# Properties

 Label 55.2.j.a Level $55$ Weight $2$ Character orbit 55.j Analytic conductor $0.439$ 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] = [55,2,Mod(4,55)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(55, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 2]))

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

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

chi := DirichletCharacter("55.4");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$55 = 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 55.j (of order $$10$$, degree $$4$$, 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: $$0.439177211117$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 7x^{14} + 25x^{12} - 57x^{10} + 194x^{8} - 303x^{6} + 235x^{4} - 33x^{2} + 121$$ x^16 - 7*x^14 + 25*x^12 - 57*x^10 + 194*x^8 - 303*x^6 + 235*x^4 - 33*x^2 + 121 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{13} - \beta_{12} + \beta_1) q^{2} + ( - \beta_{15} + \beta_{13} + \beta_{5}) q^{3} + (\beta_{8} + \beta_{6} - \beta_{4}) q^{4} + (\beta_{10} + \beta_{6} - \beta_{3}) q^{5} + ( - \beta_{11} - \beta_{9} - 2 \beta_{8} + \cdots - 2) q^{6}+ \cdots + (\beta_{11} + \beta_{9} - \beta_{6} + \cdots + 1) q^{9}+O(q^{10})$$ q + (-b13 - b12 + b1) * q^2 + (-b15 + b13 + b5) * q^3 + (b8 + b6 - b4) * q^4 + (b10 + b6 - b3) * q^5 + (-b11 - b9 - 2*b8 + b7 - b6 + b3 - 2) * q^6 + (b15 + b12 - b2 - b1) * q^7 + (-b10 + b2 - b1) * q^8 + (b11 + b9 - b6 + b4 - b3 + 1) * q^9 $$q + ( - \beta_{13} - \beta_{12} + \beta_1) q^{2} + ( - \beta_{15} + \beta_{13} + \beta_{5}) q^{3} + (\beta_{8} + \beta_{6} - \beta_{4}) q^{4} + (\beta_{10} + \beta_{6} - \beta_{3}) q^{5} + ( - \beta_{11} - \beta_{9} - 2 \beta_{8} + \cdots - 2) q^{6}+ \cdots + (\beta_{11} - 5 \beta_{9} - \beta_{8} + \cdots - 6) q^{99}+O(q^{100})$$ q + (-b13 - b12 + b1) * q^2 + (-b15 + b13 + b5) * q^3 + (b8 + b6 - b4) * q^4 + (b10 + b6 - b3) * q^5 + (-b11 - b9 - 2*b8 + b7 - b6 + b3 - 2) * q^6 + (b15 + b12 - b2 - b1) * q^7 + (-b10 + b2 - b1) * q^8 + (b11 + b9 - b6 + b4 - b3 + 1) * q^9 + (b15 - b13 + b12 + 2*b11 - b10 + 2*b9 + b8 - b7 - b6 - b5 + b4 + 1) * q^10 + (-2*b11 - 2*b9 - b8 - b6 + b4 + b3 - 2) * q^11 + (b14 + b13 + b12 - 2*b5 + b2 - b1) * q^12 + (b13 + b12 - 2*b5 + b1) * q^13 + (-b8 - 2*b7 + 2*b4 - 2) * q^14 + (b13 - b12 + 3*b8 + 2*b6 - 3*b4) * q^15 + (b11 + 2*b9 + 2*b8 - b7 - 2*b6 - b4 + 2*b3 + 2) * q^16 + (-3*b14 - b12 + 2*b5 + b1) * q^17 + (2*b15 - b14 - b13 + b10 + b5 - 3*b2 - 2*b1) * q^18 + (b11 + 2*b7 - b4 + b3 + 1) * q^19 + (-b13 - b12 - b9 + b5 + b4 - b2 - 1) * q^20 + (-2*b9 - b8 + b7 + b6 - b4) * q^21 + (-3*b15 + 2*b14 + 2*b13 + b12 + b10 + 2*b2) * q^22 + (2*b14 - 3*b13 - 2*b12 + 2*b2 + 3*b1) * q^23 + (-2*b11 + b9 + 2*b6 - b4 - 3*b3 + 1) * q^24 + (-2*b15 - 2*b11 - 3*b8 + 2*b4 - 2*b3 - 2) * q^25 + (-3*b8 + 2*b7 + b6 + 3*b4 - 2*b3 + 2) * q^26 + (2*b14 + b12 - b10 - b1) * q^27 + b5 * q^28 + (-3*b9 - b7 - b6 + b3 - 1) * q^29 + (b13 + b11 - 2*b10 - b8 + b7 + b5 - 2*b4 + b3 + 2*b2 + 2*b1 + 2) * q^30 + (-b11 + 2*b9 + b6 - 3*b4 - 2*b3 + 2) * q^31 + (-b15 - 3*b14 + b13 - 2*b12 + b10 + 5*b5 - 3*b2) * q^32 + (-b14 - 3*b13 + 2*b10 - 2*b5 - b2 + 2*b1) * q^33 + (-2*b11 - b9 + 3*b8 + 4*b7 + 4*b6 - 4*b4) * q^34 + (-b15 + b14 + b9 + b5 + b4 + b3 + b2 - b1 + 1) * q^35 + (b11 + 3*b8 - 2*b7 - b4 + b3 + 1) * q^36 + (2*b15 + b14 - b13 - b12 - b10 - b5 - b2 - 2*b1) * q^37 + (-b14 - 2*b12 + 2*b10 - 2*b5 + 2*b1) * q^38 + (5*b11 + 3*b9 + 2*b8 - 5*b7 - 5*b6 + 2*b4 + 5*b3 + 2) * q^39 + (-2*b14 + 2*b13 - b12 + 2*b10 - 3*b9 - b8 - 2*b6 + 2*b5 + b4 - 2*b2) * q^40 + (2*b11 + 2*b8 - 3*b7 + 5*b4 + 2*b3 - 5) * q^41 + (-3*b5 + b2 + 3*b1) * q^42 + (2*b15 + 2*b13 - b12 - 2*b10 + b5 - 4*b1) * q^43 + (-b11 + 4*b9 - b8 + b7 - b4 - 2*b3 + 2) * q^44 + (2*b13 + 3*b12 - b11 + b9 + b8 - 3*b5 - 2*b1 + 1) * q^45 + (-b11 - 4*b9 + b6 - b4 - 3*b3 - 4) * q^46 + (b15 - 4*b13 - 3*b10 - 4*b5 + 3*b2 + 5*b1) * q^47 + (-b15 + 2*b14 + 4*b12 - 2*b10 - 2*b5 + 3*b2 + b1) * q^48 + (-b11 - 5*b9 - 4*b8 + b7 - b6 + 4*b4 + b3 - 4) * q^49 + (2*b14 + 4*b12 + 2*b8 - 2*b6 - 3*b5 + 2*b3 - 4*b1 + 2) * q^50 + (9*b9 + 4*b8 + 3*b7 + 2*b6 - 4*b4 - 3*b3 + 3) * q^51 + (2*b15 - 2*b13 + b10 - 2*b5 - b2 - b1) * q^52 + (b15 - b14 + 5*b13 + 5*b12 + 3*b5 - 3*b2 - 8*b1) * q^53 + (-2*b11 - 2*b9 - 4*b8 - 2*b7 - 2*b6 + 2*b4 - 5) * q^54 + (2*b15 - b14 - 2*b12 + b11 + 2*b9 - 3*b8 - 2*b7 - 2*b6 + 3*b5 - b4 + 2*b3 + b1 - 1) * q^55 + (3*b11 - b9 - b8 - 1) * q^56 + (b15 - b14 + 5*b5 - 2*b2 - 5*b1) * q^57 + (-b15 + 2*b13 + 2*b5 + 2*b1) * q^58 + (2*b9 + 3*b7 - 2*b6 - 3*b3 + 3) * q^59 + (-b14 - b12 - 3*b11 + b10 - 5*b9 - 5*b8 + 3*b7 + 2*b4 + b1 - 5) * q^60 + (4*b11 + b9 - 4*b8 - 4*b7 + 3*b4 - 4) * q^61 + (b15 + b14 - b10 - b5 - b1) * q^62 + (-b15 - b13 - b10 - b5 + b2 - b1) * q^63 + (-3*b11 - 4*b9 + 3*b6 + b4 + 2*b3 - 4) * q^64 + (b15 + 2*b14 - 3*b13 + b12 - 4*b11 - b10 - 2*b9 + b8 + 3*b7 + 3*b6 - 3*b5 - 3*b4 + 2*b2 + 2*b1 - 3) * q^65 + (4*b11 + b9 + 7*b8 - 3*b7 + b6 + 4*b4 - b3 + 5) * q^66 + (-3*b15 + b14 + b13 + b12 + 3*b10 - 2*b5 + b2 + 2*b1) * q^67 + (-2*b15 + 2*b14 + 2*b13 + 2*b12 - 3*b5 + b1) * q^68 + (-b11 - 2*b8 + 4*b7 - 3*b4 - b3 + 3) * q^69 + (-b15 - 2*b13 - 2*b12 + 2*b9 + b8 - b7 - 3*b6 - b4 + b3 + b2 + b1 - 1) * q^70 + (-3*b11 + 6*b9 + 9*b8 + 3*b7 + 6*b6 - 9*b4 - 6*b3 + 9) * q^71 + (b14 - 2*b12 + 2*b10 + b5 + 2*b1) * q^72 + (-5*b15 - b14 - b13 - 5*b12 + b10 + b5 + 4*b2 + 5*b1) * q^73 + (2*b11 + 2*b8 - 4*b7 - 3*b4 + 2*b3 + 3) * q^74 + (2*b15 - 2*b14 - 4*b13 - 4*b12 - 4*b9 + b5 + 4*b4 + 2*b3 - 3*b2 + 3*b1 - 4) * q^75 + (2*b11 + b9 - b7 - b6 + b4 + 4) * q^76 + (b15 - 2*b14 + 4*b13 + b10 + b5 - 3*b2 - b1) * q^77 + (-5*b14 + b13 - 7*b12 + 12*b5 - 5*b2 - b1) * q^78 + (b11 + b9 - b6 + b4 + 5*b3 + 1) * q^79 + (2*b15 - b13 + 3*b11 + 2*b10 - b8 - b5 - 2*b4 + 3*b3 - 2*b2 - b1 + 2) * q^80 + (-8*b9 - 5*b8 - 2*b7 + 2*b6 + 5*b4 + 2*b3 - 2) * q^81 + (-2*b14 + 3*b12 - 3*b10 + 5*b5 - 3*b1) * q^82 + (b14 - 2*b12 - b10 + b5 + 2*b1) * q^83 + (b9 + 2*b7 + b6 - 2*b3 + 2) * q^84 + (-b15 - 3*b11 + 2*b10 + b8 - 2*b7 - 5*b4 - 3*b3 - 2*b2 + 3*b1 + 5) * q^85 + (3*b9 - 8*b4 + 3*b3 + 3) * q^86 + (-3*b15 + b14 + 2*b13 + 2*b12 + 3*b10 - 3*b5 + b2 + b1) * q^87 + (b15 + 3*b14 - 5*b13 - 5*b10 - 6*b5 + 5*b2 + 2*b1) * q^88 + (2*b11 + 4*b9 - 2*b8 - 6*b7 - 6*b6 + 6*b4 + 1) * q^89 + (-b15 + b14 - b13 - b12 + 2*b11 + 7*b9 - 2*b6 + b5 + 4*b4 - b3 + b2 + 7) * q^90 + (-4*b11 + b8 + 2*b7 - 4*b3) * q^91 + (-2*b15 + b14 + b12 - b10 - b5 + 3*b2 + 2*b1) * q^92 + (3*b14 + b12 - 2*b10 - 2*b5 - b1) * q^93 + (-7*b11 - 6*b9 + b8 + 7*b7 + 9*b6 - b4 - 9*b3 + 1) * q^94 + (2*b14 + b13 + 2*b12 - 2*b10 - 4*b9 - b8 + 2*b7 + 3*b6 - 2*b5 + b4 - 2*b3 + 2*b2 + 2) * q^95 + (-3*b11 - 4*b8 + 6*b7 + 6*b4 - 3*b3 - 6) * q^96 + (-b15 + b14 - 4*b13 - 4*b12 + 4*b2 + 4*b1) * q^97 + (-2*b15 + b14 + b13 + 3*b12 + 2*b10 - 4*b5 + b2 + b1) * q^98 + (b11 - 5*b9 - b8 - b7 - 3*b4 + 2*b3 - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 4 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{9}+O(q^{10})$$ 16 * q - 4 * q^4 - 2 * q^5 - 18 * q^6 + 2 * q^9 $$16 q - 4 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{9} - 6 q^{11} - 12 q^{14} - 16 q^{15} + 16 q^{16} + 6 q^{19} - 8 q^{20} + 8 q^{21} + 6 q^{24} - 16 q^{25} + 40 q^{26} + 2 q^{29} + 26 q^{30} + 8 q^{31} - 16 q^{34} + 22 q^{35} + 10 q^{36} + 30 q^{39} + 12 q^{40} - 52 q^{41} + 4 q^{44} + 12 q^{45} - 62 q^{46} - 10 q^{49} + 28 q^{50} - 42 q^{51} - 40 q^{54} - 8 q^{55} - 20 q^{56} + 2 q^{59} - 32 q^{60} - 40 q^{61} - 8 q^{64} - 40 q^{65} + 58 q^{66} + 26 q^{69} - 34 q^{70} + 36 q^{71} + 48 q^{74} - 20 q^{75} + 56 q^{76} + 38 q^{79} + 34 q^{80} + 68 q^{81} + 12 q^{84} + 58 q^{85} + 22 q^{86} + 24 q^{89} + 78 q^{90} - 20 q^{91} + 14 q^{94} + 48 q^{95} - 86 q^{96} - 72 q^{99}+O(q^{100})$$ 16 * q - 4 * q^4 - 2 * q^5 - 18 * q^6 + 2 * q^9 - 6 * q^11 - 12 * q^14 - 16 * q^15 + 16 * q^16 + 6 * q^19 - 8 * q^20 + 8 * q^21 + 6 * q^24 - 16 * q^25 + 40 * q^26 + 2 * q^29 + 26 * q^30 + 8 * q^31 - 16 * q^34 + 22 * q^35 + 10 * q^36 + 30 * q^39 + 12 * q^40 - 52 * q^41 + 4 * q^44 + 12 * q^45 - 62 * q^46 - 10 * q^49 + 28 * q^50 - 42 * q^51 - 40 * q^54 - 8 * q^55 - 20 * q^56 + 2 * q^59 - 32 * q^60 - 40 * q^61 - 8 * q^64 - 40 * q^65 + 58 * q^66 + 26 * q^69 - 34 * q^70 + 36 * q^71 + 48 * q^74 - 20 * q^75 + 56 * q^76 + 38 * q^79 + 34 * q^80 + 68 * q^81 + 12 * q^84 + 58 * q^85 + 22 * q^86 + 24 * q^89 + 78 * q^90 - 20 * q^91 + 14 * q^94 + 48 * q^95 - 86 * q^96 - 72 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 7x^{14} + 25x^{12} - 57x^{10} + 194x^{8} - 303x^{6} + 235x^{4} - 33x^{2} + 121$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 1829973 \nu^{15} - 24476532 \nu^{13} + 152757677 \nu^{11} - 557238290 \nu^{9} + \cdots - 7498263355 \nu ) / 2438578648$$ (1829973*v^15 - 24476532*v^13 + 152757677*v^11 - 557238290*v^9 + 1500352252*v^7 - 3810530967*v^5 + 7929326490*v^3 - 7498263355*v) / 2438578648 $$\beta_{3}$$ $$=$$ $$( - 608673 \nu^{14} + 4779625 \nu^{12} - 9344761 \nu^{10} - 2560494 \nu^{8} - 9430964 \nu^{6} + \cdots + 31775447 ) / 609644662$$ (-608673*v^14 + 4779625*v^12 - 9344761*v^10 - 2560494*v^8 - 9430964*v^6 + 74558203*v^4 + 726449607*v^2 + 31775447) / 609644662 $$\beta_{4}$$ $$=$$ $$( 2744937 \nu^{14} - 22433610 \nu^{12} + 99837091 \nu^{10} - 276654262 \nu^{8} + 814713604 \nu^{6} + \cdots - 771926881 ) / 2438578648$$ (2744937*v^14 - 22433610*v^12 + 99837091*v^10 - 276654262*v^8 + 814713604*v^6 - 1632321387*v^4 + 3415114020*v^2 - 771926881) / 2438578648 $$\beta_{5}$$ $$=$$ $$( 2744937 \nu^{15} - 22433610 \nu^{13} + 99837091 \nu^{11} - 276654262 \nu^{9} + \cdots - 771926881 \nu ) / 2438578648$$ (2744937*v^15 - 22433610*v^13 + 99837091*v^11 - 276654262*v^9 + 814713604*v^7 - 1632321387*v^5 + 3415114020*v^3 - 771926881*v) / 2438578648 $$\beta_{6}$$ $$=$$ $$( - 2924455 \nu^{14} + 43736988 \nu^{12} - 193036323 \nu^{10} + 488836470 \nu^{8} + \cdots - 236512683 ) / 2438578648$$ (-2924455*v^14 + 43736988*v^12 - 193036323*v^10 + 488836470*v^8 - 1053314964*v^6 + 3520413317*v^4 - 719547006*v^2 - 236512683) / 2438578648 $$\beta_{7}$$ $$=$$ $$( - 1351905 \nu^{14} + 5558819 \nu^{12} - 7678521 \nu^{10} - 13223046 \nu^{8} - 42401020 \nu^{6} + \cdots - 629543387 ) / 609644662$$ (-1351905*v^14 + 5558819*v^12 - 7678521*v^10 - 13223046*v^8 - 42401020*v^6 - 304413167*v^4 + 664868085*v^2 - 629543387) / 609644662 $$\beta_{8}$$ $$=$$ $$( - 6960689 \nu^{14} + 30587176 \nu^{12} - 61040185 \nu^{10} + 38446538 \nu^{8} + \cdots - 1738523017 ) / 2438578648$$ (-6960689*v^14 + 30587176*v^12 - 61040185*v^10 + 38446538*v^8 - 665668084*v^6 - 797037357*v^4 + 1844303058*v^2 - 1738523017) / 2438578648 $$\beta_{9}$$ $$=$$ $$( - 3962283 \nu^{14} + 31992860 \nu^{12} - 118526613 \nu^{10} + 271533274 \nu^{8} + \cdots - 383811549 ) / 1219289324$$ (-3962283*v^14 + 31992860*v^12 - 118526613*v^10 + 271533274*v^8 - 833575532*v^6 + 1781437793*v^4 - 742925482*v^2 - 383811549) / 1219289324 $$\beta_{10}$$ $$=$$ $$( 8234811 \nu^{15} - 67300830 \nu^{13} + 299511273 \nu^{11} - 829962786 \nu^{9} + \cdots - 2315780643 \nu ) / 2438578648$$ (8234811*v^15 - 67300830*v^13 + 299511273*v^11 - 829962786*v^9 + 2444140812*v^7 - 4896964161*v^5 + 7806763412*v^3 - 2315780643*v) / 2438578648 $$\beta_{11}$$ $$=$$ $$( 770947 \nu^{14} - 4704418 \nu^{12} + 13758165 \nu^{10} - 28071690 \nu^{8} + 126275508 \nu^{6} + \cdots - 243303327 ) / 221688968$$ (770947*v^14 - 4704418*v^12 + 13758165*v^10 - 28071690*v^8 + 126275508*v^6 - 152673185*v^4 - 21444456*v^2 - 243303327) / 221688968 $$\beta_{12}$$ $$=$$ $$( 10669503 \nu^{15} - 86419330 \nu^{13} + 336890317 \nu^{11} - 819720810 \nu^{9} + \cdots - 4303783 \nu ) / 2438578648$$ (10669503*v^15 - 86419330*v^13 + 336890317*v^11 - 819720810*v^9 + 2481864668*v^7 - 5195196973*v^5 + 4900964984*v^3 - 4303783*v) / 2438578648 $$\beta_{13}$$ $$=$$ $$( - 8815096 \nu^{15} + 58503253 \nu^{13} - 198965251 \nu^{11} + 429083674 \nu^{9} + \cdots + 352179707 \nu ) / 1219289324$$ (-8815096*v^15 + 58503253*v^13 - 198965251*v^11 + 429083674*v^9 - 1573766376*v^7 + 2199079808*v^5 - 1528330963*v^3 + 352179707*v) / 1219289324 $$\beta_{14}$$ $$=$$ $$( - 19639683 \nu^{15} + 162786416 \nu^{13} - 643887411 \nu^{11} + 1627587846 \nu^{9} + \cdots + 4755604557 \nu ) / 2438578648$$ (-19639683*v^15 + 162786416*v^13 - 643887411*v^11 + 1627587846*v^9 - 4886486364*v^7 + 10096782513*v^5 - 8290421402*v^3 + 4755604557*v) / 2438578648 $$\beta_{15}$$ $$=$$ $$( - 1496498 \nu^{15} + 9707543 \nu^{13} - 31032777 \nu^{11} + 60570826 \nu^{9} + \cdots - 172220447 \nu ) / 110844484$$ (-1496498*v^15 + 9707543*v^13 - 31032777*v^11 + 60570826*v^9 - 228914540*v^7 + 268358314*v^5 + 33071203*v^3 - 172220447*v) / 110844484
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{9} + 2\beta_{4} - \beta_{3} + 1$$ b9 + 2*b4 - b3 + 1 $$\nu^{3}$$ $$=$$ $$-\beta_{10} + 3\beta_{5}$$ -b10 + 3*b5 $$\nu^{4}$$ $$=$$ $$6\beta_{9} - 4\beta_{8} + \beta_{7} - 4\beta_{6} + 4\beta_{4} - \beta_{3} + 1$$ 6*b9 - 4*b8 + b7 - 4*b6 + 4*b4 - b3 + 1 $$\nu^{5}$$ $$=$$ $$\beta_{15} - 5\beta_{14} - \beta_{13} - 10\beta_{12} - \beta_{10} + 15\beta_{5} - 5\beta_{2}$$ b15 - 5*b14 - b13 - 10*b12 - b10 + 15*b5 - 5*b2 $$\nu^{6}$$ $$=$$ $$-7\beta_{11} - 22\beta_{8} + 21\beta_{7} + \beta_{4} - 7\beta_{3} - 1$$ -7*b11 - 22*b8 + 21*b7 + b4 - 7*b3 - 1 $$\nu^{7}$$ $$=$$ $$21\beta_{15} - 21\beta_{14} - 36\beta_{13} - 36\beta_{12} + 29\beta_{5} - 28\beta_{2} + 7\beta_1$$ 21*b15 - 21*b14 - 36*b13 - 36*b12 + 29*b5 - 28*b2 + 7*b1 $$\nu^{8}$$ $$=$$ $$-85\beta_{11} - 78\beta_{9} - 87\beta_{8} + 85\beta_{7} + 49\beta_{6} - 7\beta_{4} - 49\beta_{3} - 87$$ -85*b11 - 78*b9 - 87*b8 + 85*b7 + 49*b6 - 7*b4 - 49*b3 - 87 $$\nu^{9}$$ $$=$$ $$85\beta_{15} - 36\beta_{14} - 136\beta_{13} - 45\beta_{12} + 36\beta_{10} + 36\beta_{5} - 121\beta_{2} - 85\beta_1$$ 85*b15 - 36*b14 - 136*b13 - 45*b12 + 36*b10 + 36*b5 - 121*b2 - 85*b1 $$\nu^{10}$$ $$=$$ $$-342\beta_{11} - 396\beta_{9} - 230\beta_{8} + 166\beta_{7} + 166\beta_{6} - 166\beta_{4} - 529$$ -342*b11 - 396*b9 - 230*b8 + 166*b7 + 166*b6 - 166*b4 - 529 $$\nu^{11}$$ $$=$$ $$166\beta_{15} - 220\beta_{13} + 342\beta_{10} - 220\beta_{5} - 342\beta_{2} - 475\beta_1$$ 166*b15 - 220*b13 + 342*b10 - 220*b5 - 342*b2 - 475*b1 $$\nu^{12}$$ $$=$$ $$-728\beta_{11} - 1653\beta_{9} + 728\beta_{6} - 1170\beta_{4} + 651\beta_{3} - 1653$$ -728*b11 - 1653*b9 + 728*b6 - 1170*b4 + 651*b3 - 1653 $$\nu^{13}$$ $$=$$ $$728\beta_{14} + 1002\beta_{12} + 1379\beta_{10} - 2823\beta_{5} - 1002\beta_1$$ 728*b14 + 1002*b12 + 1379*b10 - 2823*b5 - 1002*b1 $$\nu^{14}$$ $$=$$ $$-4644\beta_{9} + 3748\beta_{8} - 3109\beta_{7} + 2472\beta_{6} - 3748\beta_{4} + 3109\beta_{3} - 3109$$ -4644*b9 + 3748*b8 - 3109*b7 + 2472*b6 - 3748*b4 + 3109*b3 - 3109 $$\nu^{15}$$ $$=$$ $$- 3109 \beta_{15} + 5581 \beta_{14} + 4385 \beta_{13} + 8392 \beta_{12} + 3109 \beta_{10} + \cdots - 1276 \beta_1$$ -3109*b15 + 5581*b14 + 4385*b13 + 8392*b12 + 3109*b10 - 13973*b5 + 5581*b2 - 1276*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/55\mathbb{Z}\right)^\times$$.

 $$n$$ $$12$$ $$46$$ $$\chi(n)$$ $$-1$$ $$\beta_{9}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 1.92464 + 0.625353i 1.17360 + 0.381325i −1.17360 − 0.381325i −1.92464 − 0.625353i −0.972539 − 1.33858i −0.471815 − 0.649397i 0.471815 + 0.649397i 0.972539 + 1.33858i 1.92464 − 0.625353i 1.17360 − 0.381325i −1.17360 + 0.381325i −1.92464 + 0.625353i −0.972539 + 1.33858i −0.471815 + 0.649397i 0.471815 − 0.649397i 0.972539 − 1.33858i
−1.18949 + 1.63719i 2.49233 + 0.809808i −0.647481 1.99274i −1.06421 1.96658i −4.29042 + 3.11717i −0.918552 + 0.298456i 0.183406 + 0.0595923i 3.12889 + 2.27327i 4.48555 + 0.596911i
4.2 −0.725323 + 0.998322i −0.346168 0.112477i 0.147481 + 0.453901i 0.0238439 + 2.23594i 0.363371 0.264005i 2.45903 0.798988i −2.90731 0.944641i −2.31987 1.68548i −2.24948 1.59798i
4.3 0.725323 0.998322i 0.346168 + 0.112477i 0.147481 + 0.453901i −2.11914 0.713621i 0.363371 0.264005i −2.45903 + 0.798988i 2.90731 + 0.944641i −2.31987 1.68548i −2.24948 + 1.59798i
4.4 1.18949 1.63719i −2.49233 0.809808i −0.647481 1.99274i 1.54147 + 1.61983i −4.29042 + 3.11717i 0.918552 0.298456i −0.183406 0.0595923i 3.12889 + 2.27327i 4.48555 0.596911i
9.1 −1.57360 0.511294i 1.16075 + 1.59764i 0.596764 + 0.433574i 1.09069 + 1.95203i −1.00970 3.10753i 1.31845 1.81468i 1.22769 + 1.68978i −0.278050 + 0.855749i −0.718246 3.62937i
9.2 −0.763412 0.248048i −1.03494 1.42447i −1.09676 0.796845i 1.42953 1.71943i 0.436748 + 1.34417i −0.348029 + 0.479022i 1.58326 + 2.17917i −0.0309674 + 0.0953077i −1.51782 + 0.958043i
9.3 0.763412 + 0.248048i 1.03494 + 1.42447i −1.09676 0.796845i −2.16717 0.550792i 0.436748 + 1.34417i 0.348029 0.479022i −1.58326 2.17917i −0.0309674 + 0.0953077i −1.51782 0.958043i
9.4 1.57360 + 0.511294i −1.16075 1.59764i 0.596764 + 0.433574i 0.264988 + 2.22031i −1.00970 3.10753i −1.31845 + 1.81468i −1.22769 1.68978i −0.278050 + 0.855749i −0.718246 + 3.62937i
14.1 −1.18949 1.63719i 2.49233 0.809808i −0.647481 + 1.99274i −1.06421 + 1.96658i −4.29042 3.11717i −0.918552 0.298456i 0.183406 0.0595923i 3.12889 2.27327i 4.48555 0.596911i
14.2 −0.725323 0.998322i −0.346168 + 0.112477i 0.147481 0.453901i 0.0238439 2.23594i 0.363371 + 0.264005i 2.45903 + 0.798988i −2.90731 + 0.944641i −2.31987 + 1.68548i −2.24948 + 1.59798i
14.3 0.725323 + 0.998322i 0.346168 0.112477i 0.147481 0.453901i −2.11914 + 0.713621i 0.363371 + 0.264005i −2.45903 0.798988i 2.90731 0.944641i −2.31987 + 1.68548i −2.24948 1.59798i
14.4 1.18949 + 1.63719i −2.49233 + 0.809808i −0.647481 + 1.99274i 1.54147 1.61983i −4.29042 3.11717i 0.918552 + 0.298456i −0.183406 + 0.0595923i 3.12889 2.27327i 4.48555 + 0.596911i
49.1 −1.57360 + 0.511294i 1.16075 1.59764i 0.596764 0.433574i 1.09069 1.95203i −1.00970 + 3.10753i 1.31845 + 1.81468i 1.22769 1.68978i −0.278050 0.855749i −0.718246 + 3.62937i
49.2 −0.763412 + 0.248048i −1.03494 + 1.42447i −1.09676 + 0.796845i 1.42953 + 1.71943i 0.436748 1.34417i −0.348029 0.479022i 1.58326 2.17917i −0.0309674 0.0953077i −1.51782 0.958043i
49.3 0.763412 0.248048i 1.03494 1.42447i −1.09676 + 0.796845i −2.16717 + 0.550792i 0.436748 1.34417i 0.348029 + 0.479022i −1.58326 + 2.17917i −0.0309674 0.0953077i −1.51782 + 0.958043i
49.4 1.57360 0.511294i −1.16075 + 1.59764i 0.596764 0.433574i 0.264988 2.22031i −1.00970 + 3.10753i −1.31845 1.81468i −1.22769 + 1.68978i −0.278050 0.855749i −0.718246 3.62937i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 55.2.j.a 16
3.b odd 2 1 495.2.ba.a 16
4.b odd 2 1 880.2.cd.c 16
5.b even 2 1 inner 55.2.j.a 16
5.c odd 4 2 275.2.h.d 16
11.b odd 2 1 605.2.j.d 16
11.c even 5 1 inner 55.2.j.a 16
11.c even 5 1 605.2.b.g 8
11.c even 5 2 605.2.j.h 16
11.d odd 10 1 605.2.b.f 8
11.d odd 10 1 605.2.j.d 16
11.d odd 10 2 605.2.j.g 16
15.d odd 2 1 495.2.ba.a 16
20.d odd 2 1 880.2.cd.c 16
33.h odd 10 1 495.2.ba.a 16
44.h odd 10 1 880.2.cd.c 16
55.d odd 2 1 605.2.j.d 16
55.h odd 10 1 605.2.b.f 8
55.h odd 10 1 605.2.j.d 16
55.h odd 10 2 605.2.j.g 16
55.j even 10 1 inner 55.2.j.a 16
55.j even 10 1 605.2.b.g 8
55.j even 10 2 605.2.j.h 16
55.k odd 20 2 275.2.h.d 16
55.k odd 20 2 3025.2.a.bl 8
55.l even 20 2 3025.2.a.bk 8
165.o odd 10 1 495.2.ba.a 16
220.n odd 10 1 880.2.cd.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.j.a 16 1.a even 1 1 trivial
55.2.j.a 16 5.b even 2 1 inner
55.2.j.a 16 11.c even 5 1 inner
55.2.j.a 16 55.j even 10 1 inner
275.2.h.d 16 5.c odd 4 2
275.2.h.d 16 55.k odd 20 2
495.2.ba.a 16 3.b odd 2 1
495.2.ba.a 16 15.d odd 2 1
495.2.ba.a 16 33.h odd 10 1
495.2.ba.a 16 165.o odd 10 1
605.2.b.f 8 11.d odd 10 1
605.2.b.f 8 55.h odd 10 1
605.2.b.g 8 11.c even 5 1
605.2.b.g 8 55.j even 10 1
605.2.j.d 16 11.b odd 2 1
605.2.j.d 16 11.d odd 10 1
605.2.j.d 16 55.d odd 2 1
605.2.j.d 16 55.h odd 10 1
605.2.j.g 16 11.d odd 10 2
605.2.j.g 16 55.h odd 10 2
605.2.j.h 16 11.c even 5 2
605.2.j.h 16 55.j even 10 2
880.2.cd.c 16 4.b odd 2 1
880.2.cd.c 16 20.d odd 2 1
880.2.cd.c 16 44.h odd 10 1
880.2.cd.c 16 220.n odd 10 1
3025.2.a.bk 8 55.l even 20 2
3025.2.a.bl 8 55.k odd 20 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(55, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - 2 T^{14} + \cdots + 121$$
$3$ $$T^{16} - 7 T^{14} + \cdots + 121$$
$5$ $$T^{16} + 2 T^{15} + \cdots + 390625$$
$7$ $$T^{16} - 9 T^{14} + \cdots + 121$$
$11$ $$(T^{8} + 3 T^{7} + \cdots + 14641)^{2}$$
$13$ $$T^{16} - 10 T^{14} + \cdots + 47265625$$
$17$ $$T^{16} + \cdots + 1675346761$$
$19$ $$(T^{8} - 3 T^{7} + \cdots + 121)^{2}$$
$23$ $$(T^{8} + 99 T^{6} + \cdots + 26411)^{2}$$
$29$ $$(T^{8} - T^{7} + 25 T^{6} + \cdots + 121)^{2}$$
$31$ $$(T^{8} - 4 T^{7} + 39 T^{6} + \cdots + 1)^{2}$$
$37$ $$T^{16} - 28 T^{14} + \cdots + 15768841$$
$41$ $$(T^{8} + 26 T^{7} + \cdots + 3876961)^{2}$$
$43$ $$(T^{8} + 173 T^{6} + \cdots + 212531)^{2}$$
$47$ $$T^{16} + \cdots + 59639012521$$
$53$ $$T^{16} + \cdots + 239836452361$$
$59$ $$(T^{8} - T^{7} + 129 T^{6} + \cdots + 1)^{2}$$
$61$ $$(T^{8} + 20 T^{7} + \cdots + 43681)^{2}$$
$67$ $$(T^{8} + 149 T^{6} + \cdots + 18491)^{2}$$
$71$ $$(T^{8} - 18 T^{7} + \cdots + 6305121)^{2}$$
$73$ $$T^{16} + \cdots + 1636073786281$$
$79$ $$(T^{8} - 19 T^{7} + \cdots + 101761)^{2}$$
$83$ $$T^{16} + \cdots + 45169425961$$
$89$ $$(T^{4} - 6 T^{3} + \cdots + 1871)^{4}$$
$97$ $$T^{16} + \cdots + 25937424601$$