Properties

Label 1764.2.k.m
Level 1764
Weight 2
Character orbit 1764.k
Analytic conductor 14.086
Analytic rank 0
Dimension 8
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.k (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.12745506816.4
Defining polynomial: \(x^{8} + 8 x^{6} + 55 x^{4} + 72 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{7} q^{5} +O(q^{10})\) \( q -\beta_{7} q^{5} -\beta_{5} q^{11} -3 \beta_{6} q^{13} + \beta_{3} q^{17} + ( -2 \beta_{2} - 2 \beta_{6} ) q^{19} + ( -\beta_{4} - \beta_{5} ) q^{23} -9 \beta_{1} q^{25} + \beta_{4} q^{29} + 6 \beta_{2} q^{31} + ( -4 + 4 \beta_{1} ) q^{37} + ( -\beta_{3} + \beta_{7} ) q^{41} + 8 q^{43} + 2 \beta_{7} q^{47} + 2 \beta_{5} q^{53} -14 \beta_{6} q^{55} -2 \beta_{3} q^{59} + ( -7 \beta_{2} - 7 \beta_{6} ) q^{61} + ( 3 \beta_{4} + 3 \beta_{5} ) q^{65} -12 \beta_{1} q^{67} -3 \beta_{4} q^{71} + \beta_{2} q^{73} + ( 4 - 4 \beta_{1} ) q^{79} + ( 4 \beta_{3} - 4 \beta_{7} ) q^{83} + 14 q^{85} + \beta_{7} q^{89} + 2 \beta_{5} q^{95} -7 \beta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 36q^{25} - 16q^{37} + 64q^{43} - 48q^{67} + 16q^{79} + 112q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 8 x^{6} + 55 x^{4} + 72 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 8 \nu^{6} + 55 \nu^{4} + 440 \nu^{2} + 576 \)\()/495\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + 203 \nu \)\()/165\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 533 \nu \)\()/165\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{6} + 296 \)\()/55\)
\(\beta_{5}\)\(=\)\((\)\( -46 \nu^{6} - 440 \nu^{4} - 2530 \nu^{2} - 3312 \)\()/495\)
\(\beta_{6}\)\(=\)\((\)\( 8 \nu^{7} + 55 \nu^{5} + 341 \nu^{3} + 81 \nu \)\()/297\)
\(\beta_{7}\)\(=\)\((\)\( 79 \nu^{7} + 605 \nu^{5} + 4345 \nu^{3} + 5688 \nu \)\()/1485\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + \beta_{4} + 8 \beta_{1} - 8\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{7} - 11 \beta_{6} - 5 \beta_{3}\)\()/2\)
\(\nu^{4}\)\(=\)\(-4 \beta_{5} - 23 \beta_{1}\)
\(\nu^{5}\)\(=\)\((\)\(-31 \beta_{7} + 79 \beta_{6} + 79 \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-55 \beta_{4} + 296\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(203 \beta_{3} - 533 \beta_{2}\)\()/2\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−1.28897 + 2.23256i
−0.581861 + 1.00781i
1.28897 2.23256i
0.581861 1.00781i
−1.28897 2.23256i
−0.581861 1.00781i
1.28897 + 2.23256i
0.581861 + 1.00781i
0 0 0 −1.87083 3.24037i 0 0 0 0 0
361.2 0 0 0 −1.87083 3.24037i 0 0 0 0 0
361.3 0 0 0 1.87083 + 3.24037i 0 0 0 0 0
361.4 0 0 0 1.87083 + 3.24037i 0 0 0 0 0
1549.1 0 0 0 −1.87083 + 3.24037i 0 0 0 0 0
1549.2 0 0 0 −1.87083 + 3.24037i 0 0 0 0 0
1549.3 0 0 0 1.87083 3.24037i 0 0 0 0 0
1549.4 0 0 0 1.87083 3.24037i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1549.4
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.k.m 8
3.b odd 2 1 inner 1764.2.k.m 8
7.b odd 2 1 inner 1764.2.k.m 8
7.c even 3 1 1764.2.a.m 4
7.c even 3 1 inner 1764.2.k.m 8
7.d odd 6 1 1764.2.a.m 4
7.d odd 6 1 inner 1764.2.k.m 8
21.c even 2 1 inner 1764.2.k.m 8
21.g even 6 1 1764.2.a.m 4
21.g even 6 1 inner 1764.2.k.m 8
21.h odd 6 1 1764.2.a.m 4
21.h odd 6 1 inner 1764.2.k.m 8
28.f even 6 1 7056.2.a.cy 4
28.g odd 6 1 7056.2.a.cy 4
84.j odd 6 1 7056.2.a.cy 4
84.n even 6 1 7056.2.a.cy 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.2.a.m 4 7.c even 3 1
1764.2.a.m 4 7.d odd 6 1
1764.2.a.m 4 21.g even 6 1
1764.2.a.m 4 21.h odd 6 1
1764.2.k.m 8 1.a even 1 1 trivial
1764.2.k.m 8 3.b odd 2 1 inner
1764.2.k.m 8 7.b odd 2 1 inner
1764.2.k.m 8 7.c even 3 1 inner
1764.2.k.m 8 7.d odd 6 1 inner
1764.2.k.m 8 21.c even 2 1 inner
1764.2.k.m 8 21.g even 6 1 inner
1764.2.k.m 8 21.h odd 6 1 inner
7056.2.a.cy 4 28.f even 6 1
7056.2.a.cy 4 28.g odd 6 1
7056.2.a.cy 4 84.j odd 6 1
7056.2.a.cy 4 84.n even 6 1

Hecke kernels

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

\( T_{5}^{4} + 14 T_{5}^{2} + 196 \)
\( T_{11}^{4} + 28 T_{11}^{2} + 784 \)
\( T_{13}^{2} - 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 + 4 T^{2} - 9 T^{4} + 100 T^{6} + 625 T^{8} )^{2} \)
$7$ 1
$11$ \( ( 1 + 6 T^{2} - 85 T^{4} + 726 T^{6} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 + 8 T^{2} + 169 T^{4} )^{4} \)
$17$ \( ( 1 - 20 T^{2} + 111 T^{4} - 5780 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 30 T^{2} + 539 T^{4} - 10830 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 - 18 T^{2} - 205 T^{4} - 9522 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 + 30 T^{2} + 841 T^{4} )^{4} \)
$31$ \( ( 1 + 10 T^{2} - 861 T^{4} + 9610 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 + 4 T - 21 T^{2} + 148 T^{3} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 + 68 T^{2} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 - 8 T + 43 T^{2} )^{8} \)
$47$ \( ( 1 - 38 T^{2} - 765 T^{4} - 83942 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 + 6 T^{2} - 2773 T^{4} + 16854 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 - 62 T^{2} + 363 T^{4} - 215822 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 24 T^{2} - 3145 T^{4} - 89304 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 - 110 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 - 144 T^{2} + 15407 T^{4} - 767376 T^{6} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 - 17 T + 79 T^{2} )^{4}( 1 + 13 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 - 58 T^{2} + 6889 T^{4} )^{4} \)
$89$ \( ( 1 - 164 T^{2} + 18975 T^{4} - 1299044 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 + 96 T^{2} + 9409 T^{4} )^{4} \)
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