Properties

Label 2.36.a.b
Level 2
Weight 36
Character orbit 2.a
Self dual Yes
Analytic conductor 15.519
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 36 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(15.5190261267\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 131072q^{2} \) \(\mathstrut +\mathstrut 159933852q^{3} \) \(\mathstrut +\mathstrut 17179869184q^{4} \) \(\mathstrut -\mathstrut 2838742578690q^{5} \) \(\mathstrut +\mathstrut 20962849849344q^{6} \) \(\mathstrut -\mathstrut 782281866962344q^{7} \) \(\mathstrut +\mathstrut 2251799813685248q^{8} \) \(\mathstrut -\mathstrut 24452708083441803q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 131072q^{2} \) \(\mathstrut +\mathstrut 159933852q^{3} \) \(\mathstrut +\mathstrut 17179869184q^{4} \) \(\mathstrut -\mathstrut 2838742578690q^{5} \) \(\mathstrut +\mathstrut 20962849849344q^{6} \) \(\mathstrut -\mathstrut 782281866962344q^{7} \) \(\mathstrut +\mathstrut 2251799813685248q^{8} \) \(\mathstrut -\mathstrut 24452708083441803q^{9} \) \(\mathstrut -\mathstrut 372079667274055680q^{10} \) \(\mathstrut +\mathstrut 738502081164310452q^{11} \) \(\mathstrut +\mathstrut 2747642655453216768q^{12} \) \(\mathstrut +\mathstrut 12249951000076215062q^{13} \) \(\mathstrut -\mathstrut 102535248866488352768q^{14} \) \(\mathstrut -\mathstrut 454011035446304813880q^{15} \) \(\mathstrut +\mathstrut 295147905179352825856q^{16} \) \(\mathstrut -\mathstrut 5840692944055083371214q^{17} \) \(\mathstrut -\mathstrut 3205065353912884002816q^{18} \) \(\mathstrut -\mathstrut 10098306774778877636020q^{19} \) \(\mathstrut -\mathstrut 48769226148945026088960q^{20} \) \(\mathstrut -\mathstrut 125113352333039214869088q^{21} \) \(\mathstrut +\mathstrut 96796944782368499564544q^{22} \) \(\mathstrut +\mathstrut 489049972679814652084872q^{23} \) \(\mathstrut +\mathstrut 360139018135564028215296q^{24} \) \(\mathstrut +\mathstrut 5148076382394180480787975q^{25} \) \(\mathstrut +\mathstrut 1605625577481989660606464q^{26} \) \(\mathstrut -\mathstrut 11912553524811129458996520q^{27} \) \(\mathstrut -\mathstrut 13439500139428361374007296q^{28} \) \(\mathstrut +\mathstrut 57680681810625881354441670q^{29} \) \(\mathstrut -\mathstrut 59508134438018064564879360q^{30} \) \(\mathstrut -\mathstrut 192021129092237034631174048q^{31} \) \(\mathstrut +\mathstrut 38685626227668133590597632q^{32} \) \(\mathstrut +\mathstrut 118111482550624815512221104q^{33} \) \(\mathstrut -\mathstrut 765551305563187887631761408q^{34} \) \(\mathstrut +\mathstrut 2220696844283111923686849360q^{35} \) \(\mathstrut -\mathstrut 420094326068069532017098752q^{36} \) \(\mathstrut +\mathstrut 3010234126164449766865937246q^{37} \) \(\mathstrut -\mathstrut 1323605265583817049508413440q^{38} \) \(\mathstrut +\mathstrut 1959181850253441368446078824q^{39} \) \(\mathstrut -\mathstrut 6392280009794522459532165120q^{40} \) \(\mathstrut -\mathstrut 11779659443130950895302457798q^{41} \) \(\mathstrut -\mathstrut 16398857316996115971321102336q^{42} \) \(\mathstrut -\mathstrut 15049344277571370072567952108q^{43} \) \(\mathstrut +\mathstrut 12687369146514603974923911168q^{44} \) \(\mathstrut +\mathstrut 69414943600743391538762978070q^{45} \) \(\mathstrut +\mathstrut 64100758019088666078068342784q^{46} \) \(\mathstrut -\mathstrut 139394260294904275663821581424q^{47} \) \(\mathstrut +\mathstrut 47204141385064648306235277312q^{48} \) \(\mathstrut +\mathstrut 233146227112425695351196348393q^{49} \) \(\mathstrut +\mathstrut 674768667593170023977841459200q^{50} \) \(\mathstrut -\mathstrut 934124520891949983739400936328q^{51} \) \(\mathstrut +\mathstrut 210452555691719348795010449408q^{52} \) \(\mathstrut -\mathstrut 164137655627446071138117792498q^{53} \) \(\mathstrut -\mathstrut 1561402215604044360449591869440q^{54} \) \(\mathstrut -\mathstrut 2096417302252306330106199467880q^{55} \) \(\mathstrut -\mathstrut 1761542162275154182013884301312q^{56} \) \(\mathstrut -\mathstrut 1615061101168082348565332549040q^{57} \) \(\mathstrut +\mathstrut 7560322326282355520889378570240q^{58} \) \(\mathstrut +\mathstrut 18459731811236503456378557561540q^{59} \) \(\mathstrut -\mathstrut 7799850197059903758647867473920q^{60} \) \(\mathstrut +\mathstrut 9158848164859872108837051473702q^{61} \) \(\mathstrut -\mathstrut 25168593432377692603177244819456q^{62} \) \(\mathstrut +\mathstrut 19128910131800054261096116466232q^{63} \) \(\mathstrut +\mathstrut 5070602400912917605986812821504q^{64} \) \(\mathstrut -\mathstrut 34774457490782499131636898228780q^{65} \) \(\mathstrut +\mathstrut 15481108240875495818817844543488q^{66} \) \(\mathstrut -\mathstrut 104341680355514057403064622201764q^{67} \) \(\mathstrut -\mathstrut 100342340722778162807670231269376q^{68} \) \(\mathstrut +\mathstrut 78215645951177519953973409886944q^{69} \) \(\mathstrut +\mathstrut 291071176773876046061482719313920q^{70} \) \(\mathstrut -\mathstrut 82427403953674268345461759625448q^{71} \) \(\mathstrut -\mathstrut 55062603506394009700545167622144q^{72} \) \(\mathstrut -\mathstrut 241533154698402560229433865618278q^{73} \) \(\mathstrut +\mathstrut 394557407384626759842652126707712q^{74} \) \(\mathstrut +\mathstrut 823351686226526266675632837029700q^{75} \) \(\mathstrut -\mathstrut 173487589370602068313166766407680q^{76} \) \(\mathstrut -\mathstrut 577716786808793279789850609619488q^{77} \) \(\mathstrut +\mathstrut 256793883476419067044964443619328q^{78} \) \(\mathstrut -\mathstrut 514148341183349034852628168280080q^{79} \) \(\mathstrut -\mathstrut 837848925443787647815799946608640q^{80} \) \(\mathstrut -\mathstrut 681813805109828249555084449643319q^{81} \) \(\mathstrut -\mathstrut 1543983522530059995749083748499456q^{82} \) \(\mathstrut -\mathstrut 4803384713083381637598870398273268q^{83} \) \(\mathstrut -\mathstrut 2149431026253314912592999525384192q^{84} \) \(\mathstrut +\mathstrut 16580223749343415274602968875829660q^{85} \) \(\mathstrut -\mathstrut 1972547653149834618151626618699776q^{86} \) \(\mathstrut +\mathstrut 9225093627959731735910833592412840q^{87} \) \(\mathstrut +\mathstrut 1662958848771962172201226884612096q^{88} \) \(\mathstrut -\mathstrut 12030481278409119291871425987284790q^{89} \) \(\mathstrut +\mathstrut 9098355487636637815768741061591040q^{90} \) \(\mathstrut -\mathstrut 9582914538536854506136010799625328q^{91} \) \(\mathstrut +\mathstrut 8401814555077989640184573825384448q^{92} \) \(\mathstrut -\mathstrut 30710678841110732245621064779072896q^{93} \) \(\mathstrut -\mathstrut 18270684485373693219808422320406528q^{94} \) \(\mathstrut +\mathstrut 28666493414238488155019386028413800q^{95} \) \(\mathstrut +\mathstrut 6187141219623193582794870267838464q^{96} \) \(\mathstrut -\mathstrut 65828814699540890606091570657982174q^{97} \) \(\mathstrut +\mathstrut 30558942280079860741072007776567296q^{98} \) \(\mathstrut -\mathstrut 18058375809725128675706419366624956q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
0
131072. 1.59934e8 1.71799e10 −2.83874e12 2.09628e13 −7.82282e14 2.25180e15 −2.44527e16 −3.72080e17
\(n\): 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
\(2\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3} \) \(\mathstrut -\mathstrut 159933852 \) acting on \(S_{36}^{\mathrm{new}}(\Gamma_0(2))\).