# Properties

 Label 9.28.a.b Level $9$ Weight $28$ Character orbit 9.a Self dual yes Analytic conductor $41.567$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$28$$ Character orbit: $$[\chi]$$ $$=$$ 9.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$41.5670017354$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{30001})$$ Defining polynomial: $$x^{2} - x - 7500$$ x^2 - x - 7500 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2\cdot 3^{2}\cdot 7$$ Twist minimal: no (minimal twist has level 3) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 63\sqrt{30001}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 10791) q^{2} + (21582 \beta + 101301922) q^{4} + ( - 17600 \beta + 885973050) q^{5} + (17982144 \beta + 184832599952) q^{7} + ( - 199975556 \beta - 2214659936412) q^{8}+O(q^{10})$$ q + (-b - 10791) * q^2 + (21582*b + 101301922) * q^4 + (-17600*b + 885973050) * q^5 + (17982144*b + 184832599952) * q^7 + (-199975556*b - 2214659936412) * q^8 $$q + ( - \beta - 10791) q^{2} + (21582 \beta + 101301922) q^{4} + ( - 17600 \beta + 885973050) q^{5} + (17982144 \beta + 184832599952) q^{7} + ( - 199975556 \beta - 2214659936412) q^{8} + ( - 696051450 \beta - 7464833328150) q^{10} + ( - 8106238336 \beta - 37881167834124) q^{11} + ( - 115885051776 \beta - 51510539588794) q^{13} + ( - 378877915856 \beta - 41\!\cdots\!68) q^{14}+ \cdots + ( - 78\!\cdots\!61 \beta - 86\!\cdots\!39) q^{98}+O(q^{100})$$ q + (-b - 10791) * q^2 + (21582*b + 101301922) * q^4 + (-17600*b + 885973050) * q^5 + (17982144*b + 184832599952) * q^7 + (-199975556*b - 2214659936412) * q^8 + (-696051450*b - 7464833328150) * q^10 + (-8106238336*b - 37881167834124) * q^11 + (-115885051776*b - 51510539588794) * q^13 + (-378877915856*b - 4135733843291568) * q^14 + (1475909155512*b + 34113764716850440) * q^16 + (4829442960000*b - 17345805789573954) * q^17 + (2814081001344*b + 55810238279321108) * q^19 + (17338156537900*b + 44521335383541300) * q^20 + (125355585717900*b + 1374017654425507668) * q^22 + (-179372064257408*b + 1446371537909693976) * q^23 + (-31186251360000*b - 6628747998960085625) * q^25 + (1302026133303610*b + 14354743295441494998) * q^26 + (5810682921044832*b + 64935389484490913696) * q^28 + (3892545677477312*b - 14979776236661403486) * q^29 + (-9437537521608768*b + 5183731628527747016) * q^31 + (-23200035632123664*b - 246617371131942053232) * q^32 + (-34768713191786046*b - 387882351031015702386) * q^34 + (12678641206064000*b + 126071489792015460000) * q^35 + (4481515500954624*b - 1851830454409762434658) * q^37 + (-86176986364824212*b - 937332075189678490764) * q^38 + (-138194938393914600*b - 1543039875031874650200) * q^40 + (542569103438555008*b - 740907464239739017386) * q^41 + (353437120018359936*b + 4889389311728690124620) * q^43 + (-1638728887822945960*b - 24669287355972916440216) * q^44 + (489232407491995752*b + 5750748353269100517336) * q^46 + (1793036388464342912*b - 44747499259404870161760) * q^47 + (6647372856462514176*b + 6954188910180704787945) * q^49 + (6965278837385845625*b + 75244290384445131819375) * q^50 + (-12851078941383665580*b - 303025826743631112311476) * q^52 + (80486970344586432*b - 212835637912300906738374) * q^53 + (-6515200148692262400*b - 16573435090597164079800) * q^55 + (-76786335829918200640*b - 837530065877160924354240) * q^56 + (-27024684168996270306*b - 301854097961204242273902) * q^58 + (81153339396537869824*b - 184709384821690065146412) * q^59 + (-32446692922463744256*b + 462566698714446372624710) * q^61 + (96656735767152468472*b + 1067827402280936352910536) * q^62 + (298875782050969194720*b + 845096381720945895128608) * q^64 + (-101764447274627862400*b + 197223568037573647412700) * q^65 + (35699013212104788480*b + 2075853337277606974066484) * q^67 + (114874673486784044772*b + 10653801770150175334340412) * q^68 + (-262886707046652084000*b - 2870133576278626176876000) * q^70 + (-739151259346737409920*b - 4880227927935433768834392) * q^71 + (-619525988484801057792*b + 14224978173820037578704794) * q^73 + (1803470420638961087074*b + 19449470595702056063811822) * q^74 + (1489568376644139936024*b + 12885462845301213005769128) * q^76 + (-2179481722354840041728*b - 24358794881235892694334144) * q^77 + (-2262423523592910807360*b - 24776610949337750872115560) * q^79 + (707213477015323207600*b + 27130810619037934339189200) * q^80 + (-5113955730965708073942*b - 56610724156589268690774426) * q^82 + (-440202825764870246272*b + 60236704101614367963173868) * q^83 + (4584042490968729590400*b - 25488989027087519249939700) * q^85 + (-8703329273846812193996*b - 94846560735379765584880404) * q^86 + (25527868877304539363376*b + 276918704837456901590547792) * q^88 + (21608558186896399716096*b - 102822026104848566528492778) * q^89 + (-22345605355733609609088*b - 257654509051897696625073824) * q^91 + (13044855668784082251856*b - 314439871665729128238358992) * q^92 + (25398843591486145798368*b + 269366305172362828406694432) * q^94 + (1510939733991746278400*b + 43548892256608444291825800) * q^95 + (-29604352496406812087040*b - 739169503200204797418213886) * q^97 + (-78685989404267695261161*b - 866571721971618848021799039) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 21582 q^{2} + 202603844 q^{4} + 1771946100 q^{5} + 369665199904 q^{7} - 4429319872824 q^{8}+O(q^{10})$$ 2 * q - 21582 * q^2 + 202603844 * q^4 + 1771946100 * q^5 + 369665199904 * q^7 - 4429319872824 * q^8 $$2 q - 21582 q^{2} + 202603844 q^{4} + 1771946100 q^{5} + 369665199904 q^{7} - 4429319872824 q^{8} - 14929666656300 q^{10} - 75762335668248 q^{11} - 103021079177588 q^{13} - 82\!\cdots\!36 q^{14}+ \cdots - 17\!\cdots\!78 q^{98}+O(q^{100})$$ 2 * q - 21582 * q^2 + 202603844 * q^4 + 1771946100 * q^5 + 369665199904 * q^7 - 4429319872824 * q^8 - 14929666656300 * q^10 - 75762335668248 * q^11 - 103021079177588 * q^13 - 8271467686583136 * q^14 + 68227529433700880 * q^16 - 34691611579147908 * q^17 + 111620476558642216 * q^19 + 89042670767082600 * q^20 + 2748035308851015336 * q^22 + 2892743075819387952 * q^23 - 13257495997920171250 * q^25 + 28709486590882989996 * q^26 + 129870778968981827392 * q^28 - 29959552473322806972 * q^29 + 10367463257055494032 * q^31 - 493234742263884106464 * q^32 - 775764702062031404772 * q^34 + 252142979584030920000 * q^35 - 3703660908819524869316 * q^37 - 1874664150379356981528 * q^38 - 3086079750063749300400 * q^40 - 1481814928479478034772 * q^41 + 9778778623457380249240 * q^43 - 49338574711945832880432 * q^44 + 11501496706538201034672 * q^46 - 89494998518809740323520 * q^47 + 13908377820361409575890 * q^49 + 150488580768890263638750 * q^50 - 606051653487262224622952 * q^52 - 425671275824601813476748 * q^53 - 33146870181194328159600 * q^55 - 1675060131754321848708480 * q^56 - 603708195922408484547804 * q^58 - 369418769643380130292824 * q^59 + 925133397428892745249420 * q^61 + 2135654804561872705821072 * q^62 + 1690192763441891790257216 * q^64 + 394447136075147294825400 * q^65 + 4151706674555213948132968 * q^67 + 21307603540300350668680824 * q^68 - 5740267152557252353752000 * q^70 - 9760455855870867537668784 * q^71 + 28449956347640075157409588 * q^73 + 38898941191404112127623644 * q^74 + 25770925690602426011538256 * q^76 - 48717589762471785388668288 * q^77 - 49553221898675501744231120 * q^79 + 54261621238075868678378400 * q^80 - 113221448313178537381548852 * q^82 + 120473408203228735926347736 * q^83 - 50977978054175038499879400 * q^85 - 189693121470759531169760808 * q^86 + 553837409674913803181095584 * q^88 - 205644052209697133056985556 * q^89 - 515309018103795393250147648 * q^91 - 628879743331458256476717984 * q^92 + 538732610344725656813388864 * q^94 + 87097784513216888583651600 * q^95 - 1478339006400409594836427772 * q^97 - 1733143443943237696043598078 * 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 87.1040 −86.1040
−21703.1 0 3.36807e8 6.93920e8 0 3.81056e11 −4.39681e12 0 −1.50602e13
1.2 121.102 0 −1.34203e8 1.07803e9 0 −1.13904e10 −3.25063e10 0 1.30551e11
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$

## 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 9.28.a.b 2
3.b odd 2 1 3.28.a.b 2
12.b even 2 1 48.28.a.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.28.a.b 2 3.b odd 2 1
9.28.a.b 2 1.a even 1 1 trivial
48.28.a.e 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 21582T_{2} - 2628288$$ acting on $$S_{28}^{\mathrm{new}}(\Gamma_0(9))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 21582 T - 2628288$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 1771946100 T + 74\!\cdots\!00$$
$7$ $$T^{2} - 369665199904 T - 43\!\cdots\!80$$
$11$ $$T^{2} + 75762335668248 T - 63\!\cdots\!48$$
$13$ $$T^{2} + 103021079177588 T - 15\!\cdots\!08$$
$17$ $$T^{2} + \cdots - 24\!\cdots\!84$$
$19$ $$T^{2} + \cdots + 21\!\cdots\!80$$
$23$ $$T^{2} + \cdots - 17\!\cdots\!40$$
$29$ $$T^{2} + \cdots - 15\!\cdots\!40$$
$31$ $$T^{2} + \cdots - 10\!\cdots\!00$$
$37$ $$T^{2} + \cdots + 34\!\cdots\!20$$
$41$ $$T^{2} + \cdots - 34\!\cdots\!20$$
$43$ $$T^{2} + \cdots + 90\!\cdots\!76$$
$47$ $$T^{2} + \cdots + 16\!\cdots\!64$$
$53$ $$T^{2} + \cdots + 45\!\cdots\!20$$
$59$ $$T^{2} + \cdots - 75\!\cdots\!00$$
$61$ $$T^{2} + \cdots + 88\!\cdots\!16$$
$67$ $$T^{2} + \cdots + 41\!\cdots\!56$$
$71$ $$T^{2} + \cdots - 41\!\cdots\!36$$
$73$ $$T^{2} + \cdots + 15\!\cdots\!20$$
$79$ $$T^{2} + \cdots + 43\!\cdots\!00$$
$83$ $$T^{2} + \cdots + 36\!\cdots\!28$$
$89$ $$T^{2} + \cdots - 45\!\cdots\!20$$
$97$ $$T^{2} + \cdots + 44\!\cdots\!96$$