Properties

Label 16.12.a.d
Level 16
Weight 12
Character orbit 16.a
Self dual Yes
Analytic conductor 12.293
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 16 = 2^{4} \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 16.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(12.293490889\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{109}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 64\sqrt{109}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -28 - \beta ) q^{3} \) \( + ( 3934 - 12 \beta ) q^{5} \) \( + ( -45528 - 42 \beta ) q^{7} \) \( + ( 270101 + 56 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -28 - \beta ) q^{3} \) \( + ( 3934 - 12 \beta ) q^{5} \) \( + ( -45528 - 42 \beta ) q^{7} \) \( + ( 270101 + 56 \beta ) q^{9} \) \( + ( -79540 + 693 \beta ) q^{11} \) \( + ( 525238 + 948 \beta ) q^{13} \) \( + ( 5247416 - 3598 \beta ) q^{15} \) \( + ( 715442 - 10824 \beta ) q^{17} \) \( + ( 10933300 + 4587 \beta ) q^{19} \) \( + ( 20026272 + 46704 \beta ) q^{21} \) \( + ( -17903368 + 24738 \beta ) q^{23} \) \( + ( 30939047 - 94416 \beta ) q^{25} \) \( + ( -27604696 - 94522 \beta ) q^{27} \) \( + ( -114413850 + 59556 \beta ) q^{29} \) \( + ( -32361056 + 28632 \beta ) q^{31} \) \( + ( -307172432 + 60136 \beta ) q^{33} \) \( + ( 45910704 + 381108 \beta ) q^{35} \) \( + ( 37779390 - 81564 \beta ) q^{37} \) \( + ( -437954536 - 551782 \beta ) q^{39} \) \( + ( 600607098 + 647088 \beta ) q^{41} \) \( + ( 22759916 - 324051 \beta ) q^{43} \) \( + ( 762553526 - 3020908 \beta ) q^{45} \) \( + ( 614539632 + 816420 \beta ) q^{47} \) \( + ( 883034537 + 3824352 \beta ) q^{49} \) \( + ( 4812493960 - 412370 \beta ) q^{51} \) \( + ( -1904274962 + 3514980 \beta ) q^{53} \) \( + ( -4025704984 + 3680742 \beta ) q^{55} \) \( + ( -2354062768 - 11061736 \beta ) q^{57} \) \( + ( 3006463292 - 4307463 \beta ) q^{59} \) \( + ( 4894896454 + 4588692 \beta ) q^{61} \) \( + ( -13347241656 - 13893810 \beta ) q^{63} \) \( + ( -3012688172 - 2573424 \beta ) q^{65} \) \( + ( -7351547612 + 21317583 \beta ) q^{67} \) \( + ( -10543332128 + 17210704 \beta ) q^{69} \) \( + ( -2159995544 + 20099142 \beta ) q^{71} \) \( + ( 5527819738 - 3259368 \beta ) q^{73} \) \( + ( 41287051708 - 28295399 \beta ) q^{75} \) \( + ( -9373484064 - 28210224 \beta ) q^{77} \) \( + ( -25978811632 - 41373444 \beta ) q^{79} \) \( + ( -4873980151 + 20331080 \beta ) q^{81} \) \( + ( 54113987956 + 13552251 \beta ) q^{83} \) \( + ( 60804864860 - 51166920 \beta ) q^{85} \) \( + ( -23386022184 + 112746282 \beta ) q^{87} \) \( + ( 35594145930 + 90758808 \beta ) q^{89} \) \( + ( -41689446288 - 65220540 \beta ) q^{91} \) \( + ( -11877047680 + 31559360 \beta ) q^{93} \) \( + ( 18436437784 - 113154342 \beta ) q^{95} \) \( + ( -849903838 - 64199400 \beta ) q^{97} \) \( + ( -4157458628 + 182725753 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 56q^{3} \) \(\mathstrut +\mathstrut 7868q^{5} \) \(\mathstrut -\mathstrut 91056q^{7} \) \(\mathstrut +\mathstrut 540202q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 56q^{3} \) \(\mathstrut +\mathstrut 7868q^{5} \) \(\mathstrut -\mathstrut 91056q^{7} \) \(\mathstrut +\mathstrut 540202q^{9} \) \(\mathstrut -\mathstrut 159080q^{11} \) \(\mathstrut +\mathstrut 1050476q^{13} \) \(\mathstrut +\mathstrut 10494832q^{15} \) \(\mathstrut +\mathstrut 1430884q^{17} \) \(\mathstrut +\mathstrut 21866600q^{19} \) \(\mathstrut +\mathstrut 40052544q^{21} \) \(\mathstrut -\mathstrut 35806736q^{23} \) \(\mathstrut +\mathstrut 61878094q^{25} \) \(\mathstrut -\mathstrut 55209392q^{27} \) \(\mathstrut -\mathstrut 228827700q^{29} \) \(\mathstrut -\mathstrut 64722112q^{31} \) \(\mathstrut -\mathstrut 614344864q^{33} \) \(\mathstrut +\mathstrut 91821408q^{35} \) \(\mathstrut +\mathstrut 75558780q^{37} \) \(\mathstrut -\mathstrut 875909072q^{39} \) \(\mathstrut +\mathstrut 1201214196q^{41} \) \(\mathstrut +\mathstrut 45519832q^{43} \) \(\mathstrut +\mathstrut 1525107052q^{45} \) \(\mathstrut +\mathstrut 1229079264q^{47} \) \(\mathstrut +\mathstrut 1766069074q^{49} \) \(\mathstrut +\mathstrut 9624987920q^{51} \) \(\mathstrut -\mathstrut 3808549924q^{53} \) \(\mathstrut -\mathstrut 8051409968q^{55} \) \(\mathstrut -\mathstrut 4708125536q^{57} \) \(\mathstrut +\mathstrut 6012926584q^{59} \) \(\mathstrut +\mathstrut 9789792908q^{61} \) \(\mathstrut -\mathstrut 26694483312q^{63} \) \(\mathstrut -\mathstrut 6025376344q^{65} \) \(\mathstrut -\mathstrut 14703095224q^{67} \) \(\mathstrut -\mathstrut 21086664256q^{69} \) \(\mathstrut -\mathstrut 4319991088q^{71} \) \(\mathstrut +\mathstrut 11055639476q^{73} \) \(\mathstrut +\mathstrut 82574103416q^{75} \) \(\mathstrut -\mathstrut 18746968128q^{77} \) \(\mathstrut -\mathstrut 51957623264q^{79} \) \(\mathstrut -\mathstrut 9747960302q^{81} \) \(\mathstrut +\mathstrut 108227975912q^{83} \) \(\mathstrut +\mathstrut 121609729720q^{85} \) \(\mathstrut -\mathstrut 46772044368q^{87} \) \(\mathstrut +\mathstrut 71188291860q^{89} \) \(\mathstrut -\mathstrut 83378892576q^{91} \) \(\mathstrut -\mathstrut 23754095360q^{93} \) \(\mathstrut +\mathstrut 36872875568q^{95} \) \(\mathstrut -\mathstrut 1699807676q^{97} \) \(\mathstrut -\mathstrut 8314917256q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
5.72015
−4.72015
0 −696.180 0 −4084.16 0 −73591.5 0 307519. 0
1.2 0 640.180 0 11952.2 0 −17464.5 0 232683. 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

This newform can be constructed as the kernel of the linear operator \(T_{3}^{2} \) \(\mathstrut +\mathstrut 56 T_{3} \) \(\mathstrut -\mathstrut 445680 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(16))\).