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.

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

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 \(S_{28}^{\mathrm{new}}(\Gamma_0(1))\).