Properties

Label 64.12.a.b
Level $64$
Weight $12$
Character orbit 64.a
Self dual yes
Analytic conductor $49.174$
Analytic rank $0$
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: \(0\)
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

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

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.b 1
4.b odd 2 1 64.12.a.f 1
8.b even 2 1 1.12.a.a 1
8.d odd 2 1 16.12.a.a 1
16.e even 4 2 256.12.b.e 2
16.f odd 4 2 256.12.b.c 2
24.f even 2 1 144.12.a.d 1
24.h odd 2 1 9.12.a.b 1
40.f even 2 1 25.12.a.b 1
40.i odd 4 2 25.12.b.b 2
56.h odd 2 1 49.12.a.a 1
56.j odd 6 2 49.12.c.c 2
56.p even 6 2 49.12.c.b 2
72.j odd 6 2 81.12.c.b 2
72.n even 6 2 81.12.c.d 2
88.b odd 2 1 121.12.a.b 1
104.e even 2 1 169.12.a.a 1
120.i odd 2 1 225.12.a.b 1
120.w even 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.b even 2 1
9.12.a.b 1 24.h odd 2 1
16.12.a.a 1 8.d odd 2 1
25.12.a.b 1 40.f even 2 1
25.12.b.b 2 40.i odd 4 2
49.12.a.a 1 56.h odd 2 1
49.12.c.b 2 56.p even 6 2
49.12.c.c 2 56.j odd 6 2
64.12.a.b 1 1.a even 1 1 trivial
64.12.a.f 1 4.b odd 2 1
81.12.c.b 2 72.j odd 6 2
81.12.c.d 2 72.n even 6 2
121.12.a.b 1 88.b odd 2 1
144.12.a.d 1 24.f even 2 1
169.12.a.a 1 104.e even 2 1
225.12.a.b 1 120.i odd 2 1
225.12.b.d 2 120.w even 4 2
256.12.b.c 2 16.f odd 4 2
256.12.b.e 2 16.e even 4 2

Hecke kernels

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

Hecke characteristic polynomials

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