# Properties

 Label 130.2.p.b Level $130$ Weight $2$ Character orbit 130.p Analytic conductor $1.038$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [130,2,Mod(7,130)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(130, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([3, 11]))

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

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

chi := DirichletCharacter("130.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$130 = 2 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 130.p (of order $$12$$, 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: $$1.03805522628$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 4 x^{13} - 48 x^{12} + 16 x^{11} + 8 x^{10} + 80 x^{9} + 2208 x^{8} + 760 x^{7} + 192 x^{6} + 1696 x^{5} - 2812 x^{4} - 2080 x^{3} + 128 x^{2} - 1024 x + 4096$$ x^16 - 4*x^13 - 48*x^12 + 16*x^11 + 8*x^10 + 80*x^9 + 2208*x^8 + 760*x^7 + 192*x^6 + 1696*x^5 - 2812*x^4 - 2080*x^3 + 128*x^2 - 1024*x + 4096 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{13} - 48 x^{12} + 16 x^{11} + 8 x^{10} + 80 x^{9} + 2208 x^{8} + 760 x^{7} + 192 x^{6} + 1696 x^{5} - 2812 x^{4} - 2080 x^{3} + 128 x^{2} - 1024 x + 4096$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 590814673 \nu^{15} - 4146468380 \nu^{14} - 928390744 \nu^{13} + 4089687260 \nu^{12} + 11354140000 \nu^{11} + \cdots - 16628864598016 ) / 104859315460224$$ (-590814673*v^15 - 4146468380*v^14 - 928390744*v^13 + 4089687260*v^12 + 11354140000*v^11 + 199268795464*v^10 - 36693742968*v^9 - 29269595144*v^8 - 79214106400*v^7 - 10415738723016*v^6 - 5283856068144*v^5 - 1826716185040*v^4 - 90847784423236*v^3 - 1670918027776*v^2 + 7097999986688*v - 16628864598016) / 104859315460224 $$\beta_{3}$$ $$=$$ $$( - 238350271 \nu^{15} + 6776112136 \nu^{14} + 920962592 \nu^{13} + 1417547900 \nu^{12} - 17408263760 \nu^{11} + \cdots - 1698553709056 ) / 24672780108288$$ (-238350271*v^15 + 6776112136*v^14 + 920962592*v^13 + 1417547900*v^12 - 17408263760*v^11 - 323441465840*v^10 + 59406833544*v^9 + 35999704720*v^8 + 71206342688*v^7 + 14466282723576*v^6 + 7317759717504*v^5 + 2510988293024*v^4 + 9591494753540*v^3 + 2208138097664*v^2 - 9767935909888*v - 1698553709056) / 24672780108288 $$\beta_{4}$$ $$=$$ $$( - 42599 \nu^{15} + 44468 \nu^{14} - 981520 \nu^{13} + 201412 \nu^{12} + 1784920 \nu^{11} + 1523816 \nu^{10} + 45902632 \nu^{9} - 19363096 \nu^{8} + \cdots + 953416704 ) / 3150915456$$ (-42599*v^15 + 44468*v^14 - 981520*v^13 + 201412*v^12 + 1784920*v^11 + 1523816*v^10 + 45902632*v^9 - 19363096*v^8 - 96010096*v^7 - 26627160*v^6 - 2133581824*v^5 - 769014768*v^4 - 113153212*v^3 - 850918912*v^2 + 2969694208*v + 953416704) / 3150915456 $$\beta_{5}$$ $$=$$ $$( 12187170977 \nu^{15} + 112830647620 \nu^{14} - 929509456 \nu^{13} - 44571751012 \nu^{12} - 1059059762912 \nu^{11} + \cdots - 17112277705216 ) / 419437261840896$$ (12187170977*v^15 + 112830647620*v^14 - 929509456*v^13 - 44571751012*v^12 - 1059059762912*v^11 - 5193262349936*v^10 + 1936904356776*v^9 + 1764350055088*v^8 + 36949418734592*v^7 + 257951685319032*v^6 + 85856872417632*v^5 + 49092933699296*v^4 + 110084124927812*v^3 - 359630171247376*v^2 - 170659974504448*v - 17112277705216) / 419437261840896 $$\beta_{6}$$ $$=$$ $$( - 1265139977 \nu^{15} - 1178805226 \nu^{14} + 5432135497 \nu^{13} + 22313466160 \nu^{12} + 49666391246 \nu^{11} + \cdots - 38434909738112 ) / 26214828865056$$ (-1265139977*v^15 - 1178805226*v^14 + 5432135497*v^13 + 22313466160*v^12 + 49666391246*v^11 + 15527888726*v^10 - 380049099228*v^9 - 774748798816*v^8 - 1812058232894*v^7 - 3201472688760*v^6 + 12943988393412*v^5 + 36768367556944*v^4 - 17973151399292*v^3 + 7050453741040*v^2 - 22516417908608*v - 38434909738112) / 26214828865056 $$\beta_{7}$$ $$=$$ $$( - 1638690067 \nu^{15} + 2114100244 \nu^{14} - 13423970953 \nu^{13} + 22733755088 \nu^{12} + 78329121796 \nu^{11} + \cdots - 4170710020480 ) / 26214828865056$$ (-1638690067*v^15 + 2114100244*v^14 - 13423970953*v^13 + 22733755088*v^12 + 78329121796*v^11 - 70376022734*v^10 + 587092648872*v^9 - 1140357774056*v^8 - 3663273601234*v^7 + 2485938268080*v^6 - 26143148159448*v^5 + 22634771616224*v^4 + 33755922012548*v^3 - 8792425857040*v^2 + 32037605904512*v - 4170710020480) / 26214828865056 $$\beta_{8}$$ $$=$$ $$( 931071 \nu^{15} + 170396 \nu^{14} - 177872 \nu^{13} + 201796 \nu^{12} - 45497056 \nu^{11} + 7757456 \nu^{10} + 1353304 \nu^{9} - 109124848 \nu^{8} + \cdots - 12832193536 ) / 12603661824$$ (931071*v^15 + 170396*v^14 - 177872*v^13 + 201796*v^12 - 45497056*v^11 + 7757456*v^10 + 1353304*v^9 - 109124848*v^8 + 2133257152*v^7 + 1091654344*v^6 + 285274272*v^5 + 10113423712*v^4 + 457887420*v^3 - 1484014832*v^2 + 3522852736*v - 12832193536) / 12603661824 $$\beta_{9}$$ $$=$$ $$( - 9852329051 \nu^{15} + 28705092392 \nu^{14} + 47979745276 \nu^{13} + 39990919804 \nu^{12} + 396996647936 \nu^{11} + \cdots - 27683541191168 ) / 104859315460224$$ (-9852329051*v^15 + 28705092392*v^14 + 47979745276*v^13 + 39990919804*v^12 + 396996647936*v^11 - 1811927081248*v^10 - 1873373497464*v^9 + 8436720848*v^8 - 20417839645112*v^7 + 63577459405944*v^6 + 122087056714848*v^5 + 30008228870704*v^4 + 156465164147860*v^3 - 96269892921920*v^2 - 129599682228224*v - 27683541191168) / 104859315460224 $$\beta_{10}$$ $$=$$ $$( - 5879837650 \nu^{15} - 12865387787 \nu^{14} + 11969268317 \nu^{13} + 23302404986 \nu^{12} + 345665726062 \nu^{11} + \cdots - 4094491643392 ) / 26214828865056$$ (-5879837650*v^15 - 12865387787*v^14 + 11969268317*v^13 + 23302404986*v^12 + 345665726062*v^11 + 460600090348*v^10 - 819826388034*v^9 - 422742308156*v^8 - 14389637183506*v^7 - 30910736399856*v^6 + 14854472191068*v^5 - 2833160330548*v^4 + 19411871526044*v^3 + 41087591326484*v^2 + 2695803388928*v - 4094491643392) / 26214828865056 $$\beta_{11}$$ $$=$$ $$( 28207661905 \nu^{15} - 232377364 \nu^{14} + 1044233224 \nu^{13} - 118518889004 \nu^{12} - 1347064271392 \nu^{11} + \cdots - 12479663080448 ) / 104859315460224$$ (28207661905*v^15 - 232377364*v^14 + 1044233224*v^13 - 118518889004*v^12 - 1347064271392*v^11 + 459851747240*v^10 + 197344094232*v^9 + 2510036304344*v^8 + 62172358844128*v^7 + 20879233897512*v^6 + 7105872930576*v^5 + 36088612428784*v^4 - 83570213903804*v^3 - 43054983097376*v^2 - 1158153656192*v - 12479663080448) / 104859315460224 $$\beta_{12}$$ $$=$$ $$( 38410768049 \nu^{15} + 34463647156 \nu^{14} - 14862048280 \nu^{13} - 159419819020 \nu^{12} - 1962435314384 \nu^{11} + \cdots + 6777896482304 ) / 104859315460224$$ (38410768049*v^15 + 34463647156*v^14 - 14862048280*v^13 - 159419819020*v^12 - 1962435314384*v^11 - 882470407088*v^10 + 1504274515008*v^9 + 3320585983600*v^8 + 85856534490896*v^7 + 99810274704216*v^6 + 6546737141184*v^5 + 48819093324608*v^4 - 801552820204*v^3 + 5316554174960*v^2 - 141987626500096*v + 6777896482304) / 104859315460224 $$\beta_{13}$$ $$=$$ $$( 38453963605 \nu^{15} + 1001037668 \nu^{14} + 27774368104 \nu^{13} - 190478280212 \nu^{12} - 1817356748560 \nu^{11} + \cdots + 165237149091328 ) / 104859315460224$$ (38453963605*v^15 + 1001037668*v^14 + 27774368104*v^13 - 190478280212*v^12 - 1817356748560*v^11 + 445585688672*v^10 - 765142016064*v^9 + 5254165166480*v^8 + 82927272526384*v^7 + 34020882945768*v^6 + 61465587062400*v^5 + 6442915735264*v^4 - 50792859550460*v^3 - 35245016865296*v^2 + 61918405394560*v + 165237149091328) / 104859315460224 $$\beta_{14}$$ $$=$$ $$( 38686340969 \nu^{15} - 43195556 \nu^{14} + 33462609488 \nu^{13} - 197381780260 \nu^{12} - 1825885905320 \nu^{11} + \cdots - 138661529586688 ) / 104859315460224$$ (38686340969*v^15 - 43195556*v^14 + 33462609488*v^13 - 197381780260*v^12 - 1825885905320*v^11 + 473902889680*v^10 - 1018565368008*v^9 + 5364323808592*v^8 + 83485861676672*v^7 + 32330881100952*v^6 + 73217169224496*v^5 + 10693184362208*v^4 - 66409813215484*v^3 - 30476282485264*v^2 + 45513422684288*v - 138661529586688) / 104859315460224 $$\beta_{15}$$ $$=$$ $$( 10245718021 \nu^{15} - 12016563769 \nu^{14} - 7887648452 \nu^{13} - 43717796048 \nu^{12} - 434972084266 \nu^{11} + \cdots + 7461651612160 ) / 26214828865056$$ (10245718021*v^15 - 12016563769*v^14 - 7887648452*v^13 - 43717796048*v^12 - 434972084266*v^11 + 769274901224*v^10 + 256332661716*v^9 + 693383776244*v^8 + 21113684077720*v^7 - 18846312820728*v^6 - 23666868397332*v^5 + 3471088962400*v^4 - 33147629186564*v^3 - 6593097761156*v^2 - 1552448688128*v + 7461651612160) / 26214828865056
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{15} - \beta_{12} - \beta_{10} + \beta_{9} - 4\beta_{5} + 4\beta_{3}$$ b15 - b12 - b10 + b9 - 4*b5 + 4*b3 $$\nu^{3}$$ $$=$$ $$-\beta_{12} - \beta_{10} - \beta_{9} - \beta_{7} + \beta_{6} - 6\beta_{2}$$ -b12 - b10 - b9 - b7 + b6 - 6*b2 $$\nu^{4}$$ $$=$$ $$-8\beta_{13} + \beta_{11} + 24\beta_{8} - 8\beta_{7} - 8\beta_{6} - \beta_{2} + \beta _1 + 24$$ -8*b13 + b11 + 24*b8 - 8*b7 - 8*b6 - b2 + b1 + 24 $$\nu^{5}$$ $$=$$ $$9 \beta_{15} - 9 \beta_{14} - 8 \beta_{13} - \beta_{12} - 9 \beta_{10} + \beta_{9} + 4 \beta_{8} - \beta_{7} - 9 \beta_{6} - 4 \beta_{5} - 40 \beta_{4} + 4 \beta_{3} + 40 \beta_1$$ 9*b15 - 9*b14 - 8*b13 - b12 - 9*b10 + b9 + 4*b8 - b7 - 9*b6 - 4*b5 - 40*b4 + 4*b3 + 40*b1 $$\nu^{6}$$ $$=$$ $$56\beta_{15} - 58\beta_{12} - 2\beta_{10} - 2\beta_{9} - 14\beta_{4} + 160\beta_{3} - 14\beta_{2}$$ 56*b15 - 58*b12 - 2*b10 - 2*b9 - 14*b4 + 160*b3 - 14*b2 $$\nu^{7}$$ $$=$$ $$- 56 \beta_{15} - 56 \beta_{14} - 72 \beta_{13} - 56 \beta_{12} + 274 \beta_{11} + 16 \beta_{10} - 72 \beta_{9} + 56 \beta_{8} - 72 \beta_{7} - 16 \beta_{6} + 56 \beta_{5} - 274 \beta_{2} + 56$$ -56*b15 - 56*b14 - 72*b13 - 56*b12 + 274*b11 + 16*b10 - 72*b9 + 56*b8 - 72*b7 - 16*b6 + 56*b5 - 274*b2 + 56 $$\nu^{8}$$ $$=$$ $$- 418 \beta_{14} - 32 \beta_{13} + 144 \beta_{11} + 1096 \beta_{8} - 386 \beta_{7} - 418 \beta_{6} - 144 \beta_{4} + 144 \beta_1$$ -418*b14 - 32*b13 + 144*b11 + 1096*b8 - 386*b7 - 418*b6 - 144*b4 + 144*b1 $$\nu^{9}$$ $$=$$ $$176 \beta_{15} - 176 \beta_{14} + 176 \beta_{13} - 562 \beta_{12} - 386 \beta_{10} - 386 \beta_{9} + 386 \beta_{7} - 386 \beta_{6} - 1900 \beta_{4} + 576 \beta_{3} - 576$$ 176*b15 - 176*b14 + 176*b13 - 562*b12 - 386*b10 - 386*b9 + 386*b7 - 386*b6 - 1900*b4 + 576*b3 - 576 $$\nu^{10}$$ $$=$$ $$- 352 \beta_{15} - 352 \beta_{12} + 1314 \beta_{11} + 2672 \beta_{10} - 3024 \beta_{9} + 7600 \beta_{5} - 1314 \beta_{2} - 1314 \beta_1$$ -352*b15 - 352*b12 + 1314*b11 + 2672*b10 - 3024*b9 + 7600*b5 - 1314*b2 - 1314*b1 $$\nu^{11}$$ $$=$$ $$- 4338 \beta_{15} - 4338 \beta_{14} - 2672 \beta_{13} + 1666 \beta_{12} + 13296 \beta_{11} + 4338 \beta_{10} - 1666 \beta_{9} + 5256 \beta_{8} - 1666 \beta_{7} - 4338 \beta_{6} + 5256 \beta_{5} - 5256 \beta_{3}$$ -4338*b15 - 4338*b14 - 2672*b13 + 1666*b12 + 13296*b11 + 4338*b10 - 1666*b9 + 5256*b8 - 1666*b7 - 4338*b6 + 5256*b5 - 5256*b3 $$\nu^{12}$$ $$=$$ $$-18640\beta_{14} + 18640\beta_{13} + 3332\beta_{7} - 3332\beta_{6} - 11260\beta_{4} + 11260\beta_{2} - 53184$$ -18640*b14 + 18640*b13 + 3332*b7 - 3332*b6 - 11260*b4 + 11260*b2 - 53184 $$\nu^{13}$$ $$=$$ $$- 18640 \beta_{15} + 18640 \beta_{14} + 33232 \beta_{13} - 18640 \beta_{12} + 14592 \beta_{10} - 33232 \beta_{9} - 45040 \beta_{8} + 33232 \beta_{7} + 14592 \beta_{6} + 45040 \beta_{5} + \cdots - 45040$$ -18640*b15 + 18640*b14 + 33232*b13 - 18640*b12 + 14592*b10 - 33232*b9 - 45040*b8 + 33232*b7 + 14592*b6 + 45040*b5 - 93796*b1 - 45040 $$\nu^{14}$$ $$=$$ $$- 160260 \beta_{15} + 131076 \beta_{12} + 92864 \beta_{11} + 160260 \beta_{10} - 131076 \beta_{9} + 375184 \beta_{5} + 92864 \beta_{4} - 375184 \beta_{3} - 92864 \beta_1$$ -160260*b15 + 131076*b12 + 92864*b11 + 160260*b10 - 131076*b9 + 375184*b5 + 92864*b4 - 375184*b3 - 92864*b1 $$\nu^{15}$$ $$=$$ $$- 122048 \beta_{15} - 122048 \beta_{14} + 122048 \beta_{13} + 253124 \beta_{12} + 131076 \beta_{10} + 131076 \beta_{9} + 131076 \beta_{7} - 131076 \beta_{6} - 371456 \beta_{3} + \cdots - 371456$$ -122048*b15 - 122048*b14 + 122048*b13 + 253124*b12 + 131076*b10 + 131076*b9 + 131076*b7 - 131076*b6 - 371456*b3 + 666520*b2 - 371456

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/130\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$\beta_{3}$$ $$-\beta_{5}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 −2.38987 − 0.640364i −1.18141 − 0.316559i 0.935952 + 0.250788i 2.63533 + 0.706135i −0.706135 − 2.63533i −0.250788 − 0.935952i 0.316559 + 1.18141i 0.640364 + 2.38987i −2.38987 + 0.640364i −1.18141 + 0.316559i 0.935952 − 0.250788i 2.63533 − 0.706135i −0.706135 + 2.63533i −0.250788 + 0.935952i 0.316559 − 1.18141i 0.640364 − 2.38987i
0.866025 + 0.500000i −0.640364 2.38987i 0.500000 + 0.866025i −0.606329 2.15229i 0.640364 2.38987i −0.551051 0.954448i 1.00000i −2.70334 + 1.56078i 0.551051 2.16710i
7.2 0.866025 + 0.500000i −0.316559 1.18141i 0.500000 + 0.866025i 1.44768 + 1.70418i 0.316559 1.18141i −0.401632 0.695647i 1.00000i 1.30255 0.752028i 0.401632 + 2.19970i
7.3 0.866025 + 0.500000i 0.250788 + 0.935952i 0.500000 + 0.866025i −2.23384 0.0997335i −0.250788 + 0.935952i 1.88470 + 3.26439i 1.00000i 1.78496 1.03055i −1.88470 1.20329i
7.4 0.866025 + 0.500000i 0.706135 + 2.63533i 0.500000 + 0.866025i 0.892496 2.05023i −0.706135 + 2.63533i −1.79804 3.11430i 1.00000i −3.84827 + 2.22180i 1.79804 1.32930i
37.1 −0.866025 + 0.500000i −2.63533 0.706135i 0.500000 0.866025i 0.892496 + 2.05023i 2.63533 0.706135i 1.79804 3.11430i 1.00000i 3.84827 + 2.22180i −1.79804 1.32930i
37.2 −0.866025 + 0.500000i −0.935952 0.250788i 0.500000 0.866025i −2.23384 + 0.0997335i 0.935952 0.250788i −1.88470 + 3.26439i 1.00000i −1.78496 1.03055i 1.88470 1.20329i
37.3 −0.866025 + 0.500000i 1.18141 + 0.316559i 0.500000 0.866025i 1.44768 1.70418i −1.18141 + 0.316559i 0.401632 0.695647i 1.00000i −1.30255 0.752028i −0.401632 + 2.19970i
37.4 −0.866025 + 0.500000i 2.38987 + 0.640364i 0.500000 0.866025i −0.606329 + 2.15229i −2.38987 + 0.640364i 0.551051 0.954448i 1.00000i 2.70334 + 1.56078i −0.551051 2.16710i
93.1 0.866025 0.500000i −0.640364 + 2.38987i 0.500000 0.866025i −0.606329 + 2.15229i 0.640364 + 2.38987i −0.551051 + 0.954448i 1.00000i −2.70334 1.56078i 0.551051 + 2.16710i
93.2 0.866025 0.500000i −0.316559 + 1.18141i 0.500000 0.866025i 1.44768 1.70418i 0.316559 + 1.18141i −0.401632 + 0.695647i 1.00000i 1.30255 + 0.752028i 0.401632 2.19970i
93.3 0.866025 0.500000i 0.250788 0.935952i 0.500000 0.866025i −2.23384 + 0.0997335i −0.250788 0.935952i 1.88470 3.26439i 1.00000i 1.78496 + 1.03055i −1.88470 + 1.20329i
93.4 0.866025 0.500000i 0.706135 2.63533i 0.500000 0.866025i 0.892496 + 2.05023i −0.706135 2.63533i −1.79804 + 3.11430i 1.00000i −3.84827 2.22180i 1.79804 + 1.32930i
123.1 −0.866025 0.500000i −2.63533 + 0.706135i 0.500000 + 0.866025i 0.892496 2.05023i 2.63533 + 0.706135i 1.79804 + 3.11430i 1.00000i 3.84827 2.22180i −1.79804 + 1.32930i
123.2 −0.866025 0.500000i −0.935952 + 0.250788i 0.500000 + 0.866025i −2.23384 0.0997335i 0.935952 + 0.250788i −1.88470 3.26439i 1.00000i −1.78496 + 1.03055i 1.88470 + 1.20329i
123.3 −0.866025 0.500000i 1.18141 0.316559i 0.500000 + 0.866025i 1.44768 + 1.70418i −1.18141 0.316559i 0.401632 + 0.695647i 1.00000i −1.30255 + 0.752028i −0.401632 2.19970i
123.4 −0.866025 0.500000i 2.38987 0.640364i 0.500000 + 0.866025i −0.606329 2.15229i −2.38987 0.640364i 0.551051 + 0.954448i 1.00000i 2.70334 1.56078i −0.551051 + 2.16710i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.t even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 130.2.p.b 16
5.b even 2 1 650.2.t.g 16
5.c odd 4 1 130.2.s.b yes 16
5.c odd 4 1 650.2.w.g 16
13.f odd 12 1 130.2.s.b yes 16
65.o even 12 1 650.2.t.g 16
65.s odd 12 1 650.2.w.g 16
65.t even 12 1 inner 130.2.p.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.p.b 16 1.a even 1 1 trivial
130.2.p.b 16 65.t even 12 1 inner
130.2.s.b yes 16 5.c odd 4 1
130.2.s.b yes 16 13.f odd 12 1
650.2.t.g 16 5.b even 2 1
650.2.t.g 16 65.o even 12 1
650.2.w.g 16 5.c odd 4 1
650.2.w.g 16 65.s odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} + 4 T_{3}^{13} - 48 T_{3}^{12} - 16 T_{3}^{11} + 8 T_{3}^{10} - 80 T_{3}^{9} + 2208 T_{3}^{8} - 760 T_{3}^{7} + 192 T_{3}^{6} - 1696 T_{3}^{5} - 2812 T_{3}^{4} + 2080 T_{3}^{3} + 128 T_{3}^{2} + 1024 T_{3} + 4096$$ acting on $$S_{2}^{\mathrm{new}}(130, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{2} + 1)^{4}$$
$3$ $$T^{16} + 4 T^{13} - 48 T^{12} + \cdots + 4096$$
$5$ $$(T^{8} + T^{7} + 4 T^{6} + 19 T^{5} + 18 T^{4} + \cdots + 625)^{2}$$
$7$ $$T^{16} + 29 T^{14} + 606 T^{12} + \cdots + 20736$$
$11$ $$T^{16} - 6 T^{15} + 9 T^{14} + \cdots + 2408704$$
$13$ $$T^{16} - 2 T^{15} - 14 T^{13} + \cdots + 815730721$$
$17$ $$T^{16} + 16 T^{15} + \cdots + 15513698916$$
$19$ $$T^{16} + 3 T^{14} - 146 T^{13} + \cdots + 589824$$
$23$ $$T^{16} + 6 T^{15} + \cdots + 6996987904$$
$29$ $$T^{16} + 6 T^{15} - 59 T^{14} + \cdots + 2143296$$
$31$ $$T^{16} + 464 T^{13} + \cdots + 345518244864$$
$37$ $$T^{16} + 20 T^{15} + \cdots + 856089081$$
$41$ $$T^{16} + 44 T^{15} + 719 T^{14} + \cdots + 8596624$$
$43$ $$T^{16} + 132 T^{14} + \cdots + 9764601856$$
$47$ $$(T^{8} - 26 T^{7} + 167 T^{6} + \cdots + 75232)^{2}$$
$53$ $$T^{16} + 24 T^{15} + \cdots + 7629498409$$
$59$ $$T^{16} + 46 T^{15} + \cdots + 2479944646656$$
$61$ $$T^{16} - 6 T^{15} + 165 T^{14} + \cdots + 1633284$$
$67$ $$T^{16} - 12 T^{15} + \cdots + 123184152576$$
$71$ $$T^{16} - 6 T^{15} + \cdots + 5473632256$$
$73$ $$T^{16} + 606 T^{14} + \cdots + 903408428484$$
$79$ $$T^{16} + 268 T^{14} + \cdots + 14017536$$
$83$ $$(T^{8} + 32 T^{7} + 120 T^{6} + \cdots - 5965248)^{2}$$
$89$ $$T^{16} + 24 T^{15} + \cdots + 2057050114564$$
$97$ $$T^{16} + \cdots + 478697992372224$$