Properties

Label 4600.2.e.p
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: 6.0.431642176.2
Defining polynomial: \(x^{6} + 19 x^{4} + 97 x^{2} + 64\)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( \beta_{1} - \beta_{2} ) q^{7} + ( -3 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( \beta_{1} - \beta_{2} ) q^{7} + ( -3 + \beta_{3} ) q^{9} + ( 2 + \beta_{5} ) q^{11} -\beta_{4} q^{13} + ( -3 \beta_{2} - \beta_{4} ) q^{17} + ( 4 - \beta_{3} ) q^{19} + ( -6 + \beta_{3} + \beta_{5} ) q^{21} + \beta_{2} q^{23} + ( -3 \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{27} + ( -5 - \beta_{3} + \beta_{5} ) q^{29} + ( -3 + \beta_{5} ) q^{31} + ( 2 \beta_{1} + 6 \beta_{2} + \beta_{4} ) q^{33} + ( 3 \beta_{1} - 3 \beta_{2} + \beta_{4} ) q^{37} + ( -2 - \beta_{3} + 3 \beta_{5} ) q^{39} + ( 3 + \beta_{3} ) q^{41} + 8 \beta_{2} q^{43} -2 \beta_{4} q^{47} + ( \beta_{3} + 2 \beta_{5} ) q^{49} + ( -2 - \beta_{3} + 6 \beta_{5} ) q^{51} + ( 3 \beta_{1} - \beta_{2} + \beta_{4} ) q^{53} + ( 7 \beta_{1} - 2 \beta_{2} + \beta_{4} ) q^{57} + ( 5 - \beta_{3} + \beta_{5} ) q^{59} + ( 4 - 2 \beta_{3} - \beta_{5} ) q^{61} + ( -6 \beta_{1} + 5 \beta_{2} ) q^{63} + ( 3 \beta_{1} - 3 \beta_{2} + \beta_{4} ) q^{67} -\beta_{5} q^{69} + ( -7 - \beta_{3} - 2 \beta_{5} ) q^{71} + ( -4 \beta_{1} + 6 \beta_{2} - 2 \beta_{4} ) q^{73} + ( \beta_{1} + 4 \beta_{2} + \beta_{4} ) q^{77} + 4 \beta_{5} q^{79} + ( 7 - \beta_{3} + \beta_{5} ) q^{81} + ( -\beta_{1} - 5 \beta_{2} + \beta_{4} ) q^{83} + ( -2 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} ) q^{87} + ( 4 - 4 \beta_{5} ) q^{89} + ( -2 + 3 \beta_{5} ) q^{91} + ( -3 \beta_{1} + 6 \beta_{2} + \beta_{4} ) q^{93} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{97} + ( -4 + 3 \beta_{3} - 6 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 20q^{9} + O(q^{10}) \) \( 6q - 20q^{9} + 14q^{11} + 26q^{19} - 36q^{21} - 26q^{29} - 16q^{31} - 4q^{39} + 16q^{41} + 2q^{49} + 2q^{51} + 34q^{59} + 26q^{61} - 2q^{69} - 44q^{71} + 8q^{79} + 46q^{81} + 16q^{89} - 6q^{91} - 42q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 19 x^{4} + 97 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 11 \nu^{3} + 17 \nu \)\()/8\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 6 \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} + 7 \nu^{3} - 19 \nu \)\()/4\)
\(\beta_{5}\)\(=\)\( \nu^{4} + 10 \nu^{2} + 8 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 6\)
\(\nu^{3}\)\(=\)\(-\beta_{4} + 2 \beta_{2} - 9 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{5} - 10 \beta_{3} + 52\)
\(\nu^{5}\)\(=\)\(11 \beta_{4} - 14 \beta_{2} + 82 \beta_{1}\)

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
3.07912i
2.95759i
0.878468i
0.878468i
2.95759i
3.07912i
0 3.07912i 0 0 0 2.07912i 0 −6.48097 0
4049.2 0 2.95759i 0 0 0 3.95759i 0 −5.74732 0
4049.3 0 0.878468i 0 0 0 0.121532i 0 2.22829 0
4049.4 0 0.878468i 0 0 0 0.121532i 0 2.22829 0
4049.5 0 2.95759i 0 0 0 3.95759i 0 −5.74732 0
4049.6 0 3.07912i 0 0 0 2.07912i 0 −6.48097 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.p 6
5.b even 2 1 inner 4600.2.e.p 6
5.c odd 4 1 920.2.a.h 3
5.c odd 4 1 4600.2.a.x 3
15.e even 4 1 8280.2.a.bj 3
20.e even 4 1 1840.2.a.s 3
20.e even 4 1 9200.2.a.ce 3
40.i odd 4 1 7360.2.a.by 3
40.k even 4 1 7360.2.a.cc 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.h 3 5.c odd 4 1
1840.2.a.s 3 20.e even 4 1
4600.2.a.x 3 5.c odd 4 1
4600.2.e.p 6 1.a even 1 1 trivial
4600.2.e.p 6 5.b even 2 1 inner
7360.2.a.by 3 40.i odd 4 1
7360.2.a.cc 3 40.k even 4 1
8280.2.a.bj 3 15.e even 4 1
9200.2.a.ce 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 \(S_{2}^{\mathrm{new}}(4600, [\chi])\):

\( T_{3}^{6} + 19 T_{3}^{4} + 97 T_{3}^{2} + 64 \)
\( T_{7}^{6} + 20 T_{7}^{4} + 68 T_{7}^{2} + 1 \)
\( T_{11}^{3} - 7 T_{11}^{2} + 7 T_{11} + 14 \)
\( T_{13}^{6} + 47 T_{13}^{4} + 629 T_{13}^{2} + 2500 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 64 + 97 T^{2} + 19 T^{4} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( 1 + 68 T^{2} + 20 T^{4} + T^{6} \)
$11$ \( ( 14 + 7 T - 7 T^{2} + T^{3} )^{2} \)
$13$ \( 2500 + 629 T^{2} + 47 T^{4} + T^{6} \)
$17$ \( 6889 + 1760 T^{2} + 80 T^{4} + T^{6} \)
$19$ \( ( 62 + 33 T - 13 T^{2} + T^{3} )^{2} \)
$23$ \( ( 1 + T^{2} )^{3} \)
$29$ \( ( -76 + 26 T + 13 T^{2} + T^{3} )^{2} \)
$31$ \( ( -1 + 12 T + 8 T^{2} + T^{3} )^{2} \)
$37$ \( 246016 + 13424 T^{2} + 209 T^{4} + T^{6} \)
$41$ \( ( 1 - 2 T - 8 T^{2} + T^{3} )^{2} \)
$43$ \( ( 64 + T^{2} )^{3} \)
$47$ \( 160000 + 10064 T^{2} + 188 T^{4} + T^{6} \)
$53$ \( 90000 + 9400 T^{2} + 201 T^{4} + T^{6} \)
$59$ \( ( -36 + 66 T - 17 T^{2} + T^{3} )^{2} \)
$61$ \( ( 720 - 51 T - 13 T^{2} + T^{3} )^{2} \)
$67$ \( 246016 + 13424 T^{2} + 209 T^{4} + T^{6} \)
$71$ \( ( -225 + 96 T + 22 T^{2} + T^{3} )^{2} \)
$73$ \( 4494400 + 81856 T^{2} + 496 T^{4} + T^{6} \)
$79$ \( ( 512 - 144 T - 4 T^{2} + T^{3} )^{2} \)
$83$ \( 1296 + 2680 T^{2} + 145 T^{4} + T^{6} \)
$89$ \( ( 64 - 128 T - 8 T^{2} + T^{3} )^{2} \)
$97$ \( 2500 + 6261 T^{2} + 171 T^{4} + T^{6} \)
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