# Properties

 Label 5225.2.a.u Level $5225$ Weight $2$ Character orbit 5225.a Self dual yes Analytic conductor $41.722$ Analytic rank $0$ 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: $$0$$ 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: $$x^{15} - x^{14} - 21 x^{13} + 21 x^{12} + 168 x^{11} - 165 x^{10} - 645 x^{9} + 606 x^{8} + 1239 x^{7} + \cdots - 5$$ x^15 - x^14 - 21*x^13 + 21*x^12 + 168*x^11 - 165*x^10 - 645*x^9 + 606*x^8 + 1239*x^7 - 1055*x^6 - 1160*x^5 + 793*x^4 + 485*x^3 - 171*x^2 - 93*x - 5 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + \beta_{7} q^{3} + (\beta_{2} + 1) q^{4} + \beta_{4} q^{6} + ( - \beta_{14} + 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + ( - \beta_{14} + \beta_{12} + \beta_{9} + \cdots + 1) q^{9}+O(q^{10})$$ q - b1 * q^2 + b7 * q^3 + (b2 + 1) * q^4 + b4 * q^6 + (-b14 + 1) * q^7 + (-b3 - b1) * q^8 + (-b14 + b12 + b9 - 2*b8 + b5 + b4 + b3 + 1) * q^9 $$q - \beta_1 q^{2} + \beta_{7} q^{3} + (\beta_{2} + 1) q^{4} + \beta_{4} q^{6} + ( - \beta_{14} + 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + ( - \beta_{14} + \beta_{12} + \beta_{9} + \cdots + 1) q^{9}+ \cdots + (\beta_{14} - \beta_{12} - \beta_{9} + \cdots - 1) q^{99}+O(q^{100})$$ q - b1 * q^2 + b7 * q^3 + (b2 + 1) * q^4 + b4 * q^6 + (-b14 + 1) * q^7 + (-b3 - b1) * q^8 + (-b14 + b12 + b9 - 2*b8 + b5 + b4 + b3 + 1) * q^9 - q^11 + (-b14 - b8 - b6 + b3 + 1) * q^12 + (b14 + b13 - b8 + b7 + b6 + b4 + b2 + b1) * q^13 + (b14 - b12 - b9 + b8 + b7 - b5 - b3 - b1 - 1) * q^14 + (-b13 - b12 + b11 - b9 + 2*b8 - b6 - 2*b5 - 2*b4 - b3 - b2 - b1 + 1) * q^16 + (-b8 - b7 + b6 + b5 + b4 + b3 + b2 + b1) * q^17 + (b14 + b13 - b11 + b9 - b8 + b7 + b6 + 2*b4 + b2 + b1 - 1) * q^18 + q^19 + (-b14 + b13 - b11 + b10 + b9 - 2*b8 + b7 + b6 + b5 + b3 + 2*b2) * q^21 + b1 * q^22 + (-b13 + b2 + 1) * q^23 + (b14 + b13 - b12 + b8 + b7 - b5 - b3 - b2 - 2) * q^24 + (-2*b14 + 2*b12 - b11 + b9 - 3*b8 + 2*b5 + b4 + b3 + b2 - 2*b1 + 1) * q^26 + (-b14 - b13 + b10 + b9 - b7 - b6 - b4 + b3 - b1 + 2) * q^27 + (-b13 + b11 - b9 + b8 + b7 - 2*b6 - b5 - b4 - b1 + 3) * q^28 + (-b13 - b12 + b10 - b9 + b8 + b7 + b6 - b5 - b4 - b3 - 1) * q^29 + (b14 + b13 + b11 - b10 - b9 - b8 + b7 + b4 + b3 + b2 + b1 + 1) * q^31 + (-2*b13 - b12 + b11 + b10 - 2*b9 + 5*b8 - b7 - 2*b6 - 3*b5 - 3*b4 - 2*b3 - 2*b2 - 3*b1 + 1) * q^32 - b7 * q^33 + (b13 + b12 - b11 + 2*b7 - b2 - 1) * q^34 + (-2*b14 + b13 + b12 - 2*b11 - b10 + 2*b9 - 6*b8 + b7 + b6 + 4*b5 + 2*b4 + 3*b3 + 3*b2 + b1) * q^36 + (b12 + b11 - b9 + b8 - b6 - b5 - b2 - b1 + 3) * q^37 - b1 * q^38 + (b13 - b10 + b7 + b6 + b5 + 2*b4 - b3 + b1) * q^39 + (b11 + b9 + b8 - 2*b7 + b5 - b1 + 1) * q^41 + (b14 + 2*b13 + b12 - b11 - 2*b10 + b9 - 3*b8 + 2*b6 + 2*b5 + 5*b4 + 2*b2 + 1) * q^42 + (b9 - b8 - b7 + b6 + b5 + b2 + 2*b1 + 2) * q^43 + (-b2 - 1) * q^44 + (b8 + b7 - b5 - 2*b3 - b2 - 3*b1 - 1) * q^46 + (-b12 - b10 + b9 - b8 - b7 + b4 - b2 + b1 + 1) * q^47 + (-b14 - b13 + b12 + b9 - b8 - b7 + 2*b5 + b3 + b2 + b1 + 2) * q^48 + (-2*b14 + b12 - b10 + b9 - 2*b8 + b7 + b6 + 2*b5 + b4 + b1 + 1) * q^49 + (-b9 + b8 + b7 - b5 - b4 - b1 - 2) * q^51 + (2*b14 - 2*b12 - b10 + 2*b8 - b7 + 2*b6 - b5 + b4 - b3 + 2*b2 + 2*b1 + 1) * q^52 + (-b14 - b13 + b12 + 2*b11 + 2*b10 + 4*b8 - 2*b7 - 3*b6 - b5 - 2*b4 - 3*b2 - 2*b1 + 2) * q^53 + (b14 + b13 - b10 + b9 + 2*b4 - b3 - b2 - 2*b1) * q^54 + (-b14 - b13 + 2*b11 + b10 + 4*b8 - 2*b7 - 2*b6 - 2*b5 - b4 - 2*b3 - 3*b2 - 4*b1 + 1) * q^56 + b7 * q^57 + (-b13 + b11 - b10 - 3*b9 + 4*b8 + 2*b7 - b6 - 3*b5 - b4 - 3*b3 - 2*b2 - b1) * q^58 + (-2*b13 - 2*b12 + b11 + 2*b10 - b9 + 5*b8 - 2*b7 - 3*b6 - 3*b5 - 4*b4 - b3 - 4*b2 - 4*b1 + 1) * q^59 + (-b12 - b11 - b10 - b9 + 2*b7 + 2*b6 - b5 - b4 - 2*b3 + b2 - 1) * q^61 + (-b14 + b12 - b11 + 2*b10 + b8 - b6 - b3 - b2 - 4*b1) * q^62 + (-3*b14 + 2*b13 + 2*b12 - b10 + 2*b9 - 6*b8 + 4*b5 + 3*b4 + 3*b3 + 2*b2 + 3*b1 + 4) * q^63 + (b14 - b13 - b12 + 2*b11 - 3*b9 + 4*b8 - b6 - b5 - 2*b4 - 2*b3 - b2 - 2*b1 + 1) * q^64 - b4 * q^66 + (b13 + 2*b12 - b11 - b10 - 5*b8 + b7 + 2*b6 + 3*b5 + 3*b4 + b3 + 3*b2 - b1 + 2) * q^67 + (b14 + b13 - b10 + b9 - 3*b8 + 2*b6 + 2*b5 + 2*b4 + 2*b3 + b2 + 5*b1 - 2) * q^68 + (-b14 - 2*b13 + b12 + b11 - 3*b6 - b5 - b4 + 2*b3 - 2*b2 - 2*b1 + 4) * q^69 + (b14 - b13 - b9 + 2*b8 + b7 - b6 - b5 - 2*b4 - b3 - 2*b1) * q^71 + (b14 + 3*b13 - b11 - b10 + 2*b9 - 2*b8 - b7 + 3*b6 + 2*b5 + 5*b4 + b3 + b2 + 4*b1 - 1) * q^72 + (b14 + b13 - b12 - 2*b11 - b10 + b9 + b7 + 2*b6 + b5 - b3 + 2*b2 + 3*b1) * q^73 + (b14 - b13 - b12 + b11 + b10 - 2*b9 + 2*b8 - b5 - 2*b4 - b3 - 3*b1) * q^74 + (b2 + 1) * q^76 + (b14 - 1) * q^77 + (-b14 - b13 - b12 + b10 - b9 + 2*b7 - 2*b6 - b5 - 2*b4 + 2*b3 + b2 + 1) * q^78 + (b14 + 2*b13 + b12 - b11 + b9 - 2*b8 + b7 + 3*b6 + 2*b5 + 2*b4 + b3 + 2*b2 + 2*b1) * q^79 + (-b14 - 2*b13 - b12 + b11 - b9 + 4*b8 - 3*b6 - 2*b5 - 3*b4 - b2 - b1 + 3) * q^81 + (-b13 + b10 + 2*b8 - b7 - 3*b6 - b5 - 4*b4 - 4*b1 + 5) * q^82 + (-b14 + b12 - 3*b11 + b10 + b9 - 2*b8 + b7 + 2*b6 + 2*b5 + b4 + 2*b2 - b1 - 1) * q^83 + (-2*b14 - b13 + 2*b12 - b11 - b10 + 2*b9 - 6*b8 + b7 - b6 + 3*b5 + b4 + 4*b3 + 3*b2 + 8) * q^84 + (b12 - b11 + b9 - b7 - b4 - b3 - b2 - 4*b1 - 4) * q^86 + (-2*b14 + 3*b12 + b11 - b10 + b9 - b8 + b6 + 2*b5 + b4 + b3 - b2 + 5) * q^87 + (b3 + b1) * q^88 + (-2*b10 + b7 + b6 + b5 + 3*b4 + b3 + b1) * q^89 + (b14 + b13 + b11 + 3*b10 + 2*b8 + 2*b7 - 2*b5 - b2 - 2*b1 - 4) * q^91 + (-b14 - b12 + b11 - b9 - b6 - b5 - 2*b4 + 2*b2 + 7) * q^92 + (b13 - 2*b11 - b10 - 4*b8 + b7 + 2*b6 + 3*b5 + 2*b4 + b3 + 5*b2 + 2*b1 + 1) * q^93 + (-3*b14 + b12 - 2*b11 + b10 + 3*b9 - 3*b8 + b7 - 2*b6 + b5 + 2*b3 - b1 + 1) * q^94 + (b14 - b13 + b12 + b8 - 2*b7 + b6 - b1 - 2) * q^96 + (b14 + 2*b13 + 2*b12 + b9 - 3*b8 - 2*b6 + b5 + b4 + 3*b3 - b2 - b1 + 4) * q^97 + (3*b14 - 2*b12 - b11 + b10 - b9 + 3*b8 + b7 + b6 - 3*b5 - 2*b3 + b2 + b1 - 4) * q^98 + (b14 - b12 - b9 + 2*b8 - b5 - b4 - b3 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$15 q - q^{2} + 4 q^{3} + 13 q^{4} - q^{6} + 17 q^{7} + 3 q^{8} + 15 q^{9}+O(q^{10})$$ 15 * q - q^2 + 4 * q^3 + 13 * q^4 - q^6 + 17 * q^7 + 3 * q^8 + 15 * q^9 $$15 q - q^{2} + 4 q^{3} + 13 q^{4} - q^{6} + 17 q^{7} + 3 q^{8} + 15 q^{9} - 15 q^{11} + 9 q^{12} + 3 q^{13} - 11 q^{14} + 13 q^{16} + q^{17} - 10 q^{18} + 15 q^{19} + 16 q^{21} + q^{22} + 18 q^{23} - 27 q^{24} + 15 q^{26} + 31 q^{27} + 34 q^{28} + 11 q^{29} - 2 q^{31} + 10 q^{32} - 4 q^{33} - 13 q^{34} + 8 q^{36} + 27 q^{37} - q^{38} + 4 q^{39} + 6 q^{41} + 2 q^{42} + 38 q^{43} - 13 q^{44} - 9 q^{46} + 14 q^{47} + 32 q^{48} + 28 q^{49} - 32 q^{51} + 16 q^{52} + 11 q^{53} - 11 q^{54} + 2 q^{56} + 4 q^{57} - 6 q^{58} + 13 q^{59} + 8 q^{61} + 7 q^{62} + 49 q^{63} + 9 q^{64} + q^{66} + 31 q^{67} - 26 q^{68} + 39 q^{69} - 3 q^{71} - 18 q^{72} + 18 q^{73} + 7 q^{74} + 13 q^{76} - 17 q^{77} + 18 q^{78} + 10 q^{79} + 31 q^{81} + 58 q^{82} + 16 q^{83} + 112 q^{84} - 63 q^{86} + 67 q^{87} - 3 q^{88} - 7 q^{89} - 50 q^{91} + 98 q^{92} + 26 q^{93} + 22 q^{94} - 37 q^{96} + 24 q^{97} - 46 q^{98} - 15 q^{99}+O(q^{100})$$ 15 * q - q^2 + 4 * q^3 + 13 * q^4 - q^6 + 17 * q^7 + 3 * q^8 + 15 * q^9 - 15 * q^11 + 9 * q^12 + 3 * q^13 - 11 * q^14 + 13 * q^16 + q^17 - 10 * q^18 + 15 * q^19 + 16 * q^21 + q^22 + 18 * q^23 - 27 * q^24 + 15 * q^26 + 31 * q^27 + 34 * q^28 + 11 * q^29 - 2 * q^31 + 10 * q^32 - 4 * q^33 - 13 * q^34 + 8 * q^36 + 27 * q^37 - q^38 + 4 * q^39 + 6 * q^41 + 2 * q^42 + 38 * q^43 - 13 * q^44 - 9 * q^46 + 14 * q^47 + 32 * q^48 + 28 * q^49 - 32 * q^51 + 16 * q^52 + 11 * q^53 - 11 * q^54 + 2 * q^56 + 4 * q^57 - 6 * q^58 + 13 * q^59 + 8 * q^61 + 7 * q^62 + 49 * q^63 + 9 * q^64 + q^66 + 31 * q^67 - 26 * q^68 + 39 * q^69 - 3 * q^71 - 18 * q^72 + 18 * q^73 + 7 * q^74 + 13 * q^76 - 17 * q^77 + 18 * q^78 + 10 * q^79 + 31 * q^81 + 58 * q^82 + 16 * q^83 + 112 * q^84 - 63 * q^86 + 67 * q^87 - 3 * q^88 - 7 * q^89 - 50 * q^91 + 98 * q^92 + 26 * q^93 + 22 * q^94 - 37 * q^96 + 24 * q^97 - 46 * q^98 - 15 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{15} - x^{14} - 21 x^{13} + 21 x^{12} + 168 x^{11} - 165 x^{10} - 645 x^{9} + 606 x^{8} + 1239 x^{7} + \cdots - 5$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5\nu$$ v^3 - 5*v $$\beta_{4}$$ $$=$$ $$( - 130 \nu^{14} + 267 \nu^{13} + 2498 \nu^{12} - 5165 \nu^{11} - 17984 \nu^{10} + 37341 \nu^{9} + \cdots + 1420 ) / 917$$ (-130*v^14 + 267*v^13 + 2498*v^12 - 5165*v^11 - 17984*v^10 + 37341*v^9 + 61223*v^8 - 127760*v^7 - 100496*v^6 + 212754*v^5 + 66496*v^4 - 150693*v^3 - 5275*v^2 + 25426*v + 1420) / 917 $$\beta_{5}$$ $$=$$ $$( 149 \nu^{14} - 172 \nu^{13} - 3244 \nu^{12} + 3839 \nu^{11} + 26975 \nu^{10} - 32380 \nu^{9} + \cdots - 7976 ) / 917$$ (149*v^14 - 172*v^13 - 3244*v^12 + 3839*v^11 + 26975*v^10 - 32380*v^9 - 106985*v^8 + 128798*v^7 + 205459*v^6 - 244004*v^5 - 171498*v^4 + 193237*v^3 + 46923*v^2 - 36464*v - 7976) / 917 $$\beta_{6}$$ $$=$$ $$( 317 \nu^{14} - 249 \nu^{13} - 6317 \nu^{12} + 4518 \nu^{11} + 47352 \nu^{10} - 28621 \nu^{9} + \cdots + 2251 ) / 917$$ (317*v^14 - 249*v^13 - 6317*v^12 + 4518*v^11 + 47352*v^10 - 28621*v^9 - 168032*v^8 + 74317*v^7 + 293967*v^6 - 61675*v^5 - 245341*v^4 - 16644*v^3 + 77723*v^2 + 20572*v + 2251) / 917 $$\beta_{7}$$ $$=$$ $$( - 284 \nu^{14} + 414 \nu^{13} + 5697 \nu^{12} - 8462 \nu^{11} - 42547 \nu^{10} + 64844 \nu^{9} + \cdots + 986 ) / 917$$ (-284*v^14 + 414*v^13 + 5697*v^12 - 8462*v^11 - 42547*v^10 + 64844*v^9 + 145839*v^8 - 233327*v^7 - 224116*v^6 + 400116*v^5 + 116686*v^4 - 291708*v^3 + 12953*v^2 + 53839*v + 986) / 917 $$\beta_{8}$$ $$=$$ $$( 337 \nu^{14} - 149 \nu^{13} - 7054 \nu^{12} + 3267 \nu^{11} + 56044 \nu^{10} - 26536 \nu^{9} + \cdots - 6291 ) / 917$$ (337*v^14 - 149*v^13 - 7054*v^12 + 3267*v^11 + 56044*v^10 - 26536*v^9 - 211521*v^8 + 99686*v^7 + 388431*v^6 - 172563*v^5 - 319479*v^4 + 119260*v^3 + 88551*v^2 - 15999*v - 6291) / 917 $$\beta_{9}$$ $$=$$ $$( - 55 \nu^{14} + 118 \nu^{13} + 1077 \nu^{12} - 2422 \nu^{11} - 7659 \nu^{10} + 18534 \nu^{9} + \cdots + 500 ) / 131$$ (-55*v^14 + 118*v^13 + 1077*v^12 - 2422*v^11 - 7659*v^10 + 18534*v^9 + 23670*v^8 - 65671*v^7 - 28168*v^6 + 108573*v^5 + 3384*v^4 - 76104*v^3 + 8868*v^2 + 15060*v + 500) / 131 $$\beta_{10}$$ $$=$$ $$( - 443 \nu^{14} + 536 \nu^{13} + 8851 \nu^{12} - 11217 \nu^{11} - 65615 \nu^{10} + 88387 \nu^{9} + \cdots + 7731 ) / 917$$ (-443*v^14 + 536*v^13 + 8851*v^12 - 11217*v^11 - 65615*v^10 + 88387*v^9 + 221841*v^8 - 329418*v^7 - 333755*v^6 + 592275*v^5 + 173948*v^4 - 464865*v^3 + 5549*v^2 + 102628*v + 7731) / 917 $$\beta_{11}$$ $$=$$ $$( - 508 \nu^{14} + 1128 \nu^{13} + 10100 \nu^{12} - 23428 \nu^{11} - 73690 \nu^{10} + 182710 \nu^{9} + \cdots + 10275 ) / 917$$ (-508*v^14 + 1128*v^13 + 10100*v^12 - 23428*v^11 - 73690*v^10 + 182710*v^9 + 239156*v^8 - 667481*v^7 - 320730*v^6 + 1158986*v^5 + 97156*v^4 - 868956*v^3 + 64809*v^2 + 187784*v + 10275) / 917 $$\beta_{12}$$ $$=$$ $$( - 705 \nu^{14} + 1060 \nu^{13} + 14746 \nu^{12} - 22614 \nu^{11} - 115657 \nu^{10} + 180742 \nu^{9} + \cdots + 9958 ) / 917$$ (-705*v^14 + 1060*v^13 + 14746*v^12 - 22614*v^11 - 115657*v^10 + 180742*v^9 + 420437*v^8 - 672114*v^7 - 704485*v^6 + 1169723*v^5 + 462541*v^4 - 850267*v^3 - 46982*v^2 + 155945*v + 9958) / 917 $$\beta_{13}$$ $$=$$ $$( 901 \nu^{14} - 997 \nu^{13} - 18484 \nu^{12} + 20808 \nu^{11} + 142334 \nu^{10} - 162143 \nu^{9} + \cdots - 4904 ) / 917$$ (901*v^14 - 997*v^13 - 18484*v^12 + 20808*v^11 + 142334*v^10 - 162143*v^9 - 510457*v^8 + 587309*v^7 + 853900*v^6 - 991699*v^5 - 573603*v^4 + 683198*v^3 + 70383*v^2 - 100407*v - 4904) / 917 $$\beta_{14}$$ $$=$$ $$( - 155 \nu^{14} + 142 \nu^{13} + 3190 \nu^{12} - 2979 \nu^{11} - 24657 \nu^{10} + 23174 \nu^{9} + \cdots + 897 ) / 131$$ (-155*v^14 + 142*v^13 + 3190*v^12 - 2979*v^11 - 24657*v^10 + 23174*v^9 + 88893*v^8 - 82869*v^7 - 150718*v^6 + 135738*v^5 + 107227*v^4 - 89620*v^3 - 19596*v^2 + 12550*v + 897) / 131
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5\beta_1$$ b3 + 5*b1 $$\nu^{4}$$ $$=$$ $$- \beta_{13} - \beta_{12} + \beta_{11} - \beta_{9} + 2 \beta_{8} - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + \cdots + 15$$ -b13 - b12 + b11 - b9 + 2*b8 - b6 - 2*b5 - 2*b4 - b3 + 5*b2 - b1 + 15 $$\nu^{5}$$ $$=$$ $$2 \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{9} - 5 \beta_{8} + \beta_{7} + 2 \beta_{6} + \cdots - 1$$ 2*b13 + b12 - b11 - b10 + 2*b9 - 5*b8 + b7 + 2*b6 + 3*b5 + 3*b4 + 10*b3 + 2*b2 + 31*b1 - 1 $$\nu^{6}$$ $$=$$ $$\beta_{14} - 11 \beta_{13} - 11 \beta_{12} + 12 \beta_{11} - 13 \beta_{9} + 24 \beta_{8} - 11 \beta_{6} + \cdots + 87$$ b14 - 11*b13 - 11*b12 + 12*b11 - 13*b9 + 24*b8 - 11*b6 - 21*b5 - 22*b4 - 12*b3 + 25*b2 - 12*b1 + 87 $$\nu^{7}$$ $$=$$ $$- 2 \beta_{14} + 24 \beta_{13} + 14 \beta_{12} - 15 \beta_{11} - 12 \beta_{10} + 28 \beta_{9} + \cdots - 14$$ -2*b14 + 24*b13 + 14*b12 - 15*b11 - 12*b10 + 28*b9 - 66*b8 + 10*b7 + 26*b6 + 40*b5 + 40*b4 + 82*b3 + 25*b2 + 204*b1 - 14 $$\nu^{8}$$ $$=$$ $$12 \beta_{14} - 97 \beta_{13} - 94 \beta_{12} + 108 \beta_{11} + 3 \beta_{10} - 123 \beta_{9} + \cdots + 539$$ 12*b14 - 97*b13 - 94*b12 + 108*b11 + 3*b10 - 123*b9 + 221*b8 - 4*b7 - 98*b6 - 178*b5 - 192*b4 - 111*b3 + 125*b2 - 117*b1 + 539 $$\nu^{9}$$ $$=$$ $$- 31 \beta_{14} + 219 \beta_{13} + 142 \beta_{12} - 157 \beta_{11} - 105 \beta_{10} + 278 \beta_{9} + \cdots - 149$$ -31*b14 + 219*b13 + 142*b12 - 157*b11 - 105*b10 + 278*b9 - 632*b8 + 79*b7 + 246*b6 + 387*b5 + 387*b4 + 631*b3 + 228*b2 + 1373*b1 - 149 $$\nu^{10}$$ $$=$$ $$105 \beta_{14} - 788 \beta_{13} - 737 \beta_{12} + 873 \beta_{11} + 52 \beta_{10} - 1031 \beta_{9} + \cdots + 3461$$ 105*b14 - 788*b13 - 737*b12 + 873*b11 + 52*b10 - 1031*b9 + 1842*b8 - 61*b7 - 806*b6 - 1407*b5 - 1546*b4 - 938*b3 + 613*b2 - 1046*b1 + 3461 $$\nu^{11}$$ $$=$$ $$- 328 \beta_{14} + 1811 \beta_{13} + 1263 \beta_{12} - 1420 \beta_{11} - 821 \beta_{10} + 2413 \beta_{9} + \cdots - 1419$$ -328*b14 + 1811*b13 + 1263*b12 - 1420*b11 - 821*b10 + 2413*b9 - 5349*b8 + 589*b7 + 2063*b6 + 3306*b5 + 3307*b4 + 4715*b3 + 1834*b2 + 9357*b1 - 1419 $$\nu^{12}$$ $$=$$ $$823 \beta_{14} - 6135 \beta_{13} - 5557 \beta_{12} + 6706 \beta_{11} + 599 \beta_{10} - 8148 \beta_{9} + \cdots + 22751$$ 823*b14 - 6135*b13 - 5557*b12 + 6706*b11 + 599*b10 - 8148*b9 + 14599*b8 - 634*b7 - 6352*b6 - 10765*b5 - 11966*b4 - 7575*b3 + 2872*b2 - 8882*b1 + 22751 $$\nu^{13}$$ $$=$$ $$- 2969 \beta_{14} + 14281 \beta_{13} + 10487 \beta_{12} - 11909 \beta_{11} - 6107 \beta_{10} + \cdots - 12688$$ -2969*b14 + 14281*b13 + 10487*b12 - 11909*b11 - 6107*b10 + 19588*b9 - 42604*b8 + 4316*b7 + 16321*b6 + 26549*b5 + 26579*b4 + 34686*b3 + 13850*b2 + 64362*b1 - 12688 $$\nu^{14}$$ $$=$$ $$6162 \beta_{14} - 46595 \beta_{13} - 41077 \beta_{12} + 50153 \beta_{11} + 5802 \beta_{10} + \cdots + 152169$$ 6162*b14 - 46595*b13 - 41077*b12 + 50153*b11 + 5802*b10 - 62360*b9 + 112418*b8 - 5664*b7 - 48778*b6 - 80891*b5 - 90545*b4 - 59529*b3 + 12218*b2 - 72849*b1 + 152169

## 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
 2.53606 2.47156 1.98766 1.52359 1.38157 0.916988 0.814490 −0.0619135 −0.414796 −0.451441 −1.26824 −1.31913 −2.20121 −2.21115 −2.70405
−2.53606 3.25852 4.43161 0 −8.26381 4.52539 −6.16671 7.61796 0
1.2 −2.47156 −0.237530 4.10863 0 0.587070 0.606053 −5.21162 −2.94358 0
1.3 −1.98766 −0.104882 1.95080 0 0.208469 1.24528 0.0977875 −2.98900 0
1.4 −1.52359 −1.71830 0.321328 0 2.61798 −1.47987 2.55761 −0.0474472 0
1.5 −1.38157 −2.80596 −0.0912568 0 3.87664 4.96734 2.88922 4.87342 0
1.6 −0.916988 1.46608 −1.15913 0 −1.34437 −3.11204 2.89689 −0.850619 0
1.7 −0.814490 2.50015 −1.33661 0 −2.03634 4.83914 2.71763 3.25073 0
1.8 0.0619135 −2.42872 −1.99617 0 −0.150370 0.484101 −0.247417 2.89868 0
1.9 0.414796 −0.335749 −1.82794 0 −0.139267 3.31574 −1.58782 −2.88727 0
1.10 0.451441 2.92844 −1.79620 0 1.32202 −3.27317 −1.71376 5.57574 0
1.11 1.26824 1.37964 −0.391576 0 1.74971 2.74560 −3.03308 −1.09659 0
1.12 1.31913 −0.487250 −0.259889 0 −0.642748 −0.817715 −2.98109 −2.76259 0
1.13 2.20121 −2.15466 2.84534 0 −4.74286 −2.43464 1.86078 1.64255 0
1.14 2.21115 2.94549 2.88918 0 6.51291 2.83613 1.96611 5.67589 0
1.15 2.70405 −0.205258 5.31188 0 −0.555028 2.55265 8.95548 −2.95787 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.u 15
5.b even 2 1 5225.2.a.v yes 15

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5225.2.a.u 15 1.a even 1 1 trivial
5225.2.a.v 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5225))$$:

 $$T_{2}^{15} + T_{2}^{14} - 21 T_{2}^{13} - 21 T_{2}^{12} + 168 T_{2}^{11} + 165 T_{2}^{10} - 645 T_{2}^{9} + \cdots + 5$$ T2^15 + T2^14 - 21*T2^13 - 21*T2^12 + 168*T2^11 + 165*T2^10 - 645*T2^9 - 606*T2^8 + 1239*T2^7 + 1055*T2^6 - 1160*T2^5 - 793*T2^4 + 485*T2^3 + 171*T2^2 - 93*T2 + 5 $$T_{7}^{15} - 17 T_{7}^{14} + 78 T_{7}^{13} + 197 T_{7}^{12} - 2497 T_{7}^{11} + 3534 T_{7}^{10} + \cdots + 78609$$ T7^15 - 17*T7^14 + 78*T7^13 + 197*T7^12 - 2497*T7^11 + 3534*T7^10 + 21402*T7^9 - 65992*T7^8 - 37894*T7^7 + 329737*T7^6 - 200034*T7^5 - 509711*T7^4 + 576209*T7^3 + 99459*T7^2 - 284689*T7 + 78609

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{15} + T^{14} + \cdots + 5$$
$3$ $$T^{15} - 4 T^{14} + \cdots + 3$$
$5$ $$T^{15}$$
$7$ $$T^{15} - 17 T^{14} + \cdots + 78609$$
$11$ $$(T + 1)^{15}$$
$13$ $$T^{15} - 3 T^{14} + \cdots - 6684389$$
$17$ $$T^{15} - T^{14} + \cdots + 13089$$
$19$ $$(T - 1)^{15}$$
$23$ $$T^{15} - 18 T^{14} + \cdots - 132200$$
$29$ $$T^{15} + \cdots - 1033217016$$
$31$ $$T^{15} + \cdots + 1751332969$$
$37$ $$T^{15} - 27 T^{14} + \cdots + 11533069$$
$41$ $$T^{15} + \cdots - 2328385177$$
$43$ $$T^{15} - 38 T^{14} + \cdots + 38811601$$
$47$ $$T^{15} + \cdots + 558835437$$
$53$ $$T^{15} + \cdots - 665077574267$$
$59$ $$T^{15} + \cdots - 51376062089$$
$61$ $$T^{15} + \cdots - 4019482787$$
$67$ $$T^{15} + \cdots - 1253487348767$$
$71$ $$T^{15} + \cdots - 8569180307$$
$73$ $$T^{15} + \cdots - 40852177291$$
$79$ $$T^{15} + \cdots - 6539792293$$
$83$ $$T^{15} + \cdots + 7176867851$$
$89$ $$T^{15} + \cdots - 464591789865437$$
$97$ $$T^{15} + \cdots + 799389137752$$