# Properties

 Label 45.6.a.b Level $45$ Weight $6$ Character orbit 45.a Self dual yes Analytic conductor $7.217$ 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] = [45,6,Mod(1,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("45.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 45.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: $$7.21727189158$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 5) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} - 28 q^{4} - 25 q^{5} + 192 q^{7} + 120 q^{8}+O(q^{10})$$ q - 2 * q^2 - 28 * q^4 - 25 * q^5 + 192 * q^7 + 120 * q^8 $$q - 2 q^{2} - 28 q^{4} - 25 q^{5} + 192 q^{7} + 120 q^{8} + 50 q^{10} + 148 q^{11} + 286 q^{13} - 384 q^{14} + 656 q^{16} + 1678 q^{17} + 1060 q^{19} + 700 q^{20} - 296 q^{22} - 2976 q^{23} + 625 q^{25} - 572 q^{26} - 5376 q^{28} + 3410 q^{29} - 2448 q^{31} - 5152 q^{32} - 3356 q^{34} - 4800 q^{35} + 182 q^{37} - 2120 q^{38} - 3000 q^{40} + 9398 q^{41} - 1244 q^{43} - 4144 q^{44} + 5952 q^{46} + 12088 q^{47} + 20057 q^{49} - 1250 q^{50} - 8008 q^{52} - 23846 q^{53} - 3700 q^{55} + 23040 q^{56} - 6820 q^{58} + 20020 q^{59} + 32302 q^{61} + 4896 q^{62} - 10688 q^{64} - 7150 q^{65} + 60972 q^{67} - 46984 q^{68} + 9600 q^{70} + 32648 q^{71} - 38774 q^{73} - 364 q^{74} - 29680 q^{76} + 28416 q^{77} - 33360 q^{79} - 16400 q^{80} - 18796 q^{82} - 16716 q^{83} - 41950 q^{85} + 2488 q^{86} + 17760 q^{88} - 101370 q^{89} + 54912 q^{91} + 83328 q^{92} - 24176 q^{94} - 26500 q^{95} - 119038 q^{97} - 40114 q^{98}+O(q^{100})$$ q - 2 * q^2 - 28 * q^4 - 25 * q^5 + 192 * q^7 + 120 * q^8 + 50 * q^10 + 148 * q^11 + 286 * q^13 - 384 * q^14 + 656 * q^16 + 1678 * q^17 + 1060 * q^19 + 700 * q^20 - 296 * q^22 - 2976 * q^23 + 625 * q^25 - 572 * q^26 - 5376 * q^28 + 3410 * q^29 - 2448 * q^31 - 5152 * q^32 - 3356 * q^34 - 4800 * q^35 + 182 * q^37 - 2120 * q^38 - 3000 * q^40 + 9398 * q^41 - 1244 * q^43 - 4144 * q^44 + 5952 * q^46 + 12088 * q^47 + 20057 * q^49 - 1250 * q^50 - 8008 * q^52 - 23846 * q^53 - 3700 * q^55 + 23040 * q^56 - 6820 * q^58 + 20020 * q^59 + 32302 * q^61 + 4896 * q^62 - 10688 * q^64 - 7150 * q^65 + 60972 * q^67 - 46984 * q^68 + 9600 * q^70 + 32648 * q^71 - 38774 * q^73 - 364 * q^74 - 29680 * q^76 + 28416 * q^77 - 33360 * q^79 - 16400 * q^80 - 18796 * q^82 - 16716 * q^83 - 41950 * q^85 + 2488 * q^86 + 17760 * q^88 - 101370 * q^89 + 54912 * q^91 + 83328 * q^92 - 24176 * q^94 - 26500 * q^95 - 119038 * q^97 - 40114 * q^98

## 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 0 −28.0000 −25.0000 0 192.000 120.000 0 50.0000
 $$n$$: 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 45.6.a.b 1
3.b odd 2 1 5.6.a.a 1
4.b odd 2 1 720.6.a.a 1
5.b even 2 1 225.6.a.f 1
5.c odd 4 2 225.6.b.e 2
12.b even 2 1 80.6.a.e 1
15.d odd 2 1 25.6.a.a 1
15.e even 4 2 25.6.b.a 2
21.c even 2 1 245.6.a.b 1
24.f even 2 1 320.6.a.g 1
24.h odd 2 1 320.6.a.j 1
33.d even 2 1 605.6.a.a 1
39.d odd 2 1 845.6.a.b 1
60.h even 2 1 400.6.a.g 1
60.l odd 4 2 400.6.c.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.a.a 1 3.b odd 2 1
25.6.a.a 1 15.d odd 2 1
25.6.b.a 2 15.e even 4 2
45.6.a.b 1 1.a even 1 1 trivial
80.6.a.e 1 12.b even 2 1
225.6.a.f 1 5.b even 2 1
225.6.b.e 2 5.c odd 4 2
245.6.a.b 1 21.c even 2 1
320.6.a.g 1 24.f even 2 1
320.6.a.j 1 24.h odd 2 1
400.6.a.g 1 60.h even 2 1
400.6.c.j 2 60.l odd 4 2
605.6.a.a 1 33.d even 2 1
720.6.a.a 1 4.b odd 2 1
845.6.a.b 1 39.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 2$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(45))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T$$
$5$ $$T + 25$$
$7$ $$T - 192$$
$11$ $$T - 148$$
$13$ $$T - 286$$
$17$ $$T - 1678$$
$19$ $$T - 1060$$
$23$ $$T + 2976$$
$29$ $$T - 3410$$
$31$ $$T + 2448$$
$37$ $$T - 182$$
$41$ $$T - 9398$$
$43$ $$T + 1244$$
$47$ $$T - 12088$$
$53$ $$T + 23846$$
$59$ $$T - 20020$$
$61$ $$T - 32302$$
$67$ $$T - 60972$$
$71$ $$T - 32648$$
$73$ $$T + 38774$$
$79$ $$T + 33360$$
$83$ $$T + 16716$$
$89$ $$T + 101370$$
$97$ $$T + 119038$$