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: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2.36.a.b 1
3.b odd 2 1 18.36.a.b 1
4.b odd 2 1 16.36.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.36.a.b 1 1.a even 1 1 trivial
16.36.a.a 1 4.b odd 2 1
18.36.a.b 1 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 131072 T \)
$3$ \( 1 - 159933852 T + 50031545098999707 T^{2} \)
$5$ \( 1 + 2838742578690 T + \)\(29\!\cdots\!25\)\( T^{2} \)
$7$ \( 1 + 782281866962344 T + \)\(37\!\cdots\!43\)\( T^{2} \)
$11$ \( 1 - 738502081164310452 T + \)\(28\!\cdots\!51\)\( T^{2} \)
$13$ \( 1 - 12249951000076215062 T + \)\(97\!\cdots\!57\)\( T^{2} \)
$17$ \( 1 + \)\(58\!\cdots\!14\)\( T + \)\(11\!\cdots\!93\)\( T^{2} \)
$19$ \( 1 + \)\(10\!\cdots\!20\)\( T + \)\(57\!\cdots\!99\)\( T^{2} \)
$23$ \( 1 - \)\(48\!\cdots\!72\)\( T + \)\(45\!\cdots\!07\)\( T^{2} \)
$29$ \( 1 - \)\(57\!\cdots\!70\)\( T + \)\(15\!\cdots\!49\)\( T^{2} \)
$31$ \( 1 + \)\(19\!\cdots\!48\)\( T + \)\(15\!\cdots\!51\)\( T^{2} \)
$37$ \( 1 - \)\(30\!\cdots\!46\)\( T + \)\(77\!\cdots\!93\)\( T^{2} \)
$41$ \( 1 + \)\(11\!\cdots\!98\)\( T + \)\(28\!\cdots\!01\)\( T^{2} \)
$43$ \( 1 + \)\(15\!\cdots\!08\)\( T + \)\(14\!\cdots\!07\)\( T^{2} \)
$47$ \( 1 + \)\(13\!\cdots\!24\)\( T + \)\(33\!\cdots\!43\)\( T^{2} \)
$53$ \( 1 + \)\(16\!\cdots\!98\)\( T + \)\(22\!\cdots\!57\)\( T^{2} \)
$59$ \( 1 - \)\(18\!\cdots\!40\)\( T + \)\(95\!\cdots\!99\)\( T^{2} \)
$61$ \( 1 - \)\(91\!\cdots\!02\)\( T + \)\(30\!\cdots\!01\)\( T^{2} \)
$67$ \( 1 + \)\(10\!\cdots\!64\)\( T + \)\(81\!\cdots\!43\)\( T^{2} \)
$71$ \( 1 + \)\(82\!\cdots\!48\)\( T + \)\(62\!\cdots\!51\)\( T^{2} \)
$73$ \( 1 + \)\(24\!\cdots\!78\)\( T + \)\(16\!\cdots\!57\)\( T^{2} \)
$79$ \( 1 + \)\(51\!\cdots\!80\)\( T + \)\(26\!\cdots\!99\)\( T^{2} \)
$83$ \( 1 + \)\(48\!\cdots\!68\)\( T + \)\(14\!\cdots\!07\)\( T^{2} \)
$89$ \( 1 + \)\(12\!\cdots\!90\)\( T + \)\(16\!\cdots\!49\)\( T^{2} \)
$97$ \( 1 + \)\(65\!\cdots\!74\)\( T + \)\(34\!\cdots\!93\)\( T^{2} \)
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