# Properties

 Label 225.2.h.e Level $225$ Weight $2$ Character orbit 225.h Analytic conductor $1.797$ 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] = [225,2,Mod(46,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 6]))

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

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

chi := DirichletCharacter("225.46");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 225.h (of order $$5$$, 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: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{5})$$ Coefficient field: 16.0.1130304400000000000000.3 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 5x^{14} + 5x^{12} - 10x^{10} + 205x^{8} - 700x^{6} + 1250x^{4} - 1250x^{2} + 625$$ x^16 - 5*x^14 + 5*x^12 - 10*x^10 + 205*x^8 - 700*x^6 + 1250*x^4 - 1250*x^2 + 625 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$5^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{14} + \beta_{12} - \beta_{10} + \beta_{3}) q^{2} + ( - \beta_{11} - \beta_{8}) q^{4} + (\beta_{12} + \beta_{5}) q^{5} + (\beta_{13} - \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{2} - 1) q^{7} - \beta_{4} q^{8}+O(q^{10})$$ q + (-b14 + b12 - b10 + b3) * q^2 + (-b11 - b8) * q^4 + (b12 + b5) * q^5 + (b13 - b8 - 2*b7 - b6 - b2 - 1) * q^7 - b4 * q^8 $$q + ( - \beta_{14} + \beta_{12} - \beta_{10} + \beta_{3}) q^{2} + ( - \beta_{11} - \beta_{8}) q^{4} + (\beta_{12} + \beta_{5}) q^{5} + (\beta_{13} - \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{2} - 1) q^{7} - \beta_{4} q^{8} + ( - \beta_{15} - \beta_{8} - \beta_{7} - 2 \beta_{6} - 1) q^{10} + (\beta_{14} - \beta_{12} + \beta_{9} + \beta_{5} + \beta_1) q^{11} + (\beta_{15} - \beta_{8} - \beta_{6} - 2 \beta_{2}) q^{13} + (\beta_{14} - \beta_{12} + 2 \beta_{10} - \beta_{5} - 2 \beta_{3}) q^{14} + (\beta_{11} - \beta_{7} - \beta_{6} + \beta_{2}) q^{16} + ( - \beta_{14} - 2 \beta_{10} - \beta_{9} - \beta_{5} + \beta_{4} + \beta_{3}) q^{17} + (\beta_{15} - \beta_{13} - 3 \beta_{11} - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - \beta_{2} + 3) q^{19} + ( - \beta_{14} + 2 \beta_{10} + \beta_{4} - \beta_{3} - \beta_1) q^{20} + (2 \beta_{13} + 2 \beta_{11} - \beta_{8} - 4 \beta_{7} - 4) q^{22} + (\beta_{14} - \beta_{12} - 2 \beta_{9} - 2 \beta_{5} - 2 \beta_1) q^{23} + ( - 2 \beta_{15} - \beta_{11} + 2 \beta_{8} + \beta_{7} + \beta_{2} + 1) q^{25} + (\beta_{14} - 4 \beta_{12} + \beta_{10} + 2 \beta_{9} - \beta_{4} + \beta_1) q^{26} + ( - \beta_{13} + 3 \beta_{11} + \beta_{8} + 2 \beta_{7} + 2) q^{28} + (2 \beta_{14} - 3 \beta_{5} + 3 \beta_{4} - \beta_1) q^{29} + (\beta_{15} - \beta_{13} - 2 \beta_{11} + 3 \beta_{8} + 5 \beta_{7} + 3 \beta_{6} + 3 \beta_{2} + \cdots + 4) q^{31}+ \cdots + (4 \beta_{14} - 4 \beta_{12} + 2 \beta_{9} + \beta_{5} + 2 \beta_1) q^{98}+O(q^{100})$$ q + (-b14 + b12 - b10 + b3) * q^2 + (-b11 - b8) * q^4 + (b12 + b5) * q^5 + (b13 - b8 - 2*b7 - b6 - b2 - 1) * q^7 - b4 * q^8 + (-b15 - b8 - b7 - 2*b6 - 1) * q^10 + (b14 - b12 + b9 + b5 + b1) * q^11 + (b15 - b8 - b6 - 2*b2) * q^13 + (b14 - b12 + 2*b10 - b5 - 2*b3) * q^14 + (b11 - b7 - b6 + b2) * q^16 + (-b14 - 2*b10 - b9 - b5 + b4 + b3) * q^17 + (b15 - b13 - 3*b11 - b8 + 2*b7 + 2*b6 - b2 + 3) * q^19 + (-b14 + 2*b10 + b4 - b3 - b1) * q^20 + (2*b13 + 2*b11 - b8 - 4*b7 - 4) * q^22 + (b14 - b12 - 2*b9 - 2*b5 - 2*b1) * q^23 + (-2*b15 - b11 + 2*b8 + b7 + b2 + 1) * q^25 + (b14 - 4*b12 + b10 + 2*b9 - b4 + b1) * q^26 + (-b13 + 3*b11 + b8 + 2*b7 + 2) * q^28 + (2*b14 - 3*b5 + 3*b4 - b1) * q^29 + (b15 - b13 - 2*b11 + 3*b8 + 5*b7 + 3*b6 + 3*b2 + 4) * q^31 + (-b14 + 3*b12 - b10 - 3*b9) * q^32 + (-3*b11 + 3*b6 + 2*b2) * q^34 + (-2*b12 + b10 + b5 - 2*b3 - b1) * q^35 + (-2*b15 + 2*b11 + 2*b8 - b7 - b2) * q^37 + (-b12 + b10 + 2*b9 + b5 - 2*b4 + 4*b3 + b1) * q^38 + (-b13 + b8 + 5*b7 + b6 + 2*b2 + 1) * q^40 + (b12 - b10 + 3*b9 + 4*b5 - 3*b4 - 3*b3 + 4*b1) * q^41 + (2*b15 - b7 - b6 - 2*b2 - 5) * q^43 + (4*b14 + 3*b10 + b4 - 4*b3) * q^44 + (-b13 + 5*b11 + 2*b8 - b7 - 1) * q^46 + (-2*b12 + b5 - b4 - 2*b3 + 3*b1) * q^47 + (2*b15 - b13 + b8 + b7 - b2 - 1) * q^49 + (-5*b14 + 2*b12 - 2*b10 - b9 + 2*b4 + b3 - 2*b1) * q^50 + (2*b15 + 2*b11 + 3*b6 - 2) * q^52 + (-3*b14 + 5*b12 + b5 - b4 + 5*b3 - 2*b1) * q^53 + (-b15 + 2*b13 + 3*b11 - 5*b7 - b6 - 2*b2 - 7) * q^55 + (-b14 - 2*b10 + b9 + b5 - 2*b4 + b3) * q^56 + (-b15 + b13 + 3*b11 + 3*b8 - b6 + 3*b2 - 2) * q^58 + (6*b12 - 6*b10 - b9 - 3*b5 + b4 + 5*b3 - 3*b1) * q^59 + (-2*b15 - 2*b11 - b6 + 2) * q^61 + (5*b12 - 5*b10 - 2*b9 + b5 + 2*b4 + 5*b3 + b1) * q^62 + (-3*b15 - b13 + 2*b11 + b7 - 4*b6 + b2 - 2) * q^64 + (-2*b12 + b10 + b5 + 3*b3 - b1) * q^65 + (2*b15 - 2*b13 - 8*b11 + 2*b8 + 10*b7 + b6 + 2*b2 + 3) * q^67 + (b14 + 2*b12 + b10 - 2*b4 + 2*b1) * q^68 + (3*b15 - 3*b13 + b8 + 3*b7 + 4*b6 + b2 + 6) * q^70 + (-7*b14 + 2*b1) * q^71 + (-2*b15 - 2*b13 - 9*b11 + 2*b7 + 10*b6 + 2*b2 + 9) * q^73 + (-4*b14 - b12 - 4*b10 + b9 + 2*b4 - 2*b1) * q^74 + (3*b13 - 3*b8 - b7 + 2*b6 - 3*b2 - 6) * q^76 + (b14 - b12 + 4*b10 - 4*b3) * q^77 + (3*b13 + 7*b11 + 2*b8) * q^79 + (b14 + 2*b12 - b9 - b5 + 2*b4 - 2*b3 - 2*b1) * q^80 + (-2*b15 + b13 - b8 - 6*b7 - 5*b6 + b2 + 4) * q^82 + (b14 + b10 + b9 + b5 - 3*b4 - b3) * q^83 + (b15 + 2*b13 + 3*b11 - 3*b8 - 8*b7 - 7*b6 - 2*b2 - 5) * q^85 + (8*b14 - 8*b12 + 5*b10 + 2*b9 - 5*b3 + 2*b1) * q^86 + (3*b11 + b7 - 3*b6) * q^88 + (4*b14 - 4*b12 + 6*b10 + b9 - 2*b5 - 6*b3 + b1) * q^89 + (2*b15 - 2*b8 - 6*b7 - 2*b6 - b2) * q^91 + (-2*b14 - 6*b10 - 3*b9 - 3*b5 + b4 + 2*b3) * q^92 + (-b15 + b13 + b11 + b8 + 3*b6 + b2 + 2) * q^94 + (-b14 + 4*b10 + 2*b5 - b4 + 2*b3 + b1) * q^95 + (-4*b13 + b11 - 2*b8 + 4*b7 + 4) * q^97 + (4*b14 - 4*b12 + 2*b9 + b5 + 2*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 2 q^{4}+O(q^{10})$$ 16 * q - 2 * q^4 $$16 q - 2 q^{4} + 14 q^{16} + 14 q^{19} - 30 q^{22} + 10 q^{25} + 30 q^{28} + 18 q^{31} - 20 q^{34} + 10 q^{37} - 10 q^{40} - 80 q^{43} - 32 q^{49} - 40 q^{52} - 70 q^{55} - 10 q^{58} + 32 q^{61} - 8 q^{64} - 40 q^{67} + 50 q^{70} + 60 q^{73} - 88 q^{76} + 36 q^{79} + 120 q^{82} + 20 q^{88} + 30 q^{91} + 30 q^{94} + 40 q^{97}+O(q^{100})$$ 16 * q - 2 * q^4 + 14 * q^16 + 14 * q^19 - 30 * q^22 + 10 * q^25 + 30 * q^28 + 18 * q^31 - 20 * q^34 + 10 * q^37 - 10 * q^40 - 80 * q^43 - 32 * q^49 - 40 * q^52 - 70 * q^55 - 10 * q^58 + 32 * q^61 - 8 * q^64 - 40 * q^67 + 50 * q^70 + 60 * q^73 - 88 * q^76 + 36 * q^79 + 120 * q^82 + 20 * q^88 + 30 * q^91 + 30 * q^94 + 40 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 5x^{14} + 5x^{12} - 10x^{10} + 205x^{8} - 700x^{6} + 1250x^{4} - 1250x^{2} + 625$$ :

 $$\beta_{1}$$ $$=$$ $$( - 9094 \nu^{15} + 529550 \nu^{13} - 1775155 \nu^{11} - 1261485 \nu^{9} - 4566370 \nu^{7} + 105496925 \nu^{5} - 206804175 \nu^{3} + 136237250 \nu ) / 54753875$$ (-9094*v^15 + 529550*v^13 - 1775155*v^11 - 1261485*v^9 - 4566370*v^7 + 105496925*v^5 - 206804175*v^3 + 136237250*v) / 54753875 $$\beta_{2}$$ $$=$$ $$( 18559 \nu^{14} + 486170 \nu^{12} - 1233230 \nu^{10} - 2183265 \nu^{8} - 1879805 \nu^{6} + 88524275 \nu^{4} - 87914875 \nu^{2} + 104474000 ) / 54753875$$ (18559*v^14 + 486170*v^12 - 1233230*v^10 - 2183265*v^8 - 1879805*v^6 + 88524275*v^4 - 87914875*v^2 + 104474000) / 54753875 $$\beta_{3}$$ $$=$$ $$( 85587 \nu^{15} - 683920 \nu^{13} + 729395 \nu^{11} + 734880 \nu^{9} + 21524610 \nu^{7} - 103007550 \nu^{5} + 105654425 \nu^{3} - 113406125 \nu ) / 54753875$$ (85587*v^15 - 683920*v^13 + 729395*v^11 + 734880*v^9 + 21524610*v^7 - 103007550*v^5 + 105654425*v^3 - 113406125*v) / 54753875 $$\beta_{4}$$ $$=$$ $$( 191309 \nu^{15} - 327040 \nu^{13} - 1605420 \nu^{11} - 1814665 \nu^{9} + 36243345 \nu^{7} - 11269000 \nu^{5} - 84275075 \nu^{3} + 103147500 \nu ) / 54753875$$ (191309*v^15 - 327040*v^13 - 1605420*v^11 - 1814665*v^9 + 36243345*v^7 - 11269000*v^5 - 84275075*v^3 + 103147500*v) / 54753875 $$\beta_{5}$$ $$=$$ $$( 216278 \nu^{15} - 1248275 \nu^{13} + 1298755 \nu^{11} - 360830 \nu^{9} + 45929540 \nu^{7} - 186657050 \nu^{5} + 255540700 \nu^{3} - 120294875 \nu ) / 54753875$$ (216278*v^15 - 1248275*v^13 + 1298755*v^11 - 360830*v^9 + 45929540*v^7 - 186657050*v^5 + 255540700*v^3 - 120294875*v) / 54753875 $$\beta_{6}$$ $$=$$ $$( 226756 \nu^{14} - 812710 \nu^{12} - 301695 \nu^{10} - 1956335 \nu^{8} + 45336155 \nu^{6} - 87919850 \nu^{4} + 95202125 \nu^{2} - 85476750 ) / 54753875$$ (226756*v^14 - 812710*v^12 - 301695*v^10 - 1956335*v^8 + 45336155*v^6 - 87919850*v^4 + 95202125*v^2 - 85476750) / 54753875 $$\beta_{7}$$ $$=$$ $$( - 237426 \nu^{14} + 787630 \nu^{12} + 357520 \nu^{10} + 1796010 \nu^{8} - 44976455 \nu^{6} + 87194950 \nu^{4} - 94337250 \nu^{2} - 3771750 ) / 54753875$$ (-237426*v^14 + 787630*v^12 + 357520*v^10 + 1796010*v^8 - 44976455*v^6 + 87194950*v^4 - 94337250*v^2 - 3771750) / 54753875 $$\beta_{8}$$ $$=$$ $$( 26057 \nu^{14} - 128005 \nu^{12} + 108960 \nu^{10} - 210920 \nu^{8} + 5037260 \nu^{6} - 17331500 \nu^{4} + 30950625 \nu^{2} - 22852125 ) / 4977625$$ (26057*v^14 - 128005*v^12 + 108960*v^10 - 210920*v^8 + 5037260*v^6 - 17331500*v^4 + 30950625*v^2 - 22852125) / 4977625 $$\beta_{9}$$ $$=$$ $$( - 323013 \nu^{15} + 1471550 \nu^{13} - 371875 \nu^{11} + 1061130 \nu^{9} - 66501065 \nu^{7} + 190202500 \nu^{5} - 199991675 \nu^{3} + \cdots + 54880500 \nu ) / 54753875$$ (-323013*v^15 + 1471550*v^13 - 371875*v^11 + 1061130*v^9 - 66501065*v^7 + 190202500*v^5 - 199991675*v^3 + 54880500*v) / 54753875 $$\beta_{10}$$ $$=$$ $$( - 330256 \nu^{15} + 1021010 \nu^{13} + 321835 \nu^{11} + 4065110 \nu^{9} - 59745955 \nu^{7} + 115300575 \nu^{5} - 188253675 \nu^{3} + \cdots + 81913375 \nu ) / 54753875$$ (-330256*v^15 + 1021010*v^13 + 321835*v^11 + 4065110*v^9 - 59745955*v^7 + 115300575*v^5 - 188253675*v^3 + 81913375*v) / 54753875 $$\beta_{11}$$ $$=$$ $$( 389298 \nu^{14} - 1338325 \nu^{12} - 665635 \nu^{10} - 3682905 \nu^{8} + 76510440 \nu^{6} - 149141900 \nu^{4} + 161761375 \nu^{2} - 87855125 ) / 54753875$$ (389298*v^14 - 1338325*v^12 - 665635*v^10 - 3682905*v^8 + 76510440*v^6 - 149141900*v^4 + 161761375*v^2 - 87855125) / 54753875 $$\beta_{12}$$ $$=$$ $$( - 443034 \nu^{15} + 2060985 \nu^{13} - 997060 \nu^{11} + 2317165 \nu^{9} - 91265695 \nu^{7} + 274576900 \nu^{5} - 350742825 \nu^{3} + \cdots + 260525500 \nu ) / 54753875$$ (-443034*v^15 + 2060985*v^13 - 997060*v^11 + 2317165*v^9 - 91265695*v^7 + 274576900*v^5 - 350742825*v^3 + 260525500*v) / 54753875 $$\beta_{13}$$ $$=$$ $$( 40791 \nu^{14} - 196825 \nu^{12} + 194030 \nu^{10} - 383635 \nu^{8} + 8130605 \nu^{6} - 27808800 \nu^{4} + 49697625 \nu^{2} - 36685125 ) / 4977625$$ (40791*v^14 - 196825*v^12 + 194030*v^10 - 383635*v^8 + 8130605*v^6 - 27808800*v^4 + 49697625*v^2 - 36685125) / 4977625 $$\beta_{14}$$ $$=$$ $$( 597957 \nu^{15} - 2324530 \nu^{13} + 218325 \nu^{11} - 5390820 \nu^{9} + 116417835 \nu^{7} - 286289900 \nu^{5} + 406932950 \nu^{3} + \cdots - 244363125 \nu ) / 54753875$$ (597957*v^15 - 2324530*v^13 + 218325*v^11 - 5390820*v^9 + 116417835*v^7 - 286289900*v^5 + 406932950*v^3 - 244363125*v) / 54753875 $$\beta_{15}$$ $$=$$ $$( 607697 \nu^{14} - 2843530 \nu^{12} + 1509785 \nu^{10} - 3468945 \nu^{8} + 126219210 \nu^{6} - 378889375 \nu^{4} + 538425125 \nu^{2} + \cdots - 393095875 ) / 54753875$$ (607697*v^14 - 2843530*v^12 + 1509785*v^10 - 3468945*v^8 + 126219210*v^6 - 378889375*v^4 + 538425125*v^2 - 393095875) / 54753875
 $$\nu$$ $$=$$ $$( \beta_{14} + \beta_{12} + 2\beta_{10} - \beta_{9} + 3\beta_{5} - \beta_{4} - 3\beta_{3} + 2\beta_1 ) / 5$$ (b14 + b12 + 2*b10 - b9 + 3*b5 - b4 - 3*b3 + 2*b1) / 5 $$\nu^{2}$$ $$=$$ $$\beta_{15} - \beta_{13} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{2} + 1$$ b15 - b13 + b8 + b7 - b6 + b2 + 1 $$\nu^{3}$$ $$=$$ $$2\beta_{14} + \beta_{12} + 2\beta_{10} - \beta_{5} + \beta_{4} - \beta_{3} - 2\beta_1$$ 2*b14 + b12 + 2*b10 - b5 + b4 - b3 - 2*b1 $$\nu^{4}$$ $$=$$ $$-\beta_{13} - 5\beta_{11} + 2\beta_{8} + \beta_{7} + 9\beta_{6} + \beta_{2} + 6$$ -b13 - 5*b11 + 2*b8 + b7 + 9*b6 + b2 + 6 $$\nu^{5}$$ $$=$$ $$-3\beta_{12} + 2\beta_{10} - 5\beta_{9} - 5\beta_{5} - 4\beta_{4} - 5\beta_{3} + \beta_1$$ -3*b12 + 2*b10 - 5*b9 - 5*b5 - 4*b4 - 5*b3 + b1 $$\nu^{6}$$ $$=$$ $$8\beta_{15} - 6\beta_{11} - 8\beta_{8} + 11\beta_{7} + 10\beta_{6} + 6\beta_{2} + 16$$ 8*b15 - 6*b11 - 8*b8 + 11*b7 + 10*b6 + 6*b2 + 16 $$\nu^{7}$$ $$=$$ $$13\beta_{14} - 21\beta_{12} + 40\beta_{10} + 14\beta_{9} + 19\beta_{5} + 3\beta_{4} - 46\beta_{3} + 10\beta_1$$ 13*b14 - 21*b12 + 40*b10 + 14*b9 + 19*b5 + 3*b4 - 46*b3 + 10*b1 $$\nu^{8}$$ $$=$$ $$36\beta_{15} - 29\beta_{13} - 44\beta_{11} + 11\beta_{8} - 37\beta_{7} - 18\beta_{6} + 22\beta_{2} - 48$$ 36*b15 - 29*b13 - 44*b11 + 11*b8 - 37*b7 - 18*b6 + 22*b2 - 48 $$\nu^{9}$$ $$=$$ $$-6\beta_{14} - 16\beta_{12} + 3\beta_{10} - 39\beta_{9} - 118\beta_{5} + 56\beta_{4} - 17\beta_{3} - 127\beta_1$$ -6*b14 - 16*b12 + 3*b10 - 39*b9 - 118*b5 + 56*b4 - 17*b3 - 127*b1 $$\nu^{10}$$ $$=$$ $$-85\beta_{15} + 140\beta_{13} - 210\beta_{11} - 140\beta_{8} - 165\beta_{7} + 320\beta_{6} - 55\beta_{2} + 35$$ -85*b15 + 140*b13 - 210*b11 - 140*b8 - 165*b7 + 320*b6 - 55*b2 + 35 $$\nu^{11}$$ $$=$$ $$-260\beta_{14} - 340\beta_{12} - 85\beta_{10} - 60\beta_{9} + 80\beta_{5} - 175\beta_{4} - 285\beta_{3} + 225\beta_1$$ -260*b14 - 340*b12 - 85*b10 - 60*b9 + 80*b5 - 175*b4 - 285*b3 + 225*b1 $$\nu^{12}$$ $$=$$ $$200\beta_{15} + 200\beta_{13} + 185\beta_{11} - 520\beta_{8} - 385\beta_{7} - 1005\beta_{6} + 120\beta_{2} - 1005$$ 200*b15 + 200*b13 + 185*b11 - 520*b8 - 385*b7 - 1005*b6 + 120*b2 - 1005 $$\nu^{13}$$ $$=$$ $$- 115 \beta_{14} - 415 \beta_{12} + 225 \beta_{10} + 835 \beta_{9} + 530 \beta_{5} + 820 \beta_{4} - 550 \beta_{3} - 495 \beta_1$$ -115*b14 - 415*b12 + 225*b10 + 835*b9 + 530*b5 + 820*b4 - 550*b3 - 495*b1 $$\nu^{14}$$ $$=$$ $$-1105\beta_{15} + 685\beta_{13} - 735\beta_{11} - 4150\beta_{7} - 1180\beta_{6} - 685\beta_{2} - 4885$$ -1105*b15 + 685*b13 - 735*b11 - 4150*b7 - 1180*b6 - 685*b2 - 4885 $$\nu^{15}$$ $$=$$ $$- 4125 \beta_{14} + 420 \beta_{12} - 7095 \beta_{10} - 2285 \beta_{9} - 4200 \beta_{5} + 495 \beta_{4} + 4810 \beta_{3} - 3095 \beta_1$$ -4125*b14 + 420*b12 - 7095*b10 - 2285*b9 - 4200*b5 + 495*b4 + 4810*b3 - 3095*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-\beta_{11}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
46.1
 1.85090 − 0.438393i −0.733267 − 1.75509i 0.733267 + 1.75509i −1.85090 + 0.438393i −1.13937 + 0.289499i 0.994142 + 0.627414i −0.994142 − 0.627414i 1.13937 − 0.289499i −1.13937 − 0.289499i 0.994142 − 0.627414i −0.994142 + 0.627414i 1.13937 + 0.289499i 1.85090 + 0.438393i −0.733267 + 1.75509i 0.733267 − 1.75509i −1.85090 − 0.438393i
−0.670386 + 2.06324i 0 −2.18949 1.59076i 1.69588 + 1.45739i 0 −3.32440 1.23973 0.900718i 0 −4.14384 + 2.52199i
46.2 −0.167451 + 0.515362i 0 1.38048 + 1.00297i 1.16252 1.91012i 0 1.08833 −1.62484 + 1.18052i 0 0.789737 + 0.918969i
46.3 0.167451 0.515362i 0 1.38048 + 1.00297i −1.16252 + 1.91012i 0 1.08833 1.62484 1.18052i 0 0.789737 + 0.918969i
46.4 0.670386 2.06324i 0 −2.18949 1.59076i −1.69588 1.45739i 0 −3.32440 −1.23973 + 0.900718i 0 −4.14384 + 2.52199i
91.1 −1.64259 1.19341i 0 0.655837 + 2.01846i −2.23130 0.145994i 0 −0.504300 0.0767537 0.236224i 0 3.49087 + 2.90266i
91.2 −0.757918 0.550660i 0 −0.346820 1.06740i 1.42964 + 1.71934i 0 2.74037 −0.903913 + 2.78196i 0 −0.136773 2.09036i
91.3 0.757918 + 0.550660i 0 −0.346820 1.06740i −1.42964 1.71934i 0 2.74037 0.903913 2.78196i 0 −0.136773 2.09036i
91.4 1.64259 + 1.19341i 0 0.655837 + 2.01846i 2.23130 + 0.145994i 0 −0.504300 −0.0767537 + 0.236224i 0 3.49087 + 2.90266i
136.1 −1.64259 + 1.19341i 0 0.655837 2.01846i −2.23130 + 0.145994i 0 −0.504300 0.0767537 + 0.236224i 0 3.49087 2.90266i
136.2 −0.757918 + 0.550660i 0 −0.346820 + 1.06740i 1.42964 1.71934i 0 2.74037 −0.903913 2.78196i 0 −0.136773 + 2.09036i
136.3 0.757918 0.550660i 0 −0.346820 + 1.06740i −1.42964 + 1.71934i 0 2.74037 0.903913 + 2.78196i 0 −0.136773 + 2.09036i
136.4 1.64259 1.19341i 0 0.655837 2.01846i 2.23130 0.145994i 0 −0.504300 −0.0767537 0.236224i 0 3.49087 2.90266i
181.1 −0.670386 2.06324i 0 −2.18949 + 1.59076i 1.69588 1.45739i 0 −3.32440 1.23973 + 0.900718i 0 −4.14384 2.52199i
181.2 −0.167451 0.515362i 0 1.38048 1.00297i 1.16252 + 1.91012i 0 1.08833 −1.62484 1.18052i 0 0.789737 0.918969i
181.3 0.167451 + 0.515362i 0 1.38048 1.00297i −1.16252 1.91012i 0 1.08833 1.62484 + 1.18052i 0 0.789737 0.918969i
181.4 0.670386 + 2.06324i 0 −2.18949 + 1.59076i −1.69588 + 1.45739i 0 −3.32440 −1.23973 0.900718i 0 −4.14384 2.52199i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 46.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.d even 5 1 inner
75.j odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.h.e 16
3.b odd 2 1 inner 225.2.h.e 16
25.d even 5 1 inner 225.2.h.e 16
25.d even 5 1 5625.2.a.w 8
25.e even 10 1 5625.2.a.v 8
75.h odd 10 1 5625.2.a.v 8
75.j odd 10 1 inner 225.2.h.e 16
75.j odd 10 1 5625.2.a.w 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.h.e 16 1.a even 1 1 trivial
225.2.h.e 16 3.b odd 2 1 inner
225.2.h.e 16 25.d even 5 1 inner
225.2.h.e 16 75.j odd 10 1 inner
5625.2.a.v 8 25.e even 10 1
5625.2.a.v 8 75.h odd 10 1
5625.2.a.w 8 25.d even 5 1
5625.2.a.w 8 75.j odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} + 5T_{2}^{14} + 20T_{2}^{12} + 75T_{2}^{10} + 385T_{2}^{8} + 25T_{2}^{6} + 250T_{2}^{4} + 125T_{2}^{2} + 25$$ acting on $$S_{2}^{\mathrm{new}}(225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 5 T^{14} + 20 T^{12} + 75 T^{10} + \cdots + 25$$
$3$ $$T^{16}$$
$5$ $$T^{16} - 5 T^{14} + 50 T^{12} + \cdots + 390625$$
$7$ $$(T^{4} - 10 T^{2} + 5 T + 5)^{4}$$
$11$ $$T^{16} + 30 T^{14} + 475 T^{12} + \cdots + 15625$$
$13$ $$(T^{8} - 5 T^{6} - 5 T^{5} + 310 T^{4} + \cdots + 2025)^{2}$$
$17$ $$T^{16} + 15 T^{14} + 445 T^{12} + \cdots + 17682025$$
$19$ $$(T^{8} - 7 T^{7} + 53 T^{6} - 349 T^{5} + \cdots + 11881)^{2}$$
$23$ $$T^{16} + 75 T^{14} + \cdots + 20393268025$$
$29$ $$T^{16} + 155 T^{14} + 11150 T^{12} + \cdots + 15625$$
$31$ $$(T^{8} - 9 T^{7} + 82 T^{6} - 647 T^{5} + \cdots + 641601)^{2}$$
$37$ $$(T^{8} - 5 T^{7} + 155 T^{6} - 415 T^{5} + \cdots + 9025)^{2}$$
$41$ $$T^{16} + 60 T^{14} + \cdots + 16041026265625$$
$43$ $$(T^{4} + 20 T^{3} + 115 T^{2} + 150 T - 45)^{4}$$
$47$ $$T^{16} - 25 T^{14} + \cdots + 144120025$$
$53$ $$T^{16} + \cdots + 104248286142025$$
$59$ $$T^{16} + 460 T^{14} + \cdots + 39\!\cdots\!25$$
$61$ $$(T^{8} - 16 T^{7} + 137 T^{6} + \cdots + 101761)^{2}$$
$67$ $$(T^{8} + 20 T^{7} + 395 T^{6} + \cdots + 40896025)^{2}$$
$71$ $$T^{16} + 425 T^{14} + \cdots + 67\!\cdots\!25$$
$73$ $$(T^{8} - 30 T^{7} + 715 T^{6} + \cdots + 117614025)^{2}$$
$79$ $$(T^{8} - 18 T^{7} + 353 T^{6} + \cdots + 28185481)^{2}$$
$83$ $$T^{16} + 180 T^{14} + 12225 T^{12} + \cdots + 3258025$$
$89$ $$T^{16} + 450 T^{14} + \cdots + 1272666015625$$
$97$ $$(T^{8} - 20 T^{7} + 485 T^{6} + \cdots + 86397025)^{2}$$