# Properties

 Label 375.2.g.d Level $375$ Weight $2$ Character orbit 375.g Analytic conductor $2.994$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [375,2,Mod(76,375)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(375, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("375.76");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$375 = 3 \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 375.g (of order $$5$$, degree $$4$$, 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: $$2.99439007580$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} + 20x^{14} + 156x^{12} + 610x^{10} + 1286x^{8} + 1440x^{6} + 761x^{4} + 130x^{2} + 1$$ x^16 + 20*x^14 + 156*x^12 + 610*x^10 + 1286*x^8 + 1440*x^6 + 761*x^4 + 130*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$5^{2}$$ Twist minimal: no (minimal twist has level 75) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} + \beta_{3} q^{3} + ( - \beta_{11} + \beta_{8}) q^{4} + ( - \beta_{13} - \beta_{11} + \beta_{5} - \beta_1 + 1) q^{6} + ( - \beta_{15} - \beta_{12} - \beta_{9} - \beta_{6} + \beta_{5} - \beta_1 + 1) q^{7} + ( - \beta_{15} + \beta_{14} - \beta_{12} - \beta_{8} - \beta_{7} - \beta_{4} + \beta_{3} - 2 \beta_{2}) q^{8} - \beta_{10} q^{9}+O(q^{10})$$ q + b5 * q^2 + b3 * q^3 + (-b11 + b8) * q^4 + (-b13 - b11 + b5 - b1 + 1) * q^6 + (-b15 - b12 - b9 - b6 + b5 - b1 + 1) * q^7 + (-b15 + b14 - b12 - b8 - b7 - b4 + b3 - 2*b2) * q^8 - b10 * q^9 $$q + \beta_{5} q^{2} + \beta_{3} q^{3} + ( - \beta_{11} + \beta_{8}) q^{4} + ( - \beta_{13} - \beta_{11} + \beta_{5} - \beta_1 + 1) q^{6} + ( - \beta_{15} - \beta_{12} - \beta_{9} - \beta_{6} + \beta_{5} - \beta_1 + 1) q^{7} + ( - \beta_{15} + \beta_{14} - \beta_{12} - \beta_{8} - \beta_{7} - \beta_{4} + \beta_{3} - 2 \beta_{2}) q^{8} - \beta_{10} q^{9} + (\beta_{14} + \beta_{9} + 2 \beta_{8} - \beta_{6} - \beta_{4} + \beta_{3} + 2 \beta_{2}) q^{11} + ( - \beta_{6} + \beta_1) q^{12} + ( - \beta_{15} - 2 \beta_{11} - 3 \beta_{10} - \beta_{7} + 2 \beta_{5} - \beta_1 + 1) q^{13} + ( - \beta_{13} + \beta_{12} - \beta_{11} + \beta_{9} + 2 \beta_{5} - \beta_{3} + \beta_{2} + 2) q^{14} + (2 \beta_{15} + 2 \beta_{10} + \beta_{7} + \beta_{6} - \beta_1 - 1) q^{16} + (\beta_{15} - \beta_{14} - \beta_{11} + \beta_{10} - \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{2} - \beta_1) q^{17} + ( - \beta_{13} + 1) q^{18} + ( - \beta_{13} + 2 \beta_{12} - 2 \beta_{11} - \beta_{10} - \beta_{6} + \beta_{5} + 2 \beta_{4} + \cdots + 1) q^{19}+ \cdots + ( - \beta_{15} - \beta_{12} - \beta_{9} - \beta_{6} + \beta_{2}) q^{99}+O(q^{100})$$ q + b5 * q^2 + b3 * q^3 + (-b11 + b8) * q^4 + (-b13 - b11 + b5 - b1 + 1) * q^6 + (-b15 - b12 - b9 - b6 + b5 - b1 + 1) * q^7 + (-b15 + b14 - b12 - b8 - b7 - b4 + b3 - 2*b2) * q^8 - b10 * q^9 + (b14 + b9 + 2*b8 - b6 - b4 + b3 + 2*b2) * q^11 + (-b6 + b1) * q^12 + (-b15 - 2*b11 - 3*b10 - b7 + 2*b5 - b1 + 1) * q^13 + (-b13 + b12 - b11 + b9 + 2*b5 - b3 + b2 + 2) * q^14 + (2*b15 + 2*b10 + b7 + b6 - b1 - 1) * q^16 + (b15 - b14 - b11 + b10 - b8 - 2*b7 + 2*b6 - 2*b2 - b1) * q^17 + (-b13 + 1) * q^18 + (-b13 + 2*b12 - 2*b11 - b10 - b6 + b5 + 2*b4 - b3 + b2 - 2*b1 + 1) * q^19 + (-b15 + b14 - b13 - b11 - b4 + 2*b3 - b1 + 1) * q^21 + (-b15 - b14 + b13 - b11 - 2*b10 + b8 - b4 - 2*b3 + b1 + 1) * q^22 + (2*b14 + b13 - 2*b12 + b11 + 4*b10 + 4*b7 + 2*b6 - 2*b5 - 2*b4 + 4*b3 - 2*b2 - 3) * q^23 + (2*b9 + 2*b6 - b4 - 1) * q^24 + (-2*b13 - 3*b7 - b5 - 3*b3 - 2*b2 + b1 + 3) * q^26 + (-b10 - b7 - b3 + 1) * q^27 + (-2*b15 + b14 - 3*b11 + b10 + b9 + 2*b8 - 2*b4 + b3 - 1) * q^28 + (b14 + b13 + b10 - 2*b9 + 2*b8 + b3 + b1 - 2) * q^29 + (b13 + 2*b11 - b10 + 3*b7 + b6 - b5 - b3 - b2 + 2*b1 - 1) * q^31 + (-2*b7 - b5 - b4 - 2*b3 + b1 + 1) * q^32 + (b15 - b14 - b6 - b3 + 2*b2) * q^33 + (2*b14 - 2*b12 + 2*b11 + 3*b10 - 4*b9 - 2*b8 - b7 - 2*b6 - 2*b5 + 2*b3 - 4*b2 - 2*b1 + 1) * q^34 + (b9 + b5 + b2) * q^36 + (-3*b15 + b14 - b12 + b11 - 3*b10 + b8 + 2*b7 - 2*b6 - b5 + b3 - b1 - 2) * q^37 + (-b14 + b12 + b11 - b10 + 2*b9 + b8 - 3*b7 - 2*b6 - b5 - b3 + 2*b2 + 3*b1 + 3) * q^38 + (-2*b13 + b12 - 2*b11 - 3*b10 - 3*b7 + b5 - 2*b3 + 4) * q^39 + (-3*b15 + b14 - b12 - b11 - b10 - b9 + b7 - b6 + b5 + b3 - b2 - 2*b1 - 1) * q^41 + (b15 - b14 - 2*b13 + b12 - b11 + b10 + 2*b5 + b4 + b2 - b1 + 2) * q^42 + (2*b15 - 2*b13 + 2*b12 - b9 - b6 + 4*b4 + 2*b2 + 2) * q^43 + (3*b13 - 2*b12 + 4*b11 - b10 - 2*b8 - b7 - 3*b5 - 2*b4 - b3 - 2*b2 + 4*b1 - 3) * q^44 + (2*b15 - b13 - b11 - 5*b9 - b8 + 2*b4 - b1 + 1) * q^46 + (3*b15 - 2*b13 + b10 - 2*b8 + 3*b4 - 2*b3 - 2*b1 + 1) * q^47 + (-2*b12 + 2*b10 - b9 + 2*b7 - b5 + b3 - b2 - 1) * q^48 + (-b13 + b9 - 2*b7 + b6 + 2*b5 + 2*b4 - 2*b3 + 2*b2 - 2*b1 + 1) * q^49 + (-b15 - b12 + b7 - b5 + b3 - 2*b2 + b1 - 3) * q^51 + (2*b14 + 2*b13 + 2*b11 + 3*b9 - b8 + 3*b6 - 2*b4 + 4*b3 - 4) * q^52 + (b15 - 3*b14 + b13 + b11 - b10 - 2*b9 - b8 + b4 - 8*b3 + b1) * q^53 + b11 * q^54 + (b15 - b14 + 2*b13 + 2*b11 + 2*b10 - 2*b8 - 2*b7 + 3*b6 - 2*b5 + b3 - 4*b2 + 2*b1 - 2) * q^56 + (2*b15 - b13 + 2*b12 - b7 - b5 + 2*b4 - b3 + b2 + b1 + 2) * q^57 + (-b15 + b14 + b13 + b12 + b11 - 5*b10 + b8 - 3*b7 - b6 - b5 + b4 - 4*b3 + b2 + b1 - 1) * q^58 + (-2*b14 + 2*b12 - b11 - 3*b10 + 2*b9 - b7 - 2*b6 + b5 - 2*b3 + 2*b2 + 1) * q^59 + (-2*b14 + 4*b13 - 2*b12 + 4*b11 - b10 - 2*b9 + 2*b8 - b7 - 4*b6 - 2*b5 + 2*b4 - 3*b3 + 2*b2 - 3) * q^61 + (b14 - b12 + b11 - b10 + b8 + 3*b7 + 2*b6 - b5 + b3 + 2*b1 - 3) * q^62 + (b15 - b14 + b12 + b11 - b10 - b8 + b6 - b5 - b3 + b1) * q^63 + (3*b13 - 4*b12 + 3*b11 + 4*b10 - 2*b9 - b8 + 4*b7 + b6 - 3*b5 + 3*b3 - 3*b2 - 6) * q^64 + (-b15 - b14 + b12 - b11 - b10 - b8 - 2*b7 - 2*b6 + b5 - b3 + b1 + 2) * q^66 + (-2*b15 + 2*b14 + 4*b13 - 2*b12 + 4*b10 + 3*b8 + 2*b7 - 4*b5 - 2*b4 + 6*b3 + b2 - 4) * q^67 + (2*b15 - 3*b13 + 2*b12 + 2*b7 + 5*b5 + 2*b4 + 2*b3 + 4*b2 - 5*b1 + 1) * q^68 + (2*b13 - 2*b12 + b11 + 2*b10 + 4*b7 + 2*b6 - 2*b5 - 2*b4 + 2*b3 - 2*b2 + b1 - 2) * q^69 + (-b15 - b14 + b13 - b11 - b10 - b9 - 4*b8 - b4 - 2*b3 + b1) * q^71 + (-b14 + b8 - 2*b3) * q^72 + (4*b13 + 4*b11 + 4*b10 - 2*b9 - b8 + 4*b7 + b6 - 3*b2 - 4) * q^73 + (-3*b13 + 4*b9 + 5*b7 + 4*b6 - b5 - 2*b4 + 5*b3 + 2*b2 + b1 + 3) * q^74 + (2*b15 + 3*b13 + 2*b12 + 5*b9 + 2*b7 + 5*b6 + 2*b3 - 4) * q^76 + (-b13 + 4*b12 - b11 + 4*b10 + 2*b8 + 4*b7 - 2*b6 + 4*b3 + 2*b2 - 3) * q^77 + (b13 + 3*b11 + 3*b10 + 2*b9 + b3 + b1 - 4) * q^78 + (-2*b15 - 2*b14 + 3*b13 + b11 - 3*b10 - b9 + b8 - 2*b4 + 2*b3 + 3*b1) * q^79 - b7 * q^81 + (b15 - 2*b13 + b12 + 2*b9 + 2*b6 - b5 + b2 + b1 + 3) * q^82 + (3*b15 - 3*b14 + b13 + 2*b12 + b10 - b8 - 2*b7 + 4*b6 - b5 + 2*b4 - 2*b3 - 2*b2 - 1) * q^83 + (b15 - 2*b14 + 2*b12 + b10 + b9 - b8 + b7 - b6 - 2*b3 + b2 + 3*b1 - 1) * q^84 + (-2*b14 - 2*b13 + 3*b12 - 2*b11 - b10 + 5*b9 + 2*b8 - b7 - 4*b6 + 5*b5 + 2*b4 + b3 + 9*b2 - 1) * q^86 + (b15 - b11 + b10 - 2*b9 - 2*b8 + b7 - b6 + b5 - 2*b2 - 1) * q^87 + (-2*b14 + 2*b12 - 2*b11 - 4*b10 - b9 - 3*b8 - 2*b7 + 2*b6 + 2*b5 - 2*b3 - b2 - 4*b1 + 2) * q^88 + (-b14 + b13 - b12 + b11 + 2*b10 + 2*b9 - 2*b8 + 2*b7 + b6 + b4 + 2*b3 + b2 - 4) * q^89 + (-2*b14 + 2*b12 - 2*b11 - 2*b10 + 2*b9 - 4*b7 + 3*b6 + 2*b5 - 2*b3 + 2*b2 + b1 + 4) * q^91 + (b15 - b14 - 2*b13 + 2*b12 + 2*b11 + b10 - b8 - 4*b6 + 2*b5 + 2*b4 + 4*b2 + 2*b1 + 2) * q^92 + (b13 - b7 + b5 - b3 - b2 - b1 + 3) * q^93 + (2*b15 - 2*b14 - b13 + 2*b12 + 4*b10 - b7 - 5*b6 + b5 + 2*b4 + 2*b3 + 7*b2 + 1) * q^94 + (-b14 + b13 + b11 + 2*b10 - b8 + b1 - 3) * q^96 + (2*b14 + 2*b11 - 3*b9 - b8 - 2*b3) * q^97 + (-2*b14 - 2*b13 + b12 - 2*b11 + b10 + 2*b8 + b7 - 4*b6 + 2*b4 - b3 + 4*b2 + 1) * q^98 + (-b15 - b12 - b9 - b6 + b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 2 q^{2} + 4 q^{3} - 2 q^{4} + 2 q^{6} + 16 q^{7} - 6 q^{8} - 4 q^{9}+O(q^{10})$$ 16 * q - 2 * q^2 + 4 * q^3 - 2 * q^4 + 2 * q^6 + 16 * q^7 - 6 * q^8 - 4 * q^9 $$16 q - 2 q^{2} + 4 q^{3} - 2 q^{4} + 2 q^{6} + 16 q^{7} - 6 q^{8} - 4 q^{9} - 6 q^{11} + 2 q^{12} - 8 q^{13} + 12 q^{14} - 10 q^{16} - 8 q^{17} + 8 q^{18} + 2 q^{19} + 4 q^{21} + 4 q^{22} - 2 q^{23} - 24 q^{24} + 12 q^{26} + 4 q^{27} - 28 q^{28} - 16 q^{29} + 6 q^{31} - 4 q^{32} - 4 q^{33} + 36 q^{34} - 2 q^{36} - 24 q^{37} + 38 q^{38} + 8 q^{39} - 14 q^{41} + 18 q^{42} + 40 q^{43} - 26 q^{44} + 16 q^{46} + 10 q^{47} + 10 q^{48} - 32 q^{51} - 48 q^{52} - 12 q^{53} + 2 q^{54} + 28 q^{57} - 44 q^{58} - 12 q^{59} - 28 q^{62} - 4 q^{63} - 8 q^{64} + 16 q^{66} + 12 q^{67} - 4 q^{68} + 12 q^{69} - 8 q^{71} - 6 q^{72} + 8 q^{73} + 52 q^{74} - 32 q^{76} - 18 q^{77} - 32 q^{78} + 20 q^{79} - 4 q^{81} + 32 q^{82} - 6 q^{83} - 12 q^{84} - 36 q^{86} - 14 q^{87} - 16 q^{88} - 18 q^{89} + 26 q^{91} + 36 q^{92} + 44 q^{93} + 38 q^{94} - 26 q^{96} - 8 q^{97} + 18 q^{98} + 4 q^{99}+O(q^{100})$$ 16 * q - 2 * q^2 + 4 * q^3 - 2 * q^4 + 2 * q^6 + 16 * q^7 - 6 * q^8 - 4 * q^9 - 6 * q^11 + 2 * q^12 - 8 * q^13 + 12 * q^14 - 10 * q^16 - 8 * q^17 + 8 * q^18 + 2 * q^19 + 4 * q^21 + 4 * q^22 - 2 * q^23 - 24 * q^24 + 12 * q^26 + 4 * q^27 - 28 * q^28 - 16 * q^29 + 6 * q^31 - 4 * q^32 - 4 * q^33 + 36 * q^34 - 2 * q^36 - 24 * q^37 + 38 * q^38 + 8 * q^39 - 14 * q^41 + 18 * q^42 + 40 * q^43 - 26 * q^44 + 16 * q^46 + 10 * q^47 + 10 * q^48 - 32 * q^51 - 48 * q^52 - 12 * q^53 + 2 * q^54 + 28 * q^57 - 44 * q^58 - 12 * q^59 - 28 * q^62 - 4 * q^63 - 8 * q^64 + 16 * q^66 + 12 * q^67 - 4 * q^68 + 12 * q^69 - 8 * q^71 - 6 * q^72 + 8 * q^73 + 52 * q^74 - 32 * q^76 - 18 * q^77 - 32 * q^78 + 20 * q^79 - 4 * q^81 + 32 * q^82 - 6 * q^83 - 12 * q^84 - 36 * q^86 - 14 * q^87 - 16 * q^88 - 18 * q^89 + 26 * q^91 + 36 * q^92 + 44 * q^93 + 38 * q^94 - 26 * q^96 - 8 * q^97 + 18 * q^98 + 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 20x^{14} + 156x^{12} + 610x^{10} + 1286x^{8} + 1440x^{6} + 761x^{4} + 130x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( - 11 \nu^{15} + 22 \nu^{14} - 194 \nu^{13} + 388 \nu^{12} - 1257 \nu^{11} + 2515 \nu^{10} - 3731 \nu^{9} + 7477 \nu^{8} - 5277 \nu^{7} + 10628 \nu^{6} - 3223 \nu^{5} + 6579 \nu^{4} - 565 \nu^{3} + \cdots + 9 ) / 4$$ (-11*v^15 + 22*v^14 - 194*v^13 + 388*v^12 - 1257*v^11 + 2515*v^10 - 3731*v^9 + 7477*v^8 - 5277*v^7 + 10628*v^6 - 3223*v^5 + 6579*v^4 - 565*v^3 + 1204*v^2 - 2*v + 9) / 4 $$\beta_{2}$$ $$=$$ $$( -11\nu^{14} - 194\nu^{12} - 1257\nu^{10} - 3731\nu^{8} - 5277\nu^{6} - 3223\nu^{4} - 565\nu^{2} - 2 ) / 2$$ (-11*v^14 - 194*v^12 - 1257*v^10 - 3731*v^8 - 5277*v^6 - 3223*v^4 - 565*v^2 - 2) / 2 $$\beta_{3}$$ $$=$$ $$( - 7 \nu^{15} + 32 \nu^{14} - 124 \nu^{13} + 564 \nu^{12} - 810 \nu^{11} + 3652 \nu^{10} - 2444 \nu^{9} + 10838 \nu^{8} - 3583 \nu^{7} + 15360 \nu^{6} - 2400 \nu^{5} + 9462 \nu^{4} - 595 \nu^{3} + \cdots + 12 ) / 4$$ (-7*v^15 + 32*v^14 - 124*v^13 + 564*v^12 - 810*v^11 + 3652*v^10 - 2444*v^9 + 10838*v^8 - 3583*v^7 + 15360*v^6 - 2400*v^5 + 9462*v^4 - 595*v^3 + 1716*v^2 - 47*v + 12) / 4 $$\beta_{4}$$ $$=$$ $$( 16\nu^{14} + 282\nu^{12} + 1826\nu^{10} + 5419\nu^{8} + 7680\nu^{6} + 4732\nu^{4} + 865\nu^{2} + 13 ) / 2$$ (16*v^14 + 282*v^12 + 1826*v^10 + 5419*v^8 + 7680*v^6 + 4732*v^4 + 865*v^2 + 13) / 2 $$\beta_{5}$$ $$=$$ $$( - 11 \nu^{15} - 22 \nu^{14} - 194 \nu^{13} - 388 \nu^{12} - 1257 \nu^{11} - 2515 \nu^{10} - 3731 \nu^{9} - 7477 \nu^{8} - 5277 \nu^{7} - 10628 \nu^{6} - 3223 \nu^{5} - 6579 \nu^{4} - 565 \nu^{3} + \cdots - 9 ) / 4$$ (-11*v^15 - 22*v^14 - 194*v^13 - 388*v^12 - 1257*v^11 - 2515*v^10 - 3731*v^9 - 7477*v^8 - 5277*v^7 - 10628*v^6 - 3223*v^5 - 6579*v^4 - 565*v^3 - 1204*v^2 - 2*v - 9) / 4 $$\beta_{6}$$ $$=$$ $$( - \nu^{15} + 34 \nu^{14} - 19 \nu^{13} + 600 \nu^{12} - 138 \nu^{11} + 3893 \nu^{10} - 490 \nu^{9} + 11593 \nu^{8} - 916 \nu^{7} + 16526 \nu^{6} - 893 \nu^{5} + 10291 \nu^{4} - 405 \nu^{3} + 1922 \nu^{2} + \cdots + 21 ) / 4$$ (-v^15 + 34*v^14 - 19*v^13 + 600*v^12 - 138*v^11 + 3893*v^10 - 490*v^9 + 11593*v^8 - 916*v^7 + 16526*v^6 - 893*v^5 + 10291*v^4 - 405*v^3 + 1922*v^2 - 53*v + 21) / 4 $$\beta_{7}$$ $$=$$ $$( 7 \nu^{15} + 32 \nu^{14} + 124 \nu^{13} + 564 \nu^{12} + 810 \nu^{11} + 3652 \nu^{10} + 2444 \nu^{9} + 10838 \nu^{8} + 3583 \nu^{7} + 15360 \nu^{6} + 2400 \nu^{5} + 9462 \nu^{4} + 595 \nu^{3} + \cdots + 12 ) / 4$$ (7*v^15 + 32*v^14 + 124*v^13 + 564*v^12 + 810*v^11 + 3652*v^10 + 2444*v^9 + 10838*v^8 + 3583*v^7 + 15360*v^6 + 2400*v^5 + 9462*v^4 + 595*v^3 + 1716*v^2 + 47*v + 12) / 4 $$\beta_{8}$$ $$=$$ $$( 47 \nu^{15} + 11 \nu^{14} + 828 \nu^{13} + 194 \nu^{12} + 5356 \nu^{11} + 1257 \nu^{10} + 15855 \nu^{9} + 3731 \nu^{8} + 22317 \nu^{7} + 5277 \nu^{6} + 13446 \nu^{5} + 3223 \nu^{4} + 2160 \nu^{3} + \cdots + 2 ) / 4$$ (47*v^15 + 11*v^14 + 828*v^13 + 194*v^12 + 5356*v^11 + 1257*v^10 + 15855*v^9 + 3731*v^8 + 22317*v^7 + 5277*v^6 + 13446*v^5 + 3223*v^4 + 2160*v^3 + 565*v^2 - 64*v + 2) / 4 $$\beta_{9}$$ $$=$$ $$( \nu^{15} + 56 \nu^{14} + 19 \nu^{13} + 988 \nu^{12} + 138 \nu^{11} + 6407 \nu^{10} + 490 \nu^{9} + 19055 \nu^{8} + 916 \nu^{7} + 27080 \nu^{6} + 893 \nu^{5} + 16737 \nu^{4} + 405 \nu^{3} + 3052 \nu^{2} + \cdots + 25 ) / 4$$ (v^15 + 56*v^14 + 19*v^13 + 988*v^12 + 138*v^11 + 6407*v^10 + 490*v^9 + 19055*v^8 + 916*v^7 + 27080*v^6 + 893*v^5 + 16737*v^4 + 405*v^3 + 3052*v^2 + 53*v + 25) / 4 $$\beta_{10}$$ $$=$$ $$( - 29 \nu^{15} - 32 \nu^{14} - 512 \nu^{13} - 564 \nu^{12} - 3325 \nu^{11} - 3652 \nu^{10} - 9921 \nu^{9} - 10838 \nu^{8} - 14211 \nu^{7} - 15360 \nu^{6} - 8977 \nu^{5} - 9462 \nu^{4} + \cdots - 10 ) / 4$$ (-29*v^15 - 32*v^14 - 512*v^13 - 564*v^12 - 3325*v^11 - 3652*v^10 - 9921*v^9 - 10838*v^8 - 14211*v^7 - 15360*v^6 - 8977*v^5 - 9462*v^4 - 1785*v^3 - 1716*v^2 - 40*v - 10) / 4 $$\beta_{11}$$ $$=$$ $$( 45 \nu^{15} + 23 \nu^{14} + 794 \nu^{13} + 406 \nu^{12} + 5150 \nu^{11} + 2635 \nu^{10} + 15324 \nu^{9} + 7847 \nu^{8} + 21803 \nu^{7} + 11175 \nu^{6} + 13514 \nu^{5} + 6935 \nu^{4} + 2487 \nu^{3} + \cdots + 10 ) / 4$$ (45*v^15 + 23*v^14 + 794*v^13 + 406*v^12 + 5150*v^11 + 2635*v^10 + 15324*v^9 + 7847*v^8 + 21803*v^7 + 11175*v^6 + 13514*v^5 + 6935*v^4 + 2487*v^3 + 1281*v^2 + 23*v + 10) / 4 $$\beta_{12}$$ $$=$$ $$( 45 \nu^{15} - 64 \nu^{14} + 791 \nu^{13} - 1129 \nu^{12} + 5098 \nu^{11} - 7321 \nu^{10} + 14996 \nu^{9} - 21780 \nu^{8} + 20870 \nu^{7} - 31001 \nu^{6} + 12277 \nu^{5} - 19266 \nu^{4} + \cdots - 38 ) / 4$$ (45*v^15 - 64*v^14 + 791*v^13 - 1129*v^12 + 5098*v^11 - 7321*v^10 + 14996*v^9 - 21780*v^8 + 20870*v^7 - 31001*v^6 + 12277*v^5 - 19266*v^4 + 1807*v^3 - 3595*v^2 - 69*v - 38) / 4 $$\beta_{13}$$ $$=$$ $$( -45\nu^{14} - 794\nu^{12} - 5150\nu^{10} - 15324\nu^{8} - 21803\nu^{6} - 13514\nu^{4} - 2485\nu^{2} - 17 ) / 2$$ (-45*v^14 - 794*v^12 - 5150*v^10 - 15324*v^8 - 21803*v^6 - 13514*v^4 - 2485*v^2 - 17) / 2 $$\beta_{14}$$ $$=$$ $$( - 13 \nu^{15} - 91 \nu^{14} - 230 \nu^{13} - 1605 \nu^{12} - 1498 \nu^{11} - 10404 \nu^{10} - 4486 \nu^{9} - 30930 \nu^{8} - 6443 \nu^{7} - 43958 \nu^{6} - 4052 \nu^{5} - 27219 \nu^{4} + \cdots - 39 ) / 4$$ (-13*v^15 - 91*v^14 - 230*v^13 - 1605*v^12 - 1498*v^11 - 10404*v^10 - 4486*v^9 - 30930*v^8 - 6443*v^7 - 43958*v^6 - 4052*v^5 - 27219*v^4 - 771*v^3 - 5011*v^2 - 17*v - 39) / 4 $$\beta_{15}$$ $$=$$ $$( - 45 \nu^{15} - 64 \nu^{14} - 791 \nu^{13} - 1129 \nu^{12} - 5098 \nu^{11} - 7321 \nu^{10} - 14996 \nu^{9} - 21780 \nu^{8} - 20870 \nu^{7} - 31001 \nu^{6} - 12277 \nu^{5} - 19266 \nu^{4} + \cdots - 38 ) / 4$$ (-45*v^15 - 64*v^14 - 791*v^13 - 1129*v^12 - 5098*v^11 - 7321*v^10 - 14996*v^9 - 21780*v^8 - 20870*v^7 - 31001*v^6 - 12277*v^5 - 19266*v^4 - 1807*v^3 - 3595*v^2 + 69*v - 38) / 4
 $$\nu$$ $$=$$ $$( \beta_{15} - 4 \beta_{14} - 2 \beta_{13} + 3 \beta_{12} - 4 \beta_{11} - 2 \beta_{10} + \beta_{9} - 3 \beta_{7} - \beta_{6} + 3 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - \beta _1 + 3 ) / 5$$ (b15 - 4*b14 - 2*b13 + 3*b12 - 4*b11 - 2*b10 + b9 - 3*b7 - b6 + 3*b5 + 2*b4 - 3*b3 + 3*b2 - b1 + 3) / 5 $$\nu^{2}$$ $$=$$ $$\beta_{13} + \beta_{9} + \beta_{6} - 3$$ b13 + b9 + b6 - 3 $$\nu^{3}$$ $$=$$ $$( - 3 \beta_{15} + 12 \beta_{14} + 11 \beta_{13} - 9 \beta_{12} + 22 \beta_{11} + 6 \beta_{10} - 5 \beta_{9} - 8 \beta_{8} + 4 \beta_{7} + 5 \beta_{6} - 14 \beta_{5} - 6 \beta_{4} + 14 \beta_{3} - 15 \beta_{2} + 8 \beta _1 - 14 ) / 5$$ (-3*b15 + 12*b14 + 11*b13 - 9*b12 + 22*b11 + 6*b10 - 5*b9 - 8*b8 + 4*b7 + 5*b6 - 14*b5 - 6*b4 + 14*b3 - 15*b2 + 8*b1 - 14) / 5 $$\nu^{4}$$ $$=$$ $$-7\beta_{13} - 7\beta_{9} - \beta_{7} - 7\beta_{6} + 2\beta_{4} - \beta_{3} + 14$$ -7*b13 - 7*b9 - b7 - 7*b6 + 2*b4 - b3 + 14 $$\nu^{5}$$ $$=$$ $$( 16 \beta_{15} - 54 \beta_{14} - 62 \beta_{13} + 38 \beta_{12} - 124 \beta_{11} - 22 \beta_{10} + 21 \beta_{9} + 60 \beta_{8} + 7 \beta_{7} - 21 \beta_{6} + 73 \beta_{5} + 27 \beta_{4} - 83 \beta_{3} + 78 \beta_{2} - 51 \beta _1 + 73 ) / 5$$ (16*b15 - 54*b14 - 62*b13 + 38*b12 - 124*b11 - 22*b10 + 21*b9 + 60*b8 + 7*b7 - 21*b6 + 73*b5 + 27*b4 - 83*b3 + 78*b2 - 51*b1 + 73) / 5 $$\nu^{6}$$ $$=$$ $$43 \beta_{13} + 44 \beta_{9} + 9 \beta_{7} + 44 \beta_{6} + 3 \beta_{5} - 16 \beta_{4} + 9 \beta_{3} + \beta_{2} - 3 \beta _1 - 76$$ 43*b13 + 44*b9 + 9*b7 + 44*b6 + 3*b5 - 16*b4 + 9*b3 + b2 - 3*b1 - 76 $$\nu^{7}$$ $$=$$ $$( - 103 \beta_{15} + 292 \beta_{14} + 366 \beta_{13} - 189 \beta_{12} + 732 \beta_{11} + 116 \beta_{10} - 85 \beta_{9} - 388 \beta_{8} - 106 \beta_{7} + 85 \beta_{6} - 404 \beta_{5} - 146 \beta_{4} + 514 \beta_{3} - 425 \beta_{2} + \cdots - 424 ) / 5$$ (-103*b15 + 292*b14 + 366*b13 - 189*b12 + 732*b11 + 116*b10 - 85*b9 - 388*b8 - 106*b7 + 85*b6 - 404*b5 - 146*b4 + 514*b3 - 425*b2 + 328*b1 - 424) / 5 $$\nu^{8}$$ $$=$$ $$- 4 \beta_{15} - 255 \beta_{13} - 4 \beta_{12} - 273 \beta_{9} - 66 \beta_{7} - 273 \beta_{6} - 38 \beta_{5} + 102 \beta_{4} - 66 \beta_{3} - 18 \beta_{2} + 38 \beta _1 + 440$$ -4*b15 - 255*b13 - 4*b12 - 273*b9 - 66*b7 - 273*b6 - 38*b5 + 102*b4 - 66*b3 - 18*b2 + 38*b1 + 440 $$\nu^{9}$$ $$=$$ $$( 686 \beta_{15} - 1704 \beta_{14} - 2217 \beta_{13} + 1018 \beta_{12} - 4434 \beta_{11} - 752 \beta_{10} + 316 \beta_{9} + 2420 \beta_{8} + 767 \beta_{7} - 316 \beta_{6} + 2293 \beta_{5} + 852 \beta_{4} - 3223 \beta_{3} + \cdots + 2593 ) / 5$$ (686*b15 - 1704*b14 - 2217*b13 + 1018*b12 - 4434*b11 - 752*b10 + 316*b9 + 2420*b8 + 767*b7 - 316*b6 + 2293*b5 + 852*b4 - 3223*b3 + 2378*b2 - 2141*b1 + 2593) / 5 $$\nu^{10}$$ $$=$$ $$60 \beta_{15} + 1500 \beta_{13} + 60 \beta_{12} + 1696 \beta_{9} + 457 \beta_{7} + 1696 \beta_{6} + 346 \beta_{5} - 612 \beta_{4} + 457 \beta_{3} + 200 \beta_{2} - 346 \beta _1 - 2621$$ 60*b15 + 1500*b13 + 60*b12 + 1696*b9 + 457*b7 + 1696*b6 + 346*b5 - 612*b4 + 457*b3 + 200*b2 - 346*b1 - 2621 $$\nu^{11}$$ $$=$$ $$( - 4573 \beta_{15} + 10262 \beta_{14} + 13611 \beta_{13} - 5689 \beta_{12} + 27222 \beta_{11} + 5176 \beta_{10} - 890 \beta_{9} - 14968 \beta_{8} - 4871 \beta_{7} + 890 \beta_{6} - 13169 \beta_{5} + \cdots - 16199 ) / 5$$ (-4573*b15 + 10262*b14 + 13611*b13 - 5689*b12 + 27222*b11 + 5176*b10 - 890*b9 - 14968*b8 - 4871*b7 + 890*b6 - 13169*b5 - 5131*b4 + 20309*b3 - 13505*b2 + 14053*b1 - 16199) / 5 $$\nu^{12}$$ $$=$$ $$- 606 \beta_{15} - 8832 \beta_{13} - 606 \beta_{12} - 10573 \beta_{9} - 3096 \beta_{7} - 10573 \beta_{6} - 2773 \beta_{5} + 3608 \beta_{4} - 3096 \beta_{3} - 1809 \beta_{2} + 2773 \beta _1 + 15840$$ -606*b15 - 8832*b13 - 606*b12 - 10573*b9 - 3096*b7 - 10573*b6 - 2773*b5 + 3608*b4 - 3096*b3 - 1809*b2 + 2773*b1 + 15840 $$\nu^{13}$$ $$=$$ $$( 30351 \beta_{15} - 62714 \beta_{14} - 84222 \beta_{13} + 32363 \beta_{12} - 168444 \beta_{11} - 35802 \beta_{10} - 244 \beta_{9} + 92600 \beta_{8} + 29837 \beta_{7} + 244 \beta_{6} + 76183 \beta_{5} + \cdots + 102123 ) / 5$$ (30351*b15 - 62714*b14 - 84222*b13 + 32363*b12 - 168444*b11 - 35802*b10 - 244*b9 + 92600*b8 + 29837*b7 + 244*b6 + 76183*b5 + 31357*b4 - 128353*b3 + 77413*b2 - 92261*b1 + 102123) / 5 $$\nu^{14}$$ $$=$$ $$5188 \beta_{15} + 52218 \beta_{13} + 5188 \beta_{12} + 66151 \beta_{9} + 20741 \beta_{7} + 66151 \beta_{6} + 20817 \beta_{5} - 21204 \beta_{4} + 20741 \beta_{3} + 14675 \beta_{2} - 20817 \beta _1 - 96580$$ 5188*b15 + 52218*b13 + 5188*b12 + 66151*b9 + 20741*b7 + 66151*b6 + 20817*b5 - 21204*b4 + 20741*b3 + 14675*b2 - 20817*b1 - 96580 $$\nu^{15}$$ $$=$$ $$( - 200513 \beta_{15} + 386472 \beta_{14} + 523991 \beta_{13} - 185959 \beta_{12} + 1047982 \beta_{11} + 245496 \beta_{10} + 33705 \beta_{9} - 574548 \beta_{8} - 181151 \beta_{7} + \cdots - 646739 ) / 5$$ (-200513*b15 + 386472*b14 + 523991*b13 - 185959*b12 + 1047982*b11 + 245496*b10 + 33705*b9 - 574548*b8 - 181151*b7 - 33705*b6 - 443339*b5 - 193236*b4 + 813119*b3 - 446805*b2 + 604643*b1 - 646739) / 5

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/375\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$251$$ $$\chi(n)$$ $$-\beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
76.1
 2.53767i − 1.08982i 0.0898194i − 1.53767i − 1.35083i 0.536547i − 1.53655i 2.35083i 1.35083i − 0.536547i 1.53655i − 2.35083i − 2.53767i 1.08982i − 0.0898194i 1.53767i
−2.05302 1.49161i −0.309017 0.951057i 1.37197 + 4.22249i 0 −0.784184 + 2.41347i 1.04054 1.91324 5.88835i −0.809017 + 0.587785i 0
76.2 −0.881682 0.640580i −0.309017 0.951057i −0.251013 0.772537i 0 −0.336773 + 1.03648i −3.08724 −0.947104 + 2.91489i −0.809017 + 0.587785i 0
76.3 0.0726655 + 0.0527945i −0.309017 0.951057i −0.615541 1.89444i 0 0.0277557 0.0854234i 4.36070 0.110799 0.341004i −0.809017 + 0.587785i 0
76.4 1.24400 + 0.903822i −0.309017 0.951057i 0.112618 + 0.346603i 0 0.475167 1.46241i 1.68601 0.777165 2.39187i −0.809017 + 0.587785i 0
151.1 −0.417429 1.28472i 0.809017 0.587785i 0.141788 0.103015i 0 −1.09284 0.793998i 1.59580 −2.37722 1.72715i 0.309017 0.951057i 0
151.2 −0.165802 0.510286i 0.809017 0.587785i 1.38513 1.00636i 0 −0.434076 0.315374i −2.57318 −1.61134 1.17071i 0.309017 0.951057i 0
151.3 0.474819 + 1.46134i 0.809017 0.587785i −0.292036 + 0.212177i 0 1.24309 + 0.903160i 1.49550 2.03746 + 1.48030i 0.309017 0.951057i 0
151.4 0.726446 + 2.23577i 0.809017 0.587785i −2.85292 + 2.07277i 0 1.90186 + 1.38178i 3.48189 −2.90300 2.10915i 0.309017 0.951057i 0
226.1 −0.417429 + 1.28472i 0.809017 + 0.587785i 0.141788 + 0.103015i 0 −1.09284 + 0.793998i 1.59580 −2.37722 + 1.72715i 0.309017 + 0.951057i 0
226.2 −0.165802 + 0.510286i 0.809017 + 0.587785i 1.38513 + 1.00636i 0 −0.434076 + 0.315374i −2.57318 −1.61134 + 1.17071i 0.309017 + 0.951057i 0
226.3 0.474819 1.46134i 0.809017 + 0.587785i −0.292036 0.212177i 0 1.24309 0.903160i 1.49550 2.03746 1.48030i 0.309017 + 0.951057i 0
226.4 0.726446 2.23577i 0.809017 + 0.587785i −2.85292 2.07277i 0 1.90186 1.38178i 3.48189 −2.90300 + 2.10915i 0.309017 + 0.951057i 0
301.1 −2.05302 + 1.49161i −0.309017 + 0.951057i 1.37197 4.22249i 0 −0.784184 2.41347i 1.04054 1.91324 + 5.88835i −0.809017 0.587785i 0
301.2 −0.881682 + 0.640580i −0.309017 + 0.951057i −0.251013 + 0.772537i 0 −0.336773 1.03648i −3.08724 −0.947104 2.91489i −0.809017 0.587785i 0
301.3 0.0726655 0.0527945i −0.309017 + 0.951057i −0.615541 + 1.89444i 0 0.0277557 + 0.0854234i 4.36070 0.110799 + 0.341004i −0.809017 0.587785i 0
301.4 1.24400 0.903822i −0.309017 + 0.951057i 0.112618 0.346603i 0 0.475167 + 1.46241i 1.68601 0.777165 + 2.39187i −0.809017 0.587785i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 301.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 375.2.g.d 16
5.b even 2 1 375.2.g.e 16
5.c odd 4 1 75.2.i.a 16
5.c odd 4 1 375.2.i.c 16
15.e even 4 1 225.2.m.b 16
25.d even 5 1 inner 375.2.g.d 16
25.d even 5 1 1875.2.a.p 8
25.e even 10 1 375.2.g.e 16
25.e even 10 1 1875.2.a.m 8
25.f odd 20 1 75.2.i.a 16
25.f odd 20 1 375.2.i.c 16
25.f odd 20 2 1875.2.b.h 16
75.h odd 10 1 5625.2.a.bd 8
75.j odd 10 1 5625.2.a.t 8
75.l even 20 1 225.2.m.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.i.a 16 5.c odd 4 1
75.2.i.a 16 25.f odd 20 1
225.2.m.b 16 15.e even 4 1
225.2.m.b 16 75.l even 20 1
375.2.g.d 16 1.a even 1 1 trivial
375.2.g.d 16 25.d even 5 1 inner
375.2.g.e 16 5.b even 2 1
375.2.g.e 16 25.e even 10 1
375.2.i.c 16 5.c odd 4 1
375.2.i.c 16 25.f odd 20 1
1875.2.a.m 8 25.e even 10 1
1875.2.a.p 8 25.d even 5 1
1875.2.b.h 16 25.f odd 20 2
5625.2.a.t 8 75.j odd 10 1
5625.2.a.bd 8 75.h odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} + 2 T_{2}^{15} + 7 T_{2}^{14} + 16 T_{2}^{13} + 46 T_{2}^{12} + 22 T_{2}^{11} + 125 T_{2}^{10} + 68 T_{2}^{9} + 249 T_{2}^{8} + 214 T_{2}^{7} + 455 T_{2}^{6} + 474 T_{2}^{5} + 586 T_{2}^{4} + 158 T_{2}^{3} + 93 T_{2}^{2} - 16 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(375, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 2 T^{15} + 7 T^{14} + 16 T^{13} + \cdots + 1$$
$3$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{4}$$
$5$ $$T^{16}$$
$7$ $$(T^{8} - 8 T^{7} + 4 T^{6} + 108 T^{5} + \cdots + 505)^{2}$$
$11$ $$T^{16} + 6 T^{15} + 48 T^{14} + \cdots + 27888961$$
$13$ $$T^{16} + 8 T^{15} + 62 T^{14} + \cdots + 78961$$
$17$ $$T^{16} + 8 T^{15} + 95 T^{14} + \cdots + 53860921$$
$19$ $$T^{16} - 2 T^{15} + 23 T^{14} + \cdots + 6375625$$
$23$ $$T^{16} + 2 T^{15} + 113 T^{14} + \cdots + 4389025$$
$29$ $$T^{16} + 16 T^{15} + 217 T^{14} + \cdots + 156025$$
$31$ $$T^{16} - 6 T^{15} + 97 T^{14} + \cdots + 15625$$
$37$ $$T^{16} + 24 T^{15} + \cdots + 8653650625$$
$41$ $$T^{16} + 14 T^{15} + 82 T^{14} + \cdots + 22137025$$
$43$ $$(T^{8} - 20 T^{7} + 6 T^{6} + 1760 T^{5} + \cdots + 22961)^{2}$$
$47$ $$T^{16} - 10 T^{15} + \cdots + 36687479166361$$
$53$ $$T^{16} + 12 T^{15} + \cdots + 40398990025$$
$59$ $$T^{16} + 12 T^{15} + 188 T^{14} + \cdots + 12924025$$
$61$ $$T^{16} + \cdots + 275701483356121$$
$67$ $$T^{16} - 12 T^{15} + 110 T^{14} + \cdots + 13980121$$
$71$ $$T^{16} + 8 T^{15} + \cdots + 25529328841$$
$73$ $$T^{16} - 8 T^{15} + \cdots + 757413387025$$
$79$ $$T^{16} - 20 T^{15} + \cdots + 3940125750625$$
$83$ $$T^{16} + 6 T^{15} + \cdots + 2356228681$$
$89$ $$T^{16} + 18 T^{15} + 68 T^{14} - 1034 T^{13} + \cdots + 25$$
$97$ $$T^{16} + 8 T^{15} + 50 T^{14} + \cdots + 216648961$$