Properties

Label 63.3.d.b
Level $63$
Weight $3$
Character orbit 63.d
Analytic conductor $1.717$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 63.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.71662566547\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - 3 q^{4} - \beta q^{5} + ( - \beta + 1) q^{7} + 7 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - 3 q^{4} - \beta q^{5} + ( - \beta + 1) q^{7} + 7 q^{8} + \beta q^{10} - 10 q^{11} - \beta q^{13} + (\beta - 1) q^{14} + 5 q^{16} + 3 \beta q^{19} + 3 \beta q^{20} + 10 q^{22} + 14 q^{23} - 23 q^{25} + \beta q^{26} + (3 \beta - 3) q^{28} + 38 q^{29} - 4 \beta q^{31} - 33 q^{32} + ( - \beta - 48) q^{35} + 26 q^{37} - 3 \beta q^{38} - 7 \beta q^{40} - 10 \beta q^{41} + 26 q^{43} + 30 q^{44} - 14 q^{46} + 4 \beta q^{47} + ( - 2 \beta - 47) q^{49} + 23 q^{50} + 3 \beta q^{52} - 10 q^{53} + 10 \beta q^{55} + ( - 7 \beta + 7) q^{56} - 38 q^{58} + 11 \beta q^{59} - 5 \beta q^{61} + 4 \beta q^{62} + 13 q^{64} - 48 q^{65} + 74 q^{67} + (\beta + 48) q^{70} + 62 q^{71} + 6 \beta q^{73} - 26 q^{74} - 9 \beta q^{76} + (10 \beta - 10) q^{77} - 46 q^{79} - 5 \beta q^{80} + 10 \beta q^{82} + 13 \beta q^{83} - 26 q^{86} - 70 q^{88} - 6 \beta q^{89} + ( - \beta - 48) q^{91} - 42 q^{92} - 4 \beta q^{94} + 144 q^{95} - 8 \beta q^{97} + (2 \beta + 47) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 6 q^{4} + 2 q^{7} + 14 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 6 q^{4} + 2 q^{7} + 14 q^{8} - 20 q^{11} - 2 q^{14} + 10 q^{16} + 20 q^{22} + 28 q^{23} - 46 q^{25} - 6 q^{28} + 76 q^{29} - 66 q^{32} - 96 q^{35} + 52 q^{37} + 52 q^{43} + 60 q^{44} - 28 q^{46} - 94 q^{49} + 46 q^{50} - 20 q^{53} + 14 q^{56} - 76 q^{58} + 26 q^{64} - 96 q^{65} + 148 q^{67} + 96 q^{70} + 124 q^{71} - 52 q^{74} - 20 q^{77} - 92 q^{79} - 52 q^{86} - 140 q^{88} - 96 q^{91} - 84 q^{92} + 288 q^{95} + 94 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.500000 + 0.866025i
0.500000 0.866025i
−1.00000 0 −3.00000 6.92820i 0 1.00000 6.92820i 7.00000 0 6.92820i
55.2 −1.00000 0 −3.00000 6.92820i 0 1.00000 + 6.92820i 7.00000 0 6.92820i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.3.d.b 2
3.b odd 2 1 21.3.d.a 2
4.b odd 2 1 1008.3.f.d 2
7.b odd 2 1 inner 63.3.d.b 2
7.c even 3 1 441.3.m.d 2
7.c even 3 1 441.3.m.f 2
7.d odd 6 1 441.3.m.d 2
7.d odd 6 1 441.3.m.f 2
12.b even 2 1 336.3.f.a 2
15.d odd 2 1 525.3.h.a 2
15.e even 4 2 525.3.e.a 4
21.c even 2 1 21.3.d.a 2
21.g even 6 1 147.3.f.b 2
21.g even 6 1 147.3.f.d 2
21.h odd 6 1 147.3.f.b 2
21.h odd 6 1 147.3.f.d 2
24.f even 2 1 1344.3.f.b 2
24.h odd 2 1 1344.3.f.c 2
28.d even 2 1 1008.3.f.d 2
84.h odd 2 1 336.3.f.a 2
105.g even 2 1 525.3.h.a 2
105.k odd 4 2 525.3.e.a 4
168.e odd 2 1 1344.3.f.b 2
168.i even 2 1 1344.3.f.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.d.a 2 3.b odd 2 1
21.3.d.a 2 21.c even 2 1
63.3.d.b 2 1.a even 1 1 trivial
63.3.d.b 2 7.b odd 2 1 inner
147.3.f.b 2 21.g even 6 1
147.3.f.b 2 21.h odd 6 1
147.3.f.d 2 21.g even 6 1
147.3.f.d 2 21.h odd 6 1
336.3.f.a 2 12.b even 2 1
336.3.f.a 2 84.h odd 2 1
441.3.m.d 2 7.c even 3 1
441.3.m.d 2 7.d odd 6 1
441.3.m.f 2 7.c even 3 1
441.3.m.f 2 7.d odd 6 1
525.3.e.a 4 15.e even 4 2
525.3.e.a 4 105.k odd 4 2
525.3.h.a 2 15.d odd 2 1
525.3.h.a 2 105.g even 2 1
1008.3.f.d 2 4.b odd 2 1
1008.3.f.d 2 28.d even 2 1
1344.3.f.b 2 24.f even 2 1
1344.3.f.b 2 168.e odd 2 1
1344.3.f.c 2 24.h odd 2 1
1344.3.f.c 2 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 48 \) Copy content Toggle raw display
$7$ \( T^{2} - 2T + 49 \) Copy content Toggle raw display
$11$ \( (T + 10)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 48 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 432 \) Copy content Toggle raw display
$23$ \( (T - 14)^{2} \) Copy content Toggle raw display
$29$ \( (T - 38)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 768 \) Copy content Toggle raw display
$37$ \( (T - 26)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 4800 \) Copy content Toggle raw display
$43$ \( (T - 26)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 768 \) Copy content Toggle raw display
$53$ \( (T + 10)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 5808 \) Copy content Toggle raw display
$61$ \( T^{2} + 1200 \) Copy content Toggle raw display
$67$ \( (T - 74)^{2} \) Copy content Toggle raw display
$71$ \( (T - 62)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1728 \) Copy content Toggle raw display
$79$ \( (T + 46)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 8112 \) Copy content Toggle raw display
$89$ \( T^{2} + 1728 \) Copy content Toggle raw display
$97$ \( T^{2} + 3072 \) Copy content Toggle raw display
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