# Properties

 Label 75.2.a.b 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: no (minimal twist has level 15) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} - q^{4} + q^{6} - 3 q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 - q^4 + q^6 - 3 * q^8 + q^9 $$q + q^{2} + q^{3} - q^{4} + q^{6} - 3 q^{8} + q^{9} - 4 q^{11} - q^{12} + 2 q^{13} - q^{16} - 2 q^{17} + q^{18} + 4 q^{19} - 4 q^{22} - 3 q^{24} + 2 q^{26} + q^{27} - 2 q^{29} + 5 q^{32} - 4 q^{33} - 2 q^{34} - q^{36} + 10 q^{37} + 4 q^{38} + 2 q^{39} + 10 q^{41} - 4 q^{43} + 4 q^{44} - 8 q^{47} - q^{48} - 7 q^{49} - 2 q^{51} - 2 q^{52} + 10 q^{53} + q^{54} + 4 q^{57} - 2 q^{58} - 4 q^{59} - 2 q^{61} + 7 q^{64} - 4 q^{66} - 12 q^{67} + 2 q^{68} - 8 q^{71} - 3 q^{72} - 10 q^{73} + 10 q^{74} - 4 q^{76} + 2 q^{78} + q^{81} + 10 q^{82} - 12 q^{83} - 4 q^{86} - 2 q^{87} + 12 q^{88} - 6 q^{89} - 8 q^{94} + 5 q^{96} - 2 q^{97} - 7 q^{98} - 4 q^{99}+O(q^{100})$$ q + q^2 + q^3 - q^4 + q^6 - 3 * q^8 + q^9 - 4 * q^11 - q^12 + 2 * q^13 - q^16 - 2 * q^17 + q^18 + 4 * q^19 - 4 * q^22 - 3 * q^24 + 2 * q^26 + q^27 - 2 * q^29 + 5 * q^32 - 4 * q^33 - 2 * q^34 - q^36 + 10 * q^37 + 4 * q^38 + 2 * q^39 + 10 * q^41 - 4 * q^43 + 4 * q^44 - 8 * q^47 - q^48 - 7 * q^49 - 2 * q^51 - 2 * q^52 + 10 * q^53 + q^54 + 4 * q^57 - 2 * q^58 - 4 * q^59 - 2 * q^61 + 7 * q^64 - 4 * q^66 - 12 * q^67 + 2 * q^68 - 8 * q^71 - 3 * q^72 - 10 * q^73 + 10 * q^74 - 4 * q^76 + 2 * q^78 + q^81 + 10 * q^82 - 12 * q^83 - 4 * q^86 - 2 * q^87 + 12 * q^88 - 6 * q^89 - 8 * q^94 + 5 * q^96 - 2 * q^97 - 7 * q^98 - 4 * 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
1.00000 1.00000 −1.00000 0 1.00000 0 −3.00000 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.b 1
3.b odd 2 1 225.2.a.b 1
4.b odd 2 1 1200.2.a.e 1
5.b even 2 1 15.2.a.a 1
5.c odd 4 2 75.2.b.b 2
7.b odd 2 1 3675.2.a.j 1
8.b even 2 1 4800.2.a.t 1
8.d odd 2 1 4800.2.a.bz 1
11.b odd 2 1 9075.2.a.g 1
12.b even 2 1 3600.2.a.u 1
15.d odd 2 1 45.2.a.a 1
15.e even 4 2 225.2.b.b 2
20.d odd 2 1 240.2.a.d 1
20.e even 4 2 1200.2.f.h 2
35.c odd 2 1 735.2.a.c 1
35.i odd 6 2 735.2.i.d 2
35.j even 6 2 735.2.i.e 2
40.e odd 2 1 960.2.a.a 1
40.f even 2 1 960.2.a.l 1
40.i odd 4 2 4800.2.f.bf 2
40.k even 4 2 4800.2.f.c 2
45.h odd 6 2 405.2.e.c 2
45.j even 6 2 405.2.e.f 2
55.d odd 2 1 1815.2.a.d 1
60.h even 2 1 720.2.a.c 1
60.l odd 4 2 3600.2.f.e 2
65.d even 2 1 2535.2.a.j 1
80.k odd 4 2 3840.2.k.r 2
80.q even 4 2 3840.2.k.m 2
85.c even 2 1 4335.2.a.c 1
95.d odd 2 1 5415.2.a.j 1
105.g even 2 1 2205.2.a.i 1
115.c odd 2 1 7935.2.a.d 1
120.i odd 2 1 2880.2.a.y 1
120.m even 2 1 2880.2.a.bc 1
165.d even 2 1 5445.2.a.c 1
195.e odd 2 1 7605.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a 1 5.b even 2 1
45.2.a.a 1 15.d odd 2 1
75.2.a.b 1 1.a even 1 1 trivial
75.2.b.b 2 5.c odd 4 2
225.2.a.b 1 3.b odd 2 1
225.2.b.b 2 15.e even 4 2
240.2.a.d 1 20.d odd 2 1
405.2.e.c 2 45.h odd 6 2
405.2.e.f 2 45.j even 6 2
720.2.a.c 1 60.h even 2 1
735.2.a.c 1 35.c odd 2 1
735.2.i.d 2 35.i odd 6 2
735.2.i.e 2 35.j even 6 2
960.2.a.a 1 40.e odd 2 1
960.2.a.l 1 40.f even 2 1
1200.2.a.e 1 4.b odd 2 1
1200.2.f.h 2 20.e even 4 2
1815.2.a.d 1 55.d odd 2 1
2205.2.a.i 1 105.g even 2 1
2535.2.a.j 1 65.d even 2 1
2880.2.a.y 1 120.i odd 2 1
2880.2.a.bc 1 120.m even 2 1
3600.2.a.u 1 12.b even 2 1
3600.2.f.e 2 60.l odd 4 2
3675.2.a.j 1 7.b odd 2 1
3840.2.k.m 2 80.q even 4 2
3840.2.k.r 2 80.k odd 4 2
4335.2.a.c 1 85.c even 2 1
4800.2.a.t 1 8.b even 2 1
4800.2.a.bz 1 8.d odd 2 1
4800.2.f.c 2 40.k even 4 2
4800.2.f.bf 2 40.i odd 4 2
5415.2.a.j 1 95.d odd 2 1
5445.2.a.c 1 165.d even 2 1
7605.2.a.g 1 195.e odd 2 1
7935.2.a.d 1 115.c odd 2 1
9075.2.a.g 1 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(75))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T + 4$$
$13$ $$T - 2$$
$17$ $$T + 2$$
$19$ $$T - 4$$
$23$ $$T$$
$29$ $$T + 2$$
$31$ $$T$$
$37$ $$T - 10$$
$41$ $$T - 10$$
$43$ $$T + 4$$
$47$ $$T + 8$$
$53$ $$T - 10$$
$59$ $$T + 4$$
$61$ $$T + 2$$
$67$ $$T + 12$$
$71$ $$T + 8$$
$73$ $$T + 10$$
$79$ $$T$$
$83$ $$T + 12$$
$89$ $$T + 6$$
$97$ $$T + 2$$