Properties

Label 16.38.a.b
Level $16$
Weight $38$
Character orbit 16.a
Self dual yes
Analytic conductor $138.742$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 38 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.742460999\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Defining polynomial: \(x^{2} - x - 15934380\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3 \)
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 \(\beta = 384\sqrt{63737521}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -6995700 - 9 \beta ) q^{3} + ( 2764792192950 + 3608300 \beta ) q^{5} + ( 1724221976743000 + 1206250542 \beta ) q^{7} + ( -449473689200884707 + 125922600 \beta ) q^{9} +O(q^{10})\) \( q +(-6995700 - 9 \beta) q^{3} +(2764792192950 + 3608300 \beta) q^{5} +(1724221976743000 + 1206250542 \beta) q^{7} +(-449473689200884707 + 125922600 \beta) q^{9} +(13367018177424269028 - 4023288308875 \beta) q^{11} +(\)\(26\!\cdots\!50\)\( - 90555425090676 \beta) q^{13} +(-\)\(32\!\cdots\!00\)\( - 50125714046550 \beta) q^{15} +(-\)\(44\!\cdots\!50\)\( - 3079186242321432 \beta) q^{17} +(-\)\(18\!\cdots\!60\)\( + 161622022785894675 \beta) q^{19} +(-\)\(11\!\cdots\!28\)\( - 23956564707356400 \beta) q^{21} +(\)\(13\!\cdots\!00\)\( + 4253077857910738186 \beta) q^{23} +(\)\(57\!\cdots\!75\)\( + 19952399339642970000 \beta) q^{25} +(\)\(62\!\cdots\!00\)\( + 8096937439094118630 \beta) q^{27} +(-\)\(63\!\cdots\!10\)\( + \)\(28\!\cdots\!00\)\( \beta) q^{29} +(-\)\(13\!\cdots\!12\)\( + \)\(80\!\cdots\!00\)\( \beta) q^{31} +(\)\(24\!\cdots\!00\)\( - 92157445574421583752 \beta) q^{33} +(\)\(45\!\cdots\!00\)\( + \)\(95\!\cdots\!00\)\( \beta) q^{35} +(-\)\(34\!\cdots\!50\)\( + \)\(31\!\cdots\!88\)\( \beta) q^{37} +(\)\(58\!\cdots\!84\)\( - \)\(17\!\cdots\!50\)\( \beta) q^{39} +(-\)\(63\!\cdots\!18\)\( - \)\(72\!\cdots\!00\)\( \beta) q^{41} +(\)\(12\!\cdots\!00\)\( - \)\(53\!\cdots\!19\)\( \beta) q^{43} +(-\)\(12\!\cdots\!50\)\( - \)\(16\!\cdots\!00\)\( \beta) q^{45} +(-\)\(21\!\cdots\!00\)\( + \)\(47\!\cdots\!52\)\( \beta) q^{47} +(-\)\(19\!\cdots\!43\)\( + \)\(41\!\cdots\!00\)\( \beta) q^{49} +(\)\(57\!\cdots\!88\)\( + \)\(42\!\cdots\!50\)\( \beta) q^{51} +(\)\(79\!\cdots\!50\)\( + \)\(14\!\cdots\!84\)\( \beta) q^{53} +(-\)\(99\!\cdots\!00\)\( + \)\(37\!\cdots\!50\)\( \beta) q^{55} +(-\)\(12\!\cdots\!00\)\( + \)\(54\!\cdots\!40\)\( \beta) q^{57} +(\)\(11\!\cdots\!20\)\( - \)\(60\!\cdots\!75\)\( \beta) q^{59} +(\)\(50\!\cdots\!22\)\( - \)\(36\!\cdots\!00\)\( \beta) q^{61} +(-\)\(77\!\cdots\!00\)\( - \)\(54\!\cdots\!94\)\( \beta) q^{63} +(-\)\(23\!\cdots\!00\)\( + \)\(70\!\cdots\!00\)\( \beta) q^{65} +(\)\(54\!\cdots\!00\)\( + \)\(97\!\cdots\!07\)\( \beta) q^{67} +(-\)\(45\!\cdots\!24\)\( - \)\(14\!\cdots\!00\)\( \beta) q^{69} +(\)\(37\!\cdots\!08\)\( + \)\(21\!\cdots\!50\)\( \beta) q^{71} +(\)\(98\!\cdots\!50\)\( - \)\(14\!\cdots\!36\)\( \beta) q^{73} +(-\)\(20\!\cdots\!00\)\( - \)\(65\!\cdots\!75\)\( \beta) q^{75} +(-\)\(22\!\cdots\!00\)\( + \)\(91\!\cdots\!76\)\( \beta) q^{77} +(-\)\(13\!\cdots\!40\)\( - \)\(51\!\cdots\!00\)\( \beta) q^{79} +(\)\(20\!\cdots\!21\)\( - \)\(16\!\cdots\!00\)\( \beta) q^{81} +(\)\(23\!\cdots\!00\)\( - \)\(29\!\cdots\!29\)\( \beta) q^{83} +(-\)\(22\!\cdots\!00\)\( - \)\(16\!\cdots\!00\)\( \beta) q^{85} +(-\)\(19\!\cdots\!00\)\( + \)\(36\!\cdots\!90\)\( \beta) q^{87} +(-\)\(66\!\cdots\!30\)\( + \)\(29\!\cdots\!00\)\( \beta) q^{89} +(-\)\(56\!\cdots\!92\)\( + \)\(16\!\cdots\!00\)\( \beta) q^{91} +(-\)\(67\!\cdots\!00\)\( - \)\(44\!\cdots\!92\)\( \beta) q^{93} +(\)\(49\!\cdots\!00\)\( - \)\(22\!\cdots\!50\)\( \beta) q^{95} +(\)\(30\!\cdots\!50\)\( + \)\(13\!\cdots\!48\)\( \beta) q^{97} +(-\)\(60\!\cdots\!96\)\( + \)\(18\!\cdots\!25\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 13991400q^{3} + 5529584385900q^{5} + 3448443953486000q^{7} - 898947378401769414q^{9} + O(q^{10}) \) \( 2q - 13991400q^{3} + 5529584385900q^{5} + 3448443953486000q^{7} - 898947378401769414q^{9} + 26734036354848538056q^{11} + \)\(53\!\cdots\!00\)\(q^{13} - \)\(64\!\cdots\!00\)\(q^{15} - \)\(89\!\cdots\!00\)\(q^{17} - \)\(37\!\cdots\!20\)\(q^{19} - \)\(22\!\cdots\!56\)\(q^{21} + \)\(26\!\cdots\!00\)\(q^{23} + \)\(11\!\cdots\!50\)\(q^{25} + \)\(12\!\cdots\!00\)\(q^{27} - \)\(12\!\cdots\!20\)\(q^{29} - \)\(26\!\cdots\!24\)\(q^{31} + \)\(49\!\cdots\!00\)\(q^{33} + \)\(91\!\cdots\!00\)\(q^{35} - \)\(68\!\cdots\!00\)\(q^{37} + \)\(11\!\cdots\!68\)\(q^{39} - \)\(12\!\cdots\!36\)\(q^{41} + \)\(25\!\cdots\!00\)\(q^{43} - \)\(24\!\cdots\!00\)\(q^{45} - \)\(42\!\cdots\!00\)\(q^{47} - \)\(38\!\cdots\!86\)\(q^{49} + \)\(11\!\cdots\!76\)\(q^{51} + \)\(15\!\cdots\!00\)\(q^{53} - \)\(19\!\cdots\!00\)\(q^{55} - \)\(24\!\cdots\!00\)\(q^{57} + \)\(23\!\cdots\!40\)\(q^{59} + \)\(10\!\cdots\!44\)\(q^{61} - \)\(15\!\cdots\!00\)\(q^{63} - \)\(46\!\cdots\!00\)\(q^{65} + \)\(10\!\cdots\!00\)\(q^{67} - \)\(90\!\cdots\!48\)\(q^{69} + \)\(74\!\cdots\!16\)\(q^{71} + \)\(19\!\cdots\!00\)\(q^{73} - \)\(41\!\cdots\!00\)\(q^{75} - \)\(45\!\cdots\!00\)\(q^{77} - \)\(27\!\cdots\!80\)\(q^{79} + \)\(40\!\cdots\!42\)\(q^{81} + \)\(47\!\cdots\!00\)\(q^{83} - \)\(45\!\cdots\!00\)\(q^{85} - \)\(39\!\cdots\!00\)\(q^{87} - \)\(13\!\cdots\!60\)\(q^{89} - \)\(11\!\cdots\!84\)\(q^{91} - \)\(13\!\cdots\!00\)\(q^{93} + \)\(99\!\cdots\!00\)\(q^{95} + \)\(60\!\cdots\!00\)\(q^{97} - \)\(12\!\cdots\!92\)\(q^{99} + O(q^{100}) \)

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
3992.29
−3991.29
0 −3.45869e7 0 1.38267e13 0 5.42222e15 0 −4.49088e17 0
1.2 0 2.05955e7 0 −8.29715e12 0 −1.97377e15 0 −4.49860e17 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 16.38.a.b 2
4.b odd 2 1 1.38.a.a 2
12.b even 2 1 9.38.a.a 2
20.d odd 2 1 25.38.a.a 2
20.e even 4 2 25.38.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.38.a.a 2 4.b odd 2 1
9.38.a.a 2 12.b even 2 1
16.38.a.b 2 1.a even 1 1 trivial
25.38.a.a 2 20.d odd 2 1
25.38.b.a 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 13991400 T_{3} - \)712337053132656

'>\(71\!\cdots\!56\)\( \) acting on \(S_{38}^{\mathrm{new}}(\Gamma_0(16))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 13991400 T + 899855474728862070 T^{2} + \)\(63\!\cdots\!00\)\( T^{3} + \)\(20\!\cdots\!69\)\( T^{4} \)
$5$ \( 1 - 5529584385900 T + \)\(30\!\cdots\!50\)\( T^{2} - \)\(40\!\cdots\!00\)\( T^{3} + \)\(52\!\cdots\!25\)\( T^{4} \)
$7$ \( 1 - 3448443953486000 T + \)\(26\!\cdots\!50\)\( T^{2} - \)\(64\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 - 26734036354848538056 T + \)\(70\!\cdots\!26\)\( T^{2} - \)\(90\!\cdots\!76\)\( T^{3} + \)\(11\!\cdots\!41\)\( T^{4} \)
$13$ \( 1 - \)\(53\!\cdots\!00\)\( T + \)\(32\!\cdots\!90\)\( T^{2} - \)\(87\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!89\)\( T^{4} \)
$17$ \( 1 + \)\(89\!\cdots\!00\)\( T + \)\(86\!\cdots\!30\)\( T^{2} + \)\(30\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \)
$19$ \( 1 + \)\(37\!\cdots\!20\)\( T + \)\(20\!\cdots\!78\)\( T^{2} + \)\(76\!\cdots\!80\)\( T^{3} + \)\(42\!\cdots\!21\)\( T^{4} \)
$23$ \( 1 - \)\(26\!\cdots\!00\)\( T + \)\(48\!\cdots\!10\)\( T^{2} - \)\(63\!\cdots\!00\)\( T^{3} + \)\(58\!\cdots\!09\)\( T^{4} \)
$29$ \( 1 + \)\(12\!\cdots\!20\)\( T + \)\(21\!\cdots\!18\)\( T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!81\)\( T^{4} \)
$31$ \( 1 + \)\(26\!\cdots\!24\)\( T + \)\(24\!\cdots\!66\)\( T^{2} + \)\(39\!\cdots\!64\)\( T^{3} + \)\(22\!\cdots\!21\)\( T^{4} \)
$37$ \( 1 + \)\(68\!\cdots\!00\)\( T + \)\(13\!\cdots\!90\)\( T^{2} + \)\(71\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!89\)\( T^{4} \)
$41$ \( 1 + \)\(12\!\cdots\!36\)\( T + \)\(12\!\cdots\!86\)\( T^{2} + \)\(59\!\cdots\!16\)\( T^{3} + \)\(22\!\cdots\!61\)\( T^{4} \)
$43$ \( 1 - \)\(25\!\cdots\!00\)\( T + \)\(44\!\cdots\!50\)\( T^{2} - \)\(70\!\cdots\!00\)\( T^{3} + \)\(75\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 + \)\(42\!\cdots\!00\)\( T + \)\(14\!\cdots\!70\)\( T^{2} + \)\(31\!\cdots\!00\)\( T^{3} + \)\(54\!\cdots\!69\)\( T^{4} \)
$53$ \( 1 - \)\(15\!\cdots\!00\)\( T + \)\(16\!\cdots\!70\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(39\!\cdots\!69\)\( T^{4} \)
$59$ \( 1 - \)\(23\!\cdots\!40\)\( T + \)\(64\!\cdots\!38\)\( T^{2} - \)\(79\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!61\)\( T^{4} \)
$61$ \( 1 - \)\(10\!\cdots\!44\)\( T + \)\(10\!\cdots\!26\)\( T^{2} - \)\(11\!\cdots\!24\)\( T^{3} + \)\(13\!\cdots\!41\)\( T^{4} \)
$67$ \( 1 - \)\(10\!\cdots\!00\)\( T + \)\(94\!\cdots\!30\)\( T^{2} - \)\(39\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!29\)\( T^{4} \)
$71$ \( 1 - \)\(74\!\cdots\!16\)\( T + \)\(58\!\cdots\!46\)\( T^{2} - \)\(23\!\cdots\!56\)\( T^{3} + \)\(98\!\cdots\!81\)\( T^{4} \)
$73$ \( 1 - \)\(19\!\cdots\!00\)\( T + \)\(18\!\cdots\!10\)\( T^{2} - \)\(17\!\cdots\!00\)\( T^{3} + \)\(76\!\cdots\!09\)\( T^{4} \)
$79$ \( 1 + \)\(27\!\cdots\!80\)\( T + \)\(50\!\cdots\!18\)\( T^{2} + \)\(44\!\cdots\!20\)\( T^{3} + \)\(26\!\cdots\!81\)\( T^{4} \)
$83$ \( 1 - \)\(47\!\cdots\!00\)\( T + \)\(24\!\cdots\!30\)\( T^{2} - \)\(47\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!29\)\( T^{4} \)
$89$ \( 1 + \)\(13\!\cdots\!60\)\( T + \)\(23\!\cdots\!58\)\( T^{2} + \)\(17\!\cdots\!40\)\( T^{3} + \)\(17\!\cdots\!41\)\( T^{4} \)
$97$ \( 1 - \)\(60\!\cdots\!00\)\( T + \)\(57\!\cdots\!70\)\( T^{2} - \)\(19\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!69\)\( T^{4} \)
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