Properties

 Label 725.2.b.f Level $725$ Weight $2$ Character orbit 725.b Analytic conductor $5.789$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(349,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("725.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$725 = 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 725.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: $$5.78915414654$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} + 14x^{8} + 71x^{6} + 159x^{4} + 151x^{2} + 49$$ x^10 + 14*x^8 + 71*x^6 + 159*x^4 + 151*x^2 + 49 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 14x^{8} + 71x^{6} + 159x^{4} + 151x^{2} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 3$$ v^2 + 3 $$\beta_{3}$$ $$=$$ $$( \nu^{9} + 7\nu^{7} + \nu^{5} - 44\nu^{3} - 24\nu ) / 7$$ (v^9 + 7*v^7 + v^5 - 44*v^3 - 24*v) / 7 $$\beta_{4}$$ $$=$$ $$( 2\nu^{9} + 21\nu^{7} + 65\nu^{5} + 52\nu^{3} - 13\nu ) / 7$$ (2*v^9 + 21*v^7 + 65*v^5 + 52*v^3 - 13*v) / 7 $$\beta_{5}$$ $$=$$ $$-\nu^{6} - 9\nu^{4} - 22\nu^{2} - 12$$ -v^6 - 9*v^4 - 22*v^2 - 12 $$\beta_{6}$$ $$=$$ $$\nu^{6} + 10\nu^{4} + 28\nu^{2} + 18$$ v^6 + 10*v^4 + 28*v^2 + 18 $$\beta_{7}$$ $$=$$ $$( -3\nu^{9} - 35\nu^{7} - 136\nu^{5} - 204\nu^{3} - 96\nu ) / 7$$ (-3*v^9 - 35*v^7 - 136*v^5 - 204*v^3 - 96*v) / 7 $$\beta_{8}$$ $$=$$ $$( -2\nu^{9} - 28\nu^{7} - 135\nu^{5} - 255\nu^{3} - 141\nu ) / 7$$ (-2*v^9 - 28*v^7 - 135*v^5 - 255*v^3 - 141*v) / 7 $$\beta_{9}$$ $$=$$ $$\nu^{8} + 11\nu^{6} + 39\nu^{4} + 51\nu^{2} + 21$$ v^8 + 11*v^6 + 39*v^4 + 51*v^2 + 21
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 3$$ b2 - 3 $$\nu^{3}$$ $$=$$ $$-\beta_{8} + \beta_{7} + \beta_{3} - 3\beta_1$$ -b8 + b7 + b3 - 3*b1 $$\nu^{4}$$ $$=$$ $$\beta_{6} + \beta_{5} - 6\beta_{2} + 12$$ b6 + b5 - 6*b2 + 12 $$\nu^{5}$$ $$=$$ $$8\beta_{8} - 9\beta_{7} - 2\beta_{4} - 7\beta_{3} + 10\beta_1$$ 8*b8 - 9*b7 - 2*b4 - 7*b3 + 10*b1 $$\nu^{6}$$ $$=$$ $$-9\beta_{6} - 10\beta_{5} + 32\beta_{2} - 54$$ -9*b6 - 10*b5 + 32*b2 - 54 $$\nu^{7}$$ $$=$$ $$-52\beta_{8} + 61\beta_{7} + 19\beta_{4} + 41\beta_{3} - 35\beta_1$$ -52*b8 + 61*b7 + 19*b4 + 41*b3 - 35*b1 $$\nu^{8}$$ $$=$$ $$\beta_{9} + 60\beta_{6} + 71\beta_{5} - 169\beta_{2} + 258$$ b9 + 60*b6 + 71*b5 - 169*b2 + 258 $$\nu^{9}$$ $$=$$ $$312\beta_{8} - 374\beta_{7} - 131\beta_{4} - 229\beta_{3} + 127\beta_1$$ 312*b8 - 374*b7 - 131*b4 - 229*b3 + 127*b1

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/725\mathbb{Z}\right)^\times$$.

 $$n$$ $$176$$ $$552$$ $$\chi(n)$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
349.1
 − 2.33090i − 1.88481i − 1.78154i − 1.06634i − 0.838718i 0.838718i 1.06634i 1.78154i 1.88481i 2.33090i
2.33090i 0.318289i −3.43308 0 0.741899 1.09271i 3.34037i 2.89869 0
349.2 1.88481i 1.10085i −1.55251 0 −2.07489 1.70908i 0.843434i 1.78814 0
349.3 1.78154i 3.10566i −1.17387 0 5.53284 3.64565i 1.47177i −6.64510 0
349.4 1.06634i 3.37897i 0.862915 0 −3.60314 2.91576i 3.05285i −8.41742 0
349.5 0.838718i 1.90376i 1.29655 0 −1.59672 2.46832i 2.76488i −0.624302 0
349.6 0.838718i 1.90376i 1.29655 0 −1.59672 2.46832i 2.76488i −0.624302 0
349.7 1.06634i 3.37897i 0.862915 0 −3.60314 2.91576i 3.05285i −8.41742 0
349.8 1.78154i 3.10566i −1.17387 0 5.53284 3.64565i 1.47177i −6.64510 0
349.9 1.88481i 1.10085i −1.55251 0 −2.07489 1.70908i 0.843434i 1.78814 0
349.10 2.33090i 0.318289i −3.43308 0 0.741899 1.09271i 3.34037i 2.89869 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 349.10 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 725.2.b.f 10
5.b even 2 1 inner 725.2.b.f 10
5.c odd 4 1 725.2.a.h 5
5.c odd 4 1 725.2.a.k yes 5
15.e even 4 1 6525.2.a.bm 5
15.e even 4 1 6525.2.a.bq 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
725.2.a.h 5 5.c odd 4 1
725.2.a.k yes 5 5.c odd 4 1
725.2.b.f 10 1.a even 1 1 trivial
725.2.b.f 10 5.b even 2 1 inner
6525.2.a.bm 5 15.e even 4 1
6525.2.a.bq 5 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(725, [\chi])$$:

 $$T_{2}^{10} + 14T_{2}^{8} + 71T_{2}^{6} + 159T_{2}^{4} + 151T_{2}^{2} + 49$$ T2^10 + 14*T2^8 + 71*T2^6 + 159*T2^4 + 151*T2^2 + 49 $$T_{3}^{10} + 26T_{3}^{8} + 219T_{3}^{6} + 647T_{3}^{4} + 547T_{3}^{2} + 49$$ T3^10 + 26*T3^8 + 219*T3^6 + 647*T3^4 + 547*T3^2 + 49

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + 14 T^{8} + 71 T^{6} + 159 T^{4} + \cdots + 49$$
$3$ $$T^{10} + 26 T^{8} + 219 T^{6} + \cdots + 49$$
$5$ $$T^{10}$$
$7$ $$T^{10} + 32 T^{8} + 364 T^{6} + \cdots + 2401$$
$11$ $$(T^{5} + 2 T^{4} - 35 T^{3} - 64 T^{2} + \cdots + 307)^{2}$$
$13$ $$T^{10} + 52 T^{8} + 528 T^{6} + 521 T^{4} + \cdots + 1$$
$17$ $$T^{10} + 87 T^{8} + 2563 T^{6} + \cdots + 192721$$
$19$ $$(T^{5} + 2 T^{4} - 47 T^{3} - 81 T^{2} + \cdots - 5)^{2}$$
$23$ $$T^{10} + 125 T^{8} + 4816 T^{6} + \cdots + 337561$$
$29$ $$(T + 1)^{10}$$
$31$ $$(T^{5} + T^{4} - 30 T^{3} + 43 T^{2} + 49 T - 73)^{2}$$
$37$ $$T^{10} + 90 T^{8} + 2219 T^{6} + \cdots + 3969$$
$41$ $$(T^{5} - 5 T^{4} - 13 T^{3} + 22 T^{2} + \cdots + 9)^{2}$$
$43$ $$T^{10} + 314 T^{8} + \cdots + 521345889$$
$47$ $$T^{10} + 151 T^{8} + 7423 T^{6} + \cdots + 5121169$$
$53$ $$T^{10} + 174 T^{8} + 8475 T^{6} + \cdots + 744769$$
$59$ $$(T^{5} - 11 T^{4} - 36 T^{3} + 413 T^{2} + \cdots - 2205)^{2}$$
$61$ $$(T^{5} + 5 T^{4} - 233 T^{3} - 547 T^{2} + \cdots - 16381)^{2}$$
$67$ $$T^{10} + 259 T^{8} + 18107 T^{6} + \cdots + 124609$$
$71$ $$(T^{5} + 5 T^{4} - 98 T^{3} - 241 T^{2} + \cdots + 2837)^{2}$$
$73$ $$T^{10} + 196 T^{8} + 8288 T^{6} + \cdots + 81$$
$79$ $$(T^{5} - 10 T^{4} - 146 T^{3} + \cdots - 10035)^{2}$$
$83$ $$T^{10} + 407 T^{8} + \cdots + 328080769$$
$89$ $$(T^{5} - 18 T^{4} + 86 T^{3} - 51 T^{2} + \cdots + 25)^{2}$$
$97$ $$T^{10} + 731 T^{8} + \cdots + 38574138409$$