Properties

Label 588.4.i.j
Level $588$
Weight $4$
Character orbit 588.i
Analytic conductor $34.693$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 588.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(34.6931230834\)
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.

\(f(q)\) \(=\) \( q + ( 3 - 3 \beta_{2} ) q^{3} + ( -\beta_{1} + 6 \beta_{2} ) q^{5} -9 \beta_{2} q^{9} +O(q^{10})\) \( q + ( 3 - 3 \beta_{2} ) q^{3} + ( -\beta_{1} + 6 \beta_{2} ) q^{5} -9 \beta_{2} q^{9} + ( -1 + 7 \beta_{1} - 6 \beta_{2} + 7 \beta_{3} ) q^{11} -5 \beta_{3} q^{13} + ( 15 + 3 \beta_{3} ) q^{15} + ( 52 - 4 \beta_{1} - 48 \beta_{2} - 4 \beta_{3} ) q^{17} + ( -3 \beta_{1} + 35 \beta_{2} ) q^{19} + ( 20 \beta_{1} - 48 \beta_{2} ) q^{23} + ( 52 - 11 \beta_{1} - 41 \beta_{2} - 11 \beta_{3} ) q^{25} -27 q^{27} + ( 143 - 11 \beta_{3} ) q^{29} + ( 171 + 20 \beta_{1} - 191 \beta_{2} + 20 \beta_{3} ) q^{31} + ( 21 \beta_{1} - 18 \beta_{2} ) q^{33} + ( -45 \beta_{1} + 25 \beta_{2} ) q^{37} + ( -15 \beta_{1} + 15 \beta_{2} - 15 \beta_{3} ) q^{39} + ( 72 + 18 \beta_{3} ) q^{41} + ( 362 - 3 \beta_{3} ) q^{43} + ( 45 + 9 \beta_{1} - 54 \beta_{2} + 9 \beta_{3} ) q^{45} + ( -36 \beta_{1} - 90 \beta_{2} ) q^{47} + ( -12 \beta_{1} - 144 \beta_{2} ) q^{51} + ( -243 - 9 \beta_{1} + 252 \beta_{2} - 9 \beta_{3} ) q^{53} + ( 331 + 41 \beta_{3} ) q^{55} + ( 96 + 9 \beta_{3} ) q^{57} + ( 113 - 53 \beta_{1} - 60 \beta_{2} - 53 \beta_{3} ) q^{59} + ( 40 \beta_{1} - 286 \beta_{2} ) q^{61} + ( 30 \beta_{1} - 270 \beta_{2} ) q^{65} + ( -94 + 77 \beta_{1} + 17 \beta_{2} + 77 \beta_{3} ) q^{67} + ( -84 - 60 \beta_{3} ) q^{69} + ( 778 + 44 \beta_{3} ) q^{71} + ( 634 - 53 \beta_{1} - 581 \beta_{2} - 53 \beta_{3} ) q^{73} + ( -33 \beta_{1} - 123 \beta_{2} ) q^{75} + ( 62 \beta_{1} - 761 \beta_{2} ) q^{79} + ( -81 + 81 \beta_{2} ) q^{81} + ( 755 - 101 \beta_{3} ) q^{83} + ( 68 + 28 \beta_{3} ) q^{85} + ( 429 - 33 \beta_{1} - 396 \beta_{2} - 33 \beta_{3} ) q^{87} + ( 42 \beta_{1} - 1008 \beta_{2} ) q^{89} + ( 60 \beta_{1} - 573 \beta_{2} ) q^{93} + ( -304 - 50 \beta_{1} + 354 \beta_{2} - 50 \beta_{3} ) q^{95} + ( -295 + 29 \beta_{3} ) q^{97} + ( 9 - 63 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{3} + 11q^{5} - 18q^{9} + O(q^{10}) \) \( 4q + 6q^{3} + 11q^{5} - 18q^{9} + 5q^{11} - 10q^{13} + 66q^{15} + 100q^{17} + 67q^{19} - 76q^{23} + 93q^{25} - 108q^{27} + 550q^{29} + 362q^{31} - 15q^{33} + 5q^{37} - 15q^{39} + 324q^{41} + 1442q^{43} + 99q^{45} - 216q^{47} - 300q^{51} - 495q^{53} + 1406q^{55} + 402q^{57} + 173q^{59} - 532q^{61} - 510q^{65} - 111q^{67} - 456q^{69} + 3200q^{71} + 1215q^{73} - 279q^{75} - 1460q^{79} - 162q^{81} + 2818q^{83} + 328q^{85} + 825q^{87} - 1974q^{89} - 1086q^{93} - 658q^{95} - 1122q^{97} - 90q^{99} + O(q^{100}) \)

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 \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

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
3.72311 + 6.44862i
−3.22311 5.58259i
3.72311 6.44862i
−3.22311 + 5.58259i
0 1.50000 2.59808i 0 −0.723111 1.25246i 0 0 0 −4.50000 7.79423i 0
361.2 0 1.50000 2.59808i 0 6.22311 + 10.7787i 0 0 0 −4.50000 7.79423i 0
373.1 0 1.50000 + 2.59808i 0 −0.723111 + 1.25246i 0 0 0 −4.50000 + 7.79423i 0
373.2 0 1.50000 + 2.59808i 0 6.22311 10.7787i 0 0 0 −4.50000 + 7.79423i 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 588.4.i.j 4
3.b odd 2 1 1764.4.k.q 4
7.b odd 2 1 84.4.i.a 4
7.c even 3 1 588.4.a.f 2
7.c even 3 1 inner 588.4.i.j 4
7.d odd 6 1 84.4.i.a 4
7.d odd 6 1 588.4.a.i 2
21.c even 2 1 252.4.k.f 4
21.g even 6 1 252.4.k.f 4
21.g even 6 1 1764.4.a.o 2
21.h odd 6 1 1764.4.a.y 2
21.h odd 6 1 1764.4.k.q 4
28.d even 2 1 336.4.q.i 4
28.f even 6 1 336.4.q.i 4
28.f even 6 1 2352.4.a.bt 2
28.g odd 6 1 2352.4.a.bx 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.i.a 4 7.b odd 2 1
84.4.i.a 4 7.d odd 6 1
252.4.k.f 4 21.c even 2 1
252.4.k.f 4 21.g even 6 1
336.4.q.i 4 28.d even 2 1
336.4.q.i 4 28.f even 6 1
588.4.a.f 2 7.c even 3 1
588.4.a.i 2 7.d odd 6 1
588.4.i.j 4 1.a even 1 1 trivial
588.4.i.j 4 7.c even 3 1 inner
1764.4.a.o 2 21.g even 6 1
1764.4.a.y 2 21.h odd 6 1
1764.4.k.q 4 3.b odd 2 1
1764.4.k.q 4 21.h odd 6 1
2352.4.a.bt 2 28.f even 6 1
2352.4.a.bx 2 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 11 T_{5}^{3} + 139 T_{5}^{2} + 198 T_{5} + 324 \) acting on \(S_{4}^{\mathrm{new}}(588, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 9 - 3 T + T^{2} )^{2} \)
$5$ \( 324 + 198 T + 139 T^{2} - 11 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 5560164 + 11790 T + 2383 T^{2} - 5 T^{3} + T^{4} \)
$13$ \( ( -1200 + 5 T + T^{2} )^{2} \)
$17$ \( 2985984 - 172800 T + 8272 T^{2} - 100 T^{3} + T^{4} \)
$19$ \( 473344 - 46096 T + 3801 T^{2} - 67 T^{3} + T^{4} \)
$23$ \( 318836736 - 1357056 T + 23632 T^{2} + 76 T^{3} + T^{4} \)
$29$ \( ( 13068 - 275 T + T^{2} )^{2} \)
$31$ \( 181198521 - 4872882 T + 117583 T^{2} - 362 T^{3} + T^{4} \)
$37$ \( 9545290000 + 488500 T + 97725 T^{2} - 5 T^{3} + T^{4} \)
$41$ \( ( -9072 - 162 T + T^{2} )^{2} \)
$43$ \( ( 129526 - 721 T + T^{2} )^{2} \)
$47$ \( 2587553424 - 10987488 T + 97524 T^{2} + 216 T^{3} + T^{4} \)
$53$ \( 3288793104 + 28387260 T + 187677 T^{2} + 495 T^{3} + T^{4} \)
$59$ \( 16397314704 + 22152996 T + 157981 T^{2} - 173 T^{3} + T^{4} \)
$61$ \( 41525136 - 3428208 T + 289468 T^{2} + 532 T^{3} + T^{4} \)
$67$ \( 80085604036 - 31412334 T + 295315 T^{2} + 111 T^{3} + T^{4} \)
$71$ \( ( 546588 - 1600 T + T^{2} )^{2} \)
$73$ \( 54532524484 - 283729230 T + 1242703 T^{2} - 1215 T^{3} + T^{4} \)
$79$ \( 120705520329 + 507243420 T + 1784173 T^{2} + 1460 T^{3} + T^{4} \)
$83$ \( ( 4122 - 1409 T + T^{2} )^{2} \)
$89$ \( 790420571136 + 1754996544 T + 3007620 T^{2} + 1974 T^{3} + T^{4} \)
$97$ \( ( 38102 + 561 T + T^{2} )^{2} \)
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