Properties

Label 16.10.a.a
Level 16
Weight 10
Character orbit 16.a
Self dual yes
Analytic conductor 8.241
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.24057337862\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 228q^{3} - 666q^{5} + 6328q^{7} + 32301q^{9} + O(q^{10}) \) \( q - 228q^{3} - 666q^{5} + 6328q^{7} + 32301q^{9} + 30420q^{11} - 32338q^{13} + 151848q^{15} + 590994q^{17} - 34676q^{19} - 1442784q^{21} - 1048536q^{23} - 1509569q^{25} - 2876904q^{27} + 4409406q^{29} + 7401184q^{31} - 6935760q^{33} - 4214448q^{35} + 10234502q^{37} + 7373064q^{39} + 18352746q^{41} + 252340q^{43} - 21512466q^{45} + 49517136q^{47} - 310023q^{49} - 134746632q^{51} - 66396906q^{53} - 20259720q^{55} + 7906128q^{57} + 61523748q^{59} + 35638622q^{61} + 204400728q^{63} + 21537108q^{65} - 181742372q^{67} + 239066208q^{69} - 90904968q^{71} - 262978678q^{73} + 344181732q^{75} + 192497760q^{77} + 116502832q^{79} + 20153529q^{81} + 9563724q^{83} - 393602004q^{85} - 1005344568q^{87} + 611826714q^{89} - 204634864q^{91} - 1687469952q^{93} + 23094216q^{95} - 259312798q^{97} + 982596420q^{99} + O(q^{100}) \)

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
0
0 −228.000 0 −666.000 0 6328.00 0 32301.0 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 16.10.a.a 1
3.b odd 2 1 144.10.a.j 1
4.b odd 2 1 4.10.a.a 1
5.b even 2 1 400.10.a.k 1
5.c odd 4 2 400.10.c.a 2
8.b even 2 1 64.10.a.i 1
8.d odd 2 1 64.10.a.a 1
12.b even 2 1 36.10.a.b 1
16.e even 4 2 256.10.b.b 2
16.f odd 4 2 256.10.b.j 2
20.d odd 2 1 100.10.a.a 1
20.e even 4 2 100.10.c.a 2
28.d even 2 1 196.10.a.a 1
28.f even 6 2 196.10.e.b 2
28.g odd 6 2 196.10.e.a 2
36.f odd 6 2 324.10.e.e 2
36.h even 6 2 324.10.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.10.a.a 1 4.b odd 2 1
16.10.a.a 1 1.a even 1 1 trivial
36.10.a.b 1 12.b even 2 1
64.10.a.a 1 8.d odd 2 1
64.10.a.i 1 8.b even 2 1
100.10.a.a 1 20.d odd 2 1
100.10.c.a 2 20.e even 4 2
144.10.a.j 1 3.b odd 2 1
196.10.a.a 1 28.d even 2 1
196.10.e.a 2 28.g odd 6 2
196.10.e.b 2 28.f even 6 2
256.10.b.b 2 16.e even 4 2
256.10.b.j 2 16.f odd 4 2
324.10.e.b 2 36.h even 6 2
324.10.e.e 2 36.f odd 6 2
400.10.a.k 1 5.b even 2 1
400.10.c.a 2 5.c odd 4 2

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 228 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(16))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 228 T + 19683 T^{2} \)
$5$ \( 1 + 666 T + 1953125 T^{2} \)
$7$ \( 1 - 6328 T + 40353607 T^{2} \)
$11$ \( 1 - 30420 T + 2357947691 T^{2} \)
$13$ \( 1 + 32338 T + 10604499373 T^{2} \)
$17$ \( 1 - 590994 T + 118587876497 T^{2} \)
$19$ \( 1 + 34676 T + 322687697779 T^{2} \)
$23$ \( 1 + 1048536 T + 1801152661463 T^{2} \)
$29$ \( 1 - 4409406 T + 14507145975869 T^{2} \)
$31$ \( 1 - 7401184 T + 26439622160671 T^{2} \)
$37$ \( 1 - 10234502 T + 129961739795077 T^{2} \)
$41$ \( 1 - 18352746 T + 327381934393961 T^{2} \)
$43$ \( 1 - 252340 T + 502592611936843 T^{2} \)
$47$ \( 1 - 49517136 T + 1119130473102767 T^{2} \)
$53$ \( 1 + 66396906 T + 3299763591802133 T^{2} \)
$59$ \( 1 - 61523748 T + 8662995818654939 T^{2} \)
$61$ \( 1 - 35638622 T + 11694146092834141 T^{2} \)
$67$ \( 1 + 181742372 T + 27206534396294947 T^{2} \)
$71$ \( 1 + 90904968 T + 45848500718449031 T^{2} \)
$73$ \( 1 + 262978678 T + 58871586708267913 T^{2} \)
$79$ \( 1 - 116502832 T + 119851595982618319 T^{2} \)
$83$ \( 1 - 9563724 T + 186940255267540403 T^{2} \)
$89$ \( 1 - 611826714 T + 350356403707485209 T^{2} \)
$97$ \( 1 + 259312798 T + 760231058654565217 T^{2} \)
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