# 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,4,Mod(1,2240)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2240.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$132.164278413$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2240))$$:

 $$T_{3}^{2} - 2T_{3} - 31$$ T3^2 - 2*T3 - 31 $$T_{11}^{2} + 14T_{11} - 1999$$ T11^2 + 14*T11 - 1999

## Hecke characteristic polynomials

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