Properties

Label 1764.2.e.f
Level 1764
Weight 2
Character orbit 1764.e
Analytic conductor 14.086
Analytic rank 0
Dimension 8
CM discriminant -7
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.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.157351936.1
Defining polynomial: \(x^{8} + x^{4} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} -\beta_{1} q^{4} -\beta_{5} q^{8} +O(q^{10})\) \( q + \beta_{6} q^{2} -\beta_{1} q^{4} -\beta_{5} q^{8} + ( \beta_{5} + \beta_{6} + \beta_{7} ) q^{11} + \beta_{2} q^{16} + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{22} + ( 2 \beta_{3} - \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{23} + 5 q^{25} + ( -2 \beta_{3} - \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{29} + ( -\beta_{3} - \beta_{6} + 2 \beta_{7} ) q^{32} + ( 4 \beta_{1} - 2 \beta_{4} ) q^{37} + ( -4 \beta_{1} - 2 \beta_{4} ) q^{43} + ( -\beta_{3} - \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{44} + ( 6 - 3 \beta_{1} + \beta_{2} + \beta_{4} ) q^{46} + 5 \beta_{6} q^{50} + ( 2 \beta_{3} - \beta_{5} - 5 \beta_{6} + \beta_{7} ) q^{53} + ( -6 - 3 \beta_{1} + \beta_{2} - \beta_{4} ) q^{58} + ( \beta_{1} - 2 \beta_{4} ) q^{64} + ( -2 - 4 \beta_{2} ) q^{67} + ( 2 \beta_{3} - 3 \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{71} + ( -4 \beta_{3} + 4 \beta_{5} ) q^{74} + ( -2 - 4 \beta_{2} ) q^{79} + ( -4 \beta_{3} - 4 \beta_{5} ) q^{86} + ( -8 - 3 \beta_{1} + \beta_{2} + 2 \beta_{4} ) q^{88} + ( \beta_{3} - 3 \beta_{5} + 5 \beta_{6} + 2 \beta_{7} ) q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 4q^{16} + 20q^{22} + 40q^{25} + 44q^{46} - 52q^{58} - 68q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + x^{4} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)
\(\beta_{2}\)\(=\)\( \nu^{4} \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} + 5 \nu \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} + \nu^{2} \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{5} + 4 \nu \)\()/3\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} - 5 \nu^{3} \)\()/12\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{7} + 4 \nu^{5} + 11 \nu^{3} + 20 \nu \)\()/24\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{3}\)\()/3\)
\(\nu^{2}\)\(=\)\(\beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{7} - 5 \beta_{6} - \beta_{3}\)\()/3\)
\(\nu^{4}\)\(=\)\(\beta_{2}\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{5} + 4 \beta_{3}\)\()/3\)
\(\nu^{6}\)\(=\)\(2 \beta_{4} - \beta_{1}\)
\(\nu^{7}\)\(=\)\((\)\(-10 \beta_{7} - 11 \beta_{6} + 5 \beta_{3}\)\()/3\)

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1079.1
−0.581861 + 1.28897i
−0.581861 1.28897i
−1.28897 + 0.581861i
−1.28897 0.581861i
1.28897 + 0.581861i
1.28897 0.581861i
0.581861 + 1.28897i
0.581861 1.28897i
−1.28897 0.581861i 0 1.32288 + 1.50000i 0 0 0 −0.832353 2.70318i 0 0
1079.2 −1.28897 + 0.581861i 0 1.32288 1.50000i 0 0 0 −0.832353 + 2.70318i 0 0
1079.3 −0.581861 1.28897i 0 −1.32288 + 1.50000i 0 0 0 2.70318 + 0.832353i 0 0
1079.4 −0.581861 + 1.28897i 0 −1.32288 1.50000i 0 0 0 2.70318 0.832353i 0 0
1079.5 0.581861 1.28897i 0 −1.32288 1.50000i 0 0 0 −2.70318 + 0.832353i 0 0
1079.6 0.581861 + 1.28897i 0 −1.32288 + 1.50000i 0 0 0 −2.70318 0.832353i 0 0
1079.7 1.28897 0.581861i 0 1.32288 1.50000i 0 0 0 0.832353 2.70318i 0 0
1079.8 1.28897 + 0.581861i 0 1.32288 + 1.50000i 0 0 0 0.832353 + 2.70318i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1079.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.e.f 8
3.b odd 2 1 inner 1764.2.e.f 8
4.b odd 2 1 inner 1764.2.e.f 8
7.b odd 2 1 CM 1764.2.e.f 8
12.b even 2 1 inner 1764.2.e.f 8
21.c even 2 1 inner 1764.2.e.f 8
28.d even 2 1 inner 1764.2.e.f 8
84.h odd 2 1 inner 1764.2.e.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.2.e.f 8 1.a even 1 1 trivial
1764.2.e.f 8 3.b odd 2 1 inner
1764.2.e.f 8 4.b odd 2 1 inner
1764.2.e.f 8 7.b odd 2 1 CM
1764.2.e.f 8 12.b even 2 1 inner
1764.2.e.f 8 21.c even 2 1 inner
1764.2.e.f 8 28.d even 2 1 inner
1764.2.e.f 8 84.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5} \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{4} + 16 T^{8} \)
$3$ 1
$5$ \( ( 1 - 5 T^{2} )^{8} \)
$7$ 1
$11$ \( ( 1 - 206 T^{4} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 + 13 T^{2} )^{8} \)
$17$ \( ( 1 - 17 T^{2} )^{8} \)
$19$ \( ( 1 - 19 T^{2} )^{8} \)
$23$ \( ( 1 - 734 T^{4} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 + 1234 T^{4} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 31 T^{2} )^{8} \)
$37$ \( ( 1 - 38 T^{2} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 - 41 T^{2} )^{8} \)
$43$ \( ( 1 + 58 T^{2} + 1849 T^{4} )^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{8} \)
$53$ \( ( 1 - 5582 T^{4} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 + 59 T^{2} )^{8} \)
$61$ \( ( 1 + 61 T^{2} )^{8} \)
$67$ \( ( 1 - 4 T + 67 T^{2} )^{4}( 1 + 4 T + 67 T^{2} )^{4} \)
$71$ \( ( 1 + 2914 T^{4} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 + 73 T^{2} )^{8} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )^{4}( 1 + 8 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 + 83 T^{2} )^{8} \)
$89$ \( ( 1 - 89 T^{2} )^{8} \)
$97$ \( ( 1 + 97 T^{2} )^{8} \)
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