# Properties

 Label 1875.2.b.h Level $1875$ Weight $2$ Character orbit 1875.b Analytic conductor $14.972$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Learn more

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: $$16$$ 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: $$2^{8}$$ Twist minimal: no (minimal twist has level 75) 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -3\nu^{14} - 53\nu^{12} - 345\nu^{10} - 1037\nu^{8} - 1518\nu^{6} - 1020\nu^{4} - 245\nu^{2} - 8 ) / 2$$ (-3*v^14 - 53*v^12 - 345*v^10 - 1037*v^8 - 1518*v^6 - 1020*v^4 - 245*v^2 - 8) / 2 $$\beta_{3}$$ $$=$$ $$( -4\nu^{15} - 69\nu^{13} - 430\nu^{11} - 1183\nu^{9} - 1413\nu^{7} - 483\nu^{5} + 179\nu^{3} + 45\nu ) / 2$$ (-4*v^15 - 69*v^13 - 430*v^11 - 1183*v^9 - 1413*v^7 - 483*v^5 + 179*v^3 + 45*v) / 2 $$\beta_{4}$$ $$=$$ $$( 7\nu^{15} + 124\nu^{13} + 810\nu^{11} + 2444\nu^{9} + 3583\nu^{7} + 2400\nu^{5} + 595\nu^{3} + 47\nu ) / 2$$ (7*v^15 + 124*v^13 + 810*v^11 + 2444*v^9 + 3583*v^7 + 2400*v^5 + 595*v^3 + 47*v) / 2 $$\beta_{5}$$ $$=$$ $$( -9\nu^{14} - 158\nu^{12} - 1017\nu^{10} - 2991\nu^{8} - 4185\nu^{6} - 2527\nu^{4} - 435\nu^{2} - 2 ) / 2$$ (-9*v^14 - 158*v^12 - 1017*v^10 - 2991*v^8 - 4185*v^6 - 2527*v^4 - 435*v^2 - 2) / 2 $$\beta_{6}$$ $$=$$ $$( -10\nu^{15} - 177\nu^{13} - 1154\nu^{11} - 3465\nu^{9} - 5013\nu^{7} - 3225\nu^{5} - 675\nu^{3} - 19\nu ) / 2$$ (-10*v^15 - 177*v^13 - 1154*v^11 - 3465*v^9 - 5013*v^7 - 3225*v^5 - 675*v^3 - 19*v) / 2 $$\beta_{7}$$ $$=$$ $$( 11\nu^{14} + 194\nu^{12} + 1257\nu^{10} + 3731\nu^{8} + 5277\nu^{6} + 3223\nu^{4} + 565\nu^{2} + 2 ) / 2$$ (11*v^14 + 194*v^12 + 1257*v^10 + 3731*v^8 + 5277*v^6 + 3223*v^4 + 565*v^2 + 2) / 2 $$\beta_{8}$$ $$=$$ $$( -16\nu^{14} - 282\nu^{12} - 1826\nu^{10} - 5419\nu^{8} - 7680\nu^{6} - 4732\nu^{4} - 865\nu^{2} - 13 ) / 2$$ (-16*v^14 - 282*v^12 - 1826*v^10 - 5419*v^8 - 7680*v^6 - 4732*v^4 - 865*v^2 - 13) / 2 $$\beta_{9}$$ $$=$$ $$( -19\nu^{15} - 335\nu^{13} - 2170\nu^{11} - 6440\nu^{9} - 9110\nu^{7} - 5557\nu^{5} - 945\nu^{3} + 15\nu ) / 2$$ (-19*v^15 - 335*v^13 - 2170*v^11 - 6440*v^9 - 9110*v^7 - 5557*v^5 - 945*v^3 + 15*v) / 2 $$\beta_{10}$$ $$=$$ $$( -22\nu^{15} - 387\nu^{13} - 2498\nu^{11} - 7373\nu^{9} - 10347\nu^{7} - 6237\nu^{5} - 1041\nu^{3} + 7\nu ) / 2$$ (-22*v^15 - 387*v^13 - 2498*v^11 - 7373*v^9 - 10347*v^7 - 6237*v^5 - 1041*v^3 + 7*v) / 2 $$\beta_{11}$$ $$=$$ $$( 26\nu^{15} + 459\nu^{13} + 2980\nu^{11} + 8884\nu^{9} + 12693\nu^{7} + 7957\nu^{5} + 1542\nu^{3} + 40\nu ) / 2$$ (26*v^15 + 459*v^13 + 2980*v^11 + 8884*v^9 + 12693*v^7 + 7957*v^5 + 1542*v^3 + 40*v) / 2 $$\beta_{12}$$ $$=$$ $$( 29\nu^{15} + 512\nu^{13} + 3325\nu^{11} + 9921\nu^{9} + 14211\nu^{7} + 8977\nu^{5} + 1785\nu^{3} + 40\nu ) / 2$$ (29*v^15 + 512*v^13 + 3325*v^11 + 9921*v^9 + 14211*v^7 + 8977*v^5 + 1785*v^3 + 40*v) / 2 $$\beta_{13}$$ $$=$$ $$16\nu^{14} + 282\nu^{12} + 1826\nu^{10} + 5419\nu^{8} + 7680\nu^{6} + 4731\nu^{4} + 858\nu^{2} + 6$$ 16*v^14 + 282*v^12 + 1826*v^10 + 5419*v^8 + 7680*v^6 + 4731*v^4 + 858*v^2 + 6 $$\beta_{14}$$ $$=$$ $$( -45\nu^{14} - 794\nu^{12} - 5150\nu^{10} - 15324\nu^{8} - 21803\nu^{6} - 13514\nu^{4} - 2487\nu^{2} - 23 ) / 2$$ (-45*v^14 - 794*v^12 - 5150*v^10 - 15324*v^8 - 21803*v^6 - 13514*v^4 - 2487*v^2 - 23) / 2 $$\beta_{15}$$ $$=$$ $$( -45\nu^{14} - 794\nu^{12} - 5150\nu^{10} - 15324\nu^{8} - 21803\nu^{6} - 13514\nu^{4} - 2485\nu^{2} - 19 ) / 2$$ (-45*v^14 - 794*v^12 - 5150*v^10 - 15324*v^8 - 21803*v^6 - 13514*v^4 - 2485*v^2 - 19) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{15} - \beta_{14} - 2$$ b15 - b14 - 2 $$\nu^{3}$$ $$=$$ $$\beta_{11} + \beta_{9} - \beta_{4} - 4\beta_1$$ b11 + b9 - b4 - 4*b1 $$\nu^{4}$$ $$=$$ $$-7\beta_{15} + 7\beta_{14} - \beta_{13} - 2\beta_{8} + 7$$ -7*b15 + 7*b14 - b13 - 2*b8 + 7 $$\nu^{5}$$ $$=$$ $$-7\beta_{11} + \beta_{10} - 9\beta_{9} + 3\beta_{6} + 9\beta_{4} + 21\beta_1$$ -7*b11 + b10 - 9*b9 + 3*b6 + 9*b4 + 21*b1 $$\nu^{6}$$ $$=$$ $$46\beta_{15} - 44\beta_{14} + 12\beta_{13} + 16\beta_{8} - \beta_{7} + 3\beta_{5} - 33$$ 46*b15 - 44*b14 + 12*b13 + 16*b8 - b7 + 3*b5 - 33 $$\nu^{7}$$ $$=$$ $$-4\beta_{12} + 44\beta_{11} - 13\beta_{10} + 64\beta_{9} - 34\beta_{6} - 62\beta_{4} + \beta_{3} - 123\beta_1$$ -4*b12 + 44*b11 - 13*b10 + 64*b9 - 34*b6 - 62*b4 + b3 - 123*b1 $$\nu^{8}$$ $$=$$ $$-297\beta_{15} + 273\beta_{14} - 104\beta_{13} - 106\beta_{8} + 18\beta_{7} - 38\beta_{5} - 4\beta_{2} + 181$$ -297*b15 + 273*b14 - 104*b13 - 106*b8 + 18*b7 - 38*b5 - 4*b2 + 181 $$\nu^{9}$$ $$=$$ $$60\beta_{12} - 273\beta_{11} + 122\beta_{10} - 423\beta_{9} + 286\beta_{6} + 399\beta_{4} - 18\beta_{3} + 751\beta_1$$ 60*b12 - 273*b11 + 122*b10 - 423*b9 + 286*b6 + 399*b4 - 18*b3 + 751*b1 $$\nu^{10}$$ $$=$$ $$1906\beta_{15} - 1696\beta_{14} + 803\beta_{13} + 672\beta_{8} - 200\beta_{7} + 346\beta_{5} + 60\beta_{2} - 1061$$ 1906*b15 - 1696*b14 + 803*b13 + 672*b8 - 200*b7 + 346*b5 + 60*b2 - 1061 $$\nu^{11}$$ $$=$$ $$- 606 \beta_{12} + 1696 \beta_{11} - 1003 \beta_{10} + 2728 \beta_{9} - 2167 \beta_{6} - 2518 \beta_{4} + 200 \beta_{3} - 4663 \beta_1$$ -606*b12 + 1696*b11 - 1003*b10 + 2728*b9 - 2167*b6 - 2518*b4 + 200*b3 - 4663*b1 $$\nu^{12}$$ $$=$$ $$- 12211 \beta_{15} + 10573 \beta_{14} - 5869 \beta_{13} - 4214 \beta_{8} + 1809 \beta_{7} - 2773 \beta_{5} - 606 \beta_{2} + 6402$$ -12211*b15 + 10573*b14 - 5869*b13 - 4214*b8 + 1809*b7 - 2773*b5 - 606*b2 + 6402 $$\nu^{13}$$ $$=$$ $$5188 \beta_{12} - 10573 \beta_{11} + 7678 \beta_{10} - 17457 \beta_{9} + 15629 \beta_{6} + 15819 \beta_{4} - 1809 \beta_{3} + 29186 \beta_1$$ 5188*b12 - 10573*b11 + 7678*b10 - 17457*b9 + 15629*b6 + 15819*b4 - 1809*b3 + 29186*b1 $$\nu^{14}$$ $$=$$ $$78223 \beta_{15} - 66151 \beta_{14} + 41558 \beta_{13} + 26392 \beta_{8} - 14675 \beta_{7} + 20817 \beta_{5} + 5188 \beta_{2} - 39174$$ 78223*b15 - 66151*b14 + 41558*b13 + 26392*b8 - 14675*b7 + 20817*b5 + 5188*b2 - 39174 $$\nu^{15}$$ $$=$$ $$- 40680 \beta_{12} + 66151 \beta_{11} - 56233 \beta_{10} + 111499 \beta_{9} - 109584 \beta_{6} - 99427 \beta_{4} + 14675 \beta_{3} - 183548 \beta_1$$ -40680*b12 + 66151*b11 - 56233*b10 + 111499*b9 - 109584*b6 - 99427*b4 + 14675*b3 - 183548*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
 1.53767i 1.35083i 2.53767i 0.536547i 2.35083i 0.0898194i 1.53655i 1.08982i − 1.08982i − 1.53655i − 0.0898194i − 2.35083i − 0.536547i − 2.53767i − 1.35083i − 1.53767i
2.53767i 1.00000i −4.43979 0 −2.53767 1.04054i 6.19138i −1.00000 0
1249.2 2.35083i 1.00000i −3.52640 0 −2.35083 3.48189i 3.58831i −1.00000 0
1249.3 1.53767i 1.00000i −0.364440 0 1.53767 1.68601i 2.51496i −1.00000 0
1249.4 1.53655i 1.00000i −0.360976 0 −1.53655 1.49550i 2.51844i −1.00000 0
1249.5 1.35083i 1.00000i 0.175259 0 1.35083 1.59580i 2.93840i −1.00000 0
1249.6 1.08982i 1.00000i 0.812294 0 −1.08982 3.08724i 3.06489i −1.00000 0
1249.7 0.536547i 1.00000i 1.71212 0 0.536547 2.57318i 1.99173i −1.00000 0
1249.8 0.0898194i 1.00000i 1.99193 0 0.0898194 4.36070i 0.358553i −1.00000 0
1249.9 0.0898194i 1.00000i 1.99193 0 0.0898194 4.36070i 0.358553i −1.00000 0
1249.10 0.536547i 1.00000i 1.71212 0 0.536547 2.57318i 1.99173i −1.00000 0
1249.11 1.08982i 1.00000i 0.812294 0 −1.08982 3.08724i 3.06489i −1.00000 0
1249.12 1.35083i 1.00000i 0.175259 0 1.35083 1.59580i 2.93840i −1.00000 0
1249.13 1.53655i 1.00000i −0.360976 0 −1.53655 1.49550i 2.51844i −1.00000 0
1249.14 1.53767i 1.00000i −0.364440 0 1.53767 1.68601i 2.51496i −1.00000 0
1249.15 2.35083i 1.00000i −3.52640 0 −2.35083 3.48189i 3.58831i −1.00000 0
1249.16 2.53767i 1.00000i −4.43979 0 −2.53767 1.04054i 6.19138i −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1249.16 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.h 16
5.b even 2 1 inner 1875.2.b.h 16
5.c odd 4 1 1875.2.a.m 8
5.c odd 4 1 1875.2.a.p 8
15.e even 4 1 5625.2.a.t 8
15.e even 4 1 5625.2.a.bd 8
25.d even 5 1 75.2.i.a 16
25.d even 5 1 375.2.i.c 16
25.e even 10 1 75.2.i.a 16
25.e even 10 1 375.2.i.c 16
25.f odd 20 2 375.2.g.d 16
25.f odd 20 2 375.2.g.e 16
75.h odd 10 1 225.2.m.b 16
75.j odd 10 1 225.2.m.b 16

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} + 20T_{2}^{14} + 156T_{2}^{12} + 610T_{2}^{10} + 1286T_{2}^{8} + 1440T_{2}^{6} + 761T_{2}^{4} + 130T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(1875, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 20 T^{14} + 156 T^{12} + 610 T^{10} + \cdots + 1$$
$3$ $$(T^{2} + 1)^{8}$$
$5$ $$T^{16}$$
$7$ $$T^{16} + 56 T^{14} + 1236 T^{12} + \cdots + 255025$$
$11$ $$(T^{8} - 2 T^{7} - 43 T^{6} + 90 T^{5} + \cdots + 5281)^{2}$$
$13$ $$T^{16} + 110 T^{14} + 4141 T^{12} + \cdots + 78961$$
$17$ $$T^{16} + 172 T^{14} + \cdots + 53860921$$
$19$ $$(T^{8} - 14 T^{7} + 11 T^{6} + 516 T^{5} + \cdots + 2525)^{2}$$
$23$ $$T^{16} + 174 T^{14} + 10761 T^{12} + \cdots + 4389025$$
$29$ $$(T^{8} + 2 T^{7} - 106 T^{6} - 162 T^{5} + \cdots - 395)^{2}$$
$31$ $$(T^{8} + 22 T^{7} + 169 T^{6} + 548 T^{5} + \cdots + 125)^{2}$$
$37$ $$T^{16} + 376 T^{14} + \cdots + 8653650625$$
$41$ $$(T^{8} - 8 T^{7} - 56 T^{6} + 428 T^{5} + \cdots + 4705)^{2}$$
$43$ $$T^{16} + 388 T^{14} + \cdots + 527207521$$
$47$ $$T^{16} + 472 T^{14} + \cdots + 36687479166361$$
$53$ $$T^{16} + 444 T^{14} + \cdots + 40398990025$$
$59$ $$(T^{8} + 14 T^{7} - 29 T^{6} - 886 T^{5} + \cdots - 3595)^{2}$$
$61$ $$(T^{8} + 20 T^{7} - 136 T^{6} + \cdots + 16604261)^{2}$$
$67$ $$T^{16} + 532 T^{14} + \cdots + 13980121$$
$71$ $$(T^{8} - 16 T^{7} - 78 T^{6} + \cdots - 159779)^{2}$$
$73$ $$T^{16} + 644 T^{14} + \cdots + 757413387025$$
$79$ $$(T^{8} - 30 T^{7} + 145 T^{6} + \cdots - 1984975)^{2}$$
$83$ $$T^{16} + 520 T^{14} + \cdots + 2356228681$$
$89$ $$(T^{8} + 16 T^{7} - 39 T^{6} - 1454 T^{5} + \cdots + 5)^{2}$$
$97$ $$T^{16} + 472 T^{14} + \cdots + 216648961$$
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