# Properties

 Label 75.2.g.a Level $75$ Weight $2$ Character orbit 75.g Analytic conductor $0.599$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,2,Mod(16,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("75.16");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 75.g (of order $$5$$, 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: $$0.598878015160$$ 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: yes 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$$.

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1
 0.809017 − 0.587785i −0.309017 + 0.951057i −0.309017 − 0.951057i 0.809017 + 0.587785i
−0.809017 0.587785i 0.309017 + 0.951057i −0.309017 0.951057i 0.690983 2.12663i 0.309017 0.951057i 4.47214 −0.927051 + 2.85317i −0.809017 + 0.587785i −1.80902 + 1.31433i
31.1 0.309017 + 0.951057i −0.809017 + 0.587785i 0.809017 0.587785i 1.80902 + 1.31433i −0.809017 0.587785i −4.47214 2.42705 + 1.76336i 0.309017 0.951057i −0.690983 + 2.12663i
46.1 0.309017 0.951057i −0.809017 0.587785i 0.809017 + 0.587785i 1.80902 1.31433i −0.809017 + 0.587785i −4.47214 2.42705 1.76336i 0.309017 + 0.951057i −0.690983 2.12663i
61.1 −0.809017 + 0.587785i 0.309017 0.951057i −0.309017 + 0.951057i 0.690983 + 2.12663i 0.309017 + 0.951057i 4.47214 −0.927051 2.85317i −0.809017 0.587785i −1.80902 1.31433i
 $$n$$: 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 75.2.g.a 4
3.b odd 2 1 225.2.h.a 4
5.b even 2 1 375.2.g.a 4
5.c odd 4 2 375.2.i.a 8
25.d even 5 1 inner 75.2.g.a 4
25.d even 5 1 1875.2.a.d 2
25.e even 10 1 375.2.g.a 4
25.e even 10 1 1875.2.a.a 2
25.f odd 20 2 375.2.i.a 8
25.f odd 20 2 1875.2.b.b 4
75.h odd 10 1 5625.2.a.h 2
75.j odd 10 1 225.2.h.a 4
75.j odd 10 1 5625.2.a.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.a 4 1.a even 1 1 trivial
75.2.g.a 4 25.d even 5 1 inner
225.2.h.a 4 3.b odd 2 1
225.2.h.a 4 75.j odd 10 1
375.2.g.a 4 5.b even 2 1
375.2.g.a 4 25.e even 10 1
375.2.i.a 8 5.c odd 4 2
375.2.i.a 8 25.f odd 20 2
1875.2.a.a 2 25.e even 10 1
1875.2.a.d 2 25.d even 5 1
1875.2.b.b 4 25.f odd 20 2
5625.2.a.a 2 75.j odd 10 1
5625.2.a.h 2 75.h odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + T_{2}^{3} + T_{2}^{2} + T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + T^{3} + T^{2} + \cdots + 1$$
$3$ $$T^{4} + T^{3} + T^{2} + \cdots + 1$$
$5$ $$T^{4} - 5 T^{3} + \cdots + 25$$
$7$ $$(T^{2} - 20)^{2}$$
$11$ $$T^{4} + 6 T^{3} + \cdots + 16$$
$13$ $$T^{4} + 2 T^{3} + \cdots + 361$$
$17$ $$T^{4} + 8 T^{3} + \cdots + 121$$
$19$ $$T^{4} + 4 T^{3} + \cdots + 16$$
$23$ $$T^{4} - 10 T^{3} + \cdots + 400$$
$29$ $$T^{4} + 8 T^{3} + \cdots + 841$$
$31$ $$T^{4} + 40 T^{2} + \cdots + 400$$
$37$ $$T^{4} + 15 T^{3} + \cdots + 625$$
$41$ $$T^{4} + 10 T^{2} + \cdots + 25$$
$43$ $$(T^{2} - 2 T - 44)^{2}$$
$47$ $$T^{4} + 2 T^{3} + \cdots + 16$$
$53$ $$T^{4} + 5 T^{3} + \cdots + 25$$
$59$ $$T^{4} - 4 T^{3} + \cdots + 256$$
$61$ $$T^{4} - 2 T^{3} + \cdots + 1$$
$67$ $$T^{4} - 2 T^{3} + \cdots + 16$$
$71$ $$T^{4} - 8 T^{3} + \cdots + 16$$
$73$ $$T^{4} + 10 T^{3} + \cdots + 625$$
$79$ $$T^{4}$$
$83$ $$T^{4} + 18 T^{3} + \cdots + 1936$$
$89$ $$T^{4} + 9 T^{3} + \cdots + 1681$$
$97$ $$T^{4} - 2 T^{3} + \cdots + 361$$