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Properties

Label 1764.2.t.c
Level $1764$
Weight $2$
Character orbit 1764.t
Analytic conductor $14.086$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{48})\)
Defining polynomial: \(x^{16} - x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{48}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{48}^{3} + \zeta_{48}^{11} + \zeta_{48}^{13} ) q^{5} +O(q^{10})\) \( q + ( -\zeta_{48}^{3} + \zeta_{48}^{11} + \zeta_{48}^{13} ) q^{5} + ( 2 \zeta_{48}^{4} - 2 \zeta_{48}^{12} ) q^{11} + ( \zeta_{48}^{3} - \zeta_{48}^{5} + 2 \zeta_{48}^{9} + \zeta_{48}^{13} + 2 \zeta_{48}^{15} ) q^{13} + ( -3 \zeta_{48} + 3 \zeta_{48}^{7} + 3 \zeta_{48}^{9} - 3 \zeta_{48}^{15} ) q^{17} + ( 2 \zeta_{48}^{3} - 2 \zeta_{48}^{11} + 2 \zeta_{48}^{13} ) q^{19} + ( -6 \zeta_{48}^{4} - 2 \zeta_{48}^{6} - 2 \zeta_{48}^{10} + 2 \zeta_{48}^{14} ) q^{23} + ( -\zeta_{48}^{2} + 3 \zeta_{48}^{8} - \zeta_{48}^{14} ) q^{25} + ( -2 \zeta_{48}^{2} + 2 \zeta_{48}^{6} + 2 \zeta_{48}^{10} - 4 \zeta_{48}^{12} ) q^{29} + ( 4 \zeta_{48} - 2 \zeta_{48}^{5} + 4 \zeta_{48}^{7} - 4 \zeta_{48}^{9} + 2 \zeta_{48}^{11} - 4 \zeta_{48}^{15} ) q^{31} + ( -4 - 3 \zeta_{48}^{6} + 4 \zeta_{48}^{8} + 3 \zeta_{48}^{10} + 3 \zeta_{48}^{14} ) q^{37} + ( -6 \zeta_{48}^{3} - 6 \zeta_{48}^{5} - \zeta_{48}^{9} + 6 \zeta_{48}^{13} + \zeta_{48}^{15} ) q^{41} + ( 4 - 2 \zeta_{48}^{2} - 2 \zeta_{48}^{6} + 2 \zeta_{48}^{10} ) q^{43} + ( 4 \zeta_{48} - 6 \zeta_{48}^{3} - 4 \zeta_{48}^{7} + 6 \zeta_{48}^{11} + 6 \zeta_{48}^{13} ) q^{47} + ( -3 \zeta_{48}^{2} + 8 \zeta_{48}^{4} - 8 \zeta_{48}^{12} + 3 \zeta_{48}^{14} ) q^{53} + ( 2 \zeta_{48}^{9} + 2 \zeta_{48}^{15} ) q^{55} + ( -8 \zeta_{48} + 2 \zeta_{48}^{5} + 8 \zeta_{48}^{7} + 8 \zeta_{48}^{9} + 2 \zeta_{48}^{11} - 8 \zeta_{48}^{15} ) q^{59} + ( 5 \zeta_{48} + 4 \zeta_{48}^{3} + 5 \zeta_{48}^{7} - 4 \zeta_{48}^{11} + 4 \zeta_{48}^{13} ) q^{61} + ( -4 \zeta_{48}^{4} - 3 \zeta_{48}^{6} - 3 \zeta_{48}^{10} + 3 \zeta_{48}^{14} ) q^{65} + ( -6 \zeta_{48}^{2} - 4 \zeta_{48}^{8} - 6 \zeta_{48}^{14} ) q^{67} + ( -8 \zeta_{48}^{2} + 8 \zeta_{48}^{6} + 8 \zeta_{48}^{10} + 2 \zeta_{48}^{12} ) q^{71} + ( \zeta_{48} - 6 \zeta_{48}^{5} + \zeta_{48}^{7} - \zeta_{48}^{9} + 6 \zeta_{48}^{11} - \zeta_{48}^{15} ) q^{73} + ( -8 \zeta_{48}^{6} + 8 \zeta_{48}^{10} + 8 \zeta_{48}^{14} ) q^{79} + ( -4 \zeta_{48}^{3} - 4 \zeta_{48}^{5} - 4 \zeta_{48}^{9} + 4 \zeta_{48}^{13} + 4 \zeta_{48}^{15} ) q^{83} + ( -3 \zeta_{48}^{2} - 3 \zeta_{48}^{6} + 3 \zeta_{48}^{10} ) q^{85} + ( -2 \zeta_{48} - 7 \zeta_{48}^{3} + 2 \zeta_{48}^{7} + 7 \zeta_{48}^{11} + 7 \zeta_{48}^{13} ) q^{89} + ( -2 \zeta_{48}^{2} + 2 \zeta_{48}^{14} ) q^{95} + ( 6 \zeta_{48}^{3} - 6 \zeta_{48}^{5} - \zeta_{48}^{9} + 6 \zeta_{48}^{13} - \zeta_{48}^{15} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q + 24q^{25} - 32q^{37} + 64q^{43} - 32q^{67} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(1\) \(1 - \zeta_{48}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
521.1
−0.608761 + 0.793353i
0.991445 − 0.130526i
−0.793353 − 0.608761i
−0.130526 − 0.991445i
0.793353 + 0.608761i
0.130526 + 0.991445i
0.608761 − 0.793353i
−0.991445 + 0.130526i
−0.608761 − 0.793353i
0.991445 + 0.130526i
−0.793353 + 0.608761i
−0.130526 + 0.991445i
0.793353 − 0.608761i
0.130526 − 0.991445i
0.608761 + 0.793353i
−0.991445 − 0.130526i
0 0 0 −0.923880 − 1.60021i 0 0 0 0 0
521.2 0 0 0 −0.923880 − 1.60021i 0 0 0 0 0
521.3 0 0 0 −0.382683 − 0.662827i 0 0 0 0 0
521.4 0 0 0 −0.382683 − 0.662827i 0 0 0 0 0
521.5 0 0 0 0.382683 + 0.662827i 0 0 0 0 0
521.6 0 0 0 0.382683 + 0.662827i 0 0 0 0 0
521.7 0 0 0 0.923880 + 1.60021i 0 0 0 0 0
521.8 0 0 0 0.923880 + 1.60021i 0 0 0 0 0
1097.1 0 0 0 −0.923880 + 1.60021i 0 0 0 0 0
1097.2 0 0 0 −0.923880 + 1.60021i 0 0 0 0 0
1097.3 0 0 0 −0.382683 + 0.662827i 0 0 0 0 0
1097.4 0 0 0 −0.382683 + 0.662827i 0 0 0 0 0
1097.5 0 0 0 0.382683 − 0.662827i 0 0 0 0 0
1097.6 0 0 0 0.382683 − 0.662827i 0 0 0 0 0
1097.7 0 0 0 0.923880 − 1.60021i 0 0 0 0 0
1097.8 0 0 0 0.923880 − 1.60021i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1097.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.t.c 16
3.b odd 2 1 inner 1764.2.t.c 16
7.b odd 2 1 inner 1764.2.t.c 16
7.c even 3 1 1764.2.f.b ✓ 8
7.c even 3 1 inner 1764.2.t.c 16
7.d odd 6 1 1764.2.f.b ✓ 8
7.d odd 6 1 inner 1764.2.t.c 16
21.c even 2 1 inner 1764.2.t.c 16
21.g even 6 1 1764.2.f.b ✓ 8
21.g even 6 1 inner 1764.2.t.c 16
21.h odd 6 1 1764.2.f.b ✓ 8
21.h odd 6 1 inner 1764.2.t.c 16
28.f even 6 1 7056.2.k.e 8
28.g odd 6 1 7056.2.k.e 8
84.j odd 6 1 7056.2.k.e 8
84.n even 6 1 7056.2.k.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.2.f.b ✓ 8 7.c even 3 1
1764.2.f.b ✓ 8 7.d odd 6 1
1764.2.f.b ✓ 8 21.g even 6 1
1764.2.f.b ✓ 8 21.h odd 6 1
1764.2.t.c 16 1.a even 1 1 trivial
1764.2.t.c 16 3.b odd 2 1 inner
1764.2.t.c 16 7.b odd 2 1 inner
1764.2.t.c 16 7.c even 3 1 inner
1764.2.t.c 16 7.d odd 6 1 inner
1764.2.t.c 16 21.c even 2 1 inner
1764.2.t.c 16 21.g even 6 1 inner
1764.2.t.c 16 21.h odd 6 1 inner
7056.2.k.e 8 28.f even 6 1
7056.2.k.e 8 28.g odd 6 1
7056.2.k.e 8 84.j odd 6 1
7056.2.k.e 8 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 4 T_{5}^{6} + 14 T_{5}^{4} + 8 T_{5}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( T^{16} \)
$5$ \( ( 4 + 8 T^{2} + 14 T^{4} + 4 T^{6} + T^{8} )^{2} \)
$7$ \( T^{16} \)
$11$ \( ( 16 - 4 T^{2} + T^{4} )^{4} \)
$13$ \( ( 2 + 20 T^{2} + T^{4} )^{4} \)
$17$ \( ( 26244 + 5832 T^{2} + 1134 T^{4} + 36 T^{6} + T^{8} )^{2} \)
$19$ \( ( 1024 - 512 T^{2} + 224 T^{4} - 16 T^{6} + T^{8} )^{2} \)
$23$ \( ( 614656 - 68992 T^{2} + 6960 T^{4} - 88 T^{6} + T^{8} )^{2} \)
$29$ \( ( 64 + 48 T^{2} + T^{4} )^{4} \)
$31$ \( ( 2458624 - 125440 T^{2} + 4832 T^{4} - 80 T^{6} + T^{8} )^{2} \)
$37$ \( ( 4 - 16 T + 66 T^{2} + 8 T^{3} + T^{4} )^{4} \)
$41$ \( ( 1058 - 148 T^{2} + T^{4} )^{4} \)
$43$ \( ( 8 - 8 T + T^{2} )^{8} \)
$47$ \( ( 85525504 + 1923584 T^{2} + 34016 T^{4} + 208 T^{6} + T^{8} )^{2} \)
$53$ \( ( 4477456 - 347024 T^{2} + 24780 T^{4} - 164 T^{6} + T^{8} )^{2} \)
$59$ \( ( 286557184 + 4604416 T^{2} + 57056 T^{4} + 272 T^{6} + T^{8} )^{2} \)
$61$ \( ( 3694084 - 315208 T^{2} + 24974 T^{4} - 164 T^{6} + T^{8} )^{2} \)
$67$ \( ( 3136 - 448 T + 120 T^{2} + 8 T^{3} + T^{4} )^{4} \)
$71$ \( ( 15376 + 264 T^{2} + T^{4} )^{4} \)
$73$ \( ( 1119364 - 156584 T^{2} + 20846 T^{4} - 148 T^{6} + T^{8} )^{2} \)
$79$ \( ( 16384 + 128 T^{2} + T^{4} )^{4} \)
$83$ \( ( 2048 - 128 T^{2} + T^{4} )^{4} \)
$89$ \( ( 334084 + 122536 T^{2} + 44366 T^{4} + 212 T^{6} + T^{8} )^{2} \)
$97$ \( ( 1058 + 148 T^{2} + T^{4} )^{4} \)
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