Properties

Label 588.4.i.i
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: \(\Q(\sqrt{-3}, \sqrt{-19})\)
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.

\(f(q)\) \(=\) \( q + 3 \beta_{1} q^{3} + ( -2 - 2 \beta_{1} + \beta_{3} ) q^{5} + ( -9 - 9 \beta_{1} ) q^{9} +O(q^{10})\) \( q + 3 \beta_{1} q^{3} + ( -2 - 2 \beta_{1} + \beta_{3} ) q^{5} + ( -9 - 9 \beta_{1} ) q^{9} + ( -1 - 26 \beta_{1} - \beta_{2} + \beta_{3} ) q^{11} + ( -28 + 5 \beta_{2} ) q^{13} + ( 3 - 3 \beta_{2} ) q^{15} + ( -8 - 16 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{17} + ( 85 + 85 \beta_{1} - \beta_{3} ) q^{19} + ( 96 + 96 \beta_{1} ) q^{23} + ( 3 + 7 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{25} + 27 q^{27} + ( -11 + 17 \beta_{2} ) q^{29} + ( 10 + 51 \beta_{1} + 10 \beta_{2} - 10 \beta_{3} ) q^{31} + ( 78 + 78 \beta_{1} - 3 \beta_{3} ) q^{33} + ( 77 + 77 \beta_{1} + 19 \beta_{3} ) q^{37} + ( -15 - 99 \beta_{1} - 15 \beta_{2} + 15 \beta_{3} ) q^{39} + ( -104 - 34 \beta_{2} ) q^{41} + ( -258 - 19 \beta_{2} ) q^{43} + ( 9 + 18 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} ) q^{45} + ( 98 + 98 \beta_{1} - 16 \beta_{3} ) q^{47} + ( 48 + 48 \beta_{1} - 24 \beta_{3} ) q^{51} + ( 27 + 156 \beta_{1} + 27 \beta_{2} - 27 \beta_{3} ) q^{53} + ( -153 + 27 \beta_{2} ) q^{55} + ( -252 + 3 \beta_{2} ) q^{57} + ( 3 + 636 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{59} + ( 146 + 146 \beta_{1} + 36 \beta_{3} ) q^{61} + ( 706 + 706 \beta_{1} - 38 \beta_{3} ) q^{65} + ( 9 - 433 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} ) q^{67} -288 q^{69} + ( -718 - 32 \beta_{2} ) q^{71} + ( 21 + 691 \beta_{1} + 21 \beta_{2} - 21 \beta_{3} ) q^{73} + ( -21 - 21 \beta_{1} + 9 \beta_{3} ) q^{75} + ( 93 + 93 \beta_{1} - 4 \beta_{3} ) q^{79} + 81 \beta_{1} q^{81} + ( 221 + 43 \beta_{2} ) q^{83} + ( -1032 + 24 \beta_{2} ) q^{85} + ( -51 - 84 \beta_{1} - 51 \beta_{2} + 51 \beta_{3} ) q^{87} + ( 400 + 400 \beta_{1} + 22 \beta_{3} ) q^{89} + ( -153 - 153 \beta_{1} + 30 \beta_{3} ) q^{93} + ( -86 - 298 \beta_{1} - 86 \beta_{2} + 86 \beta_{3} ) q^{95} + ( -431 - 21 \beta_{2} ) q^{97} + ( -225 + 9 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 6q^{3} - 3q^{5} - 18q^{9} + O(q^{10}) \) \( 4q - 6q^{3} - 3q^{5} - 18q^{9} + 51q^{11} - 122q^{13} + 18q^{15} + 24q^{17} + 169q^{19} + 192q^{23} - 11q^{25} + 108q^{27} - 78q^{29} - 92q^{31} + 153q^{33} + 173q^{37} + 183q^{39} - 348q^{41} - 994q^{43} - 27q^{45} + 180q^{47} + 72q^{51} - 285q^{53} - 666q^{55} - 1014q^{57} - 1269q^{59} + 328q^{61} + 1374q^{65} + 875q^{67} - 1152q^{69} - 2808q^{71} - 1361q^{73} - 33q^{75} + 182q^{79} - 162q^{81} + 798q^{83} - 4176q^{85} + 117q^{87} + 822q^{89} - 276q^{93} + 510q^{95} - 1682q^{97} - 918q^{99} + O(q^{100}) \)

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

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

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 −1.50000 + 2.59808i 0 −6.41238 11.1066i 0 0 0 −4.50000 7.79423i 0
361.2 0 −1.50000 + 2.59808i 0 4.91238 + 8.50848i 0 0 0 −4.50000 7.79423i 0
373.1 0 −1.50000 2.59808i 0 −6.41238 + 11.1066i 0 0 0 −4.50000 + 7.79423i 0
373.2 0 −1.50000 2.59808i 0 4.91238 8.50848i 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.i 4
3.b odd 2 1 1764.4.k.z 4
7.b odd 2 1 84.4.i.b 4
7.c even 3 1 588.4.a.h 2
7.c even 3 1 inner 588.4.i.i 4
7.d odd 6 1 84.4.i.b 4
7.d odd 6 1 588.4.a.g 2
21.c even 2 1 252.4.k.d 4
21.g even 6 1 252.4.k.d 4
21.g even 6 1 1764.4.a.x 2
21.h odd 6 1 1764.4.a.p 2
21.h odd 6 1 1764.4.k.z 4
28.d even 2 1 336.4.q.h 4
28.f even 6 1 336.4.q.h 4
28.f even 6 1 2352.4.a.cb 2
28.g odd 6 1 2352.4.a.bp 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.i.b 4 7.b odd 2 1
84.4.i.b 4 7.d odd 6 1
252.4.k.d 4 21.c even 2 1
252.4.k.d 4 21.g even 6 1
336.4.q.h 4 28.d even 2 1
336.4.q.h 4 28.f even 6 1
588.4.a.g 2 7.d odd 6 1
588.4.a.h 2 7.c even 3 1
588.4.i.i 4 1.a even 1 1 trivial
588.4.i.i 4 7.c even 3 1 inner
1764.4.a.p 2 21.h odd 6 1
1764.4.a.x 2 21.g even 6 1
1764.4.k.z 4 3.b odd 2 1
1764.4.k.z 4 21.h odd 6 1
2352.4.a.bp 2 28.g odd 6 1
2352.4.a.cb 2 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 3 T_{5}^{3} + 135 T_{5}^{2} - 378 T_{5} + 15876 \) acting on \(S_{4}^{\mathrm{new}}(588, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 9 + 3 T + T^{2} )^{2} \)
$5$ \( 15876 - 378 T + 135 T^{2} + 3 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 272484 - 26622 T + 2079 T^{2} - 51 T^{3} + T^{4} \)
$13$ \( ( -2276 + 61 T + T^{2} )^{2} \)
$17$ \( 65028096 + 193536 T + 8640 T^{2} - 24 T^{3} + T^{4} \)
$19$ \( 49168144 - 1185028 T + 21549 T^{2} - 169 T^{3} + T^{4} \)
$23$ \( ( 9216 - 96 T + T^{2} )^{2} \)
$29$ \( ( -36684 + 39 T + T^{2} )^{2} \)
$31$ \( 114682681 - 985228 T + 19173 T^{2} + 92 T^{3} + T^{4} \)
$37$ \( 1506681856 + 6715168 T + 68745 T^{2} - 173 T^{3} + T^{4} \)
$41$ \( ( -140688 + 174 T + T^{2} )^{2} \)
$43$ \( ( 15454 + 497 T + T^{2} )^{2} \)
$47$ \( 611671824 + 4451760 T + 57132 T^{2} - 180 T^{3} + T^{4} \)
$53$ \( 5356483344 - 20858580 T + 154413 T^{2} + 285 T^{3} + T^{4} \)
$59$ \( 161150862096 + 509422284 T + 1208925 T^{2} + 1269 T^{3} + T^{4} \)
$61$ \( 19408947856 + 45695648 T + 246900 T^{2} - 328 T^{3} + T^{4} \)
$67$ \( 32767516324 - 158390750 T + 584607 T^{2} - 875 T^{3} + T^{4} \)
$71$ \( ( 361476 + 1404 T + T^{2} )^{2} \)
$73$ \( 165260136484 + 553276442 T + 1445799 T^{2} + 1361 T^{3} + T^{4} \)
$79$ \( 38800441 - 1133678 T + 26895 T^{2} - 182 T^{3} + T^{4} \)
$83$ \( ( -197334 - 399 T + T^{2} )^{2} \)
$89$ \( 11416495104 - 87829056 T + 568836 T^{2} - 822 T^{3} + T^{4} \)
$97$ \( ( 120262 + 841 T + T^{2} )^{2} \)
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