Properties

Label 4.38.a.a
Level 4
Weight 38
Character orbit 4.a
Self dual yes
Analytic conductor 34.686
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 38 \)
Character orbit: \([\chi]\) \(=\) 4.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(34.6856152498\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 134608389910 x + 8010664803252592\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{5}\cdot 7 \)
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.

\(f(q)\) \(=\) \( q + ( -90721164 - \beta_{1} ) q^{3} + ( 1213681705398 + 1822 \beta_{1} + \beta_{2} ) q^{5} + ( 504728169053864 + 709926 \beta_{1} + 420 \beta_{2} ) q^{7} + ( 34317977529352221 - 24221340 \beta_{1} + 34830 \beta_{2} ) q^{9} +O(q^{10})\) \( q +(-90721164 - \beta_{1}) q^{3} +(1213681705398 + 1822 \beta_{1} + \beta_{2}) q^{5} +(504728169053864 + 709926 \beta_{1} + 420 \beta_{2}) q^{7} +(34317977529352221 - 24221340 \beta_{1} + 34830 \beta_{2}) q^{9} +(-3869685694838412900 - 20988995555 \beta_{1} - 1874840 \beta_{2}) q^{11} +(-23525561635929960226 - 680660687634 \beta_{1} + 36455385 \beta_{2}) q^{13} +(-\)\(97\!\cdots\!16\)\( - 6628380280974 \beta_{1} - 359845092 \beta_{2}) q^{15} +(\)\(71\!\cdots\!98\)\( + 61605599662532 \beta_{1} + 1456339550 \beta_{2}) q^{17} +(\)\(52\!\cdots\!36\)\( + 478748879151915 \beta_{1} + 7808810520 \beta_{2}) q^{19} +(-\)\(38\!\cdots\!44\)\( - 2785259500462040 \beta_{1} - 149208352020 \beta_{2}) q^{21} +(\)\(28\!\cdots\!72\)\( - 12050718366958702 \beta_{1} + 948275005820 \beta_{2}) q^{23} +(\)\(72\!\cdots\!87\)\( + 57441579694507368 \beta_{1} - 2331697407156 \beta_{2}) q^{25} +(\)\(49\!\cdots\!68\)\( + 217293634315606662 \beta_{1} - 9479454426360 \beta_{2}) q^{27} +(\)\(33\!\cdots\!66\)\( - 701696995157868170 \beta_{1} + 114545531878165 \beta_{2}) q^{29} +(\)\(72\!\cdots\!36\)\( - 3923841950161691400 \beta_{1} - 522996394615200 \beta_{2}) q^{31} +(\)\(10\!\cdots\!60\)\( + 12001490287985984940 \beta_{1} + 1286720853607530 \beta_{2}) q^{33} +(\)\(33\!\cdots\!16\)\( + 23758821250756807124 \beta_{1} - 999217379172208 \beta_{2}) q^{35} +(\)\(10\!\cdots\!34\)\( - 57017949069244204794 \beta_{1} - 4011320842862835 \beta_{2}) q^{37} +(\)\(32\!\cdots\!76\)\( - \)\(25\!\cdots\!50\)\( \beta_{1} + 12902588591571900 \beta_{2}) q^{39} +(\)\(87\!\cdots\!26\)\( + \)\(46\!\cdots\!20\)\( \beta_{1} - 8104793936473940 \beta_{2}) q^{41} +(\)\(18\!\cdots\!60\)\( + \)\(82\!\cdots\!65\)\( \beta_{1} - 8110032223098960 \beta_{2}) q^{43} +(\)\(26\!\cdots\!98\)\( + \)\(14\!\cdots\!22\)\( \beta_{1} - 112764793566228399 \beta_{2}) q^{45} +(\)\(13\!\cdots\!28\)\( - \)\(12\!\cdots\!48\)\( \beta_{1} + 538746777093986760 \beta_{2}) q^{47} +(-\)\(44\!\cdots\!03\)\( + \)\(98\!\cdots\!20\)\( \beta_{1} - 427924608670037640 \beta_{2}) q^{49} +(-\)\(29\!\cdots\!88\)\( - \)\(82\!\cdots\!70\)\( \beta_{1} - 2577359989107695160 \beta_{2}) q^{51} +(-\)\(53\!\cdots\!02\)\( + \)\(13\!\cdots\!82\)\( \beta_{1} + 7334361247153447645 \beta_{2}) q^{53} +(-\)\(16\!\cdots\!20\)\( - \)\(22\!\cdots\!30\)\( \beta_{1} - 1951991883971302140 \beta_{2}) q^{55} +(-\)\(23\!\cdots\!84\)\( - \)\(41\!\cdots\!56\)\( \beta_{1} - 18989236454951232090 \beta_{2}) q^{57} +(-\)\(18\!\cdots\!48\)\( - \)\(32\!\cdots\!95\)\( \beta_{1} + 11855666210454100640 \beta_{2}) q^{59} +(\)\(39\!\cdots\!62\)\( + \)\(14\!\cdots\!10\)\( \beta_{1} + 65581514617815233505 \beta_{2}) q^{61} +(\)\(11\!\cdots\!64\)\( + \)\(58\!\cdots\!26\)\( \beta_{1} - 47885559726681478620 \beta_{2}) q^{63} +(\)\(21\!\cdots\!56\)\( - \)\(28\!\cdots\!16\)\( \beta_{1} - \)\(35\!\cdots\!28\)\( \beta_{2}) q^{65} +(\)\(56\!\cdots\!64\)\( - \)\(98\!\cdots\!49\)\( \beta_{1} + \)\(74\!\cdots\!40\)\( \beta_{2}) q^{67} +(\)\(57\!\cdots\!08\)\( - \)\(70\!\cdots\!60\)\( \beta_{1} + \)\(13\!\cdots\!20\)\( \beta_{2}) q^{69} +(\)\(34\!\cdots\!68\)\( + \)\(21\!\cdots\!10\)\( \beta_{1} - \)\(22\!\cdots\!20\)\( \beta_{2}) q^{71} +(-\)\(28\!\cdots\!86\)\( + \)\(36\!\cdots\!76\)\( \beta_{1} + \)\(28\!\cdots\!10\)\( \beta_{2}) q^{73} +(-\)\(28\!\cdots\!04\)\( + \)\(12\!\cdots\!69\)\( \beta_{1} - \)\(13\!\cdots\!48\)\( \beta_{2}) q^{75} +(-\)\(69\!\cdots\!60\)\( - \)\(93\!\cdots\!40\)\( \beta_{1} - \)\(74\!\cdots\!80\)\( \beta_{2}) q^{77} +(-\)\(88\!\cdots\!12\)\( - \)\(12\!\cdots\!20\)\( \beta_{1} + \)\(75\!\cdots\!40\)\( \beta_{2}) q^{79} +(-\)\(12\!\cdots\!51\)\( + \)\(39\!\cdots\!40\)\( \beta_{1} - \)\(20\!\cdots\!30\)\( \beta_{2}) q^{81} +(-\)\(22\!\cdots\!68\)\( + \)\(19\!\cdots\!63\)\( \beta_{1} + \)\(14\!\cdots\!40\)\( \beta_{2}) q^{83} +(\)\(17\!\cdots\!12\)\( + \)\(47\!\cdots\!68\)\( \beta_{1} + \)\(17\!\cdots\!94\)\( \beta_{2}) q^{85} +(\)\(30\!\cdots\!16\)\( - \)\(10\!\cdots\!06\)\( \beta_{1} - \)\(95\!\cdots\!80\)\( \beta_{2}) q^{87} +(\)\(83\!\cdots\!14\)\( + \)\(78\!\cdots\!40\)\( \beta_{1} - \)\(10\!\cdots\!30\)\( \beta_{2}) q^{89} +(\)\(93\!\cdots\!84\)\( - \)\(12\!\cdots\!20\)\( \beta_{1} - \)\(14\!\cdots\!60\)\( \beta_{2}) q^{91} +(\)\(18\!\cdots\!96\)\( + \)\(17\!\cdots\!64\)\( \beta_{1} + \)\(29\!\cdots\!00\)\( \beta_{2}) q^{93} +(\)\(10\!\cdots\!88\)\( + \)\(35\!\cdots\!82\)\( \beta_{1} + \)\(15\!\cdots\!56\)\( \beta_{2}) q^{95} +(-\)\(15\!\cdots\!66\)\( + \)\(76\!\cdots\!56\)\( \beta_{1} - \)\(65\!\cdots\!90\)\( \beta_{2}) q^{97} +(-\)\(49\!\cdots\!00\)\( - \)\(67\!\cdots\!55\)\( \beta_{1} + \)\(44\!\cdots\!60\)\( \beta_{2}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 272163492q^{3} + 3641045116194q^{5} + 1514184507161592q^{7} + 102953932588056663q^{9} + O(q^{10}) \) \( 3q - 272163492q^{3} + 3641045116194q^{5} + 1514184507161592q^{7} + 102953932588056663q^{9} - 11609057084515238700q^{11} - 70576684907789880678q^{13} - \)\(29\!\cdots\!48\)\(q^{15} + \)\(21\!\cdots\!94\)\(q^{17} + \)\(15\!\cdots\!08\)\(q^{19} - \)\(11\!\cdots\!32\)\(q^{21} + \)\(86\!\cdots\!16\)\(q^{23} + \)\(21\!\cdots\!61\)\(q^{25} + \)\(14\!\cdots\!04\)\(q^{27} + \)\(10\!\cdots\!98\)\(q^{29} + \)\(21\!\cdots\!08\)\(q^{31} + \)\(31\!\cdots\!80\)\(q^{33} + \)\(10\!\cdots\!48\)\(q^{35} + \)\(31\!\cdots\!02\)\(q^{37} + \)\(97\!\cdots\!28\)\(q^{39} + \)\(26\!\cdots\!78\)\(q^{41} + \)\(54\!\cdots\!80\)\(q^{43} + \)\(80\!\cdots\!94\)\(q^{45} + \)\(41\!\cdots\!84\)\(q^{47} - \)\(13\!\cdots\!09\)\(q^{49} - \)\(89\!\cdots\!64\)\(q^{51} - \)\(15\!\cdots\!06\)\(q^{53} - \)\(50\!\cdots\!60\)\(q^{55} - \)\(69\!\cdots\!52\)\(q^{57} - \)\(54\!\cdots\!44\)\(q^{59} + \)\(11\!\cdots\!86\)\(q^{61} + \)\(34\!\cdots\!92\)\(q^{63} + \)\(65\!\cdots\!68\)\(q^{65} + \)\(16\!\cdots\!92\)\(q^{67} + \)\(17\!\cdots\!24\)\(q^{69} + \)\(10\!\cdots\!04\)\(q^{71} - \)\(85\!\cdots\!58\)\(q^{73} - \)\(84\!\cdots\!12\)\(q^{75} - \)\(20\!\cdots\!80\)\(q^{77} - \)\(26\!\cdots\!36\)\(q^{79} - \)\(37\!\cdots\!53\)\(q^{81} - \)\(68\!\cdots\!04\)\(q^{83} + \)\(52\!\cdots\!36\)\(q^{85} + \)\(91\!\cdots\!48\)\(q^{87} + \)\(24\!\cdots\!42\)\(q^{89} + \)\(28\!\cdots\!52\)\(q^{91} + \)\(54\!\cdots\!88\)\(q^{93} + \)\(32\!\cdots\!64\)\(q^{95} - \)\(46\!\cdots\!98\)\(q^{97} - \)\(14\!\cdots\!00\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 134608389910 x + 8010664803252592\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2304 \nu - 768 \)
\(\beta_{2}\)\(=\)\((\)\( 32768 \nu^{2} + 2924972544 \nu - 2940566122049024 \)\()/215\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 768\)\()/2304\)
\(\nu^{2}\)\(=\)\((\)\(645 \beta_{2} - 3808558 \beta_{1} + 8821695441174528\)\()/98304\)

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
332433.
61215.0
−393647.
0 −8.56646e8 0 1.02977e13 0 4.27767e15 0 2.83558e17 0
1.2 0 −2.31760e8 0 −1.08025e13 0 −4.54986e15 0 −3.96571e17 0
1.3 0 8.16242e8 0 4.14579e12 0 1.78638e15 0 2.15967e17 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 4.38.a.a 3
4.b odd 2 1 16.38.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.38.a.a 3 1.a even 1 1 trivial
16.38.a.d 3 4.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{38}^{\mathrm{new}}(\Gamma_0(4))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 272163492 T + 660985375731284745 T^{2} + \)\(83\!\cdots\!44\)\( T^{3} + \)\(29\!\cdots\!35\)\( T^{4} + \)\(55\!\cdots\!48\)\( T^{5} + \)\(91\!\cdots\!47\)\( T^{6} \)
$5$ \( 1 - 3641045116194 T + \)\(10\!\cdots\!75\)\( T^{2} - \)\(68\!\cdots\!00\)\( T^{3} + \)\(76\!\cdots\!75\)\( T^{4} - \)\(19\!\cdots\!50\)\( T^{5} + \)\(38\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 - 1514184507161592 T + \)\(35\!\cdots\!97\)\( T^{2} - \)\(21\!\cdots\!80\)\( T^{3} + \)\(66\!\cdots\!79\)\( T^{4} - \)\(52\!\cdots\!08\)\( T^{5} + \)\(63\!\cdots\!43\)\( T^{6} \)
$11$ \( 1 + 11609057084515238700 T + \)\(34\!\cdots\!13\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!23\)\( T^{4} + \)\(13\!\cdots\!00\)\( T^{5} + \)\(39\!\cdots\!11\)\( T^{6} \)
$13$ \( 1 + 70576684907789880678 T + \)\(10\!\cdots\!95\)\( T^{2} - \)\(90\!\cdots\!64\)\( T^{3} + \)\(17\!\cdots\!35\)\( T^{4} + \)\(19\!\cdots\!42\)\( T^{5} + \)\(44\!\cdots\!37\)\( T^{6} \)
$17$ \( 1 - \)\(21\!\cdots\!94\)\( T + \)\(72\!\cdots\!75\)\( T^{2} - \)\(14\!\cdots\!12\)\( T^{3} + \)\(24\!\cdots\!75\)\( T^{4} - \)\(24\!\cdots\!26\)\( T^{5} + \)\(38\!\cdots\!33\)\( T^{6} \)
$19$ \( 1 - \)\(15\!\cdots\!08\)\( T + \)\(45\!\cdots\!05\)\( T^{2} - \)\(61\!\cdots\!80\)\( T^{3} + \)\(93\!\cdots\!95\)\( T^{4} - \)\(67\!\cdots\!68\)\( T^{5} + \)\(87\!\cdots\!19\)\( T^{6} \)
$23$ \( 1 - \)\(86\!\cdots\!16\)\( T + \)\(51\!\cdots\!93\)\( T^{2} - \)\(75\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!79\)\( T^{4} - \)\(50\!\cdots\!44\)\( T^{5} + \)\(14\!\cdots\!27\)\( T^{6} \)
$29$ \( 1 - \)\(10\!\cdots\!98\)\( T + \)\(23\!\cdots\!95\)\( T^{2} - \)\(14\!\cdots\!60\)\( T^{3} + \)\(29\!\cdots\!55\)\( T^{4} - \)\(16\!\cdots\!38\)\( T^{5} + \)\(21\!\cdots\!29\)\( T^{6} \)
$31$ \( 1 - \)\(21\!\cdots\!08\)\( T + \)\(44\!\cdots\!21\)\( T^{2} + \)\(12\!\cdots\!68\)\( T^{3} + \)\(67\!\cdots\!31\)\( T^{4} - \)\(50\!\cdots\!68\)\( T^{5} + \)\(34\!\cdots\!31\)\( T^{6} \)
$37$ \( 1 - \)\(31\!\cdots\!02\)\( T + \)\(59\!\cdots\!07\)\( T^{2} - \)\(71\!\cdots\!80\)\( T^{3} + \)\(63\!\cdots\!19\)\( T^{4} - \)\(34\!\cdots\!78\)\( T^{5} + \)\(11\!\cdots\!13\)\( T^{6} \)
$41$ \( 1 - \)\(26\!\cdots\!78\)\( T + \)\(35\!\cdots\!71\)\( T^{2} - \)\(29\!\cdots\!12\)\( T^{3} + \)\(16\!\cdots\!51\)\( T^{4} - \)\(58\!\cdots\!58\)\( T^{5} + \)\(10\!\cdots\!41\)\( T^{6} \)
$43$ \( 1 - \)\(54\!\cdots\!80\)\( T + \)\(17\!\cdots\!29\)\( T^{2} - \)\(34\!\cdots\!80\)\( T^{3} + \)\(48\!\cdots\!47\)\( T^{4} - \)\(40\!\cdots\!20\)\( T^{5} + \)\(20\!\cdots\!07\)\( T^{6} \)
$47$ \( 1 - \)\(41\!\cdots\!84\)\( T + \)\(76\!\cdots\!05\)\( T^{2} - \)\(10\!\cdots\!12\)\( T^{3} + \)\(56\!\cdots\!35\)\( T^{4} - \)\(22\!\cdots\!96\)\( T^{5} + \)\(40\!\cdots\!03\)\( T^{6} \)
$53$ \( 1 + \)\(15\!\cdots\!06\)\( T + \)\(82\!\cdots\!23\)\( T^{2} + \)\(14\!\cdots\!60\)\( T^{3} + \)\(51\!\cdots\!99\)\( T^{4} + \)\(62\!\cdots\!14\)\( T^{5} + \)\(24\!\cdots\!97\)\( T^{6} \)
$59$ \( 1 + \)\(54\!\cdots\!44\)\( T + \)\(10\!\cdots\!69\)\( T^{2} + \)\(33\!\cdots\!64\)\( T^{3} + \)\(33\!\cdots\!11\)\( T^{4} + \)\(59\!\cdots\!84\)\( T^{5} + \)\(36\!\cdots\!59\)\( T^{6} \)
$61$ \( 1 - \)\(11\!\cdots\!86\)\( T + \)\(19\!\cdots\!95\)\( T^{2} - \)\(29\!\cdots\!40\)\( T^{3} + \)\(22\!\cdots\!95\)\( T^{4} - \)\(15\!\cdots\!26\)\( T^{5} + \)\(14\!\cdots\!61\)\( T^{6} \)
$67$ \( 1 - \)\(16\!\cdots\!92\)\( T + \)\(14\!\cdots\!77\)\( T^{2} - \)\(85\!\cdots\!00\)\( T^{3} + \)\(51\!\cdots\!79\)\( T^{4} - \)\(22\!\cdots\!68\)\( T^{5} + \)\(49\!\cdots\!83\)\( T^{6} \)
$71$ \( 1 - \)\(10\!\cdots\!04\)\( T + \)\(81\!\cdots\!45\)\( T^{2} - \)\(40\!\cdots\!60\)\( T^{3} + \)\(25\!\cdots\!95\)\( T^{4} - \)\(10\!\cdots\!24\)\( T^{5} + \)\(30\!\cdots\!71\)\( T^{6} \)
$73$ \( 1 + \)\(85\!\cdots\!58\)\( T + \)\(16\!\cdots\!75\)\( T^{2} + \)\(20\!\cdots\!76\)\( T^{3} + \)\(14\!\cdots\!75\)\( T^{4} + \)\(65\!\cdots\!22\)\( T^{5} + \)\(67\!\cdots\!77\)\( T^{6} \)
$79$ \( 1 + \)\(26\!\cdots\!36\)\( T + \)\(65\!\cdots\!09\)\( T^{2} + \)\(89\!\cdots\!76\)\( T^{3} + \)\(10\!\cdots\!31\)\( T^{4} + \)\(70\!\cdots\!16\)\( T^{5} + \)\(43\!\cdots\!79\)\( T^{6} \)
$83$ \( 1 + \)\(68\!\cdots\!04\)\( T + \)\(25\!\cdots\!53\)\( T^{2} + \)\(88\!\cdots\!20\)\( T^{3} + \)\(26\!\cdots\!19\)\( T^{4} + \)\(70\!\cdots\!16\)\( T^{5} + \)\(10\!\cdots\!67\)\( T^{6} \)
$89$ \( 1 - \)\(24\!\cdots\!42\)\( T + \)\(60\!\cdots\!75\)\( T^{2} - \)\(72\!\cdots\!80\)\( T^{3} + \)\(81\!\cdots\!75\)\( T^{4} - \)\(44\!\cdots\!22\)\( T^{5} + \)\(24\!\cdots\!89\)\( T^{6} \)
$97$ \( 1 + \)\(46\!\cdots\!98\)\( T + \)\(54\!\cdots\!87\)\( T^{2} + \)\(96\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!19\)\( T^{4} + \)\(49\!\cdots\!62\)\( T^{5} + \)\(34\!\cdots\!53\)\( T^{6} \)
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