Properties

Label 2240.2.n.j
Level $2240$
Weight $2$
Character orbit 2240.n
Analytic conductor $17.886$
Analytic rank $0$
Dimension $16$
CM discriminant -35
Inner twists $16$

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Newspace parameters

Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.n (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.968265199641600000000.1
Defining polynomial: \(x^{16} + 9 x^{14} + 44 x^{12} + 261 x^{10} + 1029 x^{8} + 1044 x^{6} + 704 x^{4} + 576 x^{2} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{24} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} -\beta_{7} q^{5} + ( \beta_{2} - \beta_{9} ) q^{7} + ( 3 + \beta_{5} ) q^{9} +O(q^{10})\) \( q + \beta_{3} q^{3} -\beta_{7} q^{5} + ( \beta_{2} - \beta_{9} ) q^{7} + ( 3 + \beta_{5} ) q^{9} + ( -\beta_{6} + \beta_{13} ) q^{11} + ( \beta_{1} - 2 \beta_{7} ) q^{13} + ( -\beta_{11} + \beta_{12} ) q^{15} + \beta_{14} q^{17} + \beta_{8} q^{21} -5 q^{25} + ( 5 \beta_{3} + \beta_{4} ) q^{27} + ( -\beta_{8} - 2 \beta_{10} ) q^{29} + ( -3 \beta_{14} - \beta_{15} ) q^{33} + ( \beta_{6} + 2 \beta_{13} ) q^{35} -\beta_{11} q^{39} + ( -5 \beta_{1} - \beta_{7} ) q^{45} + ( 4 \beta_{2} + \beta_{9} ) q^{47} -7 q^{49} + ( 5 \beta_{6} - 3 \beta_{13} ) q^{51} + ( -4 \beta_{2} + \beta_{9} ) q^{55} + 7 \beta_{2} q^{63} + ( -8 + \beta_{5} ) q^{65} -3 \beta_{12} q^{71} + ( 2 \beta_{14} - \beta_{15} ) q^{73} -5 \beta_{3} q^{75} + ( 3 \beta_{1} + 2 \beta_{7} ) q^{77} + ( -3 \beta_{11} + \beta_{12} ) q^{79} + ( 17 + 4 \beta_{5} ) q^{81} + ( -2 \beta_{3} - \beta_{4} ) q^{83} + ( \beta_{8} - 2 \beta_{10} ) q^{85} + ( -6 \beta_{2} + 5 \beta_{9} ) q^{87} + ( 5 \beta_{6} + 3 \beta_{13} ) q^{91} + ( -\beta_{14} + 2 \beta_{15} ) q^{97} + ( -10 \beta_{6} + 8 \beta_{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 56q^{9} + O(q^{10}) \) \( 16q + 56q^{9} - 80q^{25} - 112q^{49} - 120q^{65} + 304q^{81} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 9 x^{14} + 44 x^{12} + 261 x^{10} + 1029 x^{8} + 1044 x^{6} + 704 x^{4} + 576 x^{2} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 497 \nu^{15} + 2509 \nu^{13} + 3120 \nu^{11} + 29877 \nu^{9} - 70311 \nu^{7} - 1854840 \nu^{5} - 3412832 \nu^{3} - 2926144 \nu \)\()/979200\)
\(\beta_{2}\)\(=\)\((\)\( 2149 \nu^{14} + 19933 \nu^{12} + 101340 \nu^{10} + 602409 \nu^{8} + 2424393 \nu^{6} + 3235620 \nu^{4} + 3583856 \nu^{2} + 1643072 \)\()/734400\)
\(\beta_{3}\)\(=\)\((\)\( -3461 \nu^{14} - 31577 \nu^{12} - 154680 \nu^{10} - 906681 \nu^{8} - 3598197 \nu^{6} - 3654480 \nu^{4} - 924544 \nu^{2} - 478528 \)\()/587520\)
\(\beta_{4}\)\(=\)\((\)\( 1129 \nu^{14} + 10413 \nu^{12} + 50680 \nu^{10} + 296749 \nu^{8} + 1193593 \nu^{6} + 1214320 \nu^{4} + 304896 \nu^{2} + 1033792 \)\()/97920\)
\(\beta_{5}\)\(=\)\((\)\( -211 \nu^{14} - 1947 \nu^{12} - 9540 \nu^{10} - 55791 \nu^{8} - 222927 \nu^{6} - 226620 \nu^{4} - 57104 \nu^{2} - 62208 \)\()/16320\)
\(\beta_{6}\)\(=\)\((\)\( 281 \nu^{15} + 2173 \nu^{13} + 9400 \nu^{11} + 59773 \nu^{9} + 206089 \nu^{7} - 12368 \nu^{5} + 56672 \nu^{3} + 80448 \nu \)\()/39168\)
\(\beta_{7}\)\(=\)\((\)\( 1433 \nu^{15} + 12397 \nu^{13} + 57960 \nu^{11} + 347133 \nu^{9} + 1322937 \nu^{7} + 850080 \nu^{5} - 24608 \nu^{3} + 381248 \nu \)\()/195840\)
\(\beta_{8}\)\(=\)\((\)\( 4869 \nu^{14} + 42713 \nu^{12} + 208440 \nu^{10} + 1251129 \nu^{8} + 4861173 \nu^{6} + 4791120 \nu^{4} + 5233536 \nu^{2} + 2665792 \)\()/326400\)
\(\beta_{9}\)\(=\)\((\)\( 11621 \nu^{14} + 95837 \nu^{12} + 439260 \nu^{10} + 2705961 \nu^{8} + 9941577 \nu^{6} + 4710180 \nu^{4} + 4845424 \nu^{2} + 3563008 \)\()/734400\)
\(\beta_{10}\)\(=\)\((\)\( 6087 \nu^{14} + 49379 \nu^{12} + 222120 \nu^{10} + 1376067 \nu^{8} + 4966359 \nu^{6} + 1523760 \nu^{4} + 1465728 \nu^{2} + 1471936 \)\()/326400\)
\(\beta_{11}\)\(=\)\((\)\( -23527 \nu^{15} - 211299 \nu^{13} - 1027720 \nu^{11} - 6097507 \nu^{9} - 24001879 \nu^{7} - 23516560 \nu^{5} - 14834688 \nu^{3} - 19911616 \nu \)\()/1958400\)
\(\beta_{12}\)\(=\)\((\)\( -1127 \nu^{15} - 9779 \nu^{13} - 45720 \nu^{11} - 273507 \nu^{9} - 1043559 \nu^{7} - 670560 \nu^{5} + 19712 \nu^{3} - 300736 \nu \)\()/86400\)
\(\beta_{13}\)\(=\)\((\)\( 1121 \nu^{15} + 9662 \nu^{13} + 45885 \nu^{11} + 276741 \nu^{9} + 1055262 \nu^{7} + 813825 \nu^{5} + 624184 \nu^{3} + 284368 \nu \)\()/73440\)
\(\beta_{14}\)\(=\)\((\)\( -2257 \nu^{15} - 19829 \nu^{13} - 95480 \nu^{11} - 571957 \nu^{9} - 2210849 \nu^{7} - 1960240 \nu^{5} - 1404928 \nu^{3} - 558656 \nu \)\()/130560\)
\(\beta_{15}\)\(=\)\((\)\( 6439 \nu^{15} + 52163 \nu^{13} + 235560 \nu^{11} + 1462179 \nu^{9} + 5282103 \nu^{7} + 1775280 \nu^{5} + 2249216 \nu^{3} + 1757632 \nu \)\()/195840\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{15} - 2 \beta_{13} + \beta_{12} - 2 \beta_{11} - 2 \beta_{6} + 2 \beta_{1}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{10} - 3 \beta_{9} + \beta_{8} - \beta_{5} + 3 \beta_{3} - 4\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{15} + 2 \beta_{14} + 4 \beta_{13} - 5 \beta_{12} - 10 \beta_{7} + 2 \beta_{6}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(7 \beta_{10} - 9 \beta_{9} - 2 \beta_{8} + 9 \beta_{5} + 3 \beta_{4} - 15 \beta_{3} + 12 \beta_{2} - 8\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-22 \beta_{15} - 11 \beta_{14} + 5 \beta_{13} + 17 \beta_{12} - 5 \beta_{11} + 22 \beta_{7} + 63 \beta_{6} + 11 \beta_{1}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-35 \beta_{10} + 45 \beta_{9} + 5 \beta_{8} + 9 \beta_{4} + 18 \beta_{3} - 45 \beta_{2} - 81\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(38 \beta_{15} + 93 \beta_{14} + 91 \beta_{13} + 13 \beta_{12} + 91 \beta_{11} + 186 \beta_{7} - 143 \beta_{6} - 203 \beta_{1}\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(-127 \beta_{10} + 255 \beta_{9} - 190 \beta_{8} - 63 \beta_{5} - 99 \beta_{4} - 57 \beta_{3} + 396 \beta_{2} + 760\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(323 \beta_{15} - 646 \beta_{14} - 1156 \beta_{13} + 95 \beta_{12} + 170 \beta_{7} - 578 \beta_{6}\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(347 \beta_{10} - 891 \beta_{9} + 952 \beta_{8} - 605 \beta_{5} - 561 \beta_{4} + 231 \beta_{3} - 2244 \beta_{2} + 3808\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(1664 \beta_{15} + 2635 \beta_{14} + 2047 \beta_{13} - 899 \beta_{12} - 2047 \beta_{11} - 5270 \beta_{7} - 5643 \beta_{6} + 4577 \beta_{1}\)\()/4\)
\(\nu^{12}\)\(=\)\((\)\(5670 \beta_{10} - 7245 \beta_{9} - 810 \beta_{8} + 846 \beta_{4} + 1692 \beta_{3} + 7245 \beta_{2} - 7567\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(-15880 \beta_{15} - 10413 \beta_{14} + 233 \beta_{13} - 11875 \beta_{12} + 233 \beta_{11} - 20826 \beta_{7} + 47267 \beta_{6} - 521 \beta_{1}\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(-20851 \beta_{10} + 21957 \beta_{9} + 13456 \beta_{8} + 34307 \beta_{5} + 13689 \beta_{4} - 49335 \beta_{3} - 54756 \beta_{2} - 53824\)\()/4\)
\(\nu^{15}\)\(=\)\((\)\(-30349 \beta_{15} + 60698 \beta_{14} + 108580 \beta_{13} + 93775 \beta_{12} + 167750 \beta_{7} + 54290 \beta_{6}\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1119.1
0.369600 + 2.25820i
−0.369600 + 2.25820i
−0.369600 2.25820i
0.369600 2.25820i
0.141174 + 0.862555i
−0.141174 + 0.862555i
−0.141174 0.862555i
0.141174 0.862555i
1.77086 1.44918i
−1.77086 1.44918i
−1.77086 + 1.44918i
1.77086 + 1.44918i
0.676408 0.553538i
−0.676408 0.553538i
−0.676408 + 0.553538i
0.676408 + 0.553538i
0 −3.40932 0 2.23607i 0 2.64575i 0 8.62348 0
1119.2 0 −3.40932 0 2.23607i 0 2.64575i 0 8.62348 0
1119.3 0 −3.40932 0 2.23607i 0 2.64575i 0 8.62348 0
1119.4 0 −3.40932 0 2.23607i 0 2.64575i 0 8.62348 0
1119.5 0 −1.17325 0 2.23607i 0 2.64575i 0 −1.62348 0
1119.6 0 −1.17325 0 2.23607i 0 2.64575i 0 −1.62348 0
1119.7 0 −1.17325 0 2.23607i 0 2.64575i 0 −1.62348 0
1119.8 0 −1.17325 0 2.23607i 0 2.64575i 0 −1.62348 0
1119.9 0 1.17325 0 2.23607i 0 2.64575i 0 −1.62348 0
1119.10 0 1.17325 0 2.23607i 0 2.64575i 0 −1.62348 0
1119.11 0 1.17325 0 2.23607i 0 2.64575i 0 −1.62348 0
1119.12 0 1.17325 0 2.23607i 0 2.64575i 0 −1.62348 0
1119.13 0 3.40932 0 2.23607i 0 2.64575i 0 8.62348 0
1119.14 0 3.40932 0 2.23607i 0 2.64575i 0 8.62348 0
1119.15 0 3.40932 0 2.23607i 0 2.64575i 0 8.62348 0
1119.16 0 3.40932 0 2.23607i 0 2.64575i 0 8.62348 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1119.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
40.e odd 2 1 inner
40.f even 2 1 inner
56.e even 2 1 inner
56.h odd 2 1 inner
140.c even 2 1 inner
280.c odd 2 1 inner
280.n even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.n.j 16
4.b odd 2 1 inner 2240.2.n.j 16
5.b even 2 1 inner 2240.2.n.j 16
7.b odd 2 1 inner 2240.2.n.j 16
8.b even 2 1 inner 2240.2.n.j 16
8.d odd 2 1 inner 2240.2.n.j 16
20.d odd 2 1 inner 2240.2.n.j 16
28.d even 2 1 inner 2240.2.n.j 16
35.c odd 2 1 CM 2240.2.n.j 16
40.e odd 2 1 inner 2240.2.n.j 16
40.f even 2 1 inner 2240.2.n.j 16
56.e even 2 1 inner 2240.2.n.j 16
56.h odd 2 1 inner 2240.2.n.j 16
140.c even 2 1 inner 2240.2.n.j 16
280.c odd 2 1 inner 2240.2.n.j 16
280.n even 2 1 inner 2240.2.n.j 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.n.j 16 1.a even 1 1 trivial
2240.2.n.j 16 4.b odd 2 1 inner
2240.2.n.j 16 5.b even 2 1 inner
2240.2.n.j 16 7.b odd 2 1 inner
2240.2.n.j 16 8.b even 2 1 inner
2240.2.n.j 16 8.d odd 2 1 inner
2240.2.n.j 16 20.d odd 2 1 inner
2240.2.n.j 16 28.d even 2 1 inner
2240.2.n.j 16 35.c odd 2 1 CM
2240.2.n.j 16 40.e odd 2 1 inner
2240.2.n.j 16 40.f even 2 1 inner
2240.2.n.j 16 56.e even 2 1 inner
2240.2.n.j 16 56.h odd 2 1 inner
2240.2.n.j 16 140.c even 2 1 inner
2240.2.n.j 16 280.c odd 2 1 inner
2240.2.n.j 16 280.n 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 \(S_{2}^{\mathrm{new}}(2240, [\chi])\):

\( T_{3}^{4} - 13 T_{3}^{2} + 16 \)
\( T_{11}^{4} - 31 T_{11}^{2} + 4 \)
\( T_{17}^{4} - 39 T_{17}^{2} + 144 \)
\( T_{23} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( ( 16 - 13 T^{2} + T^{4} )^{4} \)
$5$ \( ( 5 + T^{2} )^{8} \)
$7$ \( ( 7 + T^{2} )^{8} \)
$11$ \( ( 4 - 31 T^{2} + T^{4} )^{4} \)
$13$ \( ( 36 + 33 T^{2} + T^{4} )^{4} \)
$17$ \( ( 144 - 39 T^{2} + T^{4} )^{4} \)
$19$ \( T^{16} \)
$23$ \( T^{16} \)
$29$ \( ( 2704 + 139 T^{2} + T^{4} )^{4} \)
$31$ \( T^{16} \)
$37$ \( T^{16} \)
$41$ \( T^{16} \)
$43$ \( T^{16} \)
$47$ \( ( 6084 + 219 T^{2} + T^{4} )^{4} \)
$53$ \( T^{16} \)
$59$ \( T^{16} \)
$61$ \( T^{16} \)
$67$ \( T^{16} \)
$71$ \( ( 144 + T^{2} )^{8} \)
$73$ \( ( -112 + T^{2} )^{8} \)
$79$ \( ( 55696 + 473 T^{2} + T^{4} )^{4} \)
$83$ \( ( -80 + T^{2} )^{8} \)
$89$ \( T^{16} \)
$97$ \( ( 2704 - 239 T^{2} + T^{4} )^{4} \)
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