# Properties

 Label 16.22.a.d Level $16$ Weight $22$ Character orbit 16.a Self dual yes Analytic conductor $44.716$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,22,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, 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("16.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 16.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: $$44.7163750859$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2161})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 540$$ x^2 - x - 540 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{8}\cdot 3\cdot 7$$ Twist minimal: no (minimal twist has level 4) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2688\sqrt{2161}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 32820) q^{3} + (204 \beta + 6844662) q^{5} + (5886 \beta + 130254040) q^{7} + (65640 \beta + 6230767581) q^{9}+O(q^{10})$$ q + (-b - 32820) * q^3 + (204*b + 6844662) * q^5 + (5886*b + 130254040) * q^7 + (65640*b + 6230767581) * q^9 $$q + ( - \beta - 32820) q^{3} + (204 \beta + 6844662) q^{5} + (5886 \beta + 130254040) q^{7} + (65640 \beta + 6230767581) q^{9} + (314445 \beta - 72717981660) q^{11} + ( - 324756 \beta + 714450208670) q^{13} + ( - 13539942 \beta - 3409891357176) q^{15} + ( - 84858072 \beta + 920310288210) q^{17} + ( - 265449285 \beta + 8390371964284) q^{19} + ( - 323432560 \beta - 96178755501024) q^{21} + (627682938 \beta + 159845962713480) q^{23} + (2792622096 \beta + 219803147959663) q^{25} + (2075280822 \beta - 886085884611720) q^{27} + ( - 4723712580 \beta + 18\!\cdots\!46) q^{29}+ \cdots + ( - 28\!\cdots\!55 \beta - 13\!\cdots\!60) q^{99}+O(q^{100})$$ q + (-b - 32820) * q^3 + (204*b + 6844662) * q^5 + (5886*b + 130254040) * q^7 + (65640*b + 6230767581) * q^9 + (314445*b - 72717981660) * q^11 + (-324756*b + 714450208670) * q^13 + (-13539942*b - 3409891357176) * q^15 + (-84858072*b + 920310288210) * q^17 + (-265449285*b + 8390371964284) * q^19 + (-323432560*b - 96178755501024) * q^21 + (627682938*b + 159845962713480) * q^23 + (2792622096*b + 219803147959663) * q^25 + (2075280822*b - 886085884611720) * q^27 + (-4723712580*b + 1871055887883246) * q^29 + (4595859000*b - 56021176731296) * q^31 + (62397896760*b - 2523130130425680) * q^33 + (66859504692*b + 19639923731212176) * q^35 + (-245612021796*b - 16681352818773610) * q^37 + (-703791716750*b - 18377525932035096) * q^39 + (-265794737520*b + 87564872161566666) * q^41 + (1278047140965*b - 7673306708264060) * q^43 + (1720360200204*b + 251727278576557662) * q^45 + (553734430452*b + 342424409644227600) * q^47 + (1533350558880*b - 633876939155943) * q^49 + (1864731634830*b + 1294766669676143448) * q^51 + (-7229502808932*b + 337652697122210790) * q^53 + (-12682198516050*b + 503855789070504600) * q^55 + (321673569416*b + 3869344735677604560) * q^57 + (-7341605116095*b - 521222625217717452) * q^59 + (-32044341559860*b + 4532998914868234382) * q^61 + (45224173167366*b + 6844149257222100600) * q^63 + (143524997516208*b + 3855741291206701524) * q^65 + (-19726418252529*b + 15232150523401387660) * q^67 + (-180446516738640*b - 15046766045364645792) * q^69 + (115464309246510*b + 4099546751415259032) * q^71 + (226051490503944*b + 12707543329687396970) * q^73 + (-311457005150383*b - 50817852431439952524) * q^75 + (-387060308442960*b + 19426885130290589280) * q^77 + (99610802523180*b - 60602176612530082448) * q^79 + (131357583788760*b - 68498060032734793191) * q^81 + (218924689501323*b - 54371468378516904900) * q^83 + (-393081522016824*b - 263994922822459877172) * q^85 + (-1716023641007646*b + 12347844638894937000) * q^87 + (1052618322986760*b - 92103999637482684486) * q^89 + (4162953147217380*b + 63213709769507333456) * q^91 + (-94814915648704*b - 69920982103000721280) * q^93 + (-105274753252734*b - 788092955533462657752) * q^95 + (5190622720371144*b + 369748892594238233570) * q^97 + (-2813974604154855*b - 130813883985288961260) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 65640 q^{3} + 13689324 q^{5} + 260508080 q^{7} + 12461535162 q^{9}+O(q^{10})$$ 2 * q - 65640 * q^3 + 13689324 * q^5 + 260508080 * q^7 + 12461535162 * q^9 $$2 q - 65640 q^{3} + 13689324 q^{5} + 260508080 q^{7} + 12461535162 q^{9} - 145435963320 q^{11} + 1428900417340 q^{13} - 6819782714352 q^{15} + 1840620576420 q^{17} + 16780743928568 q^{19} - 192357511002048 q^{21} + 319691925426960 q^{23} + 439606295919326 q^{25} - 17\!\cdots\!40 q^{27}+ \cdots - 26\!\cdots\!20 q^{99}+O(q^{100})$$ 2 * q - 65640 * q^3 + 13689324 * q^5 + 260508080 * q^7 + 12461535162 * q^9 - 145435963320 * q^11 + 1428900417340 * q^13 - 6819782714352 * q^15 + 1840620576420 * q^17 + 16780743928568 * q^19 - 192357511002048 * q^21 + 319691925426960 * q^23 + 439606295919326 * q^25 - 1772171769223440 * q^27 + 3742111775766492 * q^29 - 112042353462592 * q^31 - 5046260260851360 * q^33 + 39279847462424352 * q^35 - 33362705637547220 * q^37 - 36755051864070192 * q^39 + 175129744323133332 * q^41 - 15346613416528120 * q^43 + 503454557153115324 * q^45 + 684848819288455200 * q^47 - 1267753878311886 * q^49 + 2589533339352286896 * q^51 + 675305394244421580 * q^53 + 1007711578141009200 * q^55 + 7738689471355209120 * q^57 - 1042445250435434904 * q^59 + 9065997829736468764 * q^61 + 13688298514444201200 * q^63 + 7711482582413403048 * q^65 + 30464301046802775320 * q^67 - 30093532090729291584 * q^69 + 8199093502830518064 * q^71 + 25415086659374793940 * q^73 - 101635704862879905048 * q^75 + 38853770260581178560 * q^77 - 121204353225060164896 * q^79 - 136996120065469586382 * q^81 - 108742936757033809800 * q^83 - 527989845644919754344 * q^85 + 24695689277789874000 * q^87 - 184207999274965368972 * q^89 + 126427419539014666912 * q^91 - 139841964206001442560 * q^93 - 1576185911066925315504 * q^95 + 739497785188476467140 * q^97 - 261627767970577922520 * 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
 23.7433 −22.7433
0 −157776. 0 3.23357e7 0 8.65744e8 0 1.44329e10 0
1.2 0 92135.9 0 −1.86463e7 0 −6.05236e8 0 −1.97134e9 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
$$2$$ $$-1$$

## 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 16.22.a.d 2
4.b odd 2 1 4.22.a.a 2
8.b even 2 1 64.22.a.j 2
8.d odd 2 1 64.22.a.i 2
12.b even 2 1 36.22.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.22.a.a 2 4.b odd 2 1
16.22.a.d 2 1.a even 1 1 trivial
36.22.a.c 2 12.b even 2 1
64.22.a.i 2 8.d odd 2 1
64.22.a.j 2 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 65640T_{3} - 14536815984$$ acting on $$S_{22}^{\mathrm{new}}(\Gamma_0(16))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 65640 T - 14536815984$$
$5$ $$T^{2} + \cdots - 602941510374300$$
$7$ $$T^{2} - 260508080 T - 52\!\cdots\!64$$
$11$ $$T^{2} + 145435963320 T + 37\!\cdots\!00$$
$13$ $$T^{2} - 1428900417340 T + 50\!\cdots\!76$$
$17$ $$T^{2} - 1840620576420 T - 11\!\cdots\!56$$
$19$ $$T^{2} - 16780743928568 T - 10\!\cdots\!44$$
$23$ $$T^{2} - 319691925426960 T + 19\!\cdots\!04$$
$29$ $$T^{2} + \cdots + 31\!\cdots\!16$$
$31$ $$T^{2} + 112042353462592 T - 32\!\cdots\!84$$
$37$ $$T^{2} + \cdots - 66\!\cdots\!44$$
$41$ $$T^{2} + \cdots + 65\!\cdots\!56$$
$43$ $$T^{2} + \cdots - 25\!\cdots\!00$$
$47$ $$T^{2} + \cdots + 11\!\cdots\!64$$
$53$ $$T^{2} + \cdots - 70\!\cdots\!16$$
$59$ $$T^{2} + \cdots - 56\!\cdots\!96$$
$61$ $$T^{2} + \cdots + 45\!\cdots\!24$$
$67$ $$T^{2} + \cdots + 22\!\cdots\!56$$
$71$ $$T^{2} + \cdots - 19\!\cdots\!76$$
$73$ $$T^{2} + \cdots - 63\!\cdots\!24$$
$79$ $$T^{2} + \cdots + 35\!\cdots\!04$$
$83$ $$T^{2} + \cdots + 22\!\cdots\!64$$
$89$ $$T^{2} + \cdots - 88\!\cdots\!04$$
$97$ $$T^{2} + \cdots - 28\!\cdots\!24$$