Properties

Label 3.22.a.c
Level 3
Weight 22
Character orbit 3.a
Self dual Yes
Analytic conductor 8.384
Analytic rank 0
Dimension 2
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 22 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(8.38432032861\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{649}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3^{2}\cdot 7 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 63\sqrt{649}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 333 - \beta ) q^{2} \) \( + 59049 q^{3} \) \( + ( 589618 - 666 \beta ) q^{4} \) \( + ( 498438 - 13568 \beta ) q^{5} \) \( + ( 19663317 - 59049 \beta ) q^{6} \) \( + ( 339948056 + 182016 \beta ) q^{7} \) \( + ( 1213527924 + 1285756 \beta ) q^{8} \) \( + 3486784401 q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 333 - \beta ) q^{2} \) \( + 59049 q^{3} \) \( + ( 589618 - 666 \beta ) q^{4} \) \( + ( 498438 - 13568 \beta ) q^{5} \) \( + ( 19663317 - 59049 \beta ) q^{6} \) \( + ( 339948056 + 182016 \beta ) q^{7} \) \( + ( 1213527924 + 1285756 \beta ) q^{8} \) \( + 3486784401 q^{9} \) \( + ( 35115533262 - 5016582 \beta ) q^{10} \) \( + ( 109934561484 - 31263232 \beta ) q^{11} \) \( + ( 34816353282 - 39326634 \beta ) q^{12} \) \( + ( -24234454978 + 475236864 \beta ) q^{13} \) \( + ( -355648853448 - 279336728 \beta ) q^{14} \) \( + ( 29432265462 - 801176832 \beta ) q^{15} \) \( + ( -4144368220280 + 611332056 \beta ) q^{16} \) \( + ( -5666764520718 - 1677304320 \beta ) q^{17} \) \( + ( 1161099205533 - 3486784401 \beta ) q^{18} \) \( + ( 5980292505812 + 22384332288 \beta ) q^{19} \) \( + ( 23570290586412 - 8331896732 \beta ) q^{20} \) \( + ( 20073592758744 + 10747862784 \beta ) q^{21} \) \( + ( 117138574281564 - 120345217740 \beta ) q^{22} \) \( + ( -73254195031752 + 73613782528 \beta ) q^{23} \) \( + ( 71657610384276 + 75922606044 \beta ) q^{24} \) \( + ( -2393177123537 - 13525613568 \beta ) q^{25} \) \( + ( -1232223681984858 + 182488330690 \beta ) q^{26} \) \( + 205891132094649 q^{27} \) \( + ( -111815643477328 - 119085495408 \beta ) q^{28} \) \( + ( -899260021837026 - 1163540203264 \beta ) q^{29} \) \( + ( 2073537123587838 - 296224150518 \beta ) q^{30} \) \( + ( 5584553763472496 + 1732114342656 \beta ) q^{31} \) \( + ( -5499745757967024 + 1651516028016 \beta ) q^{32} \) \( + ( 6491525921068716 - 1846062586368 \beta ) q^{33} \) \( + ( 2433503743706826 + 5108222182158 \beta ) q^{34} \) \( + ( -6191934883974000 - 4521691532800 \beta ) q^{35} \) \( + ( 2055870844948818 - 2322198411066 \beta ) q^{36} \) \( + ( 6368132429330006 - 15038204442624 \beta ) q^{37} \) \( + ( -55667938833910332 + 1473690146092 \beta ) q^{38} \) \( + ( -1431020331995922 + 28062261582336 \beta ) q^{39} \) \( + ( -44331729560273736 - 15824277223704 \beta ) q^{40} \) \( + ( 61486010308234026 + 52192512395776 \beta ) q^{41} \) \( + ( -21000709147250952 - 16494554451672 \beta ) q^{42} \) \( + ( 144227709081135020 - 12139771405824 \beta ) q^{43} \) \( + ( 118452619567796184 - 91649782273720 \beta ) q^{44} \) \( + ( 1737945843265638 - 47308690752768 \beta ) q^{45} \) \( + ( -214013990697580584 + 97767584613576 \beta ) q^{46} \) \( + ( 418621872870798480 + 6195754260992 \beta ) q^{47} \) \( + ( -244720799039313720 + 36098546574744 \beta ) q^{48} \) \( + ( -357642698470735335 + 123751970721792 \beta ) q^{49} \) \( + ( 34043443021015587 - 2110852194607 \beta ) q^{50} \) \( + ( -334616778183877182 - 99043142791680 \beta ) q^{51} \) \( + ( -829575374121022948 + 296348356293300 \beta ) q^{52} \) \( + ( -21503982006387882 - 283816914561792 \beta ) q^{53} \) \( + ( 68561746987518117 - 205891132094649 \beta ) q^{54} \) \( + ( 1147431559447656648 - 1507174913046528 \beta ) q^{55} \) \( + ( 1015365160025284320 + 657971751305120 \beta ) q^{56} \) \( + ( 353130292175692788 + 1321772437274112 \beta ) q^{57} \) \( + ( 2697687515052145926 + 511801134150114 \beta ) q^{58} \) \( + ( -1761911665451928612 + 2106913298778112 \beta ) q^{59} \) \( + ( 1391802088837042188 - 491990170127868 \beta ) q^{60} \) \( + ( -889511564225506930 - 1794014554180608 \beta ) q^{61} \) \( + ( -2602064021838738768 - 5007759687368048 \beta ) q^{62} \) \( + ( 1185325578811074456 + 634650549532416 \beta ) q^{63} \) \( + ( 2605846006731741472 + 4767644351391840 \beta ) q^{64} \) \( + ( -16621395533088756876 + 565689197159936 \beta ) q^{65} \) \( + ( 6916915672752072636 - 7106264762329260 \beta ) q^{66} \) \( + ( -8227034333810805148 + 3370226524784640 \beta ) q^{67} \) \( + ( -463749167992163004 + 2785096352248428 \beta ) q^{68} \) \( + ( -4325586962429923848 + 4346820244495872 \beta ) q^{69} \) \( + ( 9585424990837054800 + 4686211603551600 \beta ) q^{70} \) \( + ( 8689613565575210472 + 830682487764480 \beta ) q^{71} \) \( + ( 4231310235581113524 + 4483153964292156 \beta ) q^{72} \) \( + ( 25445573134236994538 - 26617229816020992 \beta ) q^{73} \) \( + ( 40857213196837643742 - 11375854508723798 \beta ) q^{74} \) \( + ( -141314715967736313 - 798673955576832 \beta ) q^{75} \) \( + ( -34875056468046395032 + 9215330426115192 \beta ) q^{76} \) \( + ( 22714225491908012832 + 9381974200394752 \beta ) q^{77} \) \( + ( -72761576197523880042 + 10775753438913810 \beta ) q^{78} \) \( + ( -27027892797095295520 - 3613605017944320 \beta ) q^{79} \) \( + ( -23431492948174369488 + 56535499140087568 \beta ) q^{80} \) \( + 12157665459056928801 q^{81} \) \( + ( -113966859589901947998 - 44105903680440618 \beta ) q^{82} \) \( + ( 55554214638833338644 - 34423999299854848 \beta ) q^{83} \) \( + ( -6602601931692741072 - 7031879418346992 \beta ) q^{84} \) \( + ( 55796506139131484076 + 76050628806449664 \beta ) q^{85} \) \( + ( 79298433632623292604 - 148270252959274412 \beta ) q^{86} \) \( + ( -53100405029454548274 - 68705885462535936 \beta ) q^{87} \) \( + ( 29866259797357770864 + 103410217008931536 \beta ) q^{88} \) \( + ( 113460383982724032762 + 75474030898664448 \beta ) q^{89} \) \( + ( 122440293610738246062 - 17491739863937382 \beta ) q^{90} \) \( + ( 214577087342592500176 + 157144789499060736 \beta ) q^{91} \) \( + ( -169479140905068324624 + 92191305117741136 \beta ) q^{92} \) \( + ( 329762315179287416304 + 102279619819494144 \beta ) q^{93} \) \( + ( 123441557984417559888 - 416558686701888144 \beta ) q^{94} \) \( + ( -779341611765862915848 - 69983406901891072 \beta ) q^{95} \) \( + ( -324754487262194800176 + 97520369938316784 \beta ) q^{96} \) \( + ( -64165734626397873118 - 328931501848427520 \beta ) q^{97} \) \( + ( -437865368685575165307 + 398852104721092071 \beta ) q^{98} \) \( + ( 383318114113186611084 - 109008149662444032 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 666q^{2} \) \(\mathstrut +\mathstrut 118098q^{3} \) \(\mathstrut +\mathstrut 1179236q^{4} \) \(\mathstrut +\mathstrut 996876q^{5} \) \(\mathstrut +\mathstrut 39326634q^{6} \) \(\mathstrut +\mathstrut 679896112q^{7} \) \(\mathstrut +\mathstrut 2427055848q^{8} \) \(\mathstrut +\mathstrut 6973568802q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 666q^{2} \) \(\mathstrut +\mathstrut 118098q^{3} \) \(\mathstrut +\mathstrut 1179236q^{4} \) \(\mathstrut +\mathstrut 996876q^{5} \) \(\mathstrut +\mathstrut 39326634q^{6} \) \(\mathstrut +\mathstrut 679896112q^{7} \) \(\mathstrut +\mathstrut 2427055848q^{8} \) \(\mathstrut +\mathstrut 6973568802q^{9} \) \(\mathstrut +\mathstrut 70231066524q^{10} \) \(\mathstrut +\mathstrut 219869122968q^{11} \) \(\mathstrut +\mathstrut 69632706564q^{12} \) \(\mathstrut -\mathstrut 48468909956q^{13} \) \(\mathstrut -\mathstrut 711297706896q^{14} \) \(\mathstrut +\mathstrut 58864530924q^{15} \) \(\mathstrut -\mathstrut 8288736440560q^{16} \) \(\mathstrut -\mathstrut 11333529041436q^{17} \) \(\mathstrut +\mathstrut 2322198411066q^{18} \) \(\mathstrut +\mathstrut 11960585011624q^{19} \) \(\mathstrut +\mathstrut 47140581172824q^{20} \) \(\mathstrut +\mathstrut 40147185517488q^{21} \) \(\mathstrut +\mathstrut 234277148563128q^{22} \) \(\mathstrut -\mathstrut 146508390063504q^{23} \) \(\mathstrut +\mathstrut 143315220768552q^{24} \) \(\mathstrut -\mathstrut 4786354247074q^{25} \) \(\mathstrut -\mathstrut 2464447363969716q^{26} \) \(\mathstrut +\mathstrut 411782264189298q^{27} \) \(\mathstrut -\mathstrut 223631286954656q^{28} \) \(\mathstrut -\mathstrut 1798520043674052q^{29} \) \(\mathstrut +\mathstrut 4147074247175676q^{30} \) \(\mathstrut +\mathstrut 11169107526944992q^{31} \) \(\mathstrut -\mathstrut 10999491515934048q^{32} \) \(\mathstrut +\mathstrut 12983051842137432q^{33} \) \(\mathstrut +\mathstrut 4867007487413652q^{34} \) \(\mathstrut -\mathstrut 12383869767948000q^{35} \) \(\mathstrut +\mathstrut 4111741689897636q^{36} \) \(\mathstrut +\mathstrut 12736264858660012q^{37} \) \(\mathstrut -\mathstrut 111335877667820664q^{38} \) \(\mathstrut -\mathstrut 2862040663991844q^{39} \) \(\mathstrut -\mathstrut 88663459120547472q^{40} \) \(\mathstrut +\mathstrut 122972020616468052q^{41} \) \(\mathstrut -\mathstrut 42001418294501904q^{42} \) \(\mathstrut +\mathstrut 288455418162270040q^{43} \) \(\mathstrut +\mathstrut 236905239135592368q^{44} \) \(\mathstrut +\mathstrut 3475891686531276q^{45} \) \(\mathstrut -\mathstrut 428027981395161168q^{46} \) \(\mathstrut +\mathstrut 837243745741596960q^{47} \) \(\mathstrut -\mathstrut 489441598078627440q^{48} \) \(\mathstrut -\mathstrut 715285396941470670q^{49} \) \(\mathstrut +\mathstrut 68086886042031174q^{50} \) \(\mathstrut -\mathstrut 669233556367754364q^{51} \) \(\mathstrut -\mathstrut 1659150748242045896q^{52} \) \(\mathstrut -\mathstrut 43007964012775764q^{53} \) \(\mathstrut +\mathstrut 137123493975036234q^{54} \) \(\mathstrut +\mathstrut 2294863118895313296q^{55} \) \(\mathstrut +\mathstrut 2030730320050568640q^{56} \) \(\mathstrut +\mathstrut 706260584351385576q^{57} \) \(\mathstrut +\mathstrut 5395375030104291852q^{58} \) \(\mathstrut -\mathstrut 3523823330903857224q^{59} \) \(\mathstrut +\mathstrut 2783604177674084376q^{60} \) \(\mathstrut -\mathstrut 1779023128451013860q^{61} \) \(\mathstrut -\mathstrut 5204128043677477536q^{62} \) \(\mathstrut +\mathstrut 2370651157622148912q^{63} \) \(\mathstrut +\mathstrut 5211692013463482944q^{64} \) \(\mathstrut -\mathstrut 33242791066177513752q^{65} \) \(\mathstrut +\mathstrut 13833831345504145272q^{66} \) \(\mathstrut -\mathstrut 16454068667621610296q^{67} \) \(\mathstrut -\mathstrut 927498335984326008q^{68} \) \(\mathstrut -\mathstrut 8651173924859847696q^{69} \) \(\mathstrut +\mathstrut 19170849981674109600q^{70} \) \(\mathstrut +\mathstrut 17379227131150420944q^{71} \) \(\mathstrut +\mathstrut 8462620471162227048q^{72} \) \(\mathstrut +\mathstrut 50891146268473989076q^{73} \) \(\mathstrut +\mathstrut 81714426393675287484q^{74} \) \(\mathstrut -\mathstrut 282629431935472626q^{75} \) \(\mathstrut -\mathstrut 69750112936092790064q^{76} \) \(\mathstrut +\mathstrut 45428450983816025664q^{77} \) \(\mathstrut -\mathstrut 145523152395047760084q^{78} \) \(\mathstrut -\mathstrut 54055785594190591040q^{79} \) \(\mathstrut -\mathstrut 46862985896348738976q^{80} \) \(\mathstrut +\mathstrut 24315330918113857602q^{81} \) \(\mathstrut -\mathstrut 227933719179803895996q^{82} \) \(\mathstrut +\mathstrut 111108429277666677288q^{83} \) \(\mathstrut -\mathstrut 13205203863385482144q^{84} \) \(\mathstrut +\mathstrut 111593012278262968152q^{85} \) \(\mathstrut +\mathstrut 158596867265246585208q^{86} \) \(\mathstrut -\mathstrut 106200810058909096548q^{87} \) \(\mathstrut +\mathstrut 59732519594715541728q^{88} \) \(\mathstrut +\mathstrut 226920767965448065524q^{89} \) \(\mathstrut +\mathstrut 244880587221476492124q^{90} \) \(\mathstrut +\mathstrut 429154174685185000352q^{91} \) \(\mathstrut -\mathstrut 338958281810136649248q^{92} \) \(\mathstrut +\mathstrut 659524630358574832608q^{93} \) \(\mathstrut +\mathstrut 246883115968835119776q^{94} \) \(\mathstrut -\mathstrut 1558683223531725831696q^{95} \) \(\mathstrut -\mathstrut 649508974524389600352q^{96} \) \(\mathstrut -\mathstrut 128331469252795746236q^{97} \) \(\mathstrut -\mathstrut 875730737371150330614q^{98} \) \(\mathstrut +\mathstrut 766636228226373222168q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
13.2377
−12.2377
−1271.96 59049.0 −479282. −2.12776e7 −7.51077e7 6.32076e8 3.27711e9 3.48678e9 2.70641e10
1.2 1937.96 59049.0 1.65852e6 2.22745e7 1.14434e8 4.78205e7 −8.50053e8 3.48678e9 4.31669e10
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut -\mathstrut 666 T_{2} \) \(\mathstrut -\mathstrut 2464992 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(3))\).