# Properties

 Label 45.22.a.a Level $45$ Weight $22$ Character orbit 45.a Self dual yes Analytic conductor $125.765$ Analytic rank $1$ 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] = [45,22,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, 22, names="a")

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

chi := DirichletCharacter("45.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$22$$ 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: $$125.764804929$$ Analytic rank: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - 544 q^{2} - 1801216 q^{4} + 9765625 q^{5} + 1277698380 q^{7} + 2120712192 q^{8}+O(q^{10})$$ q - 544 * q^2 - 1801216 * q^4 + 9765625 * q^5 + 1277698380 * q^7 + 2120712192 * q^8 $$q - 544 q^{2} - 1801216 q^{4} + 9765625 q^{5} + 1277698380 q^{7} + 2120712192 q^{8} - 5312500000 q^{10} + 77585921744 q^{11} - 434110898702 q^{13} - 695067918720 q^{14} + 2623756304384 q^{16} - 12848917115782 q^{17} - 28605256159796 q^{19} - 17590000000000 q^{20} - 42206741428736 q^{22} - 224022192208080 q^{23} + 95367431640625 q^{25} + 236156328893888 q^{26} - 23\!\cdots\!80 q^{28}+ \cdots - 58\!\cdots\!92 q^{98}+O(q^{100})$$ q - 544 * q^2 - 1801216 * q^4 + 9765625 * q^5 + 1277698380 * q^7 + 2120712192 * q^8 - 5312500000 * q^10 + 77585921744 * q^11 - 434110898702 * q^13 - 695067918720 * q^14 + 2623756304384 * q^16 - 12848917115782 * q^17 - 28605256159796 * q^19 - 17590000000000 * q^20 - 42206741428736 * q^22 - 224022192208080 * q^23 + 95367431640625 * q^25 + 236156328893888 * q^26 - 2301410765230080 * q^28 + 51676030833142 * q^29 + 8921108838285000 * q^31 - 5874779244462080 * q^32 + 6989810910985408 * q^34 + 12477523242187500 * q^35 + 43977154002495890 * q^37 + 15561259350929024 * q^38 + 20710080000000000 * q^40 - 58168090830044570 * q^41 - 161437862491900676 * q^43 - 139749003620040704 * q^44 + 121868072561195520 * q^46 + 160064774442316592 * q^47 + 1073967286171340393 * q^49 - 51879882812500000 * q^50 + 781927496516421632 * q^52 - 2299527285858152170 * q^53 + 757675017031250000 * q^55 + 2709630532164648960 * q^56 - 28111760773229248 * q^58 - 5154256088898000016 * q^59 + 1251686105775241798 * q^61 - 4853083208027040000 * q^62 - 2306535872264142848 * q^64 - 4239364245136718750 * q^65 - 5407785329527117188 * q^67 + 23143675091620390912 * q^68 - 6787772643750000000 * q^70 + 11043230850518282368 * q^71 - 37701191520217147550 * q^73 - 23923571777357764160 * q^74 + 51524245079123111936 * q^76 + 99131406523115574720 * q^77 + 63155369968366862760 * q^79 + 25622620160000000000 * q^80 + 31643441411544246080 * q^82 + 145158253921046761428 * q^83 - 125477706208808593750 * q^85 + 87822197195593967744 * q^86 + 164537410170058702848 * q^88 - 137255030734236350514 * q^89 - 554662792011889502760 * q^91 + 403512356960269025280 * q^92 - 87075237296620226048 * q^94 - 279348204685507812500 * q^95 - 324306485265230829118 * q^97 - 584238203677209173792 * 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
−544.000 0 −1.80122e6 9.76562e6 0 1.27770e9 2.12071e9 0 −5.31250e9
 $$n$$: 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 45.22.a.a 1
3.b odd 2 1 15.22.a.a 1
15.d odd 2 1 75.22.a.b 1
15.e even 4 2 75.22.b.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.22.a.a 1 3.b odd 2 1
45.22.a.a 1 1.a even 1 1 trivial
75.22.a.b 1 15.d odd 2 1
75.22.b.c 2 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 544$$ acting on $$S_{22}^{\mathrm{new}}(\Gamma_0(45))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 544$$
$3$ $$T$$
$5$ $$T - 9765625$$
$7$ $$T - 1277698380$$
$11$ $$T - 77585921744$$
$13$ $$T + 434110898702$$
$17$ $$T + 12848917115782$$
$19$ $$T + 28605256159796$$
$23$ $$T + 224022192208080$$
$29$ $$T - 51676030833142$$
$31$ $$T - 8921108838285000$$
$37$ $$T - 43\!\cdots\!90$$
$41$ $$T + 58\!\cdots\!70$$
$43$ $$T + 16\!\cdots\!76$$
$47$ $$T - 16\!\cdots\!92$$
$53$ $$T + 22\!\cdots\!70$$
$59$ $$T + 51\!\cdots\!16$$
$61$ $$T - 12\!\cdots\!98$$
$67$ $$T + 54\!\cdots\!88$$
$71$ $$T - 11\!\cdots\!68$$
$73$ $$T + 37\!\cdots\!50$$
$79$ $$T - 63\!\cdots\!60$$
$83$ $$T - 14\!\cdots\!28$$
$89$ $$T + 13\!\cdots\!14$$
$97$ $$T + 32\!\cdots\!18$$