# Properties

 Label 5225.2.a.w 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: $$x^{15} - x^{14} - 21 x^{13} + 19 x^{12} + 170 x^{11} - 137 x^{10} - 669 x^{9} + 458 x^{8} + 1327 x^{7} + \cdots - 9$$ x^15 - x^14 - 21*x^13 + 19*x^12 + 170*x^11 - 137*x^10 - 669*x^9 + 458*x^8 + 1327*x^7 - 687*x^6 - 1256*x^5 + 353*x^4 + 519*x^3 - 37*x^2 - 75*x - 9 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - \beta_{8} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{12} - \beta_{11}) q^{6} + (\beta_{10} - 1) q^{7} + (\beta_{3} + \beta_1) q^{8} + ( - \beta_{14} - \beta_{13} + \beta_{8} + \cdots + 1) q^{9}+O(q^{10})$$ q + b1 * q^2 - b8 * q^3 + (b2 + 1) * q^4 + (b12 - b11) * q^6 + (b10 - 1) * q^7 + (b3 + b1) * q^8 + (-b14 - b13 + b8 + b7 - b3 - b1 + 1) * q^9 $$q + \beta_1 q^{2} - \beta_{8} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{12} - \beta_{11}) q^{6} + (\beta_{10} - 1) q^{7} + (\beta_{3} + \beta_1) q^{8} + ( - \beta_{14} - \beta_{13} + \beta_{8} + \cdots + 1) q^{9}+ \cdots + (\beta_{14} + \beta_{13} - \beta_{8} + \cdots - 1) q^{99}+O(q^{100})$$ q + b1 * q^2 - b8 * q^3 + (b2 + 1) * q^4 + (b12 - b11) * q^6 + (b10 - 1) * q^7 + (b3 + b1) * q^8 + (-b14 - b13 + b8 + b7 - b3 - b1 + 1) * q^9 - q^11 + (b14 + b13 - b12 - b10 + b6 - b2 - 1) * q^12 + (b14 + b13 - b12 - b7) * q^13 + (-b14 + b8 + b6 - b3 - b1 - 1) * q^14 + (-b6 + b5 + b3 + 1) * q^16 + (-b13 + b11 - b8 - b6 + 1) * q^17 + (b11 + b7 - b5 + b4 - b2 + b1 - 2) * q^18 - q^19 + (-b14 - b10 - b9 + b8 - b4 - b1 + 1) * q^21 - b1 * q^22 + (2*b14 + b12 - b11 + b10 - b8 - b7 - b6 + 2*b5 + b3 - b2 + b1 - 1) * q^23 + (-b14 - 2*b13 + b12 + b10 + b8 - b4 - 2*b3 - b1) * q^24 + (-b14 + b8 - b7 + b6 - b5 - b4 - b3) * q^26 + (b14 + b13 - b12 - b10 + b9 - 2*b8 - b7 - b5 + b4 + 2*b3 - 1) * q^27 + (-b12 + b11 - b10 + b8 - b5 - b4 - b3 - b2 - b1 - 1) * q^28 + (b13 - b12 + b11 - b10 + b9 + b8 + b6 - b5 + b2 - b1) * q^29 + (b14 + b13 + b11 - b10 + b9 - b8 - b6 - 2*b5 + b4 + b3 - b1 - 1) * q^31 + (b11 - b9 - b8 - 2*b6 + b4 + b3 + b2 - b1 + 1) * q^32 + b8 * q^33 + (b14 - b6 + b5 + b4 + b3 + 1) * q^34 + (b14 + b12 + b10 + b9 - b8 - 2*b6 + b5 + b4 + b2 - 2*b1 + 2) * q^36 + (-b14 - b13 + 2*b12 + b10 + b6 + b5 - b3 + b1 - 2) * q^37 - b1 * q^38 + (b13 - b9 + b8 + b5 + b2 - b1) * q^39 + (b12 - b11 + b8 + b7 + b6 + b5 + b4 - b2 - 1) * q^41 + (b14 + b13 - b12 - b8 - b7 - b6 + b5 + 2*b3 - 1) * q^42 + (-2*b14 + b13 + b12 - b11 + 2*b8 + b7 + b6 - b4 - b3 - b2 + b1 - 3) * q^43 + (-b2 - 1) * q^44 + (b14 - b13 + b12 - b11 - 2*b9 - 2*b8 - b7 + b6 + 2*b5 - b2 - b1) * q^46 + (-2*b14 - b13 + b11 - b9 + b8 - b5 - 2*b4 - b3 + b2 - b1 - 1) * q^47 + (-b12 + b11 + b10 - b7 + 2*b6 - 2*b3 - b2 - b1 - 3) * q^48 + (b14 - b12 + b11 - 2*b10 - b9 + b8 - b6 - b5 - b2 - b1 + 1) * q^49 + (-b14 - b13 - b11 + b8 + b6 - b4 - b2 - b1 + 1) * q^51 + (-b14 + b13 - 2*b12 + b11 - b10 + b9 + b8 + 3*b6 - b5 - 2*b4 - 2*b3 - 2*b1 - 1) * q^52 + (b13 - b10 + 2*b8 + b6 - b4 - b2 + b1 - 2) * q^53 + (b13 - b10 + b9 - 2*b6 - b5 - b4 + b3 + 2*b2 - 2*b1 + 1) * q^54 + (2*b14 + b13 - b12 - b10 + b9 - b8 - b7 - b5 - b2 - 2*b1) * q^56 + b8 * q^57 + (-b14 - b12 + b11 + b9 + 2*b8 - 2*b5 - b4 + 2*b1 - 3) * q^58 + (b14 - b12 - b6 - b5 + b4 - 1) * q^59 + (2*b14 - b13 + b12 - b11 + b9 - b6 + 2*b4 + b3 - b2 + b1 + 1) * q^61 + (b13 - 2*b12 + b11 - 2*b10 + 2*b9 - b8 + b7 - b6 - b5 + b4 - b2 - 3*b1 - 2) * q^62 + (2*b14 - b13 + b11 + 2*b10 + b9 - 2*b8 - 2*b7 - b6 + b5 + 2*b4 + 2*b1 - 1) * q^63 + (-b13 + b11 + b10 - b8 + b7 - 3*b6 + 2*b4 + 2*b3 + b1 - 1) * q^64 + (-b12 + b11) * q^66 + (-2*b14 - b13 - b9 + b7 + b6 - b3 - b2 - 2*b1 - 3) * q^67 + (b12 - b11 + b10 - b9 + b8 + b7 - b6 + b5 + b4 + 2*b3 + 2*b1 - 1) * q^68 + (-b14 + b11 - 2*b9 + 4*b8 + 2*b6 + b5 - b4 - b3 + 3*b1 - 1) * q^69 + (-b12 + b11 + b7 - b6 + b4 - b3 - 2*b2 - 3*b1 + 1) * q^71 + (-b12 - b10 - b9 - 2*b8 - b7 + 2*b3 - 2*b2 + 2*b1 - 3) * q^72 + (-b14 - 2*b13 - b12 - b9 + b8 + b5 - b3 + 1) * q^73 + (2*b14 - b13 - b12 + 2*b10 - b9 + 2*b5 - b4 - b3 - b2 - b1) * q^74 + (-b2 - 1) * q^76 + (-b10 + 1) * q^77 + (-b14 - 2*b13 + b12 + 2*b10 - b9 - 2*b6 + b5 + 2*b3 - b2 + 3*b1 - 2) * q^78 + (-b12 + b11 - 2*b10 + 2*b9 - b7 + 2*b6 - b5 - b2) * q^79 + (-2*b14 - 2*b13 + 2*b11 + 3*b8 + b7 - b5 - 2*b4 - 3*b3 - b1 + 3) * q^81 + (-3*b13 + 2*b12 + 2*b10 - b9 - b8 + 2*b7 - 2*b6 + b5 - b3 - b2 - 1) * q^82 + (-b14 - b13 + b12 - b11 + b10 - b9 - b8 - b6 - b3 - b1 + 2) * q^83 + (b14 + b12 + 2*b10 + b9 - 2*b8 - b7 - 2*b6 + 2*b4 + 3*b3 + 2*b2 + b1 - 1) * q^84 + (-b14 + 2*b13 - 2*b12 + b11 - b10 - b8 + 2*b6 - 3*b5 - b3 + b2 - 5*b1 + 2) * q^86 + (2*b14 + 2*b13 - 2*b12 + b11 - 2*b10 + b9 + b3 - 2*b1 - 2) * q^87 + (-b3 - b1) * q^88 + (2*b14 + 2*b13 - b12 - 2*b11 + 2*b8 - b7 + 2*b6 + 2*b5 + 2*b3 + b2 + 3*b1 - 1) * q^89 + (-3*b14 + 2*b12 - 2*b11 + b10 - b9 + 2*b8 + 2*b7 + 2*b6 + b2 + b1) * q^91 + (-b14 - 4*b13 + 3*b12 - 3*b11 + 2*b10 - 2*b9 + b8 + b7 + 3*b5 - 2*b4 - 3*b3 - b2 - b1 - 2) * q^92 + (-b14 + b13 + b11 - 2*b10 + b9 + 4*b8 + 3*b7 - 3*b5 - b3 - b2 + b1 - 3) * q^93 + (b14 + 3*b13 - 3*b12 - b10 + b9 + b8 - 2*b7 + 2*b6 + b5 + b3 - 2*b1 - 2) * q^94 + (b14 + 2*b13 - b12 - b11 + b10 + 3*b8 - b7 + 2*b6 - b4 - 2*b3 - 2*b2 - 2*b1 - 5) * q^96 + (-b14 - b13 - b11 + b10 - b6 + 2*b5 - b4 + b3 - 3*b1 - 1) * q^97 + (b14 - b13 + b9 - b8 - 2*b6 + 2*b5 + b4 - b2 - 2*b1 + 1) * q^98 + (b14 + b13 - b8 - b7 + b3 + b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$15 q + q^{2} - 4 q^{3} + 13 q^{4} - q^{6} - 11 q^{7} + 3 q^{8} + 19 q^{9}+O(q^{10})$$ 15 * q + q^2 - 4 * q^3 + 13 * q^4 - q^6 - 11 * q^7 + 3 * q^8 + 19 * q^9 $$15 q + q^{2} - 4 q^{3} + 13 q^{4} - q^{6} - 11 q^{7} + 3 q^{8} + 19 q^{9} - 15 q^{11} - 11 q^{12} - 3 q^{13} - 11 q^{14} + 13 q^{16} + 5 q^{17} - 12 q^{18} - 15 q^{19} + 10 q^{21} - q^{22} - 26 q^{23} - 11 q^{24} - 5 q^{26} - 19 q^{27} - 18 q^{28} + 7 q^{29} - 10 q^{31} + 12 q^{32} + 4 q^{33} + 17 q^{34} + 24 q^{36} - 31 q^{37} - q^{38} + 4 q^{39} + 2 q^{41} - 22 q^{42} - 26 q^{43} - 13 q^{44} - 23 q^{46} - 26 q^{47} - 46 q^{48} + 12 q^{49} + 12 q^{51} - 16 q^{52} - 21 q^{53} + 5 q^{54} - 10 q^{56} + 4 q^{57} - 34 q^{58} - 11 q^{59} + 20 q^{61} - 25 q^{62} - 27 q^{63} - 3 q^{64} + q^{66} - 41 q^{67} + 6 q^{68} + q^{69} + 25 q^{71} - 54 q^{72} + 6 q^{73} - 9 q^{74} - 13 q^{76} + 11 q^{77} - 28 q^{78} - 6 q^{79} + 43 q^{81} - 18 q^{82} + 20 q^{83} - 14 q^{84} + 35 q^{86} - 29 q^{87} - 3 q^{88} - 3 q^{89} + 30 q^{91} - 54 q^{92} - 2 q^{93} - 28 q^{94} - 61 q^{96} - 28 q^{97} + 2 q^{98} - 19 q^{99}+O(q^{100})$$ 15 * q + q^2 - 4 * q^3 + 13 * q^4 - q^6 - 11 * q^7 + 3 * q^8 + 19 * q^9 - 15 * q^11 - 11 * q^12 - 3 * q^13 - 11 * q^14 + 13 * q^16 + 5 * q^17 - 12 * q^18 - 15 * q^19 + 10 * q^21 - q^22 - 26 * q^23 - 11 * q^24 - 5 * q^26 - 19 * q^27 - 18 * q^28 + 7 * q^29 - 10 * q^31 + 12 * q^32 + 4 * q^33 + 17 * q^34 + 24 * q^36 - 31 * q^37 - q^38 + 4 * q^39 + 2 * q^41 - 22 * q^42 - 26 * q^43 - 13 * q^44 - 23 * q^46 - 26 * q^47 - 46 * q^48 + 12 * q^49 + 12 * q^51 - 16 * q^52 - 21 * q^53 + 5 * q^54 - 10 * q^56 + 4 * q^57 - 34 * q^58 - 11 * q^59 + 20 * q^61 - 25 * q^62 - 27 * q^63 - 3 * q^64 + q^66 - 41 * q^67 + 6 * q^68 + q^69 + 25 * q^71 - 54 * q^72 + 6 * q^73 - 9 * q^74 - 13 * q^76 + 11 * q^77 - 28 * q^78 - 6 * q^79 + 43 * q^81 - 18 * q^82 + 20 * q^83 - 14 * q^84 + 35 * q^86 - 29 * q^87 - 3 * q^88 - 3 * q^89 + 30 * q^91 - 54 * q^92 - 2 * q^93 - 28 * q^94 - 61 * q^96 - 28 * q^97 + 2 * q^98 - 19 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{15} - x^{14} - 21 x^{13} + 19 x^{12} + 170 x^{11} - 137 x^{10} - 669 x^{9} + 458 x^{8} + 1327 x^{7} + \cdots - 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5\nu$$ v^3 - 5*v $$\beta_{4}$$ $$=$$ $$( - 5 \nu^{14} + 17 \nu^{13} + 102 \nu^{12} - 329 \nu^{11} - 784 \nu^{10} + 2383 \nu^{9} + 2745 \nu^{8} + \cdots + 66 ) / 27$$ (-5*v^14 + 17*v^13 + 102*v^12 - 329*v^11 - 784*v^10 + 2383*v^9 + 2745*v^8 - 7960*v^7 - 4136*v^6 + 11952*v^5 + 1879*v^4 - 6334*v^3 - 240*v^2 + 1031*v + 66) / 27 $$\beta_{5}$$ $$=$$ $$( 22 \nu^{14} + 17 \nu^{13} - 411 \nu^{12} - 302 \nu^{11} + 2807 \nu^{10} + 1816 \nu^{9} - 8703 \nu^{8} + \cdots + 12 ) / 27$$ (22*v^14 + 17*v^13 - 411*v^12 - 302*v^11 + 2807*v^10 + 1816*v^9 - 8703*v^8 - 4558*v^7 + 12064*v^6 + 4581*v^5 - 5978*v^4 - 1420*v^3 + 867*v^2 + 86*v + 12) / 27 $$\beta_{6}$$ $$=$$ $$( 22 \nu^{14} + 17 \nu^{13} - 411 \nu^{12} - 302 \nu^{11} + 2807 \nu^{10} + 1816 \nu^{9} - 8703 \nu^{8} + \cdots - 69 ) / 27$$ (22*v^14 + 17*v^13 - 411*v^12 - 302*v^11 + 2807*v^10 + 1816*v^9 - 8703*v^8 - 4558*v^7 + 12064*v^6 + 4581*v^5 - 6005*v^4 - 1393*v^3 + 1029*v^2 - 49*v - 69) / 27 $$\beta_{7}$$ $$=$$ $$( 10 \nu^{14} + 47 \nu^{13} - 150 \nu^{12} - 854 \nu^{11} + 623 \nu^{10} + 5548 \nu^{9} - 9 \nu^{8} + \cdots + 192 ) / 27$$ (10*v^14 + 47*v^13 - 150*v^12 - 854*v^11 + 623*v^10 + 5548*v^9 - 9*v^8 - 15913*v^7 - 4580*v^6 + 19404*v^5 + 7771*v^4 - 7069*v^3 - 3030*v^2 + 395*v + 192) / 27 $$\beta_{8}$$ $$=$$ $$( - 20 \nu^{14} + 14 \nu^{13} + 381 \nu^{12} - 290 \nu^{11} - 2677 \nu^{10} + 2350 \nu^{9} + 8523 \nu^{8} + \cdots + 318 ) / 27$$ (-20*v^14 + 14*v^13 + 381*v^12 - 290*v^11 - 2677*v^10 + 2350*v^9 + 8523*v^8 - 8863*v^7 - 11846*v^6 + 14922*v^5 + 5464*v^4 - 8866*v^3 - 1311*v^2 + 1559*v + 318) / 27 $$\beta_{9}$$ $$=$$ $$( - \nu^{14} - 5 \nu^{13} + 15 \nu^{12} + 92 \nu^{11} - 62 \nu^{10} - 610 \nu^{9} - 3 \nu^{8} + 1810 \nu^{7} + \cdots - 39 ) / 3$$ (-v^14 - 5*v^13 + 15*v^12 + 92*v^11 - 62*v^10 - 610*v^9 - 3*v^8 + 1810*v^7 + 476*v^6 - 2358*v^5 - 817*v^4 + 1060*v^3 + 354*v^2 - 149*v - 39) / 3 $$\beta_{10}$$ $$=$$ $$( - 29 \nu^{14} - 4 \nu^{13} + 570 \nu^{12} + 79 \nu^{11} - 4207 \nu^{10} - 494 \nu^{9} + 14625 \nu^{8} + \cdots - 357 ) / 27$$ (-29*v^14 - 4*v^13 + 570*v^12 + 79*v^11 - 4207*v^10 - 494*v^9 + 14625*v^8 + 1595*v^7 - 24194*v^6 - 3924*v^5 + 16309*v^4 + 6020*v^3 - 2634*v^2 - 2104*v - 357) / 27 $$\beta_{11}$$ $$=$$ $$( 20 \nu^{14} - 14 \nu^{13} - 408 \nu^{12} + 263 \nu^{11} + 3163 \nu^{10} - 1891 \nu^{9} - 11655 \nu^{8} + \cdots - 183 ) / 27$$ (20*v^14 - 14*v^13 - 408*v^12 + 263*v^11 + 3163*v^10 - 1891*v^9 - 11655*v^8 + 6271*v^7 + 20702*v^6 - 9063*v^5 - 15778*v^4 + 3979*v^3 + 4146*v^2 - 425*v - 183) / 27 $$\beta_{12}$$ $$=$$ $$( 26 \nu^{14} + 25 \nu^{13} - 498 \nu^{12} - 460 \nu^{11} + 3553 \nu^{10} + 2966 \nu^{9} - 11952 \nu^{8} + \cdots - 3 ) / 27$$ (26*v^14 + 25*v^13 - 498*v^12 - 460*v^11 + 3553*v^10 + 2966*v^9 - 11952*v^8 - 8423*v^7 + 19520*v^6 + 10593*v^5 - 13972*v^4 - 5090*v^3 + 3327*v^2 + 757*v - 3) / 27 $$\beta_{13}$$ $$=$$ $$( - 55 \nu^{14} - 2 \nu^{13} + 1068 \nu^{12} + 26 \nu^{11} - 7733 \nu^{10} + 131 \nu^{9} + 26037 \nu^{8} + \cdots - 138 ) / 27$$ (-55*v^14 - 2*v^13 + 1068*v^12 + 26*v^11 - 7733*v^10 + 131*v^9 + 26037*v^8 - 1430*v^7 - 40717*v^6 + 1656*v^5 + 24881*v^4 + 3361*v^3 - 4422*v^2 - 1727*v - 138) / 27 $$\beta_{14}$$ $$=$$ $$( 35 \nu^{14} + 70 \nu^{13} - 660 \nu^{12} - 1315 \nu^{11} + 4597 \nu^{10} + 8942 \nu^{9} - 15057 \nu^{8} + \cdots + 483 ) / 27$$ (35*v^14 + 70*v^13 - 660*v^12 - 1315*v^11 + 4597*v^10 + 8942*v^9 - 15057*v^8 - 27710*v^7 + 24065*v^6 + 39618*v^5 - 16798*v^4 - 22703*v^3 + 2895*v^2 + 4177*v + 483) / 27
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5\beta_1$$ b3 + 5*b1 $$\nu^{4}$$ $$=$$ $$-\beta_{6} + \beta_{5} + \beta_{3} + 6\beta_{2} + 15$$ -b6 + b5 + b3 + 6*b2 + 15 $$\nu^{5}$$ $$=$$ $$\beta_{11} - \beta_{9} - \beta_{8} - 2\beta_{6} + \beta_{4} + 9\beta_{3} + \beta_{2} + 27\beta _1 + 1$$ b11 - b9 - b8 - 2*b6 + b4 + 9*b3 + b2 + 27*b1 + 1 $$\nu^{6}$$ $$=$$ $$- \beta_{13} + \beta_{11} + \beta_{10} - \beta_{8} + \beta_{7} - 13 \beta_{6} + 10 \beta_{5} + \cdots + 85$$ -b13 + b11 + b10 - b8 + b7 - 13*b6 + 10*b5 + 2*b4 + 12*b3 + 36*b2 + b1 + 85 $$\nu^{7}$$ $$=$$ $$- \beta_{14} - \beta_{13} + 13 \beta_{11} - 10 \beta_{9} - 11 \beta_{8} + 3 \beta_{7} - 26 \beta_{6} + \cdots + 17$$ -b14 - b13 + 13*b11 - 10*b9 - 11*b8 + 3*b7 - 26*b6 + b5 + 14*b4 + 70*b3 + 13*b2 + 155*b1 + 17 $$\nu^{8}$$ $$=$$ $$- 2 \beta_{14} - 14 \beta_{13} + \beta_{12} + 17 \beta_{11} + 12 \beta_{10} - \beta_{9} - 13 \beta_{8} + \cdots + 516$$ -2*b14 - 14*b13 + b12 + 17*b11 + 12*b10 - b9 - 13*b8 + 17*b7 - 123*b6 + 77*b5 + 29*b4 + 111*b3 + 226*b2 + 20*b1 + 516 $$\nu^{9}$$ $$=$$ $$- 14 \beta_{14} - 16 \beta_{13} - \beta_{12} + 123 \beta_{11} + \beta_{10} - 77 \beta_{9} - 95 \beta_{8} + \cdots + 202$$ -14*b14 - 16*b13 - b12 + 123*b11 + b10 - 77*b9 - 95*b8 + 46*b7 - 254*b6 + 20*b5 + 140*b4 + 527*b3 + 132*b2 + 937*b1 + 202 $$\nu^{10}$$ $$=$$ $$- 32 \beta_{14} - 138 \beta_{13} + 17 \beta_{12} + 190 \beta_{11} + 106 \beta_{10} - 20 \beta_{9} + \cdots + 3296$$ -32*b14 - 138*b13 + 17*b12 + 190*b11 + 106*b10 - 20*b9 - 129*b8 + 186*b7 - 1040*b6 + 551*b5 + 300*b4 + 946*b3 + 1479*b2 + 264*b1 + 3296 $$\nu^{11}$$ $$=$$ $$- 137 \beta_{14} - 175 \beta_{13} - 17 \beta_{12} + 1037 \beta_{11} + 21 \beta_{10} - 551 \beta_{9} + \cdots + 2061$$ -137*b14 - 175*b13 - 17*b12 + 1037*b11 + 21*b10 - 551*b9 - 761*b8 + 486*b7 - 2233*b6 + 259*b5 + 1226*b4 + 3942*b3 + 1225*b2 + 5919*b1 + 2061 $$\nu^{12}$$ $$=$$ $$- 349 \beta_{14} - 1189 \beta_{13} + 190 \beta_{12} + 1798 \beta_{11} + 840 \beta_{10} - 259 \beta_{9} + \cdots + 21900$$ -349*b14 - 1189*b13 + 190*b12 + 1798*b11 + 840*b10 - 259*b9 - 1158*b8 + 1712*b7 - 8357*b6 + 3869*b5 + 2719*b4 + 7775*b3 + 10015*b2 + 2886*b1 + 21900 $$\nu^{13}$$ $$=$$ $$- 1173 \beta_{14} - 1651 \beta_{13} - 186 \beta_{12} + 8296 \beta_{11} + 288 \beta_{10} - 3869 \beta_{9} + \cdots + 19312$$ -1173*b14 - 1651*b13 - 186*b12 + 8296*b11 + 288*b10 - 3869*b9 - 5909*b8 + 4431*b7 - 18648*b6 + 2764*b5 + 10069*b4 + 29525*b3 + 10820*b2 + 38805*b1 + 19312 $$\nu^{14}$$ $$=$$ $$- 3254 \beta_{14} - 9608 \beta_{13} + 1769 \beta_{12} + 15681 \beta_{11} + 6350 \beta_{10} + \cdots + 150072$$ -3254*b14 - 9608*b13 + 1769*b12 + 15681*b11 + 6350*b10 - 2764*b9 - 9878*b8 + 14500*b7 - 65477*b6 + 27203*b5 + 23079*b4 + 62642*b3 + 69689*b2 + 28326*b1 + 150072

## 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
 −2.47407 −2.41798 −1.93364 −1.76560 −0.761882 −0.722545 −0.367308 −0.151942 0.643169 0.678001 1.54588 1.67859 2.13710 2.13794 2.77430
−2.47407 −3.11629 4.12105 0 7.70992 −0.568753 −5.24763 6.71124 0
1.2 −2.41798 2.46128 3.84665 0 −5.95133 −1.10716 −4.46516 3.05789 0
1.3 −1.93364 −1.29385 1.73897 0 2.50184 −3.43011 0.504738 −1.32596 0
1.4 −1.76560 0.418817 1.11733 0 −0.739462 4.14636 1.55843 −2.82459 0
1.5 −0.761882 2.93875 −1.41954 0 −2.23898 0.140222 2.60528 5.63627 0
1.6 −0.722545 −1.71747 −1.47793 0 1.24095 2.23664 2.51296 −0.0502999 0
1.7 −0.367308 0.412171 −1.86509 0 −0.151394 0.0772451 1.41968 −2.83011 0
1.8 −0.151942 −3.09381 −1.97691 0 0.470080 −5.09357 0.604261 6.57165 0
1.9 0.643169 −0.586127 −1.58633 0 −0.376979 −4.99433 −2.30662 −2.65645 0
1.10 0.678001 1.35109 −1.54031 0 0.916040 1.92484 −2.40034 −1.17455 0
1.11 1.54588 −3.18552 0.389733 0 −4.92442 2.77361 −2.48927 7.14754 0
1.12 1.67859 2.69106 0.817650 0 4.51717 −2.51534 −1.98468 4.24179 0
1.13 2.13710 1.19868 2.56721 0 2.56171 −2.23006 1.21218 −1.56316 0
1.14 2.13794 −0.537061 2.57079 0 −1.14821 0.416282 1.22033 −2.71157 0
1.15 2.77430 −1.94173 5.69673 0 −5.38694 −2.77587 10.2558 0.770318 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.w yes 15
5.b even 2 1 5225.2.a.t 15

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

## Hecke kernels

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

 $$T_{2}^{15} - T_{2}^{14} - 21 T_{2}^{13} + 19 T_{2}^{12} + 170 T_{2}^{11} - 137 T_{2}^{10} - 669 T_{2}^{9} + \cdots - 9$$ T2^15 - T2^14 - 21*T2^13 + 19*T2^12 + 170*T2^11 - 137*T2^10 - 669*T2^9 + 458*T2^8 + 1327*T2^7 - 687*T2^6 - 1256*T2^5 + 353*T2^4 + 519*T2^3 - 37*T2^2 - 75*T2 - 9 $$T_{7}^{15} + 11 T_{7}^{14} + 2 T_{7}^{13} - 343 T_{7}^{12} - 841 T_{7}^{11} + 3146 T_{7}^{10} + \cdots - 191$$ T7^15 + 11*T7^14 + 2*T7^13 - 343*T7^12 - 841*T7^11 + 3146*T7^10 + 11910*T7^9 - 7884*T7^8 - 59746*T7^7 - 16943*T7^6 + 107502*T7^5 + 71441*T7^4 - 29631*T7^3 - 14373*T7^2 + 3723*T7 - 191

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{15} - T^{14} + \cdots - 9$$
$3$ $$T^{15} + 4 T^{14} + \cdots - 227$$
$5$ $$T^{15}$$
$7$ $$T^{15} + 11 T^{14} + \cdots - 191$$
$11$ $$(T + 1)^{15}$$
$13$ $$T^{15} + 3 T^{14} + \cdots - 186591$$
$17$ $$T^{15} - 5 T^{14} + \cdots - 261903$$
$19$ $$(T + 1)^{15}$$
$23$ $$T^{15} + \cdots + 15324926616$$
$29$ $$T^{15} - 7 T^{14} + \cdots - 5832$$
$31$ $$T^{15} + \cdots + 140385879$$
$37$ $$T^{15} + \cdots - 119957219817$$
$41$ $$T^{15} + \cdots + 4768760805$$
$43$ $$T^{15} + \cdots + 2065651035549$$
$47$ $$T^{15} + \cdots - 632236095$$
$53$ $$T^{15} + 21 T^{14} + \cdots - 16749$$
$59$ $$T^{15} + 11 T^{14} + \cdots + 2749729$$
$61$ $$T^{15} + \cdots - 19802477763$$
$67$ $$T^{15} + \cdots - 50650313397$$
$71$ $$T^{15} + \cdots + 26212478639$$
$73$ $$T^{15} + \cdots - 5780455047$$
$79$ $$T^{15} + \cdots - 28856420091$$
$83$ $$T^{15} - 20 T^{14} + \cdots - 12128217$$
$89$ $$T^{15} + \cdots + 111267524027057$$
$97$ $$T^{15} + \cdots - 3730533486440$$