Properties

Label 1764.3.z.k
Level $1764$
Weight $3$
Character orbit 1764.z
Analytic conductor $48.066$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(48.0655186332\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.1
Defining polynomial: \(x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 7^{4} \)
Twist minimal: no (minimal twist has level 196)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{2} - 2 \beta_{6} ) q^{5} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{2} - 2 \beta_{6} ) q^{5} + ( -6 \beta_{4} + \beta_{7} ) q^{11} + ( -3 \beta_{1} + 4 \beta_{3} + 4 \beta_{6} ) q^{13} -3 \beta_{3} q^{17} + ( 5 \beta_{1} - 5 \beta_{2} - \beta_{6} ) q^{19} + ( 26 - 26 \beta_{4} ) q^{23} + ( 27 \beta_{4} + 2 \beta_{7} ) q^{25} + ( -8 - 2 \beta_{5} ) q^{29} + ( -2 \beta_{2} + 8 \beta_{3} ) q^{31} + ( 32 - 32 \beta_{4} ) q^{37} + ( 3 \beta_{1} + 5 \beta_{3} + 5 \beta_{6} ) q^{41} + ( 10 - 7 \beta_{5} ) q^{43} + ( 12 \beta_{1} - 12 \beta_{2} - 2 \beta_{6} ) q^{47} + ( 6 \beta_{4} - 2 \beta_{7} ) q^{53} + ( 14 \beta_{1} - 6 \beta_{3} - 6 \beta_{6} ) q^{55} + ( 11 \beta_{2} + 3 \beta_{3} ) q^{59} + ( -9 \beta_{1} + 9 \beta_{2} - 8 \beta_{6} ) q^{61} + ( 24 - 24 \beta_{4} - 14 \beta_{5} - 14 \beta_{7} ) q^{65} + ( -28 \beta_{4} + 6 \beta_{7} ) q^{67} + ( -56 + 4 \beta_{5} ) q^{71} + ( 11 \beta_{2} - 3 \beta_{3} ) q^{73} + ( -60 + 60 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{79} + ( -13 \beta_{1} - 33 \beta_{3} - 33 \beta_{6} ) q^{83} + ( -54 + 6 \beta_{5} ) q^{85} + ( 11 \beta_{1} - 11 \beta_{2} - 25 \beta_{6} ) q^{89} + ( -62 \beta_{4} + 12 \beta_{7} ) q^{95} + ( -22 \beta_{1} - 33 \beta_{3} - 33 \beta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 24q^{11} + 104q^{23} + 108q^{25} - 64q^{29} + 128q^{37} + 80q^{43} + 24q^{53} + 96q^{65} - 112q^{67} - 448q^{71} - 240q^{79} - 432q^{85} - 248q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + 8 \nu \)\()/14\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} - 41 \nu \)\()/7\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{6} + 7 \nu^{4} - 28 \nu^{2} + 16 \)\()/14\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} - 20 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( -12 \nu^{7} + 49 \nu^{5} - 168 \nu^{3} + 96 \nu \)\()/14\)
\(\beta_{7}\)\(=\)\((\)\( -3 \nu^{6} + 14 \nu^{4} - 42 \nu^{2} + 24 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2}\)\()/7\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} + \beta_{5} - 14 \beta_{4} + 14\)\()/7\)
\(\nu^{3}\)\(=\)\(\beta_{1}\)
\(\nu^{4}\)\(=\)\((\)\(4 \beta_{7} - 42 \beta_{4}\)\()/7\)
\(\nu^{5}\)\(=\)\((\)\(2 \beta_{6} - 24 \beta_{2} + 24 \beta_{1}\)\()/7\)
\(\nu^{6}\)\(=\)\(-2 \beta_{5} - 20\)
\(\nu^{7}\)\(=\)\((\)\(-8 \beta_{3} - 82 \beta_{2}\)\()/7\)

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1 - \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
325.1
0.662827 + 0.382683i
−1.60021 0.923880i
1.60021 + 0.923880i
−0.662827 0.382683i
0.662827 0.382683i
−1.60021 + 0.923880i
1.60021 0.923880i
−0.662827 + 0.382683i
0 0 0 −7.33820 + 4.23671i 0 0 0 0 0
325.2 0 0 0 −4.91434 + 2.83730i 0 0 0 0 0
325.3 0 0 0 4.91434 2.83730i 0 0 0 0 0
325.4 0 0 0 7.33820 4.23671i 0 0 0 0 0
901.1 0 0 0 −7.33820 4.23671i 0 0 0 0 0
901.2 0 0 0 −4.91434 2.83730i 0 0 0 0 0
901.3 0 0 0 4.91434 + 2.83730i 0 0 0 0 0
901.4 0 0 0 7.33820 + 4.23671i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 901.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.3.z.k 8
3.b odd 2 1 196.3.h.c 8
7.b odd 2 1 inner 1764.3.z.k 8
7.c even 3 1 1764.3.d.e 4
7.c even 3 1 inner 1764.3.z.k 8
7.d odd 6 1 1764.3.d.e 4
7.d odd 6 1 inner 1764.3.z.k 8
12.b even 2 1 784.3.s.g 8
21.c even 2 1 196.3.h.c 8
21.g even 6 1 196.3.b.b 4
21.g even 6 1 196.3.h.c 8
21.h odd 6 1 196.3.b.b 4
21.h odd 6 1 196.3.h.c 8
84.h odd 2 1 784.3.s.g 8
84.j odd 6 1 784.3.c.d 4
84.j odd 6 1 784.3.s.g 8
84.n even 6 1 784.3.c.d 4
84.n even 6 1 784.3.s.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.3.b.b 4 21.g even 6 1
196.3.b.b 4 21.h odd 6 1
196.3.h.c 8 3.b odd 2 1
196.3.h.c 8 21.c even 2 1
196.3.h.c 8 21.g even 6 1
196.3.h.c 8 21.h odd 6 1
784.3.c.d 4 84.j odd 6 1
784.3.c.d 4 84.n even 6 1
784.3.s.g 8 12.b even 2 1
784.3.s.g 8 84.h odd 2 1
784.3.s.g 8 84.j odd 6 1
784.3.s.g 8 84.n even 6 1
1764.3.d.e 4 7.c even 3 1
1764.3.d.e 4 7.d odd 6 1
1764.3.z.k 8 1.a even 1 1 trivial
1764.3.z.k 8 7.b odd 2 1 inner
1764.3.z.k 8 7.c even 3 1 inner
1764.3.z.k 8 7.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{8} - 104 T_{5}^{6} + 8504 T_{5}^{4} - 240448 T_{5}^{2} + 5345344 \)
\( T_{11}^{4} + 12 T_{11}^{3} + 206 T_{11}^{2} - 744 T_{11} + 3844 \)
\( T_{13}^{4} + 776 T_{13}^{2} + 150152 \)
\( T_{19}^{8} - 1060 T_{19}^{6} + 1077998 T_{19}^{4} - 48338120 T_{19}^{2} + 2079542404 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 5345344 - 240448 T^{2} + 8504 T^{4} - 104 T^{6} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( ( 3844 - 744 T + 206 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$13$ \( ( 150152 + 776 T^{2} + T^{4} )^{2} \)
$17$ \( 26244 - 29160 T^{2} + 32238 T^{4} - 180 T^{6} + T^{8} \)
$19$ \( 2079542404 - 48338120 T^{2} + 1077998 T^{4} - 1060 T^{6} + T^{8} \)
$23$ \( ( 676 - 26 T + T^{2} )^{4} \)
$29$ \( ( -328 + 16 T + T^{2} )^{4} \)
$31$ \( 94450499584 - 481890304 T^{2} + 2151296 T^{4} - 1568 T^{6} + T^{8} \)
$37$ \( ( 1024 - 32 T + T^{2} )^{4} \)
$41$ \( ( 132098 + 740 T^{2} + T^{4} )^{2} \)
$43$ \( ( -4702 - 20 T + T^{2} )^{4} \)
$47$ \( 1420787713024 - 7189950976 T^{2} + 35193056 T^{4} - 6032 T^{6} + T^{8} \)
$53$ \( ( 126736 + 4272 T + 500 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$59$ \( 111931731844 - 1591176872 T^{2} + 22284974 T^{4} - 4756 T^{6} + T^{8} \)
$61$ \( 8688298598464 - 11625302848 T^{2} + 12607544 T^{4} - 3944 T^{6} + T^{8} \)
$67$ \( ( 7529536 - 153664 T + 5880 T^{2} + 56 T^{3} + T^{4} )^{2} \)
$71$ \( ( 1568 + 112 T + T^{2} )^{4} \)
$73$ \( 2755075464964 - 8770605128 T^{2} + 26260814 T^{4} - 5284 T^{6} + T^{8} \)
$79$ \( ( 10291264 + 384960 T + 11192 T^{2} + 120 T^{3} + T^{4} )^{2} \)
$83$ \( ( 102330818 + 25108 T^{2} + T^{4} )^{2} \)
$89$ \( 6573541223354884 - 1584251966120 T^{2} + 300734222 T^{4} - 19540 T^{6} + T^{8} \)
$97$ \( ( 310652738 + 35332 T^{2} + T^{4} )^{2} \)
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