Properties

Label 64.2.b.a
Level $64$
Weight $2$
Character orbit 64.b
Analytic conductor $0.511$
Analytic rank $0$
Dimension $2$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 64.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.511042572936\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 i q^{3} - q^{9} +O(q^{10})\) \( q + 2 i q^{3} - q^{9} -6 i q^{11} -6 q^{17} + 2 i q^{19} + 5 q^{25} + 4 i q^{27} + 12 q^{33} -6 q^{41} + 10 i q^{43} -7 q^{49} -12 i q^{51} -4 q^{57} -6 i q^{59} -14 i q^{67} + 2 q^{73} + 10 i q^{75} -11 q^{81} + 18 i q^{83} + 18 q^{89} + 10 q^{97} + 6 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 12q^{17} + 10q^{25} + 24q^{33} - 12q^{41} - 14q^{49} - 8q^{57} + 4q^{73} - 22q^{81} + 36q^{89} + 20q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(63\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
1.00000i
1.00000i
0 2.00000i 0 0 0 0 0 −1.00000 0
33.2 0 2.00000i 0 0 0 0 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 64.2.b.a 2
3.b odd 2 1 576.2.d.a 2
4.b odd 2 1 inner 64.2.b.a 2
5.b even 2 1 1600.2.d.a 2
5.c odd 4 1 1600.2.f.a 2
5.c odd 4 1 1600.2.f.b 2
7.b odd 2 1 3136.2.b.b 2
8.b even 2 1 inner 64.2.b.a 2
8.d odd 2 1 CM 64.2.b.a 2
12.b even 2 1 576.2.d.a 2
16.e even 4 1 256.2.a.a 1
16.e even 4 1 256.2.a.d 1
16.f odd 4 1 256.2.a.a 1
16.f odd 4 1 256.2.a.d 1
20.d odd 2 1 1600.2.d.a 2
20.e even 4 1 1600.2.f.a 2
20.e even 4 1 1600.2.f.b 2
24.f even 2 1 576.2.d.a 2
24.h odd 2 1 576.2.d.a 2
28.d even 2 1 3136.2.b.b 2
32.g even 8 4 1024.2.e.l 4
32.h odd 8 4 1024.2.e.l 4
40.e odd 2 1 1600.2.d.a 2
40.f even 2 1 1600.2.d.a 2
40.i odd 4 1 1600.2.f.a 2
40.i odd 4 1 1600.2.f.b 2
40.k even 4 1 1600.2.f.a 2
40.k even 4 1 1600.2.f.b 2
48.i odd 4 1 2304.2.a.h 1
48.i odd 4 1 2304.2.a.i 1
48.k even 4 1 2304.2.a.h 1
48.k even 4 1 2304.2.a.i 1
56.e even 2 1 3136.2.b.b 2
56.h odd 2 1 3136.2.b.b 2
80.k odd 4 1 6400.2.a.a 1
80.k odd 4 1 6400.2.a.x 1
80.q even 4 1 6400.2.a.a 1
80.q even 4 1 6400.2.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.2.b.a 2 1.a even 1 1 trivial
64.2.b.a 2 4.b odd 2 1 inner
64.2.b.a 2 8.b even 2 1 inner
64.2.b.a 2 8.d odd 2 1 CM
256.2.a.a 1 16.e even 4 1
256.2.a.a 1 16.f odd 4 1
256.2.a.d 1 16.e even 4 1
256.2.a.d 1 16.f odd 4 1
576.2.d.a 2 3.b odd 2 1
576.2.d.a 2 12.b even 2 1
576.2.d.a 2 24.f even 2 1
576.2.d.a 2 24.h odd 2 1
1024.2.e.l 4 32.g even 8 4
1024.2.e.l 4 32.h odd 8 4
1600.2.d.a 2 5.b even 2 1
1600.2.d.a 2 20.d odd 2 1
1600.2.d.a 2 40.e odd 2 1
1600.2.d.a 2 40.f even 2 1
1600.2.f.a 2 5.c odd 4 1
1600.2.f.a 2 20.e even 4 1
1600.2.f.a 2 40.i odd 4 1
1600.2.f.a 2 40.k even 4 1
1600.2.f.b 2 5.c odd 4 1
1600.2.f.b 2 20.e even 4 1
1600.2.f.b 2 40.i odd 4 1
1600.2.f.b 2 40.k even 4 1
2304.2.a.h 1 48.i odd 4 1
2304.2.a.h 1 48.k even 4 1
2304.2.a.i 1 48.i odd 4 1
2304.2.a.i 1 48.k even 4 1
3136.2.b.b 2 7.b odd 2 1
3136.2.b.b 2 28.d even 2 1
3136.2.b.b 2 56.e even 2 1
3136.2.b.b 2 56.h odd 2 1
6400.2.a.a 1 80.k odd 4 1
6400.2.a.a 1 80.q even 4 1
6400.2.a.x 1 80.k odd 4 1
6400.2.a.x 1 80.q even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(64, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( 36 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( 6 + T )^{2} \)
$19$ \( 4 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 100 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( 36 + T^{2} \)
$61$ \( T^{2} \)
$67$ \( 196 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -2 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( 324 + T^{2} \)
$89$ \( ( -18 + T )^{2} \)
$97$ \( ( -10 + T )^{2} \)
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