# Properties

 Label 725.2.b.g 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: $$x^{10} + 14x^{8} + 63x^{6} + 99x^{4} + 55x^{2} + 9$$ x^10 + 14*x^8 + 63*x^6 + 99*x^4 + 55*x^2 + 9 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{7} + \beta_{2}) q^{3} + (\beta_{6} + \beta_{5} - 1) q^{4} + ( - \beta_{8} + \beta_{6} - 2 \beta_{3}) q^{6} + ( - \beta_{9} - 2 \beta_{2} - \beta_1) q^{7} + (\beta_{9} - \beta_{7} + \beta_{4} - \beta_1) q^{8} + (2 \beta_{6} - \beta_{3} - 1) q^{9}+O(q^{10})$$ q + b1 * q^2 + (b7 + b2) * q^3 + (b6 + b5 - 1) * q^4 + (-b8 + b6 - 2*b3) * q^6 + (-b9 - 2*b2 - b1) * q^7 + (b9 - b7 + b4 - b1) * q^8 + (2*b6 - b3 - 1) * q^9 $$q + \beta_1 q^{2} + (\beta_{7} + \beta_{2}) q^{3} + (\beta_{6} + \beta_{5} - 1) q^{4} + ( - \beta_{8} + \beta_{6} - 2 \beta_{3}) q^{6} + ( - \beta_{9} - 2 \beta_{2} - \beta_1) q^{7} + (\beta_{9} - \beta_{7} + \beta_{4} - \beta_1) q^{8} + (2 \beta_{6} - \beta_{3} - 1) q^{9} + (\beta_{5} - \beta_{3}) q^{11} + (\beta_{9} - 2 \beta_{7} - 2 \beta_{4} - 4 \beta_{2} - \beta_1) q^{12} + ( - 2 \beta_{9} + \beta_{4} + 2 \beta_{2}) q^{13} + ( - \beta_{5} + 2 \beta_{3} + 3) q^{14} + (\beta_{8} - \beta_{6} + 1) q^{16} + (\beta_{7} + 2 \beta_{4} - \beta_1) q^{17} + (2 \beta_{9} - 3 \beta_{7} - \beta_{4} - 3 \beta_{2} - 3 \beta_1) q^{18} + ( - \beta_{6} - \beta_{5} + 1) q^{19} + (\beta_{8} - 3 \beta_{6} + \beta_{5} + \beta_{3} + 2) q^{21} + ( - \beta_{7} - 3 \beta_{2} - \beta_1) q^{22} + ( - \beta_{9} + \beta_{7} - \beta_{4}) q^{23} + ( - 2 \beta_{6} + \beta_{5} + 4 \beta_{3} + 3) q^{24} + (2 \beta_{6} - \beta_{5} - 3 \beta_{3}) q^{26} + (\beta_{9} - \beta_{7} - 4 \beta_{2} - 4 \beta_1) q^{27} + ( - 2 \beta_{9} + 2 \beta_{7} + \beta_{4} + 2 \beta_{2} + 2 \beta_1) q^{28} + q^{29} + (\beta_{8} + \beta_{6} - 2 \beta_{5} - \beta_{3} + 2) q^{31} + (\beta_{9} + 2 \beta_{4}) q^{32} + (2 \beta_{9} - \beta_{7} - 2 \beta_{4} - 2 \beta_1) q^{33} + ( - \beta_{8} - 3 \beta_{5} - 3 \beta_{3} + 3) q^{34} + (3 \beta_{8} - 4 \beta_{6} - 2 \beta_{5} + 5 \beta_{3} + 7) q^{36} + (2 \beta_{7} - 2 \beta_{4} - 2 \beta_{2} + \beta_1) q^{37} + ( - \beta_{9} + \beta_{7} - \beta_{4} + 3 \beta_1) q^{38} + ( - \beta_{8} + 2 \beta_{6} + 4 \beta_{5} - 2 \beta_{3} - 2) q^{39} + ( - 4 \beta_{8} + \beta_{6} - \beta_{5} - 2 \beta_{3}) q^{41} + ( - 3 \beta_{9} + 5 \beta_{7} + 2 \beta_{4} + 3 \beta_{2} + 4 \beta_1) q^{42} + (\beta_{9} - \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{43} + (\beta_{8} - 2 \beta_{6} + \beta_{5} + 2 \beta_{3} + 3) q^{44} + ( - \beta_{8} + 2 \beta_{6} + \beta_{5}) q^{46} + (2 \beta_{9} + \beta_{7} + 2 \beta_1) q^{47} + (2 \beta_{7} + \beta_{4} + 4 \beta_{2} + 2 \beta_1) q^{48} + (3 \beta_{8} - 2 \beta_{6} - 4 \beta_{3} - 3) q^{49} + ( - \beta_{8} + 4 \beta_{5} + \beta_{3} - 3) q^{51} + ( - 2 \beta_{9} - 5 \beta_{7} - 2 \beta_{4} - 5 \beta_{2} - \beta_1) q^{52} + (\beta_{9} - 3 \beta_{7} + 3 \beta_{2} + 2 \beta_1) q^{53} + (\beta_{8} - 6 \beta_{6} - 4 \beta_{5} + 5 \beta_{3} + 12) q^{54} + ( - 2 \beta_{8} + 6 \beta_{6} - \beta_{5} - \beta_{3}) q^{56} + ( - \beta_{9} + 2 \beta_{7} + 2 \beta_{4} + 4 \beta_{2} + \beta_1) q^{57} + \beta_1 q^{58} + ( - \beta_{8} - 2 \beta_{3} + 3) q^{59} + ( - 4 \beta_{8} - \beta_{5} + \beta_{3} - 1) q^{61} + (\beta_{9} - \beta_{7} - 3 \beta_{4} - 3 \beta_{2} + 3 \beta_1) q^{62} + ( - 3 \beta_{9} + 6 \beta_{7} - \beta_{4} + 5 \beta_{2} + 3 \beta_1) q^{63} + (2 \beta_{8} - 3 \beta_{6} - 2 \beta_{5} - 2 \beta_{3} + 2) q^{64} + (\beta_{8} - 5 \beta_{6} + 3 \beta_{3} + 6) q^{66} + (4 \beta_{7} - 2 \beta_{4} - 5 \beta_{2} - \beta_1) q^{67} + ( - 2 \beta_{7} - 2 \beta_{4} - 9 \beta_{2} + 4 \beta_1) q^{68} + (\beta_{8} + \beta_{6} - \beta_{5} - 2 \beta_{3} - 3) q^{69} + (4 \beta_{8} + 4 \beta_{6} + \beta_{5} + 2 \beta_{3} - 3) q^{71} + (6 \beta_{7} + \beta_{4} + 9 \beta_{2} + 7 \beta_1) q^{72} + (2 \beta_{9} - 2 \beta_{7} - \beta_{4} + 2 \beta_{2} - 4 \beta_1) q^{73} + ( - 2 \beta_{8} + 3 \beta_{6} + 3 \beta_{5} + 2 \beta_{3} - 3) q^{74} + ( - \beta_{8} + 3 \beta_{6} + 2 \beta_{5} - 7) q^{76} + ( - \beta_{9} - \beta_{7} + 3 \beta_{4} + 3 \beta_{2} + 3 \beta_1) q^{77} + (2 \beta_{9} - 5 \beta_{7} + 2 \beta_{4} - 6 \beta_{2} - 8 \beta_1) q^{78} + ( - 5 \beta_{8} + \beta_{6} - \beta_{5} - 2 \beta_{3} + 1) q^{79} + (4 \beta_{8} - 3 \beta_{6} - \beta_{5} + 7 \beta_{3} + 4) q^{81} + (\beta_{9} - 7 \beta_{7} - 3 \beta_{4} - 6 \beta_{2}) q^{82} + (4 \beta_{9} + \beta_{7} + 6 \beta_{4} + 6 \beta_{2} + \beta_1) q^{83} + ( - 3 \beta_{8} + 6 \beta_{6} + 4 \beta_{5} - 8 \beta_{3} - 8) q^{84} + ( - 3 \beta_{6} - \beta_{5} - \beta_{3} + 6) q^{86} + (\beta_{7} + \beta_{2}) q^{87} + ( - 2 \beta_{9} + 3 \beta_{7} + 3 \beta_{4} + 2 \beta_1) q^{88} + ( - 5 \beta_{8} + 5 \beta_{6} + 3 \beta_{5} + 2 \beta_{3} - 3) q^{89} + (\beta_{8} - 3 \beta_{6} - 4 \beta_{5} + 3 \beta_{3} - 2) q^{91} + ( - \beta_{7} - \beta_{4} - 3 \beta_1) q^{92} + ( - \beta_{9} + 5 \beta_{4} - \beta_{2} - 2 \beta_1) q^{93} + ( - \beta_{8} + \beta_{6} + 2 \beta_{5} - \beta_{3} - 6) q^{94} + ( - 2 \beta_{8} + 3 \beta_{5} + \beta_{3}) q^{96} + (4 \beta_{9} - 3 \beta_{7} + 4 \beta_{4} + 4 \beta_{2}) q^{97} + ( - 2 \beta_{9} + \beta_{7} - 4 \beta_{4} - 12 \beta_{2} - \beta_1) q^{98} + (4 \beta_{8} - 3 \beta_{6} - 3 \beta_{5} + 4 \beta_{3} + 3) q^{99}+O(q^{100})$$ q + b1 * q^2 + (b7 + b2) * q^3 + (b6 + b5 - 1) * q^4 + (-b8 + b6 - 2*b3) * q^6 + (-b9 - 2*b2 - b1) * q^7 + (b9 - b7 + b4 - b1) * q^8 + (2*b6 - b3 - 1) * q^9 + (b5 - b3) * q^11 + (b9 - 2*b7 - 2*b4 - 4*b2 - b1) * q^12 + (-2*b9 + b4 + 2*b2) * q^13 + (-b5 + 2*b3 + 3) * q^14 + (b8 - b6 + 1) * q^16 + (b7 + 2*b4 - b1) * q^17 + (2*b9 - 3*b7 - b4 - 3*b2 - 3*b1) * q^18 + (-b6 - b5 + 1) * q^19 + (b8 - 3*b6 + b5 + b3 + 2) * q^21 + (-b7 - 3*b2 - b1) * q^22 + (-b9 + b7 - b4) * q^23 + (-2*b6 + b5 + 4*b3 + 3) * q^24 + (2*b6 - b5 - 3*b3) * q^26 + (b9 - b7 - 4*b2 - 4*b1) * q^27 + (-2*b9 + 2*b7 + b4 + 2*b2 + 2*b1) * q^28 + q^29 + (b8 + b6 - 2*b5 - b3 + 2) * q^31 + (b9 + 2*b4) * q^32 + (2*b9 - b7 - 2*b4 - 2*b1) * q^33 + (-b8 - 3*b5 - 3*b3 + 3) * q^34 + (3*b8 - 4*b6 - 2*b5 + 5*b3 + 7) * q^36 + (2*b7 - 2*b4 - 2*b2 + b1) * q^37 + (-b9 + b7 - b4 + 3*b1) * q^38 + (-b8 + 2*b6 + 4*b5 - 2*b3 - 2) * q^39 + (-4*b8 + b6 - b5 - 2*b3) * q^41 + (-3*b9 + 5*b7 + 2*b4 + 3*b2 + 4*b1) * q^42 + (b9 - b4 + 2*b2 - 2*b1) * q^43 + (b8 - 2*b6 + b5 + 2*b3 + 3) * q^44 + (-b8 + 2*b6 + b5) * q^46 + (2*b9 + b7 + 2*b1) * q^47 + (2*b7 + b4 + 4*b2 + 2*b1) * q^48 + (3*b8 - 2*b6 - 4*b3 - 3) * q^49 + (-b8 + 4*b5 + b3 - 3) * q^51 + (-2*b9 - 5*b7 - 2*b4 - 5*b2 - b1) * q^52 + (b9 - 3*b7 + 3*b2 + 2*b1) * q^53 + (b8 - 6*b6 - 4*b5 + 5*b3 + 12) * q^54 + (-2*b8 + 6*b6 - b5 - b3) * q^56 + (-b9 + 2*b7 + 2*b4 + 4*b2 + b1) * q^57 + b1 * q^58 + (-b8 - 2*b3 + 3) * q^59 + (-4*b8 - b5 + b3 - 1) * q^61 + (b9 - b7 - 3*b4 - 3*b2 + 3*b1) * q^62 + (-3*b9 + 6*b7 - b4 + 5*b2 + 3*b1) * q^63 + (2*b8 - 3*b6 - 2*b5 - 2*b3 + 2) * q^64 + (b8 - 5*b6 + 3*b3 + 6) * q^66 + (4*b7 - 2*b4 - 5*b2 - b1) * q^67 + (-2*b7 - 2*b4 - 9*b2 + 4*b1) * q^68 + (b8 + b6 - b5 - 2*b3 - 3) * q^69 + (4*b8 + 4*b6 + b5 + 2*b3 - 3) * q^71 + (6*b7 + b4 + 9*b2 + 7*b1) * q^72 + (2*b9 - 2*b7 - b4 + 2*b2 - 4*b1) * q^73 + (-2*b8 + 3*b6 + 3*b5 + 2*b3 - 3) * q^74 + (-b8 + 3*b6 + 2*b5 - 7) * q^76 + (-b9 - b7 + 3*b4 + 3*b2 + 3*b1) * q^77 + (2*b9 - 5*b7 + 2*b4 - 6*b2 - 8*b1) * q^78 + (-5*b8 + b6 - b5 - 2*b3 + 1) * q^79 + (4*b8 - 3*b6 - b5 + 7*b3 + 4) * q^81 + (b9 - 7*b7 - 3*b4 - 6*b2) * q^82 + (4*b9 + b7 + 6*b4 + 6*b2 + b1) * q^83 + (-3*b8 + 6*b6 + 4*b5 - 8*b3 - 8) * q^84 + (-3*b6 - b5 - b3 + 6) * q^86 + (b7 + b2) * q^87 + (-2*b9 + 3*b7 + 3*b4 + 2*b1) * q^88 + (-5*b8 + 5*b6 + 3*b5 + 2*b3 - 3) * q^89 + (b8 - 3*b6 - 4*b5 + 3*b3 - 2) * q^91 + (-b7 - b4 - 3*b1) * q^92 + (-b9 + 5*b4 - b2 - 2*b1) * q^93 + (-b8 + b6 + 2*b5 - b3 - 6) * q^94 + (-2*b8 + 3*b5 + b3) * q^96 + (4*b9 - 3*b7 + 4*b4 + 4*b2) * q^97 + (-2*b9 + b7 - 4*b4 - 12*b2 - b1) * q^98 + (4*b8 - 3*b6 - 3*b5 + 4*b3 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 8 q^{4} - 2 q^{6} - 14 q^{9}+O(q^{10})$$ 10 * q - 8 * q^4 - 2 * q^6 - 14 * q^9 $$10 q - 8 q^{4} - 2 q^{6} - 14 q^{9} + 4 q^{11} + 26 q^{14} + 12 q^{16} + 8 q^{19} + 30 q^{21} + 38 q^{24} - 8 q^{26} + 10 q^{29} + 10 q^{31} + 18 q^{34} + 70 q^{36} - 8 q^{39} - 6 q^{41} + 38 q^{44} - 26 q^{49} - 14 q^{51} + 116 q^{54} - 16 q^{56} + 30 q^{59} - 14 q^{61} + 18 q^{64} + 70 q^{66} - 36 q^{69} - 34 q^{71} - 24 q^{74} - 68 q^{76} + 4 q^{79} + 42 q^{81} - 76 q^{84} + 62 q^{86} - 28 q^{89} - 30 q^{91} - 54 q^{94} + 12 q^{96} + 24 q^{99}+O(q^{100})$$ 10 * q - 8 * q^4 - 2 * q^6 - 14 * q^9 + 4 * q^11 + 26 * q^14 + 12 * q^16 + 8 * q^19 + 30 * q^21 + 38 * q^24 - 8 * q^26 + 10 * q^29 + 10 * q^31 + 18 * q^34 + 70 * q^36 - 8 * q^39 - 6 * q^41 + 38 * q^44 - 26 * q^49 - 14 * q^51 + 116 * q^54 - 16 * q^56 + 30 * q^59 - 14 * q^61 + 18 * q^64 + 70 * q^66 - 36 * q^69 - 34 * q^71 - 24 * q^74 - 68 * q^76 + 4 * q^79 + 42 * q^81 - 76 * q^84 + 62 * q^86 - 28 * q^89 - 30 * q^91 - 54 * q^94 + 12 * q^96 + 24 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 14x^{8} + 63x^{6} + 99x^{4} + 55x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{9} + 11\nu^{7} + 30\nu^{5} - 6\nu^{3} - 32\nu ) / 15$$ (v^9 + 11*v^7 + 30*v^5 - 6*v^3 - 32*v) / 15 $$\beta_{3}$$ $$=$$ $$( \nu^{8} + 11\nu^{6} + 35\nu^{4} + 29\nu^{2} + 3 ) / 5$$ (v^8 + 11*v^6 + 35*v^4 + 29*v^2 + 3) / 5 $$\beta_{4}$$ $$=$$ $$( -2\nu^{9} - 27\nu^{7} - 110\nu^{5} - 123\nu^{3} - 16\nu ) / 5$$ (-2*v^9 - 27*v^7 - 110*v^5 - 123*v^3 - 16*v) / 5 $$\beta_{5}$$ $$=$$ $$( -2\nu^{8} - 27\nu^{6} - 110\nu^{4} - 123\nu^{2} - 21 ) / 5$$ (-2*v^8 - 27*v^6 - 110*v^4 - 123*v^2 - 21) / 5 $$\beta_{6}$$ $$=$$ $$( 2\nu^{8} + 27\nu^{6} + 110\nu^{4} + 128\nu^{2} + 36 ) / 5$$ (2*v^8 + 27*v^6 + 110*v^4 + 128*v^2 + 36) / 5 $$\beta_{7}$$ $$=$$ $$( 2\nu^{9} + 27\nu^{7} + 115\nu^{5} + 158\nu^{3} + 51\nu ) / 5$$ (2*v^9 + 27*v^7 + 115*v^5 + 158*v^3 + 51*v) / 5 $$\beta_{8}$$ $$=$$ $$( 2\nu^{8} + 27\nu^{6} + 115\nu^{4} + 158\nu^{2} + 51 ) / 5$$ (2*v^8 + 27*v^6 + 115*v^4 + 158*v^2 + 51) / 5 $$\beta_{9}$$ $$=$$ $$( 4\nu^{9} + 54\nu^{7} + 225\nu^{5} + 286\nu^{3} + 92\nu ) / 5$$ (4*v^9 + 54*v^7 + 225*v^5 + 286*v^3 + 92*v) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} - 3$$ b6 + b5 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{9} - \beta_{7} + \beta_{4} - 5\beta_1$$ b9 - b7 + b4 - 5*b1 $$\nu^{4}$$ $$=$$ $$\beta_{8} - 7\beta_{6} - 6\beta_{5} + 15$$ b8 - 7*b6 - 6*b5 + 15 $$\nu^{5}$$ $$=$$ $$-7\beta_{9} + 8\beta_{7} - 6\beta_{4} + 28\beta_1$$ -7*b9 + 8*b7 - 6*b4 + 28*b1 $$\nu^{6}$$ $$=$$ $$-8\beta_{8} + 43\beta_{6} + 34\beta_{5} - 2\beta_{3} - 84$$ -8*b8 + 43*b6 + 34*b5 - 2*b3 - 84 $$\nu^{7}$$ $$=$$ $$43\beta_{9} - 53\beta_{7} + 32\beta_{4} - 6\beta_{2} - 161\beta_1$$ 43*b9 - 53*b7 + 32*b4 - 6*b2 - 161*b1 $$\nu^{8}$$ $$=$$ $$53\beta_{8} - 257\beta_{6} - 193\beta_{5} + 27\beta_{3} + 483$$ 53*b8 - 257*b6 - 193*b5 + 27*b3 + 483 $$\nu^{9}$$ $$=$$ $$-257\beta_{9} + 337\beta_{7} - 166\beta_{4} + 81\beta_{2} + 933\beta_1$$ -257*b9 + 337*b7 - 166*b4 + 81*b2 + 933*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
 − 2.45914i − 2.35417i − 1.22277i − 0.794018i − 0.533733i 0.533733i 0.794018i 1.22277i 2.35417i 2.45914i
2.45914i 3.33648i −4.04739 0 −8.20489 3.50903i 5.03483i −8.13211 0
349.2 2.35417i 1.80364i −3.54210 0 4.24606 0.0127981i 3.63036i −0.253109 0
349.3 1.22277i 1.05494i 0.504827 0 1.28995 4.90333i 3.06283i 1.88710 0
349.4 0.794018i 2.48525i 1.36954 0 1.97334 3.07654i 2.67547i −3.17649 0
349.5 0.533733i 0.570435i 1.71513 0 −0.304460 1.47609i 1.98289i 2.67460 0
349.6 0.533733i 0.570435i 1.71513 0 −0.304460 1.47609i 1.98289i 2.67460 0
349.7 0.794018i 2.48525i 1.36954 0 1.97334 3.07654i 2.67547i −3.17649 0
349.8 1.22277i 1.05494i 0.504827 0 1.28995 4.90333i 3.06283i 1.88710 0
349.9 2.35417i 1.80364i −3.54210 0 4.24606 0.0127981i 3.63036i −0.253109 0
349.10 2.45914i 3.33648i −4.04739 0 −8.20489 3.50903i 5.03483i −8.13211 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.g 10
5.b even 2 1 inner 725.2.b.g 10
5.c odd 4 1 725.2.a.i 5
5.c odd 4 1 725.2.a.j yes 5
15.e even 4 1 6525.2.a.bo 5
15.e even 4 1 6525.2.a.bp 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
725.2.a.i 5 5.c odd 4 1
725.2.a.j yes 5 5.c odd 4 1
725.2.b.g 10 1.a even 1 1 trivial
725.2.b.g 10 5.b even 2 1 inner
6525.2.a.bo 5 15.e even 4 1
6525.2.a.bp 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 $$S_{2}^{\mathrm{new}}(725, [\chi])$$:

 $$T_{2}^{10} + 14T_{2}^{8} + 63T_{2}^{6} + 99T_{2}^{4} + 55T_{2}^{2} + 9$$ T2^10 + 14*T2^8 + 63*T2^6 + 99*T2^4 + 55*T2^2 + 9 $$T_{3}^{10} + 22T_{3}^{8} + 155T_{3}^{6} + 411T_{3}^{4} + 367T_{3}^{2} + 81$$ T3^10 + 22*T3^8 + 155*T3^6 + 411*T3^4 + 367*T3^2 + 81

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + 14 T^{8} + 63 T^{6} + 99 T^{4} + \cdots + 9$$
$3$ $$T^{10} + 22 T^{8} + 155 T^{6} + \cdots + 81$$
$5$ $$T^{10}$$
$7$ $$T^{10} + 48 T^{8} + 740 T^{6} + 4197 T^{4} + \cdots + 1$$
$11$ $$(T^{5} - 2 T^{4} - 17 T^{3} + 14 T^{2} + \cdots - 27)^{2}$$
$13$ $$T^{10} + 108 T^{8} + 4040 T^{6} + \cdots + 185761$$
$17$ $$T^{10} + 91 T^{8} + 2663 T^{6} + \cdots + 114921$$
$19$ $$(T^{5} - 4 T^{4} - 9 T^{3} + 23 T^{2} + \cdots + 17)^{2}$$
$23$ $$T^{10} + 45 T^{8} + 728 T^{6} + \cdots + 15129$$
$29$ $$(T - 1)^{10}$$
$31$ $$(T^{5} - 5 T^{4} - 58 T^{3} + 381 T^{2} + \cdots - 251)^{2}$$
$37$ $$T^{10} + 162 T^{8} + \cdots + 13623481$$
$41$ $$(T^{5} + 3 T^{4} - 157 T^{3} + 10 T^{2} + \cdots - 19125)^{2}$$
$43$ $$T^{10} + 138 T^{8} + 5141 T^{6} + \cdots + 38809$$
$47$ $$T^{10} + 139 T^{8} + 6371 T^{6} + \cdots + 4028049$$
$53$ $$T^{10} + 206 T^{8} + 5419 T^{6} + \cdots + 81$$
$59$ $$(T^{5} - 15 T^{4} + 52 T^{3} + 85 T^{2} + \cdots + 375)^{2}$$
$61$ $$(T^{5} + 7 T^{4} - 111 T^{3} - 27 T^{2} + \cdots + 81)^{2}$$
$67$ $$T^{10} + 411 T^{8} + 48707 T^{6} + \cdots + 2819041$$
$71$ $$(T^{5} + 17 T^{4} - 178 T^{3} - 3925 T^{2} + \cdots + 5121)^{2}$$
$73$ $$T^{10} + 424 T^{8} + \cdots + 175748049$$
$79$ $$(T^{5} - 2 T^{4} - 228 T^{3} + 849 T^{2} + \cdots - 66293)^{2}$$
$83$ $$T^{10} + 919 T^{8} + \cdots + 134591395689$$
$89$ $$(T^{5} + 14 T^{4} - 360 T^{3} + \cdots + 278703)^{2}$$
$97$ $$T^{10} + 607 T^{8} + \cdots + 2311590241$$