Properties

 Label 2240.4.a.bo Level $2240$ Weight $4$ Character orbit 2240.a Self dual yes Analytic conductor $132.164$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2240 = 2^{6} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2240.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$132.164278413$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 4\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{3} + 5 q^{5} + 7 q^{7} + ( 6 + 2 \beta ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{3} + 5 q^{5} + 7 q^{7} + ( 6 + 2 \beta ) q^{9} + ( -7 + 8 \beta ) q^{11} + ( -25 + \beta ) q^{13} + ( 5 + 5 \beta ) q^{15} + ( -25 - 11 \beta ) q^{17} + ( 18 - 11 \beta ) q^{19} + ( 7 + 7 \beta ) q^{21} + ( -122 - 17 \beta ) q^{23} + 25 q^{25} + ( 43 - 19 \beta ) q^{27} + ( 13 + 6 \beta ) q^{29} + ( 60 - 45 \beta ) q^{31} + ( 249 + \beta ) q^{33} + 35 q^{35} + ( -282 - 15 \beta ) q^{37} + ( 7 - 24 \beta ) q^{39} + ( -164 - 31 \beta ) q^{41} + ( -130 - 17 \beta ) q^{43} + ( 30 + 10 \beta ) q^{45} + ( 175 - 33 \beta ) q^{47} + 49 q^{49} + ( -377 - 36 \beta ) q^{51} + ( 28 + 32 \beta ) q^{53} + ( -35 + 40 \beta ) q^{55} + ( -334 + 7 \beta ) q^{57} -616 q^{59} + ( -168 - 27 \beta ) q^{61} + ( 42 + 14 \beta ) q^{63} + ( -125 + 5 \beta ) q^{65} + ( -76 + 16 \beta ) q^{67} + ( -666 - 139 \beta ) q^{69} + 952 q^{71} + ( 338 + 86 \beta ) q^{73} + ( 25 + 25 \beta ) q^{75} + ( -49 + 56 \beta ) q^{77} + ( -507 + 62 \beta ) q^{79} + ( -727 - 30 \beta ) q^{81} + ( -188 - 150 \beta ) q^{83} + ( -125 - 55 \beta ) q^{85} + ( 205 + 19 \beta ) q^{87} + ( -108 - 11 \beta ) q^{89} + ( -175 + 7 \beta ) q^{91} + ( -1380 + 15 \beta ) q^{93} + ( 90 - 55 \beta ) q^{95} + ( 1371 - 55 \beta ) q^{97} + ( 470 + 34 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 10 q^{5} + 14 q^{7} + 12 q^{9} + O(q^{10})$$ $$2 q + 2 q^{3} + 10 q^{5} + 14 q^{7} + 12 q^{9} - 14 q^{11} - 50 q^{13} + 10 q^{15} - 50 q^{17} + 36 q^{19} + 14 q^{21} - 244 q^{23} + 50 q^{25} + 86 q^{27} + 26 q^{29} + 120 q^{31} + 498 q^{33} + 70 q^{35} - 564 q^{37} + 14 q^{39} - 328 q^{41} - 260 q^{43} + 60 q^{45} + 350 q^{47} + 98 q^{49} - 754 q^{51} + 56 q^{53} - 70 q^{55} - 668 q^{57} - 1232 q^{59} - 336 q^{61} + 84 q^{63} - 250 q^{65} - 152 q^{67} - 1332 q^{69} + 1904 q^{71} + 676 q^{73} + 50 q^{75} - 98 q^{77} - 1014 q^{79} - 1454 q^{81} - 376 q^{83} - 250 q^{85} + 410 q^{87} - 216 q^{89} - 350 q^{91} - 2760 q^{93} + 180 q^{95} + 2742 q^{97} + 940 q^{99} + O(q^{100})$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
0 −4.65685 0 5.00000 0 7.00000 0 −5.31371 0
1.2 0 6.65685 0 5.00000 0 7.00000 0 17.3137 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.4.a.bo 2
4.b odd 2 1 2240.4.a.bn 2
8.b even 2 1 560.4.a.r 2
8.d odd 2 1 35.4.a.b 2
24.f even 2 1 315.4.a.f 2
40.e odd 2 1 175.4.a.c 2
40.k even 4 2 175.4.b.c 4
56.e even 2 1 245.4.a.k 2
56.k odd 6 2 245.4.e.h 4
56.m even 6 2 245.4.e.i 4
120.m even 2 1 1575.4.a.z 2
168.e odd 2 1 2205.4.a.u 2
280.n even 2 1 1225.4.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.b 2 8.d odd 2 1
175.4.a.c 2 40.e odd 2 1
175.4.b.c 4 40.k even 4 2
245.4.a.k 2 56.e even 2 1
245.4.e.h 4 56.k odd 6 2
245.4.e.i 4 56.m even 6 2
315.4.a.f 2 24.f even 2 1
560.4.a.r 2 8.b even 2 1
1225.4.a.m 2 280.n even 2 1
1575.4.a.z 2 120.m even 2 1
2205.4.a.u 2 168.e odd 2 1
2240.4.a.bn 2 4.b odd 2 1
2240.4.a.bo 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2240))$$:

 $$T_{3}^{2} - 2 T_{3} - 31$$ $$T_{11}^{2} + 14 T_{11} - 1999$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-31 - 2 T + T^{2}$$
$5$ $$( -5 + T )^{2}$$
$7$ $$( -7 + T )^{2}$$
$11$ $$-1999 + 14 T + T^{2}$$
$13$ $$593 + 50 T + T^{2}$$
$17$ $$-3247 + 50 T + T^{2}$$
$19$ $$-3548 - 36 T + T^{2}$$
$23$ $$5636 + 244 T + T^{2}$$
$29$ $$-983 - 26 T + T^{2}$$
$31$ $$-61200 - 120 T + T^{2}$$
$37$ $$72324 + 564 T + T^{2}$$
$41$ $$-3856 + 328 T + T^{2}$$
$43$ $$7652 + 260 T + T^{2}$$
$47$ $$-4223 - 350 T + T^{2}$$
$53$ $$-31984 - 56 T + T^{2}$$
$59$ $$( 616 + T )^{2}$$
$61$ $$4896 + 336 T + T^{2}$$
$67$ $$-2416 + 152 T + T^{2}$$
$71$ $$( -952 + T )^{2}$$
$73$ $$-122428 - 676 T + T^{2}$$
$79$ $$134041 + 1014 T + T^{2}$$
$83$ $$-684656 + 376 T + T^{2}$$
$89$ $$7792 + 216 T + T^{2}$$
$97$ $$1782841 - 2742 T + T^{2}$$