# Properties

 Label 1875.2.b.e Level $1875$ Weight $2$ Character orbit 1875.b Analytic conductor $14.972$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1249,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("1875.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1875 = 3 \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1875.b (of order $$2$$, degree $$1$$, 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: $$14.9719503790$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} + 23x^{10} + 199x^{8} + 794x^{6} + 1399x^{4} + 783x^{2} + 81$$ x^12 + 23*x^10 + 199*x^8 + 794*x^6 + 1399*x^4 + 783*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 23x^{10} + 199x^{8} + 794x^{6} + 1399x^{4} + 783x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 4$$ v^2 + 4 $$\beta_{3}$$ $$=$$ $$( \nu^{11} + 20\nu^{9} + 139\nu^{7} + 377\nu^{5} + 268\nu^{3} - 93\nu ) / 72$$ (v^11 + 20*v^9 + 139*v^7 + 377*v^5 + 268*v^3 - 93*v) / 72 $$\beta_{4}$$ $$=$$ $$( 2\nu^{10} + 37\nu^{8} + 227\nu^{6} + 490\nu^{4} + 161\nu^{2} - 9 ) / 36$$ (2*v^10 + 37*v^8 + 227*v^6 + 490*v^4 + 161*v^2 - 9) / 36 $$\beta_{5}$$ $$=$$ $$( 2\nu^{11} + 37\nu^{9} + 227\nu^{7} + 490\nu^{5} + 161\nu^{3} - 45\nu ) / 36$$ (2*v^11 + 37*v^9 + 227*v^7 + 490*v^5 + 161*v^3 - 45*v) / 36 $$\beta_{6}$$ $$=$$ $$( 5\nu^{11} + 97\nu^{9} + 662\nu^{7} + 1873\nu^{5} + 1937\nu^{3} + 522\nu ) / 108$$ (5*v^11 + 97*v^9 + 662*v^7 + 1873*v^5 + 1937*v^3 + 522*v) / 108 $$\beta_{7}$$ $$=$$ $$( \nu^{10} + 23\nu^{8} + 190\nu^{6} + 677\nu^{4} + 967\nu^{2} + 342 ) / 36$$ (v^10 + 23*v^8 + 190*v^6 + 677*v^4 + 967*v^2 + 342) / 36 $$\beta_{8}$$ $$=$$ $$( -\nu^{11} - 16\nu^{9} - 71\nu^{7} - \nu^{5} + 472\nu^{3} + 409\nu ) / 24$$ (-v^11 - 16*v^9 - 71*v^7 - v^5 + 472*v^3 + 409*v) / 24 $$\beta_{9}$$ $$=$$ $$( -2\nu^{11} - 55\nu^{9} - 551\nu^{7} - 2434\nu^{5} - 4355\nu^{3} - 1683\nu ) / 108$$ (-2*v^11 - 55*v^9 - 551*v^7 - 2434*v^5 - 4355*v^3 - 1683*v) / 108 $$\beta_{10}$$ $$=$$ $$( \nu^{10} + 18\nu^{8} + 109\nu^{6} + 249\nu^{4} + 154\nu^{2} + 21 ) / 8$$ (v^10 + 18*v^8 + 109*v^6 + 249*v^4 + 154*v^2 + 21) / 8 $$\beta_{11}$$ $$=$$ $$( 2\nu^{10} + 37\nu^{8} + 233\nu^{6} + 562\nu^{4} + 377\nu^{2} + 45 ) / 12$$ (2*v^10 + 37*v^8 + 233*v^6 + 562*v^4 + 377*v^2 + 45) / 12
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 4$$ b2 - 4 $$\nu^{3}$$ $$=$$ $$\beta_{9} + \beta_{8} + \beta_{6} + \beta_{3} - 5\beta_1$$ b9 + b8 + b6 + b3 - 5*b1 $$\nu^{4}$$ $$=$$ $$-2\beta_{11} + 2\beta_{10} + \beta_{7} + \beta_{4} - 7\beta_{2} + 21$$ -2*b11 + 2*b10 + b7 + b4 - 7*b2 + 21 $$\nu^{5}$$ $$=$$ $$-10\beta_{9} - 9\beta_{8} - 10\beta_{6} + 2\beta_{5} - 15\beta_{3} + 29\beta_1$$ -10*b9 - 9*b8 - 10*b6 + 2*b5 - 15*b3 + 29*b1 $$\nu^{6}$$ $$=$$ $$26\beta_{11} - 24\beta_{10} - 12\beta_{7} - 18\beta_{4} + 48\beta_{2} - 117$$ 26*b11 - 24*b10 - 12*b7 - 18*b4 + 48*b2 - 117 $$\nu^{7}$$ $$=$$ $$86\beta_{9} + 72\beta_{8} + 92\beta_{6} - 30\beta_{5} + 144\beta_{3} - 183\beta_1$$ 86*b9 + 72*b8 + 92*b6 - 30*b5 + 144*b3 - 183*b1 $$\nu^{8}$$ $$=$$ $$-250\beta_{11} + 216\beta_{10} + 116\beta_{7} + 206\beta_{4} - 341\beta_{2} + 684$$ -250*b11 + 216*b10 + 116*b7 + 206*b4 - 341*b2 + 684 $$\nu^{9}$$ $$=$$ $$-707\beta_{9} - 557\beta_{8} - 809\beta_{6} + 322\beta_{5} - 1205\beta_{3} + 1231\beta_1$$ -707*b9 - 557*b8 - 809*b6 + 322*b5 - 1205*b3 + 1231*b1 $$\nu^{10}$$ $$=$$ $$2164\beta_{11} - 1762\beta_{10} - 1029\beta_{7} - 1995\beta_{4} + 2495\beta_{2} - 4193$$ 2164*b11 - 1762*b10 - 1029*b7 - 1995*b4 + 2495*b2 - 4193 $$\nu^{11}$$ $$=$$ $$5688\beta_{9} + 4257\beta_{8} + 6894\beta_{6} - 3024\beta_{5} + 9543\beta_{3} - 8683\beta_1$$ 5688*b9 + 4257*b8 + 6894*b6 - 3024*b5 + 9543*b3 - 8683*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times$$.

 $$n$$ $$626$$ $$1252$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1249.1
 − 2.78712i − 2.38719i − 2.13324i − 2.02791i − 0.858825i − 0.364088i 0.364088i 0.858825i 2.02791i 2.13324i 2.38719i 2.78712i
2.78712i 1.00000i −5.76803 0 −2.78712 3.15000i 10.5020i −1.00000 0
1249.2 2.38719i 1.00000i −3.69868 0 2.38719 3.31671i 4.05506i −1.00000 0
1249.3 2.13324i 1.00000i −2.55073 0 −2.13324 2.16876i 1.17484i −1.00000 0
1249.4 2.02791i 1.00000i −2.11242 0 2.02791 0.505614i 0.227977i −1.00000 0
1249.5 0.858825i 1.00000i 1.26242 0 −0.858825 3.88045i 2.80185i −1.00000 0
1249.6 0.364088i 1.00000i 1.86744 0 0.364088 2.24941i 1.40809i −1.00000 0
1249.7 0.364088i 1.00000i 1.86744 0 0.364088 2.24941i 1.40809i −1.00000 0
1249.8 0.858825i 1.00000i 1.26242 0 −0.858825 3.88045i 2.80185i −1.00000 0
1249.9 2.02791i 1.00000i −2.11242 0 2.02791 0.505614i 0.227977i −1.00000 0
1249.10 2.13324i 1.00000i −2.55073 0 −2.13324 2.16876i 1.17484i −1.00000 0
1249.11 2.38719i 1.00000i −3.69868 0 2.38719 3.31671i 4.05506i −1.00000 0
1249.12 2.78712i 1.00000i −5.76803 0 −2.78712 3.15000i 10.5020i −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1249.12 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1875.2.b.e 12
5.b even 2 1 inner 1875.2.b.e 12
5.c odd 4 1 1875.2.a.i 6
5.c odd 4 1 1875.2.a.l yes 6
15.e even 4 1 5625.2.a.o 6
15.e even 4 1 5625.2.a.r 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1875.2.a.i 6 5.c odd 4 1
1875.2.a.l yes 6 5.c odd 4 1
1875.2.b.e 12 1.a even 1 1 trivial
1875.2.b.e 12 5.b even 2 1 inner
5625.2.a.o 6 15.e even 4 1
5625.2.a.r 6 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 23T_{2}^{10} + 199T_{2}^{8} + 794T_{2}^{6} + 1399T_{2}^{4} + 783T_{2}^{2} + 81$$ acting on $$S_{2}^{\mathrm{new}}(1875, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 23 T^{10} + 199 T^{8} + \cdots + 81$$
$3$ $$(T^{2} + 1)^{6}$$
$5$ $$T^{12}$$
$7$ $$T^{12} + 46 T^{10} + 811 T^{8} + \cdots + 10000$$
$11$ $$(T^{6} - 29 T^{4} - 8 T^{3} + 184 T^{2} + \cdots - 144)^{2}$$
$13$ $$T^{12} + 112 T^{10} + 4784 T^{8} + \cdots + 121801$$
$17$ $$T^{12} + 114 T^{10} + 4705 T^{8} + \cdots + 331776$$
$19$ $$(T^{6} - 2 T^{5} - 74 T^{4} + 120 T^{3} + \cdots - 5725)^{2}$$
$23$ $$T^{12} + 139 T^{10} + 6121 T^{8} + \cdots + 518400$$
$29$ $$(T^{6} + 31 T^{5} + 379 T^{4} + 2320 T^{3} + \cdots + 6480)^{2}$$
$31$ $$(T^{6} + 2 T^{5} - 86 T^{4} + 28 T^{3} + \cdots + 3155)^{2}$$
$37$ $$T^{12} + 366 T^{10} + \cdots + 2125210000$$
$41$ $$(T^{6} - 33 T^{5} + 399 T^{4} - 2192 T^{3} + \cdots + 720)^{2}$$
$43$ $$T^{12} + 161 T^{10} + 7790 T^{8} + \cdots + 1661521$$
$47$ $$T^{12} + 404 T^{10} + \cdots + 6410244096$$
$53$ $$T^{12} + 524 T^{10} + \cdots + 207360000$$
$59$ $$(T^{6} - 8 T^{5} - 69 T^{4} + 600 T^{3} + \cdots - 2880)^{2}$$
$61$ $$(T^{6} - 34 T^{5} + 406 T^{4} + \cdots - 72001)^{2}$$
$67$ $$T^{12} + 224 T^{10} + 13840 T^{8} + \cdots + 3481$$
$71$ $$(T^{6} + 3 T^{5} - 225 T^{4} + 160 T^{3} + \cdots - 12816)^{2}$$
$73$ $$T^{12} + 434 T^{10} + \cdots + 415344400$$
$79$ $$(T^{6} + 25 T^{5} + 150 T^{4} - 395 T^{3} + \cdots + 2725)^{2}$$
$83$ $$T^{12} + 402 T^{10} + \cdots + 557715456$$
$89$ $$(T^{6} + 18 T^{5} - 219 T^{4} + \cdots - 42480)^{2}$$
$97$ $$T^{12} + 669 T^{10} + \cdots + 1042708681$$