Properties

Label 1764.4.k.z
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: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{3} \)
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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} + \beta_{3} ) q^{5} +O(q^{10})\) \( q + ( 1 + \beta_{1} + \beta_{3} ) q^{5} + ( -1 + 25 \beta_{1} - \beta_{2} + \beta_{3} ) q^{11} + ( -33 - 5 \beta_{2} ) q^{13} + ( -8 + 8 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{17} + ( 84 + 84 \beta_{1} + \beta_{3} ) q^{19} + ( -96 - 96 \beta_{1} ) q^{23} + ( -3 + 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{25} + ( 28 + 17 \beta_{2} ) q^{29} + ( -10 + 41 \beta_{1} - 10 \beta_{2} + 10 \beta_{3} ) q^{31} + ( 96 + 96 \beta_{1} - 19 \beta_{3} ) q^{37} + ( 70 - 34 \beta_{2} ) q^{41} + ( -239 + 19 \beta_{2} ) q^{43} + ( -82 - 82 \beta_{1} - 16 \beta_{3} ) q^{47} + ( 27 - 129 \beta_{1} + 27 \beta_{2} - 27 \beta_{3} ) q^{53} + ( -180 - 27 \beta_{2} ) q^{55} + ( 3 - 633 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{59} + ( 182 + 182 \beta_{1} - 36 \beta_{3} ) q^{61} + ( -668 - 668 \beta_{1} - 38 \beta_{3} ) q^{65} + ( -9 - 442 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} ) q^{67} + ( 686 - 32 \beta_{2} ) q^{71} + ( -21 + 670 \beta_{1} - 21 \beta_{2} + 21 \beta_{3} ) q^{73} + ( 89 + 89 \beta_{1} + 4 \beta_{3} ) q^{79} + ( -178 + 43 \beta_{2} ) q^{83} + ( -1056 - 24 \beta_{2} ) q^{85} + ( -422 - 422 \beta_{1} + 22 \beta_{3} ) q^{89} + ( -86 + 212 \beta_{1} - 86 \beta_{2} + 86 \beta_{3} ) q^{95} + ( -410 + 21 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 3q^{5} + O(q^{10}) \) \( 4q + 3q^{5} - 51q^{11} - 122q^{13} - 24q^{17} + 169q^{19} - 192q^{23} - 11q^{25} + 78q^{29} - 92q^{31} + 173q^{37} + 348q^{41} - 994q^{43} - 180q^{47} + 285q^{53} - 666q^{55} + 1269q^{59} + 328q^{61} - 1374q^{65} + 875q^{67} + 2808q^{71} - 1361q^{73} + 182q^{79} - 798q^{83} - 4176q^{85} - 822q^{89} - 510q^{95} - 1682q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 4 \nu^{2} - 4 \nu - 25 \)\()/20\)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{3} + 3 \nu^{2} + 27 \nu + 5 \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( 17 \nu^{3} + 8 \nu^{2} + 52 \nu - 145 \)\()/20\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 5 \beta_{1}\)\()/9\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + 41 \beta_{1} + 42\)\()/9\)
\(\nu^{3}\)\(=\)\((\)\(8 \beta_{3} - 4 \beta_{2} - 4 \beta_{1} + 57\)\()/9\)

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
−1.63746 1.52274i
2.13746 + 0.656712i
−1.63746 + 1.52274i
2.13746 0.656712i
0 0 0 −4.91238 8.50848i 0 0 0 0 0
361.2 0 0 0 6.41238 + 11.1066i 0 0 0 0 0
1549.1 0 0 0 −4.91238 + 8.50848i 0 0 0 0 0
1549.2 0 0 0 6.41238 11.1066i 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.z 4
3.b odd 2 1 588.4.i.i 4
7.b odd 2 1 252.4.k.d 4
7.c even 3 1 1764.4.a.p 2
7.c even 3 1 inner 1764.4.k.z 4
7.d odd 6 1 252.4.k.d 4
7.d odd 6 1 1764.4.a.x 2
21.c even 2 1 84.4.i.b 4
21.g even 6 1 84.4.i.b 4
21.g even 6 1 588.4.a.g 2
21.h odd 6 1 588.4.a.h 2
21.h odd 6 1 588.4.i.i 4
84.h odd 2 1 336.4.q.h 4
84.j odd 6 1 336.4.q.h 4
84.j odd 6 1 2352.4.a.cb 2
84.n even 6 1 2352.4.a.bp 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.i.b 4 21.c even 2 1
84.4.i.b 4 21.g even 6 1
252.4.k.d 4 7.b odd 2 1
252.4.k.d 4 7.d odd 6 1
336.4.q.h 4 84.h odd 2 1
336.4.q.h 4 84.j odd 6 1
588.4.a.g 2 21.g even 6 1
588.4.a.h 2 21.h odd 6 1
588.4.i.i 4 3.b odd 2 1
588.4.i.i 4 21.h odd 6 1
1764.4.a.p 2 7.c even 3 1
1764.4.a.x 2 7.d odd 6 1
1764.4.k.z 4 1.a even 1 1 trivial
1764.4.k.z 4 7.c even 3 1 inner
2352.4.a.bp 2 84.n even 6 1
2352.4.a.cb 2 84.j odd 6 1

Hecke kernels

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

\( T_{5}^{4} - 3 T_{5}^{3} + 135 T_{5}^{2} + 378 T_{5} + 15876 \)
\( T_{11}^{4} + 51 T_{11}^{3} + 2079 T_{11}^{2} + 26622 T_{11} + 272484 \)
\( T_{13}^{2} + 61 T_{13} - 2276 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 15876 + 378 T + 135 T^{2} - 3 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 272484 + 26622 T + 2079 T^{2} + 51 T^{3} + T^{4} \)
$13$ \( ( -2276 + 61 T + T^{2} )^{2} \)
$17$ \( 65028096 - 193536 T + 8640 T^{2} + 24 T^{3} + T^{4} \)
$19$ \( 49168144 - 1185028 T + 21549 T^{2} - 169 T^{3} + T^{4} \)
$23$ \( ( 9216 + 96 T + T^{2} )^{2} \)
$29$ \( ( -36684 - 39 T + T^{2} )^{2} \)
$31$ \( 114682681 - 985228 T + 19173 T^{2} + 92 T^{3} + T^{4} \)
$37$ \( 1506681856 + 6715168 T + 68745 T^{2} - 173 T^{3} + T^{4} \)
$41$ \( ( -140688 - 174 T + T^{2} )^{2} \)
$43$ \( ( 15454 + 497 T + T^{2} )^{2} \)
$47$ \( 611671824 - 4451760 T + 57132 T^{2} + 180 T^{3} + T^{4} \)
$53$ \( 5356483344 + 20858580 T + 154413 T^{2} - 285 T^{3} + T^{4} \)
$59$ \( 161150862096 - 509422284 T + 1208925 T^{2} - 1269 T^{3} + T^{4} \)
$61$ \( 19408947856 + 45695648 T + 246900 T^{2} - 328 T^{3} + T^{4} \)
$67$ \( 32767516324 - 158390750 T + 584607 T^{2} - 875 T^{3} + T^{4} \)
$71$ \( ( 361476 - 1404 T + T^{2} )^{2} \)
$73$ \( 165260136484 + 553276442 T + 1445799 T^{2} + 1361 T^{3} + T^{4} \)
$79$ \( 38800441 - 1133678 T + 26895 T^{2} - 182 T^{3} + T^{4} \)
$83$ \( ( -197334 + 399 T + T^{2} )^{2} \)
$89$ \( 11416495104 + 87829056 T + 568836 T^{2} + 822 T^{3} + T^{4} \)
$97$ \( ( 120262 + 841 T + T^{2} )^{2} \)
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