Properties

Label 4600.2.e.w
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: \(x^{10} + 18 x^{8} + 117 x^{6} + 333 x^{4} + 396 x^{2} + 144\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -\beta_{4} + \beta_{6} ) q^{7} + ( -1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -\beta_{4} + \beta_{6} ) q^{7} + ( -1 + \beta_{2} ) q^{9} -\beta_{5} q^{11} + ( \beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} ) q^{13} + ( -2 \beta_{3} - 2 \beta_{4} + 2 \beta_{9} ) q^{17} + \beta_{5} q^{19} + ( 1 - 2 \beta_{2} + \beta_{5} + \beta_{8} ) q^{21} -\beta_{4} q^{23} + ( \beta_{1} - 2 \beta_{4} + \beta_{6} ) q^{27} + ( -2 - \beta_{2} - \beta_{5} - 3 \beta_{7} ) q^{29} + ( -3 - \beta_{2} - \beta_{5} - \beta_{7} + \beta_{8} ) q^{31} + ( \beta_{4} + \beta_{6} + \beta_{9} ) q^{33} + ( 2 \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{9} ) q^{37} + ( -1 - 2 \beta_{2} + \beta_{5} - 3 \beta_{7} ) q^{39} + ( -1 + 2 \beta_{5} + \beta_{7} + \beta_{8} ) q^{41} + ( 2 \beta_{1} + 2 \beta_{4} - \beta_{9} ) q^{43} + ( \beta_{1} + 3 \beta_{3} - 3 \beta_{4} - \beta_{6} - \beta_{9} ) q^{47} + ( -3 + \beta_{5} - 2 \beta_{7} ) q^{49} + ( -2 + 2 \beta_{2} + 2 \beta_{5} + 4 \beta_{7} ) q^{51} + ( 2 \beta_{3} + 3 \beta_{4} - \beta_{6} - 3 \beta_{9} ) q^{53} + ( -\beta_{4} - \beta_{6} - \beta_{9} ) q^{57} + ( -2 \beta_{5} - \beta_{8} ) q^{59} + ( 1 + 2 \beta_{2} - 3 \beta_{5} - \beta_{8} ) q^{61} + ( 4 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{6} ) q^{63} + ( 2 \beta_{1} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{9} ) q^{67} + \beta_{7} q^{69} + ( 3 \beta_{2} + 2 \beta_{5} + 4 \beta_{7} - 2 \beta_{8} ) q^{71} + ( -2 \beta_{1} - 5 \beta_{3} - \beta_{4} + \beta_{6} + 2 \beta_{9} ) q^{73} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{6} + 2 \beta_{9} ) q^{77} + ( -1 + 2 \beta_{2} - 2 \beta_{5} - \beta_{8} ) q^{79} + ( -6 + 2 \beta_{2} + \beta_{5} + \beta_{7} + \beta_{8} ) q^{81} + ( -4 \beta_{3} - 2 \beta_{4} + 3 \beta_{9} ) q^{83} + ( -4 \beta_{1} - 3 \beta_{3} - 9 \beta_{4} + \beta_{9} ) q^{87} + ( -2 - 2 \beta_{2} - 4 \beta_{7} ) q^{89} + ( -8 - 2 \beta_{2} + \beta_{5} + 2 \beta_{8} ) q^{91} + ( -2 \beta_{1} + \beta_{3} - \beta_{6} + 2 \beta_{9} ) q^{93} + ( -\beta_{4} - \beta_{6} - 3 \beta_{9} ) q^{97} + ( 2 - 2 \beta_{2} - \beta_{5} - 2 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{9} + O(q^{10}) \) \( 10 q - 6 q^{9} - 24 q^{29} - 36 q^{31} - 18 q^{39} - 12 q^{41} - 30 q^{49} - 12 q^{51} + 2 q^{59} + 20 q^{61} + 16 q^{71} - 54 q^{81} - 28 q^{89} - 92 q^{91} + 12 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} + 18 x^{8} + 117 x^{6} + 333 x^{4} + 396 x^{2} + 144\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 4 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} + 9 \nu^{3} + 15 \nu \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{9} + 14 \nu^{7} + 61 \nu^{5} + 81 \nu^{3} \)\()/24\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{6} + 12 \nu^{4} + 39 \nu^{2} + 27 \)\()/3\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{9} + 14 \nu^{7} + 61 \nu^{5} + 93 \nu^{3} + 60 \nu \)\()/12\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{8} + 14 \nu^{6} + 63 \nu^{4} + 99 \nu^{2} + 36 \)\()/6\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{8} - 16 \nu^{6} - 81 \nu^{4} - 135 \nu^{2} - 48 \)\()/6\)
\(\beta_{9}\)\(=\)\((\)\( -3 \nu^{9} - 50 \nu^{7} - 279 \nu^{5} - 579 \nu^{3} - 336 \nu \)\()/24\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 4\)
\(\nu^{3}\)\(=\)\(\beta_{6} - 2 \beta_{4} - 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{8} + \beta_{7} + \beta_{5} - 7 \beta_{2} + 21\)
\(\nu^{5}\)\(=\)\(-9 \beta_{6} + 18 \beta_{4} + 3 \beta_{3} + 30 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-12 \beta_{8} - 12 \beta_{7} - 9 \beta_{5} + 45 \beta_{2} - 123\)
\(\nu^{7}\)\(=\)\(-3 \beta_{9} + 66 \beta_{6} - 141 \beta_{4} - 36 \beta_{3} - 192 \beta_{1}\)
\(\nu^{8}\)\(=\)\(105 \beta_{8} + 111 \beta_{7} + 63 \beta_{5} - 288 \beta_{2} + 759\)
\(\nu^{9}\)\(=\)\(42 \beta_{9} - 456 \beta_{6} + 1062 \beta_{4} + 321 \beta_{3} + 1263 \beta_{1}\)

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
2.61696i
2.29544i
1.83957i
1.36629i
0.794805i
0.794805i
1.36629i
1.83957i
2.29544i
2.61696i
0 2.61696i 0 0 0 3.83744i 0 −3.84849 0
4049.2 0 2.29544i 0 0 0 1.61758i 0 −2.26905 0
4049.3 0 1.83957i 0 0 0 3.97272i 0 −0.384010 0
4049.4 0 1.36629i 0 0 0 3.28093i 0 1.13327 0
4049.5 0 0.794805i 0 0 0 2.47193i 0 2.36829 0
4049.6 0 0.794805i 0 0 0 2.47193i 0 2.36829 0
4049.7 0 1.36629i 0 0 0 3.28093i 0 1.13327 0
4049.8 0 1.83957i 0 0 0 3.97272i 0 −0.384010 0
4049.9 0 2.29544i 0 0 0 1.61758i 0 −2.26905 0
4049.10 0 2.61696i 0 0 0 3.83744i 0 −3.84849 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.w 10
5.b even 2 1 inner 4600.2.e.w 10
5.c odd 4 1 4600.2.a.bd 5
5.c odd 4 1 4600.2.a.bf yes 5
20.e even 4 1 9200.2.a.ct 5
20.e even 4 1 9200.2.a.cv 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4600.2.a.bd 5 5.c odd 4 1
4600.2.a.bf yes 5 5.c odd 4 1
4600.2.e.w 10 1.a even 1 1 trivial
4600.2.e.w 10 5.b even 2 1 inner
9200.2.a.ct 5 20.e even 4 1
9200.2.a.cv 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 \(S_{2}^{\mathrm{new}}(4600, [\chi])\):

\( T_{3}^{10} + 18 T_{3}^{8} + 117 T_{3}^{6} + 333 T_{3}^{4} + 396 T_{3}^{2} + 144 \)
\( T_{7}^{10} + 50 T_{7}^{8} + 937 T_{7}^{6} + 8056 T_{7}^{4} + 30800 T_{7}^{2} + 40000 \)
\( T_{11}^{5} - 15 T_{11}^{3} - 6 T_{11}^{2} + 48 T_{11} + 24 \)
\( T_{13}^{10} + 80 T_{13}^{8} + 2326 T_{13}^{6} + 29425 T_{13}^{4} + 145523 T_{13}^{2} + 120409 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \)
$3$ \( 144 + 396 T^{2} + 333 T^{4} + 117 T^{6} + 18 T^{8} + T^{10} \)
$5$ \( T^{10} \)
$7$ \( 40000 + 30800 T^{2} + 8056 T^{4} + 937 T^{6} + 50 T^{8} + T^{10} \)
$11$ \( ( 24 + 48 T - 6 T^{2} - 15 T^{3} + T^{5} )^{2} \)
$13$ \( 120409 + 145523 T^{2} + 29425 T^{4} + 2326 T^{6} + 80 T^{8} + T^{10} \)
$17$ \( 147456 + 175104 T^{2} + 60480 T^{4} + 5280 T^{6} + 132 T^{8} + T^{10} \)
$19$ \( ( -24 + 48 T + 6 T^{2} - 15 T^{3} + T^{5} )^{2} \)
$23$ \( ( 1 + T^{2} )^{5} \)
$29$ \( ( 2787 - 273 T - 339 T^{2} + 12 T^{4} + T^{5} )^{2} \)
$31$ \( ( -228 - 516 T + 45 T^{2} + 93 T^{3} + 18 T^{4} + T^{5} )^{2} \)
$37$ \( 369664 + 1464320 T^{2} + 198784 T^{4} + 9088 T^{6} + 164 T^{8} + T^{10} \)
$41$ \( ( 453 + 1311 T - 237 T^{2} - 66 T^{3} + 6 T^{4} + T^{5} )^{2} \)
$43$ \( 2166784 + 1048832 T^{2} + 148192 T^{4} + 7345 T^{6} + 146 T^{8} + T^{10} \)
$47$ \( 16 + 109148 T^{2} + 155221 T^{4} + 21985 T^{6} + 290 T^{8} + T^{10} \)
$53$ \( 65536 + 290816 T^{2} + 284416 T^{4} + 17872 T^{6} + 284 T^{8} + T^{10} \)
$59$ \( ( 3004 + 1322 T - 107 T^{2} - 83 T^{3} - T^{4} + T^{5} )^{2} \)
$61$ \( ( 7648 + 12800 T + 1072 T^{2} - 200 T^{3} - 10 T^{4} + T^{5} )^{2} \)
$67$ \( 94633984 + 28909568 T^{2} + 2370112 T^{4} + 56464 T^{6} + 440 T^{8} + T^{10} \)
$71$ \( ( 8084 + 5960 T + 689 T^{2} - 179 T^{3} - 8 T^{4} + T^{5} )^{2} \)
$73$ \( 816187761 + 106077267 T^{2} + 4279401 T^{4} + 70566 T^{6} + 480 T^{8} + T^{10} \)
$79$ \( ( 1728 + 5184 T + 36 T^{2} - 153 T^{3} + T^{5} )^{2} \)
$83$ \( 5721664 + 5594384 T^{2} + 1446280 T^{4} + 41977 T^{6} + 386 T^{8} + T^{10} \)
$89$ \( ( -3104 - 4864 T - 1280 T^{2} - 44 T^{3} + 14 T^{4} + T^{5} )^{2} \)
$97$ \( 9216 + 175104 T^{2} + 830592 T^{4} + 31680 T^{6} + 324 T^{8} + T^{10} \)
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