Properties

Label 1.28.a.a
Level 1
Weight 28
Character orbit 1.a
Self dual Yes
Analytic conductor 4.619
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 28 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(4.61855574838\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{18209}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3}\cdot 3^{3} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 108\sqrt{18209}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -4140 - \beta ) q^{2} \) \( + ( -643140 - 192 \beta ) q^{3} \) \( + ( 95311648 + 8280 \beta ) q^{4} \) \( + ( 2721793950 - 147200 \beta ) q^{5} \) \( + ( 43441436592 + 1438020 \beta ) q^{6} \) \( + ( -87695981800 - 7491456 \beta ) q^{7} \) \( + ( -1597516174080 + 4626880 \beta ) q^{8} \) \( + ( 617568277077 + 246965760 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \(+(-4140 - \beta) q^{2}\) \(+(-643140 - 192 \beta) q^{3}\) \(+(95311648 + 8280 \beta) q^{4}\) \(+(2721793950 - 147200 \beta) q^{5}\) \(+(43441436592 + 1438020 \beta) q^{6}\) \(+(-87695981800 - 7491456 \beta) q^{7}\) \(+(-1597516174080 + 4626880 \beta) q^{8}\) \(+(617568277077 + 246965760 \beta) q^{9}\) \(+(19995548074200 - 2112385950 \beta) q^{10}\) \(+(69083668845972 + 9443825600 \beta) q^{11}\) \(+(-398947503588480 - 23625035616 \beta) q^{12}\) \(+(-376716900635530 + 14413771008 \beta) q^{13}\) \(+(1954170026405856 + 118710609640 \beta) q^{14}\) \(+(4252150244219400 - 427914230400 \beta) q^{15}\) \(+(-7161497892583424 + 467038103040 \beta) q^{16}\) \(+(-14876810165505870 + 645319017984 \beta) q^{17}\) \(+(-55009735113168540 - 1640006523477 \beta) q^{18}\) \(+(202282905186342380 - 2800373100480 \beta) q^{19}\) \(+(554609665713600 + 8506579320400 \beta) q^{20}\) \(+(361893656791592352 + 21655683517440 \beta) q^{21}\) \(+(-2291778392789389680 - 108181106829972 \beta) q^{22}\) \(+(1464539461560609480 + 91869621889408 \beta) q^{23}\) \(+(838747766896266240 + 303747373820160 \beta) q^{24}\) \(+(4559609393336614375 - 801296138880000 \beta) q^{25}\) \(+(-1501729627073320008 + 317043888662410 \beta) q^{26}\) \(+(-5563832564870156520 + 1186708049032320 \beta) q^{27}\) \(+(-21532828267657934080 - 1440145746583488 \beta) q^{28}\) \(+(-7773339997724279130 + 327641521921280 \beta) q^{29}\) \(+(73280705530800074400 - 2480585330363400 \beta) q^{30}\) \(+(14272277297233692512 + 5259288606220800 \beta) q^{31}\) \(+(\)\(14\!\cdots\!60\)\( + 4606950824669184 \beta) q^{32}\) \(+(-\)\(42\!\cdots\!80\)\( - 19337766414810624 \beta) q^{33}\) \(+(-75469167592967429784 + 12205189431052110 \beta) q^{34}\) \(+(-4479197692382506800 - 7481351096531200 \beta) q^{35}\) \(+(\)\(49\!\cdots\!96\)\( + 28652178919370040 \beta) q^{36}\) \(+(\)\(93\!\cdots\!90\)\( + 38848100480273664 \beta) q^{37}\) \(+(-\)\(24\!\cdots\!20\)\( - 190689360550355180 \beta) q^{38}\) \(+(-\)\(34\!\cdots\!36\)\( + 63059572235936640 \beta) q^{39}\) \(+(-\)\(44\!\cdots\!00\)\( + 247747794815952000 \beta) q^{40}\) \(+(\)\(45\!\cdots\!82\)\( + 63868156292684800 \beta) q^{41}\) \(+(-\)\(60\!\cdots\!20\)\( - 451548186553793952 \beta) q^{42}\) \(+(\)\(25\!\cdots\!00\)\( - 643846246368316992 \beta) q^{43}\) \(+(\)\(23\!\cdots\!56\)\( + 1472119359405236960 \beta) q^{44}\) \(+(-\)\(60\!\cdots\!50\)\( + 581283861039417600 \beta) q^{45}\) \(+(-\)\(25\!\cdots\!08\)\( - 1844879696182758600 \beta) q^{46}\) \(+(-\)\(57\!\cdots\!80\)\( - 606675609439012096 \beta) q^{47}\) \(+(-\)\(14\!\cdots\!20\)\( + 1074636709786871808 \beta) q^{48}\) \(+(-\)\(46\!\cdots\!07\)\( + 1313941178063001600 \beta) q^{49}\) \(+(\)\(15\!\cdots\!00\)\( - 1242243378373414375 \beta) q^{50}\) \(+(-\)\(16\!\cdots\!28\)\( + 2441317078550897280 \beta) q^{51}\) \(+(-\)\(10\!\cdots\!00\)\( - 1745415668595087216 \beta) q^{52}\) \(+(\)\(53\!\cdots\!10\)\( - 7346878234282881792 \beta) q^{53}\) \(+(-\)\(22\!\cdots\!20\)\( + 650861241876351720 \beta) q^{54}\) \(+(-\)\(10\!\cdots\!00\)\( + 15535031328808041600 \beta) q^{55}\) \(+(\)\(13\!\cdots\!20\)\( + 11561963343137876480 \beta) q^{56}\) \(+(-\)\(15\!\cdots\!40\)\( - 37037285839935029760 \beta) q^{57}\) \(+(-\)\(37\!\cdots\!80\)\( + 6416904096970179930 \beta) q^{58}\) \(+(\)\(10\!\cdots\!40\)\( - 6076200302162335040 \beta) q^{59}\) \(+(-\)\(34\!\cdots\!00\)\( - 5577406479939067200 \beta) q^{60}\) \(+(\)\(73\!\cdots\!22\)\( + 96042884669269920000 \beta) q^{61}\) \(+(-\)\(11\!\cdots\!80\)\( - 36045732126987804512 \beta) q^{62}\) \(+(-\)\(44\!\cdots\!60\)\( - 26284390368901322112 \beta) q^{63}\) \(+(-\)\(61\!\cdots\!12\)\( - \)\(22\!\cdots\!40\)\( \beta) q^{64}\) \(+(-\)\(14\!\cdots\!00\)\( + 94684042499809817600 \beta) q^{65}\) \(+(\)\(58\!\cdots\!24\)\( + \)\(50\!\cdots\!40\)\( \beta) q^{66}\) \(+(\)\(15\!\cdots\!80\)\( - \)\(32\!\cdots\!16\)\( \beta) q^{67}\) \(+(-\)\(28\!\cdots\!40\)\( - 61673569080591925968 \beta) q^{68}\) \(+(-\)\(46\!\cdots\!36\)\( - \)\(34\!\cdots\!80\)\( \beta) q^{69}\) \(+(\)\(16\!\cdots\!00\)\( + 35451991232021674800 \beta) q^{70}\) \(+(-\)\(65\!\cdots\!08\)\( + \)\(68\!\cdots\!00\)\( \beta) q^{71}\) \(+(-\)\(74\!\cdots\!60\)\( - \)\(39\!\cdots\!40\)\( \beta) q^{72}\) \(+(\)\(26\!\cdots\!30\)\( + \)\(72\!\cdots\!08\)\( \beta) q^{73}\) \(+(-\)\(12\!\cdots\!64\)\( - \)\(10\!\cdots\!50\)\( \beta) q^{74}\) \(+(\)\(29\!\cdots\!00\)\( - \)\(36\!\cdots\!00\)\( \beta) q^{75}\) \(+(\)\(14\!\cdots\!40\)\( + \)\(14\!\cdots\!60\)\( \beta) q^{76}\) \(+(-\)\(21\!\cdots\!00\)\( - \)\(13\!\cdots\!32\)\( \beta) q^{77}\) \(+(-\)\(11\!\cdots\!00\)\( + 84428481843735074136 \beta) q^{78}\) \(+(\)\(31\!\cdots\!20\)\( + \)\(10\!\cdots\!80\)\( \beta) q^{79}\) \(+(-\)\(34\!\cdots\!00\)\( + \)\(23\!\cdots\!00\)\( \beta) q^{80}\) \(+(-\)\(49\!\cdots\!39\)\( - \)\(15\!\cdots\!80\)\( \beta) q^{81}\) \(+(-\)\(32\!\cdots\!80\)\( - \)\(48\!\cdots\!82\)\( \beta) q^{82}\) \(+(\)\(85\!\cdots\!40\)\( - \)\(99\!\cdots\!92\)\( \beta) q^{83}\) \(+(\)\(72\!\cdots\!96\)\( + \)\(50\!\cdots\!80\)\( \beta) q^{84}\) \(+(-\)\(60\!\cdots\!00\)\( + \)\(39\!\cdots\!00\)\( \beta) q^{85}\) \(+(\)\(12\!\cdots\!92\)\( + 92717157064121459980 \beta) q^{86}\) \(+(-\)\(83\!\cdots\!60\)\( + \)\(12\!\cdots\!60\)\( \beta) q^{87}\) \(+(-\)\(10\!\cdots\!60\)\( - \)\(14\!\cdots\!40\)\( \beta) q^{88}\) \(+(-\)\(15\!\cdots\!90\)\( + \)\(10\!\cdots\!40\)\( \beta) q^{89}\) \(+(-\)\(98\!\cdots\!00\)\( + \)\(36\!\cdots\!50\)\( \beta) q^{90}\) \(+(\)\(10\!\cdots\!52\)\( + \)\(15\!\cdots\!80\)\( \beta) q^{91}\) \(+(\)\(30\!\cdots\!80\)\( + \)\(20\!\cdots\!84\)\( \beta) q^{92}\) \(+(-\)\(22\!\cdots\!80\)\( - \)\(61\!\cdots\!04\)\( \beta) q^{93}\) \(+(\)\(13\!\cdots\!96\)\( + \)\(30\!\cdots\!20\)\( \beta) q^{94}\) \(+(\)\(63\!\cdots\!00\)\( - \)\(37\!\cdots\!00\)\( \beta) q^{95}\) \(+(-\)\(28\!\cdots\!28\)\( - \)\(30\!\cdots\!80\)\( \beta) q^{96}\) \(+(-\)\(32\!\cdots\!30\)\( + \)\(48\!\cdots\!04\)\( \beta) q^{97}\) \(+(-\)\(88\!\cdots\!20\)\( + \)\(40\!\cdots\!07\)\( \beta) q^{98}\) \(+(\)\(53\!\cdots\!44\)\( + \)\(22\!\cdots\!20\)\( \beta) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 8280q^{2} \) \(\mathstrut -\mathstrut 1286280q^{3} \) \(\mathstrut +\mathstrut 190623296q^{4} \) \(\mathstrut +\mathstrut 5443587900q^{5} \) \(\mathstrut +\mathstrut 86882873184q^{6} \) \(\mathstrut -\mathstrut 175391963600q^{7} \) \(\mathstrut -\mathstrut 3195032348160q^{8} \) \(\mathstrut +\mathstrut 1235136554154q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 8280q^{2} \) \(\mathstrut -\mathstrut 1286280q^{3} \) \(\mathstrut +\mathstrut 190623296q^{4} \) \(\mathstrut +\mathstrut 5443587900q^{5} \) \(\mathstrut +\mathstrut 86882873184q^{6} \) \(\mathstrut -\mathstrut 175391963600q^{7} \) \(\mathstrut -\mathstrut 3195032348160q^{8} \) \(\mathstrut +\mathstrut 1235136554154q^{9} \) \(\mathstrut +\mathstrut 39991096148400q^{10} \) \(\mathstrut +\mathstrut 138167337691944q^{11} \) \(\mathstrut -\mathstrut 797895007176960q^{12} \) \(\mathstrut -\mathstrut 753433801271060q^{13} \) \(\mathstrut +\mathstrut 3908340052811712q^{14} \) \(\mathstrut +\mathstrut 8504300488438800q^{15} \) \(\mathstrut -\mathstrut 14322995785166848q^{16} \) \(\mathstrut -\mathstrut 29753620331011740q^{17} \) \(\mathstrut -\mathstrut 110019470226337080q^{18} \) \(\mathstrut +\mathstrut 404565810372684760q^{19} \) \(\mathstrut +\mathstrut 1109219331427200q^{20} \) \(\mathstrut +\mathstrut 723787313583184704q^{21} \) \(\mathstrut -\mathstrut 4583556785578779360q^{22} \) \(\mathstrut +\mathstrut 2929078923121218960q^{23} \) \(\mathstrut +\mathstrut 1677495533792532480q^{24} \) \(\mathstrut +\mathstrut 9119218786673228750q^{25} \) \(\mathstrut -\mathstrut 3003459254146640016q^{26} \) \(\mathstrut -\mathstrut 11127665129740313040q^{27} \) \(\mathstrut -\mathstrut 43065656535315868160q^{28} \) \(\mathstrut -\mathstrut 15546679995448558260q^{29} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!00\)\(q^{30} \) \(\mathstrut +\mathstrut 28544554594467385024q^{31} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!20\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(85\!\cdots\!60\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!68\)\(q^{34} \) \(\mathstrut -\mathstrut 8958395384765013600q^{35} \) \(\mathstrut +\mathstrut \)\(98\!\cdots\!92\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!80\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(48\!\cdots\!40\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!72\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(89\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(90\!\cdots\!64\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!40\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!00\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!12\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!00\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!16\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!60\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!40\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(92\!\cdots\!14\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!56\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!00\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!20\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!40\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!00\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!40\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!80\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(74\!\cdots\!60\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!80\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!44\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!60\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(89\!\cdots\!20\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!24\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!00\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!48\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!60\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(56\!\cdots\!80\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(93\!\cdots\!72\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!16\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!20\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!60\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!28\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!80\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!00\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!00\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(62\!\cdots\!40\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!00\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(99\!\cdots\!78\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(64\!\cdots\!60\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!80\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!92\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!00\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!84\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!20\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!20\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!80\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!00\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!04\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(60\!\cdots\!60\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!60\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!92\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(56\!\cdots\!56\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(65\!\cdots\!60\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!40\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!88\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
67.9704
−66.9704
−18713.6 −3.44127e6 2.15981e8 5.76560e8 6.43986e10 −1.96873e11 −1.53009e12 4.21675e12 −1.07895e13
1.2 10433.6 2.15499e6 −2.53577e7 4.86703e9 2.24843e10 2.14815e10 −1.66495e12 −2.98161e12 5.07806e13
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Hecke kernels

There are no other newforms in \(S_{28}^{\mathrm{new}}(\Gamma_0(1))\).