Properties

Label 2240.2.n.i
Level $2240$
Weight $2$
Character orbit 2240.n
Analytic conductor $17.886$
Analytic rank $0$
Dimension $16$
CM discriminant -56
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: 16.0.101415451701035401216.7
Defining polynomial: \(x^{16} - 18 x^{12} + 145 x^{8} - 72 x^{4} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{22} \)
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.

\(f(q)\) \(=\) \( q + ( \beta_{6} - \beta_{7} ) q^{3} -\beta_{11} q^{5} -\beta_{3} q^{7} + ( 3 - \beta_{1} ) q^{9} +O(q^{10})\) \( q + ( \beta_{6} - \beta_{7} ) q^{3} -\beta_{11} q^{5} -\beta_{3} q^{7} + ( 3 - \beta_{1} ) q^{9} + ( \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{13} + ( -\beta_{2} + \beta_{3} - \beta_{5} - \beta_{8} ) q^{15} + ( -\beta_{14} - \beta_{15} ) q^{19} + ( -\beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{21} + ( -2 \beta_{2} - \beta_{5} ) q^{23} + ( -\beta_{1} - \beta_{4} ) q^{25} + ( \beta_{6} - \beta_{7} + \beta_{14} - \beta_{15} ) q^{27} + ( -2 \beta_{6} - \beta_{14} ) q^{35} + ( 5 \beta_{5} - 4 \beta_{8} ) q^{39} + ( \beta_{10} - 3 \beta_{11} + 2 \beta_{12} ) q^{45} + 7 q^{49} + ( -\beta_{1} + 2 \beta_{4} - 2 \beta_{9} ) q^{57} + ( -4 \beta_{6} - 4 \beta_{7} + \beta_{14} + \beta_{15} ) q^{59} + ( -2 \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{13} ) q^{61} + ( 2 \beta_{2} - 3 \beta_{3} + \beta_{5} ) q^{63} + ( -3 + 2 \beta_{1} - 3 \beta_{4} + \beta_{9} ) q^{65} + ( -\beta_{10} - 6 \beta_{11} + 6 \beta_{12} + \beta_{13} ) q^{69} -2 \beta_{8} q^{71} + ( -\beta_{6} - 4 \beta_{7} + 2 \beta_{14} - \beta_{15} ) q^{75} + 6 \beta_{8} q^{79} + ( -1 - 3 \beta_{1} ) q^{81} + ( \beta_{6} - \beta_{7} + 2 \beta_{14} - 2 \beta_{15} ) q^{83} + ( 3 \beta_{6} + 3 \beta_{7} - 2 \beta_{14} - 2 \beta_{15} ) q^{91} + ( \beta_{2} + 3 \beta_{3} - 4 \beta_{5} + 2 \beta_{8} ) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 48q^{9} + O(q^{10}) \) \( 16q + 48q^{9} + 112q^{49} - 48q^{65} - 16q^{81} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 18 x^{12} + 145 x^{8} - 72 x^{4} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{12} + 10 \nu^{8} - 5 \nu^{4} - 508 \)\()/180\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{14} + 20 \nu^{10} - 185 \nu^{6} + 442 \nu^{2} \)\()/60\)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{14} - 130 \nu^{10} + 1079 \nu^{6} - 884 \nu^{2} \)\()/144\)
\(\beta_{4}\)\(=\)\((\)\( -11 \nu^{12} + 190 \nu^{8} - 1495 \nu^{4} + 212 \)\()/120\)
\(\beta_{5}\)\(=\)\((\)\( -13 \nu^{14} + 230 \nu^{10} - 1805 \nu^{6} + 316 \nu^{2} \)\()/120\)
\(\beta_{6}\)\(=\)\((\)\( -35 \nu^{15} - 76 \nu^{13} + 650 \nu^{11} + 1360 \nu^{9} - 5395 \nu^{7} - 10820 \nu^{5} + 5140 \nu^{3} + 2912 \nu \)\()/1440\)
\(\beta_{7}\)\(=\)\((\)\( -16 \nu^{15} + 5 \nu^{13} + 280 \nu^{11} - 90 \nu^{9} - 2180 \nu^{7} + 745 \nu^{5} + 92 \nu^{3} - 540 \nu \)\()/240\)
\(\beta_{8}\)\(=\)\((\)\( 11 \nu^{14} - 194 \nu^{10} + 1531 \nu^{6} - 268 \nu^{2} \)\()/72\)
\(\beta_{9}\)\(=\)\((\)\( \nu^{12} - 18 \nu^{8} + 141 \nu^{4} - 36 \)\()/4\)
\(\beta_{10}\)\(=\)\((\)\( 35 \nu^{15} - 76 \nu^{13} - 650 \nu^{11} + 1360 \nu^{9} + 5395 \nu^{7} - 10820 \nu^{5} - 5140 \nu^{3} + 2912 \nu \)\()/720\)
\(\beta_{11}\)\(=\)\((\)\( -29 \nu^{15} + 16 \nu^{13} + 518 \nu^{11} - 280 \nu^{9} - 4141 \nu^{7} + 2168 \nu^{5} + 1564 \nu^{3} - 176 \nu \)\()/288\)
\(\beta_{12}\)\(=\)\((\)\( 85 \nu^{15} - 17 \nu^{13} - 1510 \nu^{11} + 290 \nu^{9} + 11945 \nu^{7} - 2245 \nu^{5} - 2960 \nu^{3} - 836 \nu \)\()/720\)
\(\beta_{13}\)\(=\)\((\)\( -16 \nu^{15} - 5 \nu^{13} + 280 \nu^{11} + 90 \nu^{9} - 2180 \nu^{7} - 745 \nu^{5} + 92 \nu^{3} + 540 \nu \)\()/120\)
\(\beta_{14}\)\(=\)\((\)\( 305 \nu^{15} - 8 \nu^{13} - 5390 \nu^{11} + 200 \nu^{9} + 42385 \nu^{7} - 1840 \nu^{5} - 6700 \nu^{3} + 6256 \nu \)\()/1440\)
\(\beta_{15}\)\(=\)\((\)\( 97 \nu^{15} + 95 \nu^{13} - 1750 \nu^{11} - 1670 \nu^{9} + 14165 \nu^{7} + 13075 \nu^{5} - 7544 \nu^{3} - 2500 \nu \)\()/720\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{15} + \beta_{14} + \beta_{13} - 2 \beta_{12} - 2 \beta_{11} - \beta_{10} - \beta_{7} - 3 \beta_{6}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{8} + 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{15} + 3 \beta_{14} + 5 \beta_{13} + 6 \beta_{12} - 2 \beta_{11} - 5 \beta_{10} + 9 \beta_{7} + 7 \beta_{6}\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(-2 \beta_{9} - 6 \beta_{4} + 9 \beta_{1} + 18\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-11 \beta_{15} + 13 \beta_{14} - 7 \beta_{13} - 26 \beta_{12} - 22 \beta_{11} - 3 \beta_{10} + 25 \beta_{7} - 19 \beta_{6}\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(25 \beta_{8} + 40 \beta_{5} + 14 \beta_{3} + 10 \beta_{2}\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(41 \beta_{15} + 3 \beta_{14} + 45 \beta_{13} + 6 \beta_{12} - 82 \beta_{11} - 61 \beta_{10} + 49 \beta_{7} + 119 \beta_{6}\)\()/8\)
\(\nu^{8}\)\(=\)\((\)\(-36 \beta_{9} - 102 \beta_{4} + 63 \beta_{1} + 34\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-31 \beta_{15} + 157 \beta_{14} - 191 \beta_{13} - 314 \beta_{12} - 62 \beta_{11} + 113 \beta_{10} + 413 \beta_{7} + 69 \beta_{6}\)\()/8\)
\(\nu^{10}\)\(=\)\((\)\(319 \beta_{8} + 422 \beta_{5} - 82 \beta_{3} - 58 \beta_{2}\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(553 \beta_{15} - 345 \beta_{14} + 65 \beta_{13} - 690 \beta_{12} - 1106 \beta_{11} - 437 \beta_{10} - 423 \beta_{7} + 1219 \beta_{6}\)\()/8\)
\(\nu^{12}\)\(=\)\((\)\(-350 \beta_{9} - 990 \beta_{4} - 135 \beta_{1} - 1782\)\()/4\)
\(\nu^{13}\)\(=\)\((\)\(973 \beta_{15} + 997 \beta_{14} - 2383 \beta_{13} - 1994 \beta_{12} + 1946 \beta_{11} + 2373 \beta_{10} + 3793 \beta_{7} + 3749 \beta_{6}\)\()/8\)
\(\nu^{14}\)\(=\)\((\)\(2197 \beta_{8} + 1924 \beta_{5} - 3346 \beta_{3} - 2366 \beta_{2}\)\()/4\)
\(\nu^{15}\)\(=\)\((\)\(4097 \beta_{15} - 6429 \beta_{14} - 4995 \beta_{13} - 12858 \beta_{12} - 8194 \beta_{11} + 635 \beta_{10} - 14087 \beta_{7} + 5159 \beta_{6}\)\()/8\)

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
0.137538 0.752908i
0.137538 + 0.752908i
−0.752908 + 0.137538i
−0.752908 0.137538i
0.332046 1.81768i
0.332046 + 1.81768i
1.81768 0.332046i
1.81768 + 0.332046i
−1.81768 0.332046i
−1.81768 + 0.332046i
−0.332046 1.81768i
−0.332046 + 1.81768i
0.752908 + 0.137538i
0.752908 0.137538i
−0.137538 0.752908i
−0.137538 + 0.752908i
0 −2.97127 0 −2.14973 0.615370i 0 −2.64575 0 5.82843 0
1119.2 0 −2.97127 0 −2.14973 + 0.615370i 0 −2.64575 0 5.82843 0
1119.3 0 −2.97127 0 2.14973 0.615370i 0 2.64575 0 5.82843 0
1119.4 0 −2.97127 0 2.14973 + 0.615370i 0 2.64575 0 5.82843 0
1119.5 0 −1.78089 0 −0.615370 2.14973i 0 2.64575 0 0.171573 0
1119.6 0 −1.78089 0 −0.615370 + 2.14973i 0 2.64575 0 0.171573 0
1119.7 0 −1.78089 0 0.615370 2.14973i 0 −2.64575 0 0.171573 0
1119.8 0 −1.78089 0 0.615370 + 2.14973i 0 −2.64575 0 0.171573 0
1119.9 0 1.78089 0 −0.615370 2.14973i 0 −2.64575 0 0.171573 0
1119.10 0 1.78089 0 −0.615370 + 2.14973i 0 −2.64575 0 0.171573 0
1119.11 0 1.78089 0 0.615370 2.14973i 0 2.64575 0 0.171573 0
1119.12 0 1.78089 0 0.615370 + 2.14973i 0 2.64575 0 0.171573 0
1119.13 0 2.97127 0 −2.14973 0.615370i 0 2.64575 0 5.82843 0
1119.14 0 2.97127 0 −2.14973 + 0.615370i 0 2.64575 0 5.82843 0
1119.15 0 2.97127 0 2.14973 0.615370i 0 −2.64575 0 5.82843 0
1119.16 0 2.97127 0 2.14973 + 0.615370i 0 −2.64575 0 5.82843 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
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
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
35.c odd 2 1 inner
40.e odd 2 1 inner
40.f even 2 1 inner
56.e even 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.i 16
4.b odd 2 1 inner 2240.2.n.i 16
5.b even 2 1 inner 2240.2.n.i 16
7.b odd 2 1 inner 2240.2.n.i 16
8.b even 2 1 inner 2240.2.n.i 16
8.d odd 2 1 inner 2240.2.n.i 16
20.d odd 2 1 inner 2240.2.n.i 16
28.d even 2 1 inner 2240.2.n.i 16
35.c odd 2 1 inner 2240.2.n.i 16
40.e odd 2 1 inner 2240.2.n.i 16
40.f even 2 1 inner 2240.2.n.i 16
56.e even 2 1 inner 2240.2.n.i 16
56.h odd 2 1 CM 2240.2.n.i 16
140.c even 2 1 inner 2240.2.n.i 16
280.c odd 2 1 inner 2240.2.n.i 16
280.n even 2 1 inner 2240.2.n.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.n.i 16 1.a even 1 1 trivial
2240.2.n.i 16 4.b odd 2 1 inner
2240.2.n.i 16 5.b even 2 1 inner
2240.2.n.i 16 7.b odd 2 1 inner
2240.2.n.i 16 8.b even 2 1 inner
2240.2.n.i 16 8.d odd 2 1 inner
2240.2.n.i 16 20.d odd 2 1 inner
2240.2.n.i 16 28.d even 2 1 inner
2240.2.n.i 16 35.c odd 2 1 inner
2240.2.n.i 16 40.e odd 2 1 inner
2240.2.n.i 16 40.f even 2 1 inner
2240.2.n.i 16 56.e even 2 1 inner
2240.2.n.i 16 56.h odd 2 1 CM
2240.2.n.i 16 140.c even 2 1 inner
2240.2.n.i 16 280.c odd 2 1 inner
2240.2.n.i 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 \(S_{2}^{\mathrm{new}}(2240, [\chi])\):

\( T_{3}^{4} - 12 T_{3}^{2} + 28 \)
\( T_{11} \)
\( T_{17} \)
\( T_{23}^{2} - 56 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( ( 28 - 12 T^{2} + T^{4} )^{4} \)
$5$ \( ( 625 - 22 T^{4} + T^{8} )^{2} \)
$7$ \( ( -7 + T^{2} )^{8} \)
$11$ \( T^{16} \)
$13$ \( ( 28 + 52 T^{2} + T^{4} )^{4} \)
$17$ \( T^{16} \)
$19$ \( ( 1372 + 76 T^{2} + T^{4} )^{4} \)
$23$ \( ( -56 + T^{2} )^{8} \)
$29$ \( T^{16} \)
$31$ \( T^{16} \)
$37$ \( T^{16} \)
$41$ \( T^{16} \)
$43$ \( T^{16} \)
$47$ \( T^{16} \)
$53$ \( T^{16} \)
$59$ \( ( 8092 + 236 T^{2} + T^{4} )^{4} \)
$61$ \( ( 14812 - 244 T^{2} + T^{4} )^{4} \)
$67$ \( T^{16} \)
$71$ \( ( 32 + T^{2} )^{8} \)
$73$ \( T^{16} \)
$79$ \( ( 288 + T^{2} )^{8} \)
$83$ \( ( 26908 - 332 T^{2} + T^{4} )^{4} \)
$89$ \( T^{16} \)
$97$ \( T^{16} \)
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