Properties

 Label 75.3.f.b Level $75$ Weight $3$ Character orbit 75.f Analytic conductor $2.044$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,3,Mod(7,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("75.7");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 75.f (of order $$4$$, degree $$2$$, 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: $$2.04360198270$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta_1 q^{2} - \beta_{3} q^{3} + 8 \beta_{2} q^{4} + 6 q^{6} - 5 \beta_1 q^{7} + 8 \beta_{3} q^{8} - 3 \beta_{2} q^{9}+O(q^{10})$$ q + 2*b1 * q^2 - b3 * q^3 + 8*b2 * q^4 + 6 * q^6 - 5*b1 * q^7 + 8*b3 * q^8 - 3*b2 * q^9 $$q + 2 \beta_1 q^{2} - \beta_{3} q^{3} + 8 \beta_{2} q^{4} + 6 q^{6} - 5 \beta_1 q^{7} + 8 \beta_{3} q^{8} - 3 \beta_{2} q^{9} + 6 q^{11} + 8 \beta_1 q^{12} + 3 \beta_{3} q^{13} - 30 \beta_{2} q^{14} - 16 q^{16} - 14 \beta_1 q^{17} - 6 \beta_{3} q^{18} + 23 \beta_{2} q^{19} - 15 q^{21} + 12 \beta_1 q^{22} - 10 \beta_{3} q^{23} + 24 \beta_{2} q^{24} - 18 q^{26} - 3 \beta_1 q^{27} - 40 \beta_{3} q^{28} + 6 \beta_{2} q^{29} + 25 q^{31} - 6 \beta_{3} q^{33} - 84 \beta_{2} q^{34} + 24 q^{36} + 20 \beta_1 q^{37} + 46 \beta_{3} q^{38} + 9 \beta_{2} q^{39} - 60 q^{41} - 30 \beta_1 q^{42} + 49 \beta_{3} q^{43} + 48 \beta_{2} q^{44} + 60 q^{46} - 6 \beta_1 q^{47} + 16 \beta_{3} q^{48} + 26 \beta_{2} q^{49} - 42 q^{51} - 24 \beta_1 q^{52} - 20 \beta_{3} q^{53} - 18 \beta_{2} q^{54} + 120 q^{56} + 23 \beta_1 q^{57} + 12 \beta_{3} q^{58} - 18 \beta_{2} q^{59} - 37 q^{61} + 50 \beta_1 q^{62} + 15 \beta_{3} q^{63} + 64 \beta_{2} q^{64} + 36 q^{66} + 21 \beta_1 q^{67} - 112 \beta_{3} q^{68} - 30 \beta_{2} q^{69} + 132 q^{71} + 24 \beta_1 q^{72} - 20 \beta_{3} q^{73} + 120 \beta_{2} q^{74} - 184 q^{76} - 30 \beta_1 q^{77} + 18 \beta_{3} q^{78} + 10 \beta_{2} q^{79} - 9 q^{81} - 120 \beta_1 q^{82} + 2 \beta_{3} q^{83} - 120 \beta_{2} q^{84} - 294 q^{86} + 6 \beta_1 q^{87} + 48 \beta_{3} q^{88} - 132 \beta_{2} q^{89} + 45 q^{91} + 80 \beta_1 q^{92} - 25 \beta_{3} q^{93} - 36 \beta_{2} q^{94} - 19 \beta_1 q^{97} + 52 \beta_{3} q^{98} - 18 \beta_{2} q^{99}+O(q^{100})$$ q + 2*b1 * q^2 - b3 * q^3 + 8*b2 * q^4 + 6 * q^6 - 5*b1 * q^7 + 8*b3 * q^8 - 3*b2 * q^9 + 6 * q^11 + 8*b1 * q^12 + 3*b3 * q^13 - 30*b2 * q^14 - 16 * q^16 - 14*b1 * q^17 - 6*b3 * q^18 + 23*b2 * q^19 - 15 * q^21 + 12*b1 * q^22 - 10*b3 * q^23 + 24*b2 * q^24 - 18 * q^26 - 3*b1 * q^27 - 40*b3 * q^28 + 6*b2 * q^29 + 25 * q^31 - 6*b3 * q^33 - 84*b2 * q^34 + 24 * q^36 + 20*b1 * q^37 + 46*b3 * q^38 + 9*b2 * q^39 - 60 * q^41 - 30*b1 * q^42 + 49*b3 * q^43 + 48*b2 * q^44 + 60 * q^46 - 6*b1 * q^47 + 16*b3 * q^48 + 26*b2 * q^49 - 42 * q^51 - 24*b1 * q^52 - 20*b3 * q^53 - 18*b2 * q^54 + 120 * q^56 + 23*b1 * q^57 + 12*b3 * q^58 - 18*b2 * q^59 - 37 * q^61 + 50*b1 * q^62 + 15*b3 * q^63 + 64*b2 * q^64 + 36 * q^66 + 21*b1 * q^67 - 112*b3 * q^68 - 30*b2 * q^69 + 132 * q^71 + 24*b1 * q^72 - 20*b3 * q^73 + 120*b2 * q^74 - 184 * q^76 - 30*b1 * q^77 + 18*b3 * q^78 + 10*b2 * q^79 - 9 * q^81 - 120*b1 * q^82 + 2*b3 * q^83 - 120*b2 * q^84 - 294 * q^86 + 6*b1 * q^87 + 48*b3 * q^88 - 132*b2 * q^89 + 45 * q^91 + 80*b1 * q^92 - 25*b3 * q^93 - 36*b2 * q^94 - 19*b1 * q^97 + 52*b3 * q^98 - 18*b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 24 q^{6}+O(q^{10})$$ 4 * q + 24 * q^6 $$4 q + 24 q^{6} + 24 q^{11} - 64 q^{16} - 60 q^{21} - 72 q^{26} + 100 q^{31} + 96 q^{36} - 240 q^{41} + 240 q^{46} - 168 q^{51} + 480 q^{56} - 148 q^{61} + 144 q^{66} + 528 q^{71} - 736 q^{76} - 36 q^{81} - 1176 q^{86} + 180 q^{91}+O(q^{100})$$ 4 * q + 24 * q^6 + 24 * q^11 - 64 * q^16 - 60 * q^21 - 72 * q^26 + 100 * q^31 + 96 * q^36 - 240 * q^41 + 240 * q^46 - 168 * q^51 + 480 * q^56 - 148 * q^61 + 144 * q^66 + 528 * q^71 - 736 * q^76 - 36 * q^81 - 1176 * q^86 + 180 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 3$$ (v^2) / 3 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 3$$ (v^3) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{2}$$ 3*b2 $$\nu^{3}$$ $$=$$ $$3\beta_{3}$$ 3*b3

Character values

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

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 −1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i
−2.44949 2.44949i −1.22474 + 1.22474i 8.00000i 0 6.00000 6.12372 + 6.12372i 9.79796 9.79796i 3.00000i 0
7.2 2.44949 + 2.44949i 1.22474 1.22474i 8.00000i 0 6.00000 −6.12372 6.12372i −9.79796 + 9.79796i 3.00000i 0
43.1 −2.44949 + 2.44949i −1.22474 1.22474i 8.00000i 0 6.00000 6.12372 6.12372i 9.79796 + 9.79796i 3.00000i 0
43.2 2.44949 2.44949i 1.22474 + 1.22474i 8.00000i 0 6.00000 −6.12372 + 6.12372i −9.79796 9.79796i 3.00000i 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
5.b even 2 1 inner
5.c odd 4 2 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.3.f.b 4
3.b odd 2 1 225.3.g.b 4
4.b odd 2 1 1200.3.bg.g 4
5.b even 2 1 inner 75.3.f.b 4
5.c odd 4 2 inner 75.3.f.b 4
15.d odd 2 1 225.3.g.b 4
15.e even 4 2 225.3.g.b 4
20.d odd 2 1 1200.3.bg.g 4
20.e even 4 2 1200.3.bg.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.f.b 4 1.a even 1 1 trivial
75.3.f.b 4 5.b even 2 1 inner
75.3.f.b 4 5.c odd 4 2 inner
225.3.g.b 4 3.b odd 2 1
225.3.g.b 4 15.d odd 2 1
225.3.g.b 4 15.e even 4 2
1200.3.bg.g 4 4.b odd 2 1
1200.3.bg.g 4 20.d odd 2 1
1200.3.bg.g 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 144$$ acting on $$S_{3}^{\mathrm{new}}(75, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 144$$
$3$ $$T^{4} + 9$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 5625$$
$11$ $$(T - 6)^{4}$$
$13$ $$T^{4} + 729$$
$17$ $$T^{4} + 345744$$
$19$ $$(T^{2} + 529)^{2}$$
$23$ $$T^{4} + 90000$$
$29$ $$(T^{2} + 36)^{2}$$
$31$ $$(T - 25)^{4}$$
$37$ $$T^{4} + 1440000$$
$41$ $$(T + 60)^{4}$$
$43$ $$T^{4} + 51883209$$
$47$ $$T^{4} + 11664$$
$53$ $$T^{4} + 1440000$$
$59$ $$(T^{2} + 324)^{2}$$
$61$ $$(T + 37)^{4}$$
$67$ $$T^{4} + 1750329$$
$71$ $$(T - 132)^{4}$$
$73$ $$T^{4} + 1440000$$
$79$ $$(T^{2} + 100)^{2}$$
$83$ $$T^{4} + 144$$
$89$ $$(T^{2} + 17424)^{2}$$
$97$ $$T^{4} + 1172889$$