# Properties

 Label 12.28.a.a Level $12$ Weight $28$ Character orbit 12.a Self dual yes Analytic conductor $55.423$ Analytic rank $1$ 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] = [12,28,Mod(1,12)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(12, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 28, names="a")

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

chi := DirichletCharacter("12.1");

S:= CuspForms(chi, 28);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$12 = 2^{2} \cdot 3$$ Weight: $$k$$ $$=$$ $$28$$ Character orbit: $$[\chi]$$ $$=$$ 12.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: $$55.4226689806$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\mathbb{Q}[x]/(x^{2} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 2690444262$$ x^2 - x - 2690444262 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{7}\cdot 3^{2}\cdot 5$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2880\sqrt{10761777049}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 1594323 q^{3} + ( - 11 \beta - 751557690) q^{5} + (189 \beta - 63781057648) q^{7} + 2541865828329 q^{9}+O(q^{10})$$ q - 1594323 * q^3 + (-11*b - 751557690) * q^5 + (189*b - 63781057648) * q^7 + 2541865828329 * q^9 $$q - 1594323 q^{3} + ( - 11 \beta - 751557690) q^{5} + (189 \beta - 63781057648) q^{7} + 2541865828329 q^{9} + (200074 \beta + 34699242107916) q^{11} + (4027374 \beta + 75765958624070) q^{13} + (17537553 \beta + 11\!\cdots\!70) q^{15}+ \cdots + (50\!\cdots\!46 \beta + 88\!\cdots\!64) q^{99}+O(q^{100})$$ q - 1594323 * q^3 + (-11*b - 751557690) * q^5 + (189*b - 63781057648) * q^7 + 2541865828329 * q^9 + (200074*b + 34699242107916) * q^11 + (4027374*b + 75765958624070) * q^13 + (17537553*b + 1198225710993870) * q^15 + (-55193542*b + 15240006214515522) * q^17 + (-431583822*b + 31792627203878420) * q^19 + (-301327047*b + 101687607172532304) * q^21 + (2403589562*b + 1768502771959368168) * q^23 + (16534269180*b + 3915018874656605575) * q^25 - 4052555153018976267 * q^27 + (-176029017425*b - 7922850750148396194) * q^29 + (176583835593*b - 73037848113086950456) * q^31 + (-318982579902*b - 55321799775218960868) * q^33 + (559547230718*b - 137641558959626309280) * q^35 + (5312417763312*b + 572590949270743273886) * q^37 + (-6420934997802*b - 120795410451403154610) * q^39 + (-24787786083506*b - 924324555460526673750) * q^41 + (37733147173206*b + 4510958208746436742988) * q^43 + (-27960524111619*b - 1910358810228879800010) * q^45 + (93547997991822*b - 17346979946896935260832) * q^47 + (-24109239790944*b - 58455793873760567190039) * q^49 + (87996333462066*b - 24297492427945030581606) * q^51 + (-527622561791765*b - 109233552312581204316474) * q^53 + (-532058816456136*b - 222528605726486353312440) * q^55 + (688084013842506*b - 50687716781569054209660) * q^57 + (3426502332242264*b - 394545517797890296729044) * q^59 + (-2801696402607516*b - 971820870567519434593978) * q^61 + (480412641554181*b - 162122890930133220510192) * q^63 + (-3860229445070830*b - 4011369948747316228916700) * q^65 + (-1092996988515432*b - 7076615790967532554621132) * q^67 + (-3832098121256526*b - 2819564644898575735710264) * q^69 + (-10780690620339310*b - 12425969399925898272686760) * q^71 + (68532011052059088*b - 3059767477118668204570726) * q^73 + (-26360965641865140*b - 6241804637299143370150725) * q^75 + (-6202774569469828*b + 1162215942255631632100032) * q^77 + (-54976791526419123*b - 22445015929432600286061928) * q^79 + 6461081889226673298932241 * q^81 + (-55631423547430638*b - 17444364948833062541929452) * q^83 + (-126158937431232762*b + 42740095120259258020363020) * q^85 + (280647111148078275*b + 12631583176528841465206662) * q^87 + (529206529095572636*b + 102424790848294079209829610) * q^89 + (-242550407084107122*b + 63111820654487663083174240) * q^91 + (-281531670514138539*b + 116445921117201126111861288) * q^93 + (-25358758938971440*b + 399872688493362513346625400) * q^95 + (-1416002107607597268*b + 749083255695267526030843394) * q^97 + (508561263737096346*b + 88200817783026419347952364) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3188646 q^{3} - 1503115380 q^{5} - 127562115296 q^{7} + 5083731656658 q^{9}+O(q^{10})$$ 2 * q - 3188646 * q^3 - 1503115380 * q^5 - 127562115296 * q^7 + 5083731656658 * q^9 $$2 q - 3188646 q^{3} - 1503115380 q^{5} - 127562115296 q^{7} + 5083731656658 q^{9} + 69398484215832 q^{11} + 151531917248140 q^{13} + 23\!\cdots\!40 q^{15}+ \cdots + 17\!\cdots\!28 q^{99}+O(q^{100})$$ 2 * q - 3188646 * q^3 - 1503115380 * q^5 - 127562115296 * q^7 + 5083731656658 * q^9 + 69398484215832 * q^11 + 151531917248140 * q^13 + 2396451421987740 * q^15 + 30480012429031044 * q^17 + 63585254407756840 * q^19 + 203375214345064608 * q^21 + 3537005543918736336 * q^23 + 7830037749313211150 * q^25 - 8105110306037952534 * q^27 - 15845701500296792388 * q^29 - 146075696226173900912 * q^31 - 110643599550437921736 * q^33 - 275283117919252618560 * q^35 + 1145181898541486547772 * q^37 - 241590820902806309220 * q^39 - 1848649110921053347500 * q^41 + 9021916417492873485976 * q^43 - 3820717620457759600020 * q^45 - 34693959893793870521664 * q^47 - 116911587747521134380078 * q^49 - 48594984855890061163212 * q^51 - 218467104625162408632948 * q^53 - 445057211452972706624880 * q^55 - 101375433563138108419320 * q^57 - 789091035595780593458088 * q^59 - 1943641741135038869187956 * q^61 - 324245781860266441020384 * q^63 - 8022739897494632457833400 * q^65 - 14153231581935065109242264 * q^67 - 5639129289797151471420528 * q^69 - 24851938799851796545373520 * q^71 - 6119534954237336409141452 * q^73 - 12483609274598286740301450 * q^75 + 2324431884511263264200064 * q^77 - 44890031858865200572123856 * q^79 + 12922163778453346597864482 * q^81 - 34888729897666125083858904 * q^83 + 85480190240518516040726040 * q^85 + 25263166353057682930413324 * q^87 + 204849581696588158419659220 * q^89 + 126223641308975326166348480 * q^91 + 232891842234402252223722576 * q^93 + 799745376986725026693250800 * q^95 + 1498166511390535052061686788 * q^97 + 176401635566052838695904728 * 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
 51870.0 −51869.0
0 −1.59432e6 0 −4.03801e9 0 −7.31385e9 0 2.54187e12 0
1.2 0 −1.59432e6 0 2.53489e9 0 −1.20248e11 0 2.54187e12 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
$$2$$ $$-1$$
$$3$$ $$+1$$

## 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 12.28.a.a 2
3.b odd 2 1 36.28.a.b 2
4.b odd 2 1 48.28.a.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.28.a.a 2 1.a even 1 1 trivial
36.28.a.b 2 3.b odd 2 1
48.28.a.f 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 1503115380T_{5} - 10235921548784161500$$ acting on $$S_{28}^{\mathrm{new}}(\Gamma_0(12))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1594323)^{2}$$
$5$ $$T^{2} + \cdots - 10\!\cdots\!00$$
$7$ $$T^{2} + \cdots + 87\!\cdots\!04$$
$11$ $$T^{2} + \cdots - 23\!\cdots\!44$$
$13$ $$T^{2} + \cdots - 14\!\cdots\!00$$
$17$ $$T^{2} + \cdots - 39\!\cdots\!16$$
$19$ $$T^{2} + \cdots - 15\!\cdots\!00$$
$23$ $$T^{2} + \cdots + 26\!\cdots\!24$$
$29$ $$T^{2} + \cdots - 27\!\cdots\!64$$
$31$ $$T^{2} + \cdots + 25\!\cdots\!36$$
$37$ $$T^{2} + \cdots - 21\!\cdots\!04$$
$41$ $$T^{2} + \cdots - 53\!\cdots\!00$$
$43$ $$T^{2} + \cdots - 10\!\cdots\!56$$
$47$ $$T^{2} + \cdots - 48\!\cdots\!76$$
$53$ $$T^{2} + \cdots - 12\!\cdots\!24$$
$59$ $$T^{2} + \cdots - 89\!\cdots\!64$$
$61$ $$T^{2} + \cdots + 24\!\cdots\!84$$
$67$ $$T^{2} + \cdots + 49\!\cdots\!24$$
$71$ $$T^{2} + \cdots + 14\!\cdots\!00$$
$73$ $$T^{2} + \cdots - 40\!\cdots\!24$$
$79$ $$T^{2} + \cdots + 23\!\cdots\!84$$
$83$ $$T^{2} + \cdots + 28\!\cdots\!04$$
$89$ $$T^{2} + \cdots - 14\!\cdots\!00$$
$97$ $$T^{2} + \cdots + 38\!\cdots\!36$$