Properties

Label 4.36.a.a
Level 4
Weight 36
Character orbit 4.a
Self dual Yes
Analytic conductor 31.038
Analytic rank 1
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 36 \)
Character orbit: \([\chi]\) = 4.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.0380522535\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{4}\cdot 5\cdot 7 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 16969628 + \beta_{1} ) q^{3} \) \( + ( 93573630 + 246 \beta_{1} - \beta_{2} ) q^{5} \) \( + ( -1849765429672 - 244638 \beta_{1} + 564 \beta_{2} ) q^{7} \) \( + ( -1371898039365771 - 77158812 \beta_{1} - 15318 \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \(+(16969628 + \beta_{1}) q^{3}\) \(+(93573630 + 246 \beta_{1} - \beta_{2}) q^{5}\) \(+(-1849765429672 - 244638 \beta_{1} + 564 \beta_{2}) q^{7}\) \(+(-1371898039365771 - 77158812 \beta_{1} - 15318 \beta_{2}) q^{9}\) \(+(-140154652956383820 - 8262293541 \beta_{1} + 17576 \beta_{2}) q^{11}\) \(+(-2456900438471578858 - 145960301034 \beta_{1} + 5745423 \beta_{2}) q^{13}\) \(+(11912600874497874120 + 750084516054 \beta_{1} - 131835924 \beta_{2}) q^{15}\) \(+(\)\(20\!\cdots\!86\)\( + 12417058928292 \beta_{1} + 1633047370 \beta_{2}) q^{17}\) \(+(-\)\(11\!\cdots\!16\)\( - 48219784860123 \beta_{1} - 13362439272 \beta_{2}) q^{19}\) \(+(-\)\(11\!\cdots\!24\)\( - 414877019931448 \beta_{1} + 75977545428 \beta_{2}) q^{21}\) \(+(-\)\(32\!\cdots\!24\)\( + 1845485459987046 \beta_{1} - 296037672404 \beta_{2}) q^{23}\) \(+(-\)\(46\!\cdots\!25\)\( + 5981927839966440 \beta_{1} + 717372157860 \beta_{2}) q^{25}\) \(+(-\)\(46\!\cdots\!24\)\( - 32297545853297046 \beta_{1} - 779822285112 \beta_{2}) q^{27}\) \(+(-\)\(24\!\cdots\!66\)\( - 56580793878865506 \beta_{1} + 857835418691 \beta_{2}) q^{29}\) \(+(-\)\(10\!\cdots\!44\)\( + 334440004372376040 \beta_{1} - 18744137713440 \beta_{2}) q^{31}\) \(+(-\)\(40\!\cdots\!80\)\( + 623973325316463756 \beta_{1} + 128812730285934 \beta_{2}) q^{33}\) \(+(-\)\(13\!\cdots\!80\)\( - 3453236753480364756 \beta_{1} - 390637255582064 \beta_{2}) q^{35}\) \(+(-\)\(34\!\cdots\!62\)\( - 1297334320812122562 \beta_{1} + 198623459819283 \beta_{2}) q^{37}\) \(+(-\)\(71\!\cdots\!16\)\( + 6840061094178693230 \beta_{1} + 2971622977394220 \beta_{2}) q^{39}\) \(+(-\)\(95\!\cdots\!34\)\( + 63123754486312617384 \beta_{1} - 11777845226841724 \beta_{2}) q^{41}\) \(+(-\)\(68\!\cdots\!00\)\( - \)\(17\!\cdots\!37\)\( \beta_{1} + 16341886387601712 \beta_{2}) q^{43}\) \(+(\)\(36\!\cdots\!70\)\( + 30929043727996135974 \beta_{1} + 21657827445373431 \beta_{2}) q^{45}\) \(+(\)\(14\!\cdots\!76\)\( - \)\(23\!\cdots\!60\)\( \beta_{1} - 133055352405484248 \beta_{2}) q^{47}\) \(+(\)\(40\!\cdots\!73\)\( + \)\(19\!\cdots\!56\)\( \beta_{1} + 212273841375916584 \beta_{2}) q^{49}\) \(+(\)\(60\!\cdots\!32\)\( - \)\(22\!\cdots\!74\)\( \beta_{1} + 18936105471182664 \beta_{2}) q^{51}\) \(+(\)\(69\!\cdots\!14\)\( - \)\(76\!\cdots\!18\)\( \beta_{1} - 627221916303572269 \beta_{2}) q^{53}\) \(+(-\)\(14\!\cdots\!00\)\( - \)\(62\!\cdots\!50\)\( \beta_{1} + 1076106883834559700 \beta_{2}) q^{55}\) \(+(-\)\(23\!\cdots\!08\)\( + \)\(13\!\cdots\!12\)\( \beta_{1} - 972666146017593198 \beta_{2}) q^{57}\) \(+(-\)\(72\!\cdots\!92\)\( + \)\(24\!\cdots\!59\)\( \beta_{1} + 582701521780060576 \beta_{2}) q^{59}\) \(+(-\)\(15\!\cdots\!18\)\( - \)\(41\!\cdots\!58\)\( \beta_{1} + 2399951807059995063 \beta_{2}) q^{61}\) \(+(-\)\(20\!\cdots\!28\)\( - \)\(19\!\cdots\!30\)\( \beta_{1} - 12132436053826620396 \beta_{2}) q^{63}\) \(+(-\)\(15\!\cdots\!20\)\( - \)\(14\!\cdots\!64\)\( \beta_{1} + 14938326422972709484 \beta_{2}) q^{65}\) \(+(\)\(25\!\cdots\!28\)\( + \)\(37\!\cdots\!93\)\( \beta_{1} + 22709891622220915608 \beta_{2}) q^{67}\) \(+(\)\(88\!\cdots\!72\)\( + \)\(22\!\cdots\!92\)\( \beta_{1} - 66182008910064631812 \beta_{2}) q^{69}\) \(+(\)\(19\!\cdots\!48\)\( + \)\(11\!\cdots\!62\)\( \beta_{1} - 29755918607961419132 \beta_{2}) q^{71}\) \(+(\)\(28\!\cdots\!22\)\( - \)\(98\!\cdots\!04\)\( \beta_{1} + \)\(21\!\cdots\!78\)\( \beta_{2}) q^{73}\) \(+(\)\(28\!\cdots\!00\)\( - \)\(15\!\cdots\!65\)\( \beta_{1} + 241028779072562640 \beta_{2}) q^{75}\) \(+(\)\(12\!\cdots\!20\)\( + \)\(34\!\cdots\!56\)\( \beta_{1} - \)\(62\!\cdots\!16\)\( \beta_{2}) q^{77}\) \(+(-\)\(50\!\cdots\!48\)\( + \)\(14\!\cdots\!84\)\( \beta_{1} + \)\(31\!\cdots\!76\)\( \beta_{2}) q^{79}\) \(+(-\)\(15\!\cdots\!11\)\( + \)\(28\!\cdots\!68\)\( \beta_{1} + \)\(11\!\cdots\!02\)\( \beta_{2}) q^{81}\) \(+(-\)\(29\!\cdots\!04\)\( - \)\(10\!\cdots\!35\)\( \beta_{1} - \)\(53\!\cdots\!08\)\( \beta_{2}) q^{83}\) \(+(-\)\(38\!\cdots\!60\)\( - \)\(15\!\cdots\!52\)\( \beta_{1} - \)\(28\!\cdots\!38\)\( \beta_{2}) q^{85}\) \(+(-\)\(31\!\cdots\!68\)\( - \)\(19\!\cdots\!50\)\( \beta_{1} + \)\(97\!\cdots\!44\)\( \beta_{2}) q^{87}\) \(+(\)\(54\!\cdots\!66\)\( + \)\(32\!\cdots\!52\)\( \beta_{1} + \)\(84\!\cdots\!78\)\( \beta_{2}) q^{89}\) \(+(\)\(96\!\cdots\!44\)\( + \)\(79\!\cdots\!56\)\( \beta_{1} - \)\(87\!\cdots\!16\)\( \beta_{2}) q^{91}\) \(+(\)\(14\!\cdots\!68\)\( - \)\(11\!\cdots\!84\)\( \beta_{1} - \)\(75\!\cdots\!60\)\( \beta_{2}) q^{93}\) \(+(\)\(32\!\cdots\!20\)\( + \)\(41\!\cdots\!74\)\( \beta_{1} + \)\(16\!\cdots\!56\)\( \beta_{2}) q^{95}\) \(+(\)\(27\!\cdots\!38\)\( - \)\(11\!\cdots\!96\)\( \beta_{1} - \)\(85\!\cdots\!58\)\( \beta_{2}) q^{97}\) \(+(\)\(30\!\cdots\!20\)\( - \)\(14\!\cdots\!09\)\( \beta_{1} + \)\(60\!\cdots\!24\)\( \beta_{2}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut +\mathstrut 50908884q^{3} \) \(\mathstrut +\mathstrut 280720890q^{5} \) \(\mathstrut -\mathstrut 5549296289016q^{7} \) \(\mathstrut -\mathstrut 4115694118097313q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut +\mathstrut 50908884q^{3} \) \(\mathstrut +\mathstrut 280720890q^{5} \) \(\mathstrut -\mathstrut 5549296289016q^{7} \) \(\mathstrut -\mathstrut 4115694118097313q^{9} \) \(\mathstrut -\mathstrut 420463958869151460q^{11} \) \(\mathstrut -\mathstrut 7370701315414736574q^{13} \) \(\mathstrut +\mathstrut 35737802623493622360q^{15} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!58\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!48\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!72\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(97\!\cdots\!72\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!75\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!72\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(72\!\cdots\!98\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!32\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!40\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!40\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!86\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!48\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!02\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!00\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!10\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!28\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!19\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!96\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!42\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!00\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(70\!\cdots\!24\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!76\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!54\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!84\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!60\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(75\!\cdots\!84\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!16\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!44\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(84\!\cdots\!66\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(84\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!60\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!44\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!33\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(89\!\cdots\!12\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!80\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(94\!\cdots\!04\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!98\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!32\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!04\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(96\!\cdots\!60\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(83\!\cdots\!14\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(91\!\cdots\!60\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut -\mathstrut \) \(1597028177\) \(x\mathstrut +\mathstrut \) \(23572260890640\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 256 \nu^{2} + 4304128 \nu - 272560910336 \)\()/319\)
\(\beta_{2}\)\(=\)\((\)\( 1491456 \nu^{2} + 44832004608 \nu - 1587946449002496 \)\()/319\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut -\mathstrut \) \(5826\) \(\beta_{1}\mathstrut +\mathstrut \) \(20643840\)\()/61931520\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(16813\) \(\beta_{2}\mathstrut +\mathstrut \) \(175125018\) \(\beta_{1}\mathstrut +\mathstrut \) \(65937588343603200\)\()/61931520\)

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
19173.3
26763.9
−45936.2
0 −2.83743e8 0 4.90658e11 0 −2.46684e14 0 3.04783e16 0
1.2 0 9.85025e7 0 −2.11237e12 0 1.18095e15 0 −4.03288e16 0
1.3 0 2.36149e8 0 1.62199e12 0 −9.39810e14 0 5.73480e15 0
\(n\): 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

There are no other newforms in \(S_{36}^{\mathrm{new}}(\Gamma_0(4))\).