Properties

Label 4600.2.e.r
Level $4600$
Weight $2$
Character orbit 4600.e
Analytic conductor $36.731$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.24681024.1
Defining polynomial: \(x^{6} + 12 x^{4} + 36 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -\beta_{1} + \beta_{2} ) q^{7} + ( -1 - \beta_{3} + \beta_{4} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -\beta_{1} + \beta_{2} ) q^{7} + ( -1 - \beta_{3} + \beta_{4} ) q^{9} + ( 1 - \beta_{3} - \beta_{4} ) q^{11} + ( \beta_{1} - \beta_{5} ) q^{13} + ( -\beta_{1} - \beta_{2} + \beta_{5} ) q^{17} + ( -1 + \beta_{3} - 2 \beta_{4} ) q^{19} + ( 4 + 2 \beta_{3} - \beta_{4} ) q^{21} -\beta_{2} q^{23} + 3 \beta_{2} q^{27} + ( -1 + \beta_{4} ) q^{29} + ( 2 - 3 \beta_{3} + \beta_{4} ) q^{31} + ( \beta_{1} + 5 \beta_{2} - 2 \beta_{5} ) q^{33} + 4 \beta_{2} q^{37} + ( -5 + 2 \beta_{4} ) q^{39} + ( -2 - 3 \beta_{3} + 2 \beta_{4} ) q^{41} + ( -6 \beta_{2} + 2 \beta_{5} ) q^{43} + ( 5 \beta_{2} - \beta_{5} ) q^{47} + ( 2 - 3 \beta_{3} + \beta_{4} ) q^{49} + ( 5 - \beta_{3} - 2 \beta_{4} ) q^{51} + ( -2 \beta_{2} + 4 \beta_{5} ) q^{53} + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{5} ) q^{57} + ( -2 + 2 \beta_{3} - 2 \beta_{4} ) q^{59} + ( 9 - \beta_{3} - \beta_{4} ) q^{61} + ( 4 \beta_{1} - 4 \beta_{2} + \beta_{5} ) q^{63} + ( -4 \beta_{1} + 4 \beta_{2} + 4 \beta_{5} ) q^{67} -\beta_{3} q^{69} + ( 2 - 3 \beta_{3} + 2 \beta_{4} ) q^{71} + ( -4 \beta_{1} + 5 \beta_{2} + \beta_{5} ) q^{73} + ( -4 \beta_{2} + \beta_{5} ) q^{77} + ( -3 + 3 \beta_{4} ) q^{81} + ( -4 \beta_{1} + 4 \beta_{2} - 2 \beta_{5} ) q^{83} + ( -2 \beta_{1} - \beta_{2} + \beta_{5} ) q^{87} + ( 10 - 2 \beta_{4} ) q^{89} + ( 5 + \beta_{3} - \beta_{4} ) q^{91} + ( -2 \beta_{1} + 11 \beta_{2} - 2 \beta_{5} ) q^{93} + ( \beta_{1} + 3 \beta_{2} - 3 \beta_{5} ) q^{97} + ( -3 + 3 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9} + O(q^{10}) \) \( 6 q - 6 q^{9} + 6 q^{11} - 6 q^{19} + 24 q^{21} - 6 q^{29} + 12 q^{31} - 30 q^{39} - 12 q^{41} + 12 q^{49} + 30 q^{51} - 12 q^{59} + 54 q^{61} + 12 q^{71} - 18 q^{81} + 60 q^{89} + 30 q^{91} - 18 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 12 x^{4} + 36 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 6 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} + 6 \nu^{2} \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} + 9 \nu^{2} + 12 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} + 10 \nu^{3} + 21 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} - \beta_{3} - 4\)
\(\nu^{3}\)\(=\)\(3 \beta_{2} - 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-6 \beta_{4} + 9 \beta_{3} + 24\)
\(\nu^{5}\)\(=\)\(3 \beta_{5} - 30 \beta_{2} + 39 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4049.1
2.66908i
2.14510i
0.523976i
0.523976i
2.14510i
2.66908i
0 2.66908i 0 0 0 3.66908i 0 −4.12398 0
4049.2 0 2.14510i 0 0 0 1.14510i 0 −1.60147 0
4049.3 0 0.523976i 0 0 0 0.476024i 0 2.72545 0
4049.4 0 0.523976i 0 0 0 0.476024i 0 2.72545 0
4049.5 0 2.14510i 0 0 0 1.14510i 0 −1.60147 0
4049.6 0 2.66908i 0 0 0 3.66908i 0 −4.12398 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4049.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4600.2.e.r 6
5.b even 2 1 inner 4600.2.e.r 6
5.c odd 4 1 920.2.a.g 3
5.c odd 4 1 4600.2.a.y 3
15.e even 4 1 8280.2.a.bo 3
20.e even 4 1 1840.2.a.t 3
20.e even 4 1 9200.2.a.cd 3
40.i odd 4 1 7360.2.a.cb 3
40.k even 4 1 7360.2.a.ca 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.g 3 5.c odd 4 1
1840.2.a.t 3 20.e even 4 1
4600.2.a.y 3 5.c odd 4 1
4600.2.e.r 6 1.a even 1 1 trivial
4600.2.e.r 6 5.b even 2 1 inner
7360.2.a.ca 3 40.k even 4 1
7360.2.a.cb 3 40.i odd 4 1
8280.2.a.bo 3 15.e even 4 1
9200.2.a.cd 3 20.e even 4 1

Hecke kernels

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

\( T_{3}^{6} + 12 T_{3}^{4} + 36 T_{3}^{2} + 9 \)
\( T_{7}^{6} + 15 T_{7}^{4} + 21 T_{7}^{2} + 4 \)
\( T_{11}^{3} - 3 T_{11}^{2} - 15 T_{11} - 12 \)
\( T_{13}^{6} + 36 T_{13}^{4} + 324 T_{13}^{2} + 841 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 9 + 36 T^{2} + 12 T^{4} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( 4 + 21 T^{2} + 15 T^{4} + T^{6} \)
$11$ \( ( -12 - 15 T - 3 T^{2} + T^{3} )^{2} \)
$13$ \( 841 + 324 T^{2} + 36 T^{4} + T^{6} \)
$17$ \( 144 + 153 T^{2} + 39 T^{4} + T^{6} \)
$19$ \( ( 48 - 33 T + 3 T^{2} + T^{3} )^{2} \)
$23$ \( ( 1 + T^{2} )^{3} \)
$29$ \( ( -12 - 6 T + 3 T^{2} + T^{3} )^{2} \)
$31$ \( ( 249 - 42 T - 6 T^{2} + T^{3} )^{2} \)
$37$ \( ( 16 + T^{2} )^{3} \)
$41$ \( ( -69 - 60 T + 6 T^{2} + T^{3} )^{2} \)
$43$ \( 1024 + 6336 T^{2} + 180 T^{4} + T^{6} \)
$47$ \( 5776 + 2076 T^{2} + 93 T^{4} + T^{6} \)
$53$ \( 287296 + 23856 T^{2} + 300 T^{4} + T^{6} \)
$59$ \( ( -32 - 36 T + 6 T^{2} + T^{3} )^{2} \)
$61$ \( ( -596 + 225 T - 27 T^{2} + T^{3} )^{2} \)
$67$ \( 8667136 + 128256 T^{2} + 624 T^{4} + T^{6} \)
$71$ \( ( 203 - 60 T - 6 T^{2} + T^{3} )^{2} \)
$73$ \( 345744 + 19404 T^{2} + 309 T^{4} + T^{6} \)
$79$ \( T^{6} \)
$83$ \( 4096 + 5136 T^{2} + 264 T^{4} + T^{6} \)
$89$ \( ( -608 + 264 T - 30 T^{2} + T^{3} )^{2} \)
$97$ \( 19044 + 7245 T^{2} + 219 T^{4} + T^{6} \)
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