# Properties

 Label 2.22.a.a Level $2$ Weight $22$ Character orbit 2.a Self dual yes Analytic conductor $5.590$ 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] = [2,22,Mod(1,2)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2, base_ring=CyclotomicField(1))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("2.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 2.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: $$5.58954688574$$ Analytic rank: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - 1024 q^{2} + 71604 q^{3} + 1048576 q^{4} - 28693770 q^{5} - 73322496 q^{6} - 853202392 q^{7} - 1073741824 q^{8} - 5333220387 q^{9}+O(q^{10})$$ q - 1024 * q^2 + 71604 * q^3 + 1048576 * q^4 - 28693770 * q^5 - 73322496 * q^6 - 853202392 * q^7 - 1073741824 * q^8 - 5333220387 * q^9 $$q - 1024 q^{2} + 71604 q^{3} + 1048576 q^{4} - 28693770 q^{5} - 73322496 q^{6} - 853202392 q^{7} - 1073741824 q^{8} - 5333220387 q^{9} + 29382420480 q^{10} + 86731179612 q^{11} + 75082235904 q^{12} - 895323442786 q^{13} + 873679249408 q^{14} - 2054588707080 q^{15} + 1099511627776 q^{16} + 3257566804818 q^{17} + 5461217676288 q^{18} + 23032467644420 q^{19} - 30087598571520 q^{20} - 61092704076768 q^{21} - 88812727922688 q^{22} + 146495714575224 q^{23} - 76884209565696 q^{24} + 346495278609775 q^{25} + 916811205412864 q^{26} - 11\!\cdots\!60 q^{27}+ \cdots - 46\!\cdots\!44 q^{99}+O(q^{100})$$ q - 1024 * q^2 + 71604 * q^3 + 1048576 * q^4 - 28693770 * q^5 - 73322496 * q^6 - 853202392 * q^7 - 1073741824 * q^8 - 5333220387 * q^9 + 29382420480 * q^10 + 86731179612 * q^11 + 75082235904 * q^12 - 895323442786 * q^13 + 873679249408 * q^14 - 2054588707080 * q^15 + 1099511627776 * q^16 + 3257566804818 * q^17 + 5461217676288 * q^18 + 23032467644420 * q^19 - 30087598571520 * q^20 - 61092704076768 * q^21 - 88812727922688 * q^22 + 146495714575224 * q^23 - 76884209565696 * q^24 + 346495278609775 * q^25 + 916811205412864 * q^26 - 1130883043338360 * q^27 - 894647551393792 * q^28 - 734051633521170 * q^29 + 2103898836049920 * q^30 - 3146664162057568 * q^31 - 1125899906842624 * q^32 + 6210299384937648 * q^33 - 3335748408133632 * q^34 + 24481593199497840 * q^35 - 5592286900518912 * q^36 - 12963813600992362 * q^37 - 23585246867886080 * q^38 - 64108739797248744 * q^39 + 30809700937236480 * q^40 + 45714648841476042 * q^41 + 62558928974610432 * q^42 - 24073607797047556 * q^43 + 90944233392832512 * q^44 + 153030199143888990 * q^45 - 150011611725029376 * q^46 - 449991905173684752 * q^47 + 78729430595272704 * q^48 + 169408457631237657 * q^49 - 354811165296409600 * q^50 + 233254813492188072 * q^51 - 938814674342772736 * q^52 + 2064837217091540454 * q^53 + 1158024236378480640 * q^54 - 2488644519615417240 * q^55 + 916119092627243008 * q^56 + 1649216813211049680 * q^57 + 751668872725678080 * q^58 - 3780497099978396340 * q^59 - 2154392408115118080 * q^60 - 7619813346829729138 * q^61 + 3222184101946949632 * q^62 + 4550316391251565704 * q^63 + 1152921504606846976 * q^64 + 25690204942909643220 * q^65 - 6359346570176151552 * q^66 - 18791158016925310732 * q^67 + 3415806369928839168 * q^68 + 10489679146444339296 * q^69 - 25069151436285788160 * q^70 - 4526486567453771928 * q^71 + 5726501786131365888 * q^72 - 25571455286910443926 * q^73 + 13274945127416178688 * q^74 + 24810447929574329100 * q^75 + 24151292792715345920 * q^76 - 73999249905940031904 * q^77 + 65647349552382713856 * q^78 + 99336442530925070480 * q^79 - 31549133759730155520 * q^80 - 25188380477739579879 * q^81 - 46811800413671467008 * q^82 + 2958180217887529284 * q^83 - 64060343270001082368 * q^84 - 93471872657082583860 * q^85 + 24651374384176697344 * q^86 - 52561033166649856680 * q^87 - 93126894994260492288 * q^88 + 118802976167736540090 * q^89 - 156702923923342325760 * q^90 + 763892102998690344112 * q^91 + 153611890406430081024 * q^92 - 225313740659970099072 * q^93 + 460791710897853186048 * q^94 - 660888329121429263400 * q^95 - 80618936929559248896 * q^96 - 569053013925353654302 * q^97 - 173474260614387360768 * q^98 - 462556495295277149844 * 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
−1024.00 71604.0 1.04858e6 −2.86938e7 −7.33225e7 −8.53202e8 −1.07374e9 −5.33322e9 2.93824e10
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$

## 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 2.22.a.a 1
3.b odd 2 1 18.22.a.e 1
4.b odd 2 1 16.22.a.a 1
5.b even 2 1 50.22.a.c 1
5.c odd 4 2 50.22.b.a 2
8.b even 2 1 64.22.a.b 1
8.d odd 2 1 64.22.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.22.a.a 1 1.a even 1 1 trivial
16.22.a.a 1 4.b odd 2 1
18.22.a.e 1 3.b odd 2 1
50.22.a.c 1 5.b even 2 1
50.22.b.a 2 5.c odd 4 2
64.22.a.b 1 8.b even 2 1
64.22.a.f 1 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 71604$$ acting on $$S_{22}^{\mathrm{new}}(\Gamma_0(2))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1024$$
$3$ $$T - 71604$$
$5$ $$T + 28693770$$
$7$ $$T + 853202392$$
$11$ $$T - 86731179612$$
$13$ $$T + 895323442786$$
$17$ $$T - 3257566804818$$
$19$ $$T - 23032467644420$$
$23$ $$T - 146495714575224$$
$29$ $$T + 734051633521170$$
$31$ $$T + 3146664162057568$$
$37$ $$T + 12\!\cdots\!62$$
$41$ $$T - 45\!\cdots\!42$$
$43$ $$T + 24\!\cdots\!56$$
$47$ $$T + 44\!\cdots\!52$$
$53$ $$T - 20\!\cdots\!54$$
$59$ $$T + 37\!\cdots\!40$$
$61$ $$T + 76\!\cdots\!38$$
$67$ $$T + 18\!\cdots\!32$$
$71$ $$T + 45\!\cdots\!28$$
$73$ $$T + 25\!\cdots\!26$$
$79$ $$T - 99\!\cdots\!80$$
$83$ $$T - 29\!\cdots\!84$$
$89$ $$T - 11\!\cdots\!90$$
$97$ $$T + 56\!\cdots\!02$$