Properties

Label 3.16.a.a
Level 3
Weight 16
Character orbit 3.a
Self dual Yes
Analytic conductor 4.281
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 16 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(4.280805153\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 234q^{2} \) \(\mathstrut -\mathstrut 2187q^{3} \) \(\mathstrut +\mathstrut 21988q^{4} \) \(\mathstrut +\mathstrut 280710q^{5} \) \(\mathstrut +\mathstrut 511758q^{6} \) \(\mathstrut -\mathstrut 1373344q^{7} \) \(\mathstrut +\mathstrut 2522520q^{8} \) \(\mathstrut +\mathstrut 4782969q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 234q^{2} \) \(\mathstrut -\mathstrut 2187q^{3} \) \(\mathstrut +\mathstrut 21988q^{4} \) \(\mathstrut +\mathstrut 280710q^{5} \) \(\mathstrut +\mathstrut 511758q^{6} \) \(\mathstrut -\mathstrut 1373344q^{7} \) \(\mathstrut +\mathstrut 2522520q^{8} \) \(\mathstrut +\mathstrut 4782969q^{9} \) \(\mathstrut -\mathstrut 65686140q^{10} \) \(\mathstrut +\mathstrut 34031052q^{11} \) \(\mathstrut -\mathstrut 48087756q^{12} \) \(\mathstrut +\mathstrut 384022262q^{13} \) \(\mathstrut +\mathstrut 321362496q^{14} \) \(\mathstrut -\mathstrut 613912770q^{15} \) \(\mathstrut -\mathstrut 1310772464q^{16} \) \(\mathstrut +\mathstrut 1259207586q^{17} \) \(\mathstrut -\mathstrut 1119214746q^{18} \) \(\mathstrut -\mathstrut 2499071020q^{19} \) \(\mathstrut +\mathstrut 6172251480q^{20} \) \(\mathstrut +\mathstrut 3003503328q^{21} \) \(\mathstrut -\mathstrut 7963266168q^{22} \) \(\mathstrut +\mathstrut 11284833672q^{23} \) \(\mathstrut -\mathstrut 5516751240q^{24} \) \(\mathstrut +\mathstrut 48280525975q^{25} \) \(\mathstrut -\mathstrut 89861209308q^{26} \) \(\mathstrut -\mathstrut 10460353203q^{27} \) \(\mathstrut -\mathstrut 30197087872q^{28} \) \(\mathstrut -\mathstrut 48413458530q^{29} \) \(\mathstrut +\mathstrut 143655588180q^{30} \) \(\mathstrut +\mathstrut 130547265752q^{31} \) \(\mathstrut +\mathstrut 224062821216q^{32} \) \(\mathstrut -\mathstrut 74425910724q^{33} \) \(\mathstrut -\mathstrut 294654575124q^{34} \) \(\mathstrut -\mathstrut 385511394240q^{35} \) \(\mathstrut +\mathstrut 105167922372q^{36} \) \(\mathstrut -\mathstrut 200223317554q^{37} \) \(\mathstrut +\mathstrut 584782618680q^{38} \) \(\mathstrut -\mathstrut 839856686994q^{39} \) \(\mathstrut +\mathstrut 708096589200q^{40} \) \(\mathstrut +\mathstrut 679141724202q^{41} \) \(\mathstrut -\mathstrut 702819778752q^{42} \) \(\mathstrut +\mathstrut 279482194892q^{43} \) \(\mathstrut +\mathstrut 748274771376q^{44} \) \(\mathstrut +\mathstrut 1342627227990q^{45} \) \(\mathstrut -\mathstrut 2640651079248q^{46} \) \(\mathstrut +\mathstrut 1520672832576q^{47} \) \(\mathstrut +\mathstrut 2866659378768q^{48} \) \(\mathstrut -\mathstrut 2861487767607q^{49} \) \(\mathstrut -\mathstrut 11297643078150q^{50} \) \(\mathstrut -\mathstrut 2753886990582q^{51} \) \(\mathstrut +\mathstrut 8443881496856q^{52} \) \(\mathstrut +\mathstrut 2646053822502q^{53} \) \(\mathstrut +\mathstrut 2447722649502q^{54} \) \(\mathstrut +\mathstrut 9552856606920q^{55} \) \(\mathstrut -\mathstrut 3464287706880q^{56} \) \(\mathstrut +\mathstrut 5465468320740q^{57} \) \(\mathstrut +\mathstrut 11328749296020q^{58} \) \(\mathstrut +\mathstrut 7399371294540q^{59} \) \(\mathstrut -\mathstrut 13498713986760q^{60} \) \(\mathstrut -\mathstrut 42659617819498q^{61} \) \(\mathstrut -\mathstrut 30548060185968q^{62} \) \(\mathstrut -\mathstrut 6568661778336q^{63} \) \(\mathstrut -\mathstrut 9479308064192q^{64} \) \(\mathstrut +\mathstrut 107798889166020q^{65} \) \(\mathstrut +\mathstrut 17415663109416q^{66} \) \(\mathstrut -\mathstrut 56408026065964q^{67} \) \(\mathstrut +\mathstrut 27687456400968q^{68} \) \(\mathstrut -\mathstrut 24679931240664q^{69} \) \(\mathstrut +\mathstrut 90209666252160q^{70} \) \(\mathstrut -\mathstrut 133149677299848q^{71} \) \(\mathstrut +\mathstrut 12065134961880q^{72} \) \(\mathstrut +\mathstrut 105603350884922q^{73} \) \(\mathstrut +\mathstrut 46852256307636q^{74} \) \(\mathstrut -\mathstrut 105589510307325q^{75} \) \(\mathstrut -\mathstrut 54949573587760q^{76} \) \(\mathstrut -\mathstrut 46736341077888q^{77} \) \(\mathstrut +\mathstrut 196526464756596q^{78} \) \(\mathstrut -\mathstrut 55665674361880q^{79} \) \(\mathstrut -\mathstrut 367946938369440q^{80} \) \(\mathstrut +\mathstrut 22876792454961q^{81} \) \(\mathstrut -\mathstrut 158919163463268q^{82} \) \(\mathstrut +\mathstrut 378077412997332q^{83} \) \(\mathstrut +\mathstrut 66041031176064q^{84} \) \(\mathstrut +\mathstrut 353472161466060q^{85} \) \(\mathstrut -\mathstrut 65398833604728q^{86} \) \(\mathstrut +\mathstrut 105880233805110q^{87} \) \(\mathstrut +\mathstrut 85844009291040q^{88} \) \(\mathstrut +\mathstrut 219315065897610q^{89} \) \(\mathstrut -\mathstrut 314174771349660q^{90} \) \(\mathstrut -\mathstrut 527394669384128q^{91} \) \(\mathstrut +\mathstrut 248130922779936q^{92} \) \(\mathstrut -\mathstrut 285506870199624q^{93} \) \(\mathstrut -\mathstrut 355837442822784q^{94} \) \(\mathstrut -\mathstrut 701514226024200q^{95} \) \(\mathstrut -\mathstrut 490025389999392q^{96} \) \(\mathstrut +\mathstrut 703322682162626q^{97} \) \(\mathstrut +\mathstrut 669588137620038q^{98} \) \(\mathstrut +\mathstrut 162769466753388q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
0
−234.000 −2187.00 21988.0 280710. 511758. −1.37334e6 2.52252e6 4.78297e6 −6.56861e7
\(n\): 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
\(3\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2} \) \(\mathstrut +\mathstrut 234 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(3))\).