Properties

Label 4560.2.d.l
Level $4560$
Weight $2$
Character orbit 4560.d
Analytic conductor $36.412$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 35 x^{10} + 202 x^{8} + 362 x^{6} + 245 x^{4} + 63 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + \beta_{2} q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + q^{5} + \beta_{2} q^{7} + q^{9} + \beta_{6} q^{11} -\beta_{4} q^{13} + q^{15} -\beta_{5} q^{17} + \beta_{8} q^{19} + \beta_{2} q^{21} + ( \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{23} + q^{25} + q^{27} + ( -\beta_{6} + \beta_{7} ) q^{29} + ( -1 + \beta_{3} ) q^{31} + \beta_{6} q^{33} + \beta_{2} q^{35} + ( -\beta_{6} + \beta_{7} - \beta_{10} ) q^{37} -\beta_{4} q^{39} + ( -\beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{41} + ( -\beta_{6} + \beta_{10} ) q^{43} + q^{45} + ( -\beta_{4} + \beta_{6} ) q^{47} + ( -2 + \beta_{5} - \beta_{11} ) q^{49} -\beta_{5} q^{51} + ( \beta_{4} + \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{53} + \beta_{6} q^{55} + \beta_{8} q^{57} + ( 4 + \beta_{3} - \beta_{5} + \beta_{11} ) q^{59} + ( -5 - \beta_{5} + \beta_{11} ) q^{61} + \beta_{2} q^{63} -\beta_{4} q^{65} + ( -3 + \beta_{1} + \beta_{3} ) q^{67} + ( \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{69} + ( 1 + \beta_{3} + \beta_{5} ) q^{71} + ( 3 + \beta_{3} + \beta_{5} - \beta_{8} + \beta_{9} ) q^{73} + q^{75} + ( 2 + \beta_{3} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{77} + ( -2 - \beta_{1} + \beta_{3} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{79} + q^{81} + ( -\beta_{2} + \beta_{4} - \beta_{10} ) q^{83} -\beta_{5} q^{85} + ( -\beta_{6} + \beta_{7} ) q^{87} + ( -\beta_{2} + \beta_{7} + \beta_{10} ) q^{89} + ( 3 + \beta_{1} + 2 \beta_{3} + 2 \beta_{5} + \beta_{11} ) q^{91} + ( -1 + \beta_{3} ) q^{93} + \beta_{8} q^{95} + ( \beta_{6} - \beta_{7} - \beta_{10} ) q^{97} + \beta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 12q^{3} + 12q^{5} + 12q^{9} + O(q^{10}) \) \( 12q + 12q^{3} + 12q^{5} + 12q^{9} + 12q^{15} + 4q^{17} + 12q^{25} + 12q^{27} - 12q^{31} + 12q^{45} - 28q^{49} + 4q^{51} + 52q^{59} - 56q^{61} - 32q^{67} + 8q^{71} + 32q^{73} + 12q^{75} + 24q^{77} - 28q^{79} + 12q^{81} + 4q^{85} + 32q^{91} - 12q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 35 x^{10} + 202 x^{8} + 362 x^{6} + 245 x^{4} + 63 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -27 \nu^{10} - 946 \nu^{8} - 5482 \nu^{6} - 9742 \nu^{4} - 5871 \nu^{2} - 580 \)\()/76\)
\(\beta_{2}\)\(=\)\((\)\( -110 \nu^{11} - 3783 \nu^{9} - 19907 \nu^{7} - 27391 \nu^{5} - 8645 \nu^{3} + 556 \nu \)\()/76\)
\(\beta_{3}\)\(=\)\((\)\( 27 \nu^{10} + 927 \nu^{8} + 4836 \nu^{6} + 6550 \nu^{4} + 2242 \nu^{2} + 105 \)\()/19\)
\(\beta_{4}\)\(=\)\((\)\( -121 \nu^{11} - 4205 \nu^{9} - 23393 \nu^{7} - 37789 \nu^{5} - 19380 \nu^{3} - 2208 \nu \)\()/76\)
\(\beta_{5}\)\(=\)\((\)\( -199 \nu^{10} - 6864 \nu^{8} - 36724 \nu^{6} - 53724 \nu^{4} - 22933 \nu^{2} - 2220 \)\()/76\)
\(\beta_{6}\)\(=\)\((\)\( 245 \nu^{11} + 8437 \nu^{9} + 44733 \nu^{7} + 63333 \nu^{5} + 23560 \nu^{3} + 1052 \nu \)\()/76\)
\(\beta_{7}\)\(=\)\((\)\( -148 \nu^{11} - 5113 \nu^{9} - 27583 \nu^{7} - 41147 \nu^{5} - 17955 \nu^{3} - 1762 \nu \)\()/38\)
\(\beta_{8}\)\(=\)\((\)\( -277 \nu^{11} - 116 \nu^{10} - 9542 \nu^{9} - 4008 \nu^{8} - 50682 \nu^{7} - 21634 \nu^{6} - 72224 \nu^{5} - 32256 \nu^{4} - 27569 \nu^{3} - 13946 \nu^{2} - 1358 \nu - 1304 \)\()/76\)
\(\beta_{9}\)\(=\)\((\)\( -277 \nu^{11} + 116 \nu^{10} - 9542 \nu^{9} + 4008 \nu^{8} - 50682 \nu^{7} + 21634 \nu^{6} - 72224 \nu^{5} + 32256 \nu^{4} - 27569 \nu^{3} + 13946 \nu^{2} - 1358 \nu + 1304 \)\()/76\)
\(\beta_{10}\)\(=\)\((\)\( -479 \nu^{11} - 16566 \nu^{9} - 89894 \nu^{7} - 136674 \nu^{5} - 63631 \nu^{3} - 7396 \nu \)\()/76\)
\(\beta_{11}\)\(=\)\((\)\( -581 \nu^{10} - 20030 \nu^{8} - 106846 \nu^{6} - 154190 \nu^{4} - 61085 \nu^{2} - 4056 \)\()/76\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{10} + 2 \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{6} + 2 \beta_{4} - 3 \beta_{2}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{11} + 3 \beta_{9} - 3 \beta_{8} - 2 \beta_{5} - 3 \beta_{3} + 7 \beta_{1} - 38\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(29 \beta_{10} - 40 \beta_{9} - 40 \beta_{8} - 13 \beta_{7} - 35 \beta_{6} - 28 \beta_{4} + 63 \beta_{2}\)\()/6\)
\(\nu^{4}\)\(=\)\((\)\(-13 \beta_{11} - 57 \beta_{9} + 57 \beta_{8} + 62 \beta_{5} + 117 \beta_{3} - 199 \beta_{1} + 908\)\()/6\)
\(\nu^{5}\)\(=\)\((\)\(-811 \beta_{10} + 1112 \beta_{9} + 1112 \beta_{8} + 449 \beta_{7} + 1075 \beta_{6} + 722 \beta_{4} - 1677 \beta_{2}\)\()/6\)
\(\nu^{6}\)\(=\)\((\)\(103 \beta_{11} + 491 \beta_{9} - 491 \beta_{8} - 606 \beta_{5} - 1149 \beta_{3} + 1873 \beta_{1} - 8410\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(22889 \beta_{10} - 31438 \beta_{9} - 31438 \beta_{8} - 12955 \beta_{7} - 30827 \beta_{6} - 20164 \beta_{4} + 47043 \beta_{2}\)\()/6\)
\(\nu^{8}\)\(=\)\((\)\(-8513 \beta_{11} - 41079 \beta_{9} + 41079 \beta_{8} + 51778 \beta_{5} + 98109 \beta_{3} - 158975 \beta_{1} + 712384\)\()/6\)
\(\nu^{9}\)\(=\)\((\)\(-647551 \beta_{10} + 889742 \beta_{9} + 889742 \beta_{8} + 367595 \beta_{7} + 874255 \beta_{6} + 569606 \beta_{4} - 1329801 \beta_{2}\)\()/6\)
\(\nu^{10}\)\(=\)\((\)\(240005 \beta_{11} + 1160127 \beta_{9} - 1160127 \beta_{8} - 1466962 \beta_{5} - 2779143 \beta_{3} + 4499411 \beta_{1} - 20156686\)\()/6\)
\(\nu^{11}\)\(=\)\((\)\(18327245 \beta_{10} - 25183360 \beta_{9} - 25183360 \beta_{8} - 10408213 \beta_{7} - 24752507 \beta_{6} - 16117708 \beta_{4} + 37632183 \beta_{2}\)\()/6\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4560\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(1711\) \(1921\) \(2737\) \(3041\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2431.1
0.303940i
0.687758i
5.32019i
2.03924i
1.19434i
0.738386i
0.738386i
1.19434i
2.03924i
5.32019i
0.687758i
0.303940i
0 1.00000 0 1.00000 0 4.54425i 0 1.00000 0
2431.2 0 1.00000 0 1.00000 0 3.95229i 0 1.00000 0
2431.3 0 1.00000 0 1.00000 0 3.21222i 0 1.00000 0
2431.4 0 1.00000 0 1.00000 0 2.79053i 0 1.00000 0
2431.5 0 1.00000 0 1.00000 0 1.25182i 0 1.00000 0
2431.6 0 1.00000 0 1.00000 0 0.238175i 0 1.00000 0
2431.7 0 1.00000 0 1.00000 0 0.238175i 0 1.00000 0
2431.8 0 1.00000 0 1.00000 0 1.25182i 0 1.00000 0
2431.9 0 1.00000 0 1.00000 0 2.79053i 0 1.00000 0
2431.10 0 1.00000 0 1.00000 0 3.21222i 0 1.00000 0
2431.11 0 1.00000 0 1.00000 0 3.95229i 0 1.00000 0
2431.12 0 1.00000 0 1.00000 0 4.54425i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2431.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4560.2.d.l yes 12
4.b odd 2 1 4560.2.d.j 12
19.b odd 2 1 4560.2.d.j 12
76.d even 2 1 inner 4560.2.d.l yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4560.2.d.j 12 4.b odd 2 1
4560.2.d.j 12 19.b odd 2 1
4560.2.d.l yes 12 1.a even 1 1 trivial
4560.2.d.l yes 12 76.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4560, [\chi])\):

\( T_{7}^{12} + 56 T_{7}^{10} + 1148 T_{7}^{8} + 10480 T_{7}^{6} + 40228 T_{7}^{4} + 42864 T_{7}^{2} + 2304 \)
\( T_{31}^{6} + 6 T_{31}^{5} - 52 T_{31}^{4} - 256 T_{31}^{3} + 724 T_{31}^{2} + 2168 T_{31} - 3392 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( ( -1 + T )^{12} \)
$5$ \( ( -1 + T )^{12} \)
$7$ \( 2304 + 42864 T^{2} + 40228 T^{4} + 10480 T^{6} + 1148 T^{8} + 56 T^{10} + T^{12} \)
$11$ \( 46656 + 146016 T^{2} + 78148 T^{4} + 16720 T^{6} + 1628 T^{8} + 68 T^{10} + T^{12} \)
$13$ \( 5308416 + 3923136 T^{2} + 915844 T^{4} + 95872 T^{6} + 4892 T^{8} + 116 T^{10} + T^{12} \)
$17$ \( ( -2592 - 504 T + 820 T^{2} + 104 T^{3} - 64 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$19$ \( 47045881 + 3388346 T^{2} - 438976 T^{3} + 170031 T^{4} - 44992 T^{5} + 4652 T^{6} - 2368 T^{7} + 471 T^{8} - 64 T^{9} + 26 T^{10} + T^{12} \)
$23$ \( 7485696 + 37862784 T^{2} + 7555984 T^{4} + 499072 T^{6} + 14744 T^{8} + 200 T^{10} + T^{12} \)
$29$ \( 36864 + 102336 T^{2} + 94180 T^{4} + 32128 T^{6} + 3596 T^{8} + 116 T^{10} + T^{12} \)
$31$ \( ( -3392 + 2168 T + 724 T^{2} - 256 T^{3} - 52 T^{4} + 6 T^{5} + T^{6} )^{2} \)
$37$ \( 2657608704 + 1107646848 T^{2} + 92838340 T^{4} + 3093712 T^{6} + 48332 T^{8} + 356 T^{10} + T^{12} \)
$41$ \( 50466816 + 35407536 T^{2} + 7112260 T^{4} + 503264 T^{6} + 15452 T^{8} + 208 T^{10} + T^{12} \)
$43$ \( 7043909184 + 2716191456 T^{2} + 174292708 T^{4} + 4700032 T^{6} + 62588 T^{8} + 404 T^{10} + T^{12} \)
$47$ \( 20736 + 316800 T^{2} + 1177360 T^{4} + 200000 T^{6} + 10616 T^{8} + 184 T^{10} + T^{12} \)
$53$ \( 603979776 + 209750784 T^{2} + 24686224 T^{4} + 1204480 T^{6} + 26984 T^{8} + 272 T^{10} + T^{12} \)
$59$ \( ( 36864 + 192 T - 17120 T^{2} + 2560 T^{3} + 72 T^{4} - 26 T^{5} + T^{6} )^{2} \)
$61$ \( ( 42736 - 7808 T - 10284 T^{2} - 1072 T^{3} + 168 T^{4} + 28 T^{5} + T^{6} )^{2} \)
$67$ \( ( 205696 + 75712 T - 1512 T^{2} - 2440 T^{3} - 120 T^{4} + 16 T^{5} + T^{6} )^{2} \)
$71$ \( ( 9216 - 9408 T + 1624 T^{2} + 440 T^{3} - 96 T^{4} - 4 T^{5} + T^{6} )^{2} \)
$73$ \( ( -10784 + 24960 T - 12616 T^{2} + 2072 T^{3} - 44 T^{4} - 16 T^{5} + T^{6} )^{2} \)
$79$ \( ( -622592 + 263168 T + 21504 T^{2} - 4736 T^{3} - 336 T^{4} + 14 T^{5} + T^{6} )^{2} \)
$83$ \( 8617780224 + 1827805632 T^{2} + 127443088 T^{4} + 3821920 T^{6} + 55208 T^{8} + 380 T^{10} + T^{12} \)
$89$ \( 1201038336 + 367093680 T^{2} + 40092004 T^{4} + 2009296 T^{6} + 47516 T^{8} + 464 T^{10} + T^{12} \)
$97$ \( 7481558016 + 3424592256 T^{2} + 293339044 T^{4} + 7933936 T^{6} + 93308 T^{8} + 500 T^{10} + T^{12} \)
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