Properties

Label 1764.4.k.q
Level $1764$
Weight $4$
Character orbit 1764.k
Analytic conductor $104.079$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{193})\)
Defining polynomial: \(x^{4} - x^{3} + 49 x^{2} + 48 x + 2304\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} - 6 \beta_{2} ) q^{5} +O(q^{10})\) \( q + ( \beta_{1} - 6 \beta_{2} ) q^{5} + ( 1 - 7 \beta_{1} + 6 \beta_{2} - 7 \beta_{3} ) q^{11} -5 \beta_{3} q^{13} + ( -52 + 4 \beta_{1} + 48 \beta_{2} + 4 \beta_{3} ) q^{17} + ( -3 \beta_{1} + 35 \beta_{2} ) q^{19} + ( -20 \beta_{1} + 48 \beta_{2} ) q^{23} + ( 52 - 11 \beta_{1} - 41 \beta_{2} - 11 \beta_{3} ) q^{25} + ( -143 + 11 \beta_{3} ) q^{29} + ( 171 + 20 \beta_{1} - 191 \beta_{2} + 20 \beta_{3} ) q^{31} + ( -45 \beta_{1} + 25 \beta_{2} ) q^{37} + ( -72 - 18 \beta_{3} ) q^{41} + ( 362 - 3 \beta_{3} ) q^{43} + ( 36 \beta_{1} + 90 \beta_{2} ) q^{47} + ( 243 + 9 \beta_{1} - 252 \beta_{2} + 9 \beta_{3} ) q^{53} + ( 331 + 41 \beta_{3} ) q^{55} + ( -113 + 53 \beta_{1} + 60 \beta_{2} + 53 \beta_{3} ) q^{59} + ( 40 \beta_{1} - 286 \beta_{2} ) q^{61} + ( -30 \beta_{1} + 270 \beta_{2} ) q^{65} + ( -94 + 77 \beta_{1} + 17 \beta_{2} + 77 \beta_{3} ) q^{67} + ( -778 - 44 \beta_{3} ) q^{71} + ( 634 - 53 \beta_{1} - 581 \beta_{2} - 53 \beta_{3} ) q^{73} + ( 62 \beta_{1} - 761 \beta_{2} ) q^{79} + ( -755 + 101 \beta_{3} ) q^{83} + ( 68 + 28 \beta_{3} ) q^{85} + ( -42 \beta_{1} + 1008 \beta_{2} ) q^{89} + ( 304 + 50 \beta_{1} - 354 \beta_{2} + 50 \beta_{3} ) q^{95} + ( -295 + 29 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 11q^{5} + O(q^{10}) \) \( 4q - 11q^{5} - 5q^{11} - 10q^{13} - 100q^{17} + 67q^{19} + 76q^{23} + 93q^{25} - 550q^{29} + 362q^{31} + 5q^{37} - 324q^{41} + 1442q^{43} + 216q^{47} + 495q^{53} + 1406q^{55} - 173q^{59} - 532q^{61} + 510q^{65} - 111q^{67} - 3200q^{71} + 1215q^{73} - 1460q^{79} - 2818q^{83} + 328q^{85} + 1974q^{89} + 658q^{95} - 1122q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 49 x^{2} + 48 x + 2304\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 49 \nu^{2} - 49 \nu + 2304 \)\()/2352\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 97 \)\()/49\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 48 \beta_{2} + \beta_{1} - 49\)
\(\nu^{3}\)\(=\)\(49 \beta_{3} - 97\)

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

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
−3.22311 5.58259i
3.72311 + 6.44862i
−3.22311 + 5.58259i
3.72311 6.44862i
0 0 0 −6.22311 10.7787i 0 0 0 0 0
361.2 0 0 0 0.723111 + 1.25246i 0 0 0 0 0
1549.1 0 0 0 −6.22311 + 10.7787i 0 0 0 0 0
1549.2 0 0 0 0.723111 1.25246i 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.q 4
3.b odd 2 1 588.4.i.j 4
7.b odd 2 1 252.4.k.f 4
7.c even 3 1 1764.4.a.y 2
7.c even 3 1 inner 1764.4.k.q 4
7.d odd 6 1 252.4.k.f 4
7.d odd 6 1 1764.4.a.o 2
21.c even 2 1 84.4.i.a 4
21.g even 6 1 84.4.i.a 4
21.g even 6 1 588.4.a.i 2
21.h odd 6 1 588.4.a.f 2
21.h odd 6 1 588.4.i.j 4
84.h odd 2 1 336.4.q.i 4
84.j odd 6 1 336.4.q.i 4
84.j odd 6 1 2352.4.a.bt 2
84.n even 6 1 2352.4.a.bx 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.i.a 4 21.c even 2 1
84.4.i.a 4 21.g even 6 1
252.4.k.f 4 7.b odd 2 1
252.4.k.f 4 7.d odd 6 1
336.4.q.i 4 84.h odd 2 1
336.4.q.i 4 84.j odd 6 1
588.4.a.f 2 21.h odd 6 1
588.4.a.i 2 21.g even 6 1
588.4.i.j 4 3.b odd 2 1
588.4.i.j 4 21.h odd 6 1
1764.4.a.o 2 7.d odd 6 1
1764.4.a.y 2 7.c even 3 1
1764.4.k.q 4 1.a even 1 1 trivial
1764.4.k.q 4 7.c even 3 1 inner
2352.4.a.bt 2 84.j odd 6 1
2352.4.a.bx 2 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 \(S_{4}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{4} + 11 T_{5}^{3} + 139 T_{5}^{2} - 198 T_{5} + 324 \)
\( T_{11}^{4} + 5 T_{11}^{3} + 2383 T_{11}^{2} - 11790 T_{11} + 5560164 \)
\( T_{13}^{2} + 5 T_{13} - 1200 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 324 - 198 T + 139 T^{2} + 11 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 5560164 - 11790 T + 2383 T^{2} + 5 T^{3} + T^{4} \)
$13$ \( ( -1200 + 5 T + T^{2} )^{2} \)
$17$ \( 2985984 + 172800 T + 8272 T^{2} + 100 T^{3} + T^{4} \)
$19$ \( 473344 - 46096 T + 3801 T^{2} - 67 T^{3} + T^{4} \)
$23$ \( 318836736 + 1357056 T + 23632 T^{2} - 76 T^{3} + T^{4} \)
$29$ \( ( 13068 + 275 T + T^{2} )^{2} \)
$31$ \( 181198521 - 4872882 T + 117583 T^{2} - 362 T^{3} + T^{4} \)
$37$ \( 9545290000 + 488500 T + 97725 T^{2} - 5 T^{3} + T^{4} \)
$41$ \( ( -9072 + 162 T + T^{2} )^{2} \)
$43$ \( ( 129526 - 721 T + T^{2} )^{2} \)
$47$ \( 2587553424 + 10987488 T + 97524 T^{2} - 216 T^{3} + T^{4} \)
$53$ \( 3288793104 - 28387260 T + 187677 T^{2} - 495 T^{3} + T^{4} \)
$59$ \( 16397314704 - 22152996 T + 157981 T^{2} + 173 T^{3} + T^{4} \)
$61$ \( 41525136 - 3428208 T + 289468 T^{2} + 532 T^{3} + T^{4} \)
$67$ \( 80085604036 - 31412334 T + 295315 T^{2} + 111 T^{3} + T^{4} \)
$71$ \( ( 546588 + 1600 T + T^{2} )^{2} \)
$73$ \( 54532524484 - 283729230 T + 1242703 T^{2} - 1215 T^{3} + T^{4} \)
$79$ \( 120705520329 + 507243420 T + 1784173 T^{2} + 1460 T^{3} + T^{4} \)
$83$ \( ( 4122 + 1409 T + T^{2} )^{2} \)
$89$ \( 790420571136 - 1754996544 T + 3007620 T^{2} - 1974 T^{3} + T^{4} \)
$97$ \( ( 38102 + 561 T + T^{2} )^{2} \)
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