Properties

 Label 25.2.e.a Level $25$ Weight $2$ Character orbit 25.e Analytic conductor $0.200$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [25,2,Mod(4,25)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(25, base_ring=CyclotomicField(10))

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

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

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

chi := DirichletCharacter("25.4");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 25.e (of order $$10$$, degree $$4$$, 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: $$0.199626005053$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: 8.0.58140625.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25$$ x^8 - 3*x^7 + 4*x^6 - 7*x^5 + 11*x^4 + 5*x^3 - 10*x^2 - 25*x + 25 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + (\beta_{7} - \beta_{3} + \beta_{2} - 1) q^{3} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \cdots + 1) q^{4}+ \cdots + ( - \beta_{5} - \beta_{4} + \cdots + \beta_1) q^{9}+O(q^{10})$$ q + (b1 - 1) * q^2 + (b7 - b3 + b2 - 1) * q^3 + (-b7 - b5 - b4 + b3 - b2 - 2*b1 + 1) * q^4 + (-b6 + b5 + b3 - b2 + 1) * q^5 + (-b7 + b6 + b5 + 2*b4 - b3 - 2) * q^6 + (b6 + b4 + 2*b2 - 1) * q^7 + (b3 - 2*b2 + 2) * q^8 + (-b5 - b4 + b3 + b1) * q^9 $$q + (\beta_1 - 1) q^{2} + (\beta_{7} - \beta_{3} + \beta_{2} - 1) q^{3} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \cdots + 1) q^{4}+ \cdots + (2 \beta_{7} + 4 \beta_{5} + 2 \beta_{4} + \cdots - 2) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^2 + (b7 - b3 + b2 - 1) * q^3 + (-b7 - b5 - b4 + b3 - b2 - 2*b1 + 1) * q^4 + (-b6 + b5 + b3 - b2 + 1) * q^5 + (-b7 + b6 + b5 + 2*b4 - b3 - 2) * q^6 + (b6 + b4 + 2*b2 - 1) * q^7 + (b3 - 2*b2 + 2) * q^8 + (-b5 - b4 + b3 + b1) * q^9 + (b7 - b6 - b5 - b4 - 2*b3 + b2 - 1) * q^10 - 2*b4 * q^11 + (b7 - b6 - b4 - b3 + b2 + 2) * q^12 + (b5 + b4 + b1 - 1) * q^13 + (2*b7 - b6 + b5 + b4 - b2 + b1 + 2) * q^14 + (-2*b7 + b6 - b5 + 3*b4 - 2*b3 + b2 - b1) * q^15 + (b7 + b6 + 2*b2 + 2*b1 - 1) * q^16 + (-b7 - b5 - 3*b4 + 2*b3 - 4*b2 - b1 + 1) * q^17 + (-b7 - b4 + 3*b3 + b2 - b1 + 1) * q^18 + (-b7 + b6 + b5 + 2*b4 - 3*b3 - 2) * q^19 + (-b7 + 2*b6 + 3*b3 + b2 - 2) * q^20 + (-2*b6 - b5 - b4 + 2*b3 - 3*b2 - 2*b1 + 2) * q^21 + (2*b5 + 2*b4) * q^22 + (-b6 - b5 + b4 - 2*b3 + b2 - b1) * q^23 + (b7 + 2*b5 + b4 - 3*b3 + 3*b2 + b1 - 4) * q^24 + (2*b7 + b5 - 3*b2) * q^25 + (-b6 - 2*b5 - b4 - b3 + b2 - 2*b1 + 1) * q^26 + (-b6 - b5 - b4 + 2*b3 - 3*b2 - 3*b1 + 2) * q^27 + (-3*b7 - b5 - b4 - 3*b3 - 3) * q^28 + (b7 + 2*b6 + 2*b5 - b4 + 5*b2 + 4*b1 - 3) * q^29 + (b7 + 2*b6 - b5 - 5*b4 + 3*b3 - 2*b2 + 2*b1 + 3) * q^30 + (2*b7 - 3*b6 - 2*b5 + 4*b3 + b1) * q^31 + (-b7 - b6 + 6*b4 - 3*b3 + b2 - b1 - 2) * q^32 + (2*b6 + 2*b1) * q^33 + (-b7 - b6 + b4 + 2*b3 - b2 - 2*b1 + 3) * q^34 + (-b7 + 3*b6 + 2*b5 - b4 - b3 + 3*b2 + 2*b1) * q^35 + (2*b7 - b6 + b5 - 3*b2 + b1 + 4) * q^36 + (-b7 + b6 + b5 + b4 - 4*b3 + b1 + 2) * q^37 + (-b7 + b6 - 2*b5 - b4 - b3 - b2 - 2*b1 + 2) * q^38 + (-2*b7 + 3*b4 - b1 - 1) * q^39 + (3*b7 - 3*b6 + b5 + 3*b3 + b1 + 3) * q^40 + (-b7 - b6 + b5 + 4*b4 - 3*b3 + b2 - 3*b1 - 1) * q^41 + (b6 - 2*b4 + b1 - 2) * q^42 + (b7 - b6 - 6*b4 + 5*b3 - b2 + b1 + 3) * q^43 + (2*b7 - 2*b6 - 2*b5) * q^44 + (b7 - 2*b6 - 3*b3 + 4*b2 - 3) * q^45 + (-b7 + 4*b6 + b5 - 3*b4 + b3 + 2*b1 - 3) * q^46 + (2*b7 - 3*b5 - 3*b4 + b2) * q^47 + (b6 + b5 - 3*b2 - 3) * q^48 + (-b7 - b6 - 4*b5 - 2*b4 + 3*b3 - 3*b2 - 3*b1 + 4) * q^49 + (-4*b7 - b6 - 3*b5 + 5*b4 - 4*b3 - 3*b1 - 4) * q^50 + (2*b6 + 4*b5 + 2*b4 - 4*b3 + 4*b2 + 4*b1 - 4) * q^51 + (2*b6 + 2*b5 - b4 + 2*b3 - b2 + 2*b1 - 1) * q^52 + (2*b7 + 3*b5 + 6*b4 + b3 + 2*b2 - 2) * q^53 + (b7 + b5 + b4 - b3 - 2*b2 + 2*b1 - 1) * q^54 + (-2*b7 - 2*b5 - 2*b2 - 4*b1) * q^55 + (-b7 + 2*b6 + b5 - 2*b4 + b3 - b1 + 2) * q^56 + (b7 - 3*b6 + 5*b3 + 3*b2 + b1 + 1) * q^57 + (2*b7 - 4*b6 + 2*b5 + 5*b4 - b3 + 2*b2 - 2*b1 + 3) * q^58 + (b7 + b6 + 3*b5 - 3*b4 + b3 - 4*b2 - b1 + 3) * q^59 + (-3*b7 - 2*b6 + b5 + 4*b4 - b3 - 4*b2 - 3*b1) * q^60 + (-6*b7 + 4*b6 - 4*b5 - 2*b4 - b2 - 3*b1 - 2) * q^61 + (-b7 + b6 - b5 - 5*b4 + 3*b3 + 4*b2 - b1 - 3) * q^62 + (2*b7 - 2*b6 + 2*b4 - 2*b2 + 2) * q^63 + (-4*b7 + 4*b6 - 4*b5 - 9*b4 - 4*b2 - 2*b1 + 2) * q^64 + (2*b7 - 2*b4 - b3 - b2 + b1 + 1) * q^65 + (-2*b7 - 2*b6 - 2*b5 + 2*b4 + 2*b3 - 2*b1 + 2) * q^66 + (4*b7 + 4*b5 + 4*b4 - 2*b3 + 4*b2 + 4*b1) * q^67 + (4*b7 + b6 + 3*b4 - 3*b3 + 5*b2 + 4*b1 - 4) * q^68 + (2*b7 - b6 - 2*b5 - 4*b4 + 2*b3 - b1 + 4) * q^69 + (3*b7 - 4*b6 + 2*b5 + 5*b4 - b3 - b2 + b1 + 4) * q^70 + (2*b7 + 2*b6 + 3*b5 + 5*b4 - 6*b3 + b2 + 6*b1 - 4) * q^71 + (-3*b7 - 4*b5 - 2*b4 + 3*b3 - 2*b2 + 1) * q^72 + (2*b6 + 2*b5 + 2*b4 - 4*b3 + 5*b2 + 3*b1 - 2) * q^73 + (-3*b7 + 2*b6 - 2*b5 - b4 + b1 - 3) * q^74 + (-b7 + 2*b5 - 5*b4 + 5*b3 - b2 + 5*b1) * q^75 + (b7 + 2*b5 + b4 + b3 - b2 + b1 - 2) * q^76 + (2*b4 - 4*b3 + 4*b2 + 2*b1 - 2) * q^77 + (3*b7 - 2*b5 - 6*b4 + b3 - b2 + 5) * q^78 + (b7 - 8*b6 - 3*b5 + b4 + 3*b3 - 4*b2 - 6*b1 + 7) * q^79 + (-b6 + b5 - 4*b3 + 4*b2 + 1) * q^80 + (2*b6 - 4*b4 - 3*b3 - 2*b1 + 4) * q^81 + (3*b7 + 3*b6 - 4*b4 - 4*b3 - 2*b2 + 3*b1 - 1) * q^82 + (-4*b7 - b6 - 4*b5 - 8*b4 + 4*b3 - 8*b2 - 5*b1) * q^83 + (b7 + b6 + b5 + 5*b4 - b3 + 6*b2 + b1 - 1) * q^84 + (2*b7 - 2*b6 - 4*b5 - b4 + 4*b3 + b2 - 3*b1) * q^85 + (2*b7 - 4*b6 + 4*b5 + 5*b4 + 2*b2 + b1 - 1) * q^86 + (-3*b7 + 3*b6 - b5 + 5*b4 + b3 - 6*b2 - b1 - 3) * q^87 + (-2*b4 + 2*b3 + 2*b2 - 2) * q^88 + (-2*b7 + 7*b4 + 5*b2 - b1 - 6) * q^89 + (5*b6 + 4*b5 - 2*b4 - b3 + 2*b2 + b1 - 4) * q^90 + (b7 + b6 + b5 - b4 + b3 + b1 + 1) * q^91 + (b7 - 4*b6 + b5 + 4*b4 + b3 - 2*b2 - 3*b1 + 6) * q^92 + (-5*b7 + 4*b6 + b4 - 7*b3 - 7*b2 - 5*b1) * q^93 + (-4*b7 + 3*b6 + 4*b5 + 4*b4 + 4*b3 + b1 - 4) * q^94 + (-b7 - 3*b5 - 5*b4 + 5*b3 - 6*b2 + 5) * q^95 + (-6*b6 - 3*b5 - b4 + 4*b3 + b2 - 6*b1 + 6) * q^96 + (-4*b7 + 4*b5 + 7*b4 - 9*b3 + 4*b2 - 12) * q^97 + (2*b6 + 2*b5 - 3*b4 + 6*b3 - 2*b2 + 4*b1) * q^98 + (2*b7 + 4*b5 + 2*b4 - 2*b3 + 2*b2 + 2*b1 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 5 q^{2} - 5 q^{3} - q^{4} - 9 q^{6} + 10 q^{8} + q^{9}+O(q^{10})$$ 8 * q - 5 * q^2 - 5 * q^3 - q^4 - 9 * q^6 + 10 * q^8 + q^9 $$8 q - 5 q^{2} - 5 q^{3} - q^{4} - 9 q^{6} + 10 q^{8} + q^{9} - 5 q^{10} - 4 q^{11} + 15 q^{12} - 5 q^{13} + 13 q^{14} + 15 q^{15} + 3 q^{16} - 10 q^{17} - 5 q^{19} - 15 q^{20} - 4 q^{21} + 5 q^{23} - 20 q^{24} - 10 q^{25} + 6 q^{26} - 5 q^{27} - 15 q^{28} - 5 q^{29} + 15 q^{30} - 9 q^{31} + 10 q^{33} + 13 q^{34} + 15 q^{35} + 23 q^{36} + 30 q^{37} + 15 q^{38} - 3 q^{39} + 10 q^{40} - 4 q^{41} - 15 q^{42} - 2 q^{44} - 15 q^{45} - 19 q^{46} - 30 q^{48} + 14 q^{49} - 15 q^{50} - 4 q^{51} - 10 q^{52} - 10 q^{53} - 5 q^{54} - 10 q^{55} + 10 q^{56} + 20 q^{58} - 10 q^{60} - 9 q^{61} - 30 q^{62} + 10 q^{63} + 4 q^{64} + 5 q^{65} + 12 q^{66} + 20 q^{67} + 17 q^{69} + 30 q^{70} + 6 q^{71} + 5 q^{72} + 15 q^{73} - 12 q^{74} - 10 q^{75} - 20 q^{76} + 10 q^{77} + 25 q^{78} + 15 q^{79} + 20 q^{80} + 28 q^{81} - 45 q^{83} + 18 q^{84} - 15 q^{85} - 9 q^{86} - 20 q^{87} - 20 q^{88} - 25 q^{89} - 25 q^{90} + 6 q^{91} + 30 q^{92} - 27 q^{94} + 15 q^{95} + 16 q^{96} - 60 q^{97} - 10 q^{98} - 8 q^{99}+O(q^{100})$$ 8 * q - 5 * q^2 - 5 * q^3 - q^4 - 9 * q^6 + 10 * q^8 + q^9 - 5 * q^10 - 4 * q^11 + 15 * q^12 - 5 * q^13 + 13 * q^14 + 15 * q^15 + 3 * q^16 - 10 * q^17 - 5 * q^19 - 15 * q^20 - 4 * q^21 + 5 * q^23 - 20 * q^24 - 10 * q^25 + 6 * q^26 - 5 * q^27 - 15 * q^28 - 5 * q^29 + 15 * q^30 - 9 * q^31 + 10 * q^33 + 13 * q^34 + 15 * q^35 + 23 * q^36 + 30 * q^37 + 15 * q^38 - 3 * q^39 + 10 * q^40 - 4 * q^41 - 15 * q^42 - 2 * q^44 - 15 * q^45 - 19 * q^46 - 30 * q^48 + 14 * q^49 - 15 * q^50 - 4 * q^51 - 10 * q^52 - 10 * q^53 - 5 * q^54 - 10 * q^55 + 10 * q^56 + 20 * q^58 - 10 * q^60 - 9 * q^61 - 30 * q^62 + 10 * q^63 + 4 * q^64 + 5 * q^65 + 12 * q^66 + 20 * q^67 + 17 * q^69 + 30 * q^70 + 6 * q^71 + 5 * q^72 + 15 * q^73 - 12 * q^74 - 10 * q^75 - 20 * q^76 + 10 * q^77 + 25 * q^78 + 15 * q^79 + 20 * q^80 + 28 * q^81 - 45 * q^83 + 18 * q^84 - 15 * q^85 - 9 * q^86 - 20 * q^87 - 20 * q^88 - 25 * q^89 - 25 * q^90 + 6 * q^91 + 30 * q^92 - 27 * q^94 + 15 * q^95 + 16 * q^96 - 60 * q^97 - 10 * q^98 - 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 406\nu^{7} - 714\nu^{6} + 747\nu^{5} - 1896\nu^{4} + 2103\nu^{3} + 4949\nu^{2} + 1065\nu - 7800 ) / 1355$$ (406*v^7 - 714*v^6 + 747*v^5 - 1896*v^4 + 2103*v^3 + 4949*v^2 + 1065*v - 7800) / 1355 $$\beta_{3}$$ $$=$$ $$( 420\nu^{7} - 776\nu^{6} + 698\nu^{5} - 1924\nu^{4} + 2297\nu^{3} + 5129\nu^{2} + 1055\nu - 10265 ) / 1355$$ (420*v^7 - 776*v^6 + 698*v^5 - 1924*v^4 + 2297*v^3 + 5129*v^2 + 1055*v - 10265) / 1355 $$\beta_{4}$$ $$=$$ $$( 728\nu^{7} - 1327\nu^{6} + 1246\nu^{5} - 3353\nu^{4} + 3584\nu^{3} + 8547\nu^{2} + 2190\nu - 15715 ) / 1355$$ (728*v^7 - 1327*v^6 + 1246*v^5 - 3353*v^4 + 3584*v^3 + 8547*v^2 + 2190*v - 15715) / 1355 $$\beta_{5}$$ $$=$$ $$( -857\nu^{7} + 1666\nu^{6} - 1743\nu^{5} + 4424\nu^{4} - 4907\nu^{3} - 9470\nu^{2} - 2485\nu + 18200 ) / 1355$$ (-857*v^7 + 1666*v^6 - 1743*v^5 + 4424*v^4 - 4907*v^3 - 9470*v^2 - 2485*v + 18200) / 1355 $$\beta_{6}$$ $$=$$ $$( 891\nu^{7} - 1623\nu^{6} + 1624\nu^{5} - 4492\nu^{4} + 4991\nu^{3} + 9520\nu^{2} + 3235\nu - 18960 ) / 1355$$ (891*v^7 - 1623*v^6 + 1624*v^5 - 4492*v^4 + 4991*v^3 + 9520*v^2 + 3235*v - 18960) / 1355 $$\beta_{7}$$ $$=$$ $$( 955\nu^{7} - 1829\nu^{6} + 1942\nu^{5} - 4891\nu^{4} + 5723\nu^{3} + 9646\nu^{2} + 2415\nu - 20550 ) / 1355$$ (955*v^7 - 1829*v^6 + 1942*v^5 - 4891*v^4 + 5723*v^3 + 9646*v^2 + 2415*v - 20550) / 1355
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2}$$ -b7 - b5 - b4 + b3 + b2 $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{5} - 3\beta_{4} + 4\beta_{3} + \beta_{2} + \beta _1 + 3$$ b7 + b5 - 3*b4 + 4*b3 + b2 + b1 + 3 $$\nu^{4}$$ $$=$$ $$5\beta_{7} - 5\beta_{6} + 4\beta_{5} + 2\beta_{4} + 2\beta_{3} + 2\beta_{2} + 4\beta _1 + 2$$ 5*b7 - 5*b6 + 4*b5 + 2*b4 + 2*b3 + 2*b2 + 4*b1 + 2 $$\nu^{5}$$ $$=$$ $$4\beta_{7} - 6\beta_{6} - 7\beta_{3} + 11\beta_{2} + 4\beta _1 - 13$$ 4*b7 - 6*b6 - 7*b3 + 11*b2 + 4*b1 - 13 $$\nu^{6}$$ $$=$$ $$7\beta_{6} + 7\beta_{5} - 8\beta_{4} - 7\beta_{3} + 21\beta_{2} - 2\beta _1 - 21$$ 7*b6 + 7*b5 - 8*b4 - 7*b3 + 21*b2 - 2*b1 - 21 $$\nu^{7}$$ $$=$$ $$23\beta_{7} + 38\beta_{5} + 23\beta_{4} - 23\beta_{3} + 12\beta_{2}$$ 23*b7 + 38*b5 + 23*b4 - 23*b3 + 12*b2

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/25\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\beta_{2}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 −0.983224 − 0.644389i 1.17421 + 0.0566033i −0.357358 + 1.86824i 1.66637 − 0.917186i −0.357358 − 1.86824i 1.66637 + 0.917186i −0.983224 + 0.644389i 1.17421 − 0.0566033i
−1.98322 0.644389i 1.29224 1.77862i 1.89991 + 1.38036i −1.22570 + 1.87020i −3.70892 + 2.69469i 0.992398i −0.427051 0.587785i −0.566541 1.74363i 3.63597 2.91920i
4.2 0.174207 + 0.0566033i −0.865190 + 1.19083i −1.59089 1.15585i 0.107666 2.23347i −0.218127 + 0.158479i 3.26086i −0.427051 0.587785i 0.257524 + 0.792578i 0.145178 0.382993i
9.1 −1.35736 + 1.86824i −0.451659 + 0.146753i −1.02988 3.16963i 2.19625 + 0.420099i 0.338893 1.04301i 3.03582i 2.92705 + 0.951057i −2.24459 + 1.63079i −3.76594 + 3.53290i
9.2 0.666375 0.917186i −2.47539 + 0.804303i 0.220859 + 0.679734i −1.07822 1.95894i −0.911842 + 2.80636i 0.407162i 2.92705 + 0.951057i 3.05361 2.21858i −2.51521 0.316463i
14.1 −1.35736 1.86824i −0.451659 0.146753i −1.02988 + 3.16963i 2.19625 0.420099i 0.338893 + 1.04301i 3.03582i 2.92705 0.951057i −2.24459 1.63079i −3.76594 3.53290i
14.2 0.666375 + 0.917186i −2.47539 0.804303i 0.220859 0.679734i −1.07822 + 1.95894i −0.911842 2.80636i 0.407162i 2.92705 0.951057i 3.05361 + 2.21858i −2.51521 + 0.316463i
19.1 −1.98322 + 0.644389i 1.29224 + 1.77862i 1.89991 1.38036i −1.22570 1.87020i −3.70892 2.69469i 0.992398i −0.427051 + 0.587785i −0.566541 + 1.74363i 3.63597 + 2.91920i
19.2 0.174207 0.0566033i −0.865190 1.19083i −1.59089 + 1.15585i 0.107666 + 2.23347i −0.218127 0.158479i 3.26086i −0.427051 + 0.587785i 0.257524 0.792578i 0.145178 + 0.382993i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 25.2.e.a 8
3.b odd 2 1 225.2.m.a 8
4.b odd 2 1 400.2.y.c 8
5.b even 2 1 125.2.e.b 8
5.c odd 4 2 125.2.d.b 16
25.d even 5 1 125.2.e.b 8
25.d even 5 1 625.2.b.c 8
25.d even 5 1 625.2.e.a 8
25.d even 5 1 625.2.e.i 8
25.e even 10 1 inner 25.2.e.a 8
25.e even 10 1 625.2.b.c 8
25.e even 10 1 625.2.e.a 8
25.e even 10 1 625.2.e.i 8
25.f odd 20 2 125.2.d.b 16
25.f odd 20 2 625.2.a.f 8
25.f odd 20 4 625.2.d.o 16
75.h odd 10 1 225.2.m.a 8
75.l even 20 2 5625.2.a.x 8
100.h odd 10 1 400.2.y.c 8
100.l even 20 2 10000.2.a.bj 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.e.a 8 1.a even 1 1 trivial
25.2.e.a 8 25.e even 10 1 inner
125.2.d.b 16 5.c odd 4 2
125.2.d.b 16 25.f odd 20 2
125.2.e.b 8 5.b even 2 1
125.2.e.b 8 25.d even 5 1
225.2.m.a 8 3.b odd 2 1
225.2.m.a 8 75.h odd 10 1
400.2.y.c 8 4.b odd 2 1
400.2.y.c 8 100.h odd 10 1
625.2.a.f 8 25.f odd 20 2
625.2.b.c 8 25.d even 5 1
625.2.b.c 8 25.e even 10 1
625.2.d.o 16 25.f odd 20 4
625.2.e.a 8 25.d even 5 1
625.2.e.a 8 25.e even 10 1
625.2.e.i 8 25.d even 5 1
625.2.e.i 8 25.e even 10 1
5625.2.a.x 8 75.l even 20 2
10000.2.a.bj 8 100.l even 20 2

Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(25, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 5 T^{7} + \cdots + 1$$
$3$ $$T^{8} + 5 T^{7} + \cdots + 16$$
$5$ $$T^{8} + 5 T^{6} + \cdots + 625$$
$7$ $$T^{8} + 21 T^{6} + \cdots + 16$$
$11$ $$(T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{2}$$
$13$ $$T^{8} + 5 T^{7} + \cdots + 1$$
$17$ $$T^{8} + 10 T^{7} + \cdots + 1936$$
$19$ $$T^{8} + 5 T^{7} + \cdots + 400$$
$23$ $$T^{8} - 5 T^{7} + \cdots + 256$$
$29$ $$T^{8} + 5 T^{7} + \cdots + 483025$$
$31$ $$T^{8} + 9 T^{7} + \cdots + 1936$$
$37$ $$T^{8} - 30 T^{7} + \cdots + 116281$$
$41$ $$T^{8} + 4 T^{7} + \cdots + 13456$$
$43$ $$T^{8} + 129 T^{6} + \cdots + 246016$$
$47$ $$T^{8} + 16 T^{6} + \cdots + 65536$$
$53$ $$T^{8} + 10 T^{7} + \cdots + 8755681$$
$59$ $$T^{8} + 15 T^{5} + \cdots + 4080400$$
$61$ $$T^{8} + 9 T^{7} + \cdots + 116281$$
$67$ $$T^{8} - 20 T^{7} + \cdots + 246016$$
$71$ $$T^{8} - 6 T^{7} + \cdots + 24245776$$
$73$ $$T^{8} - 15 T^{7} + \cdots + 1$$
$79$ $$T^{8} - 15 T^{7} + \cdots + 33408400$$
$83$ $$T^{8} + 45 T^{7} + \cdots + 99856$$
$89$ $$T^{8} + 25 T^{7} + \cdots + 1392400$$
$97$ $$T^{8} + 60 T^{7} + \cdots + 301334881$$