# Properties

 Label 625.2.d.b Level $625$ Weight $2$ Character orbit 625.d Analytic conductor $4.991$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [625,2,Mod(126,625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(625, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([4]))

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

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

chi := DirichletCharacter("625.126");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$625 = 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 625.d (of order $$5$$, degree $$4$$, 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: $$4.99065012633$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 25) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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 primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{10}^{3} - 1) q^{2} + \zeta_{10}^{3} q^{3} + ( - \zeta_{10} + 1) q^{4} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{6} + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{7} + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - 2 \zeta_{10}) q^{8} + 2 \zeta_{10} q^{9}+O(q^{10})$$ q + (z^3 - 1) * q^2 + z^3 * q^3 + (-z + 1) * q^4 + (-z^3 - z) * q^6 + (z^3 - z^2) * q^7 + (-2*z^3 + z^2 - 2*z) * q^8 + 2*z * q^9 $$q + (\zeta_{10}^{3} - 1) q^{2} + \zeta_{10}^{3} q^{3} + ( - \zeta_{10} + 1) q^{4} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{6} + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{7} + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - 2 \zeta_{10}) q^{8} + 2 \zeta_{10} q^{9} + ( - 4 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 4) q^{11} + (\zeta_{10}^{2} - \zeta_{10} + 1) q^{12} + (3 \zeta_{10}^{2} - 3 \zeta_{10} + 3) q^{13} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{14} + (3 \zeta_{10}^{2} + 3) q^{16} + (2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{17} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 2) q^{18} + (3 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 3 \zeta_{10}) q^{19} + ( - \zeta_{10} + 1) q^{21} + (6 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{22} + ( - 5 \zeta_{10}^{3} + 7 \zeta_{10}^{2} - 7 \zeta_{10} + 5) q^{23} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 1) q^{24} - 3 q^{26} + (5 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 5 \zeta_{10} - 5) q^{27} + (\zeta_{10}^{3} - \zeta_{10} + 1) q^{28} + (3 \zeta_{10}^{3} + \zeta_{10} - 1) q^{29} - 3 \zeta_{10}^{2} q^{31} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 4) q^{32} + (2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{33} + ( - 4 \zeta_{10}^{2} - 2 \zeta_{10} - 4) q^{34} + ( - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{36} + ( - 2 \zeta_{10}^{2} + 3 \zeta_{10} - 2) q^{37} + (\zeta_{10}^{2} - 3 \zeta_{10} + 1) q^{38} + (3 \zeta_{10}^{2} - 3 \zeta_{10}) q^{39} + ( - 2 \zeta_{10}^{2} + 4 \zeta_{10} - 2) q^{41} + \zeta_{10}^{2} q^{42} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 3) q^{43} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10}) q^{44} + (3 \zeta_{10}^{3} + 5 \zeta_{10} - 5) q^{46} + (\zeta_{10} - 1) q^{47} + (3 \zeta_{10}^{3} - 3) q^{48} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 6) q^{49} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 4) q^{51} + ( - 3 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 3) q^{52} + ( - \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{53} + ( - 5 \zeta_{10}^{3} - 5 \zeta_{10} + 5) q^{54} + ( - \zeta_{10}^{3} + 3 \zeta_{10}^{2} - \zeta_{10}) q^{56} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 1) q^{57} + ( - 3 \zeta_{10}^{3} - \zeta_{10}^{2} - 3 \zeta_{10}) q^{58} + (3 \zeta_{10}^{2} + 6 \zeta_{10} + 3) q^{59} + (5 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} - 5) q^{61} + (3 \zeta_{10}^{2} + 3) q^{62} + ( - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{63} + (3 \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 3) q^{64} + ( - 4 \zeta_{10}^{2} - 2 \zeta_{10} - 4) q^{66} + (2 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 2 \zeta_{10}) q^{67} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2) q^{68} + ( - 2 \zeta_{10}^{3} + 7 \zeta_{10}^{2} - 2 \zeta_{10}) q^{69} + (6 \zeta_{10}^{3} + \zeta_{10} - 1) q^{71} + ( - 2 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{72} + (9 \zeta_{10}^{3} - 9 \zeta_{10}^{2} + 9 \zeta_{10} - 9) q^{73} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 1) q^{74} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 3) q^{76} + (2 \zeta_{10}^{3} - 2) q^{77} - 3 \zeta_{10}^{3} q^{78} + (5 \zeta_{10}^{3} + 5 \zeta_{10} - 5) q^{79} + \zeta_{10}^{2} q^{81} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2}) q^{82} + (2 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 2 \zeta_{10}) q^{83} + (\zeta_{10}^{2} - 2 \zeta_{10} + 1) q^{84} + (6 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 6) q^{86} + ( - \zeta_{10}^{2} - 2 \zeta_{10} - 1) q^{87} + ( - 6 \zeta_{10}^{2} - 2 \zeta_{10} - 6) q^{88} + ( - 4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} + 4) q^{89} + (3 \zeta_{10}^{2} - 6 \zeta_{10} + 3) q^{91} + ( - 7 \zeta_{10}^{3} + 9 \zeta_{10}^{2} - 7 \zeta_{10}) q^{92} + 3 q^{93} - \zeta_{10}^{2} q^{94} + ( - 4 \zeta_{10}^{3} + \zeta_{10} - 1) q^{96} + ( - 2 \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{97} + ( - 7 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 7) q^{98} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 8) q^{99} +O(q^{100})$$ q + (z^3 - 1) * q^2 + z^3 * q^3 + (-z + 1) * q^4 + (-z^3 - z) * q^6 + (z^3 - z^2) * q^7 + (-2*z^3 + z^2 - 2*z) * q^8 + 2*z * q^9 + (-4*z^3 + 2*z^2 - 2*z + 4) * q^11 + (z^2 - z + 1) * q^12 + (3*z^2 - 3*z + 3) * q^13 + (-z^3 + z^2 - z + 1) * q^14 + (3*z^2 + 3) * q^16 + (2*z^3 + 2*z^2 + 2*z) * q^17 + (2*z^3 - 2*z^2 - 2) * q^18 + (3*z^3 - 4*z^2 + 3*z) * q^19 + (-z + 1) * q^21 + (6*z^3 + 4*z - 4) * q^22 + (-5*z^3 + 7*z^2 - 7*z + 5) * q^23 + (-2*z^3 + 2*z^2 + 1) * q^24 - 3 * q^26 + (5*z^3 - 5*z^2 + 5*z - 5) * q^27 + (z^3 - z + 1) * q^28 + (3*z^3 + z - 1) * q^29 - 3*z^2 * q^31 + (-z^3 + z^2 - 4) * q^32 + (2*z^3 + 2*z^2 + 2*z) * q^33 + (-4*z^2 - 2*z - 4) * q^34 + (-2*z^2 + 2*z) * q^36 + (-2*z^2 + 3*z - 2) * q^37 + (z^2 - 3*z + 1) * q^38 + (3*z^2 - 3*z) * q^39 + (-2*z^2 + 4*z - 2) * q^41 + z^2 * q^42 + (-3*z^3 + 3*z^2 + 3) * q^43 + (-2*z^3 - 2*z) * q^44 + (3*z^3 + 5*z - 5) * q^46 + (z - 1) * q^47 + (3*z^3 - 3) * q^48 + (z^3 - z^2 - 6) * q^49 + (2*z^3 - 2*z^2 - 4) * q^51 + (-3*z^3 + 6*z^2 - 6*z + 3) * q^52 + (-z^3 - 4*z + 4) * q^53 + (-5*z^3 - 5*z + 5) * q^54 + (-z^3 + 3*z^2 - z) * q^56 + (3*z^3 - 3*z^2 + 1) * q^57 + (-3*z^3 - z^2 - 3*z) * q^58 + (3*z^2 + 6*z + 3) * q^59 + (5*z^3 + z^2 - z - 5) * q^61 + (3*z^2 + 3) * q^62 + (-2*z^2 + 2*z - 2) * q^63 + (3*z^3 - z^2 + z - 3) * q^64 + (-4*z^2 - 2*z - 4) * q^66 + (2*z^3 - 8*z^2 + 2*z) * q^67 + (-2*z^3 + 2*z^2 + 2) * q^68 + (-2*z^3 + 7*z^2 - 2*z) * q^69 + (6*z^3 + z - 1) * q^71 + (-2*z^3 - 4*z + 4) * q^72 + (9*z^3 - 9*z^2 + 9*z - 9) * q^73 + (z^3 - z^2 + 1) * q^74 + (4*z^3 - 4*z^2 + 3) * q^76 + (2*z^3 - 2) * q^77 - 3*z^3 * q^78 + (5*z^3 + 5*z - 5) * q^79 + z^2 * q^81 + (2*z^3 - 2*z^2) * q^82 + (2*z^3 + 3*z^2 + 2*z) * q^83 + (z^2 - 2*z + 1) * q^84 + (6*z^3 - 3*z^2 + 3*z - 6) * q^86 + (-z^2 - 2*z - 1) * q^87 + (-6*z^2 - 2*z - 6) * q^88 + (-4*z^3 - 4*z^2 + 4*z + 4) * q^89 + (3*z^2 - 6*z + 3) * q^91 + (-7*z^3 + 9*z^2 - 7*z) * q^92 + 3 * q^93 - z^2 * q^94 + (-4*z^3 + z - 1) * q^96 + (-2*z^3 - 3*z + 3) * q^97 + (-7*z^3 + z^2 - z + 7) * q^98 + (-4*z^3 + 4*z^2 + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{2} + q^{3} + 3 q^{4} - 2 q^{6} + 2 q^{7} - 5 q^{8} + 2 q^{9}+O(q^{10})$$ 4 * q - 3 * q^2 + q^3 + 3 * q^4 - 2 * q^6 + 2 * q^7 - 5 * q^8 + 2 * q^9 $$4 q - 3 q^{2} + q^{3} + 3 q^{4} - 2 q^{6} + 2 q^{7} - 5 q^{8} + 2 q^{9} + 8 q^{11} + 2 q^{12} + 6 q^{13} + q^{14} + 9 q^{16} + 2 q^{17} - 4 q^{18} + 10 q^{19} + 3 q^{21} - 6 q^{22} + q^{23} - 12 q^{26} - 5 q^{27} + 4 q^{28} + 3 q^{31} - 18 q^{32} + 2 q^{33} - 14 q^{34} + 4 q^{36} - 3 q^{37} - 6 q^{39} - 2 q^{41} - q^{42} + 6 q^{43} - 4 q^{44} - 12 q^{46} - 3 q^{47} - 9 q^{48} - 22 q^{49} - 12 q^{51} - 3 q^{52} + 11 q^{53} + 10 q^{54} - 5 q^{56} + 10 q^{57} - 5 q^{58} + 15 q^{59} - 17 q^{61} + 9 q^{62} - 4 q^{63} - 7 q^{64} - 14 q^{66} + 12 q^{67} + 4 q^{68} - 11 q^{69} + 3 q^{71} + 10 q^{72} - 9 q^{73} + 6 q^{74} + 20 q^{76} - 6 q^{77} - 3 q^{78} - 10 q^{79} - q^{81} + 4 q^{82} + q^{83} + q^{84} - 12 q^{86} - 5 q^{87} - 20 q^{88} + 20 q^{89} + 3 q^{91} - 23 q^{92} + 12 q^{93} + q^{94} - 7 q^{96} + 7 q^{97} + 19 q^{98} + 24 q^{99}+O(q^{100})$$ 4 * q - 3 * q^2 + q^3 + 3 * q^4 - 2 * q^6 + 2 * q^7 - 5 * q^8 + 2 * q^9 + 8 * q^11 + 2 * q^12 + 6 * q^13 + q^14 + 9 * q^16 + 2 * q^17 - 4 * q^18 + 10 * q^19 + 3 * q^21 - 6 * q^22 + q^23 - 12 * q^26 - 5 * q^27 + 4 * q^28 + 3 * q^31 - 18 * q^32 + 2 * q^33 - 14 * q^34 + 4 * q^36 - 3 * q^37 - 6 * q^39 - 2 * q^41 - q^42 + 6 * q^43 - 4 * q^44 - 12 * q^46 - 3 * q^47 - 9 * q^48 - 22 * q^49 - 12 * q^51 - 3 * q^52 + 11 * q^53 + 10 * q^54 - 5 * q^56 + 10 * q^57 - 5 * q^58 + 15 * q^59 - 17 * q^61 + 9 * q^62 - 4 * q^63 - 7 * q^64 - 14 * q^66 + 12 * q^67 + 4 * q^68 - 11 * q^69 + 3 * q^71 + 10 * q^72 - 9 * q^73 + 6 * q^74 + 20 * q^76 - 6 * q^77 - 3 * q^78 - 10 * q^79 - q^81 + 4 * q^82 + q^83 + q^84 - 12 * q^86 - 5 * q^87 - 20 * q^88 + 20 * q^89 + 3 * q^91 - 23 * q^92 + 12 * q^93 + q^94 - 7 * q^96 + 7 * q^97 + 19 * q^98 + 24 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/625\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-\zeta_{10}^{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
126.1
 −0.309017 + 0.951057i 0.809017 + 0.587785i 0.809017 − 0.587785i −0.309017 − 0.951057i
−0.190983 0.587785i 0.809017 0.587785i 1.30902 0.951057i 0 −0.500000 0.363271i 1.61803 −1.80902 1.31433i −0.618034 + 1.90211i 0
251.1 −1.30902 + 0.951057i −0.309017 + 0.951057i 0.190983 0.587785i 0 −0.500000 1.53884i −0.618034 −0.690983 2.12663i 1.61803 + 1.17557i 0
376.1 −1.30902 0.951057i −0.309017 0.951057i 0.190983 + 0.587785i 0 −0.500000 + 1.53884i −0.618034 −0.690983 + 2.12663i 1.61803 1.17557i 0
501.1 −0.190983 + 0.587785i 0.809017 + 0.587785i 1.30902 + 0.951057i 0 −0.500000 + 0.363271i 1.61803 −1.80902 + 1.31433i −0.618034 1.90211i 0
 $$n$$: e.g. 2-40 or 990-1000 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.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 625.2.d.b 4
5.b even 2 1 625.2.d.h 4
5.c odd 4 2 625.2.e.c 8
25.d even 5 2 125.2.d.a 4
25.d even 5 1 625.2.a.c 2
25.d even 5 1 inner 625.2.d.b 4
25.e even 10 2 25.2.d.a 4
25.e even 10 1 625.2.a.b 2
25.e even 10 1 625.2.d.h 4
25.f odd 20 4 125.2.e.a 8
25.f odd 20 2 625.2.b.a 4
25.f odd 20 2 625.2.e.c 8
75.h odd 10 2 225.2.h.b 4
75.h odd 10 1 5625.2.a.f 2
75.j odd 10 1 5625.2.a.d 2
100.h odd 10 2 400.2.u.b 4
100.h odd 10 1 10000.2.a.c 2
100.j odd 10 1 10000.2.a.l 2

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(625, [\chi])$$:

 $$T_{2}^{4} + 3T_{2}^{3} + 4T_{2}^{2} + 2T_{2} + 1$$ T2^4 + 3*T2^3 + 4*T2^2 + 2*T2 + 1 $$T_{3}^{4} - T_{3}^{3} + T_{3}^{2} - T_{3} + 1$$ T3^4 - T3^3 + T3^2 - T3 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1$$
$3$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - T - 1)^{2}$$
$11$ $$T^{4} - 8 T^{3} + 24 T^{2} + 8 T + 16$$
$13$ $$T^{4} - 6 T^{3} + 36 T^{2} - 81 T + 81$$
$17$ $$T^{4} - 2 T^{3} + 24 T^{2} + 32 T + 16$$
$19$ $$T^{4} - 10 T^{3} + 40 T^{2} - 25 T + 25$$
$23$ $$T^{4} - T^{3} + 51 T^{2} - 341 T + 961$$
$29$ $$T^{4} + 10 T^{2} - 25 T + 25$$
$31$ $$T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81$$
$37$ $$T^{4} + 3 T^{3} + 19 T^{2} + 7 T + 1$$
$41$ $$T^{4} + 2 T^{3} + 24 T^{2} - 32 T + 16$$
$43$ $$(T^{2} - 3 T - 9)^{2}$$
$47$ $$T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1$$
$53$ $$T^{4} - 11 T^{3} + 61 T^{2} + \cdots + 361$$
$59$ $$T^{4} - 15 T^{3} + 90 T^{2} + \cdots + 2025$$
$61$ $$T^{4} + 17 T^{3} + 139 T^{2} + \cdots + 1681$$
$67$ $$T^{4} - 12 T^{3} + 64 T^{2} + \cdots + 1936$$
$71$ $$T^{4} - 3 T^{3} + 34 T^{2} - 232 T + 841$$
$73$ $$T^{4} + 9 T^{3} + 81 T^{2} + \cdots + 6561$$
$79$ $$T^{4} + 10 T^{3} + 100 T^{2} + \cdots + 625$$
$83$ $$T^{4} - T^{3} + 31 T^{2} + 99 T + 121$$
$89$ $$T^{4} - 20 T^{3} + 240 T^{2} + \cdots + 6400$$
$97$ $$T^{4} - 7 T^{3} + 34 T^{2} - 88 T + 121$$