Properties

Label 4600.2.a.bi
Level $4600$
Weight $2$
Character orbit 4600.a
Self dual yes
Analytic conductor $36.731$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7311849298\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 3 x^{6} - 7 x^{5} + 24 x^{4} + x^{3} - 35 x^{2} + 17 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 1 + \beta_{3} ) q^{7} + ( \beta_{2} + \beta_{5} + \beta_{6} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 1 + \beta_{3} ) q^{7} + ( \beta_{2} + \beta_{5} + \beta_{6} ) q^{9} + ( -1 - \beta_{5} - \beta_{6} ) q^{11} + ( -\beta_{1} - \beta_{2} - \beta_{5} ) q^{13} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{17} + ( -1 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{19} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{21} - q^{23} + ( -1 + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{27} + ( -1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{29} + ( -\beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{31} + ( -1 - 4 \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{6} ) q^{33} + ( -3 - \beta_{2} - 2 \beta_{6} ) q^{37} + ( -1 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{6} ) q^{39} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{6} ) q^{41} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{43} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{47} + ( -2 - \beta_{1} + \beta_{4} ) q^{49} + ( -3 - \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{51} + ( -2 + \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{53} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{57} + ( -3 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{59} + ( -\beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{61} + ( 3 - 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{63} + ( 1 - 4 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} ) q^{67} -\beta_{1} q^{69} + ( -4 + \beta_{1} + 4 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{71} + ( -1 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{73} + ( -3 + 3 \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{6} ) q^{77} + ( -5 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} ) q^{79} + ( -3 + \beta_{2} - 4 \beta_{3} + \beta_{5} ) q^{81} + ( 2 - 4 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{83} + ( 4 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{87} + ( -3 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{89} + ( -1 - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{91} + ( -\beta_{1} - 5 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} ) q^{93} + ( 4 - \beta_{1} - 3 \beta_{2} + 4 \beta_{3} + \beta_{4} ) q^{97} + ( -6 - 5 \beta_{2} + 3 \beta_{3} - 4 \beta_{5} - 4 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 3q^{3} + 4q^{7} + 2q^{9} + O(q^{10}) \) \( 7q + 3q^{3} + 4q^{7} + 2q^{9} - 7q^{11} - 7q^{13} - 7q^{19} - 6q^{21} - 7q^{23} - 11q^{29} - 10q^{31} - 19q^{33} - 19q^{37} - 24q^{39} - 16q^{41} + 6q^{43} + 6q^{47} - 17q^{49} - 7q^{51} - 15q^{53} - 8q^{57} - 11q^{59} + 5q^{61} + 13q^{63} + 9q^{67} - 3q^{69} - 14q^{71} - 10q^{73} - 6q^{77} - 32q^{79} - 5q^{81} + q^{83} + 10q^{87} - 24q^{89} - 7q^{91} - 26q^{93} + 7q^{97} - 61q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 3 x^{6} - 7 x^{5} + 24 x^{4} + x^{3} - 35 x^{2} + 17 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - 11 \nu^{4} - 3 \nu^{3} + 26 \nu^{2} + 7 \nu - 6 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} - 2 \nu^{5} - 9 \nu^{4} + 15 \nu^{3} + 16 \nu^{2} - 21 \nu - 2 \)\()/2\)
\(\beta_{4}\)\(=\)\( \nu^{5} - \nu^{4} - 10 \nu^{3} + 6 \nu^{2} + 20 \nu - 6 \)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{6} - 8 \nu^{5} - 23 \nu^{4} + 63 \nu^{3} + 20 \nu^{2} - 91 \nu + 22 \)\()/2\)
\(\beta_{6}\)\(=\)\( -2 \nu^{6} + 4 \nu^{5} + 17 \nu^{4} - 30 \nu^{3} - 22 \nu^{2} + 42 \nu - 11 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{5} + \beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 6 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(9 \beta_{6} + 10 \beta_{5} - 4 \beta_{3} + 10 \beta_{2} + 15\)
\(\nu^{5}\)\(=\)\(13 \beta_{6} + 14 \beta_{5} - 9 \beta_{4} - 14 \beta_{3} + 24 \beta_{2} + 40 \beta_{1} - 7\)
\(\nu^{6}\)\(=\)\(76 \beta_{6} + 87 \beta_{5} - 3 \beta_{4} - 47 \beta_{3} + 92 \beta_{2} + 11 \beta_{1} + 90\)

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
−2.49931
−1.40334
0.189375
0.356372
1.58319
1.78665
2.98707
0 −2.49931 0 0 0 2.92777 0 3.24657 0
1.2 0 −1.40334 0 0 0 1.57117 0 −1.03063 0
1.3 0 0.189375 0 0 0 −1.65661 0 −2.96414 0
1.4 0 0.356372 0 0 0 −2.46376 0 −2.87300 0
1.5 0 1.58319 0 0 0 2.84620 0 −0.493499 0
1.6 0 1.78665 0 0 0 1.75578 0 0.192116 0
1.7 0 2.98707 0 0 0 −0.980560 0 5.92257 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(23\) \(1\)

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 4600.2.a.bi 7
4.b odd 2 1 9200.2.a.cz 7
5.b even 2 1 4600.2.a.bh 7
5.c odd 4 2 920.2.e.b 14
20.d odd 2 1 9200.2.a.dc 7
20.e even 4 2 1840.2.e.g 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.e.b 14 5.c odd 4 2
1840.2.e.g 14 20.e even 4 2
4600.2.a.bh 7 5.b even 2 1
4600.2.a.bi 7 1.a even 1 1 trivial
9200.2.a.cz 7 4.b odd 2 1
9200.2.a.dc 7 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4600))\):

\( T_{3}^{7} - 3 T_{3}^{6} - 7 T_{3}^{5} + 24 T_{3}^{4} + T_{3}^{3} - 35 T_{3}^{2} + 17 T_{3} - 2 \)
\( T_{7}^{7} - 4 T_{7}^{6} - 8 T_{7}^{5} + 41 T_{7}^{4} + 10 T_{7}^{3} - 116 T_{7}^{2} + 12 T_{7} + 92 \)
\( T_{11}^{7} + 7 T_{11}^{6} - 27 T_{11}^{5} - 198 T_{11}^{4} + 248 T_{11}^{3} + 1224 T_{11}^{2} - 1780 T_{11} + 128 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \)
$3$ \( -2 + 17 T - 35 T^{2} + T^{3} + 24 T^{4} - 7 T^{5} - 3 T^{6} + T^{7} \)
$5$ \( T^{7} \)
$7$ \( 92 + 12 T - 116 T^{2} + 10 T^{3} + 41 T^{4} - 8 T^{5} - 4 T^{6} + T^{7} \)
$11$ \( 128 - 1780 T + 1224 T^{2} + 248 T^{3} - 198 T^{4} - 27 T^{5} + 7 T^{6} + T^{7} \)
$13$ \( -2 - 161 T + 135 T^{2} + 129 T^{3} - 92 T^{4} - 13 T^{5} + 7 T^{6} + T^{7} \)
$17$ \( 52 - 184 T - 1050 T^{2} + 502 T^{3} + 105 T^{4} - 58 T^{5} + T^{7} \)
$19$ \( 1936 - 3828 T + 1104 T^{2} + 1030 T^{3} - 238 T^{4} - 53 T^{5} + 7 T^{6} + T^{7} \)
$23$ \( ( 1 + T )^{7} \)
$29$ \( 9244 + 38290 T + 15243 T^{2} - 971 T^{3} - 930 T^{4} - 56 T^{5} + 11 T^{6} + T^{7} \)
$31$ \( -225251 + 21800 T + 38532 T^{2} + 2165 T^{3} - 1243 T^{4} - 110 T^{5} + 10 T^{6} + T^{7} \)
$37$ \( 42832 + 13432 T - 10948 T^{2} - 5832 T^{3} - 700 T^{4} + 68 T^{5} + 19 T^{6} + T^{7} \)
$41$ \( 110153 + 76850 T + 8034 T^{2} - 4287 T^{3} - 955 T^{4} + 12 T^{5} + 16 T^{6} + T^{7} \)
$43$ \( 64 - 336 T - 168 T^{2} + 188 T^{3} + 64 T^{4} - 30 T^{5} - 6 T^{6} + T^{7} \)
$47$ \( 5752 - 21784 T - 39856 T^{2} + 6697 T^{3} + 1130 T^{4} - 178 T^{5} - 6 T^{6} + T^{7} \)
$53$ \( 156352 + 154304 T + 38096 T^{2} - 2704 T^{3} - 1596 T^{4} - 60 T^{5} + 15 T^{6} + T^{7} \)
$59$ \( 486592 - 430528 T + 70592 T^{2} + 16480 T^{3} - 2288 T^{4} - 252 T^{5} + 11 T^{6} + T^{7} \)
$61$ \( 2669336 - 157108 T - 209468 T^{2} + 25754 T^{3} + 2048 T^{4} - 315 T^{5} - 5 T^{6} + T^{7} \)
$67$ \( 462832 - 114808 T - 72148 T^{2} + 11872 T^{3} + 1732 T^{4} - 244 T^{5} - 9 T^{6} + T^{7} \)
$71$ \( 632317 + 554654 T + 127100 T^{2} - 211 T^{3} - 2711 T^{4} - 156 T^{5} + 14 T^{6} + T^{7} \)
$73$ \( 16 - 1460 T + 2516 T^{2} - 391 T^{3} - 466 T^{4} - 30 T^{5} + 10 T^{6} + T^{7} \)
$79$ \( 473312 + 422784 T + 29312 T^{2} - 22724 T^{3} - 2876 T^{4} + 166 T^{5} + 32 T^{6} + T^{7} \)
$83$ \( -11984 - 68664 T - 20548 T^{2} + 5720 T^{3} + 1070 T^{4} - 222 T^{5} - T^{6} + T^{7} \)
$89$ \( 311456 + 288384 T + 48656 T^{2} - 10836 T^{3} - 2260 T^{4} + 46 T^{5} + 24 T^{6} + T^{7} \)
$97$ \( -4038856 - 1487436 T + 53776 T^{2} + 45720 T^{3} + 964 T^{4} - 369 T^{5} - 7 T^{6} + T^{7} \)
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