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: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{5}\cdot 7 \)
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.

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

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

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

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:

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 \(S_{38}^{\mathrm{new}}(\Gamma_0(4))\).