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

Related objects

Downloads

Learn more about

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: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 1597028177 x + 23572260890640\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{4}\cdot 5\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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

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

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

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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4.36.a.a 3
4.b odd 2 1 16.36.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.36.a.a 3 1.a even 1 1 trivial
16.36.a.c 3 4.b odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace is the entire newspace \(S_{36}^{\mathrm{new}}(\Gamma_0(4))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 50908884 T + 78401021942610945 T^{2} + \)\(15\!\cdots\!08\)\( T^{3} + \)\(39\!\cdots\!15\)\( T^{4} - \)\(12\!\cdots\!16\)\( T^{5} + \)\(12\!\cdots\!43\)\( T^{6} \)
$5$ \( 1 - 280720890 T + \)\(50\!\cdots\!75\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!75\)\( T^{4} - \)\(23\!\cdots\!50\)\( T^{5} + \)\(24\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 + 5549296289016 T - \)\(32\!\cdots\!67\)\( T^{2} - \)\(26\!\cdots\!60\)\( T^{3} - \)\(12\!\cdots\!81\)\( T^{4} + \)\(79\!\cdots\!84\)\( T^{5} + \)\(54\!\cdots\!07\)\( T^{6} \)
$11$ \( 1 + 420463958869151460 T + \)\(35\!\cdots\!53\)\( T^{2} - \)\(12\!\cdots\!80\)\( T^{3} + \)\(99\!\cdots\!03\)\( T^{4} + \)\(33\!\cdots\!60\)\( T^{5} + \)\(22\!\cdots\!51\)\( T^{6} \)
$13$ \( 1 + 7370701315414736574 T + \)\(12\!\cdots\!15\)\( T^{2} + \)\(10\!\cdots\!92\)\( T^{3} + \)\(12\!\cdots\!55\)\( T^{4} + \)\(69\!\cdots\!26\)\( T^{5} + \)\(92\!\cdots\!93\)\( T^{6} \)
$17$ \( 1 - \)\(61\!\cdots\!58\)\( T + \)\(14\!\cdots\!35\)\( T^{2} - \)\(66\!\cdots\!36\)\( T^{3} + \)\(16\!\cdots\!55\)\( T^{4} - \)\(83\!\cdots\!42\)\( T^{5} + \)\(15\!\cdots\!57\)\( T^{6} \)
$19$ \( 1 + \)\(35\!\cdots\!48\)\( T + \)\(89\!\cdots\!65\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(50\!\cdots\!35\)\( T^{4} + \)\(11\!\cdots\!48\)\( T^{5} + \)\(18\!\cdots\!99\)\( T^{6} \)
$23$ \( 1 + \)\(97\!\cdots\!72\)\( T + \)\(80\!\cdots\!97\)\( T^{2} - \)\(91\!\cdots\!20\)\( T^{3} + \)\(36\!\cdots\!79\)\( T^{4} + \)\(20\!\cdots\!28\)\( T^{5} + \)\(95\!\cdots\!43\)\( T^{6} \)
$29$ \( 1 + \)\(72\!\cdots\!98\)\( T + \)\(60\!\cdots\!15\)\( T^{2} + \)\(22\!\cdots\!00\)\( T^{3} + \)\(92\!\cdots\!35\)\( T^{4} + \)\(16\!\cdots\!98\)\( T^{5} + \)\(35\!\cdots\!49\)\( T^{6} \)
$31$ \( 1 + \)\(30\!\cdots\!32\)\( T + \)\(68\!\cdots\!61\)\( T^{2} + \)\(96\!\cdots\!48\)\( T^{3} + \)\(10\!\cdots\!11\)\( T^{4} + \)\(75\!\cdots\!32\)\( T^{5} + \)\(39\!\cdots\!51\)\( T^{6} \)
$37$ \( 1 + \)\(10\!\cdots\!86\)\( T + \)\(59\!\cdots\!23\)\( T^{2} + \)\(20\!\cdots\!80\)\( T^{3} + \)\(45\!\cdots\!39\)\( T^{4} + \)\(62\!\cdots\!14\)\( T^{5} + \)\(45\!\cdots\!57\)\( T^{6} \)
$41$ \( 1 + \)\(28\!\cdots\!02\)\( T + \)\(31\!\cdots\!71\)\( T^{2} + \)\(13\!\cdots\!08\)\( T^{3} + \)\(88\!\cdots\!71\)\( T^{4} + \)\(22\!\cdots\!02\)\( T^{5} + \)\(21\!\cdots\!01\)\( T^{6} \)
$43$ \( 1 + \)\(20\!\cdots\!00\)\( T + \)\(15\!\cdots\!21\)\( T^{2} + \)\(95\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!47\)\( T^{4} + \)\(45\!\cdots\!00\)\( T^{5} + \)\(32\!\cdots\!43\)\( T^{6} \)
$47$ \( 1 - \)\(43\!\cdots\!28\)\( T + \)\(93\!\cdots\!45\)\( T^{2} - \)\(14\!\cdots\!56\)\( T^{3} + \)\(31\!\cdots\!35\)\( T^{4} - \)\(48\!\cdots\!72\)\( T^{5} + \)\(37\!\cdots\!07\)\( T^{6} \)
$53$ \( 1 - \)\(20\!\cdots\!42\)\( T + \)\(66\!\cdots\!87\)\( T^{2} - \)\(80\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!59\)\( T^{4} - \)\(10\!\cdots\!58\)\( T^{5} + \)\(11\!\cdots\!93\)\( T^{6} \)
$59$ \( 1 + \)\(21\!\cdots\!76\)\( T + \)\(40\!\cdots\!89\)\( T^{2} + \)\(43\!\cdots\!36\)\( T^{3} + \)\(38\!\cdots\!11\)\( T^{4} + \)\(19\!\cdots\!76\)\( T^{5} + \)\(86\!\cdots\!99\)\( T^{6} \)
$61$ \( 1 + \)\(46\!\cdots\!54\)\( T + \)\(14\!\cdots\!75\)\( T^{2} + \)\(30\!\cdots\!40\)\( T^{3} + \)\(45\!\cdots\!75\)\( T^{4} + \)\(43\!\cdots\!54\)\( T^{5} + \)\(28\!\cdots\!01\)\( T^{6} \)
$67$ \( 1 - \)\(75\!\cdots\!84\)\( T + \)\(14\!\cdots\!13\)\( T^{2} - \)\(46\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!59\)\( T^{4} - \)\(50\!\cdots\!16\)\( T^{5} + \)\(54\!\cdots\!07\)\( T^{6} \)
$71$ \( 1 - \)\(59\!\cdots\!44\)\( T + \)\(29\!\cdots\!65\)\( T^{2} - \)\(80\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!15\)\( T^{4} - \)\(22\!\cdots\!44\)\( T^{5} + \)\(24\!\cdots\!51\)\( T^{6} \)
$73$ \( 1 - \)\(84\!\cdots\!66\)\( T + \)\(49\!\cdots\!35\)\( T^{2} - \)\(19\!\cdots\!48\)\( T^{3} + \)\(81\!\cdots\!95\)\( T^{4} - \)\(22\!\cdots\!34\)\( T^{5} + \)\(44\!\cdots\!93\)\( T^{6} \)
$79$ \( 1 + \)\(15\!\cdots\!44\)\( T + \)\(80\!\cdots\!09\)\( T^{2} + \)\(76\!\cdots\!04\)\( T^{3} + \)\(21\!\cdots\!91\)\( T^{4} + \)\(10\!\cdots\!44\)\( T^{5} + \)\(17\!\cdots\!99\)\( T^{6} \)
$83$ \( 1 + \)\(89\!\cdots\!12\)\( T + \)\(61\!\cdots\!77\)\( T^{2} + \)\(24\!\cdots\!40\)\( T^{3} + \)\(89\!\cdots\!39\)\( T^{4} + \)\(19\!\cdots\!88\)\( T^{5} + \)\(31\!\cdots\!43\)\( T^{6} \)
$89$ \( 1 - \)\(16\!\cdots\!98\)\( T + \)\(16\!\cdots\!15\)\( T^{2} - \)\(21\!\cdots\!00\)\( T^{3} + \)\(28\!\cdots\!35\)\( T^{4} - \)\(46\!\cdots\!98\)\( T^{5} + \)\(48\!\cdots\!49\)\( T^{6} \)
$97$ \( 1 - \)\(83\!\cdots\!14\)\( T + \)\(11\!\cdots\!03\)\( T^{2} - \)\(57\!\cdots\!40\)\( T^{3} + \)\(39\!\cdots\!79\)\( T^{4} - \)\(98\!\cdots\!86\)\( T^{5} + \)\(40\!\cdots\!57\)\( T^{6} \)
show more
show less