# Properties

 Label 375.2.i.b Level $375$ Weight $2$ Character orbit 375.i Analytic conductor $2.994$ 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] = [375,2,Mod(49,375)]

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, 7]))

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

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

chi := DirichletCharacter("375.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$375 = 3 \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 375.i (of order $$10$$, 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 $$\Q(\zeta_{10})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} + 5x^{14} + 6x^{12} - 20x^{10} - 79x^{8} - 80x^{6} + 96x^{4} + 320x^{2} + 256$$ x^16 + 5*x^14 + 6*x^12 - 20*x^10 - 79*x^8 - 80*x^6 + 96*x^4 + 320*x^2 + 256 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: $\mathrm{SU}(2)[C_{10}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{14} q^{2} + ( - \beta_{13} - \beta_{6} - \beta_{5} - \beta_1) q^{3} + (2 \beta_{9} - \beta_{7} - 3 \beta_{4} - 3 \beta_{2} + 2) q^{4} + \beta_{8} q^{6} + ( - \beta_{14} - \beta_{13} + \beta_{10} - 2 \beta_{6} - \beta_{5} + \beta_{3} + \beta_1) q^{7} + ( - \beta_{13} - \beta_{12} - \beta_{10} + \beta_{6} + \beta_{5} - \beta_{3} + 3 \beta_1) q^{8} - \beta_{9} q^{9}+O(q^{10})$$ q + b14 * q^2 + (-b13 - b6 - b5 - b1) * q^3 + (2*b9 - b7 - 3*b4 - 3*b2 + 2) * q^4 + b8 * q^6 + (-b14 - b13 + b10 - 2*b6 - b5 + b3 + b1) * q^7 + (-b13 - b12 - b10 + b6 + b5 - b3 + 3*b1) * q^8 - b9 * q^9 $$q + \beta_{14} q^{2} + ( - \beta_{13} - \beta_{6} - \beta_{5} - \beta_1) q^{3} + (2 \beta_{9} - \beta_{7} - 3 \beta_{4} - 3 \beta_{2} + 2) q^{4} + \beta_{8} q^{6} + ( - \beta_{14} - \beta_{13} + \beta_{10} - 2 \beta_{6} - \beta_{5} + \beta_{3} + \beta_1) q^{7} + ( - \beta_{13} - \beta_{12} - \beta_{10} + \beta_{6} + \beta_{5} - \beta_{3} + 3 \beta_1) q^{8} - \beta_{9} q^{9} + ( - \beta_{11} - 3 \beta_{9} + \beta_{8} + 3 \beta_{4} + 3 \beta_{2}) q^{11} + ( - 2 \beta_{13} - \beta_{12} + \beta_{5} - \beta_{3}) q^{12} + (\beta_{14} - \beta_{13} - \beta_{10} - 2 \beta_{5} - 2 \beta_1) q^{13} + ( - 2 \beta_{11} - \beta_{9} + 2 \beta_{8} + \beta_{4} + 4 \beta_{2}) q^{14} + ( - 2 \beta_{15} - 3 \beta_{11} - 3 \beta_{9} - 2 \beta_{8} + 2 \beta_{4} - 4) q^{16} + ( - \beta_{12} - \beta_1) q^{17} + (\beta_{14} - \beta_{13} - \beta_{10} - \beta_{3} - \beta_1) q^{18} + (\beta_{11} + \beta_{9} - 3 \beta_{8} - \beta_{7} - \beta_{4} - \beta_{2} + 1) q^{19} + ( - 2 \beta_{9} + \beta_{7} - 2) q^{21} + (2 \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + \beta_{10} + \beta_{6} - 2 \beta_{5} + \beta_1) q^{22} + (\beta_{14} - \beta_{13} + \beta_{3} - \beta_1) q^{23} + ( - \beta_{15} - \beta_{11} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{4} - 2 \beta_{2} - 3) q^{24} + (\beta_{15} + \beta_{11} + 3 \beta_{9} + \beta_{8} + \beta_{7} + \beta_{4} - 2 \beta_{2} + 2) q^{26} - \beta_{6} q^{27} + ( - \beta_{14} - 2 \beta_{13} - \beta_{12} + 3 \beta_{10} - 2 \beta_{5}) q^{28} + ( - 5 \beta_{9} + \beta_{7} + 6 \beta_{4} + 6 \beta_{2} - 5) q^{29} + (\beta_{11} + \beta_{9} - \beta_{8} - \beta_{7} - 6 \beta_{4} + \beta_{2} - 1) q^{31} + ( - 2 \beta_{14} + 9 \beta_{13} + 2 \beta_{12} + 7 \beta_{6} + \beta_{5} + 2 \beta_{3} + \cdots + 2 \beta_1) q^{32}+ \cdots + ( - 2 \beta_{9} - \beta_{8} - \beta_{7} - \beta_{4} + 2 \beta_{2} + 1) q^{99}+O(q^{100})$$ q + b14 * q^2 + (-b13 - b6 - b5 - b1) * q^3 + (2*b9 - b7 - 3*b4 - 3*b2 + 2) * q^4 + b8 * q^6 + (-b14 - b13 + b10 - 2*b6 - b5 + b3 + b1) * q^7 + (-b13 - b12 - b10 + b6 + b5 - b3 + 3*b1) * q^8 - b9 * q^9 + (-b11 - 3*b9 + b8 + 3*b4 + 3*b2) * q^11 + (-2*b13 - b12 + b5 - b3) * q^12 + (b14 - b13 - b10 - 2*b5 - 2*b1) * q^13 + (-2*b11 - b9 + 2*b8 + b4 + 4*b2) * q^14 + (-2*b15 - 3*b11 - 3*b9 - 2*b8 + 2*b4 - 4) * q^16 + (-b12 - b1) * q^17 + (b14 - b13 - b10 - b3 - b1) * q^18 + (b11 + b9 - 3*b8 - b7 - b4 - b2 + 1) * q^19 + (-2*b9 + b7 - 2) * q^21 + (2*b14 - 2*b13 + 2*b12 + b10 + b6 - 2*b5 + b1) * q^22 + (b14 - b13 + b3 - b1) * q^23 + (-b15 - b11 + b9 - b8 - b7 - b4 - 2*b2 - 3) * q^24 + (b15 + b11 + 3*b9 + b8 + b7 + b4 - 2*b2 + 2) * q^26 - b6 * q^27 + (-b14 - 2*b13 - b12 + 3*b10 - 2*b5) * q^28 + (-5*b9 + b7 + 6*b4 + 6*b2 - 5) * q^29 + (b11 + b9 - b8 - b7 - 6*b4 + b2 - 1) * q^31 + (-2*b14 + 9*b13 + 2*b12 + 7*b6 + b5 + 2*b3 + 2*b1) * q^32 + (-b13 - b10 - 3*b6 - 3*b5 - b3 - 4*b1) * q^33 + (3*b9 + b4 - 1) * q^34 + (-b15 + b9 - b8 - b7 - 2*b4 - 3*b2) * q^36 + (2*b14 - b13 + b12 - 2*b10 - b5 + b3) * q^37 + (b14 + 10*b13 + 2*b12 - b10 + b5 + 2*b3 + 3*b1) * q^38 + (-b11 - 2*b9 + b8 + 2*b4 + b2) * q^39 + (-5*b9 + 4*b4 - 4) * q^41 + (-2*b13 - 2*b10 - b6 - b5 - 2*b3 - 6*b1) * q^42 + (b13 - 2*b12 + 2*b10 + b6 + 5*b5) * q^43 + (b11 + b9 + b8 - b7 - 9*b4 - 4*b2 + 4) * q^44 + (-b15 + 5*b9 - b7 - 4*b4 - 4*b2 + 5) * q^46 + (2*b14 + 2*b12 - b10 + b6 + b1) * q^47 + (-3*b14 + 4*b13 + b6 + 2*b3 + 4*b1) * q^48 + (b15 + b11 - 2*b9 + 4*b8 + 4*b7 + 4*b4 + 3*b2 - 4) * q^49 + (-b15 - b11 - b9) * q^51 - 3*b6 * q^52 + (-2*b14 + b13 - 2*b12 + b10 + b6 + b5 + b1) * q^53 + (b9 - b7 - b4 - b2 + 1) * q^54 + (b11 + b9 + b8 - b7 + 2*b2 - 2) * q^56 + (-2*b14 + 2*b13 - b12 + 3*b10 + 2*b3 + 2*b1) * q^57 + (b13 + 6*b12 + b10 - b6 - b5 + b3 - 3*b1) * q^58 + (-3*b15 - 3*b8 + 3*b4 - 6) * q^59 + (2*b11 + 4*b9 - 2*b8 - 4*b4 - b2) * q^61 + (-b14 - 3*b13 - 5*b12 + b10 + 3*b5 - 5*b3 - 2*b1) * q^62 + (2*b13 + b12 + 2*b5 + b3 + 3*b1) * q^63 + (2*b15 + 5*b11 - 4*b9 - 3*b8 + 2*b7 + 6*b4 + 10*b2) * q^64 + (2*b15 + 3*b11 + 2*b8 + 3*b4 - 1) * q^66 + (-2*b12 - b6 - b5 - 2*b1) * q^67 + (-2*b14 + 2*b13 + b12 + b10 - 2*b5 + 2*b3 + 2*b1) * q^68 + (-b11 - b9 + 2*b8 + b7 + b4 - b2 + 1) * q^69 + (4*b15 + 3*b9 + 3*b7 + 3*b4 + 3*b2 + 3) * q^71 + (-b14 + 3*b13 - b12 + 4*b6 + 3*b5 + 4*b1) * q^72 + (4*b14 - 3*b13 + 5*b6 - 2*b3 - 3*b1) * q^73 + (6*b9 - b8 - b7 - b4 - 6*b2 + 8) * q^74 + (5*b15 + 5*b11 + 6*b9 - b2 + 13) * q^76 + (-5*b14 - 5*b13 - 10*b6 + b3 - 5*b1) * q^77 + (b14 - 2*b13 + b12 + b6 - 2*b5 + b1) * q^78 + (3*b15 + 10*b9 + 3*b7 - 6*b4 - 6*b2 + 10) * q^79 - b4 * q^81 + (b14 - b13 + 4*b12 - 5*b10 - b3 - b1) * q^82 + (-4*b13 - 4*b10 - 4*b3 + b1) * q^83 + (-b15 + 2*b11 - 3*b9 - b8 + 2*b4 - 3) * q^84 + (-4*b15 - b11 + 6*b9 - 3*b8 - 4*b7 - 10*b4 - 6*b2) * q^86 + (5*b13 + b12 - b5 + b3) * q^87 + (-4*b13 - 4*b12 + 11*b5 - 4*b3 + 7*b1) * q^88 + (-b11 - 4*b9 + b8 + 4*b4 - 7*b2) * q^89 + (-3*b11 - 3*b9 - 3*b4 + 3) * q^91 + (-b13 - 2*b12 - b10 - b3 + 4*b1) * q^92 + (2*b13 - b12 + b10 + 2*b6 + 7*b5) * q^93 + (b11 + b9 - b8 - b7 - 7*b4 - 8*b2 + 8) * q^94 + (2*b15 + b9 + 2*b7 + 8*b4 + 8*b2 + 1) * q^96 + (-4*b14 + 5*b13 - 4*b12 + 4*b10 + 3*b6 + 5*b5 + 3*b1) * q^97 + (-b14 - 16*b13 - 8*b6 - 3*b3 - 16*b1) * q^98 + (-2*b9 - b8 - b7 - b4 + 2*b2 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 2 q^{4} - 2 q^{6} + 4 q^{9}+O(q^{10})$$ 16 * q - 2 * q^4 - 2 * q^6 + 4 * q^9 $$16 q - 2 q^{4} - 2 q^{6} + 4 q^{9} + 32 q^{11} + 16 q^{14} - 34 q^{16} + 10 q^{19} - 22 q^{21} - 60 q^{24} + 12 q^{26} - 10 q^{29} - 38 q^{31} - 24 q^{34} - 18 q^{36} + 16 q^{39} - 28 q^{41} + 6 q^{44} + 32 q^{46} - 32 q^{49} + 8 q^{51} + 2 q^{54} - 30 q^{56} - 60 q^{59} - 28 q^{61} + 88 q^{64} - 14 q^{66} + 16 q^{69} + 42 q^{71} + 76 q^{74} + 160 q^{76} + 60 q^{79} - 4 q^{81} - 16 q^{84} - 68 q^{86} + 42 q^{91} + 66 q^{94} + 68 q^{96} + 28 q^{99}+O(q^{100})$$ 16 * q - 2 * q^4 - 2 * q^6 + 4 * q^9 + 32 * q^11 + 16 * q^14 - 34 * q^16 + 10 * q^19 - 22 * q^21 - 60 * q^24 + 12 * q^26 - 10 * q^29 - 38 * q^31 - 24 * q^34 - 18 * q^36 + 16 * q^39 - 28 * q^41 + 6 * q^44 + 32 * q^46 - 32 * q^49 + 8 * q^51 + 2 * q^54 - 30 * q^56 - 60 * q^59 - 28 * q^61 + 88 * q^64 - 14 * q^66 + 16 * q^69 + 42 * q^71 + 76 * q^74 + 160 * q^76 + 60 * q^79 - 4 * q^81 - 16 * q^84 - 68 * q^86 + 42 * q^91 + 66 * q^94 + 68 * q^96 + 28 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 5x^{14} + 6x^{12} - 20x^{10} - 79x^{8} - 80x^{6} + 96x^{4} + 320x^{2} + 256$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{15} - \nu^{13} + 14\nu^{11} + 44\nu^{9} - \nu^{7} - 108\nu^{5} - 288\nu^{3} - 192\nu ) / 384$$ (-v^15 - v^13 + 14*v^11 + 44*v^9 - v^7 - 108*v^5 - 288*v^3 - 192*v) / 384 $$\beta_{2}$$ $$=$$ $$( -\nu^{14} - 9\nu^{12} - 10\nu^{10} + 12\nu^{8} + 63\nu^{6} + 76\nu^{4} + 48\nu^{2} + 64 ) / 192$$ (-v^14 - 9*v^12 - 10*v^10 + 12*v^8 + 63*v^6 + 76*v^4 + 48*v^2 + 64) / 192 $$\beta_{3}$$ $$=$$ $$( \nu^{15} + \nu^{13} - 46\nu^{11} - 76\nu^{9} + 65\nu^{7} + 364\nu^{5} + 256\nu^{3} - 192\nu ) / 384$$ (v^15 + v^13 - 46*v^11 - 76*v^9 + 65*v^7 + 364*v^5 + 256*v^3 - 192*v) / 384 $$\beta_{4}$$ $$=$$ $$( -\nu^{14} - 9\nu^{12} - 26\nu^{10} - 4\nu^{8} + 95\nu^{6} + 204\nu^{4} + 32\nu^{2} - 128 ) / 192$$ (-v^14 - 9*v^12 - 26*v^10 - 4*v^8 + 95*v^6 + 204*v^4 + 32*v^2 - 128) / 192 $$\beta_{5}$$ $$=$$ $$( -\nu^{15} - 5\nu^{13} + 10\nu^{11} + 100\nu^{9} + 175\nu^{7} - 112\nu^{5} - 720\nu^{3} - 960\nu ) / 384$$ (-v^15 - 5*v^13 + 10*v^11 + 100*v^9 + 175*v^7 - 112*v^5 - 720*v^3 - 960*v) / 384 $$\beta_{6}$$ $$=$$ $$( -\nu^{13} - 3\nu^{11} - 4\nu^{9} + 8\nu^{7} + 23\nu^{5} + 18\nu^{3} + 8\nu ) / 48$$ (-v^13 - 3*v^11 - 4*v^9 + 8*v^7 + 23*v^5 + 18*v^3 + 8*v) / 48 $$\beta_{7}$$ $$=$$ $$( -3\nu^{14} - 11\nu^{12} - 30\nu^{10} - 12\nu^{8} + 93\nu^{6} + 180\nu^{4} + 320\nu^{2} + 192 ) / 192$$ (-3*v^14 - 11*v^12 - 30*v^10 - 12*v^8 + 93*v^6 + 180*v^4 + 320*v^2 + 192) / 192 $$\beta_{8}$$ $$=$$ $$( 3\nu^{14} + 7\nu^{12} + 26\nu^{10} + 68\nu^{8} + 83\nu^{6} - 184\nu^{4} - 752\nu^{2} - 960 ) / 192$$ (3*v^14 + 7*v^12 + 26*v^10 + 68*v^8 + 83*v^6 - 184*v^4 - 752*v^2 - 960) / 192 $$\beta_{9}$$ $$=$$ $$( 5\nu^{14} + 9\nu^{12} - 18\nu^{10} - 100\nu^{8} - 139\nu^{6} + 96\nu^{4} + 512\nu^{2} + 512 ) / 192$$ (5*v^14 + 9*v^12 - 18*v^10 - 100*v^8 - 139*v^6 + 96*v^4 + 512*v^2 + 512) / 192 $$\beta_{10}$$ $$=$$ $$( 2\nu^{15} + 5\nu^{13} + 3\nu^{11} - 22\nu^{9} - 58\nu^{7} - 21\nu^{5} + 96\nu^{3} + 224\nu ) / 96$$ (2*v^15 + 5*v^13 + 3*v^11 - 22*v^9 - 58*v^7 - 21*v^5 + 96*v^3 + 224*v) / 96 $$\beta_{11}$$ $$=$$ $$( -\nu^{14} - 3\nu^{12} + 2\nu^{10} + 30\nu^{8} + 51\nu^{6} - 14\nu^{4} - 162\nu^{2} - 200 ) / 24$$ (-v^14 - 3*v^12 + 2*v^10 + 30*v^8 + 51*v^6 - 14*v^4 - 162*v^2 - 200) / 24 $$\beta_{12}$$ $$=$$ $$( -5\nu^{15} - 33\nu^{13} - 38\nu^{11} + 84\nu^{9} + 363\nu^{7} + 392\nu^{5} - 192\nu^{3} - 832\nu ) / 384$$ (-5*v^15 - 33*v^13 - 38*v^11 + 84*v^9 + 363*v^7 + 392*v^5 - 192*v^3 - 832*v) / 384 $$\beta_{13}$$ $$=$$ $$( -5\nu^{15} - 15\nu^{13} - 4\nu^{11} + 72\nu^{9} + 147\nu^{7} + 26\nu^{5} - 216\nu^{3} - 224\nu ) / 192$$ (-5*v^15 - 15*v^13 - 4*v^11 + 72*v^9 + 147*v^7 + 26*v^5 - 216*v^3 - 224*v) / 192 $$\beta_{14}$$ $$=$$ $$( -3\nu^{15} - 6\nu^{13} + 15\nu^{11} + 86\nu^{9} + 113\nu^{7} - 103\nu^{5} - 508\nu^{3} - 448\nu ) / 96$$ (-3*v^15 - 6*v^13 + 15*v^11 + 86*v^9 + 113*v^7 - 103*v^5 - 508*v^3 - 448*v) / 96 $$\beta_{15}$$ $$=$$ $$( 13\nu^{14} + 49\nu^{12} - 2\nu^{10} - 356\nu^{8} - 707\nu^{6} + 96\nu^{4} + 2016\nu^{2} + 2304 ) / 192$$ (13*v^14 + 49*v^12 - 2*v^10 - 356*v^8 - 707*v^6 + 96*v^4 + 2016*v^2 + 2304) / 192
 $$\nu$$ $$=$$ $$( \beta_{14} + 2\beta_{13} - \beta_{12} + 3\beta_{10} + 2\beta_{6} - \beta_{5} + \beta_{3} - \beta_1 ) / 5$$ (b14 + 2*b13 - b12 + 3*b10 + 2*b6 - b5 + b3 - b1) / 5 $$\nu^{2}$$ $$=$$ $$( 2\beta_{15} + 3\beta_{11} + \beta_{9} + \beta_{8} + 3\beta_{7} - 3\beta_{4} + 4\beta_{2} - 3 ) / 5$$ (2*b15 + 3*b11 + b9 + b8 + 3*b7 - 3*b4 + 4*b2 - 3) / 5 $$\nu^{3}$$ $$=$$ $$( -2\beta_{14} + \beta_{13} + 2\beta_{12} - \beta_{10} - 9\beta_{6} + 2\beta_{5} - 2\beta_{3} - 8\beta_1 ) / 5$$ (-2*b14 + b13 + 2*b12 - b10 - 9*b6 + 2*b5 - 2*b3 - 8*b1) / 5 $$\nu^{4}$$ $$=$$ $$\beta_{15} + \beta_{11} - \beta_{9} - \beta_{7} + 2\beta_{4} + \beta_{2} + 1$$ b15 + b11 - b9 - b7 + 2*b4 + b2 + 1 $$\nu^{5}$$ $$=$$ $$( 6\beta_{14} - 8\beta_{13} + 9\beta_{12} + 3\beta_{10} - 8\beta_{6} - 16\beta_{5} + 6\beta_{3} + 9\beta_1 ) / 5$$ (6*b14 - 8*b13 + 9*b12 + 3*b10 - 8*b6 - 16*b5 + 6*b3 + 9*b1) / 5 $$\nu^{6}$$ $$=$$ $$( 2\beta_{15} - 7\beta_{11} - 14\beta_{9} + 16\beta_{8} + 18\beta_{7} - 3\beta_{4} + 9\beta_{2} + 12 ) / 5$$ (2*b15 - 7*b11 - 14*b9 + 16*b8 + 18*b7 - 3*b4 + 9*b2 + 12) / 5 $$\nu^{7}$$ $$=$$ $$( 8\beta_{14} - 4\beta_{13} - 3\beta_{12} - \beta_{10} + 6\beta_{6} + 2\beta_{5} - 7\beta_{3} - 58\beta_1 ) / 5$$ (8*b14 - 4*b13 - 3*b12 - b10 + 6*b6 + 2*b5 - 7*b3 - 58*b1) / 5 $$\nu^{8}$$ $$=$$ $$3\beta_{15} + 10\beta_{11} + 9\beta_{9} - 4\beta_{8} - 2\beta_{7} - 2\beta_{4} + 4$$ 3*b15 + 10*b11 + 9*b9 - 4*b8 - 2*b7 - 2*b4 + 4 $$\nu^{9}$$ $$=$$ $$( -19\beta_{14} + 42\beta_{13} + 14\beta_{12} + 38\beta_{10} - 78\beta_{6} + 19\beta_{5} + 26\beta_{3} + 49\beta_1 ) / 5$$ (-19*b14 + 42*b13 + 14*b12 + 38*b10 - 78*b6 + 19*b5 + 26*b3 + 49*b1) / 5 $$\nu^{10}$$ $$=$$ $$( 27\beta_{15} - 27\beta_{11} - 114\beta_{9} + 51\beta_{8} + 3\beta_{7} + 27\beta_{4} + 114\beta_{2} - 13 ) / 5$$ (27*b15 - 27*b11 - 114*b9 + 51*b8 + 3*b7 + 27*b4 + 114*b2 - 13) / 5 $$\nu^{11}$$ $$=$$ $$( 73 \beta_{14} - 139 \beta_{13} + 62 \beta_{12} - 51 \beta_{10} + 11 \beta_{6} - 133 \beta_{5} - 62 \beta_{3} - 133 \beta_1 ) / 5$$ (73*b14 - 139*b13 + 62*b12 - 51*b10 + 11*b6 - 133*b5 - 62*b3 - 133*b1) / 5 $$\nu^{12}$$ $$=$$ $$10\beta_{15} + 18\beta_{11} + 5\beta_{9} + 5\beta_{8} + 18\beta_{7} + 3\beta_{4} - 31\beta_{2} + 36$$ 10*b15 + 18*b11 + 5*b9 + 5*b8 + 18*b7 + 3*b4 - 31*b2 + 36 $$\nu^{13}$$ $$=$$ $$( 31\beta_{14} + 67\beta_{13} - 31\beta_{12} + 68\beta_{10} - 243\beta_{6} - \beta_{5} + 136\beta_{3} - 206\beta_1 ) / 5$$ (31*b14 + 67*b13 - 31*b12 + 68*b10 - 243*b6 - b5 + 136*b3 - 206*b1) / 5 $$\nu^{14}$$ $$=$$ $$( 62\beta_{15} + 143\beta_{11} + 241\beta_{9} + 81\beta_{8} - 62\beta_{7} - 98\beta_{4} + 434\beta_{2} + 62 ) / 5$$ (62*b15 + 143*b11 + 241*b9 + 81*b8 - 62*b7 - 98*b4 + 434*b2 + 62) / 5 $$\nu^{15}$$ $$=$$ $$( - 117 \beta_{14} + 31 \beta_{13} + 162 \beta_{12} + 279 \beta_{10} + 31 \beta_{6} + 317 \beta_{5} - 117 \beta_{3} + 162 \beta_1 ) / 5$$ (-117*b14 + 31*b13 + 162*b12 + 279*b10 + 31*b6 + 317*b5 - 117*b3 + 162*b1) / 5

## Character values

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

 $$n$$ $$127$$ $$251$$ $$\chi(n)$$ $$1 - \beta_{2} - \beta_{4} + \beta_{9}$$ $$1$$

## 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
 −1.41395 + 0.0272949i −0.462894 + 1.33631i 0.462894 − 1.33631i 1.41395 − 0.0272949i −1.41395 − 0.0272949i −0.462894 − 1.33631i 0.462894 + 1.33631i 1.41395 + 0.0272949i −0.132563 − 1.40799i −0.720348 − 1.21700i 0.720348 + 1.21700i 0.132563 + 1.40799i −0.132563 + 1.40799i −0.720348 + 1.21700i 0.720348 − 1.21700i 0.132563 − 1.40799i
−1.59076 + 2.18949i −0.951057 + 0.309017i −1.64533 5.06380i 0 0.836312 2.57390i 0.470294i 8.55667 + 2.78023i 0.809017 0.587785i 0
49.2 −1.00297 + 1.38048i 0.951057 0.309017i −0.281722 0.867051i 0 −0.527295 + 1.62285i 3.94243i −1.76619 0.573870i 0.809017 0.587785i 0
49.3 1.00297 1.38048i −0.951057 + 0.309017i −0.281722 0.867051i 0 −0.527295 + 1.62285i 3.94243i 1.76619 + 0.573870i 0.809017 0.587785i 0
49.4 1.59076 2.18949i 0.951057 0.309017i −1.64533 5.06380i 0 0.836312 2.57390i 0.470294i −8.55667 2.78023i 0.809017 0.587785i 0
199.1 −1.59076 2.18949i −0.951057 0.309017i −1.64533 + 5.06380i 0 0.836312 + 2.57390i 0.470294i 8.55667 2.78023i 0.809017 + 0.587785i 0
199.2 −1.00297 1.38048i 0.951057 + 0.309017i −0.281722 + 0.867051i 0 −0.527295 1.62285i 3.94243i −1.76619 + 0.573870i 0.809017 + 0.587785i 0
199.3 1.00297 + 1.38048i −0.951057 0.309017i −0.281722 + 0.867051i 0 −0.527295 1.62285i 3.94243i 1.76619 0.573870i 0.809017 + 0.587785i 0
199.4 1.59076 + 2.18949i 0.951057 + 0.309017i −1.64533 + 5.06380i 0 0.836312 + 2.57390i 0.470294i −8.55667 + 2.78023i 0.809017 + 0.587785i 0
274.1 −2.01846 0.655837i 0.587785 0.809017i 2.02602 + 1.47199i 0 −1.71700 + 1.24748i 4.35840i −0.629102 0.865884i −0.309017 0.951057i 0
274.2 −1.06740 0.346820i −0.587785 + 0.809017i −0.598970 0.435177i 0 0.907987 0.659691i 1.11373i 1.80780 + 2.48822i −0.309017 0.951057i 0
274.3 1.06740 + 0.346820i 0.587785 0.809017i −0.598970 0.435177i 0 0.907987 0.659691i 1.11373i −1.80780 2.48822i −0.309017 0.951057i 0
274.4 2.01846 + 0.655837i −0.587785 + 0.809017i 2.02602 + 1.47199i 0 −1.71700 + 1.24748i 4.35840i 0.629102 + 0.865884i −0.309017 0.951057i 0
349.1 −2.01846 + 0.655837i 0.587785 + 0.809017i 2.02602 1.47199i 0 −1.71700 1.24748i 4.35840i −0.629102 + 0.865884i −0.309017 + 0.951057i 0
349.2 −1.06740 + 0.346820i −0.587785 0.809017i −0.598970 + 0.435177i 0 0.907987 + 0.659691i 1.11373i 1.80780 2.48822i −0.309017 + 0.951057i 0
349.3 1.06740 0.346820i 0.587785 + 0.809017i −0.598970 + 0.435177i 0 0.907987 + 0.659691i 1.11373i −1.80780 + 2.48822i −0.309017 + 0.951057i 0
349.4 2.01846 0.655837i −0.587785 0.809017i 2.02602 1.47199i 0 −1.71700 1.24748i 4.35840i 0.629102 0.865884i −0.309017 + 0.951057i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 349.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} - 3T_{2}^{14} + 48T_{2}^{12} - 341T_{2}^{10} + 1475T_{2}^{8} - 2801T_{2}^{6} + 11828T_{2}^{4} - 19723T_{2}^{2} + 14641$$ acting on $$S_{2}^{\mathrm{new}}(375, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - 3 T^{14} + 48 T^{12} + \cdots + 14641$$
$3$ $$(T^{8} - T^{6} + T^{4} - T^{2} + 1)^{2}$$
$5$ $$T^{16}$$
$7$ $$(T^{8} + 36 T^{6} + 346 T^{4} + 441 T^{2} + \cdots + 81)^{2}$$
$11$ $$(T^{8} - 16 T^{7} + 127 T^{6} + \cdots + 11881)^{2}$$
$13$ $$T^{16} - 22 T^{14} + 253 T^{12} + \cdots + 6561$$
$17$ $$T^{16} - 13 T^{14} + 83 T^{12} + \cdots + 14641$$
$19$ $$(T^{8} - 5 T^{7} + 5 T^{6} + 75 T^{5} + \cdots + 75625)^{2}$$
$23$ $$T^{16} - 67 T^{14} + 1808 T^{12} + \cdots + 130321$$
$29$ $$(T^{8} + 5 T^{7} + 70 T^{6} + 615 T^{5} + \cdots + 164025)^{2}$$
$31$ $$(T^{8} + 19 T^{7} + 172 T^{6} + \cdots + 505521)^{2}$$
$37$ $$T^{16} - 153 T^{14} + 10283 T^{12} + \cdots + 1$$
$41$ $$(T^{4} + 7 T^{3} + 69 T^{2} + 143 T + 121)^{4}$$
$43$ $$(T^{8} + 134 T^{6} + 4291 T^{4} + \cdots + 9801)^{2}$$
$47$ $$T^{16} - 63 T^{14} + \cdots + 9354951841$$
$53$ $$T^{16} - 77 T^{14} + \cdots + 4640470641$$
$59$ $$(T^{8} + 30 T^{7} + 495 T^{6} + \cdots + 13286025)^{2}$$
$61$ $$(T^{8} + 14 T^{7} + 157 T^{6} + 842 T^{5} + \cdots + 81)^{2}$$
$67$ $$T^{16} - 58 T^{14} + \cdots + 855036081$$
$71$ $$(T^{8} - 21 T^{7} + 447 T^{6} + \cdots + 829921)^{2}$$
$73$ $$T^{16} + \cdots + 562029482679921$$
$79$ $$(T^{8} - 30 T^{7} + 605 T^{6} + \cdots + 96924025)^{2}$$
$83$ $$T^{16} - 82 T^{14} + \cdots + 918609150481$$
$89$ $$(T^{8} + 255 T^{6} + 2415 T^{5} + \cdots + 10923025)^{2}$$
$97$ $$T^{16} + 22 T^{14} + \cdots + 19485170468401$$