Properties

 Label 375.2.g.a Level $375$ Weight $2$ Character orbit 375.g Analytic conductor $2.994$ 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] = [375,2,Mod(76,375)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(375, 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("375.76");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$375 = 3 \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 375.g (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: $$2.99439007580$$ 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 75) 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 + 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 + z^2 * q^6 + (4*z^3 - 4*z^2 - 2) * q^7 + 3*z^2 * q^8 - z * q^9 + (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 + 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 - 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 + (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 + (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 + (-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 + (-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 + (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 + (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 + (-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^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^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 + (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 + (-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^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 + (-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 + (-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 - 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 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} + q^{3} + q^{4} - q^{6} - 3 q^{8} - q^{9}+O(q^{10})$$ 4 * q + q^2 + q^3 + q^4 - q^6 - 3 * q^8 - q^9 $$4 q + q^{2} + q^{3} + q^{4} - q^{6} - 3 q^{8} - q^{9} - 6 q^{11} - q^{12} + 2 q^{13} - 10 q^{14} + q^{16} + 8 q^{17} - 4 q^{18} - 4 q^{19} + 10 q^{21} - 4 q^{22} - 10 q^{23} - 12 q^{24} + 18 q^{26} + q^{27} + 10 q^{28} - 8 q^{29} - 20 q^{32} - 4 q^{33} + 7 q^{34} + q^{36} + 15 q^{37} - 6 q^{38} - 2 q^{39} - 10 q^{42} - 4 q^{43} - 4 q^{44} - 10 q^{46} + 2 q^{47} - q^{48} + 52 q^{49} + 2 q^{51} - 2 q^{52} + 5 q^{53} - q^{54} - 30 q^{56} + 4 q^{57} + 8 q^{58} + 4 q^{59} + 2 q^{61} + 10 q^{62} + 10 q^{63} - 7 q^{64} - 6 q^{66} - 2 q^{67} + 2 q^{68} - 10 q^{69} + 8 q^{71} - 3 q^{72} + 10 q^{73} + 10 q^{74} + 4 q^{76} + 20 q^{77} + 7 q^{78} - q^{81} + 10 q^{82} + 18 q^{83} + 10 q^{84} + 14 q^{86} + 3 q^{87} - 18 q^{88} - 9 q^{89} - 40 q^{91} - 10 q^{92} - 20 q^{93} - 2 q^{94} - 5 q^{96} - 2 q^{97} + 13 q^{98} + 4 q^{99}+O(q^{100})$$ 4 * q + q^2 + q^3 + q^4 - q^6 - 3 * q^8 - q^9 - 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 + 18 * q^26 + q^27 + 10 * q^28 - 8 * q^29 - 20 * q^32 - 4 * q^33 + 7 * q^34 + q^36 + 15 * q^37 - 6 * q^38 - 2 * q^39 - 10 * q^42 - 4 * q^43 - 4 * q^44 - 10 * q^46 + 2 * q^47 - q^48 + 52 * q^49 + 2 * q^51 - 2 * q^52 + 5 * q^53 - q^54 - 30 * q^56 + 4 * q^57 + 8 * q^58 + 4 * q^59 + 2 * q^61 + 10 * q^62 + 10 * q^63 - 7 * q^64 - 6 * q^66 - 2 * q^67 + 2 * q^68 - 10 * q^69 + 8 * q^71 - 3 * q^72 + 10 * q^73 + 10 * q^74 + 4 * q^76 + 20 * q^77 + 7 * q^78 - q^81 + 10 * q^82 + 18 * q^83 + 10 * q^84 + 14 * q^86 + 3 * q^87 - 18 * q^88 - 9 * q^89 - 40 * q^91 - 10 * q^92 - 20 * q^93 - 2 * q^94 - 5 * q^96 - 2 * q^97 + 13 * q^98 + 4 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/375\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$251$$ $$\chi(n)$$ $$-\zeta_{10}^{3}$$ $$1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
76.1
 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 0.309017 0.951057i −4.47214 0.927051 2.85317i −0.809017 + 0.587785i 0
151.1 −0.309017 0.951057i 0.809017 0.587785i 0.809017 0.587785i 0 −0.809017 0.587785i 4.47214 −2.42705 1.76336i 0.309017 0.951057i 0
226.1 −0.309017 + 0.951057i 0.809017 + 0.587785i 0.809017 + 0.587785i 0 −0.809017 + 0.587785i 4.47214 −2.42705 + 1.76336i 0.309017 + 0.951057i 0
301.1 0.809017 0.587785i −0.309017 + 0.951057i −0.309017 + 0.951057i 0 0.309017 + 0.951057i −4.47214 0.927051 + 2.85317i −0.809017 0.587785i 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 375.2.g.a 4
5.b even 2 1 75.2.g.a 4
5.c odd 4 2 375.2.i.a 8
15.d odd 2 1 225.2.h.a 4
25.d even 5 1 inner 375.2.g.a 4
25.d even 5 1 1875.2.a.a 2
25.e even 10 1 75.2.g.a 4
25.e even 10 1 1875.2.a.d 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 1 225.2.h.a 4
75.h odd 10 1 5625.2.a.a 2
75.j odd 10 1 5625.2.a.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.a 4 5.b even 2 1
75.2.g.a 4 25.e even 10 1
225.2.h.a 4 15.d odd 2 1
225.2.h.a 4 75.h odd 10 1
375.2.g.a 4 1.a even 1 1 trivial
375.2.g.a 4 25.d even 5 1 inner
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.d even 5 1
1875.2.a.d 2 25.e even 10 1
1875.2.b.b 4 25.f odd 20 2
5625.2.a.a 2 75.h odd 10 1
5625.2.a.h 2 75.j 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 $$S_{2}^{\mathrm{new}}(375, [\chi])$$.

Hecke characteristic polynomials

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