Properties

Label 32.3.c.a
Level $32$
Weight $3$
Character orbit 32.c
Analytic conductor $0.872$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 32.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.871936845953\)
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: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 4 i q^{3} + 2 q^{5} -8 i q^{7} -7 q^{9} +O(q^{10})\) \( q + 4 i q^{3} + 2 q^{5} -8 i q^{7} -7 q^{9} -4 i q^{11} -14 q^{13} + 8 i q^{15} + 18 q^{17} -12 i q^{19} + 32 q^{21} + 40 i q^{23} -21 q^{25} + 8 i q^{27} -14 q^{29} -32 i q^{31} + 16 q^{33} -16 i q^{35} -30 q^{37} -56 i q^{39} -14 q^{41} + 28 i q^{43} -14 q^{45} + 16 i q^{47} -15 q^{49} + 72 i q^{51} + 66 q^{53} -8 i q^{55} + 48 q^{57} -52 i q^{59} + 82 q^{61} + 56 i q^{63} -28 q^{65} + 4 i q^{67} -160 q^{69} + 56 i q^{71} + 66 q^{73} -84 i q^{75} -32 q^{77} -16 i q^{79} -95 q^{81} -140 i q^{83} + 36 q^{85} -56 i q^{87} -30 q^{89} + 112 i q^{91} + 128 q^{93} -24 i q^{95} -14 q^{97} + 28 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} - 14q^{9} + O(q^{10}) \) \( 2q + 4q^{5} - 14q^{9} - 28q^{13} + 36q^{17} + 64q^{21} - 42q^{25} - 28q^{29} + 32q^{33} - 60q^{37} - 28q^{41} - 28q^{45} - 30q^{49} + 132q^{53} + 96q^{57} + 164q^{61} - 56q^{65} - 320q^{69} + 132q^{73} - 64q^{77} - 190q^{81} + 72q^{85} - 60q^{89} + 256q^{93} - 28q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(31\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
1.00000i
1.00000i
0 4.00000i 0 2.00000 0 8.00000i 0 −7.00000 0
31.2 0 4.00000i 0 2.00000 0 8.00000i 0 −7.00000 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
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.3.c.a 2
3.b odd 2 1 288.3.g.b 2
4.b odd 2 1 inner 32.3.c.a 2
5.b even 2 1 800.3.b.a 2
5.c odd 4 1 800.3.h.a 2
5.c odd 4 1 800.3.h.b 2
7.b odd 2 1 1568.3.d.b 2
8.b even 2 1 64.3.c.b 2
8.d odd 2 1 64.3.c.b 2
12.b even 2 1 288.3.g.b 2
16.e even 4 1 256.3.d.a 2
16.e even 4 1 256.3.d.c 2
16.f odd 4 1 256.3.d.a 2
16.f odd 4 1 256.3.d.c 2
20.d odd 2 1 800.3.b.a 2
20.e even 4 1 800.3.h.a 2
20.e even 4 1 800.3.h.b 2
24.f even 2 1 576.3.g.g 2
24.h odd 2 1 576.3.g.g 2
28.d even 2 1 1568.3.d.b 2
40.e odd 2 1 1600.3.b.e 2
40.f even 2 1 1600.3.b.e 2
40.i odd 4 1 1600.3.h.a 2
40.i odd 4 1 1600.3.h.c 2
40.k even 4 1 1600.3.h.a 2
40.k even 4 1 1600.3.h.c 2
48.i odd 4 1 2304.3.b.c 2
48.i odd 4 1 2304.3.b.g 2
48.k even 4 1 2304.3.b.c 2
48.k even 4 1 2304.3.b.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.3.c.a 2 1.a even 1 1 trivial
32.3.c.a 2 4.b odd 2 1 inner
64.3.c.b 2 8.b even 2 1
64.3.c.b 2 8.d odd 2 1
256.3.d.a 2 16.e even 4 1
256.3.d.a 2 16.f odd 4 1
256.3.d.c 2 16.e even 4 1
256.3.d.c 2 16.f odd 4 1
288.3.g.b 2 3.b odd 2 1
288.3.g.b 2 12.b even 2 1
576.3.g.g 2 24.f even 2 1
576.3.g.g 2 24.h odd 2 1
800.3.b.a 2 5.b even 2 1
800.3.b.a 2 20.d odd 2 1
800.3.h.a 2 5.c odd 4 1
800.3.h.a 2 20.e even 4 1
800.3.h.b 2 5.c odd 4 1
800.3.h.b 2 20.e even 4 1
1568.3.d.b 2 7.b odd 2 1
1568.3.d.b 2 28.d even 2 1
1600.3.b.e 2 40.e odd 2 1
1600.3.b.e 2 40.f even 2 1
1600.3.h.a 2 40.i odd 4 1
1600.3.h.a 2 40.k even 4 1
1600.3.h.c 2 40.i odd 4 1
1600.3.h.c 2 40.k even 4 1
2304.3.b.c 2 48.i odd 4 1
2304.3.b.c 2 48.k even 4 1
2304.3.b.g 2 48.i odd 4 1
2304.3.b.g 2 48.k even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(32, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 16 + T^{2} \)
$5$ \( ( -2 + T )^{2} \)
$7$ \( 64 + T^{2} \)
$11$ \( 16 + T^{2} \)
$13$ \( ( 14 + T )^{2} \)
$17$ \( ( -18 + T )^{2} \)
$19$ \( 144 + T^{2} \)
$23$ \( 1600 + T^{2} \)
$29$ \( ( 14 + T )^{2} \)
$31$ \( 1024 + T^{2} \)
$37$ \( ( 30 + T )^{2} \)
$41$ \( ( 14 + T )^{2} \)
$43$ \( 784 + T^{2} \)
$47$ \( 256 + T^{2} \)
$53$ \( ( -66 + T )^{2} \)
$59$ \( 2704 + T^{2} \)
$61$ \( ( -82 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( 3136 + T^{2} \)
$73$ \( ( -66 + T )^{2} \)
$79$ \( 256 + T^{2} \)
$83$ \( 19600 + T^{2} \)
$89$ \( ( 30 + T )^{2} \)
$97$ \( ( 14 + T )^{2} \)
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