Properties

Label 4704.2.a.bx
Level $4704$
Weight $2$
Character orbit 4704.a
Self dual yes
Analytic conductor $37.562$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4704.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.5616291108\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2624.1
Defining polynomial: \(x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q - q^{3} + ( 1 + \beta_{3} ) q^{5} + q^{9} +O(q^{10})\) \( q - q^{3} + ( 1 + \beta_{3} ) q^{5} + q^{9} + ( -1 - \beta_{2} ) q^{11} + ( 2 - \beta_{2} - \beta_{3} ) q^{13} + ( -1 - \beta_{3} ) q^{15} + ( 1 + \beta_{1} + \beta_{2} ) q^{17} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{19} + ( -3 + \beta_{2} ) q^{23} + ( 3 + 2 \beta_{1} + 2 \beta_{3} ) q^{25} - q^{27} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{29} + ( 2 + 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{31} + ( 1 + \beta_{2} ) q^{33} -4 \beta_{1} q^{37} + ( -2 + \beta_{2} + \beta_{3} ) q^{39} + ( 5 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{41} + ( 2 - 2 \beta_{1} + 2 \beta_{3} ) q^{43} + ( 1 + \beta_{3} ) q^{45} + ( 4 - \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{47} + ( -1 - \beta_{1} - \beta_{2} ) q^{51} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{53} + ( 2 - 5 \beta_{1} - \beta_{2} - \beta_{3} ) q^{55} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{57} + ( 3 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{59} + ( 4 + \beta_{2} - \beta_{3} ) q^{61} + ( -2 - 7 \beta_{1} - \beta_{2} + \beta_{3} ) q^{65} + ( 2 + 4 \beta_{1} + 2 \beta_{2} ) q^{67} + ( 3 - \beta_{2} ) q^{69} + ( -3 + 4 \beta_{1} + \beta_{2} ) q^{71} + ( 2 + \beta_{1} + 2 \beta_{3} ) q^{73} + ( -3 - 2 \beta_{1} - 2 \beta_{3} ) q^{75} + ( -4 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{79} + q^{81} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{83} + ( -2 + 6 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{85} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{87} + ( 7 + 5 \beta_{1} - \beta_{2} ) q^{89} + ( -2 - 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{93} + ( -6 - 8 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{95} + ( 10 - 3 \beta_{1} + 2 \beta_{3} ) q^{97} + ( -1 - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 4q^{5} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 4q^{5} + 4q^{9} - 4q^{11} + 8q^{13} - 4q^{15} + 4q^{17} - 8q^{19} - 12q^{23} + 12q^{25} - 4q^{27} + 8q^{31} + 4q^{33} - 8q^{39} + 20q^{41} + 8q^{43} + 4q^{45} + 16q^{47} - 4q^{51} + 8q^{55} + 8q^{57} + 16q^{61} - 8q^{65} + 8q^{67} + 12q^{69} - 12q^{71} + 8q^{73} - 12q^{75} - 16q^{79} + 4q^{81} - 8q^{85} + 28q^{89} - 8q^{93} - 24q^{95} + 40q^{97} - 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 2 \nu + 1 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 4 \nu - 3 \)
\(\beta_{3}\)\(=\)\( -\nu^{3} + 2 \nu^{2} + 4 \nu - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{3} + \beta_{2} + 2 \beta_{1} + 5\)\()/2\)
\(\nu^{3}\)\(=\)\(3 \beta_{3} + \beta_{2} + 4 \beta_{1} + 5\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.360409
−1.22833
0.814115
2.77462
0 −1.00000 0 −2.13503 0 0 0 1.00000 0
1.2 0 −1.00000 0 −1.04244 0 0 0 1.00000 0
1.3 0 −1.00000 0 3.04244 0 0 0 1.00000 0
1.4 0 −1.00000 0 4.13503 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4704.2.a.bx yes 4
4.b odd 2 1 4704.2.a.bz yes 4
7.b odd 2 1 4704.2.a.by yes 4
8.b even 2 1 9408.2.a.em 4
8.d odd 2 1 9408.2.a.ek 4
28.d even 2 1 4704.2.a.bw 4
56.e even 2 1 9408.2.a.en 4
56.h odd 2 1 9408.2.a.el 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4704.2.a.bw 4 28.d even 2 1
4704.2.a.bx yes 4 1.a even 1 1 trivial
4704.2.a.by yes 4 7.b odd 2 1
4704.2.a.bz yes 4 4.b odd 2 1
9408.2.a.ek 4 8.d odd 2 1
9408.2.a.el 4 56.h odd 2 1
9408.2.a.em 4 8.b even 2 1
9408.2.a.en 4 56.e even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4704))\):

\( T_{5}^{4} - 4 T_{5}^{3} - 8 T_{5}^{2} + 24 T_{5} + 28 \)
\( T_{11}^{4} + 4 T_{11}^{3} - 20 T_{11}^{2} - 48 T_{11} + 16 \)
\( T_{13}^{4} - 8 T_{13}^{3} - 4 T_{13}^{2} + 80 T_{13} + 68 \)
\( T_{19}^{4} + 8 T_{19}^{3} - 8 T_{19}^{2} - 64 T_{19} + 64 \)
\( T_{31}^{4} - 8 T_{31}^{3} - 40 T_{31}^{2} + 128 T_{31} - 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + T )^{4} \)
$5$ \( 1 - 4 T + 12 T^{2} - 36 T^{3} + 98 T^{4} - 180 T^{5} + 300 T^{6} - 500 T^{7} + 625 T^{8} \)
$7$ 1
$11$ \( 1 + 4 T + 24 T^{2} + 84 T^{3} + 302 T^{4} + 924 T^{5} + 2904 T^{6} + 5324 T^{7} + 14641 T^{8} \)
$13$ \( 1 - 8 T + 48 T^{2} - 232 T^{3} + 978 T^{4} - 3016 T^{5} + 8112 T^{6} - 17576 T^{7} + 28561 T^{8} \)
$17$ \( 1 - 4 T + 44 T^{2} - 84 T^{3} + 818 T^{4} - 1428 T^{5} + 12716 T^{6} - 19652 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 8 T + 68 T^{2} + 392 T^{3} + 1926 T^{4} + 7448 T^{5} + 24548 T^{6} + 54872 T^{7} + 130321 T^{8} \)
$23$ \( 1 + 12 T + 120 T^{2} + 780 T^{3} + 4350 T^{4} + 17940 T^{5} + 63480 T^{6} + 146004 T^{7} + 279841 T^{8} \)
$29$ \( 1 + 60 T^{2} + 128 T^{3} + 1862 T^{4} + 3712 T^{5} + 50460 T^{6} + 707281 T^{8} \)
$31$ \( 1 - 8 T + 84 T^{2} - 616 T^{3} + 3222 T^{4} - 19096 T^{5} + 80724 T^{6} - 238328 T^{7} + 923521 T^{8} \)
$37$ \( ( 1 + 42 T^{2} + 1369 T^{4} )^{2} \)
$41$ \( 1 - 20 T + 220 T^{2} - 1540 T^{3} + 9874 T^{4} - 63140 T^{5} + 369820 T^{6} - 1378420 T^{7} + 2825761 T^{8} \)
$43$ \( 1 - 8 T + 124 T^{2} - 648 T^{3} + 6710 T^{4} - 27864 T^{5} + 229276 T^{6} - 636056 T^{7} + 3418801 T^{8} \)
$47$ \( 1 - 16 T + 164 T^{2} - 1232 T^{3} + 7990 T^{4} - 57904 T^{5} + 362276 T^{6} - 1661168 T^{7} + 4879681 T^{8} \)
$53$ \( 1 + 140 T^{2} - 128 T^{3} + 9494 T^{4} - 6784 T^{5} + 393260 T^{6} + 7890481 T^{8} \)
$59$ \( 1 + 84 T^{2} - 960 T^{3} + 1350 T^{4} - 56640 T^{5} + 292404 T^{6} + 12117361 T^{8} \)
$61$ \( 1 - 16 T + 288 T^{2} - 2768 T^{3} + 27282 T^{4} - 168848 T^{5} + 1071648 T^{6} - 3631696 T^{7} + 13845841 T^{8} \)
$67$ \( 1 - 8 T + 124 T^{2} + 56 T^{3} + 3286 T^{4} + 3752 T^{5} + 556636 T^{6} - 2406104 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + 12 T + 248 T^{2} + 2380 T^{3} + 25406 T^{4} + 168980 T^{5} + 1250168 T^{6} + 4294932 T^{7} + 25411681 T^{8} \)
$73$ \( 1 - 8 T + 256 T^{2} - 1608 T^{3} + 27170 T^{4} - 117384 T^{5} + 1364224 T^{6} - 3112136 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 16 T + 284 T^{2} + 2768 T^{3} + 31366 T^{4} + 218672 T^{5} + 1772444 T^{6} + 7888624 T^{7} + 38950081 T^{8} \)
$83$ \( 1 + 204 T^{2} - 256 T^{3} + 21878 T^{4} - 21248 T^{5} + 1405356 T^{6} + 47458321 T^{8} \)
$89$ \( 1 - 28 T + 524 T^{2} - 6764 T^{3} + 72658 T^{4} - 601996 T^{5} + 4150604 T^{6} - 19739132 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 40 T + 896 T^{2} - 13608 T^{3} + 153858 T^{4} - 1319976 T^{5} + 8430464 T^{6} - 36506920 T^{7} + 88529281 T^{8} \)
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