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Properties

Label 64.12.a.f
Level $64$
Weight $12$
Character orbit 64.a
Self dual yes
Analytic conductor $49.174$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 64.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(49.1739635558\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 252q^{3} - 4830q^{5} + 16744q^{7} - 113643q^{9} + O(q^{10}) \) \( q + 252q^{3} - 4830q^{5} + 16744q^{7} - 113643q^{9} + 534612q^{11} + 577738q^{13} - 1217160q^{15} - 6905934q^{17} + 10661420q^{19} + 4219488q^{21} - 18643272q^{23} - 25499225q^{25} - 73279080q^{27} - 128406630q^{29} + 52843168q^{31} + 134722224q^{33} - 80873520q^{35} + 182213314q^{37} + 145589976q^{39} + 308120442q^{41} - 17125708q^{43} + 548895690q^{45} - 2687348496q^{47} - 1696965207q^{49} - 1740295368q^{51} + 1596055698q^{53} - 2582175960q^{55} + 2686677840q^{57} - 5189203740q^{59} - 6956478662q^{61} - 1902838392q^{63} - 2790474540q^{65} - 15481826884q^{67} - 4698104544q^{69} - 9791485272q^{71} + 1463791322q^{73} - 6425804700q^{75} + 8951543328q^{77} - 38116845680q^{79} + 1665188361q^{81} - 29335099668q^{83} + 33355661220q^{85} - 32358470760q^{87} - 24992917110q^{89} + 9673645072q^{91} + 13316478336q^{93} - 51494658600q^{95} + 75013568546q^{97} - 60754911516q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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
0 252.000 0 −4830.00 0 16744.0 0 −113643. 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 64.12.a.f 1
4.b odd 2 1 64.12.a.b 1
8.b even 2 1 16.12.a.a 1
8.d odd 2 1 1.12.a.a ✓ 1
16.e even 4 2 256.12.b.c 2
16.f odd 4 2 256.12.b.e 2
24.f even 2 1 9.12.a.b 1
24.h odd 2 1 144.12.a.d 1
40.e odd 2 1 25.12.a.b 1
40.k even 4 2 25.12.b.b 2
56.e even 2 1 49.12.a.a 1
56.k odd 6 2 49.12.c.b 2
56.m even 6 2 49.12.c.c 2
72.l even 6 2 81.12.c.b 2
72.p odd 6 2 81.12.c.d 2
88.g even 2 1 121.12.a.b 1
104.h odd 2 1 169.12.a.a 1
120.m even 2 1 225.12.a.b 1
120.q odd 4 2 225.12.b.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.12.a.a ✓ 1 8.d odd 2 1
9.12.a.b 1 24.f even 2 1
16.12.a.a 1 8.b even 2 1
25.12.a.b 1 40.e odd 2 1
25.12.b.b 2 40.k even 4 2
49.12.a.a 1 56.e even 2 1
49.12.c.b 2 56.k odd 6 2
49.12.c.c 2 56.m even 6 2
64.12.a.b 1 4.b odd 2 1
64.12.a.f 1 1.a even 1 1 trivial
81.12.c.b 2 72.l even 6 2
81.12.c.d 2 72.p odd 6 2
121.12.a.b 1 88.g even 2 1
144.12.a.d 1 24.h odd 2 1
169.12.a.a 1 104.h odd 2 1
225.12.a.b 1 120.m even 2 1
225.12.b.d 2 120.q odd 4 2
256.12.b.c 2 16.e even 4 2
256.12.b.e 2 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 252 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(64))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -252 + T \)
$5$ \( 4830 + T \)
$7$ \( -16744 + T \)
$11$ \( -534612 + T \)
$13$ \( -577738 + T \)
$17$ \( 6905934 + T \)
$19$ \( -10661420 + T \)
$23$ \( 18643272 + T \)
$29$ \( 128406630 + T \)
$31$ \( -52843168 + T \)
$37$ \( -182213314 + T \)
$41$ \( -308120442 + T \)
$43$ \( 17125708 + T \)
$47$ \( 2687348496 + T \)
$53$ \( -1596055698 + T \)
$59$ \( 5189203740 + T \)
$61$ \( 6956478662 + T \)
$67$ \( 15481826884 + T \)
$71$ \( 9791485272 + T \)
$73$ \( -1463791322 + T \)
$79$ \( 38116845680 + T \)
$83$ \( 29335099668 + T \)
$89$ \( 24992917110 + T \)
$97$ \( -75013568546 + T \)
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