# 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 $8$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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