Properties

Label 16.12.a.b
Level 16
Weight 12
Character orbit 16.a
Self dual yes
Analytic conductor 12.293
Analytic rank 0
Dimension 1
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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 36q^{3} - 3490q^{5} + 55464q^{7} - 175851q^{9} + O(q^{10}) \) \( q + 36q^{3} - 3490q^{5} + 55464q^{7} - 175851q^{9} + 597004q^{11} + 1373878q^{13} - 125640q^{15} + 10140850q^{17} + 7297396q^{19} + 1996704q^{21} + 32057464q^{23} - 36648025q^{25} - 12707928q^{27} - 13605402q^{29} - 233160800q^{31} + 21492144q^{33} - 193569360q^{35} - 257786178q^{37} + 49459608q^{39} - 221438598q^{41} + 1697758892q^{43} + 613719990q^{45} - 527509392q^{47} + 1098928553q^{49} + 365070600q^{51} + 3277379822q^{53} - 2083543960q^{55} + 262706256q^{57} + 3001908988q^{59} - 11630023610q^{61} - 9753399864q^{63} - 4794834220q^{65} + 17189000548q^{67} + 1154068704q^{69} - 26169539608q^{71} - 7039021094q^{73} - 1319328900q^{75} + 33112229856q^{77} + 4199910416q^{79} + 30693991689q^{81} + 39739936436q^{83} - 35391566500q^{85} - 489794472q^{87} + 10565331594q^{89} + 76200769392q^{91} - 8393788800q^{93} - 25467912040q^{95} - 69851645662q^{97} - 104983750404q^{99} + O(q^{100}) \)

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 36.0000 0 −3490.00 0 55464.0 0 −175851. 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.b 1
3.b odd 2 1 144.12.a.j 1
4.b odd 2 1 8.12.a.a 1
8.b even 2 1 64.12.a.c 1
8.d odd 2 1 64.12.a.e 1
12.b even 2 1 72.12.a.c 1
16.e even 4 2 256.12.b.a 2
16.f odd 4 2 256.12.b.g 2
20.d odd 2 1 200.12.a.b 1
20.e even 4 2 200.12.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.12.a.a 1 4.b odd 2 1
16.12.a.b 1 1.a even 1 1 trivial
64.12.a.c 1 8.b even 2 1
64.12.a.e 1 8.d odd 2 1
72.12.a.c 1 12.b even 2 1
144.12.a.j 1 3.b odd 2 1
200.12.a.b 1 20.d odd 2 1
200.12.c.b 2 20.e even 4 2
256.12.b.a 2 16.e even 4 2
256.12.b.g 2 16.f 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 \( T_{3} - 36 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(16))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 36 T + 177147 T^{2} \)
$5$ \( 1 + 3490 T + 48828125 T^{2} \)
$7$ \( 1 - 55464 T + 1977326743 T^{2} \)
$11$ \( 1 - 597004 T + 285311670611 T^{2} \)
$13$ \( 1 - 1373878 T + 1792160394037 T^{2} \)
$17$ \( 1 - 10140850 T + 34271896307633 T^{2} \)
$19$ \( 1 - 7297396 T + 116490258898219 T^{2} \)
$23$ \( 1 - 32057464 T + 952809757913927 T^{2} \)
$29$ \( 1 + 13605402 T + 12200509765705829 T^{2} \)
$31$ \( 1 + 233160800 T + 25408476896404831 T^{2} \)
$37$ \( 1 + 257786178 T + 177917621779460413 T^{2} \)
$41$ \( 1 + 221438598 T + 550329031716248441 T^{2} \)
$43$ \( 1 - 1697758892 T + 929293739471222707 T^{2} \)
$47$ \( 1 + 527509392 T + 2472159215084012303 T^{2} \)
$53$ \( 1 - 3277379822 T + 9269035929372191597 T^{2} \)
$59$ \( 1 - 3001908988 T + 30155888444737842659 T^{2} \)
$61$ \( 1 + 11630023610 T + 43513917611435838661 T^{2} \)
$67$ \( 1 - 17189000548 T + \)\(12\!\cdots\!83\)\( T^{2} \)
$71$ \( 1 + 26169539608 T + \)\(23\!\cdots\!71\)\( T^{2} \)
$73$ \( 1 + 7039021094 T + \)\(31\!\cdots\!77\)\( T^{2} \)
$79$ \( 1 - 4199910416 T + \)\(74\!\cdots\!79\)\( T^{2} \)
$83$ \( 1 - 39739936436 T + \)\(12\!\cdots\!67\)\( T^{2} \)
$89$ \( 1 - 10565331594 T + \)\(27\!\cdots\!89\)\( T^{2} \)
$97$ \( 1 + 69851645662 T + \)\(71\!\cdots\!53\)\( T^{2} \)
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