Properties

Label 21.14.a.a
Level $21$
Weight $14$
Character orbit 21.a
Self dual yes
Analytic conductor $22.518$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.5184950799\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 28\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 72 + \beta ) q^{2} + 729 q^{3} + ( -656 + 144 \beta ) q^{4} + ( -26280 - 740 \beta ) q^{5} + ( 52488 + 729 \beta ) q^{6} + 117649 q^{7} + ( -298368 + 1520 \beta ) q^{8} + 531441 q^{9} +O(q^{10})\) \( q + ( 72 + \beta ) q^{2} + 729 q^{3} + ( -656 + 144 \beta ) q^{4} + ( -26280 - 740 \beta ) q^{5} + ( 52488 + 729 \beta ) q^{6} + 117649 q^{7} + ( -298368 + 1520 \beta ) q^{8} + 531441 q^{9} + ( -3632640 - 79560 \beta ) q^{10} + ( -4296798 - 93940 \beta ) q^{11} + ( -478224 + 104976 \beta ) q^{12} + ( -18991006 + 218448 \beta ) q^{13} + ( 8470728 + 117649 \beta ) q^{14} + ( -19158120 - 539460 \beta ) q^{15} + ( -12533504 - 1368576 \beta ) q^{16} + ( 8321544 + 2589212 \beta ) q^{17} + ( 38263752 + 531441 \beta ) q^{18} + ( -105597652 - 4963752 \beta ) q^{19} + ( -233389440 - 3298880 \beta ) q^{20} + 85766121 q^{21} + ( -530316336 - 11060478 \beta ) q^{22} + ( -97498926 + 12654916 \beta ) q^{23} + ( -217510272 + 1108080 \beta ) q^{24} + ( 757890475 + 38894400 \beta ) q^{25} + ( -853562736 - 3262750 \beta ) q^{26} + 387420489 q^{27} + ( -77177744 + 16941456 \beta ) q^{28} + ( -1740821274 + 85588632 \beta ) q^{29} + ( -2648194560 - 57999240 \beta ) q^{30} + ( -4196898760 - 90054936 \beta ) q^{31} + ( -1677072384 - 123522816 \beta ) q^{32} + ( -3132365742 - 68482260 \beta ) q^{33} + ( 6688977792 + 194744808 \beta ) q^{34} + ( -3091815720 - 87060260 \beta ) q^{35} + ( -348625296 + 76527504 \beta ) q^{36} + ( -11168973358 + 143632656 \beta ) q^{37} + ( -19277775648 - 462987796 \beta ) q^{38} + ( -13844443374 + 159248592 \beta ) q^{39} + ( 5195581440 + 180846720 \beta ) q^{40} + ( -6253701660 + 505089780 \beta ) q^{41} + ( 6175160712 + 85766121 \beta ) q^{42} + ( 13031373164 - 379718208 \beta ) q^{43} + ( -28997651232 - 557114272 \beta ) q^{44} + ( -13966269480 - 393266340 \beta ) q^{45} + ( 22744439760 + 813655026 \beta ) q^{46} + ( 14611329468 - 1620335768 \beta ) q^{47} + ( -9136924416 - 997691904 \beta ) q^{48} + 13841287201 q^{49} + ( 146047743000 + 3558287275 \beta ) q^{50} + ( 6066405576 + 1887535548 \beta ) q^{51} + ( 86443816160 - 2878006752 \beta ) q^{52} + ( 261955934154 + 97048544 \beta ) q^{53} + ( 27894275208 + 387420489 \beta ) q^{54} + ( 276420542640 + 5648373720 \beta ) q^{55} + ( -35102696832 + 178826480 \beta ) q^{56} + ( -76980688308 - 3618575208 \beta ) q^{57} + ( 75965330736 + 4421560230 \beta ) q^{58} + ( -47222290512 + 432706712 \beta ) q^{59} + ( -170140901760 - 2404883520 \beta ) q^{60} + ( 230860277126 - 4959846216 \beta ) q^{61} + ( -513985920192 - 10680854152 \beta ) q^{62} + 62523502209 q^{63} + ( -308600410112 + 640659456 \beta ) q^{64} + ( 118879262640 + 8312531000 \beta ) q^{65} + ( -386600608944 - 8063088462 \beta ) q^{66} + ( -618939971032 - 2390900328 \beta ) q^{67} + ( 871476100992 - 500220736 \beta ) q^{68} + ( -71076717054 + 9225433764 \beta ) q^{69} + ( -427376463360 - 9360154440 \beta ) q^{70} + ( 980563309902 + 1476015420 \beta ) q^{71} + ( -158564988288 + 807790320 \beta ) q^{72} + ( -1203895891642 + 4871692872 \beta ) q^{73} + ( -466342074864 - 827422126 \beta ) q^{74} + ( 552502156275 + 28354017600 \beta ) q^{75} + ( -1611891177664 - 11949840576 \beta ) q^{76} + ( -505513987902 - 11051947060 \beta ) q^{77} + ( -622247234544 - 2378544750 \beta ) q^{78} + ( -1311247111924 - 3091143384 \beta ) q^{79} + ( 2711359641600 + 45240970240 \beta ) q^{80} + 282429536481 q^{81} + ( 737704643040 + 30112762500 \beta ) q^{82} + ( 2244780907956 - 33572095344 \beta ) q^{83} + ( -56262575376 + 12350321424 \beta ) q^{84} + ( -4725161878080 - 74202433920 \beta ) q^{85} + ( 45161642592 - 14308337812 \beta ) q^{86} + ( -1269058708746 + 62394112728 \beta ) q^{87} + ( 946187768064 + 21497556960 \beta ) q^{88} + ( 260463920292 + 10542875524 \beta ) q^{89} + ( -1930533834240 - 42281445960 \beta ) q^{90} + ( -2234272864894 + 25700188752 \beta ) q^{91} + ( 4350027485664 - 22341470240 \beta ) q^{92} + ( -3059539196040 - 65650048344 \beta ) q^{93} + ( -2759014004640 - 102052845828 \beta ) q^{94} + ( 11414417375520 + 208589665040 \beta ) q^{95} + ( -1222585767936 - 90048132864 \beta ) q^{96} + ( -6513978315802 + 3400065144 \beta ) q^{97} + ( 996572678472 + 13841287201 \beta ) q^{98} + ( -2283494625918 - 49923567540 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 144q^{2} + 1458q^{3} - 1312q^{4} - 52560q^{5} + 104976q^{6} + 235298q^{7} - 596736q^{8} + 1062882q^{9} + O(q^{10}) \) \( 2q + 144q^{2} + 1458q^{3} - 1312q^{4} - 52560q^{5} + 104976q^{6} + 235298q^{7} - 596736q^{8} + 1062882q^{9} - 7265280q^{10} - 8593596q^{11} - 956448q^{12} - 37982012q^{13} + 16941456q^{14} - 38316240q^{15} - 25067008q^{16} + 16643088q^{17} + 76527504q^{18} - 211195304q^{19} - 466778880q^{20} + 171532242q^{21} - 1060632672q^{22} - 194997852q^{23} - 435020544q^{24} + 1515780950q^{25} - 1707125472q^{26} + 774840978q^{27} - 154355488q^{28} - 3481642548q^{29} - 5296389120q^{30} - 8393797520q^{31} - 3354144768q^{32} - 6264731484q^{33} + 13377955584q^{34} - 6183631440q^{35} - 697250592q^{36} - 22337946716q^{37} - 38555551296q^{38} - 27688886748q^{39} + 10391162880q^{40} - 12507403320q^{41} + 12350321424q^{42} + 26062746328q^{43} - 57995302464q^{44} - 27932538960q^{45} + 45488879520q^{46} + 29222658936q^{47} - 18273848832q^{48} + 27682574402q^{49} + 292095486000q^{50} + 12132811152q^{51} + 172887632320q^{52} + 523911868308q^{53} + 55788550416q^{54} + 552841085280q^{55} - 70205393664q^{56} - 153961376616q^{57} + 151930661472q^{58} - 94444581024q^{59} - 340281803520q^{60} + 461720554252q^{61} - 1027971840384q^{62} + 125047004418q^{63} - 617200820224q^{64} + 237758525280q^{65} - 773201217888q^{66} - 1237879942064q^{67} + 1742952201984q^{68} - 142153434108q^{69} - 854752926720q^{70} + 1961126619804q^{71} - 317129976576q^{72} - 2407791783284q^{73} - 932684149728q^{74} + 1105004312550q^{75} - 3223782355328q^{76} - 1011027975804q^{77} - 1244494469088q^{78} - 2622494223848q^{79} + 5422719283200q^{80} + 564859072962q^{81} + 1475409286080q^{82} + 4489561815912q^{83} - 112525150752q^{84} - 9450323756160q^{85} + 90323285184q^{86} - 2538117417492q^{87} + 1892375536128q^{88} + 520927840584q^{89} - 3861067668480q^{90} - 4468545729788q^{91} + 8700054971328q^{92} - 6119078392080q^{93} - 5518028009280q^{94} + 22828834751040q^{95} - 2445171535872q^{96} - 13027956631604q^{97} + 1993145356944q^{98} - 4566989251836q^{99} + O(q^{100}) \)

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
−1.73205
1.73205
23.5026 729.000 −7639.63 9608.09 17133.4 117649. −372084. 531441. 225815.
1.2 120.497 729.000 6327.63 −62168.1 87842.6 117649. −224652. 531441. −7.49109e6
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)

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 21.14.a.a 2
3.b odd 2 1 63.14.a.a 2
7.b odd 2 1 147.14.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.14.a.a 2 1.a even 1 1 trivial
63.14.a.a 2 3.b odd 2 1
147.14.a.c 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 144 T_{2} + 2832 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(21))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2832 - 144 T + T^{2} \)
$3$ \( ( -729 + T )^{2} \)
$5$ \( -597316800 + 52560 T + T^{2} \)
$7$ \( ( -117649 + T )^{2} \)
$11$ \( -2293276854396 + 8593596 T + T^{2} \)
$13$ \( 248421977380228 + 37982012 T + T^{2} \)
$17$ \( -15698604078236352 - 16643088 T + T^{2} \)
$19$ \( -46799673266056304 + 211195304 T + T^{2} \)
$23$ \( -367159465799362236 + 194997852 T + T^{2} \)
$29$ \( -14198914849778126172 + 3481642548 T + T^{2} \)
$31$ \( -1460505601522016192 + 8393797520 T + T^{2} \)
$37$ \( 76223406498380877892 + 22337946716 T + T^{2} \)
$41$ \( -\)\(56\!\cdots\!00\)\( + 12507403320 T + T^{2} \)
$43$ \( -\)\(16\!\cdots\!32\)\( - 26062746328 T + T^{2} \)
$47$ \( -\)\(59\!\cdots\!24\)\( - 29222658936 T + T^{2} \)
$53$ \( \)\(68\!\cdots\!44\)\( - 523911868308 T + T^{2} \)
$59$ \( \)\(17\!\cdots\!56\)\( + 94444581024 T + T^{2} \)
$61$ \( -\)\(45\!\cdots\!36\)\( - 461720554252 T + T^{2} \)
$67$ \( \)\(36\!\cdots\!56\)\( + 1237879942064 T + T^{2} \)
$71$ \( \)\(95\!\cdots\!04\)\( - 1961126619804 T + T^{2} \)
$73$ \( \)\(13\!\cdots\!96\)\( + 2407791783284 T + T^{2} \)
$79$ \( \)\(16\!\cdots\!64\)\( + 2622494223848 T + T^{2} \)
$83$ \( \)\(23\!\cdots\!64\)\( - 4489561815912 T + T^{2} \)
$89$ \( -\)\(19\!\cdots\!88\)\( - 520927840584 T + T^{2} \)
$97$ \( \)\(42\!\cdots\!32\)\( + 13027956631604 T + T^{2} \)
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