# Properties

 Label 5225.2.a.y 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: $$x^{15} - 5 x^{14} - 11 x^{13} + 87 x^{12} - 4 x^{11} - 545 x^{10} + 431 x^{9} + 1480 x^{8} - 1763 x^{7} - 1609 x^{6} + 2516 x^{5} + 391 x^{4} - 1081 x^{3} - 45 x^{2} + 153 x + 15$$ x^15 - 5*x^14 - 11*x^13 + 87*x^12 - 4*x^11 - 545*x^10 + 431*x^9 + 1480*x^8 - 1763*x^7 - 1609*x^6 + 2516*x^5 + 391*x^4 - 1081*x^3 - 45*x^2 + 153*x + 15 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{5} - \beta_{4}) q^{6} + (\beta_{13} + 1) q^{7} + (\beta_{3} + \beta_1) q^{8} + (\beta_{10} + \beta_{7} + 1) q^{9}+O(q^{10})$$ q + b1 * q^2 + b5 * q^3 + (b2 + 1) * q^4 + (b5 - b4) * q^6 + (b13 + 1) * q^7 + (b3 + b1) * q^8 + (b10 + b7 + 1) * q^9 $$q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{5} - \beta_{4}) q^{6} + (\beta_{13} + 1) q^{7} + (\beta_{3} + \beta_1) q^{8} + (\beta_{10} + \beta_{7} + 1) q^{9} + q^{11} + ( - \beta_{14} - \beta_{13} + \beta_{10} - \beta_{9} + 2 \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{12} + ( - \beta_{11} + 1) q^{13} + ( - \beta_{14} + \beta_{10} + \beta_{6} + \beta_{2} + 2 \beta_1) q^{14} + (\beta_{13} + \beta_{8} - \beta_{7} + \beta_{2} + \beta_1 + 1) q^{16} + (\beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} + \beta_{6} - \beta_{5} + 2) q^{17} + (\beta_{13} + \beta_{10} + \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 2) q^{18} + q^{19} + ( - \beta_{12} - \beta_{6} + 2 \beta_{5} + \beta_{3} - \beta_{2}) q^{21} + \beta_1 q^{22} + ( - \beta_{14} - \beta_{13} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{3} + \beta_{2} + 2) q^{23} + ( - \beta_{14} - \beta_{12} + \beta_{10} - 2 \beta_{9} - \beta_{8} + 2 \beta_{5} - \beta_{4} + \beta_{2} - \beta_1) q^{24} + (\beta_{14} + \beta_{13} - \beta_{11} - \beta_{10} - \beta_{5} - \beta_{2} + \beta_1) q^{26} + (\beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} + \cdots - \beta_1) q^{27}+ \cdots + (\beta_{10} + \beta_{7} + 1) q^{99}+O(q^{100})$$ q + b1 * q^2 + b5 * q^3 + (b2 + 1) * q^4 + (b5 - b4) * q^6 + (b13 + 1) * q^7 + (b3 + b1) * q^8 + (b10 + b7 + 1) * q^9 + q^11 + (-b14 - b13 + b10 - b9 + 2*b5 - b4 + b2 - b1 + 1) * q^12 + (-b11 + 1) * q^13 + (-b14 + b10 + b6 + b2 + 2*b1) * q^14 + (b13 + b8 - b7 + b2 + b1 + 1) * q^16 + (b14 + b13 - b12 - b11 + b6 - b5 + 2) * q^17 + (b13 + b10 + b7 + b6 - b5 - b4 - b3 + b2 + 2) * q^18 + q^19 + (-b12 - b6 + 2*b5 + b3 - b2) * q^21 + b1 * q^22 + (-b14 - b13 + b11 + b10 - b9 - b3 + b2 + 2) * q^23 + (-b14 - b12 + b10 - 2*b9 - b8 + 2*b5 - b4 + b2 - b1) * q^24 + (b14 + b13 - b11 - b10 - b5 - b2 + b1) * q^26 + (b14 - b13 + b12 + b11 - b10 + 2*b9 + b8 + b7 - 2*b6 + b3 - b2 - b1) * q^27 + (b13 - b12 + b11 + b10 - b6 - b5 + 2*b2 + 3) * q^28 + (b14 - b10 + b9 + b7 - b5 + b4 + b3 - b2 + b1) * q^29 + (b14 + b13 - b11 - b2 + 1) * q^31 + (b9 - b7 + b5 + b4 + b3 + 2*b1) * q^32 + b5 * q^33 + (b14 + b12 - b11 + b4 + b3 - 2*b2 + 2*b1 - 2) * q^34 + (-b14 + b10 - b9 - b8 + b6 + b5 + b1) * q^36 + (-b13 + b12 - b8 + b5 - b4 - 2*b1 + 1) * q^37 + b1 * q^38 + (b12 - b11 - b10 - b7 + 2*b5 - b1 - 2) * q^39 + (-b14 + b12 - b9 + b6 + b5 - b4 - b3) * q^41 + (-2*b14 - b13 + b11 + b8 - b7 + 3*b5 - 2*b4 - b3 + 3*b2 - b1) * q^42 + (b10 + b9 + b8 + b2 + b1 + 2) * q^43 + (b2 + 1) * q^44 + (-b14 - b13 - b12 + b11 + b10 - b9 - 2*b8 + b7 - b4 + 2) * q^46 + (-b12 + 2*b7 - b4 - 2*b1 + 2) * q^47 + (-2*b14 - b12 + b10 - 2*b9 - 2*b8 + 2*b6 + b5 - 2*b4 - b3 + 2*b2) * q^48 + (-b14 + 2*b13 - b12 + b10 - b9 - b3 + 2*b2 - b1 + 2) * q^49 + (2*b12 - 2*b10 + b8 - b7 + b6 + b5 - b4 - b3 - 3) * q^51 + (b14 - b13 + b12 - b10 + b6 - b5 + b4 + 1) * q^52 + (b12 - 2*b10 + b9 + b8 - b5) * q^53 + (b14 - b13 + b12 + b11 - 2*b10 + 3*b9 + 3*b8 - b6 + b4 + 2*b1 - 3) * q^54 + (-2*b14 - 2*b13 + 2*b11 + b10 + b5 + b4 + 2*b3 + b2 + 4*b1) * q^56 + b5 * q^57 + (2*b14 + b13 + b12 - 2*b10 + 2*b9 + 2*b8 - b6 - 4*b5 + b4 - b3 + b2 + b1 + 2) * q^58 + (b14 + 2*b13 - b12 - 2*b11 - b10 - b7 + b5 + b4 + b3 + b1 - 1) * q^59 + (-b13 + b12 + b11 - b10 + 2*b9 + b8 - 2*b6 + b4 - b2 + 2*b1 - 1) * q^61 + (b14 + b12 - 2*b11 + 2*b6 - b2 + b1) * q^62 + (-b14 - b13 + b11 + 2*b10 - b9 - 2*b8 + 2*b7 - b5 + b4 + b1 + 2) * q^63 + (b14 + b13 + b11 - b10 + 2*b9 - b6 - b5 + b4 + b3 - b2 + 2*b1 - 1) * q^64 + (b5 - b4) * q^66 + (b11 - b8 - b5 + b4 + b3 - 2*b2 + 2) * q^67 + (2*b14 + b13 + 3*b12 - b11 - 2*b10 + b9 + b8 - b7 - 2*b5 - b3 + b2 - 2*b1 + 2) * q^68 + (-b13 + b11 - b10 + b9 - b7 + 2*b5 + b3 + b2 - 1) * q^69 + (-2*b12 - b11 - b8 - b7 + b6 + 2*b5 - b4 - b2 + 2*b1) * q^71 + (-b14 - b12 - 2*b9 - b8 - 2*b7 - b6 + 2*b5 + b2 - 2*b1 + 1) * q^72 + (-b13 - b7 - 2*b6 + b4 - b1 + 3) * q^73 + (-b11 + b10 - 2*b9 + b6 + 3*b5 - b4 - b3 - b2 + b1 - 3) * q^74 + (b2 + 1) * q^76 + (b13 + 1) * q^77 + (2*b14 + b13 - 2*b11 - 2*b10 - b7 + b5 - b4 + b3 - 3*b2 - 4) * q^78 + (b14 - b13 + b11 - b10 + b9 - b8 - b7 - b5 + 2*b4 + b3 - 1) * q^79 + (b14 + b13 + b11 + b10 + b9 - b8 + 3*b7 - 3*b5 + b4 - 2*b3 - b2 + b1 + 2) * q^81 + (-b12 - b11 + b10 - 2*b9 - 2*b8 + b7 + 2*b5 - 2*b4 - b3 - 4*b1 + 3) * q^82 + (-b13 - b9 - b8 + 2*b7 - b6 - b4 - b1 + 1) * q^83 + (-3*b14 - 2*b13 + 3*b11 + b10 - b9 - b8 - 2*b6 + 5*b5 - 2*b4 + b3 + b2 - 4) * q^84 + (b14 + 2*b13 + b11 + 2*b9 + b8 - b6 + b4 + 2*b3 - b2 + 4*b1) * q^86 + (b14 + b10 + b8 + 2*b7 - b6 + b5 - 2*b2 - 3*b1 + 3) * q^87 + (b3 + b1) * q^88 + (-b13 - b12 + b11 - 2*b9 - b7 - 2*b6 + b5 + b4 - b1 + 1) * q^89 + (b14 + 2*b13 + b12 - 2*b11 - b10 + b9 - b8 - b7 + b4 - 3*b2 + 2*b1 - 3) * q^91 + (-3*b14 - b12 + 2*b10 - 2*b9 - b8 + b7 + b6 + 3*b5 - 2*b4 - 2*b3 + 4*b2 - 2*b1 + 2) * q^92 + (2*b14 + b12 - b11 - 2*b10 + b9 - b6 + 2*b5 + b3 - 3*b2 - 2) * q^93 + (-2*b14 - 2*b13 + b11 + b10 - b9 + 2*b7 - b6 + 2*b5 - 2*b4 - 2*b3 + b2 - 2*b1 - 2) * q^94 + (-3*b14 - b13 + b11 + 3*b10 - 2*b9 - b8 + b7 + 2*b5 - b4 - 2*b3 + 2*b2 - 2*b1 + 5) * q^96 + (2*b14 + b13 + b8 - b7 + b6 + b4 - b2 + 1) * q^97 + (-4*b14 - b13 - b12 + b11 + 3*b10 - b9 - 2*b8 + b7 + 2*b6 - b4 + b3 + b2 + 4*b1 - 2) * q^98 + (b10 + b7 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 21 q^{7} + 9 q^{8} + 15 q^{9}+O(q^{10})$$ 15 * q + 5 * q^2 + 4 * q^3 + 17 * q^4 - q^6 + 21 * q^7 + 9 * q^8 + 15 * q^9 $$15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 21 q^{7} + 9 q^{8} + 15 q^{9} + 15 q^{11} + 11 q^{12} + 13 q^{13} + 9 q^{14} + 21 q^{16} + 17 q^{17} + 22 q^{18} + 15 q^{19} + 6 q^{21} + 5 q^{22} + 26 q^{23} + q^{24} + 3 q^{26} + q^{27} + 46 q^{28} + 9 q^{29} + 14 q^{31} + 18 q^{32} + 4 q^{33} - 13 q^{34} + 12 q^{36} + 9 q^{37} + 5 q^{38} - 22 q^{39} + 4 q^{41} - 6 q^{42} + 28 q^{43} + 17 q^{44} + 27 q^{46} + 14 q^{47} - 4 q^{48} + 32 q^{49} - 40 q^{51} + 14 q^{52} + 3 q^{53} - 39 q^{54} + 34 q^{56} + 4 q^{57} + 26 q^{58} + q^{59} + 2 q^{61} - 3 q^{62} + 45 q^{63} + 5 q^{64} - q^{66} + 37 q^{67} + 26 q^{68} - 7 q^{69} - 7 q^{71} + 16 q^{72} + 42 q^{73} - 43 q^{74} + 17 q^{76} + 21 q^{77} - 64 q^{78} - 10 q^{79} + 31 q^{81} + 22 q^{82} + 14 q^{83} - 32 q^{84} + 37 q^{86} + 29 q^{87} + 9 q^{88} + 15 q^{89} - 22 q^{91} + 26 q^{92} - 18 q^{93} - 44 q^{94} + 71 q^{96} + 8 q^{97} - 10 q^{98} + 15 q^{99}+O(q^{100})$$ 15 * q + 5 * q^2 + 4 * q^3 + 17 * q^4 - q^6 + 21 * q^7 + 9 * q^8 + 15 * q^9 + 15 * q^11 + 11 * q^12 + 13 * q^13 + 9 * q^14 + 21 * q^16 + 17 * q^17 + 22 * q^18 + 15 * q^19 + 6 * q^21 + 5 * q^22 + 26 * q^23 + q^24 + 3 * q^26 + q^27 + 46 * q^28 + 9 * q^29 + 14 * q^31 + 18 * q^32 + 4 * q^33 - 13 * q^34 + 12 * q^36 + 9 * q^37 + 5 * q^38 - 22 * q^39 + 4 * q^41 - 6 * q^42 + 28 * q^43 + 17 * q^44 + 27 * q^46 + 14 * q^47 - 4 * q^48 + 32 * q^49 - 40 * q^51 + 14 * q^52 + 3 * q^53 - 39 * q^54 + 34 * q^56 + 4 * q^57 + 26 * q^58 + q^59 + 2 * q^61 - 3 * q^62 + 45 * q^63 + 5 * q^64 - q^66 + 37 * q^67 + 26 * q^68 - 7 * q^69 - 7 * q^71 + 16 * q^72 + 42 * q^73 - 43 * q^74 + 17 * q^76 + 21 * q^77 - 64 * q^78 - 10 * q^79 + 31 * q^81 + 22 * q^82 + 14 * q^83 - 32 * q^84 + 37 * q^86 + 29 * q^87 + 9 * q^88 + 15 * q^89 - 22 * q^91 + 26 * q^92 - 18 * q^93 - 44 * q^94 + 71 * q^96 + 8 * q^97 - 10 * q^98 + 15 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{15} - 5 x^{14} - 11 x^{13} + 87 x^{12} - 4 x^{11} - 545 x^{10} + 431 x^{9} + 1480 x^{8} - 1763 x^{7} - 1609 x^{6} + 2516 x^{5} + 391 x^{4} - 1081 x^{3} - 45 x^{2} + 153 x + 15$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5\nu$$ v^3 - 5*v $$\beta_{4}$$ $$=$$ $$( - 16 \nu^{14} - 2151 \nu^{13} + 9337 \nu^{12} + 28321 \nu^{11} - 152439 \nu^{10} - 86367 \nu^{9} + 886590 \nu^{8} - 204807 \nu^{7} - 2159934 \nu^{6} + 1296036 \nu^{5} + \cdots - 3028 ) / 15503$$ (-16*v^14 - 2151*v^13 + 9337*v^12 + 28321*v^11 - 152439*v^10 - 86367*v^9 + 886590*v^8 - 204807*v^7 - 2159934*v^6 + 1296036*v^5 + 1984310*v^4 - 1650203*v^3 - 294512*v^2 + 358719*v - 3028) / 15503 $$\beta_{5}$$ $$=$$ $$( 896 \nu^{14} - 3568 \nu^{13} - 11273 \nu^{12} + 57342 \nu^{11} + 25437 \nu^{10} - 310444 \nu^{9} + 162099 \nu^{8} + 601589 \nu^{7} - 773252 \nu^{6} - 54982 \nu^{5} + \cdots - 16468 ) / 15503$$ (896*v^14 - 3568*v^13 - 11273*v^12 + 57342*v^11 + 25437*v^10 - 310444*v^9 + 162099*v^8 + 601589*v^7 - 773252*v^6 - 54982*v^5 + 903318*v^4 - 730656*v^3 - 49029*v^2 + 205163*v - 16468) / 15503 $$\beta_{6}$$ $$=$$ $$( 2058 \nu^{14} - 12071 \nu^{13} - 16930 \nu^{12} + 207769 \nu^{11} - 106536 \nu^{10} - 1272128 \nu^{9} + 1479090 \nu^{8} + 3265147 \nu^{7} - 5034600 \nu^{6} + \cdots - 21353 ) / 15503$$ (2058*v^14 - 12071*v^13 - 16930*v^12 + 207769*v^11 - 106536*v^10 - 1272128*v^9 + 1479090*v^8 + 3265147*v^7 - 5034600*v^6 - 2936690*v^5 + 5951043*v^4 - 151180*v^3 - 1216960*v^2 - 9117*v - 21353) / 15503 $$\beta_{7}$$ $$=$$ $$( - 2771 \nu^{14} + 10204 \nu^{13} + 46404 \nu^{12} - 192737 \nu^{11} - 267316 \nu^{10} + 1369881 \nu^{9} + 572391 \nu^{8} - 4577378 \nu^{7} - 88887 \nu^{6} + \cdots + 37572 ) / 15503$$ (-2771*v^14 + 10204*v^13 + 46404*v^12 - 192737*v^11 - 267316*v^10 + 1369881*v^9 + 572391*v^8 - 4577378*v^7 - 88887*v^6 + 7302838*v^5 - 910052*v^4 - 4881141*v^3 + 479666*v^2 + 963405*v + 37572) / 15503 $$\beta_{8}$$ $$=$$ $$( 3508 \nu^{14} - 12862 \nu^{13} - 51126 \nu^{12} + 220490 \nu^{11} + 210949 \nu^{10} - 1345835 \nu^{9} + 30514 \nu^{8} + 3476043 \nu^{7} - 1779705 \nu^{6} + \cdots + 105781 ) / 15503$$ (3508*v^14 - 12862*v^13 - 51126*v^12 + 220490*v^11 + 210949*v^10 - 1345835*v^9 + 30514*v^8 + 3476043*v^7 - 1779705*v^6 - 3350054*v^5 + 2969546*v^4 + 350687*v^3 - 897413*v^2 + 1454*v + 105781) / 15503 $$\beta_{9}$$ $$=$$ $$( - 3651 \nu^{14} + 15923 \nu^{13} + 48340 \nu^{12} - 278400 \nu^{11} - 140314 \nu^{10} + 1766692 \nu^{9} - 476298 \nu^{8} - 4974160 \nu^{7} + 2844299 \nu^{6} + \cdots + 57068 ) / 15503$$ (-3651*v^14 + 15923*v^13 + 48340*v^12 - 278400*v^11 - 140314*v^10 + 1766692*v^9 - 476298*v^8 - 4974160*v^7 + 2844299*v^6 + 6077287*v^5 - 3797680*v^4 - 2639809*v^3 + 823207*v^2 + 632068*v + 57068) / 15503 $$\beta_{10}$$ $$=$$ $$( - 3831 \nu^{14} + 11103 \nu^{13} + 64239 \nu^{12} - 188458 \nu^{11} - 378592 \nu^{10} + 1124502 \nu^{9} + 916929 \nu^{8} - 2731984 \nu^{7} - 766205 \nu^{6} + \cdots + 54009 ) / 15503$$ (-3831*v^14 + 11103*v^13 + 64239*v^12 - 188458*v^11 - 378592*v^10 + 1124502*v^9 + 916929*v^8 - 2731984*v^7 - 766205*v^6 + 2054092*v^5 + 131498*v^4 + 720525*v^3 - 521172*v^2 - 297179*v + 54009) / 15503 $$\beta_{11}$$ $$=$$ $$( 6164 \nu^{14} - 27868 \nu^{13} - 81774 \nu^{12} + 495667 \nu^{11} + 230430 \nu^{10} - 3225051 \nu^{9} + 925727 \nu^{8} + 9429078 \nu^{7} - 5427250 \nu^{6} + \cdots - 166721 ) / 15503$$ (6164*v^14 - 27868*v^13 - 81774*v^12 + 495667*v^11 + 230430*v^10 - 3225051*v^9 + 925727*v^8 + 9429078*v^7 - 5427250*v^6 - 12224615*v^5 + 7446694*v^4 + 5896449*v^3 - 1633524*v^2 - 1347638*v - 166721) / 15503 $$\beta_{12}$$ $$=$$ $$( - 6274 \nu^{14} + 26645 \nu^{13} + 82016 \nu^{12} - 457928 \nu^{11} - 210679 \nu^{10} + 2799873 \nu^{9} - 1120763 \nu^{8} - 7195859 \nu^{7} + 6154343 \nu^{6} + \cdots - 140902 ) / 15503$$ (-6274*v^14 + 26645*v^13 + 82016*v^12 - 457928*v^11 - 210679*v^10 + 2799873*v^9 - 1120763*v^8 - 7195859*v^7 + 6154343*v^6 + 6600800*v^5 - 8606052*v^4 + 143082*v^3 + 2321779*v^2 - 370041*v - 140902) / 15503 $$\beta_{13}$$ $$=$$ $$( - 6279 \nu^{14} + 23066 \nu^{13} + 97530 \nu^{12} - 413227 \nu^{11} - 478265 \nu^{10} + 2715716 \nu^{9} + 541877 \nu^{8} - 8053421 \nu^{7} + 1690818 \nu^{6} + \cdots + 24809 ) / 15503$$ (-6279*v^14 + 23066*v^13 + 97530*v^12 - 413227*v^11 - 478265*v^10 + 2715716*v^9 + 541877*v^8 - 8053421*v^7 + 1690818*v^6 + 10652892*v^5 - 3864095*v^4 - 5231828*v^3 + 1268558*v^2 + 946448*v + 24809) / 15503 $$\beta_{14}$$ $$=$$ $$( 6556 \nu^{14} - 29429 \nu^{13} - 85737 \nu^{12} + 522692 \nu^{11} + 221211 \nu^{10} - 3395752 \nu^{9} + 1156520 \nu^{8} + 9912222 \nu^{7} - 6350786 \nu^{6} + \cdots - 108038 ) / 15503$$ (6556*v^14 - 29429*v^13 - 85737*v^12 + 522692*v^11 + 221211*v^10 - 3395752*v^9 + 1156520*v^8 + 9912222*v^7 - 6350786*v^6 - 12816467*v^5 + 8859280*v^4 + 6088386*v^3 - 2386522*v^2 - 1276289*v - 108038) / 15503
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5\beta_1$$ b3 + 5*b1 $$\nu^{4}$$ $$=$$ $$\beta_{13} + \beta_{8} - \beta_{7} + 7\beta_{2} + \beta _1 + 15$$ b13 + b8 - b7 + 7*b2 + b1 + 15 $$\nu^{5}$$ $$=$$ $$\beta_{9} - \beta_{7} + \beta_{5} + \beta_{4} + 9\beta_{3} + 30\beta_1$$ b9 - b7 + b5 + b4 + 9*b3 + 30*b1 $$\nu^{6}$$ $$=$$ $$\beta_{14} + 11 \beta_{13} + \beta_{11} - \beta_{10} + 2 \beta_{9} + 10 \beta_{8} - 10 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 45 \beta_{2} + 12 \beta _1 + 85$$ b14 + 11*b13 + b11 - b10 + 2*b9 + 10*b8 - 10*b7 - b6 - b5 + b4 + b3 + 45*b2 + 12*b1 + 85 $$\nu^{7}$$ $$=$$ $$- \beta_{14} + \beta_{12} + 2 \beta_{11} + 13 \beta_{9} + 3 \beta_{8} - 11 \beta_{7} + 8 \beta_{5} + 13 \beta_{4} + 66 \beta_{3} + 3 \beta_{2} + 192 \beta _1 + 2$$ -b14 + b12 + 2*b11 + 13*b9 + 3*b8 - 11*b7 + 8*b5 + 13*b4 + 66*b3 + 3*b2 + 192*b1 + 2 $$\nu^{8}$$ $$=$$ $$14 \beta_{14} + 92 \beta_{13} - \beta_{12} + 15 \beta_{11} - 14 \beta_{10} + 29 \beta_{9} + 79 \beta_{8} - 77 \beta_{7} - 16 \beta_{6} - 20 \beta_{5} + 16 \beta_{4} + 16 \beta_{3} + 287 \beta_{2} + 108 \beta _1 + 513$$ 14*b14 + 92*b13 - b12 + 15*b11 - 14*b10 + 29*b9 + 79*b8 - 77*b7 - 16*b6 - 20*b5 + 16*b4 + 16*b3 + 287*b2 + 108*b1 + 513 $$\nu^{9}$$ $$=$$ $$- 15 \beta_{14} + \beta_{13} + 14 \beta_{12} + 31 \beta_{11} - 2 \beta_{10} + 124 \beta_{9} + 45 \beta_{8} - 93 \beta_{7} - \beta_{6} + 39 \beta_{5} + 126 \beta_{4} + 457 \beta_{3} + 48 \beta_{2} + 1267 \beta _1 + 32$$ -15*b14 + b13 + 14*b12 + 31*b11 - 2*b10 + 124*b9 + 45*b8 - 93*b7 - b6 + 39*b5 + 126*b4 + 457*b3 + 48*b2 + 1267*b1 + 32 $$\nu^{10}$$ $$=$$ $$137 \beta_{14} + 702 \beta_{13} - 15 \beta_{12} + 156 \beta_{11} - 141 \beta_{10} + 295 \beta_{9} + 581 \beta_{8} - 550 \beta_{7} - 170 \beta_{6} - 244 \beta_{5} + 178 \beta_{4} + 171 \beta_{3} + 1841 \beta_{2} + 884 \beta _1 + 3211$$ 137*b14 + 702*b13 - 15*b12 + 156*b11 - 141*b10 + 295*b9 + 581*b8 - 550*b7 - 170*b6 - 244*b5 + 178*b4 + 171*b3 + 1841*b2 + 884*b1 + 3211 $$\nu^{11}$$ $$=$$ $$- 147 \beta_{14} + 25 \beta_{13} + 137 \beta_{12} + 329 \beta_{11} - 42 \beta_{10} + 1054 \beta_{9} + 466 \beta_{8} - 721 \beta_{7} - 23 \beta_{6} + 80 \beta_{5} + 1089 \beta_{4} + 3109 \beta_{3} + 513 \beta_{2} + 8500 \beta _1 + 358$$ -147*b14 + 25*b13 + 137*b12 + 329*b11 - 42*b10 + 1054*b9 + 466*b8 - 721*b7 - 23*b6 + 80*b5 + 1089*b4 + 3109*b3 + 513*b2 + 8500*b1 + 358 $$\nu^{12}$$ $$=$$ $$1168 \beta_{14} + 5142 \beta_{13} - 147 \beta_{12} + 1393 \beta_{11} - 1243 \beta_{10} + 2609 \beta_{9} + 4163 \beta_{8} - 3830 \beta_{7} - 1524 \beta_{6} - 2421 \beta_{5} + 1695 \beta_{4} + 1553 \beta_{3} + \cdots + 20555$$ 1168*b14 + 5142*b13 - 147*b12 + 1393*b11 - 1243*b10 + 2609*b9 + 4163*b8 - 3830*b7 - 1524*b6 - 2421*b5 + 1695*b4 + 1553*b3 + 11912*b2 + 6929*b1 + 20555 $$\nu^{13}$$ $$=$$ $$- 1180 \beta_{14} + 382 \beta_{13} + 1168 \beta_{12} + 2981 \beta_{11} - 566 \beta_{10} + 8467 \beta_{9} + 4162 \beta_{8} - 5383 \beta_{7} - 328 \beta_{6} - 968 \beta_{5} + 8860 \beta_{4} + 21073 \beta_{3} + \cdots + 3428$$ -1180*b14 + 382*b13 + 1168*b12 + 2981*b11 - 566*b10 + 8467*b9 + 4162*b8 - 5383*b7 - 328*b6 - 968*b5 + 8860*b4 + 21073*b3 + 4629*b2 + 57590*b1 + 3428 $$\nu^{14}$$ $$=$$ $$9319 \beta_{14} + 36873 \beta_{13} - 1180 \beta_{12} + 11460 \beta_{11} - 10225 \beta_{10} + 21489 \beta_{9} + 29540 \beta_{8} - 26456 \beta_{7} - 12497 \beta_{6} - 21532 \beta_{5} + 14818 \beta_{4} + \cdots + 133544$$ 9319*b14 + 36873*b13 - 1180*b12 + 11460*b11 - 10225*b10 + 21489*b9 + 29540*b8 - 26456*b7 - 12497*b6 - 21532*b5 + 14818*b4 + 12994*b3 + 77735*b2 + 53005*b1 + 133544

## 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
 −2.57239 −2.27492 −1.66124 −1.59223 −0.527123 −0.442944 −0.102671 0.692644 0.776738 1.28989 1.83402 1.89451 2.40444 2.61792 2.66335
−2.57239 −0.511676 4.61720 0 1.31623 5.06037 −6.73245 −2.73819 0
1.2 −2.27492 1.91743 3.17527 0 −4.36201 −2.74127 −2.67364 0.676544 0
1.3 −1.66124 2.93460 0.759729 0 −4.87509 3.57911 2.06039 5.61190 0
1.4 −1.59223 −1.35755 0.535199 0 2.16153 0.175705 2.33230 −1.15707 0
1.5 −0.527123 1.24468 −1.72214 0 −0.656097 3.22241 1.96203 −1.45078 0
1.6 −0.442944 −1.62320 −1.80380 0 0.718985 2.12336 1.68487 −0.365237 0
1.7 −0.102671 −2.39685 −1.98946 0 0.246087 −2.47527 0.409602 2.74488 0
1.8 0.692644 1.14700 −1.52024 0 0.794464 3.60442 −2.43828 −1.68438 0
1.9 0.776738 0.738183 −1.39668 0 0.573374 −0.981439 −2.63833 −2.45509 0
1.10 1.28989 3.30063 −0.336174 0 4.25747 0.563541 −3.01342 7.89418 0
1.11 1.83402 −0.596609 1.36363 0 −1.09419 −2.23486 −1.16712 −2.64406 0
1.12 1.89451 −3.43936 1.58918 0 −6.51591 4.54180 −0.778295 8.82918 0
1.13 2.40444 1.94792 3.78133 0 4.68365 −0.665236 4.28310 0.794388 0
1.14 2.61792 2.17802 4.85352 0 5.70188 4.94832 7.47030 1.74376 0
1.15 2.66335 −1.48323 5.09344 0 −3.95037 2.27906 8.23893 −0.800022 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.y yes 15
5.b even 2 1 5225.2.a.r 15

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5225.2.a.r 15 5.b even 2 1
5225.2.a.y 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5225))$$:

 $$T_{2}^{15} - 5 T_{2}^{14} - 11 T_{2}^{13} + 87 T_{2}^{12} - 4 T_{2}^{11} - 545 T_{2}^{10} + 431 T_{2}^{9} + 1480 T_{2}^{8} - 1763 T_{2}^{7} - 1609 T_{2}^{6} + 2516 T_{2}^{5} + 391 T_{2}^{4} - 1081 T_{2}^{3} - 45 T_{2}^{2} + 153 T_{2} + 15$$ T2^15 - 5*T2^14 - 11*T2^13 + 87*T2^12 - 4*T2^11 - 545*T2^10 + 431*T2^9 + 1480*T2^8 - 1763*T2^7 - 1609*T2^6 + 2516*T2^5 + 391*T2^4 - 1081*T2^3 - 45*T2^2 + 153*T2 + 15 $$T_{7}^{15} - 21 T_{7}^{14} + 152 T_{7}^{13} - 253 T_{7}^{12} - 2119 T_{7}^{11} + 10036 T_{7}^{10} - 324 T_{7}^{9} - 76610 T_{7}^{8} + 105408 T_{7}^{7} + 195473 T_{7}^{6} - 448684 T_{7}^{5} - 87941 T_{7}^{4} + \cdots + 22429$$ T7^15 - 21*T7^14 + 152*T7^13 - 253*T7^12 - 2119*T7^11 + 10036*T7^10 - 324*T7^9 - 76610*T7^8 + 105408*T7^7 + 195473*T7^6 - 448684*T7^5 - 87941*T7^4 + 510031*T7^3 - 29409*T7^2 - 137361*T7 + 22429

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{15} - 5 T^{14} - 11 T^{13} + 87 T^{12} + \cdots + 15$$
$3$ $$T^{15} - 4 T^{14} - 22 T^{13} + 101 T^{12} + \cdots + 683$$
$5$ $$T^{15}$$
$7$ $$T^{15} - 21 T^{14} + 152 T^{13} + \cdots + 22429$$
$11$ $$(T - 1)^{15}$$
$13$ $$T^{15} - 13 T^{14} - 4 T^{13} + \cdots - 531655$$
$17$ $$T^{15} - 17 T^{14} - 23 T^{13} + \cdots + 1116225$$
$19$ $$(T - 1)^{15}$$
$23$ $$T^{15} - 26 T^{14} + 178 T^{13} + \cdots - 2682024$$
$29$ $$T^{15} - 9 T^{14} - 212 T^{13} + \cdots - 31375800$$
$31$ $$T^{15} - 14 T^{14} - 52 T^{13} + \cdots - 1388125$$
$37$ $$T^{15} - 9 T^{14} - 207 T^{13} + \cdots - 162016351$$
$41$ $$T^{15} - 4 T^{14} - 261 T^{13} + \cdots - 220545465$$
$43$ $$T^{15} - 28 T^{14} + \cdots - 212132375$$
$47$ $$T^{15} - 14 T^{14} + \cdots - 812451225$$
$53$ $$T^{15} - 3 T^{14} + \cdots - 33084572583$$
$59$ $$T^{15} - T^{14} - 435 T^{13} + \cdots - 766016655$$
$61$ $$T^{15} - 2 T^{14} + \cdots + 2601462755$$
$67$ $$T^{15} - 37 T^{14} + \cdots + 995592677$$
$71$ $$T^{15} + 7 T^{14} + \cdots - 765891181005$$
$73$ $$T^{15} - 42 T^{14} + \cdots - 23072143651$$
$79$ $$T^{15} + 10 T^{14} + \cdots + 1326572185771$$
$83$ $$T^{15} - 14 T^{14} - 441 T^{13} + \cdots + 87522675$$
$89$ $$T^{15} - 15 T^{14} + \cdots - 1998434698983$$
$97$ $$T^{15} - 8 T^{14} + \cdots + 1657280631608$$