# Properties

 Label 845.2.t.c Level $845$ Weight $2$ Character orbit 845.t Analytic conductor $6.747$ Analytic rank $0$ Dimension $16$ Inner twists $4$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [845,2,Mod(188,845)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(845, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([9, 5]))

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

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

chi := DirichletCharacter("845.188");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$845 = 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 845.t (of order $$12$$, degree $$4$$, not 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: $$6.74735897080$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{12})$$ Coefficient field: 16.0.2520509501904273801216.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 4 x^{13} - 14 x^{12} + 8 x^{11} + 8 x^{10} + 26 x^{9} + 179 x^{8} + 104 x^{7} + 40 x^{6} + \cdots + 1$$ x^16 - 4*x^13 - 14*x^12 + 8*x^11 + 8*x^10 + 26*x^9 + 179*x^8 + 104*x^7 + 40*x^6 + 74*x^5 + 6*x^4 - 20*x^3 + 2*x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 65) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{9} - \beta_{8} + \cdots + \beta_{5}) q^{2}+ \cdots + (\beta_{14} - \beta_{13} + \cdots - \beta_{7}) q^{9}+O(q^{10})$$ q + (b9 - b8 + b6 + b5) * q^2 + (b12 + b11 + b8 + b5 + b2 + 1) * q^3 + (b15 - 2*b13 + b12 + b11 + b6 + b5 + b2 - b1) * q^4 + (-b14 + b10 - b6 - b5 - b4 + b3 - b2) * q^5 + (-b14 - b12 + b9) * q^6 + (-b15 - 2*b14 + b13 - b12 + b11 - b6 - b5 - b2 + b1) * q^7 + (b14 - b10 + b6 + b5 + b4) * q^8 + (b14 - b13 + b12 + b11 + b8 - b7) * q^9 $$q + (\beta_{9} - \beta_{8} + \cdots + \beta_{5}) q^{2}+ \cdots + ( - \beta_{14} + \beta_{10} + \cdots + \beta_1) q^{99}+O(q^{100})$$ q + (b9 - b8 + b6 + b5) * q^2 + (b12 + b11 + b8 + b5 + b2 + 1) * q^3 + (b15 - 2*b13 + b12 + b11 + b6 + b5 + b2 - b1) * q^4 + (-b14 + b10 - b6 - b5 - b4 + b3 - b2) * q^5 + (-b14 - b12 + b9) * q^6 + (-b15 - 2*b14 + b13 - b12 + b11 - b6 - b5 - b2 + b1) * q^7 + (b14 - b10 + b6 + b5 + b4) * q^8 + (b14 - b13 + b12 + b11 + b8 - b7) * q^9 + (-b15 - b12 + b11 + b10 + 2*b8 + b7 - 2*b6 + b4 - 2*b3 - 1) * q^10 + (-b13 + b11 - b9 + b6 - b3 + b2 - 1) * q^11 + (b10 - b6 - b5 + b3 - 2*b2 - b1) * q^12 + (-3*b14 + 3*b10 - 4*b6 - 2*b5 - 3*b4) * q^14 + (b13 - b12 - b11 + b10 + 2*b9 - 2*b8 + b7 + 2*b4 - 2*b2) * q^15 + (b13 - 2*b11 - 2*b6 - 2*b5 - 2*b4 + 2*b3 - 2*b2 + 1) * q^16 + (b15 + 2*b13 - b11 + b9 + b8 - b7 + b6 + b5 + b2 - b1) * q^17 + (-b3 - b1 + 2) * q^18 + (-b13 - b12 - b11 - 2*b8) * q^19 + (-b15 - 2*b14 + 3*b13 - 4*b12 - 2*b11 + b9 - 4*b8 - b7 - b6 - b5 - b2 + b1) * q^20 + (b10 + b6 + b3 - b2 - b1 - 1) * q^21 + (-b15 - b14 + 2*b13 - b11 - b9 - b8 - b7 - b6 - b5 - b2 + b1) * q^22 + (-b15 + 2*b13 - b11 - b10 - b9 - b8 - b7 + b6 - b5 - b4 + b3 + 2) * q^23 - b4 * q^24 + (b14 - b10 + 2*b6 + 4*b5 + b4 + 2*b2 - b1 - 1) * q^25 + (2*b10 - b5 - b3 - 2*b2 + 2*b1 + 1) * q^27 + (-3*b15 + 5*b13 - 5*b12 - b11 - b6 - b5 + 2*b4 - 4*b3 - b2) * q^28 + (-b13 + b12 + b11 + b10 - 2*b9 + b7 - 2*b5 - b4) * q^29 + (b15 - b13 - 2*b12 + b8 + b6 + b5 + b2 - b1) * q^30 + (b14 - b5 + b4 + b3 + 1) * q^31 + (3*b9 - b8) * q^32 + (-2*b9 - b7) * q^33 + (4*b14 - 2*b10 + 3*b6 + 2*b5 + 4*b4 + 2*b2 - 2*b1 + 1) * q^34 + (b15 - b14 - 2*b13 - b12 - 2*b11 - b9 + 2*b8 + b6 + b5 + b2 - b1) * q^35 + (3*b13 - 3*b12 - 3*b11 + 4*b9 - 9*b8 + 6*b6 + b5 + 3*b4 - 3*b2) * q^36 + (-b15 + 6*b13 + b4 + 6) * q^37 + (b14 + b6 + 2*b5 + b4 - 2*b3 - 1) * q^38 + (-b14 + b10 - 3*b6 - 2*b5 - b4 - b3 - b2 + 2) * q^40 + (2*b15 - 5*b13 + 3*b12 + 2*b11 + 2*b10 - b9 + 3*b8 + 2*b7 - 3*b6 - b5 - 2*b4 + b3 - 2) * q^41 + (-2*b13 - b12 + b9 + 2*b8 - 2*b6 + b5 + 2*b4 - b3 - 3) * q^42 + (b15 + 3*b14 - 4*b13 + 4*b12 + b11 - 3*b9 + 3*b8 + b7 + b6 + b5 + b2 - b1) * q^43 + (b6 - b5 - 2*b3 + b2 - 2) * q^44 + (-4*b14 + 3*b13 + 3*b11 - b8 + 2*b7) * q^45 + (-b13 + 3*b12 - b11 + 4*b9 - 2*b8) * q^46 + (-3*b1 + 2) * q^47 + (-2*b15 - b12 - 3*b11 + 2*b9 - 3*b8 + 2*b7 - 2*b6 - 2*b5 - 2*b2 + 2*b1) * q^48 + (-5*b13 + 4*b12 + 2*b11 + 2*b6 + 2*b5 + 2*b4 + 2*b3 + 2*b2 - 1) * q^49 + (3*b15 - 6*b13 - b10 + b9 + b8 - b7 + 2*b5 + b4 + b2 - 6) * q^50 + (3*b14 + b6 - 3*b5 + 3*b4 - 3*b2) * q^51 + (b10 + 4*b6 + b5 - 2*b3 + 3*b2 - b1 - 3) * q^53 + (-b15 - 3*b13 + 3*b11 + b10 - 3*b9 + b7 + 2*b6 - b5 - 3*b3 + 2*b2 - 3) * q^54 + (-b13 - b10 - 2*b9 - b8 - b7 + 2*b6 - b5 + b4 + b2 - 1) * q^55 + (b14 - b13 + b12 + b11 + 2*b9 - 6*b8 - b7) * q^56 + (-2*b14 - b10 - 3*b6 + 3*b5 - 2*b4 + 3*b2) * q^57 + (-3*b15 - 2*b14 + 5*b13 - b12 - b11 - 3*b6 - 3*b5 - 3*b2 + 3*b1) * q^58 + (5*b14 - 6*b13 + 4*b12 - 4*b9 + 6*b8) * q^59 + (-2*b14 - b10 - 4*b6 + 3*b5 - 2*b4 + b3 + 2*b2 - 1) * q^60 + (3*b15 + b14 + 2*b13 - 2*b12 + 2*b11 + 3*b6 + 3*b5 + 3*b2 - 3*b1) * q^61 + (-b15 + 5*b13 - b12 - 2*b11 + b10 + b9 + 3*b8 + b7 - 6*b6 - 2*b5 + b3 - 3*b2 + 4) * q^62 + (-2*b13 + 2*b12 + 2*b11 + 5*b10 - 2*b9 + 5*b7 - 3*b6 - 5*b5 - 2*b4 - 3*b2) * q^63 + (4*b14 + 4*b6 + 4*b5 + 4*b4 - 2*b3 + 4*b2 - 3*b1 + 7) * q^64 + (b14 + b6 + b5 + b4 - 4*b3 + b2 + b1 - 6) * q^66 + (-b13 + b12 + b11 + b10 + 2*b9 - 2*b8 + b7 + 2*b6 + 2*b5 - b4) * q^67 + (b13 - b12 - 4*b11 + 3*b9 - 3*b8 - b6 - b5 + 3*b3 - 4*b2) * q^68 + (-3*b15 + b14 + 2*b13 - 4*b11 - 3*b6 - 3*b5 - 3*b2 + 3*b1) * q^69 + (4*b14 - 2*b10 + 8*b6 + 7*b5 + 4*b4 + b3 + 2*b2 + b1 - 7) * q^70 + (b15 + 7*b14 - 5*b13 + 3*b12 + b11 - 2*b9 + 4*b8 + b7 + b6 + b5 + b2 - b1) * q^71 + (2*b15 + 3*b14 - 4*b13 + 2*b12 - b11 + 2*b6 + 2*b5 + 2*b2 - 2*b1) * q^72 + (-b14 + 3*b10 - 2*b6 - 2*b5 - b4 - 2*b2) * q^73 + (-2*b14 + 2*b13 - 2*b12 - 2*b11 + 8*b9 - 6*b8 - b7) * q^74 + (-b15 - b13 - 4*b12 - 4*b11 - b10 + 2*b9 + 3*b8 - b7 - 6*b6 - b5 - 2*b4 - 3*b2 - 5) * q^75 + (2*b15 - 3*b13 + 2*b12 + 3*b11 - 2*b10 - b9 - 2*b7 + 5*b6 + 4*b5 - b3 + 5*b2 - 1) * q^76 + (4*b6 + b5 - b3 + 2*b2 - 3) * q^77 + (5*b14 - 2*b10 + b6 + 3*b5 + 5*b4 - 3*b2) * q^79 + (-b13 + 5*b12 + 4*b11 + b10 + 2*b8 + b7 + b6 + 3*b5 - 3*b4 + b3 + 3*b2 + 4) * q^80 + (b15 + 3*b13 - 2*b12 + 4*b11 + 4*b6 + 4*b5 + 3*b4 - 6*b3 + 4*b2 + 1) * q^81 + (-3*b15 + 4*b13 - 3*b12 + 3*b11 - 6*b9 + 7*b8 + 3*b7 - 3*b6 - 3*b5 - 3*b2 + 3*b1) * q^82 + (-4*b3 + 3*b1 + 4) * q^83 + (-b15 + b13 - 4*b12 - 3*b11 - b9 - 2*b8 + b7 - b6 - b5 - b2 + b1) * q^84 + (b15 + 3*b14 + 3*b13 + 2*b9 - 3*b8 + 3*b7 + b6 + b5 + b2 - b1) * q^85 + (-4*b10 + 7*b6 + b5 + 2*b3 - b2 + 4*b1 - 6) * q^86 + (-2*b14 + 2*b13 + b12 - b9 - 2*b8) * q^87 + (b13 - b8 + b6 - b4 + 1) * q^88 + (-b15 - 2*b13 - 3*b12 - b11 - b10 + 2*b9 + 3*b8 - b7 - 3*b6 + 2*b5 - 2*b4 - 2*b3 - 5) * q^89 + (-5*b14 + 3*b10 - 6*b6 - 4*b5 - 5*b4 + 3*b3 - b2 + 2*b1 - 1) * q^90 + (3*b14 - 2*b10 + 9*b6 + 2*b5 + 3*b4 + 2*b2 - 2*b1 + 7) * q^92 + (b15 + 3*b13 + 3*b12 + b11 + b6 + b5 + 2*b3 + b2 + 6) * q^93 + (3*b13 - 3*b12 - 3*b11 - 3*b10 + 8*b9 - 8*b8 - 3*b7 + 8*b6 + 8*b5 + 3*b4) * q^94 + (-b15 + b14 + b11 - 3*b9 + 3*b8 - 2*b7 - b6 - b5 - b2 + b1) * q^95 + (-b14 + 2*b6 - 3*b5 - b4 + 3*b3 + 5) * q^96 + (5*b14 - 5*b13 + 5*b12 + 5*b11 - 6*b9 + 4*b8 + 2*b7) * q^97 + (-2*b14 + 2*b13 - 2*b12 - 2*b11 - b9 + 11*b8 + 2*b7) * q^98 + (-b14 + b10 - b6 - b5 - b4 - b2 + b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 6 q^{3} + 8 q^{4} - 4 q^{5} + 6 q^{6}+O(q^{10})$$ 16 * q + 6 * q^3 + 8 * q^4 - 4 * q^5 + 6 * q^6 $$16 q + 6 q^{3} + 8 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{10} - 6 q^{11} - 4 q^{12} - 2 q^{15} + 8 q^{16} - 16 q^{17} + 40 q^{18} + 14 q^{19} + 2 q^{20} - 24 q^{21} - 10 q^{22} + 14 q^{23} + 2 q^{24} - 24 q^{25} + 24 q^{27} + 8 q^{28} + 14 q^{30} + 4 q^{31} + 24 q^{35} + 44 q^{37} - 4 q^{38} + 44 q^{40} - 16 q^{41} - 24 q^{42} + 6 q^{43} - 20 q^{44} - 22 q^{45} - 2 q^{46} + 32 q^{47} + 14 q^{48} - 24 q^{49} - 44 q^{50} - 48 q^{53} - 20 q^{54} - 10 q^{55} - 24 q^{58} + 22 q^{59} - 20 q^{60} - 20 q^{61} + 30 q^{62} + 96 q^{64} - 72 q^{66} - 4 q^{68} - 4 q^{69} - 136 q^{70} + 10 q^{71} + 16 q^{72} - 30 q^{75} - 6 q^{76} - 48 q^{77} + 26 q^{80} + 20 q^{81} - 20 q^{82} + 96 q^{83} + 16 q^{84} - 32 q^{85} - 92 q^{86} - 16 q^{87} + 10 q^{88} - 28 q^{89} - 28 q^{90} + 100 q^{92} + 40 q^{93} - 2 q^{95} + 60 q^{96} + 4 q^{99}+O(q^{100})$$ 16 * q + 6 * q^3 + 8 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^10 - 6 * q^11 - 4 * q^12 - 2 * q^15 + 8 * q^16 - 16 * q^17 + 40 * q^18 + 14 * q^19 + 2 * q^20 - 24 * q^21 - 10 * q^22 + 14 * q^23 + 2 * q^24 - 24 * q^25 + 24 * q^27 + 8 * q^28 + 14 * q^30 + 4 * q^31 + 24 * q^35 + 44 * q^37 - 4 * q^38 + 44 * q^40 - 16 * q^41 - 24 * q^42 + 6 * q^43 - 20 * q^44 - 22 * q^45 - 2 * q^46 + 32 * q^47 + 14 * q^48 - 24 * q^49 - 44 * q^50 - 48 * q^53 - 20 * q^54 - 10 * q^55 - 24 * q^58 + 22 * q^59 - 20 * q^60 - 20 * q^61 + 30 * q^62 + 96 * q^64 - 72 * q^66 - 4 * q^68 - 4 * q^69 - 136 * q^70 + 10 * q^71 + 16 * q^72 - 30 * q^75 - 6 * q^76 - 48 * q^77 + 26 * q^80 + 20 * q^81 - 20 * q^82 + 96 * q^83 + 16 * q^84 - 32 * q^85 - 92 * q^86 - 16 * q^87 + 10 * q^88 - 28 * q^89 - 28 * q^90 + 100 * q^92 + 40 * q^93 - 2 * q^95 + 60 * q^96 + 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{13} - 14 x^{12} + 8 x^{11} + 8 x^{10} + 26 x^{9} + 179 x^{8} + 104 x^{7} + 40 x^{6} + \cdots + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( - 19751232 \nu^{15} + 17121420 \nu^{14} - 26059230 \nu^{13} + 83661521 \nu^{12} + \cdots + 346885583 ) / 830927053$$ (-19751232*v^15 + 17121420*v^14 - 26059230*v^13 + 83661521*v^12 + 213421536*v^11 - 291607092*v^10 + 320827398*v^9 - 669559752*v^8 - 3345611220*v^7 + 402357414*v^6 - 3500827434*v^5 - 2526242976*v^4 + 1812233064*v^3 - 351690048*v^2 + 227857860*v + 346885583) / 830927053 $$\beta_{2}$$ $$=$$ $$( 21862109 \nu^{15} - 82228832 \nu^{14} + 54829334 \nu^{13} - 103879948 \nu^{12} + \cdots - 1299442096 ) / 830927053$$ (21862109*v^15 - 82228832*v^14 + 54829334*v^13 - 103879948*v^12 + 34566476*v^11 + 1086379406*v^10 - 1184756767*v^9 + 533959384*v^8 + 2002924112*v^7 - 10745716055*v^6 + 1580983978*v^5 + 1227691388*v^4 - 3966721342*v^3 + 174725614*v^2 - 109780020*v - 1299442096) / 830927053 $$\beta_{3}$$ $$=$$ $$( 43223872 \nu^{15} - 37505470 \nu^{14} + 57560755 \nu^{13} - 179855583 \nu^{12} + \cdots - 1150180157 ) / 830927053$$ (43223872*v^15 - 37505470*v^14 + 57560755*v^13 - 179855583*v^12 - 474691456*v^11 + 640910682*v^10 - 740158583*v^9 + 1484879892*v^8 + 7424054770*v^7 - 901103519*v^6 + 7771974689*v^5 + 5607770496*v^4 - 4022237444*v^3 + 780658008*v^2 - 505806810*v - 1150180157) / 830927053 $$\beta_{4}$$ $$=$$ $$( - 65367994 \nu^{15} + 99782383 \nu^{14} - 17186648 \nu^{13} + 236152604 \nu^{12} + \cdots - 75382536 ) / 830927053$$ (-65367994*v^15 + 99782383*v^14 - 17186648*v^13 + 236152604*v^12 + 511566828*v^11 - 1850743683*v^10 + 607454916*v^9 - 662108690*v^8 - 9387973873*v^7 + 10413489869*v^6 + 4111725324*v^5 - 7403218444*v^4 + 2856629574*v^3 - 1024774482*v^2 - 5243657805*v - 75382536) / 830927053 $$\beta_{5}$$ $$=$$ $$( - 73422053 \nu^{15} + 6270716 \nu^{14} - 68196002 \nu^{13} + 311340841 \nu^{12} + \cdots + 169167894 ) / 830927053$$ (-73422053*v^15 + 6270716*v^14 - 68196002*v^13 + 311340841*v^12 + 961852936*v^11 - 368779136*v^10 + 340422052*v^9 - 2481815320*v^8 - 12955692356*v^7 - 8926130569*v^6 - 13986160390*v^5 - 10020177368*v^4 - 7046955235*v^3 - 1391775484*v^2 + 904578300*v + 169167894) / 830927053 $$\beta_{6}$$ $$=$$ $$( - 247375 \nu^{15} - 413104 \nu^{14} - 54152 \nu^{13} + 961664 \nu^{12} + 5139712 \nu^{11} + \cdots + 823277 ) / 2353901$$ (-247375*v^15 - 413104*v^14 - 54152*v^13 + 961664*v^12 + 5139712*v^11 + 3985552*v^10 - 4402392*v^9 - 9886432*v^8 - 55858736*v^7 - 100715200*v^6 - 63449944*v^5 - 44943104*v^4 - 33699901*v^3 - 6219712*v^2 + 4062960*v + 823277) / 2353901 $$\beta_{7}$$ $$=$$ $$( - 91701184 \nu^{15} + 14401668 \nu^{14} + 25144234 \nu^{13} + 369822303 \nu^{12} + \cdots - 59997442 ) / 830927053$$ (-91701184*v^15 + 14401668*v^14 + 25144234*v^13 + 369822303*v^12 + 1227785712*v^11 - 1042250464*v^10 - 977440334*v^9 - 2119119307*v^8 - 15823377884*v^7 - 6192688230*v^6 + 2367959080*v^5 - 2984224507*v^4 + 2142143520*v^3 + 2771928258*v^2 + 189173704*v - 59997442) / 830927053 $$\beta_{8}$$ $$=$$ $$( 99782383 \nu^{15} - 17186648 \nu^{14} - 25319372 \nu^{13} - 403585088 \nu^{12} - 1327799731 \nu^{11} + \cdots + 65367994 ) / 830927053$$ (99782383*v^15 - 17186648*v^14 - 25319372*v^13 - 403585088*v^12 - 1327799731*v^11 + 1130398868*v^10 + 1037459154*v^9 + 2312897053*v^8 + 17211761245*v^7 + 6726445084*v^6 - 2565986888*v^5 + 3248837538*v^4 - 2332134362*v^3 - 5943848870*v^2 - 206118524*v + 65367994) / 830927053 $$\beta_{9}$$ $$=$$ $$( - 149261939 \nu^{15} + 21361763 \nu^{14} + 44723362 \nu^{13} + 599779177 \nu^{12} + \cdots - 97502912 ) / 830927053$$ (-149261939*v^15 + 21361763*v^14 + 44723362*v^13 + 599779177*v^12 + 2013691511*v^11 - 1703353444*v^10 - 1639564236*v^9 - 3436212230*v^8 - 25766966573*v^7 - 10102110998*v^6 + 3874134976*v^5 - 4854392775*v^4 + 3484507474*v^3 + 2929722678*v^2 + 307408516*v - 97502912) / 830927053 $$\beta_{10}$$ $$=$$ $$( - 6655443 \nu^{15} - 8803488 \nu^{14} - 2325954 \nu^{13} + 26513692 \nu^{12} + 127187548 \nu^{11} + \cdots - 23873676 ) / 28652657$$ (-6655443*v^15 - 8803488*v^14 - 2325954*v^13 + 26513692*v^12 + 127187548*v^11 + 79699766*v^10 - 91049707*v^9 - 255375752*v^8 - 1424770992*v^7 - 2342638594*v^6 - 1605998334*v^5 - 1139493172*v^4 - 999790034*v^3 - 157781546*v^2 + 102991260*v - 23873676) / 28652657 $$\beta_{11}$$ $$=$$ $$( - 221411843 \nu^{15} + 15075771 \nu^{14} + 90426784 \nu^{13} + 861332824 \nu^{12} + \cdots + 59813086 ) / 830927053$$ (-221411843*v^15 + 15075771*v^14 + 90426784*v^13 + 861332824*v^12 + 3050830598*v^11 - 2351422560*v^10 - 2805267148*v^9 - 4611333644*v^8 - 38847834217*v^7 - 18033221816*v^6 + 8044167129*v^5 - 10329574122*v^4 + 2900909190*v^3 + 10530926348*v^2 + 139824038*v + 59813086) / 830927053 $$\beta_{12}$$ $$=$$ $$( - 233772946 \nu^{15} + 15658393 \nu^{14} - 76858280 \nu^{13} + 981839919 \nu^{12} + \cdots + 427938242 ) / 830927053$$ (-233772946*v^15 + 15658393*v^14 - 76858280*v^13 + 981839919*v^12 + 3196541140*v^11 - 1770072138*v^10 - 874858833*v^9 - 7169504656*v^8 - 41517878554*v^7 - 23340311904*v^6 - 19974269297*v^5 - 16796208619*v^4 - 952956046*v^3 + 1266963938*v^2 - 401135677*v + 427938242) / 830927053 $$\beta_{13}$$ $$=$$ $$( 8520800 \nu^{15} + 1273720 \nu^{14} + 282688 \nu^{13} - 33030966 \nu^{12} - 124560688 \nu^{11} + \cdots - 34034380 ) / 28652657$$ (8520800*v^15 + 1273720*v^14 + 282688*v^13 - 33030966*v^12 - 124560688*v^11 + 49353940*v^10 + 69973538*v^9 + 220182783*v^8 + 1570700616*v^7 + 1127422992*v^6 + 553070786*v^5 + 893845622*v^4 + 239299684*v^3 - 96761926*v^2 + 43157922*v - 34034380) / 28652657 $$\beta_{14}$$ $$=$$ $$( 253678901 \nu^{15} + 79125423 \nu^{14} + 115310926 \nu^{13} - 990536316 \nu^{12} + \cdots - 1768355242 ) / 830927053$$ (253678901*v^15 + 79125423*v^14 + 115310926*v^13 - 990536316*v^12 - 3862540986*v^11 + 458208910*v^10 + 957416885*v^9 + 7791235618*v^8 + 48513780327*v^7 + 43787354238*v^6 + 39541250602*v^5 + 38446400390*v^4 + 15472648252*v^3 + 5326356095*v^2 + 2438655465*v - 1768355242) / 830927053 $$\beta_{15}$$ $$=$$ $$( 34034380 \nu^{15} + 8520800 \nu^{14} + 1273720 \nu^{13} - 135854832 \nu^{12} - 509512286 \nu^{11} + \cdots - 24910838 ) / 28652657$$ (34034380*v^15 + 8520800*v^14 + 1273720*v^13 - 135854832*v^12 - 509512286*v^11 + 147714352*v^10 + 321628980*v^9 + 954867418*v^8 + 6312336803*v^7 + 5110276136*v^6 + 2488798192*v^5 + 3071614906*v^4 + 1098051902*v^3 - 441387916*v^2 - 40509*v - 24910838) / 28652657
 $$\nu$$ $$=$$ $$( \beta_{15} - \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} + \beta_{7} ) / 2$$ (b15 - b13 + b12 + b11 + b10 + b7) / 2 $$\nu^{2}$$ $$=$$ $$-\beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} + \beta_{9} - 3\beta_{8}$$ -b14 + b13 - b12 - b11 + b9 - 3*b8 $$\nu^{3}$$ $$=$$ $$( -3\beta_{10} + 3\beta_{6} + 5\beta_{5} + 5\beta_{2} + 3\beta _1 + 2 ) / 2$$ (-3*b10 + 3*b6 + 5*b5 + 5*b2 + 3*b1 + 2) / 2 $$\nu^{4}$$ $$=$$ $$\beta_{15} + 6\beta_{13} + \beta_{12} + 5\beta_{11} + 5\beta_{6} + 5\beta_{5} + 4\beta_{4} - 4\beta_{3} + 5\beta_{2} + 7$$ b15 + 6*b13 + b12 + 5*b11 + 5*b6 + 5*b5 + 4*b4 - 4*b3 + 5*b2 + 7 $$\nu^{5}$$ $$=$$ $$( 11 \beta_{15} - 12 \beta_{14} + 3 \beta_{13} + 9 \beta_{12} + 11 \beta_{11} + 2 \beta_{9} + \cdots - 11 \beta_1 ) / 2$$ (11*b15 - 12*b14 + 3*b13 + 9*b12 + 11*b11 + 2*b9 - 14*b8 + 11*b7 + 11*b6 + 11*b5 + 11*b2 - 11*b1) / 2 $$\nu^{6}$$ $$=$$ $$-16\beta_{14} - 7\beta_{10} - 20\beta_{6} + 8\beta_{5} - 16\beta_{4} + 23\beta_{2}$$ -16*b14 - 7*b10 - 20*b6 + 8*b5 - 16*b4 + 23*b2 $$\nu^{7}$$ $$=$$ $$( 43 \beta_{15} - 29 \beta_{13} + 89 \beta_{12} + 105 \beta_{11} - 43 \beta_{10} - 16 \beta_{9} + \cdots + 60 ) / 2$$ (43*b15 - 29*b13 + 89*b12 + 105*b11 - 43*b10 - 16*b9 + 76*b8 - 43*b7 + 72*b6 + 132*b5 - 16*b3 + 148*b2 + 60) / 2 $$\nu^{8}$$ $$=$$ $$38 \beta_{15} - 66 \beta_{14} + 71 \beta_{13} + 46 \beta_{12} + 104 \beta_{11} + 38 \beta_{6} + \cdots - 38 \beta_1$$ 38*b15 - 66*b14 + 71*b13 + 46*b12 + 104*b11 + 38*b6 + 38*b5 + 38*b2 - 38*b1 $$\nu^{9}$$ $$=$$ $$( - 300 \beta_{14} - 175 \beta_{10} - 201 \beta_{6} + 83 \beta_{5} - 300 \beta_{4} + 92 \beta_{3} + \cdots - 284 ) / 2$$ (-300*b14 - 175*b10 - 201*b6 + 83*b5 - 300*b4 + 92*b3 + 175*b2 - 175*b1 - 284) / 2 $$\nu^{10}$$ $$=$$ $$- 279 \beta_{13} + 279 \beta_{12} + 279 \beta_{11} - 188 \beta_{10} - 233 \beta_{9} + \cdots + 467 \beta_{2}$$ -279*b13 + 279*b12 + 279*b11 - 188*b10 - 233*b9 + 733*b8 - 188*b7 - 266*b6 + 234*b5 - 279*b4 + 467*b2 $$\nu^{11}$$ $$=$$ $$( 733 \beta_{15} - 357 \beta_{13} + 1667 \beta_{12} + 2135 \beta_{11} - 468 \beta_{9} + 1778 \beta_{8} + \cdots - 733 \beta_1 ) / 2$$ (733*b15 - 357*b13 + 1667*b12 + 2135*b11 - 468*b9 + 1778*b8 - 733*b7 + 733*b6 + 733*b5 + 733*b2 - 733*b1) / 2 $$\nu^{12}$$ $$=$$ $$-1200\beta_{14} - 1200\beta_{6} - 1200\beta_{5} - 1200\beta_{4} + 966\beta_{3} - 1200\beta_{2} - 889\beta _1 - 2167$$ -1200*b14 - 1200*b6 - 1200*b5 - 1200*b4 + 966*b3 - 1200*b2 - 889*b1 - 2167 $$\nu^{13}$$ $$=$$ $$( - 3133 \beta_{15} - 5069 \beta_{13} - 887 \beta_{12} - 3133 \beta_{11} - 3133 \beta_{10} - 2246 \beta_{9} + \cdots - 5956 ) / 2$$ (-3133*b15 - 5069*b13 - 887*b12 - 3133*b11 - 3133*b10 - 2246*b9 + 8202*b8 - 3133*b7 - 8202*b6 - 2246*b5 - 6424*b4 + 2246*b3 - 5956) / 2 $$\nu^{14}$$ $$=$$ $$5222\beta_{14} - 5222\beta_{13} + 5222\beta_{12} + 5222\beta_{11} - 4099\beta_{9} + 13577\beta_{8} - 4101\beta_{7}$$ 5222*b14 - 5222*b13 + 5222*b12 + 5222*b11 - 4099*b9 + 13577*b8 - 4101*b7 $$\nu^{15}$$ $$=$$ $$( 13577\beta_{10} - 5375\beta_{6} - 32223\beta_{5} + 10448\beta_{3} - 42671\beta_{2} - 13577\beta _1 - 26848 ) / 2$$ (13577*b10 - 5375*b6 - 32223*b5 + 10448*b3 - 42671*b2 - 13577*b1 - 26848) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/845\mathbb{Z}\right)^\times$$.

 $$n$$ $$171$$ $$677$$ $$\chi(n)$$ $$-\beta_{8}$$ $$\beta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
188.1
 0.432841 + 1.61538i −0.0924980 − 0.345207i −0.545721 − 2.03666i 0.205378 + 0.766482i −0.766482 − 0.205378i 2.03666 + 0.545721i 0.345207 + 0.0924980i −1.61538 − 0.432841i 0.432841 − 1.61538i −0.0924980 + 0.345207i −0.545721 + 2.03666i 0.205378 − 0.766482i −0.766482 + 0.205378i 2.03666 − 0.545721i 0.345207 − 0.0924980i −1.61538 + 0.432841i
−2.00594 1.15813i 0.328222 0.0879467i 1.68254 + 2.91425i −1.55654 + 1.60536i −0.760248 0.203708i −1.97936 3.42835i 3.16190i −2.49808 + 1.44227i 4.98155 1.41758i
188.2 −1.36782 0.789712i −0.991530 + 0.265680i 0.247291 + 0.428320i 0.146426 2.23127i 1.56605 + 0.419621i 2.12499 + 3.68058i 2.37769i −1.68553 + 0.973141i −1.96234 + 2.93634i
188.3 −0.116595 0.0673159i 2.94457 0.788996i −0.990937 1.71635i −1.29021 1.82630i −0.396433 0.106224i −0.954850 1.65385i 0.536087i 5.44992 3.14651i 0.0274926 + 0.299788i
188.4 1.75831 + 1.01516i 1.81681 0.486813i 1.06110 + 1.83789i 1.70032 + 1.45220i 3.68871 + 0.988387i 0.809223 + 1.40161i 0.248119i 0.465739 0.268895i 1.51548 + 4.27953i
418.1 −1.75831 + 1.01516i −0.486813 + 1.81681i 1.06110 1.83789i 1.70032 + 1.45220i −0.988387 3.68871i 0.809223 1.40161i 0.248119i −0.465739 0.268895i −4.46392 0.827324i
418.2 0.116595 0.0673159i −0.788996 + 2.94457i −0.990937 + 1.71635i −1.29021 1.82630i 0.106224 + 0.396433i −0.954850 + 1.65385i 0.536087i −5.44992 3.14651i −0.273370 0.126085i
418.3 1.36782 0.789712i 0.265680 0.991530i 0.247291 0.428320i 0.146426 2.23127i −0.419621 1.56605i 2.12499 3.68058i 2.37769i 1.68553 + 0.973141i −1.56178 3.16761i
418.4 2.00594 1.15813i −0.0879467 + 0.328222i 1.68254 2.91425i −1.55654 + 1.60536i 0.203708 + 0.760248i −1.97936 + 3.42835i 3.16190i 2.49808 + 1.44227i −1.26311 + 5.02294i
427.1 −2.00594 + 1.15813i 0.328222 + 0.0879467i 1.68254 2.91425i −1.55654 1.60536i −0.760248 + 0.203708i −1.97936 + 3.42835i 3.16190i −2.49808 1.44227i 4.98155 + 1.41758i
427.2 −1.36782 + 0.789712i −0.991530 0.265680i 0.247291 0.428320i 0.146426 + 2.23127i 1.56605 0.419621i 2.12499 3.68058i 2.37769i −1.68553 0.973141i −1.96234 2.93634i
427.3 −0.116595 + 0.0673159i 2.94457 + 0.788996i −0.990937 + 1.71635i −1.29021 + 1.82630i −0.396433 + 0.106224i −0.954850 + 1.65385i 0.536087i 5.44992 + 3.14651i 0.0274926 0.299788i
427.4 1.75831 1.01516i 1.81681 + 0.486813i 1.06110 1.83789i 1.70032 1.45220i 3.68871 0.988387i 0.809223 1.40161i 0.248119i 0.465739 + 0.268895i 1.51548 4.27953i
657.1 −1.75831 1.01516i −0.486813 1.81681i 1.06110 + 1.83789i 1.70032 1.45220i −0.988387 + 3.68871i 0.809223 + 1.40161i 0.248119i −0.465739 + 0.268895i −4.46392 + 0.827324i
657.2 0.116595 + 0.0673159i −0.788996 2.94457i −0.990937 1.71635i −1.29021 + 1.82630i 0.106224 0.396433i −0.954850 1.65385i 0.536087i −5.44992 + 3.14651i −0.273370 + 0.126085i
657.3 1.36782 + 0.789712i 0.265680 + 0.991530i 0.247291 + 0.428320i 0.146426 + 2.23127i −0.419621 + 1.56605i 2.12499 + 3.68058i 2.37769i 1.68553 0.973141i −1.56178 + 3.16761i
657.4 2.00594 + 1.15813i −0.0879467 0.328222i 1.68254 + 2.91425i −1.55654 1.60536i 0.203708 0.760248i −1.97936 3.42835i 3.16190i 2.49808 1.44227i −1.26311 5.02294i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 188.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
65.f even 4 1 inner
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 845.2.t.c 16
5.c odd 4 1 845.2.o.d 16
13.b even 2 1 845.2.t.d 16
13.c even 3 1 65.2.f.b 8
13.c even 3 1 inner 845.2.t.c 16
13.d odd 4 1 845.2.o.c 16
13.d odd 4 1 845.2.o.d 16
13.e even 6 1 845.2.f.b 8
13.e even 6 1 845.2.t.d 16
13.f odd 12 1 65.2.k.b yes 8
13.f odd 12 1 845.2.k.b 8
13.f odd 12 1 845.2.o.c 16
13.f odd 12 1 845.2.o.d 16
39.i odd 6 1 585.2.n.e 8
39.k even 12 1 585.2.w.e 8
52.j odd 6 1 1040.2.cd.n 8
52.l even 12 1 1040.2.bg.n 8
65.f even 4 1 inner 845.2.t.c 16
65.h odd 4 1 845.2.o.c 16
65.k even 4 1 845.2.t.d 16
65.n even 6 1 325.2.f.b 8
65.o even 12 1 325.2.f.b 8
65.o even 12 1 845.2.f.b 8
65.o even 12 1 845.2.t.d 16
65.q odd 12 1 65.2.k.b yes 8
65.q odd 12 1 325.2.k.b 8
65.q odd 12 1 845.2.o.d 16
65.r odd 12 1 845.2.k.b 8
65.r odd 12 1 845.2.o.c 16
65.s odd 12 1 325.2.k.b 8
65.t even 12 1 65.2.f.b 8
65.t even 12 1 inner 845.2.t.c 16
195.bc odd 12 1 585.2.n.e 8
195.bl even 12 1 585.2.w.e 8
260.bj even 12 1 1040.2.bg.n 8
260.bl odd 12 1 1040.2.cd.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.f.b 8 13.c even 3 1
65.2.f.b 8 65.t even 12 1
65.2.k.b yes 8 13.f odd 12 1
65.2.k.b yes 8 65.q odd 12 1
325.2.f.b 8 65.n even 6 1
325.2.f.b 8 65.o even 12 1
325.2.k.b 8 65.q odd 12 1
325.2.k.b 8 65.s odd 12 1
585.2.n.e 8 39.i odd 6 1
585.2.n.e 8 195.bc odd 12 1
585.2.w.e 8 39.k even 12 1
585.2.w.e 8 195.bl even 12 1
845.2.f.b 8 13.e even 6 1
845.2.f.b 8 65.o even 12 1
845.2.k.b 8 13.f odd 12 1
845.2.k.b 8 65.r odd 12 1
845.2.o.c 16 13.d odd 4 1
845.2.o.c 16 13.f odd 12 1
845.2.o.c 16 65.h odd 4 1
845.2.o.c 16 65.r odd 12 1
845.2.o.d 16 5.c odd 4 1
845.2.o.d 16 13.d odd 4 1
845.2.o.d 16 13.f odd 12 1
845.2.o.d 16 65.q odd 12 1
845.2.t.c 16 1.a even 1 1 trivial
845.2.t.c 16 13.c even 3 1 inner
845.2.t.c 16 65.f even 4 1 inner
845.2.t.c 16 65.t even 12 1 inner
845.2.t.d 16 13.b even 2 1
845.2.t.d 16 13.e even 6 1
845.2.t.d 16 65.k even 4 1
845.2.t.d 16 65.o even 12 1
1040.2.bg.n 8 52.l even 12 1
1040.2.bg.n 8 260.bj even 12 1
1040.2.cd.n 8 52.j odd 6 1
1040.2.cd.n 8 260.bl odd 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(845, [\chi])$$:

 $$T_{2}^{16} - 12T_{2}^{14} + 98T_{2}^{12} - 440T_{2}^{10} + 1443T_{2}^{8} - 2552T_{2}^{6} + 3090T_{2}^{4} - 56T_{2}^{2} + 1$$ T2^16 - 12*T2^14 + 98*T2^12 - 440*T2^10 + 1443*T2^8 - 2552*T2^6 + 3090*T2^4 - 56*T2^2 + 1 $$T_{3}^{16} - 6 T_{3}^{15} + 18 T_{3}^{14} - 68 T_{3}^{13} + 196 T_{3}^{12} - 260 T_{3}^{11} + 344 T_{3}^{10} + \cdots + 16$$ T3^16 - 6*T3^15 + 18*T3^14 - 68*T3^13 + 196*T3^12 - 260*T3^11 + 344*T3^10 - 512*T3^9 - 244*T3^8 + 1000*T3^7 - 416*T3^6 + 304*T3^5 + 1056*T3^4 - 544*T3^3 + 128*T3^2 - 64*T3 + 16 $$T_{7}^{8} + 20T_{7}^{6} + 8T_{7}^{5} + 348T_{7}^{4} + 80T_{7}^{3} + 1056T_{7}^{2} - 208T_{7} + 2704$$ T7^8 + 20*T7^6 + 8*T7^5 + 348*T7^4 + 80*T7^3 + 1056*T7^2 - 208*T7 + 2704

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - 12 T^{14} + \cdots + 1$$
$3$ $$T^{16} - 6 T^{15} + \cdots + 16$$
$5$ $$(T^{8} + 2 T^{7} + \cdots + 625)^{2}$$
$7$ $$(T^{8} + 20 T^{6} + \cdots + 2704)^{2}$$
$11$ $$T^{16} + 6 T^{15} + \cdots + 16$$
$13$ $$T^{16}$$
$17$ $$T^{16} + \cdots + 181063936$$
$19$ $$T^{16} - 14 T^{15} + \cdots + 10000$$
$23$ $$T^{16} + \cdots + 1664966416$$
$29$ $$T^{16} + \cdots + 100000000$$
$31$ $$(T^{8} - 2 T^{7} + \cdots + 16900)^{2}$$
$37$ $$(T^{8} - 22 T^{7} + \cdots + 336400)^{2}$$
$41$ $$T^{16} + \cdots + 181063936$$
$43$ $$T^{16} - 6 T^{15} + \cdots + 78074896$$
$47$ $$(T^{4} - 8 T^{3} + \cdots - 164)^{4}$$
$53$ $$(T^{8} + 24 T^{7} + \cdots + 19600)^{2}$$
$59$ $$T^{16} + \cdots + 14331920656$$
$61$ $$(T^{8} + 10 T^{7} + \cdots + 13162384)^{2}$$
$67$ $$T^{16} + \cdots + 479785216$$
$71$ $$T^{16} + \cdots + 1496306311696$$
$73$ $$(T^{8} + 116 T^{6} + \cdots + 547600)^{2}$$
$79$ $$(T^{8} + 292 T^{6} + \cdots + 13719616)^{2}$$
$83$ $$(T^{4} - 24 T^{3} + \cdots - 7372)^{4}$$
$89$ $$T^{16} + \cdots + 3224179360000$$
$97$ $$T^{16} + \cdots + 188227863187456$$
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