# Properties

 Label 605.2.j.j Level $605$ Weight $2$ Character orbit 605.j Analytic conductor $4.831$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [605,2,Mod(9,605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(605, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 6]))

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

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

chi := DirichletCharacter("605.9");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.j (of order $$10$$, 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: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: 16.0.343361479062744140625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - x^{15} + 3 x^{14} - 8 x^{13} + 8 x^{12} + 7 x^{11} + 6 x^{10} + 56 x^{9} - 137 x^{8} + \cdots + 6561$$ x^16 - x^15 + 3*x^14 - 8*x^13 + 8*x^12 + 7*x^11 + 6*x^10 + 56*x^9 - 137*x^8 + 168*x^7 + 54*x^6 + 189*x^5 + 648*x^4 - 1944*x^3 + 2187*x^2 - 2187*x + 6561 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 55) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{13} + \beta_{11} + \cdots + \beta_1) q^{2}+ \cdots + ( - \beta_{15} + \beta_{13} + \beta_{4} + \cdots + 1) q^{9}+O(q^{10})$$ q + (-b13 + b11 - b8 + b5 + b4 + b1) * q^2 + (-b14 - b12) * q^3 + (-b9 - b7 - b3) * q^4 + (b15 - b6 + b1) * q^5 + 2*b15 * q^6 - 2*b8 * q^7 + (b14 + b12 + 3*b11) * q^8 + (-b15 + b13 + b4 + b3 - b2 + 1) * q^9 $$q + ( - \beta_{13} + \beta_{11} + \cdots + \beta_1) q^{2}+ \cdots + ( - 5 \beta_{14} - 5 \beta_{13} + \cdots + 5 \beta_{4}) q^{98}+O(q^{100})$$ q + (-b13 + b11 - b8 + b5 + b4 + b1) * q^2 + (-b14 - b12) * q^3 + (-b9 - b7 - b3) * q^4 + (b15 - b6 + b1) * q^5 + 2*b15 * q^6 - 2*b8 * q^7 + (b14 + b12 + 3*b11) * q^8 + (-b15 + b13 + b4 + b3 - b2 + 1) * q^9 + (b14 - b13 - b12 - b10 - b9 - b7 + b6 - b5 - b4 + 2) * q^10 + 2*b5 * q^12 + (-2*b14 + 2*b12 - 2*b2) * q^14 + (-b9 + b8 + 2*b7) * q^15 + (-3*b15 + b10 - b6) * q^16 + (-2*b10 - 2*b6) * q^17 + (b9 + 3*b8 - b7) * q^18 - 4*b2 * q^19 + (-3*b15 - 3*b13 + 4*b11 - 4*b8 + 4*b5 + 3*b4 + 3*b3 - 3*b2 + 4*b1 + 3) * q^20 + (2*b14 - 2*b13 - 2*b12 - 2*b10 - 2*b9 - 2*b7 + 2*b6 - 2*b4 - 4) * q^21 + (-b14 - b13 - b12 + b10 + b9 - b7 + b6 + b4) * q^23 + (-2*b15 + 2*b13 + 2*b4 + 2*b3 - 2*b2 + 2) * q^24 + (3*b14 + b11 - b2) * q^25 + (b10 + b6 - 2*b1) * q^27 + (6*b10 + 6*b6 - 6*b1) * q^28 + (2*b9 + 2*b7 + 2*b3) * q^29 + (-2*b12 - 2*b11 - 2*b2) * q^30 + (b13 + b4) * q^31 + (3*b14 + 3*b13 + 3*b12 - 3*b10 - 3*b9 + 3*b7 - 3*b6 - b5 - 3*b4) * q^32 + 4 * q^34 + (-4*b15 + 2*b13 - 4*b4 + 4*b3 - 4*b2 + 4) * q^35 + (b14 - b12 + 7*b2) * q^36 + (3*b9 - 2*b8 - 3*b7) * q^37 + (4*b10 + 4*b6 - 4*b1) * q^38 + (-5*b9 - b8 + b7 - 6*b3) * q^40 + (2*b14 - 2*b12 + 2*b2) * q^41 + (4*b11 - 4*b8 + 4*b5 + 4*b1) * q^42 - 2*b5 * q^43 + (-3*b14 - b13 - b12 + 3*b10 + 3*b9 - b7 + b6 + 2*b5 + 3*b4 + 5) * q^45 + (2*b15 - 2*b3 + 2*b2 - 2) * q^46 + (4*b14 + 4*b12 + 2*b11) * q^47 + (2*b9 + 2*b8 - 2*b7) * q^48 + 5*b15 * q^49 + (-4*b15 - b10 - 3*b6 + 5*b1) * q^50 + (-2*b9 - 2*b7 + 8*b3) * q^51 + (4*b13 - 4*b11 + 4*b8 - 4*b5 - 4*b4 - 4*b1) * q^53 + (-2*b14 + 2*b13 + 2*b12 + 2*b10 + 2*b9 + 2*b7 - 2*b6 + 2*b4 - 4) * q^54 + (-2*b14 + 2*b13 + 2*b12 + 2*b10 + 2*b9 + 2*b7 - 2*b6 + 2*b4 - 14) * q^56 + (-4*b13 + 4*b4) * q^57 + (-6*b14 - 6*b12 - 10*b11) * q^58 + (b9 + b7 + 4*b3) * q^59 + (4*b15 - 4*b10 - 2*b6) * q^60 + (6*b15 + 2*b10 - 2*b6) * q^61 + (2*b9 + 4*b8 - 2*b7) * q^62 + (6*b14 + 6*b12 + 2*b11) * q^63 + (-b15 + b13 + b4 + b3 - b2 + 1) * q^64 + (3*b14 + 3*b13 + 3*b12 - 3*b10 - 3*b9 + 3*b7 - 3*b6 - 4*b5 - 3*b4) * q^67 + (4*b11 - 4*b8 + 4*b5 + 4*b1) * q^68 + (b14 - b12 - 4*b2) * q^69 + (-6*b9 - 8*b8 + 6*b7 + 6*b3) * q^70 + (-3*b10 + 3*b6) * q^71 + (-7*b10 - 7*b6 + 5*b1) * q^72 - 4*b8 * q^73 + (-2*b14 + 2*b12 + 4*b2) * q^74 + (4*b15 - 3*b13 - 3*b11 + 3*b8 - 3*b5 + 2*b4 - 4*b3 + 4*b2 - 3*b1 - 4) * q^75 + (-4*b14 + 4*b13 + 4*b12 + 4*b10 + 4*b9 + 4*b7 - 4*b6 + 4*b4 - 4) * q^76 + (8*b15 - 2*b13 - 2*b4 - 8*b3 + 8*b2 - 8) * q^79 + (3*b14 + 5*b12 + 6*b11 - b2) * q^80 + (2*b9 + 2*b7 - 3*b3) * q^81 + (-6*b10 - 6*b6 + 10*b1) * q^82 + (4*b10 + 4*b6 - 2*b1) * q^83 - 12*b3 * q^84 + (-2*b14 - 4*b12 + 2*b11) * q^85 + (-2*b15 - 2*b13 - 2*b4 + 2*b3 - 2*b2 + 2) * q^86 - 4*b5 * q^87 + (-b14 + b13 + b12 + b10 + b9 + b7 - b6 + b4 + 2) * q^89 + (6*b15 - b13 + b11 - b8 + b5 + 5*b4 - 6*b3 + 6*b2 + b1 - 6) * q^90 + 2*b8 * q^92 + (-b10 - b6 - 2*b1) * q^93 + (-10*b15 + 2*b10 - 2*b6) * q^94 + (-4*b8 + 4*b7 - 4*b3) * q^95 + (-2*b14 + 2*b12 + 10*b2) * q^96 + (-3*b13 + 2*b11 - 2*b8 + 2*b5 + 3*b4 + 2*b1) * q^97 + (-5*b14 - 5*b13 - 5*b12 + 5*b10 + 5*b9 - 5*b7 + 5*b6 + 5*b5 + 5*b4) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 6 q^{4} + 3 q^{5} + 8 q^{6} + 2 q^{9}+O(q^{10})$$ 16 * q + 6 * q^4 + 3 * q^5 + 8 * q^6 + 2 * q^9 $$16 q + 6 q^{4} + 3 q^{5} + 8 q^{6} + 2 q^{9} + 40 q^{10} - 12 q^{14} - q^{15} - 14 q^{16} - 16 q^{19} + 12 q^{20} - 48 q^{21} + 4 q^{24} - q^{25} - 12 q^{29} - 6 q^{30} - 2 q^{31} + 64 q^{34} + 18 q^{35} + 30 q^{36} + 28 q^{40} + 12 q^{41} + 72 q^{45} - 8 q^{46} + 20 q^{49} - 18 q^{50} - 28 q^{51} - 80 q^{54} - 240 q^{56} - 18 q^{59} + 18 q^{60} + 20 q^{61} + 2 q^{64} - 14 q^{69} - 24 q^{70} + 6 q^{71} + 12 q^{74} - 15 q^{75} - 96 q^{76} - 28 q^{79} - 6 q^{80} + 8 q^{81} + 48 q^{84} + 2 q^{85} + 12 q^{86} + 24 q^{89} - 28 q^{90} - 44 q^{94} + 12 q^{95} + 36 q^{96}+O(q^{100})$$ 16 * q + 6 * q^4 + 3 * q^5 + 8 * q^6 + 2 * q^9 + 40 * q^10 - 12 * q^14 - q^15 - 14 * q^16 - 16 * q^19 + 12 * q^20 - 48 * q^21 + 4 * q^24 - q^25 - 12 * q^29 - 6 * q^30 - 2 * q^31 + 64 * q^34 + 18 * q^35 + 30 * q^36 + 28 * q^40 + 12 * q^41 + 72 * q^45 - 8 * q^46 + 20 * q^49 - 18 * q^50 - 28 * q^51 - 80 * q^54 - 240 * q^56 - 18 * q^59 + 18 * q^60 + 20 * q^61 + 2 * q^64 - 14 * q^69 - 24 * q^70 + 6 * q^71 + 12 * q^74 - 15 * q^75 - 96 * q^76 - 28 * q^79 - 6 * q^80 + 8 * q^81 + 48 * q^84 + 2 * q^85 + 12 * q^86 + 24 * q^89 - 28 * q^90 - 44 * q^94 + 12 * q^95 + 36 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - x^{15} + 3 x^{14} - 8 x^{13} + 8 x^{12} + 7 x^{11} + 6 x^{10} + 56 x^{9} - 137 x^{8} + \cdots + 6561$$ :

 $$\beta_{1}$$ $$=$$ $$( - 5 \nu^{15} + 15 \nu^{14} - 40 \nu^{13} + 40 \nu^{12} - 5709 \nu^{11} + 30 \nu^{10} + \cdots + 32805 ) / 261711$$ (-5*v^15 + 15*v^14 - 40*v^13 + 40*v^12 - 5709*v^11 + 30*v^10 + 280*v^9 - 685*v^8 + 840*v^7 - 90557*v^6 + 945*v^5 + 3240*v^4 - 9720*v^3 + 10935*v^2 - 1444392*v + 32805) / 261711 $$\beta_{2}$$ $$=$$ $$( - 160 \nu^{15} + 400 \nu^{14} + 3467 \nu^{12} - 1040 \nu^{11} - 4960 \nu^{10} + 6480 \nu^{9} + \cdots - 524880 ) / 785133$$ (-160*v^15 + 400*v^14 + 3467*v^12 - 1040*v^11 - 4960*v^10 + 6480*v^9 - 2480*v^8 + 39680*v^7 - 19440*v^6 - 66960*v^5 + 103680*v^4 - 19440*v^3 + 602640*v^2 - 349920*v - 524880) / 785133 $$\beta_{3}$$ $$=$$ $$( 3 \nu^{15} + 6 \nu^{14} - 31 \nu^{13} + 3 \nu^{12} - 48 \nu^{11} + 93 \nu^{10} + 81 \nu^{9} + \cdots + 13122 ) / 9693$$ (3*v^15 + 6*v^14 - 31*v^13 + 3*v^12 - 48*v^11 + 93*v^10 + 81*v^9 - 496*v^8 + 93*v^7 - 729*v^6 + 1674*v^5 + 1053*v^4 - 8920*v^3 - 10935*v + 13122) / 9693 $$\beta_{4}$$ $$=$$ $$( 9 \nu^{15} - 40 \nu^{14} + 27 \nu^{13} - 72 \nu^{12} + 72 \nu^{11} + 63 \nu^{10} - 664 \nu^{9} + \cdots - 19683 ) / 58158$$ (9*v^15 - 40*v^14 + 27*v^13 - 72*v^12 + 72*v^11 + 63*v^10 - 664*v^9 + 504*v^8 - 1233*v^7 + 1512*v^6 + 486*v^5 - 20557*v^4 + 5832*v^3 - 17496*v^2 + 19683*v - 19683) / 58158 $$\beta_{5}$$ $$=$$ $$( 31\nu^{15} + 718\nu^{10} + 22258\nu^{5} + 146286 ) / 87237$$ (31*v^15 + 718*v^10 + 22258*v^5 + 146286) / 87237 $$\beta_{6}$$ $$=$$ $$( - 31 \nu^{15} + 93 \nu^{14} - 248 \nu^{13} + 248 \nu^{12} - 501 \nu^{11} + 186 \nu^{10} + \cdots + 203391 ) / 174474$$ (-31*v^15 + 93*v^14 - 248*v^13 + 248*v^12 - 501*v^11 + 186*v^10 + 1736*v^9 - 4247*v^8 + 5208*v^7 - 20584*v^6 + 5859*v^5 + 20088*v^4 - 60264*v^3 + 67797*v^2 - 301320*v + 203391) / 174474 $$\beta_{7}$$ $$=$$ $$( - 403 \nu^{15} - 806 \nu^{14} + 1053 \nu^{13} - 403 \nu^{12} + 6448 \nu^{11} - 12493 \nu^{10} + \cdots - 1762722 ) / 523422$$ (-403*v^15 - 806*v^14 + 1053*v^13 - 403*v^12 + 6448*v^11 - 12493*v^10 - 10881*v^9 + 28336*v^8 - 12493*v^7 + 97929*v^6 - 224874*v^5 - 141453*v^4 + 266409*v^3 + 1468935*v - 1762722) / 523422 $$\beta_{8}$$ $$=$$ $$( 842 \nu^{15} + 1684 \nu^{14} - 324 \nu^{13} + 842 \nu^{12} - 13472 \nu^{11} + 26102 \nu^{10} + \cdots + 3682908 ) / 785133$$ (842*v^15 + 1684*v^14 - 324*v^13 + 842*v^12 - 13472*v^11 + 26102*v^10 + 22734*v^9 + 4150*v^8 + 26102*v^7 - 204606*v^6 + 469836*v^5 + 295542*v^4 + 5265*v^3 - 3069090*v + 3682908) / 785133 $$\beta_{9}$$ $$=$$ $$( - 1079 \nu^{15} - 2158 \nu^{14} - 6561 \nu^{13} - 1079 \nu^{12} + 17264 \nu^{11} - 33449 \nu^{10} + \cdots - 4719546 ) / 1570266$$ (-1079*v^15 - 2158*v^14 - 6561*v^13 - 1079*v^12 + 17264*v^11 - 33449*v^10 - 29133*v^9 - 79846*v^8 - 33449*v^7 + 262197*v^6 - 602082*v^5 - 378729*v^4 - 1310985*v^3 + 3932955*v - 4719546) / 1570266 $$\beta_{10}$$ $$=$$ $$( \nu^{15} - 3 \nu^{14} + 8 \nu^{13} - 8 \nu^{12} - 7 \nu^{11} - 6 \nu^{10} - 56 \nu^{9} + 137 \nu^{8} + \cdots - 6561 ) / 2154$$ (v^15 - 3*v^14 + 8*v^13 - 8*v^12 - 7*v^11 - 6*v^10 - 56*v^9 + 137*v^8 - 168*v^7 - 54*v^6 - 189*v^5 - 648*v^4 + 1944*v^3 - 2187*v^2 + 386*v - 6561) / 2154 $$\beta_{11}$$ $$=$$ $$( 646 \nu^{15} - 1615 \nu^{14} + 178 \nu^{12} + 4199 \nu^{11} + 20026 \nu^{10} - 26163 \nu^{9} + \cdots + 2119203 ) / 785133$$ (646*v^15 - 1615*v^14 + 178*v^12 + 4199*v^11 + 20026*v^10 - 26163*v^9 + 10013*v^8 + 14266*v^7 + 78489*v^6 + 270351*v^5 - 418608*v^4 + 78489*v^3 - 164997*v^2 + 1412802*v + 2119203) / 785133 $$\beta_{12}$$ $$=$$ $$( - 2 \nu^{15} + 5 \nu^{14} - 15 \nu^{12} - 13 \nu^{11} - 62 \nu^{10} + 81 \nu^{9} - 31 \nu^{8} + \cdots - 6561 ) / 2154$$ (-2*v^15 + 5*v^14 - 15*v^12 - 13*v^11 - 62*v^10 + 81*v^9 - 31*v^8 - 222*v^7 - 243*v^6 - 837*v^5 + 1296*v^4 - 243*v^3 - 3955*v^2 - 4374*v - 6561) / 2154 $$\beta_{13}$$ $$=$$ $$( - 599 \nu^{15} - 130 \nu^{14} - 1797 \nu^{13} + 4792 \nu^{12} - 4792 \nu^{11} - 4193 \nu^{10} + \cdots + 1310013 ) / 523422$$ (-599*v^15 - 130*v^14 - 1797*v^13 + 4792*v^12 - 4792*v^11 - 4193*v^10 - 3594*v^9 - 33544*v^8 + 82063*v^7 - 100632*v^6 - 32346*v^5 - 113211*v^4 - 388152*v^3 + 1164456*v^2 - 1310013*v + 1310013) / 523422 $$\beta_{14}$$ $$=$$ $$( 2170 \nu^{15} - 5425 \nu^{14} - 8855 \nu^{12} + 14105 \nu^{11} + 67270 \nu^{10} - 87885 \nu^{9} + \cdots + 7118685 ) / 1570266$$ (2170*v^15 - 5425*v^14 - 8855*v^12 + 14105*v^11 + 67270*v^10 - 87885*v^9 + 33635*v^8 - 189212*v^7 + 263655*v^6 + 908145*v^5 - 1406160*v^4 + 263655*v^3 - 2066715*v^2 + 4745790*v + 7118685) / 1570266 $$\beta_{15}$$ $$=$$ $$( - 248 \nu^{15} + 744 \nu^{14} - 1984 \nu^{13} + 1984 \nu^{12} - 4008 \nu^{11} + 1488 \nu^{10} + \cdots + 1627128 ) / 261711$$ (-248*v^15 + 744*v^14 - 1984*v^13 + 1984*v^12 - 4008*v^11 + 1488*v^10 + 13888*v^9 - 33976*v^8 + 41664*v^7 - 77435*v^6 + 46872*v^5 + 160704*v^4 - 482112*v^3 + 542376*v^2 - 1014768*v + 1627128) / 261711
 $$\nu$$ $$=$$ $$( \beta_{15} + 2\beta_{10} - \beta_1 ) / 2$$ (b15 + 2*b10 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( -2\beta_{12} - 3\beta_{11} - 3\beta_{2} ) / 2$$ (-2*b12 - 3*b11 - 3*b2) / 2 $$\nu^{3}$$ $$=$$ $$2\beta_{9} + \beta_{8} - 2\beta_{7} - 4\beta_{3}$$ 2*b9 + b8 - 2*b7 - 4*b3 $$\nu^{4}$$ $$=$$ $$( -\beta_{15} + \beta_{11} - \beta_{8} + \beta_{5} - 10\beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2$$ (-b15 + b11 - b8 + b5 - 10*b4 + b3 - b2 + b1 + 1) / 2 $$\nu^{5}$$ $$=$$ $$( 2\beta_{13} + 2\beta_{12} + 2\beta_{7} - 2\beta_{6} + 15\beta_{5} - 15 ) / 2$$ (2*b13 + 2*b12 + 2*b7 - 2*b6 + 15*b5 - 15) / 2 $$\nu^{6}$$ $$=$$ $$-5\beta_{15} - 16\beta_{10} - 16\beta_{6} + 8\beta_1$$ -5*b15 - 16*b10 - 16*b6 + 8*b1 $$\nu^{7}$$ $$=$$ $$( -26\beta_{14} + 35\beta_{11} - 35\beta_{2} ) / 2$$ (-26*b14 + 35*b11 - 35*b2) / 2 $$\nu^{8}$$ $$=$$ $$( 39\beta_{8} + 70\beta_{7} + 39\beta_{3} ) / 2$$ (39*b8 + 70*b7 + 39*b3) / 2 $$\nu^{9}$$ $$=$$ $$68 \beta_{15} - 74 \beta_{13} + 37 \beta_{11} - 37 \beta_{8} + 37 \beta_{5} + 74 \beta_{4} - 68 \beta_{3} + \cdots - 68$$ 68*b15 - 74*b13 + 37*b11 - 37*b8 + 37*b5 + 74*b4 - 68*b3 + 68*b2 + 37*b1 - 68 $$\nu^{10}$$ $$=$$ $$( 62\beta_{14} - 62\beta_{10} - 62\beta_{9} - 253\beta_{5} - 62\beta_{4} - 253 ) / 2$$ (62*b14 - 62*b10 - 62*b9 - 253*b5 - 62*b4 - 253) / 2 $$\nu^{11}$$ $$=$$ $$( -93\beta_{15} + 506\beta_{6} - 93\beta_1 ) / 2$$ (-93*b15 + 506*b6 - 93*b1) / 2 $$\nu^{12}$$ $$=$$ $$160\beta_{14} + 160\beta_{12} + 80\beta_{11} + 679\beta_{2}$$ 160*b14 + 160*b12 + 80*b11 + 679*b2 $$\nu^{13}$$ $$=$$ $$( -1198\beta_{9} - 1079\beta_{8} + 1079\beta_{3} ) / 2$$ (-1198*b9 - 1079*b8 + 1079*b3) / 2 $$\nu^{14}$$ $$=$$ $$( - 1797 \beta_{15} + 2158 \beta_{13} - 1797 \beta_{11} + 1797 \beta_{8} - 1797 \beta_{5} + 1797 \beta_{3} + \cdots + 1797 ) / 2$$ (-1797*b15 + 2158*b13 - 1797*b11 + 1797*b8 - 1797*b5 + 1797*b3 - 1797*b2 - 1797*b1 + 1797) / 2 $$\nu^{15}$$ $$=$$ $$- 718 \beta_{14} - 718 \beta_{13} - 718 \beta_{12} + 718 \beta_{10} + 718 \beta_{9} - 718 \beta_{7} + \cdots + 3596$$ -718*b14 - 718*b13 - 718*b12 + 718*b10 + 718*b9 - 718*b7 + 718*b6 + 359*b5 + 718*b4 + 3596

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/605\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$486$$ $$\chi(n)$$ $$-1$$ $$\beta_{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 1.56693 + 0.738055i −0.144291 − 1.72603i −0.897801 − 1.48120i −0.833856 + 1.51812i −0.217724 − 1.71831i 1.59696 + 0.670602i 1.13127 + 1.31158i −1.70149 + 0.323920i 1.56693 − 0.738055i −0.144291 + 1.72603i −0.897801 + 1.48120i −0.833856 − 1.51812i −0.217724 + 1.71831i 1.59696 − 0.670602i 1.13127 − 1.31158i −1.70149 − 0.323920i
−2.40079 0.780063i 0.465695 + 0.640974i 3.53725 + 2.56996i −1.81200 + 1.31021i −0.618034 1.90211i 2.03615 2.80252i −3.51992 4.84475i 0.733075 2.25617i 5.37228 1.73205i
9.2 −0.753510 0.244830i 1.48377 + 2.04223i −1.11020 0.806607i −0.228670 2.22434i −0.618034 1.90211i −2.03615 + 2.80252i 1.57045 + 2.16154i −1.04209 + 3.20723i −0.372281 + 1.73205i
9.3 0.753510 + 0.244830i −1.48377 2.04223i −1.11020 0.806607i −1.12244 1.93394i −0.618034 1.90211i 2.03615 2.80252i −1.57045 2.16154i −1.04209 + 3.20723i −0.372281 1.73205i
9.4 2.40079 + 0.780063i −0.465695 0.640974i 3.53725 + 2.56996i 2.23606 0.00508966i −0.618034 1.90211i −2.03615 + 2.80252i 3.51992 + 4.84475i 0.733075 2.25617i 5.37228 + 1.73205i
124.1 −1.48377 2.04223i −0.753510 + 0.244830i −1.35111 + 4.15829i −1.80602 + 1.31844i 1.61803 + 1.17557i −3.29456 1.07047i 5.69534 1.85053i −1.91922 + 1.39439i 5.37228 + 1.73205i
124.2 −0.465695 0.640974i −2.40079 + 0.780063i 0.424058 1.30512i 2.04481 + 0.904839i 1.61803 + 1.17557i 3.29456 + 1.07047i −2.54105 + 0.825636i 2.72823 1.98218i −0.372281 1.73205i
124.3 0.465695 + 0.640974i 2.40079 0.780063i 0.424058 1.30512i 1.49244 + 1.66512i 1.61803 + 1.17557i −3.29456 1.07047i 2.54105 0.825636i 2.72823 1.98218i −0.372281 + 1.73205i
124.4 1.48377 + 2.04223i 0.753510 0.244830i −1.35111 + 4.15829i 0.695822 2.12505i 1.61803 + 1.17557i 3.29456 + 1.07047i −5.69534 + 1.85053i −1.91922 + 1.39439i 5.37228 1.73205i
269.1 −2.40079 + 0.780063i 0.465695 0.640974i 3.53725 2.56996i −1.81200 1.31021i −0.618034 + 1.90211i 2.03615 + 2.80252i −3.51992 + 4.84475i 0.733075 + 2.25617i 5.37228 + 1.73205i
269.2 −0.753510 + 0.244830i 1.48377 2.04223i −1.11020 + 0.806607i −0.228670 + 2.22434i −0.618034 + 1.90211i −2.03615 2.80252i 1.57045 2.16154i −1.04209 3.20723i −0.372281 1.73205i
269.3 0.753510 0.244830i −1.48377 + 2.04223i −1.11020 + 0.806607i −1.12244 + 1.93394i −0.618034 + 1.90211i 2.03615 + 2.80252i −1.57045 + 2.16154i −1.04209 3.20723i −0.372281 + 1.73205i
269.4 2.40079 0.780063i −0.465695 + 0.640974i 3.53725 2.56996i 2.23606 + 0.00508966i −0.618034 + 1.90211i −2.03615 2.80252i 3.51992 4.84475i 0.733075 + 2.25617i 5.37228 1.73205i
444.1 −1.48377 + 2.04223i −0.753510 0.244830i −1.35111 4.15829i −1.80602 1.31844i 1.61803 1.17557i −3.29456 + 1.07047i 5.69534 + 1.85053i −1.91922 1.39439i 5.37228 1.73205i
444.2 −0.465695 + 0.640974i −2.40079 0.780063i 0.424058 + 1.30512i 2.04481 0.904839i 1.61803 1.17557i 3.29456 1.07047i −2.54105 0.825636i 2.72823 + 1.98218i −0.372281 + 1.73205i
444.3 0.465695 0.640974i 2.40079 + 0.780063i 0.424058 + 1.30512i 1.49244 1.66512i 1.61803 1.17557i −3.29456 + 1.07047i 2.54105 + 0.825636i 2.72823 + 1.98218i −0.372281 1.73205i
444.4 1.48377 2.04223i 0.753510 + 0.244830i −1.35111 4.15829i 0.695822 + 2.12505i 1.61803 1.17557i 3.29456 1.07047i −5.69534 1.85053i −1.91922 1.39439i 5.37228 + 1.73205i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 9.4 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 3 inner
55.j even 10 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.j.j 16
5.b even 2 1 inner 605.2.j.j 16
11.b odd 2 1 605.2.j.i 16
11.c even 5 1 605.2.b.c 4
11.c even 5 3 inner 605.2.j.j 16
11.d odd 10 1 55.2.b.a 4
11.d odd 10 3 605.2.j.i 16
33.f even 10 1 495.2.c.a 4
44.g even 10 1 880.2.b.h 4
55.d odd 2 1 605.2.j.i 16
55.h odd 10 1 55.2.b.a 4
55.h odd 10 3 605.2.j.i 16
55.j even 10 1 605.2.b.c 4
55.j even 10 3 inner 605.2.j.j 16
55.k odd 20 2 3025.2.a.ba 4
55.l even 20 2 275.2.a.h 4
165.r even 10 1 495.2.c.a 4
165.u odd 20 2 2475.2.a.bi 4
220.o even 10 1 880.2.b.h 4
220.w odd 20 2 4400.2.a.cc 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.b.a 4 11.d odd 10 1
55.2.b.a 4 55.h odd 10 1
275.2.a.h 4 55.l even 20 2
495.2.c.a 4 33.f even 10 1
495.2.c.a 4 165.r even 10 1
605.2.b.c 4 11.c even 5 1
605.2.b.c 4 55.j even 10 1
605.2.j.i 16 11.b odd 2 1
605.2.j.i 16 11.d odd 10 3
605.2.j.i 16 55.d odd 2 1
605.2.j.i 16 55.h odd 10 3
605.2.j.j 16 1.a even 1 1 trivial
605.2.j.j 16 5.b even 2 1 inner
605.2.j.j 16 11.c even 5 3 inner
605.2.j.j 16 55.j even 10 3 inner
880.2.b.h 4 44.g even 10 1
880.2.b.h 4 220.o even 10 1
2475.2.a.bi 4 165.u odd 20 2
3025.2.a.ba 4 55.k odd 20 2
4400.2.a.cc 4 220.w odd 20 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(605, [\chi])$$:

 $$T_{2}^{16} - 7T_{2}^{14} + 45T_{2}^{12} - 287T_{2}^{10} + 1829T_{2}^{8} - 1148T_{2}^{6} + 720T_{2}^{4} - 448T_{2}^{2} + 256$$ T2^16 - 7*T2^14 + 45*T2^12 - 287*T2^10 + 1829*T2^8 - 1148*T2^6 + 720*T2^4 - 448*T2^2 + 256 $$T_{19}^{4} + 4T_{19}^{3} + 16T_{19}^{2} + 64T_{19} + 256$$ T19^4 + 4*T19^3 + 16*T19^2 + 64*T19 + 256

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - 7 T^{14} + \cdots + 256$$
$3$ $$T^{16} - 7 T^{14} + \cdots + 256$$
$5$ $$T^{16} - 3 T^{15} + \cdots + 390625$$
$7$ $$(T^{8} - 12 T^{6} + \cdots + 20736)^{2}$$
$11$ $$T^{16}$$
$13$ $$T^{16}$$
$17$ $$T^{16} - 28 T^{14} + \cdots + 16777216$$
$19$ $$(T^{4} + 4 T^{3} + \cdots + 256)^{4}$$
$23$ $$(T^{4} + 7 T^{2} + 4)^{4}$$
$29$ $$(T^{8} + 6 T^{7} + \cdots + 331776)^{2}$$
$31$ $$(T^{8} + T^{7} + 9 T^{6} + \cdots + 4096)^{2}$$
$37$ $$T^{16} + \cdots + 429981696$$
$41$ $$(T^{8} - 6 T^{7} + \cdots + 331776)^{2}$$
$43$ $$(T^{2} + 12)^{8}$$
$47$ $$(T^{8} - 44 T^{6} + \cdots + 3748096)^{2}$$
$53$ $$T^{16} + \cdots + 1099511627776$$
$59$ $$(T^{8} + 9 T^{7} + \cdots + 20736)^{2}$$
$61$ $$(T^{8} - 10 T^{7} + \cdots + 4096)^{2}$$
$67$ $$(T^{4} + 87 T^{2} + 36)^{4}$$
$71$ $$(T^{8} - 3 T^{7} + \cdots + 26873856)^{2}$$
$73$ $$(T^{8} - 48 T^{6} + \cdots + 5308416)^{2}$$
$79$ $$(T^{8} + 14 T^{7} + \cdots + 65536)^{2}$$
$83$ $$(T^{8} - 44 T^{6} + \cdots + 3748096)^{2}$$
$89$ $$(T^{2} - 3 T - 6)^{8}$$
$97$ $$T^{16} + \cdots + 110075314176$$