Properties

Label 4600.2.a.bc
Level $4600$
Weight $2$
Character orbit 4600.a
Self dual yes
Analytic conductor $36.731$
Analytic rank $1$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.791953.1
Defining polynomial: \(x^{5} - 2 x^{4} - 7 x^{3} + 7 x^{2} + 9 x + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{3} + ( 1 - \beta_{2} + \beta_{4} ) q^{7} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{3} + ( 1 - \beta_{2} + \beta_{4} ) q^{7} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{11} + ( -1 + 2 \beta_{1} - \beta_{4} ) q^{13} + ( -\beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{17} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{19} + ( -2 - 2 \beta_{4} ) q^{21} - q^{23} + ( -1 - \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{27} + ( -2 - 2 \beta_{1} + \beta_{3} - \beta_{4} ) q^{29} + ( 1 + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{31} + ( -2 + \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{33} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{37} + ( 7 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{39} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{4} ) q^{41} + ( -3 + 3 \beta_{2} - 3 \beta_{4} ) q^{43} + ( -2 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} ) q^{47} + ( -\beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{49} + ( -2 - \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{51} + ( -4 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{53} + ( -4 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{57} + ( 6 - 3 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{59} + ( -6 - 2 \beta_{4} ) q^{61} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{63} + ( -2 + \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{67} + ( 1 - \beta_{1} ) q^{69} + ( \beta_{2} + 2 \beta_{4} ) q^{71} + ( \beta_{1} + 4 \beta_{3} + 5 \beta_{4} ) q^{73} + ( -5 + 3 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} ) q^{77} + ( 9 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{79} + ( -2 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} ) q^{81} + ( -3 - 3 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{83} + ( -3 - 3 \beta_{1} - 2 \beta_{2} + \beta_{4} ) q^{87} + ( \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} ) q^{89} + ( -5 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 3 \beta_{4} ) q^{91} + ( 1 + \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{93} + ( -2 - 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{97} + ( 9 - 4 \beta_{1} + \beta_{2} - 4 \beta_{3} - \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{3} + q^{7} + 4 q^{9} + O(q^{10}) \) \( 5 q - 3 q^{3} + q^{7} + 4 q^{9} - 4 q^{11} + q^{13} - 5 q^{17} + 4 q^{19} - 6 q^{21} - 5 q^{23} - 6 q^{27} - 11 q^{29} + 4 q^{31} - 13 q^{33} - 6 q^{37} + 31 q^{39} - 8 q^{41} - 3 q^{43} - 2 q^{47} - 2 q^{49} - 5 q^{51} - 18 q^{53} - 27 q^{57} + 23 q^{59} - 26 q^{61} - 5 q^{63} - 3 q^{67} + 3 q^{69} - 2 q^{71} - 4 q^{73} - 15 q^{77} + 43 q^{79} - 3 q^{81} - 30 q^{83} - 27 q^{87} + 15 q^{89} - 19 q^{91} + 15 q^{93} - 8 q^{97} + 37 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 7 x^{3} + 7 x^{2} + 9 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 6 \nu^{2} + 7 \nu + 4 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 7 \nu^{2} + 8 \nu + 7 \)
\(\beta_{4}\)\(=\)\( -2 \nu^{4} + 5 \nu^{3} + 12 \nu^{2} - 20 \nu - 11 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{4} + 2 \beta_{2} + 6 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(2 \beta_{4} - 6 \beta_{3} + 11 \beta_{2} + 11 \beta_{1} + 20\)

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
−2.08344
−0.514659
−0.336890
1.72457
3.21042
0 −3.08344 0 0 0 0.555022 0 6.50759 0
1.2 0 −1.51466 0 0 0 3.49880 0 −0.705809 0
1.3 0 −1.33689 0 0 0 −3.16736 0 −1.21273 0
1.4 0 0.724570 0 0 0 2.33840 0 −2.47500 0
1.5 0 2.21042 0 0 0 −2.22487 0 1.88594 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.bc 5
4.b odd 2 1 9200.2.a.cw 5
5.b even 2 1 4600.2.a.bg yes 5
5.c odd 4 2 4600.2.e.v 10
20.d odd 2 1 9200.2.a.cs 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4600.2.a.bc 5 1.a even 1 1 trivial
4600.2.a.bg yes 5 5.b even 2 1
4600.2.e.v 10 5.c odd 4 2
9200.2.a.cs 5 20.d odd 2 1
9200.2.a.cw 5 4.b odd 2 1

Hecke kernels

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

\( T_{3}^{5} + 3 T_{3}^{4} - 5 T_{3}^{3} - 16 T_{3}^{2} - T_{3} + 10 \)
\( T_{7}^{5} - T_{7}^{4} - 16 T_{7}^{3} + 12 T_{7}^{2} + 56 T_{7} - 32 \)
\( T_{11}^{5} + 4 T_{11}^{4} - 17 T_{11}^{3} - 88 T_{11}^{2} - 92 T_{11} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( 10 - T - 16 T^{2} - 5 T^{3} + 3 T^{4} + T^{5} \)
$5$ \( T^{5} \)
$7$ \( -32 + 56 T + 12 T^{2} - 16 T^{3} - T^{4} + T^{5} \)
$11$ \( -8 - 92 T - 88 T^{2} - 17 T^{3} + 4 T^{4} + T^{5} \)
$13$ \( -454 + 321 T + 48 T^{2} - 37 T^{3} - T^{4} + T^{5} \)
$17$ \( 32 - 72 T - 116 T^{2} - 28 T^{3} + 5 T^{4} + T^{5} \)
$19$ \( -64 - 272 T + 264 T^{2} - 53 T^{3} - 4 T^{4} + T^{5} \)
$23$ \( ( 1 + T )^{5} \)
$29$ \( -256 - 551 T - 306 T^{2} - 11 T^{3} + 11 T^{4} + T^{5} \)
$31$ \( 400 + 802 T + 159 T^{2} - 75 T^{3} - 4 T^{4} + T^{5} \)
$37$ \( 5344 + 4672 T - 576 T^{2} - 140 T^{3} + 6 T^{4} + T^{5} \)
$41$ \( -2069 - 2097 T - 671 T^{2} - 52 T^{3} + 8 T^{4} + T^{5} \)
$43$ \( 7776 + 4536 T - 324 T^{2} - 144 T^{3} + 3 T^{4} + T^{5} \)
$47$ \( 5912 + 4558 T - 33 T^{2} - 167 T^{3} + 2 T^{4} + T^{5} \)
$53$ \( -22144 - 15360 T - 2632 T^{2} - 52 T^{3} + 18 T^{4} + T^{5} \)
$59$ \( -5128 - 558 T + 1087 T^{2} + 53 T^{3} - 23 T^{4} + T^{5} \)
$61$ \( -256 + 880 T + 848 T^{2} + 236 T^{3} + 26 T^{4} + T^{5} \)
$67$ \( 512 + 320 T - 192 T^{2} - 60 T^{3} + 3 T^{4} + T^{5} \)
$71$ \( -8 + 666 T - 133 T^{2} - 67 T^{3} + 2 T^{4} + T^{5} \)
$73$ \( 52501 + 33791 T - 1001 T^{2} - 368 T^{3} + 4 T^{4} + T^{5} \)
$79$ \( 45872 - 2080 T - 3292 T^{2} + 634 T^{3} - 43 T^{4} + T^{5} \)
$83$ \( 32960 - 8816 T - 1560 T^{2} + 169 T^{3} + 30 T^{4} + T^{5} \)
$89$ \( 7120 - 12608 T + 3188 T^{2} - 154 T^{3} - 15 T^{4} + T^{5} \)
$97$ \( 1024 + 4848 T - 1512 T^{2} - 228 T^{3} + 8 T^{4} + T^{5} \)
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