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: \(12.2934908890\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{109}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 8)
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.

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

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
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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 16.12.a.d 2
3.b odd 2 1 144.12.a.p 2
4.b odd 2 1 8.12.a.b 2
8.b even 2 1 64.12.a.k 2
8.d odd 2 1 64.12.a.h 2
12.b even 2 1 72.12.a.e 2
16.e even 4 2 256.12.b.k 4
16.f odd 4 2 256.12.b.h 4
20.d odd 2 1 200.12.a.d 2
20.e even 4 2 200.12.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.12.a.b 2 4.b odd 2 1
16.12.a.d 2 1.a even 1 1 trivial
64.12.a.h 2 8.d odd 2 1
64.12.a.k 2 8.b even 2 1
72.12.a.e 2 12.b even 2 1
144.12.a.p 2 3.b odd 2 1
200.12.a.d 2 20.d odd 2 1
200.12.c.c 4 20.e even 4 2
256.12.b.h 4 16.f odd 4 2
256.12.b.k 4 16.e even 4 2

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 56 T - 91386 T^{2} + 9920232 T^{3} + 31381059609 T^{4} \)
$5$ \( 1 - 7868 T + 48841790 T^{2} - 384179687500 T^{3} + 2384185791015625 T^{4} \)
$7$ \( 1 + 91056 T + 5239889774 T^{2} + 180047463910608 T^{3} + 3909821048582988049 T^{4} \)
$11$ \( 1 + 159080 T + 362536063286 T^{2} + 45387380560797880 T^{3} + \)\(81\!\cdots\!21\)\( T^{4} \)
$13$ \( 1 - 1050476 T + 3458956762062 T^{2} - 1882621482086411612 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \)
$17$ \( 1 - 1430884 T + 16748384809766 T^{2} - 49039108076251137572 T^{3} + \)\(11\!\cdots\!89\)\( T^{4} \)
$19$ \( 1 - 21866600 T + 343123710088422 T^{2} - \)\(25\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!61\)\( T^{4} \)
$23$ \( 1 + 35806736 T + 1952928132896462 T^{2} + \)\(34\!\cdots\!72\)\( T^{3} + \)\(90\!\cdots\!29\)\( T^{4} \)
$29$ \( 1 + 228827700 T + 35907977791027054 T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!41\)\( T^{4} \)
$31$ \( 1 + 64722112 T + 51498184379920062 T^{2} + \)\(16\!\cdots\!72\)\( T^{3} + \)\(64\!\cdots\!61\)\( T^{4} \)
$37$ \( 1 - 75558780 T + 354292341022528382 T^{2} - \)\(13\!\cdots\!40\)\( T^{3} + \)\(31\!\cdots\!69\)\( T^{4} \)
$41$ \( 1 - 1201214196 T + 1274442257818453270 T^{2} - \)\(66\!\cdots\!36\)\( T^{3} + \)\(30\!\cdots\!81\)\( T^{4} \)
$43$ \( 1 - 45519832 T + 1812222731951247606 T^{2} - \)\(42\!\cdots\!24\)\( T^{3} + \)\(86\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - 1229079264 T + 5024390553242310430 T^{2} - \)\(30\!\cdots\!92\)\( T^{3} + \)\(61\!\cdots\!09\)\( T^{4} \)
$53$ \( 1 + 3808549924 T + 16648234587904299038 T^{2} + \)\(35\!\cdots\!28\)\( T^{3} + \)\(85\!\cdots\!09\)\( T^{4} \)
$59$ \( 1 - 6012926584 T + 61066799326040273366 T^{2} - \)\(18\!\cdots\!56\)\( T^{3} + \)\(90\!\cdots\!81\)\( T^{4} \)
$61$ \( 1 - 9789792908 T + \)\(10\!\cdots\!42\)\( T^{2} - \)\(42\!\cdots\!88\)\( T^{3} + \)\(18\!\cdots\!21\)\( T^{4} \)
$67$ \( 1 + 14703095224 T + 95414710392374126214 T^{2} + \)\(17\!\cdots\!92\)\( T^{3} + \)\(14\!\cdots\!89\)\( T^{4} \)
$71$ \( 1 + 4319991088 T + \)\(28\!\cdots\!82\)\( T^{2} + \)\(99\!\cdots\!48\)\( T^{3} + \)\(53\!\cdots\!41\)\( T^{4} \)
$73$ \( 1 - 11055639476 T + \)\(65\!\cdots\!62\)\( T^{2} - \)\(34\!\cdots\!52\)\( T^{3} + \)\(98\!\cdots\!29\)\( T^{4} \)
$79$ \( 1 + 51957623264 T + \)\(14\!\cdots\!78\)\( T^{2} + \)\(38\!\cdots\!56\)\( T^{3} + \)\(55\!\cdots\!41\)\( T^{4} \)
$83$ \( 1 - 108227975912 T + \)\(54\!\cdots\!06\)\( T^{2} - \)\(13\!\cdots\!04\)\( T^{3} + \)\(16\!\cdots\!89\)\( T^{4} \)
$89$ \( 1 - 71188291860 T + \)\(31\!\cdots\!82\)\( T^{2} - \)\(19\!\cdots\!40\)\( T^{3} + \)\(77\!\cdots\!21\)\( T^{4} \)
$97$ \( 1 + 1699807676 T + \)\(12\!\cdots\!50\)\( T^{2} + \)\(12\!\cdots\!28\)\( T^{3} + \)\(51\!\cdots\!09\)\( T^{4} \)
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