Properties

Label 1764.4.k.k
Level $1764$
Weight $4$
Character orbit 1764.k
Analytic conductor $104.079$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1764.k (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(104.079369250\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 6 \zeta_{6} q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + 6 \zeta_{6} q^{5} + (12 \zeta_{6} - 12) q^{11} - 82 q^{13} + (30 \zeta_{6} - 30) q^{17} - 68 \zeta_{6} q^{19} + 216 \zeta_{6} q^{23} + ( - 89 \zeta_{6} + 89) q^{25} - 246 q^{29} + ( - 112 \zeta_{6} + 112) q^{31} - 110 \zeta_{6} q^{37} + 246 q^{41} - 172 q^{43} + 192 \zeta_{6} q^{47} + ( - 558 \zeta_{6} + 558) q^{53} - 72 q^{55} + ( - 540 \zeta_{6} + 540) q^{59} - 110 \zeta_{6} q^{61} - 492 \zeta_{6} q^{65} + (140 \zeta_{6} - 140) q^{67} + 840 q^{71} + ( - 550 \zeta_{6} + 550) q^{73} + 208 \zeta_{6} q^{79} - 516 q^{83} - 180 q^{85} - 1398 \zeta_{6} q^{89} + ( - 408 \zeta_{6} + 408) q^{95} + 1586 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{5} - 12 q^{11} - 164 q^{13} - 30 q^{17} - 68 q^{19} + 216 q^{23} + 89 q^{25} - 492 q^{29} + 112 q^{31} - 110 q^{37} + 492 q^{41} - 344 q^{43} + 192 q^{47} + 558 q^{53} - 144 q^{55} + 540 q^{59} - 110 q^{61} - 492 q^{65} - 140 q^{67} + 1680 q^{71} + 550 q^{73} + 208 q^{79} - 1032 q^{83} - 360 q^{85} - 1398 q^{89} + 408 q^{95} + 3172 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 3.00000 + 5.19615i 0 0 0 0 0
1549.1 0 0 0 3.00000 5.19615i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.4.k.k 2
3.b odd 2 1 196.4.e.c 2
7.b odd 2 1 1764.4.k.e 2
7.c even 3 1 252.4.a.c 1
7.c even 3 1 inner 1764.4.k.k 2
7.d odd 6 1 1764.4.a.k 1
7.d odd 6 1 1764.4.k.e 2
21.c even 2 1 196.4.e.d 2
21.g even 6 1 196.4.a.b 1
21.g even 6 1 196.4.e.d 2
21.h odd 6 1 28.4.a.b 1
21.h odd 6 1 196.4.e.c 2
28.g odd 6 1 1008.4.a.f 1
84.j odd 6 1 784.4.a.n 1
84.n even 6 1 112.4.a.c 1
105.o odd 6 1 700.4.a.e 1
105.x even 12 2 700.4.e.f 2
168.s odd 6 1 448.4.a.d 1
168.v even 6 1 448.4.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.b 1 21.h odd 6 1
112.4.a.c 1 84.n even 6 1
196.4.a.b 1 21.g even 6 1
196.4.e.c 2 3.b odd 2 1
196.4.e.c 2 21.h odd 6 1
196.4.e.d 2 21.c even 2 1
196.4.e.d 2 21.g even 6 1
252.4.a.c 1 7.c even 3 1
448.4.a.d 1 168.s odd 6 1
448.4.a.m 1 168.v even 6 1
700.4.a.e 1 105.o odd 6 1
700.4.e.f 2 105.x even 12 2
784.4.a.n 1 84.j odd 6 1
1008.4.a.f 1 28.g odd 6 1
1764.4.a.k 1 7.d odd 6 1
1764.4.k.e 2 7.b odd 2 1
1764.4.k.e 2 7.d odd 6 1
1764.4.k.k 2 1.a even 1 1 trivial
1764.4.k.k 2 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{2} - 6T_{5} + 36 \) Copy content Toggle raw display
\( T_{11}^{2} + 12T_{11} + 144 \) Copy content Toggle raw display
\( T_{13} + 82 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$13$ \( (T + 82)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 30T + 900 \) Copy content Toggle raw display
$19$ \( T^{2} + 68T + 4624 \) Copy content Toggle raw display
$23$ \( T^{2} - 216T + 46656 \) Copy content Toggle raw display
$29$ \( (T + 246)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 112T + 12544 \) Copy content Toggle raw display
$37$ \( T^{2} + 110T + 12100 \) Copy content Toggle raw display
$41$ \( (T - 246)^{2} \) Copy content Toggle raw display
$43$ \( (T + 172)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 192T + 36864 \) Copy content Toggle raw display
$53$ \( T^{2} - 558T + 311364 \) Copy content Toggle raw display
$59$ \( T^{2} - 540T + 291600 \) Copy content Toggle raw display
$61$ \( T^{2} + 110T + 12100 \) Copy content Toggle raw display
$67$ \( T^{2} + 140T + 19600 \) Copy content Toggle raw display
$71$ \( (T - 840)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 550T + 302500 \) Copy content Toggle raw display
$79$ \( T^{2} - 208T + 43264 \) Copy content Toggle raw display
$83$ \( (T + 516)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 1398 T + 1954404 \) Copy content Toggle raw display
$97$ \( (T - 1586)^{2} \) Copy content Toggle raw display
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