# Properties

 Label 75.2.b.b Level $75$ Weight $2$ Character orbit 75.b Analytic conductor $0.599$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,2,Mod(49,75)]

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, 1]))

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

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

chi := DirichletCharacter("75.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 75.b (of order $$2$$, degree $$1$$, 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: $$0.598878015160$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} - i q^{3} + q^{4} + q^{6} + 3 i q^{8} - q^{9} +O(q^{10})$$ q + i * q^2 - i * q^3 + q^4 + q^6 + 3*i * q^8 - q^9 $$q + i q^{2} - i q^{3} + q^{4} + q^{6} + 3 i q^{8} - q^{9} - 4 q^{11} - i q^{12} - 2 i q^{13} - q^{16} - 2 i q^{17} - i q^{18} - 4 q^{19} - 4 i q^{22} + 3 q^{24} + 2 q^{26} + i q^{27} + 2 q^{29} + 5 i q^{32} + 4 i q^{33} + 2 q^{34} - q^{36} + 10 i q^{37} - 4 i q^{38} - 2 q^{39} + 10 q^{41} + 4 i q^{43} - 4 q^{44} - 8 i q^{47} + i q^{48} + 7 q^{49} - 2 q^{51} - 2 i q^{52} - 10 i q^{53} - q^{54} + 4 i q^{57} + 2 i q^{58} + 4 q^{59} - 2 q^{61} - 7 q^{64} - 4 q^{66} - 12 i q^{67} - 2 i q^{68} - 8 q^{71} - 3 i q^{72} + 10 i q^{73} - 10 q^{74} - 4 q^{76} - 2 i q^{78} + q^{81} + 10 i q^{82} + 12 i q^{83} - 4 q^{86} - 2 i q^{87} - 12 i q^{88} + 6 q^{89} + 8 q^{94} + 5 q^{96} - 2 i q^{97} + 7 i q^{98} + 4 q^{99} +O(q^{100})$$ q + i * q^2 - i * q^3 + q^4 + q^6 + 3*i * q^8 - q^9 - 4 * q^11 - i * q^12 - 2*i * q^13 - q^16 - 2*i * q^17 - i * q^18 - 4 * q^19 - 4*i * q^22 + 3 * q^24 + 2 * q^26 + i * q^27 + 2 * q^29 + 5*i * q^32 + 4*i * q^33 + 2 * q^34 - q^36 + 10*i * q^37 - 4*i * q^38 - 2 * q^39 + 10 * q^41 + 4*i * q^43 - 4 * q^44 - 8*i * q^47 + i * q^48 + 7 * q^49 - 2 * q^51 - 2*i * q^52 - 10*i * q^53 - q^54 + 4*i * q^57 + 2*i * q^58 + 4 * q^59 - 2 * q^61 - 7 * q^64 - 4 * q^66 - 12*i * q^67 - 2*i * q^68 - 8 * q^71 - 3*i * q^72 + 10*i * q^73 - 10 * q^74 - 4 * q^76 - 2*i * q^78 + q^81 + 10*i * q^82 + 12*i * q^83 - 4 * q^86 - 2*i * q^87 - 12*i * q^88 + 6 * q^89 + 8 * q^94 + 5 * q^96 - 2*i * q^97 + 7*i * q^98 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9 $$2 q + 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} - 2 q^{16} - 8 q^{19} + 6 q^{24} + 4 q^{26} + 4 q^{29} + 4 q^{34} - 2 q^{36} - 4 q^{39} + 20 q^{41} - 8 q^{44} + 14 q^{49} - 4 q^{51} - 2 q^{54} + 8 q^{59} - 4 q^{61} - 14 q^{64} - 8 q^{66} - 16 q^{71} - 20 q^{74} - 8 q^{76} + 2 q^{81} - 8 q^{86} + 12 q^{89} + 16 q^{94} + 10 q^{96} + 8 q^{99}+O(q^{100})$$ 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9 - 8 * q^11 - 2 * q^16 - 8 * q^19 + 6 * q^24 + 4 * q^26 + 4 * q^29 + 4 * q^34 - 2 * q^36 - 4 * q^39 + 20 * q^41 - 8 * q^44 + 14 * q^49 - 4 * q^51 - 2 * q^54 + 8 * q^59 - 4 * q^61 - 14 * q^64 - 8 * q^66 - 16 * q^71 - 20 * q^74 - 8 * q^76 + 2 * q^81 - 8 * q^86 + 12 * q^89 + 16 * q^94 + 10 * q^96 + 8 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 1.00000i 1.00000i
1.00000i 1.00000i 1.00000 0 1.00000 0 3.00000i −1.00000 0
49.2 1.00000i 1.00000i 1.00000 0 1.00000 0 3.00000i −1.00000 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
5.b even 2 1 inner

## Twists

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

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 4$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2}$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T - 10)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 100$$
$59$ $$(T - 4)^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 4$$