Properties

Label 4600.2.e.u
Level $4600$
Weight $2$
Character orbit 4600.e
Analytic conductor $36.731$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(x^{10} + 28 x^{8} + 260 x^{6} + 897 x^{4} + 1056 x^{2} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -\beta_{6} + \beta_{8} ) q^{7} + ( -2 + \beta_{3} - \beta_{5} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -\beta_{6} + \beta_{8} ) q^{7} + ( -2 + \beta_{3} - \beta_{5} ) q^{9} + ( \beta_{3} + \beta_{4} ) q^{11} + ( \beta_{2} - \beta_{8} - \beta_{9} ) q^{13} + ( -\beta_{1} - \beta_{2} + \beta_{6} - \beta_{9} ) q^{17} + ( -1 - 2 \beta_{4} - \beta_{5} ) q^{19} + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} ) q^{21} -\beta_{2} q^{23} + ( -3 \beta_{1} - \beta_{2} + 2 \beta_{6} + \beta_{8} ) q^{27} + ( -1 - \beta_{3} + \beta_{7} ) q^{29} + ( 4 - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{31} + ( -2 \beta_{1} + 3 \beta_{2} + 2 \beta_{6} + \beta_{8} ) q^{33} + ( -3 \beta_{2} - \beta_{6} ) q^{37} + ( -3 + 2 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{39} + ( 5 - \beta_{4} - \beta_{7} ) q^{41} + ( 2 \beta_{2} - 2 \beta_{6} - \beta_{9} ) q^{47} + ( -6 - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{49} + ( 3 - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{51} + ( -2 \beta_{1} + \beta_{2} + \beta_{6} - 2 \beta_{8} - 2 \beta_{9} ) q^{53} + ( -3 \beta_{1} - 9 \beta_{2} - \beta_{8} + \beta_{9} ) q^{57} + ( 1 - 2 \beta_{5} - \beta_{7} ) q^{59} + ( \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{61} + ( \beta_{1} - 8 \beta_{2} - \beta_{6} + \beta_{8} + 3 \beta_{9} ) q^{63} + ( 2 \beta_{1} - \beta_{2} + \beta_{6} - 2 \beta_{8} ) q^{67} + \beta_{4} q^{69} + ( 1 - 3 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{71} + ( 2 \beta_{6} + \beta_{9} ) q^{73} + ( -\beta_{1} - 3 \beta_{2} - 2 \beta_{6} - \beta_{8} + \beta_{9} ) q^{77} -2 \beta_{3} q^{79} + ( 10 - 5 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{81} + ( -9 \beta_{2} - \beta_{6} ) q^{83} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{6} + \beta_{9} ) q^{87} + ( -2 + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{89} + ( 4 + \beta_{3} - 5 \beta_{4} - 4 \beta_{5} ) q^{91} + ( 5 \beta_{1} - 7 \beta_{2} - 2 \beta_{6} - \beta_{8} + 2 \beta_{9} ) q^{93} + ( -\beta_{1} - 6 \beta_{2} + 2 \beta_{6} + 2 \beta_{8} - \beta_{9} ) q^{97} + ( 11 - 4 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 26q^{9} + O(q^{10}) \) \( 10q - 26q^{9} - 2q^{11} - 14q^{19} + 12q^{21} - 8q^{29} + 38q^{31} - 38q^{39} + 50q^{41} - 50q^{49} + 38q^{51} + 2q^{59} - 10q^{61} + 2q^{71} + 4q^{79} + 114q^{81} - 12q^{89} + 22q^{91} + 130q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} + 28 x^{8} + 260 x^{6} + 897 x^{4} + 1056 x^{2} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{9} - 12 \nu^{7} - 68 \nu^{5} - 833 \nu^{3} - 2064 \nu \)\()/1024\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{8} + 36 \nu^{6} - 52 \nu^{4} - 829 \nu^{2} - 208 \)\()/256\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{8} - 12 \nu^{6} - 4 \nu^{4} + 63 \nu^{2} - 16 \)\()/64\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{8} + 36 \nu^{6} - 52 \nu^{4} - 1085 \nu^{2} - 1488 \)\()/256\)
\(\beta_{6}\)\(=\)\((\)\( -15 \nu^{9} - 436 \nu^{7} - 4092 \nu^{5} - 12495 \nu^{3} - 4592 \nu \)\()/2048\)
\(\beta_{7}\)\(=\)\((\)\( 25 \nu^{8} + 556 \nu^{6} + 3748 \nu^{4} + 7513 \nu^{2} + 2192 \)\()/512\)
\(\beta_{8}\)\(=\)\((\)\( 7 \nu^{9} + 212 \nu^{7} + 2012 \nu^{5} + 6343 \nu^{3} + 5872 \nu \)\()/512\)
\(\beta_{9}\)\(=\)\((\)\( 25 \nu^{9} + 556 \nu^{7} + 3748 \nu^{5} + 7513 \nu^{3} + 1680 \nu \)\()/1024\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} + \beta_{3} - 5\)
\(\nu^{3}\)\(=\)\(\beta_{8} + 2 \beta_{6} - \beta_{2} - 9 \beta_{1}\)
\(\nu^{4}\)\(=\)\(10 \beta_{5} - 3 \beta_{4} - 14 \beta_{3} + 46\)
\(\nu^{5}\)\(=\)\(-\beta_{9} - 13 \beta_{8} - 28 \beta_{6} + 3 \beta_{2} + 94 \beta_{1}\)
\(\nu^{6}\)\(=\)\(2 \beta_{7} - 107 \beta_{5} + 49 \beta_{4} + 164 \beta_{3} - 485\)
\(\nu^{7}\)\(=\)\(12 \beta_{9} + 156 \beta_{8} + 328 \beta_{6} + 24 \beta_{2} - 1025 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-24 \beta_{7} + 1181 \beta_{5} - 640 \beta_{4} - 1849 \beta_{3} + 5305\)
\(\nu^{9}\)\(=\)\(-76 \beta_{9} - 1821 \beta_{8} - 3698 \beta_{6} - 683 \beta_{2} + 11341 \beta_{1}\)

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
3.36002i
3.30649i
1.93283i
1.31091i
0.568386i
0.568386i
1.31091i
1.93283i
3.30649i
3.36002i
0 3.36002i 0 0 0 1.90754i 0 −8.28974 0
4049.2 0 3.30649i 0 0 0 2.55040i 0 −7.93288 0
4049.3 0 1.93283i 0 0 0 2.38236i 0 −0.735829 0
4049.4 0 1.31091i 0 0 0 4.66212i 0 1.28151 0
4049.5 0 0.568386i 0 0 0 4.73770i 0 2.67694 0
4049.6 0 0.568386i 0 0 0 4.73770i 0 2.67694 0
4049.7 0 1.31091i 0 0 0 4.66212i 0 1.28151 0
4049.8 0 1.93283i 0 0 0 2.38236i 0 −0.735829 0
4049.9 0 3.30649i 0 0 0 2.55040i 0 −7.93288 0
4049.10 0 3.36002i 0 0 0 1.90754i 0 −8.28974 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.u 10
5.b even 2 1 inner 4600.2.e.u 10
5.c odd 4 1 920.2.a.j 5
5.c odd 4 1 4600.2.a.be 5
15.e even 4 1 8280.2.a.bs 5
20.e even 4 1 1840.2.a.v 5
20.e even 4 1 9200.2.a.cu 5
40.i odd 4 1 7360.2.a.co 5
40.k even 4 1 7360.2.a.cp 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.j 5 5.c odd 4 1
1840.2.a.v 5 20.e even 4 1
4600.2.a.be 5 5.c odd 4 1
4600.2.e.u 10 1.a even 1 1 trivial
4600.2.e.u 10 5.b even 2 1 inner
7360.2.a.co 5 40.i odd 4 1
7360.2.a.cp 5 40.k even 4 1
8280.2.a.bs 5 15.e even 4 1
9200.2.a.cu 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 \(S_{2}^{\mathrm{new}}(4600, [\chi])\):

\( T_{3}^{10} + 28 T_{3}^{8} + 260 T_{3}^{6} + 897 T_{3}^{4} + 1056 T_{3}^{2} + 256 \)
\( T_{7}^{10} + 60 T_{7}^{8} + 1268 T_{7}^{6} + 11441 T_{7}^{4} + 45568 T_{7}^{2} + 65536 \)
\( T_{11}^{5} + T_{11}^{4} - 35 T_{11}^{3} - 28 T_{11}^{2} + 172 T_{11} + 64 \)
\( T_{13}^{10} + 108 T_{13}^{8} + 3916 T_{13}^{6} + 56825 T_{13}^{4} + 285000 T_{13}^{2} + 250000 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \)
$3$ \( 256 + 1056 T^{2} + 897 T^{4} + 260 T^{6} + 28 T^{8} + T^{10} \)
$5$ \( T^{10} \)
$7$ \( 65536 + 45568 T^{2} + 11441 T^{4} + 1268 T^{6} + 60 T^{8} + T^{10} \)
$11$ \( ( 64 + 172 T - 28 T^{2} - 35 T^{3} + T^{4} + T^{5} )^{2} \)
$13$ \( 250000 + 285000 T^{2} + 56825 T^{4} + 3916 T^{6} + 108 T^{8} + T^{10} \)
$17$ \( 1024 + 8996 T^{2} + 11697 T^{4} + 3096 T^{6} + 108 T^{8} + T^{10} \)
$19$ \( ( -512 + 676 T - 180 T^{2} - 41 T^{3} + 7 T^{4} + T^{5} )^{2} \)
$23$ \( ( 1 + T^{2} )^{5} \)
$29$ \( ( -8 + 64 T + 36 T^{2} - 41 T^{3} + 4 T^{4} + T^{5} )^{2} \)
$31$ \( ( -128 - 53 T + 183 T^{2} + 72 T^{3} - 19 T^{4} + T^{5} )^{2} \)
$37$ \( 4096 + 35584 T^{2} + 21904 T^{4} + 2664 T^{6} + 97 T^{8} + T^{10} \)
$41$ \( ( 2182 + 27 T - 653 T^{2} + 212 T^{3} - 25 T^{4} + T^{5} )^{2} \)
$43$ \( T^{10} \)
$47$ \( 262144 + 1782784 T^{2} + 1080720 T^{4} + 35052 T^{6} + 341 T^{8} + T^{10} \)
$53$ \( 410953984 + 51572224 T^{2} + 2173792 T^{4} + 39856 T^{6} + 329 T^{8} + T^{10} \)
$59$ \( ( -13568 + 4560 T + 236 T^{2} - 142 T^{3} - T^{4} + T^{5} )^{2} \)
$61$ \( ( 7664 + 2092 T - 568 T^{2} - 115 T^{3} + 5 T^{4} + T^{5} )^{2} \)
$67$ \( 67108864 + 16270336 T^{2} + 1256832 T^{4} + 33316 T^{6} + 333 T^{8} + T^{10} \)
$71$ \( ( -3968 + 8139 T + 407 T^{2} - 214 T^{3} - T^{4} + T^{5} )^{2} \)
$73$ \( 1763584 + 3570752 T^{2} + 726080 T^{4} + 29652 T^{6} + 317 T^{8} + T^{10} \)
$79$ \( ( 1024 + 2432 T - 128 T^{2} - 128 T^{3} - 2 T^{4} + T^{5} )^{2} \)
$83$ \( 1698758656 + 164923648 T^{2} + 5389840 T^{4} + 74952 T^{6} + 457 T^{8} + T^{10} \)
$89$ \( ( 8192 + 2560 T - 512 T^{2} - 136 T^{3} + 6 T^{4} + T^{5} )^{2} \)
$97$ \( 2461747456 + 699082512 T^{2} + 18500872 T^{4} + 175905 T^{6} + 703 T^{8} + T^{10} \)
show more
show less