# Properties

 Label 3.20.a.b Level $3$ Weight $20$ Character orbit 3.a Self dual yes Analytic conductor $6.865$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3$$ Weight: $$k$$ $$=$$ $$20$$ Character orbit: $$[\chi]$$ $$=$$ 3.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.86450089669$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{87481})$$ Defining polynomial: $$x^{2} - x - 21870$$ x^2 - x - 21870 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 3\sqrt{87481}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta + 351) q^{2} - 19683 q^{3} + ( - 702 \beta + 386242) q^{4} + ( - 4160 \beta + 3008070) q^{5} + (19683 \beta - 6908733) q^{6} + (63936 \beta + 56946032) q^{7} + ( - 108356 \beta + 504250812) q^{8} + 387420489 q^{9}+O(q^{10})$$ q + (-b + 351) * q^2 - 19683 * q^3 + (-702*b + 386242) * q^4 + (-4160*b + 3008070) * q^5 + (19683*b - 6908733) * q^6 + (63936*b + 56946032) * q^7 + (-108356*b + 504250812) * q^8 + 387420489 * q^9 $$q + ( - \beta + 351) q^{2} - 19683 q^{3} + ( - 702 \beta + 386242) q^{4} + ( - 4160 \beta + 3008070) q^{5} + (19683 \beta - 6908733) q^{6} + (63936 \beta + 56946032) q^{7} + ( - 108356 \beta + 504250812) q^{8} + 387420489 q^{9} + ( - 4468230 \beta + 4331121210) q^{10} + (10995584 \beta - 3325035636) q^{11} + (13817466 \beta - 7602401286) q^{12} + ( - 39982464 \beta - 22036178074) q^{13} + ( - 34504496 \beta - 30350609712) q^{14} + (81881280 \beta - 59207841810) q^{15} + ( - 174233592 \beta + 59801810440) q^{16} + (582591360 \beta + 168140735874) q^{17} + ( - 387420489 \beta + 135984591639) q^{18} + (1038738816 \beta + 301059462548) q^{19} + ( - 3718431860 \beta + 3461095598220) q^{20} + ( - 1258452288 \beta - 1120868747856) q^{21} + (7184485620 \beta - 9824229663372) q^{22} + (4325606272 \beta + 1184126082984) q^{23} + (2132771148 \beta - 9925168732596) q^{24} + ( - 25027142400 \beta + 3600199539175) q^{25} + (8002333210 \beta + 23744654894682) q^{26} - 7625597484987 q^{27} + ( - 15281345952 \beta - 13342794902944) q^{28} + (14942987072 \beta + 140488625985246) q^{29} + (87948171090 \beta - 85249458776430) q^{30} + (20829313728 \beta + 20805074626856) q^{31} + ( - 64148050704 \beta - 106203054501648) q^{32} + ( - 216426079872 \beta + 65446676423388) q^{33} + (36348831486 \beta - 399673674585666) q^{34} + ( - 44571529600 \beta - 38111204008800) q^{35} + ( - 271969183278 \beta + 149638064512338) q^{36} + (1169925037056 \beta + 318997081994942) q^{37} + (63537861868 \beta - 712157321908116) q^{38} + (786974838912 \beta + 433738093030542) q^{39} + ( - 2423625810840 \beta + 18\!\cdots\!80) q^{40}+ \cdots + (42\!\cdots\!76 \beta - 12\!\cdots\!04) q^{99}+O(q^{100})$$ q + (-b + 351) * q^2 - 19683 * q^3 + (-702*b + 386242) * q^4 + (-4160*b + 3008070) * q^5 + (19683*b - 6908733) * q^6 + (63936*b + 56946032) * q^7 + (-108356*b + 504250812) * q^8 + 387420489 * q^9 + (-4468230*b + 4331121210) * q^10 + (10995584*b - 3325035636) * q^11 + (13817466*b - 7602401286) * q^12 + (-39982464*b - 22036178074) * q^13 + (-34504496*b - 30350609712) * q^14 + (81881280*b - 59207841810) * q^15 + (-174233592*b + 59801810440) * q^16 + (582591360*b + 168140735874) * q^17 + (-387420489*b + 135984591639) * q^18 + (1038738816*b + 301059462548) * q^19 + (-3718431860*b + 3461095598220) * q^20 + (-1258452288*b - 1120868747856) * q^21 + (7184485620*b - 9824229663372) * q^22 + (4325606272*b + 1184126082984) * q^23 + (2132771148*b - 9925168732596) * q^24 + (-25027142400*b + 3600199539175) * q^25 + (8002333210*b + 23744654894682) * q^26 - 7625597484987 * q^27 + (-15281345952*b - 13342794902944) * q^28 + (14942987072*b + 140488625985246) * q^29 + (87948171090*b - 85249458776430) * q^30 + (20829313728*b + 20805074626856) * q^31 + (-64148050704*b - 106203054501648) * q^32 + (-216426079872*b + 65446676423388) * q^33 + (36348831486*b - 399673674585666) * q^34 + (-44571529600*b - 38111204008800) * q^35 + (-271969183278*b + 149638064512338) * q^36 + (1169925037056*b + 318997081994942) * q^37 + (63537861868*b - 712157321908116) * q^38 + (786974838912*b + 433738093030542) * q^39 + (-2423625810840*b + 1871718915928680) * q^40 + (-1296527879552*b + 575019215140266) * q^41 + (679151994768*b + 597391050961296) * q^42 + (-1209933369216*b + 1410259476769100) * q^43 + (6581131371800*b - 7361582207025384) * q^44 + (-1611669234240*b + 1165387950346230) * q^45 + (334161718488*b - 2990047005400104) * q^46 + (-4731612028288*b - 2801545659750240) * q^47 + (3429439791336*b - 1177079034890520) * q^48 + (7281803003904*b - 4937591615096535) * q^49 + (-12384726521575*b + 20968265036900025) * q^50 + (-11467145738880*b - 3309514104207942) * q^51 + (26490147660*b + 13587208594198604) * q^52 + (3629613529152*b + 13768934428823814) * q^53 + (7625597484987*b - 2676584717230437) * q^54 + (46907634608640*b - 46015651310948280) * q^55 + (26069335672640*b + 23260586280793920) * q^56 + (-20445496115328*b - 5925753401332284) * q^57 + (-135243637522974*b + 37546460652410658) * q^58 + (572349981184*b - 27576737210917908) * q^59 + (73189894300380*b - 68124744659764260) * q^60 + (-8516769041664*b - 43596284533102810) * q^61 + (-13493985508328*b - 9096941554126056) * q^62 + (24770116384704*b + 22062059564049648) * q^63 + (175035670187040*b - 18125023109315552) * q^64 + (-28599549696640*b + 64667743959351780) * q^65 + (-141412230458460*b + 193370312464151076) * q^66 + (-47573122936320*b + 249649855438939124) * q^67 + (106986455485572*b - 257058119054517372) * q^68 + (-85140908251776*b - 23307153691374072) * q^69 + (22466597119200*b + 21715425221349600) * q^70 + (-312831875103360*b + 42457524218485272) * q^71 + (-41979334506084*b + 195357096163687068) * q^72 + (219879620692992*b - 120716855580864166) * q^73 + (91646606011714*b - 809147933720038782) * q^74 + (492609243859200*b - 70862727529581525) * q^75 + (189860815060776*b - 457834284736785112) * q^76 + (413565399899392*b + 364155455101978944) * q^77 + (-157509924572430*b - 467366042292025806) * q^78 + (-1425076089053760*b + 565795733016761720) * q^79 + (-772882372517840*b + 750553336514245680) * q^80 + 150094635296999121 * q^81 + (-1030100500863018*b + 1222625743394029974) * q^82 + (2762772599881088*b + 803871638001791892) * q^83 + (300782732373216*b + 262626232074646752) * q^84 + (1053010131039360*b - 1402375759809647220) * q^85 + (-1834946089363916*b + 1447616705997418164) * q^86 + (-294122814538176*b - 2765237625267597018) * q^87 + (5904819721788624*b - 2614705214743852848) * q^88 + (-4053972006476544*b + 2161398813184788138) * q^89 + (-1731083851564470*b + 1677965097096471690) * q^90 + (-3685747755722112*b - 3267536840656172384) * q^91 + (839474307455056*b - 1933424806346390448) * q^92 + (-409983382108224*b - 409506283880406648) * q^93 + (1140749837821152*b + 2741992840047628512) * q^94 + (1872191706045440*b - 2496561506465087880) * q^95 + (1262626082006832*b + 2090394721755937584) * q^96 + (-2571793123864320*b + 3951100617649729154) * q^97 + (7493504469466839*b - 7466269334159616201) * q^98 + (4259914530120576*b - 1288186932041546004) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 702 q^{2} - 39366 q^{3} + 772484 q^{4} + 6016140 q^{5} - 13817466 q^{6} + 113892064 q^{7} + 1008501624 q^{8} + 774840978 q^{9}+O(q^{10})$$ 2 * q + 702 * q^2 - 39366 * q^3 + 772484 * q^4 + 6016140 * q^5 - 13817466 * q^6 + 113892064 * q^7 + 1008501624 * q^8 + 774840978 * q^9 $$2 q + 702 q^{2} - 39366 q^{3} + 772484 q^{4} + 6016140 q^{5} - 13817466 q^{6} + 113892064 q^{7} + 1008501624 q^{8} + 774840978 q^{9} + 8662242420 q^{10} - 6650071272 q^{11} - 15204802572 q^{12} - 44072356148 q^{13} - 60701219424 q^{14} - 118415683620 q^{15} + 119603620880 q^{16} + 336281471748 q^{17} + 271969183278 q^{18} + 602118925096 q^{19} + 6922191196440 q^{20} - 2241737495712 q^{21} - 19648459326744 q^{22} + 2368252165968 q^{23} - 19850337465192 q^{24} + 7200399078350 q^{25} + 47489309789364 q^{26} - 15251194969974 q^{27} - 26685589805888 q^{28} + 280977251970492 q^{29} - 170498917552860 q^{30} + 41610149253712 q^{31} - 212406109003296 q^{32} + 130893352846776 q^{33} - 799347349171332 q^{34} - 76222408017600 q^{35} + 299276129024676 q^{36} + 637994163989884 q^{37} - 14\!\cdots\!32 q^{38}+ \cdots - 25\!\cdots\!08 q^{99}+O(q^{100})$$ 2 * q + 702 * q^2 - 39366 * q^3 + 772484 * q^4 + 6016140 * q^5 - 13817466 * q^6 + 113892064 * q^7 + 1008501624 * q^8 + 774840978 * q^9 + 8662242420 * q^10 - 6650071272 * q^11 - 15204802572 * q^12 - 44072356148 * q^13 - 60701219424 * q^14 - 118415683620 * q^15 + 119603620880 * q^16 + 336281471748 * q^17 + 271969183278 * q^18 + 602118925096 * q^19 + 6922191196440 * q^20 - 2241737495712 * q^21 - 19648459326744 * q^22 + 2368252165968 * q^23 - 19850337465192 * q^24 + 7200399078350 * q^25 + 47489309789364 * q^26 - 15251194969974 * q^27 - 26685589805888 * q^28 + 280977251970492 * q^29 - 170498917552860 * q^30 + 41610149253712 * q^31 - 212406109003296 * q^32 + 130893352846776 * q^33 - 799347349171332 * q^34 - 76222408017600 * q^35 + 299276129024676 * q^36 + 637994163989884 * q^37 - 1424314643816232 * q^38 + 867476186061084 * q^39 + 3743437831857360 * q^40 + 1150038430280532 * q^41 + 1194782101922592 * q^42 + 2820518953538200 * q^43 - 14723164414050768 * q^44 + 2330775900692460 * q^45 - 5980094010800208 * q^46 - 5603091319500480 * q^47 - 2354158069781040 * q^48 - 9875183230193070 * q^49 + 41936530073800050 * q^50 - 6619028208415884 * q^51 + 27174417188397208 * q^52 + 27537868857647628 * q^53 - 5353169434460874 * q^54 - 92031302621896560 * q^55 + 46521172561587840 * q^56 - 11851506802664568 * q^57 + 75092921304821316 * q^58 - 55153474421835816 * q^59 - 136249489319528520 * q^60 - 87192569066205620 * q^61 - 18193883108252112 * q^62 + 44124119128099296 * q^63 - 36250046218631104 * q^64 + 129335487918703560 * q^65 + 386740624928302152 * q^66 + 499299710877878248 * q^67 - 514116238109034744 * q^68 - 46614307382748144 * q^69 + 43430850442699200 * q^70 + 84915048436970544 * q^71 + 390714192327374136 * q^72 - 241433711161728332 * q^73 - 1618295867440077564 * q^74 - 141725455059163050 * q^75 - 915668569473570224 * q^76 + 728310910203957888 * q^77 - 934732084584051612 * q^78 + 1131591466033523440 * q^79 + 1501106673028491360 * q^80 + 300189270593998242 * q^81 + 2445251486788059948 * q^82 + 1607743276003583784 * q^83 + 525252464149293504 * q^84 - 2804751519619294440 * q^85 + 2895233411994836328 * q^86 - 5530475250535194036 * q^87 - 5229410429487705696 * q^88 + 4322797626369576276 * q^89 + 3355930194192943380 * q^90 - 6535073681312344768 * q^91 - 3866849612692780896 * q^92 - 819012567760813296 * q^93 + 5483985680095257024 * q^94 - 4993123012930175760 * q^95 + 4180789443511875168 * q^96 + 7902201235299458308 * q^97 - 14932538668319232402 * q^98 - 2576373864083092008 * 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
 148.386 −147.386
−536.316 −19683.0 −236654. −683163. 1.05563e7 1.13677e8 4.08105e8 3.87420e8 3.66391e8
1.2 1238.32 −19683.0 1.00914e6 6.69930e6 −2.43738e7 214621. 6.00397e8 3.87420e8 8.29585e9
 $$n$$: 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 3.20.a.b 2
3.b odd 2 1 9.20.a.c 2
4.b odd 2 1 48.20.a.j 2
5.b even 2 1 75.20.a.b 2
5.c odd 4 2 75.20.b.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.20.a.b 2 1.a even 1 1 trivial
9.20.a.c 2 3.b odd 2 1
48.20.a.j 2 4.b odd 2 1
75.20.a.b 2 5.b even 2 1
75.20.b.b 4 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 702T_{2} - 664128$$ acting on $$S_{20}^{\mathrm{new}}(\Gamma_0(3))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 702T - 664128$$
$3$ $$(T + 19683)^{2}$$
$5$ $$T^{2} - 6016140 T - 4576715617500$$
$7$ $$T^{2} - 113892064 T + 24397550813440$$
$11$ $$T^{2} + 6650071272 T - 84\!\cdots\!28$$
$13$ $$T^{2} + 44072356148 T - 77\!\cdots\!08$$
$17$ $$T^{2} - 336281471748 T - 23\!\cdots\!24$$
$19$ $$T^{2} - 602118925096 T - 75\!\cdots\!20$$
$23$ $$T^{2} - 2368252165968 T - 13\!\cdots\!80$$
$29$ $$T^{2} - 280977251970492 T + 19\!\cdots\!80$$
$31$ $$T^{2} - 41610149253712 T + 91\!\cdots\!00$$
$37$ $$T^{2} - 637994163989884 T - 97\!\cdots\!80$$
$41$ $$T^{2} + \cdots - 99\!\cdots\!60$$
$43$ $$T^{2} + \cdots + 83\!\cdots\!76$$
$47$ $$T^{2} + \cdots - 97\!\cdots\!76$$
$53$ $$T^{2} + \cdots + 17\!\cdots\!80$$
$59$ $$T^{2} + \cdots + 76\!\cdots\!40$$
$61$ $$T^{2} + \cdots + 18\!\cdots\!16$$
$67$ $$T^{2} + \cdots + 60\!\cdots\!76$$
$71$ $$T^{2} + \cdots - 75\!\cdots\!16$$
$73$ $$T^{2} + \cdots - 23\!\cdots\!00$$
$79$ $$T^{2} + \cdots - 12\!\cdots\!00$$
$83$ $$T^{2} + \cdots - 53\!\cdots\!12$$
$89$ $$T^{2} + \cdots - 82\!\cdots\!00$$
$97$ $$T^{2} + \cdots + 10\!\cdots\!16$$