Properties

Label 16.34.a.b
Level 16
Weight 34
Character orbit 16.a
Self dual yes
Analytic conductor 110.373
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(110.372526210\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Defining polynomial: \(x^{2} - x - 589050\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 576\sqrt{2356201}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -18959940 - 39 \beta ) q^{3} + ( -90530768250 - 125500 \beta ) q^{5} + ( 33576540033400 - 100199358 \beta ) q^{7} + ( -4010568477485427 + 1478875320 \beta ) q^{9} +O(q^{10})\) \( q +(-18959940 - 39 \beta) q^{3} +(-90530768250 - 125500 \beta) q^{5} +(33576540033400 - 100199358 \beta) q^{7} +(-4010568477485427 + 1478875320 \beta) q^{9} +(-66935907720957132 + 168326472875 \beta) q^{11} +(-1490805239129721970 - 467324482716 \beta) q^{13} +(5542640034569937000 + 5910172431750 \beta) q^{15} +(-39680574630587762670 - 332761335850632 \beta) q^{17} +(\)\(68\!\cdots\!00\)\( - 610587893846115 \beta) q^{19} +(\)\(24\!\cdots\!12\)\( + 590288754415920 \beta) q^{21} +(-\)\(13\!\cdots\!20\)\( - 20578649420360954 \beta) q^{23} +(-\)\(95\!\cdots\!25\)\( + 22723222830750000 \beta) q^{25} +(\)\(13\!\cdots\!20\)\( + 345176145382916250 \beta) q^{27} +(-\)\(83\!\cdots\!50\)\( - 760643170023021740 \beta) q^{29} +(\)\(31\!\cdots\!08\)\( - 4114562133058947000 \beta) q^{31} +(-\)\(38\!\cdots\!20\)\( - 580959425004299352 \beta) q^{33} +(\)\(67\!\cdots\!00\)\( + 4857269083705083500 \beta) q^{35} +(-\)\(52\!\cdots\!10\)\( + 25166008023558787188 \beta) q^{37} +(\)\(42\!\cdots\!24\)\( + 67001848478885553870 \beta) q^{39} +(\)\(13\!\cdots\!22\)\( - \)\(40\!\cdots\!00\)\( \beta) q^{41} +(-\)\(78\!\cdots\!00\)\( - \)\(11\!\cdots\!69\)\( \beta) q^{43} +(\)\(21\!\cdots\!50\)\( + \)\(36\!\cdots\!00\)\( \beta) q^{45} +(-\)\(27\!\cdots\!20\)\( - \)\(61\!\cdots\!48\)\( \beta) q^{47} +(\)\(12\!\cdots\!57\)\( - \)\(67\!\cdots\!00\)\( \beta) q^{49} +(\)\(10\!\cdots\!48\)\( + \)\(78\!\cdots\!10\)\( \beta) q^{51} +(-\)\(13\!\cdots\!10\)\( + \)\(25\!\cdots\!64\)\( \beta) q^{53} +(-\)\(10\!\cdots\!00\)\( - \)\(68\!\cdots\!50\)\( \beta) q^{55} +(\)\(57\!\cdots\!60\)\( - \)\(14\!\cdots\!00\)\( \beta) q^{57} +(\)\(15\!\cdots\!00\)\( - \)\(39\!\cdots\!45\)\( \beta) q^{59} +(-\)\(28\!\cdots\!18\)\( - \)\(79\!\cdots\!00\)\( \beta) q^{61} +(-\)\(25\!\cdots\!60\)\( + \)\(45\!\cdots\!66\)\( \beta) q^{63} +(\)\(18\!\cdots\!00\)\( + \)\(22\!\cdots\!00\)\( \beta) q^{65} +(-\)\(79\!\cdots\!80\)\( - \)\(35\!\cdots\!43\)\( \beta) q^{67} +(\)\(87\!\cdots\!56\)\( + \)\(90\!\cdots\!40\)\( \beta) q^{69} +(\)\(13\!\cdots\!88\)\( - \)\(23\!\cdots\!50\)\( \beta) q^{71} +(\)\(47\!\cdots\!70\)\( - \)\(73\!\cdots\!96\)\( \beta) q^{73} +(\)\(11\!\cdots\!00\)\( + \)\(33\!\cdots\!75\)\( \beta) q^{75} +(-\)\(15\!\cdots\!00\)\( + \)\(12\!\cdots\!56\)\( \beta) q^{77} +(\)\(42\!\cdots\!00\)\( + \)\(14\!\cdots\!40\)\( \beta) q^{79} +(\)\(91\!\cdots\!21\)\( - \)\(20\!\cdots\!40\)\( \beta) q^{81} +(-\)\(14\!\cdots\!60\)\( - \)\(30\!\cdots\!99\)\( \beta) q^{83} +(\)\(36\!\cdots\!00\)\( + \)\(35\!\cdots\!00\)\( \beta) q^{85} +(\)\(39\!\cdots\!60\)\( + \)\(47\!\cdots\!50\)\( \beta) q^{87} +(\)\(66\!\cdots\!50\)\( - \)\(47\!\cdots\!20\)\( \beta) q^{89} +(-\)\(13\!\cdots\!72\)\( + \)\(13\!\cdots\!60\)\( \beta) q^{91} +(\)\(66\!\cdots\!80\)\( - \)\(43\!\cdots\!12\)\( \beta) q^{93} +(-\)\(16\!\cdots\!00\)\( - \)\(30\!\cdots\!50\)\( \beta) q^{95} +(-\)\(18\!\cdots\!30\)\( + \)\(66\!\cdots\!48\)\( \beta) q^{97} +(\)\(46\!\cdots\!64\)\( - \)\(77\!\cdots\!65\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 37919880q^{3} - 181061536500q^{5} + 67153080066800q^{7} - 8021136954970854q^{9} + O(q^{10}) \) \( 2q - 37919880q^{3} - 181061536500q^{5} + 67153080066800q^{7} - 8021136954970854q^{9} - 133871815441914264q^{11} - 2981610478259443940q^{13} + 11085280069139874000q^{15} - 79361149261175525340q^{17} + \)\(13\!\cdots\!00\)\(q^{19} + \)\(48\!\cdots\!24\)\(q^{21} - \)\(26\!\cdots\!40\)\(q^{23} - \)\(19\!\cdots\!50\)\(q^{25} + \)\(27\!\cdots\!40\)\(q^{27} - \)\(16\!\cdots\!00\)\(q^{29} + \)\(62\!\cdots\!16\)\(q^{31} - \)\(77\!\cdots\!40\)\(q^{33} + \)\(13\!\cdots\!00\)\(q^{35} - \)\(10\!\cdots\!20\)\(q^{37} + \)\(85\!\cdots\!48\)\(q^{39} + \)\(27\!\cdots\!44\)\(q^{41} - \)\(15\!\cdots\!00\)\(q^{43} + \)\(43\!\cdots\!00\)\(q^{45} - \)\(54\!\cdots\!40\)\(q^{47} + \)\(24\!\cdots\!14\)\(q^{49} + \)\(21\!\cdots\!96\)\(q^{51} - \)\(26\!\cdots\!20\)\(q^{53} - \)\(20\!\cdots\!00\)\(q^{55} + \)\(11\!\cdots\!20\)\(q^{57} + \)\(30\!\cdots\!00\)\(q^{59} - \)\(57\!\cdots\!36\)\(q^{61} - \)\(50\!\cdots\!20\)\(q^{63} + \)\(36\!\cdots\!00\)\(q^{65} - \)\(15\!\cdots\!60\)\(q^{67} + \)\(17\!\cdots\!12\)\(q^{69} + \)\(26\!\cdots\!76\)\(q^{71} + \)\(94\!\cdots\!40\)\(q^{73} + \)\(22\!\cdots\!00\)\(q^{75} - \)\(30\!\cdots\!00\)\(q^{77} + \)\(85\!\cdots\!00\)\(q^{79} + \)\(18\!\cdots\!42\)\(q^{81} - \)\(29\!\cdots\!20\)\(q^{83} + \)\(72\!\cdots\!00\)\(q^{85} + \)\(78\!\cdots\!20\)\(q^{87} + \)\(13\!\cdots\!00\)\(q^{89} - \)\(26\!\cdots\!44\)\(q^{91} + \)\(13\!\cdots\!60\)\(q^{93} - \)\(33\!\cdots\!00\)\(q^{95} - \)\(36\!\cdots\!60\)\(q^{97} + \)\(92\!\cdots\!28\)\(q^{99} + O(q^{100}) \)

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
767.996
−766.996
0 −5.34420e7 0 −2.01492e11 0 −5.50153e13 0 −2.70301e15 0
1.2 0 1.55221e7 0 2.04307e10 0 1.22168e14 0 −5.31812e15 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 16.34.a.b 2
4.b odd 2 1 1.34.a.a 2
12.b even 2 1 9.34.a.b 2
20.d odd 2 1 25.34.a.a 2
20.e even 4 2 25.34.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.34.a.a 2 4.b odd 2 1
9.34.a.b 2 12.b even 2 1
16.34.a.b 2 1.a even 1 1 trivial
25.34.a.a 2 20.d odd 2 1
25.34.b.a 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 37919880 T_{3} - \)\(82\!\cdots\!96\)\( \) acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(16))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 37919880 T + 10288587693648150 T^{2} + \)\(21\!\cdots\!40\)\( T^{3} + \)\(30\!\cdots\!29\)\( T^{4} \)
$5$ \( 1 + 181061536500 T + \)\(22\!\cdots\!50\)\( T^{2} + \)\(21\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!25\)\( T^{4} \)
$7$ \( 1 - 67153080066800 T + \)\(87\!\cdots\!50\)\( T^{2} - \)\(51\!\cdots\!00\)\( T^{3} + \)\(59\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 + 133871815441914264 T + \)\(28\!\cdots\!86\)\( T^{2} + \)\(31\!\cdots\!84\)\( T^{3} + \)\(53\!\cdots\!61\)\( T^{4} \)
$13$ \( 1 + 2981610478259443940 T + \)\(13\!\cdots\!50\)\( T^{2} + \)\(17\!\cdots\!20\)\( T^{3} + \)\(33\!\cdots\!09\)\( T^{4} \)
$17$ \( 1 + 79361149261175525340 T - \)\(44\!\cdots\!50\)\( T^{2} + \)\(31\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!69\)\( T^{4} \)
$19$ \( 1 - \)\(13\!\cdots\!00\)\( T + \)\(33\!\cdots\!18\)\( T^{2} - \)\(21\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!81\)\( T^{4} \)
$23$ \( 1 + \)\(26\!\cdots\!40\)\( T + \)\(15\!\cdots\!50\)\( T^{2} + \)\(22\!\cdots\!20\)\( T^{3} + \)\(74\!\cdots\!89\)\( T^{4} \)
$29$ \( 1 + \)\(16\!\cdots\!00\)\( T + \)\(38\!\cdots\!78\)\( T^{2} + \)\(30\!\cdots\!00\)\( T^{3} + \)\(32\!\cdots\!21\)\( T^{4} \)
$31$ \( 1 - \)\(62\!\cdots\!16\)\( T + \)\(29\!\cdots\!46\)\( T^{2} - \)\(10\!\cdots\!56\)\( T^{3} + \)\(26\!\cdots\!81\)\( T^{4} \)
$37$ \( 1 + \)\(10\!\cdots\!20\)\( T + \)\(13\!\cdots\!50\)\( T^{2} + \)\(59\!\cdots\!40\)\( T^{3} + \)\(31\!\cdots\!09\)\( T^{4} \)
$41$ \( 1 - \)\(27\!\cdots\!44\)\( T + \)\(22\!\cdots\!26\)\( T^{2} - \)\(46\!\cdots\!24\)\( T^{3} + \)\(27\!\cdots\!41\)\( T^{4} \)
$43$ \( 1 + \)\(15\!\cdots\!00\)\( T + \)\(12\!\cdots\!50\)\( T^{2} + \)\(12\!\cdots\!00\)\( T^{3} + \)\(64\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 + \)\(54\!\cdots\!40\)\( T + \)\(37\!\cdots\!50\)\( T^{2} + \)\(81\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!29\)\( T^{4} \)
$53$ \( 1 + \)\(26\!\cdots\!20\)\( T + \)\(12\!\cdots\!50\)\( T^{2} + \)\(21\!\cdots\!60\)\( T^{3} + \)\(63\!\cdots\!29\)\( T^{4} \)
$59$ \( 1 - \)\(30\!\cdots\!00\)\( T + \)\(77\!\cdots\!58\)\( T^{2} - \)\(83\!\cdots\!00\)\( T^{3} + \)\(75\!\cdots\!41\)\( T^{4} \)
$61$ \( 1 + \)\(57\!\cdots\!36\)\( T + \)\(16\!\cdots\!86\)\( T^{2} + \)\(47\!\cdots\!16\)\( T^{3} + \)\(67\!\cdots\!61\)\( T^{4} \)
$67$ \( 1 + \)\(15\!\cdots\!60\)\( T + \)\(41\!\cdots\!50\)\( T^{2} + \)\(29\!\cdots\!20\)\( T^{3} + \)\(33\!\cdots\!69\)\( T^{4} \)
$71$ \( 1 - \)\(26\!\cdots\!76\)\( T + \)\(22\!\cdots\!66\)\( T^{2} - \)\(32\!\cdots\!36\)\( T^{3} + \)\(15\!\cdots\!21\)\( T^{4} \)
$73$ \( 1 - \)\(94\!\cdots\!40\)\( T + \)\(19\!\cdots\!50\)\( T^{2} - \)\(29\!\cdots\!20\)\( T^{3} + \)\(95\!\cdots\!89\)\( T^{4} \)
$79$ \( 1 - \)\(85\!\cdots\!00\)\( T + \)\(85\!\cdots\!78\)\( T^{2} - \)\(35\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!21\)\( T^{4} \)
$83$ \( 1 + \)\(29\!\cdots\!20\)\( T + \)\(37\!\cdots\!50\)\( T^{2} + \)\(62\!\cdots\!60\)\( T^{3} + \)\(45\!\cdots\!69\)\( T^{4} \)
$89$ \( 1 - \)\(13\!\cdots\!00\)\( T + \)\(47\!\cdots\!38\)\( T^{2} - \)\(28\!\cdots\!00\)\( T^{3} + \)\(45\!\cdots\!61\)\( T^{4} \)
$97$ \( 1 + \)\(36\!\cdots\!60\)\( T + \)\(42\!\cdots\!50\)\( T^{2} + \)\(13\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!29\)\( T^{4} \)
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