# Properties

 Label 5225.2.a.r 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: $$x^{15} - 5 x^{14} - 11 x^{13} + 87 x^{12} - 4 x^{11} - 545 x^{10} + 431 x^{9} + 1480 x^{8} - 1763 x^{7} + \cdots + 15$$ 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 $$q - \beta_1 q^{2} - \beta_{5} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{5} - \beta_{4}) q^{6} + ( - \beta_{13} - 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + (\beta_{10} + \beta_{7} + 1) q^{9} + q^{11} + (\beta_{14} + \beta_{13} - \beta_{10} + \cdots - 1) q^{12}+ \cdots + (\beta_{10} + \beta_{7} + 1) q^{99}+O(q^{100})$$ 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 $$x^{15} - 5 x^{14} - 11 x^{13} + 87 x^{12} - 4 x^{11} - 545 x^{10} + 431 x^{9} + 1480 x^{8} - 1763 x^{7} + \cdots + 15$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5\nu$$ v^3 - 5*v $$\beta_{4}$$ $$=$$ $$( - 16 \nu^{14} - 2151 \nu^{13} + 9337 \nu^{12} + 28321 \nu^{11} - 152439 \nu^{10} - 86367 \nu^{9} + \cdots - 3028 ) / 15503$$ (-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 $$\beta_{5}$$ $$=$$ $$( 896 \nu^{14} - 3568 \nu^{13} - 11273 \nu^{12} + 57342 \nu^{11} + 25437 \nu^{10} - 310444 \nu^{9} + \cdots - 16468 ) / 15503$$ (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 $$\beta_{6}$$ $$=$$ $$( 2058 \nu^{14} - 12071 \nu^{13} - 16930 \nu^{12} + 207769 \nu^{11} - 106536 \nu^{10} - 1272128 \nu^{9} + \cdots - 21353 ) / 15503$$ (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 $$\beta_{7}$$ $$=$$ $$( - 2771 \nu^{14} + 10204 \nu^{13} + 46404 \nu^{12} - 192737 \nu^{11} - 267316 \nu^{10} + \cdots + 37572 ) / 15503$$ (-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 $$\beta_{8}$$ $$=$$ $$( 3508 \nu^{14} - 12862 \nu^{13} - 51126 \nu^{12} + 220490 \nu^{11} + 210949 \nu^{10} - 1345835 \nu^{9} + \cdots + 105781 ) / 15503$$ (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 $$\beta_{9}$$ $$=$$ $$( - 3651 \nu^{14} + 15923 \nu^{13} + 48340 \nu^{12} - 278400 \nu^{11} - 140314 \nu^{10} + \cdots + 57068 ) / 15503$$ (-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 $$\beta_{10}$$ $$=$$ $$( - 3831 \nu^{14} + 11103 \nu^{13} + 64239 \nu^{12} - 188458 \nu^{11} - 378592 \nu^{10} + \cdots + 54009 ) / 15503$$ (-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 $$\beta_{11}$$ $$=$$ $$( 6164 \nu^{14} - 27868 \nu^{13} - 81774 \nu^{12} + 495667 \nu^{11} + 230430 \nu^{10} - 3225051 \nu^{9} + \cdots - 166721 ) / 15503$$ (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 $$\beta_{12}$$ $$=$$ $$( - 6274 \nu^{14} + 26645 \nu^{13} + 82016 \nu^{12} - 457928 \nu^{11} - 210679 \nu^{10} + \cdots - 140902 ) / 15503$$ (-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 $$\beta_{13}$$ $$=$$ $$( - 6279 \nu^{14} + 23066 \nu^{13} + 97530 \nu^{12} - 413227 \nu^{11} - 478265 \nu^{10} + \cdots + 24809 ) / 15503$$ (-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 $$\beta_{14}$$ $$=$$ $$( 6556 \nu^{14} - 29429 \nu^{13} - 85737 \nu^{12} + 522692 \nu^{11} + 221211 \nu^{10} - 3395752 \nu^{9} + \cdots - 108038 ) / 15503$$ (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 $$\nu^{6}$$ $$=$$ $$\beta_{14} + 11 \beta_{13} + \beta_{11} - \beta_{10} + 2 \beta_{9} + 10 \beta_{8} - 10 \beta_{7} + \cdots + 85$$ b14 + 11*b13 + b11 - b10 + 2*b9 + 10*b8 - 10*b7 - b6 - b5 + b4 + b3 + 45*b2 + 12*b1 + 85 $$\nu^{7}$$ $$=$$ $$- \beta_{14} + \beta_{12} + 2 \beta_{11} + 13 \beta_{9} + 3 \beta_{8} - 11 \beta_{7} + 8 \beta_{5} + \cdots + 2$$ -b14 + b12 + 2*b11 + 13*b9 + 3*b8 - 11*b7 + 8*b5 + 13*b4 + 66*b3 + 3*b2 + 192*b1 + 2 $$\nu^{8}$$ $$=$$ $$14 \beta_{14} + 92 \beta_{13} - \beta_{12} + 15 \beta_{11} - 14 \beta_{10} + 29 \beta_{9} + 79 \beta_{8} + \cdots + 513$$ 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 $$\nu^{9}$$ $$=$$ $$- 15 \beta_{14} + \beta_{13} + 14 \beta_{12} + 31 \beta_{11} - 2 \beta_{10} + 124 \beta_{9} + 45 \beta_{8} + \cdots + 32$$ -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 $$\nu^{10}$$ $$=$$ $$137 \beta_{14} + 702 \beta_{13} - 15 \beta_{12} + 156 \beta_{11} - 141 \beta_{10} + 295 \beta_{9} + \cdots + 3211$$ 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 $$\nu^{11}$$ $$=$$ $$- 147 \beta_{14} + 25 \beta_{13} + 137 \beta_{12} + 329 \beta_{11} - 42 \beta_{10} + 1054 \beta_{9} + \cdots + 358$$ -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 $$\nu^{12}$$ $$=$$ $$1168 \beta_{14} + 5142 \beta_{13} - 147 \beta_{12} + 1393 \beta_{11} - 1243 \beta_{10} + 2609 \beta_{9} + \cdots + 20555$$ 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 $$\nu^{13}$$ $$=$$ $$- 1180 \beta_{14} + 382 \beta_{13} + 1168 \beta_{12} + 2981 \beta_{11} - 566 \beta_{10} + 8467 \beta_{9} + \cdots + 3428$$ -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 $$\nu^{14}$$ $$=$$ $$9319 \beta_{14} + 36873 \beta_{13} - 1180 \beta_{12} + 11460 \beta_{11} - 10225 \beta_{10} + \cdots + 133544$$ 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
 2.66335 2.61792 2.40444 1.89451 1.83402 1.28989 0.776738 0.692644 −0.102671 −0.442944 −0.527123 −1.59223 −1.66124 −2.27492 −2.57239
−2.66335 1.48323 5.09344 0 −3.95037 −2.27906 −8.23893 −0.800022 0
1.2 −2.61792 −2.17802 4.85352 0 5.70188 −4.94832 −7.47030 1.74376 0
1.3 −2.40444 −1.94792 3.78133 0 4.68365 0.665236 −4.28310 0.794388 0
1.4 −1.89451 3.43936 1.58918 0 −6.51591 −4.54180 0.778295 8.82918 0
1.5 −1.83402 0.596609 1.36363 0 −1.09419 2.23486 1.16712 −2.64406 0
1.6 −1.28989 −3.30063 −0.336174 0 4.25747 −0.563541 3.01342 7.89418 0
1.7 −0.776738 −0.738183 −1.39668 0 0.573374 0.981439 2.63833 −2.45509 0
1.8 −0.692644 −1.14700 −1.52024 0 0.794464 −3.60442 2.43828 −1.68438 0
1.9 0.102671 2.39685 −1.98946 0 0.246087 2.47527 −0.409602 2.74488 0
1.10 0.442944 1.62320 −1.80380 0 0.718985 −2.12336 −1.68487 −0.365237 0
1.11 0.527123 −1.24468 −1.72214 0 −0.656097 −3.22241 −1.96203 −1.45078 0
1.12 1.59223 1.35755 0.535199 0 2.16153 −0.175705 −2.33230 −1.15707 0
1.13 1.66124 −2.93460 0.759729 0 −4.87509 −3.57911 −2.06039 5.61190 0
1.14 2.27492 −1.91743 3.17527 0 −4.36201 2.74127 2.67364 0.676544 0
1.15 2.57239 0.511676 4.61720 0 1.31623 −5.06037 6.73245 −2.73819 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.r 15
5.b even 2 1 5225.2.a.y yes 15

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

 $$T_{2}^{15} + 5 T_{2}^{14} - 11 T_{2}^{13} - 87 T_{2}^{12} - 4 T_{2}^{11} + 545 T_{2}^{10} + 431 T_{2}^{9} + \cdots - 15$$ 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 $$T_{7}^{15} + 21 T_{7}^{14} + 152 T_{7}^{13} + 253 T_{7}^{12} - 2119 T_{7}^{11} - 10036 T_{7}^{10} + \cdots - 22429$$ 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

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