# Properties

 Label 225.2.k.c Level $225$ Weight $2$ Character orbit 225.k Analytic conductor $1.797$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,2,Mod(49,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("225.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 225.k (of order $$6$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 12x^{14} + 102x^{12} - 406x^{10} + 1167x^{8} - 1842x^{6} + 2023x^{4} - 441x^{2} + 81$$ x^16 - 12*x^14 + 102*x^12 - 406*x^10 + 1167*x^8 - 1842*x^6 + 2023*x^4 - 441*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{14} - \beta_{10}) q^{3} + (\beta_{11} + \beta_{9} + \beta_{6} + \beta_{5} - \beta_{3} + 1) q^{4} + (\beta_{13} + \beta_{11} - \beta_{6} - \beta_{5} + 2 \beta_{2} + 2) q^{6} + (\beta_{15} + \beta_{14} + \beta_{7} - \beta_{4} + \beta_1) q^{7} + ( - \beta_{15} - \beta_{14} - 3 \beta_{12} + 2 \beta_{10} - 3 \beta_{7} + 2 \beta_{4}) q^{8} + ( - \beta_{9} - 2 \beta_{6} - \beta_{5} - \beta_{3}) q^{9}+O(q^{10})$$ q + b1 * q^2 + (b14 - b10) * q^3 + (b11 + b9 + b6 + b5 - b3 + 1) * q^4 + (b13 + b11 - b6 - b5 + 2*b2 + 2) * q^6 + (b15 + b14 + b7 - b4 + b1) * q^7 + (-b15 - b14 - 3*b12 + 2*b10 - 3*b7 + 2*b4) * q^8 + (-b9 - 2*b6 - b5 - b3) * q^9 $$q + \beta_1 q^{2} + (\beta_{14} - \beta_{10}) q^{3} + (\beta_{11} + \beta_{9} + \beta_{6} + \beta_{5} - \beta_{3} + 1) q^{4} + (\beta_{13} + \beta_{11} - \beta_{6} - \beta_{5} + 2 \beta_{2} + 2) q^{6} + (\beta_{15} + \beta_{14} + \beta_{7} - \beta_{4} + \beta_1) q^{7} + ( - \beta_{15} - \beta_{14} - 3 \beta_{12} + 2 \beta_{10} - 3 \beta_{7} + 2 \beta_{4}) q^{8} + ( - \beta_{9} - 2 \beta_{6} - \beta_{5} - \beta_{3}) q^{9} + ( - 2 \beta_{13} + \beta_{11} + \beta_{5} - \beta_{3} - \beta_{2}) q^{11} + ( - \beta_{15} - \beta_{14} + 2 \beta_{12} + \beta_{8} + \beta_{7} - \beta_{4} + \beta_1) q^{12} + ( - 2 \beta_{14} + 2 \beta_{12} - \beta_{8} - 2 \beta_{4}) q^{13} + ( - \beta_{13} - \beta_{11} - \beta_{5} + \beta_{2}) q^{14} + ( - 2 \beta_{11} + \beta_{6} + 2 \beta_{5} - \beta_{3} - 2 \beta_{2} - 1) q^{16} + (2 \beta_{15} - \beta_{14} + 3 \beta_{12} - \beta_{10} + \beta_{8} + 3 \beta_{7} - \beta_{4} - \beta_1) q^{17} + (\beta_{14} - 2 \beta_{12} - \beta_{10} + \beta_{8} + 2 \beta_{7} - 2 \beta_1) q^{18} + (\beta_{13} + \beta_{11} - 2 \beta_{9} + \beta_{2} - 1) q^{19} + (\beta_{11} + \beta_{9} + \beta_{6} - 2 \beta_{3} + 2 \beta_{2} - 1) q^{21} + (\beta_{15} - \beta_{14} + \beta_{10} - \beta_{8} - \beta_{4}) q^{22} + (\beta_{15} + \beta_{14} + 3 \beta_{12} + \beta_{10} + \beta_{4}) q^{23} + (2 \beta_{13} + 3 \beta_{9} - \beta_{5} + 6) q^{24} + (\beta_{13} + \beta_{11} + \beta_{9} - 2 \beta_{5} - \beta_{2} - 2) q^{26} + ( - 3 \beta_{15} - \beta_{12} + 2 \beta_{10} - \beta_{8} - 2 \beta_{7} + 3 \beta_{4} - \beta_1) q^{27} + (2 \beta_{15} + \beta_{12} - 2 \beta_{10} + 2 \beta_{8} + \beta_{7} - 2 \beta_{4} - 2 \beta_1) q^{28} + ( - \beta_{13} - \beta_{11} + 2 \beta_{5} - \beta_{3} - 2 \beta_{2}) q^{29} + (2 \beta_{13} - 2 \beta_{11} - \beta_{9} + \beta_{6} - 2 \beta_{5} + \beta_{3} - 2 \beta_{2} - 1) q^{31} + (3 \beta_{14} - 3 \beta_{12} - 2 \beta_{8} + 3 \beta_{4}) q^{32} + ( - 2 \beta_{15} + \beta_{14} + 5 \beta_{12} - 3 \beta_{8} + 4 \beta_{7} + \beta_{4} + \cdots + 3 \beta_1) q^{33}+ \cdots + ( - 3 \beta_{13} + 6 \beta_{11} + \beta_{9} - 4 \beta_{6} + 4 \beta_{5} - 2 \beta_{3} + \cdots + 6) q^{99}+O(q^{100})$$ q + b1 * q^2 + (b14 - b10) * q^3 + (b11 + b9 + b6 + b5 - b3 + 1) * q^4 + (b13 + b11 - b6 - b5 + 2*b2 + 2) * q^6 + (b15 + b14 + b7 - b4 + b1) * q^7 + (-b15 - b14 - 3*b12 + 2*b10 - 3*b7 + 2*b4) * q^8 + (-b9 - 2*b6 - b5 - b3) * q^9 + (-2*b13 + b11 + b5 - b3 - b2) * q^11 + (-b15 - b14 + 2*b12 + b8 + b7 - b4 + b1) * q^12 + (-2*b14 + 2*b12 - b8 - 2*b4) * q^13 + (-b13 - b11 - b5 + b2) * q^14 + (-2*b11 + b6 + 2*b5 - b3 - 2*b2 - 1) * q^16 + (2*b15 - b14 + 3*b12 - b10 + b8 + 3*b7 - b4 - b1) * q^17 + (b14 - 2*b12 - b10 + b8 + 2*b7 - 2*b1) * q^18 + (b13 + b11 - 2*b9 + b2 - 1) * q^19 + (b11 + b9 + b6 - 2*b3 + 2*b2 - 1) * q^21 + (b15 - b14 + b10 - b8 - b4) * q^22 + (b15 + b14 + 3*b12 + b10 + b4) * q^23 + (2*b13 + 3*b9 - b5 + 6) * q^24 + (b13 + b11 + b9 - 2*b5 - b2 - 2) * q^26 + (-3*b15 - b12 + 2*b10 - b8 - 2*b7 + 3*b4 - b1) * q^27 + (2*b15 + b12 - 2*b10 + 2*b8 + b7 - 2*b4 - 2*b1) * q^28 + (-b13 - b11 + 2*b5 - b3 - 2*b2) * q^29 + (2*b13 - 2*b11 - b9 + b6 - 2*b5 + b3 - 2*b2 - 1) * q^31 + (3*b14 - 3*b12 - 2*b8 + 3*b4) * q^32 + (-2*b15 + b14 + 5*b12 - 3*b8 + 4*b7 + b4 + 3*b1) * q^33 + (-b13 + b11 + 4*b3) * q^34 + (-b9 - 5*b6 - b5 + 2*b3 + 3*b2) * q^36 + (b15 + b14 - b12 - 2*b10 + 3*b8 - b7 - 2*b4 - 3*b1) * q^37 + (b15 + b14 - 2*b10 + 6*b7 + b4 - b1) * q^38 + (b11 - 2*b9 + 4*b6 + b5 + 4*b3 - b2 - 7) * q^39 + (b13 + b11 - 2*b9 + 3*b6 + b5 + 2*b3 - b2 - 2) * q^41 + (b14 - 3*b12 - 2*b10 - 3*b8) * q^42 + (2*b10 - b7 - 2*b4 + 4*b1) * q^43 + (-b13 - b11 + b9 + 2*b5 + b2 - 5) * q^44 + (-4*b13 - 4*b11 + 3*b9 + 2*b5 - 2*b2) * q^46 + (-b15 - b14 - b10 - 6*b7 + 2*b4 - 2*b1) * q^47 + (-2*b15 - 2*b14 - 5*b12 + 3*b10 + 2*b8 - 10*b7 + b4 + 2*b1) * q^48 + (-b13 - 2*b11 + 2*b9 - 3*b6 - 2*b5 - 2*b3 + b2 + 2) * q^49 + (-2*b13 - 2*b11 - 3*b9 + 5*b6 + 2*b5 + 3*b3 - b2 - 4) * q^51 + (-2*b15 - 2*b14 + b10 - 5*b7 + b4) * q^52 + (3*b15 - 3*b12 - 3*b10 - b8 - 3*b7 - 3*b4 + b1) * q^53 + (-3*b11 - b9 - 2*b6 + 2*b5 - b3 - 3*b2 - 3) * q^54 + (-6*b6 + 3*b3 + 6) * q^56 + (-2*b15 - 5*b12 + 2*b10 + 2*b8 - b7 + 4*b4 - b1) * q^57 + (2*b15 + 3*b14 - 6*b12 + 2*b10 - b8 + 3*b4) * q^58 + (3*b13 - 2*b9 + 6*b6 + 2*b3 - 3*b2 - 2) * q^59 + (2*b13 + b11 - 4*b6 - 3*b5 + b3 + 3*b2 + 4) * q^61 + (-3*b15 - 3*b12 + 3*b10 - 3*b7 + 3*b4) * q^62 + (-3*b15 - 4*b14 - 3*b12 + 2*b10) * q^63 + (-b13 - b11 - 3*b9 - b5 - 2*b2 - 7) * q^64 + (b11 + b9 + 7*b6 + 2*b5 + b3 - b2 - 4) * q^66 + (-b15 - 2*b12 - b10 - 4*b8) * q^67 + (-2*b15 - 2*b14 + 9*b12 - 2*b10 + 2*b8 - 2*b4) * q^68 + (-5*b13 + 3*b9 + 3*b6 + b5 - 3*b3 + 3) * q^69 + (3*b13 + 3*b11 - b9 + 3*b2 - 1) * q^71 + (3*b15 + b14 + 6*b12 - 5*b10 + 3*b8 - 6*b4 - 3*b1) * q^72 + (-3*b15 + 2*b14 - 2*b12 + b10 + b8 - 2*b7 + b4 - b1) * q^73 + (5*b13 + 2*b11 - 12*b6 - 7*b5 + 2*b3 + 7*b2 + 12) * q^74 + (-b13 - b11 - 3*b9 - 2*b6 - b5 + 3*b3 + b2 - 3) * q^76 + (2*b15 - 4*b14 - 3*b12 + 2*b10 + 3*b8 - 4*b4) * q^77 + (b15 - 4*b14 + 11*b12 + b10 + 4*b7 - 2*b4 - 3*b1) * q^78 + (3*b13 - b11 + b6 - 2*b5 + b3 + 2*b2 - 1) * q^79 + (-2*b13 + 4*b9 - 4*b6 + 2*b5 - 2*b3 - 3*b2 + 6) * q^81 + (-2*b14 + 3*b12 + 2*b10 - b8 + 3*b7 + 2*b4 + b1) * q^82 + (9*b7 + 3*b1) * q^83 + (-b13 - 4*b11 - 4*b9 + 2*b6 - b5 + 2*b3 - 2*b2 - 8) * q^84 + (5*b9 + 6*b6 - 5*b3 + 5) * q^86 + (-2*b15 - b12 + 2*b10 - 5*b7 + b4 + 3*b1) * q^87 + (4*b15 + 4*b14 - 5*b10 + 3*b7 + b4 - 3*b1) * q^88 + (3*b13 + 3*b11 - 3*b5 + 3) * q^89 + (2*b13 + 2*b11 - 4*b9 - 3*b5 - b2 - 6) * q^91 + (b15 + b14 + b10 + 3*b7 - 2*b4) * q^92 + (b15 - b14 + b12 - b10 + 2*b8 - 7*b7 + b4 - 4*b1) * q^93 + (b13 + 2*b11 + 4*b9 + 2*b5 - 4*b3 - b2 + 4) * q^94 + (2*b11 + 3*b9 + b6 + 2*b5 - 6*b3 - 2*b2 + 7) * q^96 + (2*b15 + 2*b14 - b10 + 7*b7 - b4 + 3*b1) * q^97 + (3*b14 - 3*b12 - 3*b10 + b8 - 3*b7 - 3*b4 - b1) * q^98 + (-3*b13 + 6*b11 + b9 - 4*b6 + 4*b5 - 2*b3 + 3*b2 + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 8 q^{4} + 16 q^{6} - 10 q^{9}+O(q^{10})$$ 16 * q + 8 * q^4 + 16 * q^6 - 10 * q^9 $$16 q + 8 q^{4} + 16 q^{6} - 10 q^{9} + 2 q^{11} + 6 q^{14} - 8 q^{16} - 8 q^{19} - 30 q^{21} + 66 q^{24} - 40 q^{26} + 2 q^{29} + 8 q^{31} + 18 q^{34} - 28 q^{36} - 50 q^{39} + 10 q^{41} - 88 q^{44} - 6 q^{49} + 22 q^{51} - 52 q^{54} + 60 q^{56} + 34 q^{59} + 26 q^{61} - 76 q^{64} - 16 q^{66} + 54 q^{69} - 32 q^{71} + 80 q^{74} - 22 q^{76} - 14 q^{79} + 34 q^{81} - 54 q^{84} + 68 q^{86} + 36 q^{89} - 68 q^{91} + 6 q^{94} + 68 q^{96} + 34 q^{99}+O(q^{100})$$ 16 * q + 8 * q^4 + 16 * q^6 - 10 * q^9 + 2 * q^11 + 6 * q^14 - 8 * q^16 - 8 * q^19 - 30 * q^21 + 66 * q^24 - 40 * q^26 + 2 * q^29 + 8 * q^31 + 18 * q^34 - 28 * q^36 - 50 * q^39 + 10 * q^41 - 88 * q^44 - 6 * q^49 + 22 * q^51 - 52 * q^54 + 60 * q^56 + 34 * q^59 + 26 * q^61 - 76 * q^64 - 16 * q^66 + 54 * q^69 - 32 * q^71 + 80 * q^74 - 22 * q^76 - 14 * q^79 + 34 * q^81 - 54 * q^84 + 68 * q^86 + 36 * q^89 - 68 * q^91 + 6 * q^94 + 68 * q^96 + 34 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 12x^{14} + 102x^{12} - 406x^{10} + 1167x^{8} - 1842x^{6} + 2023x^{4} - 441x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 3457 \nu^{14} - 28693 \nu^{12} + 210280 \nu^{10} - 306752 \nu^{8} + 709118 \nu^{6} - 1299110 \nu^{4} + 7466556 \nu^{2} - 7606359 ) / 3976425$$ (3457*v^14 - 28693*v^12 + 210280*v^10 - 306752*v^8 + 709118*v^6 - 1299110*v^4 + 7466556*v^2 - 7606359) / 3976425 $$\beta_{3}$$ $$=$$ $$( -26\nu^{14} + 74\nu^{12} + 10\nu^{10} - 10439\nu^{8} + 38576\nu^{6} - 94895\nu^{4} + 82467\nu^{2} - 18063 ) / 29025$$ (-26*v^14 + 74*v^12 + 10*v^10 - 10439*v^8 + 38576*v^6 - 94895*v^4 + 82467*v^2 - 18063) / 29025 $$\beta_{4}$$ $$=$$ $$( 5917 \nu^{15} - 90503 \nu^{13} + 784955 \nu^{11} - 3838362 \nu^{9} + 10468603 \nu^{7} - 20515435 \nu^{5} + 19722086 \nu^{3} - 16863264 \nu ) / 3976425$$ (5917*v^15 - 90503*v^13 + 784955*v^11 - 3838362*v^9 + 10468603*v^7 - 20515435*v^5 + 19722086*v^3 - 16863264*v) / 3976425 $$\beta_{5}$$ $$=$$ $$( - 7559 \nu^{14} + 69641 \nu^{12} - 572135 \nu^{10} + 1494574 \nu^{8} - 4803841 \nu^{6} + 3626995 \nu^{4} - 7975497 \nu^{2} - 5812992 ) / 3976425$$ (-7559*v^14 + 69641*v^12 - 572135*v^10 + 1494574*v^8 - 4803841*v^6 + 3626995*v^4 - 7975497*v^2 - 5812992) / 3976425 $$\beta_{6}$$ $$=$$ $$( 11836 \nu^{14} - 138574 \nu^{12} + 1170340 \nu^{10} - 4500371 \nu^{8} + 12811274 \nu^{6} - 19088030 \nu^{4} + 21981813 \nu^{2} - 814212 ) / 3976425$$ (11836*v^14 - 138574*v^12 + 1170340*v^10 - 4500371*v^8 + 12811274*v^6 - 19088030*v^4 + 21981813*v^2 - 814212) / 3976425 $$\beta_{7}$$ $$=$$ $$( 3766 \nu^{15} - 41414 \nu^{13} + 332215 \nu^{11} - 1090526 \nu^{9} + 2480389 \nu^{7} - 2137205 \nu^{5} + 467253 \nu^{3} + 2721243 \nu ) / 1325475$$ (3766*v^15 - 41414*v^13 + 332215*v^11 - 1090526*v^9 + 2480389*v^7 - 2137205*v^5 + 467253*v^3 + 2721243*v) / 1325475 $$\beta_{8}$$ $$=$$ $$( - 11836 \nu^{15} + 138574 \nu^{13} - 1170340 \nu^{11} + 4500371 \nu^{9} - 12811274 \nu^{7} + 19088030 \nu^{5} - 21981813 \nu^{3} + 4790637 \nu ) / 3976425$$ (-11836*v^15 + 138574*v^13 - 1170340*v^11 + 4500371*v^9 - 12811274*v^7 + 19088030*v^5 - 21981813*v^3 + 4790637*v) / 3976425 $$\beta_{9}$$ $$=$$ $$( - 14896 \nu^{14} + 165889 \nu^{12} - 1314040 \nu^{10} + 4313456 \nu^{8} - 9114389 \nu^{6} + 8453480 \nu^{4} - 1848168 \nu^{2} - 5535918 ) / 3976425$$ (-14896*v^14 + 165889*v^12 - 1314040*v^10 + 4313456*v^8 - 9114389*v^6 + 8453480*v^4 - 1848168*v^2 - 5535918) / 3976425 $$\beta_{10}$$ $$=$$ $$( 13477 \nu^{15} - 168383 \nu^{13} + 1451855 \nu^{11} - 6027522 \nu^{9} + 17277208 \nu^{7} - 24805735 \nu^{5} + 20660066 \nu^{3} + 11503071 \nu ) / 3976425$$ (13477*v^15 - 168383*v^13 + 1451855*v^11 - 6027522*v^9 + 17277208*v^7 - 24805735*v^5 + 20660066*v^3 + 11503071*v) / 3976425 $$\beta_{11}$$ $$=$$ $$( - 3323 \nu^{14} + 38066 \nu^{12} - 324695 \nu^{10} + 1252588 \nu^{8} - 3846136 \nu^{6} + 6436600 \nu^{4} - 8169474 \nu^{2} + 1468098 ) / 795285$$ (-3323*v^14 + 38066*v^12 - 324695*v^10 + 1252588*v^8 - 3846136*v^6 + 6436600*v^4 - 8169474*v^2 + 1468098) / 795285 $$\beta_{12}$$ $$=$$ $$( 19253 \nu^{15} - 245932 \nu^{13} + 2129695 \nu^{11} - 9130758 \nu^{9} + 26781707 \nu^{7} - 44578415 \nu^{5} + 47402299 \nu^{3} - 10338741 \nu ) / 3976425$$ (19253*v^15 - 245932*v^13 + 2129695*v^11 - 9130758*v^9 + 26781707*v^7 - 44578415*v^5 + 47402299*v^3 - 10338741*v) / 3976425 $$\beta_{13}$$ $$=$$ $$( 572 \nu^{14} - 6758 \nu^{12} + 56330 \nu^{10} - 215542 \nu^{8} + 579583 \nu^{6} - 869185 \nu^{4} + 809301 \nu^{2} - 212679 ) / 92475$$ (572*v^14 - 6758*v^12 + 56330*v^10 - 215542*v^8 + 579583*v^6 - 869185*v^4 + 809301*v^2 - 212679) / 92475 $$\beta_{14}$$ $$=$$ $$( 43568 \nu^{15} - 518857 \nu^{13} + 4432420 \nu^{11} - 17623398 \nu^{9} + 51780332 \nu^{7} - 81131165 \nu^{5} + 89205019 \nu^{3} - 6887691 \nu ) / 3976425$$ (43568*v^15 - 518857*v^13 + 4432420*v^11 - 17623398*v^9 + 51780332*v^7 - 81131165*v^5 + 89205019*v^3 - 6887691*v) / 3976425 $$\beta_{15}$$ $$=$$ $$( - 49089 \nu^{15} + 557311 \nu^{13} - 4656460 \nu^{11} + 17097154 \nu^{9} - 47712236 \nu^{7} + 67106795 \nu^{5} - 70902062 \nu^{3} - 564507 \nu ) / 3976425$$ (-49089*v^15 + 557311*v^13 - 4656460*v^11 + 17097154*v^9 - 47712236*v^7 + 67106795*v^5 - 70902062*v^3 - 564507*v) / 3976425
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{11} + \beta_{9} + 3\beta_{6} + \beta_{5} - \beta_{3} + 1$$ b11 + b9 + 3*b6 + b5 - b3 + 1 $$\nu^{3}$$ $$=$$ $$-\beta_{15} - \beta_{14} - 3\beta_{12} + 2\beta_{10} - 4\beta_{8} - 3\beta_{7} + 2\beta_{4} + 4\beta_1$$ -b15 - b14 - 3*b12 + 2*b10 - 4*b8 - 3*b7 + 2*b4 + 4*b1 $$\nu^{4}$$ $$=$$ $$-6\beta_{13} - 2\beta_{11} + 15\beta_{6} + 8\beta_{5} - 7\beta_{3} - 8\beta_{2} - 15$$ -6*b13 - 2*b11 + 15*b6 + 8*b5 - 7*b3 - 8*b2 - 15 $$\nu^{5}$$ $$=$$ $$8\beta_{15} + 11\beta_{14} - 27\beta_{12} + 8\beta_{10} - 22\beta_{8} + 11\beta_{4}$$ 8*b15 + 11*b14 - 27*b12 + 8*b10 - 22*b8 + 11*b4 $$\nu^{6}$$ $$=$$ $$-57\beta_{13} - 57\beta_{11} - 49\beta_{9} + 19\beta_{5} - 38\beta_{2} - 139$$ -57*b13 - 57*b11 - 49*b9 + 19*b5 - 38*b2 - 139 $$\nu^{7}$$ $$=$$ $$144\beta_{15} + 144\beta_{14} - 87\beta_{10} + 204\beta_{7} - 57\beta_{4} - 139\beta_1$$ 144*b15 + 144*b14 - 87*b10 + 204*b7 - 57*b4 - 139*b1 $$\nu^{8}$$ $$=$$ $$-144\beta_{13} - 253\beta_{11} - 343\beta_{9} - 588\beta_{6} - 253\beta_{5} + 343\beta_{3} + 144\beta_{2} - 343$$ -144*b13 - 253*b11 - 343*b9 - 588*b6 - 253*b5 + 343*b3 + 144*b2 - 343 $$\nu^{9}$$ $$=$$ $$631 \beta_{15} + 397 \beta_{14} + 1461 \beta_{12} - 1028 \beta_{10} + 931 \beta_{8} + 1461 \beta_{7} - 1028 \beta_{4} - 931 \beta_1$$ 631*b15 + 397*b14 + 1461*b12 - 1028*b10 + 931*b8 + 1461*b7 - 1028*b4 - 931*b1 $$\nu^{10}$$ $$=$$ $$1725\beta_{13} + 1028\beta_{11} - 3984\beta_{6} - 2753\beta_{5} + 2392\beta_{3} + 2753\beta_{2} + 3984$$ 1725*b13 + 1028*b11 - 3984*b6 - 2753*b5 + 2392*b3 + 2753*b2 + 3984 $$\nu^{11}$$ $$=$$ $$-2753\beta_{15} - 4448\beta_{14} + 10260\beta_{12} - 2753\beta_{10} + 6376\beta_{8} - 4448\beta_{4}$$ -2753*b15 - 4448*b14 + 10260*b12 - 2753*b10 + 6376*b8 - 4448*b4 $$\nu^{12}$$ $$=$$ $$19083\beta_{13} + 19083\beta_{11} + 16636\beta_{9} - 7201\beta_{5} + 11882\beta_{2} + 44023$$ 19083*b13 + 19083*b11 + 16636*b9 - 7201*b5 + 11882*b2 + 44023 $$\nu^{13}$$ $$=$$ $$-50121\beta_{15} - 50121\beta_{14} + 31038\beta_{10} - 71511\beta_{7} + 19083\beta_{4} + 44023\beta_1$$ -50121*b15 - 50121*b14 + 31038*b10 - 71511*b7 + 19083*b4 + 44023*b1 $$\nu^{14}$$ $$=$$ $$50121 \beta_{13} + 82189 \beta_{11} + 115534 \beta_{9} + 189318 \beta_{6} + 82189 \beta_{5} - 115534 \beta_{3} - 50121 \beta_{2} + 115534$$ 50121*b13 + 82189*b11 + 115534*b9 + 189318*b6 + 82189*b5 - 115534*b3 - 50121*b2 + 115534 $$\nu^{15}$$ $$=$$ $$- 215776 \beta_{15} - 132310 \beta_{14} - 496965 \beta_{12} + 348086 \beta_{10} - 304852 \beta_{8} - 496965 \beta_{7} + 348086 \beta_{4} + 304852 \beta_1$$ -215776*b15 - 132310*b14 - 496965*b12 + 348086*b10 - 304852*b8 - 496965*b7 + 348086*b4 + 304852*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-\beta_{6}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −2.28087 + 1.31686i −1.41485 + 0.816862i −1.27588 + 0.736627i −0.409850 + 0.236627i 0.409850 − 0.236627i 1.27588 − 0.736627i 1.41485 − 0.816862i 2.28087 − 1.31686i −2.28087 − 1.31686i −1.41485 − 0.816862i −1.27588 − 0.736627i −0.409850 − 0.236627i 0.409850 + 0.236627i 1.27588 + 0.736627i 1.41485 + 0.816862i 2.28087 + 1.31686i
−2.28087 + 1.31686i −0.238330 1.71558i 2.46825 4.27513i 0 2.80278 + 3.59916i 1.55662 0.898714i 7.73393i −2.88640 + 0.817746i 0
49.2 −1.41485 + 0.816862i −1.36657 1.06419i 0.334526 0.579416i 0 2.80278 + 0.389365i −0.437645 + 0.252674i 2.17440i 0.735010 + 2.90857i 0
49.3 −1.27588 + 0.736627i −0.350156 + 1.69629i 0.0852394 0.147639i 0 −0.802776 2.42219i −3.34791 + 1.93291i 2.69535i −2.75478 1.18793i 0
49.4 −0.409850 + 0.236627i 1.64411 + 0.544899i −0.888015 + 1.53809i 0 −0.802776 + 0.165713i −2.21967 + 1.28153i 1.78702i 2.40617 + 1.79175i 0
49.5 0.409850 0.236627i −1.64411 0.544899i −0.888015 + 1.53809i 0 −0.802776 + 0.165713i 2.21967 1.28153i 1.78702i 2.40617 + 1.79175i 0
49.6 1.27588 0.736627i 0.350156 1.69629i 0.0852394 0.147639i 0 −0.802776 2.42219i 3.34791 1.93291i 2.69535i −2.75478 1.18793i 0
49.7 1.41485 0.816862i 1.36657 + 1.06419i 0.334526 0.579416i 0 2.80278 + 0.389365i 0.437645 0.252674i 2.17440i 0.735010 + 2.90857i 0
49.8 2.28087 1.31686i 0.238330 + 1.71558i 2.46825 4.27513i 0 2.80278 + 3.59916i −1.55662 + 0.898714i 7.73393i −2.88640 + 0.817746i 0
124.1 −2.28087 1.31686i −0.238330 + 1.71558i 2.46825 + 4.27513i 0 2.80278 3.59916i 1.55662 + 0.898714i 7.73393i −2.88640 0.817746i 0
124.2 −1.41485 0.816862i −1.36657 + 1.06419i 0.334526 + 0.579416i 0 2.80278 0.389365i −0.437645 0.252674i 2.17440i 0.735010 2.90857i 0
124.3 −1.27588 0.736627i −0.350156 1.69629i 0.0852394 + 0.147639i 0 −0.802776 + 2.42219i −3.34791 1.93291i 2.69535i −2.75478 + 1.18793i 0
124.4 −0.409850 0.236627i 1.64411 0.544899i −0.888015 1.53809i 0 −0.802776 0.165713i −2.21967 1.28153i 1.78702i 2.40617 1.79175i 0
124.5 0.409850 + 0.236627i −1.64411 + 0.544899i −0.888015 1.53809i 0 −0.802776 0.165713i 2.21967 + 1.28153i 1.78702i 2.40617 1.79175i 0
124.6 1.27588 + 0.736627i 0.350156 + 1.69629i 0.0852394 + 0.147639i 0 −0.802776 + 2.42219i 3.34791 + 1.93291i 2.69535i −2.75478 + 1.18793i 0
124.7 1.41485 + 0.816862i 1.36657 1.06419i 0.334526 + 0.579416i 0 2.80278 0.389365i 0.437645 + 0.252674i 2.17440i 0.735010 2.90857i 0
124.8 2.28087 + 1.31686i 0.238330 1.71558i 2.46825 + 4.27513i 0 2.80278 3.59916i −1.55662 0.898714i 7.73393i −2.88640 0.817746i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.k.c 16
3.b odd 2 1 675.2.k.c 16
5.b even 2 1 inner 225.2.k.c 16
5.c odd 4 1 225.2.e.c 8
5.c odd 4 1 225.2.e.e yes 8
9.c even 3 1 inner 225.2.k.c 16
9.c even 3 1 2025.2.b.n 8
9.d odd 6 1 675.2.k.c 16
9.d odd 6 1 2025.2.b.o 8
15.d odd 2 1 675.2.k.c 16
15.e even 4 1 675.2.e.c 8
15.e even 4 1 675.2.e.e 8
45.h odd 6 1 675.2.k.c 16
45.h odd 6 1 2025.2.b.o 8
45.j even 6 1 inner 225.2.k.c 16
45.j even 6 1 2025.2.b.n 8
45.k odd 12 1 225.2.e.c 8
45.k odd 12 1 225.2.e.e yes 8
45.k odd 12 1 2025.2.a.q 4
45.k odd 12 1 2025.2.a.y 4
45.l even 12 1 675.2.e.c 8
45.l even 12 1 675.2.e.e 8
45.l even 12 1 2025.2.a.p 4
45.l even 12 1 2025.2.a.z 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.e.c 8 5.c odd 4 1
225.2.e.c 8 45.k odd 12 1
225.2.e.e yes 8 5.c odd 4 1
225.2.e.e yes 8 45.k odd 12 1
225.2.k.c 16 1.a even 1 1 trivial
225.2.k.c 16 5.b even 2 1 inner
225.2.k.c 16 9.c even 3 1 inner
225.2.k.c 16 45.j even 6 1 inner
675.2.e.c 8 15.e even 4 1
675.2.e.c 8 45.l even 12 1
675.2.e.e 8 15.e even 4 1
675.2.e.e 8 45.l even 12 1
675.2.k.c 16 3.b odd 2 1
675.2.k.c 16 9.d odd 6 1
675.2.k.c 16 15.d odd 2 1
675.2.k.c 16 45.h odd 6 1
2025.2.a.p 4 45.l even 12 1
2025.2.a.q 4 45.k odd 12 1
2025.2.a.y 4 45.k odd 12 1
2025.2.a.z 4 45.l even 12 1
2025.2.b.n 8 9.c even 3 1
2025.2.b.n 8 45.j even 6 1
2025.2.b.o 8 9.d odd 6 1
2025.2.b.o 8 45.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} - 12T_{2}^{14} + 102T_{2}^{12} - 406T_{2}^{10} + 1167T_{2}^{8} - 1842T_{2}^{6} + 2023T_{2}^{4} - 441T_{2}^{2} + 81$$ acting on $$S_{2}^{\mathrm{new}}(225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - 12 T^{14} + 102 T^{12} + \cdots + 81$$
$3$ $$T^{16} + 5 T^{14} + 4 T^{12} + 15 T^{10} + \cdots + 6561$$
$5$ $$T^{16}$$
$7$ $$T^{16} - 25 T^{14} + 451 T^{12} + \cdots + 6561$$
$11$ $$(T^{8} - T^{7} + 26 T^{6} + 107 T^{5} + \cdots + 81)^{2}$$
$13$ $$T^{16} - 64 T^{14} + \cdots + 131079601$$
$17$ $$(T^{8} + 81 T^{6} + 2214 T^{4} + \cdots + 91809)^{2}$$
$19$ $$(T^{4} + 2 T^{3} - 27 T^{2} - 80 T - 25)^{4}$$
$23$ $$T^{16} - 111 T^{14} + \cdots + 3486784401$$
$29$ $$(T^{8} - T^{7} + 41 T^{6} - 244 T^{5} + \cdots + 16641)^{2}$$
$31$ $$(T^{8} - 4 T^{7} + 58 T^{6} + 114 T^{5} + \cdots + 59049)^{2}$$
$37$ $$(T^{8} + 199 T^{6} + 9513 T^{4} + \cdots + 418609)^{2}$$
$41$ $$(T^{8} - 5 T^{7} + 50 T^{6} - 197 T^{5} + \cdots + 42849)^{2}$$
$43$ $$T^{16} - 196 T^{14} + \cdots + 205144679041$$
$47$ $$T^{16} - 186 T^{14} + \cdots + 21071715921$$
$53$ $$(T^{8} + 228 T^{6} + 13614 T^{4} + \cdots + 221841)^{2}$$
$59$ $$(T^{8} - 17 T^{7} + 287 T^{6} + \cdots + 5349969)^{2}$$
$61$ $$(T^{8} - 13 T^{7} + 172 T^{6} - 143 T^{5} + \cdots + 1)^{2}$$
$67$ $$T^{16} - 217 T^{14} + \cdots + 3486784401$$
$71$ $$(T^{4} + 8 T^{3} - 40 T^{2} - 263 T + 381)^{4}$$
$73$ $$(T^{8} + 196 T^{6} + 8478 T^{4} + \cdots + 12769)^{2}$$
$79$ $$(T^{8} + 7 T^{7} + 82 T^{6} - 93 T^{5} + \cdots + 42849)^{2}$$
$83$ $$T^{16} - 324 T^{14} + \cdots + 282429536481$$
$89$ $$(T^{4} - 9 T^{3} - 99 T^{2} + 405 T + 2025)^{4}$$
$97$ $$T^{16} - 199 T^{14} + \cdots + 824843587681$$