# Properties

 Label 5225.2.a.s Level $5225$ Weight $2$ Character orbit 5225.a Self dual yes Analytic conductor $41.722$ Analytic rank $1$ Dimension $15$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5225, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5225 = 5^{2} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5225.a (trivial)

## 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: yes Analytic conductor: $$41.7218350561$$ Analytic rank: $$1$$ Dimension: $$15$$ Coefficient field: $$\mathbb{Q}[x]/(x^{15} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{15} - 5 x^{14} - 11 x^{13} + 85 x^{12} + 6 x^{11} - 537 x^{10} + 327 x^{9} + 1556 x^{8} - 1451 x^{7} + \cdots - 13$$ x^15 - 5*x^14 - 11*x^13 + 85*x^12 + 6*x^11 - 537*x^10 + 327*x^9 + 1556*x^8 - 1451*x^7 - 2033*x^6 + 2316*x^5 + 927*x^4 - 1295*x^3 + 17*x^2 + 103*x - 13 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$1,\beta_1,\ldots,\beta_{14}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{15} - 5 x^{14} - 11 x^{13} + 85 x^{12} + 6 x^{11} - 537 x^{10} + 327 x^{9} + 1556 x^{8} - 1451 x^{7} + \cdots - 13$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 5\nu + 2$$ v^3 - v^2 - 5*v + 2 $$\beta_{4}$$ $$=$$ $$( 191 \nu^{14} - 593 \nu^{13} - 3078 \nu^{12} + 10145 \nu^{11} + 17645 \nu^{10} - 64855 \nu^{9} + \cdots + 2586 ) / 199$$ (191*v^14 - 593*v^13 - 3078*v^12 + 10145*v^11 + 17645*v^10 - 64855*v^9 - 41236*v^8 + 192627*v^7 + 24222*v^6 - 266767*v^5 + 36412*v^4 + 145493*v^3 - 35899*v^2 - 14720*v + 2586) / 199 $$\beta_{5}$$ $$=$$ $$( - 175 \nu^{14} + 784 \nu^{13} + 2269 \nu^{12} - 13520 \nu^{11} - 6966 \nu^{10} + 87304 \nu^{9} + \cdots - 8952 ) / 199$$ (-175*v^14 + 784*v^13 + 2269*v^12 - 13520*v^11 - 6966*v^10 + 87304*v^9 - 19174*v^8 - 262466*v^7 + 140264*v^6 + 369466*v^5 - 247342*v^4 - 210415*v^3 + 139935*v^2 + 29832*v - 8952) / 199 $$\beta_{6}$$ $$=$$ $$( 232 \nu^{14} - 912 \nu^{13} - 3273 \nu^{12} + 15638 \nu^{11} + 13655 \nu^{10} - 100250 \nu^{9} + \cdots + 8984 ) / 199$$ (232*v^14 - 912*v^13 - 3273*v^12 + 15638*v^11 + 13655*v^10 - 100250*v^9 - 1937*v^8 + 298645*v^7 - 107826*v^6 - 414996*v^5 + 224816*v^4 + 228950*v^3 - 134621*v^2 - 26641*v + 8984) / 199 $$\beta_{7}$$ $$=$$ $$( 182 \nu^{14} - 1086 \nu^{13} - 1715 \nu^{12} + 18797 \nu^{11} - 3573 \nu^{10} - 122071 \nu^{9} + \cdots + 18082 ) / 199$$ (182*v^14 - 1086*v^13 - 1715*v^12 + 18797*v^11 - 3573*v^10 - 122071*v^9 + 87394*v^8 + 370403*v^7 - 340210*v^6 - 530637*v^5 + 516668*v^4 + 315665*v^3 - 278544*v^2 - 52772*v + 18082) / 199 $$\beta_{8}$$ $$=$$ $$( 251 \nu^{14} - 1220 \nu^{13} - 2878 \nu^{12} + 20921 \nu^{11} + 3414 \nu^{10} - 134349 \nu^{9} + \cdots + 18215 ) / 199$$ (251*v^14 - 1220*v^13 - 2878*v^12 + 20921*v^11 + 3414*v^10 - 134349*v^9 + 70427*v^8 + 402510*v^7 - 330175*v^6 - 568345*v^5 + 530865*v^4 + 329388*v^3 - 290723*v^2 - 50386*v + 18215) / 199 $$\beta_{9}$$ $$=$$ $$( - 517 \nu^{14} + 2348 \nu^{13} + 6502 \nu^{12} - 40357 \nu^{11} - 17051 \nu^{10} + 259704 \nu^{9} + \cdots - 29641 ) / 199$$ (-517*v^14 + 2348*v^13 + 6502*v^12 - 40357*v^11 - 17051*v^10 + 259704*v^9 - 79767*v^8 - 778637*v^7 + 484130*v^6 + 1095570*v^5 - 826397*v^4 - 625100*v^3 + 463664*v^2 + 87898*v - 29641) / 199 $$\beta_{10}$$ $$=$$ $$( - 556 \nu^{14} + 2467 \nu^{13} + 6969 \nu^{12} - 42267 \nu^{11} - 17522 \nu^{10} + 271065 \nu^{9} + \cdots - 33601 ) / 199$$ (-556*v^14 + 2467*v^13 + 6969*v^12 - 42267*v^11 - 17522*v^10 + 271065*v^9 - 94230*v^8 - 810576*v^7 + 556492*v^6 + 1141490*v^5 - 952463*v^4 - 658614*v^3 + 538675*v^2 + 99121*v - 33601) / 199 $$\beta_{11}$$ $$=$$ $$( 556 \nu^{14} - 2467 \nu^{13} - 6969 \nu^{12} + 42267 \nu^{11} + 17522 \nu^{10} - 271065 \nu^{9} + \cdots + 34795 ) / 199$$ (556*v^14 - 2467*v^13 - 6969*v^12 + 42267*v^11 + 17522*v^10 - 271065*v^9 + 94230*v^8 + 810576*v^7 - 556492*v^6 - 1141490*v^5 + 952662*v^4 + 658614*v^3 - 540068*v^2 - 99320*v + 34795) / 199 $$\beta_{12}$$ $$=$$ $$( 829 \nu^{14} - 3698 \nu^{13} - 10437 \nu^{12} + 63398 \nu^{11} + 27187 \nu^{10} - 406710 \nu^{9} + \cdots + 47988 ) / 199$$ (829*v^14 - 3698*v^13 - 10437*v^12 + 63398*v^11 + 27187*v^10 - 406710*v^9 + 131592*v^8 + 1215239*v^7 - 796167*v^6 - 1704914*v^5 + 1363892*v^4 + 972414*v^3 - 768237*v^2 - 139673*v + 47988) / 199 $$\beta_{13}$$ $$=$$ $$( - 881 \nu^{14} + 4719 \nu^{13} + 9335 \nu^{12} - 81334 \nu^{11} + 443 \nu^{10} + 525338 \nu^{9} + \cdots - 72770 ) / 199$$ (-881*v^14 + 4719*v^13 + 9335*v^12 - 81334*v^11 + 443*v^10 + 525338*v^9 - 321618*v^8 - 1582924*v^7 + 1365739*v^6 + 2244404*v^5 - 2136940*v^4 - 1303991*v^3 + 1163634*v^2 + 199611*v - 72770) / 199 $$\beta_{14}$$ $$=$$ $$( 1163 \nu^{14} - 5656 \nu^{13} - 13768 \nu^{12} + 97395 \nu^{11} + 23873 \nu^{10} - 628244 \nu^{9} + \cdots + 78228 ) / 199$$ (1163*v^14 - 5656*v^13 - 13768*v^12 + 97395*v^11 + 23873*v^10 - 628244*v^9 + 270548*v^8 + 1889212*v^7 - 1355208*v^6 - 2669129*v^5 + 2221542*v^4 + 1536373*v^3 - 1232539*v^2 - 224200*v + 78228) / 199
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 5\beta _1 + 1$$ b3 + b2 + 5*b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{11} + \beta_{10} + 7\beta_{2} + \beta _1 + 15$$ b11 + b10 + 7*b2 + b1 + 15 $$\nu^{5}$$ $$=$$ $$-\beta_{14} - \beta_{13} - \beta_{9} - \beta_{8} + \beta_{5} + \beta_{4} + 8\beta_{3} + 8\beta_{2} + 29\beta _1 + 10$$ -b14 - b13 - b9 - b8 + b5 + b4 + 8*b3 + 8*b2 + 29*b1 + 10 $$\nu^{6}$$ $$=$$ $$- \beta_{14} - \beta_{13} + \beta_{12} + 9 \beta_{11} + 10 \beta_{10} - \beta_{9} - \beta_{8} + \cdots + 86$$ -b14 - b13 + b12 + 9*b11 + 10*b10 - b9 - b8 - 2*b6 + b5 + 2*b4 + b3 + 45*b2 + 13*b1 + 86 $$\nu^{7}$$ $$=$$ $$- 14 \beta_{14} - 13 \beta_{13} + \beta_{12} + 2 \beta_{11} + 2 \beta_{10} - 14 \beta_{9} - 14 \beta_{8} + \cdots + 79$$ -14*b14 - 13*b13 + b12 + 2*b11 + 2*b10 - 14*b9 - 14*b8 + b7 - 2*b6 + 11*b5 + 13*b4 + 52*b3 + 55*b2 + 180*b1 + 79 $$\nu^{8}$$ $$=$$ $$- 14 \beta_{14} - 14 \beta_{13} + 13 \beta_{12} + 67 \beta_{11} + 79 \beta_{10} - 16 \beta_{9} + \cdots + 525$$ -14*b14 - 14*b13 + 13*b12 + 67*b11 + 79*b10 - 16*b9 - 17*b8 - b7 - 28*b6 + 14*b5 + 26*b4 + 13*b3 + 286*b2 + 123*b1 + 525 $$\nu^{9}$$ $$=$$ $$- 135 \beta_{14} - 121 \beta_{13} + 15 \beta_{12} + 34 \beta_{11} + 34 \beta_{10} - 138 \beta_{9} + \cdots + 582$$ -135*b14 - 121*b13 + 15*b12 + 34*b11 + 34*b10 - 138*b9 - 138*b8 + 10*b7 - 34*b6 + 88*b5 + 119*b4 + 319*b3 + 368*b2 + 1157*b1 + 582 $$\nu^{10}$$ $$=$$ $$- 145 \beta_{14} - 142 \beta_{13} + 119 \beta_{12} + 479 \beta_{11} + 580 \beta_{10} - 185 \beta_{9} + \cdots + 3308$$ -145*b14 - 142*b13 + 119*b12 + 479*b11 + 580*b10 - 185*b9 - 195*b8 - 19*b7 - 280*b6 + 133*b5 + 241*b4 + 112*b3 + 1821*b2 + 1035*b1 + 3308 $$\nu^{11}$$ $$=$$ $$- 1126 \beta_{14} - 990 \beta_{13} + 150 \beta_{12} + 387 \beta_{11} + 383 \beta_{10} - 1197 \beta_{9} + \cdots + 4167$$ -1126*b14 - 990*b13 + 150*b12 + 387*b11 + 383*b10 - 1197*b9 - 1188*b8 + 56*b7 - 399*b6 + 627*b5 + 954*b4 + 1911*b3 + 2454*b2 + 7591*b1 + 4167 $$\nu^{12}$$ $$=$$ $$- 1335 \beta_{14} - 1273 \beta_{13} + 949 \beta_{12} + 3404 \beta_{11} + 4137 \beta_{10} - 1849 \beta_{9} + \cdots + 21219$$ -1335*b14 - 1273*b13 + 949*b12 + 3404*b11 + 4137*b10 - 1849*b9 - 1896*b8 - 243*b7 - 2463*b6 + 1072*b5 + 1964*b4 + 791*b3 + 11635*b2 + 8229*b1 + 21219 $$\nu^{13}$$ $$=$$ $$- 8760 \beta_{14} - 7610 \beta_{13} + 1278 \beta_{12} + 3722 \beta_{11} + 3634 \beta_{10} + \cdots + 29419$$ -8760*b14 - 7610*b13 + 1278*b12 + 3722*b11 + 3634*b10 - 9780*b9 - 9584*b8 + 115*b7 - 3992*b6 + 4231*b5 + 7173*b4 + 11261*b3 + 16371*b2 + 50483*b1 + 29419 $$\nu^{14}$$ $$=$$ $$- 11521 \beta_{14} - 10707 \beta_{13} + 7065 \beta_{12} + 24266 \beta_{11} + 29151 \beta_{10} + \cdots + 137637$$ -11521*b14 - 10707*b13 + 7065*b12 + 24266*b11 + 29151*b10 - 16955*b9 - 16874*b8 - 2607*b7 - 20345*b6 + 7901*b5 + 15038*b4 + 4826*b3 + 74547*b2 + 63354*b1 + 137637

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.66470 2.59653 2.59045 2.12341 1.38376 1.30094 1.10124 0.199845 0.195748 −0.324814 −1.00105 −1.44812 −1.75566 −2.15791 −2.46907
−2.66470 −2.92926 5.10062 0 7.80559 0.523619 −8.26223 5.58055 0
1.2 −2.59653 −1.22746 4.74198 0 3.18713 −3.87200 −7.11963 −1.49335 0
1.3 −2.59045 2.99450 4.71045 0 −7.75712 −1.08607 −7.02129 5.96706 0
1.4 −2.12341 0.300215 2.50886 0 −0.637479 1.66086 −1.08053 −2.90987 0
1.5 −1.38376 0.179641 −0.0852182 0 −0.248579 −4.32482 2.88543 −2.96773 0
1.6 −1.30094 −2.76347 −0.307545 0 3.59512 −4.11821 3.00199 4.63678 0
1.7 −1.10124 1.12617 −0.787279 0 −1.24018 1.81708 3.06945 −1.73174 0
1.8 −0.199845 1.67817 −1.96006 0 −0.335375 −3.54061 0.791399 −0.183735 0
1.9 −0.195748 2.69341 −1.96168 0 −0.527231 0.191992 0.775492 4.25447 0
1.10 0.324814 −3.24319 −1.89450 0 −1.05343 −0.166007 −1.26499 7.51826 0
1.11 1.00105 −1.31958 −0.997895 0 −1.32097 3.51091 −3.00105 −1.25870 0
1.12 1.44812 −2.09272 0.0970385 0 −3.03050 −1.77651 −2.75571 1.37947 0
1.13 1.75566 0.305911 1.08236 0 0.537076 2.74363 −1.61108 −2.90642 0
1.14 2.15791 2.27846 2.65657 0 4.91670 −4.96679 1.41681 2.19136 0
1.15 2.46907 −1.98081 4.09630 0 −4.89075 −1.59709 5.17592 0.923593 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.15 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$11$$ $$-1$$
$$19$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5225.2.a.s 15
5.b even 2 1 5225.2.a.x yes 15

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5225.2.a.s 15 1.a even 1 1 trivial
5225.2.a.x yes 15 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5225))$$:

 $$T_{2}^{15} + 5 T_{2}^{14} - 11 T_{2}^{13} - 85 T_{2}^{12} + 6 T_{2}^{11} + 537 T_{2}^{10} + 327 T_{2}^{9} + \cdots + 13$$ T2^15 + 5*T2^14 - 11*T2^13 - 85*T2^12 + 6*T2^11 + 537*T2^10 + 327*T2^9 - 1556*T2^8 - 1451*T2^7 + 2033*T2^6 + 2316*T2^5 - 927*T2^4 - 1295*T2^3 - 17*T2^2 + 103*T2 + 13 $$T_{7}^{15} + 15 T_{7}^{14} + 52 T_{7}^{13} - 241 T_{7}^{12} - 1775 T_{7}^{11} - 576 T_{7}^{10} + \cdots + 1813$$ T7^15 + 15*T7^14 + 52*T7^13 - 241*T7^12 - 1775*T7^11 - 576*T7^10 + 16032*T7^9 + 23706*T7^8 - 52692*T7^7 - 113783*T7^6 + 51628*T7^5 + 179731*T7^4 + 15831*T7^3 - 63105*T7^2 + 779*T7 + 1813

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{15} + 5 T^{14} + \cdots + 13$$
$3$ $$T^{15} + 4 T^{14} + \cdots + 101$$
$5$ $$T^{15}$$
$7$ $$T^{15} + 15 T^{14} + \cdots + 1813$$
$11$ $$(T - 1)^{15}$$
$13$ $$T^{15} + 13 T^{14} + \cdots + 119259$$
$17$ $$T^{15} + 11 T^{14} + \cdots - 5878079$$
$19$ $$(T + 1)^{15}$$
$23$ $$T^{15} + 10 T^{14} + \cdots + 8989816$$
$29$ $$T^{15} + \cdots - 564962440$$
$31$ $$T^{15} + \cdots - 15534356323$$
$37$ $$T^{15} + \cdots - 5312541061$$
$41$ $$T^{15} + \cdots - 1902339571$$
$43$ $$T^{15} + \cdots + 704805109$$
$47$ $$T^{15} + \cdots - 118914861797$$
$53$ $$T^{15} + \cdots + 113540087551$$
$59$ $$T^{15} + \cdots + 17679650455$$
$61$ $$T^{15} + \cdots + 16570478683$$
$67$ $$T^{15} + 35 T^{14} + \cdots - 9019249$$
$71$ $$T^{15} + \cdots - 1830004407127$$
$73$ $$T^{15} + \cdots + 6548500921$$
$79$ $$T^{15} + \cdots + 12035730685$$
$83$ $$T^{15} + \cdots + 1924224829663$$
$89$ $$T^{15} + \cdots + 1557086911875$$
$97$ $$T^{15} + \cdots - 29591370408776$$