Properties

Label 2240.2.e.f
Level $2240$
Weight $2$
Character orbit 2240.e
Analytic conductor $17.886$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{14} + 28 x^{12} + 16 x^{10} - 40 x^{8} + 610 x^{6} + 1625 x^{4} - 524 x^{2} + 1444\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q -\beta_{7} q^{3} -\beta_{10} q^{5} + ( -\beta_{6} + \beta_{8} ) q^{7} + ( 1 - \beta_{5} ) q^{9} +O(q^{10})\) \( q -\beta_{7} q^{3} -\beta_{10} q^{5} + ( -\beta_{6} + \beta_{8} ) q^{7} + ( 1 - \beta_{5} ) q^{9} -\beta_{11} q^{11} + \beta_{14} q^{13} + ( -\beta_{1} - \beta_{6} ) q^{15} + ( \beta_{2} - \beta_{10} + \beta_{14} ) q^{17} + \beta_{15} q^{19} + ( \beta_{2} + \beta_{5} + \beta_{9} + \beta_{10} ) q^{21} + ( -\beta_{4} + 2 \beta_{6} ) q^{23} + ( \beta_{5} - \beta_{13} ) q^{25} + ( -2 \beta_{7} + \beta_{8} ) q^{27} + ( 2 - \beta_{5} ) q^{29} + ( -2 \beta_{3} + \beta_{7} - \beta_{8} + \beta_{15} ) q^{31} + ( -\beta_{2} + \beta_{10} - \beta_{14} ) q^{33} + ( -\beta_{3} - \beta_{4} + 2 \beta_{7} - \beta_{8} + \beta_{11} ) q^{35} + \beta_{12} q^{37} -\beta_{11} q^{39} + 3 \beta_{9} q^{41} -\beta_{4} q^{43} + ( -\beta_{2} + \beta_{9} - \beta_{14} ) q^{45} + ( -4 \beta_{7} - \beta_{8} ) q^{47} + ( -1 + 2 \beta_{5} - 2 \beta_{9} ) q^{49} + ( -2 \beta_{1} - \beta_{11} ) q^{51} -2 \beta_{13} q^{53} + ( -\beta_{7} - 2 \beta_{8} - \beta_{15} ) q^{55} + \beta_{12} q^{57} + ( -2 \beta_{3} + \beta_{7} - \beta_{8} - 2 \beta_{15} ) q^{59} + ( \beta_{2} - \beta_{9} + \beta_{10} ) q^{61} + ( -\beta_{4} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} ) q^{63} + ( -3 + 2 \beta_{5} + \beta_{12} + \beta_{13} ) q^{65} -2 \beta_{6} q^{67} + ( -2 \beta_{2} - 3 \beta_{9} - 2 \beta_{10} ) q^{69} + ( 4 \beta_{1} - 2 \beta_{11} ) q^{71} + ( 2 \beta_{2} - 2 \beta_{10} + 2 \beta_{14} ) q^{73} + ( -2 \beta_{3} - \beta_{7} - 2 \beta_{8} ) q^{75} + ( \beta_{2} - \beta_{10} - \beta_{12} - \beta_{14} ) q^{77} + ( -2 \beta_{1} - \beta_{11} ) q^{79} + ( -1 - 4 \beta_{5} ) q^{81} + ( -\beta_{7} + 5 \beta_{8} ) q^{83} + ( 2 + 3 \beta_{5} + \beta_{12} ) q^{85} + \beta_{8} q^{87} -\beta_{9} q^{89} + ( 2 \beta_{1} - 3 \beta_{11} - \beta_{15} ) q^{91} + ( -\beta_{12} + 2 \beta_{13} ) q^{93} + ( \beta_{1} - 2 \beta_{4} + \beta_{6} - 3 \beta_{11} ) q^{95} + ( -3 \beta_{2} + 3 \beta_{10} - 3 \beta_{14} ) q^{97} + ( 2 \beta_{1} - 2 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 8q^{9} + O(q^{10}) \) \( 16q + 8q^{9} + 8q^{21} + 8q^{25} + 24q^{29} - 32q^{65} - 48q^{81} + 56q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 6 x^{14} + 28 x^{12} + 16 x^{10} - 40 x^{8} + 610 x^{6} + 1625 x^{4} - 524 x^{2} + 1444\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -84844 \nu^{14} - 2074564 \nu^{12} + 10863016 \nu^{10} - 108971684 \nu^{8} + 259974236 \nu^{6} - 1220553906 \nu^{4} - 1954893928 \nu^{2} - 140017061 \)\()/ 2498768831 \)
\(\beta_{2}\)\(=\)\((\)\( 2967085 \nu^{15} + 39930036 \nu^{13} - 191415108 \nu^{11} + 1131088376 \nu^{9} + 3306399632 \nu^{7} - 3062111390 \nu^{5} + 38410007337 \nu^{3} + 81935469158 \nu \)\()/ 39980301296 \)
\(\beta_{3}\)\(=\)\((\)\(4207655 \nu^{15} - 106481348 \nu^{13} + 576987844 \nu^{11} - 2071902984 \nu^{9} - 24668656 \nu^{7} - 5606760042 \nu^{5} - 12446679765 \nu^{3} - 93332422574 \nu\)\()/ 39980301296 \)
\(\beta_{4}\)\(=\)\((\)\( 274551 \nu^{14} - 2939696 \nu^{12} + 13430352 \nu^{10} - 23555944 \nu^{8} - 84357892 \nu^{6} + 137240250 \nu^{4} - 124066909 \nu^{2} - 3209825990 \)\()/ 768851948 \)
\(\beta_{5}\)\(=\)\((\)\( 4034 \nu^{14} - 44446 \nu^{12} + 225632 \nu^{10} - 360443 \nu^{8} - 805342 \nu^{6} + 3678039 \nu^{4} - 2426116 \nu^{2} - 13312648 \)\()/10116473\)
\(\beta_{6}\)\(=\)\((\)\( 778741 \nu^{14} - 5214580 \nu^{12} + 27045620 \nu^{10} - 25125688 \nu^{8} + 6532248 \nu^{6} + 613266210 \nu^{4} - 311886279 \nu^{2} - 1386720738 \)\()/ 1537703896 \)
\(\beta_{7}\)\(=\)\((\)\( -5290343 \nu^{15} + 30566954 \nu^{13} - 139846998 \nu^{11} - 73573152 \nu^{9} - 8711350 \nu^{7} - 2721018508 \nu^{5} - 5600946403 \nu^{3} - 4522635990 \nu \)\()/ 9995075324 \)
\(\beta_{8}\)\(=\)\((\)\( 4087383 \nu^{15} - 18581871 \nu^{13} + 75636915 \nu^{11} + 248960394 \nu^{9} - 132623279 \nu^{7} + 2173230995 \nu^{5} + 11275740801 \nu^{3} + 6038703550 \nu \)\()/ 4997537662 \)
\(\beta_{9}\)\(=\)\((\)\( -40619 \nu^{15} + 221960 \nu^{13} - 1062732 \nu^{11} - 1111276 \nu^{9} + 645500 \nu^{7} - 26060814 \nu^{5} - 81508947 \nu^{3} - 53518138 \nu \)\()/31933148\)
\(\beta_{10}\)\(=\)\((\)\(51214505 \nu^{15} - 271267084 \nu^{13} + 1118627340 \nu^{11} + 2436041896 \nu^{9} - 4344262192 \nu^{7} + 28747923386 \nu^{5} + 101267919077 \nu^{3} - 8462940866 \nu\)\()/ 39980301296 \)
\(\beta_{11}\)\(=\)\((\)\( 4924818 \nu^{14} - 28350288 \nu^{12} + 130546916 \nu^{10} + 92658541 \nu^{8} - 56159602 \nu^{6} + 2170744901 \nu^{4} + 8291019412 \nu^{2} - 774884638 \)\()/ 2498768831 \)
\(\beta_{12}\)\(=\)\((\)\( 1160909 \nu^{14} - 5994576 \nu^{12} + 20915688 \nu^{10} + 74001744 \nu^{8} - 149350828 \nu^{6} + 628515030 \nu^{4} + 2016951625 \nu^{2} - 198617650 \)\()/ 526056596 \)
\(\beta_{13}\)\(=\)\((\)\( 2650265 \nu^{14} - 16104476 \nu^{12} + 78559372 \nu^{10} + 16571240 \nu^{8} + 12504832 \nu^{6} + 1528314658 \nu^{4} + 5299245757 \nu^{2} - 371424458 \)\()/ 1052113192 \)
\(\beta_{14}\)\(=\)\((\)\( 1659481 \nu^{15} - 10635114 \nu^{13} + 49775130 \nu^{11} + 8859636 \nu^{9} - 99229442 \nu^{7} + 974013580 \nu^{5} + 2269259769 \nu^{3} - 3111082718 \nu \)\()/ 768851948 \)
\(\beta_{15}\)\(=\)\((\)\( -3799279 \nu^{15} + 24179876 \nu^{13} - 110686692 \nu^{11} - 54427560 \nu^{9} + 324888696 \nu^{7} - 2577780150 \nu^{5} - 5301496115 \nu^{3} + 7322587014 \nu \)\()/ 1537703896 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{15} - \beta_{9} + \beta_{8} - \beta_{7} - 2 \beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{13} - \beta_{11} - \beta_{6} - \beta_{5} + \beta_{4} + 2\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{15} + 2 \beta_{14} - 6 \beta_{10} + \beta_{9} + 11 \beta_{8} + 3 \beta_{7} - 4 \beta_{2}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(4 \beta_{13} + 2 \beta_{12} - 8 \beta_{11} + 4 \beta_{6} - 2 \beta_{4} + \beta_{1} - 5\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-7 \beta_{15} - 4 \beta_{14} - 16 \beta_{10} + 4 \beta_{9} + 17 \beta_{8} - 7 \beta_{7} - 3 \beta_{3} - 5 \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(9 \beta_{13} + 7 \beta_{12} - 18 \beta_{11} + 3 \beta_{6} + 16 \beta_{5} - 28 \beta_{4} + 16 \beta_{1} - 92\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-99 \beta_{15} - 92 \beta_{14} - 56 \beta_{10} + 39 \beta_{9} + 25 \beta_{8} - 85 \beta_{7} + 36 \beta_{3} + 82 \beta_{2}\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(-48 \beta_{13} + 72 \beta_{11} - 64 \beta_{6} + 150 \beta_{5} - 136 \beta_{4} - 5 \beta_{1} - 423\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-341 \beta_{15} - 462 \beta_{14} + 448 \beta_{10} - 145 \beta_{9} - 757 \beta_{8} + 91 \beta_{7} + 228 \beta_{3} + 534 \beta_{2}\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(-465 \beta_{13} - 303 \beta_{12} + 957 \beta_{11} - 327 \beta_{6} + 451 \beta_{5} - 210 \beta_{4} - 430 \beta_{1} - 584\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(152 \beta_{15} - 50 \beta_{14} + 1677 \beta_{10} - 1081 \beta_{9} - 2875 \beta_{8} + 1495 \beta_{7} + 205 \beta_{3} + 1096 \beta_{2}\)\()/2\)
\(\nu^{12}\)\(=\)\((\)\(-2012 \beta_{13} - 1766 \beta_{12} + 4427 \beta_{11} + 172 \beta_{6} - 1653 \beta_{5} + 2010 \beta_{4} - 2785 \beta_{1} + 6211\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(10541 \beta_{15} + 11792 \beta_{14} + 11678 \beta_{10} - 8293 \beta_{9} - 19633 \beta_{8} + 16121 \beta_{7} - 4980 \beta_{3} + 2676 \beta_{2}\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(-1697 \beta_{13} - 2936 \beta_{12} + 4781 \beta_{11} + 10817 \beta_{6} - 22427 \beta_{5} + 18575 \beta_{4} - 4798 \beta_{1} + 57296\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(63061 \beta_{15} + 79350 \beta_{14} - 4514 \beta_{10} + 6869 \beta_{9} + 19059 \beta_{8} + 10303 \beta_{7} - 39756 \beta_{3} - 38956 \beta_{2}\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2239.1
0.328458 1.49331i
1.61596 + 1.02509i
−0.328458 1.49331i
−1.61596 + 1.02509i
2.05580 0.953651i
−0.744612 0.556573i
−2.05580 0.953651i
0.744612 0.556573i
−0.744612 + 0.556573i
2.05580 + 0.953651i
0.744612 + 0.556573i
−2.05580 + 0.953651i
1.61596 1.02509i
0.328458 + 1.49331i
−1.61596 1.02509i
−0.328458 + 1.49331i
0 2.13578i 0 −1.94442 1.10418i 0 −2.35829 + 1.19935i 0 −1.56155 0
2239.2 0 2.13578i 0 −1.94442 + 1.10418i 0 2.35829 + 1.19935i 0 −1.56155 0
2239.3 0 2.13578i 0 1.94442 1.10418i 0 −2.35829 + 1.19935i 0 −1.56155 0
2239.4 0 2.13578i 0 1.94442 + 1.10418i 0 2.35829 + 1.19935i 0 −1.56155 0
2239.5 0 0.662153i 0 −1.31119 1.81129i 0 −1.19935 2.35829i 0 2.56155 0
2239.6 0 0.662153i 0 −1.31119 + 1.81129i 0 1.19935 2.35829i 0 2.56155 0
2239.7 0 0.662153i 0 1.31119 1.81129i 0 −1.19935 2.35829i 0 2.56155 0
2239.8 0 0.662153i 0 1.31119 + 1.81129i 0 1.19935 2.35829i 0 2.56155 0
2239.9 0 0.662153i 0 −1.31119 1.81129i 0 1.19935 + 2.35829i 0 2.56155 0
2239.10 0 0.662153i 0 −1.31119 + 1.81129i 0 −1.19935 + 2.35829i 0 2.56155 0
2239.11 0 0.662153i 0 1.31119 1.81129i 0 1.19935 + 2.35829i 0 2.56155 0
2239.12 0 0.662153i 0 1.31119 + 1.81129i 0 −1.19935 + 2.35829i 0 2.56155 0
2239.13 0 2.13578i 0 −1.94442 1.10418i 0 2.35829 1.19935i 0 −1.56155 0
2239.14 0 2.13578i 0 −1.94442 + 1.10418i 0 −2.35829 1.19935i 0 −1.56155 0
2239.15 0 2.13578i 0 1.94442 1.10418i 0 2.35829 1.19935i 0 −1.56155 0
2239.16 0 2.13578i 0 1.94442 + 1.10418i 0 −2.35829 1.19935i 0 −1.56155 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2239.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
140.c 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.e.f 16
4.b odd 2 1 inner 2240.2.e.f 16
5.b even 2 1 inner 2240.2.e.f 16
7.b odd 2 1 inner 2240.2.e.f 16
8.b even 2 1 140.2.c.b 16
8.d odd 2 1 140.2.c.b 16
20.d odd 2 1 inner 2240.2.e.f 16
28.d even 2 1 inner 2240.2.e.f 16
35.c odd 2 1 inner 2240.2.e.f 16
40.e odd 2 1 140.2.c.b 16
40.f even 2 1 140.2.c.b 16
40.i odd 4 2 700.2.g.l 16
40.k even 4 2 700.2.g.l 16
56.e even 2 1 140.2.c.b 16
56.h odd 2 1 140.2.c.b 16
56.j odd 6 2 980.2.s.f 32
56.k odd 6 2 980.2.s.f 32
56.m even 6 2 980.2.s.f 32
56.p even 6 2 980.2.s.f 32
140.c even 2 1 inner 2240.2.e.f 16
280.c odd 2 1 140.2.c.b 16
280.n even 2 1 140.2.c.b 16
280.s even 4 2 700.2.g.l 16
280.y odd 4 2 700.2.g.l 16
280.ba even 6 2 980.2.s.f 32
280.bf even 6 2 980.2.s.f 32
280.bi odd 6 2 980.2.s.f 32
280.bk odd 6 2 980.2.s.f 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.c.b 16 8.b even 2 1
140.2.c.b 16 8.d odd 2 1
140.2.c.b 16 40.e odd 2 1
140.2.c.b 16 40.f even 2 1
140.2.c.b 16 56.e even 2 1
140.2.c.b 16 56.h odd 2 1
140.2.c.b 16 280.c odd 2 1
140.2.c.b 16 280.n even 2 1
700.2.g.l 16 40.i odd 4 2
700.2.g.l 16 40.k even 4 2
700.2.g.l 16 280.s even 4 2
700.2.g.l 16 280.y odd 4 2
980.2.s.f 32 56.j odd 6 2
980.2.s.f 32 56.k odd 6 2
980.2.s.f 32 56.m even 6 2
980.2.s.f 32 56.p even 6 2
980.2.s.f 32 280.ba even 6 2
980.2.s.f 32 280.bf even 6 2
980.2.s.f 32 280.bi odd 6 2
980.2.s.f 32 280.bk odd 6 2
2240.2.e.f 16 1.a even 1 1 trivial
2240.2.e.f 16 4.b odd 2 1 inner
2240.2.e.f 16 5.b even 2 1 inner
2240.2.e.f 16 7.b odd 2 1 inner
2240.2.e.f 16 20.d odd 2 1 inner
2240.2.e.f 16 28.d even 2 1 inner
2240.2.e.f 16 35.c odd 2 1 inner
2240.2.e.f 16 140.c 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 \(S_{2}^{\mathrm{new}}(2240, [\chi])\):

\( T_{3}^{4} + 5 T_{3}^{2} + 2 \)
\( T_{11}^{4} + 15 T_{11}^{2} + 52 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( ( 2 + 5 T^{2} + T^{4} )^{4} \)
$5$ \( ( 625 - 50 T^{2} + 34 T^{4} - 2 T^{6} + T^{8} )^{2} \)
$7$ \( ( 2401 + 30 T^{4} + T^{8} )^{2} \)
$11$ \( ( 52 + 15 T^{2} + T^{4} )^{4} \)
$13$ \( ( 26 - 23 T^{2} + T^{4} )^{4} \)
$17$ \( ( 104 - 29 T^{2} + T^{4} )^{4} \)
$19$ \( ( 208 - 38 T^{2} + T^{4} )^{4} \)
$23$ \( ( 128 - 40 T^{2} + T^{4} )^{4} \)
$29$ \( ( -2 - 3 T + T^{2} )^{8} \)
$31$ \( ( 3328 - 120 T^{2} + T^{4} )^{4} \)
$37$ \( ( 416 + 44 T^{2} + T^{4} )^{4} \)
$41$ \( ( 72 + T^{2} )^{8} \)
$43$ \( ( 32 - 20 T^{2} + T^{4} )^{4} \)
$47$ \( ( 8 + 95 T^{2} + T^{4} )^{4} \)
$53$ \( ( 6656 + 164 T^{2} + T^{4} )^{4} \)
$59$ \( ( 13312 - 270 T^{2} + T^{4} )^{4} \)
$61$ \( ( 16 + 42 T^{2} + T^{4} )^{4} \)
$67$ \( ( 128 - 28 T^{2} + T^{4} )^{4} \)
$71$ \( ( 13312 + 236 T^{2} + T^{4} )^{4} \)
$73$ \( ( 1664 - 116 T^{2} + T^{4} )^{4} \)
$79$ \( ( 208 + 115 T^{2} + T^{4} )^{4} \)
$83$ \( ( 2312 + 170 T^{2} + T^{4} )^{4} \)
$89$ \( ( 8 + T^{2} )^{8} \)
$97$ \( ( 8424 - 261 T^{2} + T^{4} )^{4} \)
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