# Properties

 Label 75.2.a.a Level $75$ Weight $2$ Character orbit 75.a Self dual yes Analytic conductor $0.599$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,2,Mod(1,75)]

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([0, 0]))

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

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 75.a (trivial)

## 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: $$0.598878015160$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−2.00000 1.00000 2.00000 0 −2.00000 3.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.2.a.a 1
3.b odd 2 1 225.2.a.e 1
4.b odd 2 1 1200.2.a.c 1
5.b even 2 1 75.2.a.c yes 1
5.c odd 4 2 75.2.b.a 2
7.b odd 2 1 3675.2.a.b 1
8.b even 2 1 4800.2.a.bb 1
8.d odd 2 1 4800.2.a.br 1
11.b odd 2 1 9075.2.a.s 1
12.b even 2 1 3600.2.a.j 1
15.d odd 2 1 225.2.a.a 1
15.e even 4 2 225.2.b.a 2
20.d odd 2 1 1200.2.a.p 1
20.e even 4 2 1200.2.f.d 2
35.c odd 2 1 3675.2.a.q 1
40.e odd 2 1 4800.2.a.be 1
40.f even 2 1 4800.2.a.bq 1
40.i odd 4 2 4800.2.f.l 2
40.k even 4 2 4800.2.f.y 2
55.d odd 2 1 9075.2.a.a 1
60.h even 2 1 3600.2.a.bk 1
60.l odd 4 2 3600.2.f.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.a.a 1 1.a even 1 1 trivial
75.2.a.c yes 1 5.b even 2 1
75.2.b.a 2 5.c odd 4 2
225.2.a.a 1 15.d odd 2 1
225.2.a.e 1 3.b odd 2 1
225.2.b.a 2 15.e even 4 2
1200.2.a.c 1 4.b odd 2 1
1200.2.a.p 1 20.d odd 2 1
1200.2.f.d 2 20.e even 4 2
3600.2.a.j 1 12.b even 2 1
3600.2.a.bk 1 60.h even 2 1
3600.2.f.p 2 60.l odd 4 2
3675.2.a.b 1 7.b odd 2 1
3675.2.a.q 1 35.c odd 2 1
4800.2.a.bb 1 8.b even 2 1
4800.2.a.be 1 40.e odd 2 1
4800.2.a.bq 1 40.f even 2 1
4800.2.a.br 1 8.d odd 2 1
4800.2.f.l 2 40.i odd 4 2
4800.2.f.y 2 40.k even 4 2
9075.2.a.a 1 55.d odd 2 1
9075.2.a.s 1 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(75))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T - 3$$
$11$ $$T - 2$$
$13$ $$T + 1$$
$17$ $$T + 2$$
$19$ $$T + 5$$
$23$ $$T + 6$$
$29$ $$T - 10$$
$31$ $$T + 3$$
$37$ $$T + 2$$
$41$ $$T + 8$$
$43$ $$T + 1$$
$47$ $$T + 2$$
$53$ $$T - 4$$
$59$ $$T + 10$$
$61$ $$T - 7$$
$67$ $$T - 3$$
$71$ $$T + 8$$
$73$ $$T - 14$$
$79$ $$T$$
$83$ $$T + 6$$
$89$ $$T$$
$97$ $$T + 17$$