# Properties

 Label 50.22.b.a Level $50$ Weight $22$ Character orbit 50.b Analytic conductor $139.739$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [50,22,Mod(49,50)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

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

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

chi := DirichletCharacter("50.49");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$50 = 2 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 50.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: $$139.738672144$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 2) 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.

 $$f(q)$$ $$=$$ $$q + 512 i q^{2} + 35802 i q^{3} - 1048576 q^{4} - 73322496 q^{6} + 426601196 i q^{7} - 536870912 i q^{8} + 5333220387 q^{9} +O(q^{10})$$ q + 512*i * q^2 + 35802*i * q^3 - 1048576 * q^4 - 73322496 * q^6 + 426601196*i * q^7 - 536870912*i * q^8 + 5333220387 * q^9 $$q + 512 i q^{2} + 35802 i q^{3} - 1048576 q^{4} - 73322496 q^{6} + 426601196 i q^{7} - 536870912 i q^{8} + 5333220387 q^{9} + 86731179612 q^{11} - 37541117952 i q^{12} - 447661721393 i q^{13} - 873679249408 q^{14} + 1099511627776 q^{16} - 1628783402409 i q^{17} + 2730608838144 i q^{18} - 23032467644420 q^{19} - 61092704076768 q^{21} + 44406363961344 i q^{22} + 73247857287612 i q^{23} + 76884209565696 q^{24} + 916811205412864 q^{26} + 565441521669180 i q^{27} - 447323775696896 i q^{28} + 734051633521170 q^{29} - 31\!\cdots\!68 q^{31} + \cdots + 46\!\cdots\!44 q^{99} +O(q^{100})$$ q + 512*i * q^2 + 35802*i * q^3 - 1048576 * q^4 - 73322496 * q^6 + 426601196*i * q^7 - 536870912*i * q^8 + 5333220387 * q^9 + 86731179612 * q^11 - 37541117952*i * q^12 - 447661721393*i * q^13 - 873679249408 * q^14 + 1099511627776 * q^16 - 1628783402409*i * q^17 + 2730608838144*i * q^18 - 23032467644420 * q^19 - 61092704076768 * q^21 + 44406363961344*i * q^22 + 73247857287612*i * q^23 + 76884209565696 * q^24 + 916811205412864 * q^26 + 565441521669180*i * q^27 - 447323775696896*i * q^28 + 734051633521170 * q^29 - 3146664162057568 * q^31 + 562949953421312*i * q^32 + 3105149692468824*i * q^33 + 3335748408133632 * q^34 - 5592286900518912 * q^36 + 6481906800496181*i * q^37 - 11792623433943040*i * q^38 + 64108739797248744 * q^39 + 45714648841476042 * q^41 - 31279464487305216*i * q^42 - 12036803898523778*i * q^43 - 90944233392832512 * q^44 - 150011611725029376 * q^46 + 224995952586842376*i * q^47 + 39364715297636352*i * q^48 - 169408457631237657 * q^49 + 233254813492188072 * q^51 + 469407337171386368*i * q^52 + 1032418608545770227*i * q^53 - 1158024236378480640 * q^54 + 916119092627243008 * q^56 - 824608406605524840*i * q^57 + 375834436362839040*i * q^58 + 3780497099978396340 * q^59 - 7619813346829729138 * q^61 - 1611092050973474816*i * q^62 + 2275158195625782852*i * q^63 - 1152921504606846976 * q^64 - 6359346570176151552 * q^66 + 9395579008462655366*i * q^67 + 1707903184964419584*i * q^68 - 10489679146444339296 * q^69 - 4526486567453771928 * q^71 - 2863250893065682944*i * q^72 - 12785727643455221963*i * q^73 - 13274945127416178688 * q^74 + 24151292792715345920 * q^76 + 36999624952970015952*i * q^77 + 32823674776191356928*i * q^78 - 99336442530925070480 * q^79 - 25188380477739579879 * q^81 + 23405900206835733504*i * q^82 + 1479090108943764642*i * q^83 + 64060343270001082368 * q^84 + 24651374384176697344 * q^86 + 26280516583324928340*i * q^87 - 46563447497130246144*i * q^88 - 118802976167736540090 * q^89 + 763892102998690344112 * q^91 - 76805945203215040512*i * q^92 - 112656870329985049536*i * q^93 - 460791710897853186048 * q^94 - 80618936929559248896 * q^96 + 284526506962676827151*i * q^97 - 86737130307193680384*i * q^98 + 462556495295277149844 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2097152 q^{4} - 146644992 q^{6} + 10666440774 q^{9}+O(q^{10})$$ 2 * q - 2097152 * q^4 - 146644992 * q^6 + 10666440774 * q^9 $$2 q - 2097152 q^{4} - 146644992 q^{6} + 10666440774 q^{9} + 173462359224 q^{11} - 1747358498816 q^{14} + 2199023255552 q^{16} - 46064935288840 q^{19} - 122185408153536 q^{21} + 153768419131392 q^{24} + 18\!\cdots\!28 q^{26}+ \cdots + 92\!\cdots\!88 q^{99}+O(q^{100})$$ 2 * q - 2097152 * q^4 - 146644992 * q^6 + 10666440774 * q^9 + 173462359224 * q^11 - 1747358498816 * q^14 + 2199023255552 * q^16 - 46064935288840 * q^19 - 122185408153536 * q^21 + 153768419131392 * q^24 + 1833622410825728 * q^26 + 1468103267042340 * q^29 - 6293328324115136 * q^31 + 6671496816267264 * q^34 - 11184573801037824 * q^36 + 128217479594497488 * q^39 + 91429297682952084 * q^41 - 181888466785665024 * q^44 - 300023223450058752 * q^46 - 338816915262475314 * q^49 + 466509626984376144 * q^51 - 2316048472756961280 * q^54 + 1832238185254486016 * q^56 + 7560994199956792680 * q^59 - 15239626693659458276 * q^61 - 2305843009213693952 * q^64 - 12718693140352303104 * q^66 - 20979358292888678592 * q^69 - 9052973134907543856 * q^71 - 26549890254832357376 * q^74 + 48302585585430691840 * q^76 - 198672885061850140960 * q^79 - 50376760955479159758 * q^81 + 128120686540002164736 * q^84 + 49302748768353394688 * q^86 - 237605952335473080180 * q^89 + 1527784205997380688224 * q^91 - 921583421795706372096 * q^94 - 161237873859118497792 * q^96 + 925112990590554299688 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/50\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$\chi(n)$$ $$-1$$

## 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
1024.00i 71604.0i −1.04858e6 0 −7.33225e7 8.53202e8i 1.07374e9i 5.33322e9 0
49.2 1024.00i 71604.0i −1.04858e6 0 −7.33225e7 8.53202e8i 1.07374e9i 5.33322e9 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 50.22.b.a 2
5.b even 2 1 inner 50.22.b.a 2
5.c odd 4 1 2.22.a.a 1
5.c odd 4 1 50.22.a.c 1
15.e even 4 1 18.22.a.e 1
20.e even 4 1 16.22.a.a 1
40.i odd 4 1 64.22.a.b 1
40.k even 4 1 64.22.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.22.a.a 1 5.c odd 4 1
16.22.a.a 1 20.e even 4 1
18.22.a.e 1 15.e even 4 1
50.22.a.c 1 5.c odd 4 1
50.22.b.a 2 1.a even 1 1 trivial
50.22.b.a 2 5.b even 2 1 inner
64.22.a.b 1 40.i odd 4 1
64.22.a.f 1 40.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 5127132816$$ acting on $$S_{22}^{\mathrm{new}}(50, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1048576$$
$3$ $$T^{2} + 5127132816$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 72\!\cdots\!64$$
$11$ $$(T - 86731179612)^{2}$$
$13$ $$T^{2} + 80\!\cdots\!96$$
$17$ $$T^{2} + 10\!\cdots\!24$$
$19$ $$(T + 23032467644420)^{2}$$
$23$ $$T^{2} + 21\!\cdots\!76$$
$29$ $$(T - 734051633521170)^{2}$$
$31$ $$(T + 31\!\cdots\!68)^{2}$$
$37$ $$T^{2} + 16\!\cdots\!44$$
$41$ $$(T - 45\!\cdots\!42)^{2}$$
$43$ $$T^{2} + 57\!\cdots\!36$$
$47$ $$T^{2} + 20\!\cdots\!04$$
$53$ $$T^{2} + 42\!\cdots\!16$$
$59$ $$(T - 37\!\cdots\!40)^{2}$$
$61$ $$(T + 76\!\cdots\!38)^{2}$$
$67$ $$T^{2} + 35\!\cdots\!24$$
$71$ $$(T + 45\!\cdots\!28)^{2}$$
$73$ $$T^{2} + 65\!\cdots\!76$$
$79$ $$(T + 99\!\cdots\!80)^{2}$$
$83$ $$T^{2} + 87\!\cdots\!56$$
$89$ $$(T + 11\!\cdots\!90)^{2}$$
$97$ $$T^{2} + 32\!\cdots\!04$$
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