# Properties

 Label 3.26.a.a Level $3$ Weight $26$ Character orbit 3.a Self dual yes Analytic conductor $11.880$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3$$ Weight: $$k$$ $$=$$ $$26$$ Character orbit: $$[\chi]$$ $$=$$ 3.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$11.8799033986$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{1287001})$$ Defining polynomial: $$x^{2} - x - 321750$$ x^2 - x - 321750 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 6\sqrt{1287001}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 162) q^{2} - 531441 q^{3} + (324 \beta + 12803848) q^{4} + (22912 \beta + 285430878) q^{5} + (531441 \beta + 86093442) q^{6} + (5152896 \beta - 14843692864) q^{7} + (20698096 \beta - 11649985056) q^{8} + 282429536481 q^{9}+O(q^{10})$$ q + (-b - 162) * q^2 - 531441 * q^3 + (324*b + 12803848) * q^4 + (22912*b + 285430878) * q^5 + (531441*b + 86093442) * q^6 + (5152896*b - 14843692864) * q^7 + (20698096*b - 11649985056) * q^8 + 282429536481 * q^9 $$q + ( - \beta - 162) q^{2} - 531441 q^{3} + (324 \beta + 12803848) q^{4} + (22912 \beta + 285430878) q^{5} + (531441 \beta + 86093442) q^{6} + (5152896 \beta - 14843692864) q^{7} + (20698096 \beta - 11649985056) q^{8} + 282429536481 q^{9} + ( - 289142622 \beta - 1107799411068) q^{10} + ( - 983057152 \beta - 9093892809276) q^{11} + ( - 172186884 \beta - 6804489784968) q^{12} + (8456327424 \beta - 52571818339618) q^{13} + (14008923712 \beta - 236339484732288) q^{14} + ( - 12176376192 \beta - 151689671235198) q^{15} + ( - 2574742464 \beta - 13\!\cdots\!20) q^{16}+ \cdots + ( - 27\!\cdots\!12 \beta - 25\!\cdots\!56) q^{99}+O(q^{100})$$ q + (-b - 162) * q^2 - 531441 * q^3 + (324*b + 12803848) * q^4 + (22912*b + 285430878) * q^5 + (531441*b + 86093442) * q^6 + (5152896*b - 14843692864) * q^7 + (20698096*b - 11649985056) * q^8 + 282429536481 * q^9 + (-289142622*b - 1107799411068) * q^10 + (-983057152*b - 9093892809276) * q^11 + (-172186884*b - 6804489784968) * q^12 + (8456327424*b - 52571818339618) * q^13 + (14008923712*b - 236339484732288) * q^14 + (-12176376192*b - 151689671235198) * q^15 + (-2574742464*b - 1386723478478720) * q^16 + (-310973856000*b - 228493682174718) * q^17 + (-282429536481*b - 45753584909922) * q^18 + (1780077627648*b + 6948719247673532) * q^19 + (385841369848*b + 3998558889680112) * q^20 + (-2738460203136*b + 7888546979337024) * q^21 + (9253148067900*b + 47020249991624184) * q^22 + (-20198378016512*b + 69671149450176888) * q^23 + (-10999816836336*b + 6191279708145696) * q^24 + (13079584553472*b - 192229984003543457) * q^25 + (51201893296930*b - 383282232065537148) * q^26 - 150094635296999121 * q^27 + (61167540655872*b - 112703278385033728) * q^28 + (-126217146058624*b - 1251016745130354906) * q^29 + (153662244178302*b + 588730026817388988) * q^30 + (-1052676722058624*b + 537736051246227656) * q^31 + (692627731996416*b + 734850895408897536) * q^32 + (522436875916032*b + 4832867488454446716) * q^33 + (278871446846718*b + 14445067867763120316) * q^34 + (1130696938622720*b + 1233257975178122880) * q^35 + (91507169819844*b + 3616184855813178888) * q^36 + (295504419926016*b - 985735474619156314) * q^37 + (-7237091823352508*b - 83600313245104843512) * q^38 + (-4494039102537984*b + 27938819710224929538) * q^39 + (5640951256605216*b + 18646997230106224704) * q^40 + (5221410776755456*b - 209581677851994798534) * q^41 + (-7444916426428992*b + 125600492105611867008) * q^42 + (45176076103631616*b + 19465197949129913540) * q^43 + (-15533335619726320*b - 131194062009775850976) * q^44 + (6471025539852672*b + 80614110570904860318) * q^45 + (-66399012211501944*b + 924545251191713922576) * q^46 + (-9934991015089408*b - 369203088922474907520) * q^47 + (1368323709810624*b + 736961712126209435520) * q^48 + (-152976011168268288*b + 109490440600497259065) * q^49 + (190111091305880993*b - 574862524987934588958) * q^50 + (165264257006496000*b + 121430910948614308638) * q^51 + (91240261833091320*b - 546178738313837344528) * q^52 + (15631770768985728*b - 4247001400980921362082) * q^53 + (150094635296999121*b + 24315330918113857602) * q^54 + (-488954138065671168*b - 3639251574726155190792) * q^55 + (-367267341288709120*b + 5114478404763646248960) * q^56 + (-946006234514880768*b - 3692834305702869519612) * q^57 + (1271463922791851994*b + 6050562067716542773236) * q^58 + (-397484438615856128*b + 4216038737213096999748) * q^59 + (-205051923433390968*b - 2124998134890488401392) * q^60 + (2464520538925757952*b - 12139563991870563165730) * q^61 + (-367202422272730568*b + 48685542542480272398192) * q^62 + (1455330028814798976*b - 4192297295245847371584) * q^63 + (-760662567066516480*b + 14320819802905134303232) * q^64 + (1209171459490470656*b - 6028724634356913715836) * q^65 + (-4917502262352843900*b - 24988488675798747969144) * q^66 + (-7885549958339957760*b - 67843054021039258521028) * q^67 + (-4055693937222496632*b - 7593807188290663098864) * q^68 + (10734246211473153792*b - 37026105334951455535608) * q^69 + (-1416430879235003520*b - 52587279057336509364480) * q^70 + (18643323271852842240*b + 60201534566364999577992) * q^71 + (5845753659319240176*b - 3290299879376656827936) * q^72 + (-13385360579839137792*b + 42489004798930952398538) * q^73 + (937863758591141722*b - 13531632275282987325708) * q^74 + (-6951027494681713152*b + 102158894928827138331537) * q^75 + (25043228408851813872*b + 115692122157428338301408) * q^76 + (-32267685449300499968*b - 99712205113030992776448) * q^77 + (-27210785375613776130*b + 203691892691141127470268) * q^78 + (-27385251957433054080*b + 630380976638854108378040) * q^79 + (-32507519341027836032*b - 398546942608322133758208) * q^80 + 79766443076872509863361 * q^81 + (208735809306160414662*b - 207966360267398593225908) * q^82 + (42027911871325874432*b - 438740250914925392850756) * q^83 + (-32506938973697271552*b + 59895142968220709442048) * q^84 + (-93996787999112706816*b - 395336437252921404334404) * q^85 + (-26783722277918235332*b - 2096252946439958809243656) * q^86 + (67076966318541197184*b + 664841590048820941599546) * q^87 + (-176773665250110417984*b - 836793277788128395337856) * q^88 + (133735101954765715968*b + 1065394046187937914347322) * q^89 + (-81662416708360993182*b - 312875274181860021171708) * q^90 + (-396420239474220536064*b + 2799258737431331108730496) * q^91 + (-236043509548103826464*b + 588849246866892251653056) * q^92 + (559435569847557197184*b - 285774984810346471732296) * q^93 + (370812557466919391616*b + 520119261776239929692928) * q^94 + (667298175570421680128*b + 3873037545935561124308232) * q^95 + (-368090774519907315456*b - 390529894706999915429376) * q^96 + (-71593944273965145600*b - 2797058648649902610696958) * q^97 + (-84708326791237796409*b + 7069952605207327821305838) * q^98 + (-277644375773691962112*b - 2568383930931719617197756) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 324 q^{2} - 1062882 q^{3} + 25607696 q^{4} + 570861756 q^{5} + 172186884 q^{6} - 29687385728 q^{7} - 23299970112 q^{8} + 564859072962 q^{9}+O(q^{10})$$ 2 * q - 324 * q^2 - 1062882 * q^3 + 25607696 * q^4 + 570861756 * q^5 + 172186884 * q^6 - 29687385728 * q^7 - 23299970112 * q^8 + 564859072962 * q^9 $$2 q - 324 q^{2} - 1062882 q^{3} + 25607696 q^{4} + 570861756 q^{5} + 172186884 q^{6} - 29687385728 q^{7} - 23299970112 q^{8} + 564859072962 q^{9} - 2215598822136 q^{10} - 18187785618552 q^{11} - 13608979569936 q^{12} - 105143636679236 q^{13} - 472678969464576 q^{14} - 303379342470396 q^{15} - 27\!\cdots\!40 q^{16}+ \cdots - 51\!\cdots\!12 q^{99}+O(q^{100})$$ 2 * q - 324 * q^2 - 1062882 * q^3 + 25607696 * q^4 + 570861756 * q^5 + 172186884 * q^6 - 29687385728 * q^7 - 23299970112 * q^8 + 564859072962 * q^9 - 2215598822136 * q^10 - 18187785618552 * q^11 - 13608979569936 * q^12 - 105143636679236 * q^13 - 472678969464576 * q^14 - 303379342470396 * q^15 - 2773446956957440 * q^16 - 456987364349436 * q^17 - 91507169819844 * q^18 + 13897438495347064 * q^19 + 7997117779360224 * q^20 + 15777093958674048 * q^21 + 94040499983248368 * q^22 + 139342298900353776 * q^23 + 12382559416291392 * q^24 - 384459968007086914 * q^25 - 766564464131074296 * q^26 - 300189270593998242 * q^27 - 225406556770067456 * q^28 - 2502033490260709812 * q^29 + 1177460053634777976 * q^30 + 1075472102492455312 * q^31 + 1469701790817795072 * q^32 + 9665734976908893432 * q^33 + 28890135735526240632 * q^34 + 2466515950356245760 * q^35 + 7232369711626357776 * q^36 - 1971470949238312628 * q^37 - 167200626490209687024 * q^38 + 55877639420449859076 * q^39 + 37293994460212449408 * q^40 - 419163355703989597068 * q^41 + 251200984211223734016 * q^42 + 38930395898259827080 * q^43 - 262388124019551701952 * q^44 + 161228221141809720636 * q^45 + 1849090502383427845152 * q^46 - 738406177844949815040 * q^47 + 1473923424252418871040 * q^48 + 218980881200994518130 * q^49 - 1149725049975869177916 * q^50 + 242861821897228617276 * q^51 - 1092357476627674689056 * q^52 - 8494002801961842724164 * q^53 + 48630661836227715204 * q^54 - 7278503149452310381584 * q^55 + 10228956809527292497920 * q^56 - 7385668611405739039224 * q^57 + 12101124135433085546472 * q^58 + 8432077474426193999496 * q^59 - 4249996269780976802784 * q^60 - 24279127983741126331460 * q^61 + 97371085084960544796384 * q^62 - 8384594590491694743168 * q^63 + 28641639605810268606464 * q^64 - 12057449268713827431672 * q^65 - 49976977351597495938288 * q^66 - 135686108042078517042056 * q^67 - 15187614376581326197728 * q^68 - 74052210669902911071216 * q^69 - 105174558114673018728960 * q^70 + 120403069132729999155984 * q^71 - 6580599758753313655872 * q^72 + 84978009597861904797076 * q^73 - 27063264550565974651416 * q^74 + 204317789857654276663074 * q^75 + 231384244314856676602816 * q^76 - 199424410226061985552896 * q^77 + 407383785382282254940536 * q^78 + 1260761953277708216756080 * q^79 - 797093885216644267516416 * q^80 + 159532886153745019726722 * q^81 - 415932720534797186451816 * q^82 - 877480501829850785701512 * q^83 + 119790285936441418884096 * q^84 - 790672874505842808668808 * q^85 - 4192505892879917618487312 * q^86 + 1329683180097641883199092 * q^87 - 1673586555576256790675712 * q^88 + 2130788092375875828694644 * q^89 - 625750548363720042343416 * q^90 + 5598517474862662217460992 * q^91 + 1177698493733784503306112 * q^92 - 571549969620692943464592 * q^93 + 1040238523552479859385856 * q^94 + 7746075091871122248616464 * q^95 - 781059789413999830858752 * q^96 - 5594117297299805221393916 * q^97 + 14139905210414655642611676 * q^98 - 5136767861863439234395512 * 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 567.730 −566.730
−6968.76 −531441. 1.50092e7 4.41387e8 3.70349e9 2.02309e10 1.29237e11 2.82430e11 −3.07592e12
1.2 6644.76 −531441. 1.05985e7 1.29474e8 −3.53130e9 −4.99182e10 −1.52537e11 2.82430e11 8.60326e11
 $$n$$: 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 3.26.a.a 2
3.b odd 2 1 9.26.a.b 2
4.b odd 2 1 48.26.a.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.26.a.a 2 1.a even 1 1 trivial
9.26.a.b 2 3.b odd 2 1
48.26.a.f 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 324T_{2} - 46305792$$ acting on $$S_{26}^{\mathrm{new}}(\Gamma_0(3))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 324 T - 46305792$$
$3$ $$(T + 531441)^{2}$$
$5$ $$T^{2} - 570861756 T + 57\!\cdots\!00$$
$7$ $$T^{2} + 29687385728 T - 10\!\cdots\!80$$
$11$ $$T^{2} + 18187785618552 T + 37\!\cdots\!32$$
$13$ $$T^{2} + 105143636679236 T - 54\!\cdots\!12$$
$17$ $$T^{2} + 456987364349436 T - 44\!\cdots\!76$$
$19$ $$T^{2} + \cdots - 98\!\cdots\!20$$
$23$ $$T^{2} + \cdots - 14\!\cdots\!40$$
$29$ $$T^{2} + \cdots + 82\!\cdots\!00$$
$31$ $$T^{2} + \cdots - 51\!\cdots\!00$$
$37$ $$T^{2} + \cdots - 30\!\cdots\!20$$
$41$ $$T^{2} + \cdots + 42\!\cdots\!60$$
$43$ $$T^{2} + \cdots - 94\!\cdots\!16$$
$47$ $$T^{2} + \cdots + 13\!\cdots\!96$$
$53$ $$T^{2} + \cdots + 18\!\cdots\!00$$
$59$ $$T^{2} + \cdots + 10\!\cdots\!80$$
$61$ $$T^{2} + \cdots - 13\!\cdots\!44$$
$67$ $$T^{2} + \cdots + 17\!\cdots\!84$$
$71$ $$T^{2} + \cdots - 12\!\cdots\!36$$
$73$ $$T^{2} + \cdots - 64\!\cdots\!60$$
$79$ $$T^{2} + \cdots + 36\!\cdots\!00$$
$83$ $$T^{2} + \cdots + 11\!\cdots\!72$$
$89$ $$T^{2} + \cdots + 30\!\cdots\!20$$
$97$ $$T^{2} + \cdots + 75\!\cdots\!64$$