Properties

 Label 375.2.i.a Level $375$ Weight $2$ Character orbit 375.i Analytic conductor $2.994$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [375,2,Mod(49,375)]

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, 7]))

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

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

chi := DirichletCharacter("375.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$375 = 3 \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 375.i (of order $$10$$, 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: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{20})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ x^8 - x^6 + x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 75) Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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 primitive root of unity $$\zeta_{20}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{20} q^{2} + \zeta_{20}^{7} q^{3} - \zeta_{20}^{2} q^{4} + (\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2} - 1) q^{6} + ( - 4 \zeta_{20}^{7} + 2 \zeta_{20}^{5} - 4 \zeta_{20}^{3}) q^{7} - 3 \zeta_{20}^{3} q^{8} - \zeta_{20}^{4} q^{9} +O(q^{10})$$ q + z * q^2 + z^7 * q^3 - z^2 * q^4 + (z^6 - z^4 + z^2 - 1) * q^6 + (-4*z^7 + 2*z^5 - 4*z^3) * q^7 - 3*z^3 * q^8 - z^4 * q^9 $$q + \zeta_{20} q^{2} + \zeta_{20}^{7} q^{3} - \zeta_{20}^{2} q^{4} + (\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2} - 1) q^{6} + ( - 4 \zeta_{20}^{7} + 2 \zeta_{20}^{5} - 4 \zeta_{20}^{3}) q^{7} - 3 \zeta_{20}^{3} q^{8} - \zeta_{20}^{4} q^{9} + (2 \zeta_{20}^{2} - 2) q^{11} + ( - \zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}) q^{12} + ( - 5 \zeta_{20}^{7} + 4 \zeta_{20}^{5} - 4 \zeta_{20}^{3} + 5 \zeta_{20}) q^{13} + ( - 2 \zeta_{20}^{6} - 4 \zeta_{20}^{2} + 4) q^{14} - \zeta_{20}^{4} q^{16} + ( - 3 \zeta_{20}^{5} + \zeta_{20}^{3} - 3 \zeta_{20}) q^{17} - \zeta_{20}^{5} q^{18} + ( - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{2}) q^{19} + (4 \zeta_{20}^{4} - 2 \zeta_{20}^{2} + 4) q^{21} + (2 \zeta_{20}^{3} - 2 \zeta_{20}) q^{22} + (4 \zeta_{20}^{7} - 4 \zeta_{20}^{5} - 2 \zeta_{20}) q^{23} + 3 q^{24} + ( - \zeta_{20}^{6} + \zeta_{20}^{4} + 5) q^{26} + \zeta_{20} q^{27} + (2 \zeta_{20}^{7} + 4 \zeta_{20}^{3} - 4 \zeta_{20}) q^{28} + (\zeta_{20}^{4} + 5 \zeta_{20}^{2} + 1) q^{29} + (4 \zeta_{20}^{6} - 6 \zeta_{20}^{4} + 6 \zeta_{20}^{2} - 4) q^{31} + 5 \zeta_{20}^{5} q^{32} + ( - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{3} - 2 \zeta_{20}) q^{33} + ( - 3 \zeta_{20}^{6} + \zeta_{20}^{4} - 3 \zeta_{20}^{2}) q^{34} + \zeta_{20}^{6} q^{36} + ( - 5 \zeta_{20}^{5} + 5 \zeta_{20}^{3}) q^{37} + ( - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{3}) q^{38} + (5 \zeta_{20}^{6} + \zeta_{20}^{2} - 1) q^{39} + ( - \zeta_{20}^{6} - 2 \zeta_{20}^{4} - \zeta_{20}^{2}) q^{41} + (4 \zeta_{20}^{5} - 2 \zeta_{20}^{3} + 4 \zeta_{20}) q^{42} + ( - 6 \zeta_{20}^{7} + 2 \zeta_{20}^{5} - 6 \zeta_{20}^{3}) q^{43} + ( - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{2}) q^{44} + ( - 4 \zeta_{20}^{4} + 2 \zeta_{20}^{2} - 4) q^{46} + (2 \zeta_{20}^{7} - 2 \zeta_{20}^{3} + 2 \zeta_{20}) q^{47} + \zeta_{20} q^{48} - 13 q^{49} + ( - 3 \zeta_{20}^{6} + 3 \zeta_{20}^{4} + 2) q^{51} + (\zeta_{20}^{7} - \zeta_{20}^{5} - 5 \zeta_{20}) q^{52} + (3 \zeta_{20}^{7} + \zeta_{20}^{3} - \zeta_{20}) q^{53} + \zeta_{20}^{2} q^{54} + (6 \zeta_{20}^{6} + 6 \zeta_{20}^{4} - 6 \zeta_{20}^{2} - 6) q^{56} + (2 \zeta_{20}^{7} - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{3}) q^{57} + (\zeta_{20}^{5} + 5 \zeta_{20}^{3} + \zeta_{20}) q^{58} + 4 \zeta_{20}^{4} q^{59} + ( - \zeta_{20}^{6} - \zeta_{20}^{2} + 1) q^{61} + (4 \zeta_{20}^{7} - 6 \zeta_{20}^{5} + 6 \zeta_{20}^{3} - 4 \zeta_{20}) q^{62} + (2 \zeta_{20}^{7} + 2 \zeta_{20}^{5} - 2 \zeta_{20}^{3} - 2 \zeta_{20}) q^{63} + 7 \zeta_{20}^{6} q^{64} + ( - 2 \zeta_{20}^{6} + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{2}) q^{66} + (2 \zeta_{20}^{5} - 4 \zeta_{20}^{3} + 2 \zeta_{20}) q^{67} + (3 \zeta_{20}^{7} - \zeta_{20}^{5} + 3 \zeta_{20}^{3}) q^{68} + ( - 2 \zeta_{20}^{6} - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{2} + 2) q^{69} + (2 \zeta_{20}^{4} + 2 \zeta_{20}^{2} + 2) q^{71} + 3 \zeta_{20}^{7} q^{72} + ( - 5 \zeta_{20}^{7} + 5 \zeta_{20}^{5} + 5 \zeta_{20}) q^{73} + ( - 5 \zeta_{20}^{6} + 5 \zeta_{20}^{4}) q^{74} + (2 \zeta_{20}^{6} - 2 \zeta_{20}^{4}) q^{76} + (4 \zeta_{20}^{7} - 4 \zeta_{20}^{5} + 8 \zeta_{20}) q^{77} + (5 \zeta_{20}^{7} + \zeta_{20}^{3} - \zeta_{20}) q^{78} + (\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2} - 1) q^{81} + ( - \zeta_{20}^{7} - 2 \zeta_{20}^{5} - \zeta_{20}^{3}) q^{82} + (4 \zeta_{20}^{5} + 6 \zeta_{20}^{3} + 4 \zeta_{20}) q^{83} + ( - 4 \zeta_{20}^{6} + 2 \zeta_{20}^{4} - 4 \zeta_{20}^{2}) q^{84} + ( - 4 \zeta_{20}^{6} - 6 \zeta_{20}^{2} + 6) q^{86} + (6 \zeta_{20}^{7} - 5 \zeta_{20}^{5} + 5 \zeta_{20}^{3} - 6 \zeta_{20}) q^{87} + ( - 6 \zeta_{20}^{5} + 6 \zeta_{20}^{3}) q^{88} + (6 \zeta_{20}^{6} - \zeta_{20}^{2} + 1) q^{89} + ( - 18 \zeta_{20}^{6} + 4 \zeta_{20}^{4} - 18 \zeta_{20}^{2}) q^{91} + (4 \zeta_{20}^{5} - 2 \zeta_{20}^{3} + 4 \zeta_{20}) q^{92} + (2 \zeta_{20}^{7} - 6 \zeta_{20}^{5} + 2 \zeta_{20}^{3}) q^{93} + (2 \zeta_{20}^{6} - 4 \zeta_{20}^{4} + 4 \zeta_{20}^{2} - 2) q^{94} - 5 \zeta_{20}^{2} q^{96} + ( - 4 \zeta_{20}^{7} + 3 \zeta_{20}^{3} - 3 \zeta_{20}) q^{97} - 13 \zeta_{20} q^{98} + ( - 2 \zeta_{20}^{6} + 2 \zeta_{20}^{4}) q^{99} +O(q^{100})$$ q + z * q^2 + z^7 * q^3 - z^2 * q^4 + (z^6 - z^4 + z^2 - 1) * q^6 + (-4*z^7 + 2*z^5 - 4*z^3) * q^7 - 3*z^3 * q^8 - z^4 * q^9 + (2*z^2 - 2) * q^11 + (-z^7 + z^5 - z^3 + z) * q^12 + (-5*z^7 + 4*z^5 - 4*z^3 + 5*z) * q^13 + (-2*z^6 - 4*z^2 + 4) * q^14 - z^4 * q^16 + (-3*z^5 + z^3 - 3*z) * q^17 - z^5 * q^18 + (-2*z^4 + 2*z^2) * q^19 + (4*z^4 - 2*z^2 + 4) * q^21 + (2*z^3 - 2*z) * q^22 + (4*z^7 - 4*z^5 - 2*z) * q^23 + 3 * q^24 + (-z^6 + z^4 + 5) * q^26 + z * q^27 + (2*z^7 + 4*z^3 - 4*z) * q^28 + (z^4 + 5*z^2 + 1) * q^29 + (4*z^6 - 6*z^4 + 6*z^2 - 4) * q^31 + 5*z^5 * q^32 + (-2*z^5 + 2*z^3 - 2*z) * q^33 + (-3*z^6 + z^4 - 3*z^2) * q^34 + z^6 * q^36 + (-5*z^5 + 5*z^3) * q^37 + (-2*z^5 + 2*z^3) * q^38 + (5*z^6 + z^2 - 1) * q^39 + (-z^6 - 2*z^4 - z^2) * q^41 + (4*z^5 - 2*z^3 + 4*z) * q^42 + (-6*z^7 + 2*z^5 - 6*z^3) * q^43 + (-2*z^4 + 2*z^2) * q^44 + (-4*z^4 + 2*z^2 - 4) * q^46 + (2*z^7 - 2*z^3 + 2*z) * q^47 + z * q^48 - 13 * q^49 + (-3*z^6 + 3*z^4 + 2) * q^51 + (z^7 - z^5 - 5*z) * q^52 + (3*z^7 + z^3 - z) * q^53 + z^2 * q^54 + (6*z^6 + 6*z^4 - 6*z^2 - 6) * q^56 + (2*z^7 - 2*z^5 + 2*z^3) * q^57 + (z^5 + 5*z^3 + z) * q^58 + 4*z^4 * q^59 + (-z^6 - z^2 + 1) * q^61 + (4*z^7 - 6*z^5 + 6*z^3 - 4*z) * q^62 + (2*z^7 + 2*z^5 - 2*z^3 - 2*z) * q^63 + 7*z^6 * q^64 + (-2*z^6 + 2*z^4 - 2*z^2) * q^66 + (2*z^5 - 4*z^3 + 2*z) * q^67 + (3*z^7 - z^5 + 3*z^3) * q^68 + (-2*z^6 - 2*z^4 + 2*z^2 + 2) * q^69 + (2*z^4 + 2*z^2 + 2) * q^71 + 3*z^7 * q^72 + (-5*z^7 + 5*z^5 + 5*z) * q^73 + (-5*z^6 + 5*z^4) * q^74 + (2*z^6 - 2*z^4) * q^76 + (4*z^7 - 4*z^5 + 8*z) * q^77 + (5*z^7 + z^3 - z) * q^78 + (z^6 - z^4 + z^2 - 1) * q^81 + (-z^7 - 2*z^5 - z^3) * q^82 + (4*z^5 + 6*z^3 + 4*z) * q^83 + (-4*z^6 + 2*z^4 - 4*z^2) * q^84 + (-4*z^6 - 6*z^2 + 6) * q^86 + (6*z^7 - 5*z^5 + 5*z^3 - 6*z) * q^87 + (-6*z^5 + 6*z^3) * q^88 + (6*z^6 - z^2 + 1) * q^89 + (-18*z^6 + 4*z^4 - 18*z^2) * q^91 + (4*z^5 - 2*z^3 + 4*z) * q^92 + (2*z^7 - 6*z^5 + 2*z^3) * q^93 + (2*z^6 - 4*z^4 + 4*z^2 - 2) * q^94 - 5*z^2 * q^96 + (-4*z^7 + 3*z^3 - 3*z) * q^97 - 13*z * q^98 + (-2*z^6 + 2*z^4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 2 q^{4} - 2 q^{6} + 2 q^{9}+O(q^{10})$$ 8 * q - 2 * q^4 - 2 * q^6 + 2 * q^9 $$8 q - 2 q^{4} - 2 q^{6} + 2 q^{9} - 12 q^{11} + 20 q^{14} + 2 q^{16} + 8 q^{19} + 20 q^{21} + 24 q^{24} + 36 q^{26} + 16 q^{29} - 14 q^{34} + 2 q^{36} + 4 q^{39} + 8 q^{44} - 20 q^{46} - 104 q^{49} + 4 q^{51} + 2 q^{54} - 60 q^{56} - 8 q^{59} + 4 q^{61} + 14 q^{64} - 12 q^{66} + 20 q^{69} + 16 q^{71} - 20 q^{74} + 8 q^{76} - 2 q^{81} - 20 q^{84} + 28 q^{86} + 18 q^{89} - 80 q^{91} + 4 q^{94} - 10 q^{96} - 8 q^{99}+O(q^{100})$$ 8 * q - 2 * q^4 - 2 * q^6 + 2 * q^9 - 12 * q^11 + 20 * q^14 + 2 * q^16 + 8 * q^19 + 20 * q^21 + 24 * q^24 + 36 * q^26 + 16 * q^29 - 14 * q^34 + 2 * q^36 + 4 * q^39 + 8 * q^44 - 20 * q^46 - 104 * q^49 + 4 * q^51 + 2 * q^54 - 60 * q^56 - 8 * q^59 + 4 * q^61 + 14 * q^64 - 12 * q^66 + 20 * q^69 + 16 * q^71 - 20 * q^74 + 8 * q^76 - 2 * q^81 - 20 * q^84 + 28 * q^86 + 18 * q^89 - 80 * q^91 + 4 * q^94 - 10 * q^96 - 8 * q^99

Character values

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

 $$n$$ $$127$$ $$251$$ $$\chi(n)$$ $$\zeta_{20}^{2}$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.587785 + 0.809017i 0.587785 − 0.809017i −0.587785 − 0.809017i 0.587785 + 0.809017i −0.951057 − 0.309017i 0.951057 + 0.309017i −0.951057 + 0.309017i 0.951057 − 0.309017i
−0.587785 + 0.809017i −0.951057 + 0.309017i 0.309017 + 0.951057i 0 0.309017 0.951057i 4.47214i −2.85317 0.927051i 0.809017 0.587785i 0
49.2 0.587785 0.809017i 0.951057 0.309017i 0.309017 + 0.951057i 0 0.309017 0.951057i 4.47214i 2.85317 + 0.927051i 0.809017 0.587785i 0
199.1 −0.587785 0.809017i −0.951057 0.309017i 0.309017 0.951057i 0 0.309017 + 0.951057i 4.47214i −2.85317 + 0.927051i 0.809017 + 0.587785i 0
199.2 0.587785 + 0.809017i 0.951057 + 0.309017i 0.309017 0.951057i 0 0.309017 + 0.951057i 4.47214i 2.85317 0.927051i 0.809017 + 0.587785i 0
274.1 −0.951057 0.309017i 0.587785 0.809017i −0.809017 0.587785i 0 −0.809017 + 0.587785i 4.47214i 1.76336 + 2.42705i −0.309017 0.951057i 0
274.2 0.951057 + 0.309017i −0.587785 + 0.809017i −0.809017 0.587785i 0 −0.809017 + 0.587785i 4.47214i −1.76336 2.42705i −0.309017 0.951057i 0
349.1 −0.951057 + 0.309017i 0.587785 + 0.809017i −0.809017 + 0.587785i 0 −0.809017 0.587785i 4.47214i 1.76336 2.42705i −0.309017 + 0.951057i 0
349.2 0.951057 0.309017i −0.587785 0.809017i −0.809017 + 0.587785i 0 −0.809017 0.587785i 4.47214i −1.76336 + 2.42705i −0.309017 + 0.951057i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 349.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
25.d even 5 1 inner
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 375.2.i.a 8
5.b even 2 1 inner 375.2.i.a 8
5.c odd 4 1 75.2.g.a 4
5.c odd 4 1 375.2.g.a 4
15.e even 4 1 225.2.h.a 4
25.d even 5 1 inner 375.2.i.a 8
25.d even 5 1 1875.2.b.b 4
25.e even 10 1 inner 375.2.i.a 8
25.e even 10 1 1875.2.b.b 4
25.f odd 20 1 75.2.g.a 4
25.f odd 20 1 375.2.g.a 4
25.f odd 20 1 1875.2.a.a 2
25.f odd 20 1 1875.2.a.d 2
75.l even 20 1 225.2.h.a 4
75.l even 20 1 5625.2.a.a 2
75.l even 20 1 5625.2.a.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.a 4 5.c odd 4 1
75.2.g.a 4 25.f odd 20 1
225.2.h.a 4 15.e even 4 1
225.2.h.a 4 75.l even 20 1
375.2.g.a 4 5.c odd 4 1
375.2.g.a 4 25.f odd 20 1
375.2.i.a 8 1.a even 1 1 trivial
375.2.i.a 8 5.b even 2 1 inner
375.2.i.a 8 25.d even 5 1 inner
375.2.i.a 8 25.e even 10 1 inner
1875.2.a.a 2 25.f odd 20 1
1875.2.a.d 2 25.f odd 20 1
1875.2.b.b 4 25.d even 5 1
1875.2.b.b 4 25.e even 10 1
5625.2.a.a 2 75.l even 20 1
5625.2.a.h 2 75.l even 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - T_{2}^{6} + T_{2}^{4} - T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(375, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - T^{6} + T^{4} - T^{2} + 1$$
$3$ $$T^{8} - T^{6} + T^{4} - T^{2} + 1$$
$5$ $$T^{8}$$
$7$ $$(T^{2} + 20)^{4}$$
$11$ $$(T^{4} + 6 T^{3} + 16 T^{2} + 16 T + 16)^{2}$$
$13$ $$T^{8} - 44 T^{6} + 766 T^{4} + \cdots + 130321$$
$17$ $$T^{8} - 4 T^{6} + 166 T^{4} + \cdots + 14641$$
$19$ $$(T^{4} - 4 T^{3} + 16 T^{2} - 24 T + 16)^{2}$$
$23$ $$T^{8} - 20 T^{6} + 400 T^{4} + \cdots + 160000$$
$29$ $$(T^{4} - 8 T^{3} + 34 T^{2} - 87 T + 841)^{2}$$
$31$ $$(T^{4} + 40 T^{2} + 200 T + 400)^{2}$$
$37$ $$T^{8} + 25 T^{6} + 3750 T^{4} + \cdots + 390625$$
$41$ $$(T^{4} + 10 T^{2} - 25 T + 25)^{2}$$
$43$ $$(T^{4} + 92 T^{2} + 1936)^{2}$$
$47$ $$T^{8} - 44 T^{6} + 736 T^{4} + \cdots + 256$$
$53$ $$T^{8} + 5 T^{6} + 150 T^{4} + \cdots + 625$$
$59$ $$(T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256)^{2}$$
$61$ $$(T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1)^{2}$$
$67$ $$T^{8} - 44 T^{6} + 736 T^{4} + \cdots + 256$$
$71$ $$(T^{4} - 8 T^{3} + 24 T^{2} + 8 T + 16)^{2}$$
$73$ $$T^{8} - 100 T^{6} + 3750 T^{4} + \cdots + 390625$$
$79$ $$T^{8}$$
$83$ $$T^{8} + 76 T^{6} + 22416 T^{4} + \cdots + 3748096$$
$89$ $$(T^{4} - 9 T^{3} + 46 T^{2} - 164 T + 1681)^{2}$$
$97$ $$T^{8} - 124 T^{6} + 5806 T^{4} + \cdots + 130321$$