Properties

Label 4600.2.e.v
Level $4600$
Weight $2$
Character orbit 4600.e
Analytic conductor $36.731$
Analytic rank $0$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \(x^{10} + 18 x^{8} + 103 x^{6} + 239 x^{4} + 197 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{2} + \beta_{5} ) q^{3} + \beta_{8} q^{7} + ( -2 + \beta_{3} - \beta_{4} - \beta_{7} - \beta_{9} ) q^{9} +O(q^{10})\) \( q + ( \beta_{2} + \beta_{5} ) q^{3} + \beta_{8} q^{7} + ( -2 + \beta_{3} - \beta_{4} - \beta_{7} - \beta_{9} ) q^{9} + ( \beta_{4} + \beta_{7} + \beta_{9} ) q^{11} + ( \beta_{1} + \beta_{2} + 2 \beta_{5} ) q^{13} + ( 2 \beta_{1} + \beta_{5} + \beta_{6} ) q^{17} + ( -2 \beta_{3} + \beta_{4} - \beta_{7} + \beta_{9} ) q^{19} + ( -2 + 2 \beta_{3} ) q^{21} + \beta_{2} q^{23} + ( -2 \beta_{2} + 3 \beta_{6} + \beta_{8} ) q^{27} + ( 2 - \beta_{3} + \beta_{4} - 2 \beta_{7} ) q^{29} + ( 2 - 3 \beta_{3} - \beta_{4} + \beta_{9} ) q^{31} + ( 4 \beta_{2} + \beta_{5} - 3 \beta_{6} - 2 \beta_{8} ) q^{33} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{6} + 2 \beta_{8} ) q^{37} + ( -8 + 2 \beta_{3} - 2 \beta_{4} - \beta_{7} - \beta_{9} ) q^{39} + ( -1 - 2 \beta_{3} - 2 \beta_{4} + \beta_{7} - \beta_{9} ) q^{41} + 3 \beta_{8} q^{43} + ( -\beta_{1} + \beta_{2} + 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{8} ) q^{47} + ( 1 + 2 \beta_{4} + \beta_{9} ) q^{49} + ( \beta_{4} + \beta_{7} + 2 \beta_{9} ) q^{51} + ( 4 \beta_{1} + 6 \beta_{2} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} ) q^{53} + ( -2 \beta_{1} - 6 \beta_{2} + \beta_{5} - \beta_{6} + 2 \beta_{8} ) q^{57} + ( -5 + 3 \beta_{4} - 3 \beta_{7} + \beta_{9} ) q^{59} + ( -6 + 2 \beta_{3} ) q^{61} + ( -2 \beta_{5} + \beta_{8} ) q^{63} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{5} + \beta_{6} ) q^{67} + ( -1 - \beta_{7} ) q^{69} + ( 1 - 3 \beta_{3} + \beta_{9} ) q^{71} + ( -5 \beta_{1} + \beta_{5} - 4 \beta_{6} ) q^{73} + ( -2 \beta_{2} - 2 \beta_{6} - 3 \beta_{8} ) q^{77} + ( -8 - 2 \beta_{4} + 2 \beta_{7} + \beta_{9} ) q^{79} + ( 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{7} ) q^{81} + ( -2 \beta_{1} + 4 \beta_{2} - 3 \beta_{5} + \beta_{6} - \beta_{8} ) q^{83} + ( -\beta_{1} - 5 \beta_{2} + 3 \beta_{5} + 2 \beta_{8} ) q^{87} + ( -4 - 3 \beta_{4} + \beta_{7} - 4 \beta_{9} ) q^{89} + ( -6 + 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{7} - \beta_{9} ) q^{91} + ( -3 \beta_{1} - 5 \beta_{2} + \beta_{5} + 2 \beta_{6} + 4 \beta_{8} ) q^{93} + ( 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{5} - 2 \beta_{6} - 4 \beta_{8} ) q^{97} + ( -10 - 4 \beta_{4} - 4 \beta_{7} - \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 8 q^{9} + O(q^{10}) \) \( 10 q - 8 q^{9} - 8 q^{11} - 8 q^{19} - 12 q^{21} + 22 q^{29} + 8 q^{31} - 62 q^{39} - 16 q^{41} + 4 q^{49} - 10 q^{51} - 46 q^{59} - 52 q^{61} - 6 q^{69} - 4 q^{71} - 86 q^{79} - 6 q^{81} - 30 q^{89} - 38 q^{91} - 74 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} + 18 x^{8} + 103 x^{6} + 239 x^{4} + 197 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{9} + 4 \nu^{7} - 101 \nu^{5} - 493 \nu^{3} - 523 \nu \)\()/74\)
\(\beta_{3}\)\(=\)\((\)\( -7 \nu^{8} - 102 \nu^{6} - 366 \nu^{4} - 360 \nu^{2} - 2 \)\()/37\)
\(\beta_{4}\)\(=\)\((\)\( -7 \nu^{8} - 102 \nu^{6} - 366 \nu^{4} - 323 \nu^{2} + 109 \)\()/37\)
\(\beta_{5}\)\(=\)\((\)\( 7 \nu^{9} + 102 \nu^{7} + 329 \nu^{5} - 121 \nu^{3} - 1071 \nu \)\()/74\)
\(\beta_{6}\)\(=\)\((\)\( 11 \nu^{9} + 192 \nu^{7} + 1035 \nu^{5} + 2199 \nu^{3} + 1647 \nu \)\()/74\)
\(\beta_{7}\)\(=\)\((\)\( 16 \nu^{8} + 249 \nu^{6} + 1048 \nu^{4} + 1362 \nu^{2} + 105 \)\()/37\)
\(\beta_{8}\)\(=\)\((\)\( 33 \nu^{9} + 502 \nu^{7} + 1995 \nu^{5} + 2231 \nu^{3} - 239 \nu \)\()/74\)
\(\beta_{9}\)\(=\)\((\)\( -19 \nu^{8} - 298 \nu^{6} - 1263 \nu^{4} - 1585 \nu^{2} - 53 \)\()/37\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} - \beta_{3} - 3\)
\(\nu^{3}\)\(=\)\(-\beta_{8} + 2 \beta_{6} + \beta_{5} + 4 \beta_{2} - 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(3 \beta_{9} + 4 \beta_{7} - 9 \beta_{4} + 10 \beta_{3} + 20\)
\(\nu^{5}\)\(=\)\(13 \beta_{8} - 25 \beta_{6} - 15 \beta_{5} - 49 \beta_{2} + 35 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-40 \beta_{9} - 51 \beta_{7} + 86 \beta_{4} - 94 \beta_{3} - 171\)
\(\nu^{7}\)\(=\)\(-137 \beta_{8} + 260 \beta_{6} + 166 \beta_{5} + 499 \beta_{2} - 300 \beta_{1}\)
\(\nu^{8}\)\(=\)\(426 \beta_{9} + 534 \beta_{7} - 834 \beta_{4} + 893 \beta_{3} + 1600\)
\(\nu^{9}\)\(=\)\(1368 \beta_{8} - 2579 \beta_{6} - 1686 \beta_{5} - 4899 \beta_{2} + 2793 \beta_{1}\)

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
0.144312i
1.45894i
1.64975i
3.11721i
1.84717i
1.84717i
3.11721i
1.64975i
1.45894i
0.144312i
0 3.08344i 0 0 0 0.555022i 0 −6.50759 0
4049.2 0 2.21042i 0 0 0 2.22487i 0 −1.88594 0
4049.3 0 1.51466i 0 0 0 3.49880i 0 0.705809 0
4049.4 0 1.33689i 0 0 0 3.16736i 0 1.21273 0
4049.5 0 0.724570i 0 0 0 2.33840i 0 2.47500 0
4049.6 0 0.724570i 0 0 0 2.33840i 0 2.47500 0
4049.7 0 1.33689i 0 0 0 3.16736i 0 1.21273 0
4049.8 0 1.51466i 0 0 0 3.49880i 0 0.705809 0
4049.9 0 2.21042i 0 0 0 2.22487i 0 −1.88594 0
4049.10 0 3.08344i 0 0 0 0.555022i 0 −6.50759 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4049.10
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.v 10
5.b even 2 1 inner 4600.2.e.v 10
5.c odd 4 1 4600.2.a.bc 5
5.c odd 4 1 4600.2.a.bg yes 5
20.e even 4 1 9200.2.a.cs 5
20.e even 4 1 9200.2.a.cw 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4600.2.a.bc 5 5.c odd 4 1
4600.2.a.bg yes 5 5.c odd 4 1
4600.2.e.v 10 1.a even 1 1 trivial
4600.2.e.v 10 5.b even 2 1 inner
9200.2.a.cs 5 20.e even 4 1
9200.2.a.cw 5 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 \(S_{2}^{\mathrm{new}}(4600, [\chi])\):

\( T_{3}^{10} + 19 T_{3}^{8} + 119 T_{3}^{6} + 306 T_{3}^{4} + 321 T_{3}^{2} + 100 \)
\( T_{7}^{10} + 33 T_{7}^{8} + 392 T_{7}^{6} + 2000 T_{7}^{4} + 3904 T_{7}^{2} + 1024 \)
\( T_{11}^{5} + 4 T_{11}^{4} - 17 T_{11}^{3} - 88 T_{11}^{2} - 92 T_{11} - 8 \)
\( T_{13}^{10} + 75 T_{13}^{8} + 2107 T_{13}^{6} + 26966 T_{13}^{4} + 146625 T_{13}^{2} + 206116 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \)
$3$ \( 100 + 321 T^{2} + 306 T^{4} + 119 T^{6} + 19 T^{8} + T^{10} \)
$5$ \( T^{10} \)
$7$ \( 1024 + 3904 T^{2} + 2000 T^{4} + 392 T^{6} + 33 T^{8} + T^{10} \)
$11$ \( ( -8 - 92 T - 88 T^{2} - 17 T^{3} + 4 T^{4} + T^{5} )^{2} \)
$13$ \( 206116 + 146625 T^{2} + 26966 T^{4} + 2107 T^{6} + 75 T^{8} + T^{10} \)
$17$ \( 1024 + 12608 T^{2} + 9744 T^{4} + 1800 T^{6} + 81 T^{8} + T^{10} \)
$19$ \( ( 64 - 272 T - 264 T^{2} - 53 T^{3} + 4 T^{4} + T^{5} )^{2} \)
$23$ \( ( 1 + T^{2} )^{5} \)
$29$ \( ( 256 - 551 T + 306 T^{2} - 11 T^{3} - 11 T^{4} + T^{5} )^{2} \)
$31$ \( ( 400 + 802 T + 159 T^{2} - 75 T^{3} - 4 T^{4} + T^{5} )^{2} \)
$37$ \( 28558336 + 27983872 T^{2} + 1704064 T^{4} + 35856 T^{6} + 316 T^{8} + T^{10} \)
$41$ \( ( -2069 - 2097 T - 671 T^{2} - 52 T^{3} + 8 T^{4} + T^{5} )^{2} \)
$43$ \( 60466176 + 25614144 T^{2} + 1458000 T^{4} + 31752 T^{6} + 297 T^{8} + T^{10} \)
$47$ \( 34951744 + 21165556 T^{2} + 1547109 T^{4} + 37137 T^{6} + 338 T^{8} + T^{10} \)
$53$ \( 490356736 + 119363584 T^{2} + 4532800 T^{4} + 66736 T^{6} + 428 T^{8} + T^{10} \)
$59$ \( ( 5128 - 558 T - 1087 T^{2} + 53 T^{3} + 23 T^{4} + T^{5} )^{2} \)
$61$ \( ( -256 + 880 T + 848 T^{2} + 236 T^{3} + 26 T^{4} + T^{5} )^{2} \)
$67$ \( 262144 + 299008 T^{2} + 78336 T^{4} + 5392 T^{6} + 129 T^{8} + T^{10} \)
$71$ \( ( -8 + 666 T - 133 T^{2} - 67 T^{3} + 2 T^{4} + T^{5} )^{2} \)
$73$ \( 2756355001 + 1246938683 T^{2} + 26292185 T^{4} + 211014 T^{6} + 752 T^{8} + T^{10} \)
$79$ \( ( -45872 - 2080 T + 3292 T^{2} + 634 T^{3} + 43 T^{4} + T^{5} )^{2} \)
$83$ \( 1086361600 + 180557056 T^{2} + 7391008 T^{4} + 104529 T^{6} + 562 T^{8} + T^{10} \)
$89$ \( ( -7120 - 12608 T - 3188 T^{2} - 154 T^{3} + 15 T^{4} + T^{5} )^{2} \)
$97$ \( 1048576 + 26599680 T^{2} + 4513216 T^{4} + 85872 T^{6} + 520 T^{8} + T^{10} \)
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