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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: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
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 basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{15} + \beta_{4}) q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{15} + \beta_{4}) q^{5} + ( - \beta_{5} + \beta_1) q^{11} + (\beta_{13} - \beta_{10} + 2 \beta_{7}) q^{13} + (3 \beta_{11} - 3 \beta_{8}) q^{17} + 2 \beta_{13} q^{19} + ( - 2 \beta_{12} - 3 \beta_1) q^{23} + ( - \beta_{6} + 3 \beta_{3}) q^{25} + (2 \beta_{12} - 2 \beta_{9} - 2 \beta_{5}) q^{29} + ( - 2 \beta_{10} - 4 \beta_{7} + 4 \beta_{2}) q^{31} + ( - 3 \beta_{14} + 3 \beta_{6} + 4 \beta_{3} - 4) q^{37} + ( - 6 \beta_{15} - \beta_{11}) q^{41} + ( - 2 \beta_{14} + 4) q^{43} + ( - 6 \beta_{15} + 4 \beta_{8} + 6 \beta_{4}) q^{47} + ( - 3 \beta_{9} - 4 \beta_{5} + 4 \beta_1) q^{53} + 2 \beta_{7} q^{55} + (8 \beta_{11} - 8 \beta_{8} + 2 \beta_{4}) q^{59} + (4 \beta_{13} + 5 \beta_{2}) q^{61} + ( - 3 \beta_{12} - 2 \beta_1) q^{65} + ( - 6 \beta_{6} - 4 \beta_{3}) q^{67} + (8 \beta_{12} - 8 \beta_{9} + \beta_{5}) q^{71} + ( - 6 \beta_{10} - \beta_{7} + \beta_{2}) q^{73} + ( - 8 \beta_{14} + 8 \beta_{6}) q^{79} + ( - 4 \beta_{15} - 4 \beta_{11}) q^{83} - 3 \beta_{14} q^{85} + ( - 7 \beta_{15} - 2 \beta_{8} + 7 \beta_{4}) q^{89} - 2 \beta_{9} q^{95} + (6 \beta_{13} - 6 \beta_{10} - \beta_{7}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 24 q^{25} - 32 q^{37} + 64 q^{43} - 32 q^{67}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{48}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{48}^{7} + \zeta_{48} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{48}^{8} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{48}^{11} + \zeta_{48}^{5} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\zeta_{48}^{12} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{48}^{14} + \zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \zeta_{48}^{15} + \zeta_{48}^{9} \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -\zeta_{48}^{7} + \zeta_{48} \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( -\zeta_{48}^{14} + \zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( -\zeta_{48}^{11} + \zeta_{48}^{5} \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( -\zeta_{48}^{15} + \zeta_{48}^{9} \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( -\zeta_{48}^{14} + \zeta_{48}^{10} + \zeta_{48}^{6} \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( \zeta_{48}^{13} - \zeta_{48}^{11} + \zeta_{48}^{3} \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( -\zeta_{48}^{10} + \zeta_{48}^{6} + \zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( -\zeta_{48}^{13} + \zeta_{48}^{5} + \zeta_{48}^{3} \) Copy content Toggle raw display

Display \(\zeta_{48}^j\) in terms of \(\beta_i\)

\(\zeta_{48}\)\(=\) \( ( \beta_{8} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{2}\)\(=\) \( ( \beta_{9} + \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{3}\)\(=\) \( ( \beta_{15} + \beta_{13} - \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{4}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{5}\)\(=\) \( ( \beta_{10} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{6}\)\(=\) \( ( \beta_{14} + \beta_{12} - \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{7}\)\(=\) \( ( -\beta_{8} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{8}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{48}^{9}\)\(=\) \( ( \beta_{11} + \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{10}\)\(=\) \( ( -\beta_{14} + \beta_{12} + \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{11}\)\(=\) \( ( -\beta_{10} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{12}\)\(=\) \( ( \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{13}\)\(=\) \( ( -\beta_{15} + \beta_{13} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{14}\)\(=\) \( ( -\beta_{9} + \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{15}\)\(=\) \( ( -\beta_{11} + \beta_{7} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{48}^j\)

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(1\) \(1 - \beta_{3}\)

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 \( T_{5}^{8} + 4T_{5}^{6} + 14T_{5}^{4} + 8T_{5}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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