# Properties

 Label 75.3.c.b Level $75$ Weight $3$ Character orbit 75.c Self dual yes Analytic conductor $2.044$ Analytic rank $0$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,3,Mod(26,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("75.26");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 75.c (of order $$2$$, degree $$1$$, 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: yes Analytic conductor: $$2.04360198270$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + 4 q^{4} - 11 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + 4 * q^4 - 11 * q^7 + 9 * q^9 $$q + 3 q^{3} + 4 q^{4} - 11 q^{7} + 9 q^{9} + 12 q^{12} + q^{13} + 16 q^{16} - 37 q^{19} - 33 q^{21} + 27 q^{27} - 44 q^{28} - 13 q^{31} + 36 q^{36} - 26 q^{37} + 3 q^{39} + 61 q^{43} + 48 q^{48} + 72 q^{49} + 4 q^{52} - 111 q^{57} + 47 q^{61} - 99 q^{63} + 64 q^{64} + 109 q^{67} + 46 q^{73} - 148 q^{76} - 142 q^{79} + 81 q^{81} - 132 q^{84} - 11 q^{91} - 39 q^{93} + 169 q^{97}+O(q^{100})$$ q + 3 * q^3 + 4 * q^4 - 11 * q^7 + 9 * q^9 + 12 * q^12 + q^13 + 16 * q^16 - 37 * q^19 - 33 * q^21 + 27 * q^27 - 44 * q^28 - 13 * q^31 + 36 * q^36 - 26 * q^37 + 3 * q^39 + 61 * q^43 + 48 * q^48 + 72 * q^49 + 4 * q^52 - 111 * q^57 + 47 * q^61 - 99 * q^63 + 64 * q^64 + 109 * q^67 + 46 * q^73 - 148 * q^76 - 142 * q^79 + 81 * q^81 - 132 * q^84 - 11 * q^91 - 39 * q^93 + 169 * q^97

## Character values

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

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 0
0 3.00000 4.00000 0 0 −11.0000 0 9.00000 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.3.c.b yes 1
3.b odd 2 1 CM 75.3.c.b yes 1
4.b odd 2 1 1200.3.l.c 1
5.b even 2 1 75.3.c.a 1
5.c odd 4 2 75.3.d.a 2
12.b even 2 1 1200.3.l.c 1
15.d odd 2 1 75.3.c.a 1
15.e even 4 2 75.3.d.a 2
20.d odd 2 1 1200.3.l.d 1
20.e even 4 2 1200.3.c.a 2
60.h even 2 1 1200.3.l.d 1
60.l odd 4 2 1200.3.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.c.a 1 5.b even 2 1
75.3.c.a 1 15.d odd 2 1
75.3.c.b yes 1 1.a even 1 1 trivial
75.3.c.b yes 1 3.b odd 2 1 CM
75.3.d.a 2 5.c odd 4 2
75.3.d.a 2 15.e even 4 2
1200.3.c.a 2 20.e even 4 2
1200.3.c.a 2 60.l odd 4 2
1200.3.l.c 1 4.b odd 2 1
1200.3.l.c 1 12.b even 2 1
1200.3.l.d 1 20.d odd 2 1
1200.3.l.d 1 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(75, [\chi])$$:

 $$T_{2}$$ T2 $$T_{7} + 11$$ T7 + 11 $$T_{13} - 1$$ T13 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T$$
$7$ $$T + 11$$
$11$ $$T$$
$13$ $$T - 1$$
$17$ $$T$$
$19$ $$T + 37$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T + 13$$
$37$ $$T + 26$$
$41$ $$T$$
$43$ $$T - 61$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T - 47$$
$67$ $$T - 109$$
$71$ $$T$$
$73$ $$T - 46$$
$79$ $$T + 142$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T - 169$$
show more
show less