + \cdots + 14641)^{2}$$
$13$ $$(T^{8} - 88 T^{6} + 2216 T^{4} + \cdots + 20736)^{2}$$
$17$ $$(T^{8} - 148 T^{6} + 7490 T^{4} + \cdots + 1016064)^{2}$$
$19$ $$(T^{8} - 92 T^{6} + 2186 T^{4} + \cdots + 1296)^{2}$$
$23$ $$(T^{4} + 4 T^{3} - 38 T^{2} - 192 T - 144)^{4}$$
$29$ $$(T^{8} + 184 T^{6} + 10768 T^{4} + \cdots + 331776)^{2}$$
$31$ $$(T^{8} + 132 T^{6} + 5634 T^{4} + \cdots + 322624)^{2}$$
$37$ $$(T^{4} + 20 T^{3} + 68 T^{2} - 576 T - 2592)^{4}$$
$41$ $$(T^{8} - 68 T^{6} + 1058 T^{4} + \cdots + 5184)^{2}$$
$43$ $$(T^{8} + 132 T^{6} + 5620 T^{4} + \cdots + 419904)^{2}$$
$47$ $$(T^{8} + 52 T^{6} + 674 T^{4} + 1568 T^{2} + \cdots + 64)^{2}$$
$53$ $$(T^{4} - 8 T^{3} - 10 T^{2} + 136 T - 56)^{4}$$
$59$ $$(T^{8} + 120 T^{6} + 3444 T^{4} + \cdots + 21316)^{2}$$
$61$ $$(T^{8} - 424 T^{6} + 65672 T^{4} + \cdots + 107495424)^{2}$$
$67$ $$(T^{4} - 4 T^{3} - 54 T^{2} + 296 T - 376)^{4}$$
$71$ $$(T^{4} + 12 T^{3} - 62 T^{2} - 912 T - 1568)^{4}$$
$73$ $$(T^{8} - 356 T^{6} + 42242 T^{4} + \cdots + 5184)^{2}$$
$79$ $$(T^{2} + 72)^{8}$$
$83$ $$(T^{8} - 316 T^{6} + 31658 T^{4} + \cdots + 12194064)^{2}$$
$89$ $$(T^{8} + 232 T^{6} + 12500 T^{4} + \cdots + 454276)^{2}$$
$97$ $$(T^{4} + 100 T^{2} + 1250)^{4}$$